Researchers have discovered that some orbital paths allow for no-fuel orbital changes. But the figuring out these paths also are computationally expensive. Knot theory has been shown to find these pathways more easily, allowing the most fuel-efficient routes to be plotted

When a spacecraft arrives at its destination, it settles into an orbit for science operations. But after the primary mission is complete, there might be other interesting orbits where scientists would like to explore. Maneuvering to a different orbit requires fuel, limiting a spacecraft’s number of maneuvers. Researchers have discovered that some orbital paths allow for no-fuel orbital changes. But the figuring out these paths also are computati…

Revolutionizing Spacecraft Orbit Changes with Knot Theory

Innovative application of knot theory enables spacecraft to change orbits post-mission without expending additional fuel, opening new possibilities for space exploration

Knot theory could help spacecraft navigate crowded solar systems

It can be difficult to figure out how to move a spacecraft from one orbit to another, but a trick from knot theory can help find spots where shifting orbits becomes easy

When a spacecraft is circling a planet or sailing among a set of moons, navigating between different orbits can be tough – 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

Previously, when the likes of NASA wanted to plot a route, their calculations relied on either brute force or guesswork.

“Our new technique neatly reveals all possible routes a spacecraft could take from A to B, as long as both orbits share a common energy level.

“This makes the task of planning missions much simpler. We think of it as a tube map for space.”

In recent decades, space missions have increasingly relied on the ability to change the course of a satellite’s path through space without using fuel.

One way of doing this is to find ‘heteroclinic connections’ — the paths that allow spacecraft to transfer from one orbit to another without using fuel.

The mathematics for finding these paths is complex — usually calculated by using vast computing power to churn through one option after another or by making an ‘intelligent guess’ and then investigating it further.

This new technique uses an area of maths called knot theory to quickly generate rough trajectories — which can then be refined.

By doing so, space agencies can gain a full list of all possible routes from a designated orbit. 

They can then choose the one that best suits their mission — much as you might choose a route by studying the tube map.

The technique was tested successfully on various planetary systems — including the Moon, and the Galilean moons of Jupiter

This intriguing approach simplifies the daunting task of plotting an efficient course through congested planetary systems, like Jupiter and its many satellites, by mathematically determining the least tangled trajectory.

A breakthrough detailed in Astrodynamicssuggests that knot theory may significantly reduce the computational power or guesswork typically required for mapping orbital manoeuvres, according to the paper “Applications of knot theory to the detection of heteroclinic connections between quasi-periodic orbits.”

Danny Owen, a doctoral researcher in astrodynamics and author of this new study from the University of Surrey, claims this methodology lays out all viable routes that a spacecraft might take, provided they have the same energy level, as per the university’s press release.

Faced with the complexity of space navigation and the vast number of variables in motion, finding paths that minimize or even eliminate fuel usage is critical. Heteroclinic orbits offer such a solution but have until now been challenging to derive.

What is knot theory

Knot theory is a branch of mathematics that studies mathematical knots, which are closed loops in three dimensions that are joined at the ends so they cannot be undone. The simplest knot is a ring, or “unknot

Knot theory began in 1867 when Scottish mathematician and physicist Peter Guthrie Tait started tabulating knots after being inspired by his device for generating smoke rings. Tait hoped to create a table of knots that would be similar to a table of elements. 

Knot theory has led to many discoveries in math and beyond, and has applications in various branches of science, such as physics, molecular biology, and chemistry. For example, knot theory can help determine the chirality of flexible or partially flexible molecules. 

To characterize a knot, mathematicians project it onto a plane and count the number of times the projection crosses itself. The number of intersections that occur as the knot moves around in every possible way is a measure of the knot’s intricacy. 

What is knot theory used for?

While the geometry of a rigid molecule determines whether or not it is chiral, for flexible or even partially flexible molecules, knot theory can play a role in determining chirality

What is the knot theory of the universe?

Traditional knot theory makes sense only in three dimensions: In two dimensions only the unknot is possible, and in four dimensions the extra room allows knots to untie themselves, so every knot is the same as the unknot. However, in four-dimensional space we can knot spheres

What is the principle of knots?

General Principles of Knots

The knot will stay tight with more friction between the strands – this is referred to as knot security. You maximize friction by tying square knots with each throw (i.e., alternating directions with each throw) and pulling tight, equal tension after each throw

Knot theory, a branch of mathematics, originated as an attempt to understand the universe’s fundamental makeup. The history of knot theory can be traced back to the 18th century when French mathematician Alexandre-Théophile Vandermonde first discussed the importance of topological features in knots. In 1771, Vandermonde wrote a paper that may be the first mathematical reference to knots. In the 19th century, Carl Friedrich Gauss began mathematical studies of knots and defined the linking integral

In the 20th century, mathematicians took up the study of knot theory, and today mathematical theories of knots are used in biology, chemistry, and physics. For example, in the 1980s, biologists and chemists studying genetics discovered that DNA can sometimes become tangled

Knot theory is the study of the properties of knots, which are considered “simple closed curves”. Mathematicians consider knot theory a subfield of topology, which is the study of malleable shapes. The central problem in knot theory is to determine if two knots can be rearranged without cutting to be exactly alike. Mathematicians use the Reidemeister moves to work with knots, which include: 

• Taking out or putting in a simple twist in the knot 
• Adding or removing two crossings

Knot theory use in space exploration

Knot theory is a branch of mathematics that studies closed curves in three dimensions. It’s used in space exploration to help spacecraft find the most fuel-efficient routes between orbits, even in crowded solar systems

Knot theory can help identify “heteroclinic connections”, which are points where a spacecraft can switch orbits without using fuel. The theory can quickly generate rough trajectories that can then be refined to provide a comprehensive list of options for mission planners. This process is similar to how GPS mapping software plots the most efficient routes on Earth. 

In mathematics, a knot is an embedding of a circle in three-dimensional Euclidean space, with the ends joined so it can’t be undone. The simplest knot is a ring

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