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June 2011: The Puli Number of the Month - How far do we have to go?

384,400. About four hundred thousand km (~240,000 mi) – is it near or far? Compared to the distances in our everyday life, its a lot but still, one has to go that far to reach our closest neighbor in the heavens. The average distance of the Moon from the Earth, it takes more than a second for a ray of light and at usually three days for a spacecraft to cover it. But how does the Moon orbit us exactly?

The Earth, the Moon and the distance between them, to scale.

Firstly, it obeys Kepler's laws of planetary motion: the orbital path is an ellipse around us, and it travels faster when closer and slower when farther. It usually comes as close as 362,570 km and goes out to 405,410 km – changing its actual distance by more than 10 per cent! But since we are not alone in the Solar System, these figures are subject to change, most prominently from the influence of the Sun.

One of the effects is the change in perigee and apogee distances (the closest and farthest points from the Earth). When extreme perigees and full Moons coincide, some use the therm “Supermoon”, just as it happened this March. So how extreme it was? Not so much, it turns out: with a perigee distance of 356,575 km, it deviated by ~6000 km or 1.5% from the average. As I said, it wasn't really much.

The difference between two full Moons, one at perigee, the other at apogee.

The Sun has other effects as well: the orbit isn't really a fixed ellipse compared to the reference frame but it changes constantly. The semi-major axis, the line between the two farthest point of the ellipse is rotating, causing the perigee and apogee points to circle the Earth with a period of 8.9 years. The plane of the ellipse, or identically, the line where the ellipse intersects the orbital plane of the Earth (the node line), also rotates, with a period of 18.6 years. To make matters more complicated, the semi-major axis and the node line rotates in opposite directions! I'm sure that your heads are spinning too by now. Celestial mechanics and planetary navigation are hard!

It may be surprising but spacecrafts have to cover more than 384,400 km to reach the Moon. Unlike on Earth, straight lines are the rarest courses in space: crafts, moons, planets always orbit something. The simplest way to reach the Moon (or other bodies) is using a Hohmann transfer orbit: its an elongated (half-)ellipse around the Earth where the perigee is at the Earth (more precisely in low-Earth orbit) and the apogee is set to the distance of the desired object. Here, two maneuvers are required: one boosts the spacecraft to the transfer orbit, the second slows it down at the destination (otherwise it would swung back to the Earth on the transfer orbit). Another – at first sight counter-intuitive - fact is that the spacecraft is not going towards the Moon: it is flying towards a point where the two will meet. Like shooting at a moving target, because the Moon is orbiting the Earth of course!

A simple trans-lunar course. The red dot marks the Trans-Lunar Injection, where the spacecraft leaves low-Earth orbit with an engine burn. The transfer orbit is a bit shy of the distance of the Moon but the spacecraft would turn and fly by it because of its gravitational pull. A second burn is needed there to stay in Moon orbit.

The above fast transfer is simple and effective but it requires quite a lot of accelerating an breaking, or simply a lot of fuel. Fuel is heavy so you'd need a large rocket to launch it from the ground. But isn't there another way? There is indeed, it turns out! There are other ways, low-energy transfers that require much less fuel but much more time. These exploit various effects, the properties of the boundaries between the regions where the Earth's, Sun's and Moon's gravity dominates over the others. GRAIL for example will take such a route and will arrive at the Moon after 3.5 months of cruise. Its mileage will be a lot more than 384,400 km!


Image sources:

1.) Wikimedia Commons

2.) 3.) CC

Last Updated (Sunday, 26 June 2011 18:12)

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