Calculating near future spaceship rocket delta V in a vacuum

Question

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!

Answer 1

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.

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

You cannae break the laws o' physics.

The I_sp 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.

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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.

Answer 2

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.

Answer 3

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.