# 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

• @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