*[This Instructable was authored with my colleague, Margaret Jensen, a high school science teacher.]*

One of the wonders of the Modern World is Google Maps and other online mapping services. At your fingertips, with your computer or you phone, you can instantly see a map of anywhere in the world, and find directions to your favorite coffee shop or pizzeria. If you've played around with these sites at all, you soon discover they have a satellite layer, showing high resolution images of virtually anywhere on Earth. A fun past time for many of us (especially our students) is looking at their own homes on these satellite images, and identifying their own cars and backyard gardens and recreation sites. A common question that arises in these explorations is, "when was this picture taken?" As it turns out, you can answer that question -- this Instructable will show you how.

** If used in the classroom setting:** this activity is suitable for high school science students who have a knowledge of geometry and trigonometry; in the college setting, it is suitable for intro physics or astronomy students.

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## Step 1: Underlying Principle: Seasonal Astronomy

Our original motivation for this project was inspired by teaching introductory astronomy. Astronomy classes often spend time learning about the motions of celestial objects in the sky, and how those motions change over time. Over the course of the year, the point on the horizon where the Sun rises and sets moves north and south, and its path through the sky is higher or lower in the sky depending on the season.

The reasons to learn about the motion of the Sun appeal to historical need -- the Sun's motion is an indicator of the seasons and can be used to plan when to plant and harvest crops. Such needs are not well suited to the modern world, where few of us are farmers, and more to the point, we have calendars. The changes in the Sun's position in the sky depend on the ** date** and

**, and change how shadows are positioned on the ground. Here we show you that measuring photographs of shadows and understanding the motion of the Sun through the sky with seasons can be used to date satellite photographs (something of interest to modern students, but not historically to farmers in ancient times).**

*time of day*Examples of the Sun's motion during a day in the Northern Hemisphere are shown above for different times of year. During the *northern summer*, the Sun's position has the following properties:

- it reaches higher altitudes in the sky
- it rises and sets north of the east and west cardinal directions
- it is visible for more than 12 hours a day

During the northern winter, the Sun's position has the properties:

- it reaches lower maximum altitude during the day
- it rises and sets over the horizon at points south of the cardinal east and west directions
- it has total visibility of less than 12 hours per day

The exact reasons for these changes and differences underlie this project, but are not described here. An introductory textbook on astronomy will discuss these changes in the context of the seasons, and how they arise from the tilt of the Earth's axis and its position in its orbit around the Sun.

## Step 2: Data Sheet

Here we provide a Data Sheet for you or students to collect and maintain the information needed to complete this project. It is provided here in this step to make it easier to follow along with the instructions in the Instructable.

There are several mathematical formulae required to complete this exercise. These formulae capture the intricacies of how the astronomical motion of the Earth causes observable changes of the Sun's position in the sky. They can be worked out with careful geometry, but that is far beyond the scope of this Instructable. They are provided here by fiat, in each step of the Instructable so you can make calculations from direct measurements. Here we provide a single PDF file with all the equations in one place, for quick reference; the various equations are referred to in the steps directly.

## Step 3: Pick a Landmark

The first task is to pick a landmark that you would like to date the satellite photograph of. If you look at wide area mosaics in satellite map layers, you will notice they have been made from images taken at different times. For instance, most of the areas are virtually cloudless, and the seasons indicated by trees and ground cover are not always consistent. A prominent indicator is the differences in shadows from one side of an image to another. Consider the image of an area of New York shown in the first image above. On the left side of the image the shadows clearly point in a different direction than those on the right hand side of an image! When were these two different images taken? Students are motivated by the same question when viewing areas that are familiar and important to them.

More often than not, we pick an image of a landmark we are familiar with or interested in, like our house or a local library or school. It doesn't matter what landmark you pick, except that it ** must show a shadow on the ground** in the satellite image. Here, for illustration purposes we pick a prominent national landmark, the Washington Monument.

Screen capture the satellite image you want to work with; the larger the image the better, as you are going to take measurements of the shadow and want it to be as large as possible. You can work either directly off of your screen capture in a program like Paint or Photoshop, or print it out and use a pencil and ruler to make your measurements.

Note down the Latitude and Longitude of the landmark. This can be measured off of digital maps (like Google Earth) or on USGS Topo Maps (either on paper or various digital interfaces), or directly with a GPS or GPS phone app at the landmark location. Note this down on the worksheet.

For the Washington Monument Wikipedia reports:

- LAT = 38d 53m 22s N = 38.88944 deg N
- LONG = 77d 2m 7s W = 77.03528 deg W

## Step 4: Landmark Height: Make a Theodolite

You also need to know the height of that landmark. For the Washington Monument, I can just look the height up (h = 169 meters). For a local landmark, the height can be determined using the *tangent method*.

The tangent method requires you to stand a known distance away from the landmark, then measure the angle between the horizontal and the top of the landmark. Measuring the angle is easily accomplished by building a simple *theodolite* from a drinking straw, a protractor, a length of string, and a heavy weight like a bolt or padlock.

Tie the string around the center of the straw, then tape the straw to the long, flat edge of the protractor with the string in the center. Tie the heavy weight to the free end of the string (see second photograph above).

To use the theodolite, hold the protractor upside down, with the straw on top and the string hanging down next to the curve of the protractor. Carefully look through the straw at the top of your landmark, allowing the string to always hang straight down without swinging. When you have the top of the landmark in view, pinch the string against the protractor, and see what angle the string is at (it may be helpful to have a friend do this part).

To get the angle above the horizontal (the angle "a" on the data sheet), take 90 degrees minus the reading under your string. This subtraction step is necessary because our theodolite uses the protractor upside down!

As noted on the data sheet, you can now compute the height of landmark from the formula:

- h = B * Tan(a)

where B is the distance you were standing from the landmark when you made the measurement with your theodolite (the "baseline").

*Edit*: One of our readers correctly noted that to be as accurate as possible, one should add the eye-height of the observer, Heye, to the calculated result:

- h = B*Tan(a) + Heye

## Step 5: Map Scale

To use your satellite image, you need to know the scale. On your satellite image, find an object that has a known length -- the side of a building or the width of a driveway works well, since you can measure this directly in the real world. On your data sheet, note down the true length as the "*Measured Length in the Real World*."

If you are using a printout: use a ruler to measure the length in centimeters of the known object as it appears on the map-- note this down on your Data Sheet as "*Measured Length on Map*."

If you are using an imaging program on the computer, measure the length in pixels of the known object. The object may not be perfectly vertical or horizontal in the image, so use your cursor to find the {x1,y1} position of one end, and the {x2,y2} position of the other end. The length of the object in pixels is then given by the Pythagorean Theorem:

- length in pixels = SQRT[ (x2 - x1)
^{2}+ (y2 - y1)^{2}]

The scale K of your map then is just:

- K = TRUE LENGTH / MAP LENGTH

To find the true length of any object on the map (like shadows, see the next step), you measure the length (in centimeters on your printout, or in pixels on the computer) and multiply it by the scale.

## Step 6: Measuring Shadows

We are now ready to measure the shadow off the map! We need two pieces of information: the length of the shadow *s*, and the angle the shadow makes compared to due North, *Z* (called the "solar azimuth"). Together with the information about the height and geographic location of the landmark, you can figure out exactly where the Sun was in the sky at the moment the picture was taken!

The length of the shadow can be measured directly from the base of the shadow, where it touches the landmark, to its apex. Note the length on the Data Sheet (in centimeters or pixels), and multiply it by the scale to get the true length *s* and note it on the Data Sheet as well.

For the angle measurement, we are interested in where the Sun is located, which is exactly ** opposite** the shadow. Measure the angle between due NORTH and the shadow of the landmark.

If the shadow is to the west of North, then the Sun's angular location is:

- Z = 180 - C

If the shadow is to the east of North, then the Sun's angular location is:

- Z = 180 + C.

Record the solar azimuth *Z* on the Data Sheet.

Finding the Solar Altitude From your knowledge of the shadow and the object casting the shadow, you can figure out what angle above the horizon the Sun is at. This is a geometric reconstruction, illustrated in the figure above. You have computed hte length of the shadow s in the step above, and computed the height h of the landmark from your theodolite measurements in Step 4. From these two numbers the altitude A of the Sun is:

- A = Arctan(h/s)

Record this number on the Data Sheet.

## Step 7: Finding the Time

There are well defined ways to turn Altitude and Azimuth into the sky equivalent of latitude and longitude (what astronomers call "declination" and "right ascension"). However, most of these methods require you to also know the time, which is exactly what we don't know about the satellite picture! Instead, we will use the known, repeating motions of the Earth and sky to figure out the time of the picture.

The basic idea here is to use two simple facts:

- Noon is the time when the Sun crosses a line that runs from due north to due south (your "local meridian")
- The sky rotates by a fixed amount every hour, so if you know how far from Noon the Sun is, you know how many hours before or after noon the picture was taken

At any given moment of time, the Sun lies on an arc in the sky that runs from the point over the North Pole of Earth to a point over the South Pole of Earth, and passes through the Sun. This arc is called the "*Solar meridian*." It is located by an angle PHI measured from the meridian that passes overhead to the Solar meridian at the time the satellite picture was taken. Owing to the limitations of inline equations here, the formula for the solar meridian is given in the first image above:

- PHI = see image above!

Compute this value and record it on the Data Sheet as PHI.

Given the value of PHI for the Sun's location, and the longitude of the landmark, the TIME that the satellite photograph was taken is

- UTC = 12h - (PHI + LONGITUDE)/15.04178

Here, both PHI and LONGITUDE should be in decimal degrees. If the landmark longitude is EAST then the longitude is subtracted from PHI:

- UTC = 12h - (PHI - LONGITUDE)/15.04178

## Step 8: Finding the Date

The date is defined by which arc in the sky the Sun moves along between sunrise and sunset (the "declination" DEC), which is calculated from the measurements as

- DEC = ArcSin( sin(A)*sin(lat) + cos(lat)*cos(Z)*cos(A) )

Record this result on the Data Sheet.

With the DEC, you can calculate the Day of the Year the image was taken. There are two possible solutions:

- D1 = 81 + (365/360)*Arcsin(DEC/23.44)

or

- D2 = 81 + (365/360)*[180 - Arcsin(DEC/23.44)]

Note that the way these formulae are written, the Arcsin function should be computed to return an angle in DEGREES not in RADIANS.

This gives a *numerical day* since the start of the year. To convert it into a *date* you can use an online numbered calendar like this one at NOAA.

Why are there two different days? As the Sun moves north and south through the sky with seasons it crosses one place as summer goes to winter, and again as winter turns into summer. An alternative to using the formulae above is to simply read the Day off the graph shown above. Find your solar declination on the horizontal axis and draw a vertical line. Where it crosses the red and blue lines are the two Days the Sun is in that sky location.

## Step 9: Example: Washington Monument

Throughout this instructable we've been referring to the Washington Monument as our example for carrying out this project. The images here show my Data Sheet for this example.

Congratulations! Hopefully you can use this method to look at satellite images of your favorite locations, your school, or your house.

We hope you enjoyed this, and have good luck using it. Let us know if you have any questions in the comments. We're very interested in problems you encounter, or any suggestions for improvements you may have!

Edit: We've added to this final step a writeup of this in journal paper format which contains more or less the same information as the instructable, but a few references, and more exposition. Let us know if you find it useful!

Judges Prize in the

Maps Challenge

## 18 Discussions

11 days ago

An

inclinometer, or clinometer, is an instrumentused formeasuring angles of slope/tilt and elevation/depression of an object with respect to gravity. https://en.wikipedia.org/wiki/Inclinometer14 days ago

looks like when measuring the heights on an object using the theodolite, you forgot to add in the height of the eyeball from the ground. In my case this would add 5.5 feet.

Reply 14 days ago

That is a good point. I was imagining tall things like dorms or clock towers, which provides an opportunity to talk about errors and what matters. But on ahort things like your house this could be a huge effect. I will add a note!

Reply 13 days ago

Your edit above says to subtract the eye height. As noted here, should you not add it?

Reply 13 days ago

Yep, of course. I was addled when I edited it the first time. fixed now. :-)

13 days ago

Fun instructable, thanks so much for sharing! As @GaryG8 was saying, I believe you would *add* eye-height to object height measured with a theodolite if using eye horizon as the reference, not subtract it. Thanks again!

Reply 13 days ago

Somehow I was addled the first time I put it in the step. It says the right thing now. ;-)

13 days ago

yes you need to add into height the height from ground to the theodolite. when i am cutting down trees, i use the theodolite method to determine height so i can determine the tree won’t hit anything i care about. Forgetting to add in ground to eye could be costly.

14 days ago

This is amazing - your students are so lucky to have you.

Reply 14 days ago

We're glad you enjoyed it!

14 days ago on Step 9

I hope the Be Nice policy doesn't mean that it is taboo to point out inaccuracies in an instructable such as this. The underlying idea of this instructable, to help students and readers become aware of the fascinating movements of celestial bodies and, by extension, aware also of the whole solar system, is very worthwhile, especially when it helps to convey that this knowledge has powerful practical applications. It is also important they not think that something complex is much simpler than it actually is, especially if they might unwittingly try to apply a simplified idea in a case where an incorrect result might have serious consequences.

There are two problems here, one fundamental and another that is a matter of seemingly small detail.

The fundamental problem is that the earth's orbit is an ellipse, not a circle, and its angular movement revolving around the sun is not exactly proportional to the elapsed calendar days. The equations for the days D1 and D2 would be correct if the orbit were an exact circle. The ellipse means that D1 and D2 might be wrong by up to 4 days, depending on the day of the year. The ellipse also means the time of day, calculated by direction of a shadow, is different from true time, variously up to 15 minutes, also depending on day of the year. This variation is called "The Equation of Time." That has nothing to do with the larger differences due to time zones and daylight saving time (arbitrary conventions) which mean your watch might read more than an hour different from the time calculated by direction of a shadow.

The problems of small detail can have even greater effects than the elliptical orbit. You probably cannot get a measure of shadow length to an accuracy better than 2%, if that, from a Google Earth image. Even if measuring on the ground with a tape, the shadow edge is fuzzy, not sharp and crisp like the edge of the object that casts the shadow. More importantly, the ground on which the shadow is cast may not be exactly level, which can easily mean errors much larger than 2%. A 2% error in shadow length can translate to more than 2 weeks of error in the calculated date when we are near either the winter or summer solstice.

As said in the beginning, it is very valuable to impart the concept that the apparent and actual movements of celestial bodies, such as the sun, are known and documented to amazing precision. This formed the basis of celestial navigation for ships at sea, which is a precise technology, for over a hundred years, but has almost entirely been replaced by electronic GPS systems today. Practical celestial navigation relies on data that fully takes the earth's elliptical orbit into account.

It would be appropriate if this instructable would add some asterisks and footnotes to the effect that the calculations given are approximations because the earth's elliptical orbit differs slightly from a circle that is implied by the equations, that shadow measurement will inherently have some errors and may be distorted by sloping ground..

14 days ago on Step 9

As I recall, Eratosthenes estimated Earth's circumference using similar techniques---he posted it on

Didaktikós😁Reply 14 days ago

LOL! I have often thought about writing a circumference of the Earth Instructable. I have some worked out but haven't gone out to collect pix and data for a write-up. :-)

14 days ago

Great exercise for the school holidays. very well written.

Reply 14 days ago

Thank you! Glad you like it!

14 days ago

Wonderful Instructable!

redrok

Reply 14 days ago

Thank you!

14 days ago

looks like when measuring the heights on an object using the theodolite, you forgot to add in the height of the eyeball from the ground. In my case this would add 5.5 feet.