The Mercury Waltz

Please enjoy my delightful Mercury Waltz, a new song for the Orbital Music Box based on the planet Mercury’s 3:2 spin-orbit resonance. Songs for the Orbital Music Box are melodies that reflect the dance of moons and planets in their orbits. Mercury has a 3:2 spin-orbit resonance, meaning that it rotates on its axis 3 times for every 2 times that it orbits the sun. The whole pattern equals the length of 1 solar day – the apparent cycle of the the sun rising and setting as if standing on the body surface. In each measure of 3/4 time, one note plays on beats 1 2 3 to represent the rotations at 1/3 intervals. Another note plays on the 1st and 4th eighth notes, representing the 1/2 intervals of years. A description of the importance of these ratios is given below.

Mercury has a 3:2 spin-orbit resonance, which means that it rotates three times for every two revolutions it makes around the Sun. The way this appears on the surface of the planet is that the day is very long– it’s solar day length is 176 days, which is longer than its year! The properties of its orbit compared to Earth are given in the table below.

Orbital Period (time to orbit the Sun)365 days88 Earth days
Rotational Period (time to rotate on its axis)24 hours59 Earth days
Solar Day Length (time it appears for the sun to rise and set)24 hours176 Earth days

It’s easiest to understand Mercury with a diagram. Here we show how the planet rotates as it revolves around the Sun. [Positions 1 and 7 are separated visually but are meant to overlap in space; Figure from Planetary Sciences, de Pater & Lissauer]. At Position 1, the arrow is pointing at the Sun. If you were standing on the surface, it would appear to be noon where the Sun is directly overhead. After one Mercury year at Position 7, we see that the arrow is pointing opposite the Sun which would make it appear like midnight. This means in needs to go around the Sun a second time in order for the arrow to return to Position 1 and the Sun to appear at noon again. This cycle of noon at 1, midnight at 7, returning to noon lasts for two complete orbits (each 89 days) for a total of 176 days.

That’s how long the day would feel!

More subtle to see is the Rotational Period. The arrow is pointing to the bottom of the page at Position 1. Follow the arrow as it moves in its orbit, and you will find that two thirds of the way through its year (at Position 5) it is in the downward pointing position again. This means it took 2/3rds of a Mercury year in order to complete one rotation. Another way of saying it would be that it made 1 1/2 rotations in the time it took to complete one orbit. Therefore, it would complete three rotations in two orbits. This is why it’s called a 3:2 orbital resonance.

Here is a figure showing the amount of sunlight visible from the planet’s surface with the times at which new years and new rotations occur. The bottom axis shows the time in Earth days and the numbers along the top shows the time ratios normalized to 1. (This figure is general, the exact timing of when the rotations and new years occurred with respect to the apparent time of day would depend on where you were standing on the surface.) Therefore, the song created for Mercury is in 3/4 time, with specific notes on the third and half points of each measure to compose a delightful waltz. please enjoy!

MATLAB Handle Graphics

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