Measuring the Earth with the Sun

Measuring the Earth with the Sun
Projects and Activities Outside the Classroom
Introduction
Some Alternatives
Getting Started: The Earth is a Ball
1.Find North
Stick & Shadow
Use a Pipehenge
Compass
Use Midday
2.Measure Your Noon Sun Angle
Direct
Indirect
3.Find Your Latitude
Send Your Data to Stardome
4.Find Another Latitude
5.Find North-South Distance Between the Two Locations
6.Estimate Circumference of the Earth
Repeat in Spring
Selection of Resources/Links
* If you weren’t a hedgehog … if I weren’t a haemophiliac: 250 cartoons by
Andrew Weldon, 2006; Allen & Unwin, NSW, Australia.
How Big is the Earth?
Introduction
The aim of this project is to make an estimate of the circumference of the Earth. This is based on the
method employed by Eratosthenes over two millennia ago, but adapted for use in New Zealand
schools. Essentially, the latitudes of two locations north-south of each other and the distance between
them provides a ratio giving the Earth’s circumference.
Each of the major activities must be finished in turn before being able to find the size of the Earth:
1. Find north/midday
2. Measure noon Sun angle
3. Work out your local latitude
4. Get someone else’s latitude
5. Find north-south distance between your two locations
6. Calculate the size of the Earth
Each of these steps can be used as standalone units of study, and tailored for the learning level of
individual classes. It is hoped that teachers will make the most of the opportunities afforded at all
stages of the project, also learning about the nature of science, how science works, and the
cooperative nature of science.
Finding the circumference of the Earth using two latitudes requires measuring the altitude of the Sun
at midday at the autumnal or spring equinox, or the summer or winter solstice.
Green and blue lines indicate path of Sun at winter and summer solstices, respectively (equivalent to
the tropics of cancer and capricorn). The angle between the celestial equator and the south celestial
pole is 90°.
Your latitude can be calculated at these times of the year because you know where the Sun is in
relation to the South Celestial Pole (which is the same angle as your latitude). At the equinoxes the
noon Sun altitude will be at your co-latitude (90° − latitude). At the summer solstice it will be at your
co-latitude plus 23.5° (tilt of Earth’s axis), and at the winter solstice it will be at your co-latitude
minus 23.5° (see page 8).
Someone else a reasonable distance due north or south of you also has to find their latitude, by
measuring the midday solar altitude on the same day as you. Once you know the north-south distance
(in kilometres) between your two locations, some straightforward arithmetic solving the following
proportion will provide the desired estimate:
(angle between locations)/360° = (distance between locations)/(Earth circumference)
The observing/measuring period is a week either side (16-27 June) of the winter solstice, but as close
to 21 June as practical (taking into account cloud cover, school timetables, readiness, etc.).
David Britten
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How Big is the Earth?
Some Alternatives
Because we only know the position of the Sun at the equinoxes and solstices (without some tricky
trigonometry), finding your latitude and completing the project at other times can only be performed
in ‘practice mode’:

Use Google Earth: decimal degrees are easier to use than degrees, minutes, seconds

Use online latitude-distance calculators, e.g. http://www.geodistance.com/ and
http://www.freemaptools.com/measure-distance.htm

Use a map: old-style version of Google Earth

Use GPS: this will require some actual travel between locations!
Using these methods also gives us a second latitude (choose one as far north or south of your location
as practicable) and the distance between the two locations. Calculating the size of the Earth can now
be done as explained above.
Getting Started: The Earth is a Ball
Before thinking about the size of the Earth, students need to have a good mental model of the Earth,
and their place on it – particularly regarding north/south and the direction of the Sun. An Earth globe
should be present if possible in the classroom during all stages of “Out of This World” study, for the
children (and the teacher) to refer to.
Explore the nature of balls – their shape, comparison of different sizes, difficulty of measuring from
one side to another (compared with measuring blocks shaped as cubes, bricks or pyramids).
The ‘Earth is a Ball’ activity is a great introduction, and should be completed with younger students.
Following on from this activity, discuss how this could be used to find out how far it is all the way
around the Earth. Placing two model people on the globe north-south of each other may help children
visualise their own positions on the Earth.
Measuring from the Equator to the South Pole is a quarter of the whole circle around the Earth.
Measuring between two cities in New Zealand would be a smaller fraction of this circumference.
Ideally, every classroom should have an Earth globe permanently available, providing easy student
access to our three-dimensional world.
David Britten
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How Big is the Earth?
1. Find North
To obtain an accurate fix for your latitude, the altitude of the Sun must be measured at true midday
(local noon), i.e. as it crosses the local meridian (north-south line).
Stick & Shadow
This method has the advantage of not needing to know the time of noon. Although it is easiest using a
shadow stick to find north at the equinoxes (in March and September), it can be used at any time of
the year.
Frank Andrews, who worked for many years at Carter Observatory in conjunction with the School
Science Advisory Service, advocates using a tall stick or pole and frequent measurements. The Sun
moves 1 degree across the sky every 4 minutes (1,440 minutes ÷ 360°), so measuring the shadow
length at intervals longer than 4 minutes will give an accuracy larger than 1°.
A suggested method is to use a 2 metre stick set perpendicular on a flat surface, which provides a long
enough shadow for good measurements. The tip of the Sun’s shadow is marked and measured at 20minute intervals up till 11:45am, then at 4-minute intervals till 12:15pm, followed by further 20minute measurements (times are NZST, not daylight saving time). A line from the base of the pole
through the shortest shadow points north-south.
Plot of Sun’s shadow length, showing it is shortest at midday.
Notice that the shadow points make a curve away from the pole between March and September, and a
curve towards the pole from September to March. They make a straight line perpendicular to the
north/south line (i.e. east-west) at mid-September and mid-March (the equinoxes, when the Sun rises
and sets due east and west).
To check and improve this, take two points equally spaced in time on either side of the noon line.
Make a line joining these two points, bisect this line and join the mid-point by a straight line to the
base of the shadow stick (right angle). This should give the same north-south line.
NOTE:
The tip of the shadow is not distinct. This penumbral area is only
partially in shadow, which affects the accuracy measuring the
shadow length. A 1m stick makes about a 1cm penumbral region,
which gets longer with a longer stick.
David Britten
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How Big is the Earth?
Use a Pipehenge
The Meridian pole of a properly installed Pipehenge (http://www.pipehenge.com/new/index.html)
points north and south. A line from the blue South pole along the ground to the red North pole will
point north.
Compass
A GPS device can be used as a compass. If a magnetic compass is used, the current deviation of
magnetic north from true North must be taken into account.
Use Midday
The Sun is due north at midday, but using your clock to find noon at your location is not
straightforward.
Firstly, your clock must be set as near to NZST as possible, using radio time pips, or an online source
such as www.timeanddate.com
Secondly, NZ Standard Time is set exactly 12 hours ahead of the time at the Greenwich meridian.
However, all of NZ is less than 180° East of Greenwich, so midday is actually never at 12pm.
Thirdly, the Earth speeds up and slows down during its annual orbit around the Sun. This causes the
length of the day to get longer and shorter, and the time of midday to be earlier and later during the
year. As civil time is an averaged time to keep the hours, minutes and seconds always the same
duration, a correction called the Equation of Time needs to be applied.
Calculate Midday from Sunrise-Sunset Data
Look up the times for sunrise and sunset in your local newspaper for the day of your measurement.
The Sun will cross your local meridian at midway between these two times.
Retrieve Time of Midday from Published Data
The time (and altitude) of local solar transit can be obtained from published data, e.g. New Zealand
Astronomical Yearbook (Stardome, 2014).
David Britten
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How Big is the Earth?
2. Measure Your Noon Sun Angle
The first measurement when finding your latitude is the altitude of the Sun in the sky at midday. The
angle of the Sun can be measured either directly or indirectly.
Direct
A protractor and plumb line make a simple clinometer (for measuring angle of elevation).
To align your clinometer (or similar device) with the Sun, adjust its orientation until it makes the
smallest shadow on a flat surface below. If using a straw as in the illustration, adjust until the Sun
shines right through the straw, with the straw casting a shadow shaped as a circular ring.
WARNING: Don’t look at or near the Sun! The Sun can permanently damage eyesight.
Indirect
Measure length of stick/pole and the length of its noontime shadow.
a) Draw this on paper and measure angle with a protractor.
b) Calculate the angle from ratio of the lengths (tangent function).
Try to make a number of noon Sun measurements on successive days either side of the winter
solstice, as some days are bound to be cloudy, making observations difficult or impossible. Compare
the measurements that are made, and take the average of the reliable observations as the value to use.
David Britten
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How Big is the Earth?
3. Find Your Latitude
Your latitude is the same angle as the altitude of the South Celestial Pole, and is calculated from your
measured noon Sun angle.
The angle of the Earth’s tilt is rounded in these formulas to 23° in anticipation that student
measurements will not be accurate to less than a whole degree:
Latitude = 90 – (noon Sun angle at winter solstice + 23)
For example, if you measure the noon angle Sun at 7°, then your latitude will be:
90 – (7 + 23) = 60°S
The calculation when performing this at the summer solstice (23 December) is:
Latitude = 90 – (Noon Sun angle − 23)
At the equinoxes (23 September and 23 March) the formula becomes:
Latitude = 90 – Noon Sun angle
David Britten
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How Big is the Earth?
Send Your Data to Stardome
Your project cannot be completed without measurements from different locations up and
down the country. Sending your results to Stardome will allow other schools to complete their
calculations.
Send the following data via email to education@stardome.org.nz:

School name, suburb, city, post code. If you don’t know your post code, it can be found using
the NZ Post web pages:
* http://www.nzpost.co.nz/tools/address-postcodefinder?utm_source=sitebar_ph&utm_medium=sitebar&utm_campaign=apf
* http://www.nzpost.co.nz/sending-within-nz/how-to-address-mail/postcodes/postcode-maps

Calculated latitude (decimal degrees; not degrees, minutes, seconds)

Method used to find latitude
4. Find Another Latitude
Select the latitude of another school from the Stardome web site. Choose a school that is as close to
your longitude as possible, i.e. due north or due south of your location, which will help estimating an
accurate distance between your two locations. Choosing another school as far north or south away
from your location will also improve results.
NOTE:
Frank Andrews recalls: We tested our results by looking at the errors and noting how our result
improved with larger separations between stations. We linked up with a school in Fiji at one stage and
by using shared results collected over five days found the value for the circumference and diameter to
within ±22km. It was interesting to see how increasing the base line reduced the scatter in the results.
Once we had a baseline of more than about 900 km the increase in accuracy did not increase much by
extending the length of the line.
5. Find North-South Distance Between the Two Locations
The circumference of a sphere is a circle with a diameter that passes through the centre of the sphere.
Such circles on the Earth are called great circles. Lines of longitude are horizontal circles that are
small near the poles and larger nearer the equator. The only line of longitude that is a great circle is
the equator.
Lines of latitude are all great circles, passing through both the north pole and the south pole. The
distance between two locations lying north-south of each other is a segment of the great circle (line of
latitude) passing through both locations.
Using Google Earth (Tools/Ruler/Line) is the easiest and probably most accurate method for finding
the straight line distance between two locations. To check how close to north-south locations are, turn
on grid lines (View/Grid; Ctrl-L).
If you don’t have Google Earth, online latitude-distance calculators can be used instead:
http://www.freemaptools.com/measure-distance.htm
http://www.geodistance.com/
David Britten
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How Big is the Earth?
6. Estimate Circumference of the Earth
Calculate ratios of the angle between the two locations (difference between their latitudes) and a full
circle versus the distance between the two locations and the circumference of the Earth. The answer
should be within a few hundred kilometres of the currently accepted value.
So,
(
)
EXAMPLE:
If your two locations north/south of each other are 2 degrees (latitude) apart, and the north/south
distance is measured at 224km, then the circumference of the Earth through those locations must be:
224 × 180 = 40,320km
which is just 312km larger than the accepted value of 40,007.86km; i.e. less than 1% error.
NOTES:
It may seem surprising that the circumference of the Earth is almost exactly 40,000km.This is because
the metre was originally defined as 1/10-millionth of the distance between the north pole and the
equator along the meridian passing through Paris, making a quarter of the complete circle 10,000km.
For advanced students: the spin of the Earth causes its middle to bulge outwards slightly, creating an
oblate rather than perfectly spherical shape. The north and south poles are therefore closer to the
centre of the Earth than the equator. The equatorial circumference is about 67km greater than the
circumference measured in our project, i.e. a great circle through the north and south poles.
Repeat in Spring
Any scientific experiment needs to be duplicated to verify the conclusions. Repeat this project at the
spring equinox (September 23) to confirm and/or improve the results obtained at the winter solstice.
The formula is given on page 8.
David Britten
9
How Big is the Earth?
Selection of Resources/Links
Solstices and Equinoxes
http://scijinks.jpl.nasa.gov/solstice
Global school cooperation project
http://www.eaae-astronomy.org/eratosthenes/
K12 Measure the Earth's Circumference
http://teachers.egfi-k12.org/lesson-measure-the-earths-circumference/
The Noon Day Project
http://www.ciese.org/curriculum/noonday/overview/
The Measurement of the Earth's Circumference: Hands On Activity
http://www.juliantrubin.com/bigten/eratosthenes.html
Pipehenge
http://www.pipehenge.com/new
Earth Space Simulator (ESS) & Portable Multi Latitude Module (PML)
http://www.pipehenge.com/new/f/ess.html and http://www.pipehenge.com/new/f/pml.html
Clinometer
http://astronavigationdemystified.com/making-a-mini-clinometer/
http://nzmaths.co.nz/resource/tree-mendous-measuring
David Britten
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