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Galaxian
2012-Oct-25, 07:39 AM
A student doubting his professors decided to estimate the mass of the earth by himself,astronomically.
He has following data:
(a) Radius of earth = 6400 km.
(b) speed of light.
(c) true orbital period of Moon = 27.3 days.
(d) signals sent by Apollo astronauts reached earth in 1.3 sec.

Help him estimate the mass of the Earth.

I don't know the answer. Help me .

Jens
2012-Oct-25, 08:31 AM
I don't know the answer either, but from the information, you can find the distance from the earth to the moon, and its orbital period.

Van Rijn
2012-Oct-25, 09:18 AM
Is this a class question? A hint: Can you think of a relationship between mass and orbits?

Galaxian
2012-Oct-25, 09:23 AM
This is not a class question.

Hornblower
2012-Oct-25, 10:23 AM
In addition to what is given so far, he would need to know the value of the gravitational constant, or a means of determining it, such as the Cavendish experiment.

What does he doubt, and why?

Jeff Root
2012-Oct-25, 12:34 PM
Assuming that what the student doubts is the mass of
the Earth, then the most important information missing
from the question is Newton's law of gravity. If it is
instead Newton's law of gravity that the student doubts,
then it will be considerably more difficult to estimate
the Earth's mass.

Newton's law of gravity needs the value of the gravitational
constant which Hornblower pointed out, the orbital period
of the Moon, as given, and the distance between Earth and
Moon, which of course can be calculated from the speed of
light (which I suppose you can look up or have already
memorized) and the given time for signals to travel that
distance.

The only way I see the radius of the Earth being used is to
add to the distance you calculate from the signal travel time,
to get the distance between the center of the Earth and the
near surface of the Moon. The Moon's orbit is sufficiently
eccentric, though, that Earth's radius doesn't matter much,
in comparison. But you wouldn't know that if you had no
idea what Earth's radius is, so it is still important.

In fact, I suspect that the given figure for the signal travel
time is not an actual measurement between Earth's surface
and the Moon's surface, but a rough figure calculated from
the mean center-to-center distance. I get 1.28 seconds
travel time for that distance.

Another bit of information you need in order to apply
Newton's law of gravity is that the Moon's mass is much
less than that of the Earth. If you can also estimate the
mass of the Moon you can use it to get a more accurate
and precise value for the mass of the Earth.

-- Jeff, in Minneapolis

NEOWatcher
2012-Oct-25, 12:50 PM
The only way I see the radius of the Earth being
used is to add to the distance you calculate from the signal
travel time, to get the distance between the center of the
Earth and the near surface of the Moon.
If you only have the surface gravity, then you need it to compute the gravitational gradient. No?

ngc3314
2012-Oct-25, 12:59 PM
Either the surface gravity and radius of the Earth, or the Moon's orbital period and radius, will get you there via Kepler's third law (or at least Newton's form including the constants).