# Does the Earth's rotation affect redshift measurements?

How do you account for the spin of the earth in the mechanical sense, when measuring redshift?

Does the relatively slow speed of the earth have a major change in the measurement because of distance?

• For intergalactic redshifts the sun's velocity with respect to the cosmic microwave background must also be taken into account. It is 379 km/s. I assume that all red shifts are computed against the reference frame of the CMB. – my2cts Jul 9 '18 at 8:59
• @my2cts That is an incorrect assumption. Most redshift surveys are what are described as "heliocentric", because the convention was set before the CMB dipole was nailed down. The other part is because, if you're using redshift as a proxy for distance, then the peculiar motion of the target has a similar magnitude effect. – Sean E. Lake Jul 9 '18 at 20:22
• Heliocentric means you correct for the orbital motion of the Earth around the sun (not sure if they also correct for Earth's rotation). – Sean E. Lake Jul 9 '18 at 20:31
• @Sean E. Lake That seems like an arbitrary choice and correction to the CMB frame is elementary. So why would astronomy use the sun as a reference? – my2cts Jul 9 '18 at 20:48

It does have an effect, but whether or not you have to worry about it depends on how sensitive your measurements are. The rotational period of the Earth is about $24\operatorname{hours}$. Depending on your latitude, that could contribute to your velocity as little as $0\operatorname{m}\operatorname{s}^{-1}$ (at the poles) or as much as $464\operatorname{m}\operatorname{s}^{-1}$. Worse, the effect that this has on your observation will depend on the angle between the direction to the source when the observation is made and East along the ground. So, it's a little complicated.

That said, $464\operatorname{m}\operatorname{s}^{-1}$ corresponds to a Doppler shift between $\pm 1.55\times 10^{-6}$, so only the most sensitive measurements will be affected, and mainly if the source is near the East/West horizons.

A much more important Doppler shift to account for in most cases is the one from the orbital revolution of the Earth about the Sun. That has a speed of about $29.8\operatorname{km}\operatorname{s}^{-1}$ and depends on the angle between Earth's motion and the source (worked out from the source's position in Ecliptic coordinates and solar elongation). That effect is up to $\pm9.9\times 10^{-5}$, so even then the measurement will have to be pretty precise for it to matter.

Another redshift that I've never seen discussed in this context (because it's sooooo tiny) is gravitational redshift. Basically, when a photon climbs out of a gravity well it gets redshifted, and when it falls down one it gets blueshifted. The effect is so small that I've only seen it used in the context of whole galaxy clusters affecting cosmic microwave background photons (see: Sachs-Wolfe effect).

• 464 m/sec does not sound too difficult to measure in the RF, a cheap 24GHz automotive sensor can detect 1m/sec. – hyportnex Jul 8 '18 at 22:39
• I never said it wasn't possible. The people who do radial velocities for planets get down to m/sec levels, IIRC. I only did redshifts of galaxies, myself, where the resolution we used allowed for determinations at the level of between 1 and .1 %. – Sean E. Lake Jul 9 '18 at 0:41
• @hyportnex Having just come from a conference on exoplanetary science, next-gen (i.e. instruments being built and installed now) radial velocity measurements are aiming for errors of tens of cm/s. – alex_d Jul 9 '18 at 8:38
• @hyportnex note that microwaves a extremely well-suited for high-precision frequency detection, because you can literally count along millions of cycles. Doing that with light generally requires a laser source with which you can create interference. – leftaroundabout Jul 9 '18 at 15:59
• @leftaroundabout sure, still the most common (maybe the only) frequency shift measurement in radar is based on (ultimately) homodyne mixing the received echo with the transmitted one and only then with direct counting the shift itself. Now I did not mean to say and I did not say that radar would be applicable for astronomical Doppler shift, I only said that to measure the speed 464m/sec is no big deal at RF. A cheap sensor can easily have 1m/sec resolution and its accuracy can be in cm/sec range depending on the integration time. – hyportnex Jul 9 '18 at 16:10

The average orbital speed of the Earth is 29.78 km/s (with the rotation speed being a hundred times smaller), so that clocks in at about 1 part in 10,000 with respect to the speed of light.

When redshift is used to measure astronomical distances, the redshifts are often of the order of a few percent or more, which is a great deal higher than the precision limit $\delta z = 10^{-4}$ imposed by the orbital speed of the Earth on redshift measurements that do not account for it.

If and when astronomers measure redshifts closer to that limit, then the orbital speed of the Earth is obviously taken into account, and that precision limit is removed.

• If I recall, the uncertainty in the $\delta z$ it because they use other clues to get the distance for very distant objects. Like the apparent brightness of certain kinds of super novae. And those kinds of things are not as accurate as they can be with other aspects of radio astronomy. – user93146 Jul 9 '18 at 1:14
• @puppetsock The existence of other sources of uncertainty should be pretty obvious, to the level of a platitude. – Emilio Pisanty Jul 9 '18 at 13:36
• I should have been more complete. The usual interest in red shift in astronomy and cosmology is how it is affected by distance. The increase in red shift per distance being (related to but I forget the details) the Hubble constant. So it's perfectly possible to be accurate enough to get the difference in red shift due to the speed of the Earth's rotation, the effect with distance will be a lot bigger uncertainty. – user93146 Jul 9 '18 at 13:41

The rotation speed at the Earth is about 0.46 km/s at the equator. The spectrum of a celestial object will be blue- or red-shifted by this amount if observed close to the horizon from an equatorial observatory. For other configurations, the effect will be smaller, depending on the altitude and azimuth of the source and the latitude of the observatory.

This is a trivially small correction to cosmological redshifts, which are of order thousands or millions of km/s, however it is an importantly large correction that needs to be made to doppler measurements of velocities of stars in our own or nearby galaxies. Indeed the size of this effect is much larger than the typical radial velocity signature due to orbiting exoplanets around a star (typically of size 1-20 m/s) and so it must be corrected for in these circumstances.

In practice, because it is easy to calculate, the correction can be routinely applied to all measured redshifts, in addition for correcting for the Earth's motion around the solar system barycentre, though it is often not done in cosmological surveys because the uncertainties in the redshift measurements are usually much larger than either of these effects.