Walker constellation for hypersonic probes (RT2)

[RCgothic]

Premise/Introduction

Suppose you have a hypersonic probe aircraft (such as a spaceplane) and are using Remote Tech so that it needs to remain connected to KSC at all times. Aerodynamic forces will rip off any operational antenna other than the Reflectron DP-10, which has a maximum range of 500km. Therefore you need to provide a dense satellite relay network so that there is always an uplink satellite within 500km.

This will be a big undertaking, so I'd appreciate someone checking my math before I start building. Sorry for the lack of diagrams, I'll try and update with some later.

Walker Star Constellation.

Because of the short ranges involved, Kerbin Stationary Orbits are out, as are Molinya orbits. Walker Deltas are a little beyond my understanding (though I'd be very interested if someone could design one!), so I'm going with a Walker Star arrangement.

Walker Stars generally consist of a number of polar orbital planes, each of which contains a circle of relay probes. These orbital planes co-rotate, such that all the probes on one side are descending N-S, whilst those on the other side ascend S-N. There is a join between the two where the planes contra-rotate.

Each orbital plane provides coverage to a 'street', a parallel band which encircles the planet. The probes within each band are carefully phased with those in the next so as to maximise the width of each street, and to minimise the chance of collision over the poles. The orbital planes are evenly spaced except for the gap between the contra-rotating ones, which provides less consistent coverage because the relays cannot be placed in complementary phases.

Satellite Antennas

There's a choice to be made here: How do the probes talk to each other?

If equipped only with Reflectron DP10s, they need to stay within 500km of each other. This requires a large number of satellites per orbital phase, but as they are closer together they can tolerate a ground receiver more offset from the orbital plane, widening the streets and reducing the number of orbital planes.

If equipped with Communotron 16s then they only need to stay within line of sight of each other and within 500km of the surface point midway between. However as they are operating more at the extremity of the ground receiver's range they can tolerate less offset, requiring more orbitals. But they can also operate at higher altitudes, leaving the busy 70-100km zones clear.

Reflectron DP10

Taking a guess at sats per orbital n=9 gives:

Orbital angle between sats T= 40deg

Max in-plane angle between sat-ground SOG = T/2 =20deg

Kerbin's Radius R = 600km

DP 10 max range r = 500km

Orbital altitude (guess) a = 75km

Sat separation s = 2(R+a)sin(T/2) = 462km (in range!)

Altitude of Line of Sight L = (R+a)cos(T/2)-R = 34km (Tight, but I think I can be this precise)

Ground Station max in-plane range g = SQRT((s^2)/4 - L^2) = 233km.

These parameters all look good so far, albeit a little tricky. Now we calculate the street widths, beginning with the tolerable offset out-of-plane of a ground station for continuous contact with a particular orbital plane.

Chordal ground station offset w = SQRT(r^2 - g^2) = 442km

Angle ground station offset GOW = 2*sin^-1((w/2)/R) = 21.6deg (half-chord angle doubled so as to exploit right-angled triangle)

This is half the angle fully covered by one orbital plane, and half the maximum angle between contra-rotating planes that cannot beneficially phased. By phasing the satellites in the next plane out of phase (or almost out of phase), the areas of maximum out-of-plane coverage interlock to cover weaknesses in the neighbouring plane. In this case the equatorial sats in neighbouring planes will be out of contact with each other, but the signal can be passed north or south until the planes converge again. The maximum out-of-plane angle is calculated:

Max out-of-plane angle GOM = cos^-1(((R+a)^2 + R^2 - r^2)/(2*(R+a)*R)) = 45.7deg (Law of cosines)

Therefore the max angle between co-rotating planes is: 89deg

And contra-rotating is: 86.5deg

The minimum number of planes is 3. With a regular plane angle of 60 degrees, the seam angle is 60. Because the sats are quiteclose together there's little difference between the total coverage offset and the max offset. The total number of sats is 27. With the exception of the LoS altitude, the overlaps are all very comfortable and would look very tidy, though it does clutter 75km band.

Now let's try communatrons!

Communatron 16s

Similar to above:

Taking a guess at sats per orbital n=5 gives:

Orbital angle between sats T= 72deg

Max in-plane angle between sat-ground SOG = T/2 =36deg

Kerbin's Radius R = 600km

DP 10 max range r = 500km

Orbital altitude (guess) a = 190km

Sat separation s = 2(R+a)sin(T/2) = 928.7km (range was never going to be a problem here)

Altitude of Line of Sight L = (R+a)cos(T/2)-R = 39km (About equivalent difficulty to 4x10 DP10 sats)

Ground Station max in-plane range g = SQRT((s^2)/4 - L^2) = 466km.

So far so good. Now we calculate the street widths, beginning with the tolerable offset out-of-plane of a ground station for continuous contact with a particular orbital plane.

Chordal ground station offset w = SQRT(r^2 - g^2) = 181.2km

Angle ground station offset GOW = sin^-1((w/2)/R) = 17.4deg

Much narrower than the DP10s. Moving on:

Max out-of-plane angle GOM = cos^-1(((R+a)^2 + R^2 - r^2)/(2*(R+a)*R)) = 39.25deg (Law of cosines)

Therefore the max angle between co-rotating planes is: 56.6deg

And contra-rotating seam is: 34.7deg

The minimum number of planes is 4. With a regular plane angle of 50 degrees, the seam angle is 30. The total number of sats is 20. This is better than the DP10 equipped probes in pretty much every respect. The narrowness of the seam angle does imply I can do better though!

Refining sats per orbital n=6 gives:

Orbital angle between sats T= 60deg

Max in-plane angle between sat-ground SOG = T/2 =30deg

Kerbin's Radius R = 600km

DP 10 max range r = 500km

Orbital altitude (refined) a = 150km

Sat separation s = 2(R+a)sin(T/2) = 750km

Altitude of Line of Sight L = (R+a)cos(T/2)-R = 49.5km (much better)

Ground Station max in-plane range g = SQRT((s^2)/4 - L^2) = 378km.

So far so better. Now we calculate the street widths, beginning with the tolerable offset out-of-plane of a ground station for continuous contact with a particular orbital plane.

Chordal ground station offset w = SQRT(r^2 - g^2) = 327km

Angle ground station offset GOW = sin^-1((w/2)/R) = 31.6deg

Good. Moving on:

Max out-of-plane angle GOM = cos^-1(((R+a)^2 + R^2 - r^2)/(2*(R+a)*R)) = 41.65deg (Law of cosines)

Therefore the max angle between co-rotating planes is: 73.3deg

And contra-rotating seam is: 63.3deg

The minimum number of planes is 3. With a regular plane angle of 60 degrees, the seam angle is also 60. The total number of sats is 18.

This solution is chosen.

Phase

There are 6 satellites in each plane. This gives a phase angle between them of 60 degrees. There are three planes, so each is offset 20 degrees from the adjacent. This ensures they're will only ever be one probe passing through the pole at one time.

There is some loss of benefit in coverage through probes in adjacent planes being only 20 deg out of phase instead of the whole 30, but in this instance I'm not actually using the additional angle this effect could generate between co-rotating planes so this doesn't matter.

Conclusion

I'm going to use a constellation of 18 sats equipped with Communatron 16s in a 150km orbit, in 3 planes 60deg apart. That should give full global coverage and I reckon I can get those all in orbit in three polar launches. And let me tell you how relieved I am this didn't come out as a constellation of 80 or so like I thought it would! (But if my maths is wrong and it actually does, please tell me!

)

Can anyone do better? (Walker Delta?) Is there an issue with my maths? Sorry for the lack of diagrams, will try and add some later.

I'd be interested to hear people's thoughts, or if anyone has ever done anything similar before.

Edit:

I made an error related to the tangency of the signal to Kerbin on the street widths, resulting in an over-estimation of the coverage. See page 2 for revised details.

Edited September 24, 2015 by RCgothic

[The_Rocketeer]

I doubt you will get satisfactory coverage with fewer satellites or a delta constellation, and no way with fewer orbits. Six per orbit is also very convenient for calculating intervals since it divides both 360 degrees and 6 hours into an integer.

I haven't checked your calculations (maths is not a love of mine!), but it looks and sounds about right. I think your solution is about as straightforward as you'll get.

EDIT: An alternative I considered once was using mountain tops/highground for surface 'radio shack' antennas that kept LOS to the next peak, or had an uplink to a KSO relay. I doubt it would take less time/effort to implement than your satellite grid, but it would be an interesting alternative and would solve the transmission range and orbit creep issues.

Edited September 14, 2015 by The_Rocketeer

[Highguard]

Ground stations. Easier than orbital insertion and don't have to do station-keeping.

[Snark]

There's also the question of whether you actually need full-global coverage, or just a band around the equator. If you only need equatorial coverage, just make a ring of satellites there, won't take anywhere near as many as the multiple-polar-planes solution.

Also: You're not limited to the DP-10. You can run a Communotron-16 just fine, if you keep it stowed inside a closed cargo bay. Greatly expands your options.

Of course, if you want to stick with the DP-10 and full-global coverage just for the challenge of it, then by all means go ahead! Just sayin' there are other options.

[RCgothic]

Yeah, I like my polar expeditions so global coverage is kind of mandatory. And although I suppose I could put Communotron 16s in a cargo bay, that feels like cheating! Besides, I'm quite fired up about the challenge now.

I'm a little confused how ground stations would be any easier though. You'd have to navigate to and land at, what, 10 different locations? And there'd be massive blind spots near/on the surface caused by the horizon. At least with orbital sats if you get yourself into a terrain blind spot you'll have a sat pass overhead eventually.

With a satellite constellation I can get the satellites for each plane deployed in just one launch per each. Then it's just a case of equally spacing the six sats, which is easy. The hardest bit is getting correct phase and inclination to the adjacent plane, but again that's not very difficult with KER.

Thanks for the vote of confidence The_Rocketeer! I'll put some pictures up when it's done. Unfortunately my career mode suffered a setback with the off-screen unrecoverable death of Valentina Kerman, which has irritated me to the point of starting again from scratch (No deaths allowed in this Space Program!). I should manage deployment within about a week.

[nekogod]

Or put a second, longer range antenna on top in a fairing.

[March Unto Torment]

Christ, what the hell are you using hypersonic atmospheric probes for?

[RCgothic]

I like short flight times.

More seriously, it will allow me to launch spaceplanes into high-latititude orbits uncrewed, or as payload only, and to land science packages in remote locations. My space program is seriously risk-averse, and I figure there's no point sending a kerbal when a probe will do the same job for less mass.

And I suppose it's possible I could do the whole thing differently, but this sounded like a nice challenge!

[Snark]

I'm a little confused how ground stations would be any easier though. You'd have to navigate to and land at, what, 10 different locations? And there'd be massive blind spots near/on the surface caused by the horizon.

Certainly there are horizon issues for ground stations. However, that can be mitigated somewhat by landing them on mountaintops (or wherever elevation is high). RemoteTech's LOS calculations assume that each planet is a perfect sphere at elevation zero, and LOS is interrupted only if the path intersects that elevation zero. In other words, mountains, plateaus, etc. don't interrupt LOS. So if you can park a ground station a few km above sea level, it's visible for quite some distance (especially to a spaceplane that's flying 20 km up).

Yes, you'd have to land a fair number of them-- just how many would depend on the number of target sites.

Another challenge to consider: fly your missions without continuous comms coverage.

The RemoteTech flight computer allows you to fly at a fixed attitude (e.g. "zero roll, pitch 5 degrees up, heading whichever-direction-you-want", or whatever) at a specified throttle setting. So, let's say you want to go to Site X. First, lob a ground station on suborbital trajectory to land it somewhere reasonably close to X, ideally somewhere at high elevation. It has a long-range omni antenna that connects it to a ring of 3-4 high altitude satellites for communications back to KSC.

Then to fly your probe there, you take off from KSC, get headed in the right direction, set your bearing and throttle, and let it just cruise on autopilot until it gets close enough to the destination to regain control.

Necessary? No, not really. Just an alternate idea, in keeping with the KSP philosophy of looking at a situation and thinking "There's gotta be a more difficult way to do this."

[RCgothic]

Necessary? No, not really. Just an alternate idea, in keeping with the KSP philosophy of looking at a situation and thinking "There's gotta be a more difficult way to do this."

That's definitely a philosophy I can appreciate!

The thought of a hands-off autopilot for atmospheric flight is kind of terrifying though!

[Highguard]

That's definitely a philosophy I can appreciate!

The thought of a hands-off autopilot for atmospheric flight is kind of terrifying though!

There's always KOS (scriptable autopilot) too.

[TheBrisbyMouse]

How big of a probe are we talking? Wouldn't it be easier to just put a pod on it somewhere and have a kerbal fly it?

[Dunbaratu]

One thing I never understood is why the heck the Kerbals in RT never learned how to make ground-based connections with physical cables and seem only to know how to use radio signals and nothing else to make their comm network.

Making a low Kerbin network seems unnecessary when they could just build a few dishes on the ground around the planet.

IMO, any landed vessel on Kerbin should be considered to already have "line of sight" with KSC, even though it goes through the planet, because presumably it can hook into the ground grid network. Thus you can make a network of ground stations. (i.e. "We just got this signal from the Australian comms station...")

[The_Rocketeer]

One thing I never understood is why the heck the Kerbals in RT never learned how to make ground-based connections with physical cables and seem only to know how to use radio signals and nothing else to make their comm network.

Making a low Kerbin network seems unnecessary when they could just build a few dishes on the ground around the planet.

IMO, any landed vessel on Kerbin should be considered to already have "line of sight" with KSC, even though it goes through the planet, because presumably it can hook into the ground grid network. Thus you can make a network of ground stations. (i.e. "We just got this signal from the Australian comms station...")

Well, we had satellite networks up and working before we had the internet, and we'd never have had the internet at all if we hadn't gone crazy (as a species) over the telephone. Maybe Kerbals just aren't that interested in talking to each other for social reasons.

Besides, I think Jeb would ask: why spend decades laying cables or putting up telegraph poles to cover the whole planet when you can build 3 rockets and shoot a few dozen satellites into orbit for the same result?

[Highguard]

Interesting observation. The Victorians were running the British empire using telegraph cables around the world but the essential point was that there was 'somewhere' to run those cables to. Since there is no population outside KSC to connect with there'd be no particular reason to create a ground network unless it was easier/cheaper than a satellite constellation.

[severedsolo]

On the subject of alternatives:

I know it would add a bit of weight, but a KR-7 would also do the job - they don't break in atmosphere, and can cover the whole kerbin system.

Downside: if you aren't using RemoteTech XF you have to pick a Comm Sat and hope it doesn't go below the horizon.

[Dunbaratu]

Well, we had satellite networks up and working before we had the internet

Who said anything about the internet? Before the internet we had computers communicating with computers across cable - it just wan't done using a unified worldwide protocol yet. But for a space agency, yes they did talk to ground stations with cables long before there was an internet.

[RCgothic]

So I'm getting hit by an overheating kraken - everything overheats suddenly all at once. My launcher explodes about at about 8 min MET in a 90deg 205km x 150km parking orbit at almost exactly 90'S every time. Ideas?

Edited September 24, 2015 by RCgothic

[The_Rocketeer]

Who said anything about the internet? Before the internet we had computers communicating with computers across cable - it just wan't done using a unified worldwide protocol yet. But for a space agency, yes they did talk to ground stations with cables long before there was an internet.

Nobody said anything about the internet. Oh wait, I did.

Anyway, you're right of course, but then you're assuming a few things:

• There's somewhere else on Kerbin that Kerbals to want to connect to
• That place/those places cover the entire surface of Kerbin
• Kerbals would prefer to build infrastructure all over Kerbin than fling satellites into pretty orbital patterns

Most of all though, the answer here is pretty obvious: the physics of the Kerbol system don't allow it . To have a physical connection to other surface locations, you'd need a monstrous part count, which is forbidden by the First Law of Kerbal Dynamics: Frame Rate dissipates proportionately to the number of parts. The Second Law of Kerbal Dynamics is also relevant: That which is too far away exists only in mathematical theory. So, a cable reaching across one of Kerbin's oceans would just disappear after a certain distance. No signal.

I mean, seriously, it's like you never even studied rocketry!

@RCgothic, that's lame. I've been borrowing some of your numbers for a network of my own and didn't have any bug issues. You could try opening a support thread for it, but if I had to guess I'd say there's probably some small element of your design or structure that the Kraken hates.

In other words, try building a new rocket for the same job and see if it still happens.

[RCgothic]

It's not just the launcher. It hates the relays too. Even separate any vessel in the physics bubble going over 90'S chain detonates.

And I just now discovered two flaws in my maths. The first was just a transcription error, substantive portions unaffected.

But my street widths are wrong. I assumed I'd get the full 500km range of the transmitters, but Kerbin blocks line of sight before that. This means the streets are narrower than stated.

An 18 sat 3-plane solution does exist, but for a slightly higher orbit and unequal plane angles. I'm running the numbers now.

- - - Updated - - -

180km orbit, 63.5deg between regular planes, 53deg seam angle.

Now having a look at slightly inclined planes as a work around for having to fly over the pole.

Edited September 24, 2015 by RCgothic

[RCgothic]

I used Inventor to physically model the situation. The options to work around the polar kraken:

Various Walker Deltas are actually less efficient because you end up with more seams. I don't think complete coverage can be done with a standard delta and fewer sats.

It's possible to make a Walker Star with the crossing over the equator and the seam directly over the poles and use less DV (one orbit inclination zero, two orbits inclination +/- 63.5 deg), but it's less aesthetic than the other options.

A hybrid Star/Delta with inclination 85deg will probably avoid the kraken whilst providing total coverage and almost as good coverage as the 63.5/54deg 180km described above. It's just a little harder to phase the separate planes.

[RCgothic]

Turns out 89.5deg inclination is good enough to survive the kraken, and I was able to save my first launch from a prior quicksave by burning immediately normal.

The first six are lofted and positioned! The orbits are a slightly scrappy +/-50m from 180km, but the orbital periods are correct to about 2 thousandths of a second per orbit which should keep them nicely in line!

http://imgur.com/a/xumSn

Apparently I can't figure out how to embed an Imgur album in a post.