. Sketch the radial velocity curves of a binary consisting of two identical stars moving in circular orbits that are (a) perpendicular to and (b) parallel to our line of sight.

Binary star systems consist of two stars that are gravitationally bound and revolve around a common center of mass. There is a captivating variety within these systems; some pairs might be so close that they exchange matter, while others lie far apart but still follow the gravitational dance. Understanding these systems is essential for astronomers as they provide valuable information about star masses, size, and aspects of stellar evolution.

When we observe binary stars from Earth, their motion can create observable patterns in the light they emit. These patterns are useful in determining the properties of the systems. The exercise discussed here deals with sketching radial velocity curves, which offer insights into the movement of stars in a binary system relative to our point of view—a concept we'll delve deeper into with the discussion on the 'line of sight'.

Circular orbital motion refers to the movement of an object in a circular path around a center point due to the gravitational force exerted by another object. The importance of understanding this motion in astronomy cannot be overstated, as it pertains to planets, moons, and, as in our exercise, binary star systems.

In such a system, each star follows a circular orbit around the shared center of mass. The period of their orbits and their velocity can reveal much about the mass distribution in the system. When sketching radial velocity curves, the influence of circular motion is characterized by the smooth, regular oscillations that we see in the curves of stars with orbits perpendicular to our line of sight.

The line of sight describes an imaginary straight line along which an observer has unimpeded vision. In the context of our exercise, this concept is key to understanding why the motion of stars in a binary system can be detected differently.

For instance, if the stars' orbits are perpendicular to our line of sight, their motion towards or away from us changes their light's Doppler shift, producing visible changes in radial velocity. Conversely, if the orbits are parallel to our line of sight, this motion has no Doppler effect, since the stars never move directly towards or away from the observer. This concept ties directly to why the radial velocity would be a flat line for scenario (b) in the exercise.

Sinusoidal motion depicts the repetitive and smooth oscillations represented graphically as a sine wave. This type of motion is symmetrical and periodic, which means it completes a pattern within a regular interval, known as the period.

The scenario from our exercise where binary stars' circular orbits are perpendicular to our line of sight, results in a radial velocity graph with a sinusoidal shape. The stars' movement towards and away from us over time forms this characteristic wave because their speed, as projected along our line of sight, increases and decreases in a periodic fashion. The peaks and troughs of the curve represent maximum radial velocities where the star is moving directly towards or directly away from us.