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Use a Cross-Staff to Measure Angular Separation

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Author: Janice VanCleave

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.

Getting Started

Purpose: To use a cross-staff to measure angular separation.

Materials

  • cross-staff

Procedure

  1. 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 4.1.
    • Rest the zero end of the measuring stick against one of your cheekbones. Open the eye over this cheekbone and close the other eye. Use your open eye to sight along the length of the stick. With the bottom of the crosspiece parallel with the floor, slide the crosspiece until the left side of the door lines up with the left-hand edge of the wide side and the right side of the door lines up with the right-hand edge of the wide sight (see Figure 4.1). 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.
  2. Angular Distance between Celestial Bodies

  3. Repeat step 1 four times, for a total of five independent measurements of the same distance.
  4. 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)
  5. 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 Da = 57.3° × d (4 inches (10 cm) ÷ 20 inches (50 cm)) = 11.45°

  6. Using the method in Appendix 1, determine the random error of measurement. Record the error in column 10 of your data table.

Results

The angular separation will vary with door width. The author's measurement was 11.45°.

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