Modeling stars
How do you know when you’ve included all the important things?
Our astronomer was reviewing some old material on stars recently, including some work he’d done on the history of stellar models. A model, in this case, is a mathematical representation of important physical quantities, given others; that is, how do temperature, density, pressure, etc. vary with radius for a star of a given mass? The idea is to apply Physics to a problem, posed in a certain manner, and eventually see if one’s conclusions match the Astronomy.
The first stellar models date from the nineteenth century, and started out as almost intellectual exercises: suppose you had a great quantity of an ideal gas, acting under its own gravity; what would it look like? What would it do? Cobbling together a few equations and doing some clever bits of mathematics, it turns out that things like how the radius varies with mass drop in one’s lap and match observations pretty closely. I point out that this was before the actual power source for stars was known, and indeed before quantum mechanics had been invented. A bit later, in the first half of the twentieth century, quantum mechanics refined the picture significantly and the models matched rather well. But the actual fusion reactions taking place in a star’s core were not known, and such things as rotation (the Sun spins) and magnetic fields (much stronger on some stars than, for example, on the Earth) were ignored. The models were certainly simplified (otherwise they could not have been calculated) but the simplifications did not seem to have major effects.
Well, Sir James Jeans, one of the major astrophysicists of the time, did some calculations on the stability of these models. He found that if the energy-producing reactions depended on temperature and pressure more than a little bit, the models would blow themselves apart. That is, if they got a little hotter and denser in the core, they would expand; then contract as the energy-source shut off; get hotter and denser still; and pulsate to oblivion. He proposed a quite different sort of stellar model as an alternative. It was not generally accepted, but the ideal-gas modelers did understand his calculation to be a problem for them.
Later the stability calculation was done in a more complicated, complete way, and found not to be a problem. Our astronomer looked carefully at Jeans’ analysis and traced the difficulty to a certain simplifying assumption. In this case the simplification had changed the answer completely.
We think the question remains, and is if anything more important than before. A scientist has to simplify the situation to do any calculation at all. How do you know when you’ve included what’s important, and what things can you safely leave out?
