Determine the Orbital Radius of a planet

[damaged goods]

I've been working on this for five hours and I am on the verge of despair!

Can anybody help with this question? All I need is an equation!

There are two extrasolar planets (X and Y) that are orbiting two stars with the same mass as our Sun (1.9891 × 10^30 kg), i.e. the planets are in two separate solar systems. For the purposes of this question, you should assume that there are no other planets in either system and that both stars are at a distance of around 1000 pc from Earth.

Properties of extrasolar planets X and Y:

Planet Mass/ME *Note: ME is the mass of the Earth (5.974 × 10^24 kg)*
Planet X = 12
Planet Y = 150

Radius of star’s orbit from centre of mass of the system/km
Planet X = 2×10^4
Planet Y = 2×10^3

Orientation of the system
Planet X = face-on
Planet Y = edge-on

Question:
Determine the orbital radius for each extrasolar planet X and Y. You should state your answer in astronomical units (AU).


Can anybody provide an equation that I can use to figure this out?

Many thanks

 

[Andrew Mason]

Try starting with the definition of centre of mass. Can you write out the equation for the centre of mass of a system consisting of two masses?

AM

 

[damaged goods]

Hi Andrew, is this the equation that you mean?:

r1 = a • m2/m1+m2 = a/1 + m1/m2

where:
a is the distance between the centers of the two bodies;
m1 and m2 are the masses of the two bodies.
r1 is essentially the semi-major axis of the primary's orbit around the barycenter

Thanks

 

[Andrew Mason]

Hi Andrew, is this the equation that you mean?:

r1 = a • m2/m1+m2 = a/1 + m1/m2

where:
a is the distance between the centers of the two bodies;
m1 and m2 are the masses of the two bodies.
r1 is essentially the semi-major axis of the primary's orbit around the barycenter

Thanks

The centre of mass of a two mass system is the point r on a line between the centres of the two masses such that:

the displacement from r to mass 1 x mass1 + the displacement from r to mass 2 x mass 2 = 0. For the centre of mass:

\sum m_i\vec{r}_i = 0

Work that out for each star/planet system using the figures given. I think you are supposed to assume a circular orbit.

AM

 

[papabone]

Trouble with the ECA as well?