What is the Semimajor Axis of a Planet's Orbit in Different Reference Frames?

[cellery]

Homework Statement

There is a planet of mass m_1 orbiting a star of mass m_2. One question is "What is the semimajor axis of the planet's orbit in the coordinate system centered at the star's center", and another is "What is the semimajor axis of the planet's orbit centered at the center of mass of the system.

Homework Equations

t^2/A^3 = 4pi^2/G(m_1+m_2)
^Kepler's Third Law

The Attempt at a Solution

Basically, as far as I can visualize this problem, the semimajor axis does not change when you switch the center of the system. The planet still has to make the same orbit, so the center of said orbit should still be the same. I have an answer, but I can't tell if I'm somehow supposed to modify it for one of these scenarios.

 

[D H]

Think about this again. The orbit will have the same size and shape in any inertial frame, but the sun-centered frame is not inertial. The sun is orbiting the center of mass as well. What is the distance between the planet and the sun at apofucus and perifocus?

 

[cellery]

Alright, thanks, I'll give it a try. The problem is I just can't visualize what orbits look like if I have to account for a moving frame of reference as well. Everything changes. I don't quite see how to encorporate it into any equation, either.

But I'll see what I can do. The amount of help I need may be beyond the scope of what helpers are supposed to do.

 

[D H]

Answering these questions might help:

• What is the distance between the planet and system center of mass at apofocus and perifocus?
• What is the relationship between the distance between the planet and system center of mass and the distance between the sun and system center of mass?
• What is the distance between the planet and the sun at apofocus and perifocus?
• What does this mean in terms of the semimajor axis?

 

[cellery]

Thanks. Based on that, I was able to get an answer for each reference frame. They make sense, as far as I can tell.