Analysis of Binary Star Observations

Terms and Symbols

• a – semi-major axis of orbit (km or arcsec)
• e – eccentricity of orbit (dimensionless)
• i – inclination of orbit (deg or rad)
• q – mass ratio of stars (dimensionless)
• w – longitude of periastron (deg or rad)
• P – orbital period (days)
• T₀ – time of periastron passage (JD)
• T₁,T₂ – temperatures of stars (K)
• F₁, F₂ – rotation rates of stars (dimensionless)
• V_g – center of mass radial velocity (km/sec)
• r₁, r₂ – relative radii of stars (R/a)
• L₁/L₂ – ratio of luminosities (dimensionless)

Spectroscopic Binaries

• Get a sin(i), e, w, T₀, P, V_g
• Get q (if double-lined) and some radius and rotation information (Rossiter Effect) if eclipsing

• a and i are perfectly correlated

• Gives absolute dimensions

Eclipsing Binaries

• Get e, w, i, P, T₀
• Get r (therefore shapes), L₁/L₂
• Get F, q in favorable cases
• No information on absolute dimensions

Analyzing Light Curves

1. Develop intuition about effects of various parameters on the light curve.

2. Use other types of observations to constrain parameters.

3. Do simultaneous solutions of multiple light curves (and radial velocity curves) where appropriate.

Information in Light Curves

1. Eclipse durations give information on relative radii of stars.

2. Ratio of eclipse depths gives the ratio of the surface brightnesses of the stars for circular orbits.

3. Depths of total-annular eclipses give ratio of radii.

4. Depth of a total eclipse gives ratio of monochromatic luminosities.

5. Displacement of secondary eclipse from phase 0.5 and difference in duration of eclipses give information on e and w.

6. Shapes of the eclipses give information on the geometry of the system and limb darkening.

7. Outside eclipse variations give information on star shapes and reflection effect.

Additional Parameters for Light Curves

1. x,y – monochromatic limb darkening coefficients
Use Van Hamme tables or my ld program which interpolates
the Van Hamme tables. For detailed reflection treatment,
you also need the bolometric coefficients.

2. A – bolometric albedo
Use 1.0 for radiative envelopes and 0.5 for convective ones

3. g – gravity brightening (darkening) exponent
Use 1.0 for radiative envelopes and 0.32 for convective ones

The Wilson-Devinney Code

Modes:

-1 – X-ray binaries, q_e constraint

0 – similar to RM model, luminosities and temperatures

uncoupled

1 - Overcontact with T₂ = f(T₁)

2 - Detached

3 - Overcontact, T₂ is adjustable

4 - Semidetached, primary fills Roche lobe

5 - Semidetached, secondary fills lobe (Algols)

6 - Double contact