Rest Frequency and Observing Frequency

The rest frequency of a spectral line of interest can be calculated if it is not already tabulated. The apparent frequency (or, the observing frequency), however, needs to be calculated for each source since it depends on the relative velocity between the source and the observer. The observed frequency ($\nu_{o}$) of a given transition is related to the rest frequency of the line ($\nu_{l}$) and the radial velocity of the source w.r.t the observer (v$_{r}$) as ( $\nu_{l}-\nu_{o}$)=$\nu_{o}$v$_{r}$/c, where, c is the velocity of light. This relation is valid for v$_{r} \ll$ c, and $\theta \ll \pi$/2, where $\theta$ is the angle between the velocity vector and the radiation wave vector. The radial velocity is positive if the motion is away from the observer and the observed frequency is smaller than the rest frequency of the line. In this situation, the line is redshifted. If the velocity (v$_{r}$) is known, the observing frequency can be calculated. While dealing with extragalactic systems, one quotes the redshift rather than the radial velocity. The redshift (z) is related to the rest and observed frequencies as z = ( $\nu_{l}-\nu_{o}$)/$\nu_{o}$ and approximates to v$_{r}$/c for v$_{r} \ll$ c.

It is more useful, and common to define velocities w.r.t. the $'$local standard of rest$'$ than w.r.t. an arbitrary frame of reference. This transformation takes into account the radial velocity corrections due to the rotation of the earth about its own axis, the revolution of the earth around the Sun, and the motion of the Sun w.r.t. the local group of stars. The magnitudes of these corrections are within $\sim$ 1 km s$^{-1}$, 30 km s$^{-1}$, and 20 km s$^{-1}$ respectively. The actual value of the total correction depends on the equatorial coordinates of the source, the ecliptic coordinates of the source, the longitude of the Sun, the hour angle of the source, and the geocentric latitude of the observer.

In principle, the apparent frequency of a spectral line from a source is always changing due to the change in the radial velocity between the source and the observer. In a given observing session during a day the source can be observed from rise to set. During this period the radial component of the velocity between the source and the earth due to the rotation of the earth can (in an extreme case) change from -0.465 to +0.465 km s$^{-1}$. Consider observing a narrow spectral line (width $\sim$ 0.5 km s$^{-1}$) from this source using a spectral resolution $\sim$ 0.1 km s$^{-1}$. If no extra precautions are taken, the peak of the spectral line will appear to slowly drift across the channels during the course of the day. This drift, if not accounted for, will decrease the signal-to-noise ratio of the line, and increase its observed width in the time-averaged spectrum. Depending on the circumstances, this can completely wash out the spectral line. In order to overcome this, the continuous change in the apparent frequency is to be corrected for during an observing session so that the spectral line does not drift across frequency but stays in the same channels. This process of correction is known as Doppler Tracking. I would like to emphasize that this is important if one is observing narrow lines with high spectral resolution and that there is a significant change in the sight-line component of the earth's rotation during the observing session.