Why Are Orbits Elliptical? Rules of Circular Motion

[Red_CCF]

Hi

Can someone explain why orbits are elliptical and not circular? Does elliptical orbits still follow the rules of circular motion (ex. constant centripetal force)?

Thanks for any help that you can provide

 

[Nabeshin]

Orbits are (in general) elliptical and not circular simply because a circle is a special case of an ellipse. A circular orbit represents perfect balance between velocity and gravitational force, anything else results in an ellipse, hyperbola, or parabola (another special case where e=1). We don't observe hyperbolic or parabolic orbits because they are unbound (the objects do not orbit indefinitely but escape from solar systems). This is the case with some comets which pass through the solar system never to return again.

Of course, it is not forbidden that an orbit be circular, it simply would be a very "goldilocks" scenario in which all parameters are just right. Further, any gravitational disturbance from other bodies (inevitable, as gravitational force never decreases to zero) perturbs the orbit from a true circle.

And no, elliptical orbits do not feature a constant centripetal force. This is because the centripetal force is provided by gravity, which is a function of radius, and radius in an elliptical orbit is not constant. Some things remain unchanged when considering circular and elliptical orbits, such as the conservation of energy and angular momentum.

 

[Red_CCF]

And no, elliptical orbits do not feature a constant centripetal force. This is because the centripetal force is provided by gravity, which is a function of radius, and radius in an elliptical orbit is not constant. Some things remain unchanged when considering circular and elliptical orbits, such as the conservation of energy and angular momentum.

Thanks for the reply

If we assume a satellite orbiting Earth, if the centripetal force varies wouldn't it be possible for the Earth's gravity to pull the satellite in? Or does the satellite's centripetal force always exceeds that of the Earth's gravity? If so what else is giving the satellite a centripetal force?

 

[Nabeshin]

Thanks for the reply

If we assume a satellite orbiting Earth, if the centripetal force varies wouldn't it be possible for the Earth's gravity to pull the satellite in? Or does the satellite's centripetal force always exceeds that of the Earth's gravity? If so what else is giving the satellite a centripetal force?

The centripetal force on the satellite is the Earth's gravity. At points in the orbit where gravity is a bit stronger, the satellite moves faster (conservation of energy), and is less likely to be pulled in. The opposite happens at points in the orbit where gravity is weaker.

The whole definition of an orbit is a path in which the object is never "pulled in", so to speak. It seems you're still thinking rather circular in your conception of an orbit, judging by the fact that you are talking about the satellite being pulled in.

 

[D H]

What centripetal force? (Hint: There is no such thing in an inertial reference frame.)

Looking at orbits as a balance between centripetal and gravitational forces is not a good idea. For one thing, it only works in the special case of a circular orbit. For another, a centripetal force only appears in a rotating reference frame. In the case of a circular orbit, that frame is a frame in which the satellite appears to be stationary.

A much better point of view is to look at things from the perspective of gravitational force only. The solutions to Newton's law of gravity are conic sections: Circles, ellipses, parabolas, and hyperbolas. No centripetal force is needed.

 

[DaveC426913]

Can someone explain why orbits are elliptical and not circular?

If you play with a orbital simulator long enough it will be obvious why orbits are not circular. I programmed a simulator a ways back and, in testing it, I tried to set up orbits that were as circular as possible. My simulator used an analog method to set the direction and magnitude of the velocity: you click-drag-unclick to set it.

I can tell you, it is very difficult to get both
- the direction component exactly tangential
- the velocity component exactly right to keep it circular.

Explaining it doesn't really make it clear. But playing with a simulator makes you realize just how unlikely it is that these two unrelated components will line up just right.

 

[Nabeshin]

What centripetal force? (Hint: There is no such thing in an inertial reference frame.)

Looking at orbits as a balance between centripetal and gravitational forces is not a good idea. For one thing, it only works in the special case of a circular orbit. For another, a centripetal force only appears in a rotating reference frame. In the case of a circular orbit, that frame is a frame in which the satellite appears to be stationary.

A much better point of view is to look at things from the perspective of gravitational force only. The solutions to Newton's law of gravity are conic sections: Circles, ellipses, parabolas, and hyperbolas. No centripetal force is needed.

You are referring to the fictitious centrifugal force, DH? OP never mentioned this.

 

[D H]

You are referring to the fictitious centrifugal force, DH? OP never mentioned this.

Yes. Oops, my bad.

Centripetal force is not a particularly good term, either, for describing non-circular orbits. A much better term is "central force".

 

[Red_CCF]

Thanks for the replies

If you play with a orbital simulator long enough it will be obvious why orbits are not circular. I programmed a simulator a ways back and, in testing it, I tried to set up orbits that were as circular as possible. My simulator used an analog method to set the direction and magnitude of the velocity: you click-drag-unclick to set it.

I can tell you, it is very difficult to get both
- the direction component exactly tangential
- the velocity component exactly right to keep it circular.

Explaining it doesn't really make it clear. But playing with a simulator makes you realize just how unlikely it is that these two unrelated components will line up just right.

So it's just the way the universe works; that it is nearly impossible for an object to orbit around another object in a circular path. But how do objects know this? In Nabesin's example, the satellite would increase speed in the event that the gravity is stronger than the necessary force centripetal (center force). But how would, for example, the moon adjust to such a case? What "motivates" an object to stay in an elliptical orbit and not just fall towards the Earth?

Is there a good site where I can play around with such a simulator? I tried googling but all the ones I found uses java and my firefox crashes as soon as I click on them. Thanks!

 

[vin300]

Thanks for the replies



So it's just the way the universe works; that it is nearly impossible for an object to orbit around another object in a circular path. But how do objects know this?

Did you try searching?
http://http://en.wikipedia.org/wiki/Elliptic_orbit"
http://en.wikipedia.org/wiki/Hyperbolic_trajectory"
The bodies follow the respective trajectories if it has the velocity at a point as given by the formula.
Why they don't fall into is because the force has already increased its velocity too much as it gets close which means too much tangential kinetic energy for gravity at the closest point to pull it in.

 

[Vanadium 50]

If you solve the equations of motion, you will see that orbits are conic sections. Closed orbits are ellipses (a circle is a special ellipse) and open orbits - bodies that approach and then separate forever - are parabolas and hyperbolas.

 

[Philosophaie]

Mathematically,

from the center or focus

r = a*(1-e^2)/(1+e*cos(v))

where e is eccentricity
e = 0 Circle
0<e<1 Ellipse
e=1 Parabola
e>1 Hyperbola
http://en.wikipedia.org/wiki/Eccentricity_(orbit )

a is the semi-major axis,
v is the True Anomaly or angle from start v=0 or Argument of Perihelion,
r is the radius from center or focus to a at a certain v.

 

[russ_watters]

So it's just the way the universe works; that it is nearly impossible for an object to orbit around another object in a circular path. But how do objects know this?

Objects don't "know" anything, this is all just a matter of probability. For a given set of objects and a given distance, there are an infinite number of possible orbit eccentricities and only one circular orbit. Since an orbit is basically stable* whether circular or elliptical, it would just be extremely unlikely that an orbit would be circular. It's like playing the lottery, just with worse odds.

*With the big caveat that when you add more planets or other objects, the odds of an orbit being/staying circular drop even more as purturbations tend to increase eccentricity.

 

[Red_CCF]

Thanks for the replies. I think calculating orbits is a bit advanced for me now since I'm just starting university. I have one last question though, what is the difference between hyperbolic and parabolic orbits? To me they look the same except one has a higher eccentricity.

 

[Janus]

Thanks for the replies. I think calculating orbits is a bit advanced for me now since I'm just starting university. I have one last question though, what is the difference between hyperbolic and parabolic orbits? To me they look the same except one has a higher eccentricity.

A parabolic orbit is a special case of a hyperbolic orbit where the object's velocity is exactly equal to escape velocity. The main difference is that for parabolic oribts the asymptotes are parallel.

 

[DaveC426913]

A parabolic orbit is a special case of a hyperbolic orbit where the object's velocity is exactly equal to escape velocity. The main difference is that for parabolic oribts the asymptotes are parallel.

Of course, neither of them are actually orbits...


But to further Janus' explanation, the upsoht is that a body on a parabolic trajectory will leave the system on a course exactly 180 degrees from its entry trajectory (thus, its two asymptotes are parallel), whereas a hyperbolic trajectory has no such constraint; its exit trajectory will be anything between 0 and 180.