Introduction
Stars are suns. A large number of stars are like our sun (Sol); a great many are much smaller, while some are thousands of times larger. There are stars thousands of times dimmer and others that are thousands of times brighter the Sun. In terms of size and brightness, the Sun is in fact quite ordinary.
The most striking thing about our sun it that it is so bright. However, it appears so bright because it very close to us. The next nearest star, Proxima Centauri, is about 250,000 times further away. Stars are so far away that they appear to us as mere points of light. Yet the tiny amount of light that reaches us provides a wealth of information about objects which, for the forseable future, remain utterly beyond our physical reach.
However, before we can develop an understanding of stars we need a way to characterize them. Stars are, in detail, very complicated objects. When confronted with a complicated thing, we first try to simplify our study by abstracting from the object under study a few aspects of it that we judge to be significant. For stars, we think that the key aspects are its: temperature, luminosity, radius and mass.
Basic Properties
 | Distance |
| Temperature |
| Luminosity |
| Radius |
| Mass |
Hertzsprung-Russell Diagram
Distance
If a star is not too far away its distance from us can be measured by the method of triangulation: That is, viewing the star from different vantage points and measuring the star's apparent shift in position, relative to more distant stars. The apparent (angular) shift of an object, caused by a change of vantage point, is called parallax.
Parallax is a very common everyday phenomenon. It is what causes the Moon apparently to follow us, as we drive along a country lane, while nearby trees flit past our window. As we move along the road the apparent position of all objects, relative to more distant ones, changes. This is true of the Moon as well as things closer by. But the Moon is so far away that we cannot perceive its tiny shift backwards, due to our motion forwards, while that of nearer objects, like trees, we can easily detect. So the Moon appears to keep up with us, whilst everything else is left behind.
A star's parallax is caused by the shifting vantage of the Earth as it orbits the Sun. Figure 1 shows the position of the earth now. Figure 2, shows the position of the earth 6 months from now. Figure 3 the shows the star's parallax.
Imagine a huge circle, of radius d (in AU), drawn through the Sun and centered on the star whose distance is being measured (see Fig. 4). Because stellar parallax angles are so small we usually measure them in
arc-seconds (1 arc-sec = 1/3600 degree):
1/2pd = p/360x3600Solving for the distance to the star, d (in AU), we get
1/2pd = p/1,296,000
It is convenient to define a new unit of distance called the parsec, which is equal 206,265 AU, or 3.09 x 1013 km. This distance unit is useful because if we measure distances in parsecs and angles in arc-seconds we can use the simpler formula:d = (1,296,000/2p)/p
d = 206,265/p
Example: The nearest star to the Sun is Proxima Centauri. It has a parallax of p = 0.77 arc-sec. So its distance d from us is d = 1/0.77 or about 1.3 pc.d = 1/p
Notice that from a distance of exactly 1 parsec (206,265 AU) the radius of the Earth's orbit would have an angular size of exactly 1 arc-sec.
In August 1989 the European Space Agency (ESA) launched
the mission Hipparcos,
whose primary goal was to measure the distances to about 100,000 stars
as accurately as possible using the parallax method. One of the breathtaking
successes of this mission are the many 3-D images we now have of star clusters,
like the Hyades.
Temperature
A hot object radiates electromagnetic energy. You are a hot object therefore radiate electromagnetic energy in the form of infrared radiation (heat rays). Objects can also absorb energy. If an object absorbs all the energy that falls on it (that is, it is a perfect absorber) and can re-emit all the energy it receives (that is, it is a perfect radiator) the object is called a black body.
A black body need not be black! For example, incandescent lamps (ordinary light bulbs), as well as stars like the Sun, are excellent examples of "black bodies"!
The most important fact about such objects is that they emit a continuous spectrum of electromagnetic radiation, having the same characteristic shape. This spectrum, whose mathematical form was first derived by Max Planck in 1900, depends only on the temperature of the object.
The temperature of a star is related to its color. Red stars are cooler than Blue stars. The spectrum of light for red stars peaks at longer wavelengths than that of blue stars. If a star's spectrum peaks at a wavelength
(measured in nanometers) its surface temperature T can be calculated from Wien's Lawlmax
Example: the red giant star Betelgeuse (the upper left star in Orion) radiates most strongly at about 1000 nanometers, so its surface temperature is expected to be aboutT = 3,000,000/lmax
LuminosityT = 3,000,000/1000 = 3000 K
The luminosity of a star is the total amount of energy it radiates per second. For example, the Sun's luminosity is
L = 3.8 x 1026 watts.
How bright a star appears depends both on its luminosity and its distance from us. Therefore, we cannot assume that just because a star appears fainter than another it is intrinsically less luminous: the star that appears fainter could just be further away.
To obtain the intrinsic (or absolute) luminosity of a star we need to know:
| its distance d from us and |
| its flux (or brightness) f. |
Flux is the amount of energy passing through a given area per second. So the units are Joules per square meter per second.
As we move further from a light source the flux decreases because the energy gets spread ever more thinly over an increasingly large sphere, centered at the light source. That is, for a fixed area, for example one square meter, the amount of energy passing through that area decreases as we move further from the light source. We can relate flux, distance and luminosity provided we make some assumptions:
| The energy from the star is emitted in equal amounts in all directions; we say the energy emission isotropic. |
| The energy is not lost, that is, the energy is conserved as we move away from the star. |
Given these assumptions we deduce that the luminosity is distributed uniformly over any sphere centered at the star. So f and L must be related as follows
The flux is the energy per second passing through a square at the earth. How can we figure out the luminosity?f = L/4pd2
The star's luminosity can be written as
In the 1800s, Josef Stefan and Ludwig Boltzmann, showed that the energy, E, radiated per second per square meter can be calculated from the formulaLuminosity = Energy radiated per second per square meter x Total surface area of the star
whereE = s T4 (J/sec/m2)
is called the Stefan-Boltzmann constant, in their honor, and where T is the absolute temperature, measured in Kelvin, of the star's surface. The surface area of a spherical star iss = 5.67 x 10-8 (J/m2 sec K4)
so a star's luminosity L must beArea = 4pR2
Astronomers usually measure stellar properties in terms of those of the sun because it simplifies the formulas. For the sun we can writeL = (4pR2) (s T4)
We now divide the formula for L by the one for the sunLÄ = (4pRÄ2) (s TÄ4)
Example: The radius of Betelgeuse is roughly 1500 times that of the sun, while its surface temperature is lower by a factor of 2. We, therefore, expect its luminosity to be aboutL/LÄ = (R/RÄ)2 (T/TÄ)4
times that of the sun!L/LÄ = (1500)2 (1/2)4 = 140,000
Mass
We can find the mass of a star by
measuring the gravitational effects of one star on another. Most stars
belong to multiple star systems, like the star Mizar
in the Big Dipper. 
Stars
often come in pairs, called binary stars.
In a binary star system each star orbits about a point called the center
of mass. If you could hang the stars
at the end of long rod the center of mass would be the point along the
rod where the two stars would balance each other.
Sometimes we can see each star separately. We then have a visual binary. If we observe the stars for a long time we can trace out their orbital motion (see Fig. a, b, c, d, e) and deduce the orbital period p about the center of mass. We can also measure the size of the orbits, a. Given these data, and the more complete form of Kepler's law
we can infer the total mass of the binary system.(M1+M2)p2 = a3
Often the stars are too close to be seen as two distinct stars. Nonetheless, we can identify the binary by looking for two sets of spectral lines (one for each star) and measuring the Doppler shifts of these lines. These stars are called spectroscopic binaries (see Figs. a, b). See animation.
When a star moves away from us its light rays are stretched, that is, the wavelengths lengthen, while a star that is moving towards us has its light squeezed, that is, the wavelengths shorten. The shift in wavelength is related to the speed with which the light source is moving as follows:
v/c = Dl / l
where v
is the speed of the source and Dl is the wavelength shift.
Sometimes, just by chance, the stars are so aligned relative to us that one star passes in front of its partner. The best example of these eclipsing binaries is the star Algol (the Demon star) in the constellation Perseus. Its brightness decreases sharply as the dimmer star, Algol B, passes in front of the brighter, Algol A.
The period is measured to be 2.87 days. From a knowledge of the radial velocities of the two stars and using the total mass derived from Kepler's law we can deduce the mass of Algol A to be 3.7 solar masses and 0.8 solar masses for Algol B.
The Hertzsprung-Russell
diagram
A very important advance occurred
in the early years of the 20th century. It is a wonderful example of the
importance, to the scientific enterprise, of abstracting the right physical
aspects from an otherwise complicated physical phenomenon. In 1905-1915
Ejnar Hertzsprung (Danish) and Henry Norris Russell (American) made a scatter
plot of the luminosity of stars against their temperatures. They discovered
that the points fell into well-defined groups in this plot. This suggested
that there was a pattern to the characteristics of stars, which cried out
for an explanation.
The plot is now known as the Hertzsprung-Russell diagram (or H-R diagram) in their honor. There is a broad diagonal band in the H-R diagram called the Main Sequence. The sun is an example of a main sequence star.
