Orbital mechanics puzzle

[johndallman]

Originally Posted by DataPacRat

(Using the new beam to enhance the drive would bring that up to around 210 km/s.)

Are you sure this is worth the effort?

I hadn't really considered actively maintaining the orbits, having just assumed the cost would be too high.

You aren't going to be using a lot of thrust for the maintenance, but you don't actually have any choice about doing it. In a dynamic system this complex, nothing stays where you put it unless it's in a resonance of some kind. Trojan points are the easiest to understand and use, but 3AU range is too short to do everything with those. Your string of collimators along Saturn's and Jupiter's orbits won't stay in place: those orbits are not railroad tracks.

part of the purpose of the laser factory is to be a flashy cover for the much more important bit of industrializing they'll be doing at Saturn

Is it a good idea to draw this much attention to Saturn? Especially Saturn's atmosphere?

Originally Posted by DataPacRat

One though I expected to receive more comment by now: eccentric orbits. Eg, having a dozen collimators, each of whose apohelion is spaced at a different o'clock, so that they hang out near the planet while it's in their section of its orbit. I seem to recall having once read a proposal for a set of 'cycler' stations between Earth and Mars in such a configuration.

An Earth-Mars cycler is fortuitously possible because of some coincidental properties of the orbits of those two planets. It doesn't generalise, and with 3AU range, getting really close to the planet didn't seem worthwhile. You might be able to do something useful with somewhat eccentric orbits, but the Trojan points are just simpler.

[Fred Brackin]

Originally Posted by johndallman

Trojan points are the easiest to understand and use, but 3AU range is too short to do everything with those. .

So how about building fewer but bigger collimators? If you can get range up to a little over 7 AU you can put relay stations in Jupiter's Trojan points (where there may even be enough material you don't need to import any).?

With 7 AU collimators in Jupiter's Trojans you can reach those over half of Saturn's orbit and then probably reach anywhere in the inner system without further relays.

With one more collimator at the antipodean point in Jupiter's orbit you have full coverage and if you start things at the right point you've got a decade or more to build it before it's needed.

[DataPacRat]

Originally Posted by johndallman

Are you sure this is worth the effort?

The 180 km/s/year freighter is already just about at the limits of drives that can be built. Using trickery to put some of the engine at Saturn, creating a laser-sustained plasma inside the rocket nozzle to increase the temperature of the exhaust and thus its escape velocity... is, for the foreseeable future, the only available way to improve on that design. (Most of Our Heroes' research into drive tech is being applied to the He3-burner they plan on eventually installing into their starship.) Since most of the effort of heading to Saturn is a sunk cost, that's going to happen whether or not they install the ionosphere laser, then as long as they can work out the logistics of beam distribution, it seems more than worth the cost.

You aren't going to be using a lot of thrust for the maintenance, but you don't actually have any choice about doing it. In a dynamic system this complex, nothing stays where you put it unless it's in a resonance of some kind. Trojan points are the easiest to understand and use, but 3AU range is too short to do everything with those. Your string of collimators along Saturn's and Jupiter's orbits won't stay in place: those orbits are not railroad tracks.

How long does it take for such instabilities to become noticeable? Would it take less beam-time to keep collimators in Saturn co-orbit to stay in such an orbit, compared to stabilizing them elsewhere?

Is it a good idea to draw this much attention to Saturn? Especially Saturn's atmosphere?

The Grand Tour is going to start with Comet Halley, then hit Jupiter's moons, then Saturn and its moons, then Uranus's moons, and so on. On the surface, it's a simple publicity stunt - part of the idea is to try drawing attention to as much of the whole trip as possible, so that the more important subtle aspects get lost in the noise of all the hullabaloo.

(It may not be a /great/ plan, but it has the advantage of leveraging what resources Our Heroes have available to them, improving on their strengths, and hiding some of their weaknesses out where it's hard for anyone to get close enough to see them.)

An Earth-Mars cycler is fortuitously possible because of some coincidental properties of the orbits of those two planets. It doesn't generalise, and with 3AU range, getting really close to the planet didn't seem worthwhile. You might be able to do something useful with somewhat eccentric orbits, but the Trojan points are just simpler.

Fair enough.

Originally Posted by Fred Brackin

So how about building fewer but bigger collimators? If you can get range up to a little over 7 AU you can put relay stations in Jupiter's Trojan points (where there may even be enough material you don't need to import any).?

With 7 AU collimators in Jupiter's Trojans you can reach those over half of Saturn's orbit and then probably reach anywhere in the inner system without further relays.

With one more collimator at the antipodean point in Jupiter's orbit you have full coverage and if you start things at the right point you've got a decade or more to build it before it's needed.

To a first approximation, doubling the surface area of the collimators doubles their range. So if the design was originally 1.1 km in diameter, upping that to 1.6 km would crank up the range from 3 AU to 7 AU. Harder to build, harder to maneuver into place (especially when initially trying to pretend they're just fuel depots) - but fewer would be needed, so it might be worth the redesign.


Hm... how about a half-dozen collimators, in a rosette, each one 60 degrees apart: one at Jupiter, two in the Trojan points, one in a counter-planet orbit, and two in the counter-Trojan points? How large a range would they have to have, so that Saturn's entire orbit is covered? Putting together the answer seems like it would involve trigonometry I've been out of school too long to easily remember how to do... let's see:

Two triangles, sharing a side: ABC and BCD. A is the sun, D is Saturn, B is a Jupiter-orbit collimator, and C is halfway between B and another Jupiter-orbit collimator. Angle BAC is 30 degrees, angle ABC is 90 degrees, angle ACB is 60 degrees, and angle DCB is 90 degrees. Side AB is roughly 5 AU, and sides AC+CD is 9.5 AU. How long is side DB?

Since ABC is a 30-60-90 triangle, then side AC is 5 AU * (3^.5 / 2) = 4.3 AU (making side CD 9.5 - 4.3 = 5.2 AU); and side BC is 2.5 AU. BCD is a right triangle, and we have the two right-angle lengths, so the hypotenuse is... 5.3 AU.

So, if my math is right, then if the collimators have at least 5.3 AU range, then half a dozen in equidistant Jupiter orbits would always have one in range of Saturn. Since Jupiter's aphelion is 5.45 AU, then they'd cover the entire inner system, save for (sometimes) the bit right next to the sun, halfway inside Mercury's orbit. A 5.5 AU range isn't much bigger than 5.3, and would fill that gap. Does that seem like a feasible plan for inner-system coverage?

[Anthony]

My big problem with your 'collimators' scheme is that it's not clear they work at all; I think your beam quality gets degraded every time it bounces. Aside from that, the range at which a diffraction-limited lens can focus on a fixed size target varies linearly with the size of the lens (meaning the volume of space varies with the cube of lens size), and the weight of the focal array generally varies somewhere between the 2nd and 3rd order of lens size, so there's very little benefit to using collimators instead of just putting one really really big focal array at the start -- mostly you'd use reflectors if you need to bounce around opaque objects or hit a target from an angle other than directly from the beam source.

[DataPacRat]

Originally Posted by Anthony

My big problem with your 'collimators' scheme is that it's not clear they work at all; I think your beam quality gets degraded every time it bounces. Aside from that, the range at which a diffraction-limited lens can focus on a fixed size target varies linearly with the size of the lens (meaning the volume of space varies with the cube of lens size), and the weight of the focal array generally varies somewhere between the 2nd and 3rd order of lens size, so there's very little benefit to using collimators instead of just putting one really really big focal array at the start -- mostly you'd use reflectors if you need to bounce around opaque objects or hit a target from an angle other than directly from the beam source.

The big problem with using just one really really big focal array is that it would need really really lots of fuel and rocketry to get it there. Or, alternatively, it would require being able to use in-situ resources to make it.

Mention has been made in-setting that designs exist for an 'aerosol' lens, made up of glass beads with a highly nonlinear index of refraction, which allows a laser to put forces on them to trap them into a fresnel-like lens. A 300-meter-across version could focus a 100 MW beam, with a transmission efficiency of 96%, or focus it at 1 AU at 94% efficiency. An alternate design is a paralens, of alternating plastic and nothing, the 300 m version of which is merely 90% efficient at 1 AU. Both of them have a parameter of around 0.3 kg/m^2, about 25 tons each for the 300 meter versions.


The standard, relatively inexpensive fission-heated 'teakettle' unmanned freighter can haul about 40 tons from Earth orbit into one of Jupiter's Trojan points in around 3 years. Bringing anything there any faster would require diverting more valuable craft from trade runs currently used to bring stuff to Mars and Earth to sell. (And which also form the equivalent of second-strike nuclear submarines, to drop rocks on the heads of anyone who attacks their home base.)

[Anthony]

Originally Posted by DataPacRat

The big problem with using just one really really big focal array is that it would need really really lots of fuel and rocketry to get it there.

Same total as you'd need for multiple arrays; sure, each array is cheaper, but you need a lot more of them.

[DataPacRat]

Originally Posted by Anthony

Same total as you'd need for multiple arrays; sure, each array is cheaper, but you need a lot more of them.

Trying to figure out how big 'a lot' would be is the question that inspired this thread. If the collimators are scaled up to a range of 5.5 AU, then half-a-dozen could cover everything inwards of Jupiter; though I'm still hoping to work with 3-AU-range ones, if that doesn't require absurd numbers of them.

[johndallman]

It looks from the numbers you've given us that range is proportional to surface area. So the area of the plane of the solar system that a collimator covers is proportional to the square of surface area, and the volume is proportional to the cube of surface area.

So if mass is approximately proportional to surface area - which it looks like - and cost is proportional to mass, but effectiveness is proportional to square of surface area, then as few large collimators as practical is the cheapest option.

[Fred Brackin]

Originally Posted by DataPacRat

I'm still hoping to work with 3-AU-range ones, if that doesn't require absurd numbers of them.

It does.

Working backwards, to cover the inner system you need a ring of 6 relays with 3AU range at a distance of 3AU from the Sun. That forms a hexagon with a total circumference of 18 AU or 3 AU per side. So each relay can always reach the 2 relays on each side of it and from there there ought to be some configuration that will reach anywhere in the inner system.

Then, to get to Saturn out at c. 9.5 AU you need 3 more rings or relays each larger than the one inside of it. They have to be closer than 3AU to the other rings because you want to relay beams when the relays aren't at opposition to the nearest other relays.

It might not be quite as bad as 6+12+24+48 but it could be close.

So longer range or maybe actively maneuver the relays into the proper position and keep them there with some low thrust but zero reaction mass system like a mag sail.

[Grouchy Chris]

I can do it with 25 collimators. See my proposed array here.

This array has 17 collimators in orbit at 7.7 AU, so that one is always in range of Saturn, one always in range of Jupiter, and each is within range of its two neighbors. A single collimator is at Jupiter, and another 7 collimators orbit the Sun at 3 AU, so that one is always in range of Jupiter.

This array was designed for minimum number of collimators, at the expense of requiring a large number of connections to some targets, and also having a single point of failure at Jupiter. It also assumes minimal station-keeping for the collimators. This strikes me as doing things on the cheap and asking for trouble, but that's what the customer asked for.

Originally Posted by DataPacRat

Hm... For a possible approach, what about arranging to have a set of relays every 3AU along in Saturn's orbit, ditto Jupiter, and a single relay halfway between the planets' orbits?

That's a lot more expensive than placing a ring of relays every 3 AU between the orbits of Saturn and Jupiter, with a relay at Jupiter. In practice, the spacing has to be a bit less than 3 AU to keep both Saturn and Jupiter in range of the relays at all times.

Originally Posted by whswhs

Better to get the power straight to the Asteroid Belt. Its orbital radius is only 2.8 AU, so its circumference is just under 9 AU; you only need three relays, spaced 120° apart.

An orbital radius of 2.8 AU gives an orbital circumference of 17.6 AU.

Originally Posted by Fred Brackin

Working backwards, to cover the inner system you need a ring of 6 relays with 3AU range at a distance of 3AU from the Sun. That forms a hexagon with a total circumference of 18 AU or 3 AU per side. So each relay can always reach the 2 relays on each side of it and from there there ought to be some configuration that will reach anywhere in the inner system.

Then, to get to Saturn out at c. 9.5 AU you need 3 more rings or relays each larger than the one inside of it. They have to be closer than 3AU to the other rings because you want to relay beams when the relays aren't at opposition to the nearest other relays.

It might not be quite as bad as 6+12+24+48 but it could be close.

That's overkill. 3 more rings gets you past Saturn, when a ring inside of Saturn's orbit that can always reach it does well enough. Furthermore, you don't need the entirety of every ring. A single connection between non-adjacent rings suffices. In my design, I place that connection at Jupiter.