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Homework Statement
Imagine two planets orbiting a star with orbits edge-on to the Earth. The peak Doppler shift for each 70 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 2E+30 kg.)
Q1: Calculate the mass of the shorter period planet.
Q2: Calculate the mass of the longer period planet.
Homework Equations
(1) Kepler's 3rd law:
T^2 = (4 π^2 a^3)/(G (m_1 + m_2)) |
m_1 | primary mass
a | semi-major axis
T | orbital period
m_2 | secondary mass
G | Newtonian gravitational constant (≈ 6.674×10^-11 m^3/(kg s^2))
(orbital period and semimajor axis relation)
https://is.gd/55FdVC
(2) orbital velocity formula:
v_c = sqrt((G m)/r) |
v_c | circular velocity
m | mass of orbit center
r | orbital radius
G | Newtonian gravitational constant (≈ 6.674×10^-11 m^3/(kg s^2))
https://is.gd/ELmJxr
The Attempt at a Solution
The m of equation #2 is the same as the m_1 of equation #1. Also, the a of eq #1 is the same as the r of eq #2. So you would rearrange the smaller equation and substitute it into the larger equation. Equation #2 looks like this rearranged.
v_c = sqrt((G m)/r)
v_c^2 = (G m)/r
m = (v_c^2 r)/G
r = (G m)/v_c^2
We now substitute between the two equations.
m_1 = m
r = a
m_1 = (v_c^2 a)/G
T^2 = (4 π^2 a^3)/(G (m_1 + m_2))
= (4 π^2 a^3)/(G ((v_c^2 a)/G + m_2))
= (4 π^2 a^3)/(v_c^2 a + G m_2))
v_c^2 a + G m_2 = T^2/(4 π^2 a^3)
G m_2 = T^2/(4 π^2 a^3) - v_c^2 a
m_2 = T^2/((4 π^2 a^3) - v_c^2 a) G)
So we know the following values:
m_1 = 2E30 kg
v_c = 70 m/s
T = 7 days | 700 days
What we need to solve for are:
a = ?
m_2 = ?
So basically, I have two variables and two equations, and I can't find a single solution for that until I have a 3rd equation. What am I missing?