How Do You Calculate the Mass of a Star Based on Planetary Data?

[tibessiba]

Gravitational Force---Need Help Fast!

Homework Statement

You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be 1.8*10^7 m and its rotation period to be 22.3 hours. You have previously determined that the planet orbits from its star with a period of 402 Earth days. Once on the surface you find that the acceleration due to gravity is 12.2 m/s^2.

What is the mass of the planet? in kg

What is the mass of the star? in kg

Equations:
Mass(of planet)= g*R^2/G

The Attempt at a Solution

Ok, so I calculated the mass of the planet to be 1.48*10^25 kg using the above equation. M=[(12.2m/s^2)*(9*10^6m)^2]/6.67*10^-11 Nm^2/kg^2.

Now I am stuck with how to find the mass of the star and which equation to use to get it..

Thank you

 

[tibessiba]

Anybody?

 

[Astronuc]

Well if one knew the distance to the star from the planet, then one could apply Kepler's Law of Periods - http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html#c6

 

[tibessiba]

It states that the planet orbits 2.2*10^11 m from its star. This would them be the distance, correct??

 

[tibessiba]

So, then I use T^2=4pi^2/GM somehow? I guess I'm not exactly sure how...

 

[tibessiba]

oops, I forget r^3 in the equation..
so:

T^2=(4pi^2/GM)r^3

 

[tibessiba]

T should = 34732800 sec
r should be 2.2*10^11 m

correct??

 

[tibessiba]

Using that I calculated the mass of the star to be 1.39 * 10^64...
but that is wrong...
so where did I go wrong??

 

[tibessiba]

I calculated it again and this time I got 1.44*10^37, but that is also wrong...

What am I doing wrong?

 

[tibessiba]

If I rearrange the equation to find the mass of the star this is what I get:

M=[(2pir^3/2)^2]/GT

Is this right? It is what I have been using, but I can't come up with the correct answer.

 

[Astronuc]

It states that the planet orbits 2.2*10^11 m from its star. This would them be the distance, correct??

Yes, that is a or r in Kepler's forumula relating period with distance for one mass orbiting another mass, e.g. moon around a planet or planet about a star.

In the simplest form, one may assume that the mass of the star greatly exceeds the mass of the planet.

So $$T^2\,=\,\frac{4{\pi^2}{a^3}}{GM}$$, which can be rearranged to get

$$M\,=\,\frac{4{\pi^2}{a^3}}{GT^2}$$

So substitute in the appropriate numbers

G = 6.67 x 10^-11 N m²/kg², the universal gravitational constant,

a = 2.2 x 10¹¹ m

T = 402 d * 24 h/d * 3600 s/h = 3.47328 x 10⁷ s

Now compare the mass of the star with the mass of the planet. The mass calculated for the star might need adjusting for the mass of the planet since M is the sum of the masses, but if M >> m(planet), the M is approximately the mass of the star.

M=[(2pir^3/2)^2]/GT

This is not correct. Be careful about moving terms and exponents. T should be squared and if one brings r^3 inside the parentheses and squares those terms, then one must use r^(3/2) within the parentheses.

 

[tibessiba]

Ok... I see.

Thank you very much!