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International Journal of Current Science and Technology Vol.7, Issue, 3(A), pp. 691-694, March, 2019
ISSN: 2320-8090 RESEARCH ARTICLE
ZERO VELOCITY CURVE IN ELLIPTIC RESTRICTED THREE BODY PROBLEM WITH RADIATING PRIMARY AND OBLATE SECONDARY
Sunil Kumar Pandit
Department of Mathematics Jai Prakash University, Sadhapur, Chapra (Saran), Bihar 841301
ARTICLE INFO ABSTRACT
Article History: This paper deals with the generalized photogravitational elliptic restricted three body problem. We
Received 06th December, 2018 have supposed that bigger primary is radiating and smaller one an oblate spheroid. The stability of out Received in revised form 14th of plane libration points has been examined numerically. This model is applied on three extra solar January, 2019 planetary systems with planets in habitable zone. We conclude that these libration points are unstable Accepted 23rd February, 2019 for all three planetary systems. The Zero Velocity Curve in x-z plane is are also plotted. Published online 28th March, 2019
Key words:
Photogravitational/ ERTBP/ Oblate/ Exoplanets/ ZVC.
Copyright © 2019 Sunil Kumar Pandit,., This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION Ragos and Zagouras [11] have studied the region of stability of these equilibrium points extensively by studying periodic Radiation pressure plays a key role in the dynamical behavior solution of the system and presented the out of plane points as of sub-micrometer size grains in a star’s radiation and the saddle point of zero velocity surfaces [10]. Das etal [3] gravitational fields. The circular Restricted Three Body presented a new approach by discussing the existence and Problem (CRTBP) with the effect of the radiation pressure, stability of out of plane equilibrium points, taking the radiation when one or both primaries of the system are the source of parameter β_2 not as an independent parameter but as the radiation was discussed by Radzievsky [8,9]. The rotation of parameter dependent upon β_1, mass ratio and luminosity of celestial bodies produces an equatorial bulge, which results in the binary of P-R drag on the system. the oblateness of the body. In classical problems the primaries are taken strictly as spheres but some planets and stars are The elliptic restricted three body problem (ERTBP) sufficiently oblate to make departure form sphericity generalizes the CRTBP. by also considering the non-zero significant in the study of celestial systems. The stability of the eccentricity of the orbits of the primary. Thus the ERTBP has a triangular liberation points when PR-effect is considered in the wider applicability, although some outstanding properties of generalized photo gravitational restricted three body problem the circular model can still be adopted to the elliptical case. was studied by Kushvah and Iswar [6]. Farther they studied the Many authors ([2],[14],[7]) etc have studied the elliptical problem taking the second primary as oblate spheroid [5]. restricted three body problem in detail. Singh and Umar [13] studied the out of plane equilibrium points and their stability Simmnons etal.[12] extended the work in the circular restricted for ERTBP when both the primaries are radiating oblate problem of three bodies taking both the gravitating bodies to spheroids. be radiating. They obtained a complete solution of problem of existence and linear stability of the equilibrium points for all In this paper we have studied the equations of motion of our values of radiation pressure of both luminous bodies and all problem and found the coordinates in section 2. In section 3, values of mass ratios. It was found that there exits nine we have described Numerical application of three planetary libration points, three collinear, two triangular and four out of systems. We have examined the stability of motion and found Plane when the radiation pressure of the smaller mass is very that they are unstable. In section 4 Zero Velocity Curves have high. They have shown that L8 and L9 are always linearly been plotted in figure 4-7. Conclusions have given in section 5.nThen we write some References and at last added figures. unstable, whereas L6 and L7 are stable for small range of radiation pressure provided that both large masses are Equations of Motion luminous.
Corresponding author: Sunil Kumar Pandit The equations of Motion of an infinitesimal mass under the Department of Mathematics Jai Prakash University, Sadhapur, combined influence of oblateness of primary and radiation Chapra (Saran), Bihar 841301 pressure in a non-dimensional rotating –pulsating barycentric co-ordinate system (x, y, z) [13], are given as follows: International Journal of Current Science and Technology Vol.7, Issue, 3(A), pp. 691-694, March, 2019
″ ′ ∂Ω Stability of the Planetary Systems x − 2y = ∂ We have numerically calculated the roots characteristic Ω y″ + 2x′ = (1) equation for different values A2 for the three planetary systems shown in Table 2, 3 and 4, and found that out of plane Ω z″ = libration points are unstable for all the three planetary systems considered. Where, Φ Ω = (1 − e ) + (2)
( μ) μ where Φ = + 1 + (3)
r = (x + μ) + y + z
r = (x + μ − 1) + y + z (4)
And Figure 1 The shift in the location of the L6 and L7 for the planetary system HD 40307 g
with variation in A2 ∈ (0, 0.005]. The frame (b) shows the subregion of the graph in frame 2 (a) for smaller variation in value of A2 ∈ (0, 0.0005]. The horizontal and vertical axis 2 1 3e 3A2 represents the x- and y- coordinates of the equilibrium points. n 1 (5) a 2 2
In the differential equations dash denotes the differentiation with respect to eccentric anomaly E. A2 is the Oblateness coefficient of the second primary. q1=1-β1, is the mass reduction facer corresponding to the first primary. μ is the mass ratio, while e, a and n are the eccentricity. Semi-major axis and mean motion respectively.
Thus, the coordinates of the out of plane libration points have been approximated in the form of power series as given below:
3 3 A3 2 3e2 9 3 A5 2 * 2 2 3 (6) x 1 1 1 a q1 1 3a q1 OA2 a 2 a 9A2 z * 3A 2 1 q OA3 2 5 1 2
Numerical Application
This section deals with the numerical calculation and analysis of the triangular libration points for the three different extra solar planetary systems; HD 40307g, HD 37605 b and HD 45350 b, in the frame work of the elliptical restricted three body problem under the combined influence of oblateness and radiation pressure. The exoplanet, in these systems, is orbiting in the habitable zone of the respective stars and the eccentricity of the orbit of the exoplanet is greater than zero.
The exoplanet HD 40307g was discovered by the radial velocity method, using the European southern observatory’s HARPS apparatus [15] in the year 2012. HD 37605 b, an eccentric Jupiter orbiting close to the star in a highly eccentric orbit, was found by Hobby-Eberly Telescope (HET) in July 2004 [1]. HD 45350 b, of minimum mass 1.79 times the mass of Jupiter is orbiting the star in a highly eccentric orbit [4].
Out of Plane Libration Points of the Planetary Systems
Table 1 contains the numerical data relevant to this paper of the above mentioned three planetary systems, assuming the radius and density of the infinitesimal particle to be 2 x 10-1 cm and 1.4g cm-1.Figure 1, 2, and 3 shows the variation in the position of the libration point with A2 for the system HD 40307g, HD 37605 b and HD 45350 b respectively.
692 Zero Velocity Curve In Elliptic Restricted Three Body Problem with Radiating Primary and Oblate Secondary
Figure 6 The Zero-velocity curves in the x-z plane for the planetary system HD 37605b.
Figure 4 The Zero-velocity curves in the x-z plane, taking µ=0.0012, q1=0.999, q2=0.9888, a=0.9988, A2=0.001, c=1.2 for different value of e.
Figure 7 The Zero-velocity curves in the x-z plane for the planetary system HD 45350b.
Table 1 The numerical data ([3, 6, 15]) and the locations of the triangular Figure 5 The Zero-velocity curves in the x-z plane for the planetary system libration points of the three planetary systems. MSun and LSun denotes the Mass HD 40307 g. and Luminosity of Sun, MJ denotes mass of Jupiter and Teff denotes the effectives temperature of star.
Planetary HD 40307g HD 37605 b HD45350 b system: Mass of star: 0.75MSun MSun 1.02MSun Mass of 0.257 M 2.84M 1.79M exoplanet: J J J 693
International Journal of Current Science and Technology Vol.7, Issue, 3(A), pp. 691-694, March, 2019
µ: 3.28 x 10-5 2.70 x 10-3 1.67 x 10-3 The zero velocity curves in the x-z plane are also plotted and Teff (K) 4977 5448 5754 depicted in the figures 4-7. From figure 4 it is clear that the L: 0.23LSun 0.59LSun 1.59LSun q: 0.9368 0.8784 0.8573 curves shape is dependent on the eccentricity of the elliptic e: 0.22 0.737 0.778 orbit of the primaries, where as figure 5, 6 and 7 depicts the a(AU): 0.6 0.261 1.92 ZVC for the planetary systems respectively.
Table 2 The values of Characteristic roots for varying values of A2 References for the system HD 40307 g. 1. Cochran William D., Endl M., McArthur B., Paulson A2 λ 1,2 λ 3,4 λ 5,6 ˜0 ±121038 ±58833.4 ±134579i D.B., Smith V.V., Mac Queen P.J., Tull R.G., Good J., 0.0001 ±1.59383 ±0.32078i ±2.09455i Booth J., Shetrone M., Roman B., Stephen O., Deglman 0.0005 -0.385535±0.94875i 0.385535±0.94875i ±0.668425i F., Graver M., Soukeup M., Villarreal Jr. M.L., The 0.001 0.17293±0.999973i -0.17293±0.999973i ±0.0984227i 0.005 0.158479± i -0.158479± i ±0.0000768854i First Hobby- Eberly Telescope Planet: A Companion to HD 37605. The Astrophysical Journal, 611(2), L133- Table 3 The values of Characteristic roots for varying values of A2 L136 (2004). for the system HD 37605 b. 2. Danby, J.M.A.Stability of triangular points in the
A2 λ 1,2 λ 3,4 λ 5,6 elliptic restricted problem of three bodies. Astron.J. 69, ˜0 ±595227 ±289324 ±661819i 165172(1964). 0.0001 ±18.4387 ±9.0845 ±20.5804i 3. Das, M.K., Narang, Pankaj, Mahajan, S., Yuasa M.: On 0.0005 -4.13981 ±2.01009 ±4.71375i 0.001 ±1.74126 ±0.566399 ±2.09613i Out of Plane Equilibrium Points in Photo-Gravitational 0.005 0.686713± 0.989227i 0.686713± 0.989227i ±0.164187i Restricted Three Body Problem. J. Astrophys. Astr.30, 177-185(2009). Table 4 The values of Characteristic roots for varying values of A 2 4. Michael E., William D.C., Robert A.W., and Artie P.H.: for the system HD 45350 b. Determination of the Orbit of the Planetray Companion A2 λ 1,2 λ 3,4 λ 5,6 to the Metal-Rich Star HD 45350. Astronomical ˜0 ±342345 ±166405 ±380645i Journal. 131(6),3131-3134(2006). 0.0001 ±10.1717 ±5.13098 ±11.4284i 0.0005 ±1.79449 ±0.608057 ±2.09918i 5. Iswar B. Kushvah B.S.: Linear stability of triangular 0.001 0.767035±0.963122i -0.767035±0.963122i ±0.371623i equiliberium points in the generalized photo- 0.005 0.772475± 0.999965i 0.772475± 0.999965i ±0.101002i gravitational restricted three body problem with
Zero Velocity Curves Poynting Robertson drag. Journal of Dynamical Systems and Geometric theories. 4, 1-5(2006). The necessary conditions for capture for averaged ERTPB can 6. Kushvah B.S., Ishwar B.: Triangular equiliberium be obtained by assuming the eccentricity e to be very small and points in the generalized photo-gravitational restricted hence the Zero Velocity Curve are approximated to three body problem with Poynting Robertson drag.
Review Bull. Cal. Math. Soc. 12(1):109-114(2004). 20 x, y, z, f C* e, 7. Markeev, A.P.: Libration points in celestial mechanics 1 1 A 3A z 2 1 A 3A z 2 1 0 x2 y2 1 q 1 1 q 2 2 and cosmic dynamics. NAUKA, Moscow, 312p(1978). 2 1 3 5 2 3 5 2 n 2 r1 2 r1 r1 2r2 2 r2 r2 8. Radizievsky , V.V.: The restricted problem of three 1 A 3z 2 A 3z 2 bodies taking account of light pressure, Astron Zh , 3 1 3 1 ; 3 2 3 2 vol.27,250-256(1950). 2r1 r1 2 r2 r2 9. Radizievsky, V.V.: The space photogravitational and C* e 1 e2 C 0 restricted three-body problem., Astron Zh , vol.30,265- Here C0 is the Jacobi constant corresponding to the CRTBP. 273(1953). The figure 4 show the projection of the Zero Velocity Curve 10. Rogos, O. ,Zagouras, c.: The zero velocity surfaces in on x-z plane for varying values of eccentricity for an arbitrary the photogravitational restricted three body problem. set of values for other parameters. The figures 5-7 shows the Earth, Moon and Planets, vol.41 , 257-278,(1988a) ZVC for the three planetary system taken into consideration. 11. Rogos, O. , Zagouras, c.: Periodic solutions about the out of plane equilibrium points in the photo CONCLUSIONS gravitational restricted three body problem . Celes
The photogravitational elliptical restricted three body problem .Mech. , vol.44 ,135-154,(1988b). is studied considering the oblateness of the second primary and 12. Simmons, J.F.L., Mcdonald, A .J.C. and Brown, J.C.. radiation pressure of the first primary on the infinitesimal The three body problem with radiation particle. The out of plane libration points are obtained. A pressure. Celes. Mech.,vol.35 ,pp.145-187, (1985). semi-analytic study of three extra –solar planetary system has 13. Singh, J., Umar, A.: On ‘out of plane’ equilibrium been undertaken in this paper. The shift in the position of the points in the elliptic restricted three-body libration points with change in the values of the oblateness problem with radiating and oblate primaries 344,13- factor is depicted in Figure 1-3. It is found that the x- and y- 19(2013a). coordinates of the libration points are dependent on the 14. Szebehely V.: Theory of Orbits, The Restricted problem oblateness of the primaries. of three bodies Academic Press (New York) (1967). 15. Tuomi M., Anglada-Ecude G., Gerlach E., Jones The stability condition for the libration points are discussed. H.R.A., Reiners A., Rivera E.J., Vogt S.S., Butler R.P.: However from the numerical tabulation of roots of the Habitable-zone super- Earth candidate in a six-planet characteristic equation as shown in tables 2, 3 and 4 for all system around the K2.5V star HD 40307. Astronomy three planetary system. We conclude that the libration points and Astrophysics, 549, 48-71 (2013). are unstable for the three exoplanetary systems considered.
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