Stopping the periodic shift

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Stopping the periodic shift

As the systemic velocity of the source in the observatory frame varies with time in a predictable way, the radio-observatories tune the local oscillator frequency in order to stop the shifting of the observed frequency scale. For instance, the relation between the observatory frequency and the LSR (Local Standard of Rest) frequency is

(29)
because
(30)
Thus adding to the local oscillator frequency makes the signal frequency appear as if it was measured in the LSR frame. All the velocities are then expressed in the LSR frame instead of the observatory frame.

However, there is a single local oscillator frequency. Let's assume that the correction is done at the tuned signal frequency, i.e.,

(31)
The correction which is applied to another frequency is then
(32)
while the correction that should have been applied to another frequency is
(33)
The difference (or error) is
(34)
The correction is thus exact only in , i.e., for the tuned signal frequency1. All the other signal and image frequencies oscillates with time with a frequency amplitude which linearly increases with the distance to the reference channel. In other words, this corrects only the global frequency shift, not the dilatation around the reference channel.