This question (How many light seconds away is the JWST?), or rather one of its answers, got me thinking.
Since the communications time is about 5±0.75 seconds, it varies by about 1.5 seconds per 3 months. At times it is moving further away from us, at other times it is moving closer to us. How much of an effect does that have on the carrier frequency? Was it enough to affect the design of the telescope?
The Doppler shift caused by the speed of the JWST relative to the Earth is fairly small. Yes, it does need to be accounted for, but dealing with it is a routine matter in space communications, and such Doppler data is a very useful way of measuring the speed of a spacecraft. However, sometimes, blunders have occured, eg with the Cassini–Huygens mission.
The equation for the relativistic longitudinal Doppler effect is:
$$\frac{f_r}{f_s} = \sqrt{\frac{1-\beta}{1+\beta}}$$
where $f_r, f_s$ are the receiver and source frequencies, and $\beta$ is the radial speed, in units where the speed of light is $1$. Positive speed means the bodies are separating, causing redshift (lower frequency, longer wavelength), negative speed means they're approaching one another, causing blueshift (higher frequency, shorter wavelength).
Horizons provides radial range and range-rate data in its vector table ephemerides. You can obtain that data with a simple query URL, like this.
Of course, it's nice to see the data in a more graphical format. ;)
Here's a daily plot for the radial speed of the JWST relative to the centre of the Earth, for midnight UTC (actually TDB).
However, that plot's a little misleading because the speed of the Earth's rotation is quite substantial.
Here's a 48 hour plot, with a 1 hour time step, for the radial speed of the JWST relative to the Space Telescope Science Institute, the STScI.
You can make your own plots with this script
At these speeds, the Doppler shift is virtually linear, and quite small. Eg, at $500$ m/s, the redshift frequency ratio is ~$0.9999983322$.
The horizontal axis is in m/s, the vertical axis shows the parts per million difference from $1$.
From the JWST User Documentation at STScI:
The JWST communication subsystem provides 2-way communications with the observatory via the NASA Deep Space Network.
S-band frequencies are used for command uplink, low-rate telemetry downlink, and ranging. Ka-band frequencies are used for high rate downlink of science data and telemetry. All communications are routed through NASA's Deep Space Network, with 3 ground stations located in Canberra (Australia), Madrid (Spain), and Goldstone (USA).
There are more details on that page, but for some strange reason it makes no mention of the actual frequencies used. I can't find an official JWST page with that info, but Daniel Estévez says:
JWST uses S-band at 2270.5 MHz to transmit telemetry. The science data will be transmitted in K-band at 25.9 GHz, with a rate of up to 28 Mbps.
