Quantum equation describes galaxy mechanics


Modelling astrophysical discs reveals the emergence of Schrödinger’s equation, and the likelihood that astrophysical discs behave like subatomic particles. Andrew Masterson reports.


An artist's impression of how astrophysical discs can be understood using Schrödinger's Equation.
An artist's impression of how astrophysical discs can be understood using Schrödinger's Equation.
James Tuttle Keane, California Institute of Technology

The long-term physical evolution of astrophysical discs can be described by a fundamental equation used in quantum mechanics – raising the possibility that we should stop talking about Schrödinger’s cat and talk instead about Schrödinger’s universe.

In a paper published in the Monthly Notices of Royal Astronomical Society, California Institute of Technology planetary scientist Konstantin Batygin reveals that in refining an astrophysical modelling technique known as perturbation theory he discovered Schrödinger’s equation – the maths that describes quantum effects within an atomic-level system. He describes the finding as “astonishing”.

The discovery emerged because Batygin was researching ways to accurately describe the movement of bodies in space over time – an extremely challenging task.

On the biggest, broadest level, systems in space can be described as big things orbited by a host of smaller things. Black holes are orbited by swarms of stars, and the stars are orbited by swarms of rock, including planets.

The gravitational forces experienced by the object in the middle of any such configuration and the smaller things surrounding it mean that over time the orbiting objects form coalesce into a flat disc. However, over longer periods of time, the state of these discs is not constant: over huge distances that can stretch to hundreds of light-years, they warp and deform.

The question of how these warps develop, and how they then continue to fluctuate, is one of the biggest challenges facing astrophysics. In part this is because the complexity of the calculations required defeat either the capabilities of the computers used to model them – or the budgets of the academics who try.

To attempt an end-run around the problem, Batygin turned to a branch of mathematics called perturbation theory. It is an approach used in many fields and holds that (a la Plato) any given real-life system can be modelled into a similar but “ideal” alternative. From there, individual parameters can be altered, and the results calculated.

Perturbation theory was born out of celestial mechanics, and developed to solve the “three-body problem” – the challenge involved in accurately describing the movements of three mutually attracted objects (such as the Sun, Earth and Moon) when considered as a system.

Using perturbation theory to describe the orbits of smaller bodies around large ones in space required Batygin to posit all objects in each specific orbit as a single entity and “smear” them into the form of a concentric ring, or wire. In the model, each such ring exhibited the same gravitational force as the combined individual objects, but uniformly distributed.

In such an approach, the solar system, for instance, would be represented by the sun, followed by a wire ring for each planet, plus others for the asteroid belt and Kuiper belt. Computer simulations representing millions of years showed that these rings behaved in ways that closely mirrored the behaviour of the real composite disc surrounding the sun.

Batygin then started refining the model, realising that he could portray any astrophysical system as a centre surrounded by ever more numerous, but ever thinner, wires until, inevitably, the wires blended into a single plane.

“Eventually, you can approximate the number of wires in the disk to be infinite, which allows you to mathematically blur them together into a continuum,” he says. “When I did this, astonishingly, the Schrödinger equation emerged in my calculations.”

This was a surprise, because the equation was thought to be only applicable to phenomena occurring on a quantum scale. It is used to describe one of the most bizarre aspects of quantum mechanics – the way in which subatomic particles behave simultaneously like particles and waves, a condition known as “wave-particle duality”.

“This discovery is surprising because the Schrödinger equation is an unlikely formula to arise when looking at distances on the order of light-years,” says Batygin.

“The equations that are relevant to subatomic physics are generally not relevant to massive, astronomical phenomena. Thus, I was fascinated to find a situation in which an equation that is typically used only for very small systems also works in describing very large systems.”

The finding means that from one approach at least the smallest things in the universe – subatomic particles – and the largest things in the universe – galaxies surrounding supermassive black holes – can be said to share wave-particle duality.

And although the initial emergence of Schrödinger’s famous maths from astrophysical calculations surprised Batygin, he says he has now had sufficient time to come to terms with it, and can see a certain inevitability to what he found.

“Fundamentally, the Schrödinger equation governs the evolution of wave-like disturbances,” he says.

“In a sense, the waves that represent the warps and lopsidedness of astrophysical disks are not too different from the waves on a vibrating string, which are themselves not too different from the motion of a quantum particle in a box. In retrospect, it seems like an obvious connection, but it’s exciting to begin to uncover the mathematical backbone behind this reciprocity.”

The paper represents the second startling supposition made by the researcher.

In 2016 and 2017, he was co-author on a number of papers that suggested peculiar orbits in Kuiper belt object orbits could best be explained by the presence of a so-far-undiscovered planet that he dubbed Planet Nine.

  1. http://dx.doi.org/10.1093/mnras/sty162
  2. http://dx.doi.org/10.1093/mnras/sty162
  3. https://www.encyclopediaofmath.org/index.php/Perturbation_theory
  4. https://www.encyclopediaofmath.org/index.php/Three-body_problem
  5. https://www.konstantinbatygin.com/planet-nine-and-the-distant-solar-system/
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