While our observations can determine luminosity and spectral classes, close observation and study of binary stars can also yield mass.

While stars are just point sources far from Earth, there is still much we can learn. For example, if the star is relatively close to us, we can determine its distance using stellar parallax:

By measuring the shift angle seen from Earth, we can determine distance. In case you don't know what parallax is, hold a pen or pencil arms length from your face and close one eye at a time while looking at something far away. The shifting of the pen is the parallax.

We can also determine the brightness of a star (as well as its luminosity) if we know its distance. We can use the Inverse Square Law:

However, notice that we need to know the luminosity of the star to determine brightness. We can determine the luminosity of the star using this formula:

but without knowing brightness, it seems we cannot determine luminosity. This is where our Sun comes in. We have the benefit of a frame of reference when comparing to other stars.

With the known values in place, its just a matter of some algebra.

Another classification for stars is magnitude - how bright does the star appear to us on Earth. There are two versions of magnitude:

• Apparent magnitude - brightness seen from Earth

• Absolute magnitude - how bright the star is if 10 parsecs from Earth

The magnitude scale is graded by numbers: 0 being bright, 6 being dim. Some values of comparison:

• Our Sun - -26.7 (that's a negative)

• Sirius - -1.4

• Naked eye limit - +6.0

• Binocular limit - +10.0

• Pluto - +15.1

• The Hubble Space Telescope limit - +30.0

The magnitude scale works out to be logarithmic, and the difference between each value is 2.512. That is magnitude 1 is 2.512 times brighter than magnitude 2 and magnitude 5 is 2.512 * 2.512 * 2.512 times dimmer that magnitude 2.

By knowing the distance and apparent magnitude to a star, we can learn the absolute magnitude. Also, if we happen to know the absolute magnitude, we can determine distance. The result of this is the distance modulus:

or:

By using filters (blue, green, red) and measuring the brightness of a star in the different filters, we can determine the color index of a star (and recall from the physics section that color and temperature go hand in hand):

If the B-V value is 1, the star is white. If the B-V index is less than 1, the star is more blue and if the B-V index is more than 1, the star is more red.

All of these tools led to a correlation: a stars size relates to how bright it and how hot it is.

It is this correlation that helped to create the Hertzsprung-Russell (or H-R) diagram:

(Image Credit: Pearson Education, Addison Wesley)

Going back to luminosity, there are six classes of luminosity:

• Ia - Bright Supergiant Stars (example: Deneb)
• Ib - Supergiants (example: Antares)
• II - Bright Giants (example: Canopus)
• III - Giants (example: Capella)
• IV - Subgiants (example: Beta Cru)
• V - Main Sequence (example: Vega)

Classes can be categorized like: Ia, Ib, Ic, IIa, IIb, IIc, and so on (sub-a being brighter that sub-b).

In addition, notice the x-axis of the H-R diagram. This is the spectral class - the result of the Harvard team.

• O - O type stars are the brightest and the live the shortest
• B - B type stars are blue white and also burn bright, but not as bright as the O type
• A - A type stars are less bright, a little larger than our Sun, but still burn hotter
• F - Brighter than our Sun and a little hotter
• G - Our Sun is a G type star
• K -  dimmer that our Sun, will burn longer because temperature is lower
• M - the dimmest stars, will burn for a long time

(Image Credit: Pearson Education, Addison Wesley)

The famous pneumonic helps astronomers remember this sequence: Oh Be A Fine Girl (or Guy) and Kiss Me. Who says scientists don't have a sense of humor?

One final interesting formula: a quick and dirty correlation between a stars luminosity and magnitude is:

All that is left is to determine the mass of a star. We now know that bright stars are very large and dim stars are small, but without an accurate measure of mass, we can only speculate. This is where binary stars come into play.

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A binary star is a multiple star system bound by mutual gravitation. Both stars will revolve about a common point. You can find out more by requesting astronomy homework help from a real astronomer. By using Keplerian math, we can determine the mass by studying this movement. If we know the spectral class of the stars in question, we can assign standardized values (like the H-R diagram) that will give is the mass of all stars.

There are four main classes of binary stars:

• Visual binary stars - stars that we see on Earth as being binary
• Spectroscopic binaries - binary stars that are only seen using spectroscopy
• Eclipsing binaries - binary stars at our line of sight
• Accreting binaries - close pairs that "feed" off each other

A visual binary star system can either be a real binary, or one that looks that way because of our view on Earth. A true binary will move something like this:

(Image credit: Brooks/Cole Thomson Learning)

In order to find the mass of a visual binary:

(Image credit: Brooks/Cole Thomson Learning)

m in this case is mass and P is the period of the orbit.

A spectroscopic binary star is one that is not seen visually, but long term observation records a shift in the stars spectrum:

(Image credit: Brooks/Cole Thomson Learning)

Sometimes if the orbit of the binary star is facing us, we may not be able to see spectroscopic changes, but we will see brightness changes. Long term observation will reveal periodic changes in overall brightness:

(Image credit: Brooks/Cole Thomson Learning)

This same technique is used for the detection of exoplanets.

The final class of binary star is the accreting binary. If the pair of stars are close enough, the atmosphere of one star can pour onto another. The point of no return in this case is the outer Lagrange point (not seen in the image). If material breaches the outer Lagrange point and contacts the inner Lagrange point, material will begin to flow onto another (that is what I mean by the point of no return).

(Image credit: Brooks/Cole Thomson Learning)

A common companion star to an accreting binary system is a neutron star or a white dwarf - both covered in stellar evolution.

Some final bits: binary star system are actually quite common - Sir William Herschel discovered 10,000 binary star systems. In addition, binary stars are really part of multi-star systems. Many stars have more than one companion. A triple star system is also common.

To sum up, here is a nice chart I found that demonstrates the process of stellar classification:

(Image Credit)

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