Wednesday, May 11, 2011

Tides and their locks

Gravity causes tides. On Earth, a planet with a whole bunch of wet stuff sloshing around on the surface, this leads to the sort of sea and ocean tides that most of you have probably encountered at some point.

Tidal interactions

Tides on Earth are mainly caused by the gravitational force of the moon pulling on the Earth. Water, unlike the rock making up most of the surface of the Earth, is able to move a little bit towards the moon in response. Obviously, it's not a massive effect—we're not talking about losing chunks of ocean into space—but it's significant for the sea level to rise a few meters in certain places. The sun also has a similar effect on the Earth so if we didn't have a moon, we'd still have some tides, just on a smaller scale and rather more regularly. As it is, the reason tides are so irregular is because moon and sun aren't in sync (this is also why lunar calendars and solar calendars are so different).

I should also add that while the water on the side of the Earth closest to the moon becomes deeper due to the gravitational pull of the moon, it is conservation of angular momentum which causes the water on the side of the Earth furthest from the moon to also bulge out. I won't go into the specifics unless someone asks in the comments, but the short version is that that opposite bulge of water is require to "balance out" the bulge formed by the gravitational pull of the moon.

OK, so the only thing that really sloshes around on the Earth is water, but what would happen if the moon was bigger or the Earth was closer to the sun and had no water? Or what if we had a rocky moon orbiting close to a gas giant? Well, instead of water sloshing about, it's possible that the gravitational pull of the large planet would pull on the moon strongly enough to deform rock. This is exactly what happens with Io and Europa, Jupiter's two innermost (Galilean) moons (although Europa is more ice than rock). The tidal tug of gravity on Io is what keeps its core molten and causes so many volcanoes on its surface. It's what keeps the interior of Europa liquid (or at least, what keeps some water in liquid form below the surface) and it's also what keeps Earth's core molten (because of our moon's tidal forces). Without these tidal interactions, there would have been more than enough time for these planets and moons to cool enough for their cores to solidify. In the case of Io, I believe the smaller gravitational tugs of the other Galilean satellites, particularly Europa and Ganymede, may also play a small part in its tidal heating.

The take home message is: tidal forces cause volcanoes. It's an important point to remember if you're situating your planet/moon close to its star/primary.

Locked with tides

Let's move away from the Earth and the Jovian satellites for a moment and think about a miscellaneous rocky planet, close to it's star. Like Mercury, for example. For a long time, it was thought that Mercury was tidally locked, meaning that the same side always faces towards the sun. It turns out this isn't quite true thanks to gravitational tugs from some of the other, bigger planets. So let's ignore the other planets. We have a star and a planet forms in place around it. I've mentioned before that conservation of momentum dictates which direction planets and moons will initially rotate and orbit. Any deviations from this will be a result of later collisions. So the planet will form orbiting in the same direction that its star rotates and also rotating in this same direction. (If you're a bit confused about how an orbit and a rotation can be in the same direction, stick your thumb out and curl your fingers around. Your thumb is pointing in the direction of angular velocity of something rotating in the direction your fingers are curling. You could make a looser curl with your fingers to represent an orbit and, so long as your thumb continued pointing in the same direction, that orbit would be in the same direction as the previous rotation.)

The angular momentum of the star-planet system has to be conserved. (Angular) Momentum is mass multiplied by (angular) velocity for each body and then summed. Since the mass of the system isn't going to change much (ignore comets and spare dust/gas that might accrete), then for the planet's rotation to change (that is, get faster or slower) one of the other angular velocities has to change to compensate. This is exactly what happens when a planet becomes tidally locked around its star or when a moon becomes tidally locked around its primary (example: the Galilean moons of Jupiter); rotational angular momentum is slowly converted into orbital angular momentum resulting in a slightly faster orbit but a slower period of rotation. Given enough time, the planet will become locked in a synchronous orbit (the same side always facing its sun). This is where the "tidal" part of "tidally locked" comes from.

Now, it's possible to work out how long this process takes or, for a given time frame, what the "tidal lock radius" is; that is, the distance from the star inside of which planets will be tidally locked after that time period has elapsed. The general equation for this, as given by von Bloh et al. (2007) (and Kastings (1993), pdf sorry) is:

This isn't the most helpful equation ever. And the units are confusing
Where P0 is the original period of the planet, Q is a factor to do with the rotational properties of the planet and M* is the mass of the star. As the caption says, this isn't super useful and they use somewhat baffling units. Subbing in the values they assume for P0 and Q and assuming the same time they assume which is 4.5 Gyr (the G stands for giga—yes, just like in your computer—and means 4.5 billion years or 4 500 000 000 years), I get something close to:

See, isn't that nicer to deal with? And now we have rT in AU and M* in solar masses—yay!
So if you put in the mass of your star in units of solar masses (or the mass of your gas giant, but you still have to use solar masses) then you will learn the tidal lock radius in AU for systems which have had 4.5 Gyr in which to evolve. I think 4.5 Gyr was chosen because that's roughly the age of the Earth/solar system (actually, it's more like 4.6 Gyr, but close enough).

If you want to read more about orbital mechanics, I found a review written by the guy who originally worked this stuff out in 1977: Peale (1999) (pdf again, sorry). I haven't had the chance to read through all of it yet, and it's a bit heavy on the maths, but it looks interesting.

There is more that I want to say about tides, but I feel this post has gotten long enough so I'll leave it for a future blog. Coming up soon: the Roche limit and why Saturn has rings (and all the other gas giants too). Stay tuned!

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