Quick simple math question which I am bad at

[RainDreamer]

How do you calculate the altitude where a remote tech 2 dish cone cover the whole body which the dish is orbiting around? Like how on the RT2 tutorial page says the DTS-M1 covers kerbin at 1600km. I thought I should do something like:

(body radius * tan(dish cone angle/2)), but the result is always off by quite a bit.

Edited January 8, 2015 by RainDreamer

[stibbons]

You're close!

tan(A) is opposite over adjacent, so for dish cone angle x, tan(x/2) = radius/distance. So distance = tan(x/2)/radius.

[RainDreamer]

So, say I am using a DTS-M1 with 45 degree cone angle, and I want to cover kerbin, with its radius, according to the wiki is 600km, I would have:

tan(45/2)/600 = 0.00092975289km? What O_o Used google calculator for that. That can't be right.

[stibbons]

Apparently I'm good enough at trigonometry, but fail at basic arithmetic sorry. That should be

distance = radius/tan(x/2)

For the DTS-M1 at Kerbin that comes to 1075.5km.

[*Aqua*]

Try this web site: http://ryohpops.github.io/kspRemoteTechPlanner/

It doesn't calculate the angles but maybe it can help you a bit with the other stuff.

[RainDreamer]

Hmm, still about 600km off the one on the tutorial list. I guess we need to add the radius , because kerbin is a sphere. So, the correct formula would be (radius/tan(x/2) + radius, I supposed. There is a little ~100km inaccuracy though, hmm.

[Jimbimbibble]

It's impossible for a cone to provide total coverage of a sphere. There will always be a region near the poles where the satellite cannot provide coverage. This region is smaller the farther away you are and you only get full coverage when you are infinitely far away. However, it seems like you're interested in how far away you'd have to be to utilize the full range of the dish's coverage so that the dish cone is tangent to the planet. In that case, the expression you're looking for is r=R/sin(theta/2)-R where r is the orbital altitude, R is the planet's radius, and theta is the dish cone angle. The minus R comes from the fact that KSP measures your altitude from the surface, not the center of the planet. An expression for the maximum latitude a dish can serve (assuming the target's altitude is 0m, the sat is in an equatorial orbit, and r is at least the minimum value you calculated from the previous equation) is coverage=90-arcsin(R/(r+R)). As you can see, the only way to serve a sea level target at the pole with a sat in an equatorial orbit is if r is infinite.

[RainDreamer]

Hmm, yes, that is what I mean, the minimum altitude where the cone is fully used, not that it fully cover the whole thing.

So, using the expression you give, for the DTS-M1 at kerbin, we would have:

theta = 45 degree

R = 600km

r=(600/sin(45/2)) - 600 = -1831.59152348 km

...Something seems seriously wrong here.

Edit: Ok, loaded that expression in Wolfram|Alpha, and using degrees instead of radians, the result is: 967.87556

Still quite a bit away from the supposed answer of 1600km

Edited January 8, 2015 by RainDreamer

[GoSlash27]

RainDreamer,

I concur with jimbimbibble.

A cone with an apex angle of 45* will lie tangent to Kerbin's surface at 968km altitude.

Perhaps something's off in the tutorial or specs?

Best,

-Slashy

Edited January 8, 2015 by GoSlash27

[Jimbimbibble]

Maybe that 1600km is measured from the planet's center? That would make sense.

[LethalDose]

Maybe that 1600km is measured from the planet's center? That would make sense.

Yeah, it's all measure from the planet's center.

[RainDreamer]

Ah, that explains it. If we doesn't subtract the 600km it does come very close to 1600km, the rest are probably rounding up. I guess I should aim for a few dozen km higher after calculating then.

[Anquietas314]

Shouldn't the optimal altitude for satellites be the one that puts either (some small distance above) Kerbin's poles or the other satellites in the cluster at the upper limit of the range of the antennae? Depending on what you're optimizing for; poles if you're going for coverage (requires more satellites) or satellites if you're going for minimum number of satellites (3 if you have enough range). Obviously you still don't get full coverage of the planet's surface, but you will have coverage anywhere in a reasonable orbit. As a specific example, at Kerbin for maximum coverage you want each satellite to be able to reach, say, 100km above the north/south poles (to allow for low polar orbits without having satellites in a polar orbit). If your satellite range is 1600km, assuming Kerbin doesn't get in the way, you want the satellites to be at an altitude of sqrt(1600^2 - 700^2) - 600 = 838.74km (rounding down).

Edited January 8, 2015 by armagheddonsgw

[GoSlash27]

Ah, that explains it. If we doesn't subtract the 600km it does come very close to 1600km, the rest are probably rounding up. I guess I should aim for a few dozen km higher after calculating then.

Mathematically, you'd aim for whatever you calculate.

1,568 km is the center-apex distance from which a 45* cone will lie tangent to Kerbin's surface. Realistically, I'd round the answer to 1,570 since I always round to 3 digits. Going higher than that will just waste radiated power by shooting some of it past the surface off into space.

Best,

-Slashy

Edited January 8, 2015 by GoSlash27