Binary Stars
 Above a contact binary (modelled in Pov-Ray using a shape called a lemniscate as an approximation; the background nebula was rendered using Chris Colefax' Galaxy.inc pov-Ray extension). See the section on the Roche potential for actual mathematical plots.About half of star systems are binary star systems. A binary star system, or binary star for short, is a star system which contains two stars that orbit about one another, rotating around their common centre of mass (see below). This means that about two-thirds of stars belong to a binary system. Stars like the Sun are solitary stars. More complex systems also exist, for example, the brightest star in Gemini, alpha-Gem or Castor, is seen as a binary star in which the tow stars orbit one another in 420 years. These two stars are visual binaries, meaning that they are far enough apart (and close enough to the Solar System) to be seen as two distinct stars from Earth. However, each of these two stars is also a binary star, but one in which the two component stars are so close to one another that they look like a single star from far away, these are called close binaries. These two close binaries orbit one another in 2.9 days for Castor A and 9.2 days for Castor B. There is also a third close binary star, called Castor C, with an orbital period of 0.8 days, which also orbits the visual binary. Thus Castor is not really a single star but a system of six stars (or of three binary stars). Triple systems contain three stars, quadruple systems 4 stars and sextuple systems, like Castor, contain six stars.The binary star below is a detached binary, in which the two stars are distinctly separate, although they are so close as to appear as a single star from a distance, and so constitute a close binary system. Click on the image to enlarge.
 Supposing that we had a telescope powerful enough (or we were close enough to the stars) to see the two stars above as a visual binary. We might see two different effects - if we see the star system edge-on then we will see one star pass in front of the other, and vice versa, as they orbit about their centre of mass. The common centre of mass is a point that lies somewhere between the two stars, but closer to the more massive of the two stars, in this case the red star. If seen edge-on in this way, we see an eclipsing binary as each star eclipses the other one periodically, causing a dip in the total amount of light that reaches us from the system. If we are looking down upon the star system in plan view, more or less,  then we will not see the two stars eclipse one another, but they can still be seen to rotate around their common centre of mass, and so we have a noneclipsing binary.The two animations below show a noneclipsing binary system on the left and an eclipsing binary system on the right. The only difference is the angle at which we see the star system. If you can not view the animations (for example if you are missing the necessary plug-in, then click on the links underneath to download the animations).
 In a close binary system if two stars are very close to one another, then they will start to feel the affects of their companions gravitational field, causing tidal distortions, as material from one star is pulled closer toward the companion, forming a tidal bulge. Large giant stars will be particularly affected, as their tenuous outer layers are more easily pulled away. In the star system below the star on the right is beginning to swell as it becomes a red giant (it is a subgiant about to become a red giant), and as it expands its outer layers become more loosely bound and to the star's core and also become closer to the companion (yellow-orange dwarf) star. Thus, the more compact companion is easily able to pull on the subgiants atmosphere as it bulges toward the dwarf star. When the subgiant expands further, then some of its material will become more strongly attracted to the dwarf companion and will stream toward it in a process of mass transfer. In this case it will probably impact directly onto the smaller star, as the two stars are very close. However, when the smaller star is further away, the in-falling material may form an accretion disc of material that slowly spirals onto the recipient star, heating up as it does so. The process whereby one star gains mass at the expense of its companion, in this way, is called mass accretion.There is a limit at which each star in a binary system can only just hold on to its outer layers, and if it exceeds this maximum size then it will transfer mass onto its companion. This limit defines a distorted shape known as the Roche lobe. In the binary below, the large star on the left has just reached this limit and is said to fill its Roche lobe, if it gets any bigger (or if the stars move closer together) than it will overfill its Roche lobe and lose material to its companion, maintaining its size near to the Roche lobe radius.
 Above: the swelling star on the right has (nearly) filled its Roche lobe and is nearing the point of Roche lobe overflow. Such a system is called a semi-detached binary star. As the stars orbit one another, the bulge will remain facing the companion star.
 Below: these two binary stars have both component stars filling their Roche lobes, but only just. When this happens, the two stars make contact and so share a common envelope. If the star on the left swells up several times more, perhaps as it continues to become a red giant then the smaller companion star may be completely absorbed within the expanded red giant envelope, forming a single star envelope with two stellar cores within it. (These stars and the semidetached system above, were modelled in Pov-Ray using 'blobs' to approximate the shapes, but it is also possible to calculate the exact geometry mathematically, using so-called Roche potentials). This type of binary, in which the two stars touch and more or less merge, is called a contact binary.
 The contact binary below is a mathematically predicted shape (modeled as a solid of revolution in Pov-Ray using coordinates derived from the Roche potential).
 If we use the topmost view of this contact binary, in which the system is seen edge-on and so is eclipsing, we can simulate the light-curve that an astronomer might see when observing such a star system. The light-curve shows us how the observed or apparent light-intensity caries during the course of the stars' mutual orbit. (Again the stars orbit about their common centre of mass). We can do this by examining the average pixel intensity of each frame of the animation. When this is done, and the result plotted, then not surprisingly we obtain a light-curve that looks very much look those seen in the real case. The light-curve obtained for our eclipsing model is shown below:
 The orbital phase is the number of whole or fractional orbits completed, from the point we begin recording, which conventionally is chosen as the point at which the larger star eclipses the smaller star and the light-curve is at a minimum. The phase keeps counting indefinitely, thus the smaller star gets eclipsed at phase 0, 1, 2, 3, ... etc. At these phases the larger star is pointing straight at us.