Cover Page, Table of Contents and Others

9 772161 471005 30

International Journal of Astronomy and Astrophysics, 2019, 9, 173-353 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Table of Contents

Volume 9 Number 3 September 2019

N-Body Simulations of Gas-Free Disc Galaxies with SMBH Seed in Binary Systems R. Chan…………………………………………………………………………………………………………………173 The Analysis of Interplanetary Shocks Associated with Six Major Geo-Effective Coronal Mass Ejections during Solar Cycle 24 S. L. Soni, P. R. Singh, B. Nigam, R. S. Gupta, P. K. Shrivastava………….…………………………………………191 Re-Entry of Space Objects from Low Eccentricity Orbits C. S. Lawrence, R. K. Sharma……………………………………………….....………………………………………200 Analytical Solution for Formation Flying Problem near Equatorial-Circular Reference Orbit S. A. Altalhi, M. I. El Saftawy…………………………………………….……………………………………………217 The Distance Modulus in Dark Energy and Cardassian Cosmologies via the Hypergeometric Function L. Zaninetti………………………………………………………..……………………………………………………231 Gamma-Ray Bursts Generated by Hyper-Accreting Kerr Black Hole F. Sado…………………………………………….....…………………………………………………………………247 Gravity Constraints on the Measurements of the Speed of Light F. Ramdani…………………………………………….……………………….………………………………………265 Halo Orbits in the Photo-Gravitational Restricted Three-Body Problem S. Ghotekar, R. K. Sharma…………………..…………………………………………………………………………274 Convective Models of Jupiter’s Zonal Jets with Realistic and Hyper-Energetic Excitation Source H. G. Mayr, K. L. Chan………………………………………….…………………..…………………………………292 Models for Velocity Decrease in HH34 L. Zaninetti…………………………………………………………………………..…………………………………302 Optical Spectroscopic Monitoring Observations of a T Tauri Star V409 Tau H. Akimoto, Y. Itoh…………………………………..……………………………..…………………………………321 Parameter Inversions of Multi-Layer Media of Mars Polar Region with Validation of SHARAD Data C. Liu, Y.-Q. Jin………………....…………………………………………………..…………………………………335

The figure on the front cover is from the article published in International Journal of Astronomy and Astrophysics, 2019, Vol. 9, No. 3, pp. 231-246 by Lorenzo Zaninetti.

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For manuscripts that have been accepted for publication, please contact: E-mail: [email protected] International Journal of Astronomy and Astrophysics, 2019, 9, 173-190 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

N-Body Simulations of Gas-Free Disc Galaxies with SMBH Seed in Binary Systems

R. Chan

Coordenação de Astronomia e Astrofsica, Observatório Nacional, Rio de Janeiro, RJ, Brazil

How to cite this paper: Chan, R. (2019) Abstract N-Body Simulations of Gas-Free Disc Ga- laxies with SMBH Seed in Binary Systems. We have shown the outcome of N-body simulations of the interactions of two International Journal of Astronomy and disc galaxies without gas with the same mass. Both disc galaxies have halos of Astrophysics, 9, 173-190. dark matter, central bulges and initial supermassive black hole (SMBH) seeds https://doi.org/10.4236/ijaa.2019.93013 at their centers. The purpose of this work is to study the mass and dynamical

Received: May 1, 2019 evolution of the initial SMBH seed during a Hubble cosmological time. It is a Accepted: July 6, 2019 complementation of our previous paper with different initial orbit conditions Published: July, 2019 and by introducing the SMBH seed in the initial galaxy. The disc of the secondary galaxy has a coplanar or polar orientation in relation to the disc of the Copyright © 2019 by author(s) and Scientific Research Publishing Inc. primary galaxy and their initial orbit are eccentric and prograde. The primary This work is licensed under the Creative and secondary galaxies have mass and size of Milky Way with an initial Commons Attribution International SMBH seed. We have found that the merger of the primary and secondary License (CC BY 4.0). discs can result in a final normal disc or a final warped disc. After the fusion http://creativecommons.org/licenses/by/4.0/ of discs, the final one is thicker and larger than the initial disc. The tidal ef- Open Access fects are very important, modifying the evolution of the SMBH in the primary and secondary galaxy differently. The mass of the SMBH of the primary galaxy has increased by a factor ranging from 52 to 64 times the initial seed mass, depending on the experiment. However, the mass of the SMBH of the secondary galaxy has increased by a factor ranging from 6 to 33 times the initial SMBH seed mass, depending also on the experiment. Most of the accreted particles have come from the bulge and from the halo, depleting their particles. This could explain why the observations show that the SMBH with 6 masses of approximately 10 M  is found in many bulgeless galaxies. Only a small number of the accreted particles has come from the disc. In some cases of final merging stage of the two galaxies, the final SMBH of the secondary galaxy was ejected out of the galaxy.

Keywords Simulation, Disc Galaxy, Supermassive Black Hole, Binary Galaxies, Merger, Warped Disc Galaxies

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1. Introduction

It is well known in the literature that supermassive black holes (SMBH) exist in the majority of the galaxies, within elliptical, disc to even in dwarf galaxies [1] [2]. Several recent works in numerical simulations with SMBH with gas [3] [4] [5] show us how complex is the dynamical evolution and mass growing with gas accretion can be. Moreover, many papers have been published about simulations of binary mergers with BH seeds including complex dissipative processes but not included in the present simulations [6]-[14]. Simulations of binary mergers with BH seeds and no dissipative effects, similar to the ones presented in this work are published by several authors [15]-[22]. On the other hand, there are only few works in the literature based on simulations of interaction of gas-free disc galaxies [23]-[29], but none has treated the problem of the existence of a SMBH at the center of the galaxies. In a recent paper of Li et al. (2017), it is presented the results of the gas-free interaction of SMBHs in very eccentric galaxy orbits. Besides, there are rare works studying the evolution of such a binary galaxy in a long interval of time [28] [29] [30] in small eccentricity orbits. This work is a complementation of our previous work [29] where, there, the focus was the evolution of the discs, but here we use different initial orbit conditions and with a SMBH seed in the initial galaxy. Thus, differently, we will focus in the SMBH seed evolution in a cosmological time and covering a wider range of orbits of the galaxy binary than in the work of Li et al. (2017). Thus, the main goal of the present work is to perform numerical N-body simulations to study the time evolution of the mass and dynamics of the initial SMBH seed in the two disc galaxies. We also want to know if the tidal forces affect the evolution of the SMBH. This paper explores the scenario as follows: first, we assume a disc galaxy with the characteristics of the Milky Way (disc, bulge, halo and SMBH). Second, we let a secondary galaxy with the same characteristics orbit on prograde coplanar or polar disc (orientation in relation to the primary disc galaxy). The paper is organized as follows: in Section 2, we describe the numerical method used in the simulations. In Section 3 we present the initial conditions. In Section 4, we describe the results of the simulations. Finally, in Section 5, we summarize the results.

2. Numerical Method

The N-body simulation code used was GADGET [31]. A modified version of this basic code was made in order to introduce the SMBH gravitational interaction with the other particle. Here we have used only the N-body integration but without gas.

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The units used in all the simulations are G = 1 , [length] = 4.500 kpc, [mass] = 10 7 5.100× 10 M  , [time] = 1.993× 10 years ( H0 = 100 km/s/Mpc) and [velocity] = 220.730 km/s. Hereinafter, all the physical quantities will be referred to these

units. The Hubble time tH corresponds to 490 time units. We assumed in all the simulations the tolerance parameter θ = 0.577 . The energy is conserved to better than 6% during the entire evolution with a time step size ∆=t 1.000 × 10−3 and the softening parameter ε =8.000 × 10−4 . As mentioned above, we have utilized in this work a modified version of the GADGET code [31] in order to mimic the interaction of the galaxy particles with the SMBH particle. We have assumed that the collisions between the galaxy particles and the SMBH particle are inelastic. The collision is in such a way that they fuse with the SMBH particle with the total mass as a sum of the two ones. The Schwarzschild radius of the SMBH is defined as 2M R = bh , (1) bh c2

where M bh is the SMBH mass and c is the light velocity. We also have assumed if a galaxy particle with softening parameter  and it grazes the Schwarzschild radius of the SMBH, following the Equation (2), they are merged with one single SMBH particle (see Figure 1). Thus, the Schwarz- schild radius increases because of the additional merged galaxy mass particle with the SMBH. Definition of the condition when there is a merge between the galaxy and the SMBH particle

DR≤+bh , (2) where D is the distance between the centers of the SMBH and the galaxy particle,

 is the particle softening parameter and RBH is the Schwarzschild radius of the SMBH. This is clearly an oversimplified scenario of accretion of galaxy mass onto a SMBH which has, in reality, a much more complex physics involved. At least,

Figure 1. Schematic plot showing the initial positions and velocities of the primary and

secondary galaxies. The quantities Ra and Va are given in Table 3 and Table 4. CM denotes the center of the mass of the binary.

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this study, we can know approximately the evolution of the SMBH mass and its dynamical evolution in binary galaxies during a long-time evolution. In order to determine the number of bulge/halo/disk particles accreted onto the SMBH as a function of time, we have saved a snapshot at each time

step of 4.9 time units until the Hubble time tH = 490 of both galaxies, primary and secondary. Thus, at the end of each experiment we have 100 saved snapshot files. Each snapshot file generated by the modified GADGET code has an identification number, position, velocity and mass of each particle. In this way we can identify the bulge/halo/disk galaxy structure that each particle belongs. When a given particle is merged with the SMBH, using the condition (2), we sum the mass of this particle with the previous mass of the SMBH and set zero mass to this particle. Besides, we recalculate the new position and velocity of the SMBH after the inelastic collision and then we let evolve the system again. At the end of each experiment we count how many bulge/halo/disk particles with zero mass that certainly have merged with the SMBH. Thus, we can obtain the time evolution of the bulge/halo/disk accreted onto the SMBH.

3. Initial Conditions of the Simulations

We have utilized the self-consistent disc-bulge-halo galaxy model by Kuijken & Dubinski [32] in the simulations, the same as in our previous paper [29] (see Table 1 and Table 2). We have also introduced a SMBH seed at rest and at the center of the mass of the galaxy, in order to study its mass and dynamical evolution. Our simulations have been utilized fewer particles than other works on gas-free galaxies but without an initial SMBH seed. In order to try to answer the questions proposed here, we have run small simulations using the available computer clusters, to have, at least, an initial mass and dynamical study of the SMBH seed.

Table 1. Disc galaxy model properties.

Galaxy M d Nd Rd Zd Rt M b Nb M h Nh

G1 0.871 40,000 1.000 0.100 5.000 0.425 19,538 4.916 225,880

M d is the disc mass, Nd the number of particles of the disc, Rd the disc scale radius, Zd the disc

scale height, Rt the disc truncation radius, M b the bulge mass, Nb the number of particles in the

bulge, M h the halo mass, Nh the number of halo particles.

Table 2. Continuation of Table 1.

Galaxy m  M BH Rbh

−5 −4 −4 −10 G1 2.1764× 10 8.0000× 10 2.1764× 10 2.3597× 10

m the mass of each particle, and  is the softening of each particle. M bh the SMBH mass, Rbh the SMBH radius.

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4. The Results of the Simulations

We have run several simulations, without the secondary galaxy to check the initial structure of the galaxy model with the initial SMBH seed at rest at its center (see Figures 2-6). For the simulations with the primary and secondary galaxies we have used two clusters: SGI ICE-X and BULL-X BLADE B500, The maximum number of CPU processors have used for both clusters were 32. Each simulation took about 45 days (BULL-X BLADE B500) and 31 days (SGI ICE-X) of CPU time on average.

Figure 2. Contour plot of the primary galaxy G1 at the times t = 0 and tt= H ). The smoothing was made by averaging the 25 first and second neighbors of each pixel. The density levels in the planes XY and XZ at t = 0 are used in contour plots, in the planes

XY and XZ at tt= H .

Figure 3. Rotation curve of the galaxy G1 of the disc Vc , the angular momentum per

2 12 unit of mass J z and the velocity dispersion in the z direction Vz at the time t = 0 . Hereinafter, the coordinate R is the radius in cylindrical coordinates. The dotted line denotes the disc, the long-dashed line denotes the bulge, the short-dashed line denotes the halo, and the solid line denotes the total rotation curve.

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Figure 4. Rotation curve of the galaxy G1 of the disc Vc , the angular momentum per

2 12 unit of mass J z and the velocity dispersion in the z direction Vz at the time

tt= H . The dotted line denotes the disc, the long-dashed line denotes the bulge, the short-dashed line denotes the halo, and the solid line denotes the total rotation curve.

Figure 5. The evolution in time of the scale radius Rd . The projected particle number density on the XY plane was fitted using the sech disc approximation for each instant of time. This approximation was also used in our previous work [29]. The linear fitting pa- −−11 rameters are RdH=(0.8878 ±× 0.1993 10)[]tt +( 0.8878 ±× 0.1993 10 ) .

In Figure 2 we show the contour plot of the primary galaxy at the beginning

of the simulation ( t = 0 ) and at the Hubble time of the simulation ( tt= H ). We note that the central density in the plane XY has increased slightly after one Hubble time of simulation, since the contour levels are the same for the two moments of time. 2 12 Comparing Figure 3 and Figure 4, we note from the quantity Vz that the self-heating of the initial disc and the particle halo adds a significant source of heating in the disc. The softening can also cause the disc to heat up. Moreover,

the total rotation curves Vc and the angular momentum in the Z direction have

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Figure 6. The evolution in time of the scale height Zd . The projected particle number density on the XZ plane was fitted using the sech disc approximation for each instant of time, as used in our previous work [29]. The linear fitting parameters are −−13 −−23 ZdH=(0.8848 ×± 10 0.3456 × 10)[]tt +( 0.7696 ×± 10 0.6771 × 10 ) .

not changed at the time tt= H of the simulation. The maximum of the rotation max curve Vc = 2.5 occurs when Rmax = 2 . In Figure 5 and Figure 6 we present the temporal evolution of the scale radius

Rd and the scale height Zd . Because of the heating of the disc, the first quantity diminishes with the time while the second increases with the time. The linear fitting parameters of these two quantities are presented in the captions of these figures. In the XZ plane, the scale height has increased 8% because of the two-body relaxation heating (Figure 6). Comparing all the results presented in Figures 2-6 with our previous work

[(Chan Junqueira 2014)] at tt= H for the same quantities we can show that

they are very similar to ours here, except the rotation curve Vc (Figure 4) and

the scale hight Zd (Figure 6). In the previous paper, we have obtained that the max maximum of the rotation curve Vc = 1.2 occurs at Rmax = 0.2 and the scale

hight Zd increased only 0.2%. The differences are caused mainly because of the initial SMBH seed. All the initial conditions of the numerical experiments are presented in Table 3 and Table 4. The orbits of the initial galaxies are eccentric ( e = 0.1, 0.4 or 0.7) and the orientations of the discs are coplanar ( Θ=0 ) or polar ( Θ=90 ) to each other. The simulations always have begun with the primary and secondary galaxies at the apocentric positions. We will show only the evolution time of the SMBH of the experiments where the two discs merge or graze each other, where the tidal effects are more prominent during the evolution of the simulation (see Table 5 and Table 6). The experiments are EXP02, EXP06, EXP20 and EXP24.

Comparing the contour plots of the discs at tt= H shown in Figure 7 for the experiments EXP02 and EXP20 we can note that the merger of the primary and

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Table 3. Primary and secondary galaxy initial conditions.

EXP Θ Rp e Ra Va M bh

− 00 2.1764× 10 4

− 01 0 12 0.1 14.67 0.8732 2.1764× 10 4

− 02 0 12 0.4 28.00 0.5160 2.1764× 10 4

− 03 0 12 0.7 68.00 0.2341 2.1764× 10 4

− 04 0 15 0.1 18.33 0.7810 2.1764× 10 4

− 05 0 15 0.4 35.00 0.4615 2.1764× 10 4

− 06 0 15 0.7 85.00 0.2094 2.1764× 10 4

− 07 0 20 0.1 24.44 0.6763 2.1764× 10 4

− 08 0 20 0.4 46.67 0.3997 2.1764× 10 4

− 09 0 20 0.7 113.33 0.1814 2.1764× 10 4

− 10 0 23 0.1 28.11 0.6307 2.1764× 10 4

− 11 0 23 0.4 53.67 0.3727 2.1764× 10 4

− 12 0 23 0.7 130.33 0.1691 2.1764× 10 4

− 13 0 25 0.1 30.56 0.6049 2.1764× 10 4

− 14 0 25 0.4 58.33 0.3575 2.1764× 10 4

− 15 0 25 0.7 141.67 0.1622 2.1764× 10 4

− 16 0 30 0.1 36.67 0.5522 2.1764× 10 4

− 17 0 30 0.4 70.00 0.3263 2.1764× 10 4

− 18 0 30 0.7 170.00 0.1481 2.1764× 10 4

Θ the angle between the two planes of the discs in units of degree, Rp the pericentric distance, M1 the

primary galaxy mass, e the eccentricity, Ra the apocentric distance, Va the velocity at the apocentric

distance, M1 the primary galaxy mass, and MM21= = 0.621 is the secondary mass galaxy. In these experiments the orbits of the particles of both galaxies, primary and secondary galaxy, have clockwise rota- tions.

Table 4. Continuation of Table 3.

EXP Θ Rp e Ra Va M bh

− 19 90 12 0.1 14.67 0.8732 2.1764× 10 4

− 20 90 12 0.4 28.00 0.5160 2.1764× 10 4

− 21 90 12 0.7 68.00 0.2341 2.1764× 10 4

− 22 90 15 0.1 18.33 0.7810 2.1764× 10 4

− 23 90 15 0.4 35.00 0.4615 2.1764× 10 4

− 24 90 15 0.7 85.00 0.2094 2.1764× 10 4

− 25 90 20 0.1 24.44 0.6763 2.1764× 10 4

− 26 90 20 0.4 46.67 0.3997 2.1764× 10 4

− 27 90 20 0.7 113.33 0.1814 2.1764× 10 4

− 28 90 23 0.1 28.11 0.6307 2.1764× 10 4

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Continued

− 29 90 23 0.4 53.67 0.3727 2.1764× 10 4

− 30 90 23 0.7 130.33 0.1691 2.1764× 10 4

− 31 90 25 0.1 30.56 0.6049 2.1764× 10 4

− 32 90 25 0.4 58.33 0.3575 2.1764× 10 4

− 33 90 25 0.7 141.67 0.1622 2.1764× 10 4

− 34 90 30 0.1 36.67 0.5522 2.1764× 10 4

− 35 90 30 0.4 70.00 0.3263 2.1764× 10 4

− 36 90 30 0.7 170.00 0.1481 2.1764× 10 4

Θ the angle between the two planes of the discs in units of degree, Rp the pericentric distance, M1 the

primary galaxy mass, e the eccentricity, Ra the apocentric distance, Va the velocity at the apocentric

distance, M1 the primary galaxy mass, and MM21= = 0.621 is the secondary mass galaxy. In these experiments, the orbits of the particles of both galaxies, primary and secondary galaxy, have clockwise rota- tions.

Table 5. Characteristics of galaxy orbits and SMBH mass at tH .

EXP Disc Interaction Primary SMBH Secondary SMBH

01 Merge 58.0533 16.7611*

02 Merge 60.1646 21.8667*

03 Merge 63.4899 23.3100*

04 Merge 52.2800 33.8553

05 Merge 58.4970 14.4299*

06 Graze 60.1646 31.9683

07 Merge 59.1651 29.4152

08 Merge 67.0445 31.9678

09 Distant 57.4974 17.3160

10 Merge 59.4966 30.7468

11 Distant 58.9406 23.8649

12 Distant 62.9391 16.4281

13 Merge 60.7155 31.3022

14 Distant 61.9394 22.9770

15 Distant 61.3836 18.6481

16 Distant 58.7214 27.3059

17 Distant 64.2702 21.3118

18 Distant 60.2718 15.4290

Initial mass of SMBH of the primary and secondary galaxy is 1.1099, Primary SMBH and Secondary SMBH 7 in units of 10 M  , Graze means that the two discs touch each other for a while and then separate. Merge means that the two discs fuse. Distant means the two discs interact apart each other. The symbol (*) after the mass of the SMBH of some merging experiments means that the SMBH has been ejected out of the binary system during the evolution of the merged binary.

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Figure 7. Contour plots of the final merged discs at tt= H of the experiments EXP02 and EXP20.

Table 6. Continuation of Table 5.

EXP Disc Interaction Primary SMBH Secondary SMBH

19 Merge 53.6111 26.8622

20 Merge 52.1679 6.5489*

21 Merge 62.0517 16.5387*

22 Merge 53.3919 27.3059

23 Merge 54.1670 13.8750

24 Graze 57.9411 21.5337

25 Merge 54.8352 14.2080

26 Merge 62.0517 21.8667

27 Distant 58.1654 21.2012

28 Merge 58.8284 23.4212

29 Distant 60.4962 24.3091

30 Distant 61.3836 18.8700

31 Merge 61.8273 21.3118

32 Distant 62.3832 21.3118

33 Distant 61.7151 21.7560

34 Distant 61.6029 24.8640

35 Distant 59.3843 21.9779

36 Distant 61.0521 21.2007

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secondary discs can result in a final normal disc (EXP02) or a final warped disc (EXP20). After the fusion of discs, the final one is thicker and larger than the initial discs. In Figures 8(a)-11(a) we show the time evolution of the SMBH mass of the primary and secondary galaxy of the experiments EXP02, EXP06, EXP20 and EXP24. We also present the time evolution of the SMBH mass of the isolated galaxy in Figures 8(a)-11(a), in order to compare its SMBH mass growth to the SMBH mass growth of the primary and secondary galaxy during the evolution. In the same plots, we show the temporal evolution of the distance of

Figure 8. (a) Temporal evolution of the SMBH seed mass of the primary (long-dashed line) and secondary galaxy (short-dashed line) of the experiment EXP02. We also present the time evolution of the SMBH seed mass of the isolated galaxy. In the same plot we show the temporal evolution of the distance of the center of mass of the two galaxies (dot-dashed line). There is an arbitrary scale factor only to adjust the distance within the plot scale. (b) and (c) Time evolution of the number of accreted particles of the primary and secondary galaxy onto the SMBH. The long-dashed lines represent the halo particles. The dotted lines represent the bulge particles. The short-dashed lines represent the disk particles. (a) EXP02; (b) EXP02 (primary galaxy); (c) EXP02 (secondary galaxy).

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Figure 9. (a) Temporal evolution of the SMBH seed mass of the primary (long-dashed line) and secondary galaxy (short-dashed line) of the experiment EXP06. We also present the time evolution of the SMBH seed mass of the isolated galaxy. In the same plot, we show the temporal evolution of the distance of the center of mass of the two galaxies (dot-dashed line). There is an arbitrary scale factor only to adjust the distance within the plot scale. (b) and (c) Time evolution of the number of accreted particles of the primary and secondary galaxy onto the SMBH. The long-dashed lines represent the halo particles. The dotted lines represent the bulge particles. The short-dashed lines represent the disk particles. (a) EXP06; (b) EXP06 (primary galaxy); (c) EXP06 (secondary galaxy).

the center of mass between the two galaxies. There is an arbitrary scale factor only to adjust the distance within the plot scale of each figure. The purpose of these plots is to know if the distance approach of the primary and secondary galaxies to each other increases the mass growth of the SMBHs. Figures 8(b)-11(b) and Figures 8(c)-11(c) show the time evolution of the number of accreted particles of the primary and secondary galaxy onto the SMBHs, respectively. Theses plots show how many halo, bulge and disc particles that contribute to the growth of SMBH mass.

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Figure 10. (a) Temporal evolution of the SMBH seed mass of the primary (long-dashed line) and secondary galaxy (short-dashed line) of the experiment EXP20. We also present the time evolution of the SMBH seed mass of the isolated galaxy. In the same plot we show the temporal evolution of the distance of the center of mass of the two galaxies (dot-dashed line). There is an arbitrary scale factor only to adjust the distance within the plot scale. (b) and (c) Time evolution of the number of accreted particles of the primary and secondary galaxy onto the SMBH. The long-dashed lines represent the halo particles. The dotted lines represent the bulge particles. The short-dashed lines represent the disk particles. (a) EXP20; (b) EXP20 (primary galaxy); (c) EXP20 (secondary galaxy).

Comparing Figures 8(a)-11(a) we can notice the tidal effects in the SMBH mass of the secondary galaxy are more important (see the distance of the center of mass between the two galaxies in the plot). The approach of the galaxies to each other seems not to affect too much the primary galaxy (see Table 5 and Table 6). Comparing Figures 8(b)-11(b) and Figures 8(c)-11(c), respectively, we can note that most of the accreted particles onto the SMBH have come from the bulges and from the halos. Only a small number of the accreted particles onto the SMBH has come from the discs.

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Figure 11. (a) Temporal evolution of the SMBH seed mass of the primary (long-dashed line) and secondary galaxy (short-dashed line) of the experiment EXP24. We also present the time evolution of the SMBH seed mass of the isolated galaxy. In the same plot we show the temporal evolution of the distance of the center of mass of the two galaxies (dot-dashed line). There is an arbitrary scale factor only to adjust the distance within the plot scale. (b) and (c) Time evolution of the number of accreted particles of the primary and secondary galaxy onto the SMBH. The long-dashed lines represent the halo particles. The dotted lines represent the bulge particles. The short-dashed lines represent the disk particles. (a) EXP24; (b) EXP24 (primary galaxy); (c) EXP24 (secondary galaxy).

In some cases of final merging stage of the two galaxies, the final SMBH of the secondary galaxy is ejected out of the galaxy (see Table 5 and Table 6). From Table 5 and Table 6 we can see comparing the final SMBH mass of all the experiments that the mass of the SMBH of the primary galaxy have increased by a factor ranging from 52 to 64 times the initial seed mass, depending on the experiment. However, the mass of the SMBH of the secondary galaxy has increased by a factor ranging from 6 to 33 times in comparison to the initial seed mass, depending on the experiment. Thus, we can conclude that the tidal effects are very important, modifying the evolution of the SMBH in the primary and

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secondary galaxy differently.

5. Conclusions

We have shown the results of N-body simulations of the interactions of two gas-free disc galaxies with the same mass. Both disc galaxies have halos of dark matter, central bulges and initial SMBH seeds at their centers. We have found that the merger of the primary and secondary discs can result in a final normal disc or a final warped disc. After the fusion of discs, the final one is thicker and larger than the initial disc. The tidal effects are very important, modifying the evolution of the SMBH in the primary and secondary galaxy differently. The mass of the SMBH of the primary galaxy has increased by a factor ranging from 52 to 64 times the initial seed mass, depending on the experiment. However, the mass of the SMBH of the secondary galaxy has increased by a factor ranging from 6 to 33 times the initial SMBH seed mass, depending also on the experiment. Most of the accreted particles have come from the bulges and from the halos, depleting their particles. This could explain why the observations show that the 6 SMBH with masses of approximately 10 M  is found in many bulgeless galaxies [1]. However, only a small number of the accreted particles has come from the disc. In some cases of final merging stage of the two galaxies, the final SMBH of the secondary galaxy was ejected out of the galaxy.

Acknowledgements

The author acknowledges the financial support from Conselho Nacional de De- senvolvimento Cientfico e Tecnológico in Brazil. The author also thanks the generous amount of CPU time given by CENAPAD/UFC (Centro Nacional de Processamento de Alto Desempenho da UFC) and NACAD/COPPE-UFRJ (Núcleo de Avançado de Computação de Alto Desempenho da COPPE/UFRJ) in Brazil. In addition, this research has been supported by SINAPAD/Brazil.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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International Journal of Astronomy and Astrophysics, 2019, 9, 191-199 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

The Analysis of Interplanetary Shocks Associated with Six Major Geo-Effective Coronal Mass Ejections during Solar Cycle 24

Shirsh Lata Soni1*, Prithvi Raj Singh2, Bharti Nigam1, Radhe Syam Gupta1, Pankaj Kumar Shrivastava3

1Department of Physics, Govt. P.G. College, Satna, India 2Department of Physics, APS University, Rewa, India 3Department of Physics, Govt. P.G. Model Science College, Rewa, India

How to cite this paper: Soni, S.L., Singh, Abstract P.R., Nigam, B., Gupta, R.S. and Shrivasta- va, P.K. (2019) The Analysis of Interplane- A Coronal Mass Ejection (CME) is an ejection of energetic plasma with tary Shocks Associated with Six Major magnetic field from the Sun. In traversing the Sun-Earth distance, the ki- Geo-Effective Coronal Mass Ejections dur- nematics of the CME is immensely important for the prediction of space ing Solar Cycle 24. International Journal of Astronomy and Astrophysics, 9, 191-199. weather. The objective of the present work is to study the propagation https://doi.org/10.4236/ijaa.2019.93014 properties of six major geo-effective CMEs and their associated interplanetary shocks which were observed during solar cycle 24. These reported CME Received: May 9, 2019 events produced intense geo-magnetic storms (Dst > 140 nT). The six CME Accepted: August 27, 2019 Published: August 30, 2019 events have a broad range of initial linear speeds ~600 - 2700 km/sec in the LASCO/SOHO field of view, comparing two slow CMEs (speed ~579 km/sec Copyright © 2019 by author(s) and and 719 km/sec), three moderate speed CMEs (speed ~1366, 1571, 1008 Scientific Research Publishing Inc. km/sec), and one fast CME (speed ~2684 km/sec). The actual arrival time of This work is licensed under the Creative Commons Attribution International the reported events is compared with the arrival time calculated using the License (CC BY 4.0). Empirical Shock Arrival model (ESA model). For acceleration estimation, we http://creativecommons.org/licenses/by/4.0/ utilize three different acceleration-speed equations reported in the previous Open Access literatures for different acceleration cessation distance (ACD). In addition, we compared the transit time estimated using the second-order speed of CMEs with observed transit time. We also compared the observed transit time with transit time obtained from various shock arrival model. From our present study, we found the importance of acceleration cessation distance for CME propagation in interplanetary space and better acceleration speed for transit time calculation than other equations for CME forecasting.

Keywords

Coronal Mass Ejection (CME), IP Shock, Geomagnetic Strom

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1. Introduction

Coronal Mass Ejection is the most energetic process of solar atmosphere. The CME can be defined as an ejection of plasma with magnetic field from the Sun to the interplanetary space. And its effect on Earth’s environment and space weather. Kinematics of CMEs in space depends upon the initial seed additionally affected by ambient solar wind conditions [1] [2] [3]. The propagation of coronal mass ejections has variations continuously due to their internal energy and interaction with other into interplanetary [4] [5]. The Earth directed CME (i.e. halo or partial halo CME) affect the magnetosphere of Earth. These CME knows as geo-effective CME, the geo-effectiveness of CME identified by geo-magnetic storm disturbances index; Dst (or horizontal component of geo-magnetic dis- turbance field; SYM/H index). The geo-effectiveness is high, if the value of Dst is more negative. In the interplanetary medium, the CME went through acceleration and deceleration due to solar wind speed and finally come to speed nearly equal to speed of solar wind [6] [7]. But as we know that the speed of solar wind shows variation during the 11 year period of solar cycle. Estimation of the arrival of CME to near Earth is very important for predicting the space weather. There is no certain method for model to calculate the arrival time of CME at 1 AU accurately. However, Gopalswamy, in 2001, estimated the transit time of 47 Earth directed CME events observed during the period of 1996 to 2000 following the Empirical shock arrival (ESA) model. For estimation of transit time, they proposed a formula for acceleration (a = 2.193 − 0.0054u) related to the initial speed (u) of CMEs [7]. While the transit time for 83 halo CME events at 1 AU investigated by Michalek et al. 2004 [8]. Among these 83 events, an equation was obtained between effective acceleration and initial speed as a = 4.11 − 0.0063u for 49 CME events with several fast. Another acceleration equation a = 3.35 − 0.0074u, was obtained for extreme events including very fast CMEs (~2684 km/sec), few very slow speed CMEs (~400 km/sec) and two main cases were chosen to representing events for which: 1) there is no acceleration of CMEs and 2) accelerating CMEs at 1 AU. In the present work, we examine the propagation of six geo-effective CME events having a wide range of linear initial speed (~600 to 2700 km/sec) and produced intense geo-magnetic storms having value of Dst index more than (−140 nT). We compare the estimated arrival times with the actual arrival times and also transit time obtained using Drag Based Model (DBM) Vrsnak et al., 2013 [2].

2. Data Selection

We studied the set of six major geo-effective CME events observed by SOHO/ LASCO during the solar cycle 24. The six CME events generated geo-magnetic storms of high intensity Dst > 140 nT. These selected CMEs events are associated with C, M and X class X-ray flares. The detail of selected CMEs, associated flare and geo-magnetic storms are listed in Table 1. Geo-effective CME details are also obtained from the online catalogues of SOHO/LASCO at https://cdaw.gsfc.nasa.gov/CME_list,

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Table 1. Details for major geo-effective CMEs with associated flare. Last column shows the detail of generated interplanetary shock with DST value.

CME Speed Flare IP Shock (km/s) Date/Time TRise TPeak TEnd Intensity Location Date/Time Dst IP Shock TT

22-10-2011 10:25 1005 15:14 15:29 15:20 M1.3 N29W91 24-10-2011 18:31 −147 63.17

07-03-2012 00:24 2684 00:02 00:40 00:24 X5.4 N17E15 08-03-2012 11:03 −145 33.4 15-03-2015 01:48 719 01:15 03:20 02:13 C1.3 S19W25 17-03-2015 04:45 −223 50.95 21-06-2015 02:36 1366 01:02 02:00 01:42 M2.0 N12E16 22-06-2015 18:33 −204 39.59 16-12-2015 09:24 579 08:34 09:23 09:03 C6.6 S13W04 19-12-2015 16:16 −155 78.86 06-09-2017 12:24 1571 11:53 12:10 12:02 X9.3 S09W42 07-09-2017 22:38 −142 34.23

http://www.lesia.obspm.fr/cesra/highlights/highlight07-5.html. The geomagnetic storms details are obtained from: http://wdc.kugi.kyoto.u.ac.jp/Dst_realtime /index.html. And Omni web from: https://omniweb.gsfc.nasa.gov/form/dx1.html. X-ray solar flare data (start time, peak time, last time and intensity) obtained from: https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-features/solar-fla res/x-rays/goes/xrs/. The arrival time of CMEs and their associated interplanetary shocks/ICME are determined from the time difference between the first detection (appearance) time of a CME in C2 coronagraph on-board at SOHO/LASCO and IP shock/ ICME arrival time in ACE/Wind, identified by variation in solar plasma parameters (density, temperature, velocity, etc.) and magnetic field strength at 1 AU.

3. Results and Discussion

Gopalswamy et al., 2001 and Kim et al., 2007 described a procedure to estimate

transit time of CME at 1 AU [7] [9]. The total transit time is given by T = T1 + T2

where T1 is the time of travel up to the acceleration cessation distance d1 up to 1

AU (in Equation (1)) and T2 is travel time for reaming distance d2 at the constant speed (in Equation (2)). For estimating acceleration, we are using three different equations (Equations (3)-(5)) given by Gopalswamy 2001, Michalek et al. 2004. To obtain the effective interplanetary acceleration from the linear initial speed of CMEs and arrival time with three acceleration cessation distances (ACD = 0.7 AU, 0.6 AU and 0.5 AU) and then CME travels the remaining distance (0.3 AU, 0.4 AU and 0.5 AU) with constant speed respectively. Therefore, it is good for study to compare the calculated travel time of these CMEs with the

observed travel time. The transit time equations for T1 and T2 are given by Go- palswamy et al., 2001 are the following:

2 −+u() u +2 ad1 T = (1) 1 a d T 2 = 2 (2) 2 ()u+ 2 ad1

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where u is the linear speed and a is the acceleration at interplanetary medium. We calculate interplanetary acceleration of selected CMEs using with different following equations, Gopalswamy et al. (2001) au=−∗2.193() 0.0054 (3)

Michalek et al. (2004) au=−∗4.11() 0.0063 (4)

Michalek et al. (2004) au=−∗3.35() 0.007 (5)

In the equations above, u is the initial speed of CME. Now we compare the transit times calculated using these above equations with the actual transit time of reported CME events.

3.1. ESA Model: Arrival Time Using Linear Speed of CMEs

As we have three equation for calculating transit time, so firstly we calculate Transit time using Equations (1) and (2) with acceleration speed Equation (3) given by Gopalswamy et al.; 2001 at different acceleration cessation distances (0.7 AU, 0.6 AU and 0.5 AU). The graphical representation of transit time for various ACD 0.7 AU, 0.6 AU and 0.5 AU against CME speed is plotted in Fig- ures 1(a)-(c) respectively. For comparison, we also plotted transit time profiles estimated by using ESA model for three different acceleration cessation distances. The differences between estimated transit time for various acceleration cessation distances and actual arrival time are reported in Table 2. The first column indicates the sequence of selected CME events as mentioned in Table 1. And column second represents the difference between actual transit time and estimated transit time results for different ACD using Equation (3). From Table 2, it seems that the error in transit time value if less (0.3 - 4 hour) for event 4th and 6th, while CME events 1st, 4th and 6th have less error for 0.7 AU. This transit time estimation method is repeated for other acceleration-speed equations i.e. (4) and (5) given by Michalek et al. (2004). In the table, columns third and fourth present the difference between actual arrival time and the arrival time

Table 2. Difference between actual transit time with various acceleration cessation distances (0.7 AU, 0.6 AU and 0.5 AU) for Equations (3)-(5). The blue shaded values show the minimum estimated transit time deviation (~0.6 - 10 hr) from actual transit time.

(∆TIP Shock = difference between actual transit time with transit time obtained from various acceleration-speed equations).

No. of ∆TIP Shock Using Equation (3) ∆TIP Shock Using Equation (4) ∆TIP Shock Using Equation (5) Event 0.7 AU 0.6 AU 0.5 AU 0.7 AU 0.6 AU 0.5 AU 0.7 AU 0.6 AU 0.5 AU

1 0.76 7.57 10.324 4.77 13.84 22.186 12.033 1.51 6.323

2 15.29 15.49 15.792 15.18 15.39 15.72 14.41 14.75 15.19 3 −31.75 −27.22 −23.44 −17.03 1.529 −36.51 −31.54 −27.39 −23.34 4 −0.38 0.893 −0.38 0.791 2.568 1.733 −4.484 −2.09 0.18 5 48.46 46.33 45.21 50.3 47.651 42.88 40.46 46.349 43.68

6 0.61 1.48 6.4 1.51 3.64 3.07 −2.42 −0.83 0.79

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(a)

(b)

(c) Figure 1. Comparison between actual transit time (TT) of IP shock and estimated transit time for various acceleration cessation distances (0.7 AU, 0.6 AU and 0.5 AU) obtained from different acceleration-speed Equations (3)-(5) in plot (a), (b) and (c) respectively.

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calculated by using Equations (4) and (5) respectively. On the other hand acceleration-speed Equation (5) gives the approximately nearly consistent transit time value with the actual transit time (approximately 0.8 - 15 hr) 6th except event 3rd and 5th events. From the table, we can see that Equation (3) produced

the minimum error (∆TIP Shock = 0.6 - 10 hr) for three CME events (1st, 4th and 6th). We get the actual arrival time values of IP shock from Equation (3) for event 1st, 2nd, 4th and 6th but the deviation is more for other two events (event 3rd and 5th), this is may be due to the their slow linear speed (719 km/sec and 579 km/sec respectively).

3.2. ESA Model: Transit Using Second Order Speed of CMEs

In this section of observation, we obtained the arrival time using second-order speed of CMEs instead of linear speed reported in the LASCO/SOHO catalogue. So here we are estimating transit time from Equation (3) by using two different second-order speed: 1) at 20 solar radius and 2) at final distance at different acceleration cessation distance (0.7 AU) using ESA model with acceleration-speed Equation (3). In Figure 2(a) and Figure 2(b) shows the transit times calculated

(a)

(b) Figure 2. Transit time calculated using second-order speed (a) at final distance (b) at 20 solar radii.

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using second order speed at 20 solar radius and at final distance respectively. The transit time error (within ±11 hr) for the last four events, while the transit time error (~0.2 - 7 hr) for last three events using second-order speed at a final distance. It seems from the investigation above that the speed at final speed gives the arrival time nearly consistent with the actual arrival time.

3.3. Comparison with Other Models

In this section, we compare the actual arrival time with transit time obtained by other models (Table 3). Here we are using three different shock arrival prediction models (1) constant Speed Model, (2) Drag Based Model (DBM) proposed by Vrsnak et al. (2013) and (3) transit time prediction model given by Schwenn et al. (2005). The arrival time error for IP shock obtained from the models above with actual transit time are listed in Table 4. In the constant speed model, it has been predicted that CME travels the entire Sun Earth distance at the same speed (initial speed of CME) to reach at 1 AU. And also plot these different obtain transit time of IP-shocks in Figure 3. The Drag Based Model assumes that the CME speed dragged due to interaction of ICME and ambient solar wind. In

DBM, the given parameters are: starting radial distance of CME (r0), CME speed

at r0 (v0), asymptotic solar wind and drag parameter. The DBM tool is accessible in the website http://oh.geof.unizg.hr/index.php/en/spaceweather-tools. After putting all the values for reported CMEs, we have got the transit times at 1 AU. We have taken average ambient solar wind speed as 500 km/sec, which is the average speed of plasma flow recorded by in-situ instrument. Schwenn et al. (2005) proposed a relationship between the arrival time and linear speed of CMEs as:

TVrr =−∗203 20.77 ln ()CME

where Trr is arrival time and VCME is linear speed of CME. In this case of arrival time of interplanetary shock, minimum transit time error is given by DBM model for four events (less than 6 hr).

Figure 3. Comparison between observed (actual) transit time and calculated transit time using different models (Schwenn Model, Drag Based Model, Constant Speed Model).

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Table 3. Transit time error for second order speed at final distances in second column and second order speed at 20 solar radii for all three reported CME events.

Second-Order Speed (at Final Distance) Second-Order Speed (at 20 Solar Radii) No. of Event 0.7 AU 0.6 AU 0.5 AU 0.7 AU 0.6 AU 0.5 AU

1 26.73 26.67 26.75 23.64 23.58 23.67

2 15.71 15.67 15.70 17.19 17.16 17.18

3 −18.44 −18.56 −18.42 −11.33 −11.43 −11.28

4 −0.27 −0.34 −0.25 2.60 2.54 2.62

5 −0.95 −1.12 −0.98 −7.51 −7.71 −7.60

6 7.29 7.23 7.28 7.30 7.25 7.30

Table 4. Difference between the actual transit time and estimated transit time with various models (transit time error) for of IP shocks associated with selected CMEs.

∆T IP Shock (in Hours)

Schwenn Model Drag Based Model Constant Speed Model

3.75 15.7 21.71

−5.61 5.14 17.87

−15.42 −6.47 −7

−15.99 −0.79 5.12

7.99 12.44 6.89

−15.91 −3.34 7.7

4. Conclusion

For the present study, we presented the estimation of arrival time of six major coronal mass ejections which produced intense geo-magnetic storms more than Dst > 140 nT observed during solar cycle 24. The reported CME events have different linear speed such as ~579 - 2684 km/sec. The interplanetary acceleration values, calculated from the speed of CME using various acceleration-speed equations, are utilized in the ESA model. All the plots regarding transit time of CMEs at 1 AU obtained by using ESA model for three different acceleration cessation distance (0.5 AU, 0.6AU and 0.7AU). Study demonstrated that each event acts differently in the interplanetary space. The CME propagation is also governed by the speed of CME, interplanetary acceleration/deceleration and acceleration cessation distances. Summarizing the above study and analysis, it is seen that the result of all comparison of arrival times for reported six CME, the arrival time (transit time) error is minimum for the acceleration Equation (3) for ESA model. In addition, the transit time are compared with various models (constant speed model, Schwenn Model and Drag Based Model). Especially, the minimum arrival time error is obtained for Drag Based Model (DBM) for the acceleration-speed Equation (4) for ESA model. Presented study also shown that the linear speed provide minimum transit time error, instead of second order speed at final dis-

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tances or at 20 solar radii.

Acknowledgements

We are grateful to Solar Geo-physical Data team, Kyoto and OMNI data team for their open data source policy. Authors are thankful to SOHO/LASCO CME catalogue (generated and maintained by CDAW data centre by NASA). We thank Prof. Bhuwan Joshi, USO, Physical Research Laboratory, Ahmedabad, and Pro. Hari Om Vats, scientist, Physical Research Laboratory, Ahmedabad India for his great support to us.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References [1] Vrsnak, B. and Gopalswamy, N. (2002) Influence of the Acceleration Drag on the Motion of Interplanetary Ejectas. Journal of Geophysical Research: Space Physics, 107, 1009. https://doi.org/10.1029/2001JA000120 [2] Vrsnak, B., Žic, T., Vrbanec, D., Temmer, M., Rollett, T., Möstl, C., Veronig, A., Čalogović, J., Dumbović, M., Lulić, S., Moon, Y.-J. and Shanmugaraju, A. (2013) Propagation of Interplanetary Coronal Mass Ejections: The Drag Based Model. So- lar Physics, 285, 295-315. https://doi.org/10.1007/s11207-012-0035-4 [3] Shanmugaraju, A. and Vršnak, B. (2014) Transit Time of Coronal Mass Ejections under Different Ambient Solar Wind Conditions. Solar Physics, 289, 339-349. https://doi.org/10.1007/s11207-013-0322-8 [4] Manoharan, P.K. (2006) Evolution of Coronal Mass Ejections in the Inner Helios- phere: A Study Using White-Light and Scintillation Images. Solar Physics, 235, 345-368. https://doi.org/10.1007/s11207-006-0100-y [5] Manoharan, P.K., Gopalswamy, N., Yashiro, S., Lara, A., Michalek, G. and Howard, R.A. (2004) Influence of Coronal Mass Ejection Interaction on Propagation of In- terplanetary Shocks. Journal of Geophysical Research: Space Physics, 109, A06109. https://doi.org/10.1029/2003JA010300 [6] Gopalswamy, N., Lara, A., Manoharan, P.K. and Howard, R.A. (2005) An Empirical Model to Predict the 1-AU Arrival of Interplanetary Shocks. Advances in Space Re- search, 36, 2289-2294. https://doi.org/10.1016/j.asr.2004.07.014 [7] Gopalswamy, N., Lara, A., Yashiro, S., Kaiser, M.L. and Howard, R.A. (2001) Pre- dicting the 1-AU Arrival Times of Coronal Mass Ejections. Journal of Geophysical Research: Space Physics, 106, 29207-29217. https://doi.org/10.1029/2001JA000177 [8] Michalek, G., Gopalswamy, N., Lara, A. and Manoharan, P.K. (2004) Arrival Time of Halo Coronal Mass Ejections in the Vicinity of the Earth. Astronomy and Astro- physics, 423, 729-736. [9] Kim, K.H., Moon, Y.J. and Cho, K.S. (2007) Prediction of the 1-AU Arrival Times of CME-Associated Interplanetary Shocks: Evaluation of an Empirical Interplane- tary Shock Propagation Model. Journal of Geophysical Research: Space Physics, 112, A05104. https://doi.org/10.1029/2006JA011904

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International Journal of Astronomy and Astrophysics, 2019, 9, 200-216 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Re-Entry of Space Objects from Low Eccentricity Orbits

Cynthia Sharon Lawrence, Ram Krishan Sharma

Department of Aerospace Engineering, Karunya Institute of Technology and Sciences, Coimbatore, India

How to cite this paper: Lawrence, C.S. and Abstract Sharma, R.K. (2019) Re-Entry of Space Objects from Low Eccentricity Orbits. This paper deals with the re-entry predictions of the space objects from the International Journal of Astronomy and low eccentric orbit. Any re-entering object re-enters the Earth’s atmosphere Astrophysics, 9, 200-216. with a high orbital velocity. Due to the aerodynamic heating the object tends https://doi.org/10.4236/ijaa.2019.93015 to break into multiple fragments which later pose a great risk hazard to the Received: April 29, 2019 population. Here a satellite is considered as the space object for which the Accepted: August 27, 2019 re-entry prediction is made. This prediction is made with a package where the Published: August 30, 2019 trajectory path, the time of re-entry and the survival rate of the fragments is

Copyright © 2019 by author(s) and done. The prediction is done using DRAMA 2.0—ESA’s Debris Risk Assess- Scientific Research Publishing Inc. ment and Mitigation Analysis Tool suite, MATLAB and Numerical Predic- This work is licensed under the Creative tion of Orbital Events software. The predicted re-entry time of OSIRIS 3U Commons Attribution International License (CC BY 4.0). was found to be on 7th March 2019, 7:25 (UTC), whereas the actual re-entry http://creativecommons.org/licenses/by/4.0/ time was on 7th March 2019, 7:03 (UTC). The trajectory path found was Open Access 51.5699 deg. (Lat), −86.5738 deg. (Long.) with an altitude of 168.643 km. But

the actual trajectory was 51.76 deg. (Lat), −89.01deg. (Long.) with an altitude of 143.5 km.

Keywords Re-Entry, Space Objects, Low Eccentricity Orbits, DRAMA 2.0, Risk Event Statistics

1. Introduction

Space objects refer to astronomical objects as well as the artificial space objects, i.e. naturally occurring or man-made objects in space. Both the astronomical and artificial objects tend to enter the Earth’s atmosphere, especially the man-made objects. Man-made objects like satellites and rockets are being sent to space for communication, navigation and other space missions. They tend to decay after a certain orbital lifetime. Spent rocket stages, old satellites and fragments from

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disintegration, erosion and collisions are considered as debris. In particular, an accurate estimation of orbital decay of objects during the final stages of re-entry is of considerable importance. This helps to predict the re-entry time and location and thus plan proper hazard assessment and mitigation strategies. The database available for the prediction of orbital lifetime and re-entry of debris objects is the set of two line elements (TLEs). In general, the physical parameters of the objects like mass, area of cross section, shape and dimensions are not available accurately. Further, the atmosphere in which the objects decay varies significantly. A low Earth orbit lies between the altitudes 150 and 2000 kilometer, with a period of about 88 minutes to 127 minutes. Some of the important studies carried out in the area reported in this study are in References [1]-[11]. In this paper, a method to carry out the re-entry predictions of space debris entering from low eccentricity Earth orbit, employing the orbital data in the form of TLEs, is presented. The ballistic coefficient and eccentricity of the re-entering objects are considered as uncertain parameters. The Earth’s zonal

harmonic terms J2 to J6 are included along with the drag perturbation [1] [2]. The re-entry time estimation in each case is computed using NPOE software and MATLAB. The influence of luni-solar perturbations, Earth’s oblateness and atmospheric drag are considered to predict the re-entry time. The re-entry trajectory path along with the debris risk analysis is determined using Debris Risk Assessment and Mitigation Analysis (DRAMA) software. The orbital data and other details of the space objects which re-enter are taken from space-track.org maintained by US Air Force. NPOE is an interactive computer program for computers which can model important orbital events and predict the long-term evolution of satellites in Earth orbits. Program NPOE implements a special perturbation solution of orbital motion using a variable step size Runge-Kutta-Fehlberg (RKF78) integration method to numerically integrate Cowell’s form of the system of differential equations. Orbital events are predicted using Brent’s method for finding the root of a nonlinear equation. MATLAB has also been used to predict the long-term behaviour of the Earth’s satellites subjected to various perturbations. The program actualizes a special perturbation solution of orbital motion as same as NPOE using the variable step size Runge-Kutta-Fehlberg (RKF78) integration method to numerically solve Cowell’s form of the system of differential equation subjected to the central body gravity and other external forces which is otherwise called as orbital initial value problem (IVP).MATLAB is used to plot the graphs of the orbital events. DRAMA is a comprehensive tool [3] for the compliance analysis of space mission with space debris mitigation standards provides with distinct tools to enable the assessment of debris mitigation strategies for the operational and disposal phases of a mission as well as the estimation of the risk caused by objects sur- viving a re-entry of the spacecraft. The following tools that are available within DRAMA are:

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1) ARES: Assessment of Risk Event Statistics The ARES provides an assessment of collision-related events between an operational spacecraft and trackable objects orbiting the Earth, the statistical collision probability, the mean number of conjunction avoidance manoeuvres and the fuel consumption associated to those manoeuvres. 2) MIDAS: MASTER (Based) Impact Flux and Damage Assessment In MIDAS user-defined BLEs can be provided and flux computations are now performed using the MASTER-2009 model. Debris and meteoroid collision flux and damage analysis are done using MIDAS. 3) OSCAR: Orbital Spacecraft Active Removal OSCAR offers the possibility to select between different standardized methods (ISO, ECSS) to generate forecasts of future solar and geomagnetic activity. OSCAR allows for the simulation of drag augmentation devices as a new disposal system. 4) CROC: Cross Section of Complex Bodies Computes the cross-section of complex bodies for different aspect angle conditions. 5) SARA: (Re-Entry) Survival and Risk Analysis It is used for the simulation of the re-entry into the Earth’s atmosphere. This prediction package is used to study the re-entry of OSIRIS 3U satellite. OSIRIS-3U is a CubeSat mission launched on 14 August 2017. Orbital Satellite for Investigating the Response of the Ionosphere to Stimulation and Space Weather is the acronym of OSIRIS which is a three-unit CubeSat developed by the students of the Penn State University. The mission of OSIRIS-3U is to investigate the radio wave interaction in the ionosphere, particularly the interaction of high-power radio waves.

2. Working with NPOE and MATLAB

Newton’s law [4] describes the force between two bodies of masses acting on each other that are at a particular distance. The same theory is used on the two-body problem, where the earth has a bigger body mass than the satellite. The law of gravitation is used here is as, −GMm F = r 2 where, G =6.673 × 10−11 m 3⋅⋅ kg −− 1 s 2 , the gravitational constant; M = the mass of centre object; m = the mass of orbiting object; r = the distance between the two masses. Transforming into the vector form, −GM rr= r3 The mass of the satellite (m) is neglected due to its small mass.

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In this study eccentricity and ballistic coefficient are considered as uncertain parameters [5]. The ballistic coefficient depends on the mass of the object, drag coefficient and the effective area. Of these, drag coefficient and effective area

have more significant uncertainties. For the ballistic coefficient ( M= mCd A), 2 the drag coefficient ( Cd ) and the drag area are 2.20.03 m [6]. As mentioned in the introduction, NPOE and MATLAB script was used to implement a special perturbation solution of orbital motion using a variable step size Runge-Kutta-Fehlberg (RKF78) integration method to numerically solve Cowell’s form of the system of differential equation subject to the central body gravity and other external forces.

a()()()()()() rv,,t= r rr ,,  t =+++ ag r a d rv,,t asm r , tt r ,

The satellite’s acceleration due to Earth’s gravity field is calculated with a vector equation derived from the gradient of the potential function expressed as

arg ()(),,tt= ∇φ r

This acceleration vector is a combination of pure two-body or point mass gravity acceleration and the gravitational acceleration due to higher-order non-spherical terms in the Earths geo-potential. The acceleration experienced by the satellite due to atmospheric drag [7] is computed in NPOE using the following vector expression: 1 CA a() rv,,tt= − ρ () r , v v d d 2 rrm The acceleration contribution of the Sun and Moon represented by point masses is given by

r r  rr  ar(),t =−+−µµmb−− em sb −− + es  sm m 3 3 s 33  rmbem−− r  rr sbes −− 

Generally, in the low earth orbit, the space objects are more prone to the gravity and atmospheric drag. But when the altitude exceeds approximately 1000 km, the solar effects from the Sun increase. Here, OSIRIS 3U is at an altitude between 180 km - 210 km and so the solar radiation pressure along with the luni-solar perturbation is being neglected. In this case, the variation of re-entry time is determined based on varying the ballistic coefficient (BC) [8] and the following graphs Figures 1-4 have been plotted, respectively. It is found based on the data plotted (Figures 1-4) that the ballistic coefficient does not show much variation in the re-entry time as expected. The graphs show that the orbital lifetime with BC = 60 kg/m2 and BC = 80 kg/m2 show very less variation. Hence, it was neglected as being an uncertain parameter and BC = 70 kg/m2 was taken for the prediction of re-entry time of OSIRIS 3U. The observed semi-major axis and the osculating eccentricity from the TLEs are given in Table 1.

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Figure 1. Change in semi major axis for BC = 60 kg/m2.

Figure 2. Change in eccentricity for BC = 60 kg/m2.

Table 1. Osculating orbital elements.

TLE Epoch (UTC) Semi Major Axis (km) Eccentricity 04.02.19 20:31 6669.964476 0.0005899031 09.02.19 13:25 6662.310370 0.0002481707 10.02.19 21:01 6664.590117 0.0004735118 11.02.19 22:34 6661.128219 0.0007581420 13.02.19 21:10 6656.298443 0.0011430093 15.02.19 19:43 6653.712256, 0.0011598911 18.02.19 19:44 6641.503586 0.0012821558 22.02.19 12:08 6649.046280 0.0011109638

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Figure 3. Change in eccentricity for BC = 80 kg/m2.

Figure 4. Change in semi major axis for BC = 80 kg/m2.

The osculating eccentricity and ballistic coefficient taken as uncertain parameters were found to be 0.0080354104 and 70 kg/m2, respectively. Using the estimated values, the re-entry time was predicted to be on 7th March 7:25 (UTC) as shown in the graphs (Figure 5 and Figure 6).

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Figure 5. Change in semi major axis for BC = 70 kg/m2.

Figure 6. Change in eccentricity for BC = 70 kg/m2.

3. Working with Drama 3.1. ARES 3.1.1. Annual Collision Probability (ACP) The Annual Collision Probability (ACP) [3] is modelled by means of an analogy with the laws of kinetic gas theory. The mean number of collisions encountered

by an object with a collision section Ac which moves through a stationary me-

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dium of uniform particle density D, at a constant velocity v during a given time ∆t is expressed as:

c= vDAc ∆ t When simulation was performed on OSIRIS 3U with the ARES tool for the ACP functionality, the functionality returned with the value as in Table 2.

Table 2. ACP.

Functionalities Values

ACP_d 0.7141E−05

ACP_w 0.6424E−03

Flux_d 0.6910E−01

Flux_w 0.1011E+02

where, ACP_w—probability of collision with any object of the whole population; ACP_d—probability of collision with detectable objects; Flux_d—flux due to the detected population [1/km2/year]; Flux_w—flux due to the whole population [1/km2/year].

3.1.2. Avoidance Schemes Assessment Mean number of Manoeuvres per Year The probability of collision for an object-to object encounter is computed by 22 R Rx− 1 − 1 − δδrrT1C P= ∫∫e2 ddyx − 2π det() C R −−Rx22 where, R—sum of the two object radii; δ r —vector between a point in the integration area and the point where the near-miss is predicted.

Figure 7. Predicted manoeuvres per year.

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Figure 7 shows the yearly mean number of avoidance manoeuvres predicted for OSIRIS 3U. The number of manoeuvres is a function of accepted collision probability level. According to the same figure, if OSIRIS 3U operators wanted to avoid all collisions with a probability greater than 1e-006, they would need to perform approximately 2.8 manoeuvres. Risk Reduction and False Alarm Rate Figure 8 and Figure 9, shows how much the number of manoeuvres contri- butes to mitigating the collision risk. If OSIRIS 3U performs the 2.8 manoeuvres that are required to prevent all close encounters with a probability of collision greater than 1e−006, the collision risk would approximately be ACP_w - ACP_d (Table 2).

Figure 8. Risk for OSIRIS 3U.

Figure 9. Remaining risk for OSIRIS 3U.

Finally, Figure 10 shows the accepted collision probability level along with the false alarm rate. False Alarm rates are high, due to the uncertain position of the objects that are involved in the collisions.

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Figure 10. False alarm rate for OSIRIS 3U.

3.1.3. Required ∆V

The required ∆V j (for each population group) is obtained by integration of ∆V

Fj over the manoeuvring area. The total ∆V ( ∆VT ) is obtained by adding the contribution of each population group:

∆=∆VVTj∑ j Figure 11 shows how much of ∆V is required to reach the ACPL values. The x-axis represents the collision avoidance strategy. A value of 0 means a cross-track manoeuvre, while values greater than 0 represents along-track manoeuvres.

Figure 11. Required ∆V for OSIRIS 3U.

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It is observed that when sooner the manoeuvres are performed the more the cheaper they are. However, manoeuvres that performed much before the close approach have large uncertainties associated. The less the time to an event, the better the uncertainties are known, and therefore, the better the prediction.

3.1.4. Propellant Mass Fraction for Avoidance Manoeuvres The propellant mass fraction burned during the expected avoidance manoeuvres, during the satellite lifetime, is linked to the required ∆V to perform those manoeuvres and the propulsion system characteristics. The computation of the propellant mass fraction requires the specific impulse

( Isp ). The ratio between the propellant mass ( mp ) to be burned by a known ∆V

and the initial satellite mass ( mo ) is given by: −∆V mp Ig PMF = =1 − e sp am m  o 

Figure 12 shows the propellant mass fraction required to reach the ACPL values.

Figure 12. Required propellant mass for OSIRIS 3U.

3.2. MIDAS

For the given simulation time ∆t and the impact flux F, the number of impacts

( nimp ) and the probability of collision ( Pcoll ) was computed.

Number of Impacts (NOI)— nimp = FA ⋅ ⋅∆ t −nimp Probability of collision (POC)— Pecoll =1 − The orbit of OSIRIS 3U is the target orbit and is defined by the following parameters:

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Semi-major Axis (a) = 6646.4 km. Eccentricity (e) = 8.39E-3. Inclination (i) = 51.6305 deg. Right ascension of the ascending node (Ω) = 162.247 deg. Argument of Perigee (ω) = 350.489 deg. And the objects considered are of range: 0.1000E+00 m—Lower Threshold 0.1000E+03 m—Upper Threshold The flux was considered for a diameter range 1 mm < d < 20 cm for a time span between May 1st, 2001 and May 1st, 2050. All sources of debris and the meteoroid were also considered. Figure 13, npen (m) is the average number of penetrations of the respective oriented plate by all particles with mp> m

where, mp = particle mass.

Figure 13. Mass vs number of impacts.

Figure 14 npen (d) is the average number of penetrations of the respective oriented plate by all particles with dp> d

where, dp = particle diameter. Figure 15 PNP (m) is the probability that no penetration of the oriented plate by a particle with mp > m will occur. It is based on the average cumulated number of penetrations in this mass class.

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Figure 14. Impactor diameter vs number of impacts.

Figure 15. Mass vs probability of collision.

Figure 16, PNP (d) is the probability that no penetration of the oriented plate by a particle with dp > d will occur. It is based on the average cumulated number of penetrations in this diameter class. On comparing the overall number of impacts on the reference sphere with those on the Sun oriented surface there are more impacts on the oriented surface than on the sphere.

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Figure 16. Impactor diameter vs probability of collision.

3.3. OSCAR

The residual lifetime of OSIRIS 3U was estimated by OSCAR for the minimum cross-section in flight direction. From Figure 17, it can be concluded that the end of orbital lifetime for this orbit was founded to be in May 2019, whereas the actual orbital lifetime ends in March 2019. The lifetime margin was considered as 1% and the disposal option was not considered. Figure 18, provides the evolution of singly averaged eccentricity of OSCAR and Figure 19, provides the evolution of singly averaged inclination of OSCAR.

Figure 17. Date vs altitude.

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Figure 18. Date vs eccentricity.

Figure 19. Date vs inclination.

3.4. CROC

The cross-sectional [9] computation is made using projections determined by using the given point of view direction and the Z-Buffer Algorithm for visible surface determination. The Z-Buffer Algorithm is an array of n × n of pixels. For each pixel there is a record of the depth of object within the pixel that lies closest to the observed. This method is used for hidden surface detection. The cross section of a satellite with respect to the aspect angle θϕ=0, = 0 was found to be 19,900.0 mm2 and the perturbation of atmospheric drag acts on this surface. For an angle of θϕ=45, = 15 the surface area is 45411 mm2.

3.5. SARA

Due to the large orbital velocity, the spacecraft experiences high mechanical

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loads and heating rates while entering the atmosphere and during the aero-braking process the potential and kinetic energy of the spacecraft is con- verted into thermal energy that is consumed by the spacecraft, resulting in thermal and mechanical loads destroying the spacecraft either completely or partially [6] [9] [10]. The spacecraft is destroyed either by melting or evaporation or chemical reaction. For thermal or mechanical destruction, the properties of the spacecraft materials are considered. Object-oriented method was used to analyse and the breakup altitude was set for a range between 75 km and 85 km. The solar panel breakup altitude was assumed to be at 95 km. Along with the re-entry trajectory dynamics, the aerodynamic, aero thermo- dynamic and thermal analyses were also performed for OSISRIS 3U re-entry. The initial conditions of the trajectory [11] and the spacecraft model was defined in terms of osculating orbital elements and by the spacecraft components, speci- fied by their shape, size and material. The output of the analysis comprises the mass, cross-section, velocity, incident angle and impact location. The trajectory path was found to be 51.5699 deg. (Lat), −86.5738 deg. (long.) with an altitude of 168.643 km at a velocity of 7.51313 km/s with an approximate temperature of 300 K. It was found that no objects survived upon re-entry and so there was no ground impact.

4. Conclusion

The predicted re-entry time was found to be on 7th March 2019, 7:25 (UTC), whereas the actual re-entry time was on 7th March 2019, 7:03 (UTC). The trajectory path found was 51.5699 deg. (Lat), −86.5738 deg. (Long.) with an altitude of 168.643 km. But the actual trajectory was 51.76 deg. (Lat), −89.01 deg. (Long.) with an altitude of 143.5 km.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References [1] Sharma, R.K. (1990) On Mean Orbital Elements Computation for Near Earth Ob- jects. Indian Journal of Pure and Applied Mathematics, 21, 468-474. [2] Sharma, R.K. and Mani, L. (1985) Study of RS-1 Orbital Decay with KS Differential Equations. Indian Journal Pure and Applied Mathematics, 6, 833-842. [3] Brauna, V., Gelhaus, J., Kebschull, C., Sanchez-Ortiz, N., Oliveira, J., Dominguez, R., Wiedemann, C., Krag, H. and Vorsmann, P. (2013) DRAMA 2.0—ESA’s Space Debris Risk Assessment and Mitigation Analysis Tool Suite. 64th International As- tronautical Congress, Beijing, September 2013. [4] Hintz, G.R. (2015) Orbital Mechanics and Astrodynamics Techniques and Tools for Space Missions. Springer, Berlin, 24-26.

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[5] Mutyalarao, M. and Sharma, R.K. (2010) Optimal Re-Entry Time Estimation of an Upper Stage from Geostationary Transfer Orbit. Journal of Spacecraft and Rockets, 47, 686-690. https://doi.org/10.2514/1.44147 [6] Kummer, A.T. (2012) System Design and Instrumentation Development for the Osiris-3U CubeSat Mission. MS Thesis, The Pennsylvania State University, State College. [7] Perini, L.L. (1974) Orbital Lifetime Estimates. The John Hopkins University, Balti- more, ANSP-M-8, Copy 55. https://doi.org/10.2172/4269348 [8] Gondelach, D.J., Armellink, R. and Lidtke, A.A. (2017) Ballistic Coefficient Estima- tion for Reentry Prediction of Rocket Bodies in Eccentric Orbits Based on TLE Da- ta. Hindawi Mathematical Problems in Engineering, 2017, Article ID: 7309637. https://doi.org/10.1155/2017/7309637 [9] Garzon, M.M. (2012) Development and Analysis of the Thermal Design for the Osi- ris-3U CubeSat Mission. The Pennsylvania State University, State College. [10] Song, S., Kim, H. and Chang, Y.-K. (2018) Design and Implementation of 3U Cu- beSat Platform Architecture. International Journal of Aerospace Engineering, 2018, Article ID: 2079219. https://doi.org/10.1155/2018/2079219 [11] De Lafontaine, J. and Garg, S.C. (1982) A Review of Satellite and Orbit Decay Pre- diction. Proceedings of the Indian Academy of Sciences, 5, 197-258.

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International Journal of Astronomy and Astrophysics, 2019, 9, 217-230 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Analytical Solution for Formation Flying Problem near Equatorial-Circular Reference Orbit

Shaheera A. Altalhi1, Magdy Ibrahim El Saftawy1,2

1King Abdul-Aziz University, Jeddah, Saudi Arabia 2National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt

How to cite this paper: Altalhi, S.A. and El Abstract Saftawy, M.I. (2019) Analytical Solution for Formation Flying Problem near Equatori- The relative motion between multiple satellites is a developed technique with al-Circular Reference Orbit. International many applications. Formation-flying missions use the relative motion dy- Journal of Astronomy and Astrophysics, 9, namics in their design. In this work, the motion in invariant relative orbits is 217-230. considered under the effects of second-order zonal harmonics in an equatori- https://doi.org/10.4236/ijaa.2019.93016 al orbit. The Hamiltonian framework is used to formulate the problem. All Received: March 29, 2019 the possible conditions of the invariant relative motion are obtained with dif- Accepted: August 27, 2019 ferent inclinations of the follower satellite orbits. These second-order condi- Published: August 30, 2019 tions warrantee the drift rates keeping two, or more, neighboring orbits from

drifting apart. The conditions have been modeled. All the possibilities of Copyright © 2019 by author(s) and Scientific Research Publishing Inc. choosing mean elements of the leader satellite orbit and differences in mo- This work is licensed under the Creative menta between leader and follower satellites’ orbits are presented. Commons Attribution International License (CC BY 4.0). Keywords http://creativecommons.org/licenses/by/4.0/ Open Access Invariant Relative Orbits, Formation Flying Satellites, Relative Motion

1. Introduction

As the geostationary Earth orbits (GEO) belt becomes more crowded it is in- creasingly difficult to acquire slots for new satellites. Consequently, many or- ganizations choose to collocate their spacecraft in the same slot. Also for missions which a single satellite cannot accomplish, as global position satellite system (GPS), the needed of formation flight began. The formation flight concept is the use of several small satellites, which work together in a group (or constellation) to accomplish the objective of one larger, usually more expensive, satellite. This increases the likelihood of mission success in the event of a malfunction Hughes [1]. Formation flights have invariant rela-

DOI: 10.4236/ijaa.2019.93016 Aug. 30, 2019 217 International Journal of Astronomy and Astrophysics

S. A. Altalhi, M. I. El Saftawy

tive orbits for their satellites to ensure that they will not separate over time. The invariant relative orbits have been studied for a long time, as the earlier work of Clohessy and Wiltshire [2] in addition to the studies of Tschauner and Hempel [3]. These models introduced conditions on the initial relative position and velocity so that the relative orbits result to be periodic, which are closed orbits. Recently, Schaub and Alfriend [4], Abd El-Salam et al. [5] passing through Li and Li [6] until Abd El-Salam and El-Saftawy [7] in which they discussed the invariant relative orbits due to the influence of the perturbative effects of the as- phericity of the Earth, the relativistic corrections and the direct solar radiation

pressure. Rahoma [8] also, discussed the J2 invariant relative orbits with the effect of lunisolar attraction. In this paper, we extend the works of Schaub and Alfriend [2] and Abd El-Salam et al. [5] model by introducing an atlas for the curves of invariant relative orbits’ conditions. This atlas will be presented using Mathematica program to calculate and plot graphics of the initial conditions of invariant relative orbits. Those graphics will be shown as curves in 2D; in the case of the orbit of the leader satellite is equatorial.

2. Hamiltonian System

There are several ways to derive the equations of motion for any such system. We emphasized on the Hamiltonian structure of this system. The Hamiltonian formulation allows additional conservative forces to add to the Hamiltonian, thus the addition of complexity to the model can be incorporated with ease. Non-conservative forces can add in the momenta equations of motion. The Ha- miltonian equations of motion allow us to directly use control and simulation techniques.

After expressing the Hamiltonian, as a series in power of J2 (The second geopotential zonal harmonic) up to the second order, and using Lie-Deprit-Kamel perturbation method Kamel [9] to eliminate, in successive, the short and long periodic terms, the transformed Hamiltonian, H**, for different orders 0, 1, and 2, are obtained by El-Saftawy et al. [10]. J 2 H**=++ H ** JH ** 2 H** (1) 0 212 2 where, µ 2 H ** = − η (1.1) 0 2 2,0 ** 2 HA1= 11η 3,3 ()32 − (1.2) A2 ** 1 11  4 H2 =( −5517ηη1,9 +− 2463,7 3456 ηη4,6 + 135 5,5 ) 128 µ 2  2 +(11520η1,9 −++ 2976 η3,7 8064 ηη4,6 672 5,5 ) (1.3) +−ηη + − η + η ( 47041,9 576 3,7 15364,6 288 5,5 )

3 $2 +A22()5ηη 3,7 − 3 5,5 ( 35 −+ 40  8) 2

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−−ij With, ηij,= LL 12,  = SinI (I is the inclination of the orbit), μ is the gravitational parameter of the planet and zero order quantities defined as: µ 42r A = e , 11 4

µ 64rJ = e 4 A22 2 . 32J2

2 where li and Li are the Delaunay elements ( La1 = µ , LL21=()1 − e,

LL32= Cos I), re is the equatorial radius of the Earth, and J2, J4 are the second and fourth geopotential zonal harmonic respectively. The problem of designing invariant relative orbits for spacecraft flying forma- tions is outlined as follows: 1) Compute the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly. 2) These secular drift rates are set equal between two neighboring orbits. 3) Having both orbits drift at equal angular rates on the average, they will not separate over time due to the influence of the perturbative effects of the asphe- ricity of the Earth up to the desired order of magnitude (or the accuracy) to the equations of motion. Using the canonical equations of motion, ∂∂** **  HH lLii=, =−= , i 1, 2, 3 (2) ∂∂Lli

Since the argument of mean latitude θ is the sum of the mean anomaly and

the argument of perigee (l1 + l2). Evaluating the derivatives yields the sum of the argument of perigee and the mean anomaly rate of changes. Follows, the rate of change of mean latitude, θ , and the secular drift rates of the longitude of the    ascending node, l3 can be calculated, i.e. θ =ll12 + . So, using Equation (1) in Equation (2), the result can written in the form: 2 J n 2 J n  2 θ  2 l3 θ = ∑ Dn and l3 = ∑ Dn (3) n=0 n! n=1 n!

θ l3 With, Dn and Dn are published by Abd El-salam et al. [5] and given by: θ 2 D0 = µ K0

l3 D1= AZ 11 1 2 θ D1= 11 ∑ KA 0 i=1 33A2 57 θ = 11 + D2 2 ∑∑Zii AZ22 128µ ii=26 2 = 33A2 8 12 θ = 11 + D2 2 ∑∑KKiiA22 128µ ii=39 2 =

where Ki and Zi are function of the action variable and introduced in Ap- pendix. To prevent the satellites from drifting apart over time, the average secular

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 growth needs to be equal. So, it would be desirable to match all three rates ll12,    and l3 between the satellites in each formation. So θ and l3 of all satellites in the formation should be equal.    θθiii=l1, + l 2, = l 1, j + l 2, jj = ∀≠ ij (4)  l3,ij= l 3, ∀≠ij

Denoting the reference means orbit elements with the subscript “0”. Using   Taylor expansion for the drift rate θi and l3,i of a neighboring orbit “i” about the reference orbital elements, retaining the terms up the second-order derivatives, can be simplified as:

∂θ ∂∂ θθ   i ii δθi =++η−1,1  δ L1 ∂L1 ∂∂ LL 23 xx= 0xx= 00xx= ∂∂θθ ∂ θ   iii  ++η−1,0 δη−−1,1 − η 2,1 δ I ∂∂LL23 ∂ L3  xx= 00xx= xx= 0 

 2 22 2 1 ∂θ ∂∂ θθ  ∂ θ  ++ iη  ii ++2 i 2−2,2 222 2  ∂L1 ∂∂ LL 23∂L2∂L3  xx= 0xx= 00xx= xx= 0

22  2 ∂∂θθ2 1 ∂θ ++ηii δη + i 2 −−1,1  ()L1 2,0 2 ∂∂LL12 ∂∂ LL13  2  ∂L2 xx= 00xx=  xx= 0 222  ∂∂θθ2 1 ∂θ2 ++ii22δη + η i δ 222()−−1,1 4,2 ()I ∂∂LL23 ∂∂LL332  xx= 00xx= xx= 0   ∂∂22θθ ∂ 2 θ  +η ii ++2 i −2,1 222  ∂∂LL23∂∂LL23  xx= 00xx= xx= 0 ∂∂22θθ ii ++η−−1,0  ()δ L1 ()δη 1,1 ∂∂LL12 ∂∂ LL13  xx= 00xx=  (5) ∂∂22θθ −−η iiδ δη −−3,1  2 ()I ()1,1 ∂L3 ∂∂LL23 xx= 00xx=  ∂2θ ∂∂22θθ  −+η i ii+η δδ −3,2  2 −2,1 ()()LI1  ∂L3 ∂∂LL23 ∂∂LL13   xx= 0 xx= 00 xx=

With  = CosI, similarly:  ∂l ∂∂ ll  δηlL =++3,i3, ii3, δ 3,i −1,1  1 ∂L1 ∂∂ LL 23 xx= 0xx= 00xx= ∂∂ll ∂ l   3,ii3, 3,i  ++η−1,0 δη−1,1 −η− 2,1 δ I ∂∂LL23 ∂ L3  xx= 00xx= xx= 0   2 222 11∂l ∂∂ ll ∂ l ++ 3,iiη 3, +2 3,ii2 3,  22−2,2  2  22∂∂LL12∂L3 ∂∂LL23  xx= 00xx= xx= 00xx=

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22  2 ∂∂ll 2 1 ∂l ++η3,ii3,  ()δηL + 3,i −−1,1   1 2, 0 2 ∂∂LL12 ∂∂ LL13 2  ∂L2 xx= 00xx=  xx= 0 222  ∂∂ll2 1 ∂l2 ++3,ii223, δη + η 3,i δ 222()−−1,1 4,2 ()I ∂∂LL23 ∂∂LL332  xx= 00xx= xx= 0 

 22 2 ∂∂ll ∂ l ++η 3,i 2 3,ii+ 2η 3,  −2,1 2  2 −2,1  ∂L2 ∂L3 ∂∂LL23  xx= 0 xx= 00xx=

22 ∂∂ll ++η 3,ii3,  ()δL δη −−1,0   1 ()1,1 ∂∂LL12 ∂∂ LL13 xx= 00xx=  ∂∂22ll −+η 3,ii3, δ δη −−3,1  2 ()I ()1,1 (6) ∂L3 ∂∂LL23 xx= 00xx=

 22 2 ∂∂ll  ∂ l  −ηη 3,ii ++3, 3, i ()()δδLI  −−3,2   2 2,1 1 ∂L3 ∂∂LL23 ∂∂LL13   xx= 00xx= xx= 0   where δθii= θ − θ0 is the difference between the drift rates of the argument of mean latitude of the reference orbit and one of the neighboring orbits, and  θθi = 0 . XX= 0   And δl3,ii= ll 3, − 3,0 is the difference between the drift rates of the ascending node of the reference orbit and one of the neighboring orbits, and  ll3,i = 3,0 . XX= 0 Now, the conditions satisfying the invariance property for the relative orbits are:   δθii=−= θ θ0 0 (7)   δl3,ii=−= ll 3, 3,0 0 (8) Substituting the included derivatives of the last two equations into Equations (5) and (6) and after the needed mathematical manipulations we will get:

222 Aθ()δ L++ AIA θθ δ() δ I + A θ δη + A θ() δ L δη LL11 1 I II ηη ()1,−−1 L1η 1() 1, 1 (7') ++AAθθδδη I+ Aθ+ Aθδδ I L =0 ηηI () ()1,− 1 L11LI ()()1

222 All33()δ L++ AIAll33 δ() δ I + A l 3 δη + A() δ L δη LL11 1 I II ηη ()1,−−1 L1η 1() 1, 1 (8') ++AAll33()δδη I() + AAll33+()()δδ I L =0 ηηI 1.− 1 L11 LI 1 Al3 Aθ Multiplying Equation (7') by LL11 and Equation (8') by LL11 and subtract- ing we will get: 2 aL1δ 1 δη 1,−− 1+ a 2() δη 1, 1 + aL 3 δ 1 + a 4 δη 1,− 1 += a 5 0 (9)

with the coefficients ai ’s are: a =θθll33 −  1 L1η LL11 L 1η LL11

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a =θθl3 − l3 2 ηη LL11 ηη LL11

aI=+θθθ ()δδ l3 −+  ll33() I 3 L11IL1 L11L LL1 L1 IL1

a = θθ++δδI l3 − θ ll33I 4 ηηI () LL11 LL11 ηIη()

22 a=+−+θθ()()δδ II  l3  θll33()()δδII  5  I II  LL11 L11LI II 

and the derivatives of θ ’s and l3 ’s are in the appendix. Solving Equation

(9), for δ L1 , we get:

2 a2()δη 1,−− 1++ aa 4 δη 1, 1 5 δ L1 = − aa1δη 1,− 1+ 3

Substituting the last results in Equation (7') we get the quartic equation:

432 bbbbb1()()()()δη 1,−−−− 1++++= 2 δη 1, 1 3 δη 1, 1 4 δη 1, 1 5 0 ,

where the coefficients bi ’s are functions of Li ’s and given by: b=−+ a22θ aa  θθ a 1 2LL11 21L1η 1 ηη

b=2 aaθ −+ aa aaθ − aa  θθ+ δ I 2 24 LL11 ()23 14 L1η 21L1 IL 1()

++2 θθδ +θ a1ηI η ()I2 aa13 ηη b=+ a222 aaθ −+ aa aa θθ + a 3() 4 25 LL11 ()43 51 L1η3 ηη −+aa aa θ + θδδI +2aa θθ + I ()23 14 L11 IL () 13ηηI () ++2 θθδδ2 aI1 I ()()II I b=2 aaθ − aa  θ −+ aa aa θθ+ δ I 4 45LL11 53L1η () 43 51 L1 IL 1()

++2  θθδ +θ δδ +θ 2 a3ηηI ()I 2aa31I ()() III I

2 b=−++ a22θ aa   θθ()()()δI a θ δδ I+  θ I 5 5LL11 53L1 IL 1 3I II

Solving the resulting equation we will get four roots of δη1,− 1 : δη =AA − A, ()1,− 1 1,2 1 2 3 (10) δη =AAA + . ()1,− 1 3,4 1 2 4

and four roots of δ L1 : a(A−222 AA + AA − AA)( +aa A −− A A ) + δ = − 2 5 12 23 13 4 1 2 3 5 ()L1 1 , aa11(AAA−− 2 3) + 3 a(A−222 AA − AA + AA)( +aa A −− A A ) + δ = − 2 5 12 23 13 4 1 2 3 5 ()L1 2 , aa11(AAA−− 2 3) + 3 (10') a(A+222 AA − AA − AA)( +aa A +− A A ) + δ = − 2 6 12 23 13 4 1 2 3 5 ()L1 3 , aa11(AAA+− 2 3) + 3 aa(A+++222 AA AA AA) +( A +−AA) +a δ = − 2 6 12 23 13 4 123 5 ()L1 4 , aa11(AAA+− 2 3) + 3

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where,

b2 1 1 A1 = − , A2=BBB 123 ++, A3=2BBBB 1234 −−− , 4b1 2 2 1 A =2BBBB −−+ , AAAA=++222, AAAA=++222, 42 1234 51 23 61 24 b2 2b 213C C C =2 − 3 = 1 = 4 = 2 B1 2 , B2 , B3 13 , B4 , 4b1 3b1 3bC14 32× b1 8A2 bb3 4bb 8 =−+2 =−+2423 − C1 b 33 bb 24 12 bb 15, C2 32 , bb11b1

3 22 C3=−++−2 b 3 9 bbb 234 27 bb 14 27 b 2 b 5 72 bbb 135,

1 3 3 CC= +−4 C + C2 43( () 1 3) .

3. Modeling of Invariant Relative Orbit Conditions for near Equatorial-Circular Case

When we choose the leader orbit to be circular equatorial then, we will obtain

four solutions for δη1,− 1 (which we redefined as δη for simplicity), as in Equ-

ation (10), and four solutions for δ L1 as in Equation (10'). Here, we will present the plots of these solutions, which we obtained in the last section (Equa-

tions (10) and (10')) to compare the effect of J4 on the conditions of invariance for the formation. Before we introduce the graphs, it is important to mention that in all figures, we provided a set of curves in each condition for the inclination of each member of the formation with respect to the leader orbit (−≤22δ I ≤) as example. By means that the follower satellites’ orbits, in the formation, will be inclined by a range of 2˚ with respect to the leader satellite orbit, of course we can extend this range. Also, we will introduce a comparison

between the effects of J4 in the formation. The variation of the formation relative to the leader orbit in δη is related to the variation in eccentricity for the followers through: e δη=−= δ e MAG() eδ e , 1− e2

i.e. the variation in the follower’s orbit in η1,− 1 (δη ) is scaled by MAG() e in their variations in eccentricities δ e . It is important to note that the scale function MAG() e is always negative for circular and elliptical orbits. In our

case the eccentricity of the leader orbit, e0 , is zero, then δ e= ee − 0 is always positive.

While the variation in the δ L1 , for the followers, is scaled by mag() a for the variation of the semi-major axis through the relation 1 δδL1 = a = mag() a δ a . It is important to note that the scale function 2 µa mag() a is always positive, but δ a maybe positive, negative or zero.

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3.1. The First Solution (Circular Formation)

The first solution of Equations (10) for δ e and (10') for δ a for the orbits of the followers with respect to the leader orbit, for inclination range [−2˚, +2˚] in

case of J2 and the net effects of J2 and J4 can represent in the following curves. Figure 1(a) and Figure 1(b) show the first choice of the invariant relative

conditions, in the equatorial-circular case. In this choice, δη and δ L1 get Zero for all δδae, and δ I values. That is mean, the formation for the followers orbit will be in the same orbital plane of the leader satellite (In Plane Circular Formation) with their eccentricities and semi-major axis was scaled by MAG() e and mag() a of the leader satellite. The scale function MAG() e , for this solution, is equal zero whatever the choosing the value of δ e . Also the scale function mag() a never equal zero, then δ a must equal zero. That can me conclude that the follower satellite’s must be in the same orbit of the leader one.

In this case, the effect of J4 has no significant variation in the formation.

3.2. The Second Solution

Figure 2(a) and Figure 2(b) show the second choice of the invariant relative formation, in the equatorial-circular case. In this solution, the choice of eccen- tricites and semi-major axis for the followers orbit is not affect the formation in

case of J2 effect whatever choosing the inclination for folowers orbits. But the J4 effects change the choosing of eccentricities slightly while the choosing the semi-major axis still unaffected whatever chosing the inclinations for the followers orbits. As we see in the vertical axis. The semi-magor axis and eccentri- cites of the followers orbits must be greater than those for the leader one.

(a)

(b)

. Figure 1. (a) The formation under the effect of J2 only for the 1st solution in the circular

equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 1st. solution in the circular equatorial of the leader orbit.

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(a)

(b)

Figure 2. (a) The formation under the effect of J2 only for the 2nd solution in the circular

equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 2nd solution in the circular equatorial of the leader orbit.

The effects of J4 are changing slightly while the choosing of eccentricies is not for choosing the semi-major axis.

3.3. The Third Solution

Figure 3(a) and Figure 3(b) show the third choice of the invariant relative formation, in the equatorial-circular case. In this solution, the value of the function MAG() eδ e <−1.3 which gives the limits for the eccentricities through the inqualety ee42+−1.69 1.69 0 . The solution for this inqualety has only one positive value 0.839936. With respect to choosing the semi-major axis, it is not depend on choosing the inclination, by mean that the formation will be in plane of the leader one.

In the case of the formation, under the effects of J2 and J4, the function MAG() eδ e is modified the limit of the eccentricities to be 0.774904.

3.4. The Fourth Solution

Figure 4(a) and Figure 4(b) show the fourth choice of the invariant relative formation, in the equatorial-circular case. In this choice, the formation for def- ferent inclination is distributed about the leader orbit (with δ I = 0 ) and the function MAG() eδ e is increasing for positive δ I while it is decreasing for negative δ I . For choosing the semi-major axis, it increases by increasing the semi-major axis of the leader orbit whatever choosing δ I .

In this formation choics, the effect of J4 is significant for choosing the eccentricities for the followers as it clear from the first of Figure 4(b). In the first of

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S. A. Altalhi, M. I. El Saftawy

(a)

(b)

Figure 3. (a) The formation under the effect of J2 only for the 3rd solution in the circular

equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 3rd solution in the circular equatorial of the leader orbit.

(a)

(b)

Figure 4. (a) The formation under the effect of J2 only for the 4th solution in the circular

equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 4th solution in the circular equatorial of the leader orbit.

Figure 4(b), the function MAG() eδ e is positive and increasing while MAG() e is always negative. And δ e , in our case, must be positive or zero. For that this solution is not mathematically accepted.

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Also, the second of Figure 4(b) shows that this choice is not continuous for orbits with semi-major axis greater than 2.5 earth radii. And the function mag() aδ a is decreasing by increasing the semi-major axis.

4. Conclusions

The problem formulated using the oblate Earth model, truncating its potential

series at J4 to the equations of motion, and then the canonical equations of motion and the Hamiltonian formed. In order to keep the relative motion invaria- ble, eight-second order conditions between the differences in the semi-major axis a and the inclination I are obtained. These conditions guarantee that the drift rates of neighboring orbits are equal on the average. The resulting orbits require less control and maintenance fuel. Then we studied the curves of these conditions in equatorial-circular case. The plots of these cases are presented as

relations between δη or δ L1 and the semi-major axis of the leader satellite orbit, at different δ I . In the first choice, we can conclude that the follower satellite’s must be in the same orbit of the leader one (on orbit formation). In the second choice, whatever the choosing the inclination for the followers it is not affect the choosing the semi-major axis (In plane with different eccentrisi- ties).

The third choice, under the effects of J2 and J4, the value of the function MAG() eδ e <−1.3 which gives the limits for the eccentricities through the in- equality ee42+−1.69 1.69 0 . The solution for this inqualety has only one positive value 0.839936. With respect to choosing the semi-major axis, it does not depend on choosing the inclination, by mean that the formation will be in plane of the leader one. The fourth choice, the function MAG() eδ e is positive and increasing while MAG() e is always negative. And δ e , in our case, must be positive or zero. For that, the eccentricities of the followers must be negative and that is not mathematically accepted. Also, the second of Figures 4(b) shows that this choice is not continuous for orbits with semi-major axis greater than 2.5 earth radii. And the function mag() aδ a is decreasing by increasing the semi-major axis.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References [1] Hughes, S.P. (1999) Formation Flying Performance Measures for Earth-Pointing Missions. MSc Thesis, Blacksburg, Virginia. [2] Clohessy, W.H. and Wiltshire, R.S. (1960) Terminal Guidance System for Satellite Rendezvous. Journal of the Aerospace Sciences, 27, 653-658. https://doi.org/10.2514/8.8704

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[3] Tschauner, J. and Hempel, P. (1965) Rendezvous Zueinem in Elliptischer Bahn Umlaufenden Ziel. Acta Astronautica, 11, 104-109.

[4] Schaub, H. and Alfriend, K. (2001) J2 Invariant Relative Orbits for Spacecraft For- mations. Celestial Mechanics and Dynamical Astronomy, 79, 77-95. https://doi.org/10.1023/A:1011161811472 [5] Abd El-Salam, F.A., El-Tohamy, I.A., Ahmed, M.K., Rahoma, W.A. and Rassem, M.A. (2006) Invariant Relative Orbits for Satellite Constellations: A Second Order Theory. Applied Mathematics and Computation, 181, 6-20. https://doi.org/10.1016/j.amc.2006.01.004 [6] Li, X. and Li, J. (2005) Study on Relative Orbital Configuration in Satellite Forma- tion Flying. Acta Mechanica Sinica, 21, 87-94. https://doi.org/10.1007/s10409-004-0009-3 [7] Abd El-Salam, F.A. and El-Saftawy, M.I. (2012) Second Order Constraints in the Theory of Invariant Relative Orbits Including Relativistic and Direct Solar Radia- tion Pressure Effects. Indian Journal of Science and Technology, 5, 1-14. [8] Rahoma, W.A. (2013) Lunisolar Invariant Relative Satellite Orbits. American Jour- nal of Applied Sciences, 10, 307-312. https://doi.org/10.3844/ajassp.2013.307.312 [9] Kamel, A.A. (1969) Expansion Formulae in Canonical Transformations Depending on a Small Parameter. Celestial Mechanics, 1, 190-199. https://doi.org/10.1007/BF01228838 [10] El-Saftawy, M.I., Ahmed, M.K.M. and Helali, Y.E. (1998) The Effect of Direct Solar Radiation Pressure on a Spacecraft of Complex Shape, II. Astrophysics and Space Science, 259, 151-171. https://doi.org/10.1023/A:1001577425093

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Appendix

2 J n 2 J n 2 J n θ = 2 θ θ = 2 n θ = 2 θ L1 ∑n , η ∑θ , I ∑n , n=0 n! n=1 n! n=1 n!

2 J n 2 J n 2 J n θ = 2 θ θ = 2 θ θ = 2 θ LL11 ∑n , ηη ∑n , II ∑n , n=0 n! n=1 n! n=1 n!

2 J n 2 J n 2 J n θ = 2 θ θ = 2 θ θ = 2 θ L1η ∑n , Iη ∑n , IL1 ∑n . n=1 n! n=1 n! n=1 n!

2 J n 2 J n 2 J n l3 = 2 l3 l3 = 2 l3 l3 = 2 l3 L1 ∑n , η ∑n , I ∑n , n=1 n! n=1 n! n=1 n!

2 J n 2 J n 2 J n l3 = 2 l3 l3 = 2 θ l3 = 2 l3 LL11 ∑n , ηη ∑n , II ∑n , n=1 n! n=1 n! n=1 n!

2 J n 2 J n 2 J n l3 = 2 l3 l3 = 2 l3 l3 = 2 l3 L1η ∑n , Iη ∑n , IL1 ∑n . n=1 n! n=1 n! n=1 n!

With,

θ θ ∂D0  0 = , ∂L1

θθ θ θ ∂∂DDnn ∂ D n  n = +ηη1,−− 1 + 1, 1 , ∂∂LL1 2 ∂ L3

θθ θ ∂∂DDnn n = LL11+  , ∂∂LL23

θ θ ∂Dn n = −L2 , ∂L3

2 θ θ 1 ∂ D0 0 = , 2 ∂∂LL11

2θ 222 θθθ θ 11∂Dnn ∂∂∂ DDDnn2 n =+η2,− 2  ++2 22∂∂LL11  ∂∂ LL22 ∂∂ LL 33 ∂∂ LL 23 , 22θθ ∂∂DDn n ++η1,− 1  ∂∂LL1132∂∂ LL

22θθθ2 θ 1 2 ∂∂∂DDDn 2 nn  n =L1  ++2 , 2 ∂∂LL22 ∂ LL 33∂ ∂∂ LL 23

2 θ θ 1 2 ∂ Dn n = η0,− 2 , 2 ∂∂LL33

2θ 2 θ 2 θ 22 θθ θ ∂Dn2 ∂ D n ∂ D n ∂∂ DD nn n = LL2 ++2 + 1  +  , ∂∂LL22 ∂∂ LL 33 ∂∂ LL 23 ∂∂ LL 12 ∂ LL 13∂ 

22θθ θ ∂∂DDn n n = −η−−1, 1 +  ,  ∂∂LL33 ∂ LL23∂ 

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22θθθ 2 θ ∂∂DDnn ∂ D n n = −ηη1,−− 2  ++1,1 , ∂∂LL33 ∂ LL2∂3 ∂∂ LL 13

∂l3 ∂∂ ll 33 l3 Dnn DDn  n =++η1,− 1  , ∂L1  ∂∂ LL2 3

∂∂l33l l3 DDn n n =L1  +  , ∂∂LL23

∂ l3 l3 Dn n = −L2 , ∂L3

∂22l3 ∂∂∂ lll 33322 l3 11Dn DDD nn2 n n = + η2,− 2  ++2  22∂∂LL11  ∂∂ LL22 ∂∂ LL 33∂∂ LL 23 , 22l33l ∂∂DDn n +η1,− 1 +  ∂∂LL1123 ∂∂ LL

∂∂∂222ll333l l3 1 22DDDnnn  n =L1  ++2  , 2 ∂∂LL22 ∂∂ LL 33 ∂ LL 23∂ 

∂2 l3 l3 1 22 Dn n = L2 , 2 ∂LL33∂

∂2l3 ∂ 2 l 3 ∂ 2 l 3 ∂∂ 22 ll 33 l3 Dn 2 Dn D n DD nn n =LL2  ++2 ++1  , ∂∂LL22 ∂∂ LL 33 ∂∂ LL 23 ∂∂ LL 12 ∂∂ LL13

∂∂2 ll332 l3 DDn n n =−+η−−1, 1 ,  ∂∂LL33∂ LL 23∂ 

∂∂22ll33 ∂ 2 l 3 l3 DDn nn D n =−ηη1,−− 2   ++1,1  , ∂∂LL33 ∂ LL 2∂ 3 ∂ LL13∂

with,

−3 2 2 KL01= , K1 =()93 − η4,3 , K2 =()15 − 3 η3,4 ,

42 42 K3 =(23907 +− 1782 3897)η1,10 , K4 =(1839 +− 162 433)η2,9 ,

42 42 K5 =−−+( 902 7452 5026)η3,8 , K6 =(11274 +− 588 3990)η4,7 ,

42 42 K7 =(4203 +− 3734 5921)η5,6 , K8 =−+−( 225 1570 1825)η6,5 ,

42 42 K9 =−+−( 525 450 45)η4,7 , K10 =(525 −+ 450 45)η6,5 ,

42 42 K11 =−+−( 1925 1350 105)η3,8 , K12 =(945 −+ 630 45)η5,6 ,

and,

3 3 Z1 = −6η3,4 , Z2 =(324 + 7356 )η1,10 , Z3 =−−( 1656 328 )η3,8 ,

3 3 Z4 =(768 + 4608 )η4,7 , Z5 =(628 − 180 )η5,6 ,

3 3 Z6 =(700 − 300 )η3,8 , Z7 =−+( 420 180 )η5,6 .

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International Journal of Astronomy and Astrophysics, 2019, 9, 231-246 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

The Distance Modulus in Dark Energy and Cardassian Cosmologies via the Hypergeometric Function

Lorenzo Zaninetti

Physics Department, via P. Giuria 1, Turin, Italy

How to cite this paper: Zaninetti, L. Abstract (2019) The Distance Modulus in Dark Energy and Cardassian Cosmologies via the The presence of the dark energy allows both the acceleration and the expan- Hypergeometric Function. International sion of the universe. In the case of a constant equation of state for dark ener- Journal of Astronomy and Astrophysics, 9, gy we derived an analytical solution for the Hubble radius in terms of the 231-246. https://doi.org/10.4236/ijaa.2019.93017 hypergeometric function. An approximate Taylor expansion of order seven is derived for both the constant and the variable equation of state for dark Received: June 11, 2019 energy. In the case of the Cardassian cosmology, we also derived an analytical Accepted: September 1, 2019 Published: September 4, 2019 solution for the Hubble radius in terms of the hypergeometric function. The astronomical samples of the distance modulus for Supernova (SN) of type Ia Copyright © 2019 by author(s) and allows the derivation of the involved cosmological in the case of constant eq- Scientific Research Publishing Inc. uation of state, variable equation of state and Cardassian cosmology. This work is licensed under the Creative

Commons Attribution International License (CC BY 4.0). Keywords http://creativecommons.org/licenses/by/4.0/ Cosmology, Observational Cosmology, Distances, Redshifts, Radial Open Access Velocities, Spatial Distribution of Galaxies, Magnitudes and Colors, Luminosities

1. Introduction

The name dark energy started to be used by [1] in order to explain both the expansion and both the acceleration of the universe. In a few years the dark energy was widely used as a cosmological model to be tested. Many review papers have been written; we select among others a general review by [2] and a theoretical review by [3]. The term wCDM has been introduced to classify the case of constant equation of state and we will use in the following wzCDM to classify the variable equation of state. The Cardassian cosmology started with [4] and was introduced in order to model both the expansion and the acceleration of

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L. Zaninetti

the universe, the name from a humanoid race in Star Trek. As an example [5] derived the cosmological parameters for the original Cardassian expansion and the modified polytropic Cardassian expansion. The cosmological theories can be tested on the samples of Supernova (SN) of type Ia. The first sample to be used to derive the cosmological parameters contained 7 SNs, see [6], the second one contained 34 SNs, see [7] and the third one contained 42 SNs, see [8]. The above historical samples allowed to derive the cosmological parameters for the expanding and accelerating universe. At the moment of writing the astronomical research is focused on value of the distance modulus versus the redshift: the Union 2.1 compilation contains 580 SNs, see [9], and the joint light-curve analysis (JLA) contains 740 SNs, see [10]. The above observations can be done up to a limited value in redshift z ≈ 1.7 , we, therefore, speak of evaluation of the distance modulus at low redshift. This limited range can be extended up z ≈ 8 , the high redshift region, analyzing the Gamma-Ray Burst (GRB) and, as an example, [11] has derived the distance modulus for 59 calibrated high- redshift GRBs, the so-called “Hymnium” GRBs sample. This paper reviews in Section 2.1. The ΛCDM cosmology evaluates the basic integral of wCDM cosmology in Section 3, introduces a Taylor expansion for the basic integral of wzCDM cosmology in Section 4 and analyzes the Cardassian model in Section 5. The parameters which characterize the three cosmologies are derived via the Levenberg-Marquardt method in Section 6.

2. Preliminaries

This section reviews the ΛCDM cosmology and the adopted statistics.

2.1. The Standard Cosmology

In ΛCDM cosmology the Hubble distance DH is defined as c DH ≡ . (1) H0

The first parameter is ΩM 8πGρ Ω= 0 M 2 , (2) 3H0

where G is the Newtonian gravitational constant, H0 is the Hubble constant

and ρ0 is the mass density at the present time. The second parameter is ΩΛ Λc2 Ω≡ Λ 2 , (3) 3H0 where Λ is the cosmological constant, see [12]. These two parameters are con-

nected with the curvature ΩK by

ΩM +ΩΛ +ΩK =1. (4)

The comoving distance, DC is z dz′ DD= (5) CH∫0 Ez()′

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where Ez() is the “Hubble function”

32 Ez()()()= ΩM 1 + z +ΩK 1. + z +ΩΛ (6)

In the case of ΩK , we have the flat case.

2.2. The Statistics

The adopted statistical parameters are the percent error, δ , between theoretical value and approximated value, the merit function χ 2 evaluated as 2 N 2 yyi,, theo− i obs χ = ∑  (7) i=1 σ i

where yi, obs and σ i represent the observed value and its error at position i 2 and yi, theo the theoretical value at position i, the reduced merit function χred , the Akaike information criterion (AIC), the number of degrees of freedom NF= n − k where n is the number of bins and k is the number of parameters and the goodness of the fit as expressed by the probability Q.

3. Constant Equation of State

In dark matter cosmology, wCDM, the Hubble radius is 1 dzH ( ;ΩM ,, w Ω=DE ) , (8) 3 33+ w ()()11+zz ΩM +ΩDE +

where w parametrizes the dark energy and is constant, see Equation (3.4) in [13] or Equation (18) in [14] for the luminosity distance. In flat cosmology

ΩM +ΩDE =1, (9) and the Hubble radius becomes 1 dzH ();,Ω=M w . (10) 3 33+ w ()()()1+zz ΩMM + 11 −Ω +

The indefinite integral in the variable z of the above Hubble radius, Iz , is Ω= Ω Izzwdzwz()();,MM∫ H ;,d. (11)

3.1. The Analytical Solution

In order to solve the indefinite integral we perform a change of variable 1+=zt13 11 Ω= Iz() t;,M w ∫ d.t (12) 3 w 23 −t(() −1 +ΩMM tt −Ω )

The indefinite integral is

w 11−− 1 t −()1 −Ω − −11 −− M 221 F ,ww ;1 ; 26 6 ΩM Iz t;,Ω= w , ()M 6 (13) ΩM t

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where 21F()abcz ,;; is the regularized hypergeometric function, see Appendix B. This dependence of the above integral upon the hypergeometric function has been recognized but not developed by [15]. We now return to the variable z, the redshift, and the indefinite integral becomes

w −zzz32 +3 + 311 +() −Ω 11−−11 1 ( ) M −221 F , −ww ;1 −− ; 26 6 −Ω M Iz() z;,Ω=M w . (14) 6 32 ΩM zzz +3 ++ 31

We denote by Fz();,ΩM w the definite integral

F()( z;,ΩM w ==Ω−=Ω Iz z z ;, MM w)( Iz z 0;,. w) (15)

3.2. The Taylor Expansion

We evaluate the integrand of the integral (11) with a first series expansion, TI

about z = 0 , denoted by I and a second series expansion, TII , about z = 1, denoted by II . The order of expansion for the two series is 7. The integration of

TI in z is denoted by IzI ,7 and gives i=7 i IzI,7() z;,Ω= M w∑ cIi, z (16) i=1

and the coefficients, cIi, , are reported in Appendix A. The integral, IzII ,7 of

the second Taylor expansion about z = 1, TII is complicated and we limit

ourselves to order 2, IzII ,2 , see Appendix A. The two definite integrals,

FzI ,7();,Ω M w and FzII ,7();,Ω M w are

FII,7()( z;,Ω= M w Iz,7 z =Ω−=Ω z ;,M w)( Iz I,7 z 0;,,M w) (17)

and

FII ,7()( z;,Ω= M w IzII ,7 z =Ω−=Ω z;,M w)( IzII ,7 z 0;,.M w) (18)

The percent error, δ , between the analytical integral F and the two

approximations, FI ,7 and FII ,7 is evaluated as F δ =−×1I ,7 100 (19) I F

F δ =−×1II ,7 100. (20) II F

On inserting the astrophysical parameters as reported in Table 1 we have

δδI= II at z ≈ 0.58 , see Figure 1. The above value in z will, therefore, be the boundary between region I and region II for the Taylor approximation of the definite integral

FzII ,7();Ω M , w , 0.58 ≤≤ z 1.4 Fz7M();,Ω= w  (21) FzI ,7( ;Ω M , w ), 0 << z 0.58

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2 2 Table 1. Numerical values from the Union 2.1 compilation of χ , χred and Q, where k stands for the number of parameters.

2 2 Cosmology SNs k parameters χ χred Q

ΛCDM 580 3 H 0 = 69.81 ; Ω=M 0.239 ; Ω=Λ 0.651 562.61 0.975 0.658

wCDM

Hypergeometric 580 3 H 0 =(70.02 ± 0.35) ; Ω=M (0.277 ± 0.025) ; w =−±( 1.003 0.05) 562.21 0.974 0.662 solution

wCDM Taylor 580 3 H =(70.02 ± 0.47) ; Ω=(0.282 ± 0.07) ; w =−±()1.01 0.2 562.21 0.974 0.662 approximation 0 M

wzCDM Taylor 580 4 H =()70.08 ± 0.31 ; Ω=(0.284 ± 0.01) ; w =−±( 1.03 0.031) ; w =()0.1 ± 0.018 ; 562.21 0.976 0.651 approximation 0 M 0 1

Cardassian 58k0 3 H 0 =(70.15 ± 0.38) ; Ω=M (0.305 ± 0.019) ; n =−±( 0.081 0.01) 562.35 0.974 0.661

Figure 1. Numerical values of δ I (full red line) and δ II (dashed blue line) as function of the redshift, parameters as in Table 1.

4. Variable Equation of State

The dark energy as function of the redshift is assumed to be z wz() = w + w , (22) 011+ z

where w0 and w1 are two parameters to be fixed by the fit. The Hubble radius in wzCDM cosmology is 1 dH ( z;Ω=M01 ,, ww) (23) wz1 ++−3 3 3ww0133 1+ z ()()()1+zz ΩMM + 11 −Ω + e

which is the same as Equation (20) in [14]. The above integral does not yet have an analytical expression and we evaluate the integrand with a first series expansion about z = 0 and a second series expansion about z = 1. Also here the order of the two series expansion is 7. The integration in z is denoted by

IwzI ,7 and gives

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i=7 i IwzI,7( z;Ω= M ,, w 0 w 1 ) ∑ cIi, z (24) i=1

and the first five coefficients, cIi, , are reported in Appendix C. The integral,

IwzII ,7 of the second Taylor expansion about z = 1 is complicated and we limit

ourselves to order 2, IwzII ,2 , see Appendix C. The two definite integrals,

FwzI ,7( z;Ω M ,, w 0 w 1 ) and FwzII ,7( z;Ω M ,, w 0 w 1 ) are

FwzII,7( z;Ω M ,, w 0 w 1 )( ==Ω−=Ω Iwz,7 z z;M ,, w 0 w 1 )( Iwz I,7 z0;,,M w 0 w 1 ) , (25)

and

FwzII ,7( z;Ω==Ω−=Ω M ,, w 0 w 1 )( IwzII ,7 z z;M ,, w 0 w 1 )( IwzII ,7 z0;,,M w 0 w 1 ) .(26)

Finally the definite integral, Fwz , is

FwzII ,7( z;Ω M , w 0 , w 1 ) , 0.58≤≤ z 1.4 Fwz7( z;Ω= M01 ,, w w )  (27) FwzI ,7( z;Ω M , w 0 , w 1 ) , 0<< z 0.58

The above definite integral can also be evaluated in a numerical way,

Fwznum ( z;ΩM01 ,, w w ) .

5. Cardassian Cosmology

In flat Cardassian cosmology the Hubble radius is 1 dH () z;Ω=M ,, wn , (28) 33n ()()()1+zz ΩMM + 11 −Ω +

where n is a variable parameter, n = 0 means ΛCDM cosmology, see Equation (17) in [14]. The indefinite integral in the variable z of the above Hubble radius, Iz , is Ω= Ω Izzndznz()();,MM∫ H ;,d. (29)

Also here in order to solve the indefinite integral we perform a change of variable 1+=zt13 11 Ω= Iz() t;,M n ∫ d.t (30) 3 nn23 −t ΩMM +Ω t + tt

The indefinite integral is n−1 −1 67n − t ()Ω−1 − −− M 221 F 1 2,() 6n 6 ; ; 66n −ΩM Iz t;,Ω= n , ()M 6 (31) ΩM t

where 21F()abcz ,;; is the regularized hypergeometric function. We now return to the original variable z as function of z which is − 3 n 1 11+z Ω− −1 67n − ()() ()M −2 F 1 2, −−() 6n 6 ; ; 21−Ω 66n M  Iz() z;,Ω=M n . (32) 6 3 Ω+M ()1 z

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We denote by Fzc ();,ΩM n the definite integral

Fc ()()() z;,Ω=M n Iz z =Ω−=Ω z ;, MM n Iz z 0;,. n (33)

6. The Distance Modulus

The luminosity distance, dL , for wCDM cosmology in the case of the analytical solution is c dL( zcH;, 0M ,Ω= , w) ()() 1 + z F z ; ΩM , w , (34) H0

where Fz();,ΩM w is given by Equation (15) and in the case of the Taylor approximation is c dL,7 ( zcH;,0 ,Ω= M , w) ()() 1 + z F7 z ; Ω M , w , (35) H0

where Fz7M();,Ω w is given by Equation (21). The distance modulus in the case of the analytical solution for wCDM is

()m−=+ M 25 5log10(d L ( zcH ; ,0 ,Ω M , w)) , (36)

and in the case of the Taylor approximation −=+ Ω ()m M 7 25 5log10(d L,7 ( zcH ; ,0 , M , w)) . (37)

In the case of variable equation of state, wzCDM, the numerical luminosity distance is c dL,num ( zcH;,,0Ω=+Ω M01 ,, ww) () 1 zFwznum ( z ;M01 ,, ww) , (38) H0

where Fwznum ( z;ΩM01 ,, w w ) is the definite numerical integral and the Taylor approximation for the luminosity distance is c dL,7( zcH;,, 0Ω=+Ω M01 ,, ww) () 1 zFwzz7( ; M01 ,, ww) , (39) H0

where Fwz7( z;Ω M01 ,, w w ) is given by Equation (27). In wzCDM, the numerical distance modulus is −=+ Ω ()m M num 25 5log10(d L,num ( zcH ; ,0 , M , w 0 , w 1 )) , (40)

and the Taylor approximated distance modulus is −=+ Ω ()m M 7 25 5log10(d L,7 ( zcH ; ,0 , M , w 0 , w 1 )) . (41) In the case of Cardassian cosmology the luminosity distance is c dL( zcH;, 0M ,Ω= , n) ()() 1 + z Fc z ; ΩM , n , (42) H0

where Fzc ();,ΩM n is given by Equation (33) and the distance modulus is

()m−=+ M 25 5log10(d L ( zcH ; ,0 ,Ω M , n)) . (43)

The cosmological parameters unknown are three, H0M,Ω and w, in the case

of wCDM and four, Hw0,,Ω M0 and w1 , in the case of wzCDM. In flat

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Cardassian cosmology the number of parameters is three, H0M,Ω and n. In the presence of a given sample for the distance modulus, we can map the chi-square as given by Formula (7), see Figure 2 in the case of wCDM with hypergeometric solution. The above cosmological parameters are obtained by a fit of the astronomical data for the distance modulus of SNs via the Levenberg-Marquardt method (subroutine MRQMIN in [16]) which minimizes the chi-square as given by Formula (7). Table 1 presents the above cosmological parameters for the Union 2.1 compilation of SNs and Figure 3 reports the best fit. As a practical example of the utility of the cosmological parameters determination, we report the distance modulus in an explicit form for the Union 2.1 compilation in wCDM.

2 Figure 2. Map of the χ in wCDM cosmology when H0 =(70.02 ± 0.35) .

Figure 3. Hubble diagram for the Union 2.1 compilation. The solid line represents the best fit for the exact distance modulus in wCDM cosmology as represented by Equation (36). Parameters as in third line of Table 1; Union 2.1 compilation.

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 1  ()mM−=+5 5 × ln 4281.52() 1 +z ln() 10  

− (44) 1 32 1.003  21F 0.1661, ;1.1661;− 2.6101( zzz + 3 ++ 3 1)  2 ×−3.8 +3.4146  6 zzz32+3 ++ 31   

when 0<

−   32 0.16666 +5ln − 4273.59() 1 +z 3.62142( zzz+ 3++ 3 1) (45)  

− 32 1.081  ×21F 0.15417,1 2;1.1541;− 2.2786( zzz+ 3 ++ 3 1) − 3.304 ( ) 

when 0<

Figure 4. Hubble diagram for the JLA compilation. The solid line represents the best fit for the exact distance modulus in wCDM cosmology as represented by Equation (36). Parameters as in the third line of Table 2; JLA compilation.

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Figure 5. Hubble diagram for the Union 2.1 compilation + the “Hymnium” GRBs sample. The solid line represents the best fit for the exact distance modulus in wCDM cosmology as represented by Equation (36). Parameters as in second line of Table 3.

2 2 Table 2. Numerical values for the JLA compilation of χ , χred and Q, where k stands for the number of parameters.

2 2 Cosmology SNs k parameters χ χred Q

ΛCDM 740 3 H 0 = 69.39 ; Ω=M 0.18 ; Ω=Λ 0.537 625.74 0.849 0.99

wCDM

Hypergeometric 740 3 H 0 =()69.71 ± 0.5 ; Ω=M (0.293 ± 0.021) ; w =−±( 0.996 0.08) 627.908 0.851 0.998 solution

wCDM Taylor 740 4 H =(69.99 ± 0.29) ; Ω=(0.133 ± 0.13) ; w =−±( 0.709 0.18) 625.69 0.848 0.998 approximation 0 M

wzCDM Taylor 740 4 H =(69.99 ± 0.29) ; Ω=()0.3 ± 0.009 ; w =−±( 1.05 0.027) ; w =(0.097 ± 0.01) ; 628.76 0.854 0.998 approximation 0 M 0 1

Cardassian 740 3 H 0 =(70.036 ± 0.44) ; Ω=M (0.301 ± 0.019) ; n =−±( 0.055 0.0045) 628.73 0.863 0.999

2 2 Table 3. Numerical values from the Union 2.1 compilation + the “Hymnium” GRBs sample of χ , χred and Q, where k stands for the number of parameters.

2 2 Cosmology SNs k parameters χ χred Q

ΛCDM 639 3 H 0 = 69.80 ; Ω=M 0.239 ; Ω=Λ 0.651 586.08 0.921 0.922

wCDM

Hypergeometric 639 3 H 0 =()70.12 ± 0.4 ; Ω=M (0.294 ± 0.024) ; w =−±( 1.04 0.04) 585.42 0.92 0.924 solution

wzCDM numerical 639 4 H =()70 ± 0.32 ; Ω=()0.3 ± 0.011 ; w =−±( 1.05 0.033) ; w =()0.1 ± 0.01 ; 585.59 0.922 0.92 integration 0 M 0 1

Cardassian 639 3 H 0 =(70.10 ± 0.42) ; Ω=M (0.299 ± 0.019) ; n =−±( 0.063 0.0095) 585.43 0.92 0.924

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7. Conclusions

Constant equation of state In the case of wCDM cosmology, we found a new analytical expression for the Hubble distance in terms of the hypergeometric function, see Equation (13). As a consequence an analytical expression for the luminosity distance and the distance modulus is derived. Two approximate Taylor expansions for the Hubble distance about z = 0 and z = 1 of order 7 are also derived. The

derivation of the value of w, ΩM and H0 , here considered as a parameter to be found, is given for the Union 2.1 compilation, the JLA compilation and the Union 2.1 compilation plus the “Hymnium” GRBs sample, see Tables 1-3. As an example, in the case of the Union 2.1 compilation, we have derived

H0 =(70.02 ± 0.35) , Ω=M (0.277 ± 0.025) and w =−±( 1.003 0.05) . Variable equation of state In the case of wzCDM cosmology the Hubble distance, Equation (23) is evaluated numerically and with a Taylor expansion of order 7, see Equation (24).

The four parameters w0 , w1 , ΩM and H0 are reported in Tables 1-3. As an example, in the case of the Union 2.1 compilation, we have found

H0 =()70.08 ± 0.31 , Ω=M ()0.284 ± 0.01 , w0 =−±( 1.03 0.031) , and

w1 =()0.1 ± 0.018 . High redshift The inclusion of the “Hymnium” GRBs sample allows to extend the calibration of the distance modulus up to z = 8 (see Table 3). As an example, the Union 2.1 compilation + the “Hymnium” GRBs sample gives

H0 =()70 ± 0.32 , Ω=M ()0.3 ± 0.011 , w0 =−±( 1.05 0.033) , and

w1 =()0.1 ± 0.01 . Cardassian cosmology A new solution for the Hubble radius for Cardassian cosmology is presented in terms of the hypergeometric function, see Equation (reficardz). As an example, in the case of the Union 2.1 compilation, we have derived

H0 =(70.15 ± 0.38) , Ω=M (0.305 ± 0.019) and n =−±( 0.081 0.01) .

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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Appendix A. Taylor Expansion When W Is Constant

The coefficients of the Taylor expansion of IzI ,7() z;,Ω M w about z = 0

cI ,1 = 1, (A.1)

cwI ,2 =34 Ω−M 34 w − 34, (A.2)

9Ω22w c=−32 Ω ww22 −Ω+38 ww + +58 + M , (A.3) I ,3 M M 8 71ww 932 35 45 w45ΩΩww2 135 22 c =−− −− +MM − I ,4 64 64 64 64 16 64 (A.4) 243ΩΩΩ23w 117 w3 135 33 ww 71 Ω −+++MMM M, 64 64 64 64 93w 63 27 ww34 27 309 w 2309ΩΩww2 927 22 c = ++ + + −MM + I ,5 80 128 80 640 320 80 320 729Ω23w 351 ΩΩ ww3 8133 2349 Ω 24 w 27 Ω w4 +M − MM −+ M − M (A.5) 80 80 16 320 16 81ΩΩΩ34w 567 44 ww 93 −+MM − M, 8 128 80 3043w 231 27 w5 141 ww 34 63 14175Ω45w c =−−− − − − M I ,6 2560 512 2560 256 512 512 5103Ω55w301w2 301 ΩΩ ww2 903 22 +M −+MM − 512 256 64 256 3807ΩΩ23wwww 18333 2115 ΩΩ33 5481 24 −++−MM MM (A.6) 256 256 256 256 315ΩΩΩΩwwww434442 945 6615 2673 5 ++−MMMM − 64 32 512 256 3267ΩΩw5 6885 35 ww 3043 Ω +++MM M, 2560 256 2560 2689ww 8165 81 w 171 w 3 1665 w 4 48259 w 2 429 c = + ++ + + + I ,7 2240 35840 2240 224 7168 35840 1024 95985ΩΩ46ww 1968356 24057 Ω66 w 61479 Ω26 w +−+MM M + M 1024 256 1024 5120

1053Ωw6 23085 ΩΩΩ36 www 6075 45 2187 55 −−M MMM +− 1280 448 64 64 8019Ω25w 9801ΩΩww5 2065535 144855 Ω24 w +−M MM−+ M 224 2240 224 3584 8325Ωw4 24975 Ω34 w 24975 ΩΩ44 ww 4617 23 −−M M + MM + 896 448 1024 224 2223ΩΩww3 2565 33 48259 Ω w2 144777 Ω22 w −−−MM M + M (A.7) 224 224 8960 35840 2689wΩ − M . 2240 The integral of the Taylor expansion of order 2 about z = 1 is

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N Iz = , (A.8) II ,2 D where w w w ww N =×Ω(38Mwz −× 68 w Ω+×Ω MM 38z − 3 wz 8 −×Ω 148 M (A.9) ww w +6wz 8 − 3 8 −Ω 3MM z + 14 × 8 + 14 Ω ) z and 32 33++ww33 D =−( 2 ΩMM + 28 +Ω ) . (A.10)

B. The Hypergeometric Function

The regularized hypergeometric function, 21F()abcz ,;; , as defined by the Gauss series, is

∞ ()()absss ab aa()()++11 bb 2 21F()abcz ,;; =∑ z =++1 z z + s=0 ()c s! c cc()+1 2! s (B.1) Γ()c∞ Γ+Γ+()() as bs = ∑ z s Γ()()a Γ bs=0 Γ+() c ss! = + where z x iy , ()a s is the Pochhammer symbol = + +− ()()()as aa1 a s 1, (B.2) Γ()z is the Gamma function defined as ∞ Γ=()ze−−tz tt1 d, (B.3) ∫0 z is a complex variable defined on the disk z < 1 that should not be confused with the redshift, see [21] [22] [23] [24] [25]. The following relationship

−a x 21F()()abcx ,;;=−− 1 x21 F ac , bc ;; (B.4) x −1 connect the the hypergeometric function with x in (−1, 1) to one with x in 1 −∞, , see more details in [26]. 2

C. Taylor Expansion When W Is Variable

The coefficients of the Taylor expansion of IwzI ,7( z;Ω M ,, w 0 w 1 ) about z = 0

cI ,1 = 1, (C.1) 3 33 cw= Ω− w −, (C.2) I ,24 0 M 44 0 9Ω22w c=58 + wwww − 14 + 14 Ω− Ω+38 w22 − 32 Ω w + M0, (C.3) I ,3 0 1 1 M 0 M 0 M 0 8 35 71w 17ww 17 Ω 71wΩ 45 w2 9 ww c =−−+0 1 − 1M +0 M − 0 + 01 I ,4 64 64 32 32 64 64 32 45ΩΩΩΩwwww2 135 22 243 23 117 3 +−M0 M0 − M0 + M0 (C.4) 16 64 64 64 135Ω33w 9 w 3 9 ΩΩ ww 27 2 ww +M0 −− 0 M01 + M01, 64 64 8 32

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27w3 63 9ww2ΩΩ2349ΩΩ24ww 27 4 27 22 c =0 +−1M +M0 −M0 +1M I ,5 80 128 40 320 16 160 81ΩΩ34w 567 43 w 27 ww 2 309 w2 3 729Ω2w3 −M0 + M0 − 01 +−+ 0 ww M0 8 128 160 320 401 80 351ΩΩw3 81 33 ww 93 129ww 9 2 27w4 −M0 − M0 + 0 −11 ++0 (C.5) 80 16 80 160 160 640 351Ω ww2 129w Ω 309ΩΩw2 927 22 ww 93 Ω ++M01 1M−+−M0 M0 0 M 160 160 80 320 80 81ΩΩ32ww 729 22 ww 9 +M01 − M01 − Ω2 ww +Ω3. ww 32 160 4 M01 M01 The integral of the Taylor expansion of order 2 about z = 1 in the case wzLCDM cosmology Nwz Iwz = , (C.6) II ,2 Dwz where 3 w1 4 12++ 3ww01 3 12++ 3ww01 3 32w1 Nwz =×Ω+×Ω−Ωe( 6 2 M0zw 3 2 M1zw 6eM 2z

12++ 3ww01 3 12++ 3ww01 3 12++ 3ww01 3 +×Ω−×Ω−×Ω62 Mzw122 0 M 62 M1 w

12++ 3ww01 3 12++ 3ww01 3 32w1 −×6 2 zw0−×3 2 zw 1M+28e Ω 2 (C.7)

12++ 3ww01 3 12++ 3ww01 3 12++ 3w03w1 −28 × 2 Ω−×M 6 2z + 12 × 2 w0

12++ 3ww01 3 12++ 3ww01 3 +×6 2 wz1 +28 × 2 )

and

32 33ww01++ 33 ww 01 32w1 Dwz =64( −ΩMM 2 + 2 + Ω e) . (C.8)

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International Journal of Astronomy and Astrophysics, 2019, 9, 247-264 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Gamma-Ray Bursts Generated by Hyper-Accreting Kerr Black Hole

Feyiso Sado

Department of Physics, Addis Ababa University, Addis Ababa, Ethiopia

How to cite this paper: Sado, F. (2019) Abstract Gamma-Ray Bursts Generated by Hy- per-Accreting Kerr Black Hole. Interna- The observed properties of Gamma-Ray Bursts such as rapid variability of tional Journal of Astronomy and Astro- X-ray light curve and large energies strongly signature the compact binary, physics, 9, 247-264. disk accreting system. Our work particularly highlights the extremely rotat- https://doi.org/10.4236/ijaa.2019.93018 ing, disk accreting black holes as physical source of the flares variability and Received: April 27, 2019 X-ray afterglow plateaus of GRBs. We investigate the compact binary mergers Accepted: September 2, 2019 (neutron star - neutron star and neutron star onto black hole) and gravita- Published: September 5, 2019 tional core collapse of super massive star, where in both cases hyper-accreting

Copyright © 2019 by author(s) and Kerr hole is formed. The core collapse in a powerful gravitational wave ex- Scientific Research Publishing Inc. plained as a potential source for the radiated flux of hard X-rays spectrum. This work is licensed under the Creative We described the evolution of rapidly rotating, accreting BH in general rela- Commons Attribution International tivity and the relativistic accretion flow in resistive MHD for viscous radia- License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ tion. We compute the structure of accretion disk, the accretion luminosity of Open Access the dynamical evolution of inner accretion disk and precisely determine their

radiation spectra, and compare to observational data of X-ray satellites. Fi- nally, we obtained the resulting disk radiation basically explained as the X-ray luminosity of the central source, such as LMC X-1 and GRO J1655-40. These results are interestingly consistent with observational data of galactic X-ray source binary systems such as X-ray luminosities of Cygnus X-1 and Seyfert galaxies (NGC 3783, NGC 4151, NGC 4486 (Messier 87)) which are powerful emitters in X-ray and gamma-ray wavebands of the observed X-ray variability with typical luminosity.

Keywords Relativistic Disk Accreting BH-Gamma Rays, Bursts-Radiation, X-Ray Luminosity

1. Introduction

Different theories for the gamma-ray bursts (GRBs) progenitor systems emerged

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using the observational results, with the leading models involving a compact object merger for the short bursts [1] [2] [3] [4] and a massive star core collapse (collapsar model) [5] [6] to a compact object (black hole or neutron star) origin for the long bursts. A new intimation into the progenitors that emit GRBs came from the compact stellar and host galaxy coincidence of GRB 980425 with the supernova SN 1998 bw [7] that resembles the ordinary Type Ic SN 1994I or the weak version of hypernova, SN 2002ap [8] and recently observed gravitational wave [9]. Beside collapsar and merger progenitor types, the accretion-induced collapse of a rapidly rotating white dwarf and neutron star scenarios is also extensively studied [10]. Long duration GRBs associated with type Ib/c supernovae (SNe) are powered by collapsars [11]. The compact binary mergers are the promising sources of short GRBs [12] [13]. Therefore, main classes of possible progenitors models have been proposed for the origin of gamma-ray bursts are two neutron stars or neutron star-black hole mergers and massive star gravitational core collapse (hypernova). These prospective progenitor system activities capable of

producing GRBs involving accretion of a massive ( ~ 0.1M  ) disk onto a new- born black hole can result from the explosion of a massive star core collapse, or following the coalescence of binary compact stellar remnants. In both cases a spinning black hole is formed with torus system, either from the super massive stellar core collapse or from a tidally disrupted neutron star, form a temporary accretion disk or torus which ultimately fall into the black hole, generating a fraction of its gravitational energy that power GRB AGN. The observed X-ray light curves (Swift-XRT) of GRBs widely compact the sources producing them [14] [15] [16]. For instance the plateau and the overly- ing x-ray flare(s) acceptably powered by compact progenitors, identified as gravitational core collapse of a rapidly rotating super massive star [16] [17] or merged core of compact objects such as double NS, NS-BH or WD-BH [18] [19]. Based on several observational results the central binary sources of energetic burst emission was identified as gravitational core collapse of super massive star or merged core of 2NSs or NS-BH [20] as engines for short gamma-ray bursts, which are very efficient at converting high photon energy into luminous radiation. The intrinsic glowing flow of gamma rays emission from rotating, strongly magnetized disk accreting BH formed in dying super-giant stars or compact mergers [21] [22] typical disclosed, where collapsar or merger explained as prime candidates that form disk accreting black hole in AGN. The entire scenarios basically explained as the observed AGN radiation spectrum. Thus, Gam- ma-Ray Bursts are primal result of accretion onto black holes. Our subject of study, a viscous, strongly magnetized disk accreting black hole discussed in [23] and it was shown that the accretion disk largely characterized by a strong magnetic field, differential rotation and shear-induced turbulent stresses. The accretion torus dynamics is potentially driven by shear stress or convection that results in luminous radiation where angular momentum transport ensures the turbulent disk formation. Previous work has shown that lu-

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minous radiation of AGN and X-ray binaries are consistent with magnetized disk shear instability. Moreover, the spectrum of Cyg X-1 [24] associated with Galactic X-ray sources and large luminosity of AGN can be explained by torus disk accretion process [25]. It is also argued that X-ray flares are produced by magnetar [26] [27]. Magnetar engine deeply explored in [28] and the resulting spin-down luminosity compared with that of X-ray emission to shade light on light-curve features. These observed X-ray variability with thermal hard X-ray spectrum component has generally been interpreted as thermal emission from turbulent accretion disk [29]. But the physical sources of luminous burst radiation have not yet been well settled. Thus, we are intended to study the evolution of rapidly rotating, accreting BH in full general relativity and the relativistic accretion flow in MHD and its numerical calculation with accurate GRMHD code [30]. We approach the outflow radiation using MHD conserved equations to construct spectrum of radiative heating due to viscous and magnetic dissipation in general relativistic Kerr geometry. We compute the structure of accretion disk, the accretion luminosity of the dynamical evolution of inner accretion disk and precisely determine their radiation spectra. The aim of this paper is to explore relativistic disk accreting Kerr black hole as the potential central source of gamma ray bursts. We begin with the detailed discussions of merger of compact binaries involving neutron stars and super massive star gravitational core collapse. Particularly, investigating super massive star gravitational core collapse leads to newly formed rapidly spinning, black holes and compact core mergers (perhaps NS-NS and NS-BH). Finally, we infer the radiative heating flux density due to viscous and magnetic energy dissipation in general relativistic Kerr Black Hole. In Section 2 we present a brief description of general relativistic resistive MHD formulations. The relativistic accretion flow in resistive MHD and the resulting radiation basically explained as the X-ray luminosity of the central source explored in Section 3. Finally, in Section 4 we provide a summary and draw the concluding remarks.

2. Relativistic MHD Disk Model

Luminous disk accretion onto rotating black holes general governed by magnetohydrodynamic (MHD) equations. This spinning, magnetized black hole with thin asymmetric turbulent relativistic disk widely simulated and MHD equations are numerically solved [25] [30]. Then, traditionally the equations of relativistic MHD are given in the conserved form: the mass continuity equation µ ∇=µ ()ρσu , (1) where ρ is rest mass density, u µ is the 4-velocity of the fluid and σ is source or sink. The energy-momentum conservation µν ∇=µT 0, (2)

where, ν = 0,1,2,3 , then stress-energy tensor is µν µ ν T =(ρ + ρε ++P b2 ) uu. (3)

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The Einstein tensor Gµν and Tµν the total stress-energy tensor are related as

GTµν=8, π µν (4)

and the Ricci tensor ν Tµν=−= RG µν . (5)

The Maxwell's equations calculated from Faraday tensor F µν follows µν ∇=µ F 0, (6)

µν µ ∇=µ FJ, (7)

i i ij using a covariant derivative of a vector ( AA;,k= k +Γ jk A) we obtain ∂∂ −12 2 µ ν P µν µ 2 ν λ gµµ() Pc+ρρ uu + g +Γνλ () Pc + uu =0, (8) ∂∂xx

is the four-vectorial mass flux-density conservation equations of relativistic fluid including gravity. Taking the dominant part of the radial component we can retrieve

002 012 112 uuΓ+00 2 uu Γ+01 uu Γ=11 0,

for simplicity we take the equatorial plane, Br , Bθ , vr and vθ . ∂ρ +∇⋅()ρυ =0, ∂t

∂∂ρ 1 2 +=()r ρυr 0, (9) ∂∂ttr 2 this follows from mass accretion rate MA =−⋅∫ ρυ d . The momentum conservation equation for magnetized plasma where matter interacts with electromagnetic field is calculated from partial time derivative of ( γρυ ) and the continuity equation ∂v v γρ22+ + ⋅∇ = −∇ −ρ + × + ρ − ⋅ ()Pc ()vv P Fg jB E2 () Ej, (10) ∂t c

including all magnetorotation instability in resistive MHD: corotation field

2 EJ⋅=η J +BBr φ , (11)

the 2nd term is from magnetic tension force. Similarly, the equation of energy conservation follows from the partial time derivative of the total internal energy density (energy per unit mass) of the fluid 1 PB2 ε= ρυ 2 + + +Φρ , (12) 2γ −π 18 g

comprising kinetic, the internal specific enthalpy, magnetic and potential energies, respectively. Thus, the energy conservation equation is then ∂ε +∇⋅( S +P v +ρ Φg v) =∇⋅() vt ⋅ − h, (13) ∂t where

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112 γ P S=ρ() γυ +v +B ××() vB, 21γµ− 0 is the total energy flux and consists of the macroscopic transport of the total energy with velocity, the work done by the pressure and magnetic forces and vector h is the net thermal heat flux exchanged by the element of fluid per unit time per unit area, m h= v′2vε () trv,, d, A ∫ 2 represents energy transport equation. The pressure is

1 2 2 PB=() Γ−12ε − ρ() γυ − with partial differential 2

∂vPα αβ ∂ =()Pvβ + T αβ v β . ∂∂xxββ

These fundamental general relativistic MHD conservation equations can be expressed as a hyperbolic, first-order, flux-conservative partial differential equation ∂U +∇⋅FS()()UU = , (14) ∂t where U denotes a state vector as function of MHD conserved variables ii ( ρυ,,PB , ) and F is the flux vector, with the five-dimensional state vector = ρ U DS,,j EB , , which are a system of hyperbolic partial differential equations. The flux vector F is Dvi ii i Svj+− Pδγ jj bB F = , (15) Evii+− Pυ b0 B i γ − γρ v i  ik ki υυBB−

0 i i i ii where S j is Poynting flux and b= γ Bvk , b= Bγγ + Bvvk are magnetic field in fluid’s rest frame. The source term S is also 0  µν ∂gν j δ Tg−Γ µ νµ δ j ∂x S = , (16) µ 00∂ lnα µν α TTµ −Γνµ ∂x  0 summarizes the evolution equations for the magnetohydrodynamic variables *2 0 *2 with conserved variables: D = γρ , Sj=ρ h γυ jj − α bb, Eh=ργ − P, * υυiBB j− ij. Here, hP=++1  ρ specific enthalpy from stress-energy tensor µ µν 2 µ and b= uFν from which b= bbµ = 2 P magnetic pressure is obtained. Hence, compactly set by Jacobian determinant ∂∂UU +=AS()UU(), (17) ∂∂tx

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where A()U is Jacobian matrix. For column matrix of flux we can have ∂∂FF J=+=22AS() Uy(), F ∂∂tx2 in the sense of linearized perturbations, that is lower terms of Equation (15), simplified over index i = 1, 2, 3 gives us

Dv1  +− γ Dv21 P B ρvv−− BB 2 12 1 ρvv13− BB 1 3 F = 0 , (18) 1  Ω3 −Ω 2 ()()E+ Pυ11 − BvB ⋅− Dv1  υυ12BB− 21

imprecisely maintain the flow equation in cylindrical coordinate

∂ρvr ∂ 22 BBr φ +()ρvφφ −+= B Pr , ∂∂tr r including resistive term in the magnetized plasma. Therefore the conservative formulation Equation (14) has the general synthetic solution of the form = yx() y0 ∫ fxd, (19) µµ= µ = 2 where f()()() U∫ U SU and ()()U y1 exp() kx d x. The integration can be solved analytically or numerically, where partial differential equation solver can be applied. Such numerically calculations are largely applied to MHD equations [30].

3. Disk Accretion Luminosity

The amount of energy dissipation and angular momentum transfer typical determine the accretion disk efficiency to convert gravitational energy into luminous radiation. The conservation of angular momentum prevents matter from falling directly into the hole in directions perpendicular to the rotation axis by centrifugal forces. The gas can, however, crumple along the rotation axis of the in falling torus so that a luminous disk of debris forms surrounding the hole, which is largely characterized by strong magnetic field, differential rotation and shear-induced turbulent stresses. The matter in this turbulent accretion disk can only fall into the rotating black hole if it loses angular momentum by turbulent stresses (strong magnetic or viscous forces) acting on the disk. Matter accreted from a geometrically thin disk reach the most stable inner circular orbit of ra-

dius rI and continues freely falling into the chasm of black hole. Luminous radiation generated when debris of disk accreted into the black hole. 2 Using viscosity priscription trφ =αα() PP gB + ≈Σ v s [23] for gravitationally unstable rotating, relativistic thin and axisymmetric disk, the vertically averaged

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surface density ( Σ ) is defined as the mass per unit surface area of the disk, given by integrating the gas density ρ in the z-direction H Σ=()()()rt,ρρ rzt ,, d z = 2 rt , H , (20) ∫−H where, H is the scale height, when the gravitational force balanced with the vertical pressure gradient υ 2 Hr≈ s  , (21) ΩK

is the disk half-thickness at radius r. Since rH , the condition for

height-integrated thin disk. We must have ΩKs υ and so the rotation of the disk is highly supersonic. Here for axisymmetric flows, cylindrical coordinates ()rz,,φ are employed with the z-axis chosen as the axis of rotation and the central plane of the disk lies in the equatorial plane of the kerr hole at z = 0 . It follows that the rate of mass flowing inward is readily integrated from mass conservation in equilibrium 1 ∂∂ ()rΣ+υυ() Σ=0, (22) rr∂∂rz z

integrating to rΣυr = constant. Here Συr is the inward flux of material and the mass accretion rate will be  Mr=2, πΣ−()υr (23)

a small inflow radial “drift” velocity υr is negative near the horizon, so that matter is being accreted. Since the fluid particles can experience magnetic resistive and viscous dissipation, the constraint equation is  M d d2 11 +()r Htrφ + r() v rr Bφφ −= v B 0. (24) 4dπ r d r rr d The whole accretion torus within accretion radius (outer edge of the disk) rotates about the hole with specific angular momentum of a circular disk

22Br  ≈Ωrd = GMHd r = r vφφ − r B , (25) M

rd is the radius where the outer edge of the disk forms. Similarly, the angular momemtum conservation with viscous-stress tensor uniquely determines viscous accretion disk 2 B 1 χ 2 ρρDtV=−∇ P + +() BB ⋅∇ +Ft −Ω z +∇⋅ , (26) 2µµoo c where, d ∂ ∂ ∂ ∂ ∂ ∂∂ Dt= = + vβφ = + vvvrz + + = + v ⋅∇. dtt∂ ∂ xβ ∂ t ∂ r ∂ z ∂∂φ t

The general form, including the viscous-stress tensor term ( tij ), simply expressed as ∂ 2 Tij χ 2 1 B ρρDvti= g i + + ρF −Ωz +()BBP ⋅∇ −∇ + , (27) ∂jcµµoo2

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where g is force density and

Tij=−+ Ptδ ij ij ,

or we can typically express as

ρDvti= −∇ P + ∇ ⋅ t, (28) where the viscous stress tensor (force density) t has components 2 ==ηρ ∇+∇−∇⋅δ ttij ji (),, Ti v j j v i ij v (29) 3

η is the dynamic or shear viscosity coefficient, usual known in kinetic mo-

lucler theory. Where we get the component trφ —a tangential viscous force per unit area exerted by the disk inside accretion radius:

dΩk 2 trφ =ηη = ∇ v, (30) r dr acts over an area ( 2πrH ) of the disk for ∇⋅v =0 from mass continuity equation. And Equation (28) will be

∇P η 2 Dv =−+∇v, (31) ti ρρ η where =καν = vH is kinematic viscosity. The disk tori in accretion radius ρ s rotates more rapidly and experiences a backward torque acting on the disk outside capture radius due to friction between adjacent layers of fluid elements generates a torque that carries angular momentum outwards.

3 dΩk τη=π+−=π+−()22rH rtrφφφ r() v B rr v B Hr r() vφφ Brr v B , (32) dr is the torque generated by shear viscosity and magnetic forces, resulting in viscosity dominated turbulent magnetohydrodynamical stresses, where shear flows are stabilized by either differential rotation or strong magnetic fields. The gas spiral inwards and gradually loses angular momentum with transport rate  2 J=χρ Mvφφ −+ tr v φX through viscous force while the fluid outside capture radius gains the angular momentum. where χ is resistive (friction) coefficient

in the fluid flows and Mv φ is thrust: a force applied perpendicular to the area 2 and trφ ~ αα() PP gB+=Σ v s. Thus, the φ component of the momentum equation is

vvr φ 11 ρ Dt vφ+ = ∂r()rtφ r +∂ zz t φφ + t r. (33) rr r

Here ttrrφφ= are the non-negligible components of tij in the disk's coor-

dinates. Hence, using Dt in (33) gives

1122 ∂t()ρρrvφφ +∂r() r v v r =∂ rr() r t φ, (34) rr integrating over z gives

1122 ∂t()ρρrvφφd z +∂r() r v v r d, z =∂ rr() r W φ (35) ∫∫rr

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= where Wrrφφ∫ tzd with increased pressure (stiff equation of state [31]). Using surface density, Equation (20), we can rewrite

1122 ∂Σt()rvφφ +∂Σ r() r v v r =∂ rr() r W φ. (36) rr The gradient of specific angular momemtum is the internal torques generated by shear viscosity and magnetic forces, resulting in viscosity dominated turbulent magnetohydrodynamical stresses. The radial component of angular momentum also determined as

2 vφ 11 ρρD v − =−Ω−∂+∂r 2 P rt +∂ t − t . t r r r() rr z rz φφ (37) r rr

The strongest viscous force exerted between two adjacent annulus of the disk

is trφ component of the stress tensor, thus we shall assume that ttrr =φφ = 0 . Equation (27) then simplifies to v2 ρρ∂ +vv ∂ =φ −Ω2 r −∂ P, ()t rr r r (38) r integrating v2 Σ ∂ +vv ∂ =Σφ −Ω2 r −∂ W, ()t rr r r (39) r where W= ∫ Pzd . Equation (39) states simply the radial hydrostatic pressure balance. Vertical gradients in the azimuthal field create vertical magnetic pressure gradients for an incompressible flow, are balanced by vertical fluid pressure gradients. Using energy conservation equation we obtain the radiation energies emitted by accretion torus of rapidly spinning black hole. The basic equation for thin disk follows, from energy conservation Equation (13) is ∂ε dQ +∇⋅( S +P v +ρρ Φg v) =∇⋅() vT ⋅ − h − , (40) ∂ttd but consider various formulations of viscous stress tensor

Tij=−+ Ptδ ij ij ,

∇⋅()vT ⋅ =∇⋅() −Pδij + t ij v j =− PI +t,

gives pressure ( −∇ ⋅()Pv ) and viscous ( ∇⋅()vT ⋅ ) work. Heating due to viscous dissipation reduces kinetic energy.

∂ ()uTi ij ∇⋅()vT ⋅ = . ∂x j

Thus, the work on the disk due to forces on disk area is

∇ ⋅()vT ⋅ = −∂j()()Pv j + ∂ i t ij v j , (41) substituting into (40) we get

∇⋅( Sv +P +ρ Φg v) =−∇⋅ h −∂j()()Pv j +∂ i t ij v j , (42)

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for axisymmetric thin disk approximation vz = 0 , imply that 1 ∂=∂i()t ij v j r() rt rφφ v , r simple form for internal energy equation ∂ ()ρε ∂Q +∇⋅K =∇⋅() vT ⋅ +ρ −∇⋅h, ∂∂tt advection first term and the work done by ∇⋅K due to radiation ∇⋅h , where radiation is due to viscous dissipation ∇⋅()vT ⋅ . Hence, 1 ∇⋅++Φ=∂( SvPρ g v) r() rt rφφ v −∂ r() Pv r −∂ zz h . r Indeed, the steady-state mass conservation equation ensure  vP2 ∇⋅v ρρ + +P + Φg =0, 21Γ−

where we are left with gradient of magnetic & turbulent viscosities result in gradient of energy flux density. Thus, 11 ∂r()rt rφφ v −∂ r() Pv r −∇⋅ B ×() v × B =∂zz h . (43) r µ0

Equation (43) encompasses the differential rotation term ( tvrφφ), strong magnetic field ( B××() vB), meridional circulation term ( grad() Pv− B ) and shear

induced turbulence term ( trφ ) in the relativistic radiative MHD balanced with

gradient of energy flux density ( hz ). The heat diffuses toward the top and bottom surfaces of the disk where it is radiated away. Actually, the energy dissipated into heat due to viscousity is radiated in the vertical direction and radiation emitted along vertical midline from both surfaces of the disk (e.g. turbulent transport across sheared flows). Thus, we have vertical viscous dissipation per

unit area (energy flux density hhz= vis + h B ) of disk surfaces given by τ ∂h = ∂Ω, (44) zz4πr r with (30) and assuming Keplerian tangential velocity this becomes

9 η 2 ∂hH =η Ω2 + ∇×B , zz 84K π with the help of (20) and (30), we generally determine vertical radiation energy flux density due to viscous dissipation as

9 2 η 2 h =κν ΣΩ + ∇× B , (45) zK84π is the rate of energy dissipation per unit volume due to the work done by the viscous forces. We know that torque is rate of change of angular momentum

dJ 22 τ = =M Ω r =−π2, r Ht φ (46) dt r with (32) we will determine equality as

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 12 M(() rdI−=π() r) 3,η H() GMr

12 where, ()()r=Ω= r2 GMr and solving for η

M 12 12 η ()r = ()GMr− () GMr I 12 dI 3πH() GMrd 12 (47) M r =1, − I 3πHr d substituting into (45) we express radiation energy flux in terms of mass accretion rate as 1 3 r 2 Mv dΩ  = Ω−2 I =−KI − hMzK1 1, (48) 8ππHrd 4d r  is the rate of energy dissipation due to viscous forces. The maximum amount of heat is radiated away from the surfaces of the disk before matter accreted into the kerr hole. To determine the power of this radiation, consider a mass of lu-

minous torus M d falling from disk into the gravitational field of a rapidly ro-

tating massive black hole with mass M H . Therefore, integrating (48) over the two faces of the disk determine the total accretion power

rd Lacc =2 ×π 2hrz d, r ∫rI

establishes a maximal black hole disk accretion luminosity GM M M BH d d  2 Lacc = ∝==πE()22 r() Ht Ωrφ , (49) 2rrII

the energy per unit time E dissipated in an annulus of width H.

The relativistic disk model parameters are: the black hole mass, M H , and  Kerr rotation parameter a, the mass accretion rate, M d , the inner rI and out-

er rd radii of the disk. Figure 1 summarizes disk luminosity with range of mass 10 accretion rate, MMd = 0.1  , and initial rI = 10 m . Radiation luminosity drops  −12 as we go away from the black hole; for disk accretion rate of MMd = 10 yr 1 28 16 , we obtain the accretion luminosity as less as 10 erg/s at rI = 10 m , shown in −9 Figure 1. But, for disk accretion rate of MMd = 10 yr , we obtain very high accretion luminosity comparable to Eddington luminosity (~1037 ergs/s) at 10  −9 17 rI = 10 m . An accretion rates MMd ~ 10 yr ~ 10 g s generate luminosity ~1037 ergs/s. When accretion rate increases, the luminosity linearly grows. The 7  19 spectral flux was computed for a model with MMH = 10  , M = 10 g s ,  4 θ = 45 , D = 10 kpc and rrIg= 10 and obtained an accretion disk spectra

with variations in black hole spin ( a= Jc GM H = 0.35 , 0.7 and 0.998.), i.e., rotation influenced flux as shown in Figure 2. These indicate that highly spinning black hole binaries emit shortest wavelength, energetic radiation (see Figure 2).

1 25 M  yr= 6.3 × 10 g s .

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Figure 1. The accretion luminosity versus disk accretion rate at different values of in-

nermost stable circular orbit radius ( rI ).

Figure 2. Flux density spectrum versus shortest wavelength emitted by a thin accretion disk around a rotating black hole at different values of Kerr rotation parameters a = 0.37, 0.7 and 0.998.

Thus, expressing the flux vector MHD Equation (14) in terms of the density terms, we formulate the general partial differential equations as dy +=χ f() xy, 0, (50) dx where y represents flux terms and x represents density terms such as mass, momentum and energy densities. Setting f() x, y= xy arbitrary function of source or sink, we obtain the radiation flux as

2 yy=0 exp() −χ x , (51) 2  where, L00=π=4 r y GMM r and χ represents shear induced turbulant stresses with magneto-rotational instability enhance disk turbulence. In general, Poynting and radiation fluxes largely contribute to the luminosity

LL=++0 Snuc S poyn + S Ohm , (52)

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with specific energy density ratio

4φρ 22 Br=()vv + φ . (53) B2

The ratio B is the magnetization function of magnetic flux.

4. Results and Discussion

We have studied the evolution of accreting BH in full general relativity including, imperfect MHD for viscous radiation. The relativistic accretion flow in resistive MHD and the resulting radiation basically explained as the X-ray luminosity of the central source, such as LMC X-1 and GRO J1655-40. LMC X-1 is a luminous X-ray source in the Large Magellanic Cloud (LMC). These results are interestingly consistent with observational data of galactic X-ray source binaries such as Cyg X-1, Cyg X-3 and Seyfert galaxies (NGC 3783, NGC 4151, NGC 4486 (Mes- sier 87)). They are powerful emitters in X-ray and γ-ray wavebands. The spectrum of the observed hard X-ray flux exponential decays with energy is shown in Figure 3. The large luminosity of AGN (1046 erg/s) with the shortest wavelength can be explained by torus disk accretion process and accounts for observed X-ray rapid variability in the erratic light curves. The thin disk accretion onto extremely spinning magnetized BH is the likely source of high energy from binary X-ray sources. Typical X-ray source binary systems: Cyg X-1, LMC X-1, Cen X-3 (X-ray binary pulsar) and Cyg X-3 (X-ray luminosity of ~1031 J/s). The emission of X-ray flux is of synchrotron nature. For instance, the hard X-ray spectrum of the binary X-ray source Hercules X-1, as observed by the Ginga satellite displays synchrotron absorption at 35keV in high-intensity X-ray interaction with matter, shown in Figure 4.

Figure 3. The luminosity flux spectrum of MHD disk radiation with magnetic, dynamical and gas pressure components. Steep decaying energy density (i.e. magnetic flux densities). 2 B 2 Where  = =1.5 × 10−92 dyne cm magnetic flux densities,  ~ ρvP+ dynami- B 25 P gas

cal energy density and XR is the X-ray spectrum of X-ray binary Black Hole (Sco X-1). The X-ray differential energy spectrum in 10 - 103 KeV.

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Figure 4. Frequency distribution of radiated energy flux density. The emission of X-ray and γ-ray fluxes are of synchrotron radiation type.

4.1. Thin Accretion Disk Radiation 30  Thin accretion disk luminosity Lacc ~ 10 Mrd I as blackbody radiation with 1  3 4 inner disk temperatures Td~ () Mr dI . A turbulent accreting disk emission spectrum consistent with the broad blackbody spectra (see 4) and also match to the synchrotron spectrum. For instance, the spectrum of Cyg X-1 [29] is the su- 5/3 perposition of synchrotron radiation power law decay Et∝ and blackbody 43 0.8 spectrum with spectral luminosity Lν ≈×10 ergs ν in the high frequency range. The luminosity spectrum of the disk can be approximated as a blackbody (Figure 4). The typical gas temperatures of 107 K at inner most radius made the flux is emitted from the inner face of the disk. At the inner disk boundary the local spectra contain an excess of high-energy photons in X-ray range (1017 Hz) display huge X-ray emission which can be fitted to observe X-ray data. As a result, the spectra obtained from a thin accretion disk around a Kerr black hole contains an energy maximum in the frequency range of the strong thermal X-ray sources. The radiation spectrum from an accretion disk around a Kerr black hole consistent with the observed spectrum of X-ray sources candidates. The spectral 7  19  flux were computed for MHB = 10 M , M = 10 g s , θ = 45 , D = 10 kpc 4 and rrIg= 10 and obtained an accretion disk spectra with variations in black

hole spin ( a= Jc GM H = 0.35 , 0.7 and 0.998.), i.e., rotation influence flux. For

extremelly spinning ( a= Jc GM H = 1) Kerr black hole, rrIg= . The black hole mass and spin, the accretion rate, the disk inclination angle and the inner disk radius are defining parameters of disk luminosity. The X-ray spectra of black hole binaries observed so far, LMC X-1, GRO J1655-40, PSO J334+01 spectrum significantly infer disk accreting Kerr hole. The model then anticipates that geometrically thin, optically thick disks in black hole binaries radiate X-rays (3 × 1016 Hz). The structure of the radio lobes observed in super-critical PSO J334+01 spectrum significantly infer massive BH coalescence. At high frequencies, scattering opacity effects are useful and result in a modified Wien spectrum. For

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high accretion rate ( M ) and viscous alpha spectrum peak appears in the soft

X-ray range, where disk becomes optically thin. The high-energy photons at rI and the disk rotation shifts up to the photon frequencies determine all flux frequency band of a thin accretion disk (see Figure 4) and at high γ-ray energy the spectrum is steepy exponential decay form. For example, the X-ray continuum of binary X-ray pulsars is characterized by a power-law of photon index of α = 0.3 - 2 with an exponential cuttoff at high energies (MeV), because light-curve features (plateau and the flare) are constrainted by high energy emission. Therefore, a significant amount of the total flux is emitted in the soft X-ray range. Most of the flux is emitted in the UV band (ν ≥×2 1015 Hz ) up to the 17 5 X-ray range ( ν ≥ 10 Hz ) at typical gas temperature of TI = 10 K from 8 MMH = 10  . These estimates are sufficient for modeling the majority of accretion disk spectra from observed black hole X-ray bursts. A black hole with mass 8 47 MMH = 10  emits X-ray luminosities of 10 erg/s identified.

4.2. Conclusion

We have summarized that high-energy GRBs generated by rotating black hole-accretion disk system. Basic analyses of Figure 1 deduce that accretion luminosity drops as we go away from the central Kerr black hole. This probes the energy injection that generates plateau phase as well as x-ray flares. The radiation flux as function of emitted photon frequency and energy density explained the engine as accreting compact binary galactic X-ray emitters. The observed luminosity in X-ray ranges for active galactic nuclei tell us, hard X-ray emission from galactic compact binary systems, typical X-ray source binaries, scorpius X-1 [32]. Early X-ray light curve components with large luminosities imply accretion disk radiation of these progenitors where, the total flux is emitted in the soft X-ray energy range shown in Figure 3. Thin disk radiates locally as a blackbody above critical frequency as shown in Figure 3, indicate that the total flux is emitted in the soft X-ray range at high frequency due to Poynting, magnetic and radiation pressure fluxes as studied in Equation (51) and the result shown in Figure 3. The spectral evolution resembles that of high energy particle acceleration. Particularly, synchrotron and multi-colour blackbody contribution (see

Figure 4). The electron energy distribution in the most stable inner radius ( rI ) gets hotter due to transition of bulk kinetic energy into thermal energy, following a power-law distribution. The synchrotron spectrum slopes as ν 2 at low ()12− p frequencies and scales as (ν ), where p ≈ 2.4 is the power-law index for

the energy distribution of electrons. High Lorentz factor γ c electrons cool more rapidly, and this causes a break in the synchrotron spectrum at a critical

cooling frequency ν c .

Acknowledgements

I thank Addis Ababa University, Jijiga University and Adama Science and Tech- nlogy University for all support made.

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Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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International Journal of Astronomy and Astrophysics, 2019, 9, 265-273 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Gravity Constraints on the Measurements of the Speed of Light

Faiçal Ramdani

Laboratory of Geophysics & Natural Hazards, Geophysics, Natural Patrimony & Green Chemistry Research Center (Geopac), Mohamed V University, Rabat, Morocco

How to cite this paper: Ramdani, F. Abstract (2019) Gravity Constraints on the Mea- surements of the Speed of Light. Interna- The speed of light in a vacuum is a constant of special relativity, electromag- tional Journal of Astronomy and Astro- netic wave theories, and astrophysical distances. However, several measure- physics, 9, 265-273. ments of its speed (c) at locations on the Earth’s surface seem to vary at dif- https://doi.org/10.4236/ijaa.2019.93019 ferent times during the last century. Efforts have been made on instruments Received: June 23, 2019 performance to achieve a unique viable value in any spacetime referential. Accepted: September 2, 2019 The time-variability on c-values obtained is here addressed inside the gravity Published: September 5, 2019 field (g) in which the measurements of c have been estimated. It appears a

Copyright © 2019 by author(s) and correlation of c and g both daily (tidal) and yearly (no-tidal) variations which Scientific Research Publishing Inc. suggest that the gravity acceleration control the c-variability everywhere in a This work is licensed under the Creative spacetime referential. Implications of this model provide a sensitivity con- Commons Attribution International License (CC BY 4.0). stant of c from g, and the estimates of c on planets of the solar system where g http://creativecommons.org/licenses/by/4.0/ values are known. It is deduced an upper limit of gravity in black hole that Open Access can cancel the speed of light in the horizon.

Keywords Gravity, Time, Speed of Light, Solar System, Black Hole

1. Introduction

Velocity of light was intensively studied in the last century since it was a basis for Special relativity and electromagnetic waves theories. A variety of instruments have been used to improve the accuracy of c-measurements [1]. Efforts were focused on reducing errors due to instruments and to immediate environments such as temperature, pressure, clocks, length of ray path which was fully discussed as possible sources for no regular values obtained from the speed of light. However, no identical c-values are maintained [2] despite the fact that the same instruments are used under the same laboratory conditions. To solve these issues

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most studies computed the average of a set of measurements, and proposed it as a final result of the speed of light [1] [3] [4]. This was done at several epochs from the beginning of the last century until the last value of c (1983) accredited by the International bureau of Weight and length which in the way defined the meter from the adopted c-value (299,792,458 m/s). Such relativistic parameter has direct implications on the accuracy of space geodetic techniques [5]. Models of variable speed of light (VSL) and anisotropy were developed later to explain cosmological issues [6] [7]. A review of the c-fluctuations [2] [8] where data are available shows that the values of c change but not in an arbitrary form, and may be re-found after a certain time. This variation of c is small and has not been of special interest. A systematic decreasing c with time has not been successfully achieved [9] since varying instruments were used on varying epochs, and because of further observations have indicated an increasing value in the same laboratory. The impact of gravity on velocity of light has not been fully studied, but its possible existence was, however, raised since 1911 [10]. The problematic connection of gravity and speed of light is addressed as it appears complex when time variation is included (Figure 1). The variable g is of a low level (μgal) due to tidal forces and ocean loading, and about 1 mgal due to tectonic processes, while spatial variations in gravity are observed between inland and offshore zones, plate boundaries, Mountains and plateau areas, even using similar instruments to distant areas [11].

2. Experimental Data

Absolute gravity data from several regions provided by worldwide data centers, BGI (Bureau Gravimetrique International) and NOAA (National Oceanic and Atmospheric Administration) show the spatial variations of gravity. Figure 2(a) shows three years of annual variations of short and long wavelengths related to tidal and no-tidal forces in Boulder station. The variation is periodic and the average trend is there horizontal but it can be decreasing in other sites as in a

Figure 1. 3D-deviations of c and g from cm and gm values. Absolute g-values are observed in the site of c-measurements but not synchronized with c-values. The time is as- signed to c-measurements.

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(a)

(b)

(c) Figure 2. Yearly variations of the absolute gravity is shown in varying setting, Boulder (a) where the trend curve (dashed line) is horizontal, Darwin Australia observations with a line of trending variation (b) and in Conception (Chile) which shows evidence for periodic variation at large period of gravity observations (c).

Norway experiment [12]. Much longer period of observations with limited data (Figure 2(b)) shows again oscillatory variation in Darwin station (Australia) and even with a large number of observations as in Conception-Chile where the underlying subduction made it low the gravity anomaly (Figure 2(c)). Speeds of light measurements were made without taking into account the state of gravity in the area of c-experiments. When several measurements are achieved, and it can last several days or weeks, the gravity has also changed due to tidal forces, atmospheric and ocean loading. Resulting successive c-values are therefore of prime importance, not the average of the values obtained from the experiment. While each c-value may be affected by flow in clocks, the variation between values is not clock-dependent. As no synchronized values of c and g are available since 1911 when equipment of such experiment did not exist, there are, however, some successive experiments of c-estimates. Since successive they are a function of time in the same site. It is, therefore, possible to compare the evolution of c with time and the absolute gravity recorded in the same site. Variation model appears in the c-fluctuations by using two sets of measurements, one due to Froomer [13] and other measurements carried out in Boulder (US bureau of standard) during two weeks [14]. The variation seems to be periodic to both cases (Figure 3(a) and Figure 3(b)) which vary around a c-mean. The c-values found in Chile where low gravity is known provided high average of 3.05 × 108 m/s [15]. The accuracy of this value is not of importance here, but the distances

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(a)

(b)

(c) Figure 3. Successive c-measurements in NPL (UK) (a) and US Boulder station (b) and Chile (c). Trend variations by c-fluctuations appear to be periodic.

and time delay of their successive observations, then in other terms the time-variations of c. Figure 3(c) shows the results of distance-times measurements and it also fits with a periodic trend. Other high value of c estimated in Caracas experiments (3.009 × 108 m/s) using one-way method. [16] agrees with low gravity environment but it lacks the accuracy of the two-way method. More reliable set of c-measurements in Sweden. [17] shows daily variations of c that is of tidal type when compared morning and afternoon observations (Figure 4(a) and Figure 4(b)). During the 11 days of measurements appears a peak to peak period about 3 - 4 days the origin of which is open to question in the absence of synchronized gravity data in the same site of c-measurements.

3. Gravity-Time-Velocity Model

As c-values vary with time g also vary, and the variation seems to be periodic. This suggests that the speed of light is not affected by gravity by bending only, but the value of speed also is sensitive to the time and place where it is measured. These examples show the form of variability in the values of both c and g around

c0 and g0. The amount of sensitivity of c from gravity may be experimented. The correlation of their variability (not the values) suggests a relationship between c and g in term of causality. The fluctuations in c are assumed to be due to gravity

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(a)

(b) Figure 4. Daily variations of c-estimates from Sweden with a peak to peak period of about 3 - 4 days are indicated in (a); variation during one day between morning and afternoon is shown in (b) where a trend variation is indicated (dashed lines).

which is time varying. In this simple model gravity is the acceleration that affects time variation of c.

c()() gt, =⋅+ Ag t c0 (1)

where A is constant (sec). Experimental data of c using geodimeter and Laser methods [3] [4] [8] are compared with absolute gravity data measured in the same site (Figure 5(a) and Figure 5(b)). Since c estimates are taken as average of several measurements, they are then representative for the site where an absolute gravity is observed (Table 1). Time variations of g and then of c in the same site is here assumed to be of second order. Since the values determined by geodimeter methods are unreliable because of the large margin of errors (Figure 5(a)), it is more appropriate to take as values those determined by Laser whose

errors appear acceptable (Figure 5(b)). The trend of the cL values can be expressed by the equation

CgL =−+50.446 299792953.5 (2)

where C0 = 299,792,953.5 m/s is the c-velocity in vacuum without gravity,

CC0 −=50.44 g =∆ C (3) where ∆C is the variation of c in a gravitational field g. In this relationship, it appears the sensitivity A of c when g varies. A is too small to be significant in Earth surface measurements where the range of variation of g is limited (9.76 - 9.83 m/s2). When gravity varies substantially in outer space, such a relationship can gain more attention. Table 2 shows updated values of c in varying gravity field of solar planets while a black hole upper limit

gravity value gBH is obtained for c = 0 since at more higher values of gBH this will make c negative according to Equation (2).

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(a)

(b) Figure 5. C-variations versus gravity velocity (g) is plotted in varying sites at times of c-measurements using geodimeter (a) and laser (b) methods. Resulting trending curves are decreasing to both cases.

Table 1. C-measurements at varying epochs using geodimeter and laser and associated estimates of absolute gravity recorded in the site of c-observations.

Methods Time C (m/s) C-error (m/s) g (m/s²)

Geodimeter 1953 299,792,400 110 9.811856

1953 299,792,200 130 9.818074

1955 299,792,400 400 9.816501

1967 299,792,500 50 9.810565

1971 299,792,375 60 9.819047

Laser 1972 299,792,460 6 9.794161

1974 299,792,459 0.8 9.811856

1978 299,792,458.8 0.2 9.811856

1979 299,792,458.1 1.9 9.80616

1983 299,792,458.6 0.3 9.800849

−2 gBH = 5943218.401 m⋅ s With near 0-velocity the photons cannot escape from black holes, or probably

are accumulated in the horizon. The upper limit of cmax from which c slowly decreases by a constant A due to change in gravity may be reached at g ≈ 0.

−1 cAmax =299792953.5 m ⋅=− s ; 50.44

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The difference between c-adopted (1983) and cmax is 495.51 m/s. cmax is the speed in a vacuum without gravity. The variation of c is inversely proportional to gravity, which induces that velocity increases at low gravity field at a fixed time. In turn, when gravity is big then the speed of light is reduced, and may be vanished at infinite gravity environment as it is the case around black holes. Figure 6 shows the decreasing values of c in the solar system, the slower speed is found in the sun because of its high gravity.

Table 2. Results of c-estimates in planets of solar system where g-values (Nasa) are known, and gravity limit in black hole.

Planet g (m/s²) c (m/s)

Sun 274 299,779,132.9

Jupiter 23.1 299,791,788.3

Neptune 11 299,792,398.7

Saturn 9 299,792,499.5

Earth 9.8 299,792,459.2

Uranus 8.7 299,792,514.7

Venus 8.9 299,792,504.6

Mars 3.7 299,792,766.9 Mercury 3.7 299,792,766.9

Moon 1.6 299,792,872.8

Pluto 0.7 299,792,918.2

Black hole 5,943,555.78 0

(a)

(b) Figure 6. Speed of light fluctuations model in the solar system with sun (a); and without sun (b) based on gravity shows linear decreasing of the speed of light from solar planets including earth (full blue circle) with respect to sun (yellow).

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4. Discussion and Conclusions

The spacetime variation of c explains the issues in the last century to obtain a unique value even when similar instruments were used. The varying spatial sites where c was estimated in addition to the timing of the observation constrained the comparison between results. Even though this study used limited but the more reliable available values of c and gravity, it has introduced a control of gravity in electromagnetic wave propagation in term of space-time fluctuation of c. A synchronized observation of gravity and c-velocity may confirm the hypothesis while the use of the average of consecutive c-measurements has no physical significance. Each determination is a natural value of c at a given time and at a given value of gravity which is also space dependent. Since gravity changes on Earth are periodic in the short term, long term tectonic factors can cause a variation of gravity in the site of c-measurements. Observations from Figure 4 exhibit the short-term daily c-variation that may be caused by short term gravity variation. For this reason, the use of the means of c-values may be valid in a particular site because of the periodicity of g, but the c-values may change in another site when gravity is different. Measuring the speed c in the Moon may confirm if the suggested c-value (Table 2) is valid and if the coefficient A is universal or planet-dependent. Second-order variation due to time should be introduced to avoid excessive use of averages. This will result in a spatio-temporal variation of c and a global mapping of c which proves to be necessary as it exists for g. The correlation between c and g could be highlighted in a referential of time. This will lead to better decipher the degree of sensitivity of c when g varies both daily

and yearly. Laser methods suggest the subtraction of about 50 g from cmax to determine a realistic value of c. In black hole setting, no need to cancel time to stop

photons from moving in the horizon, the limit value gBH seems sufficient. Speed fluctuation of c may improve the model of c-invariant while the exact impact of g on c needs further synchronized measurements of (c, g, t) parameters. By introducing the impact of gravity on the speed of light the constancy of many parameters will be refined.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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[4] Evenson, K.M., Wells, J.S., Petersen, F.R., Danielson, B.L., Day, G.W., Barger, R.L. and Hall, J.L. (1972) Speed of Light from Direct Frequency and Wavelength Mea- surements of the Methane-Stabilized Laser. Physical Review Letters, 29, 1346-1349. https://doi.org/10.1103/PhysRevLett.29.1346 [5] Muller, J., Soffel, M. and Klioner, S.A. (2008) Geodesy and Relativity. Journal of Geodesy, 82, 133-145. https://doi.org/10.1007/s00190-007-0168-7 [6] Magueijo, J. (2003) Cosmology “without” Constants. Astrophysics and Space Science, 283, 493-503. https://doi.org/10.1023/A:1022560802810 [7] Cahill, R.T. (2006) A New Light-Speed Anisotropy Experiment: Absolute Motion and Gravitational Waves Detected. Progress in Physics, 4, 73-92. [8] Evenson, K.M. (1975) The Development of Direct Optical Frequency Measurement and the Speed of Light. ISA Transactions, 14, 209-216. https://doi.org/10.7567/JJAPS.14S1.65 [9] Barrow, J.D. (2005) Cosmological Constants and Variations. Journal of Physics: Conference Series, 24, 253-267. https://doi.org/10.1088/1742-6596/24/1/031 [10] Einstein, A. (1911) On the Influence of Gravity on the Propagation of Light. Anna- len der Physik, 35, 898-908. https://doi.org/10.1002/andp.19113401005 [11] Hinderer, J., Crossley, D. and Xu, H. (1994) A Two Year Comparison between the French and Canadian Superconducting Gravimeter Data. Geophysical Journal In- ternational, 116, 252-266. https://doi.org/10.1111/j.1365-246X.1994.tb01796.x [12] Memin, A., Rogister, Y., Hinderer, J., Omang, O.C. and Luck, B. (2011) Secular Gravity Variations at Svalbard (Norway from Ground Observations and Grace Sa- tellite Data). Geophysical Journal International, 184, 1119-1130. https://doi.org/10.1111/j.1365-246X.2010.04922.x [13] Froome, K.D. (1958) A New Determination of the Free-Space Velocity of Electro- magnetic Waves. Proceedings of the Royal Society of London. Series A, Mathemati- cal and Physical Sciences, 247, 109-122. https://doi.org/10.1098/rspa.1958.0172 [14] Florman, E.F. (1955) A Measurement of the Velocity of Propagation of Very-High-Frequency Radio Waves at the Surface of the Earth. Journal of Research of the National Bureau of Standards, 54, 335. https://doi.org/10.6028/jres.054.038 [15] Ortiz, M. and Montecinos, A.M. (2015) How to Measure the Speed of Light with a Dinner Budget. Revista Brasileira de Ensino de Física, 37, 1502. https://doi.org/10.1590/S1806-11173711649 [16] Greaves, E.D., Rodriguez, A.M. and Ruiz-Camacho, J. (2009) A One Way Speed of Light Experiment. American Journal of Physics, 77, 894-896. https://doi.org/10.1119/1.3160665 [17] Bjerhammar, A. (1972) A Determination of the Velocity of Light Using the Twin Superheterodyne Principle. Tellus, 24, 481-495. https://doi.org/10.3402/tellusa.v24i5.10661

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International Journal of Astronomy and Astrophysics, 2019, 9, 274-291 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Halo Orbits in the Photo-Gravitational Restricted Three-Body Problem

Saurav Ghotekar, Ram Krishan Sharma

Department of Aerospace Engineering, Karunya Institute of Technology and Sciences, Coimbatore, India

How to cite this paper: Ghotekar, S. and Abstract Sharma, R.K. (2019) Halo Orbits in the Photo-Gravitational Restricted Three-Body We study halo orbits in the circular restricted three-body problem Problem. International Journal of Astron- (CRTBP) with both the primaries as the sources of radiation. The posi- omy and Astrophysics, 9, 274-291. https://doi.org/10.4236/ijaa.2019.93020 tioning of the triangular equilibrium points is discussed in a rotating coordinate system. Received: April 11, 2019 Accepted: September 2, 2019 Keywords Published: September 5, 2019 Halo Orbits, Circular Restricted Three-Body Problem, Lagrangian Points, Copyright © 2019 by author(s) and Scientific Research Publishing Inc. Radiation Pressure, Lindstedt-Poincare Method This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ 1. Introduction Open Access Halo Orbits have been the topic of interest for the past few decades among the research society. It is a periodic, three-dimensional orbit around the Lagrangian points in a three-body problem. The name “Halo” was first used by Farquhar [1] in his doctoral thesis (1968). It was first proposed for the Apollo mission for placing the satellite in an orbit around L2 Lagrangian point in the Earth-Moon system. An enormous contribution was made by Richardson [2] towards the construction of Halo orbits using Lindstedt-Poincare method. His method explained the general idea about Halo orbits. The first and simplest periodic exact solution to the three-body problem is the motion on collinear ellipses found by Euler [3]. Euler studied the motion of the Moon assuming the Earth and the Sun orbit each other on circular orbits and the Moon to be massless. This approach is known as the restricted three-body problem. The Lagrangian points are an equilibrium solution to the restricted three-body problem. Szebehely [4] has an ex- cellent thesis on the Lagrangian points and the restricted three-body problem. The classical model of the three-bodies is not accurate for studying any system

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in the solar family as it does not consider the perturbing forces such as oblateness and radiation pressure as given by Dutt and Sharma [5]. Some of the significant works in the Photogravitational Restricted Three-Body Problem were done by Radzievskii [6], Bhatnagar and Chawla [7], Eapen and Sharma [8]. The first person to study about the solar radiation pressure was Radzievskii and he proposed that the maximum force due to radiation pressure that acts in the radial direction can be given by

FFps= ()1, − q

where q is a pressure reduction factor which is defined in terms of particle radius a, density δ and radiation pressure efficiency factor x as 5.6× 10−3 qx=−⋅1 (c.g.s. units) aδ q =1 −ε .

ε is a variable, depending upon the specification of the third body which is also a radiating body. A star radiates or shines due to nuclear fusion of hydrogen into helium in its core which then releases energy that traverses through the interior of the star and then radiates into space. The eventual existence of star is directly dependent on its mass. The characteristics of a star like its diameter and temperature change over time while its rotation and movement are dependent on its environment. When two or more stars that are bound by gravity and move around each other in a stable orbit in the same plane is called a binary or multi-star system. This paper considers that both the primaries as source of radiation and the following calculations and results are made. So, there will be two pressure radia-

tion terms viz. q1 and q2 representing the radiation of the larger and smaller primaries, respectively. Lindstedt-Poincare method is used to find the solution for the equations of motion of CRTBP through successive approximations. Numerical methods are used with this analytical solution as the initial value to generate the required Halo orbits. Changes in the Halo orbits for various values of radiation pressure are also analyzed.

2. Circular Restricted Three-Body Problem

In the Circular Restricted Three-Body Problem, two bodies revolve around their center of masses in circular orbits under the influence of their mutual gravitational attraction and a third body moves in a plane defined by the motion of the two revolving bodies. The problem of the motion of the third body is called the Circular Restricted Three-Body Problem, henceforth referred to as CRTBP. In the CRTBP, the mass of the third body is considered negligible compared to the primaries and its presence does not disturb the circular motion of the primaries (Figure 1).

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Figure 1. Circular restricted three-body problem in normalized units.

The restricted three-body problem has five equilibrium points, called Lagran- gian or Libration Points. These points are the points of zero velocities and the objects placed in these points remains stable. Out of the five Lagrangian points,

three are collinear (L1, L2, L3) and the other two points (L4, L5) form the equila- teral triangle. Although the Lagrangian point is just a point in empty space, its peculiar characteristic is that it can be orbited. The circular restricted three-body problem (CRTBP) is a special case of the restricted three-body problem where bigger and smaller primaries move in circular motion around their common center of mass. The five equilibrium points of CRTBP are known as the Lagrangian points where the gravitational forces due to two primaries and the centrifugal force on a spacecraft are balanced. Another assumption is made that the universe is strictly limited to these three bodies under consideration, so there is no other gravitational influence in the system. Though the limitations that have been introduced seem to restrict the problem to the point of impracticality, there are many situations in the solar system to which the CRTBP applies.

3. Equations of Motion

The analytical approximation of three dimensional, periodic orbits about collinear points is performed. The equations of motion are developed from a La- grangian approach described in Richardson [2]. The analytical approximation solution for the equations of motion is calculated through computer-based program. The equations of motion for the RTBP is written as (Szebehely [4], Sharma [9], Simmons [10]) ∂Ω xy−=2  (2.1) ∂x ∂Ω +=2xy  (2.2) ∂y ∂Ω z = (2.3) ∂z where

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xy22+ ()1− µ q µq Ω= +1 + 2 , 2 rr12

12 12 = +µ 2 ++22 = −+µ 2 +22 + r1 (() x yz) , r2 (() x1 yz) .

Here the oblateness is neglected so and q1 and q2 are the radiation pressure terms of larger and smaller primary, m µ = 2 mm12+

where m1 and m2 are masses of larger and smaller primary respectively.

Here m2 = µ and m1 =1 − µ .

Location of Lagrangian Points

These are the equations for the location of the Lagrangian points including the perturbing force of radiation pressure of both the primaries. The equations for the location of Lagrangian points have been referred from Nishanth and Sharma [11].

For L1

5 4 2 23 2 x+2()µ − l x +−( l 42 lµµ +) x +( µ l() l −+ µ µ q21 − lq) x

22 2 2 3 3 +(µl +20() lq1 + µµ q 2) x + q 21 −= lq

For L2

5 4 2 23 2 x+2()µ − l x +−( l 42 lµµ +) x +( µ l() l −− µ µ q21 − lq) x

22 2 2 3 3 +(µl +20() lq1 − µµ q 2) x − q 21 −= lq

For L3

5 4 2 23 2 x+2() l −µ x +( µ − 42 µ l + l) x +( µµ l() −− l µ q21 − lq) x

22 2 2 3 3 +(µl −20() lq1 + µµ q 2) x − q 21 −= lq

where l =1 − µ .

For L4 1 +312 x =−+µ , y = , 2 2

For L5 1 −312 x =−+µ , y = . 2 2 The location of Lagrangian points varies with the variation of radiation pressures. This can be observed after solving the equations. The equations will give the positions of Lagrangian points.

4. Analytical Computation of Halo Orbits

For the computation of the Halo orbits, the origin should be transferred to the

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Lagrangian points L1 and L2. The transformation is given by Xx=+±−µ ξ 1, Yy= ,

Zz= .

Hence the equation of motion can be again written as ∂Ω ξ ()X−=2, nY  ∂X ∂Ω ξ ()Y+=2, nX  ∂Y ∂Ω ξ Z = . ∂Z where xy22+ ()1− µ q µq Ω= +1 + 2 , 2 RR12

222 RX1 =()()()ξξ +1 ++ Y ξ Z ξ,

222 RX2 =()()()ξξ ++ Y ξ Z ξ .

upper sign in the above equation depict L1 and lower sign depict L2. The distance between smaller primary and larger primary is considered to be unity. The usage of Legendre polynomials can result in some computational advan- tages when non-linear terms are considered. The non-linear terms are expanded by using the following formula as given by Koon [12]. m 11∞ ρ Ax++ By Cz = ∑ Pm , 222 ρ ()()()xAyBzC−+−+− DDm=0   D

where ρ 2=++xyz 2 22, DA=++222 B C. The above formula is used for expanding the non-linear terms in the equations of motion. The equations of motion after substituting the values of the non-linear terms and by some algebraic steps by defining a new variable, cm after expanding up to m = 2, we get ∂ ∞    m X X−2 nY −+() 12 c2 X = ∑ cmmρ P , (3.1) ∂X m≥3 ρ ∂ ∞    m X Y+21 nX +−() c2 Y = ∑ cmmρ P , (3.2) ∂Y m≥3 ρ ∂ ∞   m X Z+= cZ2 ∑ cmmρ P , (3.3) ∂Z m≥3 ρ

with

()m−22  X Pm=∑ ()34 +− kP 2m−−22k  , k =0 ρ

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m m+1 1 ()()−−11q µξ m 39µµAA = +± µ +22 − cm 3 m+1()1.23 ξ()1 ξ 22ξξ

Neglecting the non-linear higher order terms in Equations (3.1)-(3.3), we get   X−2 nY −+() 1 2 c2 X = 0, (3.4)   Y+2 nX +−() c2 1 Y = 0, (3.5)  Z+= cZ2 0. (3.6)

It is clear that the z-axis solution, obtained by putting X = Y = 0, does not de-

pend upon X and Y and c2 > 0. Hence, we can conclude that the motion in Z-direction is simple-harmonic. Meanwhile, the motion in XY-plane is coupled. A fourth-degree polynomial is obtained which gives two real and two imaginary roots as Eigen values.

()c−+198 c22 − nc α = 2 22, 2

()c−−2 98 c22 − nc λ = 2 22. 2 Thus, the solution to the linearized Equations (3.4)-(3.6) will be of the given form as derived by Thurman and Worfolk [13] as Xt=++ Aeααtt A e− A cosλλ t + A sin t , () 12aa 3 a 4 a

Y t=−+ kAeααtt kA e− − kA sinλλ t + kA cos t , () 11aa 12 2 3 a2 4 a Z t= Acos ct+ A sin ct , () 5aa 26 2 where 21c +−α 2 21c ++λ 2 k = 2 ; k = 2 . 1 2α 2 2λ AA, ,, A Here 12aa 6 a are arbitrary constants. Since we are interested in pe- AA= = 0 riodic solutions we take 12aa. As mentioned earlier, there is a necessity to introduce frequency and amplitude terms to perform the Lindstedt-Poincare method. Then the solution to the linear problem can be written in terms of amplitudes, phases and the frequencies

(λ and c2 ), as

Xt()()=−+ Ax cosλφ t ,

Y()() t= kA2 x sinλφ t+ ,

Zt()()= Az sinλψ t + .

Here an assumption is made that λ = c2 .

4.1. Amplitude Constraint

When we consider the periodic 3D orbits, the amplitudes Ax and Az are related by non-linear algebraic relationship given by Richardson [2].

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22 lA12xz+ lA +∆=0.

where l1 and l2 depend upon the roots of the characteristic equation of the linear equation. The correction term, ∆=λ 22 −c arises due to the addition of frequency term in Equation (3.6).

4.2. Phase Constraint

For periodic 3D orbits, the in-plane phase (ϕ) and out-of-plane phase (ψ) are related as mπ ψφ−=,m = 1, 3 2

When Ax is greater than certain value, the third-order solution bifurcates. This bifurcation is manifested through the phase-angle constraint. The solution

branches are obtained according to the value of m. For m = 1, Az is positive and

we have the northern Halo (z > 0) and for m = 3, Az is negative and we have the northern Halo (z < 0).

4.3. Lindstedt-Poincare Method

We can find better approximations to the non-linear problem in a neighborhood of the equilibrium point solutions by using the perturbation techniques of Lindstedt-Poincare. The equations of motion with non-linear terms up to third order approximation as in Richardson [2] and Thurman and Worfolk [13] are   X−2 nY −+() 12 c2 X

3 222 2 2 2 =c34(2 X −− Y Z)( +2 c 2 X − 3 Y − 3 ZX) +О() 4, 2

  3 222 YnXc+2 +()2 − 1 Y =− 3 cXY 34 − cXYZY( 4 −−) +О()4, 2

 3 222 ZcZ+23 =−3 cXZ − c 4( 4 X −− Y ZZ) +О()4. 2 A new independent variable τ = ωt is introduced as discussed in Equation d (5.1). Here on, ' refers to . dτ

n ω=+<1∑ νωnn ,1 ω n>1

We choose the value of ωn such that the secular terms get removed with successive approximations. 22 Under the following assumptions ω1 = 0 and ω21=sAxz + sA 2, most of the secular terms are removed.

The coefficients s1 and s2 are given in Appendix. The equations of motion changes again.

2 ωωX′′−2 nY ′ −+() 12 c2 X (3.7) 3 222 2 2 2 =c34(2 X −− Y Z)( +2 c2 X − 3 Y − 3 ZX) +О() 4, 2

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2′′ ′ 3 222 ωωY+2 Xc +()2 − 1 Y =− 3 cXYcXYZY 34 −( 4 −−) +О()4, (3.8) 2

2′′ 3 222 ω ZcZ+23 =−3 cXZ − c 4( 4 XY −− ZZ) +О()4. (3.9) 2 We continue the perturbation analysis by assuming the solutions of the form, where ν is a small parameter which takes care of the non-linearity.

23 XX()()()()τ=+++ ν123 τν X τν X τ, (3.10)

23 YY()()()()τ=+++ ντν Y23 τν Y τ, (3.11)

23 ZZ()()()()τ=+++ ν123 τν Z τν Z τ. (3.12)

We substitute these Equations (3.10)-(3.12) in Equations (3.7)-(3.9) and equating the coefficients of the same order of ν , we get the first, second and third order equations respectively.

4.3.1. First-Order Equations The analytical solution can be found using mathematical software like Maxima and Mathematica. The first-order equation is obtained by taking the coefficients of the term ν . It is given as

X1′′−2 Y1 ′ −+() 1 2 cX21 = 0,

YXc11′′+2 ′ +−()21 1 Y = 0,

Z1′′ += cZ21 0. The periodic solution to the equations above is

Xt1 ()()=−+ Ax cosλφ t , (3.13)

Y1 ()() t= kAx sinλφ t + , (3.14)

Zt1 ()()= Az sinλψ t + . (3.15)

where, k = k2.

4.3.2. Second-Order Equations To get the second-order equations we segregate the terms of ν2,

X2′′−2 Y2 ′ −+() 12 cX2122 =γ ,

YXc22′′+2 ′ +−()22 1, Y22 =γ

Z2′′ += cZ22γ 23. where

τ12=+=+ λτ φ;. τ λτ ψ γ= γ τ + γ τ ++ α ωλ2 + ω λ 21 1cos2 1 2 cos2 2 1  2Axx1 2 A1 nk ,

γγτγτ22= 3sin 1 + 4 sin 2 1 ,

γ23= γ 5sin τ 2 + γ 6 [sin() ττ 1 ++ 2 sin( ττ2 − 1 )].

We need to set the value ω1 = 0 , to remove the secular terms. The particular

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solution of the second-order equations is found using mathematical software like Maxima and Mathematica.

X 2 ()τ=++ ρ20 ρ 21cos 2 τρ 1 22 cos 2 τ 2 , (3.16)

Y2 ()τ= σ21sin 2 τσ 1 + 22 sin 2 τ 2 , (3.17)

Z2 ()()()τ= κ21sin ττ 1 ++ 2 κ 22sin ττ 2 − 1 . (3.18)

The coefficients are given in Appendix I.

4.3.3. Third-Order Equations To get the third-order equations we segregate the terms of ν3,

X3′′−2 Y3 ′ −+() 12 cX2133 =γ ,

YXc33′′+2 ′ +−()22 1, Y33 =γ

Z3′′ += cZ23γ 33. where,

γ31=+−+ ν 1 2Akx ωλ2 () λcos 2 τ1 γ 7 cos τ 1

+γ8cos()() 2 ττ 21 ++ γ 9 cos 2 ττ 21 − ,

γ32=+−+ ν 2 2Akx ωλ2 () λ1 sin τ1 β 3 sin 3 τ 1

+β4sin()() 2 ττ 21 ++ β 5 sin 2 ττ 21 − , 2 γ33= ν 3 ++Az ()2 ωλ2 sin τ2 δ 3 sin 3 τ 2  +δ4sin()() 2 ττ 12 ++ δ 5 cos 2 ττ 12 − . From the above equations, it is not possible to remove the secular terms by

setting ω2 = 0 . Hence, the phase relation is used here to remove the secular terms. π ψφ=+=pp, 0,1,2,3 2 Hence, the solution of the third-order equation will be

X 3 ()τρ= 31cos3 τ 1 , (3.19)

Y3 ()τ= σ31sin 3 τσ 1 + 32 sin τ 1 , (3.20)  p 2 ()−=1κτ31 sin 3 1 ,p 0, 2 Z τ = 3 ()  p−1 (3.21)  2 ()−=1κτ32 cos 3 1 ,p 1, 3 The coefficients are given in Appendix I. Thus, the third-order analytical solution is developed.

4.3.4. Final Approximation A A Applying the mapping, A → x and A → z , will remove ν from all x ν z ν equations. We now combine the solutions up to third order to get the final solution.

X()τ=++ ρ20 ρ 21cos2 τ 1 ρ 22 cos2 τ 2 + ρ 31 cos3 τ 1 −+Atx cos() λφ

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Y ()τ= σ21sin 2 τσ 1 + 22 sin 2 τσ 2 + 31 sin 3 τσ 1 ++ 32 sin τ 1 kAx sin () λφ t +

Atz sin ()()()λψ++ κ21sin ττ 1 + 2 + κ 22sin τ 2 − τ 1  p  2  +−()1κτ31 sin 3 1 ,p = 0, 2 Z ()τ =  Atz sin ()()()λψ++ κ21sin ττ 1 + 2 + κ 22sin τ 2 − τ 1  p−1  2  +−()1κτ32 cos 3 1 ,p = 1, 3

The coefficients are given in the Appendix I.

5. Analytical Construction of Halo Orbits

The output from the analytical computation of Halo orbit is used to construct Halo orbits through computer-based codes and is plotted using software such as MATLAB. Input parameters are given such as mass ratio (µ), mean distance be-

tween the two primaries, radiation pressures (q1 and q2), amplitude in

Z-direction (Az). Halo orbits are plotted for the classical case of Antares-Sun-Proxima Centauri

system around L1 Lagrangian point. µ = 0.0011314, q1 = q2 = 1, Az = 110,000 km (Figures 2-5).

Figure 2. X vs Y—classical.

Figure 3. X vs Z—classical.

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Figure 4. Y vs Z—classical.

Figure 5. 3D halo orbit—classical.

6. Numerical Computation

Differential Correction (DC) schemes use the STM for targeting purposes. One application is the iterative process to isolate a trajectory arc that connects two points in solution space. Differential Correction techniques can be used to quickly obtain a solution with the desired parameters in a wide range of problems. A sufficiently accurate guess for the initial state is always required. Diffe- rential corrections are often used to obtain periodic solutions to the non-linear differential equations in the CRTBP. A common assumption, making use of the symmetry in this problem, is that the desired solution is symmetric about the x-z plane. The initial guess for finding the solution is taken from the analytic solution. Since the halo orbits are symmetric about x-z plane, (y = 0), and they intersect this plane perpendicularly i.e. ( xy= = 0 ), the initial state vector takes the form, T Xt00()(= x 0,0, z 0 ,0, y 0 ,0)

The final state vector which again lies on the same plane, takes the form as given below and crosses the x-z plane perpendicularly.

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T Xt12() 12 = ( x00,0, z ,0, y 0 ,0)

Then the orbit will be periodic with period Tt= 2 12. The transition matrix at

t12 can be used to adjust the initial values of a nearby periodic orbit. Using Runge-Kutta fourth order method, the equations of motion are integrated until y changes sign. Then the step size is reduced and the integration goes forward again. This is repeated until y becomes almost zero, and the time at this point is

defined as t12. The orbit is considered periodic if x and z are nearly zero

at t12. If this is not the case, x and z can be reduced by correcting two of the three initial conditions and integrate again.

6.1. Numerical Computation of Halo Orbits

The following plots are 2D and 3D plots of the halo orbits for perturbed models. The variation in size, shape of the halo orbits is observed by comparing it with the plots of classical case and the effects of radiation pressures on the halo orbit is observed. The numerical computation of halo orbits is referred from Chi- dambararaj P. and Sharma R. K. [14].

The input parameters are µ = 0.0011314, q1 = 0.98, q2 = 0.97 (Figures 6-10).

Figure 6. X vs Y.

Figure 7. Y vs Z.

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Figure 8. X vs Z.

Figure 9. 3D halo orbit (q1 > q2).

Figure 10. 3D halo orbit (q2 > q1).

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6.2. Perturbing Effects on the Orbit

The effects due to radiation pressure can be seen from Figure 11 below and it is seen that with an increase in radiation pressure, the orbit moves towards the larger source of radiation irrespective of the size of the primary. The oblateness

coefficient are taken as A2 = 0, μ = 0.0011314 and the two cases of radiation

pressures are studied when q1 = 0.98, q2 = 0.97 and when q1 = 0.96, q2 = 0.97. With increasing perturbing forces, it is seen that the orbit will move towards the primary with larger perturbing force and thus it doesn’t remain Halo orbit.

Hence it is to be seen that the value of Az should be taken care such that the mission objectives are optimally obtained (Table 1). It has been observed that the orbit length shortens and the width increases closer to the primary whose radiation pressure is larger. The size factor also comes into effect on the Halo orbit. This can be observed from the 3D plots above as both the plots are not mirror images of each other. This tells that the orbit doesn’t vary with the radiation pressure only. The variation of the orbit shape and size is affected by the mass of the primary as we increase the radiation pressure. Case: When the radiation pressure for both the primaries is equal and not equal to 1. Around the L1 Lagrangian point for the Sun-Proxima system.

For µ = 0.109509, q1 = q2 = 0.98 Halo period = 2.4662.

Figure 11. 3D halo orbit.

Table 1. Radiation pressure parameters, mass ratio and halo period.

q1 q2 μ Halo Period

1 1 0.0011314 6.1178

0.98 1 0.0011314 6.1827

0.98 0.97 0.0011314 6.2234

0.96 0.97 0.0011314 6.0348

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7. Summary and Conclusions

The radiation pressures of larger as well as smaller primary are added. The Lindstedt-Poincare method gives approximate good initial values for the numerical solution. The numerical solution makes use of the adaptive Runge-Kutta fourth-order method as integrator. In the vicinity of L1 point, it has been seen that with increase in radiation pressure, the Halo orbit increases in orbital period. It has been seen that the Halo orbits are very sensitive to initial conditions.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References [1] Farquhar, R. (1968) The Control and Use of Libration Point Satellites. [2] Richardson, D. (1980) Analytical Construction of Periodic Orbits about the Colli- near Points. Celestial Mechanics, 22, 241-253. https://doi.org/10.1007/BF01229511 [3] Euler, L. (1767) Themoturectilineorum corpore se mutuoattrahentium, Novi com- mentarii academium scientistum Petropolitanæ 11. Oeuvres, Seria Secunda tome XXV Commentaries Astronomy, 144-151, 286. [4] Szebeheley, V. (1967) Theory of Orbits. Academic Press, New York. [5] Dutt, P. and Sharma, R.K. (2011) Evolution of Periodic Orbits in the Sun-Mars Sys- tem. Journal of Guidance, Control and Dynamics, 34, 635-644. https://doi.org/10.2514/1.51101 [6] Radzievskii, V. (1950) The Restricted Problem of Three Bodies Taking Account of Light Pressure. The Astronomical Journal, 27, 250-256. [7] Bhatnagar, K.B. and Chawla, J.M. (1979) A Study of the Lagrangian Points in the Photo-Gravitational Restricted Three-Body Problem. Indian Journal of Pure and Applied Mathematics, 10, 1443-1451. [8] Eapen, R.T. and Sharma, R.K. (2014) A Study of Halo Orbits at the Sun-Mars L1 Lagrangian Point in the Photogravitational Restricted Three-Body Problem. Sprin- ger, Berlin, 437-441. https://doi.org/10.1007/s10509-014-1951-6 [9] Sharma, R.K. and Rao, P.S. (1975) Collinear Equilibria and Their Characteristics Exponents in the Restricted Three-Body Problem When the Primaries Are Oblate Spheroids. Celestial Mechanics, 12, 189-201. https://doi.org/10.1007/BF01230211 [10] Simmons, J.F.L., Mcdonald, A.J.C. and Brown, J.C. (1985) The 3-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187. https://doi.org/10.1007/BF01227667 [11] Nishanth, P. and Sharma, R.K. (2017) Mars Interplanetary Trajectory Design via Lagrangian Points in the Restricted Three-Body Problem. Technical Report, ISRO. [12] Koon, W.S., Lo, M.W. and Marsden, J.E. (2011) Dynamical Systems, the Three-Body Problem and Mission Space Design. Springer, Berlin. [13] Thurman, R. and Worfolk, P.A. (1996) The Geometry of Halo Orbits in the Circular Restricted Three Body Problems. Tech. Rep. GCG 95, Univ. Minnesota, Minneapo- lis.

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[14] Chidambararaj, P. and Sharma, R.K. (2016) Halo Orbits around Sun-Earth L1 in Photogravitational Restricted Three-Body Problem with Oblateness of Smaller Pri- mary. International Journal of Astronomy and Astrophysics, 6, 293-311. https://doi.org/10.4236/ijaa.2016.63025

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Appendix I

Coefficients for the second and third-order equations and solution [10] [11] 3Ac2 γ =x 3 k 2 + 2, 1 4  3Ac2 γ = z 3 , 2 4

2 γ31=2,Akx ωλ() − n λ

3Ack2 γ = x 3 , 4 2

2 γ51= 2,Az ωλ 3AAc γ = xz3 , 6 2

3c3 22 2 α1 =()2, −−kAxz A 4  α ρ = − 1 , 20 2 ()nc+ 2 2

22 44nλγ41− γ( λ +− nc2) ρ21 = , 222 22 ()(n−4λλ ++ n 42 − cc22)

22 γλ22(4 +−nc) ρ = − 22 2 , 22 22 ()(n−4λλ ++ n 42 − cc22)

22 4nλγ14− γ( 42 λ ++ nc2) σ 21 = , 222 22 ()(n−4λλ ++ n 42 − cc22)

4nλγ 2 σ 22 = , 222 22 ()(n−4λλ ++ n 42 − cc22) γ κ = − 6 , 21 3λ 2 γ κ = 6 , 22 λ 2

3c3 ν1 =−Axz(42 ρ20 ++ ρ21 kA σ 21 ) +() κκ21 + 22 2

3cA4 x 2 22 +()k −+2 AAxz 2, 2 

2 33c34c Akx 3 22Az ν2 =−Akxx(2ρ20 −− k ρσ21 21 ) +k −1,A + 2 242

2 3c3 33cA4 zz1 22A ν3 =AAzx() ρρ22 −+20 () κκ21 + 22 +kA −2,x + 2 224

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3132 γ7 =−cA3xx()2 ρσ 21 −− k 21 cA 4 ()23 + k , 22

332 γ=c( kA σκ +− A2, A ρ) − c A A 822 3x 22 z 21 x22 4 xz

332 γ=−c( kA σκ −+ A2, A ρ) − c A A 9 223 x22 z 22 x22 4 xz

3332 β3=c 3 Axx() σρ 21 −− k 21 c 4 kA() k +4, 28 33 β=c A() σρ −− k c kA A2 , 428 3x 22 22 4 xz 33 β=c A() σρ ++ k c kA A2 , 528 3x 22 22 4 xz

3 3 δρ3 =−+( Aczz4 4, Ac3 22 ) 8

3322 δρ4 =−−−c3( Az 21 Ak x 21) c 4 Axz A() k +4, 28

3322 δρ5=c 3( Az 21 −+ Ak x 22) c 4 Axz A() k +4, 28

22 69nλ() β3+ ζβ 4 −( λ +−nc23)() γ + ζγ 4 ρ = 31 2 , 22 22 ()(n−9λλ ++ n 92 − cc22)

22 6nλ() γ3+ ζγ 4 −( 92 λ ++nc2)()β3 + ζβ 4 σ 31 = , 222 22 ()(n−9λλ ++ n 92 − cc22) kβ σ = − 5 . 32 2nλ

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International Journal of Astronomy and Astrophysics, 2019, 9, 292-301 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Convective Models of Jupiter’s Zonal Jets with Realistic and Hyper-Energetic Excitation Source

Hans G. Mayr1, Kwing L. Chan2

1NASA Goddard Space Flight Center, Greenbelt, USA (Retired) 2State Key Laboratory for Lunar and Planetary Sciences, Macau University of Science and Technology, Macau, China

How to cite this paper: Mayr, H.G. and Abstract Chan, K.L. (2019) Convective Models of Jupiter’s Zonal Jets with Realistic and Hy- Numerical simulations of Jupiter’s zonal jets are presented, which are gener- per-Energetic Excitation Source. Interna- ated with realistic and hyper energetic source. The models are three dimen- tional Journal of Astronomy and Astro- sional and nonlinear, applied to a gas that is convective, stratified and com- physics, 9, 292-301. https://doi.org/10.4236/ijaa.2019.93021 pressible. Two solutions are presented, one for a shallow 0.6% envelope, the other one 5% deep. For the shallow model (SM), Jupiter’s small energy flux Received: July 10, 2019 was applied with low kinematic viscosity. For the deep model (DM), the Accepted: September 15, 2019 energy source and viscosity had to be much larger to obtain a solution with Published: September 18, 2019 manageable computer time. Alternating zonal winds are generated of order Copyright © 2019 by author(s) and 100 m/s, and the models reproduce the observed width of the prograde equa- Scientific Research Publishing Inc. torial jet and adjacent retrograde jets at 20˚ latitude. But the height variations This work is licensed under the Creative of the zonal winds differ markedly. In SM the velocities vary radially with al- Commons Attribution International License (CC BY 4.0). titude, but in DM Taylor columns are formed. The dynamical properties of http://creativecommons.org/licenses/by/4.0/ these divergent model results are discussed in light of the computed meri- Open Access dional wind velocities. With large planetary rotation rate Ω, the zonal winds are close to geostrophic, and a quantitative measure of that property is the

meridional Rossby number, Rom. In the meridional momentum balance, the 2 ratio between inertial and Coriolis forces produces Rom = V /ΩLU, U zonal, V meridional winds, L horizontal length scale. Our analysis shows that the meridional winds vary with the viscosity like ν1/2. With much larger viscosity and meridional winds, the Rossby number for DM is much larger,

Rom(DM) >> Rom(SM). Compared to the shallow model with zonal winds varying radially, the deeper and more viscous model with Taylor columns is much less geostrophic. The zonal winds of numerical models in the literature tend to be independent of the energy source, in agreement with the present results. With 104 times larger energy flux, the zonal winds for DM only increase by a factor of 3, and the answer is provided by the zonal momentum budget with meridional winds, VU/L = ΩV, yielding U = ΩL, independent of

the source. The same relationship produces the zonal Rossby number, Roz =

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U/ΩL, of Order 1, which is commonly used as a dimensionless measure of the zonal wind velocities.

Keywords Convective Atmosphere, Zonal-Mean Variations, Radial Zonal Winds, Taylor Column Zonal Winds, Energy Invariant Zonal Winds, Geostrophic Balance, Meridional Wind Dynamics

1. Introduction

The alternating wind bands observed on Jupiter (e.g., Smith et al. [1] [2]; Porco et al. [3]) have been simulated with numerical models that can be placed in two groups. One is confined to the stable region in the clouds and accounts for horizontal cascading of turbulence (e.g., Rhines [4]; Marcus et al. [5]; Showman et al. [6]). The other class of models extends below the cloud top and accounts for convective energy transport from the interior (e.g., Busse [7]; Mayr et al. [8]; Sun et al. [9]; Ingersoll et al. [10]; Zhang and Schubert [11]; Christensen [12]; Aur- nou and Olson [13]; Heimpel and Aurnou [14]; Chan and Mayr [15] [16]; Cai and Chan [17]). In this paper, we present simulations of Jupiter’s alternating wind bands generated by convection. The numerical models are nonlinear and three dimensional, applied to a gas that is stratified and compressible. Two solutions are discussed, one for a shallow envelope 0.6% of planetary radius, the other one 5% deep more commensurate with reality. For the shallow model (SM), the small planetary energy flux from the interior was applied together with the corresponding low kinematic viscosity. For the deep model (DM), the applied energy and viscosity had to be much larger to achieve sufficient fast thermal relaxation with manageable computer time, conceptually similar to the deep convective models that have appeared in print. The mean zonal winds generated with SM and DM differ markedly, varying radially with altitude and aligned along Taylor columns, respectively, and they are discussed in light of the computed meridional winds.

2. Numerical Models

The numerical models discussed are based on a series of earlier studies (Chan and Sofia [18] [19]; Chan et al. [20]; Chan [21] [22]; Chan and Mayr [15]). Jupi- ter’s atmosphere is treated as an ideal gas, and a uniform energy flux is applied at the bottom. The energy is carried by convection over 95% of the layer, and it is emitted by radiative diffusion from a thin stable layer at the top. The vertical component of the Coriolis force is accounted for, which figures prominently in the momentum budget. A uniform kinematic viscosity is adopted to dissipate the kinetic energy generated by convection, with the Prandtl number set to 1/3.

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Stress-free and impenetrable boundary conditions are applied at the top and bottom.

2.1. Shallow Model (SM)

The numerical code employs the transformed spectral procedure with associated vector spherical harmonics and solves the time-dependent 3D nonlinear Navier Stokes equations (Chan et al. [20]; Chan and Mayr [15]). The solution procedure is carried out in two stages. For each time step, the linearized equations are solved implicitly in spectral space, and the nonlinear terms are then computed explicitly in physical space. The shallow model (SM) extends into the convection region with ∆r = 0.6% depth of the planetary radius. Jupiter’s energy flux, F = 5.4 W/m2, is applied at the bottom boundary, but the smaller Solar input is ignored. The kinematic viscosity, ν = 5.6 m2/s, is employed, which produces the dimensionless Ekman number, E = ν/Ω∆r2 = 1.7 × 10−7, with planetary rotation rate, Ω = 1.778 × 10−4 rad/s (9.8 hours). SM applies triangular truncation of spherical harmonics up to degree 20 (T20) with limited latitudinal resolution, and 68 radial grid levels for the 0.6% (430 km) shell. With low kinematic viscosity and small Ekman number, the model ran 1 year to reach thermal relaxation, and we present in the following the time average zonal mean variations of the computed zonal wind velocities. Figure 1 is a composite of the zonal wind properties that characterize the SM simulation of Jupiter’s alternating wind bands. Except for the color code, Figure 1(b) is taken from Chan and Mayr [15] and shows the winds at 4 different altitude levels, identified in the underlying scale of the fractional radius. In qualita- tive agreement with observations, alternating wind bands are generated. A dominant prograde equatorial jet is produced with velocities close to 70 m/s, which is within a factor of 2 of the observed values. Adjacent to the equatorial jet, strong retrograde winds are generated at 20˚ latitudes, in agreement with the Voyager observations (Smith et al. [2]). But the alternating jets at higher latitudes are too wide and too few in number. The Taylor-Proudman theorem predicts that for a fluid that is geostrophic and incompressible, the zonal winds become aligned along Taylor columns (TC), and this is the prevailing picture of convective models of the Jupiter atmosphere (e.g., Christensen [12]; Aurnou and Olson [13]; Heimpel and Aurnou [14]). In typical TC models, the tangent cylinder to the inner boundaries determines the width of the equatorial jet, which is defined by the location of the maximum retrograde zonal winds. As illustrated in Figure 1(a), the angular span for the intersections of the tangent cylinder extends to 6˚ latitudes. In contrast, the equatorial jet in Figure 1(b) has a much larger 20˚ span, which demonstrates that SM does not produce a TC pattern at low latitudes. Figure 1(b) shows that the zonal winds at different levels vary with altitude. This picture is brought into focus in Figure 1(c), where the wind pattern of the southern hemisphere is displayed along an expanded altitude scale.

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Figure 1. Composite of the zonal wind properties that characterize the shallow model (SM) simulation of Jupiter’s alternating wind bands. Except for the color code, (b) is taken from Chan and Mayr [15] and shows the winds at 4 different altitude levels, identified in the underlying scale of the fractional radius. As illustrated in (a), the angular span for the intersections of the tangent cylinder extends to 6˚ latitudes. In contrast, the equatorial jet in (b) has a much larger 20˚ span, which demonstrates that SM does not produce a Taylor column pattern parallel to the rotation axis. This picture is brought into focus in (c), where the zonal wind pattern of the southern hemisphere is displayed on an expanded altitude scale.

The zonal winds of SM are not aligned along the rotation axis but vary radially with altitude, and the question is whether this property will survive in deeper models with realistic planetary parameters and sufficient low viscosity. Short of the results from such a computationally demanding study, it is instructive to examine a simulation from a numerical model, stratified and compressible, which is much deeper but employs a much larger energy source and viscosity.

2.2. Deep Model (DM)

Considering Ohmic dissipation associated with Jupiter's magnetic field and measured conductivity, Liu et al. [23] estimated that the zonal winds cannot penetrate below 0.96 radius. This is on the order of depth employed in present convective models (e.g., Christensen [12]; Aurnou and Olson [13]; Heimpel and Aurnou [14]), referred to as CAOH. CAOH simulate the prograde equatorial jet and alternating wind bands with large viscosity, applying much larger energy flux than Jupiter’s. Following CAOH, a deep model (DM) was constructed with relative depth Δr = 5%, which was presented by Chan and Mayr [16]. For this model, a grid point formulation was employed to integrate the time-dependent 3D nonlinear

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Navier Stokes equations across the globe (Chan [22]). The model was run with an energy flux, F = 7.8 × 104 W/m2, a factor of 1.4 × 104 larger than that of Jupi- ter. For the kinematic viscosity the value ν = 2.3 × 105 m2/s was chosen, which produces the Ekman number E = ν/ΩΔr2 = 1.0 × 10−4. With this large viscosity, the model ran 2 months to produce the numerical results. Analogous to Figure 1, we present in Figure 2 the zonal winds from the deep model (DM). In agreement with observations, multiple alternating jets are generated that extend to high latitudes. The prograde equatorial jet has a velocity of about 200 m/s, and the adjacent retrograde jets are 20o wide in agreement with Voyager observations. But unlike the latitudinal variations of the computed zonal winds, on Jupiter the equatorial jet dominates. In contrast to SM, the zonal winds from DM clearly show the pattern of Taylor columns. Illustrated in Figure 2(a), the angular span for the intersect- ing tangent cylinder is ±20˚ wide, determined by the 5% depth of the model. This agrees with the latitudes of the retrograde jets in Figure 2(b), which define the width of the equatorial wind pattern. Around the equator, the winds at different altitude levels vary between −150 and +150 m/s, aligned parallel to the tangent cylinder. Away from the equator, the deeper zonal winds (positive or negative) occupy lower latitudes, brought into focus in Figure 2(c), which is consistent with alignment along the rotation axis. The zonal winds of DM form Taylor columns, in substantial agreement with the convection models of CAOH.

Figure 2. Composite of the zonal wind properties, similar to Figure 1, but for the deep model (DM). Figure 1(b) is taken from Chan and Mayr [16] and shows the winds at 4 different altitude levels, identified in the underlying scale of the fractional radius. (a) shows that the angular span of the tangent cylinder extends to 20˚ latitudes, in agreement with the width of the equatorial jet in (b). The zonal winds form a Taylor column pattern, brought into focus in (c).

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3. Discussion

For model simulations of a convective Jovian atmosphere that is both stratified and compressible, it is remarkable that the resulting alternating wind bands are formed with such different altitude patterns. In the shallow model the zonal velocities vary radially, but in the deeper model the variations are aligned along the rotation axis to form Taylor columns. Apart from the different vertical domains, the applied energy source and related viscosity must come into play. The Taylor-Proudman theorem applies if the zonal winds are in geostrophic balance. For the shallow model (SH), geostrophy was explicitly demonstrated by comparing the meridional pressure gradient with the Coriolis force (Chan and Mayr [15]). A quantitative measure of geostrophic balance is the meridional

Rossby number, Rom, which is defined as the ratio between inertial and Coriolis forces. Among the nonlinear inertial accelerations in the meridional momentum equation that describes the mean zonal wind, U, the term V∂V/∂θ is the largest, θ latitude and V mean horizontal meridional wind. Compared with the Coriolis

force term ΩU, the meridional Rossby number then can be estimated, Rom = V2/LΩU, with the characteristic horizontal length scale, L = λ/2π (λ, horizontal wavelength) that is related to the planetary radius, r. Figure 3 shows the meridional winds, on the left for the shallow model (SM), on the right for the deep model (DM). At any given latitude, the converging and diverging velocities have opposite directions in the northern and southern hemispheres. Apart from that, the different wind patterns mirror those of the zonal velocities (Figure 1, Figure 2): a dominant single cell circulation for SM with dominant equatorial jet, and a multi-cellular circulation for the alternating wind bands of DM. Obeying flow continuity, the meridional velocities are much larger

Figure 3. Computed meridional winds, left for the shallow model (SM), right the deep model (DM). Converging and diverging velocities have opposite directions in the northern and southern hemispheres. Obeying flow continuity, the meridional velocities are much larger at the top of the domain (black versus red/green), where the ambient densities are much smaller (identical in both models). Note that the maximum wind velocities for SM are very small less than 0.10 m/s, in contrast to DM with velocities close to 30 m/s (figures are taken from Chan and Mayr [16]).

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at the top of the domain (black versus green/red), where the ambient densities are much smaller (identical in both models). Apart from the contrasting wind patterns, the meridional winds feature large differences in magnitude. For SM at the top of the domain, the maximum wind velocities are very small less than 0.10 m/s, in contrast to DM with velocities close to 30 m/s. Given the wind velocities, the corresponding model parameters are listed in Table 1 for L = r. The applied viscosities, ν, are sufficiently small in both models to assure that rotation dominates and the Ekman numbers are very small, E << 1. This holds also for the deep model with much larger ν. The meridional Rossby

numbers are very small as well, Rom << 1, demonstrating that both model results

are approximately in geostrophic balance. But Rom is much larger for DM with Taylor columns. Our model results reveal an intriguing relationship between the viscosity and meridional winds. As shown in Table 1, last column, the dimensionless numbers for the two models, V2/νΩ differ only by a factor of two, which is remarkable considering that the input parameters for the energy source and viscosity differ by orders of magnitude. The chosen viscosity apparently determines the magnitude of the meridional wind, V varying with (νΩ)1/2. Another intriguing property of the numerical results is the invariance of the zonal velocities in relation to the energy source. As shown in Table 1, the winds increase from 70 m/s (SM) to 200 m/s (DM), produced by an energy flux a factor of 1.4 × 104 larger. This trend is observed in planetary atmospheres. And the Ju- piter models in the literature, with energies far exceeding the planetary value, all feature zonal wind velocities comparable to those observed. Mayr et al. [24] addressed this problem with a simplified 2D scale analysis of the zonal momentum budget, where the Coriolis force, ΩV, is balanced by the viscous stress of the zonal winds, UK/L2, with K the eddy viscosity and L the horizontal scale of the circulation. Applying mixing length theory, K = VL, one obtains U = ΩL, which produces for the Jovian circulation zonal winds of order 100 m/s. Essentially, the solution is provided by the nonlinear zonal momentum budget. For the zonal-mean circulation, the inertial force, V∂U/∂λ, dominates, and the balance with the Coriolis force yields, VU/L = ΩV, to produce U = ΩL, independent of the energy source. The same relationship produces the zonal

Rossby number, Roz = U/ΩL, of order 1, which is commonly used as dimensionless measure of the zonal wind velocities.

Table 1. The chosen model parameters for SM and DM are listed: ∆r (m) depth of convection region, F (W/m2) planetary energy flux from interior, and ν kinematic viscosity. The model generated zonal and meridional wind velocities, U (m/s), V (m/s), are shown for the top of the atmosphere. The Ekman number, E = ν/ΩΔr2, is a measure of the importance of planetary rotation. And 2 the meridional Rossby number, Rom = V /rΩU, is a measure of geostrophic balance for the zonal winds. The model invariant ratios, V2/νΩ, indicate that the meridional winds, V vary with (νΩ)1/2.

2 2 2 2 2 Δr (m) F (W/m ) ν (m /s) U (m/s) V (m/s) E = ν/ΩΔr Rom = V /rΩU V /νΩ

Shallow SM 4.3 × 105 5.4 5.6 70 0.1 1.7 × 10−7 1.1 × 10−8 10.0

Deep DM 3.6 × 106 7.5 × 104 2.3 × 105 200 30 1.0 × 10−4 3.4 × 10−4 22.0

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In geostrophic balance, the zonal winds are produced by latitudinal temperature/pressure variations. But the source that produces the temperature variations is also generating the meridional circulation that redistributes or dissipates the kinetic energy. The zonal velocities thus tend to be independent of the energy source.

4. Summary

Numerical simulations of Jupiter’s zonal jets are discussed, which are generated with models that are three dimensional and fully nonlinear, applied to a gas that is convective, stratified and compressible. Solutions are presented for shallow and deep atmospheric envelopes, generated with realistic and hyper-energetic source. In the shallow model (SM) with realistic energy source, the zonal winds vary radially with altitude, in contrast to the energetic deep model (DM) where the winds are aligned along the rotation axis to form Taylor columns (TC). In agreement with observations, both models produce prograde equatorial jets of order 100 m/s. Both models also reproduce the observed width of the equatorial jet with adjacent retrograde jets at 20˚ latitude—a natural outcome for SM, but determined by the chosen 5% depth of DM with TC. The dynamical properties of these divergent model results are discussed in light of the meridional winds, which are small in magnitude compared with the zonal winds. But unlike the rotational zonal winds, the meridional winds have divergence, and thus are involved with energy and momentum transport, which is of central importance for understanding the zonal mean circulation.

The Rossby number, Rom, for the meridional momentum balance is the quantitative measure of geostrophy, and it is a quadratic function of the meridional

winds. For DM with large viscosity and TC, the meridional winds and Rom are orders of magnitude larger compared to SM. DM is much less geostrophic. Ranking geostrophy cannot explain the difference between SM and DM. For Taylor columns to form, Taylor-Proudman requires that the gas is also incompressible, in addition to geostrophic. But both models treat the atmosphere as compressible. In models like DM with large viscosity, the enhanced energy transport by the meridional winds has the capacity to reduce the vertical variations in the latitudinal temperature distribution to produce a barotropic environment that favors the formation of TC. The question is whether deeper models, with realistic energy flux and low viscosity, will produce zonal winds that vary radially with altitude like SM. The numerical results presented highlight an important property of planetary atmospheres, the invariance of the zonal winds in relation to the energy source. With 104 times larger source, the velocities of DM increase only by a factor of 3. And the Jupiter models in the literature with energies far exceeding the planetary value all feature zonal winds comparable to those observed. Following up on an earlier paper (Mayr et al. [24]), the solution of this problem is provided by the nonlinear zonal momentum budget with meridional winds. It produces zonal winds varying with the planetary rotation rate and horizontal scale of the circu-

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lation, independent of the energy source.

Acknowledgements

Funded by the Science and Technology Development Fund, Macau SAR (File No. 0045/2018AFJ). The reviewer’s comments contributed significantly to improve the presentation of the paper. This work was supported by the State Key Laboratory for Lunar and Planetary Sciences, Macau University of Science and Technology.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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https://doi.org/10.1029/2000GL012643 [13] Aurnou, J.M. and Olson, P.I. (2001) Strong Zonal Winds from Thermal Convection in a Rotating Spherical Shell. Geophysical Research Letters, 28, 2557-2559. https://doi.org/10.1029/2000GL012474 [14] Heimpel, M. and Aurnou, J. (2007) Turbulent Convection in a Rapidly Rotating Spherical Shell: A Model for Equatorial and High Latitude Jets on Jupiter and Sa- turn. Icarus, 187, 540-557. https://doi.org/10.1016/j.icarus.2006.10.023 [15] Chan, K.L. and Mayr, H.G. (2008) A Shallow Convective Model for Jupiter’s Alter- nating Wind Bands. Journal of Geophysical Research, 113, E10002. https://doi.org/10.1029/2008JE003124 [16] Chan, K.L. and Mayr, H.G. (2008) Convective Models of Jupiter’s Wind Bands: Transition from Deep to Shallow Envelopes. 40th Annual Meeting of Division of Planetary Sciences of American Astronomical Society, Ithaca. [17] Cai, T. and Chan, K.L. (2012) Three-Dimensional Numerical Simulation of Con- vection in Giant Planets: Effects of Solid Core Size. Planetary and Space Science, 71, 125-130. https://doi.org/10.1016/j.pss.2012.07.023 [18] Chan, K.L. and Sofia, S. (1986) Turbulent Compressible Convection in a Deep At- mosphere: III Tests on the Validity and Limitation of the Numerical Approach. The Astrophysical Journal, 307, 222-241. https://doi.org/10.1086/164409 [19] Chan, K.L. and Sofia, S. (1989) Turbulent Compressible Convection in a Deep At- mosphere: IV Results of Three-Dimensional Computations. The Astrophysical Journal, 336, 1022-1040. https://doi.org/10.1086/167072 [20] Chan, K.L., Mayr, H.G., Mengel, J.G. and Harris, I. (1994) A “Stratified” Spectral Model for Stable and Convective Atmospheres. Journal of Computational Physics, 113, 165-176. https://doi.org/10.1006/jcph.1994.1128 [21] Chan, K.L. (2001) Rotating Convection in f-Planes: Mean Flow and Reynolds Stress. The Astrophysical Journal, 548, 1102-1117. https://doi.org/10.1086/318989 [22] Chan, K.L. (2006) A Finite-Difference Convective Model for Jupiter’s Equatorial Jet. In: Kupka, F., et al., Eds., Proceedings IAU Symposium No. 239: Convection in As- trophysics, Cambridge University Press, New York, 230-232. https://doi.org/10.1017/S174392130700049X [23] Liu, J., Goldreich, M. and Stevens, D.J. (2008) Constraints on Deep-Seated Zonal Winds Inside Jupiter and Saturn. Icarus, 196, 653-664. https://doi.org/10.1016/j.icarus.2007.11.036 [24] Mayr, H.G., Chan, K.L., Harris, I. and Schatten, K. (1991) What Maintains the Zon- al Circulation in Planetary Atmospheres. The Astrophysical Journal, 367, 361-366. https://doi.org/10.1086/169634

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International Journal of Astronomy and Astrophysics, 2019, 9, 302-320 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Models for Velocity Decrease in HH34

Lorenzo Zaninetti

Physics Department, via P.Giuria 1, Turin, Italy

How to cite this paper: Zaninetti, L. (2019) Abstract Models for Velocity Decrease in HH34. International Journal of Astronomy and The conservation of the energy flux in turbulent jets that propagate in the in- Astrophysics, 9, 302-320. terstellar medium (ISM) allows us to deduce the law of motion when an in- https://doi.org/10.4236/ijaa.2019.93022 verse power law decrease of density is considered. The back-reaction that is

Received: July 24, 2019 caused by the radiative losses for the trajectory is evaluated. The velocity de- Accepted: September 16, 2019 pendence of the jet with time/space is applied to the jet of HH34, for which Published: September 19, 2019 the astronomical data of velocity versus time/space are available. The intro-

Copyright © 2019 by author(s) and duction of precession and constant velocity for the central star allows us to Scientific Research Publishing Inc. build a curved trajectory for the superjet connected with HH34. The bow This work is licensed under the Creative shock that is visible in the superjet is explained in the framework of the Commons Attribution International theory of the image in the case of an optically thin layer. License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/ Open Access Keywords

Herbig-Haro Objects, Bok Globules, Bipolar Outflows

1. Introduction

The equation of motion plays a relevant role in our understanding of the physics of the Herbig-Haro objects (HH) after [1] [2]. A common example is to evaluate the velocity of the jet in HH34 as 300 km/s, see [3], without paying attention to its spatial or temporal evolution. A precise evaluation of the evolution of the jet’s velocity with time in HH34 has been done, for example, by [4]. It is, therefore, possible to speak of proper motions of young stellar outflows, see [5] [6] [7] [8]. The first set of theoretical efforts exclude the magnetic field: [9] have modeled the slowing down of the HH 34 superjet as a result of the jet’s interaction with the surrounding environment, [10] have shown that a velocity profile in the jet beam is required to explain the observed acceleration in the position-velocity diagram of the HH jet, [11] found some constraints on the physical and chemical parameters of the clump ahead of HHs and [12] reviewed some important

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understanding of outflows from young stars. The second set of theoretical efforts include the magnetic field: [13] analysed the HH 1-2 region in the L1641 molecular cloud and found a straight magnetic field of about 130 micro-Gauss; [14] analysed HH 211 and found field lines of the magnetic field with different orientations; [15] analysed HH 111 and found evidence for magnetic braking. These theoretical efforts to understand HH objects leave a series of questions unanswered or partially answered, as follows:  Is it possible to find a law of motion for turbulent jets in the presence of a medium with a density that decreases as a power law?  Is it possible to introduce the back reaction into the equation of motion for turbulent jets to model the radiative losses?  Can we model the bending of the super-jet connected with HH34?  Can we explain the bow shock visible in HH34 with the theory of the image? To answer these questions, this paper reviews in Section 2 the velocity observations of HH34 at a 9 yr time interval, Section 3 analyses two simple models as given by the Stoke’s and Newton’s laws of resistance, Section 4 applies the conservation of the energy flux in a turbulent jet to find an equation of motion, Section 5 models the extended region of HH34, the so called “superjet”, and Section 6 reports some analytical and numerical algorithms that allow us to build the image of HH34.

2. Preliminaries

The velocity evolution of the HH34 jet has recently been analysed in [SII] 2, (672 nm), frames and Table 1 in [4] reports the Cartesian coordinates, the velocities, and the dynamical time for 18 knots in 9 years of observations. To start with time, t, equal to zero, we fitted the velocity versus distance with the following power law α v() xx;,00 v= v 0 ×() xx 0 , (1)

where v and x are the velocity and the length of the jet, v0 is the velocity at

xx= 0 and α with its relative error is a parameter to be found with a fitting procedure. The integration of this equation gives the time as a function of the position, x, as given by the fit xxx−+αα1 − t = − 00, (2) ()α −1 v0

where x0 is the position at t = 0 . The fitted trajectory, distance versus time, is

ααln()(x0−− ln tv 0 ++ tv 00 x ) α −1 xtx();,00 v = e , (3)

and the fitted velocity as function of time is α ααln()x−− ln( t() − 1 vx +) 1 0 00 vtx();, v= v e α −1 . 00 0 (4) x0

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Table 1. Numerical values for the physical parameters of HH34.

Knot x (pc) v (km/s) Time (yr)

1 0.002594 171.01 0.

2 0.004021 202.15 7.484

3 0.007214 171.23 25.131

4 0.009434 176.13 37.903

5 0.012545 164.04 56.303

6 0.016017 163.24 77.381

7 0.018125 141.03 90.415

8 0.019580 162.00 99.502

9 0.021046 165.07 108.71

10 0.02622 156.08 141.76

11 0.02745 156.08 149.75

12 0.03096 142.03 172.63 13 0.03528 148.03 201.20 14 0.03723 148.03 214.17

15 0.04050 148.569855 236.15

16 0.04420 138.293167 261.17

17 0.04880 134.082062 292.60

18 0.05684 143.003494 348.25

The adopted physical units are pc for length and year for time, and the useful conversion for the velocity is 1 pc year= 979682.5397 km s . The fit of Equation (1) when x is expressed in pc gives −± vx() = 0.000107 x0.0998 0.01618 pc yr , (5)

from which we can conclude that the velocity decreases with increasing distance, see Figure 1. The time is derived from Equation (2) and Table 1 reports the basic parameters of HH34. This time is more continuous in respect to the dynamical time reported in column 6 of Table 1 in [4].

3. Two Simple Models

When a jet moves through the interstellar medium (ISM), a retarding drag force

Fdrag , is applied. If v is the instantaneous velocity, then the simplest model assumes

n Fvdrag ∝ , (6)

where n is an integer. Here, the case of n = 1 and n = 2 is considered. In classical mechanics, n = 1 is referred to as Stoke’s law of resistance and n = 2 is referred to as Newton’s law of resistance.

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Figure 1. Observational points of velocity in pc/yr versus distance in pc (empty circles) and best fit as given by Equation (1) (full line).

3.1. Stoke’s Behaviour

The equation of motion is given by dvt() = −Bv() t . (7) dt The velocity as function of time is −Bt vv= 0e, (8)

where v0 is the initial velocity. The distance at time t is vve−Bt x==−+ st() x 00. (9) 0 BB The time as function of distance is obtained by the inversion of this equation 1 xB−− Bx v t =−−ln 00. (10) Bv0 The velocity as a function of space is

v() x;,, x00 v B=−++ xB Bx0 v 0. (11) The numerical value of B is vv− B = − 01, (12) xx01−

where v1 is the velocity at point x1 ; the data of Table 1 gives B = 0.0009549 (Figure 2).

3.2. Newton’s Behaviour

The equation of motion is

dvt() 2 = −Av() t . (13) dt The velocity as function of time is v v= vt() = 0 , (14) Atv0 +1

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Figure 2. Observational points of velocity in pc/yr versus distance in pc (empty circles), best fit as given by Equation (1) (full line), Stokes behaviour as given by Equation (11) (dashed line) and Newton behaviour as given by Equation (17) (dot-dash-dot-dash).

where v0 is the initial velocity. The distance at time t is

ln()Atv0 + 1 x= st() = + x. (15) A 0 The time as function of distance is obtained by the inversion of the above equation e1xA− Ax0 − t = . (16) Av0

The velocity as function of the distance is

v0 v() xx;,,00 v A = . (17) exA− Ax0 The numerical value of A is 1 v A = − ln0 , (18) xx01−  v 1

where v1 is the velocity at point x1 ; the data of Table 1 gives A = 5.68381834.

4. Energy Flux Conservation

The conservation of the energy flux in a turbulent jet requires a perpendicular section to the motion along the Cartesian x-axis, A Ar() = π r2 (19)

where r is the radius of the jet. Section A at position x0 is 2 α Ax()00= π xtan  (20) 2

where α is the opening angle and x0 is the initial position on the x-axis. At position x, we have

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2 α Ax() = π xtan . (21) 2 The conservation of energy flux states that

113 ρρ()()x vAx3 = ()()() xvx Ax[ B (22) 2200 0

where vx() is the velocity at position x and vx00() is the velocity at position

x0 , see Formula A28 in [16]. More details can be found in [17] [18]. The density is assumed to decrease as a power law δ x0 ρρ= 0  (23) x

where ρ0 is the density at xx= 0 and δ a positive parameter. The differential equation that models the energy flux is

δ 3 1dx0 2 1 32 xt() x−= vx00 0. (24) 2dxt 2

The velocity as a function of the position, x,

2 δ 2 x 3 x0 xv 00 x vx() = δ . (25) x0 x x Figure 3 reports the velocity as a function of the distance and the observed points. We now have four models for the velocity as a function of time and Table 2 reports the merit function χ 2 , which is evaluated as

Figure 3. Observational points of velocity in pc/yr versus distance in pc (empty stars).

The theoretical fit is given by Equation (25) (full line) with parameters x0 = 0.00259 pc ,

v0 = 191.27 km s and δ = 1.7 .

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Table 2. The values of the χ 2 for four models of velocity of HH34.

Model χ 2

Power law fit (no physics) 1479

Stoke’s behaviour 3813

Newton’s behaviour 3317

Turbulent jet 2373

N 2 2  χ =∑ yyi,, theo − i obs (26) i=1

where yi, obs represents the observed value at position i and yi, theo the theoretical value at position i. A careful analysis of Table 2 allows us to conclude that the turbulent jet performs better in respect to the Stokes’s and Newton’s behavior. The trajectory, i.e. the distance as function of the time,

1 tv33 3ln()() 3− 3ln 5 −−δ ln0 δ −53 x0 xtr();,00 vo ,δ = x e , (27)

and the velocity as function of time

−−2δ ()δδ−5511−−() −+21δ 21δ − 33 33 1 δ − 3 tv00 tv vtr();, v ,δδ= 35 () 5 − δ −5 3 x 00 033  x0 xx00   −δ (28) 11tv33 tv33 −3ln()() 3 + 3ln 5 −+δδ ln00−3ln()() 3 + 3ln 5 −+ ln δδ−−55xx33 × 00 v0 ee  

Figure 4 reports the trajectory as a function of time and of the observed points. The rate of mass flow at the point x, mx () , is

2 α m ()() x;, v alpha=ρ v x π x tan  (29) 2

and the astrophysical version is

m ( xx;,0 v 0,km s , M ,α ) (30) −−8 43 23δδ2 23+23 = 7.9252910nx () tan()α 2 x0v 0,km s M  yr

where α is the opening angle in rad, x and x0 are expressed in pc, n is the -3 number density of protons at xx= 0 expressed in particles cm , M  is the

solar mass and v0,km s is the initial velocity at point x0 expressed in km/s. This rate of mass flow as function of the distance x increases when δ < 2 , is constant when δ = 2 , and decreases when δ > 2 .

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Figure 4. Observational points of distance in pc versus time in years (empty stars). The theoretical curve is given by Equation (27) (full line) with the same parameters as in Figure 3.

The Back Reaction

Let us suppose that the radiative losses are proportional to the flux of energy

1 3 − ρ ()()()xvx Ax, (31) 2 where  is a constant that is thought to be  1. By inserting in the above equation the considered area, Ax() , and the power law density here adopted the radiative losses are δ 2 1 x0 32α −π ρ0 vxtan . (32) 22x  By inserting in this equation the velocity to first order as given by Equation

(25), the radiative losses, Q() xx;00 , v ,,δ  , are 1 2 Q() xx;,,, v δρ=−πv32x ()tan() α 2x , (33) 0 0 2 00 0

The sum of the radiative losses between x0 and x is given by the following integral, L, x L()( x;,,, x00 vδ = Q x ;,, x00 v delta ,d) x ∫x0 2 (34) 1 32α =−π ρ00vx 0tan () xx −0 . 22 The conservation of the flux of energy in the presence of the back-reaction due to the radiative losses is δ 11x vx32 x−+ vx 33 0 vx32 ρρ = vx32 00 00  0 000 (35) 22x The real solution of the cubic equation for the velocity to the second order,

vc () x;,δ xv00 , , is

3 22+δ 42 − δ −+ 2 δδ − vc ()() x;,δ x00 , v=−−− x x01 x0 x vx 0 x 0. (36)

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Figure 5 reports the effect of introducing the losses on the velocity as function of the distance for a given value of  , i.e. the velocity decreases more quickly. The presence of the back-reaction allows us to evaluate the jet’s length, which can be derived from the minimum in the corrected velocity to second order as a function of x, ∂v() x;,δ xv , c 00= 0, (37) ∂x which is −v(δδ x −  xx − +22  x −+ δ) x−δδ3 + 23 x −+ 53 3 0 000 = 23 0. (38) 31( +−()xx0 )

The solution for x of the above minimum allows us to derive the jet’s length,

x j , δδxx−22 +− x = 00 . (39) j ()δ −1

Figure 6 reports an example of the jet’s length as a function of the parameter δ .

Figure 5. Velocity to the second order as function of the distance, see Equation (36), when  = 0 (full line) and  = 0.1 (dashed line), other parameters as in Figure 3.

Figure 6. Length of the jet as a function of δ , and the other parameters are as in Figure 3.

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5. The Extended Region

To deal with the complex shape of the continuation of HH34 (e.g. see the new region HH173 discovered by [19]), we should include the precession of the source and motion of the host star, following a scheme outlined in [20]. The ()()()(1 1 11) various coordinate systems are x = ()xyz,, , x = ()xyz,, , , x()()()()3= ( xyz 3,, 33) . The vector representing the motion of the jet is represented by the following 13× matrix: xt()  G = 0, (40) 0 where the jet motion L(t) is considered along x axis. ()1 The jet axis, x, is inclined at an angle Ψ prec relative to an axis x , and therefore the 33× matrix, which represents a rotation through z axis, is given by:  cos()()Ψprec −Ψ sinprec 0   F =ΨΨsin()()prec cosprec 0 . (41)  0 01 

()1 The jet is undergoing precession around the x axis and Ω prec is the angular velocity of precession expressed in radians per unit time. The transformation from ()1 the coordinates x fixed in the frame of the precessing jet to the nonprecessing ()2 coordinate x is represented by the 33× matrix 10 0   P=0 cos()() Ωprec tt −Ω sinprec . (42)  ΩΩ 0 sin()()prectt cos prec ()2 The last translation represents the change of the framework from ( x ), ()3 which is co-moving with the host star, to a system ( x ), in comparison to which the host star is in a uniform motion. The relative motion of the origin of the coordinate system ( xyz()()()3,, 33) is defined by the Cartesian components

of the star velocity vvvxyz,,, and the required 13× matrix transformation representing this translation is

vtx =  B vty . (43)  vtz

On assuming, for the sake of simplicity, that vx = 0 and vz = 0 , the translation matrix becomes 0 =  B vty . (44) 0

The final 13× matrix A representing the “motion law” can be found by

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composing the four matrices already described;  cos()Ψ prec xt()   A=+ B() P ⋅⋅ F G = vty +cos()() Ωprec t sin Ψprec xt() . (45)  ΩΨ sin()()prect sin prec xt()

The three components of the previous 13× matrix A represent the jet’s motion along the Cartesian coordinates as given by an observer who sees the star moving in a uniform motion. The point of view of the observer can be modeled by introducing the matrix E, which represents the three Eulerian angles ΘΦΨ,, , see [21]. A typical trajectory is reported in Figure 7 and a particularised point of view of the same trajectory is reported in Figure 8 in which a loop is visible.

Figure 7. Continuous trajectory of the superjet connected with HH34: the three Eulerian angles characterising the point of view are Φ=0 , Θ=0 and Ψ=0 . The precession  is characterised by the angle Ψ=prec 10 and by the angular velocity Ω =  = prec 0.00496551674 year . The star has velocity vy 31.107 km s , the considered time is 29,000 yr and the other parameters are as in Figure 3.

Figure 8. Continuous trajectory of the superjet connected with HH34: the three Eulerian angles characterising the point of view are Φ=100 , Θ=77 and Ψ=135 . The other parameters as in Figure 7.

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6. Image Theory

This section summarises the continuum observations of HH34, reviews the transfer equation with particular attention to the case of an optically thin layer, analyses a simple analytical model for theoretical intensity, reports the numerical algorithm that allows us to build a complex image and introduces the theoretical concept of emission from the knots.

6.1. Observations

The system of the jet and counter jet of HH34 has been analysed at 1.5 μm and 4.5 μm, see Figure 3 in [3]. The intensity is almost constant, 16−− 1 2 I1.5 ≈×8 10 erg ⋅ s arcsec for the first 12" of the jet and 16−− 1 2 I1.5 ≈×3 10 erg ⋅ s arcsec for the first 20" of the counter jet. At larger distances, the intensity drops monotonically. At a distance of 414 pc as given by [4] the conversion between physical and angular distance is 1 pc= 498.224'' . For example, at 1.5 μm, the emission is mainly due to the [Fe II]1.64 μm line.

6.2. The Transfer Equation

For the transfer equation in the presence of emission only see, for example, [22] or [23], is

dIν =−+kIννρρ j ν, (46) ds

where Iν is the specific intensity, s is the line of sight, jν is the emission

coefficient, kν is a mass absorption coefficient, ρ is the density of mass at position s, and the index ν denotes the frequency of emission. The solution to Equation (46) is

jν −τν ()s Iνν()τ =()1e − , (47) kν

where τν is the optical depth at frequency ν :

dτρνν= ks d. (48)

We now continue to analyse a case of an optically thin layer in which τν is

very small (or kν is very small) and where the density ρ is replaced by the concentration Cs() of the emitting particles:

jν ρ = KC() s , (49)

where K is a constant. The intensity is now s Iν ()() s= K C s'd s' optically thin layer, (50) ∫s0

which in the case of constant density, C, is

Iν () s= KC ×−() s s0 optically thin layer. (51)

The increase in brightness is proportional to the concentration of particles integrated along the line of sight.

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6.3. Theoretical Intensity

The flux of observed radiation along the centre of the jet, Ic is assumed to scale as Q() xx; , v ,, b I() xx; , v ,, b ∝ 00 , (52) c 00 x2 where Q, the radiative losses, is given by Equation (33). The explicit form of this equation is

23 2 1 (−+1 ()x − x0) xv 00 π()tan()αρ 2 0 I() xx; , v ,, b = − . (53) c 00 2 x2 This relation connects the observed intensity of radiation with the rate of energy transfer per unit area. A typical example of the jet of HH34 at 4.5 μm is reported in Figure 9.

6.4. Emission from a Cylinder

A thermal model for the image is characterised by a constant temperature and density in the internal region of the cylinder. Therefore, we assume that the number density C is constant in a cylinder of radius a and then falls to 0, see the simplified transfer Equation (51). The line of sight when the observer is situated at the infinity of the x-axis and the cylinder’s axis is in the perpendicular position is the locus parallel to the x-axis, which crosses the position y in a Cartesian x-y plane and terminates at the external circle of radius a. A similar treatment for the sphere is given in [24]. The length of this locus in the optically thin layer approximation is =×22 − ≤< lab 2( a y) ;0 ya . (54)

The number density Cm is constant in the circle of radius a and therefore the intensity of the radiation is

Figure 9. Observational points of intensity at 4.5 μm (empty stars) and theoretical curve  as given by Equation (53) (full line). when  = 1 100 , ρ0 = 1 , α = 2.86 and other parameters as in Figure 3.

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= ××22 − ≤ < I0am C2( a y) ;0 ya . (55)

A typical example of this cut is reported in Figure 10 and the intensity of all the cylinder is reported in Figure 11.

6.5. Numerical Image

The numerical algorithm that allows us to build a complex image in the optically thin layer approximation is now outlined.  An empty, value = 0, memory grid ()i,, jk which contains 4003 pixels is considered.  The points which fill the jet in a uniform way to simulate the constant density in the emitting particles are inserted, value = 1, in ()i,, jk  Each point of ()i,, jk has spatial coordinates xyz,, which can be represented by the following 13× matrix, A,

Figure 10. 1D cut of the intensity, I, when a = 0.01 pc .

Figure 11. 2D map of the intensity of a jet which has length 0.1 pc and radius of 0.01 pc.

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x  Ay= . (56) z

The orientation of the object is characterised by the Euler angles ()ΦΘΨ,, and therefore by a total 33× rotation matrix, E, see [21]. The matrix point is represented by the following 13× matrix, B, B= EA ⋅ . (57)  The intensity 2D map is obtained by summing the points of the rotated images. A typical result of the simulation is reported in Figure 12, which should be compared with the observed image as given by Figure 13.

Figure 12. 2D intensity map of HH34, parameters as in Figure 8.

Figure 13. Three-color composite image of the young object HH34.

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6.6. The Mathematical Knots

The trefoil knot is defined by the following parametric equations: xt=sin()() + 2sin 2 t (58)

yt=cos()() − 2cos 2 t (59)

zt= −sin() 3 (60)

with 02≤≤πt . The visual image depends on the Euler angles, see Figure 14. The image in the optically thin layer approximation can be obtained by the numerical method developed in Section 6.5 and is reported in Figure 15. This 2D map in the theoretical intensity of emission shows an enhancement where two mathematical knots apparently intersect.

Figure 14. 3D view of the trefoil when the three Eulerian angles which characterise the point of view are Φ=0 , Θ=90 and Ψ=0 .

Figure 15. Image of the trefoil with parameters as in Figure 14, the side of the box in pc is 1 and the radius of the tube in pc is 0.006.

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7. Conclusions

Laws of motion: We analysed two simple models for the law of motion in HH objects as given by the Stoke’s and Newton’s behaviour, see Section 3. A third law of motion is used for turbulent jets in the presence of a medium whose density decreases with a power law, as given by Equation (23). The model that is adopted for the turbulent jets conserves the flux of energy. For example, Equation (25) reports the velocity as function of the position. The χ 2 analysis for observed theoretical velocity as function of time/space, see Table 2, assigns the smaller value to the turbulent jet. Back reaction: The insertion of the back reaction in the equation of motion allows us to introduce a finite rather than infinite jet’s length, see Equation (39). The extended region: The extended region of HH34 is modeled by combining the decreasing jet’s velocity with the constant velocity and precession of the central object, see the final matrix (45). The theory of the image: We have analysed the case of an optically thin layer approximation to provide an explanation for the so called “bow shock” that is visible in HH34. This effect can be reproduced when two emitting regions apparently intersect on the plane of the sky, see the numerical simulation as given by Figure 12. This curious effect of enhancement in the intensity of emission can easily be reproduced when the image theory is applied to the mathematical knots, see the example of the trefoil in Figure 15.

Acknowledgements

Credit for Figure 13 is given to ESO.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References [1] Herbig, G.H. (1950) The Spectrum of the Nebulosity Surrounding T Tauri. The As- trophysical Journal, 111, 11-14. https://doi.org/10.1086/145232 [2] Haro, G. (1952) Herbig’s Nebulous Objects near NGC 1999. The Astrophysical Journal, 115, 572-573. https://doi.org/10.1086/145576 [3] Raga, A.C., Reipurth, B. and Noriega-Crespo, A. (2019) The HH34 Jet/Counterjet System at 1.5 and 4.5 μm. Revista Mexicana de Astronomia y Astrofisica, 55, 117-123. https://doi.org/10.22201/ia.01851101p.2019.55.01.12 [4] Raga, A.C., Noriega-Crespo, A., Rodrguez-González, A., et al. (2012) The Kinemat- ics of HH 34 from HST Images with a Nine-Year Time Baseline. The Astrophysical Journal, 748, 103. https://doi.org/10.1088/0004-637X/748/2/103

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[5] Raga, A.C., Noriega-Crespo, A., Carey, S.J. and Arce, H.G. (2013) Proper Motions of Young Stellar Outflows in the Mid-Infrared with Spitzer (IRAC). I. The NGC 1333 Region. Astronomical Journal, 145, 28. https://doi.org/10.1088/0004-6256/145/2/28 [6] Noriega-Crespo, A., Raga, A.C., Moro-Martn, A., Flagey, N. and Carey, S.J. (2014) Proper Motions of Young Stellar Outflows in the Mid-Infrared with Spitzer II HH 377/Cep E. New Journal of Physics, 16, Article ID: 105008. https://doi.org/10.1088/1367-2630/16/10/105008 [7] Guzmán, A.E., Garay, G., Rodrguez, L.F., Contreras, Y., Dougados, C. and Cabrit, S. (2016) A Protostellar Jet Emanating from a Hypercompact H II Region. The Astro- physical Journal, 826, Article No. 208. https://doi.org/10.3847/0004-637X/826/2/208 [8] Raga, A.C., Reipurth, B., Esquivel, A., Castellanos-Ramrez, A., Velázquez, P.F., Hernández-Martnez, L., Rodrguez-González, A., Rechy-Garca, J.S., Estrella-Trujillo, D. and Bally, J. (2017) Proper Motions of the HH 1 Jet. Revista Mexicana de Astro- nomia y Astrofisica, 53, 485-495. [9] Cabrit, S. and Raga, A. (2000) Theoretical Interpretation of the Apparent Decelera- tion in the HH 34 Super Jet. Astronomy & Astrophysics, 354, 667-673. [10] López-Martn, L., Raga, A.C., López, J.A. and Meaburn, J. (2001) Theory and Ob- servations of a Jet in the σ Orionis Region: HH 444. Revista Mexicana de Astrono- mia y Astrofisica Conference Series, 10, 61-64. [11] Viti, S., Girart, J.M., Garrod, R., Williams, D.A. and Estalella, R. (2003) The Mole- cular Condensations Ahead of Herbig-Haro Objects. II. A Theoretical Investigation of the HH 2 Condensation. Astronomy & Astrophysics, 399, 187-195. https://doi.org/10.1051/0004-6361:20021745 [12] Raga, A.C., Reipurth, B., Cantó, J., Sierra-Flores, M.M. and Guzmán, M.V. (2011) An Overview of the Observational and Theoretical Studies of HH 1 and 2. Revista Mexicana de Astronomia y Astrofisica, 47, 425-437. [13] Kwon, J., Choi, M., Pak, S., Kandori, R., Tamura, M., Nagata, T. and Sato, S. (2010) Magnetic Field Structure of the HH 1-2 Region: Near-Infrared Polarimetry of Point-Like Sources. The Astrophysical Journal, 708, 758-769. https://doi.org/10.1088/0004-637X/708/1/758 [14] Lee, C.F., Rao, R., Ching, T.C., Lai, S.P., Hirano, N., Ho, P.T.P. and Hwang, H.C. (2014) Magnetic Field Structure in the Flattened Envelope and Jet in the Young Protostellar System HH 211. The Astrophysical Journal, 797, L9. https://doi.org/10.1088/2041-8205/797/1/L9 [15] Lee, C.F., Hwang, H.C. and Li, Z.Y. (2016) Angular Momentum Loss in the Envelope-Disk Transition Region of the HH 111 Protostellar System: Evidence for Magnetic Braking? The Astrophysical Journal, 826, Article No. 213. https://doi.org/10.3847/0004-637X/826/2/213 [16] De Young, D.S. (2002) The Physics of Extragalactic Radio Sources. University of Chicago Press, Chicago. [17] Zaninetti, L. (2016) Classical and Relativistic Flux of Energy Conservation in As- trophysical Jets. Journal of High Energy Physics, Gravitation and Cosmology, 2, 41-56. https://doi.org/10.4236/jhepgc.2016.21005 [18] Zaninetti, L. (2018) Classical and Relativistic Evolution of an Extra-Galactic Jet with Back-Reaction. Galaxies, 27, 134. https://doi.org/10.3390/galaxies6040134 [19] Bally, J. and Devine, D. (1994) A Parsec-Scale “Superjet” and Quasi-Periodic Struc- ture in the HH 34 Outflow? The Astrophysical Journal, 428, L65. https://doi.org/10.1086/187394

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[20] Zaninetti, L. (2010) The Physics of Turbulent and Dynamically Unstable Her- big-Haro Jets. Astrophysics and Space Science, 326, 249-262. https://doi.org/10.1007/s10509-009-0255-8 [21] Goldstein, H., Poole, C. and Safko, J. (2002) Classical Mechanics. Addison-Wesley, San Francisco. [22] Rybicki, G. and Lightman, A. (1991) Radiative Processes in Astrophysics. Wi- ley-Interscience, New-York. [23] Hjellming, R.M. (1988) Radio Stars IN Galactic and Extragalactic Radio Astronomy. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-3936-9_9 [24] Zaninetti, L. (2009) Scaling for the Intensity of Radiation in Spherical and Aspheri- cal Planetary Nebulae. Monthly Notices of the Royal Astronomical Society, 395, 667-691. https://doi.org/10.1111/j.1365-2966.2009.14551.x

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International Journal of Astronomy and Astrophysics, 2019, 9, 321-334 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Optical Spectroscopic Monitoring Observations of a T Tauri Star V409 Tau

Hinako Akimoto, Yoichi Itoh

Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo, Hyogo, Japan

How to cite this paper: Akimoto, H. and Abstract Itoh, Y. (2019) Optical Spectroscopic Mon- itoring Observations of a T Tauri Star V409 We report the results of optical spectroscopic monitoring observations of a T Tau. International Journal of Astronomy Tauri star, V409 Tau. A previous photometric study indicated that this star and Astrophysics, 9, 321-334. experienced dimming events due to the obscuration of light from the central https://doi.org/10.4236/ijaa.2019.93023 star with a distorted circumstellar disk. We conducted medium-resolution (R Received: July 24, 2019 ~10,000) spectroscopic observations with 2-m Nayuta telescope at Ni- Accepted: September 16, 2019 shi-Harima Astronomical Observatory. Spectra were obtained in 18 nights Published: September 19, 2019 between November 2015 and March 2016. Several absorption lines such as Ca Copyright © 2019 by author(s) and I and Li, and the Hα emission line were confirmed in the spectra. The Ic-band Scientific Research Publishing Inc. magnitudes of V409 Tau changed by approximately 1 magnitude during the This work is licensed under the Creative observation epoch. The equivalent widths of the five absorption lines are Commons Attribution International License (CC BY 4.0). roughly constant despite changes in the Ic-band magnitudes. We conclude http://creativecommons.org/licenses/by/4.0/ that the light variation of the star is caused by the obscuration of light from Open Access the central star with a distorted circumstellar disk, based on the relationship

between the equivalent widths of the absorption lines and the Ic-band magnitudes. The blue component of the Hα emission line was dominant during the observation epoch, and an inverse P Cygni profile was observed in eight of the spectra. The time-variable inverse P Cygni profile of the Hα emission line indicates unsteady mass accretion from the circumstellar disk to the central star.

Keywords Star Formation, Pre-Main Sequence Stars, T Tauri Stars

1. Introduction

Young stellar objects (YSOs) are generally variable stars. Some objects show periodic variability, while others exhibit irregular variability. Some objects also exhibit episodic variations. For example, FU Orionis outbursts represent the most

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extreme case of variability in these objects [1]. Most members of this class have undergone outbursts of 4 - 6 magnitude in optical brightness [2] [3]. In [4], the authors proposed the movement of cool spots on the stellar surface due to stellar rotation, unsteady accretion of circumstellar material onto the star, and obscuration of light from the photosphere by a distorted circumstellar disk as causes of the light variation of YSOs. Thus far, numerous photometric studies on the variability of YSOs have been conducted. In [5], the photometric monitoring of 24 YSOs was reported, and the authors observed one or two (quasi-) sinusoidal curve(s) with a period of 1.2 to 12 days in the optical light curves of 20 objects. They suggested spot movement on the stellar surface due to stellar rotation as the cause of the light variation. Also, numerous studies have also been reported on the variability of YSOs observed by spectroscopy. In [6], spectroscopic monitoring observations of a weak-lined T Tauri star, V410 Tau, were reported. The authors obtained spectra with a wavelength coverage of 3850 Å - 9050 Å and a resolution of ~12,000 and identified hot spots on the photosphere by producing Doppler images of the object. Spectroscopic observations are useful for further understanding of the brightness variations of YSOs. [7] conducted the photometric observation of HD 288313. This object showed light variations, which was considered to be attributed to change of coverage of cool spots. Distribution of the chromospheric active regions was examined by spectroscopic observations. The Hα equivalent width shows rotational modulation only at occasional epochs. It was proposed that the chromospheric active regions spread across the stellar surface. AA Tau is another example of YSOs exhibiting light variation. In [8], the authors reported high-resolution spectroscopic observations of AA Tau, which exhibits light variation due to obscuration of the photosphere by the inner edge of a magnetically-warped disk. When the star is faint, a red-shifted absorption line appears on the broad emission line of the Balmer series. The redshifted absorption line indicates infalling motion of cold material in front of the photosphere, suggesting an accretion flow of circumstellar material onto the photosphere. A classical T Tauri star, RW Aur A, is an irregular variable with a large amplitude. [9] searched for periodicities in the variations of the brightness and colors of RW Aur A over three decades. With the spectroscopic observations, they insisted the accretion of the magnetosphere. Study of the circumstellar environments of YSOs helps in the discussion of the formation of planets [10]. In particular, objects that display large photometric dimming events caused by circumstellar disks give us the opportunity to study the evolution of the circumstellar environments of young stars. Photometric observations of V409 Tau were carried out with the Kilodegree Extremely Little Telescope North (KELT-North) [10]. V409 Tau is a Class II object [11] with the spectral type of K8--M0 [11] or M1.5 [12]. They observed two separate dimming events; one from January 2009 to October 2010 and another from March 2012 until at least September 2013. In the latter event, the depth of

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the dimming was 1.4 magnitude in the V-band. They estimated the upper limit of the duration of the 2009 dimming event to 630 days. The interval between the beginnings of the two dimming events was 1130 days. By combining data from the Combined Array for Research in Millimeter Astronomy, the All Sky Auto- mated Survey, the Catalina Real-time Transient Survey, and the Wide Angle Search for Planets, the authors constructed the spectral energy distribution, which was fitted with the star and disk SED model with a disk inclination angle of 81 degrees. They indicated that an almost edge-on circumstellar disk was distorted and that the dimming was caused by the obscuration of light from the central star by the distorted disk. V409 is a UX Orionis candidate [10]. UX Ori type variability is observed in Herbig Ae/Be stars and some T Tauri stars with early K spectral types [1]. The variability is characterized by large-amplitude light variations with no evident veiling or effective temperature variations. This is interpreted to be caused by variable obscuration by circumstellar dust. [1] constructed an accretion disk model in which a large amount of accretion material heats gas in the circumstellar disk, making the disk flaring. They calculated the probability of observations of the star through the circumstellar disk. The result shows that high mass-accretion rate stars are more likely to be observed through the circumstellar disk than low mass-accretion rate stars. They suggested that accretion is driving UX Ori variability. In this paper, we present the spectroscopic monitoring observations of V409 Tau, which exhibits irregular variability. A previous photometric study [10] suggested obscuration of the photosphere by a distorted circumstellar disk as the cause of the variability.

2. Observations and Data Analysis

We conducted a series of spectroscopic observations of V409 Tau over 18 nights between November 2015 and March 2016. Observations were carried out with the medium- and low-dispersion long-slit spectrograph (MALLS) mounted on the Nasmyth platform of the Nayuta telescope at Nishi-Harima Astronomical Observatory, Japan. With a grating of 1800 lines/mm and a 0.8'' slit, we obtained spectra with a wavelength resolution of ~10,000 between 6280 Å and 6720 Å. This wavelength range was chosen for investigating the Hα emission line which is an index of accretion, and metallic absorption lines originating from the photosphere. The exposure time ranged from 900 - 1200 s, and 1 - 17 spectra were taken each night. The goal was to reach S/N of 20 on good observation conditions. Flat frames and comparison frames were acquired with a halogen lamp and an Fe-Ne-Ar lamp in the instrument, respectively. Dark frames were also taken. Details of the observations are shown in Table 1. The image analysis software IRAF (Imaging Reduction and Analysis Facility) was used for image processing. First, the overscan and dark current were sub- tracted from the raw data. Next, we performed flat-fielding with the normalized

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Table 1. Observation date, exposure time, and S/N.

Date (JST) Exp. Time (s) S/N Date Exp. Time (s) S/N

2015 Nov 29 3600 23 2016 Jan 7 6000 11

2015 Nov 30 3600 19 2016 Jan 10 8400 16

2015 Dec 1 3600 25 2016 Jan 12 4800 2

2015 Dec 4 3600 20 2016 Jan 13 4800 4

2015 Dec 7 3600 29 2016 Jan 16 7200 5

2015 Dec 19 3600 8 2016 Mar 7 1200 8

2015 Dec 27 6000 5 2016 Mar 11 1200 2

2015 Dec 29 3600 10 2016 Mar 12 4500 11

2016 Jan 1 20,400 39 2016 Mar 26 1800 20

flat frames, wavelength calibration and distortion correction with comparison frames, and background subtraction. The spectrum was extracted and binned based on the slit width. We combined the spectra acquired for each night, and finally, we normalized the continuum level of the spectrum. We shifted the wavelengths of the spectra so that the wavelengths of the five absorption lines described below matched to the wavelengths of the lines in vacuum.

3. Results

We obtained spectra for 18 nights. The spectra exhibited the Hα emission line and several absorption lines. The Hα line displayed an inverse P Cygni profile. Among the spectra, those acquired for eight of the nights exhibited a signal-to-noise (S/N) ratio greater than 15, enabling clear confirmation of the absorption lines (Figure 1). The equivalent widths of five absorption lines (Ca I at 6439 Å, Ca I and Co I at 6450 Å, Ca I at 6463 Å, Li and Fe I at 6708 Å, and Ca I at 6718 Å) were measured using the SPLOT task in IRAF. The lines were fitted with a Gaussian function, and the error in the equivalent width was estimated from the root-mean-square of the continuum region adjacent to the line in the spectrum.

Table 2 shows the equivalent widths of the absorption lines and the Ic-band

magnitudes of the object. The Ic-band magnitudes were taken from the database of the Kamogata/Kiso/Kyoto wide-field survey (KWS) [13]. The light curve of

V409 Tau is shown in Figure 2. If the Ic-band magnitude was not available for a particular observation date, we used photometric data taken within one day of that date. We do not further discuss the spectrum taken in March 2016 because no photometric data are available within 30 days from the observation date. During the observation period, V409 Tau changed by 1 magnitude in the

Ic-band; however, no significant changes were observed in the equivalent widths of the absorption lines.

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Figure 1. Optical spectra of V409 Tau. The horizontal axis represents the wavelength and the vertical axis presents the normalized flux + constant. Spectra with S/N ratios greater than 15 taken in eight nights are shown. Prominent absorption lines are Ca I at 6439 Å, Ca I and Co I at 6450 Å, Ca I at 6463 Å, Li and Fe I at 6708 Å, and Ca I at 6718 Å. The Hα emission line shows an inverse P Cygni profile that changes with time. A spectrum of an M0 dwarf (HD156274) is also shown [33].

Figure 2. KWS light curve of V409 Tau. The filled circles indicate the Ic-band magnitudes. The open circles indicate the V-band magnitudes.

Table 2. Observation date, Ic-band magnitude, and equivalent widths of the absorption lines.

Date Ic ()mag EW6439 ()Å EW6450 ()Å EW6463 ()Å EW6708 ()Å EW6718 ()Å

+0.07 +0.13 +0.19 +0.10 +0.07 2015 Nov 29 11.45± 0.48 0.38−0.06 0.39−0.09 0.59−0.19 0.53−0.06 0.29−0.12

+0.11 +0.11 +0.16 +0.14 +0.79 2015 Nov 30 11.19± 0.15 0.42−0.09 0.37−0.08 0.49−0.13 0.56−0.11 0.26−0.10

+0.05 +0.08 +0.08 +0.07 +0.20 2015 Dec 1 11.19± 0.15 0.43−0.05 0.41−0.07 0.53−0.08 0.59−0.06 0.68−0.22

+0.19 +0.33 +0.12 +0.09 +0.71 2015 Dec 4 10.97± 0.12 0.51−0.11 0.57−0.21 0.52−0.15 0.59−0.08 0.27−0.02

+0.10 +0.10 +0.40 +0.09 +0.50 2015 Dec 7 11.14± 0.13 0.41−0.08 0.41−0.09 0.48−0.11 0.63−0.08 0.36−0.04

+0.04 +0.01 +0.07 +0.04 +0.03 2016 Jan 1 11.59± 0.24 0.38−0.04 0.45−0.10 0.45−0.07 0.43−0.04 0.18−0.08

+0.26 +0.11 +0.20 +0.27 +0.04 2016 Jan 10 12.01± 0.36 0.16−0.05 0.36−0.09 0.54−0.23 0.50−0.27 0.35−0.27

+0.11 +0.25 +0.36 +0.13 +0.46 2016 Mar 26 - 0.45−0.09 0.41−0.13 0.43−0.10 0.59−0.11 0.27−0.13

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4. Discussion 4.1. Absorption Lines

V409 Tau showed a 1-magnitude variation in the Ic-band during the observation epoch. To specify the cause of the light variation, we constructed three simple models, in which the variation of the equivalent widths of the absorption lines was investigated. Since the SED of V409 Tau indicates little contribution of the circumstellar disk in the optical wavelength range [10], we assume that the absorption lines in the optical region originate from the photosphere. First, we investigated the possibility that the luminosity changes resulted from the coverage of cool spots on the stellar surface. The spots have a temperature approximately 1000 K lower than the temperature of the photosphere [5]. As- suming a constant stellar radius and an effective temperature of 3800 K [14], the

radiation of the central star, Iλ , is the sum of the blackbody radiation, BTλ () , of the normal photosphere and that of the cool spots, as follows;

IBλλ=()()()3800 ⋅− 1ββ +Bλ 2800 ⋅ , (1)

where β ( 01≤≤β ) represents the coverage of the cool spots. The calculated

radiation was multiplied by the transmittance of the Ic-band filter [15]. We set

β = 0 for the brightest epoch (11 magnitude in the Ic-band) and calculated the

Ic-band magnitude by changing β . For the faintest epoch (12 magnitude in the

Ic-band), we set β = 0.5 . The equivalent width of an absorption line is the sum of the equivalent widths of the absorption lines for the 3800-K and 2800-K spectra. EW= EW ()()()3800 ⋅− 1ββ +EW 2800 ⋅ (2)

The model spectra were calculated using the BT-NextGen model [16]. We assumed the surface gravity to be log g = 3.5 and the metallicity to be the same as the Sun. The spectra for temperatures of 3800-K and 2800-K were calculated and combined with the coverage factor, β , and we smoothed the model spectrum to the spectral resolution of the observed spectra. Table 3 shows the calculated equivalent widths of the absorption lines. The equivalent widths decrease or are constant with increasing the spot coverage.

Table 3. Equivalent widths of the absorption lines expected for luminosity changes caused by cool spots on the stellar surface.

β 0.0 0.1 0.2 0.3 0.4 0.5

Ic ()mag 11.00 11.35 11.45 11.56 11.83 11.99

EW6439 ()Å 1.04 0.99 0.94 0.89 0.84 0.79

EW6450 ()Å 0.25 0.23 0.21 0.19 0.17 0.15

EW6463 ()Å 0.88 0.83 0.79 0.74 0.69 0.65

EW6708 ()Å 0.79 0.79 0.79 0.79 0.79 0.78

EW6718 ()Å 0.30 0.30 0.29 0.29 0.29 0.29

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We next considered unsteady material accretion from the circumstellar disk onto the photosphere. When the material accretion rate increases, the boundary layer connects to the inner part of the disk [17] and emits continuum light in the ultraviolet and optical wavelengths [18]. We assumed constant intensities of the photosperic absorption lines. When the intensity of continuum increases by α , the equivalent width EW ′ is S() I+=α EW()1 + V , where I is the intensity of continuum, S is the intensity of absorption line, and VI= α . We set V = 0 and used the 3800 K spectrum in the BT-NextGen model when the star is 12 magnitude. The EWs of the absorption lines were calculated until V increased

and the Ic-band magnitude reached 11 magnitude. We also considered the obscuration of light from the photosphere by the distorted disk. AA Tau-like variables show periodic dips in their light curves, which are thought to be caused by the obscuration of light from the photosphere caused by warps in the inner disk. The periods of the dips are typically five to ten days, corresponding to a distance of 0.1 AU between the central star and the inner disk for Keplerian rotation [19]. In the case of AA Tau, the luminosity drops by 1.2 magnitude with a period of 8.2 days [8] [20]. It is proposed that the inner disk is distorted by the magnetic field and thus obscures the photosphere. As light from the central star passes through the disk, the intensity decreases uniformly within a certain wavelength range. As the absorption line weakens, the adjacent continuum light also weakens. As a result, the equivalent width of the absorption line does not change when the photosphere is obscured. Figure 3 shows the relationship between the equivalent widths of the absorption lines and the broadband magnitudes. The observed equivalent widths are roughly constant at all magnitude. In the case of the cool spots, the equivalent widths decrease slightly or are constant with decreasing luminosity. In the case of accretion, the equivalent widths increase with decreasing luminosity. In the case of the disk, the equivalent widths are constant. Figure 3 indicates that the cause of the 2015 dimming event is the case where the coverage of cool spots on the stellar surface has changed or the case where the light from the photosphere is obscured by a distorted disk. V409 Tau has a periodic photometric variation of 4.754 days, corresponding to the rotation period of the photosphere with star spots [21]. This period is significantly shorter than the duration of the 2015 dimming event. The cool spot model indicates that the large portion of the photosphere is covered by the spot when the star was faint. The temperature of the spot is as low as 2800 K, so that the spectrum would show absorption features at 6650 Å and 6680 Å. However, no clear absorption features appeared in the observed spectra. Therefore we claim the occultation by the disk as the cause of the light variation.

Figure 4 is a ()V− IIcc, color-magnitude diagram. The Ic-band data are taken from the KWS database. We acquired V-band photometric data from ASAS-SN [22] taken on the same day as the KWS data. The arrow in the figure indicates interstellar extinction. Photometric variation can be interpreted as the

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Figure 3. Relationship between the equivalent widths of the absorption lines and the

Ic-band magnitudes. The equivalent widths of the five absorption lines at 6439 Å, 6450 Å, 6463 Å, 6708 Å, and 6718 Å were measured (filled circles). Dashed-dotted line: equivalent widths expected if the luminosity changes result from variations in the coverage of cool spots on the stellar surface. Solid line: equivalent widths expected if the luminosity changes result from unsteady material accretion from the circumstellar disk onto the photosphere. Dotted line: equivalent widths expected if the luminosity changes result from obscuration by the distorted disk. The equivalent widths of the obscuration model are shifted to match the equivalent width of the cool spot model at 11 magnitude. The measured equivalent widths are roughly constant for all absorption lines regardless of the

Ic-band magnitude.

Figure 4. ()V− IIcc, color-magnitude diagram. We used Ic-band photometric data from KWS and V-band photometric data from ASAS-SN. The arrow indicates interstellar extinction. Photometric variation can be interpreted as the intersteller extinction.

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intersteller extinction. This further supports that the dimming event occurred due to the obscuration by the disk. [10] presented the light variation of V409 Tau. They reported that the star experienced two dimming events in 2009-2010 and 2012-2013. The authors discussed the number of obscuring objects and their distances from the central star. In the discussion, they presented the possibility that obscuring objects of 2009 and 2012 are different. This is because V409 Tau did not experience significant dimming in 2002 and 2006. Other occultation happened in the 1960s [23]. They suggested that the obscuring object is located at farther 10.7 AU from the central star if the dimming events are attributed to different obscuring objects. They argued that if it is a single feature, it implies that the duration and depth is changing. In this case, the distance between the central star and the obscuring object was determined as 1.7 AU. In this paper, the duration of the dip cannot be precisely estimated due to the

sparsity of the photometric data. According to the KWS data, the Ic-band magnitude of the star increased to 11.2 mag on September 22, 2014 and then varied within the range of 0.5 mag. The magnitude dropped to 12.9 mag on December 28, 2015 and then returned to 11.2 mag on October 26, 2017. We assume a single obscuration object, with dimming events ending on September 22, 2014 and

October 26, 2017. If we adopt 0.57M  and 1.11R [10] for the mass and radius of the central star, respectively, the distance between the center star and the obscuration object is 1.76 AU. In our observations, the maximum duration of the dimming event was ~668 days (December 28, 2015 to October 26, 2017), similar to that of the 2009-2010 dimming event. However, the beginning of the 2012-2013 dimming event is not consistent with the orbital period of the obscuration object. For a precise determination of the shape and number of obscuring objects and the distance from the central star, more frequent photometric observations are required.

4.2. Hα Emission Line

The Hα emission line of V409 Tau shows an asymmetric profile with time variation. The Hα line had an S/N > 5 in the spectra acquired for 13 nights. Among these spectra, nine displayed an inverse P Cygni profile (Table 4). In [24], the authors reported on low- and high-resolution spectroscopy of a T Tauri star, T Cha. The Hα emission line profile changed from pure emission to an inverse P Cygni profile in less than one day. The authors claimed that a red-shifted absorption component appeared in the optically thin Hα emission when an episodic mass accretion was caused. [25] claimed that the redshifted absorption arises from an accretion flow from the inner disk to the stellar surface along the magnetospheric field lines. As suggested by the relatively high projected rotational velocity of the star [26], T Cha is seen in an almost edge-on geometry, as is the case for V409 Tau [10]. We propose that the redshifted absorption for V409 Tau arises from the accretion flow and is overlaid on the broad emission line.

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Table 4. Observation date and equivalent widths of the blueward and redward components in the Hα emission line.

date (JST) EWblue ()Å EWred ()Å inverse P Cygni

+0.23 +0.10 2015 Nov 29 0.63−0.16 0.01−0.01 yes

+0.33 +0.28 2015 Nov 30 1.22−0.25 0.32−0.17 yes

+0.22 +0.09 2015 Dec 1 0.99−0.16 0.06−0.04 yes

+0.38 +0.20 2015 Dec 4 0.90−0.25 0.18−0.13 yes

+0.33 +0.38 2015 Dec 7 1.03−0.25 0.43−0.25 yes

+1.18 +1.02 2015 Dec 19 2.59−0.85 0.88−0.54 no

+0.97 +0.78 2015 Dec 29 1.58−0.65 0.59−0.39 no

+0.23 +0.09 2016 Jan 1 1.28−0.17 0.03−0.02 yes

+0.66 +0.54 2016 Jan 7 3.64−0.52 0.42−0.30 no

+0.55 +0.43 2016 Jan 10 1.07−0.40 0.48−0.27 yes

+0.71 +0.65 2016 Mar 7 1.15−0.54 0.60−0.43 yes

+0.54 +0.53 2016 Mar 12 1.42−0.40 0.58−0.36 no

+0.31 +0.22 2016 Mar 26 0.78−0.24 0.12−0.10 yes

In [27], the shape of the Hα emission line of 63 YSOs was analyzed. The authors measured the equivalent widths of the blueward and redward components of the line. For the relatively weak emission line stars [10 Å < EW(Hα) < 70 Å], the equivalent widths of the blueward and redward components are comparable. In contrast, among the strong emission line stars [EW(Hα) ≥ 70 Å], the redward components are stronger than the blueward components for many objects. We measured the equivalent widths of the redward and blueward components of the Hα emission line of V409 Tau between −500 km/s and +500 km/s (Figure 5). We set 6563 Å to 0 km/s. We defined redward component as the velocity range from 0 km/s to +500 km/s and blueward component from 0 km/s to −500 km/s. We did not measure the equivalent width of the absorption line. The total equivalent width of the redward and blueward components of V409 Tau varies from 0.64 Å to 4.08 Å. For the Hα emission line of V409 Tau, the blueward component is dominant. The equivalent widths of both the blueward and redward components vary with time. The time-variable inverse P Cygni profile of the Hα emission line of V409 Tau indicates unsteady mass accretion from the circumstellar disk to the central star. Based on the correlation between the mass of T Tauri stars and the mass accretion rate [28], the mass accretion rate of V409 −8 Tau is estimated to be 10M  yr . Assuming that the inner radius of the accre-

tion disk is five times the stellar radius [29], RRs = 1.11  and MMs = 0.57 

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[10] [30], we determined that the accretion luminosity is 0.127L . Thus, the luminosity of the object increases by 0.4 magnitude when the accretion phenomenon occurs. Because continuum excesses caused by accretion shocks primari- ly contribute to the blue and ultraviolet flux [31], we expect that the light varia-

tion due to unsteady mass accretion is less than 0.4 magnitude in the Ic-band.

Figure 6 shows the relationship between the Ic-band magnitudes and the equivalent widths of Hα emission line. Except for one data, one may find a ten- dency that the equivalent widths increase with decreasing the broad-band brightness. This implies light variation due to veiling. However, most of the Hα lines show the inverse P Cygni profile. Since we are measuring only the emission line part, it is necessary to construct a detailed model that takes self-absorption into account (e.g. [32]).

Figure 5. Equivalent widths of the redward and blueward components of the Hα emission line of V409 Tau. The open circle indicates the inverse P Cygni profile. The blueward components are dominant for all spectra, and the inverse P Cygni profile appeared when the Hα emission was weak.

Figure 6. Equivalent widths of the Hα emission line of V409 Tau and Ic-band magnitudes. Most of the Hα lines show the inverse P Cygni profile (open circles).

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5. Summary

We conducted optical spectroscopic monitoring of a T Tauri star V409 Tau. Medium-resolution spectra were obtained over 18 nights in 2015 and 2016.

While the Ic-band magnitude of the object changed by 1 magnitude during the observations, the equivalent widths of the absorption lines remained nearly constant. We constructed three simple models to investigate the variation of the equivalent widths of the absorption lines. We concluded that the light variation arises from the distorted disk. If only one obscuring feature is present, the distance between the central star and the obscuring feature is estimated as 1.76 AU. The Hα emission line profile of V409 Tau showed a time-variable inverse P Cygni profile, indicating unsteady mass accretion from the circumstellar disk to the central star.

Acknowledgements

H. A. is supported by the Iue Memorial Foundation.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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International Journal of Astronomy and Astrophysics, 2019, 9, 335-353 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717

Parameter Inversions of Multi-Layer Media of Mars Polar Region with Validation of SHARAD Data

Chuan Liu, Ya-Qiu Jin*

Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai, China

How to cite this paper: Liu, C. and Jin, Abstract Y.Q. (2019) Parameter Inversions of Mul- ti-Layer Media of Mars Polar Region with HF (high frequency) radar sounder technology has been developed for several Validation of SHARAD Data. International missions of Mars surface/subsurface exploration. This paper presents a model Journal of Astronomy and Astrophysics, 9, 335-353. of rough surface and stratified sub-surfaces to describe the multi-layer struc- https://doi.org/10.4236/ijaa.2019.93024 ture of Mars polar deposits. Based on numerical simulation of radar echoes from rough surface/stratified interfaces, an inversion approach is developed Received: July 9, 2019 to obtain the parameters of Polar Layered Deposits, i.e. layers thickness and Accepted: September 22, 2019 Published: September 25, 2019 dielectric constants. As a validation example, the SHARAD radar sounder data of the Promethei Lingula of Mars South Polar region is adopted for pa- Copyright © 2019 by author(s) and rameters inversion. The result of stratification is also analyzed and compared Scientific Research Publishing Inc. This work is licensed under the Creative with the optical photo of the deep cliff of Chasma Australe canyon. Dielectric Commons Attribution International inversions show that the deposit media are not uniform, and the dielectric License (CC BY 4.0). constants of the Promethei Lingula surfaces are large, and become reduced http://creativecommons.org/licenses/by/4.0/ around the depth of 20 m - 30 m, below where most of the deposits are nearly Open Access pure ice, except a few thin layers with a lot of dust.

Keywords Radar Sounder, Inversion of Multi-Layer Parameters, Stratified Media, Mars Polar Region, SHARAD

1. Introduction

The physical properties of Mars polar deposits have been studied for several decades. Some studies show that North Polar Layered Deposit (NPLD) and South Polar Layered Deposit (SPLD) might be rich in water ice [1] [2] [3]. The stratified NPLD and SPLD media were formed due to varying amounts of dust impurity mixed with the water ice [4] [5] [6]. The varying impurity ratio is likely

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C. Liu, Y. Q. Jin

related to historical climate change [5]. Techniques to study the dielectric properties of the regolith media in NPLD and SPLD are important for the study of Mars climate. One such technique is the inversion of dielectric constants using HF radar to penetrate through the regolith. HF radar waves can penetrate through the Mars regolith media several kilo- meters. The MARSIS (Mars Advance Radar for Subsurface and Ionospheric Sounding) onboard the Mars Express operates in 4 bands centered at 1.8, 3, 4, 5 MHz with a bandwidth of 1 MHz, and can penetrate through the media as deep as 4 km [7]. The HF radar sounder of the SHAllow RADar (SHARAD) onboard Mars Reconnaissance Orbiter (MRO) operates with a 20 MHz central frequency with a bandwidth of 10 MHz. Its vertical resolution is higher than MARSIS, and its penetration depth is usually less than about 1 - 2 km [8] [9] (As water ice with a small loss tangent, the SHARAD signals may reach depths more than 2 km [1] [2]). China is planning to explore Mars multi-layer structure using HF and VHF radar as well [10] [11]. To study the radar sounder data, Mouginot et al. [12] adopted the MARSIS data (1 - 5 MHz) to retrieve global surface reflectivity with kilometer-scale surface roughness to estimate the Mars surface dielectric constant. Nouvel et al. [13], Lauro et al. [14], Mouginot et al. [15] particularly studied the surface dielectric constants of NPLD and SPLD. Grima et al. [1], Zhang et al. [16] [17], Alberti et al. [18] evaluated the average dielectric constant within 2 km depth of Mars polar deposits, using the radar echo time delays through the surface/subsurface of the regolith media. However, the study of the Mars cratered rough surface with multi-stratified interfaces, the development of the physical parameters inversion, and the validations using HF radar data are remained to be further studied [10]. Based on numerical simulations of the radar sounder echoes from the one-layer model with rough surface/subsurface media, Ye and Jin [11] found that under the Kirchhoff approximation with a mean zero slope, the received echo at nadir direction preserves the functional dependence of the surface reflectivity. It leads to the inversion of the surface dielectric permittivity derived from the ratios of the received echo powers and the medium reflectivity. Furthermore, Liu and Jin [10] proposed a numerical approach of radar echoes from rough surface/multi-subsurface and inversions. Based on [10] [11], this paper presents a model of parallel-stratified media to describe the multi-layer structure of Mars polar region. A relationship between the received radar sounder echoes and reflectivities of rough surface/multi-subsurface is presented. The inversions of the thickness and dielectric permittivity of each layer where the radar wave can reach are designed. As data validation, inversions are applied to SHARAD data on Promethei Lingula of Mars SPLD.

2. Model of the Stratified Media and Inversion of Dielectric Constants

1) A Model of Parallel Stratified Media

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The SHARAD radar sounder data has shown that there are multi-layer structures in Mars Polar regions [6]. When the sub-layer thickness varies little within one radar footprint, the multi-layer structure from the top to hundreds meters below can be modeled as a parallel-stratified media, as shown in Figure 1. As the electromagnetic (EM) wave of radar sounder is vertically incident upon the top surface through ionosphere, there would be multi-reflection and transmission through the media. Based on the difference of time delay of each echo from media interfaces, and separating the rough surface clutter and the echoes of the interfaces, the echoes from each interface can be identified. Suppose that multiple reflection and transmission between interfaces are neglected. It means that the echo from the n-th interface experienced one-reflection, round trip of 2(n − 1) transmissions (i.e. including round trip attenuation) through previous (n − 1) layer media. This assumption is based on small difference on final surface reflectivity caused by underlying multi-layer structures. Thus, as the incident radar power through ionosphere is directly on the top sur- ()a − A face PP00= e , the echo from the n-th interface is written as n−1 = 2 − ′′ Pn Pr0 n∏  t mexp() 4 kmm d (1) m=1

()a where P0 is denoted as the transmitted power of dipole antenna (with nota- ()a − A tion (a)), and similarly, PPnn= e denotes the n-th reflected peak power re-

ceived by the radar antenna, i.e. observation. In Equation (1), tm is the trans-

mittivity between the (m − 1)-th and the m-th media, rn is the reflectivity of

the n-th layer, dm is the thickness of the m-th layer, km′′ is the imaginary part

of the wave number of the m-th layer, exp()− 4kdmm′′ is the round-trip attenuation in the m-th layer [19], e− A is the one-way attenuation through ionosphere layer. The wave number of the m-th layer is written as

1 ε m′′ kmmmmm= k00εεεε = k′ +≈ i ′′ k 0 ′ + ik 0 (2) 2 ε m′

Figure 1. A model of rough surface and stratified media.

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Here k0 is the wave number of free space. ε m′ and ε m′′ are the real and im-

aginary parts of the m-th layer dielectric constant, respectively. km′′ is written as

1 ε m′′ kkm′′ = 0 (3) 2 ε m′

The transmittivity satisfies

trmm=1 − (4) and the layer thickness can be expressed as

c τ m dm = (5) ε m 2

where c is the light speed in free space, and τ m is the time delay as EM wave propagates through the m-th medium twice, i.e. the time delay between two echoes from two successive interfaces. Substituting Equations (3)-(5) into Equation (1), the reflected power from the n-th interface is derived as n−1 2 ε ′′ P= Pr()1 −− r exp k cτ m nn00∏ m m′ m=1 ε m (6) n−1 2 =Pr1 −− r exp k cτδ tan 00n∏ ()( m mm) m=1

where tanδm= εε mm′′ ′ is the (attenuation) loss-tangent of the m-th layer. 2) Calculation of Loss Tangent Since roughness of underlying interfaces is totally unknown, the model makes all underlying interfaces as plane-stratified, as shown in Figure 1. Equation (6)

presents a set of total n equations, where Pn and τ m can be found from the ra- ()a dar data Pn , but the incidence power directly upon the top surface P0 , the at- − A tenuation e , the reflectivities rr1,, n , and the loss tangents tanδδ11 , , tan n−

are to be solved. Considering the ionosphere attenuation is wrapped into P0 , there are totally 2n independent unknowns in Equation (6). It has been known from Mars studies that the loss tangents of NPLD and SPLD are actually very small, as usually 0.001 - 0.005 [1] [3]. Some extreme cases of the media with high conductivity (dipolar and conductive) are excluded. In radar sounder technology for exploring multi-layering structure, whole media should be at a very low loss to make the wave penetration to reach the interfaces.

Certainly, the reflectivity rnn ( =1, , n − 1) is a main factor to affect Pn ,

comparing with τδmm, tan and exp(−kc0 τδmm tan ) . To reduce the number of unknowns and reach final inversion, all small loss tangents of the n-layers are trivial and seen as the same within one illuminated area (e.g. 1.81 km in the next example). The low value of the loss tangent makes such approximation reasona- ble. Thus, Equation (6) is simplified as

n−1 n−1 2 Pnn=−− Pr00exp k c tanδτ∑ m∏()1 r m (7) m=1 m=1

Taking natural log of both sides of Equation (7), it gives

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nn−−11 lnPn=− kc00 tanδτ∑∑ mn + lnP ++ ln r 2 ln() 1 −r m (8) mm=11=

It means that the echo from each interface is a linear function of the time delay

lnPn =−() kc0, tanδτtotal n ++b ξ (9)

n−1 where ττtotal, n= ∑ m is the time delay from the n-th interface echo to the top m=1 surface echo, b is an unknown constant, ξ is a random variable to take account

of different reflectivities, rn , of the interfaces. Equation (9) can be seen as a linear regression model with the regress or τ . Using the radar range echoes from all interfaces and their respective ranges, the linear fitting is obtained with the least square method. Then the loss tangent can be calculated by the slope of linear function in Equation (9). 3) Solution of Dielectric Constant of Each Layer Since the loss tangent is obtained, the number of unknowns now becomes n + 1. The set of Equation (7) can be directly solved, and the reflectivity is written as P r = n n n−1 n−1 (10) 2 P00exp−− kc tanδτ∑ mm∏()1 r m=1 m=1

Equation (10) can be solved, iteratively. Reflectivity is a function of the dielectric constants of layering media. Because the roughness of all sub-interfaces is totally unknown, it is practicable to model all sub-interfaces between layers as flatly stratified within a limited area. This is a good and workable assumption within a limited area illuminated by the radar waves, even if a small error might be caused due to small scale roughness of the interfaces. It is also noted that in derivation of Equation (1), multiple-reflection and transmission are neglected. Thus, the reflectivity from the (n − 1)-th layer to the n-th layer is derived based on a half-space model, i.e. the reflectivity of this interface is written as

2 εε− r = nn−1 n  (11) εεnn+ −1

Since the loss tangent is very small, it yields εεmm≈ ′ , Equation (11) becomes 2 2 εε′′= −1 nn−1 (12) 1± rn

But, there would be two solutions from Equation (12) due to the term ± rn . Using the phase change of the echoes, an unique solution may be obtained. The echo phase from the n-th interface is written as nn−−11 ϕϕn =+++00kc∑∑ τm 2 ϕtm,, ϕ rn (13) mm=11=

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where ϕ0 is the phase of EM incidence, ϕtm, denotes the phase from each

transmission, and ϕrn, the phase of each reflection. As EM wave is vertically incident from the (n − 1)-th layer to the n-th layer, the reflection coefficient and transmission coefficient are, respectively, written as

εεnn− −1 Rn = (14) εεnn+ −1

2 ε n Tn = (15) εεnn+ −1

Since the loss tangent is very small, it can be seen that if εεnn′′> −1 , it makes

Rn > 0 and ϕrn, = 0 ; otherwise, if εεnn′′< −1 , it makes Rn < 0 and ϕrn, = π .

In transmission, the phase keeps unchanged, i.e. ϕtm, = 0 .

As incident upon the top surface, ϕr,1 = 0 . Equation (13) gives ϕϕ01= . Thus, all phases due to reflections from all interfaces can be calculated from the data of radar range echoes as

n−1 ϕrn,= ϕϕ n −− 10kc∑ τm (16) m=1

Based on these approximations, it yields the dielectric constant of each layer as  2 2 −=10εϕ′ n−1,rn 1− rn  ε ′ = n  2 (17) 2 −=1 εϕ′− π + n 1,rn 1 rn

4) Calculation of Ionospheric Attenuation and Dielectric Constant of the Sur- face Medium Propagation through the ionosphere causes phase distortions and attenuation. The SHARAD Reduced Data Record (RDR) data has already corrected the phase distortion using Phase Gradient Autofocus (PGA) method [20]. The ionosphere is excited by solar radiation, and its effect during daytime is much stronger than during night [21] [22] [23]. To avoid the additional error caused by phase distortion, only data acquired during the night are specifically used in the following example. Moreover, ionospheric attenuation is taken into account using an uniform constant e− A when the SHARAD data acquired with the similar solar ze- nith angle (SZA). Since the SHARAD data, as available, have not been absolutely calibrated [20], the simulated echo power from the interfaces used in the inversion [10] [24], ()()aa PP1,, n , should be adjusted to match the observations PP1 ,, n on the radar receiver. It gives

()a − A Pnn=≡= Pe CP n () n 1, (18)

where calibration constant C ≡ e− A can be seen to take account the one-way

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ionospheric attenuation.

It has been studied [25] that there is purely CO2 ice covering an area of the South Pole of Mars, and its dielectric constant is known as about 2.2. Thus, ()a based on the observation data P.11()ε = 22 , as available, and our simulated

data P.11()ε = 22 at this South Pole location, C of Equation (18) is obtained and applied to the whole inversion region. The top surface is modeled as a rough surface, described by the known DEM data. From the radar equation [26], the echo from the top surface (with ε ) un-

der radar EM wave incidence (nadir incidence θi = 0 ) is written as

PP10()()ε= γε (19) where γ is the backscattering coefficient of rough surface. The Kirchhoff approximation (KA) of rough surface scattering requires the curvature radius of the surface much larger than the radar wavelength [19], it is a gentle rough surface on most areas on Mars. It has been discussed [11] that based on derivations of rough surface scattering with surface mean zero slope and the KA, the received echoes power at nadir direction is simply proportional to the surface reflectivity. It presented the inversion that the ratio of the received echo powers from one unknown and another tested surface permittivity can present inversions of the surface reflectivity, and dielectric unknown. Thus, it is derived as [11] 2 ′′ 2 i2kri ⋅ ′ k0 ∫∫ edS γε() = S r ()ε (20) πA 1

where ki is the incident wave vector, r′′ is the distance vector from the pixel center of integral to the nadir point. 0 0 Suppose that the top surface has a test value ε1 , the simulation gives P11()ε . ()a The ratio of the observation P11()ε with an unknown ε1 over the simulation 0 0 CP11()ε with assumed ε1 gives [10] PPPCr()aa()ε() () ε γε() () ε 11= 11 = 01 = 11. ()a 0 0 00 (21) P11()ε P11()ε CP 0 γε() 1 Cr 11() ε

where C was defined in Equation (18), and actually can be evaluated in the next approach. Substituting Equation (11) into Equation (21), it gives the inverted

ε1′ , as follows 2    2 ε1′ = −1. (22) P()a ε 11() 0 1− r11()ε CP ε 0 11()

It is noted that this inversion is applicable for surfaces with gentle roughness and zero mean slope. Indeed, there are large areas on the SPLD to fit this description as shown from DEM data. Our approach focuses on those cases and

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ignores those cases with steep slopes or highly varying roughness.

Using the inverted ε1′ and Mars Orbiter Laser Altimeter (MOLA) elevation

data, the backscattering coefficient γε()1 can be calculated in our numerical simulation [10] [27]. It yields the incident power on the top surface, of Equation (10) as follows,

P11()ε P0 = (23) γε()1

()a Substituting the inverted ε10′, P and observation P2 into Equations (10), it

gives r2 . Then, Equation (17) gives ε 2′ . Sequentially, it yields ε n′ ()n = 3, , etc. The thickness of each layer can be then calculated by Equation (5).

3. Inversion of Dielectric Constants at the Promethei Lingula of Mars South Polar Region

1) SHARAD Radar Echoes Data from the Promethei Lingula As shown in Figure 2, the Promethei Lingula around Mars South Pole is a part of SPLD. Figure 3 presents one SHARAD observation over this region. The radargram in Figure 3 is composed of about 2500 echoes after pre-summing on board the spacecraft. The radar flight distance is about 90.5 km during 2500 times vertical sounding. A stratified structure can be well identified in SHARAD radargram. Figure 4 is the optical photo of High Resolution Image Science Ex- periment (HiRISE) around the cliff of Chasma Australe canyon on the west edge of the Promethei Lingula, which clearly shows the existence of stratifications of layering media [28].

Figure 2. Location of SHARAD track 17485_01 data in Promethei Lingula on a MOLA elevation map.

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Figure 3. A portion of SHARAD track 17485_01 data from stratified media in Promethei Lingula.

Figure 4. The stratified structure on HiRISE photo of Chasma Australe canyon eastern cliff. (ESP_023590_0975_RED.abrowse.jpg on HiRISE website).

We choose the track 17485_01 of SHARADRDR data on PDS Geosciences node (filename: r_1748501_001_ss11_700_a.dat on website http://pds-geosciences.wustl.edu/), which passes by Promethei Lingulanear the Chasma Australecanyon during the night (SZA is about 112˚), for inverting the parameters of multi-layer media. The vertical resolution of the data is 15m in vacuum and about 8.5 m in pure ice. The latitude and longitude of each frame are indicated in the SHARAD RDR data, and the along-track distance between each frame can be calculated. Especially, the distance between each frame in Figure 3 is about 36.2 m (0.00056˚ in latitude and 0.002˚ in longitude). Accord- ing to the radargram and optical image, the topography and interfaces below the top surface look almost flat for one radar footprint. It validates our flatly stratified media model for Promethei Lingula. 2) Echoes from the Surface/Sub-Surfaces and Interface Locations In the radargram, the nadir surface echo, off-nadir clutters of rough surface, and echoes from layering interfaces must be identified and treated, separately. While the strongest peak is often from the nadir echo, the surface roughness and DEM geometry can make bright off-nadir returns [29]. Here, we specifically choose Promethei Lingula, where whole surface slopes are rather small and make the brightest echoes from nadir direction. However, it is always difficult to separate the echoes from underlying interfaces and clutter of rough top surface, because both amplitudes might be in the same order. As the radar is moving, ap- proaching or leaving, the clutters of surface roughness might show variation as a point or short arc in radargram. In contrast, the time delay of the echoes from

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flat interfaces has little change as radar is moving, and usually shows straight line in radargram. Based on this intuition, we define a threshold S to decide if the echo is from the subsurface or not: 1 mn Sij(),,=∑∑si() ++ qj p (24) 2n qmpn=−=− 1 theij -th sample of frame is local maximum where Sij(), =  . 0 theij -th sample of frame is not local maximum For example, let n = 25 and m = 1, and hence S indicates the ratio of local maximum from totally nearby 50 frames with the similar time delay (not exceeding 1 sampling interval). If S > 0.7 , it means that more than 70% of adjacent frames have reflector with the same time delay, and the reflector is judged to be from the interface. Otherwise, the isolated reflector is seen as the surface clutter. In this way, the surface echoes and interface echoes are distinguished from frame 69,301 - 69,350 of the track 17485_01, which extends about 1.81 km. Finally, the stratification of multi-interfaces is shown in Figure 5. Not all the reflectors in Figure 5(a) are kept in Figure 5(b). Some short lines and bright points are erased, and only are kept those long lines. If a few adjacent pixels on the long line happen to be weak and break the continuity of the extracted interface, the weak pixels are artificially fixed and judged as interface to avoid layering discontinuity. Linear interpolation of the interface line is used to decide the interface ranges of weak pixels. And the power of the nearest pixel is judged as the echo power of the interface.

Thus, the surface echoes, the echoes from underlying interfaces, i.e. Pnn ,= 2,, can be obtained. Sometimes, the echoes from different locations of the same interface might be quite different. It might be caused by different interface- topography, or happens to be mixed by the surface clutters. Dimmer or brighter radar echoes may also be caused by the change of interface reflectivity or the change of the interface time delay, for the time delay changes lead to different interference in the radar signal. To avoid such fluctuations of the interface echoes to affect final inversion, the echoes from the same interface is taken as an averaged value.

Figure 5. Location of multi-interfaces extracted from the SHARAD data. (a) SHARAD radargram from track17485_01; (b) stratification Location of multi-layers.

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3) Dielectric Constant of the Surface Medium Figure 6(a) shows the surface radar echoes power acquired from SHARAD in South Polar Region, where the central white circle is no-data region. The surface echo (the strongest peaks in each frame) from each area is taken from SHARAD data are used to generate Figure 6(a), which covers most part of SPLD. Similar figures had been done by Grima et al. [30] (Mars surface reflectivity map of SHARAD) and Mouginot et al. [12] (Mars surface reflectivity map of MARSIS). Using the MOLA (Mars Orbiter Laser Altimeter) elevation data, the surface echoes from this rough surface can be numerically simulated [10] [27]. At the 0 beginning, we take a proposed dielectric constant ε1 = 3 over whole area, 0 which is similar to the water ice. And ε1 = 3 is used for the echoes simulation ′ 0 around Mars South Polar region. The simulated P11()ε is obtained as shown in Figure 6(b). ()a P11()ε The ratio of Figure 6(a) over Figure 6(b), i.e. ()a 0 of Equation (21), is P11()ε

shown in Figure 6(c). It has been studied [25] that there is thick purely CO2 ice cap covering the South Pole, which can be also seen from blue colors of Figure

6(a) and Figure 6(c). The dielectric constant of CO2 is much lower than water

ice and rocks. So the echo power of CO2 cap is much less than other places. The constant of C, Equation (18), is actually obtained from these figures.

Figure 6. (a) The SHARAD surface echoes data; (b) simulated echoes with dielectric constant 3; (c) ratio of SHARAD echoes/simulated echoes; (d) dielectric constant of Mars South Polar Region.

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Using the algorithm aforementioned, the dielectric constant of Mars surface

over the South polar region, ε1′ , can be inverted, as shown in Figure 6(d). The

inverted results are influenced by the seasonal CO2 ice, because the data used in the inversion are all acquired during autumn and winter, and the Mars polar re-

gions are covered by a thin CO2 ice layer less than 1 m thick at that time [31].

Some studies show that the radar echo power decreases when a thin CO2 ice

covers on the top player [24]. Since the covering of seasonal CO2 ice is variable and uncertain for data acquired in different time, which is much thinner in terms of SHARAD resolution, the inverted dielectric property of the first layer

can be seen as the effective dielectric constant including thin CO2 ice layer as available. The inverted results are the equivalent dielectric constants of a cluster

of thin layers comprised of CO2 ice and water ice with dust. The ε1′ of most areas of SPLD is between 3 and 4. These results show that SPLD is mainly comprised of water ice, which is consistent with previous studies [7] [32]. However,

there are still some areas of SPLD in Figure 6(d) having ε1′ larger than 4, in-

cluding the Promethei Lingula. It can be seen that the areas with large ε1′ are all located in the outer part of SPLD, which does mean that these areas may contain more impurity than the center of SPLD. Certainly, low resolution of MOLA data (except the data near the pole, which has a high resolution) as available, to describe complex rough surface might cause inversion error. The typical Root Mean Square heights over the SPLD at SHARAD scales is ~0.30 m [30], which is smaller than the MOLA resolution of 1 m, which might cause an error in inversion. 4) Calculation of Loss Tangent To reduce the fluctuation of different reflectivities of the interfaces ( ξ in Equation (9)), the echoes powers of the interfaces with the same time delay are averaged. Using the least square method to make a linear fitting for the average echoes power with different time delay, a linear equation of 5 lnPn =−× 1.11 10τ total, n + 4.3 is obtained, as shown in Figure 7. From Equation 5 (9), it yields tanδ =×= 1.11 10()kc0 0.00088 . The confidence interval of loss tangent with the confidence level of 95% is [0.0004, 0.0014].

Figure 7. Relationship between each echo power and its time delay.

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Figure 8. Dielectric constant of each layer (along track).

Figure 9. Dielectric constant of each layer (across track).

Figure 10. Dielectric constant range of each layer.

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The hypothesis of linear regression model to well fit the data is tested using F distribution. It takes F = 13.1 for total 42 data points in the linear fitting. Set- ting the significant level α = 0.01 , it gives F ()1,40= 7.31 and FF> ()1, 40 . α α Thus, the linear regression is good to fit the data. The loss tangent is much smaller than previous result [1] [3], which contained both absorption of martials and signal loss at each interface reflection [1], while in our inversion, the signal losses of the interface reflections are totally removed, as shown in Equation (8). 5) Echoes Phase Estimation and Unique Solution Determination The sampling interval of the SHARAD data is 0.075 μs [20], which corresponds to the depth of 0.75 wavelength in free space according to Equation (5). It would be difficult to obtain enough accurate phase information as the radar wave propagates through the layers. Firstly, we interpolate each frame data using sinc function [33]. Then, the local maximum of the interpolated data is judged as the new interface instead of the interface extracted before. Thus, the time delay of each interface can be estimated accurately. Then, the phases are calculated by Equation (16). Due to surface clutter interference, there are noises in the phases which make errors during the phase evaluation. We take an average of the phases of echoes from the same interface to reduce the errors. 69350 ϕϕn= Im ln∑ exp ()i nf, (25) f =69301

where ϕnf, is the phase of frame f and n-th interface. Substituting Equation (25) into Equation (16), the phase of reflection is calcu-

lated. Considering that ϕrn, usually is not exactly equal to 0 or π due to the phase accuracy, Equation (17) is changed as  2 2 ππ −1,εϕ′ ∈− n−1,rn  1− rn  22 ε ′ = n  2 (26) 2 ππ   − εϕ′ = −π − π 1n−1,rn  ,, + 22   1 rn 6) Inversions and Validation

The dielectric constant of surface medium ε1′ inverted from the frame 69,301 - 69,350 (83˚S 102˚E in Figure 6(d)) of the track17485_01 is about 5 (see the location of Layer 1 at elevation 2100-2200 in Figure 8 and Figure 9. Note that the layer 0 on the top of Figure 8 and Figure 9 is Mars atmosphere). This result may be overestimated as discussed in part C of section II. However, it is possible that Layer 1 of Promethei Lingulacontain more impurity than the center

of SPLD. Thus we choose ε1′ = 5 .

Substituting ε1′ into Equation (25), and changing the loss tangent from 0.0004 to 0.0014, the varying range of the layered dielectric constants are iteratively calculated, as shown in Figure 10. Since the loss tangent is quite small, the error caused by using the same loss tangent is not larger than 25% for the inverted dielectric constants in Figure 10. The dielectric constant of Layer 7 is 2.5, which is much lower than ice. As the error is taken into account, this layer might

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be seen as pure ice. Using Equation (5), the thickness of each layer is calculated, as shown in Fig- ure 8. There won’t be echo from the place below Layer 9, since EM wave there- has become very weak due to transmission and reflection. Thus, the dielectric constant below 1500 m in Figure 8 is actually unknown, and the thickness of layer 9 cannot be inverted. The inversion accuracy depends on the evaluation of interface locations and echo powers. The phase accuracy might be also interfered by surface clutter. Suppose the depth of each interface do not change across the track, as shown in Figure 9. The red point in Figure 9 is the nadir of track 17485_01 and the dashed line is the stratification in Figure 8. Then the position of each interface exposed on the cliff can be calculated using MOLA elevation. To validate our inversions, the calculated interface exposed on the cliff is projected to a map-projected optical image, as shown in Figure 11. Note that Figure 8 gives the vertically layering structure, while Figure 11 shows a slant section of the cliff photographed vertically downwards It can be seen that the inverted multi-layering structure is well described on the cliff of Chasma Australe canyon near track 17485_01, which was indicated in a HiRISE optical image, even not exactly the same matching. The optical layers are much finer than the radar resolution, and they cannot be a 1:1 correlation. The layers in radargram correspond to the packets of thinner layers in optical image. Many layers can be seen on optical image but cannot be detected by radar, because this HF radar technology is capable only to detect the interfaces with a significant change of dielectric media. The inversion model presented is based on the implicit assumption that there is no more than one such interface within a SHARAD vertical resolution (10 - 15 m). If SHARAD reflections are caused by merged reflections of packets of thinner layers with different dielectric constants, the inversed layer-depths and dielectric properties are understood on the average or effective sense for radar sounder echoes. Moreover, details of the technology parameters and measurements, such as SNR for each echo and high resolution elevation data might further improve the inversions.

Figure 11. Cliff stratification of layering media overlapped on a HiRISE photo (a portion of PSP_006264_0970_RED.JP2).

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The inversion results show that the dielectric deposits in Promethei Lingula are not uniformly stratified, seen along the dashed line indicated in Figure 9. Most layers seem nearly pure ice (e.g. blue colors), but some thin layers including the surface layer are with a lot of dust (e.g. yellow colors). As the deposit is seen as a component of ice and dust, its dielectric constant is written as [34]

113 ε=−+()1 cc εε33 (27) dust ice dust dust

where cdust is the dust fraction, and can be calculated as 11 33 εε− ice cdust = 11 (28) 33 εεdust− ice

Taking the dust basalt ( εdust = 8 ), the dust fraction is calculated using Equa- tion (28), as shown in Figure 11. Layer 3 and Layer 5 have a dust load of 60% - 70%, which means these layers may be a compact frozen ground (permafrost) rather than dirty ice. However, SHARAD reflections might be caused by thinner layers, on the order of a meter thick or so [35] [36]. Packets of low dust thin layers with a few thin permafrost layers might be another possible existence of Lay- ers 3 and 5. Thus, it is possible to overestimate the dust fraction in the inversion. Making a dielectric average of top 8 layers media, it yields 88 εε= ∑∑ddii i (29) ii=11=

It gives the dielectric constant of regolith impurity, ε = 3.6 , which corresponds to the dust fraction about 12%.

4. Conclusions

This paper presents a model of rough surface and stratified interfaces to describe the multi-layer structure of Mars Polar Layered Deposit. The range echoes of HF radar sounder from rough surface/subsurface is numerically calculated. And under the Kirchhoff approximation with a mean zero slope, the received echo at nadir direction preserves the functional dependence of the surface reflectivity. In radargram to show the radar range echoes, the nadir surface echo and the echoes from layering interfaces are separated from off-nadir surface clutters. As the surface dielectric constant is derived from the ratio of the received echo peak power, the inversion approach is designed to obtain the dielectric constant and layer thickness of next layers, sequentially. As a validation example, the SHARAD data are adopted to inversions of the Promethei Lingula stratified media of Mars South Polar region. The vertical profile of the dielectric constants of layering media is obtained. Correspondingly, the layered structure is obtained, which is visually similar to the optical image on the cliff of Chasma Australe canyon. The inversion results show that the surface media of Promethei Lingula has larger dielectric constant with impurity, while as the media below the surface might contain much less impurity and more pure water ice. As more details of the technology parameters and measurements can

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be taken into account, the inversion accuracy can be further improved. The inversion model is based on the assumption that there is no more than one interface within a SHARAD vertical resolution. As SHARAD reflections might be caused by packets of thinner layers, multi-layer model under this resolution won’t be able to see them, individually, which are merged with other stronger adjacent reflections. The inversed layer-depths and dielectric properties are understood on the average or effective sense for radar sounder echoes.

Acknowledgements

This work was supported by the National Key Research and Development Pro- gram of China 2017YFB0502703. The MOLA elevation data and SHARAD Reduced Data Record (RDR) data are all from PDS Geosciences Node (website: http://pds-geosciences.wustl.edu/). The HiRISE optical data are from the HiRISE website (http://hirise.lpl.arizona.edu/).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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