Astronomy help

I'm doing a bit of homework for Astronomy class, and one of the questions has me utterly stumped.

The question:

Consider a planet with a mass of 10 MJupiter (10 x the mass of Jupiter) orbiting around a 3-MSun (3 x the mass of our Sun) star with a (orbital) period of 1.5 years. Calculate the distance between the center of the star and the center of mass.

I've looked all over the book, my notes, the internet, and I'm still having trouble. I can't find the formula that directly correlates the orbital period of a star, the mass of its planet, its orbital period, and the average distance of the center of the star from its center of mass.

I know that a more massive planet = a larger orbital radius for the star around the center of mass, but is there a formula or something that states the ratio of Planet-mass to Star-mass and relates that to how large the orbital radius of the star is?

Help!

I read "anatomy help" at first. If that was the case, I would've been interested.

Planets are gay. Take a weight lifting class. It's really the only class you can take your shirt off in and not be asked to leave.

And it's one of the classes that gives us more redshirts.

13 Spider Bloody Chain said:

I'm doing a bit of homework for Astronomy class, and one of the questions has me utterly stumped.

The question:

Consider a planet with a mass of 10 MJupiter (10 x the mass of Jupiter) orbiting around a 3-MSun (3 x the mass of our Sun) star with a (orbital) period of 1.5 years. Calculate the distance between the center of the star and the center of mass.

I've looked all over the book, my notes, the internet, and I'm still having trouble. I can't find the formula that directly correlates the orbital period of a star, the mass of its planet, its orbital period, and the average distance of the center of the star from its center of mass.

I know that a more massive planet = a larger orbital radius for the star around the center of mass, but is there a formula or something that states the ratio of Planet-mass to Star-mass and relates that to how large the orbital radius of the star is?

Help!

Click to expand...

Can't say I'm familiar with any of this () but you likely have all the formulas you need in your text, if not your notes. Try looking at the variables you have, what formulas you have that use them (the formula doesn't need to be using all of them at once) and what those formulas generate.

In other words, plug-and-chug until you get what you want. 

First of all, I don't know if anything I will tell you is true. Second: Kepler's laws may help. At least in this solar system, the time of orbit and the distance of the planet from the sun is a constant ratio, and mass has little to do with it.

That is all.

The planet and the star both follow an elliptical orbit, but the eccentricity of the ellipse is not given in the problem, so I think it does not matter, and it can be safely assumed that both bodies follow a circular path around the center of mass.

Newton's third law states that the force on both objects is equal but opposite in direction. The direction of the force/acceleration on a body in circular motion is always orthogonal to the direction of the motion, therefore the two bodies always move in opposite directions, so the orbital period is the same for both.

Let's name the mass of the star M1 and its orbit radius R1. The planet has mass M2 and orbit radius R2
The orbital period for both is T = 1.5 year

For uniform circular motion, the centripetal force F = (4 x pi^2 x M x R) / (T^2)
The force is the same for both, so that means that M1 x R1 = M2 x R2
From that follows that R1/R2 = M2/M1.

This centripetal force must equal Newton's law of gravitation F = (G x M1 x M2) / (R^2)
In this R is the distance between the planet and the star.
Because the planet and the star are at the opposite sides of the center of mass, R = R1 + R2.

The two expressions of the force give us: (G x M1 x M2) / ((R1+R2)^2) = (4 x pi^2 x M1 x R1) / (T^2)
Combining this with the earlier result R1/R2 = M2/M1, we have a system of two equations and two unknown variables, R1 and R2, so you should be able to solve the rest and fill in the values if necessary. (The value of R1 is the answer to the problem.)

PS. I cannot be held responsible for flaws in this argument, use at your own risk

Another group of folks at another forum suggested using the centripetal force equation. I will think on this.

Meanwhile, a big thank you to the folks who actually helped out and didn't just spam! 

Edit: I took a look at your argument, dekelt, but I have to wonder where you got the 4pi^2 x M x R/ T^2 equation from. I googled for centripetal force formulas and didn't get anything like it.

13 Spider Bloody Chain said:

I took a look at your argument, dekelt, but I have to wonder where you got the 4pi^2 x M x R/ T^2 equation from. I googled for centripetal force formulas and didn't get anything like it.

Click to expand...

You're right, normally (as on wikipedia) the cetntripetal force is expressed F = (M x V^2) / R, but both are correct.

It should be easy to see that V x T equals the distance traveled in one orbital period. The length of this orbit can also be calclated from the circumference of the circle: 2 x pi x R.
Therefore, V x T = 2 x pi x R --> V = (2 x pi x R) / T.
If you substitute this expression for V in the formula for the centripetal force, you get F= (M x 4 x pi^2 x R) / (T^2), like I wrote.

Ah well, I managed to find a reasonable answer through another method. But thanks for the help anyways!