Moon's Orbital Radius Increasing

In an unexpected and unprecented move, the European Space Agency announced this morning that their data shows that the Moon has started to increase its mean orbital radius at a rate considerably higher than previously believed.  It's been known for some time that the distance from the Earth to the Moon is increasing, but the rate of increase is worrying.  In other words, the Moon is moving further away from the Earth faster than we thought, and in a few year's time will leave the Earth's orbit completely.  

This finding comes after a series of measurements of the Earth-Moon distance (carried out using a more accurate process than this one), using a laser beam pointed at a network of mirrors which Neil Armstrong placed on the Moon's surface over 40 years ago, during the Apollo 11 mission.  By measuring how long it takes for a beam of light to travel to the Moon and back, ESA scientists have been able to determine that the distance from the Earth to the moon has, over the last three years, increased by an average of 140 metres per year.  

However, more alarming is the fact that the rate of increase is also going up - the Moon is, on average, moving further away at a faster rate with time.  The increase over the last six months is about 1.4% more than the average increase over the previous six months.
 

Exact forecasts vary, but ESA scientists are all agreed that within 14 years the Moon's orbit will have extended so far that it will leave the Earth's gravitational field completely, and head off into space.  The team of scientists have proposed various reasons for the Moon's recent moves, and the most common suggestions are related to the recent tectonic activity on Earth - the tsunami of 2004, the volcano in Iceland during 2009-10, and possibly the recent earthquakes near New Zealand, an in particular Japan, which has left to a shortening of the length of the Earth's day.  The recent earthquakes have coincided with the moon coming towards a particularly close approach (perigee) and the theory proposes that this has caused the moon to increase its speed while making this close approach, which will lead to it reaching a larger distance at its furthest point 14 days later.

Other scientists have yet to confirm the team's findings, which have sparked considerable controversy in astronomical circles.  Teams in the southern hemisphere have carried out measurements into the exact time for the moon's orbit and have not noticed a significant change in this - either an increase or a decrease, and therefore have concluded that the moon's orbital radius has not changed.  Other teams are preparing to carry out their own measurements using Armstrong's mirrors, and will be sharing their results later next week.

Other astronomy related articles you may find interesting:

Calculating the height of geostationary satellites
Calculating the angle of elevation of geostationary satellites
The Earth-Moon distance is increasing (published 1 April 2011)
What are constellations?

Calculating (and explaining) escape velocity

Calculating the Earth-Moon distance

This post follows up my previous post on geostationary satellites.  Long before we were launching satellites (even non-geostationary ones), our natural satellite, the Moon, was orbiting the Earth.  As the moon goes around the earth, its phase (shape) changes, and in fact, the word "month" derives from "moonth", the time taken for the moon to go from new to full to new again.  This time is the time taken for one complete orbit around the Earth - the different phases of the moon are a result of us seeing a different amount of the lit half of the moon (I once based a very neat science lesson on this principle - in fact I used it in my interview lesson  and subsequently got the job).

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

We can use physics, and our knowledge of the mass of the Earth, the value of pi and the time the Moon takes to complete one orbit, to work out how far it is from the Moon to the Earth.

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:

Calculation of the Earth-Moon distance:

G is the universal gravitational constant, 6.67300 × 10^-11 m³ kg^-1 s^-2

M is the mass of  the Earth, 5.9742 × 10²⁴ 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).

Other astronomy related articles you may find interesting:

Calculating the height of geostationary satellites
Calculating the angle of elevation of geostationary satellites
The Earth-Moon distance is increasing (published 1 April 2011)
What are constellations?

Calculating (and explaining) escape velocity
Measuring g using a pendulum