On the shape of the local bubble

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.


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 ( ) 0.097 0.013 keV kT = ± or ( ) 6 1.1252 0.15 10 K T = ± × , see [3]. Recently, the following features of the LB have been discussed: the variations of the polarization 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 60 Fe in deep-sea measurements on Earth has triggered the study of the LB-sun interaction, 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]; hydrodynamical 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 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.

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

An Exponential Profile
The density is assumed to have the following exponential dependence on z in Cartesian coordinates: where b represents the scale. In spherical coordinates, the density has the fol- which has a jump discontinuity at 0 r r = when 0 θ > . Given a solid angle ∆Ω , the total mass swept,

A Gaussian Profile
The density is assumed to have the following Gaussian dependence on z in Cartesian coordinates: where b represents a parameter. In spherical coordinates, the density is ( ) and presents a jump discontinuity at x t x t − = π ∫ (7)

The Inverse Square Dependence
The density is assumed to have the following dependence on z in Cartesian coordinates, In this paper, we will adopt the following density profile in spherical coordinates where the parameter 0 z fixes the scale and 0 ρ is the density at    2  4  0  3  0  0  0 0  2  3  3   3  2  4  0  0  0  0  0  2  3  3   0   ln cos  1  2  3 cos cos cos cos ln cos 2 cos cos cos cos

Navarro-Frenk-White Profile
The usual Navarro-Frenk-White (NFW) distribution has a dependence on r in spherical coordinates of the type where b represents the scale, see [14] for more details. The NFW profile along the axis z can be obtained by substituting r with ( ) The piece-wise density is and has a jump discontinuity at

The Thin Layer Approximation
The conservation of the momentum in spherical coordinates along the solid angle ∆Ω in the framework of the thin layer approximation states that ( ) ( ) In the first phase from 0 r = to 0 r r = 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.

Motion with a Constant Density
In the case of constant density of the ISM, the analytical solution for the trajectory is and the velocity is where 0 r and 0 v are the position and the velocity when

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 There is no analytical solution.

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

Motion with an Inverse Square Dependence
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 International Journal of Astronomy and Astrophysics There is not an analytical solution for this differential equation.

Motion with a Navarro-Frenk-White Profile
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 A first approximated solution of this differential equation can be given as a series of order 4 ( ) 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  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 [17] [18] [19] [20]. We choose an approximation of degree 2 in the numerator and degree 1 in the denominator about the point 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.

Astrophysical Results
The adopted astrophysical units are pc for length and yr for time; while the initial velocity 0 v is expressed in pc•yr −1 . The astronomical velocities are evaluated in km•s −1 and therefore

How to Start
The starting equations for the evolution of the SB [15] [21] are defined by the following parameters: * N , which is the number of SN explosions in 5.0 × 10 7 yr; OB Z , which is the distance of the OB associations from the galactic plane; 51 E , which is the energy in 10 51 erg and is usually chosen equal to one; 0 v , which is the initial velocity, which is fixed by the bursting phase, 0 t ; 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 and its velocity * 51 5 2 5 0 7 In the following, we will assume that the bursting phase ends at

The Astronomical Data
The LB has been recently observed in the X-ray in the 0.1 -1.2 keV region by [3], whose Figure 7 reports six configurations of the LB along great-circle cuts through the Galactic pole and the Galactic plane. As a target of the simulation, we have chosen the cut characterized by galactic longitude, l, between 120˚ and 300˚. An observational percentage reliability, obs  , is introduced over the whole range of the polar angle θ , obs num obs obs, 100 1 , where num r is the theoretical radius of the local LB, obs r 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 Results
The numerical solution is reported as a cut in the x-z plane: see Figure 3 for an exponential density profile as given by Equation (2); see Figure 4 for a Gaussian International Journal of Astronomy and Astrophysics  International Journal of Astronomy and Astrophysics density profile as given by Equation (5); see Figure 5 for an inverse square density profile as given by Equation (9); and see Figure 6 for a NFW density profile as given by Equation (13).
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 Figure 7.

60 Fe-Signature
Some radioisotopes on Earth, such as 60 Fe (half life of 1.5 × 10 6 yr [22]), were measured in a deep-sea ferromanganese crust: the concentration of 60 Fe increased 2.8 Myr ago, see [6]. These measurements have triggered some simulations that can explain the LB in the framework of SN explosions [12] [23] [24].
The encounter between the advancing shell of the LB and the Sun is here simulated in 2D assuming a constant density, see Equation (18)     The distance LB-Sun, D, is reported in Figure 9 as function of the elapsed time.

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

Conclusions
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 [3]. The best result is obtained adopting the NFW profile with a percentage reliability of obs 82.69% =  . Similar results are obtained in the framework of the magnetic field model, see Figure 2 in [13]. The 60 Fe-signature is compatible with the model that we have developed here and Figure 9 reports the distance between the Sun and the LB. A simulation of the International Journal of Astronomy and Astrophysics   exploding bubbles is reported in Figure 11. A more precise simulation of the exploding bubbles can be done when more detailed observations of the network, such as that reported in Figure 10, are available.