The
New and Improved Hubble Space Telescope
(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:
- a city
- a football
stadium
- a house
- a person
Make assumptions
about the typical size for each of these objects. What do your results tell
you about the ability of robot spacecraft to detect intelligent life, or life
of any kind, through photographs taken from hundreds or thousands of miles
away?
| 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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