Rotation Rate of Mercury

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/Astronomy/mercury.pdf