Finding Eccentricity of Orbit Given Masses, Positions, and Velocities

In summary, the conversation is about finding the eccentricity of an orbit with given masses, cartesian position components, and cartesian velocity components for particles 1 and 2. The approach involves using reduced mass from the center of mass frame and the equation ε = \sqrt{1 + \frac{2 E L^2}{\mu k^2}} where E is energy, L is angular momentum, μ is reduced mass, and k is a constant. The speaker has two questions about the method and asks for guidance to correct it or if there is a better method. The conversation also mentions a task to find the orbit's characteristics with only one arbitrary point of position and velocity.
  • #1
jgoldst1
1
0

Homework Statement


Find the eccentricity of an orbit given the masses, cartesian position components, and cartesian velocity components for particles 1 and 2. The case is reduced to the xy plane.

Homework Equations


I am attempting this problem using reduced mass from the center of mass frame.
ε = [itex]\sqrt{1 + \frac{2 E L^2}{\mu k^2}}[/itex]
where
E = energy
L = [itex]\mu r^2 \dot{\theta}[/itex]
μ = [itex]\frac{m1m2}{m1+m2}[/itex]
k = Gm1m2
r = distance between the two particles

The Attempt at a Solution


I have two general questions. 1) Is the method below correct? If no, I would appreciate guidance to correct the method. 2) If there a better method?

If I knew the velocity, energy, and angular momentum of the reduced mass "particle", I could input the information into the relevant equation.

Is the velocity v of the reduced mass "particle" the difference between the velocities of particles 1 and 2? Similarly, is the position r of the "particle the difference between the positions of particles 1 and 2?

Given the velocity, would the energy of the "particle" be [itex] E = \frac{1}{2}μv^2- \frac{Gm1m2}{r} [/itex] ?

Would the angular momentum L of the "particle" be μ* r x v? Where I would take the cross product of the "particle's" position and velocity components then find the square L^2?

Thank you.
 
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  • #2
Hello!
One of my assignments for a discipline named planetary systems was to write a program and a paper about the orbit of Eris.
One of the tasks was to find the orbit's characteristics with only one arbitrary point of position and velocity.

Take a look ;)

My best regards, Iris.
 

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What is the formula for finding eccentricity of an orbit given masses, positions, and velocities?

The formula for finding eccentricity of an orbit is e = (2L^2)/(μmke^2 - 1), where L is the specific angular momentum, μ is the reduced mass of the system, m is the mass of the orbiting object, k is the gravitational constant, and e is the eccentricity.

What are the required parameters for calculating eccentricity of an orbit?

The required parameters for calculating eccentricity of an orbit are the masses of the objects involved in the orbit, their positions in relation to each other, and their velocities relative to each other.

Can eccentricity be negative?

No, eccentricity cannot be negative. It is a dimensionless quantity that ranges from 0 to 1, where 0 represents a perfectly circular orbit and 1 represents a highly elliptical orbit.

How does eccentricity affect the shape of an orbit?

Eccentricity affects the shape of an orbit by determining how elongated or circular the orbit is. A higher eccentricity means a more elongated and elliptical orbit, while a lower eccentricity means a more circular orbit.

What is the significance of calculating eccentricity of an orbit?

Calculating eccentricity of an orbit is significant because it provides valuable information about the shape and stability of the orbit. It also allows scientists to make predictions about the behavior of the orbiting objects and their interactions with each other.

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