Photometric Binary Stars

In the photometric binaries, a periodic variation in the total brightness is caused by the motions of the components in a double system. Usually the photometric binaries are eclipsing variables, where the brightness variations are due to the components passing in front of each other. A class of photometric binaries where there are no actual eclipses are the ellipsoidal variables. In these systems, at least one of the components has been distorted into an ellipsoidal shape by the tidal pull of the other one. At different phases of the orbit, the projected surface area of the distorted component varies. The surface temperature will also be lower at the ends of the tidal bulges. Together these factors cause a small variation in brightness.

The inclination of the orbit of an eclipsing binary must be very close to 90°. These are the only spectro-scopic binaries for which the inclination is known and thus the masses can be uniquely determined.

The variation of the magnitude of eclipsing variables as a function of time is called the lightcurve. According to the shape of the lightcurve, they are grouped into three main types: Algol, j Lyrae and W Ursae Majoris type (Fig. 9.6).

Algol Stars. The Algol-type eclipsing variables have been named after j Persei or Algol. During most of the period, the lightcurve is fairly constant. This corresponds to phases during which the stars are seen separate from each other and the total magnitude remains constant. There are two different minima in the lightcurve, one of which, the primary minimum, is usually much deeper than the other one. This is due to the brightness difference of the stars. When the larger star, which is usually a cool giant, eclipses the smaller and hotter component, there is a deep minimum in the lightcurve. When the small, bright star passes across the disc of the giant, the total magnitude of the system does not change by much.

The shape of the minima depends on whether the eclipses are partial or total. In a partial eclipse the lightcurve is smooth, since the brightness changes smoothly as the depth of the eclipse varies. In a total eclipse there is an interval during which one component is completely invisible. The total brightness is then constant and the lightcurve has a flat bottomed minimum.

Sun O

Sun O

Time in hours

Fig. 9.6. Typical lightcurves and schematic views of Algol, f Lyrae and W Ursae Majoris type binary systems. The size of the Sun is shown for comparison

Time in hours

Fig. 9.6. Typical lightcurves and schematic views of Algol, f Lyrae and W Ursae Majoris type binary systems. The size of the Sun is shown for comparison

The shape of the minima in Algol variables thus gives information on the inclination of the orbit.

The duration of the minima depends on the ratio of the stellar radii to the size of the orbit. If the star is also a spectroscopic binary, the true dimensions of the orbit can be obtained. In that case the masses and the size of the orbit, and thus also the radii can be determined without having to know the distance of the system.

¡3 Lyrae Stars. In the f Lyrae-type binaries, the total magnitude varies continuously. The stars are so close to each other that one of them has been pulled into ellipsoidal shape. Thus the brightness varies also outside the eclipses. The f Lyrae variables can be described as eclipsing ellipsoidal variables. In the f Lyrae system itself, one star has overfilled its Roche lobe (see Sect. 11.6) and is steadily losing mass to its companion. The mass transfer causes additional features in the lightcurve.

W UMa Stars. In W UMa stars, the lightcurve minima are almost identical, very round and broad. These are close binary systems where both components overfill their Roche lobes, forming a contact binary system.

The observed lightcurves of photometric binaries may contain many additional features that confuse the preceding classification.

- The shape of the star may be distorted by the tidal force of the companion. The star may be ellipsoidal or fill its Roche surface, in which case it becomes drop-like in shape.

- The limb darkening (Sects. 8.6 and 12.2) of the star may be considerable. If the radiation from the edges of the stellar disc is fainter than that from the centre, it will tend to round off the lightcurve.

- In elongated stars there is gravity darkening: the parts most distant from the centre are cooler and radiate less energy.

- There are also reflection phenomena in stars. If the stars are close together, they will heat the sides facing each other. The heated part of the surface will then be brighter.

- In systems with mass transfer, the material falling onto one of the components will change the surface temperature.

All these additional effects cause difficulties in interpreting the lightcurve. Usually one computes a theoretical model and the corresponding lightcurve, which is then compared with the observations. The model is varied until a satisfactory fit is obtained.

So far we have been concerned solely with the properties of binary systems in the optical domain. Recently many double systems that radiate strongly at other wavelengths have been discovered. Particularly interesting are the binary pulsars, where the velocity variation can be determined from radio observations. Many different types of binaries have also been discovered at X-ray wavelengths. These systems will be discussed in Chap. 14.

The binary stars are the only stars with accurately known masses. The masses for other stars are estimated from the mass-luminosity relation (Sect. 8.7), but this has to be calibrated by means of binary observations.

Time

Time

If the effective temperatures of the stars are TA and TB and their radius is R, their luminosities are given by

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