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.
- Your little finger at arm's length subtends about one degree.
- 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 of an object is the angle between lines
extending from your eye to each end of an object.
- The angular size, actual size, and distance to an object are simply related (at least for small angles) by
- 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*