Asteroseismic studies of stellar binaries and multiple star systems primarily rely on photometry, which involves measuring the variations in a star's brightness over time. These brightness variations are caused by stellar oscillations, which can be analyzed to infer information about the star's properties.

Asteroseismic studies of binary and multiple star systems investigate various types of stellar oscillations, including radial pulsations (where the star expands and contracts along its radius), non-radial pulsations (where the star oscillates in specific patterns across its surface), and mixed-mode pulsations (a combination of radial and non-radial oscillations). Each type of oscillation provides unique information about the star's properties.

Determining the masses of stars in binary and multiple star systems through asteroseismology involves combining information from orbital periods, oscillation amplitudes, and oscillation frequencies. By analyzing these parameters together, astronomers can infer the masses of the individual stars in the system.

Asteroseismic techniques offer several advantages in studying binary and multiple star systems. They provide more accurate mass measurements, enable the detection of hidden companions that may not be visible through other methods, and allow astronomers to probe the internal structure and dynamics of stars.

The Kozai mechanism is the phenomenon where the orbital period of a binary star system changes due to the gravitational influence of a third star. This mechanism can cause the eccentricity of the binary orbit to vary over time, leading to changes in the orbital period.

Eclipsing binary light curves are particularly useful for studying tidally interacting binary star systems. By analyzing the variations in the light curve during eclipses, astronomers can infer information about the tidal interactions between the stars, such as the amount of mass transfer and the degree of synchronization between their orbits.

Asteroseismic studies of multiple star systems aim to achieve multiple goals, including determining the masses and radii of the individual stars, investigating the dynamical interactions between the stars, and understanding the formation and evolution of multiple star systems.

Asteroseismology can help in identifying stellar companions that are too faint to be detected directly by analyzing the variations in the star's radial velocity. The gravitational influence of the unseen companion causes the star to wobble, which can be detected through precise radial velocity measurements.

Asteroseismic studies of binary and multiple star systems face several challenges, including the faintness of the stars, the complexity of the stellar oscillations, and the difficulty in separating the signals from different stars. These challenges require careful data analysis and advanced modeling techniques to extract meaningful information.

Rotation splitting of oscillation frequencies is an asteroseismic technique used to study the internal rotation of stars in binary and multiple star systems. By analyzing the frequency differences between oscillation modes that are sensitive to different parts of the star's interior, astronomers can infer information about the star's rotation profile.

The primary reason for the observed frequency differences between oscillation modes in binary and multiple star systems is the gravitational influence of the companion stars. The gravitational interaction between the stars causes perturbations in the stellar oscillations, leading to frequency shifts that can be detected and analyzed.

Asteroseismic studies can contribute to understanding the formation and evolution of binary and multiple star systems by providing information about the initial conditions of the system, tracing the orbital evolution of the stars, and studying the mass transfer between the stars.

Magnetic Doppler imaging is an asteroseismic technique used to study the effects of magnetic fields on stellar oscillations. By analyzing the variations in the oscillation frequencies and amplitudes caused by the presence of magnetic fields, astronomers can infer information about the strength and distribution of magnetic fields in stars.

Asteroseismic studies of binary and multiple star systems face several limitations, including the limited number of observable stars, the difficulty in interpreting the complex oscillation patterns, and the lack of comprehensive theoretical models for binary and multiple star systems.