Suzy B Software 2/Suzy B Software CD-ROM 2 (1994).iso/adult_ed/lectures/lect12a.txt

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 ----- 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.