# 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
• Is a minimum time trajectory more efficient than a minimum energy one? – Narasimham Sep 22 '17 at 6:46

## 1 Answer

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.

• 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