----- The following copyright 1991 by Dirk Terrell
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 Lecture #12  "A Look at Close Binary Stars  Part 1"

   The sun is an typical star in many respects. There is nothing 
particularly special about the sun's size, temperature, mass, or luminosity. 
But there is one area in which the sun differs from the majority of other 
stars and that is in its lack of a stellar companion. In fact, recent 
estimates place the percentage of 'stars' that are multiple systems as high 
as ninety percent. This situation turns out to be rather helpful to 
astrophysics because binary stars are the most accurate source of 
information on physical properties of stars such as mass and radius. 

   There are several ways in which we can discover the binary nature of a 
star. The simplest way is to see the two stars orbit one another, usually in 
a period of many years. These are the visual binaries and a typical example 
is 61 Cygni, the first star to have its parallax measured. With binoculars 
the components of 61 Cygni are easily seen as stars of magnitudes 5.2 and 
6.0. The system is about 11 light years from Earth and the stars orbit each 
other with a period of about 650 years. Unfortunately, the amount of 
information that we can get from visual binaries is limited, mainly because 
these stars must be relatively nearby to be resolved.

   Most systems are far enough away so that even the largest telescopes 
cannot resolve the components, but we can still deduce that there is more 
than one star in the system by studying its spectrum. As the two components 
orbit one another, their velocities with respect to observers on Earth 
undergo periodic changes, alternately towards the earth and then away. These 
changes can be detected as the shifting of the spectral lines caused by the 
Doppler effect. If the spectral lines of both stars are visible in the 
spectrum, the system is called a double-lined spectroscopic binary. If only 
one star's lines appear, the system is said to be single-lined. In a 
double-lined system we can determine the minimum masses of the stars but we 
can get the actual masses only if the inclination of the orbit is known.

   If the two stars pass in front of one another as seen from Earth then we 
have the relatively uncommon, but extremely useful case of an eclipsing 
binary. With each eclipse, the brightness of the system drops. A plot of the 
brightness of the system versus time is the light curve, and the eclipse 
that causes the greatest drop in brightness is called the primary eclipse. 
Most often the time coordinate is measured as the phase, which is the 
fraction of the orbital period that the stars have completed since the 
middle of the last primary eclipse. For example, if half a cycle has passed, 
the system is at phase 0.5. Eclipses, especially total ones where the light 
of one star is completely blocked, can provide a wealth of information about 
the properties of the stars in a binary system. Most eclipsing systems have 
periods ranging from a few hours to a few days, although some are much 
longer. For example, the rather enigmatic Epsilon Aurigae has a period of 
over 27 years! Here we will look at close binaries- systems where the 
separation of the two components is small and the evolution of each star 
affects the other. 

   Light curves come in many different shapes. In order to understand them 
we must first look at the shapes that the stars themselves assume. In early 
attempts to understand light curves it was assumed that the shapes of the 
stars could be modeled as ellipsoids, and in some systems the ellipsoid 
approximation is quite sufficient, but in others it breaks down. A better 
way to model the stars' shapes is known as the Roche model. However, because 
of its mathematical complexity, the application of the Roche model to 
calculating and analyzing light curves did not come until 1971 when Robert 
Wilson and Edward Devinney developed the now widely used Wilson-Devinney 
(WD) method of light curve analysis, and others developed models applicable 
to certain special cases.

   The Roche model is based on a few simplifying assumptions about the two 
stars. One is that the stars act gravitationally as point masses. That is, 
each star attracts the other as if all of its mass were concentrated at its 
center. This is not a bad approximation since stars are highly centrally 
condensed. We also assume that the stars' orbits are circular and that the 
stars rotate once every orbital period, just as the moon rotates once every 
time it orbits the earth. Thus we say that the stars have synchronous 
rotation. These assumptions make the mathematics of the problem considerably 
simpler, but they do not apply to all eclipsing binaries. For example, we 
know of some systems that have eccentric (elliptical) orbits and some where 
the components do not have synchronous rotation. In 1979 Wilson modified the 
WD method to include the effects of eccentric orbits and non-synchronous 
rotation, but we will mainly be concerned with systems that satisfy the 
original assumptions.

   In the Roche model we expect the surfaces of the stars to lie along 
surfaces of constant potential energy (equipotentials). Anyone who has taken 
a bath should be able to understand the idea of equipotentials. When you 
plug the drain and put water in a tub, it assumes some level based on the 
amount of water. The surface of the water is one of constant potential 
energy. If you add more water to the front of the tub, the water doesn't 
pile up at the front and stay low at the back, it flows quickly so that all 
of the water is at the same level. This level will be higher than it was 
before you added the water, but it will be another surface of constant 
potential energy (higher potential energy than before). Since the surfaces 
of stars are fluids, we expect them also to assume the shape of 
equipotential surfaces. If one part of a star's surface were to find itself 
at a higher potential than the rest of the surface, flows would quickly 
bring all of the surface to the same potential just as the water in a bathtub.

   If the stars' surfaces are to coincide with equipotential surfaces, what 
do these equipotential surfaces look like? Near the mass point at the center 
of either star the dominant force acting on a particle is the gravity of 
that star, and we expect the equipotentials to be essentially spherical. 
Farther away, however, the gravity of the other star and the centrifugal 
force (arising from the rotation of the binary as a whole) become 
significant. The result is that the equipotentials around one star are no 
longer spherical, but have a bulge towards the other star, much as the 
oceans of the earth bulge towards the moon because of its gravitational 
attraction. It seems logical that there should be a place where the gravity 
of one star is just balanced by the gravity of the other star and the 
centrifugal force. Indeed such a point (called the inner Lagrangian,or L1, 
point) does exist, located on the line joining the centers of the two stars 
and closer to the less massive star. The equipotential surface for each star 
that includes the L1 point is called that star's Roche lobe. As we shall 
see, the L1 point plays a very important role in the evolution of the binary 
system. 

   Adoption of the Roche model leads very naturally to a method of 
classifying binaries based on the degree to which each star fills its Roche 
lobe. If both stars are smaller than their their Roche lobes, the binary is 
called a detached system, and if the stars are much smaller than their 
lobes, they will have spherical shapes. A typical example is RS 
Chamaeleonis, where two A8 IV stars orbit each other in a period of about 
1.7 days. Figure 1 shows RS Cha at phases 0.00, 0.05, 0.10, 0.15, 0.25, 
0.35, 0.40, 0.45, and 0.50, starting at upper left and moving left to right 
and top to bottom. The circle to the right shows the size of the sun 
compared to the binary. The bottom of the figure shows, from top to bottom, 
the light curves of the system in the infrared, visible, and ultaviolet 
parts of the spectrum. The vertical lines indicate phase 0.0 (left line) and 
phase 0.5 (right line) for the light curves. The horizontal lines indicate 
the 50% brightness level for each light curve, where 100% brightness occurs 
when the system is at phase 0.25, corresponding to the middle picture of RS 
Cha. If the stars are larger, perhaps only slightly smaller than their 
lobes, they will be more distorted as in the case of MR Cygni.