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Transparency and stability of low density stellar plasma related to Boltzmann statistics , inverse stimulated Bremsstrahlung and dark matter

The rate of stimulated inverse bremsstrahlung is calculated for low electron density stellar plasmas and the condition under which the plasma becomes transparent is presented. The stability of low density stellar plasma is analyzed for a star with a spherical symmetry in equilibrium between the gravitational attractive forces and the repulsive pressure forces of an ideal electron gas where the analysis is developed by the use of Boltzmann statistics. Fundamental and surprising results are obtained by which the radius and the total mass of the star are inversely proportional to the square root of the electron density in the star center. The total gravitational forces of the star with very low electron and mass densities are very large (!) due to the extreme large star volumes. The absorption and emission of radiation for extremely low density star plasmas vanishes over all the entire electromagnetic spectrum. The present results are supported by numerical calculations. Similar effects are predicted for low density stellar plasmas which have different structures and the properties of such plasmas might show certain similarities with those of dark matter.


Introduction
The stellar plasmas statistics under non-relativistic conditions can be analyzed for two different asymptotic regions: a) The Rutherford-Born approximation is valid for very fast electrons giving classical collision frequency which scales as 3 1 / v with electron velocity v [e.g. Ginzburg 1961;Piel 2010;Chen 1990;Fitzpatrick 2015]. The use of Born approximation is valid under the conditions: , where e is the electron charge, the Planck constant (divided by 2π ), e m the electron mass and ω the frequency of the electromagnetic field.( e.g. Marcuse 1962;Krainov 2000;Meyer-ter-Vehn 2008). (In the present work we use the MKS units and treat the plasma with one ionic component but the analysis can easily be generalized to more ionic components). Very dense stellar plasmas satisfy Fermi-Dirac statistics where the average electron velocity is depending only on the electrons density and is independent of temperature [e.g. Ashcroft & Mermin 1967;Seeger 1991]. For this case by averaging the dependence on 3 1 / v over Fermi-Dirac distribution we obtain strong absorption and strong reflection for light incident on such plasmas. Such conditions are valid for fast electrons with high densities. b) In the present work we are interested in the case of relatively slower electrons and smaller densities for which 2 0 / 4 1 Ze v πε > . The absorption in this case is equal to bremsstrahlung stimulated absorption minus the stimulated bremsstrahlung emission. In zeroth order approximation the total rate of stimulated radiation for one electron vanishes [Krainov 2000]. We calculate the rate of stimulated inverse absorption in first order approximation by taking into account the difference in the number of electrons for stimulated emission with those for stimulated absorption and find the conditions under which the plasma becomes nearly transparent. Very high densities of plasmas producing white dwarfs, which are related to Fermi-Dirac statistics, were treated quite long ago [Chandrasekhar 1967;Edington 1959;Kippenhann, Weigert &Weiss 2013]. In the present work we treat the transparency of plasmas with relatively low densities at enough high temperatures at which Boltzmann statistics is valid. While the mass density of a star with the present plasma conditions is relatively low its stability conditions, derived in the present work, show that it can extend on extremely large volumes so that its gravitational forces become very significant. Due to the transparency of such stars we raise the idea that such star plasmas might have implication to dark matter.
Dark matter is thought to be non-baryonic, possibly being composed of some as-yet undiscovered particles [e.g. Bertone and Hooper 2018;Trimble 1987]. Its presence is implied in astrophysical observations, due to the existence of gravitational effects that cannot be explained unless more matter is present that cannot been seen. The name dark matter refers to the fact that it does not appear to interact with electromagnetic radiation and thus it is invisible ('dark') to the electromagnetic spectrum. Usually stars are observed either by radiation emitted by them or by reflected and transmittance of light incident on them from other sources. It is interesting to note by following the present analysis that stellar plasmas with very low densities do not emit any significant radiation and their transmittance is nearly equal 1 (reflectance nearly zero). We find that the stability of a star under spherical symmetry, which is based on the present plasmas conditions, can extend on extremely large volumes so that its gravitational forces become very large. While we developed the present analysis for stars with spherical symmetry similar results are predicted for star plasmas which have different structures. We study the possibility that such plasmas might be related to dark matter.
The present paper is arranged as follows: By using the theories on stimulated inverse bremsstrahlung, we treat in section 2 the transmittance of monochromatic electromagnetic field through stellar plasmas with relatively slow electrons and low densities. We show that under certain conditions analyzed in the present work these plasmas have extremely low transmittance and reflectivity. In section 3 we study the stability conditions of such plasmas showing that that although their mass density is quite low they can extend on extremely large volumes and therefore they produce very significant gravitational forces. Due to these properties we find that such plasmas might be related to dark matter. In section 4 we demonstrate our results by numerical calculations. In the Appendix we develop alternative method for calculating the star radius which gives similar result to that developed in the main article. In Section 5, we summarize our results.

Optical properties of low densities stellar plasmas related to inverse stimulated bremsstrahlung and to plasma transparency
The differential cross section for stimulated emission or absorption for one electron-ion collision (neglecting quantum corrections) was given by [Krainov 2000;Berestetskii, Lifshitz & Pitaevsky 1998 ;Manakov 2015 where p is the electron momentum.
We notice that , e a w is proportional to 1 / p in comparison to the dependence on 3 1 / p for fast electrons. We notice also that e w for stimulated emission is equal to stimulated absorption a w so that the total stimulated radiation T w for one electron-ion collision in zeroth order approximation vanishes, i.e. [Krainov 2000]: 0 ! T e a w w w = − = .
(5) In order to get the total rate of inverse stimulated absorption of the plasma we need to multiply , Here , In Eq.  1.06 10 So, for Carbon plasma ( 6 Z = ) and Oxygen plasma ( 8 Z = ) for 5 0 10 T K = it is a good approximation but for Hydrogen plasma ( 1 Z = ) it is only a fair approximation.

The stability of stellar plasmas with low densities under Boltzmann statistics
We study here the stability properties of a star which includes plasma with relatively low densities. For simplicity of discussion we treat a star with spherical symmetry where its stability is related to gravitational forces without any other external perturbations e.g. magnetic fields etc. While this topic was studied under high electron densities using Fermi-Dirac statistics [Chandrasekhar 1967;Edington 1959; Kippenhan, Weigert &Weiss 2012] we study here the problem for low densities using Boltzmann statistics.
The gravitational force at a distance r from the star center is due entirely to the mass r M interior to this distance: where G is the constant of gravitation. Assuming that φ is the gravitational potential then According to the hydrostatic equation where P is the star plasma pressure and ρ the star density, both are functions of distance r from the star center. Inserting Eq. (13) into Eq. (14) we get: This equation describes the increase of the plasma pressure as we move to larger values of r balancing the attractive gravitational forces.
Assuming that the star plasma behaves as an ideal gas then the pressure P is given by Assuming also that the gradient of temperature is small relative to the gradient of the electron density e n (isothermal process) then we get: If there are κ nucleons for each electron, then the mass density is given approximately by This equation describes the decrease of the plasma density ( ) ( ) e i n r n r = as a function of the change of the exponential function of the potential ( ) r φ . Here ( ) r φ is changing from zero to large negative values, as function of the distance r from the star center. Our aim in the following analysis is to find the change of ( ) ( ) e i n r n r = as function of the distance r from the star center. In these calculations we assume that 0 n is the density of ionized electrons in the star center which is taken as experimental parameter.
In order to find the dependence of the electron density ( ) e n r on the distance r from the star center we need to take into account the Poisson equation for the potential φ which for a star with spherical symmetry has the form: We note that on the right side of Eq. (21)  According to Eq. (19) we get: Inserting Eq. (18) We define 0 ( ) ( ) e n r r n θ = , and divide Eq. (24) by 0 n then we get: We divide Eq. (25), for ( ) r θ by ξ . Then, this equation can be written as: Eq. (27) We get here the interesting result that the radius of the star R increases inversely proportional to the square root of its electron density in the star center.
The total mass star M of the star is proportional to its volume, and to the average mass density Here we substituted the value of the star radius according to Eq. (28) and inserted an averaged value for the electron density e n which is proportional to its density o n in the star center. We find that the star mass is increasing with temperature as it is proportional to 3/2 T . We find also the astonishing result that the star mass is inversely proportional to the square root of the density of the electrons in its center due to increase in the volume proportional to 3 R .
The above effects can be summarized as follows: for low densities plasma the balance between the ideal gas radiation pressure and gravitational forces leads to extremely large value for the star radius so that although the mass density is quite small the star volume is extremely large so that it leads to very large star mass with strong gravitational forces. We demonstrate these effects in the next

Numerical Calculations
We demonstrate our results in the following 3 examples. . It has a radius which is larger approximately by a factor 1.75 relative to solar radius. The radius of the present star becomes smaller for Carbon or Oxygen plasma for which 12 κ = or 16, respectively. The radius is also decreased for lower temperature as according to Eq. (28) it is proportional to T . We have to take into account that for partial ionization the parameter κ increases inversely proportional to the part of ionization and thus the star radius is decreased further.

Star with electron density
Substituting in (29 T .
It also decreases for larger values of κ for Carbon and Oxygen plasma and also for partial ionization.
The above effects become much stronger for smaller densities as analyzed in the following examples.
For completely ionized plasma we assume in Eq. (10) the approximation So, for completely ionized hydrogen plasma for which: 2 κ = , the present star mass is smaller approximately by factor 143 than the solar mass. For plasmas with values of κ larger than 2 this value can be reduced but in any case it gives very large mass for low electron density star due to the extremely large volume.
For completely ionized plasma we assume in Eq. (10) For completely ionized Hydrogen plasma: 2 κ = , so that the radius of such star has a radius which is larger approximately by a factor 1750 relative to the Solar Radius ! . This radius becomes smaller for plasmas for which 2 κ > and for lower temperatures.
So, for completely ionized hydrogen plasma for which: 2 κ = , the present star mass is smaller approximately by factor 4.5 than the solar mass !. For plasmas with values of κ larger than 2 this value can be reduced, but for electrons densities which are smaller than that of 20 0 10 n = the star mass becomes larger. In any case it gives large mass for low electron density star due to the extremely large volumes. For completely ionized plasma we assume in Eq. (10) For this case Hydrogen, Carbon and Oxygen plasmas become transparent over all the entire significant electromagnetic spectrum.
We demonstrated the present ideas by analyzing 3 examples. The present effects become even stronger, if the density of electrons is reduced further. We assumed in our analysis a star with spherical symmetry under the opposing forces: gravitational forces and radiation pressure for plasma behaving as an ideal gas. The assumption of a star with spherical symmetry is assumed here mainly due to mathematical convenience. I expect that similar effects will occur in low density star plasmas in which the plasmas will have different various structures. It seems that the present analysis is relevant to dark matter.

Summary Discussions and Conclusions
In the present work we have shown that the absorption of low ionized stellar plasmas with one ionic component with Z atomic number, satisfying Boltzmann statistics, in which the condition κ = ). We find that for the present transparent plasmas the mass density is very small but, as analyzed in the present article, the stability conditions for such plasmas leads to extremely large volumes so that gravitational forces become very large. In the present work we assumed that the star plasma behaves as an ideal gas where the pressure e B P n k T = was given by Eq. (16), where B k is the Boltzmann constant. The stability of the star with low electron densities is obtained by considering it to be in equilibrium between two opposing forces: The repulsive ideal gas electron pressure and the attractive gravitational force. For a star with spherical symmetry the balance between these opposite forces leads to Eq. (19). In this equation the density of electrons is given by an exponential function of the potential with a small coefficient in the exponent. In principle, this equation can be solved by taking into account the Poisson equation for the potential given in Eq. (20). Since these equations are difficult to solve we preferred in the present analysis to transform Eq. (20) to a differential equation for the density of electrons ( ) e n r as function of the distance r from the star center. After some calculations we obtained the differential equation for ( ) e n r in Eq. (23).
After some additional transformations we obtained the differential equation (27)   proportional to the electron density as given by Eq. (18). The star volume, however, increases proportional to 3 R leading the total mass of the star to increase inversely proportion to the square root of the electron density 0 n at the star center as given by Eq. (29)! We added in the Appendix a calculation for the star radius by using a series expansion of the exponential of Eq. (19). This calculation gives an order of magnitude for the star radius which is similar to that given by Eq. (28).
Usually one would think that bremsstrahlung phenomena cannot be ignored when electrons are colliding with ions but we should take into account that, under the conditions presented in the present work, bremsstrahlung stimulated emission involves also bremsstrahlung stimulated absorption so that the summation of these two opposing effects leads to inverse stimulated absorption as given by Eq. (9).
Since this absorption is proportional to the multiplication of the density of electrons ( ) , as demonstrated by the numerical calculations in section 4. Trying to relate the present work to certain phenomena about dark matter we need to use the following considerations. In the above analysis we obtained extremely large volumes for plasma with low electron density, where we got the result that as we decrease the density of the electrons the volumes increases very much. The astonishing result from the present analysis is that a star with low mass density extends over extremely large volumes so that the total mass is very large and gravitational forces become very significant. These phenomena which seem to be quite strange follow from the fact that in order that the gravitational forces of low density star will be in equilibrium with the radiation pressure of ideal electron gas they should extend on larger and larger volumes as the mass density is decreased. The present analysis of a star with spherical phenomena was made for simplicity of calculations but it does not imply that such plasmas will have spherical symmetry since similar effects will occur on other different structures of the plasmas in which the explicit calculation should be different. Finally, following an article [Ben-Aryeh 1967] which was published quite long ago, the emissivity (the ratio between the experimental radiation and the black body radiation) is equal to the absorbance so that for transparent plasma the emissivity vanishes and we will not observe any radiation emitted from such plasma.
and express the potential function as a series expansion. Then this equation can be written as: The solution for ( ) y φ can be estimated by using the following series expansion: We obtain the coefficients , , , , a b c d ⋅⋅⋅ corresponding respectively to 1,0,1, 2,3 n = − by equating to zero the sum of terms with the same exponent of y . We get: Here we used the comparisons for n y only for 1,0,1,2, n = − ⋅⋅⋅ but the comparison can be made for larger values of n with somewhat lengthy calculations.
Since z is a small number we get for the solution of ( ) y φ : (A8) So this estimation for the star radius is larger by factor 6 2.45 relative to the estimation made in Eq.
(28) but both estimations lead to the same orders of magnitudes.