Rotation keeps in step
(appeared on 9th Oct 2013)

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A near satellite is not free to spin as it wishes while it goes round the mother planet, says S.Ananthanarayanan.

A day on the moon lasts as long as a lunar month. This is because the moon spins round only once as it goes round the earth. But the planet, Mercury spins round thrice in the time it takes to go twice round the sun. Near satellites tend to slow their speed of rotation to match their period of orbit. But the way Mercury has settled in seems to be a little different

Dr. Benoit Noyelles of the University of Namur, Belgium, in collaboration with Drs. Julien Frouard of the University of São Paulo, Rio Claro, Brazil, and Valeri Makarov and Michael have presented before the American Astronomical Society their model of how this end-state of the rotation-orbit relationship came about, where they draw inferences of the early state of the interior of the planet.

Tidal forces

When a satellite goes around a parent body, the latter exerts greater gravitational force on the nearer side of the satellite than on the farther side. The gravitational stress causes a bulge, albeit small, along the line connecting the two objects. But as the satellite is spinning, and deformation of a satellite is slow to adjust, this bulge moves away, while the greater gravitational pull is at the part of the satellite that now comes in line. The parent body thus tries to draw out the part that is now nearest and also exerts a force on the bulge that is moving ahead, which tends to slow the rotation of the satellite.

The result, over millennia, is to slow down the rotation and finally keep the same part of the satellite always facing the parent body, with a permanent, though minute bulge in the shape of the satellite. This is the case with our own moon, which always shows the same face to the earth. But the moon does turn around, as it goes round the earth and over the length of a lunar month, all parts of the moon experience night and day, in respect of light from the sun. The time the moon takes to spin once around is hence the same as the time to go once round the earth.

This is also the case with satellites of other planets. It is not the case with the planets themselves, because the planets are at large distances away from the sun and the gravitational effects of the sun, to cause deformation of the planets is not appreciable. But we know that the effect is large with respect to the earth and the moon, as ocean tides are largely because of the moon. In fact, ocean tides should gradually slow down the speed of rotation of the earth, while the speed of orbit, and hence of rotation, of the moon should slightly increase, in reaction.

But in the case of the planet Mercury, which is the innermost planet of the solar system, the tidal effect of the sun in not small and negligible. The speed of rotation has thus been slowed down, over the ages, and compared to the 365 rotations of the earth for each orbit of the sun, Mercury goes round only one and a half times during each orbit. The question is, why one and a half times, why not only once?

Elliptical orbits

The answer lies in the shape of the orbit. All the planets of the solar system, except Mercury and Pluto, move in almost circular orbits. The orbits of Pluto and Mercury are like distinct ovals, Pluto’s is more so, but that of Mercury is also nearly the same, with the length of the oval one and a half times the breadth. But as Pluto is so far away from the Sun, there is no meaning of any tidal force of the sun on Pluto, in any case.

Mercury, in this elongated orbit, moves fast when it is near the sun, but slows down to a crawl when it is at a distance. When it is quickly moving past, when nearer the sun, it feels strong tidal forces and its speed of rotation during that time would be very nearly the same as the short time it takes to move around the sun. But once it has moved round the sun, it sets out on its long journey through the remaining part of the oval, and as it moves away, its speed of orbit reduces. But with reduced gravitational pull, the speed of spin, which was fixed while it was near the sun,

does not change, and the planet continues to turn at nearly the same speed, as it slowly proceeds along the orbit, and ends up turning round, or rotating, more than once. It would appear that it would take a specific composition of the material of the planet, for deformation to induce a single rotation for each orbit, in a case of a non-circular orbit.

The settling in of the speed of rotation to be the same as the speed of orbit is known as resonance, after the same word which indicates two musical notes of the same, or mutually multiple, pitch. In the case of Mercury, where the planet turns one and a half times round in an orbit, or three times in two orbits, so that the same side as before would be facing the sun after the second orbit, there is asymmetry and this is also resonance. Understanding how Mercury came to stop at this resonance and not go on to the state of one rotation for each orbit, which is a more stable condition, would help understand how molten and deformable the planet was, at different times after it first fell into the orbit around the sun.

This is the work that the group which has presented its results to the American Astronomical Society has done. Given the ovality of Mercury’s orbit, the fact that the planet has been trapped in the 3:2 spin-orbit resonance has come about because Mercury reached a not fully molten, which is a relatively cold, state quite early in its life and also that the segregation of matter, as a soled crust and a liquid core, happened after the resonance had been established.

The study also shows that such a 3:2 resonance may occur in exo-planets, or planetary systems of other suns, where the orbits are even more markedly oval. This insight would be useful in estimating how far such exo-planets may have conditions to support life. NASA’s current MESSENGER mission and the European/Japanese mission in the next decade, to study the surface of Mercury, would help refine the tidal models of Mercury and exo-planets

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