The Theory Behind Longitude

Your coordinates while at sea depend on two dimensions, Latitude and Longitude. Latitude is your position on the surface of the Earth based North/South, while Longitude is your position on the Earth based East/West.  Previously, we have discussed the math and simple calculations to find your Longitude without going into the theory behind the calculations.

Longitude is time dependent, and longitude will also make time zones make sense as well. As with before, a Earthcentric view of the solar system makes longtitude easier to explain, but do remember that it is the Earth that revolves around the sun (Heliocentric Solarlarsystem). So, back to the Earthcentric view:  The sun revolves around the Earth in 24 hours, and the Earth is a sphere.

As a sphere, the earth can be divided into degrees, and we all know from geometry class that a sphere (and a circle) is divided into 360 degrees. 

This means that the sun revolves around the entire Earth in 24 hours, and therefore makes the journey of 360 degrees in 24 hours. This can be simmered down to a speed of 15 degrees per hour, and 0.25 degrees per minute of time. This is where the "Hours x 15" and the "Minutes x 0.25" comes from.

Now, the sun, while visible from sunrise until sunset over a vast expanse of the Earth, is only actually directly overhead in one very particular position. This position is called the "Meridian" and it is where the sun is directly overhead, also known as "Local Aparent Noon". Noon is a very important part in this story, and leads to the reason that the sighting is called a "Noon Site"

When viewed from the suns perspective, the Earth is slowly turning beneath it, and there is only a small sliver of Earth that is located directly under the sun. This small sliver is called the Meridian and it is the small segment of Earth that is currently experiencing "Noon". Every minute, this meridian moves 0.25 degrees to the West, slowly making its way around the Earth until it reaches its starting point for the day.  

The starting point, which is degree 0 and known as the Prime Meridian, is located over Greenwich, which gives the reason behind the name of that time zone of Greenwich Mean Time, also abbreviated as GMT. In the name of science, and to move away from any places name, this very same meridian and timezone is also called UTC which stands for Universal Time Coordinated.  

Now, you are not standing on the sun looking down on the Earth slowly rotate under you. Instead you are standing on the Earth, watching the sun slowly rise in the East, move directly overhead, and then set in the West. At some point in the day, the sun will be located directly overhead, and that very moment will be when the sun's meridian is shining down on your little sliver of Earth. At this point, the sun is at it's zenith (highest point) in the sky and this time is your local noon.  

You don't need a sextant to figure out your longitude, only a clock and a shadow, but a sextant does help. 

When your shadow is pointing directly at True North (not magnetic North) the sun has made its journey from 0 degrees to be directly overhead of you. The time it took to reach you is proportional to the number of degrees you are from 0 degrees. So, if it took exactly 1 hour for the sun to be directly overhead, then you are exactly 15 degrees to the West of 0 degrees. This would mean that your longitude is 15W.  You would also be located in the next timezone, known as UTC-1.

If your local noon occurs at 1 hour and 1 minute after the sun has left 0 degrees, then you would be located at 15.25 degrees West of the Prime Meridian. Now, you won't see coordinates listed as decimals, instead they are listed in the format that is Degrees:Minutes:Seconds; where 60 seconds is 1 minute, and 60 minutes is 1 degree.  (it's easy to think of it in the same format as time, Hours:Minutes:Seconds, but since its coordinates, the Hours are called Degrees). 0.25 degrees can be multiplied by 60 to convert it into minutes: 0.25 x 60 = 15 minutes. This means that if your local apparent noon occurs at 1 hour and 1 minute after the sun has left 0 degrees, then your longitude would be 15 degrees and 15 minutes West (also written as 15*15'W).

Now, thinking of the time as "Time since the sun has left 0 degrees" may be helpful at first, it doesn't help with the actual calculations. To make the math easier, simply have a 24 hour clock set to UTC time and look at that clock at your local apparent noon. 1 hour and 1 minute after the sun left 0 degrees would be 13:01. If you are located further West, local noon will occur later on the clock. If you are located in the Eastern Hemisphere, then your local noon will occur before the clock says 12:00.

I know this might seem rather simple, and that is because it is. The sun moves across the sky at a set and specific speed of 15 minutes of longitude every 1 minute of time. If you check the time of your local noon, you can easily find your longitude. 

All you need to do is measure the time difference from your local noon and Noon of UTC. The difference in hours is multiplied by 15, and the difference in minutes is multiplied by 0.25.  

If your local noon occurs at 16:55, then your difference from 12:00 is 04:55.  
04 x 15 = 60 degrees
55 x 0.25 = 13.75 (the whole numbers are degrees, the decimal is going to be minutes) 

13.75 = 13 degrees, 0.75 x 60 = 45 minutes

60 degrees + 13 degrees + 45 minutes = 73*45' W

If your local noon occurs at 08:47, then your time difference from 12:00 is 3 hours and 13 minutes (03:13). 
03 x 15 = 45 degrees
13 x 0.25 = 3.25

3.25 = 3 degrees, 0.25 x 60 = 15 minutes

45 degrees + 3 degrees + 15 minutes = 48*15'E

Just that easily, with an accurate measure of time, you can find your longitude on this large round Earth. 

There was mention of timezones earlier, and this is how they come into play. Every hour, the sun is generally overhead an area of 15 degrees. So, at 12:00 UTC, the sun is over the area of 0*W to 14*59'W.  
At 13:00, the sun is over the area of 15*W to 29*59'W
At 14:00, the sun is over the area of 30*W to 44*59'W

Every hour, the sun has moved over 15 degrees, so the Earths 24 timezones are divided up into 15 degree increments. Every hour, the sun has moved 15 degrees West, and a new timezone is experiencing their Noon.  

Reefing When Ocean Crossing

The biggest distinction between ocean crossing and coastal sailing is the lack of help. If something breaks close to shore, there are plenty of people around who can help you. They will tow you back to a marina where an army of skilled labor exists to get you back out there sailing! In the ocean, if something breaks, you better know how to fix it yourself with the supplies you are carrying on board your yacht because no one is coming to help you!

This lack of aid is precisely why you should reef early. If you see a storm coming, don't wait for it to hit you to then start adjusting the sail plan! Reef now and wait for it to pass over. Once it has passed and completely cleared you, then shake out the reefs! This will ensure that you and your gear are exposed to minimal risk during the ocean crossing.

Maddie and I sleep in the V-Berth up in the bow and our third crew member sleeps in the quarter berth. At night, we rotate watches accordingly. Maddie has first watch from 9 to midnight. I have second watch from midnight to 4:30AM and the third crew member has the morning watch from 4:30AM until we all get up. Since we are up in the bow, we hear the sound of the stem cutting through the water, and we can also feel how much we are heeling over. I have frequently poked my head out of the forward hatch to see dark and stormy looking clouds all around us and notice that we are full sail! I assume that the crew member on watch is keeping an eye on them and tracking their movements with the compass. A storm that is not coming at you is not a storm you need to worry about. Then we pick up speed and begin to heel over to an extreme angle and I hear a desperate cry from the helm.

"It's time to reef!"

Yes, it is time to reef, and it was time to reef a long time ago! The two of us run up to the mast and begin taking in the sails while Maddie works the sheets and helm to keep us safe. I quickly tuck in a few reefs in the main while he lowers the jib entirely. After all of this, he will usually say something like, "We reefed at the perfect time" and I don't understand what he is talking about.

Now, he knows how I reef. I track the storms on the horizon and if I find one that is coming at us, get ready to reef. I will also reef if everything looks fine but the temperature just dropped significantly. I will reef at the first hint of anything getting stronger, and the reefing is so easy to do!

The boat is stable, the winds are light and manageable, and I easily tuck in a reef or two in the main without really needing the winch handle. The jib is easy to pull over the deck by tugging on the lazy sheet and releasing the halyard. There is no fuss about it, this is the perfect time to reef!

Once we are reefed, we wait for the storm to strike and when it does, nothing happens! We do not heel, we do not panic, and most of all, we do not risk ourselves or our boat in the process.

Sport Racers Adjusting to Ocean Cruising

Our third crew member is a racer at heart. Everything he does is for the sake of speed and performance, and he believes there is always something more that could be done!

We got to see his true colors when we were sailing from Bimini to Florida. The winds were picking up and squalls were raging all around us. I decided that this would be a good time to see how he handles the situation, since when he is on watch and I'm sleeping, he will be making these same choices.

We were full sail and doing around 6 knots before the winds picked up. The air temperature dropped and I wanted to reduce sail, but I also wanted to see what he would do. He began trimming the sails more and more, raising our speed to 7 knots, then 7.5 knots. The wind got stronger and we started heeling.

His response was to sheet the sails in harder and get our speed up to 8.6 knots. I figured this was fast enough and as the squalls were approaching, he would want to reef. Instead he said "I'm sure there is more in the sails" and he continued trimming.

He managed to get our speed up to 9.7 knots with winds so intense we were heeling over well past 25 degrees. As we were zipping though the water and the squalls continued to gain on us, I waited to hear him say the magic words: "Lets reef" but they never came.

More wind means more speed to a racer, and reefing is the act of slowing down. So I stepped in as captain and declared that we would now reef. The jib came down and two reefs went into the mainsail to balance out the staysail. I was hanging onto the mast and boom as we were violently being tossed in the seas as I cranked on the clew line to tuck in the second reef and create a nice flat sail, all the while thinking "I wish we did this when the weather was still calm."

When I got back to the cockpit, the racer declared: "You reefed just in time, our speed is still 8 knots!"

Maybe this is when a racer reefs, but as a cruiser, I reef the moment I think I might need to reef. This experience taught me that I will have to declare when we reef all the time as he will wait until it's no longer safe to reef before considering the option.

His helming skills are wonderful and he can trim for maximum performance at any moment, making him a wonderful crew member. But as a racer, he always wants more speed!

We are a cutter with a small electric motor, so when it comes to ocean crossing, we consider ourselves to be engineless. Since we rely so heavily on sails for propulsion, we carry an extensive wardrobe for Wisdom. We have a Full-Batten Mainsai, Battenless Mainsail, Trysail, Light-Air Mainsail, Staysail, Jib, and Drifter. I have specially ordered each sail with painstaking detail, so I know which sail plan is best for what point of sail. This means that I also know what speeds we will get out of each sail combination; he doesn't though, and this is just endless trimming potential!

In the beginning of our voyage, we would have sails set that I knew would work best for the conditions we were in and the conditions that were coming. Ocean crossing is an endurance sport, so it is best not to wear yourself out in the beginning! He would come out of the cabin with the navigation software on his phone and declare that we are not going fast enough and demand a sail change.

I would explain that the sails we have up are moving us along at 3 knots, and a sail change would require a lot of work and give us the same speed. This answer was not to his satisfaction and he would argue and argue until sails were changed. I like cruising because I like to listen to the wind and the waves. When someone begins yelling at me, I can't hear the wind and waves anymore.

Since he had so much energy, I showed him our speed on the navigation equipment and told him how to switch the sails over. I let him change the sails over and over and show him that all his efforts in the heat of the day under the blazing sun yielded no change in our speed.

I figured that this was a one time thing, and that now he would listen to me when I said that the correct sails are set for the conditions, but it wasn't. This guy is like the Energizer Bunny, he keeps going and going and going! Everyday, he would poke his head out of the cabin and begin yelling at me that we weren't going fast enough. I would tell him that it's the winds fault (we had very light airs) and he would insist on the same song and dance of sail changes. I quickly learned that the moment he forcefully put his hand on the companionway slats to lift himself up and begin yelling was the perfect time to tell him to go to the bow and hank on a different sail!

This went on for quite some time and I figured that this is just how he is. Every morning, he would complain about the slow speed of the sailboat in light airs. If wind came and we began sailing quickly, he was certain that more speed was available. I even explained to him about hull speed and calculated our maximum speed to show him that we could not possibly move any faster than we are at the moment, but still, he would insist on a sail change for "more speed."

I grew tired of this, and then on the 10th day, he broke! He came out of the cabin with a cup of coffee in his hand and a book. He sat quietly and read and didn't mention a thing! At this time, we actually did need to change sails, but he argued against it. I was shocked!

We were trying to beam reach with the wind just ahead of the beam flying only the Drifter. I said that we should lower the drifter and raise all the sails: Mainsail, Staysail, and Jib. He then told me we are on a broad reach. We both looked at the Monitor Windvane which was set at a close reach as it struggled to fight the lee helm of the drifter with no mainsail.

I looked at the situation and realized: he's tired and doesn't want to change the sails. I just woke up so I didn't want to start working just yet either. So we both just sat back and relaxed. We listened to the wind and the waves and enjoyed ocean cruising. It only took 10 days to reach this point of nirvana.

The Law of Averages

Averages are an incredible mathematical phenomenon, but they can also play into social experiments as well. If you ask a large enough sample size an obscure question and average the answers, the average answer will be correct! Now, the correct answer comes out of a series that is mostly composed of incorrect answers, and yet the average lies somewhere in the middle where the truth also lurks!

This wonder of averages is important to remember when ocean crossing. There will be fast times and slow times. Some days, you will sail farther than 100 miles in a 24 hour period! Other days, you might not even make 20 miles.

If you focus on the minute details, you will be focusing on the incorrect information. Instead, you should look at the averages.

In a day, you will have times when you sail fast and times when you sail slow. If you are pleased by the fast times, you will be disappointed when things slow down. Or, you could look at how many miles you sailed that day and be pleasant.

Now, looking at each day is also rather nit-picky. Some days you will sail farther than others, but over the course of the journey, you will find that you will sail on average 100 miles per day. Your boat might be faster and average a higher number, or your boat might be slower and you average a smaller number, but it will be in the neighborhood of 100 miles per day.

To sail 100 miles in 24 hours, you need to maintain a speed of around 4.5 knots (4.16 knots to be exact). But instead of worrying about how fast you are going at each and every moment, simply turn off your instruments and relax. Whether you trim your sails for maximum performance or you sit back and relax as you lazily scoot along, you will still get there, and it all averages out in the end.

Celestial Navigation for Dummies (in the Western Hemisphere)

Celestial Navigation is an utmost skill needed for any ocean voyager. Compasses can lie, electronic equipment can fail, but the stars will always be there! Yes, a cloudy day or night will obscure your view of the stars, but at some point, something will become visible in the sky and if you know how to use it, you can get a bearing on where you are.

The easiest way to find your position is to take a Noonsite. This will give you your latitude and your longitude with the least amount of math. All you need to get your noon site is a sextant and a clock (which is set at UTC time).

A noon site requires you to know how to use a sextant, which is really easy to learn. All you are doing is measuring the angle between the sun and the horizo.

When you take your noon site, you will receive two pieces of information that you will later translate into coordinates on a map. First you will have your time, which gives you your longitude. Second, you will have your sextant measurement, which will give you your latitude. 

Longitude

Longitude is a factor of time on Earth. To make this explanation simpler, we need to use an Earthcentric view of the universe.  

Remember that the Sun revolves around the Earth once every 24 hours. The Earth can be divided into 360 degrees of longitude, and in 24 hours, 360 degrees will pass by. 

360 degrees / 24 hours = 15 degrees per hour. 

At noon over Greenwich (where UTC is the time zone) the sun is directly overhead (their local noon).
In one hour, the sun will be directly over 15 degrees W.  
In two hours, the sun will be directly over 30 degrees W.
n three hours, the sun will be directly over 45 degrees W, and so on.

This means that the time of your noon site is going to b a cryptic form of your longitude, and with really simple math, can be converted into your East or West coorditantes.

When you take your noon site, look at the time (it is helpful to set a 24 hour clock to UTC so that you don't make any errors in calculation here, as each hour is 15 degrees!). Then you will simply subtract your time by 12, as this will give you the difference in time from Greenwich to your local noon.

Now take your time difference and separate it into two columns, hours and minutes. The hours will be multiplied by 15 and the minutes by 0.25; the answer will then be added to give you your coordinates.

Decimals are easy to convert into minutes by simply multiplying the decimal by 60.

So, for example, say its 4:55 PM UTC when the sun is directly overhead and you take your noon site. The math will be as follows:

   16  :  55
 -12. :  00

    4  :  55
x15   x0.2

   60  +  13.7     (0.75 degrees x 60 = 45 minutes)
  60* +  13*45'
  73*45'W

Just that simple, you now know that your latitude is 73 degrees and 45 minutes West.

Another example would be local noon (when the sun is directly overhead) at 2:38 PM UTC.

The math would be as follows:

  14  :  3
-12  :  00
   2  :  38
x15  x0.25
  30  +  9.5     (0.5 x 60 = 30 minutes)
 30* +  9*30'

  39*30'

Thats all there is too it! Longitude has nothing to do with the actual number that is displayed on your sextant but everything to do with "when" you measured the sun at its highest point in the sky.

Latitude

Ok, latitude is the North & South value of your coordinates and really weighs heavily on the reading you record with your sextant. To properly calculate your latitude using the sun (by day, obviously), you will need to use the tables in the Nautcal Almanac. These tables can seem long and complicated, and very confusing, but they are not all that bad once you learn what to look for. f you don't want to use a book, you can always download the latest Nautical Almanac for free! A simple Google Search will pull up a few options, and you will certainly be able to download your own copy.

I personally use a print version from 2017 (I bought it in late 2016 and have not updated my copy as of late 2018) as the angle of the sun over the horizon doesn't change all that much from one year to the next. Yes, your reading will not be as accurate as possible, but at the same time, you are measuring the sun on a pitching and rolling deck of a boat out in the ocean! There are going to be errors in your measurement, so a few tiny errors in your calculations will only compound into a slight bit of error in your final coordinates.

It is important to keep something in perspective here, the goal of basic celestial navigation is to ind land, not to find your exact position on the earth. Advanced celestial navigation will allow you to pinpoint your exact position by using three distinct celestial bodies and finding their intersected lines of position. That is very accurate and also a lot of work to do, which is why basic celestial navigation is just fine for ocean cruising.

The goal is to be less than 25nm off from your actual position. A GPS will tell you your true position, and you can then do some simple math to figure out how far off you are. It is good to practice that way you can get your error way down to less than 25nm. I personally tend to get us within 4nm of our true position with simple math and an outdated Nautical Almanac, meaning it can be done and it's not difficult to do. A 25nm error or less will allow you to find land, and once you reach land, you can then use visual navigation to get yourself in to port. By constantly practicing, you will know how good you are at celestial navigation should your electronic navigation equipment fail, forcing you to rely solely on your basic celestial navigation skills.

On the other hand, if you do not have a Nautical Almanac, you can also just measure the angle from the horizon to Polaris, the North Star, and that is your latitude; no math involved!

An important factor that your Nautical Almanac will tell you is a value called Declination. This is the angle of the sun relative to the horizon. You know how shadows are taller in the winter and shorter in the summer, this is because the sun is further overhead in the summer and lower on the horizon in the winter.

The Nautical Almanac is nothing more than an overfilled calendar. On the top corner of the page is the date range of that page, so simply leaf until you find today's page. Then look on the left side of the page for today's date. Then in the box of today, look for the hour you took your noon site. Don't worry too hard about the (time) minutes that you took your sight, simply round up or down to the nearest whole hour. Now, the declination will be displayed to you as Degrees, Minutes, and Decimal of Minutes.

Simply write this number down and begin to do your simple math!

You will take your sextant reading and substract it from 90 degrees, as the sun is directly overhead when you took your noon site. Then you will add or substract your declination to this number. The answer will be your coordinates for latitude.

You might be wondering what I mean when I say add or substract; which one is it? Well, it depends. If the sun is between you and the equator, you will add. If you are between the sun and the equator, you will substract. To keep the math and theory here simple, since this is just "Basic Celelstial Navigation" let's suggest that you do both and see which one is closer to where you actually are.

For example, you are sailing through the Bahamas, meaning that your latitude is somewhere between 27N and 22N. You get your sextant reading and you find today in the Nautical Almanac. It says your declination is 23*24.7' That is a pretty accurate value presented to you, but should you add or substract it from your recording? Well do both and see which one fits your assumed position best! One will be close to where you actually are and the other will be wrong by almost 24 degrees! Now you know for your present location that you should either add or substract next time.

This may sound really confusing, but I guarantee you that it is very simple to carry out this calculation. Lets do an example:

Declination is 23*24.7'
extant recording is 87*04'

We start by subtracting our reading from 90. Since each degree is 60 minutes, it makes the math easier if you just write out 90* as 89*60'

  89* 60'
-87* 04'

   2* 56'

Now, we aren't sure if we should add or substract, so lets do both!

     2*     56'
 -23*     24.7'

-21*      31.3'  A negative value would indicate that you are south of the equator, and thus in the Southern Hemisphere as you sail through the Bahamas. 

21*31.3'S

    2*   56'
  +23*  24.7'

25*    80.7'     (80.7' - 60' = 20.7') Each degree is 60', so to turn a number like 80 into degrees and minutes, simply substract 60 from it. The remaining number is minutes, and the 60 you took off becomes 1 degree.
25*    1* 20.7'    Now add the values together and that will be a possible N value for you as you sail through the Bahamas.

26*20.7'N

Being how you are in the Bahamas, the value of 26*20.7'N seems more plausible than 21*31.3S, so you know for the time being that you need to be adding your declination to the value, instead of subtracting it.

 

Finding Your Coordinates

It is important to know where you are before you leave, that way you have a rough idea of what the answer should be while you are doing your calculations. If you find that you are way off, you know you made some clerical error and can figure out how to make adjustments to erase this error in tomorrows noon site. 

Remember, Longitude is a factor of time, Latitude is a factor of the sun at noon with some minor addition or subtraction. 

With this, you can figure out where you are in the ocean using only a sextant and a clock!