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

In summary, an ellipse has two foci and for a planet in such an orbit, one focus is occupied by the star while the other is empty. According to Ptolemy, the line connecting the planet and the empty focus moves at a constant angular velocity. However, this is not true for highly eccentric orbits and is not a good approximation. The distance and speed of the planet at the point closest to the star is different from the point where the planet is equidistant from both foci. This results in a non-constant angular velocity. Additionally, the perihelion and aphelion distances are not equal, disproving the assumption that the angular velocity is constant in an elliptical orbit. These calculations do not support
  • #1
greswd
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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?
 
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  • #2
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.
 
  • #3
Do you have a reference for Ptolemy using elliptical orbits to model planetary motion?
 
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  • #4
mfb said:
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?
 
  • #5
I don't see how you converted v2 sin θ now, but the two are not equal, correct.
 

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

1. What is Ptolemy's Theory of the Empty Focus of an Elliptical Orbit?

Ptolemy's Theory of the Empty Focus of an Elliptical Orbit is a scientific theory proposed by Greek astronomer Claudius Ptolemy in the 2nd century AD. According to this theory, the Earth is at one of the two foci of an elliptical orbit of a planet, with the other focus being empty. This theory was used to explain the observed retrograde motion of planets in the sky.

2. How does this theory differ from previous theories of planetary motion?

Ptolemy's theory differed from previous theories, such as the geocentric model proposed by Aristotle, by incorporating the concept of an elliptical orbit with an empty focus. This allowed for a more accurate prediction of planetary positions and motions in the sky.

3. Was Ptolemy's Theory of the Empty Focus of an Elliptical Orbit widely accepted during his time?

Yes, Ptolemy's theory was widely accepted during his time and was the dominant theory of planetary motion for over a thousand years. It was also incorporated into the famous geocentric model of the solar system, known as the Ptolemaic system.

4. How did Ptolemy's theory eventually fall out of favor?

Ptolemy's theory faced challenges from other astronomers, such as Nicolaus Copernicus, who proposed a heliocentric model of the solar system. This theory was eventually proven to be more accurate and was widely accepted, leading to the downfall of Ptolemy's theory and the geocentric model.

5. Is Ptolemy's Theory of the Empty Focus of an Elliptical Orbit still relevant in modern astronomy?

While Ptolemy's theory is no longer accepted as a valid explanation for planetary motion, it still holds historical and educational significance in the field of astronomy. It also paved the way for further advancements and understanding of the solar system and beyond.

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