How Do You Calculate Total Energy in an Elliptic Orbit?

Differentiate to get dAM/dt = 0 = mva + mvr (dr/dt)v = \frac{h}{mr}Use this in the first equation to eliminate v and solve for E:E(r) = \frac{1}{2}m\left(\frac{h^2}{m^2r^2}\right) + \frac{1}{2}mv_r^2 - \frac{GMm}{r}= -\frac{GMm}{2a}In summary, it is possible to derive the total energy in an elliptic orbit by using the equations for kinetic and potential energy, as well as the conservation of
  • #1
Päällikkö
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How does one derive the total energy in an elliptic orbit:
[tex]E= - \frac{GMm}{2a}[/tex]
where a is the semi-major axis?
I did manage to get the result for the special case of circular orbit, as
[tex]v = \sqrt{\frac{GM}{R}}[/tex]

But the problem is that I can't figure out a way to express v in an elliptic orbit.

If at all possible, give hints (that is: not a direct answer), as I'd rather try it myself first :).
 
Last edited:
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  • #2
Not really, no.

I can't get the given equation:
[tex]v=\sqrt{2\mu\left({1\over{r}}-{1\over{2a}}\right)}[/tex]
without assuming the result (total energy) I'm trying to get.

EDIT: Either I am getting paranoid or someone replied, but deleted his/her message :).
 
  • #3
Would a formula for elliptical velocity be too much help ?
 
  • #4
Fermat said:
Would a formula for elliptical velocity be too much help ?
Isn't that the equation I wrote in my second message?

I can get it by assuming
[tex]E= - \frac{GMm}{2a}[/tex]
but that's the equation I want to prove.
 
Last edited:
  • #5
Päällikkö said:
...

EDIT: Either I am getting paranoid or someone replied, but deleted his/her message :).
Sorry for the confusion. That was me, then I thouight that maybe that was too much of hint - that you didn't really want !
 
  • #6
Help, anyone?
 
  • #7
Päällikkö said:
How does one derive the total energy in an elliptic orbit:
[tex]E= - \frac{GMm}{2a}[/tex]
where a is the semi-major axis?
I did manage to get the result for the special case of circular orbit, as
[tex]v = \sqrt{\frac{GM}{R}}[/tex]

But the problem is that I can't figure out a way to express v in an elliptic orbit.

If at all possible, give hints (that is: not a direct answer), as I'd rather try it myself first :).
The speed can be resolved into two orthogonal components - radial and tangential. Since [itex]v^2 = v_t^2 + v_r^2[/itex] (Pythagoras) the total kinetic energy is simply the sum of the tangential and radial kinetic energies. Use that, together with the fact that angular momentum and total energy is constant, to derive the relationship between radius, speed and total energy.

[tex]E(r) = KE + PE = \frac{1}{2}mv^2 - \frac{GMm}{r} = \frac{1}{2}mv_t^2 + \frac{1}{2}mv_r^2 - \frac{GMm}{r}[/tex]

Use the fact that the radial KE (middle term) is 0 when r is maximum or minimum (ie. when r = a or r=b)

AM
 

FAQ: How Do You Calculate Total Energy in an Elliptic Orbit?

1. What is an elliptic orbit?

An elliptic orbit is a type of orbit in which an object, such as a planet or satellite, follows an elliptical path around another object, usually a larger body like a star or planet. This means that the distance between the two objects varies throughout the orbit, with the closest point being the periapsis and the farthest point being the apoapsis.

2. How does energy play a role in elliptic orbits?

In elliptic orbits, energy is constantly being exchanged between kinetic energy (the energy of motion) and potential energy (the energy associated with an object's position). As the object moves closer to the central body, its kinetic energy increases and its potential energy decreases, and vice versa as it moves farther away.

3. What is the relationship between energy and the shape of an elliptic orbit?

The shape of an elliptic orbit is determined by the amount of energy the object has. The more energy an object has, the more elongated its orbit will be. This is because higher energy levels allow the object to reach greater distances from the central body before being pulled back in by gravity, resulting in a more elliptical shape.

4. Can an object have a perfectly circular orbit in an elliptic orbit system?

No, an object cannot have a perfectly circular orbit in an elliptic orbit system. This is because a circular orbit requires a specific balance of energy and velocity, while an elliptic orbit involves constantly changing energy and velocity as the object moves closer and farther away from the central body.

5. How does the energy of an object in an elliptic orbit change over time?

The energy of an object in an elliptic orbit remains constant over time, but the distribution of energy between kinetic and potential energy changes as the object moves along its orbit. This is known as the conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another.

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