----- The following copyright 1991 by Dirk Terrell
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 Lecture #12  "A Look at Close Binary Stars  Part 2"

   We know from the theory of stellar evolution that stars expand as they 
evolve. Now, a single star can expand as it pleases, but in a binary system 
a star is limited to its Roche lobe. To see this consider what happens as 
the star expands. A battle in the life of every star is that between the 
inward pull of gravity and the outward push of gas pressure. Normally the 
two just balance. But what happens when the star expands to the size of its 
Roche lobe? Everywhere along the surface of the star the pressure is 
balanced by gravity, except at L1 where the effective gravity vanishes. At 
the L1 point we find gas under pressure on one side and essentially a vacuum 
on the other, and the gas expands into the domain of the other star. Any 
attempt by the star to expand beyond its Roche lobe is halted by rapid flow 
through the L1 point. Returning to the bathtub analogy, it is like trying to 
overfill the tub- the water flows out the overflow hole as fast as it comes 
from the spout and stays at the same level.

   A system where one star fills its Roche lobe and the other is smaller 
than its lobe is called a semi-detached system. A very famous example of 
such a system is Algol, the Demon Star, in Perseus. Algol is actually a 
triple system, but two of the stars form an eclipsing pair with a period of 
2.87 days. The primary eclipses are easily seen by naked eye as the system 
drops from magnitude 2.3 to 3.5, as a B8 main sequence star is eclipsed by a 
K subgiant. However, when the much cooler K star is eclipsed by the hotter 
B8 star, only a small part (2-3% in the visible part of the spectrum) of the 
total light of the system is blocked, so that the secondary eclipse is very 
shallow. There are many systems that have deep primary eclipses and shallow 
(sometimes barely even noticeable) secondary eclipses and we refer to these 
systems collectively as the Algols. Algols make up a relatively large class 
of eclipsing binaries not because there are a disproportionately large 
number of them in space, but because their very deep primary eclipses make 
them easy to discover. Inspection of the computer generated pictures shows 
that the Algol secondary is quite distorted while the primary is virtually 
spherical. This means that the primary, although roughly the same size as 
the secondary, must be much smaller than its Roche lobe, because tidal 
distortions are visible only when a star is large compared to the lobe. 
Therefore we conclude that the primary must have a larger Roche lobe than 
the secondary and the only way for this to happen is for the primary to be 
more massive than the secondary. But how can the primary, which is on the 
main sequence, be more massive than the secondary, which has already evolved 
away from the main sequence? Stellar evolution theory tells us that high 
mass stars evolve more quickly than low mass stars. Is stellar evolution 
theory wrong, or is there another explanation to this dilemma (known as the 
Algol paradox)? The resolution of the Algol paradox came from the work of J. 
Crawford, F. Hoyle, D.Morton, B. Paczynski, and others who showed that the 
Algol secondary was once the more massive star in the system. Being the more 
massive star it evolved first and expanded to fill its Roche lobe, 
transfering mass to the star that is now the primary. Mass transfer occured 
rapidly and as time went on the originally less massive primary became the 
more massive star. After the mass ratio reversed, the rate of mass transfer 
slowed considerably and the system found itself in the state we now see.

   What happens if both stars fill their Roche lobes? In this situation it 
is possible for both of the stars to be bigger than the lobes and we have an 
overcontact binary. (In this case, the overflow is clogged and the water can 
rise above the level of the overflow hole.) In an overcontact system both 
stars exchange mass and energy through a connecting neck. Most overcontact 
binaries have a thin connecting neck, that is, the components are just a 
little bigger than their Roche lobes, as in the case of the prototype for 
overcontact systems, W UMa. A few of them, such as DK Cyg, are connected by 
a much thicker neck. Notice the nearly equal eclipse depths, which is 
typical of the light curves of W UMa systems.

   When he extended the WD model to include non-synchronous rotation, Wilson 
discovered that another type of binary is possible and he called it a double 
contact system. In such a system both components fill their limiting lobes, 
but are not in contact with one another. This situation can occur if one of 
the stars rotates faster than synchronously. Systems where one of the stars 
is rapidly transfering matter to the other (systems that will eventually 
become Algols) provide a means by which one star can fill its Roche lobe and 
the other can rotate faster than synchronously. The matter falls from the L1 
point and may impact the surface of the mass-gaining star and cause the star 
to spin faster, much like squirting water on a pinwheel to make it spin. The 
systems RZ Sct, V356 Sgr, and Beta Lyr are, most likely, double contact 
systems. Notice the lenticular shape of the RZ Sct primary which arises from 
its rapid rotation, nearly seven times faster than the synchronous rate.

   Now let us look more carefully at the illustrated light curves. For DK 
Cyg and RS Cha examine the light curves between eclipses. In RS Cha this 
part of the light curve is relatively flat, while for DK Cyg brightness 
variation is evident even when no eclipses are occuring. The system is 
bright when the two stars are seen broadside and  dim when seen along the 
elongated ends. This variation in brightness is caused by the tidal 
distortion of the stars and is called ellipsoidal variation. In some 
systems, such as HDE 226868 (Cygnus X-1), there are no eclipses and the 
brightness variations are due solely to ellipsoidal variation.

   Another feature of some light curves is that the secondary eclipse seems 
to be at the top of a hill, such as in Algol and RZ Sct. Known as the 
reflection effect, this feature arises because the side of the secondary 
star which faces the primary star is heated by the radiation of the primary. 
Since one side of the star is hotter than the other, it is brighter and it 
is this hotter and brighter side that we are looking at during secondary 
eclipse. Of course, the secondary also heats up the primary but since Algol 
secondaries are much cooler than the primaries, the effect of the secondary 
on the primary is very small. If the temperatures of the two stars are 
nearly the same, the two stars heat each other by the same amount.

   Just as with solar eclipses, binary eclipses can be total, annular, or 
partial. In a total eclipse one star completely covers the other and during 
this time the brightness of the system remains constant. The secondary 
eclipse of DK Cyg is total, as can be seen in the pictures and in the flat 
bottoms of the secondary eclipses. The primary eclipse of DK Cyg is annular 
-- one star is contained within the disk of the other. If the brightness of 
a star were constant across the disk of then annular eclipses would also 
have flat bottoms, but stars exhibit a phenomenon known as limb darkening 
whereby the edge of the disk appears darker than the center because we are 
looking at higher and cooler (thus dimmer) layers of the star. Because of 
limb darkening, annular eclipses have rounded bottoms. In the infrared (top) 
light curve of DK Cyg the primary eclipse has a flat bottom because the limb 
darkening in the infrared is very small, but in the other two the eclipse 
has a round bottom because limb darkening is greater in the visible and 
ultraviolet parts of the spectrum. All of the other systems have partial 
eclipses, but W UMa is very close to having a total primary eclipse.

   We have now discussed all of the possible morphological types of binaries 
and seen how their shapes and light curves differ. Eclipsing binaries 
provide us with a very effective way of determining the sizes, shapes, 
temperatures, and other properties of stars, but there are many systems for 
which we do not have adequate light curves. With the recent development of 
quality photometers at modest costs, observations of eclipsing binaries will 
provide yet another way for amateur astronomers to make significant 
contributions to the study of the sky.