The mainstay of astronomy has always been the observation and understanding of stars. All different aspects of it. From their formation, subsequent evolution and eventual death. This endeavor is what has led astronomy towards a gradual understanding of the Universe and its ingredients. And ultimately, we hope will lead to an understanding of ourselves and where we fit into the greater mosaic that constitutes the Universe.
The properties intrinsic to the star give us clues to the birth and evolutionary history of the star itself. These might be its brightness, temperature, mass, size, etc. Properties extrinsic to it, such as its position and velocity, give us clues about where the stars fit into the greater scheme to form galaxies and nebulae, as well as how new generations of stars may be born out of the ashes of the old ones. But the problem is a fairly complicated one. The first complication is that the intrinsic and non-intrinsic properties of a star are intimately mixed. If we wanted to measure the brightness of a star (an intrinsic property) we also need its distance from us (a extrinsic property), for more distant the star, the fainter it is. The second complication is that unlike the case of a human, whom we can observe over our own lifetime, being born, growing up and eventually dying, the lifetime of a star is far too long for us to observe a single star being born, evolve and then die. This would need observing it for millions of years. What we have instead is a momentary glimpse of a large number of stars at different stages of evolution. So this would be like understanding human evolution by observing for a moment an entire population of individuals at different stages of their life-cycle. This is more complicated than may appear at the first glance. Consider this analogy for a while longer. Many societies are skewed towards the younger population. So in our observation of the typical society we will see a lot of teenagers. Now remember that while a large number of infants die at childbirth it is unlikely we will actually observe this very often, precisely because it happens so quickly. So we will see a large number of infants older than a few months. Except of course that we don't know that they are new born. We didn't actually see them in the process of being born, the number of women giving birth at any instant being a small fraction of the total population. We don't even know that women give birth. We will also be very unlikely to see a person actually dying for the same reasons. We will see two groups of humans who need considerable assistance to survive. If we knew their lifecycles we'd realize these two groups are the new born and the very old. In the absence of this knowledge we may be tempted to come to the wholly incorrect conclusion that these are closely related parts of the human lifecycle. It is important therefore to be very clear about the problems of this approach. Unfortunately it is only approach afforded to us by nature and we must make the most of it.
As mentioned before, brightness and distance are the two properties of
the star that go together. The flux, F(r), at a position r is
defined so that the energy passing through a small area 
, at
that position, in a small time interval, 
, is 
. Now consider a point source of radiation at the origin
that is isotropic. That is the amount of radiation is
independent of the direction of radiation. Then the flux F depends
only on the distance r from the source and not on the
direction. Then the energy crossing the area per unit time
is, 
. But consider two spheres around the source,
centered on the source, with radii 
and 
, 
. Then the energy crossing the two spheres per unit time is
and 
, respectively. But no energy is created or lost between the
spheres. So these two energies must be the same, 
. This
implies,
This rule, that the flux of an isotropic sources drops as the inverse
square of the distance to the source, follows directly from
conservation of energy and is the reason why the Sun is a lot brighter
than anything else in the sky.
This means that when we measure the brightness of a star in the sky we
are measuring a quantity that is a combination of the intrinsic
brightness of the star and its distance from us. The quantity
we want to measure is the luminosity of the star. This is
defined as the total energy passing a sphere equal to the radius of
the star per unit time. That is, 
, if 
is the radius of the star. Then by the inverse square rule, the flux
we measure on Earth will be,
where, d is that distance of the star from Earth. Thus, only by
measuring both the distance of the star and the flux of its radiation
on Earth can we measure its luminosity.
Measurement and categorization of stars according to the light we
receive from them had an early start. Hipparchos, who lived around 120
B.C. built the first catalogue of stars. He devised a system of
categorizing stars by their brightness on Earth's sky called
magnitudes. This scheme assigns a value of 1 for the brightest stars
and 6 for the faintest ones, with 2-5 values for the stars in
between. Pogson gave a more rigorous version of this scale that is
still used today, but it is not very useful and we will describe it
only for completeness. If the flux from two stars, of magnitude 
and 
, measured on Earth be 
and 
then,
Note that the magnitude decreases as the received flux
increases. Obviously this measure of magnitude depends on the
distance of the star from us by the inverse square rule. This is why
the magnitude m is referred to as the apparent magnitude of
the star. The apparent magnitude of the brightest star, the Sun, is
-26.8. The negative number means that it is very bright relative to
the next brightest stars of magnitude 1. The faintest stars we can see
using the largest telescopes are of the 
or 
magnitude. This difference of 50 magnitudes translates to a difference
of a factor of 
in flux! While some of it is due to intrinsic
differences in brightness of stars, a lot of it is due to the
different distances. Suppose we took a star of apparent magnitude m
and brought it to a distance D, from its known distance of d, then
its magnitude now is,
This magnitude is called the Absolute
magnitude and is a measure of the intrinsic brightness of the star.
In 1879 Josef Stefan, an Austrian working in Vienna, showed
experimentally that the flux from a black body was proportional to the
fourth power of its temperature. Ten years later another Austrian,
Ludwig Boltzmann, proved this relation. It is now known as the
Stefan-Boltzmann law that states that the flux from a black body is
connected to its temperature very simply as, 
, where
the constant 
is called the Stefan-Boltzmann constant. Now in
as much as stars are close to black bodies their temperatures and flux
are connected by this law. Then the luminosity of the star is,
This means that if we can measure the luminosity of a star and its
temperature we can measure its radius.
Measuring the distance of a star is extremely hard. There is only one
direct way of measuring it and as of now it has been applied to only
the nearest set of stars. But this forms the fundamental way of
determining the distances between stars and in effect fixes the
scale and size of the Universe. So we will get into it in some
detail. This method is called the method of parallaxes. Remember
when we discussed the Copernican Revolution that one of the objections
raised to having the Earth move was the alleged absence of the motion
of stars. In fact it took another 200 years before Bessel would
discover the small motions of the stars due to Earth's motion about
the Sun. If the angle subtended by the different positions of the star
over intervals of six months be 
degrees, then the distance,
d, to the star is given by,
where, 
is the radius of the Earth's orbit around the Sun. This
distance is usually referred to as 1 Astronomical Unit (or 1 AU). It
is called the baseline for the parallax of
degrees. Usually the angle is expressed in arcseconds (1
degree = 60 arcminute = 3600 arcseconds). Then the distance becomes,
Astronomers frequently express distances in units of 206265 AU,
which is called a parsec. The name can be understood as an
abbreviation of the distance that produces a parallax of one
arc-second for a baseline of 1 AU. With this definition, the
distance of an object is simply the inverse of its parallax (in
arc-seconds) over six months of Earth's revolution about the
Sun. Unfortunately the star with the biggest parallax, i.e., the
smallest distance (don't forget the bigger the parallax the
closer the star) is Sirius, whose parallax is 0.377 arcseconds, which
translates to a distance of 2.65 parsecs. In 1989, European Space
Agency launched a satellite, Hipparchos to measure parallaxes correct
to 2 milliarcseconds! This would mean accurate distances to
stars 500 parsecs away. Unfortunately the satellite didn't fire its
booster rocket at the appropriate time which was designed to get it
onto a more circular orbit. This meant that the satellite was stuck on
a highly elliptical orbit, that had it come plunging further into the
Earth's atmosphere that it was designed for. Still it achieved a
significant fraction of its designated targets through massive
restructuring of the programmed design. It is hoped that when these
distances are published it will solidify further the distance scale of
the Milky Way Galaxy and the Universe.
There is one complication to the measurement of distances using the trigonometric parallax. The ``fixed'' stars aren't very fixed. If we look at the same star over the interval of several years we will find that it has moved in the sky. This is known as proper motion, because this motion belongs to the star as opposed to the apparent motion due to parallax. In most cases the movement is extremely small. Seldom more than an arcsecond a year, the record being held by Barnard's star at 10.25 arcseconds per year. The star is named after Edward Emerson Barnard, the American astronomer who pioneered celestial photography. He was born in 1857 in Nashville, Tennessee and studied at the Vanderbilt University. He was an astronomer in Lick and Yerkes Observatories and a professor at the University of Chicago. Aside from the star with the largest proper motion he also discovered Jupiter's fifth satellite and 16 comets. This means that to measure the distance of an object by trigonometric parallax we need at least three pictures separated by six months of the same object in the sky in order to separate the proper motion of the star and its apparent motion due to parallax.
Notice that we have expressed distances in terms of the Astronomical Units without bothering to say what it actually is. Of course in principle we can calculate the parallax of the Sun against the distant stars from two positions on the Earth's surface whose distance is known. Then we can evaluate the distance of the Sun from Earth. But this method is error-prone because of the extreme brightness of the Sun. It is much more convenient to use Newtonian Gravity and dynamics to express the motion of the planets in terms of AU. Then if we measure the distance to any planet (or the Moon) we can get the Sun's distance. These days we use laser ranging of the Moon and timing radar observations of Venus to get accurate distances. So we can leave all the distances in terms of parsecs or AU with the knowledge that we can translate these distances to everyday units (like kilometers, miles, cm, etc.) using the known distances to Venus and the Moon.
As we said before the distances to which it is possible to measure
parallax is very small compared to the typical distances between
stars. How can we measure the distances to the distant stars? The
short answer is we cannot, directly. Fortunately there are indirect
means of measuring the distances to these stars. One of the most
important is the method of standard candle. This one harks back to the
previous subsection on brightness of stars. Remember what we said
about the ``inverse square law'' of brightness. The further we went
from a source of the fainter it appeared. This means of course that if
we did know how bright the source would be if it were at some
particular distance we can tell how far the source is given its
observed brightness. Unfortunately in the past, people assumed all
stars are of the same brightness and came to grossly incorrect
conclusions on the size of the Galaxy and the Universe. In fact there
is a factor of about 
(a billion) difference in the intrinsic
brightness of the faintest and the brightest stars.
Astronomers are usually very interested in finding out the numbers of bright stars to the number of faint stars. This distribution of stars in brightness is called the luminosity function. If we look at the sky there are apparently more bright stars than there are faint stars. However this is an illusion. Because bright stars are more conspicuous we can see them from further away than we can faint stars. Most of the stars we see nearby are faint where there are very few bright stars. So we need to remember that there are many faint stars that we can see to every bright star we see when we count the number of faint stars to bright stars. This bias in counting faint stars is an example of a selection bias that occurs in all samples of data, whether it is stars, birds or people. Whenever statistical studies are to be made these biases must be carefully removed. Once it is removed the luminosity function of stars are clearly weighted heavily towards faint stars. That is, there are many more faint stars for every bright star. Contrary to what is apparently suggested by looking at the sky.