Binary Systems

Guide

Binary Systems

Binary star systems can be classified according to their specific observational characteristics:

Optical double: These stars are not binaries and are not gravitationally bound, but simply two stars that lie along the same line of sight.
Visual binary: Both stars in the binary can be resolved independently, and it is possible to monitor the motion of each member.
Astrometric binary: If one of the members is significantly brighter than the other, both stars may not be observable directly. The existence of the unseen companion can be deduced by observing the oscillatory motion of the visible component.
Eclipsing binary: If the orbital plane is located along the line of sight ( $i \approx 90^\circ$ ), one star may periodically pass in front of the other, blocking the light of the eclipsed component. Regular dips in lightcurves sign that the observed system is an eclipsing binary. The dips in the light curve can further be used to estimate the inclination of the system. A grazing eclipse will have a sharp dip, whereas a partial eclipse will have a flat bottom. Using measurements of duration of eclipses, the radii of each member can also be found. The deeper (primary) minima occurs when the hotter of the two stars is eclipsed behind the cooler star. The other minima is called the secondary minima.
Spectroscopic binary: If a star has two superimposed, independent, and discernible spectra, it can be decomposed into the spectra of the associated binary stars. Due to opposite radial velocity, the two spectra will be oppositely Doppler shifted. These are also known as spectroscopic binaries.
Photometric binary: A periodic variation in the total brightness can be observed, caused by the motions of the component stars. Usually, these are eclipsing binaries, but can also be ellipsoidal variables—stars which have been distorted into an ellipse by the tidal pull of the other. According to the shape of the lightcurve, they can be grouped into three main types: Algol type, $\beta$ Lyrae type, and W Ursae Majoris type.

The mass function of a binary system inclined at an angle $i$ , with stars of mass $m_1$ and $m_2$ is defined as

\tag{4.3.1} \frac{m_2^3}{(m_1 + m_2)^2} \sin^3 i

As shown in equation (3.2.14), it is equal to

\boxed{ \frac{m_2^3}{(m_1 + m_2)^2} \sin^3 i = \frac{P}{2 \pi G} v_{1r}^3 }

Roche Lobe

Roche lobe is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. It is an approximately teardrop-shaped region bounded by a critical gravitational equipotential, with the apex of the teardrop pointing towards the other star star (the apex is at the $L_1$ Lagrangian point of the system).

Close to each star, surfaces of equal gravitational potential are approximately spherical and concentric with the nearer star. Far from the stellar system, the equipotentials are approximately ellipsoidal and elongated parallel to the axis joining the stellar centers. A critical equipotential intersects itself at the $L_1$ Lagrangian point of the system, forming a two-lobed figure-eight with one of the two stars at the center of each lobe. These lobes are called Roche lobes.

$L_1$ is the gravitational capture equilibrium point. It is a gravity cut-off point of the binary star system. It is at the lowest potential of all Lagrange points. When a star “exceeds its Roche lobe”, its surface extends out beyond its Roche lobe and the material which lies outside the Roche lobe can “fall off” into the other object’s Roche lobe via the first Lagrangian point. In binary evolution this is referred to as mass transfer via Roche-lobe overflow.

The Roche lobe is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. (Source: Wikipedia)

Consider a binary system with two stars of masses $M_1$ and $M_2$ , having orbital separation $a$ . Approximation of the lobe (of $M_1$ ) as a sphere of radius $r_1$ with mass ratio $q = M_1 / M_2$ is given by the Eggleton formula:

\tag{4.3.2} \frac{r_1}{a} = \frac{0.49 q^{2/3}}{0.6q^{2/3} + \ln (1 + q^{1/3})}

Exoplanets

Exoplanets are planets orbiting some other star, other than the Sun. The first exoplanet found in 1992 orbited a pulsar. The first exoplanet orbiting a normal star was found in 1995 around $\beta$ Pictoris. Rogue planets are ones which have escaped from planetary systems, and are wandering in space.

Detection

There are various methods for detecting exoplanets

Astrometric methods are based on perturbations of the proper motion or radial velocity of the star, by the planet. If the planet is large enough, these can be detected.
Periodic doppler shifts in star;s spectra, because of the changing radial velocity of the star, can be detected, and the existance of an exoplanet can be deduced.
A planet causes the distance of a star to vary slightly, which in the case of regularly variable stars, changes the periodically of the brightness variation of the star in a regular manner.
~200 exoplanets have been found by direct imaging of the system.
If a star passes in front of a background star, its gravity will bend the light, making the background star look brighter. If the star has exoplanets, the brightening is slightly stronger.
When a planet is between the star and the observer, it occults a part of the stellar disk, reducing its brightness. This event is called a transit. Periodic transits confirm the existance of an exoplanet. The transit depth is defined as

\tag{4.3.4} \text{Transit Depth} = \left( \frac{R_\text{planet}}{R_\text{star}} \right)^2

Stellar Classification Stellar Evolution