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Flat-Earth-Academy: Longitude, Latitude, and the Sun

Basics of Navigation
Do you know where you are?
Find out with a sextant!

(Page's URL: latlongsun.htm)

In this page, we'll look at how you might find out your latitude and longitude... in the before GPS, before Google era.

Alas, we must learn to walk before we can try to run. There's quite a bit of the rather dull "learn to walk" stuff, I fear... but I hope you'll struggle through it, get to the "running" part.

Latitude and longitude? "They" "drew" imaginary lines all over the earth. Any spot on the earth can be "named" by giving its lat/long. The Greenwich Observatory in London, England is at about lat 51 degrees North (of the equator), longitude 0 degrees. Quito, in Ecuador, is about 0.1 degree South, 78 degrees West (of Greenwich). The North pole is at 90 degrees North, and does not have a meaningful latitude number! (To get lat/long the easy way: Call something in maps.google.com. Right click somewhere. The lat/long of the place you clicked will pop up in an info box. Just click on the lat/long numbers to copy them to your clipboard.)

Help with lat/long at Wikipedia

Apologies to readers in the southern hemisphere. You will have to turn bits of this page "upside down" to make them right for you.

Apologies to a few more readers: if you are north of the equator, but still south of latitude 24, parts of this won't be quite right, for parts of your year. But don't despair. It will mostly be right, and I think you'll be able to spot the places where you need to make allowances at those times for where you are.

Imagine standing with your back to the sun, in the middle part of the day. At least north of the equator, and preferably north of latitude 24.

The sun will be roughly south of you, won't it?

North/ East/ South/ West isn't as easy as you might think it is! But a general idea of them will do for this. Whew.

Riddle: I let you blindfold me, but *I* get to "magic" the two of use to any place on earth. You then "spin" to face which ever way you wish to. I will be able to tell you what direction you are facing. How?

So, you're standing there, past early morning, before early evening. North of Latitude 24. No further north than the top of England. All of these very "soft". One step north of England won't ruin things.

With your back to the sun.

You are facing roughly north.

The sun will have risen to your right, won't it? In, roughly speaking, the east. And it will set in the west.

You know this!

And you know that the sun is "higher in the sky" in the middle of the day.

And, for the same time of DAY, it is higher in the sky in summer than it was at that time, in that place, in the winter.

Theory and Practice

The page you are reading will cover some theory. If you want to see what happened when I tried to use it in practice, I have a page for you with the details of an attempt to measure a latitude and longitude from where the sun was in the sky.

So far, so good?

A five year old knows the first few things I've listed, and a seven year old may well be able to tell you about the sun being higher in the sky in summer than in winter.

HOW high in the sky??

How could you "put a number on" how high the sun is in the sky?

You can't say it is "50cm" high, can you?

By the way: DO NOT LOOK AT THE SUN!

I'm going to say "Do not look at the sun" a number of times in this. Just don't. Don't interpret anything I say here to suggest that you should. Well. Unless you can get ahold of a sextant, and if you can, be sure to use the filters properly.

Going back to "How high in the sky?"

IMAGINE...but don't do it facing the sun, and pointing your left arm at the sun, and your right arm at the horizon.

Your arms would make an ANGLE! (To someone looking at you from one side of you.) It would be a good measure of "how high the sun is".

While we're at it... "left" and "right"...

So... we can say "how high" with an angle.

Now imagine pointing one arm to where the sun rose, and the other arm at the point on the horizon precisely below where it is now.

Again... your arms make an angle. But this one is most easily seen by a drone hovering directly above you, looking down on your head.

Those two angles are the heart of the secret to figuring out where you are in the planet

Figuring out what lat/ long lines you are at on the planet.... without using Google! Or GPS!

Oh. You also need to know the date, and the time of day.

More better

This next bit may start to give you a headache, or tempt you to give this up as a waste of time.

I hope it won't. I hope you will find the complexity amusing. There won't be test on it!

You don't need to understand it all to be able to get a rough idea of your lat/long.

Every angle has a vertex. (The pointy bit.) And two arms.

For our "how far left/right" number, we were at the vertex. One arm pointed towards the sun. (Well, to the point on the horizon below it.) The other, could point to where the sun rose, but for a variety of reasons, that's not a very good direction to reference the sun from

"South" is better.

For rough calculations of your lat/long, "any old south" will do.

"Any old south"???

There are two basic "souths", and a third thing that is related, and is important to precise lat/long determinations.

You have probably heard of "true" north and "magnetic" north? (See Wikipedia, if not.)

And that makes for two "souths".

Most people will tell you that the sun is to the south of you at noon.

It is. Sort of. Depending on what you mean by "south". Depending on what you mean by "noon"!

We'll come back to this. Sorry! But first...

What is "noon"??

"Noon" is when your clock says 12:00.

However, there's another sort of "noon". When the sun is at the highest in the sky that it will achieve in the day is "solar noon". At that time, it will be exactly towards "true" south, by the way. (It's all connected... which eventually, if you think about all of this long enough, you can see.)

So! You have a way to find true south for yourself.

Anyone can get a compass, and somehow look up the difference between magnetic and true north for where you are, and determine true south that way.

Ummm... but... you need to know your lat/long to get the magnetic/true difference. And you're not doing it yourself. And the point of all this was to let you work out your lat/long.

Back to "where is the sun"?...

So, we could say where the sun is, both "up and down" and "left and right" with the two angles mentioned.

But how can we measure them?

Remember I asked you to imagine yourself standing with your back to the sun?

Your shadow!

It shows the "towards the sun" part of the "left/right" angle very clearly, and the "towards south" part is easy too.

But the "how high" angle is pretty easy. The shorter your shadow, the higher the sun is in the sky.

I'm just getting going on this page. I will try to come back to the matter of turning the length of your shadow into an angle. But if you work that out for yourself before I get to it, you could well have had some fun doing that. Give it a try!

A big pain: The "tilt" of the earth's axis

You know the earth spins?

If you start a stopwatch on Monday when the sun is as high as it will get in the sky that day, and wait very nearly (but not exactly) 24 hours, you will find that noon Tuesday has arrived. The earth spins once every (approximately) 24 hours.

To make something like a bicycle wheel spin, you need an axle.

The earth doesn't have "an axle", but there is a line "around" which it spins. That line is called the axis of the earth's rotation. (Spinning.)

If you could slip a HUGE sheet of plywood into the earth, so that it came out all around the earth at the earth's equator, that rigid sheet would be at right angles to the earth's axis.

An imaginary rigid "sheet" like that is called a "plane". The phrase "the plane of the earth's equator" refers to something like what I described a moment ago.

If you "tip" the axis or the plane of the equator, the other tips too. Obvious, I hope?

Hang in there! We're getting to the point!...

The earth's rotation is in a certain plane. The earth's orbit (around the sun) is also in a plane. A different plane. The amount of difference between the two changes (according to a strict pattern) over the course of the year.

One day in the summer day is longer than any other. On that day, the two planes are at their most out-of-line, with the North Pole tipped TOWARDS from the sun. On the shortest day, six months later, the North Pole is tipped as far AWAY from the sun as it gets. These are the two "solstices".

Half way between the two, there are two days when the two plane are NOT out of line. The plane of the earth's equator (90 degrees "away" from the axis of its rotation) is, on those days, the same as the plane of its orbit around the sun.

T.H.I.N.K.

Always try to think when reading things.

If you think about the following, I believe this "the earth tips" stuff will be easier to believe in...

1) I said that the sun's position in the sky can tell you where you are on the earth, your lat/long.

2) You know that the sun's position in the sky, for the same time of day, changes over the course of the year. If your viewing from the same place "over the course of the year", SOMETHING else must be going on!

That "something else" it the tilt of the earth's axis, which, compared to the sun, points one way at one time, another way at another.

Whew! We can almost all that behind for now.

Before we move on, I will just mention that the change gives rise to the "thing" called "declination". We'll need that later.

Take a break!

This would be a good time to take a break. Or at least a moment to skim through what you've read so far, pin down what's clear to you, what isn't quite as clear as you hope it will be by the time you're done.

Back to work

Remember that back at the beginning, I said we'd "learn to walk", and then get to the more fun "running"? Good news! We're nearing the end of the "walking".

We know a bit about...

Saying "where" the sun is in the sky.

Details of the pattern of where it is, at different times of the day, and of the year.

We know a bit more about the ordinal directions (North, etc.)

We know a bit about time... for instance, that "solar noon" is when the sun is as high in the sky as it will get that day.

Time

England has one time zone. If two people in England what to watch the ten o'clock news when it is first broadcast, they could use a telephone to remind each other "switch the TV on now."

If one of those people is in London, and the other in Bristol (about 100 miles west of London), if they had both set their watches to solar noon, they would find that their watches did not agree. (Solar noon in Bristol comes slightly after solar noon in London.

Does this seem strange??? You know that "noon" (be it solar or any other) in, say New York, is a long time after noon in London, don't you?

In our every day lives, we use a "standard time", that is shared across wide regions.

For calculating our position on the globe, we need to look at time the way an astronomer does. "Time" for the astronomer is about the sun "moving across" the sky. (Well, appearing to. "Of course", is it more to do with the earth spinning while the sun "sits still". (It doesn't, but that's a story for another day.)

While I'm being difficult: Here's another "bet you didn't know that": The number of seconds (as used by you and me and the scientists, too... in most instances) between solar noon on, say 4 July and noon on 5 July is not the same as the number of seconds between the noons of two other days in the year.

Don't worry much about what I just said... but it will come back to be a nuisance later.

At last!

[image latlonsun_alti_at_diff_lat.jpg]

The diagram shows three people... one each at A, B and C.

The sun's rays arrive from so very nearly "the same" direction that even if it were a point source, you wouldn't see the divergence. (Also, as the sun is bigger than the earth, they are even closer to all being parallel.)

Look closely! You can see that the angles... "x", "y" and "z"... between the direction the sun's rays are coming from and the LOCAL horizons at A and B and C are different

Shezam!, as Forest Gump would have said.

That is the basis of how the position of the sun in the sky can tell you your latitude!

The rest of the story...

For the moment, "the rest of the story" will have to come another time. How soon that is will depend on part on what interest is shown in this.

Feel free to send an email with comments. That would help get me back to this!

Contact details below.

If you can't wait, have a look at Bowditch's American Practical Navigator, the 1888 edition. (If shows a blank page, use the controls on the webpage to turn to a different page.)

Want a chuckle?

The above theory is all well and good. You may be amused by my page about my attempt to measure a latitude and longitude from where the sun was in the sky. (Yes- that's the page I mentioned earlier, at the top of the page you are currently reading.)

Sketches of things to come..

I have so far badly neglected the complementary question: "How do you find your longitude? (I will explain more clearly another time, but for now...)

The "simple" answer isn't hard to understand... when you start to think about it correctly. (Don't worry that you are being dim if the following poor explanation doesn't suffice.)

However, the "simple" answer depends on something I'd rather not have to depend upon: Knowing the time at Greenwich England, when you aren't there.

First: You figure out precisely when your local solar noon is/ was.

To go with that, you need to know the time in Greenwich England at the moment you had your local solar noon. (That's the bit I don't want to have to rely upon.)

In our modern world, that, of course, it not hard to know. In "the good old days", it meant carrying a terribly accurate clock with you from Greenwich, to do it in the "obvious" way.

Apologies to readers in the southern hemisphere. You will have to turn bits of this page "upside down" to make them right for you.

The "un-obvious way" was to carry the time with you from a place for which you already knew the longitude. From the longitude there, and the time of solar noon there, you could work out what the relevant solar noon at Greenwich had been.

So... you have the time of local solar noon where you are... on a clock that showed noon when Greenwich had it's solar noon.

Let's say that your solar noon came 12 hours after Greenwich's solar noon.

In that case, you are half way around the world from Greenwich, aren't you?

(Ponder and explore that until you see it is true.)

That's the simple case.

What if your solar noon was six hours after Greenwich's? You are one quarter of the way around the world, to the west, from Greenwich.

And you'd use the same logic for any other differences in time. The arithmetic is a little messier, but the principle is the same.

As I said- the basic principles are "simple", when you pull away the tedious details which you NEED to allow for, if you are going to get and accurate answer.

And here's something I didn't know, which will matter, if you are "cheating" by using "GPS time"... GPS time and UTC time are NOT the same! In July 2022, they differ by about 18 seconds. And yes, 18 seconds matters here! Yikes. (I'll say more about all of that, too, when I take this page to a "second edition". Sorry I can't do it now.)

I learned about this when a kind friend who knows about these things sent me somehting I've edited slightly, but was quite like...

Did you know that GPS does not actually use UTC? GPS time
"started" on 1/1/1980, at the time that UTC time was THEN. But
GPS time has since diverged from UTC time by 18 seconds. This
is because GPS time does not add leap seconds. So to set your
clocks from a raw GPS signal you need to know the current leap
second adjustment. I'm not sure, but I think the GPS system
transmits the current leap second adjustment
in it's data channel.

Leap seconds is also an interesting topic when discussing NTP.
There are different ways an NTP implementation will adjust
a leap second. Some just add or remove a second at midnight
but this can introduce horrible unexpected bugs. Other
implementations speed up or slow down hours before to slowly
get there. What's interesting is when we're approaching a leap
second event NTP servers on the internet can and do diverge,
potentially introducing more strange bugs.

A Google search tells me...

"NTP is short for Network Time Protocol. And that NTP an internet protocol used to synchronize with computer clock time sources in a network. It belongs to and is one of the oldest parts of the TCP/IP suite. The term NTP applies to both the protocol and the client-server programs that run on computers."

I assume it was THIS "NTP" that my friend was referring to.

There ARE situations and systems where disagreements over "the time" of "only" a second DO matter! Even if... especially if!... they "only" occur for short periods at infrequent itervals.

----------
I believe there are ways to determine latitude that do NOT depend on a clock. They provide other ways of knowing what the time is in Greenwich various places in the world. They avoid the need for a clock that can keep accurate time even over long periods, and periods where the clock is being "tossed about" on a journey.

I will try to research them, write them up.

For extra credit!

You can do it! Let's say you are on the equator. And you make a 60 second error in when solar noon occured. How many miles will this change your idea of your latitude by?

That first problem isn't very hard if you look up the circumference of the earth... are you going to fuss about the fact that the earth isn't a perfect sphere? You might want to... ! (I wouldn't, though. But like so many things here- when you think carefully about "everyday" thigs, it turns out they aren't as simple as you thought. In a very real sense, the earth doesn't have a diameter. (The diameter around the equator is different from the pole-to-pole diameter.)

But! What if you are at latitude 80 and make a 60 second error in the time of solar noon. How many miles wrong will your latitude answer be in that case? (Hint: You have to figure out the circumference of the lie of latitude for 80 degrees. Help: Use 3959 miles for the radius of the earth. (That's roughly right, again pretending the earth is a perfect sphere.)

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Another "unimportant" tangent... Have you heard of NAUTICAL miles? (A nautical mile is 1.15 statute miles long.)

The nice thing about nautical miles is that if you move one nautical mile along a line of latitude, you will have travelled one "minute"... as in 1/60th of a degree. ("arc-minute" (not minute-as-60th-of-an-hour(time)), and "degree-as-a-measure-of-angle" (not temperature.)

It's a pity that nothing so simple works to connect distance and movement east and west, but navigators often want to convert from changes between angles of latitude (north/south movement) to distances. Using nautical miles makes that really simple.

A difficulty in respect of determining solar noon...

I said a long way above that solar noon was when the sun is as high as it will be that day in your sky. And this is true!

But how high the sun is in the hour either side of solar noon doesn't change much. The precise moment when it is at the highest place is hard to pin down.

Happily, the sun rises (and falls) quite quickly at other times.

I'm picking numbers which wouldn't be likely to arise, but, say the sun was 40 degrees above the horizon (and rising) at 10:10am, and 40 degrees above the horizon (and falling) at 14:30 that afternoon.

That would let you say that solar noon happened at 12:05, by the same clock that you used for the "10:10" and "14:30".

What if you can't see the horizon?

What if, for instance, you are doing this on land, with the sea nowhere in sight?

There's a "trick" for that.

You put a saucer of water on the ground and measure the angle between the sun in the sky and the reflection of the sun in the saucer!

If that is, say, 64 degrees... then had you had a horizon to use, you would have seen an angle of 32 degrees!

If you do diagrams, you should be able to reassure yourself that what I say is true.

How do you measure the angle?

The "length of shadow" system I mentioned earlier is quite legitimate.

Not easy to use on a sailing ship, admittedly.

The sextant was invented to measure the sorts of angles of which we have spoken. It can also be used to measure angles you need if doing mapping by triangulation.

But sextants are not easily made or obtained... although simple, low resolution ones are available for use by people wanting to learn about using a sextant. They are remarkably accurate for the $30 they cost.

Issues of measurement

"Measurement" is central to "doing things scientifically". And while science may not be able to give us world peace, or even peace of mind for one individual, or some of the more important things in life, science HAS made a big difference to the quality of life of much of mankind.

Some of what we've been discussing here hints at some of the issues of measurement.

Do you want a quick, easy, inexpensive way to measure something accurately?

Good luck to you!

Generally speaking, if it is quick and easy, it probably will only give an approximate answer. What accurate and inexpensive? Probably possible... if you can accept "difficult to use".

Devising your own instruments can be a very engrossing, satisfying pastime.

Real world issues in measuring how high the sun is in the sky

Let's go back to the question of determining your latitude from the position of the sun in the sky at solar noon.

If the basic elevation above the horizon is, say, 40 degrees, then that's a very good start. Some of the things which must be taken into consideration if you want a highly accurate result don't matter very much. I am going to list them so that you can appreciate how painstaking the people who did these things "as their day job" were. Remember- not only did they have to do all of the following "right", the did it without computers, or even calculators.

Remember what I told you about the angle between the plane of the earth's equator and the plane of the earth's orbit? (The "declination")

This number goes from zero to +23.5 back to zero and then to -23.5 and then back to zero over the course of a year.

So, other things being equal, you can get reading of 20 to 66 degrees from the same place at solar noon, over the course of the year.

Obviously, you must correct for declination. There are tables to tell you the number to use.

When you checked the angle... did you measure to the middle of the sun? With a sextant, it is easier to "line up" the top or bottom edge of the sun with "the mark" you need to line it up with to get a reading.

As the process of converting the angle observed to a latitude inferred, you need to put in a correction for which way you did the measuring.

Your instrument is unlikely to show "zero" when it should. But the error it is likely to have is likely to be constant. Rather like a speedometer in a car, if it always gives a speed that is, say, 3 mph faster than the true speed.

Happily checking the sextant for this "problem" is easy, and compensating for it is also easy.

Apologies to readers in the southern hemisphere. You will have to turn bits of this page "upside down" to make them right for you.

You might say "Why not adjust the sextant so when it shows zero, the angle really is zero?"

It is a Bad Idea to try this sort of thing. As long as it is constant, having a small error in the instrument is not a problem.

If you try to adjust the sextant to give zero, you may well, unconsciously, say "that's good enough" when it is still slightly off zero.

Measure what the "bit it is of" is, and allow for it.

In Bowditch, 1888, this is the "I.C." figure that is used in calculations.

---
If you are using the horizon you can see from a dock or ship, the height of your eye above sea level will matter.

(The angle you would get from your sextant will be larger if you are higher, other things being equal.)

---
You have to allow for the fact that when the light from the sun passes from space into the earth's atmosphere, it bends! This is an example of refraction.

Doing it "properly"

Yes! Doing it "all" is a bit tedious. But there's no rocket science involved.

When Nathaniel Bowditch wrote the first edition of is American Practical Navigator, he wasn't a university professor trying to count angels dancing on the head of a pin.

The book was suppose to be a Guide to How To Do It.

If you get a copy, try reading it. Not every bit has to do with what we've been talking about, but many are, or are at least related to them.

The 1888 edition of Bowditch's American Practical Navigator is available, free, online to anyone with the curiosity to explore it. You can view it in a web browser or download a copy.

Article 281 is particularly useful. I learned half of what I know from it!

Yes! There are more "modern" ways of doing these things. But I can't build my own GPS system. I can come close to "doing it" the old fashioned way. I find that a challenge that I have been enjoying struggling with. As bits of the whole fall into place, I get pleasure from "solving" that part of the "puzzle". And if I finally get everything right, I will be very pleased when MY figure for where my house is matches up with what my GPS unit says.

Why your part of the world is warmer in summer

I bet you were told, and believed, that places are warmer in summer because they are closer to the sun when in the summer because of the effects of declination. Warmer because the earth has "tipped towards" the sun.

No. No. No.

How do schools get away with peddling these ignorant ideas?

In the summer the days are longer, the nights are shorter. That alone would make the days warmer than in winter. Duh.

Also, in summer, the angle between the sun's rays and the "blanket" of air around the earth is closer to 90 degrees than is the case in summer. So the radiation from the sun gets to the earth's surface more directly. So less heat/ light is lost along the way.

When, as in winter, the light is entering the blanket of air obliquely (not at 90 degrees) it has to travel farther through the air before reading the surface.

In addition, more heat and light "bounce off" the atmosphere when the light falls obliquely.

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Just saying.

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It's both! Flat-Earth-Academy.com is something I started years ago. For a variety of reasons, I can't offer you httpS:// access there. (As you are not asked to input any information, that's moot, but it "worries" search engines.) So I'm moving to my new, all singing, and will do the httpS:// dance site, "WYWTK.com", and Flat-Earth-Academy is gradually acquiring pages there. (Well, HERE, as what you are reading is one of my "wywtk/fea" pages.)

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