# Why are orbits elliptical?

1. May 20, 2005

### cj

Why are orbits elliptical?
Any ideas?

Last edited by a moderator: Jan 19, 2015
2. May 20, 2005

### dextercioby

What orbits...?

Daniel.

3. May 20, 2005

### tony873004

If they were round, a better question would be "Why are they round?"

Round is a state of perfection, and nothing is perfect.

But, most orbits in our solar system are close to circular.

Pertabutions from the other planets is one of the main reasons that planets can never acheive a perfectly circular orbit.

4. May 20, 2005

### dextercioby

Nope.Kepler's problem for Coulomb potential admits either hyperbolic or elliptical trajectories.

Daniel.

5. May 20, 2005

### neurocomp2003

pick up a standard astronomy text and find out =] eg Carroll and Ostlie

6. May 20, 2005

### cj

Providing for the possibility is one thing; seeing why is another.

Again, why are some (all in our solar system) planet's
orbits elliptical?

7. May 20, 2005

### cj

I guess this is one reason why discussion boards exist.

8. May 20, 2005

### tony873004

I don't understand the "nope" part . But I agree with the rest.

I imagine you're responding to my post.

It's kinda the point I was trying to make. Everything is either hyperbolic or elliptical because circular or parabolic are perfect conditions that only exist on paper. If you think you have a perfectly circluar orbit, try expressing its eccentricity accurate to 15 digits.

9. May 20, 2005

### cj

So it's the gravitational effect from other planets that
cause this deviation from perfection?

10. May 20, 2005

### dextercioby

Here's the eccentricy

$$\epsilon=\sqrt{1+\frac{2EM^2}{m\alpha^{2}}}$$

,where

$$U_{C}=-\frac{\alpha}{r}$$

Daniel.

P.S.U judge from that formula what the eccentricity may be.

11. May 21, 2005

### neurocomp2003

look up kepler's proof for the eccentricity of planetary orbit online. Its like a 1-2page proof.

Some things to consider.
[1] Gravitation of other planets(i think this is minimal if my memory serves me right)
[2] When the Object started orbiting a particular point.
If the point(axis) of rotation is not directly in the center than the orbit will not be perfectly circular...the sun is not located in the center of earths rotation.

Last edited: May 21, 2005
12. May 21, 2005

### Andrew Mason

13. May 21, 2005

Staff Emeritus
If you take the idealized case of one gravitating star and one planet of negligible mass compared to the star, with an arbitrary direction and speed for initial condition, the orbit is a conic section with the star at one focus, no other perturbations required. The fundamental reason is that the inverse square law of gravitation is a recirocal quadratic, and a reciprocal quadratic relationship in polar coordinates generates a conic section. To see this, convert the focus-directrix definitions of the conic sections to polar coordinates with the origin at the focus.

14. May 21, 2005

### DaveC426913

No. They would be elliptical even without other planets.

An orbit has two components: orbital speed and radial distance. The only way you would get a circular orbit is of these two (otherwise independent) values were just right.

15. May 21, 2005

### SpaceTiger

Staff Emeritus
People often rag on the fact that planetary motions aren't circular, that they're really elliptical, but even the latter is not really true. Truth is, the planetary perturbations are extremely complicated and the resulting shape of the orbit is not an ellipse. In fact, when doing high-precision orbit determination, we actually express the eccentricity as a function of time, implying that the shape of the orbit is not described by a single ellipse. In addition, the orbits precess, implying that the orientation of the "ellipse" is also changing with time.

It really just depends on how precise you need to be. For many cases, it's perfectly alright to approximate a planet's orbit as circular. In others, you may want to approximate it as an ellipse (as Kepler did), and in others you'll need to go to higher order (as Einstein did with Mercury).

16. Jul 5, 2005

### amt

There is still no clear and concise answer to the original question though.

17. Jul 5, 2005

Staff Emeritus
I thought my answer in post #13 was concise. Perhaps it wasn't clear? The conic section/focus property falls out of simultaneously satisfying the conservation of angular momentum and the inverse square law of gravity. This is essentially what Newton showed in the first 13 propositions of his Principia.

18. Jul 6, 2005

### Chronos

Many good technical arguments... and correct. The 'picture' explanation is that orbits circle the center of mass of the system. But the complicating factor is the center of mass of any system moves as the bodies move. Under GR, this introduces a dragging effect. If you trace out a path that conserves momentum [with respect to the center of mass] over time, you get an ellipse [in a euclidean coordinate system].

19. Jul 11, 2005

### Felix83

This may be wrong, but here's how I imagine it:
The sun provides a certain centripital force with its gravity, which changes depending on your distance. If a planet comes along and it is moving with a certain velocity and has a certain mass, you could do the math and figure out the radius of the circle it would move in...if it did move in a circle that is. For a certain specific velocity and mass of a planet, there is only ONE radius that has the exact centripital force to keep it moving in a circle around the sun. However what if as it approaches the sun, the radius is slightly larger than what it should be for a circular orbit? Imagine the planet moving in a straight line. As is passes the sun, it gets pulled sideways towards the sun slightly, but not enough to hold it in a circular path. So now it is slowly picking up velocity in the direction perpendicular to its original path - in addition to its original velocity. Right as the planet passed the sun, the pull of gravity was perpendicular to its direction, so it couldn't slow the planet down. As it moves away, the angle becomes smaller, and the sun beings to slow the planet down and eventually stop it and reverse direction. Now it is being accelerated towards the sun, and its sideways velocity is slowing. Now that I've tried, it's a little hard to put into words, but it makes perfect sense to me. Draw it out on paper, draw the sun and draw a circular orbit path. Now start with a radius a little larger and same velocity, now just imagine the forces that the sun is applying on the planet.

20. Aug 9, 2005

### Locrian

Thanks spacetiger, that's an interesting perspective, and not one I've really heard before.