Calculating the mass of a planet?

Question

Let's say a planet is orbiting an unknown mass. If we know the amount of time it takes for the planet to orbit, the distance it is from the mass, and the mass of the planet, how can we calculate how many times the mass' mass is the planet?

Could you also please tell me how to calculate the gravitational force between the planet and the mass? I've forgotten the formula/process.

Answer 1

The problem is a little bit tricky. The issue is not finding the mass of the unknown object; the problem is finding the mass of the planet which you are sitting on!

Let's say:
- Mass of the Unknown Mass = m1
- Mass of the planet = m2

1) Force of gravity = G*m1*m2/R^2

2) Centripetal force of planet = m2*w^2 * R
= m2*(2piR/T)^2 /R
= 4*pi^2 * m2 *R/T^2

3) Since these forces are equal,
G*m1*m2/R^2 = 4*m2*pi^2 * R/T^2
G*m1 = 4*pi^2*R^3/T^2

m1 = [(4*pi^2)/G] (R^3/T^2)

So what happened? We can find the mass of the "unknown" mass, but we can't find the mass of our own planet. It's cancelled out of the equation! So how can we find the ratio of the two, m2/m1 ?

Well, if you already know m2, then just divide it by m1, gotten from the formula. But what if you don't know m2?

There's still a way. The problem is that we pretended, in equation 2), that only m2 was orbiting m1. That is equivalent to assuming that m2 is infinitesimal compared to m1. But if it's infinitesimal, we can't measure it. So we cheated ourselves. But we can do this better: We recognize that since both m1 and m2 are massive, both are moving. In fact, both are moving in circles, with the same period T, but with different radii. The heavy m1 will be moving in a tiny circle of radius R1, and the lighter planet will be moving in a big circle of radius R2. The R in equation 1) is R = R1 + R2. So the right equations are:
1') F-grav = G*m1*m2/R^2 (unchanged)
2')
a) F-c = m1*(2*pi*R1/T)^2/R1
b) F-c = m2*(2*pi*R2/T)^2/R2

3')
a) G*m1*m2/R^2 = F-grav = F-c = m1*(4*pi^2*R1/T^2)
= 4*pi^2*m1*R1/T^2
m2 = R1*(4*pi^2 * R^2)/(G*T^2)
b) and likewise:
m1 = R2*(4*pi^2 * R^2)/(G*T^2)

So m2/m1 = R1/R2 = (R - R2)/R2

where R is the total distance between the centers of the planet and mass, and R2 is the radius of the planet's orbit (which is NOT centered at the center of the mass, but somewhere further out).

So, we were told that we know R, but how do we find R2? Well, you have to measure it. How? One way: measure the Doppler shift for a spectral line from a distant star. It will change over time because sometimes the planet is moving towards and sometimes away from the star. The maximum difference between the speeds is 2 * orbital speed
= 2 * (2*pi*R2/T), so that allows us to measure
R2 = (Speed-difference*T)/(4*pi)

and that allows us to calculate
m2/m1 = (R - R2)/R2

So we can do it; but we need more information than is stated in the problem directly. Either we have to know the mass of the planet m2, or we have to do a measurement to get R2, such as a Doppler-shift measurement.

Answer 2

all you need to do is use this formula
T^2 = 4.(pi)^2.D^3 /G.M
where T is the time of orbit ,D is the distance between the 2 center of gravities G is newtons gravitational constant (=6.7 x 10^-11) and M is the unknown mass (the mass being orbited)
then the gravitational force is equal to = G.M.m/D^2
same notation as above and m is the mass of the planet

the first equation is just a combination of newtons law of gravitation and angular force

Answer 3

The centripedal and gravitational forces must balance for the orbit to occur.

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