Why Stars and Planets Exist
"How on Earth did it ever happen?" "It will happen again as sure as the Sun will rise tomorrow morning!" These two commonplace expressions proclaim our faith in the abiding stability of the two astronomical bodies most essential to our existence.
But why do planets and stars exist at all, maintaining their mechanical stability over cosmically vast periods of time? Both the Earth and the Sun are four and a half billion years old, and modern scientific investigation suggests that their masses and sizes have not changed very much in all that time. Why are these bodies so exactly in equilibrium? More precisely, why is the inward-directed force of gravity so perfectly balanced by the outward-directed force of material pressure? In these bodies, the pressure comes about from the fast microscopic motions of the constituent atoms and molecules. But even our highly mobile atmosphere does not either suddenly collapse or fly off into space upon the slightest beat of a butterfly's wing. Nor do earthquakes below the surface and meteorite impacts from above provoke cataclysms of the whole Earth, even though they do cause vigorous vibrations inside it.
Four centuries ago, the great physicist Galileo performed experiments showing that an object dropped on the Earth accelerates downward at a rate of 32 feet per second every second. If the enormous pressures inside the Earth were for some reason to be suddenly removed, the surface layers would collapse to the center in about an hour under the force of gravity. Similarly, if the pressures below the surface were to mount uncontrollably, the Earth would explode in just as little time.
Everyday experience tells us that when a violent displacement is quickly made in a compressible object of any kind, the material immediately alters its pressure in response to the change of density. If the density increases, so does the pressure. The steepness of the change in pressure relative to the change in density is measured by a dimensionless quantity that physicists have denoted by the Greek letter γ (gamma).
August Ritter in 1879 proved a remarkable theorem. He showed that if γ exceeds the value 4/3 inside a large astronomical body, the structure of the body persists in dynamical stability. His explanation is straightforward. If γ is large and the density of the body increases due to a rapid compression of the material, the pressure rises so steeply that the force of gravity, though also increased, is resisted completely. (Recall Isaac Newton's famous law of gravity which states that the force of gravity between any two particles of matter increases as the square of the distance between them diminishes.) Conversely, if γ is large and the material in the body suddenly expands, the pressure falls off so steeply that gravity, though weakened, is able to bring the body back to its original equilibrium state.
Ritter's important theorem applies to all physical situations in which mechanical displacements are so rapid that there is no time for a large amount of heat to be leaked out of the system. These constitute "adiabatic" changes of the system. Inside the Earth, the value of γ for hard rocks is very large, and so the Earth remains stable. Even the Earth's moving atmosphere, and likewise the entire Sun, both of which are completely gaseous, have γ equal to 5/3, making them stable, too.
What about other bodies in the cosmos, such as the very brilliant supergiant stars, in which γ is close to 4/3 and heat losses are extremely rapid and large in the outer layers? GISS scientist Richard Stothers has recently shown from new calculations that all of the nonadiabatic influences on dynamical stability add up to precisely zero. Although nonadiabatic factors do affect whether or not the body vibrates around an average state of equilibrium, they do not influence the average equilibrium state itself.
What happens to a star in which γ drops below 4/3? If the outer parts are unstable, they explode. On the other hand, an unstable core collapses. Will the Sun ever become dynamically unstable? Probably yes, but only in its outer layers. Not to worry, though, the Sun will not reach this state for another six billion years.
Stothers, R.B. 1999. Criterion for the dynamical stability of a nonadiabatic spherical self-gravitating body. M. Notices Royal Astron. Soc. 305, 365-372.