Exoplanet systems / Binary Stars

Inclination angles

Inclination is the angle between the line of sight and the normal of the plane of orbit

• An inclination of 0° is a face-on orbit, meaning the plane of the exoplanet's orbit is perpendicular to the line of sight with Earth.
• An inclination of 90° is an edge-on orbit, meaning the plane of the exoplanet's orbit is parallel to the line of sight with Earth.

Transits

Usually during exoplanet transits, the velocity vector is assumed to not change. However there are exceptions (hot Jupiters). In that case trigo must be used.

Also, transits are a special kind of occultation where the object is not obscured fully

Transit curves

Same area is obscured during primary and secondary minimum

• Reduction of brightness during primary and secondary minimum can be used to determine the ratio of surface temperatures. Also, larger dip occurs when the hotter star is obscured

Duration of transit can be used with the orbital velocity/radius (obtained from doppler spectroscopy) to determine radii of the stars

Periodicity of curve determines orbital period

Difference in duration of complete eclipse and partial eclipse (1st contact to 4th contact vs 2nd contact to 3rd contact) gives the ratio of the radii

Period and orbital velocity can determine orbital radius using d=s/t

Doppler shifts

Velocity of host star can be approximated as $v=\frac{m}{M}\sqrt{\frac{GM}{r}}$ (for exoplanets)

Galaxies/star clusters

Tully-Fisher relation (for spirals/lenticulars): $L \propto v^4$, also $L \propto m$ for given class of galaxy

Also can relate emission line width to asymptotic rotation velocity (non-relativistic): $v=\frac{c \Delta \lambda}{2 \lambda}$