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MEASURING ANGLES
Fig 1: Islamic Astronomers of the Middle Ages in an
Observatory in Istanbul.
Fig 2: Tycho Brahe in Denmark. His observations were
later used by Kepler.
Image can be found in the book “History of the Modern World” by
Palmer, Colton & Kramer at http://highered.mcgrawhill.com/sites/0072316551/ or
http://www.unf.edu/classes/freshmancore/core1images/muslimastro
nomers-MS-1.jpg
Image from http://buhlplanetarium2.tripod.com/Buhlexhibits.htm or
http://casswww.ucsd.edu/public/tutorial/images/history/brahe_quadrant.gif
or http://zebu.uoregon.edu/~imamura/121/lecture-3/brahe.html or
http://commons.wikimedia.org/wiki/Image:Tycho_Brahe_in_his_laborator
y.png
Have you ever thought how we determine the positions of stars, or the sizes of anything? If you want to
know the size of Jupiter (or even just the moon), you can hardly fly to it with your ruler in your hand
and measure it. However, the moon has a definite size in the sky. And what’s the size of the entire sky?
This seems a ridiculous question, however if we think in terms of angles it gets easier. One whole circle
has 360 degrees. A huge cloud that occupies about one 10th of the sky then has a diameter of 180o/10,
i.e., 18o. Similarly, you can measure the diameter of the moon, which turns out to be about 0.5o. So in
Astronomy we always think in angles - i.e., in “angular sizes.” Thousands of years ago we observed and
measured angles in the sky, and today we still do the same - and ironically we still use pretty much the
same methods. In this Lab you will learn the simplest and most basic method of how to measure angles.
Measuring Angles  Lab 1  1
PART I
There are several methods of measuring angles and rather sophisticated instruments have been designed.
Yet it is surprising how few people actually know how to determine angles with their own body parts…
Maybe it is not quite as accurate, but at least it provides rather good estimates. This is how you do it –
stretch out your arm. Your fist will cover an angle of about 10o, your index finder 1o, and you whole
hand about 20o.
Let’s check if this hypothesis. What is the angular size of your finger?
a)
Extend your arm in front of you. Measure the distance between your
eye and the tip of your index finger.
cm
b)
Measure the width of your index finger (across your finger nail).
cm
c)
Calculate the angular size of your finger in degrees. Consult the TOOLKIT to figure out which
trigonometric function to use. Write the trigonometric function into the box. SHOW your
calculation; ZERO points otherwise!
Angle
width of finger
Distance from eye to finger
d)
Rewrite your answer using the rules of significant figures
e)
How could you determine the uncertainty in the angular size? Make a suggestion.
degrees
Your estimated uncertainty (guess this) in the width of your finger is
f)
Rewrite your answer using significant figures and error estimates
(Your answer should have the form of 13.4 ± 0.5 or so.)
f)
Does the One-Degree-per-Finger Rule apply to you?
(In Astronomy we often use estimates. If your value falls within 30%
of the expected value, your estimate is considered reasonable.)
2  Lab 1  Measuring Angles
degrees
±
degrees
DETERMINING THE HEIGHT OF THE CAMPUS CENTER USING YOUR FINGERS
Fig 3: Woodcut from 1533: surveyors are using a cross-staff to measure angles.
See http://www.math.nus.edu.sg/aslaksen/teaching/heavenly.html or http://www.reformation.org/sir-francis-drake.html
Whether you use your fingers, or a cross-staff like in the above picture, the basic method is the same.
While the measurements with the cross-staff are more accurate, your finger is a hell of a lot ‘handier.’
a)
b)
Chose a tall landmark – any building will do. How many finger widths
is that building? Estimate the accuracy of your measurement. (Your
answer should have the form of 13.4 ± 0.5 or so.)
±
fingers
Go roughly halfway to that building. How many finger widths is the
building at half that distance?
±
fingers
Go roughly quarter the way to that building.
±
fingers
What is the angular size of the campus center from building?
±
degrees
What is the angular size at half the distance to that building?
±
degrees
What is the angular size at quarter the distance to that building?
±
degrees
c) Explain in words how angular size and distance are related
d) Abbreviate what you said (e.g., angular size increases/decreases as distance increases/decreases)
Measuring Angles  Lab 1  3
e) Insert three points into the plot below.
Max #
of degrees
Angular Size
(in degrees)
Distance
Quarter the Way to
the tall building
Half-Way to the
tall building
All the way to the
tall building
Is this plot consistent with your statement in (d)?
f) Now figure out the exact relationship between the angular size, the linear size and the distance. Use
your toolkit to choose the right formula. Write it down below.
g) To figure out the physical height of the building we need to know how far it is. Either look up the
exact distance to that building, measure it (paste it out), or estimate it.
distance = _________________meters
h) What is the physical height of the building in meters? Show your calculation.
e) Estimate the uncertainty in that measurement.
f)
The height of the building is
meters
±
h) Do you think this is a good method to determine the height of any building? EXPLAIN.
4  Lab 1  Measuring Angles
meters
DO EITHER OPTION A OR B
PART II — A
(Sunset Labs ONLY –or– do as a Homework Assignment)
There is a rule of thumb (no pun intended) that the sun moves “one finger every ten minutes.” Your
assignment is to check this statement.
a)
Find a spot where you can watch the Sun. Bring a watch and write
down the time from the watch.
b)
Using your fingers determine the height of the Sun above the horizon.
fingers
c)
Estimate the uncertainty (fractions of fingers are okay)
fingers
d)
This corresponds to ... ±... degrees:
e)
Predict the time the sun will set according to the 10 min per finger rule.
f)
Come back at Sunset; record the time
g)
Was it close to your prediction?
(A measurement is generally regarded as “reasonable” if it falls within
one third of the predicted value. For example if the predicted value is
100, any measurements within 60 and 130 are considered reasonable.)
h)
Calculate the theoretical value, i.e., determine how many minutes it
takes the sun to move 1o. (HINT: The Sun takes 24 hours to move
around the sky. This corresponds to 360o. So you need to figure out
how many minutes it takes the Sun to move 1o.) SHOW calculation!
i)
Is the scheme of “10 minutes per finger” reasonable? (A measurement
is reasonable if it falls within one third of the predicted value.)
k)
If it is not reasonable suggest a different wording of the “rule”
±
degrees
Measuring Angles  Lab 1  5
DO EITHER OPTION A OR B
PART II — B
There is a rule of thumb (no pun intended) that the Sun and the Stars move “one finger every ten
minutes”. Your assignment is to check this statement.
a)
Take a pencil and piece of chalk, and go outside. Find a tree, a lamppost, or some other distinct
mark on the horizon and position yourself so that the Sun (or star) is directly above that landmark.
Use the chalk to mark the exact position where you are standing (you will return to that position).
T he position of the S un (or
star) w hen you co me back
some time later
Initial position of
the Sun (or star)
T he Land M ark
Us e your fingers to
meas ure the dista nce
the Sun (star) moved.
Chalk Circle
Horizon
Towards W est
Time of first observation.
b)
Come back about half an hour later to exact same location. The Sun will appear to have moved.
Time of second observation.
How many fingers did the Sun move?.
fingers
Estimate the uncertainty (fractions of fingers are okay)
fingers
Transform this to degrees.
±
degrees
How many degrees did the sun move in 10 minutes
±
degrees
c)
Calculate the theoretical value, i.e., determine how many minutes it takes the Sun to move 1o.
(HINT: The Sun takes 24 hours to move around the sky. This corresponds to 360o. So you need to
figure out how many minutes it takes the Sun to move 1o.) SHOW the calculation!
d)
Then determine how many degrees the sun should have moved in 10 minutes.
6  Lab 1  Measuring Angles
DO EITHER OPTION A OR B
PART III — A
(FALL SEMESTER)
Go outside and look at the night sky and try to find the constellations below (look towards the
Southwest, all the way up to the Zenith). You’ll see Deneb and Vega above you (Vega in you Zenith),
and Altair a little further South. These are the three brightest stars in the summer sky and are often
referred to as the Summer Triangle. Use your fist and fingers to determine the size of the Summer
Triangle. How far is Vega from Deneb, Deneb from Altair, and Altair from Vega? Use your fist and
fingers to determine the distances. ___________ ___________ ___________
Deneb is in the constellation Cygnus (or Swan), which is sometimes called the Northern Cross. The star
at the other end of the cross is Alberio, a rather spectacular double star. A little further to the south
you’ll see Altair in Aquila, west of that you’ll see Hercules, and further in the Southwest will be
Ophichus, the Serpent Holder. Scorpio (below Serpens) will be on the Southwest Horizon, but you
might only see Scorpio’s brightest Antares, which will be a little above the Horizon.
Cygnus X-1 is
an elliptical
galaxy with a
dust-ring. It
has a mass
black hold in
its center. It is
in Cygnus,
between
Deneb and
Altair. How
many finger
widths is it
from Deneb
and Alberio?
________
________
The Ring Nebula, M57, is worth
checking. You’ll even see it in a
small telescope. It is between the
two fainter stars in Lyra. How far
are these two stars from Vega and
Altair? _______ ________
M13 is one of
the larger
globular clusters.
It is rather
spectacular and
can be easily
seen with a small
telescope.
M5 –
Globular
Cluster
The M16
challenge
M12 (top) & M10
More globular clusters
Measuring Angles  Lab 1  7
DO EITHER OPTION A OR B
PART III — B
(SPRING SEMESTER)
Go outside, look at the night sky, and try to find the constellations below (look the Sout, all the way up
to Zenith). You can recognize Orion by a box of four stars with three diagonal stars in the middle
(Orion’s Belt). Rigel is the blue star at the bottom left, and Betelgeuse is the orange star at the top left.
The 3 belt stars point down towards Sirius, the brightest star in the sky, and up towards Aldebaran, a red
giant. You have now recognized three stars of the Winter Hexagon: Sirius, Rigel, and Aldebaran.
Continue to Capella, which should be more or less in your Zenith. Then go East Castor and Pollux (the
Heads of Gemini), then to Procyon and back to Sirius. Use your finger and fist to determine the
separation between Rigel and Betelgeuse, and between Castor and Pollux: _______ ______
The Rosette
Nebula (bottom
red dot) and
Cone Nebula
The Orion Nebula
is one of the most
photographed
nebulae in the
sky. It is also the
classical example
of a star
formation region.
It is in the Sword
______ degrees
down from
Orion’s Belt.
8  Lab 1  Measuring Angles
The Horsehead nebula
is pretty, but might be
difficult to see with a
small telescope.
LAB REPORT
1) Objective of the Lab.
2) Summarize the main results of this Lab.
3) Critically comment on this method of measuring angles. How accurate is this method? When would
you use it? [Be original; do not just propose to measure heights of buildings or mountains.]
Measuring Angles  Lab 1  9
FOR EXTRA CREDIT
DETERMINE YOUR DISTANCE BETWEEN YOU AND ANY TALL BUILDING OF
YOUR CHOICE
a)
Find a spot from which you can see the horizon and a tall building.
Write down your location.
b)
Write down the location of the tall building.
c)
From your location measure the angular height of that tall building.
d)
Make an educated guess of the number of floors of that building.
e)
Make an educated guess (in meters) of the height of each floor.
f)
Calculate the physical height (in meters) of that building.
g)
Using the measured angular height and the estimated physical height,
calculate the distance (in meters) to that building.
A n g le
h e ig h t
d is ta n c e
h)
Transform this to miles.
i)
Take a large map1 on which you see both the tall building and your
location. Using a ruler measure the distance in inches.
j)
Convert that distance to miles.
k)
Compare the measured distance (g) with the calculated distance (j). Is
the measured distance close to the distance obtained from the map?
l)
Finally comment on this method of determining distances.
1
A Google map is fine.
10  Lab 1  Measuring Angles