The sizes of the two Roche lobes depends on the mass and separations of the orbiting systems.This figure shows the Roche lobes for three mass combinations.The masses are the best estimates for the mass in three observed close binary systems.
which must be perpendicular to the surfaces. The equipotential surfaces must therefore also be surfaces of constant density. (Again, this is analogous to the situation on Earth. By the equation of hydrostatic equilibrium, we can only have a pressure gradient in the direction of the gravitation force. That would be perpendicular to the potential lines.)
In discussing the evolution of close binary systems, we divide them into three classes (Fig. 12.3): (1) detached, in which each star is totally contained within its own Roche lobe; (2) semidetached, in which the photosphere of one star exactly fills its side of the Roche lobe; and (3) contact binaries, in which both stars are at or over the Roche lobe.
So far in this book, we have discussed the evolution of isolated stars. However, we have already seen that approximately half of all stars are in binary systems. If the binaries are completely detached, and there is no mass transfer, then the evolution will not be altered by the presence of the companion. However, mass transfer in semidetached or contact systems can influence stellar evolution.
In general, the more massive star in a binary system will evolve off the main sequence first. When that star becomes a red giant, it may become large enough to fill its Roche lobe. In that case mass transfer to the companion will take place. This can alter the evolution of the companion. The degree to which it alters the evolution depends on the nature of the companion. As the more massive star continues to lose mass, its Roche lobe shrinks, but the Roche lobe for the companion grows. This means that mass transfer
will take place until the masses of the stars are about equal. Some slow mass transfer may continue after that point.
At some point the star that was losing mass will become a white dwarf or some other collapsed object. In the following sections we look at examples of each type of collapsed object that we have discussed - white dwarf, neutron star and black hole.
We first consider systems in which the first star to evolve off the main sequence becomes a white dwarf. Eventually, the white dwarf s companion goes through its evolution off the main sequence. The companion becomes a red giant and fills its Roche lobe. At this point mass transfer starts back in the other direction from the original mass transfer. Now mass is falling in on the white dwarf. Not all the infalling matter strikes the white dwarf. Because of its angular momentum, some of the material goes into orbit around the white dwarf. This orbiting material forms a disk, called an accretion disk (Fig. 12.4). The disk forms because material can fall parallel to the axis of rotation but not perpendicular to that axis. (We will discuss disk formation in more detail in Chapter 15.)
As material falls in, its potential energy decreases, so its kinetic energy increases. The increase in kinetic energy will not equal the potential energy decrease, because some energy will be radiated away. We can expect roughly half of the change in potential energy to show up as kinetic energy. But this increase in kinetic energy will not produce much radiation on its own. This is where the accretion disk helps. As the faster moving gas strikes the accretion disk, it slows down, but its temperature increases. The now heated gas can then radiate.
Artist's conception of mass transfer leading to an accretion disk. [STScI/NASA]
We can estimate the energy available from the change in potential energy as mass falls in. If the mass starts at distance r1 from the white dwarf, and ends up a distance r2 from it, the luminosity is given by (see Problem 12.5)
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