I have recently read that we can only know the masses of stars in binary systems, because we use Kepler's third law to indirectly measure the mass. However, it is not hard to find measurements for the mass of stars not in binary systems. So how is the mass of these stars determined?
Astrophysics – How to Determine the Masses of Single Stars
astronomyastrophysicsmassstarsstellar-physics
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There are numerous distance indicators used for within the galaxy. The most common way is by using intrinsic magnitude. By knowing how bright an object would be if we were close, we can determine how far away it is by how dim it is. There are many types of stars where we have a rough idea of how bright they should be due to characteristics of the star:
Cephied Variables: The original type of variable star that was used by Hubble to determine the distance to the Andromeda Galaxy.
RR Lyrae Variable: Like the Cephied variable, but usually dimmer.
Type 1a Supernova: These guys, unlike the first two, are cataclismic variables. Essentially a binary white dwarf slowly accretes matter from its binary till it reaches the Chandrashankar Limit, after which point it explodes in a very characteristic way (since the mass at the time of explosion is roughly constant).
Main Sequence Stars: Generally less accurate than the first 3, there are some types of main sequence stars which are used to find distances in a similar way.
There are a few other ways we can measure distances:
Perpendicular Movement: For example there is a "light echo" from SN 1987A which is essentially light from the supernova interacting with dust around the old star. Since this echo should be expanding at the speed of light, we can tell how far away the nova is by the angular velocity of the light.
Relative Velocity in a Moving Cluster: (see dmckee's answer)
Tulley-Fisher relation: A relationship between the luminosity of the galaxy and it's apparent width. Can be used as a decent distance calculator.
Faber-Jackson Relation: Similar to Tulley-Fisher, relates luminosity with radial velocity dispersion rate.
EDIT: Some more information about redshifts.
The whole relationship between redshift and distance was in fact established by Hubble by relating distance to Cephied variables (I believe) with redshift. Later on it was made more precise using supernova, which are brighter and can be seen from much father away (I think recent supernova can be occasionally seen around Z=2, while Cephieds are all Z<1). Within a galaxy, redshift cannot be used directs since the "peculiar velocity," the velocity within the galaxy, completely overshadows the effects of universe expansion on which Hubble's Law is based. Redshift within the galaxy is useful for certain other techniques.
EDIT: corrected a few minor errors.
In general, yes you need to know the orbital inclination angle $i$ in order to fully solve the orbit. The radial velocity amplitude $K$ is just modified to $K \sin i$ (where $i=0$ is a face-on orbit). Combining this with the orbital period and Keplerian orbits gives you the "mass function" $$ \frac{M_1^3 \sin^3 i}{\left(M_1 + M_2\right)^2} = \frac{K_{2}^3 \sin^3 i\ P_{orb}}{2\pi G},$$ where the right hand side can be measured from radial velocity data in a spectroscopic binary. If you have a velocity amplitude for both stars, then there is a similar expression with the labels reversed. Without $i$ this can then only tell you the mass ratio $M_1/M_2$.
There are several ways to break this degeneracy depending on what kind of binary system it is.
In a visual binary system where you can observe the orbits, then the orbital path of both objects can be observed and the inclination of the orbit is directly measured. However, radial velocity amplitudes are not usually measurable (too small) and one relies on the absolute size of the orbit, which in turn requires a distance (parallax) estimate.
In an eclipsing binary, then the shape and depth of the eclipses can be uniquely solved to give the inclination and hence the masses of the individual stars.
In non-eclipsing close binary systems, or when one component is not seen, then ellipsoidal modulation of the seen component depends on the mass ratio and the inclination. Together with the radial velocity curve, this can then give unique masses for the components.
In general it is not possible to get any more than a mass ratio for the components of a double lined spectroscopic binary system (SB2), or the "mass function" (see above) of a single lined spectroscopic binary system (SB1).
To make further progress in these general cases you need an estimate of the primary mass. This can be done with reference to stellar evolutionary models. In principle, for an SB2, the mass ratio and the combined appearance of an object in the Hertzsprung-Russell diagram contain enough information to determine the masses of the individual components and the age of the system. In practice this is hard and there are degeneracies. A better way is to fit a combination of spectral type templates to the measured spectrum and hence estimate the spectral types and hence masses.
In an SB1 you really are stuck. The spectral type and position in the HR diagram give you $M_1$, but you will only have a lower limit to the unseen secondary mass. This is why it is difficult to estimate the masses of black holes in binaries - you need to know the inclination.
Best Answer
The Hertzsprung–Russell diagram is the key to determining masses of individual stars. For stars on the main sequence, their properties are essentially determined by their mass. Age and metallicity are also interrelated factors, but of considerably less importance than mass. That is, if you tell me the mass of a star on the main sequence, I can tell you its temperature, luminosity, radius, etc., to reasonably good accuracy. This means that if you are able to measure the luminosity and temperature of a star, I can put it on a Hertzsprung–Russell diagram, and tell you how massive it is. Of course, calibrating this relationship in the first place required measuring the masses of stars directly using stars in binary systems, as you mention.
[Edit: I did not notice that the star you linked to specifically was Arcturus, for which this does not directly apply.] For a giant like Arcturus, masses are often determined in a bit more complicated manner. The Hertzsprung–Russell diagram still provides a guide, in developing models of stellar evolution that can produce the observed patterns of non-main-sequence stars on the HR diagram. As Arcturus is no longer on the main sequence, these stellar evolution models are used to find the mass that produces the combination of temperature, luminosity, and radius observed, from which a mass can be inferred.