Mass of stars companion from radial velocity

[ChrisBaker8]

Homework Statement

The solar-like star HD209458 with a mass of 1.14 solar masses exhibits radial velocity variations with a period of 3.52 days and an amplitude of 84m/s. What is the mass of its companion and what type of object is it?

Homework Equations

M/m = r/R = v/V

[M,R,V = star, m,r,v = companion]

The Attempt at a Solution

I've just worked this out myself so if it's off please tell me, but I have to assume the star and planet (or body) are both moving in circular orbits, and the are on opposite sides of the centre of mass, and rotating around it in their orbits (stars one much smaller) with equal periods, as shown in this clip:

http://en.wikipedia.org/wiki/File:Planet_reflex_200.gif

from Wikipedia (Radial Velocity).

Now, I'm assuming the data given means that (in the above animation, assuming the viewer is to the left of the centre of mass at infinity) the star will be moving 42m/s faster towards the observer (or 42m/s slower away from the observer, to be more accurate) at the bottom of the orbit, and 42m/s faster away from the observer at the top of the orbit.

The orbit of both takes 3.52 days. So if I know the star is traveling (forgetting the whole system is moving) at 42m/s around the circle, and I know the period, I can calculate the circumference and therefore radius of the stars orbit. But from there I don't know. The companion could be close and large, or far away and small. I'm not sure how I can work this out.

 

[mgb_phys]

Sounds reasonable, I'm guessing you have to assume that it's not an eclipsing binary and you have to that it's orbiting it's compact companion.

 

[ChrisBaker8]

I'm not sure what that means. I still have no way of working out the mass of the companion. It could be the same size/mass of the star and an equal radius away on the other side, or it could be a much smaller planter and further away, like in the animations. How can I tell which is the case if the question is looking for the mass of the companion?

I know M, R and V but not m, r or v

 

[mgb_phys]

i think you are over-thinking the question and should just assume it is a circular orbit around a more massive companion.

 

[ChrisBaker8]

I... assume the star is orbiting around an unmoving (within the system) object of infinite mass? ?

 

[mgb_phys]

I don't see you have the info to do anything else !

Keplers third law period = sqrt ( 4 pi^2 r^3 / GM )
You can get r from the radial speed and period.

 

[ChrisBaker8]

period = sqrt ( 4 pi^2 r^3 / GM )

304128 s = $$\sqrt{(4\pi^{2}r^{3})/(6.673.10^{-11}.M)}$$

Is this using M as the given star mass and r as the radius of the orbit of its companion?

 

[cepheid]

Is this using M as the given star mass and r as the radius of the orbit of its companion?

Yes. M is the mass of the central star, R is the radius of the object orbiting around it. This formula is just a result of Newton's Law of Gravitation.

 

[ChrisBaker8]

right, I used K3L to work out r, which was 7.078 x 10^9 m

therefore mass of companion = 6.154 x 10^26 kg

this makes it what...a big planet? Jupiter is x10^27, so I'll assume so

 

[ChrisBaker8]

wait this makes the body a Hot Jupiter, doesn't it?