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If a planet is tidally-locked, can it have an obliquity (an axial tilt greater than 0°)?
e.g. the Earth's rotation is being slowed by tidal forces. A long way in the future it will be tidally locked. As it slows, will the obliquity also be eroded or will other forces (such as the orbit of the moon) allow it to maintain an obliquity?
Well, I can't do the math, but if an intuitive answer is OK:
If a planet is tidally-locked, can it have an obliquity?
I'm going to cheat and give you an answer using moons, because many moons are are tidally locked to their planets.
Io is a funny one, it's slightly eccentric orbit and proximity to Jupiter causes it to resurface pretty regularly. It's axial tilt isn't listed and might not be known.
Europa. Axial tilt 0.1°
Ganymede. Axial tilt 0–0.33°
Callisto. Axial tilt zero
Titan, to Saturn, Axial tilt zero
Triton, to Neptune, Axial tilt zero
Our Moon is a bit of an oddball, it has an Axial Tilt that's 1.5424° to it's orbit. click here for picture. Why our Moon varies a bit more than the others, I'm not sure, but it might be the proximity to the sun which keeps the Moon from orbiting more directly over Earth's equator and causes some imperfection in it's tidal lock. (Kind of a guess, I'm not sure on this part)
But, simple answer, Tidal locking with solid bodies by definition, the same side of the planet or moon faces the object it orbits all of the time. Obliquity doesn't make much sense if the planet's primary rotation is in sync with it's orbit around an object. (Tidal Locked planets do in fact rotate). So, it seems counter intuitive that a planet could have a high obliquity and at the same time, be tidally locked to the object it orbits. Mercury and Venus, work as examples for this common sense argument too. Mercury is, in a sense tidally locked but due to it's eccentric orbit it's in a 3:2 ratio. Mercury has 0 obliquity and Venus appears to be on it's way to being tidally locked to the sun and it has a 177.36 degree obliquity, but that's essentially 2.64 degrees, if Venus wasn't rotating in the opposite direction of the other planets. As Venus moves into being tidally locked to the sun, it's rotation should approach 0 (or 180) obliquity, the slowing of it's rotation in comparison to it's rotation matching it's orbit suggests that this is the only likely outcome.
If you want to check the axial tilt of other moons, more listed here.
Now, there may be some factors that could, just maybe, lead to exceptions, perhaps A planet with very high inclination might be a wild card. Or a rotation that's in a 2:1 resonance with the tidal locking. A binary star system could have some odd orbits too (See Pluto's and Charon's, satellite Nix here. Now, Nix isn't spherical, so it's prone to much more pronounced oddities, but a planet around a binary star could exhibit some strange orbital resonances, but I suspect using a binary star system to find an exception isn't what you had in mind.
I'm 90%-95% confident in saying that obliquity should tend towards zero as the planet becomes tidally locked to a single star that it orbits.
A long way in the future it will be tidally locked. As it slows, will the obliquity also be eroded or will other forces (such as the orbit of the moon) allow it to maintain an obliquity?
This is a tricky one because if the sun wasn't going to go Red Giant, the Earth would eventually become tidally locked to the Moon (in 10-50 billion years), and not the sun. If the Earth didn't have a moon, then it could get tidally locked to the sun, in, oh, I don't know, 100-300 billion years as a guess. The Moon is the dominant tidal force on the Earth, so in it's current set-up, the Earth can't get tidally locked to the sun. If the Moon was smaller, then maybe somebody could run the numbers on this question. If the Moon was 1/3rd it's current mass then the Lunar and Solar tidal forces would be similar, and . . . then it becomes an interesting question.
Curious sidebar, while the Moon currently stabilizes Earth's obliquity, that's not always going to be the case. As the Moon moves further from the Earth, the Earth's obliquity is expected to show much more pronounced fluctuations, but that doesn't have anything to do with the Earth being tidally locked to the sun.
Hope that helps. Corrections welcome.
The Obliquity of earth has been oscillating(approximately) with a mean period of 41,040 years. Theoretically, it has been justified that in absence of the moon, earth obliquity might become unstable. It is also a known fact that the moon is slowly moving away. By the time earth is tidally locked, the moon's effect might be significantly less, resulting in obliquity shifts of about 20-25 degrees.
