Understanding Earth Tilt, Sun's Position and Lattitude Calculation

I've been trying to figure out how to calculate the latitude of my location with just the shadow of a stick cast by the sun. I am doing this under the assumption that I am stranded in some place with no almanac or any other data. (not even day or month).

I want to find the latitude roughly. But I have problems picturing the globe along with it's tilted axis and the angles formed by the shadow.

Could you help me understanding the concept and give me a definitive picture of angles that can be calculated. A perspective like a side-view of the earth and sun (orbiting plane - perpendicular to the screen) would help me understand much better.

Here is my efforts to understand so far -

I am not an expert nor have a good background in maths (my maths teacher ruined my life). So please explain me in simple words.

• Without knowing the month of the year you are more or less stuck. Say your stick casts no shadow. It may be the June solstice and you are 23.4 degrees North of the equator or the December solstice and you are 23.4 degrees South of the equator. Or it might be an equinox, and you are on the equator. Unless you know the date you can't be more accurate than that with a simple stick. – James K Sep 25 '15 at 16:33
• If you have figured out the answer, could you please post it? It is perfectly acceptable to post an answer to your own question. – Donald.McLean Sep 30 '15 at 14:34

I've been trying to figure out how to calculate the latitude of my location with just the shadow of a stick cast by the sun. I am doing this under the assumption that I am stranded in some place with no almanac or any other data. (not even day or month).

You really cannot find your latitude with only the sun and sticks at your disposal and no declination data or time data. Maybe if you could record your shadow over a period of a day, you could tell if you are in the northern or southern hemisphere by observing which way the shadow moves. If it moves clockwise, you are in northern hemisphere.

And if you could repeat and observe for 365 days (from the day you are stranded, assuming you are very accurately measuring the angles) you could find if you are within the tropic of cancer or above it, as you might not have the sun right on top of your stick over the year if you are above the 23.5 N parallel.

And that's as far as you can get. To determine the precise location, you will need the sun's declination at any given time. May be you will get lucky, if the day you are measuring the angle and the equinox matches, but you wouldn't know it.

Could you help me understanding the concept and give me a definitive picture of angles that can be calculated.

A picture is worth thousand words. And an interactive demonstration... It's worth at least a hundred 2D diagrams.

• One can determine latitude my making observations over the course of a year. – David Hammen Sep 26 '15 at 14:27
• Interactive helped me a lot. Thank you so much. – Venkatesh MG Oct 18 '15 at 19:20
• I have been looking so long for an animation like that. Thanks a bunch! – Ramon Martinez Jan 30 '17 at 15:06

Assuming you aren't north of the Arctic Circle or south of the Antarctic Circle, you can determine your latitude my making observations throughout the course of a day, and over the course of a year. You'll need

• A rather straight stick,
• A fairly flat piece of ground,
• A plumb bob (which you can make out of string and a rock),
• Some small pebbles to mark the tip of the shadow of the stick over the course of a year at solar noon, and
• A trigonometry table or a calculator.

Use your plumb bob to ensure your stick is as close to vertical as you can make it. You'll want to place the stick in exactly the same place every day.

The first thing you'll want to find is the north-south line that passes through the base of the stick. To do this, place a pebble at the tip of the stick's shadow when the shadow is at its shortest. If you do this perfectly, you'll have the north-south line on day number one. You almost certainly won't do this perfectly, so you'll need to repeat this for a few days. Because the Sun rises more or less in the east and sets in the west, you also know which way is north and which is south. You have a compass.

You'll also have a very rough idea of your latitude. If the Sun sets to the left of the north-south line, you know you are somewhere north of 23.44 south latitude. If it sets to the right, you are south south of 23.44 north latitude.

If you want a better idea of your latitude, keep doing this until the day you see the Sun rise exactly in the East. This happens twice a year, typically on March 20 and September 23. Use a string to measure the length of the stick's shadow when the tip of the shadow crosses the north-south line. You do not need a ruler; all you need to know is the ratio of the shadow's length to the stick's length. Now it's a matter of trigonometry: $$\phi = \arctan\left(\frac {\text{shadow length}}{\text{stick length}}\right)$$