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

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Author: Janice VanCleave


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.

Design Your Own Experiment

  1. Design and experiment to test the accuracy of the cross-staff in measuring angular separation between stars in a constellation (group of stars that appears to make a pattern in the sky). For example, you might measure the angular separation of some of the stars in the Big Dipper which is an asterism (group of stars with a shape within a constellation) in the Ursa Major constellation. See separation A shown in Figure 4.2 from Alkaid to Dubhe, separation B from Megrez to Dubhe, and separation C from Dubhe to Polaris. Record your measurements in a table like Table 4.2. If it is too dark to read your cross-staff, stand with the light from a building behind you. Make five measurements for each separation and average them. Compare your average with known (true) angular separations given and record the difference. If the difference is more than the known value, record a positive (+) error. If the difference is less, record a negative (–) error. For more information about the angles between the stars of the Big Dipper, see Terence Dickinson's, Night Watch (Willowdale, Ontario: Firefly Books, 1998), p. 30.
  2. Angular Distance between Celestial Bodies

  3. Design an experiment to determine if the angular separation of celestial bodies changes. Perhaps you can repeat the measurements between the stars of the Big Dipper at different times in one night or at the same time on different nights.
  4. Use your cross-staff to measure the angular diameter of a celestial body such as the Moon. Since the Moon's diameter appears to be relatively small, use the smaller notch, S2, on the left side of the crosspiece. If it is not a full moon, measure the Moon's height rather than its width.

Get the Facts

The cross-staff was the first practical instrument for measuring the altitude of the Sun from the deck of a ship. The cross-staff measured the distance between the horizon and the Sun quite well, but it had problems. The user had to look directly at the Sun, risking eye damage or blindness. Another problem was aligning the crosspiece so that one end appeared to touch the Sun while the other touched the horizon. In 1594, John Davis (1550–1605), a British navigator, published his book The Seaman's Secrets, in which he described an improved instrument called the backstaff. What advantages did the backstaff have over the cross-staff? How were these two instruments different? For information about these and other navigational instruments, see Richard Moeschl's Exploring the Sky (Chicago: Chicago Review Press, 1993), pp. 115–123. Another resource is Dennis Fisher's Latitude Hooks and Azmuth Rings: How to Build and Use 18 Traditional Navigational Instruments (New York: TAB/McGraw-Hill, 1994).

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