Special case of rotation period

Question

Rotation period is either prograde, like the Earth, or retrograde, like Venus. How likely is it for a body orbiting its parent object (either a planet around a star or a moon around a planet) to have a rotation period that is neither prograde nor retrograde?

By that I mean, this body has an infinite rotation period, and behaves like the Hubble space telescope when imaging distant stars, it doesn't rotate relative to distant stars.

Seen from the surface of the parent body, this object would look like it spins retrograde. (Seen from the Earth's surface, the Moon looks like it doesn't spin, yet it spins prograde.)

Seen from the object's surface, the night sky would be identical each night, and each day/night cycle is also a year.

Is this kind of infinite rotation period likely to exist and what would be the ideal conditions for it to be stable for a long period of time? What could be the shortest stable year or day duration for such a body?

i.e. avoiding short term tidal locking. (for instance, Earth won't ever be tidally locked to the Sun because the Sun will become a supernova billions of years before it could happen.)

Answer 1

This is extremely unlikely. And if it happens, it's a state of limited duration.

To explain "space of measure zero" David Hammen mentions in his comment, broken down for this example: there is exactly ONE state (exactly matching rotation) while there are infinitely many other possible rotation periods: just count all numbers, fractional and irrational included, and compare that to the amount of 0s among that.

Also it wouldn't be able to exist over an extended period of time due to every planet being subject to tides thus subject to slow secular or even longer-term changes in its rotation. And even a tiny bit slower or faster rotation would lead to the night sky changing slowly over the years.

Further, if you look at a planet where each side faces always the same side of the sky, you realize two things:

• it is a VERY SLOW rotator, thus the the day-night cycle would match its annular cycle. That's slower than any planet in the solar system.
• it would be on the "retrograde side" or rotational motion as it is slower than tidally-locked (tidally-locked means one revolution around its own axis during one year). This makes it even less likely to occur as usually planets and planetesimal get prograde spin during the initial phases of formation (and is changed by collisions in some cases in the later part of formation).

The temporal evolution of the spin would be such, that tides would spin-up its own rotation at the expense of orbital rotation (thus it would move slightly inward in order to start rotating faster - the speed depends on the strength of the tidal interaction with its host star. For comparison: on Earth, 1/3 of the tides is currently caused by the Sun)