Question

Another spectral correction is due to the Doppler effect. If the atoms you are observing are in a gas, they are likely in motion, some moving toward you and some away. This will shift some frequencies up and some down, making the spectral line appear to have a width. Compute the expected Doppler broadening ?lof the Lyman alpha line from a hot star with temperature 10,000K.   Use the fact that the average thermal kinetic energy is given by (3/2) kT, where k is Boltzman’s constant. Use this to compute the RMS velocity (square root of the average squared velocity) of one component of velocity (that toward or away from observer), and use that value in the Doppler formula. Let ?l be the expected shift either up or down from the central value. What is the percentage shift?

Answer 1

From the thermal energy estimates, we get
  
And so, the velocity is
  
a)
And from the Non-relativistic Doppler shift of wavelength, we get

where, the plus sign is for redshift and the minus sign is for blue shift. So, the range of wavelengths that can be expected is
  
And putting the expression of the velocity we get the range of the wavelength as
  
b)
And for a hydrogen atom,
  
And temperature of the outer atmosphere of the sun is
  T = 6000 K.
And for a given
  
By putting the given values and the values for the constants, we get