Determining Mass from Orbital Period and Radius

[procon4]

Homework Statement

What is the mass of a planet (in kg and in percent of the mass of the sun), if:

its period is 3.09 days,
the radius of the circular orbit is 6.43E9 m,
and the orbital velocity is 151 km/s.

Homework Equations

I'm unsure what formulas to use, though these seem relevant.

F= ma

accel. centripetal = v^2/r

Total Energy = -G*(mass of planet)*(mass of sun)/2*radius

The Attempt at a Solution

I thought I should use Force of gravity = mass of the planet times the centripetal acceleration, but the mass of the planet cancels out. I can't ID the right relationship between the period, radius and mass, so I'm not sure what to do.

Thanks for any help.

JS

 

[blacknovember]

Consider using vis viva equation as applied to circular orbits

 

[procon4]

That's a really good suggestion--I'm surprised that equation isn't in our textbook. The problem is that the mass of the star around which the planet orbits is not given. So I guess there must be some relationship between period, orbital radius, and mass, but I'm not sure what it is.

I would have expected an energy-related equation could work, but I haven't found one that doesn't either require the star's mass or in which the mass of the planet doesn't cancel out.

 

[blacknovember]

The mass of the sun is a known quantity which you can lookup. You could derive vis viva from what the question gives you though...

edit:I don't think you even need the period TBH

 

[fcb]

Use Keplers law of period and the mass turns out to be 2.207610x10³⁰

 

[fcb]

110% mass of the sun.

 

[fcb]

You can also use orbital velocity and work it out from there.

 

[procon4]

So just to clarify the situation here, the star at the center of the planet's orbit is not the sun. But another problem was that I needed to find the mass of the star, not the planet. To do that, I just used the F=ma equation, with F being the force of gravity, m being the mass of the planet, and a =v^2/r. The mass of the planet cancels out and you're left with the mass of the star.

The answer fcb posted is correct. Thanks everyone.

This question was called "Hot Jupiter," from Mastering Physics Ch. 12.