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The shape of the local bubble is modeled in the framework of the thin layer approximation. The asymmetric shape of the local bubble is simulated by introducing axial profiles for the density of the interstellar medium, such as exponential, Gaussian, inverse square dependence and Navarro-Frenk-White. The availability of some observed asymmetric profiles for the local bubble allows us to match theory and observations via the observational percentage of reliability. The model is compatible with the presence of radioisotopes on Earth.

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 [^{60}Fe in deep-sea measurements on Earth has triggered the study of the LB-sun interaction, see [

These models leave some questions unanswered, or only partially answered, as 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 approximation 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, discusses the ^{60}Fe-signature and explores the interaction of many bubbles.

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 ∈ [ 0, π ] , and the azimuthal angle φ ∈ [ 0,2 π ] .

The following profiles are considered: exponential, Gaussian, inverse square dependence and Navarro-Frenk-White.

The density is assumed to have the following exponential dependence on z in Cartesian coordinates:

ρ ( z ; b , ρ 0 ) = ρ 0 exp ( − z / b ) , (1)

where b represents the scale. In spherical coordinates, the density has the following piecewise dependence

ρ ( r ; r 0 , b , ρ 0 ) = ( ρ 0 if r ≤ r 0 ρ 0 exp ( − r cos ( θ ) b ) if r > r 0 (2)

which has a jump discontinuity at r = r 0 when θ > 0 . Given a solid angle Δ Ω , the total mass swept, M ( r ; r 0 , b , θ , ρ 0 , Δ Ω ) , in the interval [ 0, r ] is

M ( r ; r 0 , b , θ , ρ 0 , Δ Ω ) = ( 1 3 ρ 0 r 0 3 − b ( r 2 ( cos ( θ ) ) 2 + 2 r b cos ( θ ) + 2 b 2 ) ρ 0 ( cos ( θ ) ) 3 e − r cos ( θ ) b + b ( r 0 2 ( cos ( θ ) ) 2 + 2 r 0 b cos ( θ ) + 2 b 2 ) ρ 0 ( cos ( θ ) ) 3 e − r 0 cos ( θ ) b ) Δ Ω (3)

The density is assumed to have the following Gaussian dependence on z in Cartesian coordinates:

ρ ( z ; b , ρ 0 ) = ρ 0 e − 1 2 z 2 b 2 , (4)

where b represents a parameter. In spherical coordinates, the density is

ρ ( r ; r 0 , b , ρ 0 ) = ( ρ 0 if r ≤ r 0 ρ 0 e − 1 2 z 2 b 2 if r > r 0 (5)

and presents a jump discontinuity at r = r 0 when θ > 0 . The total mass swept, M ( r ; r 0 , b , θ , ρ 0 ) , in the interval [ 0, r ] is

M ( r ; r 0 , b , θ , ρ 0 ) = ( 1 3 ρ 0 r 0 3 + ρ 0 ( − r b 2 ( cos ( θ ) ) 2 e − 1 2 r 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r b ) ) − ρ 0 ( − r 0 b 2 ( cos ( θ ) ) 2 e − 1 2 r 0 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r 0 b ) ) ) Δ Ω (6)

where erf ( x ) is the error function, defined by

erf ( x ) = 2 π ∫ 0 x e − t 2 d t . (7)

The density is assumed to have the following dependence on z in Cartesian coordinates,

ρ ( z ; z 0 , ρ 0 ) = ρ 0 ( 1 + z z 0 ) − 2 . (8)

In this paper, we will adopt the following density profile in spherical coordinates

ρ ( r ; r 0 , b , ρ 0 ) = ( ρ 0 if r ≤ r 0 ρ 0 ( 1 + r cos ( θ ) z 0 ) − 2 if r > r 0 (9)

where the parameter z 0 fixes the scale and ρ 0 is the density at z = z 0 . The above density presents a jump discontinuity at r = r 0 when θ > 0 . The mass M 0 swept in the interval [ 0, r 0 ] is

M 0 = 1 3 ρ 0 r 0 3 Δ Ω (10)

The total mass swept, M ( r ; r 0 , z 0 , θ , ρ 0 , Δ Ω ) , in the interval [ 0, r ] is

M ( r ; r 0 , z 0 , θ , ρ 0 , Δ Ω )

= ( 1 3 ρ 0 r 0 3 + ρ 0 b 2 r ( cos ( θ ) ) 2 − 2 ρ 0 b 3 ln ( r cos ( θ ) + b ) ( cos ( θ ) ) 3 − ρ 0 b 4 ( cos ( θ ) ) 3 ( r cos ( θ ) + b ) − ρ 0 b 2 r 0 ( cos ( θ ) ) 2 + 2 ρ 0 b 3 ln ( r 0 cos ( θ ) + b ) ( cos ( θ ) ) 3 + ρ 0 b 4 ( cos ( θ ) ) 3 ( r 0 cos ( θ ) + b ) ) Δ Ω (11)

The usual Navarro-Frenk-White (NFW) distribution has a dependence on r in spherical coordinates of the type

ρ ( r ; r 0 , b , ρ 0 ) = ρ 0 r 0 ( b + r 0 ) 2 r ( b + r ) 2 , (12)

where b represents the scale, see [

ρ ( r ; r 0 , b , ρ 0 , θ ) = ρ 0 r 0 ( b + r 0 ) 2 r cos ( θ ) ( b + r cos ( θ ) ) 2 , (13)

The piece-wise density is

ρ ( r ; r 0 , b , ρ 0 θ ) = { ρ 0 if r ≤ r 0 ρ 0 r 0 ( b + r 0 ) 2 r cos ( θ ) ( b + r cos ( θ ) ) 2 if r > r 0 (14)

and has a jump discontinuity at r = r 0 when θ > 0 . The total mass swept, M ( r ; r 0 , b , ρ 0 θ ) , in the interval [ 0, r ] is

M ( r ; r 0 , b , θ , ρ 0 , Δ Ω ) = ( 1 3 ρ 0 r 0 3 + ρ 0 ( ( b + r cos ( θ ) ) ln ( b + r cos ( θ ) ) + b ) ( b + r 0 ) 2 r 0 ( cos ( θ ) ) 3 ( b + r cos ( θ ) ) = − ρ 0 ( ( b + r 0 cos ( θ ) ) ln ( b + r 0 cos ( θ ) ) + b ) ( b + r 0 ) 2 r 0 ( cos ( θ ) ) 3 ( b + r 0 cos ( θ ) ) ) Δ Ω (15)

The conservation of the momentum in spherical coordinates along the solid angle Δ Ω in the framework of the thin layer approximation states that

M 0 ( r 0 ) v 0 = M ( r ) v , (16)

where M 0 ( r 0 ) and M ( r ) are the swept masses at r 0 and r, and v 0 and v are the velocities of the thin layer at r 0 and r. This conservation law can be expressed as a differential equation of the first order by inserting v = d r d t :

M ( r ) d r d t = M 0 ( r 0 ) v 0 . (17)

In the first phase from r = 0 to r = r 0 the density is constant and the explosion is symmetrical. In the second phase the density is function of the polar angle θ and therefore the shape of the advancing expansion is asymmetrical. The equation of motion for the four profiles is now derived.

In the case of constant density of the ISM, the analytical solution for the trajectory is

r ( t ; t 0 , r 0 , v 0 ) = 4 r 0 3 v 0 ( t − t 0 ) + r 0 4 4 , (18)

and the velocity is

v ( t ; t 0 , r 0 , v 0 ) = r 0 3 v 0 ( 4 r 0 3 v 0 ( t − t 0 ) + r 0 4 ) 3 / 4 , (19)

where r 0 and v 0 are the position and the velocity when t = t 0 , see [

In the case of an exponential density profile for the ISM, as given by Equation (2), the differential equation that models momentum conservation is

( 1 3 r 0 3 − b ( ( r ( t ) ) 2 ( cos ( θ ) ) 2 + 2 r ( t ) b cos ( θ ) + 2 b 2 ) ( cos ( θ ) ) 3 e − cos ( θ ) r ( t ) b + b ( r 0 2 ( cos ( θ ) ) 2 + 2 r 0 b cos ( θ ) + 2 b 2 ) ( cos ( θ ) ) 3 e − r 0 cos ( θ ) b ) d d t r ( t ) = 1 3 r 0 3 v 0 . (20)

There is no analytical solution.

In the case of a Gaussian density profile for the ISM, as given by Equation (5), the differential equation that models momentum conservation is

( 1 3 ρ 0 r 0 3 + ρ 0 ( − r ( t ) b 2 ( cos ( θ ) ) 2 e − 1 2 ( r ( t ) ) 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r ( t ) b ) ) − ρ 0 ( − r 0 b 2 ( cos ( θ ) ) 2 e − 1 2 r 0 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r 0 b ) ) ) d d t r ( t ) = 1 3 ρ 0 r 0 3 v 0 . (21)

In the case of an inverse square density profile for the ISM, as given by Equation (9), the differential equation that models the momentum conservation is

( 1 3 ρ 0 r 0 3 + ρ 0 z 0 2 r ( t ) ( cos ( θ ) ) 2 − 2 ρ 0 z 0 3 ln ( r ( t ) cos ( θ ) + z 0 ) ( cos ( θ ) ) 3 − ρ 0 z 0 4 ( cos ( θ ) ) 3 ( r ( t ) cos ( θ ) + z 0 ) − ρ 0 z 0 2 r 0 ( cos ( θ ) ) 2 + 2 ρ 0 z 0 3 ln ( r 0 cos ( θ ) + z 0 ) ( cos ( θ ) ) 3 + ρ 0 z 0 4 ( cos ( θ ) ) 3 ( r 0 cos ( θ ) + z 0 ) ) d d t r ( t ) − 1 3 ρ 0 r 0 3 v 0 = 0. (22)

There is not an analytical solution for this differential equation.

In the case of a Navarro-Frenk-White density profile for the ISM, as given by Equation (13), the differential equation that models momentum conservation is

( 1 3 ρ 0 r 0 3 + r 0 ρ 0 ( ( b + r ( t ) cos ( θ ) ) ln ( b + r ( t ) cos ( θ ) ) + b ) ( b + r 0 ) 2 ( cos ( θ ) ) 3 ( b + r ( t ) cos ( θ ) ) − r 0 ρ 0 ( ( b + r 0 cos ( θ ) ) ln ( b + r 0 cos ( θ ) ) + b ) ( b + r 0 ) 2 ( cos ( θ ) ) 3 ( b + r 0 cos ( θ ) ) ) d d t r ( t ) = 1 3 ρ 0 r 0 3 v 0 . (23)

A first approximated solution of this differential equation can be given as a series of order 4

r ( t ; t 0 , r 0 , v 0 , b ) = r 0 + v 0 ( t − t 0 ) − 3 2 ( b + r 0 ) 2 v 0 2 ( t − t 0 ) 2 r 0 cos ( θ ) ( b + r 0 cos ( θ ) ) 2 + 1 2 ( b + r 0 ) 2 v 0 3 ( ( cos ( θ ) ) 3 r 0 2 − cos ( θ ) b 2 + 9 r 0 2 + 18 b r 0 + 9 b 2 ) ( t − t 0 ) 3 r 0 2 ( cos ( θ ) ) 2 ( b + r 0 cos ( θ ) ) 4 (24)

To find a second approximate solution for this differential equation of the first order in r, we separate the variables and we integrate. The following non-linear equation is obtained

N D = t − t 0 , (25)

where

N = − 6 ( b + r 0 ) 2 ( b + r 0 cos ( θ ) ) ( 1 / 2 r cos ( θ ) + b ) ln ( b + r 0 cos ( θ ) ) + 6 ( b + r 0 ) 2 ( b + r 0 cos ( θ ) ) ( 1 / 2 r cos ( θ ) + b ) ln ( b + r cos ( θ ) ) − 6 cos ( θ ) ( r − r 0 ) ( − 1 / 6 r 0 3 ( cos ( θ ) ) 4 − 1 / 6 b r 0 2 ( cos ( θ ) ) 3 + 1 / 2 r 0 ( b + r 0 ) 2 cos ( θ ) + b ( b + r 0 ) 2 ) , (26)

and

D = v 0 r 0 2 ( cos ( θ ) ) 4 ( b + r 0 cos ( θ ) ) . (27)

In this case, it is not possible to find an analytical solution for the radius, r, as a function of time. Therefore, we apply the Padé rational polynomial, see [

N N D D = t − t 0 , (28)

where

N N = − ( r 0 − r ) ( 4 ( cos ( θ ) ) 3 r 0 3 + 2 ( cos ( θ ) ) 3 r 0 2 r + 12 ( cos ( θ ) ) 2 r 0 2 b + 8 r 0 cos ( θ ) b 2 − 2 cos ( θ ) b 2 r − 9 r 0 3 − 18 b r 0 2 + 9 r 0 2 r − 9 r 0 b 2 + 18 r 0 b r + 9 b 2 r ) , (29)

and

D D = 2 cos ( θ ) v 0 ( b + r 0 cos ( θ ) ) ( 2 cos ( θ ) r 0 2 + r 0 r cos ( θ ) + 4 b r 0 − b r ) . (30)

The resulting Padé approximant for the trajectory, the radius r 2,1 , is the second approximated solution

r 2,1 ( t ; t 0 , r 0 , v 0 , b ) = N N N D D D . (31)

where

N N N = ( cos ( θ ) ) 3 t r 0 2 v 0 − ( cos ( θ ) ) 3 r 0 2 t 0 v 0 − ( cos ( θ ) ) 3 r 0 3 − 6 ( cos ( θ ) ) 2 r 0 2 b − cos ( θ ) b 2 t v 0 + cos ( θ ) b 2 t 0 v 0 − 5 r 0 cos ( θ ) b 2 + 9 r 0 b 2 + 18 r 0 2 b + 9 r 0 3 + A , (32)

and

D D D = 2 ( cos ( θ ) ) 3 r 0 2 − 2 b 2 cos ( θ ) + 9 b 2 + 18 b r 0 + 9 r 0 2 , (33)

and

A = ( b + r 0 cos ( θ ) ) 2 cos ( θ ) ( ( 3 r 0 + v 0 ( t − t 0 ) ) 2 r 0 2 ( cos ( θ ) ) 3 − 2 ( 3 r 0 + v 0 ( t − t 0 ) ) ( − 3 r 0 + v 0 ( t − t 0 ) ) r 0 b ( cos ( θ ) ) 2 + ( − 3 r 0 + v 0 ( t − t 0 ) ) 2 b 2 cos ( θ ) + 54 r 0 v 0 ( b + r 0 ) 2 ( t − t 0 ) ) . (34)

The two approximations that we have used here cover the range in time after which the percent error is ≈10%: 1360 yr for the Taylor series and 194,285 yr for the Padé approximant. We conclude that in our case the Padé approximant has a wider radius of convergence in respect to the Taylor series.

The adopted astrophysical units are pc for length and yr for time; while the initial velocity v 0 is expressed in pc∙yr^{−1}. The astronomical velocities are evaluated in km∙s^{−1} and therefore v 0 = 1.02 × 10 − 6 v 1 where v 1 is the initial velocity expressed in km∙s^{−1}.

The starting equations for the evolution of the SB [^{7} yr; Z OB , which is the distance of the OB associations from the galactic plane; E 51 ,

which is the energy in 10^{51} erg and is usually chosen equal to one; v 0 , which is the initial velocity, which is fixed by the bursting phase, t 0 ; the initial time in yr, which is equal to the bursting time; and t, which is the proper time of the SB. The radius of the SB is

R = 111.56 ( E 51 t 7 3 N * n 0 ) 1 5 pc , (35)

and its velocity

V = 6.567 1 t 7 2 / 5 E 51 N * n 0 5 km / s . (36)

In the following, we will assume that the bursting phase ends at t = t 7 , 0 (the bursting time is expressed in units of 10^{7} yr) when N S N SNs are exploded

N S N = N * t 7,0 × 10 7 5 × 10 7 . (37)

The two following inverted formula allow us to derive the parameters of the initial conditions for the SB in terms of r 0 expressed in pc and v 0 expressed in km∙s^{−1}

t 7,0 = 0.05878 r 0 v 0 , (38)

and

N * = 2.8289 × 10 − 7 r 0 2 n 0 v 0 3 E 51 . (39)

The LB has been recently observed in the X-ray in the 0.1 - 1.2 keV region by [

ϵ obs = 100 ( 1 − ∑ j | r obs − r num | j ∑ j r obs , j ) , (40)

where r num is the theoretical radius of the local LB, r obs is the observed radius of the local LB, and the index j varies from 1 to the number of available observations. The observational percentage of reliability allows us to fix the theoretical parameters.

The numerical solution is reported as a cut in the x-z plane: see

density profile as given by Equation (5); see

The theory of the asymmetrical expansion already developed is independent of the azimuthal angle φ and therefore the 3D advancing surface of a LB can be obtained by rotating a cut in x-z plane, see

Some radioisotopes on Earth, such as ^{60}Fe (half life of 1.5 × 10^{6} yr [^{60}Fe increased 2.8 Myr ago, see [

1) r 0 the initial radius of the LB,

2) r e the radius of the LB when encounters the LB,

3) r a the actual radius of the LB,

4) D the distance between the sun and the LB, D = r a − r e , and they are reported in

1) t 0 the time at which the radius of the LB is r 0 ,

2) t F 60 e the time at which ^{60}Fe was deposited on the Earth,

3) t a the actual time of the LB,

4) t e the time of the encounter between LB and Sun, t e = t a − t F 60 e .

The distance LB-Sun, D, is reported in

The LB is a part of other bubbles which show a Swiss-cheese structure, see

Two factors allow the comparison of different models which simulate the LB: the observational percentage reliability, see Equation (40); and acceptable observational cuts of the LB, see [^{60}Fe-signature is compatible with the model that we have developed here and

exploding bubbles is reported in

Credit for

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

Zaninetti, L. (2020) On the Shape of the Local Bubble. International Journal of Astronomy and Astrophysics, 10, 11-27. https://doi.org/10.4236/ijaa.2020.101002