One of the more helpful aids to identifying constellations and locating objects in the sky is the ability to relate their positions to angular distances or angular separations from well known or readily identified reference positions or objects. To become familiar with this technique we will establish some readily available standards or "yardsticks" and use these standards to measure some distances in the sky.
Recall that an angle is formed whenever two straight lines intersect. An angle may be thought of as the fraction of a circle (a piece of pie?) encompassed by the two lines. The angle is the "opening" as measured from the intersection of the two lines. Angles are generally measured in degrees, but they can also be measured in radians or even in hours! For our astronomical applications the straight lines are the imaginary lines coming from the objects or positions of interest and intersecting at the eye of the observer.
Recall also that a full circle has a total of 360 degrees, a half circle has 180 degrees, and a quarter circle has 90 degrees. An angle of 90 degrees is often referred to as a right-angle.
Any object, such as a pencil may be used as an angular standard since when held at arm's length, straight lines from either end of the pencil will intersect at your eye and always produce the same angle (assuming the length of your arm doesnŐt change and you donŐt chew on your pencil too much). In the diagram below, S is the length of the pencil, L is the length of your arm, and theta is the associated angle.
Since you may not always have a pencil with you or you may occasionally have to sharpen it, we will adopt more constant "standards". For our angular standards we will adopt various parts of your hand when viewed arm's length. Specifically, let us adopt as standard and measure the angles associated with:
To determine the angles associated with each of these "standards" you will need to measure for each the distance S plus the distance L (note that L may change for different orientations of your hand). Work in pairs and help each other obtain the required measurements. You may then determine the associated angle by any of the following methods:
2. Make a scale drawing of the SL triangle on graph paper letting one or two blocks (or whatever) represent one centimeter. Measure the angle theta with a protractor. (The angle measured should be the same as in method 1. Why?).
3. Compute the angles using the "skinny triangle" approximation. In this approximation, as long as S is much, much less than L, we may compute the angle using
theta = 57.30 ( S / L )
where the angle theta will be in degrees. Your instructor will show you why this is true.
Note that other more rigorous methods of computing such angles are available and you should free to use other methods as long as you understand them. You should also feel free to compare the results obtained using different methods and to measure the angles subtended by other aspects of your hand or other objects as well.
Summarize ALL results in a table and describe the techniques that you use. Your final results should be a table showing the angles you derived for each of the specified anatomical standards.
A simple ruler held at arm's length may be used as a simple cross staff. One merely needs to calibrate the device so that one knows how many degrees corresponds to each centimeter measured on the ruler. A simple way to achieve such a calibration is outlined in the next paragraph. (You should be aware that more rigorous procedures are also possible.)
Note that if one stands 17 feet away from a wall an 8 1/2 x 11 inch sheet of paper subtends an angle of 3.09 degrees along the long side. Call this the viewing angle V. (Your instructor will show you how such a result may be computed.) If you now stand 17 feet from the wall and hold a ruler at arm's length (use both hands, or use only one hand, but be consistent) you may determine the apparent length of the paper as measured along the ruler. Call this length S cm. Your personal calibration factor for this device, call it C, is the number of degrees that corresponds to one centimeter at your arms length. It may be computed simply as C = V/S.
Derive your personal calibration factor, C.
Describe the procedures you use and present your results.
If you now measure the apparent distance between two stars (or points on the sky) to be a length d (cm) as measured against your ruler when held at arm's length, then the angular separation between these stars (or points) may be computed as theta = C x d
You have now calibrated an instrument that you may use to measure angular distances on the sky.
The size of the delta Cephei triangle...
The distance between Delta Cephei and the constellation of Cassiopeia...
c. Delta Cep to Gamma Cas
Measure the approximate altitudes for the following objects...
Note that the altitude for a celestial object is the angular vertical distance from the object to the nearest horizon. For the altitude observations make sure you also record the time to the nearest minute.