How much fuel does a ship use in the Expanse?

So, I've been binge watching The Expanse on Syfy, and I love how much detail the screenwriters and authors put into making the ships and the physics as real as possible in the show. But, there was one detail that still kind of confuses me: gravity on ships like the Donnager and the Rocinante.

I do understand how it works. While the engines are on, the ship is accelerated to simulate gravity (in this case, accelerated at 9.8 meters per second, simulating Earth's gravity). But although constantly accelerating engines are used today, it is far less efficient than lining up a transfer through one quick burn. And that's just for a lightweight probe.

So, considering the fast but inefficient engines of The Expanse, just how much fuel would one ship need to burn in order to sustain a constant sense of gravity for the entire duration of a trip (such as from Earth to Ceres)? Sure, a massive vessel like the Donnager could probably hold enough fuel, but there's no way a ship as tiny as the Rocinante could pull off a journey that far, right?

• I think the idea of the Epstein drive is that they're extremely efficient, to a science-fictional degree, and that's why it's possible to maintain 1g thrust all the way. Whether the math works out on that, I can't say, but "considering the fast but inefficient" enginges part of the question seems to be starting off on the wrong foot (the strategy of going full burn may well be inefficient, but the engines themselves are fabulous enough to make up for it). Commented Jul 17, 2016 at 19:17
• See space.stackexchange.com/questions/4001/…. You have to accelerate the fuel needed for later acceleration, and you need fuel to come to a stop (or at least slow down considerably) at your destination. Commented Jul 17, 2016 at 19:18
• Assuming the engines are 100% efficient, a ship that creates gravity by burning continually in one direction, the slowing down by burning in the opposite direction at the half-way point could use as little as a few hundred kilograms of fuel to get across the solar system. Of course, such engines are the stuff of science fiction. Commented Jul 17, 2016 at 19:23
• @Valorum - A hundred kilograms of fuel compared to how much payload mass? Commented Jul 18, 2016 at 21:26
• Seven. It uses seven fuel. Commented Jan 19, 2017 at 14:50

So, considering the fast but inefficient engines of The Expanse

The entire point of the engines in The Expanse, called "Epstein drives", is that they're not "fast and inefficient". They are in fact extremely efficient. And, of course, completely fictional and indistinguishable from magic – most of the physics in the show and book series is realistic, but the existence of the highly efficient Epstein drives is the main hand-waving they had to do in order to present a colonized solar system (while still confining humanity to the solar system).

I don't remember how much the TV series talked about the Epstein drives, but the technology is introduced on the very first page of the first book in the series (Leviathan Wakes) and immediately explains how the solar system was colonized.

There is also a short story called Drive, which is a prequel focusing on the invention of the Epstein drive. It can be read online for free on Syfy's web site: http://www.syfy.com/theexpanse/drive/prequel.php – it might make you able to suspend your disbelief on this matter.

While the engines are on, the ship is accelerated to simulate gravity (in this case, accelerated at 9.8 meters per second, simulating Earth's gravity).

The Epstein drive can sustain higher accelerations without problems too, to reach the destination faster. The limitation of acceleration here is the human body, but there are steroid cocktails that let humans survive very high burns for short periods of time. Humans raised on Earth can survive slightly higher than 9.8 m/s² without trouble, while those from the asteroids prefer even lower than that.

Of course, ships travelling on a sustained burn need to flip around at the half-way point and then burn in the opposite direction to slow down, which gives the pasengers the same simulated gravity throughout the journey (with the exception of the flipping).

• There are not indistinguishable from magic whatsoever. Fusion can get an isp of 2 million. Assuming 40% of energy goes directly to thrust a 200 ton ship needs 8 kilograms of deuterium and 9 tons of water (assuming water is the bulk of propellent) to maintain 1G for 24 hours. That's physics. That's possible in real life. Far from magic. Having the fusion detonation inside the ship, &having a tiny exhaust plume is magic and mostly for dramatic effect so ships can fly close by. The propellent efficiency taking up a small amount of space while going that fast isn't magic. Commented Jan 20 at 21:43

The Tsiolkovsky rocket equation gives a general formula, applicable to any time of rocket that shoots exhaust out the back to accelerate in the forward direction, which can tell you the ratio of initial mass including fuel to final mass once fuel has been expended, given the values of the "effective exhaust velocity" (given for different types real and of hypothetical rockets here and here), the acceleration rate during the rocket burn, and the change in the rocket's velocity between the beginning and end of the burn. In this answer I did some substitutions and rearrangements on the relativistic version of the Tsiokolvsky rocket equation, to get a formula that would give you the initial/final mass ratio expressed in terms of effective exhaust velocity V, acceleration A, and distance traveled D between the beginning and end of the rocket burn (expressed in units where the speed of light is 1, like seconds and light-seconds):

e^((1/V) * atanh((A * sqrt((D)^2 + (2 * D/A)))/(sqrt(1 + (A * sqrt((D)^2 + (2 * D/A)))^2))))

As I noted in the answer, this equation is for continuous acceleration in one direction, but if you want to figure out the mass ratio given the assumption that you accelerate for a distance of D/2 and then decelerate for another D/2, you'd just substitute D/2 into the above equation and square it, giving:

(e^((1/V) * atanh((A * sqrt((D/2)^2 + (2 * D/(2A))))/(sqrt(1 + (A * sqrt((D/2)^2 + (2 * D/(2A))))^2)))))^2

So, you can just copy and paste this equation into this handy online calculator, hit the "execute" button, and then fill in the desired values for V, A, and D to get the mass ratio. Handy units for travel within the solar system might be hours and light-hours, in which case a 1g acceleration works out to A=0.00011776 and 1 astronomical unit (AU) works out to D=0.138612. Earth has an average orbital distance of 1 AU and Ceres has an average distance of 2.7675 AU, so the closest approach is about 1.7675 AU (or D=0.245 light-hours) and the farthest is about 3.7675 AU (D=0.522 light-hours).

For a maximally efficient antimatter rocket where the all the exhaust shoots out at the speed of light, we have V=1, and with A=0.00011776 and D=0.522 the second long equation tells you the initial mass including fuel would only be 1.0158 times larger than the final mass at the end of trip, so you only have 0.0158 tons of fuel for every ton of payload mass.

But an antimatter rocket is a pretty far-future technology, more realistic would be some kind of torch ship which has both high acceleration and high change of velocity, compared to either existing systems like ion drives (high change in velocity compared to chemical rockets, but very low acceleration so the time to achieve that change in velocity is large) or chemical rockets (high thrust but much lower change in velocity than an ion drive, for a given fuel mass). Aside from really advanced technologies like antimatter drives, the most likely technology would be some kind of nuclear drive; the chart here shows that the exhaust velocities for various types of nuclear fusion reactions as fractions of light speed, and the giant rocket engine list I posted earlier also shows a number of nuclear-thermal rockets that use fission (look for ones with 'NTR' in the 'code' column of the chart). That chart gives effective exhaust velocities in meters/second, but you can divide by 299792458 m/s to get the effective exhaust velocity as a fraction of light speed; I wouldn't bother with anything that has an effective exhaust velocity much lower than 0.004 times light speed (about 1.2 million meters/second) or 0.003 (about 900,000 meters/second), since plugging these in for V in the equation I gave while keeping the other values of A=0.00011776 and D=0.522 indicates a mass ratio of about 50 for V=0.004 (meaning you need about 49 tons of fuel for every ton of payload) and 186 for V=0.003 (185 tons of fuel for every ton of payload).

Looking it over, there are very few examples on the chart that have an exhaust velocity in the range of 1 million m/s or higher and have a high thrust in the thousands of Newtons range (necessary if you want 1 g acceleration); in fact it looks like none of the nuclear-thermal fission designs would be in anywhere near the right range, so a nuclear fusion drive (labeled either 'Fusion' or 'Pulse' in the 'code' column) might be the most likely alternative to antimatter. However, near the bottom of the chart there's one called "NSWR (90% UTB) MAX" which turns out to be a nuclear fission based design using a nuclear salt water reaction, with this particular option assuming a very enriched uranium salt where "the 2% uranium bromide solution used uranium enriched to 90% U235 instead of only 20%". In this case the theoretical exhaust velocity would be 4700000 meters/second, and if you divide by 299792458 m/s you conclude this works out to V=0.0157, plugged into the equation that gives a mass ratio of only 2.7, or 1.7 tons of fuel for every ton of payload. So, that could be a good option for a type of technology that fits the basic parameters and isn't as advanced as a fusion or antimatter rocket.

• But there was an interstellar ship in the story that was under construction for Tau Ceti. Perhaps antimatter is still on the table. Commented Jul 19, 2016 at 0:49
• Well, the page on the nuclear salt water reaction engine does preface the idea of 90% enriched uranium by saying "Zubrin then goes on to speculate about a more advanced version of the NSWR, suitable for insterstellar travel." Maybe I was too hasty in ruling out antimatter on the basis of interstellar travel (I'll edit that line), but it still seems like a ridiculously advanced technology given what I've seen in summaries of the overall tech level of The Expanse series. Commented Jul 19, 2016 at 1:12
• The interstellar generation ship bound for Tau Ceti was not supposed to continuously burn like the other Epstein drive outfitted ships in the series (which accelerate for half the journey, flip, and then decelerate) – it was instead designed to burn until it reached a certain speed, cut the drive, and then spin up the main section of the ship to create centrifugal "gravity". Commented Aug 10, 2016 at 10:20
• The rocket engine list link you supplied has a brief discussion of the Eastern drive. projectrho.com/public_html/rocket/… Commented Aug 10, 2016 at 10:52
• +1 I was thinking of matter-antimatter engines as well. Presently, antimatter created in particle accelerators is very expensive to produce and store, but who knows several hundred years from now? Commented Mar 22, 2017 at 16:54

The answer to a quantity of fuel question is quite defined in the series — very little.

Fuel

The fuel is described as "pellets" for fusion reactors. Pellets are small enough to have a "crate" of them just lying around on the ship and transported by a person grabbing the crate. So not something loaded by the ton.

The reactor is described as making small mass of fuel fuse and release large quantities of energy.

In first book Rocinante is described having enough fuel pellets loaded for "thirty years" of reactor use.

Drives

The torch drives, described in the series as older technology with current use limited to very small craft, also need significant quantities of propellant (which typically seems to be water in the setting). They seem to work on principle of fusion drive heating up and ejecting propellant to generate thrust.

The current Epstein drives seem to need either very little or none propellant. They are described as being a variation of fusion drive that can somehow convert energy into thrust highly efficiently. The details are purposely omitted, it's one magical thing that setting hinges on.

Economics

The price of fuel in the book is loosely compared to be less than imported cheese in outer planets region.

Whenever restocking of Rocinante comes up it seems to be distant third: repairs and ammunition, then human consumables, then tech consumables like propellant and fuel.

While ships are mentioned to be expensive themselves, the fact of viable space piracy in the setting seems to indicate that running the ship if you already have one is affordable and doesn't depend on any controlled supply of fuel.

Travel

The adopted model of space travel seems to hold humans as limiting factor.

Ships are described to be able to easily generate accelerations which will simply kill any human crew. So the limiting factor of travel speed is crew training — military is trained to (and can be ordered to) sustain high acceleration for prolonged time, civilians not so much.

Still it seems to be sufficient enough for ships to "just go" where they need to, rather than consider/use gravitation of planets for acceleration and care about their convenient position.

Overall

Setting states that Epstein drives used for space propulsion are incredibly efficient at using tiny fuel mass for thrust. without (?) use of propellant Later books seem to say that Epstein drives still do need propellant and occasional resupply of it, just much much less than torch drives.

The Epstein drive used in the Expanse is just within the window of what's theoretically possible for a fusion drive, according to this article. In "Drive", Epstein's original prototype gets up to 5% of c before it runs out of fuel, so we may assume that the Nauvoo will get to 2.5% of c and then coast for a few centuries before decelerating at its destination.

Well they use fusion reactors. In hydrogen-hydrogen fusion 0.71 % of the fuel is converted to energy (E=mc^2*0.0071) . This is still a lot of energy due to the c^2 term being there. How this energy is utilised to produce thrust is an unsolved issue, but it seems to me that a ship with a few tons of fuel could virtually travel with 1 g for years before running out of fuel. This off course is dependent on the engine's effectiveness which is unknown to us.