## Determination of binary parameters

If we are interested in determining the masses and radii of stars, then we have to turn almost right away to binary stars, since it is only by measuring orbital motion under gravity, and by measuring the shape and depth of eclipses, that we are able to determine these quantities to a good accuracy - one or two per cent in favourable cases; see Hilditch (2001). Analysis of the spectrum of an isolated star can determine such useful quantities as the star's surface temperature, gravity and composition. This is done by comparing the observed spectrum, preferably not just in the visible region of wavelengths but also in the ultraviolet (UV) and infrared (IR), with a grid of computed spectra for a range of temperatures, gravities and compositions. However, we do not get a mass from this process, or a radius, only the combination that gives the gravity - except in the special case of white dwarfs, where there is expected to be a tight radius-mass relation (Section 2.3.2) so that both mass and radius are functions only of gravity.

If we have an accurate parallax, as from the Hipparcos satellite, we can get closer to determining the mass of an isolated star, because the distance, the temperature (from spectral analysis), and the apparent brightness give us the radius; and hence the gravity (also from spectral fitting) gives us the mass. However, even if the parallax is good to ~1%, the gravity is much less accurate, because spectra are usually nothing like so sensitive to gravity as they are to temperature. Perhaps an accuracy of ~25% is achievable.

The parameters of binary systems are generally obtained from astrometric, or spectro-scopic, or photometric observations, and in favourable cases by a combination of two, or even all three, of these methods. Note that terms such as 'astrometric' and 'photometric', coined originally to refer to observations in the visible portion of the electromagnetic spectrum, are now generally used to cover all parts of the spectrum, for instance radio and X-rays. If the two components of a binary are so far apart in the sky as to be resolvable from each other, which means at visual wavelengths more than ~0.1" (0.5 iirad) apart, then the system is a 'visual binary' or 'VB', and careful astrometry, sometimes over a century or more, can reveal the orbit. Visual binaries tend to have long periods because short-period orbits are generally not resolvable. Only for systems within ~5 pc of the Sun (about 50 in number) could a separation of 0.2" correspond to a period of year. The upper limit of well-determined