The following has been generously provided by Leos Ondra (ondra@bm.cesnet.cz) from the Czech Republic. Leos is an amateur astronomer of extraodinary skill and talent. His web site http://www.bm.cesnet.cz/~ondra is a must-visit for anyone who has looked upward to the night sky.

Despite bright future of astroseismology, today it's impossible to determine mass of a star unless it orbits another one in a suitable binary system. But once you have all the essential ingredients, prepared in advance by top professional observers, weighing stars may be as easy as weighing apples.

Capella, the brightest star of the winter constellation Auriga the Charioteer, is the best imaginable example. First, it is a double-lined spectroscopic binary, that is, absorption lines of both components are present in the spectrum so you can simply find the ratio between the components' masses. Second, its orbit is perfectly circular which makes the task particularly easy. Third, the orbit is measured with amazing accuracy by means of long-baseline interferometry. This adds the other piece of valuable information, namely inclination of the orbit.

The first of the enclosed diagrams shows radial velocity curves. While the blue dots represent data for the cooler star of the binary, the yellow G8 III giant, the red symbols belong to the other components, the G1 III giant. Using the scale of the vertical axis, calibrated in kilometers per second, you can easily get semi-amplitudes K₁ and K₂ of these perfect sinusoids. The ratio K₁/K₂ equals inverse ratio between the components' masses, M₂/M₁.

The radial velocity curves of Capella's components are perfect sinusoids, proving that the orbit is circular. Adapted with permission from a recent paper by D.J. Barlow, F.C. Fekel and C.D. Scarfe, PASP 105, 476, 1993. Based on measurements made by the authors at the McDonald Observatory and the Kitt Peak National Observatory plus older data from the Dominion Astrophysical Observatory (by Batten et al., PASP 103, 623, 1991). Copyright 1993 Astronomical Society of the Pacific.

The other essential ingredient in weighing Capella is this orbit measured with the Mark III Long Baseline Optical Interferometer (located on Mt. Wilson near Los Angeles, California, and operated by the Remote Sensing Division of the Naval Research Laboratory with funding from the Office of Naval Research). Adapted with permission from Hummel et al., Astron. J. 107, 1859, 1994.
Astron. J. 107, 1859, 1994.

If we observed the orbital plane of Capella exactly edge-on, that is, (inclination i = 90 degrees), the semi-amplitudes K1 and K2 would also equal orbital velocities v1 and v2 of the stars around the common center of gravity. In the general case you have to apply the relation v1 = K1/sin(i) (and similarly for the other star). The great majority of double-lined spectroscopic binaries offer no way to find the inclination i and all you can calculate is the lower limit of individual masses. Well-behaved eclipsing binaries, where the inclination is always close to 90 degrees and its exact value can be found from light curve analysis, are a rare exception. Bright and nearby, Capella provides us with another possibility to exploit optical interferometry. The second diagram shows the relative orbit measured with the Mark III. The ratio between the axes of this neat ellipse (note it spans only about 0.1 arcsecond and the giants are drawn to scale!) equals cos(i).

Having now the true orbital velocities v1 and v2, simply multiply them by the orbital period P (104.0233 days) to get circumferences and thus radii r1 and r2 of the orbits around the binary's center of gravity. The sum r1+r2 of the radii is nothing but the major semi-axis a of Kepler's third law

M₁+M₂ = 4pi²a³/P²G

where M1+M2 is the total mass of Capella, pi the well-known mathematical constant (3.141592...) and G the gravitational constant. Be sure to use correct units while inserting major semiaxis (meters) and period (seconds) into the equation. The gravitational constant expressed in the same system of units (SI) is 6.672E-11 N m²kg^-2. Then you get the total mass in kilograms (remember that one solar mass is 1.989E+30 kg).

Finally, you have to calculate the component masses when you know both their sum and ratio, but that is really an easy job. If you work carefully, it's possible to weigh Capella with error of only a few percent. Now have a look at the correct masses.

Weighing a star (Results)

Quantity	Value	Source
Semi-amplitude K₁ (cooler G8 III star)	26.05 ą 0.10 km s^-1	1
Semi-amplitude K₂ (hotter G1 III star)	27.40 ą 0.30 km s^-1	1
Ratio K₁/K₂ = M₂/M₁	0.9507	lab
Interferometric orbit major semiaxis A	56.47 ą 0.05 milliarcseconds	2
Interferometric orbit minor semiaxis B	41.42 milliarcseconds	lab
Cos(i) = B/A Sin(i)	0.7335 0.6797	lab lab
Inclination i	137.18 ą 0.05 °	2
Orbital velocity v₁ = K₁/sin(i)	38.33 km s^-1	lab
Orbital velocity v₂ = K₂/sin(i)	40.31 km s^-1	lab
Orbital period P	104.0233 ą 0.0008 days 8987613 ą 69 seconds	1
Orbit radius r₁ = v₁P/2pi	5.4822E+10 m	lab
Orbit radius r₂ = v₂P/2pi	5.7663E+10 m	lab
Major semiaxis a = r₁+r₂	1.1249E+11 m 0.7519 AU	lab
Total mass M₂+M₁ = 4pi²a³/P²G	1.0426E+31 kg 5.24 solar masses	lab
Mass M₁ (cooler G8 III star)	2.69 solar masses 2.69 ą 0.06	lab 2
Mass M₂ (hotter G1 III star)	2.55 solar masses 2.56 ą 0.04 solar masses	lab 2
Distance d = a[AU] / A[arcseconds]	13.3 parsecs 43.4 light years	lab

References:

[1] Barlow, D.J., Fekel, F.C. and Scarfe, C.D., PASP 105, 476, 1993

[2] Hummel, C.A. et al., Astron. J. 107, 1859, 1994