# Positions and Sizes of Cosmic Objects

Astronomers use angular measure to describe the apparent size of an object in the night sky. An angle is the opening between two lines that meet at a point and angular measure describes the size of an angle in degrees, designated by the symbol °. A full circle is divided into 360° and a right angle measures 90°. One degree can be divided into 60 arcminutes (abbreviated 60 arcmin or 60'). An arcminute can also be divided into 60 arcseconds (abbreviated 60 arcsec or 60"). The angle covered by the diameter of the full moon is about 31 arcmin or 1/2°, so astronomers would say the Moon's angular diameter is 31 arcmin, or the Moon subtends an angle of 31 arcmin. If you extend your hand to arm's length, you can use your fingers to estimate angular distances and sizes in the sky. Your index finger is about 1° and the distance across your palm is about 10°. ### The Small-Angle Formula

The angular sizes of objects show how much of the sky an object appears to cover. Angular size does not, however, say anything about the actual size of an object. If you extend your arm while looking at the full moon, you can completely cover the moon with your thumb, but of course, the moon is much larger than your thumb, it only appears smaller because of its distance. How large an object appears depends not only on its size, but also on its distance. The apparent size, the actual size of an object, and the distance to the object can be related by the small angle formula:

D = θ d / 206,265

D = linear size of an object
θ = angular size of the object, in arcsec
d = distance to the object

Example:

A certain telescope on Earth can see details as small as 2 arcsec. What is the greatest distance you could see details as small the the height of a typical person (1.6 m)?

d = 206,265 D / θ = 206,265 × 1.6 m / 2 = 165,012 m = 165.012 km

This is much less than the distance to the Moon (approximately 384,000 km) so this telescope would not be able to see an astronaut walking on the moon. (In fact, no Earth based telescope could.)

### Practice Questions

1. The average distance to the Moon is approximately 384,000 km. The Moon subtends and angle of 31 arcminutes, or about 1/2°. Use this information and the small-angle formula to find the diameter of the moon in kilometers.
2. At what distance would you have to hold a quarter (which has a diameter of about 2.5 cm) for it to subtend and angle of 1°?