_{1}

^{*}

In order to model the Fermi bubbles we apply the theory of the superbubble (SB). A thermal model and a self-gravitating model are reviewed. We introduce a third model based on the momentum conservation of a thin layer which propagates in a medium with an inverse square dependence for the density. A comparison has been made between the sections of the three models and the section of an observed map of the Fermi bubbles. An analytical law for the SB expansion as a function of the time and polar angle is deduced. We derive a new analytical result for the image formation of the Fermi bubbles in an elliptical framework.

The term super-shell was observationally defined by [

The name Fermi bubbles starts to appear in the literature with the observations of Fermi-LAT which revealed two large gamma-ray bubbles, extending above and below the Galactic center, see [

This section reviews the gas distribution in the galaxy. A new inverse square dependence for the gas in introduced. In the following we will use the spherical coordinates which are defined by the radial distance r, the polar angle θ, and the azimuthal angle φ.

The vertical density distribution of galactic neutral atomic hydrogen (H I) is well-known; specifically, it has the following three component behavior as a function of z, the distance from the galactic plane in pc:

n ( z ) = n 1 e − z 2 / H 1 2 + n 2 e − z 2 / H 2 2 + n 3 e − | z | / H 3 . (1)

We took [^{−3}, H 1 = 127 pc , n 2 = 0.107 particles cm^{−3}, H 2 = 318 pc , n 3 = 0.064 particles cm^{−3}, and H 3 = 403 pc . This distribution of galactic H I is valid in the range 0.4 ≤ r ≤ r 0 , where r_{0} = 8.5 kpc and r is the distance from the center of the galaxy.

A recent evaluation for galactic H I quotes:

n H = n H ( 0 ) exp − z 2 2 h 2 , (2)

with n H ( 0 ) = 1.11 particles cm^{−3}, h = 75.5 pc , and z < 1000 pc see [

n ( z ) = n 0 sech 2 ( z 2 h ) , (3)

where n_{0} is the density at z = 0 , h is a scaling parameter, and sech is the hyperbolic secant ( [

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 . (4)

In the following we will adopt the following density profile in spherical coordinates

ρ ( z ; z 0 , ρ 0 ) = { ρ 0 r < r 0 ρ 0 ( 1 + r cos ( θ ) z 0 ) − 2 r 0 < r (5)

where the parameter z_{0} fixes the scale and ρ 0 is the density at z = z 0 . Given a solid angle Δ Ω the mass M_{0} swept in the interval [ 0, r 0 ] is

M 0 = 1 3 ρ 0 r 0 3 Δ Ω (6)

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 z 0 2 r ( cos ( θ ) ) 2 − 2 ρ 0 z 0 3 ln ( r cos ( θ ) + z 0 ) ( cos ( θ ) ) 3 − ρ 0 z 0 4 ( cos ( θ ) ) 3 ( r 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 ) ) Δ Ω (7)

The density ρ 0 can be obtained by introducing the number density expressed in particles cm^{−}^{3}, n_{0}, the mass of hydrogen, m_{H}, and a multiplicative factor f, which is chosen to be 1.4, see [

ρ 0 = f m H n 0 . (8)

An astrophysical version of the total swept mass, expressed in solar mass units, M ⊙ , can be obtained introducing z 0, p c , r 0, p c and r 0, p c which are z_{0}, r_{0} and r expressed in pc units.

This section reviews the equation of motion for a thermal model and for a recursive cold model. A new equation of motion for a thin layer which propagates in a medium with an inverse square dependence for the density is analyzed.

The starting equation for the evolution of the SB [_{51}, the energy in 10^{51} erg, v_{0}, 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 the proper time of the SB. The SB evolves in a standard three component medium, see formula (1).

The 3D expansion that starts at the origin of the coordinates; velocity and radius are given by a recursive relationship, see [

In the case of an inverse square density profile for the interstellar medium ISM as given by Equation (4), the differential equation which models momentum conservation is

( 1 3 ρ 0 r 0 3 + ρ 0 z 0 2 r ( t ) ( c o s ( θ ) ) 2 − 2 ρ 0 z 0 3 l n ( r ( t ) c o s ( θ ) + z 0 ) ( c o s ( θ ) ) 3 − ρ 0 z 0 4 ( c o s ( θ ) ) 3 ( r ( t ) c o s ( θ ) + z 0 ) − ρ 0 z 0 2 r 0 ( c o s ( θ ) ) 2 + 2 ρ 0 z 0 3 l n ( r 0 c o s ( θ ) + z 0 ) ( c o s ( θ ) ) 3 + ρ 0 z 0 4 ( c o s ( θ ) ) 3 ( r 0 c o s ( θ ) + z 0 ) ) d d t r ( t ) − 1 3 ρ 0 r 0 3 v 0 = 0, (9)

where the initial conditions are r = r 0 and v = v 0 when t = t 0 . We now briefly review that given a function f ( r ) , the Padé approximant, after [

f ( r ) = a 0 + a 1 r + ⋯ + a p r o b 0 + b 1 r + ⋯ + b q r q , (10)

where the notation is the same of [_{i} and b_{i} are found through Wynn’s cross rule, see [

1 v ( r ) = N N D D , (11)

where

N N = ( cos ( θ ) ) 5 r 0 4 r + ( cos ( θ ) ) 4 r 0 4 z 0 + ( cos ( θ ) ) 4 r 0 3 r z 0 + ( cos ( θ ) ) 3 r 0 3 z 0 2 − 3 ( cos ( θ ) ) 3 r 0 2 r z 0 2 + 3 ( cos ( θ ) ) 3 r 0 r 2 z 0 2 − 6 ( cos ( θ ) ) 2 ln ( r cos ( θ ) + z 0 ) r 0 r z 0 3 + 6 ( cos ( θ ) ) 2 ln ( r 0 cos ( θ ) + z 0 ) r 0 r z 0 3 − 3 ( cos ( θ ) ) 2 r 0 2 z 0 3

+ 3 ( cos ( θ ) ) 2 r 2 z 0 3 − 6 cos ( θ ) ln ( r cos ( θ ) + z 0 ) r 0 z 0 4 − 6 cos ( θ ) ln ( r cos ( θ ) + z 0 ) r z 0 4 + 6 cos ( θ ) ln ( r 0 cos ( θ ) + z 0 ) r 0 z 0 4 + 6 cos ( θ ) ln ( r 0 cos ( θ ) + z 0 ) r z 0 4 − 6 cos ( θ ) r 0 z 0 4 + 6 cos ( θ ) r z 0 4 − 6 ln ( r cos ( θ ) + z 0 ) z 0 5 + 6 ln ( r 0 cos ( θ ) + z 0 ) z 0 5 (12)

and

D D = r 0 3 v 0 ( cos ( θ ) ) 3 ( r r 0 ( cos ( θ ) ) 2 + cos ( θ ) r 0 z 0 + cos ( θ ) r z 0 + z 0 2 ) . (13)

The above result allows deducing a solution r 2,1 expressed through the Padè approximant

r ( t ) 2 , 1 = A N A D , (14)

with

A N = 3 ( cos ( θ ) ) 2 r 0 3 + 2 r 0 t v 0 z 0 cos ( θ ) − 2 r 0 t 0 v 0 z 0 cos ( θ ) + 10 c o s ( θ ) r 0 2 z 0 + 2 t v 0 z 0 2 − 2 t 0 v 0 z 0 2 − 2 r 0 z 0 2 − ( ( r 0 c o s ( θ ) + z 0 ) 2 ( 9 ( c o s ( θ ) ) 2 r 0 4 − 12 c o s ( θ ) r 0 2 t v 0 z 0 + 12 c o s ( θ ) r 0 2 t 0 v 0 z 0 + 4 t 2 v 0 2 z 0 2 − 8 t t 0 v 0 2 z 0 2 + 4 t 0 2 v 0 2 z 0 2 + 18 c o s ( θ ) r 0 3 z 0 + 42 r 0 t v 0 z 0 2 − 42 r 0 t 0 v 0 z 0 2 + 9 r 0 2 z 0 2 ) ) 1 / 2 , (15)

and

A D = z 0 ( 4 r 0 cos ( θ ) − 5 z 0 ) . (16)

A possible set of initial values is reported in

The above parameters allows to obtain an approximate expansion law as function of time and polar angle

r ( t ) 2 , 1 = B N B D , (17)

with

n_{0}[particles/cm^{3}] | 1 |
---|---|

E_{51} | 1 |

N^{*} | 5.87 × 10^{8} |

r_{0} | 220 pc |

v_{0} | 3500 km/s |

z_{0} | 12 |

t | 5.95 × 10^{7} yr |

t_{0} | 36948 yr |

B N = 31944000 ( cos ( θ ) ) 2 + 18.8632 t cos ( θ ) + 5111040 cos ( θ ) + 1.0289 t − 101376 − ( ( 220 cos ( θ ) + 12 ) 2 ( 21083040000 ( cos ( θ ) ) 2 − 24899.49 t cos ( θ ) + 3219955200 cos ( θ ) + 0.0073517 t 2 + 4210.27 t − 102871295 ) ) 1 / 2 , (18)

and

B D = 10560 c o s ( θ ) − 720. (19)

This section introduces a test for the reliability of the model, analyzes the observational details of the Fermi bubbles, reviews the results for the two models of reference and reports the results of the inverse square model.

An observational percentage reliability, ϵ obs , is introduced over the whole range of the polar angle θ,

ϵ obs = 100 ( 1 − ∑ j | r obs − r num | j ∑ j r obs , j ) many directions , (20)

where r num is the theoretical radius, r obs is the observed radius, and the index j varies from 1 to the number of available observations. In our case the observed radii are reported in

The exact shape of the Fermi bubbles is a matter of research and as an example in [

The thermal model is outlined in Section 3.1 and

The cold recursive model is outlined in Section 3.2 and

The inverse square model is outlined in Section 3.3 and

numerical solution as a cut in the x-z plane.

A rotation around the z-axis of the above theoretical section allows building a 3D surface, see

This section reviews the transfer equation and reports a new analytical result for the intensity of radiation in an elliptical framework in the non-thermal/thermal case. A numerical model for the image formation of the Fermi bubbles is reported.

The transfer equation in the presence of emission only in the case of optically thin layer is

j ν ρ = K C ( s ) , (21)

where K is a constant, j ν is the emission coefficient, the index ν denotes the frequency of emission and C ( s ) is the number density of particles, see for example [

I ν = K ′ × l , (22)

where K ′ is a constant and l is the length along the line of sight interested in the emission; in the case of synchrotron emission see Formula (1.175) in [

A real ellipsoid, see [

z 2 a 2 + x 2 b 2 + y 2 d 2 = 1 , (23)

in which the polar axis of the Galaxy is the z-axis.

We are interested in the section of the ellipsoid y = 0 which is defined by the following external ellipse

z 2 a 2 + x 2 b 2 = 1. (24)

We assume that the emission takes place in a thin layer comprised between the external ellipse and the internal ellipse defined by

z 2 ( a − c ) 2 + x 2 ( b − c ) 2 = 1 , (25)

see _{m}, remains constant up to (0, a − c) and then falls again to 0. The length of sight, when the observer is situated at the infinity of the x-axis, is the locus parallel to the x-axis which crosses

the position z in a Cartesian x-z plane and terminates at the external ellipse. The locus length is

l I = 2 a 2 − z 2 b a (26)

when ( a − c ) ≤ z < a

l I I = 2 a 2 − z 2 b a − 2 a 2 − 2 a c + c 2 − z 2 ( b − c ) a − c (27)

when 0 ≤ z < ( a − c ) .

In the case of optically thin medium, according to equation (22), the intensity is split in two cases

I I ( z ; a , b ) = I m × 2 a 2 − z 2 b a (28)

when ( a − c ) ≤ z < a

I I I ( z ; a , c ) = I m × ( 2 a 2 − z 2 b a − 2 a 2 − 2 a c + c 2 − z 2 ( b − c ) a − c ) (29)

when 0 ≤ z < ( a − c ) ,

where I m is a constant which allows to compare the theoretical intensity with the observed one. A typical profile in intensity along the z-axis is reported in

I I ( z = a − c ) I I I ( z = 0 ) = r = 2 a − c b c a . (30)

As an example the values a = 6 kpc , b = 4 kpc , c = a 12 kpc gives r = 3.19 . The knowledge of the above ratio from the observations allows to deduce c once a and b are given by the observed morphology

c = 2 a b 2 a 2 r 2 + b 2 . (31)

As an example in the inner regions of the northeast Fermi bubble we have r = 2 , see [

A thermal model for the image is characterized by a constant temperature in the internal region of the advancing section which is approximated by an ellipse, see equation (24). We therefore assume that the number density C is constant and in particular rises from 0 at (0, a) to a maximum value C_{m}, remains constant up to (0, −a) and then falls again to 0. The length of sight, when the observer is situated at the infinity of the x-axis, is the locus parallel to the x-axis which crosses the position z in a Cartesian x-z plane and terminates at the external ellipse in the point (0,a). The locus length is

l = 2 a 2 − z 2 b a ; − a ≤ z < a . (32)

The number density C_{m} is constant in the ellipse and therefore the intensity of radiation is

I ( z ; a , b , I m ) = I m × 2 a 2 − z 2 b a ; − a ≤ z < a . (33)

A typical profile in intensity along the z-axis for the thermal model is reported in

The source of luminosity is assumed here to be the flux of kinetic energy, L_{m},

L m = 1 2 ρ A V 3 , (34)

where A is the considered area, V the velocity and ρ the density, see formula (A28) in [

L = ϵ L m , (35)

where ϵ is a constant of conversion from the mechanical luminosity to the observed luminosity. A numerical algorithm which allows us to build a complex image is outlined in Section 4.1 of [

Law of motion. We have compared two existing models for the temporal evolution of the Fermi bubbles, a thermal model, see Section 3.1, and an autogravitating model, see Section 3.2, with a new model which conserves the momentum in presence of an inverse square law for the density of the ISM. The best result is obtained by the inverse square model which produces a reliability of ϵ obs = 90.71 % for the expanding radius in respect to a digitalized section of the Fermi bubbles. A semi-analytical law of motion as function of polar angle and time is derived for the inverse square model, see Equation (17).

Formation of the image. An analytical cut for the intensity of radiation along the z-axis is derived in the framework of advancing surface characterized by an internal and an external ellipsis. The analytical cut in theoretical intensity presents a characteristic “U” shape which has a maximum in the external ring and a minimum at the center, see Equation (30). The presence of a hole in the intensity of radiation in the central region of the elliptical Fermi bubbles is also confirmed by a numerical algorithm for the image formation, see

Credit for

Zaninetti, L. (2018) The Fermi Bubbles as a Superbubble. International Journal of Astronomy and Astrophysics, 8, 200-217. https://doi.org/10.4236/ijaa.2018.82015