# The proper motion of a star for 1 year is equal to its annual parallax. Determine the star’s tangential speed (in km / s) relative to the Sun.

By definition, the annual parallax of a star π is the angle at which a segment r is seen from this star, equal to the average distance of the Earth from the Sun (1 AU). If we express the distance from the Sun to the star L in parsecs, and the parallax in arc seconds, then π ″ = 1 AU. / L (pc). The proper motion µ is the angular velocity of the star across the sky. If during time T the star has passed the distance D perpendicular to the line of sight, then µ ″ = D (AU) / L (pc) T. By the condition of the problem, the displacement of the star in time T, equal to 1 year µ ″ T = π ″, hence it turns out that during this time the star has passed the distance D equal to 1 AU. This can also be directly verified from the definition of these quantities. The tangential speed of the star in relation to the Sun is vT = D / T = 1 AU. / 1 year = 1.5⋅108 km / 3.156⋅107 s = 4.74 km / s.