## Spectroscopic Binaries

The spectroscopic binaries (Fig. 9.5) appear as single stars in even the most powerful telescopes, but their

spectra show a regular variation. The first spectroscopic binary was discovered in the 1880's, when it was found that the spectral lines of Z UMa or Mizar split into two at regular intervals.

The Doppler shift of a spectral line is directly proportional to the radial velocity. Thus the separation of the spectral lines is largest when one component is directly approaching and the other is receding from the observer. The period of the variation is the orbital period of the stars. Unfortunately, there is no general way of determining the position of the orbit in space. The observed velocity v is related to the true velocity v0 according to v = v0 sin i,

where the inclination i is the angle between the line of sight and the normal of the orbital plane.

Consider a binary where the components move in circular orbits about the centre of mass. Let the radii of the orbits be a1 and a2. From the definition of the centre of mass m1a1 = m2a2, and writing a = a1 + a2, one obtains am2

The true orbital velocity is

where P is the orbital period. The observed orbital velocity according to (9.3) is thus vi =

2nal sin i

Fig. 9.5. Spectrum of the spectroscopic binary k Arietis. In the upper spectrum the spectral lines are single, in the lower one doubled. (Lick Observatory)

Substituting (9.4), one obtains

Solving for a and substituting it in Kepler's third law, one obtains the mass function equation:

If one component in a spectroscopic binary is so faint that its spectral lines cannot be observed, only P and v1 are observed. Equation (9.6) then gives the value of the mass function, which is the expression on the left-hand side. Neither the masses of the components nor the total mass can be determined. If the spectral lines of both components can be observed, v2 is also known. Then (9.5) gives

and furthermore the definition of the centre of mass gives mi =

m2V2 vi

When this is substituted in (9.6), the value of m2 sin3 i, and correspondingly, mi sin3 i, can be determined. However, the actual masses cannot be found without knowing the inclination.

The size of the binary orbit (the semimajor axis a) is obtained from (9.5) apart from a factor sin i. In general the orbits of binary stars are not circular and the preceding expressions cannot be applied as they stand. For an eccentric orbit, the shape of the velocity variation departs more and more from a simple sine curve as the eccentricity increases. From the shape of the velocity variation, both the eccentricity and the longitude of the periastron can be determined. Knowing these, the mass function or the individual masses can again be determined to within a factor sin3 i.

From accurate studies of the spectra of nearby stars, several planet-sized companions have been found. In the years 1995-2002, about 100 extrasolar planets were observed, with masses in the range of 0.1 up to 13 Jupiter masses.

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