By engaging with all the videos within this series, you will effectively complete a full undergraduate course in astronomy, equipping yourself with the knowledge and skills necessary to navigate the night sky with confidence, learning all the basics and many advanced topics! We previously discussed binary stars, focusing on spectroscopic and eclipsing binaries. A planet was discovered orbiting Proxima Centauri, our closest stellar neighbor. Today, we’ll explore how we determine the mass of this distant planet and its star. To measure the mass and distance of Proxima b from its star, we use Newton’s law of gravitation. The gravitational force between two objects is proportional to their masses and inversely proportional to the square of the distance between them. This force also equals mass times acceleration, linking observed motions to masses. Using Newton’s version of Kepler’s third law, we relate the planet’s orbital period to the sum of the star’s and planet’s masses, assuming the planet’s mass is much smaller. This gives us a good approximation. For simplification, we assume the planet’s orbit is circular. The gravitational force needed to maintain this orbit equals the planet’s mass times the centripetal acceleration. This leads to the equation: the orbital velocity squared of the planet is proportional to the mass of the star divided by the orbital radius. In reality, both the star and the planet orbit their common center of mass. By balancing the moments, we derive that the product of the star’s mass and its orbital velocity equals the product of the planet’s mass and its orbital velocity. Using these linked equations, we establish relationships among the variables and measure the unknowns. To determine mass, we first measure the orbital period from the sinusoidal pattern in radial velocity data. The data’s peaks and troughs indicate the star’s radial speed. The star’s mass is inferred from its spectral type using established mass-luminosity relationships. Plugging measured data (orbital period, radial speed, and spectral type) into equations yields the distance between the star and the planet (orbital radius), the planet’s orbital speed, and the planet’s mass. For Proxima b, the data suggests it has about 1.25 times Earth’s mass and orbits 5% of an astronomical unit from Proxima Centauri, implying a close orbit. Assuming a similar density to Earth, Proxima b’s radius is estimated to be about 10% larger. This is based on the assumption that the planet, being close to its star, is likely rocky. Studying binary stars helps derive their masses and establish a crucial mass-luminosity relationship for understanding stellar evolution. This relationship shows that most visible stars are less massive and dimmer than our sun, with M-type and K-type stars being the most common. Despite the star’s violent flares, Proxima b orbits within its habitable zone, making it an interesting subject for study. Overall, the segment emphasizes clear definitions, underlying geometry, and practical observing guidance so viewers can connect the concept to the real sky.