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And how much fuel would you need to carry? I'm trying to determine what kind of engine you'd need to accelerate a good-size spacecraft (thousands to millions of tons) at a constant 1g in a race across the solar system, and what kind of weight-to-fuel ratio you'd end up with (as well as what ratio would be ideal). Obviously fusion rockets are currently the most plausible and efficient method of propulsion we have in "harder" science fiction (aside from nuclear propulsion and lasers), so I need to know what kind of reactor you'd need to be packing to push a racing vessel equipped with crew, shielding and weapons down the cosmic cul-de-sac, so to speak.

I'd really appreciate it if anyone could tell me how to solve for this problem, because I do need to apply it to other constant rates of thrust (most of the racers are altered transhumans who can handle abnormal g-loads and one aptly-named "Machinehead" has cut himself down to a cybernetically-augmented head in a jar of suspension fluid in order to push the smallest possible craft at a constant 43gs).

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ "Obviously fusion rockets are currently the most plausible and efficient method of propulsion we have in "harder" science fiction (aside from nuclear propulsion and lasers)". Fusion rockets are a form of nuclear propulsion. $\endgroup$ – AngelPray Jan 7 '17 at 18:39
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    $\begingroup$ Before anyone can really give an answer, there are two important points: (1) There aren't any functional, production fusion reactors in existence so it's very tricky to work out how big an example of a technology that we can't currently create would be and (2) Miniaturisation is often a part of technology progress. Calder Hall was the world's first commercial power plant and produced less power than one of the Ford-class carrier's A1B reactors. One was an entire facility, the other can fit (with another) on to a ship. $\endgroup$ – Matt Bowyer Jan 7 '17 at 18:44
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    $\begingroup$ I'm not certain this is answerable given we don't know how much fuel a fusion reactor could be made to pump out. Current man-made fusion reactors lose energy so obviously you can't use them. Natural fusion reactors are quite dangerous to put on-board a spacecraft. $\endgroup$ – Bellerophon Jan 7 '17 at 18:46
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    $\begingroup$ Hard science answer isn't possible yet - these reactors are not developed yet, and all we are getting is negative energy balance. Our, at best, slightly positive one. There are no scientific papers on scalability of this solution, because we don't really have the solution yet. $\endgroup$ – Mołot Jan 7 '17 at 19:16
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    $\begingroup$ The closest answer I can give you is that the inventor of the polywell fusion reactor (not known for it's efficiency by the way) claimed that the power output scaled with the 7th power of machine radius... But do you really need a fractional-Petawatt reactor output though? Flinging 1 tonne per second out the end of a 1 million tonne ship at 0.3% light speed every second will generate a constant 1g acceleration and requires at bare minimum of 500 terawatts but flinging 100 tonnes a second at 1/100th the speed will generate the same acceleration but requires only 5 terawatts. $\endgroup$ – Samwise Jan 7 '17 at 23:11
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Warning: We haven't reached hard science level for such question, so I'd treat it as science based, tops.

enter image description here

For inspiration of design I'd suggest looking at "Stellarator":

Quite possibly a realistic fusion reactor would include even more twisted design to contain plasma in magnetic trap. With possibility to direct part of plasma through nozzle.

Realistic engine would have to be huge. Big part of energy is being produced in form of neutrons that escape any magnetic containment. In a normal power plant they would be used to heat water and produce electricity... you would presumably have to make it big enough, that such neutrons would hit some Li-6 and H-1, thus producing you further fuel (deutrium and tritium).

Needless to say, regardless how efficient such process would be, you'd need huge radiators to dispose waste heat.

As nicely put by project rho:

enter image description here

http://www.projectrho.com/public_html/rocket/basicdesign.php#radiators

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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variables and constants

$a$ - is the craft acceleration, in m/s2
$g_0$ - is the acceleration at the Earth's surface, in m/s2
$I_{sp}$ - is the specific impulse of engines, in seconds
$\Delta v$ - delta-v, value you have to define
$m_0$ - dry mass of the craft, kg

Equations

$\Delta v = g_0 I_{sp} \ln\left(\frac{m_{fuel}+m_0}{m_0} \right)$ - Tsiolkovsky rocket equation

Q/A

how much fuel would you need to carry?

$m_{fuel} = m_0\left(\exp \left(\frac{\Delta v}{g_0 I_{sp}}\right) - 1\right)$

what kind of weight-to-fuel ratio you'd end up

$\frac{m_{fuel}}{m_0} = \exp \left(\frac{\Delta v}{g_0 I_{sp}}\right) - 1$

well as what ratio would be ideal

cite from wiki specific impulse :

Theoretically, for a given delta-v, in space, among all fixed values for the exhaust speed the value $v_e = 0.6275 \Delta v$ is the most energy efficient for a specified (fixed) final mass, see energy in spacecraft propulsion.

Ideal what what ideal what?

Obviously fusion rockets are currently the most plausible and efficient method of propulsion we have in "harder" science fiction...

No, it is not obvious, it depends on the situation(goals) and definition of efficiency(which parameter is optimized).

An example energy wise mass-drivers are very efficient, as they use planets and other very massive bodies as reactive mass.

Because of conservation of momentum and connection between momentum and kinetic energy, kinetic energy in the system will be distributed not evenly:

$$\Delta v_{planet}M_{planet} = \Delta v_{craft}m_{craft} = p_0$$

$${E_p} = \frac{p_0^2}{2\cdot M_{planet}} ,\, {E_c} = \frac{p_0^2}{2\cdot m_{craft}} ,\, \frac{E_p}{E_c} = \frac {m_{craft}}{M_{planet}}$$

Obvious here is they have to have a massive body, but if the goal is to reach a certain speed, which may be perfectly fine for travel cross the solar system.

Can be the system be applied to the race it depends on rules and the goals of the race - if it is about to maneuvering in dense star war asteroid fields then with proper tech it is the only viable option because the owner will win the race.

If it is about of reaching $\alpha$-Centauri (or Oort cloud for the inner system race) it might be also a valid option.

Reactor energy

$\eta$ - engine efficiency, defined as $\frac {\text{kinetic energy of exchaust}}{\text{Total energy produced by reaction used in the engine}}$

$P_r$ - reactor power, total

$$P_r = \frac{1}{2} \left(g_0 I_{sp}\right)^2\cdot\dot{m}_{fuel}, \, \dot{m}_{fuel} = \frac{a}{g_0 I_{sp}}\cdot \left(m_0 + m_{fuel}\right)$$

$$ \begin{align} P_r &= \frac{1}{2\eta} a\cdot g_0 I_{sp} \left(m_0 + m_{fuel}\right) \\ &=\frac{1}{2\eta} a\cdot g_0 I_{sp} m_0 \exp \left(\frac{\Delta v}{g_0 I_{sp}}\right) \end{align} $$

enter image description here

For an thermonuclear rocket engine only charged products are useful, not charged and any gamma radiation during the reaction are the waste.

Thus, D+T, D+D are so so fuels, D+3He - also have its own problems(needs sophisticated setup, to prevent D+D react first, as D+D is faster reaction than D+3He)

$\eta$ for D+D fuel is less than about 0.66, $I_{sp}$ is about 826'000 seconds

So for one 100'000 tons (dry mass) ship, with acceleration of $g_0$, with delta-v 100km/s, D+D fuel

$P_r=$ 6e+15 W or 6000 TW

$m_{fuel} \approx 0.0124\cdot m_0$, or 1.24% of dry mass of the ship, or 1240 tons

The race will go on about 2h 50min, max distance covered in the time will be about 509'683 km with resulting speed 100 km/s.

Reactor size

ITER projected to be a 500MW reactor, plasma volume 840 cubic meters (https://www.iter.org/factsfigures)

So about 0.6 MW/m3, it is with D+T fuel. Lawson criterion for D+D is about 2 orders of magnitude higher(http://hyperphysics.phy-astr.gsu.edu/hbase/NucEne/lawson.html), it means when established stable state it has to produce 2 orders of magnitude more per volume to work, so we may expect higher output from a D+D reactor (it must if it works).

so for a 100'000 tons ship, with a 6000TW reactor, we might expect something like a cube $2150 \times 2150 \times 2150$ meters (or an equivalent volume of another shape), for a D+D reactor $460 \times 460 \times 460$ meters (or an equivalent volume of another shape).

Pro tip

Use lesser acceleration with thermonuclear engine or use massdriver catapult to put your cyborgs on the edge of their acceleration capabilities.

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