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# Use a Cross-Staff to Measure Angular Separation (page 2)

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Author: Janice VanCleave

### Why?

Like a circle, the sphere of sky surrounding Earth measures 360°. But only about half the sphere is visible above the horizon (imaginary line where the sky appears to touch Earth). So the sky you see covers an arc (segment of a circle) of about 180° from one side of the horizon to the other. As an observer from Earth, you look at celestial bodies from a great distance. The apparent distance between celestial bodies is how large the linear measurement between the bodies appears to be from Earth. The angular separation or angular distance is the apparent distance expressed in radians or degrees. Altitude is the angular distance above the horizon.

The cross-staff is an instrument used to determine angular separation. At a specific distance from an object, you measure the apparent width of an object (the width of the sight of the crosspiece) and the distance of the crosspiece from your eye. The ratio of apparent width to distance multiplied by 57.3° expresses the angular separation in degrees.

### Try New Approaches

1. Does the width of the cross-staff's sight affect the results? Repeat the experiment twice, using the medium, "M," and small, "S," sights. Compare your calculations.
2. Assess the accuracy of your cross-staff by determining the true angular separation between the sides of the door (DA). Do this by using a ruler to measure the actual width of the door, da, and using the sighting distance, ds, of 4 yards (4 m) and this equation:
DA = 57.3° ×( da ÷ ds)
3. For example, if door measurement da equals 29 inches, then

DA = 57.3° ×(29 inches ÷ 144 inches)
= 11.54°
4. Determine the relative error of your measurements using the method in Appendix 2.
1. Does sighting distance (ds) affect angular separation? Repeat the original experiment, collecting data for distances of 2, 4, 6, and 8 yards (m) using the wide sight. Determine the relative error for each sighting distance measurement.
2. Repeat the previous experiment using the medium and small sights of the cross-staff. Draw a line graph of your findings. Put the sighting distance on the horizontal axis. Put angular separation on the vertical axis. Use points and lines of different colors for the wide, medium, and small sights.