Kepler's 3rd law and a binary system

  • #1
402
14

Homework Statement


I am having a issue with the question, when I check the solution to the problem, I cant seem to make sense of how they have derived there equation.

Q: By analyzing the superposition of frequencies and amplitudes in the radial velocity data for a star 55 Cancri A, the first three exoplanets in the system were deduced. The star was found ti have the following observed parameters: 55 Cancri b corresponds to ##V_{obs}=71.8ms^m{-s}## and a ##P=14.6## days, 55 Cancri c corresponds to ##V_{obs}=10.0ms^{-s}##, ##P=43.9## days and 55 Cancri d corresponds to ##V_{obs}=47.2^m{-s}##, and ##P=5218## days. Assuming the orbits are coplanar and circular, determin, (i) which planet is furthest from the star, (ii) which planet has the lowest mass and (iii) which planet has the highest mass.

There equation for the solution is

$$M_p=\left(\frac{PM_s^2}{2\pi G}\right)\cdot \frac{V_{\left\{obs\right\}\:}}{sin\left(i\right)\:}$$


Homework Equations



Mass function
##\frac{PV_{obs}^3}{2\pi G}=\frac{M_2sin^3\left(i\right)}{\left(M_1+M_2\right)}##
link to full derivation: https://imagine.gsfc.nasa.gov/features/yba/CygX1_mass/binary/equation_derive.html

The Attempt at a Solution



binary_orbit.png
[/B]
I am slightly confused by there solution because if I were to rearrange the equation mass function to make ##M_2## the subject I get the following

##M_2=\left(\frac{PM_1^2}{2\pi G}\right)^{\frac{1}{3}}\cdot \frac{V_{obs}}{sin\left(i\right)}##

which it the same as the equation given in the solution, but my assumption is that ##M_1>>M_2##

but this contradicts the diagram that I have shown because m1 is orbiting m2 which make m2 the larger of the two stars. This make me think maybe I am misunderstanding either the question, or the physics.

Any advice would be much appreciated and thank in advance
 

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Answers and Replies

  • #2
35,646
12,212
Can't you just swap m1 and m2 in the diagram? It shouldn't matter, the situation is symmetric.
 
  • #3
phyzguy
Science Advisor
4,923
1,862
I think you are over-thinking this. Ignore the fact that it is a binary star - the stellar companion is so far away that it has a negligible influence on the time scales we are talking about. Just consider it a single star with three planets orbiting it, all of which are much less massive than the star. Then what do you get?
 

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