1. Rotation Rate of Mercury Lab 9

2. Mercury Closest planet to Sun, ~ 0.4 AU Very small, even Ganymede is larger Very eccentric orbit ~0.308 - 0.467 AU Sidereal rotational period = 58.7 days (rotation is the length of time for an object to spin once on its axis ) Mercury has rotation of three times every two orbits Sidereal year = 88 days 1:1 resonance not possible because orbit very eccentric

3. Mercury’s rotation It takes Mercury about 59 Earth days to spin once on its axis (the rotation period), and about 88 Earth days to complete one orbit about the Sun However, the length of the day on Mercury (sunrise to sunrise) is 176 Earth days

4. 2:3 resonance A point initially pointing toward the Sun will point in the same direction after one rotation (59 days or 2/3 of the orbital period), but that point will no longer be directed toward the Sun It takes three rotations of the planet during two orbits of the planet about the Sun, or 88 x 2=176 days, for the mark to get back to the same position.

5. Mercury spins on its axis every 59 days But the length of a day on Mercury is about three times this

6. In two revolutions of Mercury around the Sun, the planet rotates three times on its axis

7. Explanation Rotation period of Mercury is 59 days, which is exactly two-thirds of the planet's orbital period Because there are exactly three rotations for every two revolutions, we say that there is a 3:2 spin-orbit resonance in Mercury's motion Resonance just means that two characteristic times— here Mercury's day and year—are related to each other in a simple way A simpler example of a spin-orbit resonance is the Moon's orbit around Earth This rotation is synchronous with the revolution, so the resonance is said to be 1:1

8. Explanation of Mercury’s Rotation The 3:2 spin-orbit resonances didn’t occur by chance. Tidal forces due to the Sun’s gravity are responsible in a very subtle way. Tidal forces try to synchronize the rotation rate with the instantaneous orbital speed. But tidal forces decrease with distance, so the perihelion distance won out. At perihelion the rotation rate and orbital speed is the same, but not so at other points, so we end up with this 3:2 spin-orbit resonance.

9. Doppler Shift object that is moving away from you has a longer wavelength than it had when it was emitted - a redshift object that is moving towards you has a shorter wavelength than it had when it was emitted - a blueshift

10. 2 kinds of velocity 2 motions of Mercury produce Doppler shift –Orbital velocity –Rotation on its axis Edge of planet rotating towards us has an orbital velocity faster than the rest of the planet So echo of pulse has a higher frequency

11. Calculate! Edge of planet rotating away from us has an orbital velocity slower than the rest of the planet So echo of pulse has a lower frequency Difference in echoes can be calculated to give rotational velocity of surface of Mercury From that we can calculate period of rotation

12. Detailed handbook http://www.phys.unt.edu/courses/Astrono my/mercury.pdf