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?
[Physics] How to estimate the mass of a star
astrophysicsmassstars
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If you have $L$ and you have $T$, then nothing more complicated than Stefan's law is required. If $T$ is the effective temperature of the star then this gives an exact answer.
$$ R = \left(\frac{L}{4\pi \sigma_B T^4}\right)^{1/2}$$, where $\sigma_B = 5.67\times 10^{-8}$ in SI units.
If on the other hand you are trying to solve the structure from first principles then you need to learn about polytropes and the solutions of the Lane-Emden equation. A star supported solely by radiation pressure can be treated as a $n=3$ polytrope, which has no analytic solution.
On p.155-162 of Clayton's "Principles of stellar evolution and nucleosynthesis" you can find a treatment using polytropes and some tables with solutions for various values of $n$. The radius of a star is $$ R = \left[ \frac{(n+1)K}{4\pi G}\right]^{1/2} \rho_c^{(1-n)/2n} \alpha_1,$$ where $\rho_c$ is the (here unknown) central density, $n=3$ and $K$ is the constant in the polytropic equation of state (the exact value of $K$ depends on what proportion of the gas pressure is due to radiation pressure) and for a $n=3$ polytrope $\alpha_1=6.9$.
The mass is given by $$ M = -4\pi \left[ \frac{(n+1)K}{4\pi G}\right]^{3/2} \rho_c^{(3-n)/2n} \alpha_1^2 \left(\frac{d\phi}{d\alpha}\right)_{\alpha_1},$$ where $-\alpha_1^2 (d\phi/d\alpha)_{\alpha_1} = 2.02$ for a $n=3$ polytrope.
In the standard model, the ratio of normal gas pressure to total pressure is $\beta$, such that $\beta=0$ for a star solely supported by radiation pressure. It can be shown that the mass of such a star is given by $$ M = 18 \frac{(1 - \beta)^{1/2}}{\mu^2 \beta^2} M_{\odot},$$ where $\mu$ is the mean number of mass units per particle. Thus if you know $M$ and the composition, this gives you $\beta$.
The value of $K$ is then given by $$ K = \left[ \frac{9N_0^{4} k_B^{4}c}{4\mu^4 \sigma_B} \frac{(1-\beta)}{\beta^4}\right]^{1/3}$$ and $N_0$ is Avogadro's number.
This value of $K$ enables you to derive $\rho_c$ from the second polytropic relation and then substitute this into the first polytropic relation to get $R$. Good luck!
I haven't seen the term 'apparent flux' before. Flux is always 'apparent' in the sense that it depends on the distance from you to the source. Your equation for flux received $A(f) = \frac{FR^2}{D^2}$ is only true if $F$ is the flux at the surface of the star.
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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.