Calculating the Ratio of Planetary Orbital Periods

[lizzyb]

Homework Statement

Two planets A and B, where B has twice the mass of A, orbit the Sun in elliptical orbits. The semi-major axis of the elliptical orbit of planet B is two times larger than the semi-major axis of the elliptical orbit of planet A.

What is the ratio of the orbital period of planet B to that of planet A?

Homework Equations

$$T^2 = (\frac{4 \pi^2}{G M}) r^3$$

The Attempt at a Solution

$$M_B = 2 M_A$$
$$a_B = 2 a_A$$
$$\frac{T_B}{T_A} = \frac{\sqrt{\frac{4 \pi^2}{G 2 M_A} 8 a_A^3}}{\sqrt{\frac{4 \pi^2}{G M_A} a_A^3}} = \sqrt{\frac{8}{2}} = \sqrt{ 4 } = 2$$
but that was wrong. ?

 

[Andrew Mason]

Homework Equations

$$T^2 = (\frac{4 \pi^2}{G M}) r^3$$

The Attempt at a Solution

$$M_B = 2 M_A$$
$$a_B = 2 a_A$$
$$\frac{T_B}{T_A} = \frac{\sqrt{\frac{4 \pi^2}{G 2 M_A} 8 a_A^3}}{\sqrt{\frac{4 \pi^2}{G M_A} a_A^3}} = \sqrt{\frac{8}{2}} = \sqrt{ 4 } = 2$$
but that was wrong. ?

Planetary Mass is not a factor. Kepler's Third law states that $T^2/a^3$ is the same for all planets. The M in your equation is the mass of the sun.

AM

 

[lizzyb]

thank you.