**SCIENCE** – Apparent distance between celestial objects is the linear size of the space objects observed from Earth. Angular separation or angular distance is the apparent distance expressed in radians or degrees.

In this experiment, you will create a cross-staff, similar to the instruments used by navigators and astronomers in ancient times. You will use this tool to measure angular separation and angular diameter of celestial objects, including the Moon.

**Learning Objectives**

The goal of this School of Nature practice is to create and use a cross-staff for measuring angular separation.

**Materials**

- Ballpoint pen
- Stapler
- A4 paper
- Long ruler
- Scissors

**Procedure**

- Copy or photocopy the cross-staff pattern (see Figure 1).
- Cut along the solid lines of the pattern.
- Fold along the dotted lines.
- Position the T-shape at the top so that its sides A and B meet the sides A and B at the bottom. Use the stapler to attach side A first and then side B.
- Cut along the part marked “cut” and then insert the long ruler into it and between the top and bottom parts stapled together. The sides marked “W,” “M,” and “S” should face the 0 mark on the ruler. This paper is the cross-shaped part with wide viewing angle, “W,” of 10 cm; medium viewing angle, “M,” of 5 cm; and narrow viewing angle, “S,” of 2.5 cm (see Figure 2). The markings on each side of the cross-shaped part indicate close-range viewing and will be used in the following experiments.
- Use the cross-staff to measure angular separation. You can perform the measurements indoors or outdoors.

- Stand 4 meters away from a closed door or another object. Record the wide viewing angle in column 1 of the Angular Separation Data Table (Wide Viewing Angle) as shown in Table 1.
- Place the zero mark of the ruler opposite your temple. Close one eye above your temple and use the open eye to look along the ruler. Slide the cross-shaped part until the left side of the door aligns with the left side of the wide viewing angle and the right side of the door aligns with the right side of the wide viewing angle (see Figure 3). The wide viewing angle is d1, equal to 10 cm. Record this wide viewing angle measurement in column 2 of the data table.
- Read the number on the ruler at the base of the side labeled “touch.” This measurement is d2, the distance from your sight to your eye. Record this measurement in column 3 of the data table for Experiment 1.

- Repeat step 6 four times to obtain a total of 5 independent measurements at the same distance.
- Calculate the angular separation between the left and right sides of the door in degrees (0°) using your 5 measurements and the following equation: D = 57.3° x (d1 : d2) where D is the angular separation, d1 is the wide viewing angle, and d2 is the viewing distance from the eye. D is expressed in degrees, while d1 and d2 must use the same unit, either inches or centimeters. Note: d1 : d2 indicates a unitless quantity. When an angle measurement does not use a unit, it is expressed in radians. To convert an angle to degrees, use the conversion 57.3° per 1 radian. For example, for the wide viewing angle, if d1 = 10 cm and d2 = 50 cm, then D = 57.3° X 10 cm : 50 cm = 11.45°
- Use the method in Appendix 1 to determine the measurement errors. Record the measurement errors in column 10 of your data table. Results: The angular separation will vary with the width of the door. The value obtained by the author is 11.45°.

**Why?**

Similar to a circle, the celestial dome surrounding the Earth also spans 360°. However, only about half of the sphere is visible above the horizon (the imaginary line that marks the boundary between the sky and Earth).

So, the sky we see covers roughly half a circle (a portion of the full circle) from one horizon to the other. As an observer on Earth, we perceive celestial objects to be at a considerable distance. The apparent distance between celestial objects is the linear size between these objects as seen from Earth. Angular separation or angular distance is the apparent distance expressed in radians or degrees.

The cross-staff is a tool used to determine angular separation. At a certain distance from an object, we measure the apparent width of the object (the wide viewing angle of the cross-shaped part) and the distance of the cross-shaped part from the eye. The ratio of the apparent width to the distance is multiplied by 57.3° to express the angular separation in degrees.