Using kepler's laws to find mass of planet

[jhson114]

There's a planet with a moon that orbits the plant in 30 days and moves at a distance of 370million meters from the center of the planet. i need to find the mass of the planet and I've used the following equation that was derived from the keplers laws: P = 2pi(a^(3/2))/(GM)^(1/2). P is the time it takes to or it and A is the average distnace from the planet. I've tried using P in days and years, and a in meters and AU. But i can't seem to arrive at a correct answer. maybe my equation is wrong? thanks in advance.

 

[SpaceTiger]

There's a planet with a moon that orbits the plant in 30 days and moves at a distance of 370million meters from the center of the planet. i need to find the mass of the planet and I've used the following equation that was derived from the keplers laws: P = 2pi(a^(3/2))/(GM)^(1/2). P is the time it takes to or it and A is the average distnace from the planet. I've tried using P in days and years, and a in meters and AU. But i can't seem to arrive at a correct answer. maybe my equation is wrong? thanks in advance.

If P is in years, M is in solar masses, and a is in AU, it's simply:

$$P^2=\frac{a^3}{M_p}$$

For your version of the equation, you should use some other consistent set of units (like meters, kilograms, and seconds; or centimeters, grams, and seconds).

 

[OlderDan]

There's a planet with a moon that orbits the plant in 30 days and moves at a distance of 370million meters from the center of the planet. i need to find the mass of the planet and I've used the following equation that was derived from the keplers laws: P = 2pi(a^(3/2))/(GM)^(1/2). P is the time it takes to or it and A is the average distnace from the planet. I've tried using P in days and years, and a in meters and AU. But i can't seem to arrive at a correct answer. maybe my equation is wrong? thanks in advance.

Your equation looks right. Do you know the mass of the moon? The mass in this equation is really the reduced mass of the planet moon combination.

If the orbit is elliptical, a is the semimajor axis of the ellipse. Since you are only given one distance from the planet, I assume you are supposed to be treating the obit as circular with a = 370million meters.

Edt
If reduced mass is relevant, then the distance would be to the center of mass, not to the other object, so this was not a helpful idea. I see you got it worked out now with a simpler calculation.

 

[tony873004]

If you can assume the orbit is circular, and the planet's mass >> moon's mass, you could also re-write the formula for circular velocity:

velocity = sqrt(Gravitational Constant * Mass / radius)

into

M = v^2 r / G

G = gravitational constant = 6.673e-11
compute your velocity from your radius and your period (in seconds).

 

[jhson114]

i don't know the mass of the moon and we are to assume that its a circular orbit. the equation P^2 = a^3/M doesn't seem right because i get a 2.24e-6 as mass. with my previous equation is it right to AU and Years for A and P and M is KG?

 

[tony873004]

You got the right answer except for one thing. That way gives solar masses and not kilograms. The Sun is 1.98911e30 kg.

The method I gave you wants meters, seconds, and gives kilograms.

Try it both ways to see if you get approximately the same answer. You should.

 

[SpaceTiger]

i don't know the mass of the moon and we are to assume that its a circular orbit. the equation P^2 = a^3/M doesn't seem right because i get a 2.24e-6 as mass.

That's about the mass of the Earth in solar masses, so if you plug that in with the distance in AU, you'll get your period in years. If you use your equation, you need to put in, for example, the distance in meters and the mass in kilograms, you'll get out the period in seconds.

 

[SpaceTiger]

Sounds like you might need to brush up on your http://oakroadsystems.com/math/convert.htm .

 

[jhson114]

okay. i got it. i had all the units messed up. thanks guys :)