Ships must practice celestial navigation
114 comments
·January 10, 2025throw0101a
madphilosopher
The main principle of celestial navigation is pretty easy to visualize.
Pick a celestial body that's in your sky right now, like the Sun. At any given time, the Sun is directly over a single point on the globe (the GP, or Geographic Position). So if you measure the Sun as being directly over your head, you know where you are exactly on the globe, after consulting your clock and almanac.
But, if you measure the Sun at a non-overhead angle, then you and everyone else with that same measurement must be on a circle whose centre is the Sun's GP. (Visualize the circle as the edge of a flashlight beam being pointed directly downward at the GP.) The rest of celestial navigation is refinements to figure out where you are on that circle.
Onavo
You can also do it at night using star charts (or as you call it, an "almanac"). That's how most of the digital celestial navigation solutions work, they use the positions of bright stars to determine a fix based on observatory data.
You can try it yourself here:
nonrandomstring
Might I differ on "easy", from second hand experience of watching my father go through his advanced yachtsmanship RYA Astro Navigation exams and cursing at trig functions? In practice its a lot of paper and compasses.
pbhjpbhj
The USA who just threatened to invade a few NATO allies? People working with USA for the next few years seems pretty foolhardy. Surely everyone else in NATO needs to be getting together and building it defense system that exclude USA.
throw0101a
For (e.g.) eLoran, each chain is independent of every other chain. So the network chain(s) run in the EU are not dependent on the chains in US/CA, are not dependent on the chains in Russia, or the Middle East:
* https://en.wikipedia.org/wiki/File:NGA-Atlantic_Loran.png
The chains run by Japan are not dependent on the chains run by South Korea, would not be dependent on chains run by AU or NZ:
* https://en.wikipedia.org/wiki/File:NGA-Pacific_Loran.png
* https://en.wikipedia.org/wiki/Loran-C#Limitations
India, China, and Pakistan could all run their own infrastructure:
* https://commons.wikimedia.org/wiki/File:Loranstationscrkl.jp...
The only agreement being the technical standards (frequencies) and timing offset for near-by chains.
(And I'm Canadian.)
AceyMan
It wasn't "the USA", it was our incoming president, who everyone knows speaks for no one but himself.
thatguy0900
The president speaks for the united states. Brushing his comments off like that is absurd. You act like he got the job in a raffle, the majority of the voting population voted for him, explicitly asking him to speak for them.
belter
The USA technical infrastructure can be shutdown entirely any time EU wants. Hint: ASML Machines and Remote Disable...
scottLobster
I'm pretty sure the threatened invasions are just distractions to change the conversation from the H1B debacle.
If for no other reason than Canada is a country a lot of Americans actually care about (many have relatives there), and without a formal declaration of war congress could step in at any time and declare the whole thing illegal, enabling the military to refuse orders relevant to the invasion.
But as an American who has been a little sick of Europe mooching off of our military overwatch (see various European nations running out of bombs during the Libya campaign), I'm all for an independent European military command with independent capacity. The Cold War is over, the Russian tank hordes that once threatened to roll across Western Europe haven't managed to roll halfway across Ukraine with even reluctant, intermittent, indirect western support. We don't need to be under some monolithic military command anymore, Europe does not (or at least should not) need US strategic overwatch to fend off Russia.
As for the "European militarization has historically led to world wars" argument, the UK, France, and Russia all have nukes. Germany could probably build a few in a long weekend if sufficiently motivated. We aren't going to see a WWI or WWII rematch unless the AI "revolution" actually turns out to be more than smoke and mirrors for dumb money and enables perfect missile defense or something.
So yeah, please get an ex-US NATO off the ground so we can focus on China.
forgotoldacc
Even if it's just a distraction, countries usually distract from their internal problems by targeting un-allied countries as the source of their problems. For example, when China has some internal issue, they start complaining about Japan or Taiwan. When Iran has problems, they threaten the US.
When countries start threatening those who aren't their active enemies, it's a first step into normalization of the rhetoric and building up support for inhuman actions. Iraq wasn't even in anyone's mind until suddenly America needed to invade and "liberate" it just for some president's personal ego, and the overwhelming majority became convinced that Iraq was involved in 9/11 (it wasn't) and steamrolling the entire country was doing the world a public service (it wasn't)[1]. Nobody was even thinking about Greenland or Denmark in the US until someone with a big ego started talking about liberating it. Now the same political faction that had 89% support for the Iraq War is starting up the same thing.
[1] https://www.pewresearch.org/politics/2023/03/14/a-look-back-...
fastasucan
>But as an American who has been a little sick of Europe mooching off of our military overwatch
Thousands of european soldiers have died because of US war on terror the last decades, not the other way around.
wklauss
> But as an American who has been a little sick of Europe mooching off of our military overwatch
As a European (now living in the US) I think this perspective might be an oversimplification. Access to EU soil has been incredibly important for US military strategy over the last 50 years. Having bases in Germany, Spain and Italy is instrumental in US campaigns in Middle East or Asia. Not to mention the amount of European lives lost in conflicts initiates or led by US interests.
dageshi
My impression is the US has pretty consistently tried to dissuade the EU from developing any kind of capable military organisation and to instead do it as part of NATO.
Although I have to assume that with Trump as president for a second time a lot of people in europe are going to have to worry that the US can't be relied on in the way it could in the past so I think you might get your wish.
Where that leaves NATO afterwards, I don't know.
jltsiren
> So yeah, please get an ex-US NATO off the ground so we can focus on China.
If the US does not care about its European allies, it no longer has the economic power to "focus on China". From an European perspective, China is far away and not particularly threatening. If there are no specific reasons to support the US, it's better to not take sides and trade with both sides.
BRICS is already a serious challenge to the Western hegemony. If the US thinks that "the West" has no longer a reason to exist, it will be seriously outnumbered by those who don't share its ambitions.
vlovich123
> and without a formal declaration of war congress could step in at any time and declare the whole thing illegal, enabling the military to refuse orders relevant to the invasion.
I’m pretty sure that war requires congressional approval BEFORE an invasion full stop. Congress in recent history has been fairly cavalier about letting the executive launch military action and looking the other way, but it’s not actually supposed to work this way.
> The Cold War is over, the Russian tank hordes that once threatened to roll across Western Europe haven't managed to roll halfway across Ukraine with even reluctant, intermittent, indirect western support
This is a gross mischaracterization of the situation that significantly underplays what actually happened I think. Without what Biden did right before the war saying “it’s going to happen” and mustering broad domestic and international support as well as using sanctions to freeze Russian assets and use them to pay Ukraine for reparations, Ukraine wouldn’t exist today.
We’re talking about $70B of military HW and $23B in terms of economic and humanitarian aid and another ~87B for the Ukrainian government to keep the lights on. That’s from the US alone. The international community has also contributed another $100B.
> We aren't going to see a WWI or WWII rematch unless the AI "revolution" actually turns out to be more than smoke and mirrors for dumb money and enables perfect missile defense or something.
It’s a slower burn. Putin isn’t going to try to take everything at once. It’s the Hitler annexation strategy over a longer time period. A little Georgia here, a little Crimea there, now it’s the entirety of Ukraine. Partly because his country is weaker but also because war is more expensive to prosecute than 100 years ago due to technology and the resistance is much better prepared for such an attempt.
Where do you think Eastern and Western European civilians will flee if any of those countries is drawn into conflict with Russia? Conflict forces desperate immigration which then creates anti-immigration counter responses in domestic populations here in the US.
China is important but what are we going to do there? Do you think we’re going to successfully defend an invasion of Taiwan when it happens? Cause that’s going to be their first military action. And if people are complaining about supporting Ukraine, how do you think they’ll fair regarding Chinese nationalists?
Finally, I’m pretty sure the troops and equipment we need there are also fairly different. If China is delivering huge numbers of land troops to Taiwan in the first place I’d say the battle is very lost. It’s going to be a sea/land siege so if the US gets directly involved it’ll be a meeting of the navies.
piokoch
This is very simplistic way of interpreting what Trump said. I would advice to read a bit about the significance of the GIUK (Greenland, Island, UK) in view of warming climate and growing importance of Arctics - melting ice might provide a much better and faster route for trade.
The problem is who will control the entrance and exit to that route and GIKU is one of them. And here we are on the key problem: Denmark for years totally neglected building any security/military infrastructure on Greenland, which suddenly might become a crucial element of the World economy and trade.
Do you really want sadistic regimes like Russia or China to take control over key parts of the potential trade routes (80% of global trade goes through the sees and oceans)?
Whatever you think about Trump, he will be gone in 4 years, someone else can be elected if Trump/Vance fails to deliver. Do you think we can get rid of Putin and his dreams on conquests Europe that easily? Or Xi, who is speeding with extending military potential of China, including nuclear weapons, aircraft carriers? Try to guess why Xi is doing this.
nazgob
Denmark is in NATO, US base is already in Greenland and I'm sure Danes would be open to add few more. There is no need to invade it. The moment it happens, Europe is lost to the US as an ally and NATO is gone.
pjc50
You've not addressed threatening the territorial integrity of Denmark. If you're going to say we've misinterpreted him, please quote him. Verbatim, not cleaned up.
Propelloni
Without touching the argument if the end justifies the means, the master of the deal apparently has communicated very badly what his intentions are. Obviously, nobody in Europe understood him.
But let's just wait and see, actions speak louder than words, although given past actions that is not instilling great confidence in me.
olau
Not to pick on your point about Denmark, but the fact is that there has been no aggressors and thus none to defend it against. The US did try to have a sort of covert base on Greenland many years ago, and in the end it failed.
Yes, someone could invade. But they didn't.
And if they did, they probably wouldn't have been able to stay there for long enough to actually get anything out of it long term, without it turning into a war against NATO.
It is my understanding that it has been agreed a few years ago that additional monitoring should be put in place. I don't think anything else than general tech incompetence of the Danish defense forces has prevented that from happening yet.
But on the hand, as far as I understand the matter, there is still too much ice for it to become a major trade route.
erulabs
The question was “Greenland and Panama” and the answer was “I wouldn’t rule anything out”. It requires a pretty hard squint to convert that into a threat “to invade a few NATO allies”. This sort of intentional misunderstanding of an exaggeration does so much more harm than good.
A western defense system that excludes the USA is naive at best.
FireBeyond
It’s not a hard squint.
If these countries were our allies the answer would be “I have no idea why you would even be thinking that question”, not “we won’t rule anything out”.
dzhiurgis
> Now if only the US (and others) would get their act together and build out a backup system to GNSS
They are moving towards quantum navigation (esp subs)
throw0101a
> They are moving towards quantum navigation (esp subs)
How does that help the merchant marine that is part of the logistical supply chain? Are container ships going to get this quantum nav boxes too? The US pays airlines a retainer to be a reserve fleet [1]: will they get these boxes as well in case of emergency?
What happens to all the civilian infrastructure that need navigation and timing signals?
Considering only the "military" ramifications of GNSS disruption is myopic.
dylan604
> Now if only the US (and others) would get their act together and build out a backup system to GNSS. China, for example, has built out an eLoran system:
What prevents other countries from using these other systems?
nayuki
Encryption. For example, GPS's P(Y) code is encrypted and only for military use. https://en.wikipedia.org/wiki/Global_Positioning_System
dylan604
But that’s what the three letter agencies are supposed to do. Surely there’s a good guy back door to that encryption /s
throw0101a
> What prevents other countries from using these other systems?
Nothing.
Nothing also prevents other countries from using China's BeiDou GNSS or their eLoran network.
knifie_spoonie
The US government can apparently turn off their GPS signals over certain parts of the world:
https://idstch.com/geopolitics/denial-gps-services-kargil-wa...
Almondsetat
the Earth is the center of the universe, though, just like any other point in space
arter4
>As The American Practical Navigator (aka “Bowditch”) states, “No navigator should ever become completely dependent on electronic methods. The navigator who regularly navigates by blindly pushing buttons and reading the coordinates from ‘black boxes’ will not be prepared to use basic principles to improvise solutions in an emergency.”
I wonder if this mindset is also applied, for example, to the rest of the military. Does the Army regularly practice land navigation? I know they get at least one landnav class, but it is a perishable skill. If you don't practice, you'll soon forget about it.
I guess this could also be useful to civilians. Being able to do stuff without relying too much on electronics.
nradov
Some Army units, particularly ground combat units, regularly practice land navigation with map and compass. I don't think they typically spend much time on celestial navigation beyond the basics of finding heading based on constellations. They're not usually carrying sextants.
i_am_proteus
There's no real need for celestial navigation on land, the same way there's no need for celestial navigation in most coastal waters: if you can see lines of bearing to known landmarks/navigation markers, you can obtain a fix.
lupire
Ships are far more isolated than land crews, and direction-finding is much harder at sea than on land. If you're part of an organization that cares where you are and wants you back, you are pretty easy to find your general location venture off on a land journey and get stuck. A single human might be hard to find under a rock or snow, but an army unit that wants to fund is easy to spot.
paganel
The problem with getting lost as an army unit on land is that you might inadvertently get in your enemy's sights before anyone on your side might have had the chance to find you, and at that moment it is game over for said unit. There have been documented a few of these such cases in the war in Ukraine.
cromulent
There is a lot of GPS jamming happening these days, especially in the Baltic around Kaliningrad. Makes a lot of sense.
null
cguess
It's really funny to watch a bunch of people contradict the US Navy when it comes to navigation at sea. No, a cell phone isn't going to work in the middle of the Pacific and no, the US Navy doesn't use Google maps. Go get on an actual boat sometime and sail out of sight of land, you lose cell signal way before you're even over the horizon.
myself248
A cellphone's GPS only relies on the network for a quickstart. With no network, it takes longer to get the initial fix, but then works perfectly fine.
throw0101a
> A cellphone's GPS only relies on the network for a quickstart.
The point is not to rely on GPS/GNSS:
Tor3
Not always true. I have devices with GPS which not only relies on a network for getting ephemeris data for the satellites, but they can't function without it. In other words, they don't even try to get ephemeris data through the GPS signal. So, if there's no network, it doesn't work. One of my devices is even a tablet without a SIM card, it only has wifi.. so if I'm near a cafeteria with wifi I can get a fix, and keep it for a while, but without it it's totally stuck. Forever.
In any case, as others have said, in the ocean a cellphone is useless for navigation. At least use a standalone satellite navigator. And learn to use paper maps and a sextant, it's actually a lot of fun.
wakawaka28
Ships do use GPS and so do phones and other devices. What exactly are you trying to say?
rickcarlino
Have there been any computer vision systems that can approximate celestial navigation using common sensors like a camera, An electronic compass, and a tilt sensor? Something like a computer vision based auto sextant. This is an idea I have thought about for a while but I have zero background in this area.
kragen
Yes, starcams are standard equipment on satellites including Cubesats because they provide more precise attitude information than any other available sensor. There are numerous free-software packages for star tracking; a quick search finds https://github.com/spel-uchile/Star_Tracker. I don't know if there's, like, an Android app that will do the required sensor fusion to tell you where on Earth you are.
tgsovlerkhgsel
I looked into it briefly, and while it may be easy at sea with a horizon reference, it becomes surprisingly hard without it (which also complicates testing). The article touches on this with the mention of the bubble sextant required at night, presumably because you can't see the horizon. You'd have the same problem over land.
The stars only give you the angle at which you're looking at the sky. If you know the angle to the center of the earth (or can relate the sky measurement to that) and the exact time, you know where you are, within the inaccuracy of your measurements. Being 1/60th of a degree (1 angular minute) off places you 1 nautical mile off.
A bubble sextant solves this by including a literal 2D bubble level. "When properly used, the bubble sextant gives good results, by which we mean an accuracy to within 5 miles." (https://www.usni.org/magazines/proceedings/1939/june/averagi...).
The obvious solution that immediately springs to mind is a tilt sensor/accelerometer. However, once you get to a useful accuracy, those start getting expensive - several hundred dollars for a 0.01 degree system, which would give you an error of about 1 km.
You'd then need a sufficiently rigid mount for everything, proper calibration (remember, some angle on your mount being 0.05 degrees off completely destroys all usefulness of your system!), and then a camera and a plate solver (software that turns a photo into sky coordinates) should give you a useful fix. I think you could build a system that gets you within a few nautical miles for about $1000 (parts, not labor).
gmueckl
The closest unit I can think of right now was the SR-71's celestial navigation unit. I don't know how it worked internally, but it supposedly navigated the spy planes to targets across the globe before GPS existed.
nine_k
A number of ICBMs used / uses a similar approach. In space, stars are always visible well, and terrestrial navigation aids may have been jammed or destroyed when ICBMs are put to use.
ttepasse
From one SR-71 memoir I remember the factoid that sometimes the electromechanical star tracker was sensitive enough to catch stars in the daylight while the plane was still on the ground. Makes one wonder what is possible now with modern digital photographic sensors and processing power. And miniaturisation of course, the 60s celestial navigation unit was the size of a fridge.
bigfatkitten
Many aircraft, mostly numbers had similar systems.
walrus01
I'm not aware of any specifically, but one of the instruments that can be used as a data point to further reduce error when navigating in a zero-GPS/zero-radio-signal environment is an INS. The very highest precision ones are quite classified and used on submarines.
Polizeiposaune
Not exactly what you're looking for, but spacecraft can use optical star trackers which compare what they see to a catalogue of known stars to determine orientation.
See, for instance:
https://www.rocketlabusa.com/space-systems/satellite-compone...
null
cguess
Yes, there's one mentioned in the article, STELLA. Here's a paper describing it https://adsabs.harvard.edu/full/1999naos.symp..239K
tgsovlerkhgsel
That's an amazing paper:
TL;DR star tracking is a thoroughly solved problem, the hard problem is finding "where is down" without looking at the horizon (which may not be visible). The solution to that is coupling into the ship's inertial navigation system, which still may drift up to an arcminute per hour, but if you feed the star tracker data back into the INS for corrections, it gets better.
giraffe_lady
It's wild that the state of art on this is still basically how they left it at the end of the apollo program.
gertrunde
Somewhat related - https://oceangloberace.com/
A round-the-world yacht race where modern technology is not permitted. (Where 'Modern' = pretty much anything that wasn't in common use in 1970's/1980's).
A fair few of the boats that have taken part had previously run the Whitbread Race in 80's.
GPS isn't permitted outside of emergency situations, and neither are mobile phones/computers (which are sealed in a bag by race organizers for the duration of each leg).
Celestial navigation is required iirc.
Other banned items include: - Carbon Fiber - Digital Music (Only cassette tapes permitted!)
_xerces_
Curious how navigation at night was not possible without expensive equipment, sounds like they were relying only on starts in the morning and evening? Are the measuring something like angle of those morning/evening stars or their set/rise times with respect to the sun?
UniverseHacker
It is not true- the authors sound very inexperienced with celestial navigation. There are many ways including the lunar distance method to get a position at night with regular equipment. The math is more complex than a simple noon solar sighting, but it can be done with just a regular cheap plastic sextant and a watch.
It’s also no big deal to go 12 hours with no position. If you know your speed and heading you can accurately estimate your position much longer than that.
Overall, they also made it sound almost impossibly difficult for a large team of professionals, when solo and otherwise short handed recreational sailors have been reliably sailing around the world with celestial navigation for more than a century- through all possible conditions.
danielvf
Note that they were staying roughly 2 miles within the actual track, while having the bulk of the work being done by a combo of officers and newbs that they had just trained. That's high accuracy standards for celestial nav, not even counting that this is most of other people's first time doing this in anger.
throw0101a
> Curious how navigation at night was not possible without expensive equipment, sounds like they were relying only on starts in the morning and evening?
As a sibling comment notes, it is possible. There are tables for lunar distance:
* https://thenauticalalmanac.com/Lunar_Distance_Tables.html
* https://en.wikipedia.org/wiki/Lunar_distance_(navigation)
* https://www.starpath.com/resources2/brunner-lunars.pdf
The planets Venus, Mars, Jupiter, and Saturn can be used, as well as several dozen planets (lookup tables in an almanac)
* https://en.wikipedia.org/wiki/Nautical_almanac
* https://thenauticalalmanac.com
Two US military videos explaining the theory (ground points/GP, circle of position, etc):
* USAF: https://www.youtube.com/watch?v=UV1V9-nnaAs
* Army: https://www.youtube.com/watch?v=G4DRBi66cOA
The USAF has a video because that's how planes used to do navigation outside of radio range—sextants on the ceiling of the cockpit:
quercusa
And just how expensive is a bubble sextant?
ianburrell
I discovered that real Tamaya marine sextants are available on eBay for $100-150. They were 30x that new.
My understanding is that the sextants are coming from breaker yards in India where the sextants were left on ships and salvaged.
wrycoder
I have one I paid $150 for. But bubble sextants are usually only used on aircraft.
doodaddy
> CIC was to notify Lieutenant Commander Stanton if the Essex deviated more than 10 nautical miles (nm) from the planned track.
Pretty impressive! USS Essex would cover 10nm in about half an hour, so not much tolerance for going off-course.
fmajid
So does that mean USN ships are issued with a precision mechanical chronometer for longitude, like John Harrison's original marine chronometer that won the Royal Navy's Longitude Prize?
ls612
An electric timekeeping device which is not networked would presumably be good enough for this purpose wouldn’t it? The concern is ewar not so much the ship not having electricity.
fmajid
In this application, I would also be concerned about resilience in the event of an EMP strike. Not sure a battery-powered quartz timepiece would survive such, even if it would survive a failure of the ship's electrical systems (not sure if that would matter, as I doubt the ship's propulsion could survive such).
HPsquared
If it doesn't require external communication, it's very easy to harden against EMP. Just put it in a Faraday cage.
bodhiandphysics
you can just put a quartz watch in a faraday cage. That'll be more accurate than any mechanical timepiece. Mechanical clocks have a nasty habit of breaking.
y33t
Believe it or not, some people still use slide rules and books on trig tables for this very purpose. They use them too. No sense in having them if you aren't competent at using them.
cafard
> when celestial navigation is all but impossible without advanced, expensive equipment (i.e., a bubble sextant).
Compared to what the Navy usually steers by, how advanced and expensive are bubble sextants?
5555624
A new Tamaya sextant can run about $3,000 and Chinese-made Astra sextants are about a third of that. (Bubble sextants aren't used at sea.)
That's the only "expensive" equipment needed for celestial navigation or at least that was the case when I learned it over 40 years ago.
knallfrosch
Probably they use a $500,000 computer that fills a room and could be replaced with a $50 Android smartphone with offline Google Maps.
HPsquared
Said Android smartphone relies on clear airwaves and billions worth of GPS satellites.
nayuki
> analog navigation techniques became relevant again
I'm not sure exactly what methods were used in this navigation exercise, but if they write down numbers with finite precision at any point in the process, then the method has at least some digital component to it. Note that digital means the use of digits, not necessarily any involvement of computers or electronics.
For example, if they take a reading on a sextant, write down a number, and manually transfer it to a coordinate on paper, then that is a semi-digital process. If they take a number and then look up some kind of trigonometric table, that is definitely digital and not analog. But if the navigation process entirely consists of analog mechanical linkages and at no point any number is read out, then I would deem it 100% analog.
> using the radian rule, steering 1 degree off base course for 12 hours at a speed of 16 knots results in nearly 3.5 nm left or right of track (565 yards per hour)
This brings up a laughable feature of the US customary measurement "system", a hodge-podge of units with no coherent logic to it: 1 nautical mile ≈ 2025.371... yards, an awkward number that isn't even whole. This is because 1 nautical mile = 1852 metres exact and 1 yard = 0.9144 metre exact. Converting between these units would be a pain. (Whereas at least 1 mile = 1760 yards exact.)
Analogously, let's say you're piloting an airplane at an altitude of 15000 feet and have a horizontal distance of 7 nautical miles to your landing site. What would be the descent angle if you flew down in a straight line? The answer is not immediately obvious because you can't do trigonometry on different units. Whereas if I said you're 4572 m high and 12964 m horizontal distance away, then the angle is arctan(4572/12964) ≈ 19.4°. And even if distances are reported in kilometres, even a child knows that 1 km = 1000 m.
brookst
What a wonderfully pedantic takedown of the conflation of digital and electronic.
floren
Much written, nothing said
nayuki
Can't tell if that's a compliment or a snark. Do you have anything else to add to the discussion?
Btw, digital logic can be made out of LEGO, fluidics, electromagnetic (not electronic!) relays, steel plates (old railway signaling logic).
brookst
It was genuine admiration for sticking to a prescriptive language thing long after that ship sailed.
BTW you can do digital logic on… actual digits/fingers. But when someone says “digital signal processor” that’s not usually what they mean. Sadly.
y33t
Knots and nmi are not US customary units. 1nmi is 1 arcminute along the Earth's surface -- 1/60 of 1 degree. Knots are simply nmi/hr. Navigation is really just the application of spherical geometry. Not sure why they're referring to yards, I've never heard of those used in a navigational context.
If you want odd American units you should check out surveying. Units vary by geographical region.
nayuki
> Knots and nmi are not US customary units.
Uh yeah they are. They are _customarily used_ in the US, so they count. They're literally used on all ships and planes. Any unit that is broadly used in the USA is USC. Further, they are indeed listed under the USC page: https://en.wikipedia.org/wiki/United_States_customary_units .
y33t
Sorry, I meant that they aren't exclusively US units like US has its own unique pound, etc. Nmi are widely used because elevation of a celestial body can be directly translated to it's Geographical Point's distance from your location with some mental math. Handy.
xerox13ster
Go ahead and tune to your favorite station on an analog FM radio without using any digits, please.
Do you mind sharing what that station is with the class, or where it is on the dial?
nayuki
> tune to your favorite station on an analog FM radio ... dial
I mean, when an ad says "Listen to 123.4 FM!", they are literally conveying that information to you digitally; they are telling you the number to tune to.
A fully analog radio has markings to give you a rough guide of where the frequencies are, e.g. 99.0 MHz, 99.1 MHz, 99.2 MHz, etc. When you take the number in your mind and map it onto the markings, you are performing a digital-to-analog conversion (DAC). And because the dial is analog, you can make it fall between any marking you want; there are no steps that you're forced to take.
Meanwhile, a digital tuner would let you tune to 99.1 MHz and 99.3 MHz with nothing in between.
Training for this was discontinued, but brought back in 2016:
* https://www.npr.org/2016/02/22/467210492/u-s-navy-brings-bac...
Now if only the US (and others) would get their act together and build out a backup system to GNSS. China, for example, has built out an eLoran system:
* https://rntfnd.org/2024/10/03/china-completes-national-elora...
An old USAF video explaining how the theory works (it assumes a geocentric worldview: the Earth is the centre of the universe (but it's not flat :)):
* https://www.youtube.com/watch?v=UV1V9-nnaAs