The Local Bubble
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International Journal of Astronomy and Astrophysics, 2020, 10, 11-27 https://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717 On the Shape of the Local Bubble Lorenzo Zaninetti Physics Department, Turin, Italy How to cite this paper: Zaninetti, L. Abstract (2020) On the Shape of the Local Bubble. International Journal of Astronomy and The shape of the local bubble is modeled in the framework of the thin layer Astrophysics, 10, 11-27. approximation. The asymmetric shape of the local bubble is simulated by in- https://doi.org/10.4236/ijaa.2020.101002 troducing axial profiles for the density of the interstellar medium, such as Received: December 19, 2019 exponential, Gaussian, inverse square dependence and Navarro-Frenk-White. Accepted: February 3, 2020 The availability of some observed asymmetric profiles for the local bubble al- Published: February 6, 2020 lows us to match theory and observations via the observational percentage of Copyright © 2020 by author(s) and reliability. The model is compatible with the presence of radioisotopes on Scientific Research Publishing Inc. Earth. This work is licensed under the Creative Commons Attribution International Keywords License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ ISM: Bubbles, ISM: Clouds, Galaxy: Disk Open Access 1. Introduction The local bubble (LB) is a low-density region that surrounds the Sun. Because it is emitting in the X-rays, it is also called Local Hot Bubble (LHB), see [1] [2]. In the framework of thermal ionization equilibrium, the temperature is kT =(0.097 ± 0.013) keV or T =(1.1252 ±× 0.15) 106 K , see [3]. Recently, the following features of the LB have been discussed: the variations of the polariza- tion degree P, see [4]; and the polarization from the interstellar medium, due to irregular dust grains aligned with the magnetic field, see [5]. The presence of 60Fe in deep-sea measurements on Earth has triggered the study of the LB-sun inte- raction, see [6]. We now select some theoretical efforts that model the LB, as follows: the one-dimensional hydrocode with non-equilibrium ion evolution and dust, see [7] [8]; different tests to explain the FUSE data, see [9] [10]; the use of the parallel adaptive mesh refinement code EAF-PAMR, see [11]; hydrodynam- ical simulations of the LB, see [12]; and the study of the 3D structure of the magnetic field, see [13]. These models leave some questions unanswered, or only partially answered, as DOI: 10.4236/ijaa.2020.101002 Feb. 6, 2020 11 International Journal of Astronomy and Astrophysics L. Zaninetti follows: - Can we model the LB in the framework of the thin layer expansion of a shell in an interstellar medium (ISM) with symmetry in respect to the equatorial plane of the explosion? - Can we compare the data of the theoretical expansion, which is a function of the latitude, with the observed profiles of expansion of the LB? - What is the range of reliability of the Taylor expansion and Padé approxima- tion of the theoretical expansion in the framework of the thin layer? - Can we model the LB-Sun interaction? To answer these questions: Section 2 analyzes four profiles of density for the interstellar medium (ISM); Section 3 derives four equations of motion for the LB; and Section 4 discusses the results for the four equations of motion in terms of reliability of the model, it also introduces the interaction of many bubbles, dis- cusses the 60Fe-signature and explores the interaction of many bubbles. 2. The Density Profiles A point in Cartesian coordinates is characterized by x, y and z, and the position of the origin is the center of the LB. The same point in spherical coordinates is characterized by the radial distance r ∈∞[0, ], the polar angle θ ∈π[0, ] , and the azimuthal angle ϕ ∈π[0,2 ]. The following profiles are considered: exponential, Gaussian, inverse square dependence and Navarro-Frenk-White. 2.1. An Exponential Profile The density is assumed to have the following exponential dependence on z in Cartesian coordinates: ρ( zb; , ρρ00) = exp( − zb) , (1) where b represents the scale. In spherical coordinates, the density has the fol- lowing piecewise dependence ρ00if rr≤ ρρ(rr;00 ,, b ) = r cos(θ ) (2) ρ00exp−>if rr b which has a jump discontinuity at rr= 0 when θ > 0 . Given a solid angle ∆Ω , the total mass swept, M( rr;00 ,,, bθρ ,∆Ω) , in the interval [0,r] is M( rr;00 ,,, bθρ ,∆Ω) 222 b r cosθ++ 2rb cos θρ 2 b r cos(θ ) 1 ( ( ( )) ( ) ) 0 − =ρ 3 − b 00r 3 e (3) 3 (cos(θ )) 222 b r cosθ++ 2rb cos θρ 2 b r0 cos(θ ) ( 00( ( )) ( ) ) 0− + b ∆Ω 3 e (cos(θ )) DOI: 10.4236/ijaa.2020.101002 12 International Journal of Astronomy and Astrophysics L. Zaninetti 2.2. A Gaussian Profile The density is assumed to have the following Gaussian dependence on z in Car- tesian coordinates: 1 z2 − 2 b2 ρ( zb;, ρρ00) = e , (4) where b represents a parameter. In spherical coordinates, the density is ρ if rr≤ 00 ρρ= 1 z2 (rr;00 ,, b ) − (5) 2 b2 ρ00e if rr> and presents a jump discontinuity at rr= 0 when θ > 0 . The total mass swept, M( rr;00 ,,, bθρ) , in the interval [0,r] is M( rr;00 ,,, bθρ) 2 2 1 r (cos(θ )) 23− θ 13 rb 2 2 1b π 212 cos( )r =ρρr +− e b + erf (6) 00 023 3cos(θθ) 22cos( ) b ( ) ( ) 2 2 1 r0 (cos(θ )) 2 − 3 rb 2 1b π 212 cos(θ )r0 −ρ − 0 e 2 b + erf ∆Ω 0 23 cos(θθ) 22cos( ) b ( ) ( ) where erf ( x) is the error function, defined by 2 x 2 erf ( xt) = e−t d . (7) π ∫0 2.3. The Inverse Square Dependence The density is assumed to have the following dependence on z in Cartesian coordinates, −2 z ρ( zz;,00 ρρ) = 0 1 + . (8) z0 In this paper, we will adopt the following density profile in spherical coordinates ρ00if rr≤ −2 ρρ(rr;00 ,, b ) = r cos(θ ) (9) ρ001+>if rr z0 where the parameter z0 fixes the scale and ρ0 is the density at zz= 0 . The above density presents a jump discontinuity at rr= 0 when θ > 0 . The mass M 0 swept in the interval [0,r0 ] is 1 Mr=ρ 3 ∆Ω (10) 03 00 The total mass swept, M( rr;00 , z ,,θρ 0 ,∆Ω) , in the interval [0,r] is M( rr;00 , z ,,θρ 0 ,∆Ω) DOI: 10.4236/ijaa.2020.101002 13 International Journal of Astronomy and Astrophysics L. Zaninetti 243 1 ρρbr ρθ0brln( cos( ) + b) b =+− ρ r3 002 − 3 00 2 33 (cos(θ)) (cos( θ)) (cos( θθ)) (rb cos( ) + ) (11) 243 ρρbr ρθ00brln( cos( ) + b) b −00 + + 0 ∆Ω 22 33 θ θ θθ+ (cos( )) (cos( )) (cos( )) (rb0 cos( ) ) 2.4. Navarro-Frenk-White Profile The usual Navarro-Frenk-White (NFW) distribution has a dependence on r in spherical coordinates of the type 2 ρ rbr( + ) ρρ= 00 0 (rr;00 ,, b ) 2 , (12) rb( + r) where b represents the scale, see [14] for more details. The NFW profile along the axis z can be obtained by substituting r with rzcos(θ ) = 2 ρ rbr( + ) ρ ρθ= 00 0 (rr;00 ,, b , ) 2 , (13) rcos(θθ)( br+ cos( )) The piece-wise density is ρ00if rr≤ 2 ρ(rr; ,, b ρθ) = ρ00rbr( + 0) (14) 00 > 2 if rr0 rcos(θθ)( br+ cos( )) and has a jump discontinuity at rr= 0 when θ > 0 . The total mass swept, M( rr;00 ,, bρθ) , in the interval [0,r] is M( rr;00 ,,, bθρ ,∆Ω) 2 1 ρθ0((br+cos( )) ln( br + cos( θ)) ++ bbr)( 00) r = ρ r3 + = (15) 3 00 3 (cos(θθ)) (br+ cos( )) 2 ρθ0((br+ 0cos( )) ln( br +0 cos( θ)) ++ bbr)( 00) r − ∆Ω 3 θθ+ (cos( )) (br0 cos( )) 3. The Thin Layer Approximation The conservation of the momentum in spherical coordinates along the solid an- gle ∆Ω in the framework of the thin layer approximation states that M00( r) v 0= Mrv( ) , (16) where Mr00( ) and Mr( ) are the swept masses at r0 and r, and v0 and v are the velocities of the thin layer at r0 and r. This conservation law can be ex- dr pressed as a differential equation of the first order by inserting v = : dt dr Mr( ) = M( r) v. (17) dt 00 0 DOI: 10.4236/ijaa.2020.101002 14 International Journal of Astronomy and Astrophysics L. Zaninetti In the first phase from r = 0 to rr= 0 the density is constant and the explo- sion is symmetrical. In the second phase the density is function of the polar an- gle θ and therefore the shape of the advancing expansion is asymmetrical. The equation of motion for the four profiles is now derived. 3.1. Motion with a Constant Density In the case of constant density of the ISM, the analytical solution for the trajec- tory is 4 34 r( tt;,,0 r 0 v 0) = 4 rv 00( t −+ t 0) r 0, (18) and the velocity is 3 rv00 vtt( ;,,00 r v 0) = , (19) 3434 (4rv00( t−+ t 0) r 0) where r0 and v0 are the position and the velocity when tt= 0 , see [15] [16]. 3.2. Motion with an Exponential Profile In the case of an exponential density profile for the ISM, as given by Equation (2), the differential equation that models momentum conservation is 22 2 b rt cosθθ++ 2rtb cos 2 b cos(θ )rt( ) 1 (( ( )) ( ( )) ( ) ( ) ) − 3 − b r0 3 e 3 (cos(θ )) (20) 222 b r cosθθ++ 2rb cos 2 b r0 cos(θ ) ( 00( ( )) ( ) ) − d1 +=b 3 3 e.