Final answer:
The apparent angular separation of two stars is the angle measured by an observer between each star and can vary from different locations. Planetary motion can affect this separation, as does atmospheric refraction on different planets. Distances between stars are determined using parallax, standard candles, or redshift measurements.
Step-by-step explanation:
Apparent Angular Separation
The apparent angular separation of two stars is the angle measured by an observer between lines of sight to each star. This separation can differ from different observation points due to the vast distances to stars, making the geometry sensitive to the observer's location in space. To determine if two stars are a visual binary system, measurements such as common proper motion, doppler shifts for radial velocity, and repeated observations over time to look for orbital motion, could be performed with a good observatory.
Impact of Planetary Motion and Atmospheres
Planetary motion can affect the apparent separation due to the change in the observer's point of view over time. This is particularly significant within a planetary system, where the relative positions of planets and the stars change with orbital motion. The atmospheres of the two planets could cause the light from the stars to refract differently, potentially altering the apparent separation depending on the atmospheric composition, density, and temperature.
Determining the Distance Between Stars
To determine the distance between stars, various methods are used, such as parallax measurements for relatively nearby stars. For stars farther away, astronomers may use standard candles like cepheid variables or red giants, or even the redshift of starlight for the most distant stars.