### Angular vs. Linear Measurement

• Linear Measurement = actual size

• The linear size of an object is an intrinsic property of the object, as measured by a ruler.

• Angular measurement = apparent size

• The angular size of an object is the angle between lines extending from your eye to each end of an object.

• The apparent size of an object depends on its distance from you.

• The greater the distance to a given object the smaller its angular size.

• One way to measure the apparent size of an object is to attach a protractor to your nose and measure the angle subtended by the object. You could measure the angular diameter of the Moon this way.

• You could also measure the "angular separation" between two objects (e.g. two stars on the night sky) by the same method.

• You carry angular measurement tools with you all of the time.

• Your fist at arm's length subtends roughly ten degrees.

• The Moon, which is about 3500 km in diameter and 400,000 km away subtends 1/2 degree. You should be able to cover it with your little finger!

• The Sun is 1.4 million kilometers in diameter but 150 million kilometers away. It also subtends 1/2 a degree.

• The angular size, actual size, and distance to an object are simply related (at least for small angles) by

• To get the right answer you must be careful to use the same units for size and distance (for example kilometers). In general you must be careful not to mix units in equations.
• The 57.3 is a conversion from units of radians to the more common units of degrees.
• If you know two of the quantities (e.g. distance and angular size) you can find the third (actual size).
• If you triple the distance to an object its angular size will decrease by 1/3.

• The two statements above capture the essence of the manner in which mathematics gets used in this course.

### Altitude

• In an angular sense, the altitude of an object is the angle you would measure between an object and the horizon directly below it.

• This usage of the term altitude should not be confused with the everyday definition of altitude -- namely the height an object is above the ground.

• The point directly overhead is known as the zenith. The altitude of the zenith is 90 degrees.

• The altitude of the celestial pole in the sky is equal to your latitude.

• An observer at the North Pole sees the North Star at the zenith (altitude = 90 degrees)

• An observer at the Equator sees the North Star on the horizon (altitude = 0 degrees)

Updated September 4, 2009