How Can I Derive the Kepler Equation?

[Philosophaie]

I am trying to derive the Kepler equation:

M = E - e * sin(E)

where M=Mean Anomaly, e=Eccentricity and E=Eccentric Anomaly.

If you drop a perpendicular down from the object to the Perihelion-axis you can take:

a * cos(E) = a * e + Recl * cos (TA)

where Recl is the Ecliptic radius to the object from the Sun and TA=True Anomaly.

I am having a hard time equating M and TA because one is on the Ecliptic Plane and the other is on the Orbiting Plane.

Any hints are appreciated.

 

[tfr000]

Mean anomaly and true anomaly are usually both measured in the plane of the orbit, so I guess I don't understand the question.

 

[WhatIsGravity]

Try going at it using the conservation of momentum/inertia..

 

[D H]

I am having a hard time equating M and TA because one is on the Ecliptic Plane and the other is on the Orbiting Plane.

This is not the case, which is probably why you are confused. Mean anomaly is not an angle. It is merely the mean anomaly at some epoch time plus the product of time since that epoch and mean motion. There is no meaningful angle you can draw that represents mean anomaly. Kepler's equation relates mean anomaly to eccentric anomaly (which is an angle). Both eccentric anomaly and true anomaly are measured on the orbital plane rather than on the ecliptic.

 

[BobG]

Orbit of a planet or the orbit of a moon orbiting a planet or the orbit of a satellite orbiting the Earth?

If you're talking about a planet's orbit, the ecliptic plane of that planet (not the ecliptic plane, which normally refers to the Earth's ecliptic plane), then the ecliptic plane is the orbit plane of that planet. Using ecliptic plane in a generic question about orbits really creates a lot of confusion, probably for yourself, as well, since you seemed to believe they were referring to two separate planes. If you're talking about the Moon's orbit, then the Moon's orbital plane definitely is not the same as the ecliptic plane (and the orbital plane of a satellite will not be the ecliptic plane).

But, as DH said, it's going to be hard to find a geometric comparison between Mean Anomaly and Eccentric Anomaly. Mean Anomaly refers to the time domain. It's your location in time relative to perigee (the time you were at perigee). Eccentric Anomaly refers to the physical domain and represents an actual physical angle relative to perigee.