# Calculate distance between two planets in elliptical orbits

emilinus

## Homework Statement

Two planets are orbiting the sun in elliptical orbits in the same plane. The orbit of planet Q has a semi-major axis of 64.8 AU and eccentricity of 0.445. Planet R has a semi-major axis of 42.5 AU and eccentricity of 0.825. The perihelion point of Planet R's orbit is rotated relative to that of Planet Q by 124 degrees. Determine the closest and greatest distance that Planet R and Q get from each other in AU.

## Homework Equations

P^2 is proportional to a^3
c^2=a^2-b^2 (where b is the semi-minor axis)
c=ae
Circumference (orbital distance) = 2pi*sqrt((a^2+b^2)/2))

## The Attempt at a Solution

P = sqrt*(a^3)
PQ= 277.0661022 yrs
PR= 521.6299378 yrs
cQ = (64.8*0.445) = 28.836 AU
cR = (42.5*0.885) = 35.0625 AU
bQ = sqrt*(64.8^2-28.836^2) = 58.03 AU
bR = 24.02 AU

So I'm not sure that I understand the set up of the orbits but either way I don't know where to go from here. How I understand it is that when both are at their perihelion points, there is an angle of 124 degrees between them where planet R's orbit is rotated from that of Planet Qs so that they aren't perfectly aligned but are still within the same plane (I hope that makes sense). With that I have two side lengths (both of their semi-major axis lengths) and an angle so I can get a distance between them. What next (if that's even a correct start)?

Mentor
Do you know the polar form for a planetary orbit?

emilinus
No that wasn't provided. I do have mean angular motion though. n=2pi/P and for Q = .012045292 and for R = .022677567. I'm not sure what the units are though...is it radians/year?

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Mentor
Well, if it's a mean then it could be for anything ;-)

It's easy to rotate a polar equation through an angle simply by adding an offset to the angle. That's why I asked about the polar form. You should have it somewhere in your text. It looks like:

$$R(\theta) = \frac{p}{1 + e\; cos(\theta)}$$

Where p is the semi latus-rectum for the orbit: p = a(1 - e2)

I don't know what tools you're expected to have on hand to find the maximum separation. Since the major axes of the two ellipses are not collinear, it will not be simply a matter of checking at the axes ends.

emilinus
Do you think it would be possible to use radians/year and go through different amounts of time to find the least and greatest amount of angular distance between the two planets? Then maybe I could plug the angles into that polar form equation to find the distance (using cosine law).

Mentor
I don't know what other information you were given other than what's in the problem statement that you provided. For example, I don't see any information on initial positions of the planets, or the mass of their Sun. I haven't checked to see if the orbital periods will be harmonically related, so that they might repeat the same patterns and "avoid" certain relative positions.

If it is assumed that the orbits are not harmonically related, then over time they will take on all relative positions allowed by their trajectories. So you don't even need to worry about particular mean angular velocities; just step the angles by small increments, and capture the maximum separation.

Even easier would be to use a "canned" solver to minimize the distance function (a function of two variables). What course is this for? Are you expected to have this sort of software available?

emilinus
The Sun I believe it supposed to be this solar system's Sun. No there's no software available. It's for an intro astronomy course in university.

The only other info I was given was that the use of spreadsheet might be helpful in solving the problem. So I'm assuming I just have to go through and find the closest and farthest angle differences...but wouldn't they just be 180 and 0 if the planets take all possible trajectories? That seems too easy.

Mentor
The Sun I believe it supposed to be this solar system's Sun. No there's no software available. It's for an intro astronomy course in university.

The only other info I was given was that the use of spreadsheet might be helpful in solving the problem. So I'm assuming I just have to go through and find the closest and farthest angle differences...but wouldn't they just be 180 and 0 if the planets take all possible trajectories? That seems too easy.

Yes, that would be too easy :-) It's not angular separation that you're looking for, it's physical distance. Remember, the planets move independently, so one can be anywhere on its own orbit and the other anywhere on its own orbit. So you're looking for the most widely separated locations between the two orbits. Due to the rotation of one of the ellipses, this won't occur when the planets are at the ends of their axes, but somewhere else on their orbits.

There are two ellipses, one of which is rotated 124 degrees about its focus (essentially the Sun). So their major and minor axes are not aligned. At least you should be able to plot the two ellipses using a spreadsheet's plotting functions so that you can see what you're up against. You'll then be able to locate by eye the approximate locations of the most distantly separated points.

emilinus
I don't know how to use a spreadsheet plotting function...

I'm going to attempt to draw out the two ellipses properly to visualize where the farthest and closest points should be and then I suppose I'll have to sort of guess what angle that will be and use the calculated R lengths to solve for the distance between them (with a bit of trial and error to get the values as large and small as possible).

Mentor
The PolarPlotter2 Excel add-in is handy. It can be obtained http://www.andypope.info/charts/polarplot3.htm".

First create your data to be plotted. A column of angles and another two columns containing your R(angle) results. With the add-in installed, there will be an "Insert Polar Plot" selection on the "Insert" pull-down menu. Select it and follow the "wizard". easy-peasy.

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emilinus
Thanks! I can't seem to figure out how to rotate one of the ellipses but I can try to visualize it.

Mentor
Thanks! I can't seem to figure out how to rotate one of the ellipses but I can try to visualize it.

For the polar plot, just subtract the offset angle inside the cos() function:

R(Θ) = p/(1 + e(cos(Θ - Φ))

emilinus
That's very cool, thank you again. It actually matches my drawing, which is nice. I'm wondering though, because their orbits actually overlap, would their closest points be if they theoretically collided? From what I understand this isn't possible...and so my new dilemma lies in where to assume the closest point would be.

Mentor
That's very cool, thank you again. It actually matches my drawing, which is nice. I'm wondering though, because their orbits actually overlap, would their closest points be if they theoretically collided? From what I understand this isn't possible...and so my new dilemma lies in where to assume the closest point would be.

If the planet's periods are not locked into some fixed harmonic ratio (which can happen when the mutual gravitational perturbations are just right), then they are doomed to collide if they are in the same plane. So your minimum distance will likely be zero, as you've surmised. It's possible that the overlapping orbits are the result of the problem's author not checking to see what he had wrought.

mysterypengui
I'm actually doing the same question (unless I'm mistaken, I'm in the same class), and the numbers are pseudo-randomly generated, meaning any orbit is possible. When you have orbits that don't intersect, how would you find longest/shortest distance?

Mentor
I'm actually doing the same question (unless I'm mistaken, I'm in the same class), and the numbers are pseudo-randomly generated, meaning any orbit is possible. When you have orbits that don't intersect, how would you find longest/shortest distance?

Your choices, if the ellipses are not conveniently oriented so that the solution is obvious, is either a graphical approach (plot and measure with a ruler) or numerical (computer) method. I don't know of any convenient, direct mathematical approach.

zuberous
Your choices, if the ellipses are not conveniently oriented so that the solution is obvious, is either a graphical approach (plot and measure with a ruler) or numerical (computer) method. I don't know of any convenient, direct mathematical approach.

I still don't understand this problem, don't you clearly need to know the closest (perihelion) and furthest (aphelion) points to the sun aswell to make this easier?

we have both the semi major axis's and eccentricity's and the rotation at the perihelion point (for one planet)
how do we get on about finding the min and max distances?

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Mentor
I still don't understand this problem, you clearly need to know the closest (perihelion) and furthest (aphelion) points to the sun aswell to make this easier?

any detailed help would be beneficial!

The perihelion and aphelion points may or may not help depending upon the relative orientation of the two ellipses (draw a few and see... make sure that they share a focus).

If you know that one ellipse is rotated with respect to the other, and if that angle of rotation is not something trivial like 180 degrees, you'll have to plot them and "eyeball" the closest and furthest distances between their trajectories. You 'll then have to decide if you can get away with measuring from the plot, or solving numerically for the distances.