Orbital mechanics puzzle

[DataPacRat]

Here's a conundrum I can only barely begin to solve; this seems to be a likely forum to find someone who might know a solution, or might know how to work out a solution, or might know someone else who might know how to work out a solution... Any takers?

* Our Heroes have built a laser at Saturn. It has a range of 3 AU.
* Our Heroes can place beam collimators into various orbits, around the sun or around other planets. Each collimator can accept a beam, from Saturn or from another collimator, and send it on for another 3 AU.
* Our Heroes don't want to use excessive numbers of beam collimators, because they're somewhat expensive to build and take a while to get placed into their orbits.
* Our Heroes want to be able to apply this beam to as much of the Solar System as they can; concentrating on the routes rockets take between various planets, especially between planets in neighbouring orbits.

* How should the collimators be placed, so that as they move around in their orbits, the paths of 3-AU-or-less neighbouring collimators will provide the greatest coverage?

[johndallman]

Which planets are the sources and destination of the traffic of most interest?

I have to say, Saturn is not a good place to have done this. Since your minimum distance from Earth is about 8AU, and from Mars about 7.4 AU, you are going to need several relays to get your beam into the inner system. Down in the inner system where things move really fast, the Venus Trojan points are traditional; using both of them gives you redundancy and means one will almost always be in a good position. They're also in range of each other, but that's also true of the Earth and Mars Trojan point pairs.

Your relay points into the inner system are more of a problem. Jupiter's orbit is 4AU inside Saturn's, which means you can't use it directly. I think you're going to have to have the first relay being actively positioned; there doesn't seem to be a convenient orbit for it that will keep them 2.5-3AU in from Saturn for any extended period. You need another relay at a radius of about 4AU, which again has to be actively positioned, and another at about 1.5 AU which needs to stay approximately aligned with Saturn, which also has to be actively positioned.

Once you've relayed into the inner system, the Trojan relays give you complete coverage out to about 3.5AU and conditional coverage that can stretch as far as 4.6AU.

I'm not asking if the beam is for propulsion or a weapon since Niven's rule tells us it will serve for both if it's capable of either. But the relays will get attacked soon once it is used as a weapon.

[MatthewVilter]

Another important question might be what kind of transfer orbits people generally use (which depends of how much delta-V is available). It sounds like the tech your heroes are working with might be advanced enough that people just fly more or less from point a to point b but I guess in either case the only thing consistent about travel paths is going to be their start and end points.

I agree that it is pretty hard to setup a relay from Saturn to the inner system. I would actually go for an actively positioned system. You could use solar sails to keep a line of collimators in between Saturn and the inner system but I think they end up being prohibitively large. On the other hand I think you could make the sails much smaller and more effective if you power them with the laser! If the laser is powerful enough and the collimators are light enough then you only need to spend a small percentage of its time on station keeping blasts. It also helps that you only need to station keep collimators in high (outer system) orbits because the absolute difference in velocity between their desired path and what would be their normal orbit is small.

[DataPacRat]

Originally Posted by johndallman

Which planets are the sources and destination of the traffic of most interest?

Our Heroes' home base is the asteroid (442) Eichsfeldia; the most regular trade route is Eichsfeldia to Phobos to Luna and back.

Depending on how various sub-plots go, the Saturn laser is going to be emplaced during a Grand Tour of all the planets of the solar system - from Saturn, to Uranus, Neptune, Pluto, then back to Mercury to work outwards again. Part of the hope is to use the beam during the trip.

I have to say, Saturn is not a good place to have done this.

Our Heroes are applying a technique to use some interesting properties of Saturn's ionosphere as a lasing medium. The other sites where this trick can be pulled are politically infeasible, so it's either a Saturn-based beam or no beam at all.

Since your minimum distance from Earth is about 8AU, and from Mars about 7.4 AU, you are going to need several relays to get your beam into the inner system. Down in the inner system where things move really fast, the Venus Trojan points are traditional; using both of them gives you redundancy and means one will almost always be in a good position. They're also in range of each other, but that's also true of the Earth and Mars Trojan point pairs.

My initial musings for the inner solar system were to have at least a pair of relays somewhere along Mars's orbit, such as its Trojan points, or the Trojan points and the counter-Mars point; but I'm open to other approaches.

Your relay points into the inner system are more of a problem. Jupiter's orbit is 4AU inside Saturn's, which means you can't use it directly. I think you're going to have to have the first relay being actively positioned; there doesn't seem to be a convenient orbit for it that will keep them 2.5-3AU in from Saturn for any extended period. You need another relay at a radius of about 4AU, which again has to be actively positioned, and another at about 1.5 AU which needs to stay approximately aligned with Saturn, which also has to be actively positioned.

Once you've relayed into the inner system, the Trojan relays give you complete coverage out to about 3.5AU and conditional coverage that can stretch as far as 4.6AU.

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?

I'm not asking if the beam is for propulsion or a weapon since Niven's rule tells us it will serve for both if it's capable of either. But the relays will get attacked soon once it is used as a weapon.

The former is the plan, the latter will be a nice potential benefit. :)

[DataPacRat]

Originally Posted by MatthewVilter

Another important question might be what kind of transfer orbits people generally use (which depends of how much delta-V is available). It sounds like the tech your heroes are working with might be advanced enough that people just fly more or less from point a to point b but I guess in either case the only thing consistent about travel paths is going to be their start and end points.

Our Heroes' typical inner system vessel uses the design seen here; with a typical load of fuel and cargo, it gets a delta-vee of around 180 km/s per year. (Using the new beam to enhance the drive would bring that up to around 210 km/s.)

I agree that it is pretty hard to setup a relay from Saturn to the inner system. I would actually go for an actively positioned system. You could use solar sails to keep a line of collimators in between Saturn and the inner system but I think they end up being prohibitively large. On the other hand I think you could make the sails much smaller and more effective if you power them with the laser! If the laser is powerful enough and the collimators are light enough then you only need to spend a small percentage of its time on station keeping blasts. It also helps that you only need to station keep collimators in high (outer system) orbits because the absolute difference in velocity between their desired path and what would be their normal orbit is small.

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

The Saturn installation is going to start with a 40 MW beam, but will also include a factory on one of the moons to build equipment to bring it up to 80-100 MW, and then multiple 100 MW beams to use on multiple craft at once. It'll take something of a redesign to use even .5 or 1 gigawatt on a single target.

[Fred Brackin]

Originally Posted by DataPacRat

The Saturn installation is going to start with a 40 MW beam, but will also include a factory on one of the moons to build equipment to bring it up to 80-100 MW, and then multiple 100 MW beams to use on multiple craft at once. It'll take something of a redesign to use even .5 or 1 gigawatt on a single target.

Huh? That might sound like a lot if you say it all at once but it's not a l0ot compared to the usual range of space combat weapons or laser propulsion systems.

I have trouble imagining that building a conventional 100 MW laser is so difficult and/or expensive tat it'd be worth going to Saturn to take advantage of some natural quirk of charged particle dynamics much less the very significant problems involved in piping the beam back into the inner system.

Just because it's handiest I look at Gurps Spaceships and even a Very Rapid Fire 100 MJ laser (hopefully similar to what you may intend as a 100 MW continuous beam) is only 150 million. It's entirely likely that big collimating mirrors (and for 3 AU they need to be very large indeed) will be a large percentage of that number.

I would think it would be much cheaper to build a conventional laser closer to where you need it.

Forget combat uses too. The collimating mirrors are going to be _extremely_ hard to defend.

[DataPacRat]

Originally Posted by Fred Brackin

Huh? That might sound like a lot if you say it all at once but it's not a l0ot compared to the usual range of space combat weapons or laser propulsion systems.

I have trouble imagining that building a conventional 100 MW laser is so difficult and/or expensive tat it'd be worth going to Saturn to take advantage of some natural quirk of charged particle dynamics much less the very significant problems involved in piping the beam back into the inner system.

Just because it's handiest I look at Gurps Spaceships and even a Very Rapid Fire 100 MJ laser (hopefully similar to what you may intend as a 100 MW continuous beam) is only 150 million. It's entirely likely that big collimating mirrors (and for 3 AU they need to be very large indeed) will be a large percentage of that number.

I would think it would be much cheaper to build a conventional laser closer to where you need it.

Forget combat uses too. The collimating mirrors are going to be _extremely_ hard to defend.

This is a hard SF setting, with limited accelerations and relatively uncommon interplanetary travel. Our Heroes are going to be the first folk to set foot on many of the moons they'll be hitting. One part if a larger plan has involved a trip to Saturn anyway, so since they'll be in the neighbourhood, the costs vs benefits of getting the laser-making factory set up are pretty favourable.

According to the numbers I'm currently working with, the collimators will have a diameter of about 1 km. They will, as you say, be quite fragile; their main defense is simply that it's so bleeping hard to get /to/ them


(Spoiler: part if the purpose of the laser factory is to be a flashy cover for the much more important bit if industrializing they'll be doing at Saturn, to wit setting up some He3-extracting aerostats to build up fuel reserves for a future interstellar rocket that'll be lucky to hit 5%c, but will make a lovely way to get out of the Solar system if the New Cold War ever turns hot. The folk they're hiding their secrets from don't hold the idiot ball, so the cover is going to need to be actually a good plan in and of itself.)

[whswhs]

Originally Posted by DataPacRat

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?

Saturn's mean orbital radius is 10 AU, I believe, which means that its orbit is about 63 AU long. Building 20 relays is going to multiply your costs a lot. And it's going to be difficult to control the alignment of all those relays; even if you send a signal on a straight line to one of the opposed relays, instead of bouncing it around the orbit, you'll have a delay of 10,000 seconds, or roughly 3 hours, each way. I'm thinking that you'll lose a lot of power from cumulative slight misalignments. Even if each relay passes on 90% of the power that reaches it, by the time you get to the opposite side you'll have lost two-thirds of the power.

And how much is there out in Saturn's orbit that you need to supply power to, anyway?

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. You might even go for six, and still save construction costs and power losses. And there almost surely are things in the Asteroid Belt that you'll want to illuminate.

Bill Stoddard

[DataPacRat]

Originally Posted by whswhs

Saturn's mean orbital radius is 10 AU, I believe, which means that its orbit is about 63 AU long. Building 20 relays is going to multiply your costs a lot. And it's going to be difficult to control the alignment of all those relays; even if you send a signal on a straight line to one of the opposed relays, instead of bouncing it around the orbit, you'll have a delay of 10,000 seconds, or roughly 3 hours, each way. I'm thinking that you'll lose a lot of power from cumulative slight misalignments. Even if each relay passes on 90% of the power that reaches it, by the time you get to the opposite side you'll have lost two-thirds of the power.

And how much is there out in Saturn's orbit that you need to supply power to, anyway?

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. You might even go for six, and still save construction costs and power losses. And there almost surely are things in the Asteroid Belt that you'll want to illuminate.

Bill Stoddard

I can't find anything in your post to disagree with. :)

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.

What I don't know is just how many such cyclers would be needed to cover, say, five or ten years of planetary motion...

[MatthewVilter]

Eccentric orbits are quite a posability. I don't know the specifics mechanically but I know this kind of thing has been done.

The first thing is getting an orbit that is in resonance (at least the "...their orbital periods being related by a ratio of two small integers." part) with Saturn so that each time* Saturn flies through a given part of it's orbit the appropriate satellite is swinging up to meet it. The other trick is to have the orbit be wide and fast enough that each satellite is with Saturn long enough that you actually save on the number of satellites you need to buy. The second part is what I'm not sure about.

*Actually thinking about this just leads me back to actively maintaining the orbits. I just don't think it makes sense to set up a full network, to cover the ~29.5 year orbit of Saturn, all at once. And if you instead plan launch the collimators as needed it suddenly becomes obvious that (for the first 30 years anyway) it just makes more sense to power one satellite continuously then to continuously be launching one satellite after another.


Tomorrow I might try to come up with the amount of power needed to maintain the relevant pseudo-orbits but I am by no means an expert on the math and physics involved...