02 July 2014

Mercury's Orbit

Mercury has a unique orbit around the Sun.  It was long believed that Mercury had a 1:1 resonance with the Sun, much like the Moon has a 1:1 resonance with Earth, i.e. the same face is always facing the Sun.  However it was discovered that Mercury has a 3:2 resonance, i.e. for every three complete rotations, Mercury completes two full orbits around the Sun.  This resonance, its proximity to the Sun, and its highly eccentric orbit leads to some strange phenomena.

Mercury, by far, has the highest eccentricity of all the planets (e=0.206). What does this mean?  An orbit that is eccentric means that the orbit is elliptical, or oval. The closer the eccentricity of an orbit is to zero, the more circular the orbit is. For reference, the orbital eccentricity of the Earth is 0.017.

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Eccentricity leads us to mention Johannes Kepler's three laws of planetary motion, specifically Kepler's Second Law:
  • A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
This is just a fancy way of saying the closer the planet is to the Sun, the faster it goes.
 
Isaac Newton later proved this with physics and calculus in his three laws of motion,
 
We also know that Mercury is 0.4 AU, on average, from the Sun. Because it is so close, it is by far the fastest moving planet in terms of orbital speed.  And from its highly eccentric orbit, when Mercury is at perihelion, it is moving its fastest.

Before we move on to the funky stuff, let's discuss more about Mercury's sidereal rotation and sidereal period, or its day and its year, A sidereal day is how long it takes a planet to rotate once on its axis completely, or how long it takes a star to appear at the same spot in the sky.  For Earth, this is about 23 hours and 56 minutes.  On Mercury, it takes about 58.5 Earth days.  A sidereal year is how long it takes to complete one complete orbit around the Sun.  Earth takes about 365.25 days to complete one sidereal year. Mercury has a sidereal year of approximately 88 days.  So for every three sidereal days on Mercury, Mercury completes two full orbits, hence 3:2 resonance.

But because of its unique position in the solar system, funny things happen on Mercury. As Mercury approaches perihelion, its orbital angular velocity increases.  At around four days before perihelion, the orbital angular velocity equals the rotational angular velocity. When these two angular velocities are equal, the Sun appears stationary in the Mercury sky.  As Mercury gets closer to perihelion, the angular orbital velocity increases and becomes larger than the rotational velocity. The Sun then appears to move backwards in the sky!  In some cases, it is possible that the Sun could actual set in the east.  At perihelion, the orbital angular velocity begins to decrease. Four days after reaching perihelion, the orbital rotational velocity once again equals the rotational orbital velocity. The Sun again appears stationary then again begins its apparent movement to the west.

Another interesting fact about Mercury's orbit, is that the time from true noon to true noon, what we call a solar day, is larger than a Mercury year. One Earth, our solar day is 24 hours. On Mercury, it is 176 Earth days, or two Mercury years.  Mercury has one of the slowest rotational angular velocities in the solar system, comparable to that of Venus, which is a topic for later.

Mercury rotation characteristic - notice the ellipitical orbit
 
In conclusion, because of its proximity to the Sun, its highly eccentric orbit, and its tidally locked 3:2 resonance with the Sun, strange things happen in the sky of Mercury.
 

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