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For scenaristic reasons, my colonization spaceship needs to reach Proxima Centauri within 10 years of ship time. How do they do that?

I'm starting to get a headache over this. At first I thought the only way to go was to accelerate at a constant 1G, which also provides gravity for the crew. I thought the fuel was no problem for a civilization able to mine the gas giants for it. But I realized that this wasn't compatible with the 10 years schedule I want, since the ship would get there in under 4 years of ship time.

Should I use another propulsion method or flight plan? What could possibly lenghten this trip?

An important element is that the trip was planned to last up to 10 years of ship time, it didn't take this long because of unplanned maneuvers or accidents.

Bonus:

  • How to handle the problem of gravity onboard the ship if I don't use a constant 1G acceleration?
  • About what amount of fuel would the spaceship need? Just to get an idea. The ship is to host 10,000 people tightly packed (again for scenaristic reasons), so it's rather large.
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    $\begingroup$ Looks like it was covered in some other answers, but "a preposterous amount of fuel" is the simple answer to the second bonus question. A fuel tank so huge it makes the ship itself seem like a tiny mote. $\endgroup$
    – jdunlop
    Apr 18, 2022 at 4:05
  • $\begingroup$ Practically impossible to do under a century. (Generation ships make for more interesting storylines anyway.) $\endgroup$ Apr 18, 2022 at 17:39

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Proxima Centauri is approximately 4x1013 kilometres away from us (or 40 trillion, if you like). To cross it in 10 years (from the point of view of an outside observer) requires an average speed of ~0.423 c, or ~127000 km/s. Thing is that unless you can accelerate instantaneously, which you can, you need to work your way up to that speed, and because you've spent so much time below it your peak speed now needs to be so high that you can't possibly ignore relativistic effects. There's a useful web page that's been copied about a bit called The Relativistic Rocket, from which I will take some or more of the equations below.

Because you can't ignore relativity, the meaning of "10 years" has also kinda gone out of the window because the time experienced by the crew of the ship will be less than the time seen by someone at Sol or Proxima Centauri observing the ship.

To travel in a co-ordinate time of $t$ over a distance of $d$ accounting for relativistic effects, you'd need a continuous acceleration of $$a = {d\over{\left({t\over2}\right)^2 - \left({d\over 2c}\right)^2}}$$ which in this case will be ~.2 standard gravities. The proper time experienced by the ship's crew will be less... $T = \frac{c}{a}\sinh^{-1}\left( \frac{at}{c} \right)$ or ~7.13 years. You can rerun the numbers yourself if you wanted 10 years shipboard time!

Your peak speed at $h = \frac{t}{2}$ will be $$v = \frac{ah}{\sqrt{1 + (ah / c)^2}}$$ which in this case will be a fairly brisk .72c. I hope you've got good shielding technology.

The good old rocket equation tells us that than when the delta-V is much greater than the exhaust velocity of your rocket, the mass-ratio of your rocket becomes enormous, requiring ridiculous amounts of fuel. Plausible fusion rockets will have exhaust velocities of no more than 0.09 c, and even in that case the amount of useful oomph will be low because only a small proportion of the exhaust will be travelling at that speed. Naturally, the relativistic rocket equation is even less forgiving that the regular kind.

Relativistic delta-V is tricky, but there's another useful measure known as rapidity which is $r = \tanh^{-1}\left(\frac{v}{c}\right)$ where $v$ is your regular speed and $c$ is the speed of light. The peak rapidity of the ship will therefore be ~.902. The total delta-r of the flight will be twice this figure (to go from "stopped" to r=.902 and then back to zero again), and so the relevant relativistic rocket equation will be $$\exp \left( \frac{\Delta rc}{v_e} \right) = \frac{m_0}{m_1}$$ where $\Delta r$ is your total change in rapidity, $v_e$ is your exhaust velocity (say, 0.09c), $m_0$ is the initially fully fuelled mass of your starship and $m_1$ is the mass of your starship with empty fuel tanks. (if you were running some numbers yourself, you may find that $\tanh^{-1}$ is called something like atanh on whatever fancy calculator you're using)

This gives you an impractical mass ratio over over 2900 just to get up to top speed. If you didn't want to whiz past your target at nearly three quarters of the speed of light, you need an eye-watering mass ratio of over 37000. By comparison, Ariane 5 has a mass ratio of 40 which is one of the highest anyone has managed to achieve and it does that using staging. Even if you had many fusion stages it seems implausible that you'd hit your target mass ratio and stil lhave a useful payload mass.

I thought the fuel was no problem for a civilization able to mine the gas giants for it.

Sure, you can mine a whole load of 3He and make some pretty good fusion rockets, but it just ain't enough. If you want a rocket to get you to the speeds you're after, you need antimatter, and quite a lot of it.

For complicated reasons, the exhaust velocity of a beam-core antimatter rocket (sometimes called a pion rocket) is about .33c. For even more complicated reasons, the rocket equation doesn't quite work for antimatter rockets, so you need a different and much more complex one (some details here, scroll down a bit). The equivalent delta-V to the delta-r calculated above is $\Delta v = c \tanh(\Delta r)$, or about ~.947c, Throw that into the Frisbee delta-V equation, and out comes a mass ratio of a little over 33. If your ship had a dry mass of 100000 tonnes, you need somewhere in the region of 1.6 million tonnes of antimatter, plus the same amount again of regular matter (much easier to obtain and handle) for your reaction mass.

If your tech level was even higher, you might try to create an antimatter fuelled photon drive, maybe something like Winterberg's ambiplasma pinch graser. With an exhaust velocity of c, you'd have a positively sensible mass ratio... $\exp\left(\frac{aT}{c}\right) - 1$ (where $T$ is the proper time of the trip!) or about 5. You'd now need "only" 250000 tonnes of antimatter for you 100000 tonne spaceship.

How you'd go about creating such ridiculously large quantities of antimatter is outside of the scope of this answer, but it won't be easy.

Should I use another propulsion method or flight plan?

Firstly, you should not use a rocket. Trying to travel interstellar distances in a sensible period of time with anything other than a magical matter-to-energy conversion rocket or reactionless drive is a mug's game.

You should probably use a beamed propulsion system. Mass beam propulsion is probably the best type to use, with a particular favorite of mine being the late Jordin Kare's High-acceleration Micro-scale Laser Sails for Interstellar Propulsion (sometimes called "Sailbeam" for short).

Diagram of how sailbeam works, with a laser acceleration stage driving multiple sails at a distant spacecraft with a magnetic deflector

Basically, you shoot very high velocity projectiles at your spaceship, which vaporises them and bounces them off a magnetic nozzle to provide acceleration. You can have a huge propulsion array left behind in the solar system using good old solar power to drive it, and have a much more svelte starship and no requirement for half a million tonnes of antimatter to remain well behaved for a decade. Mass beams also reduce the need for the sort of giant planet-frying laser arrays needed to push giant laser sail starships.

Ideally you want your boost phase acceleration to be as high as possible, whilst maintaining crew comfort and ship integrity. If you can manage more than 1G, that's great, because that keeps the boost phase short and reduces the need to accurately fire a relativistic machinegun across interstellar distances. You could build up acceleration to let crew and cargo acclimatise.

How to handle the problem of gravity onboard the ship if I don't use a constant 1G acceleration?

Spin "gravity" will be just fine. There are plenty of other places to read about it, but I'll link one of mine from this site: How fast can a ship with rotating habitats be accelerated?.

The key thing is that the design of a habitat which uses centrifugal forces to provide artificial gravity can be tailored to suit the external gravitational field or rocket acceleration, and a suitably designed hinged rotating section can provide a constant 1G internal artificial gravity for an external acceleration of anywhere between 0 and 1G inclusive.

I also suspect that you can have a lower-than-Earthlike acceleration on your ship without seriously ill effects, but you probably wouldn't want it any lower than the gravity of your destination planet (if there is one). You can also have a higher spun gravity, so you can do things like slowly reduce apparent gravity to ease acclimatisation after a >1G boost phase.

About what amount of fuel would the spaceship need? Just to get an idea. The ship is to host 10,000 people tightly packed (again for scenaristic reasons), so it's rather large.

If you're using mass-beam propulsion to accelerate you up to speed, you need no reaction mass for the boost phase of your flight. Now, it is possible to use a mass-beam system to decelerate too, but it is tricky and risky so maybe don't do that.

Instead, you can use a magnetic parachute.

A chart showing starship deceleration rate and current velocity given time after activation of magnetic parachute

This diagram is taken from The Magnetic Sail: Final Report to the NASA Institute of Advanced Concepts and shows that you can get a 1G deceleration rate without using a rocket at all, if your magnetic brake is good enough, and your target star has a strong enough stellar wind. The deceleration rate at Proxima Centauri is not something I'm going to compute for you right now, but it should be possible to perform a sort of "magnetocapture" manoever (the stellar wind equivalent of an aerocapture) to enter orbit around the target star, and your magnetic sail can then be used as a means of slow propulsion in-system.

What's left, then, is to carry enough fusion-fuel to run a reactor to power the magnetic deflector nozzle during the acceleration phase, the magnetic parachute during the deceleration phase, and life support and shipboard functions for the whole flight. You can then add enough fusion fuel and reaction mass to be able to use a fusion rocket to get around a bit more promptly in the target system, but honestly it might be better to carry some in-system transport spacecraft and treat the starship as a big space station instead.

A sensible mass ratio for a fusion spacecraft is probably somewhere between 1.2 and 2.5, depending on your patience and mission timescales (crawling through the Project Rho realistic fusion designs section for information on ships that are mostly designed by scientists and engineers aiming for plausibility). A thousand-tonne in-system spacecraft might, therefore, need somewhere between 200 and 1500 tonnes of fuel and reaction mass.


addition in response to comments

As far as scaling beam propulsion goes, that depends very much on the kind of beam you're using. A single big laser sail could be driven by a gigantic phased array, and you might be able to add new array elements as you wish, but that big sail could be incredibly large (a thousand kilometer across, and probably more!). A magnetic sail on the other hand can be made much larger much more easily, and it is possible to use multiple microsail launchers to target a very large spacecraft magnetic sail making it easier to scale the launch system "horizontally" as well as being able to use a smaller number of much more capable microsail drivers that push larger microsails, or push them for longer (so they get faster).

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  • $\begingroup$ Alright, excellent. I'll go for your solution if you can just confirm it could be scaled in a fictional world to work for a 10'000 people strong ship? $\endgroup$
    – dyarob
    Apr 18, 2022 at 17:27
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    $\begingroup$ @dyarob I'm just doing some rocket equation edits, but I'll see about doing some scaling a little later. $\endgroup$ Apr 18, 2022 at 17:28
  • $\begingroup$ Could you point me in the right direction with the research I'd need to do to be able to talk about this tech starting considering I have poor knowledge in rocket science? Would you be able to give me a couple more details about how it would work for my ship, like size of the array or er anything really since right now I can hardly picture it. I can start working on the story with only a vague idea of what it should look like and do the research along the way. $\endgroup$
    – dyarob
    Apr 18, 2022 at 17:31
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    $\begingroup$ @dyarob I've probably finished with the edits now. There are lots of links in my answer, but you might consider reading Mass Beam Propulsion: An Overview and the full Kare paper first. Rocket science is famously complex, so its hard to talk much more about it in one answer than I already have done. Feel free to post more questions, though ;-) $\endgroup$ Apr 20, 2022 at 20:16
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If you want to accelerate a ship of that size (mass really, I guess somewhere close to 10^8 kg from your description) to relativistic velocities, you won't use chemical fuel. Fusion might do it, but only with a ridiculous fuel to payload ratio. Even if you use antimatter you'll probably need a million tons of it or something in that ballpark. Both options are inconvenient at best.

If you insist on rocket propulsion, you'll need some sort of Clarke-Tech. Conversion drives or something like Revelations Space's Conjoiner Drives are needed. Sublight warp drives are another option.

The sane thing to do would of cause be the use of external propulsion. You send a small probe with your most advanced propulsion system and a packet capable of bootstrapping the industry ahead. It establishes itself in the target system and duplicates the same external propulsion system you used for the acceleration phase. On the technical side, your options are laser pushed light sails (have fun scaling that one up), Kare Sailbeams or Fusion Highways.

What you are proposing is a generation ship. Even optimistic projections don't show them moving faster than 0.1c with definitively possible technology.

How to handle the problem of gravity onboard the ship if I don't use a constant 1G acceleration?

  • rotation can be used to generate gravity
  • freeze them cryonically
  • only take uploaded minds with you (helps a lot with the mass issue) (just run your setup in a simulation onboard the vessels computers)

About what amount of fuel would the spaceship need? Just to get an idea. The ship is to host 10,000 people tightly packed (again for scenarists reasons), so it's rather large. depends on the propulsion system. About 99% of its mass if you insist on fusion or something like that. Very little if you use the external concepts. The Clarke-Tech pretty much implies direct matter to energy conversion, so calculate how much energy your trip needs, convert that to mass via E = m * c^2 and double it to consider inefficiencies.

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  • $\begingroup$ Your answer makes me question if I got my premises wrong. I just assumed the biggest obstacle was getting enough fuel, and with being able to mine the gas giants virtually infinitely in 2300... Would the fuel amount be more manageable, say, if the ship only initially accelerated to .5c and then went on like that before deccelerating at the end of the journey, or would it still be unrealistic and I have to figure out another way of interstellar travel? Assume ultralight materials all over. $\endgroup$
    – dyarob
    Apr 16, 2022 at 9:21
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    $\begingroup$ @dyarob I can tell you that you got you premises wrong. Your entire question shows that you don't know much about the physics of spaceflight in general. It's fine, we all started there. I could explain stuff to you, but that would be tedious and you would be limited by my own lackluster understand. I think you should do your own reading. Atomic Rockets, ToughSF and Orion's Arm are some of the best sources for an aspiring Sci-Fi author. $\endgroup$ Apr 16, 2022 at 12:08
  • $\begingroup$ Upvote for pointing to Atomic Rockets; it's a valuable SF resource! $\endgroup$ Apr 16, 2022 at 20:04
  • $\begingroup$ You can’t have significant inefficiencies, because any fusion or total conversion energy that’s not turned into kinetic energy will get turned into heat instead, and unless you’re 99.9%+ efficient that heat will vaporise the ship. $\endgroup$
    – Mike Scott
    Apr 17, 2022 at 0:32
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    $\begingroup$ @MikeScott Unless you have access to Magmatter that is. It's one pathway to such a conversation drive and offers you perfect full spectrum mirrors as well. A little mirror right behind the reaction zone will keep all that heat and radiation nicely away from your vessel. Or you go for a mag-orion based approach where the reaction happens hundreds of meters behind the vessel and you use magnetic fields to interact with its plasma. ToughSF's Epstein drive concept might be of interest to you. He ran the numbers on a torchship without radiators. $\endgroup$ Apr 17, 2022 at 5:21
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If you're using a traditional rocket (i.e. anything that accelerates forwards by expelling something from the ship backwards), then the rocket equation applies:

$$M_0 = M_1 \exp(\Delta v/v_e)$$

$M_0$ is your rocket's "wet mass", i.e. the mass of the rocket plus the propellant at the beginning of the journey

$M_1$ is the "dry mass", the mass of the rocket at the end of the journey

$v_e$ is the exhaust velocity, the speed that the propellant is expelled from the rocket

$\Delta v$ is the delta-v of the journey; in relativistic contexts this is the integral of proper acceleration times proper time (acceleration times time as experienced by the rocket)

And $\exp$ is the natural exponential function (e to the power of x).

One way to get a 10 year journey to Proxima (4.25 light-years away) is to accelerate at 1g for 0.42 years proper time (0.43 years back on Earth), coast for 9.15 years proper time (10.02 Earth years), and then decelerate for another 0.42 years proper time to arrive at the destination. (Max velocity is 40.6% of the speed of light; total time experienced by crew 9.99 years, total time according to Earth observer is 10.88 years.) You can spin for gravity during the coasting phase, just design your habitat module to be able to handle gravity from both the thrust direction and the radial direction.

With a fusion engine, exhaust velocities are around 10% of light speed IIRC, and the rocket equation tells you that you need... about 5480 kg of propellant for every 1 kg of rocket and payload at the end of the journey. There's no possible way to carry such an insane amount of propellant. So you'll need to invoke some fictional engine technology.

With a photon rocket that can directly convert mass-energy of your fuel to an enormous planet-scorching laser (do not point your exhaust within an AU of anyone or anything you love), you would need about 1.37 kg of mass-energy convertible fuel per kg of rocket. Much more manageable.

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I will refer to my own question and the answers therein. If you want to accelerate to some decent proportion of the speed of light you need a lot of energy. If you use fusion which converts only a few percent of the mass into energy used for acceleration the rocket equation will tell you that you are not getting close to the speed of light. You need some antimatter or similar that allows you to convert almost the entire mass into energy.

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Circuitous route

/Should I use another propulsion method or flight plan? What could possibly lenghten this trip?/

Keep your 1g; remember to flip around partway thru to decelerate at 1g.

Keep your 10 years.

Make your voyage longer. Your people do not go straight from point A to point B because there is stuff in the way. Or they want to visit some other point and go there first.


@Trish was skeptical so here is a real world and a fiction world scenario for the circuitous route plan.

1. Slow down using Centauri A and B.

centauri system

Centauri system from wikipedia. Sloppy red arrow mine.

The Centauri system is a binary star (Centauri A and B) and this is orbited by Proxima, which is the destination in the OP. Instead of a straight line to Proxima, the ship heads for the binary then uses gravity assist to change course while shedding velocity and so saving on fuel.

https://en.wikipedia.org/wiki/Gravity_assist

Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed or redirect its path. The "assist" is provided by the motion of the gravitating body as it pulls on the spacecraft.1 Any gain or loss of kinetic energy and velocity by a passing spacecraft is correspondingly lost or gained by the gravitational body, in accordance with Newton's Third Law.

A more scifi (but still real world) idea of slowing down an incoming craft using the binary is a parachute - a reverse solar sail using photon pressure to decelerate.

https://www.spaceflightinsider.com/missions/how-to-slow-down-an-interstellar-spacecraft-at-alpha-centauri/

This could be done in combination with the gravity assist deceleration which is not fiction.


One fictional scenario is a rescue mission. Trish pointed out there are no known bodies between our system and Centauri. Maybe in the fiction, the colony ship in the OP is not the first. Another left some years ago and has run into trouble en route. The new ship needs to intercept its predecessor and rescue the crew and passengers. I like the idea of the bored engineering crew messing around with the broken ship and salvaging some parts, which come in handy later.

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  • $\begingroup$ there is nothing between the Sol system and the Proxima Centauri system. It's named for it is the very closest star to our system. $\endgroup$
    – Trish
    Apr 17, 2022 at 0:14
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    $\begingroup$ @Trish you seem very certain about this fictional world that someone else is building, and that can be built in any way that advances the story. $\endgroup$
    – Willk
    Apr 17, 2022 at 1:19
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    $\begingroup$ If it isn't in the question, insert real world. $\endgroup$
    – Trish
    Apr 17, 2022 at 1:40
  • $\begingroup$ your path around AC A&B to Proxima doesn't work out with the acceleration profile you want. $\endgroup$
    – Trish
    Apr 20, 2022 at 22:24
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Well, here's the problem with your setup: Accelerating at 1G, you reach the Speed of Light in around a year. So, your best option for extending your trip is to slow the acceleration, to give e.g. moon-like gravity. Although, after some quick calculations, even that's a bit too much. The exact acceleration for reaching Light Speed in 10 years is 0.94998497350876 m/s^2, comparable to gravity on Pluto.

This will still provide plenty of 'gravity', don't you worry.

And for the fuel, I'll give you this highly technical answer: You'll need a lot.

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    $\begingroup$ Your calculation assumes 10 years of accelleration, but that is not realistic. Take into consideration your ship will need to "turn its tail" and decelerate as well. You don't want to reach your target at near light speed, or a relative fraction of light speed. $\endgroup$
    – Goodies
    Apr 15, 2022 at 23:05
  • $\begingroup$ @Goodies Yes, true. In that case, it would be nearing Moon gravity levels, which then swaps halfway through the trip. $\endgroup$
    – Murphy L.
    Apr 16, 2022 at 19:08
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    $\begingroup$ There is no such thing as the amount of acceleration to reach light speed unless you ignore relativity... which you should not be doing if you are dealing with speeds near light speed. $\endgroup$ Apr 16, 2022 at 20:01
  • $\begingroup$ @MattDickau Yes, which means you have even less time before you reach too fast to continue acceleration. I was using the Newtonian t*a=s formula for this answer, which certainly breaks down near Light Speed. $\endgroup$
    – Murphy L.
    Apr 16, 2022 at 21:02
  • $\begingroup$ Even a rough estimate on non relativistic numbers should tell you that you need to calculate relativistic. Under Newtonian rules, you'd need an average speed of 42% of light speed. $\endgroup$
    – Trish
    Apr 17, 2022 at 0:10

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