A Sense of Where You Are, Part 2

In Part 1 of this celestial navigation mini-series, we focused on concepts, definitions, units and tools. In this post we’ll focus on the easier of the two coordinates: latitude.

As a quick refresher from Part 1, latitude is a measurement of distance from the equator. It ranges from 0 degrees at the equator to 90 degrees at the North or South poles. Lines of latitude are also referred to as “Parallels” because they are parallel and equidistant from each other from north to south. Furthermore, they are measured in degrees and minutes (60 minutes in a degree) and one minute equals one nautical mile.

Not only is the measurement of latitude easier, historically its measurement or estimation has been performed far longer than the measurement of longitude. The key reason for this is that when measuring latitude, you don’t need to know the exact time of day. As you can imagine, keeping accurate time on a rolling and pitching ship was a tremendous challenge for early mariners, we’ll talk more about that in Part 3 (longitude). As early as 300 BC the Greeks were studying latitude during their explorations. In the 9th century CE, Muslim navigators invented the Arabian Kamal to measure latitude in their exploration of the Indian Ocean.

In the northern hemisphere a fun experiment that can be done far from the ocean is to carefully observe the changing height of Polaris (the North Star) when traveling. Even north-to-south travel of a couple hundred miles can be detected. You’ll notice that when you are further north, Polaris will be higher in the sky and the converse when you are further south. Actually, with a sextant a quick way to estimate latitude is to “shoot” Polaris and accurately measure its height from the horizon in degrees. That height is roughly equal to your latitude. The diagram below (courtesy of All About Sailing…) shows this relationship nicely.

As said earlier, this is a good estimation of latitude as Polaris is actually not exactly over our North Pole. The time-lapse photo of the north sky below was done with a camera shutter speed measured in hours. You can see the trails of the outer stars circling Polaris. Polaris is that innermost trail that moves very little but the wobble is clear.

Determining latitude from the noonday sun is slightly more complex but uses the same concepts. When doing a noon sight of the sun, you’re measuring the highest angle that the sun reaches in the sky from your position on a specific day of the year.

As said before, the beauty for early mariners is that they didn’t need the exact time of day when the sun reached it’s highest point in the sky. They just roughly estimated when it would happen and began taking periodic altitude measurements with their sextant say a half-hour before and continued doing so until measurements indicated that the sun was getting lower in the sky. They then looked back through their recorded measurements and the largest number (in degrees and minutes) is what they were looking for.

This is the concept of Local Apparent Noon (LAN). As you know, we’ve divided the earth into hourly time zones and logically there are 24 of them. If you do some quick math, it becomes clear that each time zone must be 15 degrees of longitude wide (360 degrees in a circle divided by 24 hours = 15 degrees). In theory the sun should pass overhead (i.e., reach it’s highest point in the sky) at 12:00 PM local time at the exact center of each time zone. However, given that it’s highly unlikely that you’ll be measuring latitude at exactly the center of your local time zone, we use the concept of Local Apparent Noon (LAN) to indicate when the sun reaches its highest point in the sky at your specific position on the earth.

Another concept to think about that makes getting the LAN altitude of the sun easier is that the sun follows a sine curve through the sky. What this means is that when the sun is at the horizon (rising or setting), its vertical movement appears to be very rapid—for those who have studied trigonometry, think about the slope of a sine curve at the x-axis of a graph. If you’ve ever watched a complete sunrise or sunset and been startled at how quickly the sun appears to be moving vertically through the sky, you’ve seen this in action. The opposite happens at the sun’s highest point in the sky (i.e., your LAN). When the sun gets close to your LAN, it appears to just “hang” in the sky with no appreciable change in altitude for 5 or 10 minutes. As a flash-forward, keep this concept in mind because we’ll revisit it in Part 3 (longitude)—what makes determining the sun’s altitude easier, makes determining its exact time of passage (the time of your LAN) all the more difficult.

After you’ve measured that angle with your sextant at your LAN, you need to make corrections to the measurement for several “real world” things such as sextant error, the diameter of the sun and the height of your eye above the horizon. These corrections get you from “Height of Sextant” to “Height Observed”. In the case of the noon sight I was taking when I first sighted land on February 23rd, my Height of Sextant was 61 degrees 14.2 minutes and the corrected “Height Observed” was 61 degrees 26.5 minutes—not a huge difference but significant. To flash-forward, if I didn’t correct I’d be off by an additional 12 miles due to those factors.

From there you need to consult the Nautical Almanac for the year. The Nautical Almanac is a several hundred page book of tables that track the position of sun, moon and key stars for every day of the year. Below is an example of one page of the Nautical Almanac for 2020. This page has sun and moon data for February 21, 22, 23. I’ve marked-up the page with some specifics that I was looking for. First, I took the sight on Sunday, February 23, 2020, the day that I made landfall in the Virgin Islands. Second, the time the sun passed overhead was closest to 4:00 PM (1600 in 24-hour time) GMT (Greenwich Mean Time), all celestial navigation is done using GMT.

To determine my latitude, what I’m looking for is the “Declination” of the sun when I observed it highest in the sky. “Declination” is defined as a celestial object’s geographic position over the earth. In this case, the “celestial object” is the sun and, said another way, the declination is an observer’s latitude where the sun would appear directly overhead (at “Zenith” in astronomical-speak). To help understand declination better, think about these examples: at the equinoxes (roughly March and September 21) the sun’s declination is zero degrees (it’s right over the equator), on the solstices (June and December 21) the sun’s declination is 23.5 degrees which defines the Tropic of Cancer and Tropic of Capricorn.

Since the sight I did was in February (the Northern Hemisphere’s winter), the sun’s declination is south (the sun is in the Southern Hemisphere). In the Nautical Almanac, declination is abbreviated “Dec”. From the image above, the declination I needed to use on that day was South 9 degrees, 51.6 minutes.

The picture below shows the quick calculations I did and the rationale for the calculations.

I took 90 degrees and subtracted the height of the sun (from my sextant) and also subtracted the declination of the sun. To show my math…

90 degrees
-61 degrees 26.5 minutes (Height Observed from sextant)
-9 degrees, 51.6 minutes (Declination from Nautical Almanac)
18 degrees, 41.9 minutes (my North latitude)

My GPS latitude at that time was North 18 degrees, 43.3 minutes. Given that a degree of latitude equals a nautical mile, I was off by 1.4 nautical miles. A very good sight I must say, especially for rolling and pitching on the open ocean. As comparison, when I first started learning I’d be in a fixed location in my backyard and regularly be off by tens and even hundreds of miles.

One thing that you may have noticed in this post is that not only is there math involved, but it’s sexagesimal math. That is, base-60 as opposed to our standard base-10. The sexagesimal system originated with the Sumerians in the 3rd millennium BC, was passed down to the Babylonians, and we use it today to measure time, angles and geographic coordinates. In many ways, 60 is a superior number in that it can be divided evenly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. It’s also the lowest number that can be divided by every integer between 1 and 6.

In Part 3 of A Sense of Where You Are, we’ll focus on longitude and sum it all up. I hope reading this offers a better perspective and sense of wonder about both the abilities of pre-GPS mariners and the technical marvel of GPS. Pre-GPS mariners would regularly sail and navigate by making one, or sometimes no, sights per day to fix their position. If cloud cover obscured their sun and stars, they would rely on dead reckoning until the clouds broke. Modern GPS recalculates all of this (and longitude) every second regardless of the weather.

In the meantime, Hazel James and I have sailed to the British Virgin Islands. We cleared customs and immigration in Spanish Town on the island of Virgin Gorda and are now in the Gorda Sound for a few days. While in Spanish Town, and upon the execellent advice of my brother and sister-in-law, we explored The Baths—a fascinating group of rock formations that I’ll feature in a future post.

Lest anyone think that sailing and cruising like this is all fun and games… when I was in the marina pictured immediately above, I had to replace the cranking (starting) battery in HJ. Fortunately there were some good chandleries (marine stores) in Spanish Town. Upon starting HJ’s diesel engine to test the new battery, I found that the engine wasn’t pumping raw water (sea water) through her cooling system. After a bit of diagnosis I had to replace the impeller in her raw water pump. Finally, here at Leverick Bay Marina in Gorda Sound last night, a stiff breeze started blowing in the late evening from the North—an atypical and unprotected wind direction for the marina. Even with all fenders out, it made for a rolling, pitching, noisy and largely sleepless night. We came through it fine but it’s good to see the sun up. I also need to figure out what we’re going to do today since we’re supposed to get more of the same. I don’t want anyone thinking that this life is like home or hotel living. You’ve got to remember the good during the bad and the bad during the good. In the next several days, when the weather settles down and the trade winds kick-back-in, we plan to sail to the low-lying island of Anegada about 15 nautical miles north of us.