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.

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

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:
- a city
- a football stadium
- a house
- a person

English |
Metric |
||||

12 inches | @ 30 feet = | 1.9 degrees | 10 cm | @ 10 meter = | 34.4 minutes |

1" | @ 30 feet = | 9.55 minutes | 2 cm | @ 10 meter = | 6.87 minutes |

1/8" | @ 30 feet = | 1.2 minutes | 1 cm | @ 10 meter = | 3.43 minutes |

1/16" | @ 30 feet = | 36 seconds | @ 10 meter = | ||

1/32" | @ 30 feet = | 18 seconds | 1 mm | @ 10 meter = | 20.6 seconds |

1/64" | @ 30 feet = | 9 seconds | 0.1 mm | @ 10 meter = | 2.1 seconds |

1/1000" | @ 30 feet = | 1.1 seconds | 0.01 mm | @ 10 meter = | 0.21 seconds |

1/10,000" | @ 30 feet = | 0.1 seconds | 0.005 mm | @ 10 meter = | 0.1 seconds |

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