# Earth orbit

I asked this question to Cecil about a year ago, haven’t gotten a reply so i figured i would ask again on the message boards.

I read in a column of the Straight Dope that the Earth gains anywhere from 10-100tons per day.

If you go and add that up over the course of a year, decade, or century it is an unbelieveable number, so i have to think…what is this doing to Earth’s rotation and orbit?

Is it pulling us farther from the sun? Moving us closer? Slowing us down? Speeding us up?

Possibly nothing, if the sun is gaining mass at a proportional rate.

The latest info:

That’s 5,972,000,000,000,000,000,000 tons.

So at about 100 tons a day that’s essentially a 0% percent gain even in a billion years.

The orbit of a satellite depends only on the mass of the primary, the distance between primary and secondary and the gravitational constant.

The gravitational force is: f = Gm[sub]p[/sub]m[sub]s[/sub]/d[sup]2[/sup]
where G = gravitational constant, m[sub]p[/sub] = mass of primary, m[sub]s[/sub] = mass of satellite, d = distance between centers of mass.

The acceleration of the secondary is a = f/m[sub]s[/sub]

the mass of the satellite cancels and the resulting acceleration of the satellite toward the primary is:

a = Gm[sub]p[/sub]/d[sup]2[/sup]

This is true for the case where the satellite is much less massive than the primary. Actually the earth and the sun orbit around their common center of mass. For a given weight of the earth this common center is a certain distance from the center of the sun. As the earth gains mass, if the sun doesn’t, the common center of mass shifts further from the center of the sun and this distance of the earth to the center gets shorter.

To carry this further requires going into the angular momentum of the system which I’m not up to at the moment. However if either or both bodies gain mass I would expect the rotation to gradually decrease in order to keep the angular momentum constant.

Just to show the math a little more:
(100 tons per day) times (365.25 days per year) times (1,000,000,000 years) equals: 36,525,000,000,000 tons, which when added to the earth’s mass gives a total of 5,972,000,036,525,000,000,000 tons. That’s a change of less than one one millionth of a percent over a billion years. So, whatever effect it has is pretty minor.

Actually, the Sun is losing mass, mostly via the solar wind but also the mass equivalent of the energy it’s radiating into space. Some mass does fall into the sun (the stray comet, for example) but I believe the net effect is a loss.

What this means is that the Earth is very slowly moving away from the Sun. Which is desirable, since the Sun is gradually heating up. Not that this is going to make much difference in the long run – the Sun will eventually heat up enough to make the Earth uninhabitable and the slow recession of the Earth won’t delay this for any significant amount of time (maybe a few decades, at a guess).

One more time, the earth’s orbit is unaffected by the mass change because that is determined only the mass of the sun. The earth’s rotation would gradually slow in order to maintain the same angular momentum. However that effect would be hidden by the slowing of the rotation as a result of the lunar tidal bulge.