Name: __________________

Lab Partners: __________________

Instructor's Signature: __________________

Lab Title: Angular Diameter, Size, and Distance

Equipment:

Medium sized styrofoam balls

Optical benches, magnetic holders

Meter sticks, tape measures

Coins (provided by students)

Moon (or close approximation)

Introduction:

In this lab we learn about the concept of angular diameter and make use of it to determine the size of the moon. The angular size (or angular diameter) of an object is simply the angle between one side of the object and the other.

angular size

D object

L

In the diagram shown above, L represents the distance from the observer to the object, and D represents the actual size of the object. If the distance L is much greater than D, then the actual angular size is determined from the simple relation:

q = angular size = D/L

Note: If the object being measured is spherical, then the angular size is the same as the angular diameter of the object.

To determine the angular size of an object, we will take advantage of the fact that two round or spherical objects can have the same angular diameter, even if they are at very different distances from the observer.

d D

l

L

In the diagram above, both the small and the larger object have the same angular diameter. Thus:

q = d/l = D/L

In this lab, we will measure d and l for the small nearby object and use this last equation to determine the angular diameter. Knowing q, we can then measure the distance L to a more distant object and calculate its diameter D.

Procedure (Part 1):

1. Using calipers, measure the diameter of a coin, and of a styrofoam ball. The instructor will show you how to use calipers if you aren’t familiar with them.

2. Place and optical bench on one end of the table, and the styrofoam ball near the other end (the bench should be lined up with the ball). Tape the coin in a magnetic holder and place it on the optical bench.

3. Adjust the position of the coin until the styrofoam ball is just covered by the dime when viewing with one eye. Record the distance of the coin from the eye of the observer, and the distance from the styrofoam ball to the eye in the data table.

Note the fuzziness around the coin. This is due to the diffraction of light waves around the edges of the coin. Since there is no way to change the nature of light, astronomers must find ways to compensate for this effect in order to improve their precision.

4. Determine the angular diameter of the coin using the previous equation. Note that units of the angle will be in radians. Since the dime covers far less than 1 radian (57.3 degrees), your answer will be a very small number.

q_coin = ____________________ radians.

5. Note that the previous equation can also be written as:

D = L x q

Determine D, the diameter of the sphere, using the value of q that you determined in step 4, the distance from your eye to the styrofoam ball (L), and the version of the equation shown here. Enter your results in Table I.

6. Determine the percent difference between the value D you determined in step 5, and the value of D you determined from using the calipers. Enter your results in Table I.

% difference = | D_calipers – D_angular| / D_calipers x 100

7. Change places with your partner(s) and repeat steps 4,5,6 for each partner

Procedure (Part 2 - The Moon):

1. Working with a partner, hold a coin in front of the full moon until it is completely covered by the coin. You may want to use the optical bench and holder (like a telescope) to improve your accuracy. If the moon isn’t visible, your instructor will place a large ball a suitable distance away to simulate the moon.

2. Determine the distance from the observer’s eye to the coin in meters. Enter this result, and all other results to follow, in Table II.

3. Determine the angular diameter of the coin. Change places with your partner(s) and repeat. Record all your values.

4. Using the fact that the average distance to the Moon is 3.844 x 10⁸ meters, determine the diameter of the Moon.

5. Determine the percent difference between the value you determined in part 4, and the accepted value of the Moon’s diameter (3.476 x 10⁶ m). Repeat for each partners values.

There is a good chance that your value will not be very close to the actual value, so don’t panic. There are good experimental reasons why your values may be somewhat off. Can you guess what they might be ?

Table 1.

	Partner #1	Partner #2	Partner #3	Partner #4
Distance from eye to coin (cm)
Distance from eye to ball (cm)
Diameter of ball (from step 5, cm)
Diameter of ball (using calipers, cm)
% difference

Table II

	Partner #1	Partner #2	Partner #3	Partner #4
Distance from eye to coin (cm)
Angular diameter of coin (radians)
Distance to Moon (from step 4, m)
% difference with given value

Procedure (Part 3):

In this part of the lab, you will measure the angular diameter of the Moon by timing the transit of the Moon past a reference object (e.g. a wire). From this transit time, and knowledge of the Earth’s rotation rate, the angular diameter of the Moon can then be calculated. As in part II, the diameter of the Moon can then be calculated using the known distance of the Moon from the Earth. It may be necessary to complete this part of the lab at home.

1. Set up a spotting telescope on a tripod. If no scope is available, mount a pipe, or a cardboard tube, or even binoculars on a fence, chair, or other suitable object so that it can point at the Moon, and not be disturbed while the Moon is observed. The telescope, or sighting tube must have some sort of cross hairs. These can be built into the scope, or simply taped on the outside. Your instructor will show you how, if you don’t know how to do this.

2. Center your scope, or sighting tube, on the Moon. Without touching it, watch as the Moon drifts oout of view. This drift is due , of course, to the rotation of the Earth. The Moon’s motion around the Earth also causes a drift, but this motion is imperceptible over small time intervals.

3. Turn the scope, or the cross hair arrangement, so that the Moon’s drift path is perpendicular to one of the hairs.

4. Move the scope so that the Moon is almost in contact with the perpendicular cross hair. When it comes into contact with the hair, start timing. Stop timing when the Moon just passes the hair.

5. If you find that the Moon didn’t move perpendicularly in step 4, readjust the scope. Repeat step 4 at least two more times. Average your best times.

	#1	#2	#3	Average
Times (sec)

6. Since the Earth turns through a full circle in 23h, 56m, and 4s (86,164s), the Moon will appear to move through a full circle (2p radians) in the sky in that same time period. We can describe its angular speed as

Angular speed = angular distance / time = q / T

Thus the Moon’s angular speed is 2p/86,164 = 7.2884 x 10^-5 rads/sec.

Since this value is constant, we can rewrite the above equation to find the angular diameter of the Moon:

q = angular speed x time

where we use the angular speed from above and the time you measured in step 5.

q_moon = ____________________ radians.

7. Using the angular diameter you calculated in step 6, and the known distance to the Moon, calculate the liner diameter of the Moon as you did in Part II of this lab.

D_moon = ____________________ meters.

8. Determine the % difference between the value from step 7, and the accepted value given in Part II.

% difference = ____________________.

Name: __________________

Lab Partner: __________________

Pre-lab Exercises: Angular Diameter and Distance

Show all of your work as well as the answers below. If you need extra space, use the reverse side, or add scratch paper and staple it to the pre-lab sheets.

1. Radians and degress are different angular measures, just as feet and cm are different measures of linear distance. Angles can be expressed in degress, minutes, and seconds, or in radians. For example, the angular measure from the horizon in the north to the horizon in the south is 3.14 radians (also known as pi). We can also say that the angle from horizon to horizon is 180 degrees.

a. Using this information, how many degrees are equivalent to one radian ?

b. A pair of close binary stars are separated by 2 x 10^-6 radians in the sky. What is their angular separation in degrees ?

2. Recall that the circumfrence of a circle is given by the formula C = 2pr, where r is the radius of the circle.

a. What formula would you use to find the length of a quarter circle ?

b. If you picked p/2 x r, you are correct. If so, notice that you are getting the distance by multiplying an angle in radians (p/2) by the radius. What would be the length of a quarter circle of radius 100 m ?

c. A part of a circumfrence of a circle is called an arclength. The arc length of the quarter circle in part(b) above is 157m. The angle (p/4) in this case is called the angle subtended by the arc length. Obviously, the smaller the arc length of a given circle, the smaller the angle subtended. Note that a simple formula for the angle subtended by an arc length is :

Angle subtended = q = arc length / radius

What angle is subtendedby an arc length of 10 m for a circle of radius 100 m ? Draw a diagram showing the entire circle, an arc length of ten meters, and the angle subtended.

3. In Part 1b above, the binary stars are separated by 2 x 10^-6 radians. If their known distance from Earth is 25 light years, what is their actual separation (in light years) from one another ?

4. Many textbooks on astronomy claim that yoour fist subtends an angle of five degrees when held at arm’s length.

a. How can this generalization be true, since everyone has different size hands and arms ?

b. Make any measurements you need and verify, or disprove this claim by determining the actual angle subtended by your fist when held at arm’s length. Explain what measurements you made and show your work.