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  . In the framework of thermal ionization equilibrium, the temperature is or , see . Recently, the following features of the LB have been discussed: the variations of the polarization degree P, see ; and the polarization from the interstellar medium, due to irregular dust grains aligned with the magnetic field, see . The presence of 60Fe in deep-sea measurements on Earth has triggered the study of the LB-sun interaction, see . We now select some theoretical efforts that model the LB, as follows: the one-dimensional hydrocode with non-equilibrium ion evolution and dust, see  ; different tests to explain the FUSE data, see  ; the use of the parallel adaptive mesh refinement code EAF-PAMR, see ; hydrodynamical simulations of the LB, see ; and the study of the 3D structure of the magnetic field, 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 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 , the polar angle , and the azimuthal angle .
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:
where b represents the scale. In spherical coordinates, the density has the following piecewise dependence
which has a jump discontinuity at when . Given a solid angle , the total mass swept, , in the interval is
2.2. 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 when . The total mass swept, , in the interval is
where is the error function, defined by
2.3. 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 fixes the scale and is the density at . The above density presents a jump discontinuity at when . The mass swept in the interval is
The total mass swept, , in the interval is
2.4. 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  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 when . The total mass swept, , in the interval is
3. The Thin Layer Approximation
The conservation of the momentum in spherical coordinates along the solid angle in the framework of the thin layer approximation states that
where and are the swept masses at and r, and and v are the velocities of the thin layer at and r. This conservation law can be expressed as a differential equation of the first order by inserting :
In the first phase from to the density is constant and the explosion is symmetrical. In the second phase the density is function of the polar angle 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 trajectory is
and the velocity is
where and are the position and the velocity when , see  .
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
There is no analytical solution.
3.3. 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
3.4. 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
There is not an analytical solution for this differential equation.
3.5. 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
Figure 1 reports a comparison between numerical and series solution.
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
Figure 1. Numerical solution (full line) and series solution (dashed line). The parameters are , , , and .
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    . We choose an approximation of degree 2 in the numerator and degree 1 in the denominator about the point to the left-hand side of Equation (25). The resulting equation of second degree is
The resulting Padé approximant for the trajectory, the radius , is the second approximated solution
Figure 2 reports a comparison between the numerical and the series solution.
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.
4. Astrophysical Results
The adopted astrophysical units are pc for length and yr for time; while the initial velocity is expressed in pc∙yr−1. The astronomical velocities are evaluated in km∙s−1 and therefore where is the initial velocity expressed in km∙s−1.
4.1. How to Start
The starting equations for the evolution of the SB   are defined by the following parameters: , which is the number of SN explosions in 5.0 × 107 yr; , which is the distance of the OB associations from the galactic plane; ,
Figure 2. Numerical solution (full line) and Padé approximant (dashed line). The parameters are , , , and .
which is the energy in 1051 erg and is usually chosen equal to one; , which is the initial velocity, which is fixed by the bursting phase, ; 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
In the following, we will assume that the bursting phase ends at (the bursting time is expressed in units of 107 yr) when SNs are exploded
The two following inverted formula allow us to derive the parameters of the initial conditions for the SB in terms of expressed in pc and expressed in km∙s−1
4.2. The Astronomical Data
The LB has been recently observed in the X-ray in the 0.1 - 1.2 keV region by , 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, , is introduced over the whole range of the polar angle ,
where is the theoretical radius of the local LB, 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.
4.3. The Results
Figure 3. Geometrical section of the LB in the x-z plane with an exponential profile (green points) and observed profile (red stars). The parameters are , , , and . The observational percentage reliability is and .
Figure 4. Geometrical section of the LB in the x-z plane with a Gaussian profile (green points) and observed profile (red stars). The parameters are , , , and . The observational percentage reliability is and .
The theory of the asymmetrical expansion already developed is independent of the azimuthal angle and therefore the 3D advancing surface of a LB can be obtained by rotating a cut in x-z plane, see Figure 7.
Some radioisotopes on Earth, such as 60Fe (half life of 1.5 × 106 yr ), were measured in a deep-sea ferromanganese crust: the concentration of 60Fe increased 2.8 Myr ago, see . These measurements have triggered some simulations that can explain the LB in the framework of SN explosions   . 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 following distances are involved:
1) the initial radius of the LB,
2) the radius of the LB when encounters the LB,
3) the actual radius of the LB,
4) D the distance between the sun and the LB, , and they are reported in Figure 8. The times of the 2D simulation are
Figure 5. Geometrical section of the LB in the x-z plane with an exponential profile (green points) and observed profile (red stars). The parameters are , , , and . The observational percentage reliability is and .
Figure 6. Geometrical section of the LB in the x-z plane with a NFW profile (green points) and observed profile (red stars). The parameters are , , , and . The observational percentage reliability is and .
Figure 7. 3D surface of the LB with parameters as in Figure 6, NFW profile. The three Euler angles are , and .
1) the time at which the radius of the LB is ,
2) the time at which 60Fe was deposited on the Earth,
3) the actual time of the LB,
4) the time of the encounter between LB and Sun, .
Figure 8. A sketch of the LB sun encounter. The black circle is the initial radius, the blue circle is the radius when the Sun is reached and green circle is the actual radius.
The distance LB-Sun, D, is reported in Figure 9 as function of the elapsed time.
4.5. Collective Effects
The LB is a part of other bubbles which show a Swiss-cheese structure, see Figure 10. We simulate this network with the multiple explosion of N bubbles in 2D assuming a constant density, see Equation (18). We choose and the time is allowed to vary in a random way in the interval , the position of the explosion on the two Cartesian axis is randomly generated in the interval , see Figure 11.
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 . The best result is obtained adopting the NFW profile with a percentage reliability of . Similar results are obtained in the framework of the magnetic field model, see Figure 2 in . The 60Fe-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
Figure 9. Distance between Sun and the LB as function of time. The parameters of the LB are , , , and .
Figure 10. Map of the galactic environment of the sun with side of ≈528.12 pc.
Figure 11. Network of seven bubbles, , , , and .
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.
Credit for Figure 10 is given to the University of Bologna, see https://www.sslmit.unibo.it/zat/images/cartography/M-Way_2.htm.
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