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# Resonance between Venus and Earth

This is a summary of Gold and Soter, Atmospheric tides and the resonant rotation of Venus, Icarus 11 356-366 (1969).

Tidal Deformation

Imagine we had a moon made of water or some other dense fluid. It's small enough that the center is fluid (rather than solidifying under the pressure), and let's ignore the fact that it would probably freeze on the side away from the Sun, there would be a cloud of evaporated haze around the whole thing, etc. This moon has its own gravity, fairly weak but enough to hold the thing together and keep a vaguely spherical shape.

The bits of fluid that are nearer the Earth would be in a "lower orbit", but are not moving as fast as they need to in order to keep in a circular orbit. Thus they tend to follow an elliptical path, and fall a bit closer to the Earth. Likewise, the bits of fluid on the far side of the moon are moving faster than they ought to, and move away a bit. The moon's own gravity prevents it from flying apart, but the equilibrium shape is not a perfect sphere, it's more like a Rugby ball (prolate spheroid) with the pointed ends facing towards and away from the Earth. This is tidal deformation.

Tidal Deformation of Plastic and Solid Bodies

For something (a moon or planet) that is rotating and solid (rather than fluid), it's not so simple. Solids can deform too, but they might deform more slowly or take a while to change shape. It's even messier if the planet is part liquid and part solid, as is the case for many planets and large moons in our solar system. In the case of the Earth, there is deformation caused by the Sun's tidal influence, just like what I just described, but it's only about 20 centimeters. (See http://en.wikipedia.org/wiki/Earth_tide and look for "solar semi-diurnal")

Geoid Irregularity

There's a much bigger irregularity in the shape of the Earth that results from the magma mass concentrations, continental plates, etc. There is a 140-meter difference in the diameter of the Earth measured through the equator, depending on where you measure it. In other words, it is more accurate to treat the Earth's equator as an ellipse rather than a circle, and the shape of the Earth as a triaxial ellipsoid rather than an oblate spheroid. The longer axis is roughly aligned with 0 and 180 degrees longitude; you can see the actual (much more elaborate) shape at http://en.wikipedia.org/wiki/Geoid

Here is an article from 1985 about the shape of Venus and how it differs from the spheroid and triaxial ellipsoid: http://adsabs.harvard.edu/abs/1985BAICz..36..129B The shape of Venus is shown to be pretty irregular, with variations on a similar scale to those seen on Earth. Figure 1 is a slice through the equator; figures 2 through 6 are slices perpendicular to the equator at various longitudes.

Offset of Tidal Bulges by Planet's Rotation, and Momentum Transfer

For a solid planet that has no atmosphere and no moons, but is rotating, the tidal bulge will point not towards the Sun, but a bit in the direction of the planet's rotation. The angle will depend on the rotation rate and various physical properties of the planet's composition. Since the bulges are not in line with the Sun, and since the Sun pulls a little harder on the bulge that is on the sunward side, there is a torque, and this torque acts to slow the rotation down (over a very long, i.e. geologic, timescale). This torque acts on the diurnal bulges, which are small in amplitude compared to the irregular shape of the planet — but because the diurnal bulges are always pointed in the same direction compared to the Sun, the torque is constantly acting in the same direction. The torque on irregular shape of the planet goes both ways, canceling itself out every rotation. However, once the smaller always-slowing torque slows it down far enough, the irregular shape of the planet will determine what side ultimately ends up facing the Sun.

This is the situation for our Moon, which was slowed down long ago by the Earth's torque on the Moon's tidal bulges. It is also the case for the planet Mercury, but Mercury did not slow down all the way. It rotates 3 times for each two times it orbits the Sun (not 1:1) apparently because this resonance is more compatible with its eccentric orbit, which at times gets to be as high as an eccentricity of 0.45.

Atmospheric Tides

For a planet that has an atmosphere, there is a tidal deformation of the atmosphere due to gravity, and that causes a slowing torque like what was already described. However, there is an even greater thermal deformation. Due to heating of the dayside and the resulting winds, the atmosphere bulges at different points. The Sun pulls on these bulges too, causing extra wind that wouldn't be there otherwise, and this wind ultimately pushes on the solid parts of the Earth. In the Earth's case, the biggest bulges point towards and away from the Sun, but are a bit off-center, in the opposite direction from the direction of rotation — that is, oriented in such a way that the torque exerted on these bulges tends to speed up the Earth's rotation! The tidal bulges are directly observable through barometric readings: there is a twice-daily oscillation of atmospheric pressure, of magnitude 1.16 millibars, with maxima occurring just before 10 AM and just before 10 PM local time, observable in any place near the equator.

Given what we hear most of the time about tides, and the Earth-Moon mutually slowing each other down, etc. any proposed "tidal effect" that would cause a rotating body to speed up would seem to be impossible, like a perpetual motion machine. So it's important to note that, if there is an effect causing the the planet's rotation to speed up, it will only speed up to the point where the tidal bulges point at right angles to the direction of the Sun. At that point the torque is zero, and there is no more accelerating effect.

The Earth's Situation

The relative strength of the Sun's torque on these atmospheric bulges, as compared to the torque on the tidal bulge in the solid mass of the planet, depends on how fast the planet is turning (thus, position of the bulges) and on the relative density of its atmosphere, the atmosphere's fluid properties, etc. In the case of the Earth, they pretty much cancel each other out. This was first shown by Lord Kelvin. If that were all, the Earth's rotation rate would pretty much remain constant. However, because the Earth has a moon, and the Moon does not cause a thermal bulge in the Earth's atmosphere, the Moon influences the Earth only through solid tides and the ocean tides, in both cases causing a bulge that lags behind the Moon's position and thus generating a coupling (torque) that causes the Earth to slow down. This is why scientists predict that (if the solar system were to last long enough) the Earth and Moon would eventually end up as a tidally locked double planet (like Pluto and Charon, or many double asteroids) where they both rotate at the same rate as the Moon's orbital period. This will in fact not happen, because it would take more than 5 billion years for the Earth and Moon to become locked and the Sun will turn into a red giant (destroying the Earth and Moon) before then.

Atmospheric Tides on Venus

In the case of Venus, there is no moon, so it's just the first two effects, the Sun's torque on the deformed solid mass of Venus, and the Sun's torque on the atmospheric tidal bulges. This is what the Gold-Soter paper looks at: they show that these two effects cancel out if Venus is turning slowly enough and if the atmosphere is heavy enough, etc. They are not able to prove it because some of the needed measurements were not available — but they work through all the math.

There are some cool diagrams showing how strong the Sun's slowing influence is (the horizontal T_q line in figures 3 and 5) as compared to the Earth's influence during inferior conjunction events. The Earth's torque on Venus is shown by the vertical spikes that get stronger as Venus' rotation approaches the "synchronous" point (at which it points the same side towards the Sun all the time). Note the numbers on the bottom: 225 is the Venus year in Earth days, 365 is Earth year; -243.16 is the proposed resonance close to Venus' rotation period (243 days, retrograde); 584 is the number of days between Earth-Venus conjunctions.

In the first chart (figure 3) there is no atmospheric tidal effect. Venus gets slowed down by the Sun until it has synchronous rotation — just like the Moon or Mercury.

In the second chart (figure 5), there is an atmospheric tide caused by the Sun heating Venus' atmosphere. This causes a torque that counteracts the solid tide torque. They hypothesize that the relative strengths of the two tidal effects could be similar to that with Earth, with the critical value (B-A)/C being about 2.2e-5. This allows the Earth's tidal influence on Venus to play a role, strong enough to lock Venus into a resonance with Earth (indicated by the "capture" label).

Conclusions

Gold and Soter conclude that if Venus has thermally driven (i.e. heated by the Sun) atmospheric tides with a large enough semidiurnal component (twice per day, i.e. two bulges spaced 180 degrees apart) and if the bulges have a "phase lead" (i.e are located somewhat prior to noon and midnight local time, as on the Earth), then the torque of the Sun on the atmosphere tide can counteract the torque of the Sun on the solid-body tide, and with these forces thus balanced, the extremely small torque of the Earth on the irregularly-shaped Venus would be enough to lock it into the observed resonance, such that Venus always shows the same side to Earth at each inferior conjunction.

They do not suggest experiments to demonstrate this, but an obvious line of investigation would be to try to measure the variation in barometric pressure throughout the Venus day, i.e. trying to locate the atmosphere bulges to see if they are on the leading or trailing side of the subsolar and anti-solar points. This is hard to do at the surface of Venus, but might be easier to deduce from the high-altitude atmospheric tides, which are of much greater amplitude (see the http://en.wikipedia.org/wiki/Atmospheric_tide article and search for "exponentially"). We've sent enough spacecraft to Venus that this should be known by now, but I couldn't find anything.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2012 Jun 07. s.27