Final answer:
The velocity of the star relative to the Sun, measured perpendicular to the line of sight, is approximately 47.4 kilometers per second. It is calculated using the star's proper motion and its distance of 10 parsecs.
Step-by-step explanation:
To calculate the velocity of a star relative to the Sun, perpendicular to the line of sight, we use its proper motion and distance. The star in question is 10 parsecs away and has a proper motion of 10 arcseconds per year. Since 1 parsec corresponds to a parallax of 0.1 arcsecond, and proper motion is measured in arcseconds, we can use these values to determine the star's transverse velocity.
First, we convert the angular motion from arcseconds to radians: 10 arcseconds * (1/3600 degrees/arcsecond) * (π/180 radians/degree) = 4.8481 x 10-5 radians/year. Then, because 1 parsec = 3.086 x 1016 meters, we can calculate the actual velocity: velocity = 4.8481 x 10-5 radians/year * 10 parsecs * 3.086 x 1016 meters/parsec = 1.495 x 1012 meters/year. This roughly converts to 47.4 kilometers per second.