- #1

phantomvommand

- 265

- 39

- Homework Statement
- this isn't really homework, but I've noticed some apparent inconsistencies between the different versions of Kepler's 3rd law for various orbits

- Relevant Equations
- Conservation of Energy

Conservation of Angular Momentum

Period = Area / dA/dt

In a binary elliptical/circular orbit, the R^3 refers to the distance between the 2 planets. In the picture below, ##T^2 = kr^3##.

However, for an elliptical orbit about the sun (which is assumed to be fixed as it is so heavy), and where the sun is located at a focus of the ellipse, ##T^2 = ka^3##, a is the semi-major axis.

The idea that R^3 refers to the distance between the 2 planets does not agree with the 'elliptical orbit about sun' case. Is there any resolution to this?

Also, for a much more general case, such as when 2 plants are both in elliptical orbit about a common centre of mass that is not at the centre of either ellipse, what is the R in the R^3? I have tried working it out, but the algebra is very difficult to resolve, if even resolvable.

My method is as such:

X1------------Y1--C---X2----------------------Y2

X1X2 and Y1Y2 represent the perigee/apogee of 2 elliptical orbits. X1X2 is 1 ellipse, and Y1Y2 another. C is the common centre of mass. One can assign a velocity for each of the 4 points X1, X2, Y1 and Y2, and assign a length for X1Y1, Y1C, CX2, X2Y2, and write down a conservation of energy equation for when planet 1 is at X1 (and plant 2 is at Y2), and when planet 1 is at X2 (and planet 2 is at Y1).

Once can also equate the periods of planet 1 and planet 2. Their periods can be found using total area of ellipse / dA/dt, and dA/dt = L/2m, although L is different for both of the planets. L is also equivalent for planet 1 at X1 and X2 (same for planet 2). Somehow, R can be found, but the algebra is too complex for me.

All help is appreciated.

However, for an elliptical orbit about the sun (which is assumed to be fixed as it is so heavy), and where the sun is located at a focus of the ellipse, ##T^2 = ka^3##, a is the semi-major axis.

The idea that R^3 refers to the distance between the 2 planets does not agree with the 'elliptical orbit about sun' case. Is there any resolution to this?

Also, for a much more general case, such as when 2 plants are both in elliptical orbit about a common centre of mass that is not at the centre of either ellipse, what is the R in the R^3? I have tried working it out, but the algebra is very difficult to resolve, if even resolvable.

My method is as such:

X1------------Y1--C---X2----------------------Y2

X1X2 and Y1Y2 represent the perigee/apogee of 2 elliptical orbits. X1X2 is 1 ellipse, and Y1Y2 another. C is the common centre of mass. One can assign a velocity for each of the 4 points X1, X2, Y1 and Y2, and assign a length for X1Y1, Y1C, CX2, X2Y2, and write down a conservation of energy equation for when planet 1 is at X1 (and plant 2 is at Y2), and when planet 1 is at X2 (and planet 2 is at Y1).

Once can also equate the periods of planet 1 and planet 2. Their periods can be found using total area of ellipse / dA/dt, and dA/dt = L/2m, although L is different for both of the planets. L is also equivalent for planet 1 at X1 and X2 (same for planet 2). Somehow, R can be found, but the algebra is too complex for me.

All help is appreciated.