What is the condition for balancing pelectron and pgrav?

Question

What is the condition for balancing [latex]p_{electron }[/latex] and [latex]p_{grav}[/latex] ?

Accepted Answer

We set

[latex]p_{electron }= p_{grav}[/latex] , (36.9)

and using eqns 36.5 and 36.8 this implies that

[latex]p_{electron} =\frac{2}{3} u =\frac{\hbar ^{2} }{5m_{e} } (3π^{2} )^{2/3} n^{5/3} [/latex]. (36.5)

[latex]p_{grav} = \frac{Ω}{3V} =-\frac{G }{5}\left(\frac{4π }{3} \right) ^{1/3} M^{2/3} ρ^{4/3} ,[/latex] (36.8)

[latex]ρ =\frac{4G^{3} M^{2} m^{3}_{e} }{27π^{3} \hbar^{6} } \left(\frac{Am_{p} }{Z} \right) ^{5}.[/latex] (36.10)

Question 36.1: What is the condition for balancing pelectron and pgrav?...