Ptolemy's Theory of the Empty Focus of an Elliptical Orbit

[greswd]

An ellipse has two foci. For a planet in such an orbit, the star is at one of the foci. The other is empty.

According to Ptolemy, if we draw a line connecting the planet and the empty focus, we will find that the line moves at a constant angular velocity.

Is this true, or is it a crude approximation like epicycles?

 

[mfb]

It is not true, and for highly eccentric orbits it is not even a good approximation. Take the point closest to the star, and compare it to the point where the planet has the same distance to both focal points. The distance is different by a factor of approximately 2, the speed is different by a factor that diverges for very eccentric orbits, and the point closest to the sun has a right angle between motion and the (empty focus - planet) line while the angle is close to 0 or pi for the other point. Therefore, the angular velocity $\omega = \frac{v \sin \theta}{r}$ won't be constant.

 

[nasu]

Do you have a reference for Ptolemy using elliptical orbits to model planetary motion?

 

[greswd]

It is not true, and for highly eccentric orbits it is not even a good approximation. Take the point closest to the star, and compare it to the point where the planet has the same distance to both focal points. The distance is different by a factor of approximately 2, the speed is different by a factor that diverges for very eccentric orbits, and the point closest to the sun has a right angle between motion and the (empty focus - planet) line while the angle is close to 0 or pi for the other point. Therefore, the angular velocity $\omega = \frac{v \sin \theta}{r}$ won't be constant.

The perihelion is at a distance a (1-e). The other point is at a distance of a.
v₁ a (1-e) = v₂ sinθ a

From the other focus, the planet is at aphelion. The distance is a (1+e)
ω₁ = v₁ / a (1+e)
ω₂ = v₂ sinθ / a

v₁ (1-e) / a should be equals to v₁ / a (1+e)

but they are not. are my calculations correct?

 

[mfb]

I don't see how you converted v₂ sin θ now, but the two are not equal, correct.