I'm confused about 1st Kepler's law and the 2 body problem

In summary, the conversation discusses Kepler's first law and the motion of two bodies in orbit. The expert explains that any two-body attractive potential problem can be simplified to a single body in a central potential, resulting in ellipses as the trajectories. This is due to the conservation of angular momentum and can be further studied in classical mechanics texts. The person asking the question clarifies their understanding and agrees that both bodies are rotating around the barycenter.
  • #1
fluidistic
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Hi PF,
According to Kepler's first Law any planet in orbit of the Sun describes an elliptical orbit.
I don't know why I can't go on wikipedia since days. But if I remember well I've seen that when there are 2 bodies like 2 stars for example then the center of mass will remain quiet (since there's no external force) while the 2 bodies would rotate around it.
So I'm confused and don't understand why the motion of a satellite should be an ellipse. Probably due to conservation of angular momentum. But then I don't understand why the 2 bodies are rotating around their barycenter.
Or maybe it's a combination of the 2 situation. I'm all confused. Could you please explain to me what's happening?
 
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  • #2
Any two-body attractive potential problem is equivalent to that of a single body in a central potential [tex]V(r)[/tex] (i.e. independent of angle [tex]\theta[/tex]). Since the potential is spherically symmetric, angular momentum is conserved. Since the original potential is a function of [tex]\frac{1}{r_c^2}[/tex] (where [tex]r_c[/tex] denotes the distance between the centers of mass of the two bodies), all radially bounded trajectories are ellipses. The analysis is quite lengthy and can be found in Goldstein's Classical Mechanics. I think even Landau's Mechanics has a derivation.

EDIT: Ah crap, I messed up my TeX. I'll fix it asap.
 
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  • #3
Thank you covariance. I didn't study this stuff yet and I'll do that in my 3rd year.
If I understood well the 2 bodies are rotating around their center of mass and it's equivalent to say that one of them has an elliptical orbit around the other body.
EDIT : I could enter in wikipedia. Now I visualize it, it's true! I mean if one chose as origin a body then the other body seems to orbit elliptically. While in fact both bodies are rotating around the barycenter of the 2 bodies. Very nice indeed.
 
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1. What is Kepler's First Law?

Kepler's First Law, also known as the Law of Ellipses, states that all planets move in elliptical orbits with the Sun at one focus. This means that the distance between a planet and the Sun varies throughout its orbit, with the closest point being the perihelion and the farthest point being the aphelion.

2. How does the 2 body problem relate to Kepler's First Law?

The 2 body problem refers to the mathematical problem of determining the motion of two objects, such as a planet and its orbiting star, under the influence of their mutual gravitational attraction. Kepler's First Law is used to describe the shape of the resulting orbit in this scenario, as it states that the planet and star will follow an elliptical path around their common center of mass.

3. What are the implications of Kepler's First Law?

Kepler's First Law has significant implications for our understanding of planetary motion and the structure of our solar system. It shows that planets do not move in perfect circles as previously believed, but rather in elliptical orbits, and that the Sun plays a central role in the motion of all planets.

4. Is Kepler's First Law applicable to all objects in the universe?

While Kepler's First Law was originally derived to describe the motion of planets around the Sun, it can be applied to any two bodies with a significant difference in mass. This means that it can also be used to describe the orbits of moons around planets, or even objects in binary star systems.

5. How was Kepler's First Law discovered?

Johannes Kepler discovered this law in the early 17th century through his observations of the planets, particularly Mars. He noticed that the planet's orbital speed varied at different points in its orbit, leading him to propose the idea of an elliptical orbit instead of a circular one. This was a major breakthrough in our understanding of planetary motion and laid the foundation for future advancements in astrophysics.

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