Bessel functions – Bean Thinking

Second shot coffee and cake — How would you describe the gravitational attraction between a Long black, a hot chocolate and a piece of cake?

Not a question of how many coffees are acceptable before lunch, but an astronomical conundrum with consequences for your cup.

It starts with gravity. Perhaps you remember that Newton came up with a set of equations describing the laws of gravity. You may even remember the essence of those equations, that the force between two masses is proportional to their product and inversely proportional to the square of the distance between them. If we wanted to phrase it mathematically, the force, F, is given by:

F = GMm/(r x r)

Where G is a constant and r the distance between the masses M and m.

Which is all very well, but suppose we have three masses, or four? M, m and M’, m” for example. If we happened to drop an apple (mass = m) between the moon (mass = M*) and the Earth (mass = M), how exactly, and where exactly, would it fall? How do we add an extra mass into the equation?

It is one of those problems that can seem far removed from your coffee cup, but in fact, the connection is quite close.

The Orion Nebula, M42, can just be seen with the naked eye in the sword of Orion, it is known as a birth place for stars. This image was obtained using the Hubble Space telescope. A separate dust cloud also in Orion was observed for 11 years as a possible host for planetary formation. Credit:
NASA, ESA, M. Robberto ( Space Telescope Science Institute/ESA) and the Hubble Space Telescope Orion Treasury Project Team

But although you may not often drop an apple somewhere between the Earth and the Moon, the question became relevant recently when astronomers observed a dusty disc, the sort of environment that is capable of planet formation, surrounding a three star system. The stars are found in the constellation Orion, which is visible in the evening at this time of year (autumn/winter) from the Northern Hemisphere.

Although these dusty discs are thought to be a host to planetary formation, astronomers have yet to observe any planets actually forming out of the dust. It is thought that in some cases, the gravitational perturbations caused by multiple stars at the heart of the dust clouds could lead to the formation of planets. And so the system in Orion, with three stars in the centre of the dust cloud was perfect to observe the effect of the three stars on the integrity of the disc. Over 11 years, the astronomers recorded the system and then included modelling into understanding how the planetary disc was breaking up. But of course, to do this, they would have needed to understand how the gravitational force is affected by having 3 or more interacting masses.

To solve the problem requires mathematical functions known as a “Bessel functions”. These functions were first described by the astronomer Friedrich Wilhelm Bessel in 1817 who used them for exactly this sort of problem. But they don’t just apply to describing the gravity between three or more objects. They can be used amongst other things to understand heat transfer, to model the microwave fields in a microwave oven and to understand vibrations on your coffee.

The beat of a drum or the resonance on our coffee – the mathematical description of the resonance patterns on coffee is shared with the mathematical description of the gravitational force between three or more objects.

Because when you see a series of concentric circles on the surface of your coffee where the table underneath the cup is vibrating, or when you see more complex patterns as you drive a take away cup over a rough table surface, these patterns can be described using exactly the same Bessel functions as would have been used to model the star system in Orion.

And so there is a direct link between the maths describing the planetary formation in a star system visible in our night sky and the patterns of your coffee cup. But if you want to drink your coffee while gazing at Orion, you may want to stick to decaff, or wake before dawn.

A lawyer, an accountant and and emperor walk into a cafe…

Strata, geology — This is not a resonance in a coffee cup but the concentric circle pattern is similar to a resonance that you could frequently see.

Have you ever noticed concentric rings on the surface of your coffee, forming as the table under the coffee cup vibrates slightly? Perhaps you have seen more complicated patterns. You may have observed, as you have played with your coffee, that some patterns are more stable than others. The one that is formed from concentric circles is fairly easy to form and to see. A more complex one looks like a chequer board, you may perhaps of seen others. These patterns are what are known as ‘resonances’ on the surface of the coffee and they are the consequence of standing waves being set up on the coffee surface. Many people who have gone through an undergraduate physics degree will immediately be reminded of Chladni figures and there is a good reason for this. Ernst Chladni (1756 – 1827) was a pioneer in investigating such resonances, one of the reasons that he has been described as “the father of experimental acoustics”.

And yet Chladni was not a physicist in the way that we now think of the term. In fact, by training he was a lawyer, a consequence of following his father’s rather insistent ‘advice’. Obediently, Chladni had trained in law and had started working as a lawyer in 1782 when his father died. Chladni appears to have taken this event as an opportunity to start to investigate the scientific problems that he was actually interested in and so re-invented himself as an acoustician testing the theories of music developed by people like Bernoulli and Euler¹.

transmission lines, electrical noise — Like strings on a guitar. Resonances on a string can be used to make musical notes.

Did Chladni drink coffee in eighteenth century coffee houses while admiring the resonances in the cup? Sadly what comes down to us in history is not his coffee habit but his experiments with sand covered metal plates secured onto wooden rods. Chladni caused resonances on these plates by rubbing them with a violin bow. By exciting resonances similar to those you can see on the surface of your coffee, Chladni was able to test theories about the sounds made by curved metal surfaces (e.g. bells). Indeed, these experiments became so important to understanding acoustic theory that Chladni started a European tour demonstrating his plates and their relevance to designing musical instruments. It was presumably through one of these tours that he met an Emperor of the time, Napoleon Bonaparte.

But despite this great experimental progress, the mathematics used to understand these resonance patterns, was developed by another physicist with a non-typical career path, Friedrich Bessel (1784-1846). Bessel had trained as an accountant but with the good fortune of timing, he had apprenticed into an exports company. At this time, such companies would have been interested in the problem of longitude and so Bessel gained an opportunity to indulge his interest in astronomy. As a consequence of this work, particularly his work on the orbit of Halley’s comet, Bessel secured a job in an astronomical observatory and it was there that he started the work that would eventually lead us to be able to describe, mathematically, the resonances on the surface of your coffee.

Did Bessel drink coffee? Had he seen Chladni demonstrate his plates? We don’t know the answer to those questions and in many ways it is not relevant because Bessel’s mathematics did not concern such resonances at all. Instead, almost to underline the idea that everything is connected, particularly with physics and coffee, Bessel was working on the problem of how to calculate the gravitational attraction between multiple objects.

Kettle drum at Amoret — The note made by a drum is a function of the size and shape (therefore resonance pattern) of the drum and also the gas filling the drum. Would this drum-table sound the same if banged on Venus as on Earth?

Perhaps you remember from school Newton’s famous description of the gravitational attraction between two bodies as being F = GMm/r² (where F is the force, G the gravitational constant, M and m the masses of the two bodies and r the distance between them). That’s all very good but what if there were three bodies, or four, or…

It was this problem that Bessel was working on and by so doing he solved the problem of Chladni’s patterns. The maths that describes the many body problem also describes the way that these resonances form. Those patterns in your coffee are described by the same maths as allows us to calculate complex gravitational problems.

And so perhaps it is not quite correct to title this post as a lawyer, an accountant and an emperor walk into a café, but it would be fair to say that each time you catch those resonances in your coffee cup, the influence and interests of these investigators of nature are infused within the brew.

You can find a sketch of Chladni entertaining Bonaparte with his metal plates here.

¹Harmonius Triads, Physicists, musicians and instrument makers in nineteenth century Germany, MIT Press, 2006