Nonsymmetrical Orbits?

[Dman979]

I was wondering why they are not possible when not in 'bolic orbits.

I'm making a few simplifications here: uniform gravity wells, spherical planets, objects in orbit with no pull on their parent body, and so on.

As I understand it, at a given point on your orbit some distance from your Ap or Pe (X), there is another point on the orbit the same distance away from the Ap or Pe (Y). X and Y are the same distance from the surface of the planet.

Or: Mirroring your orbit through the Ap and Pe, the two points are the same distance from the ground.

Is it possible to have an orbit where the two points are not the same height above ground-level? For instance, can an oval orbit have the Ap and Pe 90 degrees apart, instead of 180?

I don't think so, but I want to know why.

[Bill Phil]

Above standard sea level, or local ground level?

Most orbits are measured from the center of mass of the major body. In simplified mechanics.

You're in orbit right now. But your AP is right next to the sea level and your PE is ~11 km. Above the center of the Earth (~). In this case PE is farther from sea level than AP.

[ZetaX]

Consider a point P where you are moving tangentially in regard to the planet, i.e. not moving closer or further away from it. You get some orbit after that point, however it actually looks like.

Now assume back in P time suddenly reverted: because Newton's laws are exactly the same if time runs backwards, the orbit is the same as if you completely reverted your movement. And as the laws are also unchanged if mirrored in that way, your backwards-time orbit will be symmetric to the forward-time one. Thus your total orbit will be symmetrical in this regard.

Therefore your orbit has that axis of symmetry if you have such a point. Now you can check that any point P of smallest distance, i.e. the Pe, works for this.

Or you could just do all the work of solving the differental equations to check that orbits are conics ;-)

[maltesh]

Kepler's first law (rephrased and paraphrased) All orbits are conic sections (Circles, ellipses, parabolas, and hyperbolas), with the central body at one of the foci.

An ellipse is defined as "Pick two points (F1,F2), these are the foci. Pick a length that is longer than the distance between these two points (2a) ; this is eventually going to be the length of the Major axis. Find all points (p) such that the distance from the P to F1 added to the distance P to F2 is equal to 2a. The curve traced out by all those points is an ellipse."

From there, it's not difficult to see that the closest point on the ellipse boundary to F1 (the periapse) is always going to be on the line passing through both foci, and on the opposite side of F2 from F1, and that the furthest point from F1 on the boundary always going to be on that same line, on the same side of F1 as F2 is. It's also, by necessity, going to be symmetric across the major axis.

[Dman979]

Let's say that my orbit is ^^ Can my parent body be located at the very center of the elipse? I know it can be closer to either of the long ends, that's KSP. Could it be in the middle on the Y-axis and offset on the X-axis?

[Kibble]

Thats how orbits could be shaped if gravitational acceleration didn't decrease with the square of distance. But it does.

[Dman979]

In KSP gravity stays the same until you reach the SOI change. I have never made an orbit like that.

[NFUN]

In KSP gravity stays the same until you reach the SOI change. I have never made an orbit like that.

It changes. Ever notice how it takes a lot of dV to get from a 100km orbit to a Munar encounter, but not much more to get to Minmus's orbit? That is a result of decreasing gravitation force.

[NovaSilisko]

Thats how orbits could be shaped if gravitational acceleration didn't decrease with the square of distance. But it does.

Since you mention it, gravity that falls off linearly with the distance is fun. Orbital velocity is the same at any altitude. Orbits take a sort of trefoil form and slowly precess.

[maltesh]

http://www.coloring-pages.ws/albums/simple-shapes/normal_oval.jpg

Let's say that my orbit is ^^ Can my parent body be located at the very center of the elipse? I know it can be closer to either of the long ends, that's KSP. Could it be in the middle on the Y-axis and offset on the X-axis?

If it's a Keplerian Orbit, the parent body has to be at one of the Foci, and if the foci are in the very center of an ellipse, then that ellipse is a circle.

In the image you've drawn, the major axis is vertical, and the minor axis is horizontal. The foci are /always/ going to be on the major axis of an ellipse.

[dharak1]

While on the topic of gravity, what would orbits look like in other situations? What if gravitational acceleration dropped off with the cube of distance?

[pincushionman]

There actually are ways to get gravity to give you weird orbital shapes under certain conditions. They are:

1) Multiple gravitating bodies. This causes odd Lagrange point orbits at L4 and L5, and other wacky stuff like horseshoe orbits.

2) General Relativity. Around a black hole, for instance, that trefoil orbit shown on the previous page? A very real thing.

[kerbiloid]

Because gravitational and electromagnetic forces are central and conservative forces.

http://en.wikipedia.org/wiki/Central_force

https://en.wikipedia.org/wiki/Conservative_force

In N-dimensional space central force is proportional: F ~ 1/R^(N-1)

As Kerbals and humans live in 3d space, gravitational and electromagnetic forces are F ~ 1/R^2

As R^2 is symmetrical function, any orbit caused by these forces is also symmetrical (of course if we mean a single material point body).

So, all simple orbits are conics: elliptic, parabolic, hyperbolic.

Also:

http://www.dartmouth.edu/~phys44/lectures/Chap_4.pdf

[Brotoro]

While on the topic of gravity, what would orbits look like in other situations? What if gravitational acceleration dropped off with the cube of distance?

If you deviate slightly from 1/r² by changing the exponent a little (to 2.1, for example), the elliptical orbit will precess, with the apoapsis point being further ahead ('ahead' meaning in the same direction that the object is orbiting) each time the satellite returns to apoapsis. The satellite traces out a rosette pattern. If the exponent is slightly LESS than 2, the apoapsis moves backward each orbit, also making a rosette. The greater the deviation of the exponent is from 2, the wider is the spacing between the 'petals' of the rosette. For an inverse cube situation, I recall that the orbits are very non-stable, and the slightest perturbation would cause the satellite to crash into the parent body or get ejected away.

The orbit of the planet Mercury does advance its aphelion point by a very tiny amount with each orbit because gravity does not, in fact, match the perfect 1/r² case of Newtonian physics...but General Relativity does correctly predict the observed motion.

[Dman979]

It changes. Ever notice how it takes a lot of dV to get from a 100km orbit to a Munar encounter, but not much more to get to Minmus's orbit? That is a result of decreasing gravitation force.

I'm not quite sure that's why. It takes more ∆-V to get to a higher orbit, and orbits are different than encounters.

I'd bet that the ∆-V for an orbit at Minmus-height is larger than the one for an orbit at Mun height.

My understanding was that the more elliptical the orbit, the faster the speed at Pe, so a better Oberth Effect. If you circularize at Ap, the Pe goes up a lot at the beginning of the burn, but goes up slower and slower as the burn continues.

Does anyone know how the game processes gravity?

If it's a Keplerian Orbit, the parent body has to be at one of the Foci, and if the foci are in the very center of an ellipse, then that ellipse is a circle.

In the image you've drawn, the major axis is vertical, and the minor axis is horizontal. The foci are /always/ going to be on the major axis of an ellipse.

So it's never possible to have the axes switched?

Edited April 9, 2015 by Dman979

[cpast]

I'm not quite sure that's why. It takes more ∆-V to get to a higher orbit, and orbits are different than encounters.

I'd bet that the ∆-V for an orbit at Minmus-height is larger than the one for an orbit at Mun height.

My understanding was that the more elliptical the orbit, the faster the speed at Pe, so a better Oberth Effect. If you circularize at Ap, the Pe goes up a lot at the beginning of the burn, but goes up slower and slower as the burn continues.

But the dV *difference* between Mun height and a height 500km higher is less than the *difference* between the Mun height and a height 500km lower. Also, the orbital velocity decreases with distance, which wouldn't happen if gravity was constant force to the SOI boundary (orbital velocity would *increase* with altitude were that the case). It's wu

Does anyone know how the game processes gravity?

Yes. The game computes either the gravitational force or the gravitational acceleration caused by the body at the center of your current SOI at your altitude (using standard Newtonian gravity). What changes at the SOI boundary is which body KSP looks to as the sole source of gravity (as opposed to real life, where a spacecraft can be affected by multiple bodies at once).

So it's never possible to have the axes switched?

Nope. Never. Not with point-mass planets and Newtonian gravity, at least.

[GoSlash27]

DMan969,

If you're assuming an elliptical orbit with the parent body at a focus, then your question is one of geometry rather than orbital mechanics.

Is it possible to have an ellipse that's asymmetrical about the major axis? No.

Is it possible to have an ellipse where the closest and furthest points from a focus do not lie along the major axis? No.

HTHs,

-Slashy

[Whirligig Girl]

I would like to see someone try different equations for gravity. In addition to invers square, how about exponential, cubic, or even constant? Are orbits possible in constant gravity?

[ZetaX]

I would like to see someone try different equations for gravity. In addition to invers square, how about exponential, cubic, or even constant? Are orbits possible in constant gravity?

As long as it's spherically (or even just radially) symmetric, you will always have circular orbits. They might just be unstable.

[Brotoro]

I would like to see someone try different equations for gravity. In addition to invers square, how about exponential, cubic, or even constant? Are orbits possible in constant gravity?

I found an orbit simulator Java App at: http://astro.u-strasbg.fr/~koppen/body/TwoBodyOld.html

It was a bit of a pain to run because I had to temporarily disable some Java security settings, but here are some runs.

The "force exponent" for normal gravity is -2, and would result in repeating elliptical orbits.

Below is the result for force exponent -2.1. The satellite starts at the triangle marker with an upward velocity. You can see that the apoapsis point advances around the perimeter of the rosette with each orbit.

With a force exponent of -2.5 (shown below), the apoapsis moved 180 degrees around the parent body with each orbit:

Another example:

The next one shows a force exponent closer to zero than -2. The apopais point is behind the previous apoapsis for each orbit:

And the next one has force exponent -1 (so it's a 1/r force)

This final one has a force exponent of 1, so the satellite is obeying Hooke's Law (as if it was connected to the parent object by a spring). This makes the weird case of an ellipse with the parent body in the center that somebody wanted earlier.

[cantab]

To the original question: From Newton's Law of Universal Gravitation, you can work out that in the case of two point masses orbiting each other with negligible outside influence the orbits will be conic sections - circles, ellipses, parabloae, or hyperbolae. All of those are symmetric.

[maltesh]

[Enter NEWTON, HALLEY]

HALLEY: I wonder, what would be the mathematical description of the path of a body moving under an inverse square law of attraction?

NEWTON: It would be an ellipse.

HALLEY: How do you know this thing?

NEWTON. I have CALCULATED it.

[NEWTON drops mic, walks out of room. HALLEY is stunned, looks around and picks up a letter.]

HALLEY: Is this from Leibniz?

NEWTON: GET OUT.

[Dman979]

So, to sum it up, the 8th grade theory of seasons (the earth moves closest at summer, furthest at winter, equal at spring and summer) is not possible because of orbital mechanics.

[pincushionman]

So, to sum it up, the 8th grade theory of seasons (the earth moves closest at summer, furthest at winter, equal at spring and summer) is not possible because of orbital mechanics.

This is actually true…if you're in the Southern Hemisphere. It will be true in the North in something like 10,000-16,000 years from now. But that's getting a little OT.

[Deutherius]

So, to sum it up, the 8th grade theory of seasons (the earth moves closest at summer, furthest at winter, equal at spring and summer) is not possible because of orbital mechanics.

If you mean the weird elliptical orbit with the Sun being in the center instead of in the focal point of the ellipse - no. Also, that would net you two close passes per year - and we don't have two summers. The usual theory is that summer occurs when the Earth is in perihelion and winter happens when we get to aphelion.

Not to mention that the seasons are being caused by the Earth's axial tilt and not the distance from the Sun