The stars Sirius, Capella, and Algol, which were used as examples in the previous lecture, are binary systems which are both apparently bright and intrinsically bright. However, intrinsically bright systems are atypical: to get an accurate idea of whether average stars are in binary systems, we need an accurate census of nearby stars.
Of the 33 nearest stars to the Earth, 16 are in binary systems. A more accurate survey yields:
A question to ponder: Why are so many stars born as twins?
The generalization of Kepler's third law to systems other than the Solar System:
MA + MB = a3 / P2
In the Solar System, where MA = mass of Sun = 1, and MB = mass of a planet << 1, Kepler's third law reduces to the familiar form ``P squared equals a cubed''
For a visual binary, measure the time taken for the stars to trace a complete ellipse on the sky.
For a spectroscopic binary, measure the time between maxima in the observed radial velocity.
For an eclipsing binary, measure the time between the large intensity dips (or equivalently, the time between the small intensity dips -- but the large dips are easier to recognize).
For a visual binary, you can measure the angular separation of the stars; if you also know the distance to the binary system, you can compute the physical separation of the stars.
If you know P & a for a system, the SUM of the masses can be computed from Kepler's third law.
For a visual binary, you can compute the RATIO of the masses (MA / MB) by measuring the relative distances of each star from the system's center of mass. (For instance, if the stars are equally distant from the center of mass, their masses must be identical.)
Knowing both the sum MA + MB and the ratio MA / MB, you can find the masses individually.
Example - the Sirius system:
Gosh! This means that Sirius B, though its radius is smaller than that of the Earth, is as massive as the Sun! (This means the density is about two TONS per cubic centimeter.)
Computed masses range from 0.08 solar masses to 50 solar masses. Masses vary systematically along the main sequence. The coolest, dimmest, smallest stars (M stars) have a mass of 0.08 solar masses; the hottest, brightest, biggest stars (O stars) have a mass of 50 solar masses.
Thus, just as temperature, luminosity, and radius vary systematically along the main sequence, so does mass. Main sequence stars HOTTER than the Sun are MORE LUMINOUS, LARGER, and MORE MASSIVE than the Sun. On the other hand, main sequence stars COOLER than the Sun are LESS LUMINOUS, SMALLER, and LESS MASSIVE than the Sun.
In the supergiant and giant regions of the H-R diagrams, stars of different masses are scrambled together. (That is, two supergiant stars or two giant stars with the same luminosity and temperature can have widely different masses).
The luminosity of a main sequence star depends VERY STRONGLY on its mass. In other words, if you change the mass of a star by only a little bit, you change its luminosity by a lot. (For math fans only: L = M3.5)
Example:
A plot of Luminosity (the vertical axis) versus Mass (the horizontal axis) is shown below.
Question to ponder: What determines these values? Why are temperatures limited to a relatively small range? Do stars have an internal `thermostat'? The lowest-mass star is still 80 times more massive than the planet Jupiter; do there exist objects of intermediate mass? If so, will they be stars or planets?
