Z/A in the Chandrasekhar formula

[marcus]

A white dwarf---a cooling remnant of a star, no longer fusing---is protected from collapse by the pauli exclusion principle which says that no two electrons can occupy the same state (position and momentum). So crowding electrons together forces them to compensate by having high momenta and a pressure arises called "electron degeneracy pressure" which would exist even if the temperature were zero.

Chandrasekhar discovered a formula for the largest mass which can support itself against gravitational collapse by the pressure of its electrons.

this formula has a term (Z/A)²

how to understand it? well for a given atom Z is the number of electrons! and A is the atomic weight. So Z/A gives a good idea of
how electron-rich it is and how well it will resist collapse. In effect it is an "electrons per weight" number, giving a notion of "pressure per weight" as well.

That's all I have to say about it for now. I tried to understand the formula better today. If anyone has some insight or wants to try to explain it go ahead. Its interesting. The formula is:


0.20 (Z/A)² (hc/Gm_prot²)^3/2

which is a pure number expressing the mass as a multiple of the proton mass m_prot

In other words, if you want a complete expression for the mass you multiply the number times m_prot and get

0.20 (Z/A)² (hc/Gm_prot²)^3/2 m_prot

That's verbatim from Frank Shu, I use a version rewritten using hbar in place of h.

 

[marcus]

There is an important length called the (angular format) Compton wavelength of the proton

Λ_prot

and in natural units (c=G=hbar=1) it is simply the
reciprocal of the mass.

the mass of the proton happens to be 1/13E18 and so

Λ_prot = 13E18

In natural units the Chandrasekhar limit is expressed not in proton masses but in Planck masses and the previous formula
essentially simplifies to

pi (Z/A)² Λ_prot²

it gives 1.3E38 for the case Z/A=1/2

If you prefer replace the lamba by the reciprocal of the proton mass, it doesn't make any numerical difference

pi (Z/A)² (13E18)²

One solar mass is 0.93E38 in natural units, so if you like to express things in solar masses you can work out what 1.3E38 is.