One of my photographs of the moon, taken through a telescope. The darkening at the bottom of the image is the edge of the telescope's field of view |

Back to the two key equations that we'll need, which are the force on a body moving in a circular path:

where

And Newton's Law of Gravity

Equating the two, and rearranging to find r, gives us

This is the same equation used for geostationary satellites, and describes the basic relationship between the distance between two bodies (a planet and a moon, for example, or a star and a planet). This gives it great power as it can be used in many different situations.

Turning to the current situation, then:

G is the universal gravitational constant, 6.67300 × 10

^{-11}m^{3}kg^{-1}s^{-2}^{}M is the mass of the Earth, 5.9742 × 10

^{24}kg

T is the time to complete one orbit, which for the Moon is 27.32166 days, which is 2,360,591 seconds.

Plugging the numbers into the formula above gives the distance as 383,201 km

However, this is not the distance to the Moon from the Earth's surface. Newton's law of gravity gives the distance between the centres of gravity of the two bodies. I'm ignoring the radius of the Moon (which is perhaps an oversight on my part, you decide) but we must subtract the radius of the Earth from this value, to give the orbital height. Radius of Earth = 6378.1 km, so the distance to the Moon is calculated as = 376,823 km, or, if you prefer, (at 1.61 km to the mile), 234,147 miles.

Previously, I've learned that the Moon is about a quarter of a million miles away, so I'm glad the method I've used shows a figure which is 'about right' without any checking. Looking at other sources, it looks like my figures are close enough, considering the assumptions I've made. One key assumption I've made is to suggest that the moon travels in a circular orbit, and it doesn't. It has an elliptical orbit, which means the distance from Earth to Moon changes during the orbit - so I've calculated an average distance. Still, my figure is pretty close, and not a million miles away (and next time somebody reliably informs you that their opinion is not a million miles away, you can tell them that not even the Moon is that far away).

## No comments:

## Post a Comment