Vis-viva equation (Orbital Velocity) with massive satellite?

[taylorules]

Wikipedia states that "In the vis-viva equation the mass m of the orbiting body (e.g., a spacecraft ) is taken to be negligible in comparison to the mass M of the central body."
I'm wondering how the velocity is determined if the satellite's mass is non-negligible. For example, in a binary star system where both stars have comparable masses, would the vis-viva equation be accurate?
Thanks.

 

[TurtleMeister]

Have you found an answer to your question yet? I was waiting for someone to reply to you because I found the vis-viva equation interesting. So don't take my post as an authoritative answer.

v is defined as the relative speed of the two bodies. If the difference in mass of the two bodies is not significant then simply adding the two masses should work.

But since I could not find a reference of the equation being used this way, I am not sure that it is correct.

Keep in mind that the relative speed of the two bodies is not the same thing as the speed of one body about the common center of mass of the two bodies. In the latter case, the speed of one body will be different from the speed of the other body if there masses are different.

 

[taylorules]

v is defined as the relative speed of the two bodies.

Thank you! That was the problem.
Because Body1's speed = Body2's speed * (Mass2/Mass1), and v = Body1's speed + Body2's speed, I found Body2's speed = (Mass1 * v) / (Mass1 + Mass2).
The problem was that I thought v represented Body2's speed, not the combined speed of both.
Thanks

 

[TurtleMeister]

Glad I could help. If you write out the equation using your method, you will find that the (M1+M2) cancels out:

This is the speed of each body relative to the barycentre of the two bodies.

 

[PJC01]

The vis-viva equation, also known as the orbital velocity equation, is a fundamental equation used in orbital mechanics to calculate the velocity of an orbiting body around a central body. The equation takes into account the masses of both bodies, as well as their distance from each other.

In the case of a massive satellite orbiting a central body, the vis-viva equation assumes that the mass of the satellite is negligible compared to the central body. This is a reasonable assumption for most satellite orbits around planets or other large celestial bodies, as the mass of the satellite is much smaller than that of the central body.

However, in a binary star system where both stars have comparable masses, the vis-viva equation may not be accurate. In this case, the masses of both stars would need to be taken into account in the equation to accurately determine the orbital velocity of the satellite.

In general, the vis-viva equation is most accurate when the mass of the orbiting body is significantly smaller than the central body. For systems where the masses are comparable, more complex equations and calculations would need to be used to determine the orbital velocity.

Overall, the vis-viva equation is a useful tool in orbital mechanics, but its accuracy may vary depending on the specific system being studied.