When a spacecraft is circling a planet or sailing among a set of moons, navigating between different orbits can be tough 鈥� but a trick from knot theory may help. It can be used to identify points called heteroclinic connections, where a craft can transfer from one orbit to another without burning any of its precious fuel.
鈥淭hese heteroclinic connections, a lot of the time we know there are some for a pair of orbits, but there鈥檚 not a good way to know exactly how many,鈥� says at the University of Surrey in the UK. 鈥淚t鈥檚 just a case of getting lots of data and hoping for the best.鈥� This requires either huge amounts of computer power or a 鈥済uess-and-check鈥� method.
Owen and , also at the University of Surrey, came up with a way to find all of the heteroclinic connections for any pair of orbits. In knot theory, the number of times two lines cross is denoted by a figure called the linking number. The researchers realised that the linking number of a pair of orbits should be equal to the number of heteroclinic connections. Any time two gravitationally stable orbits cross, that is an opportunity for a spacecraft to transfer between them.
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By representing each orbit as a series of closed loops, like rubber bands, the researchers could find the linking number and therefore spot each heteroclinic connection. 鈥淚f you plot two lines in 3D space, chances are that they鈥檙e not going to intersect because they鈥檙e infinitesimally narrow,鈥� says Owen. 鈥淪o you run these simulations of closed loops, and if the linking number changes, you know that they have passed through each other.鈥�
Any spot where the linking number changes is a place where a spacecraft could pass from one orbit to the other without firing its thrusters or using any fuel. This would be most useful for shifting between different orbits around an object to view a larger area of its surface, or for hopping between small bodies such as moons 鈥� orbital changes that have long been difficult to calculate and often energy-intensive.
Astrodynamics
