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How do we estimate the mass of a single star? I guess we need the luminosity the surface temperature, radius, distance, etc. But we know nothing about the reality, because we can measure the real gravitational forces by only a single star: the Sun. How can we know that the models we've created are good?

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Estimating the mass of a "single star" can be a very difficult task, though perhaps your question is too pessimistic.

There are a number of suggested relationships linking the mass of a star to its luminosity. These can of course come from stellar evolution models, but they can also be empirically calibrated using stars in resolved binary systems of known distance, where Kepler's laws can be used to measure the masses of both components as well as their orientation. Another possibility is close, eclipsing binaries, where the inclinations, radii and masses of both components can be found. Of course to estimate the luminosity of any star requires a precisely known distance (often not known) as well as measurements of brightness, preferably in several wavelength ranges and a spectral type, so that one can account for any extinction by the interstellar medium.

Examples of this approach and these calibrated relationships include the well-known papers by Delfosse et al. http://adsabs.harvard.edu/abs/2000A%26A...364..217D See also http://en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation

This can work because stars spend most of their lives on the main sequence, where their luminosity evolves only slowly with time. There is a "sweet spot" between about 0.1 and 0.9$M_{\odot}$ where this method can work well. At higher masses, many stars may have evolved away from the main sequence and so the relationship between luminosity and mass becomes age dependent. Measuring the ages of stars is as difficult (if not more so) than measuring their masses, so this presents a problem. Often the only solution is to obtain a model-dependent mass by studying the star's position on a Hertzsprung-Russell diagram and attempting to identify a particular "mass track" of an evolving star that matches the luminosity and temperature of the star in question. Unfortunately, as well as the model-dependence (different models, containing different physics yield different masses), there are also problems of degeneracy where two mass tracks can cross. In addition the tracks can depend on the chemical composition of the star.

There are also major problems at very low masses. There are a lack of well studied calibrating binary systems, but more importantly, very low-mass objects also have an age-dependent mass-luminosity relation because they may still be contracting towards the main sequence (or just cooling in the case of brown dwarfs). The models here are very uncertain and mass tracks lie close together in the HR diagram. So estimating masses for very low-mass objects could be uncertain by a factor of two. http://adsabs.harvard.edu/abs/2012EAS....57...45J

There are other ways to verify and test mass calibrations, depending on what other information is available. For instance if the radius of a star is known, perhaps because it is close enough to have an interferometrically measured radius or estimated from $L/T_{\rm eff}^4$, then model stellar atmospheres can make predictions about what spectral absorption lines might look like at different gravities. i.e One can estimate surface gravity from the spectrum and then estimate mass from the known radius. Unfortunately, estimates of gravity are not usually precise enough for this to give meaningful constraints, though white dwarf masses are routinely estimated in this way. In white dwarfs of known radial velocity (admittedly usually in binary systems), the masses can also be estimated from the gravitational redshift of absorption lines, but this can be useful for calibrating other relationships.

An emerging hope is that the technique of asteroseismology - studying the pulsation of stars - will directly yield masses of calibrate some of the other empirical relationships. New, precise photometry from satellites such as Kepler have begun to make such estimates possible for solar-type stars and red giants (e.g see http://adsabs.harvard.edu/abs/2014ApJ...785L..28E ). There is still the need for detailed photometry and spectroscopy though, to deal with various degeneracies and composition dependence.

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A binary star system, which is quite common, will allow us to determine mass with great accuracy, using Kepler's Third Law of Planetary Motion which is as follows: $$ \frac{T^2}{r^3} = \frac{4 \pi^2}{GM_{sum}} $$ Using the orbital period, $T$, we can determine the acceleration and affect that stars have on one another to determine mass. Once we have the masses of these stars, we note their spectroscopic characteristics like luminosity, metallicity, temperature, radius, composition, etc...

Using the Hertzsprung-Russell Diagram, we can plot a star with known characteristics to roughly estimate its mass.

Hertzsprung-Russell Diagram

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    $\begingroup$ "In the full formulation under Newton's laws of motion, M should be replaced by M+m, where m is the mass of the orbiting body. Consequently, the proportionality constant is not truly the same for each planet. Nevertheless, given that m is so small relative to M for planets in our solar system, the approximation is good in the original setting." - according to your link. Since we are talking about 2 stars as you mentioned and not 2 planets, we have to use the m + M form. Now we have 2 variables and one equation, and so we cannot determine the mass. Correct me if I am wrong... $\endgroup$ – inf3rno Oct 17 '14 at 21:50
  • $\begingroup$ yeap, it is a little more complex than that: imagine.gsfc.nasa.gov/YBA/cyg-X1-mass/binary-formula.html $\endgroup$ – Wolphram jonny Oct 17 '14 at 22:05
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    $\begingroup$ @inf3rno For a spectroscopic binary system, the orbits of both stars around their centre of mass can be determined, so that both their masses can be derived (or more precisely, a mass range, due to the unknown inclination of their orbital plane) $\endgroup$ – Pulsar Oct 17 '14 at 22:06
  • $\begingroup$ One possibility: if we would know the distance between the Sun and the center of the galaxy and the orbital period, then we could estimate the mass of that super massive black hole, and after that the other stars... After that we could add the masses to this Hertzsprung-Russell diagram. $\endgroup$ – inf3rno Oct 17 '14 at 22:06
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    $\begingroup$ For visual binary systems (i.e. resolved orbits), both the masses can be determined as well as the orientation of the orbit. An accurate distance is required. The problem is finding systems that are far enough apart to be resolved, but close enough together to show appreciable orbital motion in decades or less. Particularly problematic for low-mass systems. Eclipsing binaries also have well-estimated orbital inclinations and so the masses of both components can be measured accurately. These two types of object are the "gold-standard" for calibrating other relationships. $\endgroup$ – Rob Jeffries Oct 17 '14 at 22:53

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