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# Doppler profile

The Doppler profile is a spectral line profile which results from the thermal motion of the emitting atom or molecule. When thermal motion causes a particle to move towards the observer, the emitted radiation will be shifted to a higher frequency. Likewise, when the emitter moves away, the frequency will be lowered. For non-relativistic thermal velocities, the doppler shift will be:

$f = f_0\left(1+\frac{v}{c}\right)$

where f is the observed frequency and f0 is the rest frequency, v is the velocity towards the observer, and c is the speed of light.

Since there is a distribution of speeds both toward and away from the observer in any volume element of gas, the net effect will be to broaden the observed line. The distribution of speeds towards and away from an observer is derived from the Maxwell distribution. If P(v)dv is the fraction of particles with velocity component v to v+dv along a line of sight, then:

$P(v) = \sqrt{\frac{m}{2\pi kT}}\,e^{-mv^2/2kT}$

where m is the mass of the emitting particle, T is the temperature and k is the Boltzmann constant. The probability distribution is seen to be just a normal distribution with variance

$\textrm{var}(v)=\frac{kT}{m}$

From the equation for the shift, it follows that f is normally distributed as well. The doppler profile is then:

$D(f,m,\sigma)=\frac{e^{-(f-f_0)^2/2\sigma^2}}{\sigma \sqrt{2\pi}}$

where the width of the line is given by σ:

$\sigma=\frac{f_0}{c}\sqrt{\frac{kT}{m}}$

(This is easily proven using the properties of the characteristic function of a normal distribution.)

03-10-2013 05:06:04