BINARY STARS
In 1650 the Italian Jesuit Giovanni Riccioli observed that Mizar, one of two stars in the handle of Ursa Major long used as a test of visual acuity, itself had a very close companion. The odds of two stars of similar brightness appearing so close together is too small not to suspect that the stars are actually physically close together, not like the chance grouping of Mizar and the fainter more distant of the pair Alcor. In the following years many such double stars were discovered and by 1804 William Herschel discovered that the fainter component of Castor (Castor B) had moved relative to the brighter component (Castor A), the first proof that gravity worked outside the solar system and the stars were gravitationally bound, visual binary stars.

In 1889 Pickering found that the spectral lines of the brighter component of Mizar (Mizar A) were usually double and the spacing varied with a regular 104 day period, the first spectroscopic binary to be discovered.

Immediately after the discovery of the spectroscopic binary Mizar, Algol the demon star, beta Persei, was discovered to be a spectroscopic binary introducing the most important class of binary stars the eclipsing binary. Algol normally shines at V=2.09 but at intervals of 2d20h49m it fades to a third of its normal brightness then after a few hours returns to normal brightness.

Today we know that binary systems are quite common, over half the stars within 20 pc are in binary or even more complex systems.
(Of 100 nearby systems with a star like the sun, 57 are single, 38 double, 4 triple and one quadruple system.)

Observational Data

Visual binary star orbits are found by measuring the position angle (like azimuth) and separation in arcsec of the dimmer star usually called the secondary or "B" with respect to the primary or "A" over as long a time base as possible. This results in a relative orbit which will be an ellipse but not necessarily with the primary at one of the foci. This is because the orbit plane is likely to be inclined to the plane of the sky, but a simple transformation (rotation) yields the orbit inclination (and line of nodes). The only problem now is the system distance (1/parallax) since we need the semimajor axis of the relative orbit a in AU not arcsec (a"). If the distance r=1/p" is known, then a=ra" and we just apply (m1+m2)=(ra")3/p2 to get the sum of the masses.

If the rather more difficult astrometric task of measuring the true orbits is carried out then the motion about the center of mass can be measured and from the seesaw law m1r1=m2r2 we can get the mass ratio as well as the sum and, hence, the individual masses.

Visual binary drawbacks: You need the distance for the sum of the masses, the mass ratio is hard to get.

Spectroscopic binary star orbits are not measured directly but can be inferred from the spectral lines. Rather than an orbit we get a radial velocity curve, a graph of the radial velocity of both components as a function of time. Ideally this looks like two sine waves 180 degrees out of phase. The ratio of the velocity amplitudes gives the mass ratio right away, a difficult problem for the visual binaries. The period of the system is just the period of the radial velocity curves and since we have the period and velocity we have the size of the orbit

a=VP/2p
where V is in AU/year (just divide the amplitude in km/s by 4.74), and P is in years. (Did you know that the earth's orbital speed is just the circumference of the earth's orbit per year, that is, 2p AU per year? Multiply by 4.74 and you get 30 km/s.) Unfortunately, there is a BIG PROBLEM here because we are not at all likely to be in the binary orbit plane. There will be an orbital inclination i (i=90 for edge on orbit) so we really only get a component of the radial velocity V times sin(i) where i is completely unknown! So we really observe ao which is a times sin(i). This does not affect the mass ratio at all but the formula for the sum of the masses
(M1+M2)=[ao/sin(i)]3/P2
tells a sad story: The masses go as the cube of something we don't know. If we assume we are in the orbit plane so sin(i)=1 we only obtain a lower limit to the sum of the masses.

Spectroscopic binary drawbacks:
The mass ratio is easy but the sum of the masses is only a lower limit.

Spectroscopic binary problems:
Often the orbits are not circular. this complicates the analysis but the problem is quite tractable.
Some spectroscopic binary systems only show lines from one component (one might be too dim to see) so at best you get the mass function the ratio of the unseen component times sin3(i) divided by the sum of the masses. Eclipsing binary

Eclipsing binary stars are usually spectroscopic binaries but with a marvelous difference: In order for an eclipse to occur we must lie close to the binary orbital plane so sin(i) will be pretty close to unity. In addition to the radial velocity curve, we get a light curve that tells us a real story about the system! The figure is a model eclipsing binary system and below is a light curve and five "spectrograms". The first spectrum shows only the red giant spectrum, the blue dwarf is eclipsed. Next the blue dwarf has swung around the giant and is approaching and two new spectral lines appear. The middle shows both spectra and the giant is partially (annular) eclipsed (the bottom lines in the figure overlap. The light curve shown has an obvious primary and secondary minimum, and each minimum has an obvious points of (1) first contact, (2) totality, (3) end of totality, (4) last contact. Since we know the orbital speed, the eclipse duration (3)-(1) or (4)-(2) yields the radius so data such as the above yields the individual masses and radii of the two stars. But a careful analysis of the light curve can give a highly accurate value for the orbital inclination, orbital eccentricity, distribution of light across the stars (limb darkening, starspots), shape of the star (they are often so near to each other that they are tidally distorted) and whether one star might heat the other star's photosphere (hot spot). No doubt about it, a careful analysis of eclipsing binary data is well worth the effort and many astronomers devote a lifetime to the study of these systems.

But wait, there's more! You might think that an eccentric orbit with very small, perhaps contact periastron would make for a pretty messy system. You are correct, but anybody who has worked on eclipsing binary stars will tell you that this just makes the analysis more enjoyable. Why isn't this in the "drawbacks" section? Careful analysis indicates just how the stars distort near periastron and now for a special bonus for patient observers: the distortions cause the angle between the major axis (the line of apsides) of the orbit and the observer to slowly rotate and this observed apsidal motion can be compared with predictions by theoretical stellar models so giving an insight to the actual structure of stars. A real bonus.
See APSIDAL MOTION for more details.

Eclipsing binary drawbacks:
I can't think of any offhand. Too bad they don't announce their distance too!

Observational selection
Most stars are binaries and the separation ranges from contact binaries to hundreds of AU. The maximum separation is determined by tidal distortion by the galaxy and chance encounters with other stars. Consider two stars like the sun. If they are separated by one AU then their period is 0.707 years and their orbital speed is about p AU per .707 years or 21 km/s. Hence, the system will be a spectroscopic binary. But at a distance of 1pc the separation is one arcsec so it is just barely a visible binary as well. But if the system is more distant, the components cannot be resolved. The stars can be "seen" out to distance modulus ten or greater so such pairs or anything with less separation can only be seen spectroscopically. The farther apart the stars are the slower the orbital speed (it falls as the square root of separation) and most visual binaries have separations of order 100 AU and periods of hundreds of years.

So in general it is safe to say a star will either be a visual or a spectroscopic binary but not both. If a spectroscopic binary is resolved, and the inclination estimated, then both the masses and the distance are determined. As technology progresses, higher resolution both in the visual and the spectroscopic instrumentation may bring out a significant number of these binaries.

It is also pretty safe to say that eclipsing binary stars are also spectroscopic binaries even though distance is not a matter for such stars. For a star to be an eclipsing binary the odds are that the two stars are close together since the closer they are the farther off the orbital plane the observer can be and still enjoy an eclipse. Odds are that the period will be under a week though there are a few long period eclipsing binaries.