1. IB Physics Astronomy Elective
The study of binary and variable stars reveals vital information about stars

2. The Problem
Looking through a telescope at the stars there is very little information we can gain from them.
To be sure, we know what colour they are and we can see that some are more luminous than others.
If we use a spectrograph we can tell what elements they are made up from. From these facts alone, it is difficult to tell just how much mass they contain.
By looking at pairs of stars that orbit one another we can try to answer the question, how much mass do the stars have?

3. Binary Stars Binary stars can be of two fundamental types:
Visual Binaries
Optical Doubles

4. Visual Binaries
Visual Binaries are stars that are clearly gravitational associated with one another. They orbit each other around a common centre called the barycentre.
Visual binaries can be seen optically through a telescope. Only a small portion of binary stars are visual binaries.
In order to see a visual binary, the stars must be separated by fairly wide distances, and the orbital periods are usually very long.

5. Optical Doubles (false binaries)
Optical Doubles are stars that appear to lie close together, but in fact do not, they only appear to us from our earthly observation to be close together. One of the stars in the pair is actually behind the first star and very far away. The stars of an optical double are not gravitationally bound.

6. William Herschel
William Herschel began looking for optical doubles in 1782 with the hope that he would find a measurable parallax, by comparing a close star to the more distant star in an optical double.
Herschel did not find any optical binaries, but he did catalogue hundreds of visual binaries.
In 1804 Herschel had so many measurements of visual binaries that he concluded that a pair of stars known as Castor were orbiting one another.
This was an important discovery, because it was the first time observational evidence clearly showed two objects in orbit around each other outside of the influence of our own Sun and Solar System.

7. Spectroscopic Binaries
It is also possible to detect binary stars using a spectroscope.
If two stars are orbiting each other they will both produce a spectrum.
If the stars are close to being the same brightness it is possible to see different spectral lines from both stars.
These stars are of particular interest because it can be used to determine the radial velocity of the orbit of the two stars.

8. Doppler Effect
Stars appear red shifted when receding away from the earth and blue shifted as they approach.
This effect is caused by the Doppler effect which distorts arriving light waves from the stars depending on the direction if their motion.
A Spectroscopic binary will alternate between blue and red shifted spectral lines.
Spectroscopic binaries are not detectable if we are seeing the star head on because no Doppler shifts would be present in the spectrum.

9. Eclipsing Binaries
An eclipsing binary star system contains two or more stars in orbit around their centre of mass.
These are usually systems that cannot be resolved with a telescope.
Photoelectric photometry provides a means to gather information about these systems.

10. Eclipsing Binaries and Stella Mass
The amount of light we receive depends on whether we are seeing the light from the two separate stars, or whether we are seeing the light from one star being eclipsed by the other.
The diagram is the light curve of an eclipsing binary system with total eclipses. It shows how the light of the system varies with time. The light curve repeats itself after the stars have gone once around the barycentre.

11. From the light curve, we can measure the time it takes for one star to cross the disk of the other one.
This time depends on the relative sizes of the two stars, and also on how fast one star crosses in front of the other.
This is equivalent to determining the distance from one end of your sidewalk to the other if you know how long it takes you to walk that distance and how fast you walk.
Therefore, the other piece of information that we need in order to calculate the diameter of the stars is their orbital speeds.
These are determined from the spectrum of the stars, or the light that we receive at each wavelength. As a star moves toward or away from us in its orbit, its spectrum shifts back and forth in an amount directly related to the orbital speed of the star (this is called Doppler shift).
Eclipsing binary systems comprise less than 5% of all stars, so this method yields radii for a small number of stars. Nevertheless, this method is very useful because we can assume that stars of similar spectral type have similar diameters.

12. Astrometric Binaries
Astrometric binaries deviate from proper motion ie. They ‘wobble’ around an invisible companion

13. Variable Stars
Variable Stars are stars that vary in their light output.
The origins of these light variations define the classification system of variable stars.
There are two kinds of variable stars;
intrinsic in which variation is due to physical changes in the star or stellar system and
extrinsic in which variability is due to the eclipse of one star by another or the effects of stellar rotation eg eclipsing binary.

14. Extrinsic Variables
Within the extrinsic group there are two classes: eclipsing binary and rotating stars.
We have dealt previously with eclipsing binaries.
In rotating variable stars, variation in brightness is usually small and results in the rotation of the star exposing dark or bright spots, or patches ("star spots") on its surface.  Rotating variable stars are often binary systems. 

15. Intrinsic Variables
There are four main classes of variable stars. Within the intrinsic group of variables there are two classes: pulsating and eruptive.

16. In pulsating variables, variation in brightness is caused by periodic expansion and contraction in the star's surface layers.  These pulsations may be radial (in which case the star retains its spherical shape) or non-radial (resulting in periodic deviations from a spherical shape). 
Pulsating variables themselves can be divided up into different types according to their pulsation period, mass, and evolutionary status:

17. Cepheids:
Cepheids: These stars have periods of 1-70 days with amplitudes of variation from 0.1 to 2.0 magnitudes. 
Cepheids obey a strict period-luminosity relationship with Cepheids of higher luminosities having longer periods, both depending on the radius. 
Therefore, by measuring the period of a Cepheid variable, one can obtain its luminosity, and by measuring its apparent brightness one may deduce how far away it is. 
We will look at the importance of Cepheid Variables in determining astronomical distances later.

18. Other Intrinsic Variables
RR Lyrae stars: These stars have periods of days with amplitudes of variation from 0.3 to 2 magnitudes.  These pulsating variables are white giant stars of spectral class A. 
RV Tauri stars: These stars have periods of days with amplitudes of variation up to 3.0 magnitudes.  They are yellow supergiants generally of spectral classes from G to K. 
Long Period Variables (Miras): These stars have periods of days with amplitudes of variation from 2.5 to 5.0 magnitudes.  They are giant red variables with spectral classes ranging through M, C, and S. 
Supernovae: These  stars show sudden, dramatic, and final magnitude increases as a result of a catastrophic stellar explosion.  Thus, there is no period, and amplitudes of variation are 20+ magnitudes.
Novae: These close binary systems consist of a main sequence, Sun-like star and a white dwarf.  They increase in brightness by 7 to 16 magnitudes in a matter of one to several hundred days.  After the outburst, the star fades slowly to its initial brightness over several years or decades. 

19. The Period-Luminosity Relation
In 1912, Henrietta Swan Leavitt of Harvard College Observatory catalogued over 1777 variable stars in the nearby Magellanic Clouds. Of this impressive count, 25 were found to be of Cepheid-type and were located in the Small Magellanic Cloud (SMC).
When Leavitt arranged the Cepheids by period, a surprising fact arose: ordering these stars in terms of increasing period revealed that they had also been ordered by increasing brightness.
Thus, the resulting sort showed the brighter the Cepheid, the longer it took to vary.

20. Typical Cepheid Variable Curve

21. Other Cepheid Light Curves

23. Measuring Astronomical Distances using Cepheids
Upon finding this impressive relation, Leavitt reported that
"A straight line can be readily drawn among each of the two series of points corresponding to maxima and minima, thus showing that there is a simple relation between the brightness of the variable and their periods."
Therefore, by measuring the period of a Cepheid variable, one can obtain its luminosity, and by measuring its apparent brightness one may deduce how far away it is. 

24. This extraordinary finding proved to be a means of determining cosmic distances; a parameter which is crucial in our understanding of the Universe.
Because of the work of this incredible woman, the foundation of modern astronomy had been set, and the field was about to go through a wonderful revolution.