As we see (6.4.27) is completely identical in form to (6.4.22). Apart from the minor difference of a factor 2 in the coefficient kns , the quantity Mns is the mass of the neutron star measured in units of baryon masses, just as Mwd was the mass of the white dwarf measured in the same units. Also the factor 2 difference has a nice interpretation. We might write:

where, by definition, NB is the number of baryons per fermion of the Fermi gas. In the case of white dwarfs the Fermi gas is made of electrons and for each electron there are two baryons in a star originally made of helium. In the neutron star case, the Fermi gas is made by the neutrons and each of them counts for one baryon. Similarly Rns , just as Rwd , is the radius of the star measured in units of Compton wave-length of the fermionic particles out of which the relevant Fermi gas is made. For neutron stars we measure the radius in neutron wave-lengths just as for white dwarfs we measure it in Compton wave lengths of the electron.

With the above understanding the formula for the Fermi pressure takes a general form valid for both white dwarfs and neutron stars:

6.4.2 The Equilibrium Equation

Let us now recall that in Newton’s theory the central pressure of a star of mass M, radius R and constant density is given by (6.3.48). Multiplying and dividing by the same quantities the central pressure can be rewritten in the form given below:

where by comparison with the original definition (6.3.48), the constant K turns out to have the following value:

If a neutron-star or a white-dwarf could be conceived as a constant density object, then the equilibrium equation would just be obtained by equating Pc as given in (6.4.31) with the Fermi pressure P0 as given in (6.4.30). Obviously the constant density approximation is too rough, since we know that density varies with depth already in normal stars and even more in compact ones. Yet let us remark that on dimensional grounds, as long as we are in a classical non-relativistic theory, namely as long as we are not supposed to use neither the speed of light c, nor Planck’s constant , the only combination of available parameters G, M and R that has the dimension of a pressure is precisely GMR4 2 , as can be verified by looking at Table 6.1 It follows that solving the pressure differential equation (6.3.37) with any kind of equation of state for the stellar matter:

the final result for the central or average pressure in a spherical star of mass M and radius R must be:

where α is a dimensionless, numerical coefficient. A very reasonable physical assumption, justified by numerical solutions of the pressure equation in presence of typical equations of state like:

is the following: the coefficient α ≈ 1 is of order unity. In this case we can draw very important and interesting conclusions. Equating the average gravitational pressure (6.4.34) with the Fermi pressure (6.4.30), we obtain a determination of the star

By taking a second derivative, (6.4.45) is reduced to:

Introducing a rescaled variable z = r/β (6.4.46) is turned into the following standard form:

which goes under the name of Lane-Emden equation. The boundary conditions that correspond to the considered physical situation are easily found:

The solution to (6.4.48)–(6.4.49) cannot be written in an explicit analytic form for generic values of n, admitting only a numerical determination. Yet for a few values of n θ (z) is an elementary function. In particular, as the reader can check, we have:

where we have named z0 the first zero of the θ (z) function and I0 the following integral:

For other values of n the above data have to be determined numerically through a computer programme. We also remark that the second identity in (6.4.51) follows from the fact the θ (z) is assumed to satisfy the differential equation (6.4.48).

What is the relevance of the integral I0? This is easily understood, if we calculate the mass M of the star which is supposedly described by the considered equation of state. We immediately find:

The above relation follows from the simple consideration that the first zero of θ (z), namely z0, defines the radius R ≡ βz0 of the star and shows that something very special happens for the polytropic index n = 3. Indeed for such a value the total mass of the star is independent from the central density ρ0. We are therefore led to consider whether n = 3 has a special theoretical interpretation. This is indeed the case. For n = 3 the equation of states takes the form:

which is precisely the limiting one for an extremely high density degenerate gas of fermions. How to see it? This is simple. At very high density (xF - 1) we can drop the subleading term in (6.4.14) keeping only the fourth order one. Relying on the form (6.4.11) of the Fermi momentum we conclude that p ρf4/3, where ρf is the density of the fermionic particles composing the gas, electrons in the white dwarf case, neutrons in the neutron-star case. If NB is the number of baryons per Fermi-particle, then we have:

ρ

ρf (6.4.54)

NB mp

and we conclude that p ρ4/3. So we conclude that the equation of state for degenerate compact stars of very high density is indeed of the polytropic type with polytropic index n = 3 γ = 43 . So we can calculate the mass M from formula (6.4.52) if we evaluate the index κ from (6.4.14) and (6.4.11). By direct substitution we find:

which inserted in (6.4.52) yields:

which is the celebrated Chandrasekhar formula for the upper mass limit of whitedwarfs and neutron stars. To evaluate it explicitly we just need the value of I0[3] which can be calculated from the numerical solution of the Lane-Emden equation in the case n = 3 (see Fig. 6.11). We find:

Inserting this value of I0[3] and those of the fundamental constants in (6.4.56) we find:

where we have denoted by MChwd the upper mass-limit for white dwarfs and by MChns the same for neutron stars.