How Do You Calculate the Mass of a Star Based on Its Planet's Orbit?

[od943]

Homework Statement

A distant star has a single planet circling it in
a circular orbit of radius 3.33 × 1011 m. The
period of the planet’s motion about the star
is 836 days.
What is the mass of the star? The
value of the universal gravitational constant
is 6.67259 × 10−11 N · m2/kg2.
Answer in units of kg.

Homework Equations

v = 2 π r / T
ac = v2 / r
Fc = mac
Kepler's laws?

The Attempt at a Solution

uhhhh...have i have no idea where to start

 

[od943]

wait i figured it out...you use v=(gm/r)^(1/2) and substitute in 2(pi)r/T (in seconds) for v...then solve for M
the answer i ended up with was 4.187530248*10^30

 

[ideasrule]

Nice.

 

[qraal]

wait i figured it out...you use v=(gm/r)^(1/2) and substitute in 2(pi)r/T (in seconds) for v...then solve for M
the answer i ended up with was 4.187530248*10^30

Well done! And your method is exactly right, Kepler's laws being unnecessary in this case, though just as useful if you know how.

You can, of course, combined the equations to get...

T² = 4.π².r³/GM

...and manipulated that as needed. You can see where Kepler's law comes from then, too.