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The most well-known repeat ground track orbits are geostationary and geosynchronous orbits, having orbital periods in a 1:1 ratio with the revolution of the Earth.

Faster repeat ground track orbits are however sometimes used. Molniya orbits do for instance use a 2:1 ratio, spending most of their foot print time over two locations in at high latitude. A significant motivation for using these orbits is their comparatively low delta-v costs to geostationary orbits (~2/3 the cost from LEO).

Are there commonly used satellite orbits with even shorter periods than these half-day ones?
Used for their repeating ground track properties, that is. There are of course many satellites with shorter periods, but those tend to have drifting sinusoidal tracks.

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Sun-synchronous orbits can be set to go over the same part of Earth every day with shorter periods. Wikipedia has a nice table, but it is entirely possible to have up to 16 orbits/ day, where they all will pass over the same ground track every day.

The 16 orbits/day would require too low (274 km) of an orbit to be practical for long term use, however. It does seem like at least a few satellites have used this in the past, namely the Microsat-R, which was the subject of the Indian Anti-Satellite test a few years back. (All these were done with queries at https://www.space-track.org/)

For 15 orbits/ day, there are many satellites in that particular orbit, with a required altitude of 567 km. Just to name a few, HJ-1A and HJ-1B. There are quite a few others. This seems to be the practical limit to repeated ground tracks. Among them were many cubesat-sized Earth observation satellites, which is an excellent place to put them.

It is worth noting that for any orbit, you will only repeatedly pass over the same track at most once a day, although you can orbit more than once and have repeated ground tracks. The exception is for a perfectly equatorial orbit, which will pass over the same point at every orbit. There are no satellites that are exactly equatorial with a low altitude (All of them are geostationary), so this is most likely correct.

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    $\begingroup$ @uhoh The following argument should be enough: After n orbits, the satellite is back at the same inertial location. If 1/n divides the length of a revolution, the Earth has then completed exactly one revolution, meaning the ground location too has the same inertial location. That's a repeat, and it's independent of the parity of n. $\endgroup$ – SE - stop firing the good guys Nov 17 '20 at 18:34
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    $\begingroup$ For odd numbers of orbits per day, won't there be both an upward-going and a downward-going pass over the same position on the equator exactly 12 sidereal hours apart? $\endgroup$ – uhoh Nov 17 '20 at 18:35
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    $\begingroup$ @uhoh That's a good point, but that's more of a "point" than a "track", and can only be directly inferred from parity in case of an equatorial ground point. $\endgroup$ – SE - stop firing the good guys Nov 17 '20 at 18:37
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    $\begingroup$ "...will only repeatedly pass over the same spot at most once a day..." How is a spot not a point? $\endgroup$ – uhoh Nov 17 '20 at 18:38
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    $\begingroup$ @uhoh I should change that to track, that's what I intended. Good catch... $\endgroup$ – PearsonArtPhoto Nov 17 '20 at 18:39

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