A star (not Barnard's star) at a distance of 10 pc is observed to have a proper motion of 0.5 arcsec / year. What is its transverse speed in AU / year?

The star will have a transverse speed of 315950.9 AU/year

Explanation:

d = 1/p

d = distance to star, measured in parsecs

p = parallax, measured in arcseconds = 0.5 arcsec/year

So, d = 1/0.5 = 2 parce

1 parsec = 3.26 light years

2 parce = 6.52 light years

⇒Transverse speed in AU / year = Distance/parallax

distance = 10pc = 2060000 AU

Transverse speed in AU / year = 2060000 Au/6.52 light years

Transverse speed = 315950.9 AU/year

Therefore, A star (not Barnard's star) at a distance of 10 pc observed to have a proper motion of 0.5 arcsec / year. Will have a transverse speed of 315950.9 AU/year