WS 10.1, Problem 2: White (I still don't know why the plural is spelled this way) Dwarfs

A white dwarf can be considered a gravitationally bound system of massive particles.

a) What is the relationship between the total kinetic energy of the electrons that are supplying
the pressure in a white dwarf, and the total gravitational energy of the WD?

This part is pretty much just a bunch of algebraic manipulation of the Virial Theorem.

$K=-\frac{1}{2}U$

The above equation, though, is only useful for relating the energies of one electron.  In order to find the relationship for the whole white dwarf, you have to multiply by the total number of electrons.  

$N_e\frac{1}{2}m_e v^2 = \frac{3}{10}\frac{GM^2}{R}$

but because of the relationship between momentum and velocity, 

$N_e\frac{1}{2}m_e \frac{p_e}{m_e}^2 = \frac{3}{10}\frac{GM^2}{R}$

$N_e\frac{p_e^2}{m_e} = \frac{3}{5}\frac{GM^2}{R}$

b)  Express the relationship between the kinetic energy of electrons and their number density n (Hint: what is the relationship between an object's kinetic energy and its momentum?)

So, we start with the last equation from part (a).  

$KE=N_e\frac{p_e^2}{m_e}$

In the problem, we're given the following information

• $\Delta p\Delta x>\frac{h}{4\pi}\Rightarrow \Delta p\sim \frac{1}{\Delta x}$

• $\Delta p\approx p$

• Volume\sim \Delta x^3$

If number density is number over volume, 

$n_e=\frac{N_e}{V}=\frac{N_e}{\Delta x^3}\sim \Delta x^{-3}$

We went through all of that so that we could find a relationship between the $p_e^2$ in the equation from part (a) and the number density.  

$p_e^2\sim n_e^{2/3}$

The total number of electrons, $N_e$, is equal to the number of protons, which is equal to 

$\frac{M_*}{m_p}$

After all of that, we can write 

$KE=\frac{M_*n_e^{2/3}}{m_p m_e}$

c) What is the relationship between $n_e$ and the mass M and radius R of a WD?

We kind of started solving this part in part (b).

$n_e=\frac{N_e}{V}$

We can get rid of a lot of constants in this equation to end up with 

$n_e\sim \frac{M}{R^3}$

d) Substitute back into your Virial energy statement, aggressively yet carefully drop constants, and relate the mass and radius of a WD.

First, let's recall the the equation from part (a).

$N_e\frac{p_e^2}{m_e} = \frac{3}{5}\frac{GM_*^2}{R_*}$

Based on all of the relationships we've found in parts (b) and (c), we can rewrite this equation as 

$\frac{M_*n_e^{2/3}}{m_p m_e}=\frac{3}{5}\frac{GM_*^2}{R_*}$

First, let's get rid of the constants we don't really need to have a basic understanding of the relationships between these quantities 

$M_*=\frac{M_*^2}{R_*}$

Finally, after simplifying, we find that the mass and radius of a White Dwarf are inversely proportional 

$M\sim R^{-1}$