# 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