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## Conversion of source velocity/redshift to sky frequency

Sometimes confusion may arise on how to calculate the sky frequency for a given redshift or source velocity. This stems from the fact that optical and radio astronomers calculate radial source velocities from different formulas. (Remember that velocities are not directly measured but calculated from wavelength or frequency shifts). Optical astronomers deduce radial velocities from the measurement of a wavelength shift: where c is the speed of light, ( ) the unshifted and (f) the shifted wavelength (frequency), respectively. The astronomical redshift z is defined as At low redshifts some radio astronomers use a different convention which presumably comes from the fact that frequency shifts rather than wavelength shifts are measured. The radio convention is As a consequence, radio ``velocity'' and optical recession velocity are not the same for a given redshift: This means that in order to calculate the correct sky frequency from a given velocity one needs to know if the velocity was calculated using the optical or radio convention. An optical velocity (or redshift z) requires another formula than a radio velocity obtained e.g. from 21 cm HI measurements. The sky frequency for a given source velocity and line rest frequency is

• optical convention:  For a given velocity the sky frequencies calculated with the two formulas above can differ significantly. For example, calculating the sky frequency in the 230 GHz range with the inappropriate formula (optical instead of radio convention, or vice versa) at a source velocity of 10000 km/sec results in an error of about 250 MHz. At 20000 km/sec, this error has increased to about 1 GHz!

At the IRAM 30m telescope the conversion is done in the following way: the drive program calculates the velocity of the telescope relative to the local standard of rest, takes the source velocity (from the source catalogue), and then calculates the sky frequency applying the optical convention: If you want to observe e.g. a high-redshift object with redshift z at line rest frequency , the recommended procedure is to set the source velocity v (in the source catalogue) to zero and give the corrected frequency (calculated with the optical formula) to the drive program by using the OBS command:
RECEIVER rx_name /FREQUENCY LSB.

You might alternatively calculate the optical source velocity , give this velocity in the source catalogue, and use the line rest frequency in the OBS command:
RECEIVER rx_name /FREQUENCY LSB.

If you need to use the radio convention, it is best to set the source velocity v (in the source catalogue) to zero and give the corrected frequency (calculated with the radio formula) using the OBS command:
RECEIVER rx_name /FREQUENCY LSB
instead of the line rest frequency.

Instead of giving frequencies directly in the RECEIVER command, you can also define a line with the corrected frequency in your line catalogue, and use the OBS command:    