Final answer:
The student's question involves using Kepler's third law (option a) of planetary motion to determine the mass located within a star's orbit around the galactic center, often revealing the presence of a black hole.
Step-by-step explanation:
Understanding Orbital Dynamics and Mass Determination of Cosmic Objects
The question involves applying Kepler's third law of planetary motion which is crucial for understanding the dynamics of objects orbiting a central mass, such as a star orbiting the center of the galaxy. According to Kepler's third law, the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This can be adapted for stars orbiting a galactic center where the central mass is much more significant.
To determine the mass within a star's orbit around the galactic center, we use the modified form of Kepler's third law which relates the mass of the object to the orbital speed and radius of the orbit. You provided an example where a star is orbiting at a speed of 6200 km/s at a distance of 20 light-hours from the galactic center. Applying Kepler's third law, we could calculate the mass located inside its orbit, typically revealing a mass indicative of a central black hole.
We also briefly mention the application of the Doppler effect which helps measure the radial velocities of stars or gas in galaxies by observing shifts in their spectral lines. This is important for determining the speed of these objects as they move towards or away from us. Finally, we have Hubble's law, which relates the recession velocity of galaxies to their distance from Earth, but it is not directly relevant to the question of orbital dynamics around a galactic center.