How one of Calculus’ Oldest Problems can Help Spaceflight 10x in Efficiency
The Brachistochrone Problem:
Johann Bernoulli proposed the “Brachistochrone Problem” in 1696, and with, that Johann Bernoulli pretty much created the whole field of calculus of variations. The Brachistochrone problem was simple: “Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time”, but its answer was not immediately obvious to Johann Bernoulli. Johann Bernoulli tried to find the most efficient curve by measuring the pathway of refracting light, which was known by then to always take the most efficient pathway, but Johann Bernoulli had a major measuring error. Johann’s brother, Jakob Bernoulli, originally proposed that the curve would follow that of a cycloid, but he also had a proof error. Upon hearing of the problem, Newton solved it within a day by using the tangent of a cycloid to form another cycloid, a tautochrone curve:
The solution:
Newton’s solution was revolutionary in that it used a path for the bead that would use all of its momentum by the time it reached the second point, and in essence meant that the curve would swoop down below the second point and the bead would have to slide upwards until it stopped exactly at the second point, which was never apparently obvious beforehand:
How will this be useful?:
Now back in the 21st century, orbital mechanics has a brachistochrone problem of sorts in that we don’t know what the most efficient route for a spacecraft is with the propulsion methods we currently utilize, or at least we don’t know how to implement the most efficient path. In the future technology like nuclear propulsion and plasma-based propulsion will allow for constant acceleration through an orbital maneuver which will allow for easier implementation of a brachistochrone curve. For an orbital transfer to happen to somewhere like Mars for example, the spacecraft first has to burn prograde to the orbit, or in the direction of the orbit; and then once it reaches its target, it has to burn retrograde, or in the exact opposite direction, for it to kill enough of its momentum to be captured and land on Mars.
How will it work?
The idea for a brachistochrone maneuver entails using a method of constant acceleration to be able to burn prograde until exactly halfway to the target, or the apogee, then the spacecraft can turn around to burn retrograde to kill enough of the momentum to be easily be captured by the target’s gravity. Such a maneuver would not only be the fastest possible route, but it could also be the most efficient for slow-accelerating engines such as ion-drives.
What can the Brachistochrone maneuver prevent:
Orbital mechanics is very hard, and to be able actually reach an intended target requires fully accurate precision. Interplanetary travel has been described as “firing an arrow at the moon and hitting a grape on the surface”, in that one small screw up of 0.1 extra m/s can end up sending your spacecraft hurtling into deep space, never encountering its intended destination. We don’t quite have the right technology to be implementing brachistochrone maneuvers yet, and even though we have engines that work best under constant acceleration such as ion-drives, we don’t have the computing power to be able to measure and feel when exactly to switch directions. Such a computer would have to be implemented on-board the spacecraft as the vast distance of space would make communication time of one on earth too long.
How we can start implementing Brachistochrone maneuvers:
The uprising of quantum computing, and dynamic compiling programing languages, such a Julia, can allow for large amounts of data to be interpreted efficiently to determine the correct route of a spacecraft. As more powerful methods of propulsion become available it will also become easier to use brachistochrone trajectories. For now, we can start simulating what brachistochrone maneuvers could look like to start getting ready, and important data to use for that is Spacex’s falcon booster suicide burn data. The landing of the falcon boosters in similar to a brachistochrone burn in that it kills all of the momentum by the time it reaches its target. We can also use Nasa’s Orion’s new reaction control systems to continues to innovate on deep space RCS that could quickly respond to adversity.
The future is efficiency:
Future methods of space travel our supposed to be powerful enough that propulsion would come second to the worry of time. Things in space are far away, and it is worth making sure that a spaceflight can be done in the most quick and efficient way possible in order to minimize the amount of time wasted aboard a spacecraft. Brachistochrone maneuvers are going to literally be the fastest way to get from point A to B, but whether or not that can become a reality will come down to whether or not it can be computed in a appropriate amount of time.
Learn More:
https://www.nasa.gov/vision/space/travelinginspace/future_propulsion.html
