Math need for interstellar voyage

[Richy]

Hi there

I'm working on a short story placed on an interstellar ship. But I need you guys to help me with the math and authentic technology involved.

The story is taking place in the 22th century. I'm talking of an interstellar colony ship carrying 10'000 colonists in cryo-sleep and a crew of 12. I'm imagine a taildragger ship with a length of about 2 km. It's mass could be around 10'000 tons (e.g. at least 500kg per colonist of lifesupport equiptment + 5000 tons of crew habitat, reactor, engines, fuel, structure, etc). Plausible?

The ship should accelerate continuosly until reaching max speed (between 0.6-0.8c), then turn around and deccelerate the other half of the journey. The entire traveling time should fit within of one to two decades. The destination is our closest star with a distance of 4.3 ly.

What kind of technology could be feasable by then to achive this criterias?
And how would the timing of such a trip look like? How long would it take to leave our solar system? Is it even possible to "spiral-in" directly on a planet in the destination system or use swing-bys at departure and arrival?

To be honest, inspiration came a bit from James Cameron's "ISV Venture Star" from Avatar. But in contrary to said ship, I imagine a ship that constatly thrust on lower levels until it reaches max speed, then turns around and constantly thrusts to brake down for arrival. I thought about ion-drives and a low-g acceleration with the help of a solar sail for acceleration and braking. But they are far to weak, to get a realistic setup for the framework I want to use. Any other ideas, e.g. VASIMR?

Thanks for you support!

Richy

 

[BLANDCorporatio]

The story is taking place in the 22th century. I'm talking of an interstellar colony ship carrying 10'000 colonists in cryo-sleep and a crew of 12. I'm imagine a taildragger ship with a length of about 2 km. It's mass could be around 10'000 tons (e.g. at least 500kg per colonist of lifesupport equiptment + 5000 tons of crew habitat, reactor, engines, fuel, structure, etc). Plausible?

Even without doing the math, no

Ok, that was a bit flippant, so I should explain.

Assuming you don't go for magic, plausible tech will obey the [ame="http://en.wikipedia.org/wiki/Rocket_equation"]rocket equation[/ame].

It says the deltaV a ship can produce is equal to the exhaust velocity of its engine times the logarithm of its mass ratio (the ratio of total mass, including propellant, to the 'dry' mass: the ship without any propellant).

You can 'cheat' a little by having the 'propellant' be outside the ship with a solar sail or similar device to capture thrust by 'pushing against' something else (in the acceleration phase, the ship in Avatar was pushed along by a battery of lasers mounted on a planet, for example). That way you don't have to carry the propellant.

Solar sails are weak for the speeds you consider (they won't work when you get far from a star, and won't accelerate you too much before you do). Laser batteries to push your ship are a very good idea because ...

suppose you have to carry all the propellant you need on the ship.

Your deltaV requirement is at least 1.2c (accelerate to 0.6c, break from 0.6c; planetary and stellar escape velocities are negligible).

A Vasmir engine is estimated at 294 km/s at the excellent "So you wanna build a rocket" website.

Plugging in the rocket equation-

1.2*300000 = 294*log(mass ratio)

If the log of the mass ratio is in the thousands, you do not want that mass ratio. Typically you'd like a mass ratio in the tens, at most, so a log of 5 should already get you worried.

Ok, suppose instead you use the best engines there can be in terms of exhaust velocity (they shoot light out, so escape velocity is c).

This makes the mass ratio very tolerable:

1.2c = c*log(mass ratio)

which makes the mass ratio a nice if rather expensive 3.32. Roughly, this means for every ton of ship (includes payload, colonists, life support, the ship's structure like propellant tanks ...) you need 2.32 tons of propellant.

And because your deltaV budget requires more deltaV than the best engine possible (as far as we know today), mass ratios will be bad.

But still, 3.32 mass ratio is decent. There's another problem though.

Unless you use magic, engine power is the product of exhaust velocity by thrust.

This means, if the engine has a given power (say a jiggawatt), then the bigger the exhaust velocity, the smaller the thrust.

Light-based engines got (almost) no thrust, is what I'm saying. 300MW of engine power buys you 1Newton of thrust from an engine that shoots light out. If your ship has tens of thousands of tons, it won't accelerate to 0.6c any time soon.

So ok, let's beef up the engine in the million terrawatt range so as to have both high exhaust velocity and decent thrust. But then you run into another problem: usually engines are not 100% efficient. This means, if the engine produces those millions of terrawatts in mechanical power, at least a fraction of that is going to be dissipated as heat. If your engine is 90% efficient (awesome), and produces 90 million terrawatts of power, it still dissipates 10 million terrawatts of heat (horrible). So you need to take care of that and have big-mass radiators.

One way to skimp on this a little is go the Avatar route and have your ship accelerated by something on the outside pushing on it (that outer battery of lasers). That way the ship's deltaV requirement reduces to the deltaV needed to break, say 0.6c.

Then you'd have:

0.6c = c*log(mass ratio)

which gives about 1.8 (0.8 tons of propellant for each ton of ship- very good!) if you use a c-exhaust velocity engine OR

0.6c = v*log(3.32)

if you can tolerate the mass ratio from before (2.32 tons of propellant for each ton of ship, which is decent) giving you an exhaust velocity of about 0.5c (which also means, for the same engine power that your c-exhaust engine gives, this one would give you twice the thrust).

If you can tolerate higher mass ratios (10 is a 'good' number for a rocket without staging, claims the "So you wanna build a rocket" website), then you can reduce exhaust velocity even further and gain better thrust without that much engine power.

In other news, do you have some time to talk about the Lord? Lol, I'm evangelizing that startship concept because it is just crazy cool in my opinion. Won't get you there fast, but it sure means travelling in style.

 

[fsci123]

Feel free to ignore my feeble mind but...

10k people at 0.8c... Thats a stretch. The only propulsion that can do anything close is pure antimatter. But the heat and radiation produced will destroy the vessel unless it features some of the largest radiators ever made(hundreds of miles in area). Solar sails on this vessel will be even larger. It will have to be thousands of miles wide and it will have to spin to avoid a collapse.

Maybe im just a pessimistic person. But having 10k colonists in a craft 10,000 tons is a little bit unrealistic. I remember reading that it takes 10ktons for just a handful of people. You could slow it down a little(0.2c or 0.4c). Or you could reduce the amount of people or convert everybody into embryos. Or both.

Goodluck

 

[BLANDCorporatio]

10k people at 0.8c... Thats a stretch. The only propulsion that can do anything close is pure antimatter.

Infidel

Black holes are the future.

 

[fsci123]

Infidel

Black holes are the future.

Yeah but imagine the energy needed to create the black hole and maintain it behind the ship. How would people on earth react to knowing there is a black hole hovering around the planet.:suicide: How would you create a black hole? Last time i checked some of the methods are either highly dangerous or require materials that would immediately make black hole propulsion obsolete /cough/Monopoles/cough/

 

[Richy]

Thanks for the inputs!

I'm not that surprised of the pessimistic outlook of my idea.
But still, the story takes place in more than 100 years from now. And say, when the ship was launched, there were at least 50 years of R&D in interstellar technology. Not like the "free-time" science like today. More like R&D in the space race in the cold war, when there was a real goal in sight (then the moon, now say a second earth or something like that).

I don't want to say, I'll just use "magic", but I impose, that half a century of dedicated research brings new ways (maybe even from the mysterious worlds of quantum mechanics), that allow us to round some of the numbers or optimize some engines beyond what seems realistic today.

For example the problem with the weight: I imagine that new materials, like artificially created diamatiod structures, new composites or nano materials allow the construction of extremely light spacecraft. Crew quarters could (and should) be built without any metal structures (to avoid Bremsstrahlung), and could have very low density but are still extremely stable.
Or the invention of an optical transistor could convert our copper heavy electronics to light optic-only computers, that are not only lighter, but also less prone to disturbances by solar flares or cosmic radiation.

Of course I'm also willing to adjust some of my numbers, but 10'000 sounds more dramatic, than say 100.

 

[BLANDCorporatio]

Cutting the ship's weight is good, it means overall there's less payload to lug around.

But the problem with the mass ratio remains. You aren't escaping the rocket equation just by making the rocket out of super-cardboard. If its mass ratio needs to be 10, then it needs to be 10, meaning you still need 9kg of propellant for each kg of ship. And I was giving you the 'nice' version there that doesn't account for relativistic effects (which start to become significant at about 0.6c). You're looking at even worse mass ratios.

(EDIT: and this cannot be stressed enough, mass ratio is not a weight problem. Your rocket could weigh, total, 1kg- so super light. Is it capable to have a mass ratio of 1000? that means the "rocket", including the propellant tank, weighs 1 gram. All the rest is propellant. You'd discover this means the tank is too flimsy to contain the stuff, especially if you want some measure of thrust to accelerate you quickly to 0.6c. The rocket's structure would simply crumble.)

And you're certainly not escaping the engine power problem. If you want an engine that gives you good thrust, as well as high exhaust velocity so as to use less propellant, you must use a very high power engine. Some of that power will be dissipated as heat, which you cannot use for any purpose. You just have to radiate it out.

Also, these engines we've mentioned (antimatter, and more exotic stuff like black holes): this isn't anywhere near what we could do today. A fairly safe bet is it would not be doable with several decades of 'intensive cold war research'. (Engine-wise, the cold war brought nothing fundamentally new to practical use; chemical engines is what we've been stuck with for hundreds of years).

In other words, if you want a 'realistic' look at what tech in 100 or 200 years time might look, if we decided to prioritize space: go to that "So you wanna build a rocket" website. See that engine list. Pick from those.

Yes, it's not looking good.

PS:

I don't want to say, I'll just use "magic", but I impose, that half a century of dedicated research brings new ways (maybe even from the mysterious worlds of quantum mechanics), that allow us to round some of the numbers or optimize some engines beyond what seems realistic today.

Lol, but that's exactly what 'magic' means in this context. Weird thingies we don't know of today that might exist so as to make superadvanced tech possible.

Again, if that's what you want to use, fine. But you asked for plausibility in the original post. Without 'magic', it's NO.

 

[Fizyk]

1.2c = c*log(mass ratio)

which makes the mass ratio a nice if rather expensive 3.32. Roughly, this means for every ton of ship (includes payload, colonists, life support, the ship's structure like propellant tanks ...) you need 2.32 tons of propellant.

For a relativistic rocket (and 0.6-0.8 c is relativistic), it's a bit different: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

 

[BLANDCorporatio]

For a relativistic rocket (and 0.6-0.8 c is relativistic), it's a bit different: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

This is certainly true. I kept things simple, because they look bad already even with the classical rocket equation (and said, eh, at 0.6c the mass is 1.25 times the rest mass; tolerable ). The relativistic rocket equation (which yes, is more accurate here) produces results that are more discouraging.

 

[Fizyk]

The relativistic rocket equation (which yes, is more accurate here) produces results that are more discouraging.

True, and it's actually even worse than you would think at first glance You not only need to use a different equation for deltaV - you also need to calculate deltaV differently.

Your deltaV requirement is at least 1.2c (accelerate to 0.6c, break from 0.6c; planetary and stellar escape velocities are negligible).

In the relativistic case, it would be better to speak of "delta-rapidity" instead of deltaV. Rapidity is a nice thing connected to velocity like that: [math]v = c \tanh (\eta)[/math], where [math]\eta[/math] is the rapidity. It is nice, because with velocities you need formulae like [math]V = \frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}[/math], and with rapidities it's just [math]\eta = \eta_1 + \eta_2[/math].

In this sense it behaves just like velocity in Newtonian/Galilean mechanics, so what you actually need for accelerating to 0.6 c and braking again to 0 is twice the delta-rapidity needed for just accelerating to 0.6 c.

Rapidity for 0.6 c is 0.693. Double that, and you need 1.386 for braking, which gives:

[math]\frac{M}{m} = \exp(1.386) = 4[/math]
Not a big difference, but it is there. For 0.8 c it's more obvious:

[math]\frac{M}{m} = \exp(2 \;\textrm{atanh}(0.8)) = 9[/math]
You would get only 4.95 in the Newtonian case for deltaV = 1.6c.

 

[BLANDCorporatio]

irt. Fyzik:

Thanks, that's a more elegant way to calculate it than what I had in mind. (Use the relativistic rocket equation to compute the mass ratio for 0.6c then square the result for braking). I don't have a 'feel' for rapidities yet, but I should get one.

---------- Post added at 10:58 AM ---------- Previous post was at 10:44 AM ----------

some of the methods [for black hole creation] are either highly dangerous or require materials that would immediately make black hole propulsion obsolete /cough/Monopoles/cough/

I am intrigued by your ideas and would like to subscribe to your newsletter. How would magnetic monopoles help with propulsion? But yeah, it sounds like that might be a cool bit of speculative engine-building.

PS: on keeping the BH near the ship: similar containment problems happen with other fuels like antimatter, and there'd be similar solutions for the BH case. In fact, I'd argue BHs might have a few advantages in this regard, as the charge on them doesn't need to be 'too big' and won't make the BH explode anyway.

 

[statickid]

PS: on keeping the BH near the ship: similar containment problems happen with other fuels like antimatter, and there'd be similar solutions for the BH case. In fact, I'd argue BHs might have a few advantages in this regard

In that particular paper they argue "ease" of containment as one of the biggest advantages to BHs. I mainly skimmed the document once, but I'm pretty sure they mentioned reflecting some of the radiation it produces to nudge it around and keep it in a container

 

[Richy]

Ok, if fuel mass is a problem. What about a magnetic bussard ramjet, to scoop in interstellar gas as energy source for fusion reactors and reaction mass for acceleration, say fuel. Would the additional use of a magnetic parachute that grabs on interstellar matter/gas for deceleration improve our situation here?

 

[statickid]

It looks like it will be up to you how much fiction you want in your science fiction.

Some aspect of your story must have flexibility.

It looks to me like crew size and travel distance are important to the story. It's possible that the either the timeline needs to be longer or the fiction needs to increase as if by magic.

 

[BLANDCorporatio]

What about a magnetic bussard ramjet, to scoop in interstellar gas as energy source for fusion reactors and reaction mass for acceleration, say fuel. Would the additional use of a magnetic parachute that grabs on interstellar matter/gas for deceleration improve our situation here?

The magsail/parachute for breaking seems like a good idea. How about having an outside propeller (like the laser battery thing in the Avatar ext. universe), and magsail brake then?

Bussard ramjets seem to have a maximum velocity to them that's quite a bit less than 0.6c. When I first encountered the concept, this "maximum" was explained (probably somewhat mistakenly, but for now that's how I understand it) as an artifact of momentum conservation. You have the fuel/propellant caught in your scoop, you then chuck it out (in more or less the same relative direction it had when it hurtled towards you) so any momentum the ship gains is obtained by accelerating that incoming fuel/propellant even further. But, said fuel/propellant is already moving fast to begin with.

"Proper" explanations of this drag seem to require effects like brehmsstrallung (which happens because you need to electrically charge then focus the rapidly incoming fuel/propellant; basically, the classical Bussard Ramjet doesn't just accelerate the incoming fuel/propellant; it also needs to stop it first). But the drag is there.

One concept of Bussard Ramjet uses proton-proton fusion to convert the fuel/propellant into thrust for the ship. Apparently, because of the 'drag' from before, the max speed of this ship would be 0.12c.

If you don't need to stop the fuel/propellant as it passes through the ship, or you can stop it but retrieve the radiated energy ("Bussard scramjet"), limits may be less bad. As far as I know, here be dragons as this is all very speculative ("magic"). However, in this case it seems to me the "momentum conservation" argument I first heard of becomes more important.

In the end, there's stuff coming at the ship's scoop with velocity A. Magic happens. The stuff flies out from behind the ship with velocity B, which is hopefully greater than A (and thus the ship obtains thrust). If A and B are already large when compared to c, there won't be much difference between them and not much thrust. But at least there seems to be no "terminal velocity" here, in the sense that you could accelerate as close to c as you wanted, eventually.

 

[fsci123]

Ok, if fuel mass is a problem. What about a magnetic bussard ramjet, to scoop in interstellar gas as energy source for fusion reactors and reaction mass for acceleration, say fuel. Would the additional use of a magnetic parachute that grabs on interstellar matter/gas for deceleration improve our situation here?

Fusing pure hydrogen is close to impossible. And even if one could fuse it and use the excess energy for thrust, there would be problems gathering enough hydrogen. Last time i checked, the solar system and surrounding stars are located in bubble nearly devoid of free hydrogen.


Beware, Im not an evil person bent on destroying your dreams, and to prove it i would offer an optimistic view on mag-sails. Magsails can decelerate your vessel... so feel free to use them. But the mag sail has to be made of a superconductor loop that is several miles in diameter.

Note: When in operation magsails produce a radio howl. This could be featured in your story...

 

[Thorsten]

True, and it's actually even worse than you would think at first glance

Ah, but the propellant gets relativistic before the rocket does, so if you're using a near-c matter stream, then you get the relativistic mass corrections on the propellant, so a given propellant mass can by you an arbitrary amount of momentum, the relation is no longer

Delta p= Delta m v

but

Delta p = gamma Delta m_0 v

with

gamma = 1/sqrt(v^2/c^2)

so for a near-c matter stream gamma can be arbitrarily high and the momentum gain from a kg of matter may be hundred times that of the non-relativistic limit.

Provided you have the energy to accelerate your matter stream to near-c. Doesn't work with light as propellant, because that has no rest mass, so the relativistic momentum is different.

Ok, if fuel mass is a problem. What about a magnetic bussard ramjet, to scoop in interstellar gas as energy source for fusion reactors and reaction mass for acceleration, say fuel.

The scooping up inevitably causes a drag, because you need to accelerate the fuel from the rest frame of the universe into the rest frame of the reactor. That drag needs to be compensated by the thrust, you can react part of the hydrogen and use that to propel the rest. If you want to do p-p fusion, you need a 100.000 km sized reactor (there's a reason stars are large...). If you can't do p-p fusion, the terminal velocity comes out embarrassingly low.

I've done the calculation (I have a 160 page collection of equations amusing myself with fusion-powered spacecraft concept studies) - under favourable assumptions, I get a terminal velocity of 450 km/s for a ramjet.

An efficient fusion-powered design can give you a Delta v of about 8000 km/s, so that's somewhat better. If you really want to go relativistic, it has to be anti-matter powered. Then you basically need to carry enough antimatter to provide the energy, and the relativistic effect on the propellant mass will help you with the reaction mass problem.

---------- Post added at 06:01 AM ---------- Previous post was at 01:20 AM ----------

Just looking up a few numbers... The only antimatter-powered ship I've ever calculated is for a completely different purpose, but it might give you the order of magnitude.

It has 30.000 tons mass, carries a 48 PW anihilation reactor and 121 tons of antimatter (the mass of the confinement system is 1200 tons though, you can't assume there's no overhead), of which the reactor comsumes 4.7 tons/day.

There's 1600 tons reaction mass aboard, reaction mass is accelerated to 200.000 km/s in the main thruster, making use of the relativistic mass effects and the ship has an optimum acceleration of 2 g, reaching a total Delta v of 15.000 km/sec.

So if I reassign mass which I've used for other purposes for an interstellar voyage, it could potentially carry a total of 23.000 tons of antimatter and reaction mass. That reaches into the relativistic range for the ship, and gives you a Delta eta of 1.17 or a Delta v of ~248.000 km/h and would get you to 0.4 c and back. Provided you can refuel at the destination.

You can always scale the mass up - since for 10.000 people you probably need ~40.000 tons of space and life support, the ship needs to be huge - at least 10 times the mass of my design, that gets you well into getting 1500 tons of antimatter from somewhere...

So that's the order of magnitude. To get you to 0.8 c, even such an antimatter powered ship needs a propellant fraction (antimatter + reaction mass) of close to 90%. So we're talking a half-million ton behemoth.

 

[Fizyk]

Ah, but the propellant gets relativistic before the rocket does, so if you're using a near-c matter stream, then you get the relativistic mass corrections on the propellant, so a given propellant mass can by you an arbitrary amount of momentum, the relation is no longer

Delta p= Delta m v

but

Delta p = gamma Delta m_0 v

with

gamma = 1/sqrt(v^2/c^2)

so for a near-c matter stream gamma can be arbitrarily high and the momentum gain from a kg of matter may be hundred times that of the non-relativistic limit.

Indeed, but still you get the highest Isp from emitting just light, IIRC. You can for example annihilate matter and antimatter and emit gamma rays (provided that you can appropriately focus them and change other products of the annihilation into gamma rays, too). I'll check that though, because I'm not so sure. The way I derived the rocket equation used the assumption that it is matter you are emitting.

 

[Thorsten]

Indeed, but still you get the highest Isp from emitting just light, IIRC. You can for example annihilate matter and antimatter and emit gamma rays (provided that you can appropriately focus them and change other products of the annihilation into gamma rays, too).

Yes, that's correct. In the ultrarelativistic limit, the momentum of whatever you emit is just equal to the energy (in natural units), so in that limit you do not need any reaction mass, just an energy source, and emitting photons gets you there. So does emitting particles at 0.9999c. All that matters for the final velocity then is the total amount of energy which you carried and how much of that you could convert into momentum. If you carry that much energy though, the ship is effectively losing mass just as well, because any energy carrier out of which you can take that kind of energy is getting lighter in the process - antimatter loses all its rest mass if you convert it into photons. I didn't mean to say that light doesn't work - just that the equation doesn't apply. Sorry of that wasn't clear.

provided that you can appropriately focus them and change other products of the annihilation into gamma rays, too

Well, you can't - you'll never catch the neutrinos from the meson decays. In general I figured building an antimatter reactor which captures a fraction of the anihilation energy to drive a particle accelerator providing thrust is technologically easier than any mirror which would allow you to directly utilize the anihilation reaction for thrust - normal matter won't survive that, keeping a gas/plasma mirror stable is tricky, gamma-rays don't really reflect but just do small-angle scattering,....

If you could solve all that, you'd get some more oomph from antimatter. Not that much though.

 

[BLANDCorporatio]

This is fascinating, but Thorsten, what equation would you use to compute deltaV based on exhaust velocity, mass ratio, and whatever other parameters needed?

Since that's a bit ill-defined, some more specific scenarios.

Ship A is happy with always having a velocity less than, say, 0.2c in the galaxy frame. It has an engine that chucks light out.

Ship B, like ship A, is content with traveling "slow". Its engine however chucks out mass at relativistic speeds.

Ship C ain't got time for that and flies "fast" (over 0.6c or whatever in the galaxy frame), and uses an engine that chucks out light.

Ship D is in a hurry too, but its engine chucks out matter.

The mission all ships (A to D have) is start somewhere at a velocity that may as well be 0 and accelerate until they reach their target velocity (some small fraction of c for A and B, some large fraction of c for C and D).