# Brachistochrone variation for Earth-to-Mars Orbit

Wouldn't a Brachistochrone curve to Mars orbit be a much faster and more efficient way to send personnel and equipment to mars as compared to the Hohmann transfer if we used an Ion engine?

The below picture is from here, but with the addition of the blue line to indicate a trajectory that seems more efficient to me.

• Brachistochrone curve is defined to be the shortest path in terms of time taken between two points A and B. The condition being the only force acting is gravity. The moment you introduce another force you're changing the problem. Because if you add any kind of propulsion system then you are minimising your curve on travel time, with a constraint in the maximum available thrust. Which is not exactly a Brachistochrone curve. – Astroynamicist Sep 20 '17 at 0:21
• @Astroynamicist Be careful about the distinction with the "classical" Brachistochrone curve (under a uniform gravity field) with the general Brachistochrone problem, which is a class of problems in the field of calculus of variations (or variational calculus). The condition is whatever you say it is, whether it's a uniform gravity force, thrust, whatever. The solution will look different depending on the inputs, but it's still a Brachistochrone problem and can be solved in the same fashion. – Tristan Sep 20 '17 at 14:21
• @Tristan I got your point. so any minimum time problem is considered a Brachistochrone problem. So, if I have an optimal trajectory that minimizes delta-v then is that a Brachistochrone, considering it will take the minimum time for my constraint? – Astroynamicist Sep 20 '17 at 21:34
• @Astroynamicist A trajectory that minimizes delta-v will likely not minimize transit time. It still falls under a class of problems covered by calculus of variations, though -- namely, constraint minimization of a functional. – Tristan Sep 21 '17 at 14:36
• Does this answer your question? Are ion thruster trajectories classified as brachistochrones? I think @imallett's answer does very nicely! – uhoh Apr 27 '20 at 1:27

## There can't be a "Brachistochrone curve to Mars orbit"

A Brachistochrone curve or something analogous would be a non sequitur in orbital mechanics as it is defined by a constraint, such as a track or rail whose shape is varied. Without a constraint to optimize, you only have the orbits that respond to gravity and propulsive forces.

If there were a physical track in space, you might optimize its shape to minimize time in an analagous way by perhaps falling towards the Sun and then veering later, but you would have to attach the track to something much bigger and immovable, which is impractical.

## The use of the term is relegated to fiction on Earth and science on Kerbin

2020 update: Comments point to the use of Torchships; What exactly is a Brachistochrone anyway? which while it doesn't give any useful math or science, says:

A "Brachistochrone" is a minimum transit time / maximum deltaV mission. Torchships use this because they have lots of deltaV to spare.

Detail of artwork by Philippe Bouchet AKA "Manchu" for Robert Heinlein's Time For the Stars, Torchship "Lewis & Clark"

Likewise Scott Manley's video Brachistochrone Trajectories For Spaceships Explained talks about torchships but without anything more than $$x = \frac{1}{2} a t^2$$ in the math or science departments.

So while "Brachistochroneness" might be invoked on Kerbin, I don't see how it can be applied to propulsive spaceflight.

• Digging a little deeper, for a given set of constraints, there would be a brachistochrone path, i.e., a path for which the traversal time is the shortest. (Note, brachistochrone comes from βράχιστος χρόνος, which literally means "shortest time"). You would apply the same variational calculus concepts that you would use to get the "traditional" brachistochrone, with the a priori understanding that the solution would probably not look anything like the traditional cycloid solution. Ultimately, it's just a constraint minimization problem, albeit with different constraints and inputs. – Tristan Sep 19 '17 at 14:37
• @Tristan would you consider expanding on that? Are you talking about the "normal" brachistochrone problem but with a $-\mathbf{\hat{r}}/r^2$ force instead of $-g\mathbf{\hat{z}}$? So for example a given initial impulse followed by constrained motion from Earth's to Mars' orbit might be less time than a Hohman ellipse? OK that sounds interesting! – uhoh Sep 19 '17 at 15:04
• Ultimately it depends on how you set it up, but yes. The solution technique will be the same regardless. Unless specific constraints are given, I suspect the solution will be entirely uninteresting (i.e., a hyperbolic orbit that intersects at the right place and time) – Tristan Sep 19 '17 at 16:15
• A more general sense of "brachistochrone" is in fact used to describe very-high-delta-V orbit-ish paths. – Nathan Tuggy Sep 19 '17 at 18:03
• @NathanTuggy OK, well "orbit-ish torchship trajectories" are about as antithetical to constrained trajectories in conservative force fields as I can imagine, but it's not like mathematical terms like brachistochrone curve have well defined meanings that allow us to effectively communicate or anything like that, right? – uhoh Sep 20 '17 at 0:31

First: As (somewhat confusingly, IMO) stated in the other answer, the use of "brachistochrone" is somewhat of a misnomer, potentially. I will use the term "constant-acceleration minimum-time transfer".

Second: "Efficiency", in spaceflight, is a somewhat complicated and many-sided thing. Efficiency of cost? Total spacecraft mass lifted into orbit? Amount of energy? Duration? Fuel?

Simply to travel along that blue line, which makes a sharp angle with the orbit of both Earth and Mars, and which travels backwards relative to the orbits of the planets (on the trip from Earth to Mars) requires an immense amount of propellant compared to a more normal trajectory like the one on the original drawing. Having an ion drive doesn't change that fact (though it does make the amount of delta-V somewhat more achievable. Keep in mind that ion drives are so low in thrust that they may take months just to escape from the orbit of Earth -- in which time Earth and Mars will both have moved a significant distance in their orbits.) A constant-acceleration minimum-time transfer along that line is even worse -- taking thousands of times more delta-V than even just a basic trajectory along that line.

If time is of the essence, and you have access to technology that is currently only science fiction (imagine an engine that is as powerful as exploding several thermonuclear warheads a second... inside the ship's rocket nozzle) then this is practical. Otherwise, the need to save propellant, fuel, or money building the engine will make more limited options more efficient.