HOW TO BUILD A STAR

Copyright (C) 1995 by Sten Odenwald

( Part III )

Solar Dynamo Theory provides a mathematical framework for understanding how sunspots form, how periodic polarity reversals occur, and to what they depend on. One of the basic equations describing this process is, During a sunspot cycle, the entire magnetic field of the sun changes its shape, beginning with a field that looks like that of a familiar bar magnet, but changing to one that looks more like a donut shape along the sun's equatorial zone. This equation describes how the stellar magnetic field changes its shape from a polar geometry, B_p , into a toroidal shape, B_u : The basic process of the sun's 22-year field reversal. When solved for a particular stellar case, the equation shows how the stellar magnetic field evolves, and predicts, among other things, the duration of the sunspot cycle and the latitude distribution of the spots on the star's surface. The quantity G is called the 'turbulent eddy diffusivity' while R represents the radius of the region producing the field. The value of G depends on how rapidly magnetic fields can be transported from one place on the sun's surface to another. The faster this occurs, the shorter will be the sunspot cycle. Amazingly, this theory also works well in explaining why the polarity of the earth's magnetic field reverses every 250,000 years! The same equations are used, only the values for G and R change to reflect earth's smaller size and the conductivity of its iron core.

Most known stars rotate, some barely at all, while others, such as the so-called 'emission-line B-type stars', spin fast enough to deform their shapes into a distinctly oval shape. In particularly extreme cases, not only is the star deformed, but it spins-off matter along its equator where the centrifugal force wins over gravity and launches streamers of hot gas into space. Stellar rotation can produce a whole host of effects including sunspot cycles, surface deformation and convection. To include the rotation of a star into its mathematical description, we have to re-write all of the equations in terms of a rotating coordinate system. Since the shape is no longer a perfect sphere, instead of the temperature, density and composition only depending on the distance from the star's center, they now also depend on stellar latitude and longitude angles and are represented by a set of mathematical functions called Spherical Harmonic Legendre Polynomials. The affect of stellar rotation on the structure and evolution of stars is so complicated to describe mathematically, that only with the advent of fast computers have actual, realistic, calculations been attempted.

In addition to the slow, million-year long changes that stars experience during the course of their evolution, any amateur astronomer will tell you that some stars, usually the red ones, undergo visible changes in brightness within a few days or weeks. Stars vary in brightness in this way because they are passing through an unstable period towards the end of their lives. This phenomenon does not involve the expansion and contraction of the star's entire body from core to surface, but only the outer layers nearest the stellar surface. When the layers expand, the star's surface cools slightly and the star dims in brightness. When the layers collapse, they heat up slightly and the star brightens. Stellar variability can be described, mathematically, once a particular stellar model has been computed giving the initial dimensions of the unstable layers, their temperatures and compositions. A set of equations are then used to calculate the amplitude of the oscillation and its period, the result is an equation that looks like this,

Stellar winds appear to be a common feature of many types of stars throughout their lives, especially for the bloated red supergiants such as Betelgeuse which looses 1.4 solar masses of material every million years. Since at this rate, Betelgeuse will lose its entire remaining mass in about 20 million years, it must be well on its way to some major change in its life, perhaps a supernova explosion. One of the equations used to describe this outflow of matter from the surface of a star, including the effects of magnetic fields and rotation is,

Stellar winds can be detected around other stars by the affect that they have on the star's spectrum. Unusually broadened spectral lines from key elements, or other peculiarities in the profiles of these lines can indicate the presence of hot, ionized gas being ejected from a star. If the stellar winds are cool and dense enough, dust grains can condense out of the gas like raindrops. Although the surface of a star is usually very hot, exceeding 2,500 K in most cases, at a sufficiently great distance from the star, temperatures within the outflowing matter will be cool enough for carbon, or silicon atoms to stick together forming dust grains. This process of condensation can be described by equations that follow the growth of dust grains, and describe what observers on earth will see as they look at a star with such a dusty envelope surrounding it. For some stars like the infrared source IRC+10216, carbon dust grains are condensing in the atmosphere of this star in such numbers that the star itself is optically invisible. All that one can observe is the infrared emission from the heated dust grains which now form a dense cocoon around the star.

All of these equations, when combined together in a computer program, and after extensive de-bugging, can be used to create theoretical models of objects that run through their evolution, lose mass through stellar winds, evolve to become white dwarfs or neutron stars, and otherwise look surprisingly like the stars we see in the night sky. In theory, it would be nice to have a single program that could evolve a star from a collapsing gas cloud to, say, its eventual demise as a white dwarf or supernova; a program that would follow detailed changes in surface magnetic fields and solar wind output. In practice, however, this is not necessary or even desirable. If you are studying the collapse of a star's dense core prior to the supernova phase, the presence of absence of spots on the star's surface is not likely to make much of a difference physically or observationally. You might, however, be interested in whether or not the star was rotating, or how the convection patterns occurring at a particular location within the star are influencing the chemical composition of the core region. Both of these make a measurable difference in the properties of the left-over remnant, or in the chemical composition of the gas ejected into interstellar space.

A single computer program attempting to follow a star as it evolves from birth to supernova, yet giving detailed predictions for surface magnetic fields and spot distributions would have to follow the minute to minute changes in these fields while handling the million-year changes due to its evolution. It would also have to correctly keep up with the microsecond to microsecond changes in stellar structure during the supernova detonation itself. Even at a temporal resolution of one minute, there will be 10 trillion of these timesteps during the full life of such a star posing a daunting computational and bookkeeping problem. The solution? Theoreticians tailor their programs for studying the physics of interest, not the entire evolutionary process. If you want to study the supernova, begin the model with a 'realistic' composition provided to you by a stellar evolution model. Ignore stellar winds and surface magnetic fields. Once you have run your computer models spanning the last milliseconds of a supernova's life, you can patch them into the results from other models by arguing that the starting conditions you began your computations with, are compatible to the conditions predicted from the evolution models spanning 100 million years at thousand-year intervals. Like a giant patchwork quilt, astronomers use many interwoven, and interdependent, theories to assemble a complete view of a star's life; a view that no single one of the theories can describe completely.

One issue that all mathematical prognosticians must face, is one that may well thwart any practical attempt to construct stellar models of arbitrarily high predictive power. It is in the very nature of the mathematical approach that it will never lead to a perfect match between observation and theory for all length scales and time intervals. The reason? It's related to why meteorologists will never be able to tell you that, for example, five months from today, at 3:35 PM there will be a rain shower over the town of Adams, Massachusetts which will last 1 hour and 45 minutes. To make a prediction that specific, it is very likely that meteorologists will need to measure the state of the earth's atmosphere today, within every cubic inch over the entire globe, throughout its entire 100 kilometer thickness. In addition to the literally astronomical data storage requirements, the computer will not even be allowed to round-off any of the intermediate numbers it computes, and it will have to complete the calculation before the target hour passes.

Mathematicians tell us that nearly all the equations we create to represent nature are inherently unstable for use in forecasting. They are not unstable because they are incomplete, though that certainly contributes to faulty predictions, they are unstable because the data we feed them always are incomplete. When you construct a mathematical representation of a physical system, you begin by selecting the quantities for the variables in the model at a particular starting time. You start the stellar evolution calculation with, for example, a surface temperature of 6,000 K, a total stellar mass of 2.000 times the mass of the sun, and a composition approximated by treating hydrogen and helium separately, lumping all the elements heavier than helium together into one number, and that the star is of the same composition through out. The equations then tell you how each of the parameters change with each time step you evolve the model into the future, or past. The only problem is that the values that the variables take on at the end of the computation can be very sensitive to their values when you started the calculation. For the weather problem, it has been jokingly said that to know the weather pattern on one spot on the earth a few years into the future will depend on how vigorously a butterfly was stirring up the atmosphere a thousand miles away last year! For stellar evolution calculations, fortunatly,it appears that what you wind up with as a stellar model is not too sensitive to where you start out, provided you only want to know a star's size, luminosity, surface composition and temperature. Our curse, that we can never study the interior of a distant star or photograph its surface and surroundings, becomes our blessing since from our vantage point on earth that which we want to know about a star and can measure, can be summed up in a short list of numbers. A few million years difference in age between two stars like our sun, amounts to an observational difference between them that is, largely, not measurable in terms of temperature, luminosity or spectral features.

So where does this put the classical goal of science as a means of predicting and accurately portraying natural phenomena? For astronomy, it says that there are limits to our knowledge about the physical world. Within those limits we can hope to learn a great deal about the stars and the distant galaxies, but none of this knowledge will be certain. This will probably come as a bitter pill for many non-scientists as they may still founder on the wishful dreams of obtaining absolute knowledge, untarnished truths, and some scheme for distinguishing clearly between right and wrong answers. In science, we are accustomed to laws that may be overturned by the very next observation, theories that may be incomplete, or data that may not only be uncertain, but even wrong and misleading. This is not the arena that so many people might imagine science to be. Scientists do search for objective truths, but those truths are not written in capital letters and inscribed in stone. It is not that scientists have to change their methodology so that Truths can be revealed, it is that society has to learn that absolute truths about the physical world probably do not exist. As Jacob Brownowski states so poignantly in his essay 'Knowledge or Certainty'

"...Science is a very human form of knowledge. We are always at the brink of the known, we always feel forward for what is to be hoped. Every judgement in science stands on the edge of error, and is personal. Science is a tribute to what we can know although we are fallible..."