and 
, is:
Here,
is Boltzmann's
constant, and 
is Planck's
constant. c is of course the speed of light, and e = 2.718 is the
base of natural logarithms.
Writeup 2: The Physics of Stars
Hot materials radiate electromagnetic radiation. That is, anything with a non-zero temperature glows. Think of a glowing ember in a fire, for example. Cooler objects, like the objects in the room, are glowing as well, however, the electromagnetic radiation they emit is much longer wavelengths than our eyes are sensitive to: they emit in the infrared part of the spectrum. With infrared goggles, you can in fact see in the dark.
(Most of what we see around us is shining by reflected light. The distinction between reflected light and emitted light is crucial).
This radiation is called ``black-body radiation'', and is crucial for understanding how stars and planets shine. The basic formulas were first derived by Max Planck in 1900; it was his great insight that light comes in packets called photons that allowed him to match the experimental data. Black is a color that absorbs (i.e., does not reflect) all photons that fall upon it; thus they appear dark by reflected light. The statements below refer strictly to a perfect black-body, one that absorbs all photons which fall upon it. In this limit, the spectrum of a black-body depends only on its temperature, and not what it is made of.
The amount of energy given off by a blackbody of temperature T of
surface area A, with wavelengths between and , is:
Here, is Boltzmann's
constant, and is Planck's
constant. c is of course the speed of light, and e = 2.718 is the
base of natural logarithms.
The above equation gives the spectrum of black-body radiation as a
function of wavelength. The total amount of energy per unit time
(i.e., the luminosity), including all wavelengths, is:
where 
is called the Stefan-Boltzmann constant. Those of you who know some
calculus can have fun deriving the fact that the integral of the
black-body spectrum is proportional to 
; getting the numerical
value of the Stefan-Boltzmann constant right is quite a bit more
work.
The black-body spectrum has a characteristic shape; at long
wavelengths, it rises steadily to a peak wavelength, and then drops
precipitously. The peak wavelength, where more radiation is coming
out than any other, depends only on the temperature of the black
body, and is given by:
(Again, those of you who know calculus will have fun deriving this.)
Stars are pretty good approximations to black-bodies, as are planets, and we will find ourselves using these formulas in trying to understand their properties.
The Earth goes around the Sun once per year; we observe the heavens on
a moving platform. Thus our line of sight to the stars is constantly
changing. What this means is that the direction to the stars as we
perceive it is shifting, more so for the more nearby stars. We can
use this shift, or parallax, as a measure of the distances of
the stars. Over six months, the Earth moves from one side of its
orbit to another, a distance of 2 AU (One AU, or Astronomical
Unit, is the mean distance from the Sun to the Earth, about 
). The nearest stars are about 4 light years away;
one light year is about 
cm, or about 60,000 AU. Draw a very
long skinny isosceles triangle, with its apex at the star, and its
base the s = 2 AU diameter of the Earth's orbit. Its height is the
distance d from the Sun to the star. The change in perspective we see
the star move is the same as the apex angle of that triangle, 
. This is
a very small angle. We could use trigonometry to get a
relationship between the size of the angle, and the distance to the
star, but far easier is to use the small-angle approximation:
where is measured in radians. Can't get much simpler
than that! Indeed, it is because the simplicity of this equation that
astronomers and mathematicians are enamored of radians. For
reference, there are 
radians in a circle (
), or
degrees in a radian. Astronomers often measure
angles in arcseconds (60 arcminutes in a degree, and 60 arseconds in
an arcminute); there are roughly 200,000 arcseconds in a radian.
When we work this out for the nearest stars, we find they have a tiny parallax, about 1/200,000 radians, or about one arcsecond. Tiny, but measurable, and indeed, this is the way in which distances to nearby stars are measured. We will find ourselves using the small-angle formula in other contexts as well.
For more distant stars, we've already seen another way to measure their distances, from the inverse-square law relating brightness and luminosity.
One answer is that stars shine by black-body radiation because they
are hot. This is correct, but not complete. All that energy has to
come from somewhere, and if there is no internal energy source in a
star, it will gradually cool off. The geological record tells us that
the oceans on Earth have remained liquid for 4 billion years, so the
luminosity of the Sun has remained close to constant. When one drops
something under gravity, it gains kinetic energy, which turns into
heat when it hits the ground. Perhaps then this
gravitational potential energy as the material that makes up the
Sun collapsed is adequate to power the Sun. The gravitational
potential energy of a mass M of radius R (of constant density) is:
where 
is
Newton's Gravitational Constant. We may have a homework problem in
which you will show that this amount of energy is far from adequate to
power the Sun for billions of years.
What is really going on is thermonuclear fusion. The Sun
(like the universe as a whole) is made up mostly of Hydrogen and
Helium. In the interior of a star, it is tremendously hot, and is
completely ionized. The positively charged protons (hydrogen
nuclei) repel each other, but if they can come close enough
together, they can fuse in a series of reactions to make helium
nuclei. It is so hot, that they are moving at enormous speeds, and
so can indeed crash into each other to fuse. In the core of the
Sun, the reactions involved are as follows (the p-p chain):
Here p and n are protons and neutrons, pn is a deuterium
nucleus, ppn is a Helium-3 nucleus, and ppnn is a Helium-4 nucleus
(also known as an 
particle). 
is a positron (an
anti-electron), 
is an electron neutrino, and 
is a
photon. The net reaction is:
So hydrogen nuclei have been transformed into helium nuclei. Cool
enough, but there is more. If you add up the mass of four hydrogen
nuclei, you get a value about 0.7% more than the mass of one helium
nucleus (the positrons and neutrinos are negligible in comparison).
Where did the mass go? Einstein had the answer; it was transformed
into energy, by perhaps the most famous formula in physics:
This, then, is the energy source of stars; the transmutation of
hydrogen into helium, and the subsequent release of energy.
Before answering this question, it is an appropriate time to define a star. A star is a ball of gas held together by its own gravity, which is undergoing thermonuclear fusion in its center.
Indeed, gravity holds together all large objects in the universe,
from the Earth and its moon, to stars, to galaxies. If gravity is
always pulling things together, why don't objects held together by
gravity just collapse altogether (into a black hole)? The answer is
that there is something holding them up against gravity. In the case
of the Earth, it is the tensile strength of the material of which the
Earth is made that is holding it up. In the case of stars, there is a
gas pressure P associated with its temperature T and the number
density (number per unit volume) of particles (atoms, atomic nuclei,
electrons...) in the gas: the perfect gas law states:
where k is the Boltzmann constant introduced above (you may be
familiar with this equation in the form chemists like to write it:
where N is the number of moles of the gas in question, and R is
k divided by Avogadro's number).
Thus a star like the Sun is in hydrostatic equilibrium. Gravity is holding it together, but the pressure of the gas (because it is dense and hot) is keeping it from collapsing further. And what makes it, and keeps it, hot? The continual generation of energy in the core via thermonuclear fusion.
Well, what about Jupiter? (It has 1/1000 the mass of the Sun, or 300 times the mass of the Earth). It is gaseous, so has no tensile strength. It has no thermonuclear activity in its core, so it is cold, and therefore doesn't have much internal gas pressure. It is held up by a rather different sort of internal pressure, called degeneracy pressure. As you may have learned in a chemistry class, Wolfgang Pauli realized that electrons (and more generally, all sub-atomic particles of half-integer spin) cannot be crowded together too closely. More specifically, two electrons cannot have the same set of quantum numbers; this is called the Pauli exclusion principle. A gas with electrons is called degenerate when the electrons are so tightly packed that the Pauli exclusion principle keeps them from getting any closer together; this exerts an outward pressure which counteracts gravity.
Here's another way to think about it: the Heisenberg Uncertainty Principle states that one cannot measure both the position and momentum of a particle with high accuracy at the same time. When you make a gas very dense, you are confining each particle to a small volume, i.e., you're narrowing down its position. Therefore its momentum must be quite uncertain, so it can't be sitting still, so it has a lot of motion associated with it, and the net effect of the (random) motion of all those particles comes a pressure. We can actually use these ideas to figure out the basic properties of a planet held together by gravity and held up by degeneracy pressure; we'll take a stab at that in a future homework.
Michael Strauss