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We'll start this lesson with a short review of what we've been able to determine about stars from our observations of magnitudes, luminosities, parallaxes, spectra, radii. We'll find that the luminosities of stars have a HUGE range in values -- from 10-3 times that of the Sun to over 106 times that of the Sun! Over a range of 9 powers of 10. The radii of stars go from 1/1000 times the Sun to about 1000 times the radius of the Sun, 6 powers of 10. Temperatures of stars, on the other hand, range from a low of about 3000 K to maybe 60,000 K, or only a factor of 20. Why the differences? Aren't we missing something?
Stars like company, close company. Over 50% of the stars (some estimates say closer to 70% of the stars) in the sky are part of a binary or multiple star system. How do we know this? How do we find out whether or not two stars that look close together in the sky are actually gravitationally bound? Can some stars be so close together that even the most powerful telescopes cannot resolve them?
After completing this lesson, you should be able to
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We also classify stars by their evolutionary stage and their luminosity class.
That is, main sequence (V) (dwarf), subgiant (IV), giant (III), bright giant or
dim supergiant (II), and supergiant (I).
Spectral Types: O B A F G K M
Then subclasses: For example F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 G0 etc.
The spectral classifications range from O3 to M8.
Something is definitely missing. Let's say we are describing someone in our
class. We know where she is sitting, we know the color of her clothes, how far
away she is from her friends, her body temperature. But, we don't have the
full story! How tall is she? How much does she weigh? (Her age will have to
wait for the next chapter.)
We could never know the exact mass of a star if it were not for binaries. At least 50 per cent of the stars in the sky (and probably even more) are members of binary or multiple star systems. To determine masses, we must rely on our old friend, Newton's version of Kepler's third law:
but, if we express the period in years, the semi-major axes in AU's, and the masses in terms of our Sun's mass, then we can simplify it tremendously:
We now have the sum of the masses. We have, however, only one equation with two unknowns. We assume we can measure the distance each star is from the center of mass of the system, so that we know a1 and a2. We can call in another simple principle of physics: the teeter-totter principle.
The symbols a1 and a2 represent the distances
of each star to the center of mass of the system. We now have two equations with
two unknowns.
Wait a minute? How did we measure the distances? (Think: parallax!)
Before you continue with your study of visual binaries, take some time to review just how orbits work. What happens when you vary the eccentricities of orbits? the masses of the stars? the distances they are apart? What does it mean when we talk about "center of mass"? Start with this simple example of Kepler's Third Law and watch how the period changes as you vary the mass of the primary star and the distance of the secondary star.
We can observe visual binaries over a number of years and watch them as
they orbit the center of mass. We can then determine the angular
separations. Then, we use something very powerful, the small-angle formula.
When angles are extremely small, then the sine and tangent of the angle are
approximately equal to the angle itself. (If you don't believe this, put in a
small angle, in radians, in your calculator and take the sine and tangent of
it.) Using the small angle formula, we determine the distance to the star, we
measure the angular size of the separation, and then use the formula:
We can also calculate the semimajor axis in those cases where we can measure the
orbital velocity, v. We can, without too much pain, assume the orbit
is a circle of radius a, making the circumference of the orbit
2*pi*a. We have measured the period, p. We know that the
velocity of the orbit is equal to the distance traveled in one orbit divided by
the time it took it to orbit (p). That's 2*pi*a/p.
Then, simply solve for a:
[semi-major axis] =
[period*velocity]/[2*pi].
What if the stars are too close together or too far away to see the individual
stars? Then we must rely on another procedure to unravel the system. We go to
our telescopes with their spectrographs and obtain spectra of the stars. We
monitor the stars over time and watch the absorption lines shift from blue to
red (red to blue). We get the motion along the line of sight for the two stars.
We must be able to see the absorption lines from both of the stars
(double-lined spectroscopic binary) for this to work. By watching the relative
shifts of the two sets of absorption lines, we get an idea of the ratios of the
masses of the two stars -- the relative velocities of the two stars
around their center of mass are inversely proportional to their
relative masses. Check out this
simulation of a spectroscopic binary and confirm this for yourself.
The Doppler effect gives motion only along our line of sight (radial motion).
We really cannot know the exact inclination angle unless the binary system
eclipses. Again, to see for yourself, work a few parameters using the
Eclipsing Binaries simulation.
The light curve from eclipsing binaries also gives us information about the radii of the two stars. We know the velocity of the stars, and we know how long the eclipses last: # of kilometers/second * # of seconds == # of kilometers. Astronomers have also been able to measure the angular diameters of a few 10's of stars that lie along the ecliptic by timing occultations of the stars by the Moon. A technique known as "speckle interferometry" (look it up!) has also yielded a few more angular diameters. Of all the stars, only Betelgeuse is large enough for us to actually see the disk of the star.
From the text: