Angular momentum and eccentricity

In summary, the conversation discusses the relationship between angular momentum and orbital eccentricity. While there may not be a direct relationship, there is an indirect one represented by the formula L=\sqrt{\frac{a(1-e^2)}{m_1+m_2}}. It is also mentioned that orbits have two degrees of freedom and can be described by the semi-major axis and eccentricity. The angular momentum is affected by both of these factors, as well as the masses and gravitational constant. A formula from Wikipedia is also mentioned, which states that the energy is related to the semi-major axis and the combined mass of the objects in orbit. A clarification is made that the value of c in the formula depends on the masses.
  • #1
Alexrey
35
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I'm trying to find the relationship between angular momentum and orbital eccentricity but so far I haven't really found anything. I did find an indirect relationship, though, which looked like it should come out to,[tex]L=\sqrt{\frac{a(1-e^{2})}{m_{1}+m_{2}}},[/tex] but I may be completely wrong. Anyone know the correct answer?
 
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  • #2
Even if you remove as many variables as possible, orbits always have two degrees of freedom, they can be written as semi-major axis and eccentricity. The angular momentum will depend on both, together with the masses and the gravitational constant.

I found this formula at wikipedia:
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The energy is [itex]E = \frac{-G(M+m)}{2a}[/itex]

Therefore, [itex]e^2 = 1-\frac{c}{a} L^2[/itex] with [itex]c=\frac{M+m}{GM^2m^3}[/itex] and [itex]L=\sqrt{\frac{1-e^2}{ca}}[/itex] where c just depends on the masses.
 
  • #3
Thanks very much.
 

FAQ: Angular momentum and eccentricity

What is angular momentum?

Angular momentum is a physical quantity that measures the rotational motion of an object around a fixed point or axis. It is calculated by multiplying the object's mass, velocity, and distance from the axis of rotation.

How is angular momentum related to eccentricity?

Angular momentum and eccentricity are related through Kepler's Second Law, which states that the line joining a planet to the sun sweeps out equal areas in equal times. This means that as a planet's eccentricity increases, its angular momentum also increases.

Can angular momentum and eccentricity change over time?

Yes, both angular momentum and eccentricity can change over time due to external forces such as gravitational interactions with other objects. In the case of a planet orbiting a star, its eccentricity can change due to the influence of other planets in the system.

How does angular momentum affect an object's orbit?

The angular momentum of an object affects its orbit by determining the shape and size of the orbit. Objects with higher angular momentum will have larger and more elliptical orbits, while objects with lower angular momentum will have smaller and more circular orbits.

What is the difference between orbital and spin angular momentum?

Orbital angular momentum is the measure of an object's rotational motion around a fixed point or axis, while spin angular momentum is the measure of an object's internal rotation. In the case of a planet, its orbital angular momentum is determined by its orbit around the sun, while its spin angular momentum is determined by its rotation on its own axis.

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