Geolocation through the heavens

Finding our exact position in time and space has, historically, always been a matter of great importance to us. For people to work toward a common objective, they have to—more or less simultaneously—meet at particular places and moments in time. But how did our ability to do this improve throughout the ages? As it turns out, building clocks and watching the heavens have taken us a long way in that direction.

Without clocks, there’d be far less gatherings of this kind

Although I have no evidence to back this up, I suppose the first means of organizing meetings depended on oral communication and time estimation based on the sun’s position in the sky; basically people agreed meeting at a convenient place and time. If you think about it, picking a mutually convenient time is quite a hard thing to pull off when there are many people and no clocks.

Perhaps the best you can do in this case, is pick among three events: sunrise, (local) noon, and sunset. Sunrise is particularly useful in this regard, as you can just look for first light; i.e., the exact moment sun-rays pass over your horizon. However, that’s not as accurate as you might think, as it depends on your elevation and position on earth. For instance, if you’re on top of a mountain, you’ll see first light earlier than someone at sea level.

Can you imagine depending on the sun’s position instead of clocks?

At this point most people give up, because they realize that, yes indeed, elevation is a parameter that influences how accurately people can meet in the absence of clocks. Big deal, there’s probably tons of other parameters that influence the error term, right? Well, I beg to differ; I think difference in elevation, h, is a very important parameter, so examining it more closely will reveal something interesting.

Intellectual boldness isn’t as cool as climbing a mountain, I know

Let’s assume a worst-case scenario of two young lovers, Alice and Bob, who want to meet. They’re madly in love, but live in separate villages—Alice at the base and Bob near the top of a mountain—where clocks are unheard of. They agree to meet at a small grove halfway up the mountain at first light, but our protagonists live at different elevations. How much longer will Bob wait until his beloved shows up?

The moment he sees first light, Bob’s gaze traces an imaginary line that grazes over the edge of his visible horizon, extending into deep space. At the same time, Alice still anxiously awaits for first light to appear in the east. In fact, she won’t won’t see first light until a time t after Bob starts climbing down the mountain.

Mountaintops see first light before the sea does

So how big is t? Well, if you think about it, it’s basically the extra angular distance, θ, the sun has to traverse before appearing in Alice’s field of view. To find θ, we have to calculate the angle at which the gazes (upon first light) of our two protagonists meet. If you work out the geometry, it turns out that

θ ~ 60·√(2h/R_earth) deg ~ 1 deg,      (1)

where R_earth ~ 5,000 km (ballpark numbers people, don’t freak out) is the earth’s radius.

This is turning out to be quite a romantic tale, isn’t it?

So basically t is equal to the time needed for the sun to traverse about 1 degree in the sky. Since it does a complete rotation (360 degrees) in 24 hours, the sun has an (average) angular speed ω ~ (360/24) deg/hour ~ 10 deg/hour. Putting it all together, we get

t = θ/ω ~ 0.1 hours ~ 5 min.      (2)

This is pretty remarkable: Bob has to wait about 5 minutes before Alice comes along. And that’s essentially the kind of accuracy restricting you in the absence of clocks.

Feeling good about having clocks yet?

When you do have clocks, it all comes down to the accuracy with which they’re synchronized. Especially for early pendulum-type clocks, the problem is that, given sufficient time, their readings diverge from the actual positions of the sun in the sky. (This is mostly caused by daily changes in temperature, hence length, and therefore oscillation period.) Errors of these kind keep piling up, unless you do something to correct it mechanically.

When the time-keeping error reaches 5 minutes or so, clock synchronization becomes pointless. So, how slowly should clocks diverge to remain synchronized to a sufficient degree of accuracy? This basically depends on how long you want to keep them synchronized: Any uncorrected clock has a finite rate at which it piles up its own time-keeping error, hence it can be used to an accuracy ≲ 1 minute (i.e., better than no clocks at all) for a finite period of time.

Heavy things swinging back-and-forth once made for great clocks

The point at which accurate clockwork became essential, was during the so-called Age of Discovery (AoD) in 15^th–18^th century Europe (about which I recently wrote), when European colonizers sailed across oceans in search of profit. When you’re traversing the seas, you’d like know your position to at least 10% accuracy. Since oceans span a good fraction (~ 10%) of the earth’s circumference, I’d say the minimum required accuracy is s ~ 1,000 km. But how can you determine your position on earth?

Can you imagine crossing oceans without the Internet or GPS?

Your longitude is pretty easy to determine using the stars (e.g., the North Star), so latitude is the tricky part. To determine latitude, you can compare the time between two fixed celestial events; e.g., local noon, and noon defined at some reference position like your hometown. As in Alice and Bob’s case, we need to calculate some kind of angle θ. Assuming a purely east-to-west (or vice-versa) voyage, it turns out that

θ ~ 60·(s/R_earth) deg ~ 10 deg.      (3)

That’s actually worse than word-of-mouth accuracy (~ 1 deg). Given that trans-oceanic voyages with sailing ships lasted for ~ 100 days, an assumed ~ 10% accuracy means knowing your rough position every week or so. Ideally, you’d like a fairly accurate report on a daily basis, which means increasing the accuracy to ~ 1%; Reports on an hourly basis—which requires an accuracy t ~ 0.1 min ~ 10 sec—puts you at a clear advantage, requiring a level of sophistication well beyond word-of-mouth complexity.

Surprisingly enough, competent sailing is complicated stuff

Our ability to coordinate our actions over huge distances is thus enabled by accurate time-keeping, which, historically, was achieved by looking at the heavens. Today, we can basically determine our position within s ~ 1 m using satellites in (geosynchronous) orbit. Comparing that to an accuracy t ~ 10 sec achieved during the AoD—which is equivalent to s ~ 10 km—modern geolocation methods are more accurate by a factor of 10,000: That’s millisecond precision, which even professional videogamers can’t consciously perceive.

We’ve always looked at the heavens to find our position in the Cosmos

So, next time you use Google Maps, give a little thought to the accuracy involved in achieving the miraculous precision required. Remember that this is all made possible by scientific study of the heavens. More important still, remember that enjoying the fruits of applied scientific knowledge is a privilege, not a right.