Satellite Mass: Newsletter Challenge (02/01/11)

Dear members, with assumtions we have ....

For a small satelite to stay in orbit at 2000 km it has to move at 4427 m/s. However it moves at 6500 km/s. Therefore 2073 km faster than it should. So it moves faster because it is heavy and the attraction force between two objects come into play. The resultant angular velocity * mass of the satelite must therefore = F = mass satelite * mass earth * G / (r*r) Solving for r we get 13533.34 km This r is between the centers of the 2 objects. Taking into account the earths radius of 6371 km and assuming the 2000 km distance between their surfaces then the radius of the satelite must be 5162 km .Assuming it is made up of roughly the same stuff as earth and knowing that the mass is directly proportional to the cube of the radius and having earths mass as 6x10^24 kg we have the mass of our satelite = 0.53*mass earth = 3.1382x10^24kg

Mass is irrelevant.

A.W.

Mass is relevant for barycentric orbit.

Mass is relevantly insignificant for geocentric orbit

The orbit is fixed by the speed and the level about the surface of Earth, ie depends not on the mass.

edouard_robinet@web.de

It seems the mass is irrelevant as long as the pull of Earth's gravity at 2000 km altitude equals the centrifugal force e.g. g (@ 2000 km) equals v**2/r

Hi Billy,

It is relevant, as the "pull of Earth's gravity at 2000 km altitude" depends on both masses and distances from the barycenter (Gm_em_s/(r_e+r_s)²), while the satellite's centrifugal force around the barycenter depends only its own mass and distance (m_sv_s²/r_s).

The values given in the question do not correlate with a negligible satellite mass, but with some 7.52*10²³ kg, or 12.5% of Earth's mass.

-J

Thanks for the equation and the lesson. I posted without doing the detailed math first and oversimplified the problem (8366/6366 = roughly 3/4, 3/4² is a little more than 1/2, and (6500 m/s)²/8366000 m is a little more than 1/2 of 9/8 m/s². Later after looking at the velocity again I realized that 6.5 km/s looked a little slow for orbital velocity at that altitude. I also originally assumed that the satellite was man-made and its mass was insignificant compared to the earth. (A faulty assumption).

This reminds me of what one of my professors told me about 30 years ago: When we ASSUME we make an ..... ;)

Satellite mass is 0 kg

what with a virtuell coordinate system with the zero-points in the center of the satellit rotating around the center of the earth/the barycenter of the earth-satellite system?

The 'official' answer is as bad as the second equation given.

It should be ΣF=ma=(mv^2)/r, otherwise r also cancels out.

It is pretty clear that this question's originator did not do any sums, otherwise he/she would have noticed that no satellite can geocentrically orbit at 2000 km above Earth at 6.5 km/s orbital speed. This requires a mix of coordinates systems plus a fairly heavy satellite. See #89 above.

-J

Agreed Jorrie

The ma=(mv^2)/r^2 is probably just a typo, but you'd think a Challenge setter would be little more careful!

As you say, the 'official' answer is poor. If "you can't determine the mass as it cancels" is all that is expected, there's no need to specify both the height and the speed. Just need to say the satellite mass is << Earth mass, and either height = 2000km or speed = 6.5km/s. Giving both figures, which don't agree for a low mass satellite, has given rise to the discussion.

Also the answer appears to be just as written when the Challenge was set, with no comments on the points raised in the thread.

Cheers.......Codey

That's the way it works Codie.

You lodge the question AND your answer - then the Q is posted, then your answer - on the appointed date.

One obviously has to be a good question composer - and have the right answer.

Neither I would class as "easy"

Or 'try not to be too cruel' on those who give it a go, if you ain't done it.

OK, I didn't think I was being cruel exactly, but fair point, I might take you up on that sometime.

But presumably there's no reason the setter can't follow the thread and perhaps add a few comments to the answer, if necessary.

Codey

They sometimes do and I'm sure 'Anonymous Poster' will be a great boon.

But clearly the setter here/there was "throwing a rat into a pit of hungry pythons to see which would 'win'"

It's 'entertainment' and it's not bad at that.

I enjoyed it and I learned a great deal about who knows what.

Agreed. It was a poor question and answer. As you and others have pointed out, the mass would have to be huge. I bet the tidal forces from it would be pretty destructive here on earth.