Question

White dwarf stars are essentially plasmas of free electrons and free protons

(hydrogen atoms that have been stripped of their electrons). Their densities

are typically 5 × 10^9 kg/m3 (i.e., 10^6 times the average density of Earth).

Their further collapse is prevented by the fact that the electrons are highly

degenerate, that is, all the low-lying states are filled and no two identical

electrons can be forced into the same state.

(a) Estimate the temperature of this system. (Assume nonrelativistic electrons and that p2

f /2m = (3/2)kT .)

(b) Now suppose that the gravity is so strong that the electrons are forced

to combine with the protons, forming neutrons (with the release of a

neutrino). If the temperature remains the same, what would be the density

of this degenerate neutron star?

Answer 1

Density of star=$5*10^{9}kg/m^3$

Mass of star, $M=N(m+2m_p)\equiv2Nm_p$

Where; m=mass of electron, m_p= mass of proton,

Electron density, $n=\frac{N}{v}=\frac{\rho}{2m_p}$

=$\frac{5*10^9}{2*1.67*10^{-27}}=1.49*10^{36}$

a) $T=\frac{P^2}{3mk}$

$=(\frac{3n}{8π})^{\frac{2}{3}}\frac{h^2}{3*9.1*10^{-31}*1.38*10^{-23}}$

$=(\frac{3*1.49}{8π})^{\frac{2}{3}}\frac{10^{24}*h^2}{3*9.1*10^{-31}*1.38*10^{-23}}$

$=\frac{0.316*10^{24}*10^{54}}{3*9.1*1.38}*(6.62*10^{-24})^2$

$=3.67*10^9K$

b) $T=\frac{P^2}{3m_nk}$

Where m_n= mass of neutron

$T=(\frac{3n}{8π})^{\frac{2}{3}}\frac{h^2}{3*1.67*10^{-27}*1.38*10^{-23}}$

$=\frac{0.316*10^{24}*10^{54}}{3*1.67*1.38}*(6.62*10^{-24})^2$

$=2*10^6K$