Worksheet 14.1, Problem 2: White Dwarfs

1. A white dwarf can be considered a gravitationally bound system of massive particles. 
   
   1. (a)  Express the kinetic energy of a particle of mass m in terms of its momentum p instead of the
      usual notation using its speed v.
      
   2. $$\frac{1}{2}mv^2 = \frac{p^2}{2m}$$
   3. (b)  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?
      
   4. We know  $$K = - \frac{1}{2} U$$ and $$U = \frac{GM^2}{r}$$ and $$K = \frac{p^2}{2 m_e}$$  Combining these gives:  $$\frac{GM^2}{r} = \frac{p^2 N}{2m_e}$$  Where  $$N = \frac{M_{\star}}{m_p + m_e} = \frac{M_{\star}}{m_p \left( 1 + \frac{m_e}{m_p} \right) } \approx \frac{M_{\star}}{m_p}$$   So \[  $$\frac{GM^2}{r} = \frac{p^2 M_{\star}}{2m_e m_p}$$  

   1. (c)  According to the Heisenberg uncertainty Principle, one cannot know both the momentum and
      position of an election such that ∆p∆x > $\frac{h}{4\pi}$ . Use this to express the relationship between the 4π  kinetic energy of electrons and their number density ne (Hint: what is the relationship between an object’s kinetic energy and its momentum? From here, assume p = ∆p and then use the Uncertainty Principle to relate momentum to the volume occupied by an electron assuming Volume ~ (∆x)3.)   $$V \sim x^3 \sim  n_e^{-1}$$   $$px = \frac{ h}{4 \pi}$$ $$p = \frac{h}{ 4 \pi x } = \frac{h n_e^{ \frac{1}{3}}}{4 \pi }$$ For 1 electron $$K = \frac{p^2}{2m_e} = \frac{h n_e^{\frac{2}{3}}}{32 \pi^2 m_e }$$ For many $$K = \frac{h^2 n_e^{\frac{2}{3}}}{32 \pi^2 m_e } \cdot \frac{M}{m_p}$$

   
   

   1. (d)  Substitute back into your Virial energy statement. What is the relationship between ne and the mass M and radius R of a WD?   $$K = - \frac{1}{2} U$$   $$\frac{h^2 n_e^{\frac{2}{3}}}{32 \pi^2 M_e } \cdot \frac{M}{m_p} = \frac{GM^2}{2r}$$ $$\frac{16M}{R} = \frac{h^2 n_e^{\frac{2}{3}}}{\pi^2 m_e m_p }$$

   
   

   1. (e)  Now, aggressively yet carefully drop constants, and relate the mass and radius of a WD. $$h^{\frac{2}{3}} = \frac{M}{R}$$ Where $$n = \frac{m}{V} = \frac{m}{\frac{4}{3} \pi R^3 }$$    $$\left( \frac{M}{R} \right)^{\frac{2}{3}} = \frac{M}{R}$$ $$R = \frac{1}{M^{\frac{1}{3}} }$$

   
   

   1. (f)  What would happen to the radius of a white dwarf if you add mass to it? 
      
   2. 
      
   3. The relationship is inverted, so it would decrease. 
   4. 
      

   UPDATE: I know my latex is not compiling in some places; I have no idea why. No matter how many different ways I type it, it won't compile (but just for some?). Do you have any suggestions?

I worked with Barra, April, and Sean on this problem.