6.3 Eclipsing binaries and spectroscopic binaries

Binary star systems offer a unique window into stellar physics. By observing how two stars orbit each other, astronomers can determine crucial properties like mass and size. These systems come in various types, from widely separated pairs to those so close they share atmospheres.

Analyzing binary stars involves studying their light curves and spectra. Eclipsing binaries reveal size and temperature information through periodic dips in brightness. Spectroscopic binaries show orbital motion through Doppler shifts in spectral lines. Together, these methods unlock the secrets of stellar evolution and structure.

Binary Star Systems

Geometry of eclipsing binaries

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• Two stars orbit common center of mass with orbital plane aligned with Earth's line of sight
• Total brightness varies over time with periodic dips due to eclipses
• Primary eclipse occurs when brighter star obscured causing deeper dip in light curve
• Secondary eclipse happens when fainter star obscured resulting in shallower dip
• Light curve shapes differ for detached binaries (flat regions between eclipses) and contact binaries (continuous variation from tidal distortion)
• Eclipse duration depends on stellar sizes and orbital parameters (semimajor axis, eccentricity)
• Eclipse depth relates to temperature and size differences between stars (hot Jupiter vs Sun-like star)

Properties from binary light curves

• Analyze light curves using photometric measurements and period determination techniques
• Estimate stellar radii from eclipse duration and orbital velocity
• Determine mass ratio through relative eclipse depths and spectroscopic data
• Estimate temperatures using eclipse depth ratios and color information (B-V index)
• Calculate orbital inclination from eclipse shapes and durations
• Account for limb darkening effects on light curve shape during ingress and egress
• Consider reflection effects causing brightening of facing stellar surfaces
• Analyze ellipsoidal variations from tidal distortion in close binaries (Algol system)

Principles of spectroscopic detection

• Detect binary systems through spectral line shifts caused by Doppler effect
• Measure radial velocities from periodic changes in line-of-sight velocity
• Classify as single-lined (SB1) or double-lined (SB2) spectroscopic binaries based on visible spectral lines
• Require high-resolution spectrographs to detect small velocity changes (m/s precision)
• Plot radial velocity curve (velocity vs time) revealing sinusoidal shape for circular orbits
• Determine orbital period from time between successive velocity maxima or minima
• Identify non-circular orbits from deviations in radial velocity curve shape

Mass determination in spectroscopic binaries

• Apply mass function formula: $f(M) = \frac{(M_2 \sin i)^3}{(M_1 + M_2)^2} = \frac{P K_1^3}{2\pi G}$
• Analyze radial velocity curve to determine $K_1$ and $P$ through curve fitting
• Estimate eccentricity from deviation of radial velocity curve from perfect sine wave
• Calculate semi-major axis using $a \sin i = \frac{K_1 P}{2\pi}$ for circular orbits
• Determine mass ratio for SB2 systems: $q = \frac{M_2}{M_1} = \frac{K_1}{K_2}$
• Calculate minimum mass $M_2 \sin i$ for SB1 systems
• Constrain inclination by combining with eclipsing binary data when available
• Estimate errors through propagation of measurement uncertainties (Monte Carlo simulations)