Question

1. Derive the expression for the mean temperature in a star:

∝ M^2/3 < ρ > ^1/3

2. Show that superposition of two linearly polarised light waves having different amplitudes and a finite phase difference can be used to produce elliptically plane polarised waves.

Answer 1

The Virial theorem for stars (not in nuclear reactions) and filled with a gas in hydrostatic equilibrium states that: 
      Internal energy Ei = - Ω /2  or  - Eg / 2
                     where Ω or Eg = Gravitational Energy.

The derivation of Virial theorem is obtained starting from the equation:
          dP/dm = - G m /(4 π r^4)
Where P is the pressure at the point on the spherical surface with a radius r from the center of the star.  m is the mass enclosed in the spherical surface of radius r.

Gravitational Energy of the star = Eg = - G M^2 / R
           M = mass of the star and  R = radius of the star

Let   ρ = average density of star = M/[4π R^3 /3]
      so R = [ 3 M / (4πρ) ]^1/3

Internal Energy Ei
  = 1/2 * G M^2 * [ 3 M / (4πρ) ]^-1/3
  = 1/2 * G M^(5/3) ρ^(1/3) * (3/4π)^1/3

Internal Energy (of a mono-atomic ideal gas or gas in the form of ions) 
 = Ei = 3/2 * k_B T * N = 3/2 * k_B * T * [M /μ m_H]
     k_B = Boltzmann's constant.  
      T = average temperature of the star
      N = number of molecules/particles of gas
      M = mass of the gas
      m_H = mass of the particle of gas
     basically N is proportional to the mass M of star

So we get:   Ei = constant * M^(5/3) * ρ^(1/3) = constant * T * M
 
    Thus  T = constant * M^(2/3) * ρ^(1/3)