(Dis) Orientation: 4. Navigation

In which we discuss navigating by the stars. This is the 4th section in a chapter (previously titled Lost in Space) on how I get my bearings in the night sky, and in life.

Author’s note: this is the fourth section in a six part series. You can find the first section here, the second section here, and the third section here.

Prized possessions are often black, with gold lettering. At least that’s how I feel about my 2002 Bicentennial Edition of Bowditch: The American Practical Navigator. Weighing in at over 5 pounds and 878 pages, and complete with red and blue ocean current map, 36 tables covering logarithms to dew points, and glossy color photographs of 52 foot waves at force 12 (~70 knot) wind speeds: this book is a wonder to behold. So I’ve struggled to understand why friends who I show it to are less impressed - a polite “that’s cool, Luke,” and we move on. But I realize now it’s not so much the heft of that book that makes it so meaningful to me. I’ve slowly come to terms with the extent to which I’ve made a hero of Nathanial Bowditch, the young, independently thinking 19th century navigator of limited means who found dangerous errors in the navigational tables and methods of the day, figured out how to correct them, and rewrote the book helping others find their way. 

I vividly remember 12-year-old me reading a child’s biography, titled Carry On Mr. Bowditch. My dad, still, on occasion, says this to me. “Dad, I think this will work, I’m going to go for it.” “Carry On, Mr. Bowditch.” Or “Dad, I don’t know if I can do it.” “Carry on, Mr. Bowditch.” Even looking at the table of contents now stirs my memory. Chapters like “Discovery,” “Book Sailing,” and the final chapter, “Man against the fog”: what I always wanted to be. Consider what the Salem Marine society wrote in their eulogy of Dr. Bowditch: “as long as ships sail, the needle point to the north, and the stars go through their wonted courses in the heavens, the name of Dr. Bowditch will be revered as of one who helped his fellow-men in a time of need, who was and is a guide to them over the pathless ocean.” That’s it. I want to be that. It’s what I’ve been trying to steer toward. I guess I’m realizing the navigational marker Mr. Bowditch has been, subtile but strong, like a stick anchored in desert island sand. 

This section is about using the heavens to find our way through the pathless oceans of this planet. We’ve been through a lot of heady stuff thus far, but our goal here is to take the wavy abstract ideas of the previous sections and use them as firm, practical grounding to get our bearing. The key comes from the fact that the stars look different from each spot on earth. Rather than, like in the last section, threatening to capsize those of us still barely managing to stay afloat, this fact we’ll find to be a great miracle. Because if the appearance of the sky is unique to each place on earth, then we can what it looks like to figure the specific place that view maps to – to know where we stand, even when far from home. Maybe you don’t plan on being stranded on a desert island without GPS, but people don’t usually plan to be lost – I didn’t – so just in case, let’s look at how it works. 

The principles of navigating by the stars are surprisingly straightforward. We need to figure out our position in two directions: north - south, and east - west. Our position north or south of the equator is traditionally the easiest. To determine this, remember that the north star is directly above the north pole: a literal pole sticking out of the northern top of the earth, if tall enough (just 323 light years or two thousand trillion miles tall) would hit it. So standing on north pole, Polaris is directly overhead - always. Saying this with numbers (don’t cringe, you got this!): when you’re visiting Santa or petting polar bears or whatever at the north pole, latitude 90°, Polaris would be straight overhead, an angle of 90° up from the horizon. 

Now, imagine sailing from there due south, toward the equator. Polaris would no longer be straight above you. Halfway down the planet, at 45° latitude, it would appear half way down the sky, a 45° angle above the horizon. And when you reached the equator, 0° latitude, it would appear 0° above the horizon, right on it. So amazingly, the angle of the north star with the horizon is the same as your latitude, measure that angle, and you know where you are (at least in the northern hemisphere). 

By the same principle, just with a little more math, and maybe a sketch or two, you can work out your latitude from any star, which means this works anywhere on the planet. When a star reaches its highest point in its east to west march across the sky, you can measure how many degrees it is above the horizon. Knowing the star’s declination tells you where that star lies relative to the north star, and thus your latitude. And though it’s easiest to make these measurements at night, when it’s dark, in the day, you can use the sun, as long as you know its declination from the time of the year. (The sun moves between declination +23.5° around June 21 to -23.5° around December 21.)

Figuring out your east - west direction is a little harder because, as you’re sure to remember, the earth is spinning. The trick is having a really good watch, well, two, actually. One watch is your reference time, set to the time in your home port – say in Greenwich, England, at exactly 0° longitude – before you left. Noon on this reference watch tells you when the sun is highest overhead there. But now, as you sail west across the Big Blue Beautiful Sea for many many miles, you’ll find the sun is now highest overhead at a different time. Noon, on your local watch, then, is different from noon in Greenwich. Now here’s the move: we know how fast the earth is spinning: one full revolution, 360°, every 24 hours. So every hour of time different we observe on our watches is 360°/24 hours = 15° different in longitude. That is, when we’re 15 degrees west of Greenwich, our little part of earth will be pointed directly at the sun 1 hour before Greenwich is pointed directly at the sun.

Thinking about it another way: just like if you drove 2 hours at 50 mile per hour you can calculate that you drove 100 miles, here we’re using the time we observe, say 2 hours different between our watches, and the speed of earth’s spin, 15° per hour, to compute we’re at longitude 30°. 

I should mention that, like with latitude, you doesn’t strictly have to be the sun object straight overhead; you could use stars for this too, but it’s a bit harder since you have to correct for the little complication that the sun wanders among the stars at varying speeds, so a day according to the stars is different from a solar day. I also might mention that determining when the sun is highest in the sky can be a bit tough, but one good method is to use a stick as a sort of makeshift sundial, tracing out the length of the shadow with time. When the shadow is shortest, that’s when the sun is highest in the sky. This works especially well on sandy beaches when you’re stranded on desert islands (and perhaps wondering why the rum is gone).

 So that’s it. If you can see the sun or the stars and get the timing right, you can figure out your location anywhere on the planet, no matter how the winds may blow you. 

A pretty useful technique for any journey, really. Like if you are setting out to be an astronomer. The path on the map looks something like this: have a mom who can pick you up late from school so you can participate in after school activities like science olympiad. Get assigned to an astronomy event. Feel good about yourself, that you can identify constellations while ice skating in your back yard. Show sufficient aptitude to get put in more advanced placement math and science classes. Have clear expectations that college is what you’ll do next. When you get to college, explore a couple of physics classes, and meet a professor who takes an interest in you. Have her advocate for you, and get drawn in to her research. Not know what else to do, so apply for grad school. Having good grades, money to apply, and a connection to a big name research professor who actually knows you all help. Pass your quals, write and defend your dissertation, all 207 pages of it. Collect that coveted hardcover black book with your name stamped in gold letters on this side: Almost Dark Galaxies in the ALFALFA Survey – A Dissertation – Leisman. Learn to write job applications. Meet Bruce at a conference. Apply, be energetic and say all the right things, feel like a good fit, have your spouse get a position close enough. Feel like you fit in! Life is great! You made it! You won the academic game! Of course, there are more oceans beyond the Americas, so be a little aware you’ll need to get grants, and avoid disillusionment and administrative traps, and eventually solve the two-body-problem. But these are probably small issues. Sounds like a plan. 

If only the real journey was just like we mapped it. I learned too late the frequency with which one should double check one’s positioning relative to the north stars of life. As my textbook says, ”although celestial navigation is easy in principle, [several] considerations make it more difficult in practice.” 

What makes it so hard? For one, you need really accurate clocks. What we today take for granted was what historian Dava Sobel calls “utterly unattainable,” a dilemma that “stumped the wisest minds of the world for the better part of human history.” Without accurate clocks that functioned at sea, sailors were unable to know their longitude, and thus were quickly lost as soon as they were out of sight of land. In 1714 England’s parliament offered a king’s ransom (over $15 million in today’s money) for anyone that could produce a method for determining longitude. The prize was ultimately claimed in 1773 “after 40 struggling years of political intrigue, international warfare, academic backbiting, scientific revolution, and economic upheaval.” The person responsible was a surprise, no famous royal astronomer or renowned watchmakers, rather “a man of simple birth… with no formal education or apprenticeship to any watchmaker,” named John Harrison. Harrison was able, through years of effort, to construct mechanical clocks that were able to keep time on a rocking ship to within 1 second a month, transforming exploration of the seas; as Sobel writes Harrison “wrestled the world’s wearabouts from the stars, and locked the secret in a pocket watch.”

I remember visiting Harrison’s clocks, now on display in the Old Royal Observatory in Greenwich, England, at the prime meridian of the world. It was the summer after my first year as an assistant professor, and I was exhausted, but had just spent a week in Cambridge England at a faculty development retreat, charting the course for a successful academic career. It was a particularly poignant retreat; I remember our mentor, a quiet white haired professor who had been at the institution for longer than I’d been alive, tearing up as we dug into the words of regret in a Robert Frost poem; he remembered missed basketball games, graduations, the sacrifices of family he’d made for career, what if I’d taken that other road? If I only had more time. Anyway, after the retreat I thought I’d take a few days to explore England, YOLO, ya know? And a visit to the observatory seemed mildly interesting, maybe a couple hour activity max. I remember the sloping lawns and the pole where they would drop a red ball precisely at noon so each ship in the harbor could set it’s clock. And standing with other tourists and influencers on the prime meridian of the world, made visible through a small sign, and a polished brass strip fitted among large cobblestone. But I remember most stepping into the part of the museum with Harrison’s clocks, large Rube Goldberg like boxes, nothing like what I’d expected: wonders of gears and springs and intricately fashioned metal, shiny golden beacons against the relative dark of the room. For all the clocks ticking around, inside that building felt timeless. We ended up spending the whole day at the observatory, appreciating the irony of moments drawn out like accordions in a place designed so precisely around time. 

It got me thinking, perhaps our watches have improved, but I wonder, are we any better at keeping time? A PhD thesis takes six years. As I know now, disillusionment at least 4 more. I’ve been away from home for more than a decade: my parents, forever 35 in my mind, have begun to gray. I’ll move back someday, right? I remember working on the factory floor at Amway during high school, talking to the others there. “I took this job as a temporary thing, to get me through. I thought I’d be here maybe a year. It’s been 20 years now, my kids are grown.” That conversation was spooky to me then. It’s still haunting now. I recently – or was it a year ago now? – read Toward the End of Time, one of John Updike’s later novels. A slower moving novel that often paused to watch flowers open and close, there were still lines that stuck: “The year is like a life - it is later than you think, the main business over and done with before you fully begin. There is a kind of tidal retard in our perception of forward motion.” How is it that we are so easily fooled by time?

Time isn’t the only tricky part. It can be quite a challenge to precisely measure the position of the stars. It’s a good question when you sit down and ponder it - how do you actually measure the number of degrees from the horizon to the star, how high it soars in the sky above? You need a tool that marks degrees. A rough way to do this is to use what you were born with: your balled up fist is about 10 degrees when you hold it at arms length, your thumb covers about two. So if I stand outside here in Chicago and face north, stretch my arms out in front of my, and carefully stack, fist on fist like they teach you in baseball, and then, leap frogging, fist over fist, and then a thumb, I reach Polaris, four fists and a thumb above the horizon. Like measuring horses. If you kept going, hand over hand till your arms were straight above you, and then behind your back where you can’t really reach and then dislocating your arm to get back where you started, a full circle should take about 36 hands, 360 degrees. At least that’s what they say, I’ve never tried it, and don’t really recommend. Anyway, as you’re probably figuring out, there are more precise tools than just your hands. A slightly curved ruler at the end of a stick in a T shape, called a cross staff, is definitely more precise, since you can measure to the nearest centimeter or even millimeter, and calculate how that corresponds to a degree.¹ 

A sextant, for measuring heights above the horizon. Image from MarineInsight.com.

Over time people have of course developed more clever tools. A sextant is one of the best tools for sighting stars by hand, it adds the addition of lenses that function like a small telescope, a clever use of mirrors that makes the object appear like it’s right on your horizon when it’s positioned correctly, making for an even more precise measurement. After all, it’s easier to compare things on top of each other, rather than when they are in very different parts of the sky. Anyway, point is, aligning different views gives a very careful measurement of the angle on the sky, precision better than a needle held at arm’s length, at least in principle. I’ve tried aligning a sextant and can’t get anything close to this: it takes lots of practice and skill to use these tools well. The difficulties using a sextant are abundant, with names like graduation error, prismatic error, centering error, index error, or collimating error. And of course, as Bowditch notes, “one of the principal sources is the observer… the sextant may not be rocked properly, a false horizon may have been used… the height of the eye may be wrong,” and on and on. Then again, something our physics lab students struggle to grasp, “human error” usually just means you messed up. The key to doing good science, to navigating well, is mitigating these errors: understanding what about the setup makes this hard, and led to the error? What’s limiting your performance, your precision? It can be tough to explore, but the answer leads to improvement in the future. And it’s not just physics. Business books highlight systems thinking – the way all the pieces link together. And I’ve read enough – seen enough – experienced enough – of the US penal system and other systemic structures to see the parallels in our need for understanding actions and “errors” in context. 

Of course, I don’t need to preach about how doing something with precision – getting it just right – is hard, or the extent to which our jittery hands just get in the way. Making a three-point basket, parallel parking your truck in that spot that might fit a Mini Cooper. I hear recently that Hemingway rewrote the ending to A Farewell to Arms 39 times. An interesting conflict of my time is the advent of AI chatbots to solve the world’s problems. “Write a letter to students emphasizing that signing a contract is a binding agreement” my colleague asked it the other day. The response was this nice letter that our manager approved it, and yet when I read it later, I found it severely lacking. It had the worlds, but the feeling of the message was off, and thus it completely missed the mark. Even this tool has its own prismatic error, and has to be rocked properly. Not that I meant to make you think of your workplace when you’re trying to relax, but from hammering nails to wording business pitches, you’ve had experience with the need for precision for things to sink in. 

Precise time, and precise measurement are both important, yes. But they are meaningless if you haven’t considered one other tricky issue: have you actually properly identified the stars? The American Practical Navigator dryly states “A basic requirement of celestial navigation is the ability to identify the bodies observed.” No shit, Sherlock. But then again, it can be tricky in practice. Let’s take for granted you’ve learned to read sky charts and the basic names and shapes and tricks that get you around. Start with a few things you know, and then let one star point you to another. Like if you’re trying to find the north star, you first make out the big dipper, one of the most recognizable patterns in the sky, you can use the two end stars of the dipper, the side where water might first drip down if you just raise the handle, a line drawn from the bottom through the top star and continued about four times the distance will reach Polaris. Ok, we know roughly where it is, but to find its precise altitude, we need to find it in the little eyepiece of our sextant; in the limited field of view through it’s little spotting telescope, are you sure you have the right star? The American Practical Navigator observes “this procedure is difficult. If the body is lost while it is being brought down, it may not be recovered without starting over again.” 

I think I’ve spent a lot of my life navigating after stars without checking to make sure they are what I think they are. Consider my heroes. It’s not a question I’ve given a lot of thought to. If you asked me before I sat down to write this, I’d probably say I didn’t have any. But on examination I know they, though unconsidered, exists: people I model my life after, people I’d do almost anything to please. Looking carefully, I see an oversized influence of my parents, yes, and a few specific teachers, and friends. I also see, quietly lurking, all those characters from books of my childhood: not just bits and pieces of Professor Haan and Elder Ardie, but also Aslan and Ishmeal and Jane Eyre and Sherlock Holmes. Trying to sort it out is a bit of a constantly shifting mess: was Mom my north star then? Or that girl I liked sophomore year? Or the constellation formed by the lot of them, me perhaps hoping that steering their general direction would get me close enough?

It was from my parents I learned to wrestle with the issue that we, humans, know how we want to act, but often don’t act that way; and that we, humans, are aware that each of us can’t all possibly be the center of everything, are aware of good greater than ourselves, and yet act most often, most primarily, most primally, in our own self interests. It seems a situation we ourselves can’t solve. Maybe that’s it, no solution. But my parents and mentors and the heroes I watched showed me about hope. It was the hint of hope in the pages of my mom’s Bible, black, with gold lettering on the cover, with all the stuff falling out of the pages, clipped obituaries, notes, letters, pictures, etc. that helped us feel its worth. And it was more than a hint of hope we’d hear in their voices singing hymns, my dad on his guitar after a family dinner. “Amazing grace,” we’d sing, and “Be still, my soul”: “the waves and winds still know: His voice who ruled them, while He dwelt below.” And indeed, how precious did that grace appear as a brooding 15 year old high school freshman, or a boundary pushing 21 year old college student, or a brain-tied 32 year old professor? 

In fact, at 32 years old I’d never been more lost in my life, and I couldn’t even see it. I’d done a lot of “dead reckoning” in my life: steering toward the next landmark without an eye to my celestial global positioning. Here I was at the top of my game in a coveted professor role. My mom, my thesis advisor, and all those around me pleased with me, or at least my accomplishments. But what I didn’t anticipate was how I wasn’t really pleased with me, or my accomplishments. I was here, it felt, by happenstance. I was exhausted, lonely: work was my life the 80 hours a week when I wasn’t commuting 3 hours to see my wife. I think I’d wanted to so hard to be the captain and hero of my life, I’d lost touch with its navigator. 

And a more subtile, but important thing: one of the hardest unanticipated things about being a professor was the way it bloated my pride. Sure, it’s not a gold plated castle, but there’s authority that comes with the position – it’s not called the ivory tower for nothing. You’re thrust in the position of having to answer everything, of being the guidestar for many young lives. “Prof. Leisman, I did bad on this test, what am I going to do?” “Professor, this girl won’t talk to me anymore, what should I do?” Or, maybe 20 minutes before class, “What do you think about suicide? I’ve been thinking about it a lot lately.” Lord, help me. I don’t get paid enough for this.

You function, you tell stories, you share your experiences, and pretty soon you get comfortable in it because humans are adaptable after all. And then you get to thinking you actually know something and have something to say. But that prideful road, for all its appeal, can lead to a sort of willful blindness to the jerk you’re becoming, a fog to hid the true position of the stars. Ha, I remember in high school AP calculus we made these bright red T-shirts that said “don’t be a j(x)=d3x/dt3”

\(j(x) = \frac{d^3x}{dt^3}\)

where the equation is fancy math notation for something called a jerk, the change in the change in the change of some quantity. I think even at the time we found some irony in the pretentiousness of such a shirt, but oh how far that path can go.

I’m sure you’ve talked with such people; maybe, like me, been such a person. Several months ago I had to meet with a high up executive at a large education company in Iowa. As we ate together he talked incessantly about his accomplishments, his knowledge of jazz, his interview process for an even more important position at an even bigger company. I was repulsed and exhausted by the conversation, his words know-it-all demeanor like maggots on my cheese. And yet, reflecting later, I understood, I thought. We’re so often what we’re shaped to be, products of our environments, performing only as we know how. I suspect much better people than me or this executive have fallen susceptible to this achilles of pride. Then again, I’d like to hope that not every person falls in this trap, or at least that those deceived by its foliage might still somehow climb out.

For me, a somewhat saving grace came in the form of economic headwinds and a department being downsized. Someone needed to be cut, and I was the junior member, the bottom of the totem. I could have fought it, perhaps, or jumped ship to a different role that fanned my pride, but I was exhausted, done, and noticing that the stars looked different, wondering how I’d gone wrong. Around that time I re-encountered a passage from East of Eden, I novel I remembered as one of my favorites though it had been many years since I read it. Steinbeck writes: “I remember the death[ of a]… man devoted to making [others] brave and dignified and good in a time when they were poor and frightened… How can we go on without him?” Like Nathanial Bowditch. Something I want to be. A hint of starlight through the fog by which I might begin to reset my heading. Because a more prized possession might be that something so good might someday be said about me: “Dedicated to making all people braver, more dignified, more good” – golden letters carved in black granite headstone. And just maybe, below that, just maybe: hope, realized – “Was lost, but now I’m found.”

1. The cross staffs we make in our intro astronomy labs have sticks that are 57.3 cm long, which, through the magic of geometry means 1cm on the ruler corresponds to 1 degree on the sky. You can actually do this math! We know there are 360 degrees in a circle, so if we want a circle to be 1 degree =1 cm then we need a 360 cm circle. Geometry says that the distance around a circle = 2𝜋 times its radius (where 𝜋 is a special math number close to 3.14159 that no one has ever written down in full but shows up whenever circles are involved). Algebra allows us to rearrange the equation: dividing both sides by 2𝜋 and plugging in numbers gives that the radius of the circle = (360 cm) / (2𝜋) = 57.3 cm.