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The 1984 issue
of Astronomy and Astrophysics Abstracts lists 9361 papers
written by 10,863 authors covering 2,200 major topics in astronomy.
Under the simple heading of 'Stars', for example, you would
find forty-five sub-topics ranging from 'Star Catalogs' to
'Stellar Winds' and spanning some 1189 individual papers.
The activity implied by all of these papers is quite enormous.
Each paper represents months or years of work based, in part,
on the findings of still earlier research programs that have
probably appeared in the literature over the years. A single
paper may have thirty or more references to data or ideas
presented by other researchers.
What is ultimately
the result of all this activity? As astronomers, we hope that
it will eventually culminate in a thorough, detailed understanding
of all the major collections of matter in the universe, their
properties at a single instant in time, as well as their evolution
from birth to death. These 'major collections' include: interstellar
molecules, dust grains, asteroids, planets, solar systems,
stars, nebulae, star clusters, galaxies, clusters of galaxies,
and the universe itself. This represents a range in length
scale of 10^36 and a mass range of 10^52! Not only would we
like to know, by recourse to basic physical law, the answers
to general questions like "How do stars with 5.5 times
the mass of the sun evolve?" but, we would also like
to be able to explain specific objects in our universe such
as, "What mechanisms are producing the jets in SS 433
and 3C 273?"
As a chronicle
of the progress in our knowledge over the centuries, let's
consider the subject of stellar structure and evolution. It
is safe to say that for the last century, more articles have
been written on this subject and related issues than any other.
That means that more time and effort has gone into understanding
stellar structure and evolution than any other field in astronomy.
It is, what you might call, a mature discipline whose basic
theoretical and observational ingredients are reasonably well
understood at the present time. That we can know so much about
objects that are so far away is a testament to the power of
the scientific method and human technological inventiveness.
For thousands
of years, stars were simply lights burning silently in the
depths of the heavens; any discussions about what they were
revealed more about the human imagination than about nature.
It wasn't until Joseph Fraunhofer invented the spectroscope,
and began to examine the light from the sun, the planets,
and several bright stars, that the first step was taken towards
answering the age old question, "What is a star?".
Just as for the sun, each bright star that was examined with
the spectroscope revealed a rainbow of colors crossed by a
pattern of dark lines. It was quickly discovered that the
lines could be matched by a number of commonly known elements
available in the laboratory. By 1864 Father Angelo Secchi
at the Vatican Observatory began a program of systematically
classifying the spectra of 4000 stars. Sir William Huggins
and William Miller carefully studied the light from Sirius,
Aldebaran and Beta Pegasi, identifying the elements hydrogen,
sodium and magnesium from the dozens of spectral lines detected.
The first issues
of The Astrophysical Journal, published in 1895, covered new
developments in spectroscopy, both the theoretical principles
on which it was based, and improvements leading to the design
of even more powerful and sensitive spectroscopes. By this
time, thanks to the pioneering efforts of Gustav Kirchoff
and Robert Bunsen, the solar spectrum had been resolved into
twenty thousand spectral lines corresponding to thirty-nine
elements. The state of the art of understanding the sun was
candidly summarized by E.J. Wilczynski from the University
of Berlin: "Almost every student of solar physics has
his own theory, and usually he himself is the only one that
believes in it." Although much of the work in solar physics
had involved a careful study of sunspots and the spectroscopic
features of the sun's surface, the internal structure of the
sun was also becoming a lively topic of discussion. By the
later decades of the 19th century, older ideas about the solar
interior, involving a 'liquid body with clouds suspended over
its surface', were being quickly replaced by the more modern
view of a fully incandescent, gaseous ball which rotated at
an increasing rate as you moved from its poles to its equator.
It might amuse you to know that among the early speculations
about the solar interior, Sir John Hershell announced that
the surface of the sun was covered by living, luminescent
organisms thousands of miles long!
The origin of
the sun's tremendous energy supply also made its appearance
into the arena of acceptable inquiry. Hermann von Helmholtz
in 1854, had proposed that the gravitational energy lost by
the sun during its slow contraction, will show up as a comparable
quantity of heat energy which could provide the 'missing energy
source' for the sun. To produce the measured solar power of
400 trillion trillion watts, the sun's radius would only need
to decrease by about thirty meters each year. Not much of
a change when you consider that the radius of the sun is 700
million meters. But, this means that about 30 million years
ago, the sun was twice its present size and that in another
30 million years, it will be a burned-out red cinder, incapable
of supporting life on earth. Between 1878 and 1883, Helmholtz's
idea remained popular and was even refined to obtain an age
of 4.3 million years. Fortunately, for us, this cartoon sketch
does not represent the real world.
In 1906, Karl
Schwarzschild published a fundamental paper in astronomy,
describing the appearance of an incandescent, stable ball
of gas in considerable detail, using basic principles in physics.
Not only did he show that the sun's limb should be darkened
to the precise degree observed, but went on to prove that
the distribution of matter within the sun could be determined
once you could specify the exact dependency of the gas pressure
on its temperature and density. He also discovered that, under
certain conditions, energy would be transported from the center
of the star outwards, either by the convective boiling motion
of matter, or by the streaming of radiation from the core
to the surface. Sir Arthur Eddington continued this work by
including the affects of radiation pressure, showing that
stars that are mechanically stable are only possible for certain
combinations of mass and luminosity. An amazing discovery
indeed! Even for the creation of stars, nature followed a
set of very specific rules favoring certain stellar properties
over others.
Between 1913
and 1917, Henry Norris Russel and Ejnar Hertzsprung claimed
from their study of star sizes, that blue stars were the hottest
as well as the largest, while red dwarf stars were the smallest.
They proposed that a star began its life as a hot blue star
and, by contaction, wound up as a dull red dwarf. Eddington,
mentioned above, further discovered that the core temperatures
of all the 'main sequence' dwarf stars that Hertzsprung and
Russel had been studying, were actually very similar, about
20 to 30 million degrees, and that this temperature didn't
depend on the star's mass or size. Instead of evolving from
blue to red as they cooled, Eddington proposed that gas clouds
would contract until their central temperatures reached about
20 million degrees at which time they would stop contracting
and become a stable star. This explanation re-ignited interest
in two older questions, "What was the process that stopped
the contraction of the star at this temperature?", and
"Where did the energy come from if not from Helmholtz's
mechanism of gravitational contraction?" The answers
could not emerge from the physical principles understood at
that time, but had to wait for the 20th century discovery
of nuclear dissintigration and fusion.
The British
astronomer R. d'E. Atkinson was the first to suggest, in 1931,
that the capture of a proton by an atom could liberate enough
energy to light the sun. Eight years later, Hans Bethe and
C. von Weizacker presented the same idea but marshalled better
evidence for it in their study of thermonuclear fusion process
known as the carbon-nitrogen-oxygen, or 'CNO cycle'. The CNO
cycle was soon found to work well for stars like our own sun
where internal temperatures had been estimated to be about
20 million degrees. Yet, the majority of the sars in the sky
were less luminous than the sun. The red dwarf stars like
Kruger 60A whose core temperature was only 16 million degrees
was a case in point. That slight temperature difference translated
into a 100-fold reduction in energy production and a predicted
luminosity for Kruger 60A about 100 times fainter than it
was known to be. So, what is it that powers stars cooler than
the sun? The answer was provided by Hans Bethe who showed
that a fusion reaction which converted hydrogen into helium,
but not involving the CNO reaction, would work at these low
temperatures. More advanced burning cycles have been studied
since then which are capable not only of supplying even greater
quantities of energy to support a star against gravitational
collapse, but reactions capable of creating all of the known
elements in the periodic table, in the cores of very massive
stars.
The first stellar
models that showed, in detail, how a star evolves from the
hydrogen fusion phase called the 'main sequence', through
the red giant phase did not become available until electronic
computers were developed. Prior to the advent of computers,
the computations had to be done by hand using desk calculators.
This led to trade-offs between using a crude model of the
star's interior and taking many steps in time, or using a
moderatly detailed model of the interior but taking only a
handfull of time steps.
In 1955, R.
Harm and Martin Schwarzschild published 15 'models', some
calculated by hand, others by using the electronic computer
at the Princeton Institute for Advanced Study. The models
presented the star's interior in three zones: the core, the
outer envelope and the intermediate zone where convection
would likely occur. Radiation pressure was ignored, as were
differences in chemical composition between the zones, and
no internal energy source was treated in detail. It took one
full year of laborious work on a desk calculator to construct
the hand-calculated models which were computed for a total
of 127 time steps. The models specified the changes in 18
quantities in each of the three zones. In contrast, the computer
generated model was followed for 37 time steps and required
less than a day to compute. Continuing in a steady progression
as faster computers were developed, present-day computers
can calculate complete stellar models in less than one minute!
The computational
extension of the models from the hydrogen burning phase to
later stages began in ernest in 1961 with the appearance of
several papers announcing detailed, independent studies of
5, 10 and 15 solar mass stars by Chushiro Hayashi, Robert
Cameron and Emil Polak. They used IBM 650 and 7090 computers,
splitting each star into a dozen or more internal shells.
Their program followed the evolution of each star's structure,
shell by shell, through the helium burning stage. For the
most massive stars, the carbon and neon fusion stages were
followed as well. They watched as the stars swelled to enormous
dimensions and became red supergiants, as their cores collapsed
switching first to helium burning, then to carbon and neon.
By 1964, the
role of neutrinos in producing added pressure in the dense
cores of more massive stars was discovered and incorporated
into the models. John Cox and Edwin Salpeter also examined
the evolution of stars where electron degeneracy pressure
was important. A similar calculation for stars 4 to 8 times
the mass of the sun done by David Arnett in 1969 showed that
if the carbon burning cycle was triggered in a core that was
degenerate, the entire star would blow up in a 'Carbon Detonation
Supernova'. Whether anything was left behind other than an
expanding cloud of gas seemed to depended very critically
on the density of the star's core before the detonation, and
just how much pressure the neutrinos escaping from the star's
core produced in the overlaying matter. Depending on the core's
density and mass, what would be left behind the star after
this explosion would be: nothing, a white dwarf, or possibly
a neutron star.
Since the 1960's,
computer models have become more sophisticated. Periodic revisions
have been made in the number of nuclear reactions that are
considered, as well as updates in the reaction rates and energy
yields based on more exact theoretical calculations supplemented
by experimental results. The detailed role of convective mixing
in transporting energy from place to place within the star
and changeing the composition of the star is also being studied,
as are the roles of rotation and mass loss. As the models
become more refined, they are used to an ever increasing degree
in explaining the observed details of known stars. Some stars
show an overabundance of certain elements over others which
cannot be entirely explained by temperature effects alone.
This suggests that convective mixing seems to be the culprit,
wherein the elements in deeper layers in the star are mixed
with the visible surface layers. Then again, for the peculiar
A-type stars, convection may be supressed by strong magnetic
fields that have been measured on the surfaces of these stars,
so that atomic diffusion driven by radiation pressure may
be a more important factor.
A related area
of study concerns the evolution of binary stars. The presence
of a nearby star can alter the evolution of both stars, especially
if matter is being pulled from one companion and dumped onto
the other. The gravitational stresses that result inside a
star with a close companion can alter convection patterns
and mix enriched hydrogen gas with hydrogen depleted material
in the core, so that one star, essentially, gets to re-live
its youthful, hydrogen burning phase all over again as though
it had just been born.
The final stages
in the evolution of stars is also of great theoretical interest.
Exactly how do planetary nebulae form? How are neutron stars
and black holes produced from supernova explosions? Do all
supernovae produce identifiable remnants? Although we are
tantalizingly close to answering these questions and can do
so in general terms, the details are still a bit vague.
I have spoken
about mathematical models for stars, but I have not really
described for you what I mean by this terminology. How do
you reduce a pinpoint of light in the sky into a collection
of equations, and what would these equations look like? The
basic equations defining the structure and evolution of a
star have been known for nearly a century. They describe what
determins whether a star is stable, or subject to gravitational
collapse. They describe how energy is transported from the
core of the star to its surface, and how the density and temperature
of the gas varies from the core to the surface. This theoretical
model must also describe how much energy is liberated by the
various possible fusion reactions occuring in each gram of
matter in the core. When we express all these relationships
and interdependencies in symbolic form, we get the 'equations
of stellar structure' which look like this:
But these equations
are not enough because you also have to specify how the pressure
inside a star, which supports it against gravitational collapse,
is dependent on the values for other physical quantities like
a star's chemical composition, M, temperature, T, and density,
D, which may, in turn, change from place to place inside the
star. The amount of energy released in the thermonuclear fires
in the star's core, E, also depend on these quantities as
does K, the stellar opacity. The equation linking the pressure
to the other variables is called the 'Equation of State' by
the astronomical cognoscenti, and its form can change as the
star evolves or as you dissect the star and examine various
layers within it. The pressure due to light radiation and
high temperature gas is usually expressed by, For high gas
densities near 10^5 grams/cc, we also have to include electron
degeneracy pressure, P_e, caused when electrons are squeezed
together into a small volume. The opacity of a star determines
how transparent it will be to its own emitted light radiation.
Since radiation pressure is in many cases the most important
internal support for a star, its accurate specification is
crucil. Depending on the kind of interaction involved between
matter and the light streaming out from the star's deep interior,
the mathematical description of the transparency of the star's
matter takes-on a variety of different forms. The sum total
of these will determine how opaque the star is at a particular
point in its interior, and how much radiation pressure will
result. To write down all the different forms of the matter-radiation
interaction that contribute to a star's opacity would easily
fill a book of this size!
Although gravity
is the ultimate source of energy for heating a star's interior,
it is the nuclear reactions that provide the energy from which
the star's internal pressure is ultimately derived. A complicated
network of interdependent equations is required to account
for the energy released by fusion reactions and how they change
the internal element composition of a star. These describe
how rapidly one element is converted into another by fusion
or radioactive decay, and shows how the rate of energy release
depends on the local temperature and density of the star.
To assemble these equations, one must first write down all
the important pathways by which the conversion from one element
to another occurs, and the energy released at each step. For
example, when the cores of stars more massive than the sun
reach temperatures exceeding 100 million degrees, the so-called
Triple Alpha reaction becomes important in supplying the thermal
pressure needed to prevent further gravitational collapse.
In this fusion reaction, two helium nuclei fuse into a single
beryllium nucleus; then, after an additional helium nucleus
fuses with the beryllium, one obtains a single carbon nucleus
as nuclear 'ash'. The reaction also produces a considerable
amount of energy.
At still higher
temperatures appropriate to pre-supernova conditions where
temperatures exceed 5 billion degrees, one encounters reactions
that convert carbon into oxygen, oxygen into magnesium and
silicon, and finally silicon into iron. All these reactions
are very temperature sensitive. For instance, in Triple Alpha
fusion, the reactions produce 10 times more energy at 105
million degrees than at 100 million degrees! Where does a
star get the high energys and temperatures to allow these
reactions to proceed? The answer is from the gravitational
collapse of the core of the star under its own weight. Just
as a rock gains speed and kinetic energy as it falls to the
ground unsupported, the matter inside the core of a star,
if unsupported by a counter-balancing pressure, will continue
its fall towards the stellar core. In so doing, it gains kinetic
energy which appears as an increase in temperature of the
gas.
The change in
the chemical composition of a star as it 'burns' one element
and leaves behind another as a nuclear 'ash' can be represented
by yet another set of equations. Modern nuclear reaction networks
such as those used to study the last years of a star about
to become a supernova, incorporate over 250 nuclear species
and their isotopes, along with their highly interdependent
equations of interconversion. Having considered the interior
of the star and what goes into describing its inner workings,
what of its outer layers?
How does a star
look to a distant observer? All you see through the eyepiece
of the most powerful telescope is the radiation emitted by
the surface of the star. The interior is completely hidden
from view. Not only that, but the light produced in the star's
dense core requires millions of years to reach its surface,
before it can start its journey to earth. There are models
available for predicting the strengths and shapes of the atomic
spectral lines emitted by the surface gases, but these models
depend on the temperature, density, composition and surface
gravity of the star. You can obtain pedictions for these quantities
at a particular instant in the life of a star using your stellar
evolution model. These 'stellar atmosphere' models are very
complicated; to merely write down the necessary equations
would fill up several books this size. The most sophisticated
model now in routine use is the one developed by Robert Kurutz
at the Center for Astrophysics in Cambridge, and his co-workers.
His model contains 1,760,000 spectral lines for elements between
hydrogen and nickle, and computes the expected spectrum shape
and line intensities for most kinds of stars commonly studied
in detail.
In addition
to high temperature plasmas of charged atoms, stars are known
to contain magnetic fields. A detailed study of the sun reveals
a strong surface field of about 1 gauss, and sunspots where
the fields are thousands of times stronger, along with a periodic
22-year cycle of magnetic polarity reversal, better known
as the Sunspot Cycle. Other phenomena related to stellar magnetic
fields include prominences, flares and coronal holes. Magnetic
fields have been detected on nearly 100 stars, mostly of the
peculiar A-type, which have surface fields 100 to 30,000 times
stronger than the sun's. Sunspot cycles have also been observed
on a number of nearby stars. Thanks to the rapid influx of
data from satellite observations of the sun, and long-term
studies from ground-based earth observatories, the detailed
description of the role of magnetic fields in our sun has
evolved rapidly from crude 'back of the envelope' calculations
to highly sophisticated theoretical models. Presumably, the
physics of the magnetic fields on more distant stars can also
be described by this same theory, or simple modifications
of it.
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 coccoon 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 occuring 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 separatly, 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 vigoriously 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 accuratly 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 enscribed 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..."
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