## Tidal Theory

Tidal Wave (Source)

We’re used to thinking of various astronomical bodies having gravity and gravitationally interacting with each other, and it’s certainly true that they do, and that these interactions can be exploited to detect extrasolar planets through their pull on their parent stars (either by astrometry or Doppler spectroscopy). So far in our study of extrasolar planetary systems, we have treated astronomical bodies as point masses — that is, the entirety of their gravity is assumed to come from a single point, and that the entirety of their matter exists at that point. But this is, of course, an oversimplification. The consideration of astronomical bodies as extended objects introduces a range of new effects that are collectively called “tides.”

We may be familiar with the concept of the tides in the ocean, where the gravitational pull of the Moon on the oceans of Earth causes them to be raised twice a day, and so this is where we will start our consideration of tides. The gravity of Earth’s moon pulls on Earth, causing it to stretch into a more elliptical shape. A diagram illustrating this is shown below.

We see then that the rise over the side of Earth opposite the Moon is caused by the distortion of the shape of Earth. It’s important to understand that this is a very general phenomenon: It is not exclusive to the Earth-moon system, but any two-body system can be represened this way. Furthermore, while it is not shown in the above image for simplicity, the primary body (Earth in the case of the Earth-Moon system, the star in the case of a hot Jupiter system) will also raise a tide on the secondary mass.

The tidal deformation of the primary body is not an instantaneous process. There is inertia to overcome and so if the primary mass is rotating, the tidal bulge will be dragged away from the line connecting the two bodies. A diagram of this is shown below.

In the above image, we assume a prograde orbit for the satellite, and $r$ is a line joining the centre of the two bodies. Note that the rotation of the primary forces its long axis to lead ahead of the orbiting body. For a circular orbit, the long axis of the planet will lead ahead of $r$ by some constant angle, even while the planet rotates. This is why there are two lunar tides each day. Aside from causing boats to rise and descend in the water, relative to fixed points on land, these tidal effects have some important implications.

Firstly, it affects the orbits of the two bodies. In the case of the Earth-Moon system, the oceanic tidal bulge facing the Moon has a stronger gravitational attraction on the Moon than the tidal buge on the opposite hemisphere on Earth, simply because it is closer (and gravitational attraction scales inversely with $r^2$). Since the nearer tidal bulge is pushed ahead of the $r$ line, the net gravitational attraction of the Moon toward Earth is also displaced from $r$. Since the tidal bulge is pushed ahead of the Moon in its orbit, the gravitational attraction of the Moon toward Earth has a small component in the direction of its orbital motion. This causes the Moon to accelerate (it’s orbital velocity increases), which then causes its orbital altitude (the semi-major axis) to increase. Any readers casually aware of orbital mechanics is familiar with this effect. If you’re unfamiliar with orbital mechanics, a full description of it is beyond the scope of this writing, but I would encourage you to try Kerbal Space Program.

Conversely, if the tidal bulge lagged behind the $r$ line, the orbit of the Moon would decay. This can be done either by having the Moon in a retrograde orbit as is the case for Neptune’s moon Triton, or an orbit that is closer to the planet than the geosynchronous orbit radius, as is the case for Mars’ moon Phobos. Triton and Phobos are thus doomed.

Let’s define two factors (given by Dobbs-Dixon et al. 2004, Ferraz-Mello, et al. 2008), $\hat{s}$ and $\hat{p}$, as the strength of the stellar and planetary tides, respectively,

$\displaystyle \hat{s}\equiv\frac{9}{4}\frac{k_0}{Q_0}\frac{m_1}{m_0}R_0^5,\qquad\hat{p}\equiv\frac{9}{2}\frac{k_1}{Q_1}\frac{m_0}{m_1}R_1^5$

Where $k_0$ and $k_1$ is the Love number of the primary and secondary, respectively, $Q_0$ and $Q_1$ is the tidal dissipation factor of the primary and secondary (a quantity that encapsulates how a body responds to tidal deformation, but it is very hard to measure, and usually poorly known), respectively, $m_0$ and $m_1$ as the masses of the two bodies, and $R$ as their radii. The Love number quantifies the mass concentration of the body and will not be expanded on significantly here. The separation between the two bodies will change with a rate of

$\displaystyle \dot{a}=-\frac{4}{3}na^{-4}\hat{s}\left[(1+23e^2)+7e^2(\hat{p}/2\hat{s})\right]$

Where $a$ is the semi-major axis of the orbit, $e$ is the eccentricity of the orbit, and $n$ is the

Secondly, the tidal deformation of a body will be resisted by inertia, causing its rotation to slow (and its synchronous orbit radius to expand outward). This will continue until either the body’s spin period mathes its orbital period (as is the case for the Moon), or is some integer ratio of the orbital period other than 1:1 (as is the case for Mercury), as is often the case for bodies in eccentric orbits.

For fluid bodies (such as gas giant planets or stars) in circular orbits, the end result of tidal spindown will result in a perfectly synchronous rotation, via a process called tidal synchronisation. In eccentric orbits, the planet moves around the star at varying speeds, making a synchronous rotation impossible. For such planets, the end result of tidal spindown is “pseudosynchronisation,” with a planet in a pseudosynchronous rotation such that the rotation rate of the planet matches the angular velocity around the star at periapsis (where the planet moves fastest), but with the rotation of the planet being faster than the angular velocity elsewhere. In rigid bodies (such as the terrestrial planets), tidal synchronisation is achieved in the form of a spin-orbit resonance, such that the long axis of the body is aligned with $r$ at each periapsis. It doesn’t need to be the same hemisphere of the body, as we see in the case of Mercury, which presents alternating hemispheres toward the sun at each periapsis.

Thirdly, tidal deformation of a body in a non-synchronous, eccentric orbit will cause the orbital eccentricity to tend to zero. This process, called “tidal circularisation,” is responsible for the circular orbits we see for the hot Jupiter population. As a body swings around periapsis, the tidal deformation intensifies and inertia resists this motion. In addition to causing heating to the object, it also saps some of its orbital energy, causing it to slow down. As this happens at periapsis more strongly than anywhere else, the net result is that the eccentricity of the orbit tends to zero.

The rate of the change in the eccentricity is given by

$\displaystyle \dot{e}=-\frac{2}{3}nea^{-5}\hat{s}\left[9+7(\hat{p}/2\hat{s})\right]$

Tidal deformations are a significant source of heating of a form that is collectively called “tidal heating.” An extreme example of it is seen at Io, where Jupiter’s other large moons perturb Io, pumping up its eccentricity, which is subsequently damped by tidal interaction with Jupiter, causing heating, and some rather spectacular volcanism.

Understanding the influences of tides on planetary systems is a vital component of understanding their evolution, formation and habitability. Tidal interactions between moons and planets can completely melt those moons, and tidal interaction between planets can tidally heat those planets and potentially completely melt them, too. Tides can also dominate the final rotational behaviour of a planet, with effects for its climate that are still being examined.