Calculating Solar Mass using peak Doppler shifts

[bbbl67]

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?

 

[gneill]

Two planets with the same maximum velocity but different periods would seem to indicate that one or both orbits are elliptical, not circular. You'll need to use a more general form of the orbit velocity equation. Unfortunately that will introduce another variable, the perihelion distance (where velocity on orbit is maximized).

 

[bbbl67]

Two planets with the same maximum velocity but different periods would seem to indicate that one or both orbits are elliptical, not circular. You'll need to use a more general form of the orbit velocity equation. Unfortunately that will introduce another variable, the perihelion distance (where velocity on orbit is maximized).

Is this solvable as it is now, do you think?

 

[gneill]

Is this solvable as it is now, do you think?

I must admit that at the moment I can't think of a way to proceed.

 

[TSny]

If there was only one planet, then the star and planet can both orbit around the center of mass ("barycenter") in circular orbits. If the Doppler shift is associated with spectral lines emitted by the star due to its orbital motion, then you can work out the mass of the planet using the concept of reduced mass, etc.
https://en.wikipedia.org/wiki/Gravitational_two-body_problem

But, I don't know how to handle two planets unless you just assume they can be treated independently.

@bbbl67, it is important to note that the orbital speed of the star equals (2π/T)⋅r_s, where r_s is the radius of the star's orbit. r_s is related to the distance of separation, a, of the planet and star by a factor that depends on the masses of the planet and the star (see the link above). This gives you another relation to work with.

 

[haruspex]

I find it hard to believe there is enough information here.

The trouble is that the velocity is given by $v_i^2=\mu_i(\frac 2{r_i}-\frac 1{a_i})$, where $\mu_i=M+m_i$, i=1,2.
But we have no idea about r_i. We do not know how the semi-major axis of each aligns with our view of the system.

There is another difficulty. If we assume both m_i<<M, then clearly there is no way to find their masses, since both velocity and period only depend on M. It follows that at least one mass is comparable to M. But that makes it a three body problem in which Kepler no longer applies.

 

[haruspex]

If there was only one planet, then the star and planet can both orbit around the center of mass ("barycenter") in circular orbits. If the Doppler shift is associated with spectral lines emitted by the star due to its orbital motion, then you can work out the mass of the planet using the concept of reduced mass, etc.

Doesn't that assume a roughly circular orbit?

 

[TSny]

Doesn't that assume a roughly circular orbit?

Yes, I am assuming the planet and star orbit in circles about the center of mass. https://upload.wikimedia.org/wikipedia/commons/f/f2/Orbit2.gif

The problem statement isn't clear. I might be totally misinterpreting the problem. But, I can get an answer with this interpretation.