# Are ion thruster trajectories classified as brachistochrones?

Usually, when performing trajectory optimizations for chemical rockets, the limiting factor is the $\Delta v$ budget. That means velocity change is the optimization factor, with things like transfer time being a secondary consideration. Some 'perfect' transfers, like an infinite apoapsis bi-elliptic transfer even have infinite transfer times.

By contrast, for example when discussing more advanced forms for propulsion, the $\Delta v$ budget is no longer setting a limit and instead minimizing the transfer time is the most important thing to consider. Typically, the spacecraft in question have a high continuous acceleration, and the (subjectively) more intuitive approach is to thrust towards the target and start to slow down half way. This is a brachistochrone transfer.

Here is what I am thinking: A spacecraft using an ion engine is not usually limited by $\Delta v$ either, and it also uses continuous thrust. However, it has a very low thrust, meaning that the trajectory looks nothing like the simple high-thrust transfers. Here is for instance the trajectory of the spacecraft Dawn:

image: JPL

So, given the similarities

• Continuous thrust
• Not limited by $\Delta v$
• Optimization is for transfer time

Are ion engine trajectories also classified as brachistochrones?

• I don't know brachistochrones at all, so let me see if I understand your question here. If low thrust were brachistochrones, then there exists at least one metric whereby continuous thrusting leads to better results on that metric although initially it seems like it wouldn't? – ChrisR Mar 30 '17 at 5:35
• Not in generality. Given the way you have asked the question it would just take one example of an ion engine trajectory that is not brachistochrone. A simple circle to circle spiral is such an example. Whether it meets the third criteria of "optimised for transfer time" is another matter but you have stated that as an assumption rather than a requirement for the answer. Whilst you have put an image of an interplanetary transfer you also have not stated this as an assumption. Presumably you wish to exclude ion thruster trajectories that aim to keep one in the same place, such as NSSK? – Puffin Mar 30 '17 at 9:28
• I tried to read the Wikipedia article on Brachistochrone curve to see if I could figure this out but gave up fairly quickly. I think it would be great if you or someone could add a link to a description of a brachistochrone transfer - what that means, and how one would be able to determine if a given transfer orbit were or weren't a brachistochrone transfer. Describing it in words in the body of the question would also be fine. Thanks! – uhoh Jul 6 '17 at 10:46
• The hard sci-fi resource Atomic Rockets (projectrho.com/public_html/rocket/torchships.php) uses the term "brachistochrone orbit" to describe powered transfers with fusion rockets. – ORcoder Feb 16 '18 at 21:45
• I think also that the term "brachistochrone" is not really used/familiar/understood in real-world mission design. It's caused controversies before. – ikrase Apr 27 '20 at 3:48

This question turns out to be surprisingly involved to answer.

### 1. "Brachistochrone" is a 17th-century term for a particular physics problem.

The term appears to have originated in 1694[src] (or several years later, by source), with Johann Bernoulli, in the course of originating (or at least popularizing) the Brachistochrone Problem (whose solution is the brachistochrone curve) by way of a challenge issued within the 1682–1782 scientific journal Acta Eruditorum. You can try to read it (in Latin) if you want more particulars.

The Brachistochrone Problem is not about spacecraft. A typical 1855 definition ran (spelling original):

The curve upon which a body moves in the least possible time from one given point to another, is called the brachystochrone.

This makes sense because "brachistochrone", whatever its coinage, undoubtedly comes from Greek "βράχιστος χρόνος" ("brákhistos khrónos", "shortest time").

### 2. This term was then adapted to describe constant-thrust or constant-acceleration trajectories for spacecraft in a sci-fi context.

The application to space travel appears much later. In fact, after searching through many science fiction, fiction, and literature databases—if admittedly the only extant ones being variously shoddy—as far as I can tell, references to "brachistochrone" only appear very recently, and in connection with the Atomic Rockets site. This suggests that Atomic Rockets invented (or at least popularized, from a presumed coinage in the golden age of sci-fi) the usage of this term; I have contacted the author for comment.

In any case, the term is only very rarely used, even in modern sci-fi, and it is completely absent in technical materials. Constant-thrust trajectories tend to be called . . . "constant-thrust trajectories", to the limited extent they are discussed (owing undoubtedly to the limited extent to which they are conceivable with realistic technology) in the scientific literature.

### 3. This adaptation is problematic because these trajectories are not actually "brachistochrone" in general.

Again, "brachistochrone" literally means "shortest time".

The shortest possible travel time of a rocket to traverse a distance in free space is given by the rocket expending all of its propellant as fast as possible, and then coasting. A constant-thrust rocket will not achieve this unless that constant thrust is also the maximum the rocket can ever produce. This may be necessary for a surface takeoff, but otherwise it puts unnecessarily high G-forces on the payload (such as any astronauts) that only increase as the propellant is expended.

Toning down the G loading of a powerful rocket gives rise to constant-acceleration trajectories, where the rocket throttles back (which may be unrealistic in itself) progressively, as propellant is expended, so that 1 G (or whatever) is maintained. This is also called "brachistochrone"—but it obviously isn't, since the rocket is not expending propellant at the maximum rate.

There also remains the issue that the shortest travel time to a destination means always accelerating toward it at the greatest possible rate . . . and pointedly, never decelerating.

Essentially, you have to pile a lot of caveats onto the term to make it make sense for space travel. Something like, the brachistochrone being the shortest-time trajectory, subject to a maximum acceleration of [X] gees, with the criterion that the spacecraft stops at the end, and also the interesting acceleration comes from a rocket, not solely gravity.

### 4. Point of order: current ion drives' trajectories have not been constant-thrust, and are very much still Δv-limited.

You are correct that ion drives are capable of thrusting for long periods in principle, namely because they tend to have a very low maximum mass flow rate. However, current ion spacecraft haven't technically demonstrated constant-thrust trajectories yet.

For example, the full version of the very graphic you linked shows that the trajectory only was thrusting for about half the time. Much of this must have been at reduced power/thrust, too; Dawn is solar powered, and the maximum draw power of its drive (3⨯ NSTAR surplus, each 2.1 kW) greatly exceeds the solar panels' capacity (e.g. at Ceres, about 1.5 kW). Those two transfers also used about 84% of Dawn's Δv; even at reduced power, continuous thrust would have expended all propellant.

### Conclusion

Fundamentally, this is a terminological issue. You can classify, or refuse to classify, various trajectories as "brachistochrone", but that's going to be rather arbitrary because "brachistochrone" is not, as far as I can tell, a technical term to describe spacecraft trajectories. Instead, it appears to be an informal science-fiction term—and also one that doesn't really make a whole lot of literal sense.

Ion drives have a low mass flow rate, resulting in long burn durations. With a significantly higher mass ratio and nuclear power, truly continuous-thrust trajectories could be achieved with the likes of ion engines (although current spacecraft have not yet achieved this). Speculative propulsion technologies could make such trajectories even more appealing.

• +1 I like your answer better than mine so I've voted to close that question as duplicate of this. – uhoh Apr 27 '20 at 1:34
• @uhoh Thanks, but while our answers seem to have quite some similarities, I would say the questions are fairly different (although separately, that question is rather under-posed, in my view) . . . – imallett Apr 27 '20 at 4:14
• In Stack Exchange when we mark a question as duplicate it really means that the question is answered by answers to the other question, even if the questions are not the same. I've started the process but it will still take four more votes to happen so others will review it as well. I think you've explained that there are no real "brachistochrone orbits" in spaceflight more convincingly than I have, so future readers should be directed to this answer. – uhoh Apr 27 '20 at 7:52