Download The Earth--Our Observing Platform

PBL (Problem Based Lab) #1: The Earth--Our Observing Platform
Name ________________ period _____
Purpose: To provide activities to help one better appreciate what we see from the Earth as we look toward the sky, and realize modern
physical science was born from efforts to make sense of those observations. These activities will both help you "see the big picture"
and reinforce specific concepts / skills needed to in both introductory physics and astronomy courses: angular distances in the sky,
parallax shifts, use of similar triangles / basic trigonometry in problem solving, unit conversions, percent error, significant figures, etc.
Introduction: The modern scientific conceptual framework has been constructed over the last twenty five hundred years. This
progressed slowly until people began appreciating a certain way of proceeding—which came to be called the scientific method. In
this regard we go back to the first decade of the 17 th century, when Francis Bacon in England began writing about what this method
might make possible, and when Johannes Kepler in central Europe / Galileo in Italy really began to use it in a relatively modern way.
While today we recognize that the understanding embodied in this conceptual framework has made possible the seemingly miraculous
technological advances that continue to transform human society, we need to appreciate something else: that our understanding
ultimately rests on observation. So it seems entirely appropriate to look at the development of that framework in terms of making
sense out of the observations. (Of course observation often involves measurement. For more on that topic, refer to (handout #1):
“A Summary of Everything You Always Wanted to Know About Measurement”)
In trying to understand our place in the universe, we have looked to the sky. What we see is very dependent on the Earth itself,
particularly its own motions. Handout #2 “What We See From Our Observing Platform—A Summary” lists ten important
observations. Time doesn't permit covering the whole story of how the scientific method was used to make sense of these
observations and figure out how the Earth fit into the bigger celestial scheme of things. Our focus here is on four specific problems of
both physics / astronomy and historical interest: A) What is the shape of the Earth? B) How big is the Earth? C) How far away
from the Sun is it? D) Is the Earth moving? It took people over two millennia to fully solve these problems--the last two proved
especially difficult. Work on them not only resulted in the birth of modern physical science in the 17 th century, but it had critically
important historical ramifications. Most notably, while Columbus knew the correct answer to problem A), his mistaken answer to
problem B) (he believed the Earth was around 25% smaller in circumference than it actually is) influenced his decision to sail west!
A) What is the shape of the Earth? Procedure / Activities / Questions
1) Imagine you're on the surface of a spherically shaped planet looking toward a very very very distant bright star. At point B in our
representation below, this bright star is on the horizon (altitude angle = 0 o). As you move you on this curved surface from points
a) A to B to C or b) C to B to A (select one) this bright star becomes increasingly visible--rising higher and higher in the sky
(increasing altitude) as you move. After using a ruler and a protractor, by drawing in the horizons and measuring the altitude of the
bright star for positions A and C, explain your answer.
Note: Many people actually having the experience of seeing a bright star rising
higher in the sky as they move on the surface of the Earth and appreciate they're
living on the surface of a sphere. Historically, one of the more famous of these
people was the Greek philosopher and astronomer Posidonius (135-51 BCE). His
experience of watching the bright southern star Canopus rise higher in the sky as he
moved from Rhodes (where he thought it "culminated" on the horizon) to Alexandria
(where he thought it culminated at altitude = 7.5 o) is remembered today because he
used this information to determine the Earth's size. In this respect he followed in the
footsteps of Erathostenes (276- 195 BCE) of Alexandria who used a bright star
known as the Sun and the overhead direction (known as the zenith) rather than the
horizon for reference. (See part B below for details. )
2) Imagine you watch a lunar eclipse one night for several hours, paying particular
attention to the shape of the Earth’s shadow. From photographs documenting this
event (shown at the right) is the Earth’s shadow (select one)
a) always straight b) sometimes straight, sometimes curved c) always curved ?
3) Consider two competing hypotheses: #1 The Earth is shaped like a sphere #2 The Earth is shaped like a flat disk. Use each to
construct related models for making sense out of your observations / the photographs in 1) above. Produce drawings below to
illustrate your models.
4) Experiment with disks and spheres in sunlight as directed by your instructor. Comment on shapes of shadows produced by both.
Based on predictions you can make based on each model, explain why model #1 does a better job of explaining the observations / the
photographs in 2) above.
B) How big is the Earth? Procedure / Activities / Questions
Eratosthenes estimated the circumference of the Earth using knowledge of the Sun’s position in the sky from two places at the same
moment (see figure below). On a certain day, he knew, the Sun is directly overhead in Syene (modern Aswan). At noon, it shines
directly into the bottom of wells. On the same day at noon, the sun in Alexandria is not directly overhead, but casts a shadow. He
reasoned from this observation that the Earth must be a sphere. The shadow angle was reportedly 1/50th of a circle (360 o / 50 = 7.2o).
He also had the distance from Alexandria to Syene surveyed by pacing off the distance. The result was approximately 4530 to 5000
stadia (836 km to 925 km). He reasoned that if the Sun is very far away, its rays are parallel to each other by the time they reach the
Earth. Given all that, he also knew how the circumference of the Earth could be calculated. His result was close to the accepted
modern answer, though it is difficult to be sure just how close.
For an imaginary observer at the center of the Earth, the 7.2 o angle between directions to Alexandria and Syene is
to 360o (all the way around) as the distance between Alexandria and Syene (=AS) is to the circumference C of the Earth.
7.2
= AS
or
C = 50 AS
360
C
In general, if the angle between the direction to the Sun and the zenith is (zenith angle) θ, and the distance between the location where
the Sun observed at that zenith angle and the point on the Earth where the Sun is overhead at that exact moment (called the subsolar
point) is D, then
θ
=
D
or
C = 360 D
360
C
θ
How can you determine the zenith angle θ of the Sun? Answer: by measuring the length of the shadow of a vertically oriented meter
stick. Consider the figure below at the left where distance L represents the shadow length, and ADJ = 1 the one meter length.
Note the sides of the right triangle are labeled HYP (for
hypotenuse), OPP (for opposite, that is the side opposite angle θ),
and ADJ (for adjacent, that is the side adjacent to angle θ).
If you don't know of any other way, here's a way to determine
angle θ from your data: make a scale drawing and use a protractor
to measure angle θ.
Of course there's an easier way if you recall basic trigonometry:
the tangent of angle θ = OPP / ADJ or tan θ = L / 1.00 meters
Gettting θ requires finding the angle whose tangent is L / 1.00 m,
which we write as θ = tan-1 (L / 1.00 m).
Procedure / Data : (Be sure and record units! Remember: Measurements recorded w/o units are worthless!)
Your method will be similar to that of Eratosthenes:
1. On a sunny day, take a meter stick, hold it loosely between your fingers so it hangs vertically, just touching the ground. This will be
your vertical reference line. Measure the length of the shadow it casts as accurately as you can. The end of the shadow is a bit fuzzy.
Do the best you can, trying to measure to the nearest mm. One or two meter sticks lying flat on the ground will be needed. Note the
time to the nearest second. A link to a time service (http://nist.time.gov or http://tycho.usno.navy.mil) and a stopwatch will be needed.
date
time
shadow length
2. For the exact time of your shadow determination, using the latitude and longitude of the school as provided by your instructor, go to
http://www.jgiesen.de/sunearth/index.htm and determine the latitude and longitude of the subsolar point = where the sun is overhead
at the zenith. Then go to http://www.movable-type.co.uk/scripts/latlong.html and determine the distance between these two locations.
school
subsolar point
distance between school & subsolar point
lat=
long=
lat=
long=
Note the theoretically expected value for the length of the shadow cast by a meter stick is given. Record it here=____________
Data Handling / Calculations: (Show all your work; round final results to the proper # of significant figures.)
1. Compare your measured shadow length with the theoretically expected value for the length of the shadow cast by a meter stick by
computing the percent error.
2. Determine the Sun's zenith angle θ at the moment of your shadow length measurement.
3. Using Eratosthenes' method, calculate the circumference of the Earth, and its diameter.
4. Compare your diameter result above to the accepted value by computing the percent error.
C) How far away from the Sun is the Earth? Procedure / Activities / Questions
Today the average Earth--Sun distance is well established. Referred to as the astronomical unit (au) , its value is 149.6 x 10 6 km. We
begin our investigation of how it is determined by considering small angular distances. In an extraordinary coincidence--one that
makes solar eclipses especially spectacular--the angular sizes of the Sun and the Moon in the sky as seen from the Earth are very
nearly equal. We now proceed to measure one of these….
1) As directed by your instructor, using the pinhole projection method, make measurements that will enable you to determine the
angular diameter of the Sun. Using the diagram below as your guide, record values for d and r. d = ________ r=_________
2) Guided by the math behind Erathostenes' method, determine the angular diameter of the Sun. Show all your work, final result, and
uncertainty of the measurement in the space provided below. (Hint: either 1) assume angle θ is so small that linear distance d is very
nearly equal to a (curved) part of a circle! or 2) use basic trigonometry)
3) For the historical record, Aristarchus (310-230 BCE) was the first to attempt determining the size of the Sun. His method for
getting its size in comparison to that of the Earth--involving the relative positions of Earth-Moon-Sun at the time of a half moon--was
fundamentally correct. His result--based on inaccurate measurements--was much too small. Assuming the modern day value for the
Sun's diameter, use your result above to calculate the distance to the Sun. Show all your work, final result, and uncertainty of the
measurement in the space provided below. (Hint: pay attention to significant figures!)
4) If the orbit of the Earth around the Sun (or Sun around Earth in the simplest ancient geocentric model) is circular, the Earth--Sun
distance will be constant. Design an experimental procedure that in principle would enable determining whether the Earth--Sun
distance changes.
5) In actuality, as Kepler established in 1609, the Earth's orbit is slightly elliptical and the Earth--Sun distance varies by 3%. Using
equipment similar to what you used today, would the experimental procedure you came up with in 4) above be able to verify this?
Explain.
6) Study the chart below of Earth--Sun distances on particular dates. To how many significant figures is Earth's orbit circular?
Give the radius of this orbit rounded to the proper # of significant figures.
Date
Distance (x 108 km)
Date
Distance (x 108 km)
1 January
1.4710
1 July
1.5208
1 February
1.4741
1 August
1.5183
1 March
1.4823
1 September
1.5097
1 April
1.4949
1 October
1.4977
1 May
1.5073
1 November
1.4848
1 December
1.4751
1 June
1.5169
D) Is the Earth moving? Procedure / Activities / Questions
The earliest and very simplest Earth-centered (geocentric) Aristotelian model that people used to explain celestial motions had the
Earth stationary and celestial bodies moving around it in circles. It was fatally flawed by its inability to explain the (sometimes
observed) retrograde motion of planets. This model was refined (and complicated!) by adding more circles (epicycles, etc.). Known
as the Ptolemaic system, it was able to explain retrograde motion--but it seemed terribly contrived. Many who used it thought of it as
simply a way to explain the data--a mathematical model divorced from any physical reality. The simplest Sun-centered (heliocentric)
Copernican model very naturally explained retrograde motion, and predicted planetary positions as well as the Ptolemaic model. But
its requirement that the Earth move around the Sun gave it what many felt was a fatal flaw: ancient observers, before the invention of
the telescope, could detect no corresponding observed parallax shift as Earth moves.
(This diagram not to scale—angle P is actually less than one second of arc!)
On January 1st, Earth is at position A--note relative apparent position of nearby
star as seen against the background of more distant stars. On July 1st, Earth is
at position B--note relative apparent position of nearby star: it has moved with
respect to its apparent position six months earlier. There is a parallax shift.
Angle P is called the parallax (it is one half of the overall angular shift).
1) As directed by your instructor, view the parallax shift of your finger held at arms length when viewed first with your right eye
then with your left. Use some object against a distant wall in the classroom for background reference. Describe what you saw.
2) Compare the parallax shift involved in 1) above to the actual parallax shift due to the Earth's annual revolution around the Sun.
Additional Problems (Show all your work and round to the proper number of significant figures.)
1) Since (360 / 7.5) = 48, Posidonius thought the distance around the Earth (its circumference) was 48 x the distance between
Alexandria and Rhodes. If that distance is 3750 stadia, and one stadium = 1/10 mile, compute the circumference of the Earth.
Compare this to the accepted value (in miles) and compute the percent error.
2) The true latitude difference between Alexandria and Rhodes is 5.23o. If Posidonius had measured this number for the altitude
difference in Canopus position, instead of 7.5 o, he would have computed the Earth’s circumference to be what? Percent error is what?
3) Ancient distance determinations involving units of stadia are fraught with uncertainty since they are ultimately based on another
unit whose length is uncertain: the roman pace. If a stadium = 125 roman paces, and a roman pace = 296 mm, figure out
a) how many meters there are in one stadium? b) how many miles there are in one stadium?
4) Suppose two students each make three determinations of Earth's circumference. Student A repeats what Erathostenes did, with the
following results: 24900 miles, 25800 miles, 23800 miles. Student B repeats what Posidonius did, with the following results: 17990
miles, 18002 miles, 18014 miles. Which students' results have... more accuracy? ...more precision? Explain.
5) Bonus problem: Imagine you’ve lazy and don’t wish to travel very far to determine the size of the Earth. While lying on the beach
(remember you’re lazy) with the Sun about to set, you realize that you have an opportunity to make such a determination. Here’s what
you do: You start a stopwatch just as the top of the Sun disappears below the smooth, unobstructed western horizon provided by a
calm ocean. You then stand, lifting your eyes by your height h (assume it = 1.70 m), and stop the watch when you again see the top of
the Sun disappear below the horizon. (Imagine your stopwatch reads 11.1 sec.) a) Draw a diagram that shows the geometry involved.
b) Explain how you could a triangle and approximation to estimate the size of the Earth. (The answer obtained is within 20% of the
actual value for the Earth’s radius). c) Compute an actual value for the Earth’s radius.
6) Assuming the Earth's orbit around the Sun is circular with a period of 365.25 days, determine its average orbital speed in km / hr.
7) Of course, besides revolving around the Sun, the Earth also rotates on its axis once a day. For someone located on the equator,
determine the rotational speed in km / hr.
8) News Item: Associated Press, June 1, 2008, New York -- Like a bolt out of nowhere, Usain Bolt is now the world's fastest man.
The Jamaican sprinter set the world record in the 100 meter dash Saturday night with a time of 9.72 seconds at the Reebok Grand
Prix...Determine Bolt's average speed during this race a) with respect to a co-ordinate system fixed to the ground there at the track,
b) with respect to a co-ordinate system fixed to the Sun. (Note: get a numerical answer for part a); comment on how you'd go about
working part b) without getting a numerical answer.)