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# Angular Separation: Angular Distance between Celestial Bodies

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Author: Janice VanCleave

### Angular Separation: Angular Distance between Celestial Bodies

The apparent distance between celestial bodies is how large the linear measurement between bodies appears to be from Earth. Angular separation or angular distance is the apparent distance expressed in radians or degrees.

In this project, you will build a cross-staff resembling the instrument used by early navigators and astronomers. You will use it to measure both angular separation and angular diameter of celestial bodies including the Moon.

### Materials

pen
white copy paper
scissors
stapler
yardstick (meterstick)

### Procedure

1. Trace or photocopy the cross-staff pattern (see Figure 2.1).
2. Cut along the solid lines.
3. Fold the paper away from you along the dashed lines.
4. Position the T-shaped top so that its A and B sides meet the A and B sides of the bottom. Staple the A sides together first, then staple the B sides.
5. Slip the yardstick (meterstick) through the rectangular cutout section and between the top and bottom sections that are stapled together. The side labeled 'W," "M," and "S" should face the zero end of the measuring stick. The paper is the crosspiece. It has a wide sight, "W," 4 inches (10 cm) across; a medium sight, "M," 2 inches (5 cm) across; and a small sight, "S," 1 inch (2.5 cm) across (see Figure 2.2). The notches in each side of the crosspiece are smaller sights and will be used in a later experiment.
6. Use the cross-staff to measure angular separation. You can take measurements inside or outdoors.
• Stand 4 yards (4 m) from a closed door or other object. Record this sighting distance in column 1 of an Angular Separation (Wide Sight) Data table like Table 2.1.
• Rest the zero end of the measuring stick against one of your cheekbones. Close the eye over the cheekbone and use your open eye to sight along the length of the stick. Slide the crosspiece until the left side of the door lines up with the left-hand edge of the wide sight and the right side of the door lines up with the right hand edge of the wide sight (see Figure 2.3). The width of the wide sight is d1 and is equal to 4 inches (10 cm). Record this sight width in column 2 of your data table as shown.
• Read the value on the measuring stick where the bottom of the labeled side of the crosspiece touches the measuring stick. This measurement is d2, the distance from the sight to your eye. Record measurement d2 in column 3 of your data table for trial 1.
7. Repeat step 6 four times, for a total of five independent measurements of the same distance.
8. Calculate the angular separation between the left and right sides of the door in degrees, 0°, using the average of your five measurements for d2 and this equation:
Da = 57.3° × (d1 ÷ d2)
9. where Da is the angular separation, d1 is the width of the sight, and d2 is the distance from the sight to your eye. Da is expressed in degrees. Both d1 and d2 must be expressed in the same unit—either inches or centimeters. Note: d1 ÷ d2 yields a number without a unit of measurement. When no unit of measurement is indicated in giving the measure of an angle, the angle is understood to be expressed in radians. To express the angle in degrees, the conversion 57.3° per 1 radian is used.

For example, for the wide sight, if d1 = 4 inches (10 cm) and d2 = 20 inches (50 cm), then

D = 57.3° × 4 inches (10 cm) ÷ 20 inches (50 cm) = 11.45°
1. Using the method in Appendix 1, determine the measurement error. Record the error in column 10 of your data table.
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