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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:
- <math>
f = f_0\left(1+\frac{v}{c}\right)
<math>
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:
- <math>
P(v) = \sqrt{\frac{m}{2\pi kT}}\,e^{-mv^2/2kT}
<math>
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
- <math>
\textrm{var}(v)=\frac{kT}{m}
<math>
From the equation for the shift, it follows that f is normally distributed as well. The doppler profile is then:
- <math>
D(f,m,\sigma)=\frac{e^{-(f-f_0)^2/2\sigma^2}}{\sigma \sqrt{2\pi}}
<math>
where the width of the line is given by σ:
- <math>
\sigma=\frac{f_0}{c}\sqrt{\frac{kT}{m}}
<math>
(This is easily proven using the properties of the characteristic function of a normal distribution.)
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