Question

Imagine two planets orbiting a star with orbits edge-on to the Earth. The peak Doppler shift for each 75 m/s, but one has a period of 7 days and the other has a period of 700 days. The star has a mass of one solar mass. Assume 1 solar mass equals 2*10^30 kg.

1.) Calculate the mass of the shorter period planet. (Hint: See Mathematical Insight Finding Masses of Extrasolar Planets)

2.) Calculate the mass of the longer period planet.

Answer 1

first, we find distance or semi major axis for each planet

P² = a³ ( Kepler's third law)

for this to work, the period has to be in years,

7 days = 0.01916 years

so,

a = 0.071615 AU

so,

velocity = 2r / T

velocity = 2 * 0.071615 * 1.5e11 / ( 7 * 24 * 3600)

I have converted days to seconds

so,

velocity = 1.116e5 m/s

so,

mass of smaller planet is

M = mass of star * velocity of star / velocity of planet

M = 2e30 * 75 / 1.116e5

M = 1.344e27 Kg

__________________

Follow the exact same procedure for larger planet

a = 1.5429 AU,

and

v = 24043.5 m/s

so,

M = 6.238e27 Kg