Calculating the moon's mass from Earth distance and diameter

[wasi-uz-zaman]

hi, i can calculate the distance of moon from the Earth and its diameter - but i could't get the formula by which i can calculate the mass of the moon - although i can calculate the mass of the moon by assuming it has same average density that of Earth -

 

[mfb]

The orbital period of the moon depends on its distance and the sum of masses of Earth and moon - if you can determine this sum with sufficient precision, you can subtract the mass of Earth and get the mass of moon.
Satellites orbiting the moon are a better way to determine its mass - they give direct access to the gravitational acceleration at a specific distance, together with the gravitational constant this can be used to calculate its mass.

 

[wasi-uz-zaman]

but how do i calculate the sum of Earth and moon mass?

 

[mfb]

Measure distance, orbital period and the gravitational constant.

$M+m=\frac{4\pi^2a^3}{GT^2}$ with the semi-major axis a (for a circular orbit, this would be the distance)

 

[BobG]

I'm curious as to how you calculated the distance between the Moon and Earth. That distance might not be the semi-major axis of the Moon's orbit. It might be the sum of the Moon's semi-major axis and the Earth's semi-major axis, as measured from their combined center of mass. (The 'a' in the previous equation is actually the sum of the semi-major axes, or the distance you most likely calculated.)

In practice, calculating the mass and the semi-major axis of planets was an almost impossible task even after Newton turned Kepler's Third Law into a formula. You had a formula containing three unknown variables (the universal gravitational constant, the mass, and the semi-major axis) and the only known was the orbital period.

In fact, that's why the Earth's semi-major axis for it's orbit around the Sun was measured in astronomical units, with one AU being the distance between the Sun and the Earth. You could measure Jupiter's semi-major axis in AU's, but had no way to convert that into a more traditional measure such as kilometers.