Description
ASTR 1010 – Spring 2022
Week 5 Lab
The Size of Earth
Eratosthenes of Cyrene (276-194 BCE) was a scientist, artist, and librarian in North Africa. He
was a student of Zeno (the founder of the Stoic school of philosophy) and devised a
mathematical method for determining prime numbers. But he is best known today for making
the first accurate estimate of Earth?s circumference using shadows, footsteps, and a little math.
In this lab you will reproduce Eratosthenes?s experiment (without having to walk as far as he
did), estimating the size of Earth (and the uncertainty in your estimate) by measuring the
number of paces it takes your lab group to travel between two points.
Materials
?
?
Tape measure
Clipboard
ASTR 1010 – Spring 2022
Week 5 Lab
Instructions
?
?
?
Work in groups of 3-4. Groups of 1 or 2 are not allowed under any circumstances.
Complete your work for all group members on a single printout of this lab. Printouts will
be available at your lab session. (1 pt) Make sure the names of all group members
are at the top of the Google Doc.
When you are finished, scan the completed lab pages to a single PDF and have each
group member submit it on Canvas.
Part 1: Warm-up questions
1. (1 pt) If the Sun was directly above the town of Syene at solar noon on the summer
solstice, at what latitude must Syene be located?
2. (2 pts) In your own words explain the difference between precision and accuracy. Can a
measurement be precise without being accurate? Can it be accurate without being
precise?
ASTR 1010 – Spring 2022
Week 5 Lab
Part 2: Experimental approach
Details of Eratosthenes?s method for measuring the circumference of Earth were given in his
treatise “On the Measurement of the Earth”, which is now lost. However, some details of these
calculations appear in works by other authors. Apparently, Eratosthenes compared the noon
shadow at Midsummer (June 21st) between Syene (now Aswan on the Nile in Egypt) and
Alexandria, 500 miles to the North on the Mediterranean Sea. He assumed that the Sun was so
far away that its rays were essentially parallel, and then with a knowledge of the distance
between Syene and Alexandria, he gave the length of the circumference of the Earth as
250,000 stadia (1 stadium = the length of a Greek stadium).
We still do not know how accurate this measurement is because we still do not know the exact
length of a Greek stadium. However, scholars of the history of science have suggested a value
for the stadium and estimate that Eratosthenes?s measurement was ~17% too small.
Unfortunately, in Renaissance times, the length of a Greek stadium was under-estimated as
well, yielding an even smaller circumference for the Earth. This small value led Columbus to
believe that the Earth was not nearly as large as it is, so when he sailed to the New World, he
was quite confident that he had sailed far enough to reach India.
1. (5 pts) Complete the diagram below, following the presentation by the TA at the
beginning of the lab session.
a. Draw one line for the direction of the Sun?s rays at Syene on the summer
solstice, and one for the direction of the Sun?s rays at Alexandria on the same
day. Label as ?sunlight?
b. Indicate the direction of the zenith at both Alexandria and Syene.
c. Label the zenith angle (a, the angle of the Sun from the zenith) at Alexandria.
d. Show the relationship between
and Alexandria.
a
and the difference in latitudes between Syene
ASTR 1010 – Spring 2022
Week 5 Lab
Eratosthenes realized that the difference in latitudes between the two cities could be related to
the walked distance between them in a way that revealed the circumference of the Earth. The
relationship he wrote down was:
a
?
360
=
?????? ????????
????h’? ?????????????
2. (2 pts) Explain in words why the fraction on the left side of the equation (a/360o) must
equal the fraction on the right side of the equation (Distance/Circumference).
3. (2 pts) Rearrange the equation above to solve for the circumference of Earth:
Earth?s circumference =
To reproduce Eratosthenes?s experiment, we need to select two locations that are separated in
latitude, and know the difference in latitude (a), and the distance between them. Eratosthenes
measured the first quantity by using shadows cast at the two locations, and relied on others to
tell him the second quantity. For this lab we will do the reverse. We have selected the two
locations, and it is your job to measure the distance between them!
4. (1 pt) Verify your answer to #2 above with the TA or LA before proceeding to the next
section. Have the TA or LA record that your answer is correct.
ASTR 1010 – Spring 2022
Week 5 Lab
Part 3: Measuring the distance to Baseline road
Your experiment will use SBO as the starting point, and Baseline road as the end point.
?
?
The latitude of SBO has been determined (via astronomical measurement using one of
the telescopes) to be 40.00372 degrees (+40? 00′ 13.4″) North.
The latitude of Baseline road was determined to be precisely 40 degrees North Latitude
in the late 1800?s.
Unfortunately, in recent years due to traffic
control necessity, the course of Baseline Road
has been altered just south of SBO. As shown
in the satellite image to the right, Baseline
curves gently north between Broadway Blvd
and 30th Street. The white line is our best
estimate for exactly 40 degrees North latitude
based upon the course of Baseline Road east
and west of this bend. Perhaps realizing that
they had altered a geographically (and
astronomically) important landmark, the city of
Boulder (or maybe RTD?) has painted a red
line on the sidewalk near the bus stop in order
to mark the exact location of the 40th parallel.
(Notice how it splits the rock to the east.)
To measure the north-south distance between
SBO and Baseline Road, you will need to:
?
Plan a route.
?
Walk that route, carefully counting your
paces.
?
Calibrate the size of your paces using a
tape measure.
Each member of your lab group must make these measurements (both pacing between SBO
and Baseline and “calibrating” their paces by stepping off 100 yards). Each participant will then
use Eratosthenes?s equation from Part 2 to determine how many paces you would need to walk
to get all the way around the Earth. By calibrating your paces, you will then determine the
circumference of the Earth.
ASTR 1010 – Spring 2022
Week 5 Lab
The rest is up to you! If you need help be sure to ask the LAs or TA for some pointers. Each
individual in each group must make their own measurements using the method agreed to
by the group. Good luck. Keep thinking and stay safe! Especially when crossing Baseline and
other streets?the cars do not know that you are conducting an historical reenactment!
Finally, DO NOT USE SMARTPHONES TO ACCOMPLISH THIS.
Eratosthenes did not have a phone.
1. (3 pts) Describe the route you took in enough detail that we can understand your path.
You may insert a marked up Google map if you would like, but the route must still be
described in words.
2. (3 pts) Why did you choose the route that you did? What procedures did you use to
make sure your measurements were as accurate as possible?
ASTR 1010 – Spring 2022
Week 5 Lab
3. (5 pts) Fill in the table below as you walk. Return to SBO when you are done!
Number of Steps
(record for each group member,
always in the same order)
Direction
traveled
(N,S,E,W)
Comments
(not necessary, but here
if you need it)
ASTR 1010 – Spring 2022
Week 5 Lab
Part 4: Compute the circumference of Earth
Return to SBO if you haven?t already.
1. (4 pts) Considering ONLY the number of steps that are relevant for measuring
a,
how
many total steps did each member of your group take to get from SBO to Baseline
road. Report ONLY the number of steps that are relevant for measuring a.
Name: __________________________
# Steps: ______________
Name: __________________________
# Steps: ______________
Name: __________________________
# Steps: ______________
Name: __________________________
# Steps: ______________
2. (5 pts) Pick one member of your group, and use Eratosthenes?s equation to calculate
the circumference of the Earth, in units of both meters and kilometers. Show all work,
and describe any additional measurements or assumptions you had to make.
ASTR 1010 – Spring 2022
Week 5 Lab
3. Write down what each member of your group calculates for the circumference (in
kilometers) based on their paces. You don?t need to show the work again.
Name: __________________________
Circumference: ______________ km
Name: __________________________
Circumference: ______________ km
Name: __________________________
Circumference: ______________ km
Name: __________________________
Circumference: ______________ km
4. (2 pts) Calculate your group’s average value for the Earth?s circumference. Show your
work.
ASTR 1010 – Spring 2022
Week 5 Lab
Part 5: Estimating uncertainty
Perhaps the most important aspect of scientific investigation is assessing uncertainty in an
answer. Webster’s dictionary defines error as “the difference between an observed or
calculated value and the true value”. We don’t know the true value; otherwise there would be
no reason to make the measurement. We wish our measurements to be both ACCURATE and
PRECISE.
Accuracy relates to how closely the results of the experiment are to the true result. Thus,
accuracy speaks to whether our chosen methods actually work to allow a measurement of the
quantity we seek to determine, whether all assumptions have been accounted for and whether
these assumptions do not compromise the measurement. Errors in setting up an accurate
experiment are called systematic errors, and more and more precise measurements cannot
reduce these types of errors.
Precision, on the other hand, refers to the actual measurement process itself. Greater
precision in measurement can be accomplished by using a more accurate measuring device or
by repeating measurements several times. Uncertainties in precision are called measurement
or random uncertainties and repeated measurement can reduce these uncertainties (e.g.,
independent measurements by equally precise measuring tools or people) but never eliminate
them. However, be warned, precise measurements do not yield an accurate result if the
experimental setup is inaccurate; i.e., systematic and measurement errors are independent of
one another and both must be dealt with to obtain the best value for the true result.
ASTR 1010 – Spring 2022
Week 5 Lab
Any scientific measurement has inherent uncertainties and errors (precision in measurement
and errors in experimental setup) that limit the ultimate precision of the result. All scientific
experiments have these limitations, which must be quoted with the result. For example, even
political polling reports results and uncertainties. (e.g. ?54% with an uncertainty of 3 %?) But
beware – systematic errors are not reported and can be much larger in some cases! (e.g. What
if only men were polled? Only rich people were polled?) In this experiment, think about the
experimental setup, the specific methods that you and your group employed and the
uncertainties and errors which may have limited the ultimate precision of your result.
1. (2 pts) Typically, an experimental result is listed as: [value obtained] ? [precision].
Through a comparison of your final results on the circumference of the Earth with the
results from the other members of your group, estimate the precision of your
measurement. Give your group?s result in this format
Circumference = _______________ ? _______________ km
2. (4 pts) If we repeated your measurements many times, sometimes it?d be a little too
high and sometimes a little too low. Think back on how you got your data ? what kinds
of errors do you think might have been random, and how big are they? (List at least
two.)
3. (2 pts) Why does averaging many results reduce the measurement uncertainty (i.e.
produce a more precise measurement)?
ASTR 1010 – Spring 2022
Week 5 Lab
4. (4 pts) Systematic errors are ones that no matter how many times we repeat the
measurement, we will be consistently inaccurate. Think back on how you got your data
? what kinds of errors do you think might have been systematic, and how big are they?
(List at least two.)
5. (3 pts) Why does averaging many results not reduce the effects of systematic errors (i.e.
not produce a more accurate measurement)?
ASTR 1010 – Spring 2022
Week 5 Lab
The Size of Earth
Eratosthenes of Cyrene (276-194 BCE) was a scientist, artist, and librarian in North Africa. He
was a student of Zeno (the founder of the Stoic school of philosophy) and devised a
mathematical method for determining prime numbers. But he is best known today for making
the first accurate estimate of Earth?s circumference using shadows, footsteps, and a little math.
In this lab you will reproduce Eratosthenes?s experiment (without having to walk as far as he did),
estimating the size of Earth (and the uncertainty in your estimate) by measuring the number of
paces it takes your lab group to travel between two points.
Materials
?
?
Tape measure
Clipboard
ASTR 1010 – Spring 2022
Week 5 Lab
Instructions
?
?
?
Work in groups of 3-4. Groups of 1 or 2 are not allowed under any circumstances.
Complete your work for all group members on a single printout of this lab. Printouts will
be available at your lab session. (1 pt) Make sure the names of all group members are
at the top of the Google Doc.
When you are finished, scan the completed lab pages to a single PDF and have each
group member submit it on Canvas.
Part 1: Warm-up questions
1. (1 pt) If the Sun was directly above the town of Syene at solar noon on the summer
solstice, at what latitude must Syene be located?
2. (2 pts) In your own words explain the difference between precision and accuracy. Can a
measurement be precise without being accurate? Can it be accurate without being
precise?
ASTR 1010 – Spring 2022
Week 5 Lab
Part 2: Experimental approach
Details of Eratosthenes?s method for measuring the circumference of Earth were given in his
treatise “On the Measurement of the Earth”, which is now lost. However, some details of these
calculations appear in works by other authors. Apparently, Eratosthenes compared the noon
shadow at Midsummer (June 21st) between Syene (now Aswan on the Nile in Egypt) and
Alexandria, 500 miles to the North on the Mediterranean Sea. He assumed that the Sun was so
far away that its rays were essentially parallel, and then with a knowledge of the distance
between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000
stadia (1 stadium = the length of a Greek stadium).
We still do not know how accurate this measurement is because we still do not know the exact
length of a Greek stadium. However, scholars of the history of science have suggested a value
for the stadium and estimate that Eratosthenes?s measurement was ~17% too small.
Unfortunately, in Renaissance times, the length of a Greek stadium was under-estimated as well,
yielding an even smaller circumference for the Earth. This small value led Columbus to believe
that the Earth was not nearly as large as it is, so when he sailed to the New World, he was quite
confident that he had sailed far enough to reach India.
1. (5 pts) Complete the diagram below, following the presentation by the TA at the beginning
of the lab session.
a. Draw one line for the direction of the Sun?s rays at Syene on the summer solstice,
and one for the direction of the Sun?s rays at Alexandria on the same day. Label
as ?sunlight?
b. Indicate the direction of the zenith at both Alexandria and Syene.
c. Label the zenith angle (?, the angle of the Sun from the zenith) at Alexandria.
d. Show the relationship between ? and the difference in latitudes between Syene
and Alexandria.
ASTR 1010 – Spring 2022
Week 5 Lab
Eratosthenes realized that the difference in latitudes between the two cities could be related to
the walked distance between them in a way that revealed the circumference of the Earth. The
relationship he wrote down was:
?
360
?
=
?????? ????????
????h’? ?????????????
2. (2 pts) Explain in words why the fraction on the left side of the equation (a/360o) must
equal the fraction on the right side of the equation (Distance/Circumference).
3. (2 pts) Rearrange the equation above to solve for the circumference of Earth:
Earth?s circumference =
To reproduce Eratosthenes?s experiment, we need to select two locations that are separated in
latitude, and know the difference in latitude (?), and the distance between them. Eratosthenes
measured the first quantity by using shadows cast at the two locations, and relied on others to
tell him the second quantity. For this lab we will do the reverse. We have selected the two
locations, and it is your job to measure the distance between them!
4. (1 pt) Verify your answer to #2 above with the TA or LA before proceeding to the next
section. Have the TA or LA record that your answer is correct.
ASTR 1010 – Spring 2022
Week 5 Lab
Part 3: Measuring the distance to Baseline road
Your experiment will use SBO as the starting point, and Baseline road as the end point.
?
?
The latitude of SBO has been determined (via astronomical measurement using one of
the telescopes) to be 40.00372 degrees (+40? 00′ 13.4″) North.
The latitude of Baseline road was determined to be precisely 40 degrees North Latitude
in the late 1800?s.
Unfortunately, in recent years due to traffic
control necessity, the course of Baseline Road
has been altered just south of SBO. As shown
in the satellite image to the right, Baseline
curves gently north between Broadway Blvd
and 30th Street. The white line is our best
estimate for exactly 40 degrees North latitude
based upon the course of Baseline Road east
and west of this bend. Perhaps realizing that
they had altered a geographically (and
astronomically) important landmark, the city of
Boulder (or maybe RTD?) has painted a red line
on the sidewalk near the bus stop in order to
mark the exact location of the 40th parallel.
(Notice how it splits the rock to the east.)
To measure the north-south distance between
SBO and Baseline Road, you will need to:
?
Plan a route.
?
Walk that route, carefully counting your
paces.
?
Calibrate the size of your paces using a
tape measure.
Each member of your lab group must make these measurements (both pacing between SBO
and Baseline and “calibrating” their paces by stepping off 100 yards). Each participant will then
use Eratosthenes?s equation from Part 2 to determine how many paces you would need to walk
to get all the way around the Earth. By calibrating your paces, you will then determine the
circumference of the Earth.
ASTR 1010 – Spring 2022
Week 5 Lab
The rest is up to you! If you need help be sure to ask the LAs or TA for some pointers. Each
individual in each group must make their own measurements using the method agreed to
by the group. Good luck. Keep thinking and stay safe! Especially when crossing Baseline and
other streets?the cars do not know that you are conducting an historical reenactment!
Finally, DO NOT USE SMARTPHONES TO ACCOMPLISH THIS.
Eratosthenes did not have a phone.
1. (3 pts) Describe the route you took in enough detail that we can understand your path.
You may insert a marked up Google map if you would like, but the route must still be
described in words.
2. (3 pts) Why did you choose the route that you did? What procedures did you use to
make sure your measurements were as accurate as possible?
ASTR 1010 – Spring 2022
Week 5 Lab
3. (5 pts) Fill in the table below as you walk. Return to SBO when you are done!
Number of Steps
(record for each group member,
always in the same order)
Direction
traveled
(N,S,E,W)
Comments
(not necessary, but here
if you need it)
ASTR 1010 – Spring 2022
Week 5 Lab
Part 4: Compute the circumference of Earth
Return to SBO if you haven?t already.
1. (4 pts) Considering ONLY the number of steps that are relevant for measuring ? , how
many total steps did each member of your group take to get from SBO to Baseline road.
Report ONLY the number of steps that are relevant for measuring ?.
Name: __________________________
# Steps: ______________
Name: __________________________
# Steps: ______________
Name: __________________________
# Steps: ______________
Name: __________________________
# Steps: ______________
2. (5 pts) Pick one member of your group, and use Eratosthenes?s equation to calculate the
circumference of the Earth, in units of both meters and kilometers. Show all work, and
describe any additional measurements or assumptions you had to make.
ASTR 1010 – Spring 2022
Week 5 Lab
3. (4 pts) Write down what each member of your group calculates for the circumference (in
kilometers) based on their paces. You don?t need to show the work again.
Name: __________________________
Circumference: ______________ km
Name: __________________________
Circumference: ______________ km
Name: __________________________
Circumference: ______________ km
Name: __________________________
Circumference: ______________ km
4. (2 pts) Calculate your group’s average value for the Earth?s circumference. Show your
work.
ASTR 1010 – Spring 2022
Week 5 Lab
Part 5: Estimating uncertainty
Perhaps the most important aspect of scientific investigation is assessing uncertainty in an
answer. Webster’s dictionary defines error as “the difference between an observed or calculated
value and the true value”. We don’t know the true value; otherwise there would be no reason to
make the measurement. We wish our measurements to be both ACCURATE and PRECISE.
Accuracy relates to how closely the results of the experiment are to the true result. Thus,
accuracy speaks to whether our chosen methods actually work to allow a measurement of the
quantity we seek to determine, whether all assumptions have been accounted for and whether
these assumptions do not compromise the measurement. Errors in setting up an accurate
experiment are called systematic errors, and more and more precise measurements cannot
reduce these types of errors.
Precision, on the other hand, refers to the actual measurement process itself. Greater precision
in measurement can be accomplished by using a more accurate measuring device or by
repeating measurements several times. Uncertainties in precision are called measurement or
random uncertainties and repeated measurement can reduce these uncertainties (e.g.,
independent measurements by equally precise measuring tools or people) but never eliminate
them. However, be warned, precise measurements do not yield an accurate result if the
experimental setup is inaccurate; i.e., systematic and measurement errors are independent of
one another and both must be dealt with to obtain the best value for the true result.
ASTR 1010 – Spring 2022
Week 5 Lab
Any scientific measurement has inherent uncertainties and errors (precision in measurement and
errors in experimental setup) that limit the ultimate precision of the result. All scientific
experiments have these limitations, which must be quoted with the result. For example, even
political polling reports results and uncertainties. (e.g. ?54% with an uncertainty of 3 %?) But
beware – systematic errors are not reported and can be much larger in some cases! (e.g. What
if only men were polled? Only rich people were polled?) In this experiment, think about the
experimental setup, the specific methods that you and your group employed and the
uncertainties and errors which may have limited the ultimate precision of your result.
1. (2 pts) Typically, an experimental result is listed as: [value obtained] ? [precision]. Through
a comparison of your final results on the circumference of the Earth with the results from
the other members of your group, estimate the precision of your measurement. Give
your group?s result in this format
Circumference = _______________ ? _______________ km
2. (4 pts) If we repeated your measurements many times, sometimes it?d be a little too high
and sometimes a little too low. Think back on how you got your data ? what kinds of
errors do you think might have been random, and how big are they? (List at least two.)
3. (2 pts) Why does averaging many results reduce the measurement uncertainty (i.e.
produce a more precise measurement)?
ASTR 1010 – Spring 2022
Week 5 Lab
4. (4 pts) Systematic errors are ones that no matter how many times we repeat the
measurement, we will be consistently inaccurate. Think back on how you got your data
? what kinds of errors do you think might have been systematic, and how big are they?
(List at least two.)
5. (3 pts) Why does averaging many results not reduce the effects of systematic errors (i.e.
not produce a more accurate measurement)?
ASTR 1010 – Spring 2022
Homework 04
Due Thursday, February 10 at 11:59pm on Canvas
1. MasteringAstronomy
(15 points)
Complete the MA assignment entitled ?Homework 04 – MA? currently available on Canvas.
Your score on the assignment is your score for this question.
2. Lunar Stargazer (15 points)
You are a stargazer on the lunar nearside, on the center of the Moon?s disk as seen from
Earth.
a. (2 pts) How often do you see the Sun rise, roughly? Support your answer.
b. (2 pts) How often do you see Earth set, roughly? Support your answer.
c. (3 pts) You go stargazing one week after ?Full Earth?. Draw a diagram showing the Sun,
Earth, and Moon in the appropriate positions as viewed from above, and shade them
appropriately. Label Earth with an ?E? and the Moon with an ?M? so we aren?t confused.
d. (2 pts) Based on your answer in part c, draw the phase of Earth as viewed from the
Moon one week after ?Full Earth?, shading the dark part (i.e. draw a circle that
represents Earth as viewed from the Moon, and shade it appropriately).
e. (3 pts) Your friend on Earth is excited about the upcoming total solar eclipse. Sketch
how Earth would look to you (on the Moon) at a given instant during this eclipse (like
part d, draw a circle for Earth and shade it appropriately).
f.
(3 pts) The constellation Gemini is visible from Earth near the meridian at midnight in the
entire month of February. It is visible for you on the Moon, too, during this same time.
Which way will Gemini appear to move in your sky over the course of a lunar night?
3. Feedback (0 points?always welcome, almost never required)
a. How long did this HW take you to complete?
b. Is there any part of the HW that caused you particular trouble?
c. Did you work with others to complete the assignment?
d. How have the lab sessions and lab assignments been going?
e. Any other comments?
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