## Activity: "Name That Angle"

(an activity to explain and explore angular resolution)

by Scott Hildreth, Astronomical Society of the Pacific/Chabot College

We can help students grasp the significance of Hubble's extremely keen eye, and learn a bit about angles and math in the process, with the following in-class activity.

### Background

Picture the sky overhead as a large bowl, spanning 180 degrees from one horizon to the other. One degree on the sky then represents 1/180th of the visible hemispherical bowl. The Sun and Moon each span about one-half of one degree in our sky. Looking through binoculars or telescopes allows us to magnify a small section of the sky. To measure sizes or distances in that small section, astronomers divide a degree into 60 smaller pieces called arcminutes, and further subdivide minutes into 60 smaller sections called arcseconds. One second of arc on the sky is 1/3600th of one degree.

### Activity overview

Place a ruler on the classroom wall, have students stand a fixed distance away, and determine the smallest things that they can see from that distance. Translate the size of the object and its distance to an equivalent "angular resolution." Within the classroom, most students will be able to see small shiny objects about 0.5 centimeter (1/4 inch) in diameter from 10 meters (30 feet) away. These objects represent an angle of about 1.7 arcminutes. A grain of sand or salt, seen from 10 meters away with binoculars or a small telescope, represents an angle of about 10 arcseconds. Compare this to the HST's resolution of 0.1 arcseconds, which is 1,000 times sharper!

### Materials

A meter stick or ruler (with big numbers if possible), tape, a tape measure, and an assortment of very small items (which students can be asked to bring), such as single grains of dust, Cream of Wheat, salt, sand, coarse pepper, white pepper corns, small white peas or beans, coins, marbles, and balls.

### Suggested procedure

Fold a long piece of tape in two, or put a piece of double-sided tap along the ruler, under the markings so that the small objects will adhere temporarily to its surface. Place the ruler on a wall, at eye-level for the students. Have students work in small groups, with two members acting as independent observers, and the rest of the group as judges who evaluate whether the observers correctly identify where on the ruler an object is placed. Judges place an object somewhere along the tape "in secret," and then ask the observers to view the ruler from 10 meters, and separately write down where the object is located on the ruler. For larger objects, the observers should agree with each other, and with the judges. For smaller objects, at the limit of the student's resolution, observers may disagree slightly with each other, and with the judges, so repeated measurements should be taken.

Use Table 1 to translate the size of the target to an equivalent angular size at 10 meters (30 feet). For small angles, like these, under 2 degrees, you can safely interpolate between the values in the table for target objects with sizes between those listed. For example, a grain of rice 3 millimeters wide held 10 meters away has an angular resolution of (3) x (20.6 arc seconds) = 61.8 arc seconds -- just about one minute of arc.

Once the limit of a student's resolution is reached, students can then walk forward slowly toward the ruler, ultimately reaching a distance where the smaller objects can be located. Measurements of this distance can then be used to calculate the angular size of the target, using the following approximate formulae. Please note that size and distance need to be expressed in the same units, that is, both in inches, or both in centimeters.

```
Angle (in degrees) = (57.3) x Size/Distance
Angle (in minutes) = (3440) x Size/Distance
Angle (in seconds) = (206,400) x Size/Distance
```

### Extensions with more math

Once students have gathered data on what angle they can resolve, have them use mathematical ratios to create an "analogy" expressing the angle using much larger distances and more common objects. For example, something with a resolution of one arc second means that an observer could see a dime held about two miles away.

Help students create their analogies by developing a ratio equation:

```
Angular Size (of smaller object @ distance 1) = Angular Size (of larger object @ distance 2)
```
As older students develop their skills and comfort with ratios and units, you can encourage them to make reasonable estimates of distances as they answer questions like:
• If you can just resolve an angle of 1 arcminute, how far away is an automobile seen at night when its headlights just appear as two separate sources? (Use an approximate distance of two meters, or six feet, between the headlights.)
• If you were using a telescope with resolution like HST's, how close would you have to be to Earth to resolve:
1. a city