Eratosthenes, angles, and satellites

[henryrasia]

Hello all! I would like to ask this question on the KSP forums and it's about geometry.

In case you don't know, Eratosthenes was the Greek man who calculated the circumference of the Earth by investigating angles of elevation at different latitudes. Summarized in this diagram:

I'm trying to resolve an unsettlingly similar problem, but instead of the sun satellites. The question basically is: What is the mathematical relationship between angle of elevation (aka altitude in the horizontal coordinate system) and latitude from the point of zenith (theta in the diagram).

But here's the catch: I can't assume rays are parallel as Eratosthenes did! This is because the Sun can be considered infinitely far away, when a small satellite has a definitive answer.

This adds an annoying extra angle that I need to account for but I do not know how. This is a simplified diagram of my problem:

Trust me, my WIP version is WAY messier.

See that missing angle between the pink and blue lines? That's the one I can't figure out. BTW pink lines are parallel, the circle is the celestial body, and everything else I can answer questions for. The answer should be as general as possible, but the final solution should be phi in terms of epsilon. Known distances are: radius of circle, distance from S to Sat2, and angle epsilon. If you could help me I'd be very grateful, as my end of year research project depends on this and I'm stuck! Also, if you could please show and explain the work (or simply reasoning) that'd be perfect!

Thanks in advance,

Henry

Edited February 5, 2015 by henryrasia
Removed values. This is supposed to be a general problem!

[PakledHostage]

Angle S-V-Sat2 is epsilon plus 90° (141.87°). Angle phi is 29.41 degrees. The missing angle between the pink and blue lines is 180° - phi - (epsilon + 90°) = 8.72°

[SuperFastJellyfish]

It looks as though all you need to do is subtract those other two surrounding angles from 90, if that is a zenith line and a tangential line creating a right angle in total. I'm not an expert, and that's all i got.

[henryrasia]

PakledHostage thanks for your quick answer! Regrettably, however, the question is to find phi in terms of epsilon (or any other known value, written above). I just mentioned the missing angle because that sucker is the one that's keeping me from Eratosthenes' simple answer. I'll edit the picture to reflect that. Sorry for the confusion!

BTW, the answer needs to be general, that picture is just one of the possible angle combinations, for better visualization.

Thanks again!

[PakledHostage]

Regrettably, however, the question is to find phi in terms of epsilon (or any other known value, written above).

Maybe you can help us understand better by telling us what you are trying to do? I don't think I understand your question well enough to answer any better than I did earlier.

[henryrasia]

This is a smaller part of a bigger project that I'm doing for the IB Extended Essay (if you know what that is), which is a very extensive research paper (well, as extensive as it can be for a highschooler ) due as a draft (for me) at the end of this school year.

Bottom line is, I need an equation with phi at one side and known numbers/variables on the other. This will be used later when I decide specifically what elevation angle (altitude?) and other parameters to use in my simulation. I previously arrived at the obviously wrong conclusion that phi=2*epsilon, it was then that I figured out the mystery angle was messing up my beautifully simple application of Eratosthenes' experiment. And now I need to figure out what is the true equation. The real killer? I need to explain everything on the paper, so using geogebra directly is not an option...

I've asked here after a lot of thinking myself. And the furthest I've got is having a gut feeling that the solution might rest with using auxiliary lines drawn to suit our needs. But I haven't figured out if that's even true yet. It might very well be hiding under my nose but I can't see it .

But hey! You're the Figaro GPS guy! How are you doing? Maybe you could actually help me with the second part of my project, which revolves around GPS line of sight and such! But that's enough for now, I don't want to bore you.

Henry

[PakledHostage]

Maybe I'm a bit thick, but I still don't understand what you are asking? I have a sneaking suspicion that you're over complicating something though. Do you want to find the elevation (angle above the horizon) of a satellite that is located in an equatorial orbit at a known orbital radius? What are you going to be doing with the equation that you're trying to find?

[henryrasia]

Right now I just want an equation that looks something like this (À is supposed to be pi):

Φ=ε OR Φ=2ε OR Φ=ε*(À/2)-À

OR some equation like that (the true one, of course) that lets me figure out Φ given an ε.

If it's hard to visualize, imagine that point V can slide around the surface of the circle, but only between U and the S-Sat2 line. That freedom of position is what's going to change the value of ε, and therefore that of Φ (and that's why I need a general formula). Sat2 can also get closer or farther from the circle, which will change the positions of the red and blue lines, and therefore will also change the value of angle ε.

I'm considering distance S-Sat2 to be of a known value A, and the elevation angle to be of a known value ε. I also know the radius of the circle ® if that's of any use.

That's what I need, a general formula for Φ in terms of ε. (but not exclusively of ε)

___________________________________________________________________________________

Now why am I even posing this question.

My project is, basically, to figure out mathematically what is the least amount of GPS satellites around the Moon needed for full coverage at all times.

The mathematical way doesn't seem to be working out, so instead I've began a simple algorithm that's going to randomly generate orbits and see if they match the requirements. For that I'll use spherical coordinates in conjunction with Keplerian elements to determine visibility of satellites on the surface. But since the visibility cone on the ground has a safety margin elevation angle above the horizon (which I have yet to decide exactly how wide it'll be), I need to know what latitudes and longitudes, deviating from zenith, will be able to see the satellite.

Maybe I'm overcomplicating this, but it's the only method I've come up with that can feasibly give me results that I can write about in the paper. All other methods are basically guessing, which is not exactly scientific. I realize that this is brute-force computing, but if you have any better ideas please let me know!

Sorry for the long wall of text, but I hope things are clearer now,

Henry

Edited February 5, 2015 by henryrasia

[PakledHostage]

Right now I just want an equation that looks something like this (À is supposed to be pi):

Φ=ε OR Φ=2ε OR Φ=ε*(À/2)-À

OR some equation like that (the true one, of course) that lets me figure out Φ given an ε.

OK. If you know the distance along your line S-Sat2 (orbital radius of Sat2), you know the spherical body's radius and you restrict yourself to a 2D problem with the satellite in an equatorial orbit then you've got a basic high school trigonometry problem. You know the length of two sides of your triangle and one of the angles (i.e. you measure ε and the angle S-V-Sat2 is ε + 90). From that, you can figure out the relationship between Φ and ε. I won't tell you how to do it because doing your homework for you doesn't do you any favours. Sorry.

[Hint: See Solving SSA Triangles]

[henryrasia]

I won't tell you how to do it because doing your homework for you doesn't do you any favours. Sorry.

Hey! This isn't a mere HOMEWORK, this is RESEARCH!

Seriously, though, I'll try my hand at it and report back.

BTW, I don't think he kind of orbit of Sat2 is important at all. I only need to know the angles at a particular moment, which happens to be when it's right on the edge of the safety margin.

[PakledHostage]

BTW, I don't think he kind of orbit of Sat2 is important at all. I only need to know the angles at a particular moment, which happens to be when it's right on the edge of the safety margin.

OK. I assumed that Φ was to be the latitude of the observer at V. Just be sure that you restrict yourself to the 2D case if you're going to solve it using an SSA triangle.

[GoSlash27]

You have 2 "phi"s in that diagram.

If you know phi at the planet center and S is given, then you have angle center>sat2>u.

Given that, distance from center to Sat2 is tan(thetaC+S)

If point V is given, then cos(thetaC)R= gives you a reference point in a right triangle involving V and Sat2.

Having defined that point (P), angle P>Sat2>V= your missing angle.

This is probably something that I could explain better in photoshop...

Best,

-Slashy

- - - Updated - - -

HTHs,

-Slashy

[Yasmy]

First, that's no epsilon.

ξ = xi

ε = epsilon

Let's call the missing angle beta, β.

Then β = 90 - Æ - ξ

or β = À/2 - Æ - ξ if you prefer angles in radians.

Just add up all the angles in a 90 degree or 180 degree section around point V.

As far as I can tell, that is what you asked for.

If you need the value of ξ in terms of r, (Sat2-S) and Æ, check out PakledHostage's answer.

You know two sides of a triangle and the angle between them.

[henryrasia]

ξ = xi

ε = epsilon

Oops, pardon my ignorance...

But thank you guys for the answers! The question was to describe phi in terms of epsilon, radius, and altitude of Sat2. PakledHostage made me see that the problem is a silly SSA triangle problem where I should apply the law of sines. The mystery angle ended up being completely negated! I'm trying to simplify my resulting equation, as it appears it does not change when radius or altitude change, which is odd to me

So yes, the answer was simple, but all I need to do now is simplify it, but I think I can do that on my own. I'll come back if I have problems.

Thanks again! It was very helpful!

[PB666]

I have looked at your problem.

You don't necessarily need to know r but you need to know the ratio of the GPS's altitude to r, which you did not provide as a given. Label this kr

Another hint, the chord of angle = 2 * sin halfangle. Psi forms a chord of the angle between the reciever, the celestials center and the satellite where they intercept the celestials surface. This r * cos psi falls short the surface and the chord can be used to solve for that shortfall (q). The angle between 'down' and reciever for the satellite (I will call gamma) forms a triange = ((kr+ q)/sin gamma * cos gamma = r sin psi. Epsilon (E) = 90 - gamma - psi. You can solve by substitution. You will probably find the problem is slightly recursive.