What is the relationship between mass, speed and distance of a planet orbiting the sun?

Question

After reading this fascinating story about a new exoplanet, I was wondering about how mass, speed and distance determine a circular orbit of a planet around a star.

Given the mass of the sun and star, and the distance between them, is there only one possible orbital velocity?

Given any 2 of the following, is it possible to calculate the 3rd?

• Relative mass of the planet to the sun
• Orbital velocity
• Distance between the 2 masses

And, given 1 of the above, are there an infinite number of possible values for the other 2?

Answer 1

Yes, it's possible to calculate the 3rd if any two or provided. Also, For a given property (like say orbital velocity) Indeed, there are infinite number of values..!

The orbital velocity is calculated by equating the centripetal and gravitational force:

$$\frac{M m}{M+m} \frac{v^2}r=\frac{GMm}{r^2}$$ Where $M$ is the mass of the Sun, $m$ is the mass of the planet, $r$ is the distance between them, and $G$ is the gravitational constant. (The quantity $\frac{M m}{M+m}$ is the reduced (effective) mass.)

For circular orbits, $v=\sqrt{G(M+m)\over r}$ (constant)

Using Newton's generalization of Kepler's third law, the orbital period is given by: $T=2\pi\sqrt{r^3\over G(M+m)}$

See other methods of determining these in olden days...