# Labs from YSI 94 : Small Angle Calculations.

April Whitt
CARA Yerkes Summer Institute, August 1994

This is a Teacher's Guide and a lab.

## Objectives

Students will:
• review concepts of circle, radius, diameter, circumference and the equations used to calculate them
• measure small angles in degrees without using a protractor
• calibrate one hand to use in estimating angles in the sky
• determine the angular size of Jupiter in the night sky

## Materials Needed

meter stick, blackboard and chalk, calculator

## Procedures

The small angle approximation is a method of doing problems that would ordinarily require trigonometry, but which may be solved by simple arithmetic as long as the angle involved is no more than a few degrees.

### Measuring a degree

1. Measure the distance from the chalkboard to the back of the room. Consider this distance to be the radius of a large vertical circle.
2. Compute the circumference of the circle by first doubling the radius, then multiplying by (pi, = 3.1416).
3. One degree is defined as 1/360 of a circle, so divide the circumference by 360.
4. Make a pair of marks on the chalkboard the appropriate distance apart to represent one degree when seen from the back of the room.
Example: Assume the distance from the chalkboard to the back of the room is 8.3 meters. 8.3 x 2 = 52.15 meters. This is the circumference of the circle. Dividing by 360 gives 0.144 meters, or 14.4 cm. Place two marks 14.4 cm apart on the chalkboard to represent one degree as seen from the back of the room.

### Calibrating a Hand

1. Once a degree has been measured, make a series of chalk marks on the chalkboard one degree apart, covering at least 10 degrees.
2. Have students stand at the back of the room and find the angular sizes of parts of their hands held at arm's length. In particular, find a way to estimate accurately 1° and 10° . Typically, a fist will measure about 10° and some finger widths will measure 1° .

### Determining the angular size of a moon, planet, star, etc.

1. What is needed are the size and distance of a celestial body. Think of the distance to the object as the radius of a large circle. Multiply by 2 to find the circumference and divide by 360 to find the size of a degree. Find the ratio of the object size to the size of a degree of arc to find the degree measurement of the object. Multiply by 60 to convert to minutes and by 60 again to convert to seconds, if desired.
Example: Jupiter's diameter is 142,800 miles, and its distance from the Earth at opposition is 628,700,000 miles. For a circle centered on the Earth reaching to Jupiter, the circumference is 3.95 billion miles. A degree is 1/360th of that distance or about 11 million miles. Therefore, Jupiter's apparent size is 0.013 degrees. Since that are 60 minutes in a degree, this is equivalent to .78 minutes of arc. As an extension to the example, in a 100 power telescope Jupiter would appear to be 78 minutes of arc in diameter. Since the full Moon appears to be 30 minutes of arc to the unaided eye, we see that Jupiter at 100 power would be over twice the size of the unmagnified Moon.

Example: Find the angular size of Alpha Centauri, assuming a diameter of 1 million miles (close to that of the sun) and a distance of 4.5 light year (ly) (1 ly = 5.8 x 10 miles). The circle centered at the Earth reaching to Alpha Centauri has a circumference of 1.64 x 10 miles, so the size of a degree of arc is 4.55 x 10 miles. Alpha Centauri is thus 2.2 x 10 degrees or 0.008 seconds of arc. Even at 100 power the apparent size of the disk would be less that a second of arc, which is not resolvable to the eye.

## Extensions

The small angle technique is applicable to finding the length of a parsec. A parsec is defined as the distance to a hypothetical star whose parallax is one second of arc. From such a star the angular distance from the earth to the sun would appear to be one second of arc. The arc length is 1 AU (93,000,000 mi) and the are 60 x 60 x 360 seconds in a full circle. Therefore there are 1,296,000 AU in a full circle around a hypothetical star one parsec away. Dividing by gives the diameter of the circle which is 412,530 AU. Dividing by 2 gives the radius of the circle which is the distance to a hypothetical star one parsec away: 206,265 AU. Multiplying by 93,000,000 gives the equivalent distance in miles.

Important Disclaimers and Caveats: