Calculating near future spaceship rocket delta V in a vacuum

A generation spaceship leaves Earth around the year 2060 on a journey to colonize Alpha Centauri A (ACA). In this fiction, fusion power is achieved in 2040, improved over 20 years, and used within the solar system. The trip to ACA will take 110 years. The ship will accelerate halfway, flip, and decelerate for the second half.

Linear distance can be expressed as (if acceleration is constant): $$s = v_0 t + {1\over 2} a t^2$$

• s = linear distance traveled (m) = 2.06575e+16 m = 2.1835 Lyrs (half of 4.367 light years to Alpha Centauri)
• $$v_0$$ = initial linear velocity (m/s) = Mars mean orbital velocity (m/s) = 24070
• t = time (s) = 1.734e+9 s = half of 110 year trip
• a = acceleration (m/s2) = $$((2)( 2.06575e+16 - 24070(1.734e+9) ))/(1.734e+9)^2 = 0.01371296016 {m\over s^2}$$ = about 0.0014 g felt from thrusters.

Total ship mass = ~ 125,000 kg. F=ma. So constant 2356 N thrust needed from the ship drives (rounded to whole number).

But, if chemical rockets are used at journey’s start, how much could initial velocity be increased?

As of 2020, the thrust of the Merlin 1D Vacuum is (981 kN) with a specific impulse (ISP) of 348 seconds

Say we used a similar but improved engine (in the year 2060) to get 700 ISP.

• Mount 4 of these on the rear of the ship and burn for 360 seconds. (based on a Falcon Heavy 1st stage burn length)
• The ship is in a vacuum, orbiting Mars (Mars mean orbital velocity (m/s) = 24070)

How much initial velocity would we gain beyond initial orbital 24070 m/s?

I tried to figure this out, but I'm having trouble finding out how ISP converts to delta V in a vacuum, as most rocketry resources deal with leaving Earth's surface. Thank you for your help!

• For properly formatted formulas, please refer to this tutorial math.meta.stackexchange.com/q/5020
– L.Dutch
Mar 28, 2021 at 18:30
• 125t for a generation ship for 110years - u kidding me, right? Mar 28, 2021 at 18:39
• @MolbOrg my story would like to challenge the concept that generation ships MUST be huge city size cylinders... I believe compact is possible. look at the size of the ISS? polar research stations? yes they get resupplied often, but they don't have fusion tech and onsite indoor farming. It's a complex situation but I find it very interesting and important to our future. how big must the first Mars colony be to survive? bigger is better, but for pioneers establishing the first foothold, they need to get the biggest bang for their bucks Mar 28, 2021 at 18:48
• @StarfishPrime unfortunately I've been to prison, and met people doing life sentences who have been locked up 40+ years with no chance of parole. I was amazed at how they adapted to their tiny restricted irritating world, with no hopes of leaving, yet became positive and productive (after they overcame their initial despair). Many found solace and purpose in helping others or spiritual/religious beliefs. I think prison life applies very well to interstellar travel, and humans are much more adaptable and resilient than I ever imagined. Mar 28, 2021 at 18:55
• @KoonW and whilst I think of it, the multi-decade flight time to somewhere nearby runs the risk of someone else inventing a fusion rocket that's ten times better, or even a sensible beam or pellet propulsion system that's a hundred times better or more. What's worse than condemning your children to a tiny cell for most of their lives? Doing it with the knowledge that it was probably pointless, though at least there's a chance they'll get a welcome party when they finally arrive. Mar 28, 2021 at 19:17

3 Answers

The rocket equation works just fine whether for a rocket leaving a gravity well, or already in deep space. Things are easier in space, things like gravity drag are much less problematic and aerodynamic drag is nonexistent. Just use the equations you've already found and you'll get what you need.

Say we used a similar but improved engine (in the year 2060) to get 700 ISP.

You cannae break the laws o' physics.

The Isp of chemical engines is strictly limited by the mass of the exhaust products, and the temperature of the combustion. The latter cannot be higher than the melting point of your engine, so it tops out at 3000K, more or less. The former is limited by the chemical reactions of your fuel.

One of the most energetic rocket fuels tested was an unpleasant combination of lithium, hydrogen and fluorine, which produced a range of extremely corrosive exhaust species and required hard to handle and quite expensive chemicals and even that couldn't break 600 seconds. Anything more energetic requires metastable fuels which are a) not really that much better b) hideously dangerous and c) might not actually be practical to manufacture store and use, ever.

For 700 seconds then, you'll be needing some kind of nuclear thermal rocket, most likely.

if chemical rockets are used at journey’s start, how much could initial velocity be increased?

Not by much. Your chemical rockets give you a lot of thrust, but that low specific impulse means you can't use it for long.

From the good old rocket equation, $$\Delta_v = v_e \ln{\frac{M}{M_e}}$$ where $$v_e$$ is your exhaust velocity... lets go with the implausible performance you originally asked for, and so get an exhaust velocity of 7 km/s. $$M/M_e$$ is the mass ratio of your ship... if we give your ship an also moderately implausible mass ratio of 100 (so for a miniscule 125 tonne ship you'd a good 12375 tonnes of fuel or reaction mass, and that doesn't take into account the weight of the boost stage), you get a maximum $$\Delta_v$$ of a little over 32 km/s.

Your given flight profile gives a top speed of more like 23000 km/s, of about 8% of C.

Your rocket boost has reduced the flight time by about 27 days. Pretty small beans compared to the 110 year expected duration.

There are a bunch of things you can do to improve your flight performance, and none of them involve chemical rockets, which are frankly a weird choice once you've mastered all the technologies required to fly humans in a fusion rocket to another star. Its like using a horse to give a jet aircraft a bit of a boost to help get it off the runway sooner.

Saying how you could or should make your flight is a bit beyond the scope of this answer, but if you feel like asking more questions on the subject I'm sure you'll find plenty of suggestions.

• great info! what if engine cooling tech improves vastly in the next 40 years? would this allow higher ISP? although from what ive learned, engine cooling tech is quite sophisticated already. just a curiosity, as, seeing the numbers, it seems chem rockets are a waste in this case. thanks again! Mar 30, 2021 at 3:18
• @KoonW I don't see it helping enough to make up for the rest of the downsides. Ultimately, you're making a multi-decade flight so if you want to do rocket staging to help speed things up you may as well use high-Isp rockets to do so, eg. a disposable fusion-powered boost stage like the Daedalus Project proposed. The best answer is of course "don't use rockets", but that's for another question, I think ;-) Mar 30, 2021 at 8:49

Sorry to be so blunt, but as this was a concept I once considered heavily, you really aren't understanding the complexity or magnitude of the problems of interstellar spaceflight at all. Nothing about this concept works from an actual realism standpoint. Even well thought out generation ship designs are extremely unlikely to ever happen.

To start with, your ship is several orders of magnitude smaller than it would need to be. Proposals for unmanned probes have a payload mass larger than your intended spacecraft, with their dry mass(including all structural materials) being at least several times larger. Add in the fuel and you're talking about at least hundreds of thousands of tonnes just for a simple unmanned probe design.

Interstellar colonization like this requires a self contained environment to last a century, while efforts like Biosphere 2 couldn't even last a couple of years without massively cheating via pumping in outside air. The second attempt failed within literal months. While there actually was some impressive research done, it shows just how challenging it really is to build a self contained habitat. The only way you could plausibly have interstellar designs work is if they can be built sustainably within the solar system first, which gives much less of a reason to leave.

From a population standpoint, you need at least 160 people just to make the journey itself, let alone to colonize the destination when you get there. Most more realistic estimates suggest tens of thousands of people(though I will also note that the design shown here also doesn't give enough room for fuel either).

If you really want something to give your design an extra kick at no cost, a vastly better solution is the magnetic sail. Instead of having to carry all of the fuel, most of the deceleration can be done by creating a magnetic field against the interstellar medium and using that to slow down. A mission profile for what that might look like can be found here.

• i was hoping that fusion power would greatly reduce the fuel mass needed, help power lots of automation, and help get there much faster... for genetic diversity i am considering stored embryos and exowomb tech improved beyond current prototypes tested with animals. i didnt bring up genetics in this question because i was focusing on possibilities of chem rockets for initial delta V. but thank you for the links and feedback! Mar 30, 2021 at 3:27
• also bluntness is appreciated - i dont want to write a story that people are put off by because it offends their senses of reason :). and i love to learn about these subjects myself Mar 30, 2021 at 3:32

delta-v = 9.8 x ISP x ln((dry_mass+fuel_mass)/dry_mass)

for regular chemical rockets, the proportion of dry mass to wet mass is around 1:20-1:25

so plugging numbers, like 700-sec isp, and taking quite optimistic 1:100 (it is in space, no need to get through the earth atmosphere, something like that)

• 700 isn't unrealistic if we consider some hybrid chemical nuclear propulsion stage

we get 31.6 km/s - not a lot.

However, you do have fusion, direct fusion engine, or its indirect version. And the good thing about those is that you can vary their isp, basically the equivalent of regulating trust on a typical chemical rocket.

this way you can increase acceleration by increasing the rate of reactive mass "consumption", not the energy or fuel, but reactive mass. So whatever effect you wished to have you may get a better result with your fusion engine, instead of chemical rockets.

resulting speed at a flipping point is around 24'200 km/s so a plus-minus 100 km/s won't do a significant difference overall, so chemical rocket is only if is some part of the plot, but not as a by design solution, in most case scenarios. Get some kick from an orbit, 1km/s solid booster is enough.

Acceleration and mass

The overall mass of the ship is a little bit too little, so it on the short side, maybe consider it an order of magnitude more, even if it just for few pairs of individuals.

You need some bootstrapping equipment, and if you do not have handwavium or a bit more advanced tech, then it may take a hundred tons or a few on its own.

As acceleration rate - it realistic, but if as an example you need to adjust it for some reason having 1-2 gens arriving at a particular age, then you can do that safely, as it is on the conservative side of things.

• i tried to focus this question around the benefit of initial chem rockets, but youre right, and i have a lot of reading on this site regarding what is needed for self sufficiency on a generation ship. there seems to be a lot of good Qs on that, that i have saved for reading when i get a chance. Mar 30, 2021 at 3:34
• great to know about "throttling" fusion drives, giving a lot more flexibility. i need to learn more about possible generator and drive tech too, also bookmarked a lotta pages on that. thank you! Mar 30, 2021 at 3:35
• @KoonW adressing some other of your comments, realism of interstellar space travel, or space activities, is a challenging and big topic, so a little bit of handwavium does not hurt if there is a good story. And squezing people and their necessities, is more challenging aspect of your setting, which can be rich on details of all sorts, and then propulsion is just a fraction, which u can state and do not go in to details. Small 80-100y gen ship, is more a suprising novelty, which at the same time may have strong connections with visions of starship(spacex) to mars and it on mars, space stations Mar 30, 2021 at 6:35