You are so far the most knowledgeable person on the subject that is at Subsim that I've had absolutely nobody to talk to on the subject. What is the time, date and location of your graphics. I don't think the average person can understand them and I think they deserve to. The scale is also too small to read the text.

Let me try from an astronomy program I use called Carte du Ciel. I'm not going to be able to pick the same location on Earth, but I can pick one at about 35 degrees latitude. I'll have to download into my Linux installation and I'll do it tonight.

First let's set the observing location:
https://goo.gl/ubQmih

Now let's look at the sky in Carte du Ciel, whole sky view:
https://goo.gl/nkVs4b

A couple of things of note. The altitude is the altitude of the center of the object, not the limb. The moon at zero altitude is half visible. Then look at the altitude at 29'58.5", essentially that's 30 minutes high. Since the moon is 30' wide that means that to the eye, the entire moon is visible with 15' of clear space below. You won't see that without a telescope, if you're lucky.

But look at that thing below the altitude. Nathaniel, you know what that is, but I'm explaining interesting stuff to others. The Geometric altitude is where the moon REALLY is. You see, our atmosphere acts like a lens, bending light that enters it from an angle. For objects on the horizon, and this thing qualifies, you can see that atmospheric lensing raises the moon from only 1 minute above the horizon (half visible) to 30 minutes above the horizon where you can see the entire Moon with half a moon diameter of black space below the moon! Rule of thumb: on the horizon all objects are raised by about half a degree.

https://goo.gl/TP1Egm
And in the far corner of the sky sits the Sun. Its altitude is 1 degree 39' 10.3", but its REAL altitude is 1 degree 19' 13.4". Now the moon isn't perfectly full, but pretty close, and you can see that with the Sun 1 degree 39 minutes above the horizon (you can just see all if it IF you have a perfect horizon plus perfect sky) and the Moon half a degree in the sky (there is a quarter degree of sky under it, not really enough to see even with a telescope unless your viewing conditions are perfect) they are essentially rising and setting as a unit. Considering that every day the moon's rise time is 50 minutes earlier and we're only 3 minutes apart comparing rise time of sun to set time of the moon this is pretty close to an ideal situation. You can possibly measure that they are not EXACTLY 180 degrees apart, but I daresay that no instrument of a submarine or ship would establish that. Measuring positions of objects on the horizon is basically impossible anyway.

And that's why you mentioned shooting noon positions. That's much more accurate. Straight up, you don't have to worry about atmospheric refraction. Since you are looking through the minimum thickness of atmosphere, seeing conditions affect you the least they possibly can for that particular weather condition. BUT the time of culmination, when the Sun is highest, depends on your longitude and YOU DON'T KNOW THAT. So you don't know exactly when to look to find the time of local noon. Of course navigators have sneaky and effective ways to deal with that uncertainty;

Nathaniel, take it away! How about explaining how to do a noon sight and how to use that information to develop your longitude? Latitude is a piece of cake. Just measure the altitude of Polaris. If you're being fussy you can use a polar alignment scope to correct for the offset of Polaris from the actual pole, but that's less than a degree. If you're south of the equator then the bet is off!

Did you know: the rate of continental drift in the Atlantic Basin, the rate the moon is increasing its distance from Earth and the rate your fingernails grow are all just about exactly the same? Coincidence? Or conspiracy?

I'm using Stellarium to produce the graphics. There's a version available for Ubuntu and a link for the source for Linux, so I think you should be able to get it running on your system. It's not as accurate as the program I use for serious calculations: USNO's MICA. But, it's great for being able to visualize various scenarios. And, unlike MICA, it's free! (Although, if you're in need of a program which will produce accurate astronomical data, MICA is the way to go. However, the USNO website can be used to produce most of the same data at no cost. Also, NB: MICA only runs on Windows and Mac.)

If you really want to discuss celestial, perhaps we could meet up on Discord for a voice chat. There's a link to the Subsim discord up top ↑. It might save us both a lot of typing. :) I must admit, I'm a total noob when it comes to Discord, but I hear it's similar to Teamspeak, which I've used extensively in the past.

Navigation, celestial in particular, is a passion of mine. So, if you'd like to be bored to death by the minutia of the subject...I'm your man. :D

I'm really interested here in getting the Subsim rank and file to understand some of the underlying principles happening in celestial navigation in a way that won't necessarily enable them to do it, but so that they don't feel like idiots whenever someone around them happens to mention it. Doing the chat wouldn't really accomplish that.

Now, is Greenwich Hour Angle the same as the Right Ascension of an astronomical object? I'm inclined to say close but no cigar, because astronomers use the Right Ascension to find an object. You use it for navigation.

So on an astronomical star chart, Right Ascension doesn't move for 25 or 50 years and then everybody changes charts. I imagine the Greenwich Hour Angles are real time adjusted for the very hour of calculation so the Greenwich Hour Angle of a star isn't the same two consecutive days at the same time.

Now there's a way to connect Hour Angles or Right Ascension with Longitude on Earth. Hour lines build to the west, starting at Greenwich. A star overhead at noon, will be 15 degrees away in an hour. So one hour in time is equal to 15 degrees of longitude on Earth. One degree of distance is equal to 4 minutes in time.

Let's say you are looking at the moon in a telescope with the perfect magnification to show the disk of the moon with no black space around. That means that roughly your field of view is half a degree. (notice how daintily I avoid saying 30 minutes, confusing angles and time. That would be bad.....) Now aim the telescope so the moon is just about to touch the field of view on one side. How long until it vanishes from view on the other side of your field of view?

Okay, the moon is half a degree wide. It's going to move half a degree to fill up your field of view and then another half degree to leave. So total movement to travel completely across your eyepiece is one degree. Moving 4 degrees takes one minute, so the moon will pass you completely by in a quarter of that time or 15 seconds! In 15 seconds you will go from seeing nothing to black space to seeing nothing but moon to seeing nothing but black space againi. It's hauling butt! Not really. Actually what you are seeing is the rotation of the Earth moving the aiming point of your telescope.

There's that reality trumping math thing biting us on the arse yet again! Math is an imperfect descriptive language like English. It can describe fallacy or truth with equal aplomb, and is only useful if we verify and correct, especially for chaotic change. There is no formula for when to add for a leap second. Yet.

Right ascension is not used in navigation, but a related concept is: "SHA" or Sidereal Hour Angle. To understand both, you need to understand the "first point of Aries", usually referred to as just "Aries"*. This is the point at which the ecliptic (the path that the Sun seems to follow in its apparent course through the stars throughout the year) crosses the celestial equator (the projection of the Earth's equator onto the "celestial sphere") when the Sun is ascending in declination. Ultimately, it's just a completely arbitrary point from which everyone agrees to measure positions.

Right ascension is the apparent arc distance of a body from Aries, measured from west to east. It is usually measured in hours, minutes and seconds. It's most handy for astronomers, because of something called "sidereal time". When Aries is on the observer's meridian, local sidereal time equals 00:00. About 6 hours later*, Aries will be on the observer's western horizon (setting) and sidereal time will be 06:00. Now, if a body has an RA of 6h0m0s* it will be on the observer's meridian - the best time to view it through a telescope. So, if we want to view a body with an RA of say...15h27m43s, we simply watch our local sidereal clock until that time, point our telescope at our meridian at the proper elevation, et voilà! There it is.

Now, SHA is also measured from Aries. But, unlike RA, it is measured from east to west in degrees, arc minutes and arc seconds. In celestial, it is usually only used for stars. The reason for this is to keep the almanac at a reasonable size. You see, there are 57 stars traditionally used in navigation (plus Polaris). They were selected based on their magnitude and distribution in the sky, such that, for an observer anywhere on Earth, the probability would be high that the positions of enough stars visible at twilight would be available for a "fix".

You are correct that GHA (Greenwich Hour Angle) is tabulated for each hour on the daily pages of the almanac, but even that is not enough precision for navigation. The rate of change in GHA varies between stars, planets and the Sun and Moon, but in every case it's fast enough to matter. Therefore, tables are included towards the back of the book which allow for interpolation down to each second of each minute of each hour of every day in a year. However, listing the GHA for all 57 stars, even hourly, would result in a massive tome...much too bulky to be practically usable...let alone even carried on a small vessel, where space and weight are at a premium. So instead, the GHA of Aries is listed, along with the SHAs of the 57 stars. (See here for a sample "opening" of an almanac, courtesy of Navsoft.) To obtain the GHA of a star, one needs to add the SHA of the star to the GHA of Aries (subtracting 360° when exceeding that value), and then interpolate for minutes and seconds. Longitude is then added (for east longitudes) or subtracted (for west longitudes) to obtain LHA, or Local Hour Angle. (Again, subtracting 360° when necessary.)

LHA along with declination (also listed on the daily pages of the almanac) are the two variables needed to calculate the position a body occupies at a given location. In practice, the position is calculated for an "assumed position" (AP) which is then compared to an actual observation (corrected for various factors) and an "azimuth" (direction) and "intercept" (distance) are obtained, allowing the navigator to draw a "line of position" (LOP) relative to the AP. The result is the intersection (if you're lucky) of the lines at the fix. As you mentioned earlier, the more likely scenario due to errors is a series of lines describing an area (a triangle for three observations) which contains the "most probable position" (MPP).

Since RA and SHA are both measured from Aries, just in different directions, SHA can be obtained from RA by converting RA to degrees (by multiplying by 15) and then subtracting it from 360. And RA can be obtained from SHA by subtracting SHA from 360. (Dividing the resultant degrees by 15 to obtain h:m:s, of course.) It should be noted that this is rarely necessary, due to the aforementioned fact that these two values are seldom used in the same pursuit. The main exception being when a navigator is plotting the positions of the Sun, Moon and planets on a "star finder" for planning purposes.

You may be wondering why the SHAs and declinations are listed on each three-day page of the almanac. It's because they are changing more rapidly than every 25 to 50 years due to "proper motion". Some more than others, but enough that it makes a difference for navigational purposes. However, there are a few pages in the almanac which list the SHAs and declinations for many additional stars for six month periods in case those stars need to be used in an emergency where the traditional stars are not visible due to cloud cover or some other reason.

Finally, see here for an animation I made which attempts to explain how SHA and declination are measured. As I said, I sympathize with visual learners, being one myself.

...And that is probably way more than you ever wanted to know about that. ;)

*Not to be confused with the Zodiac sign or constellation. Although, Aries gets its name from the constellation, because when it was first described, that's where it resided. Now, due to precession, its apparent position has moved.

*A "sidereal day" is about 4 minutes shorter than a solar day. Therefore, if you look at a particular star each day at the same time, it will appear to have shifted position to the east by about four minutes of time, or about one degree. This is due to the motion of Earth revolving around the Sun.

*Note that when referring to minutes and seconds of time, we use "m" and "s". When referring to arc minutes and seconds, we use ' and ". That's how we avoid confusion.

About the first point in Aries, which isn't in Aires any more. It is definitely not an arbitrary point (of course! It has to be measured and confirmed.) The first point in Aries is that point where the Sun's path crosses the celestial equator in its path from south to north. This happens on the first day of spring for us northern hemisphere denizens.:salute:

All this mess with the first point in Aries not being in Aries any more comes from the roots of astronomy in astrology, 3000 years ago. Because the Earth wobbles on its axis, the north pole makes a circle in the sky with a period of 26,000 (roughly) years. This MOVES the first point in Aries, the place where the Sun crossed the celestial equator 3000 years ago to the point where the "first point in Aries" is two constellations away now in Pisces!

For a little fun, go see the total solar eclipse, find an astrology obsessed friend and travel to the eclipse. Before the eclipse, ask them which constellation the Sun is in. Then during the eclipse, have them identify clearly that the Sun is not in its astrological constellation at all! It is two constellations over. But wait! It gets worse. Astrology says that during the year the Sun passes through 12 constellations. That is dead wrong. The Sun passes through 15 different constellations, spending the most time in the largest of them, Ophiuchus, which doesn't rate a seat at the Zodiac table!

So astrology, which as its basis says that the positions of the stars, sun, moon and planets guide our lives, doesn't even bother to track the stars, sun, moon and planets. So much for any credibility. The shock on an astrology enthusiast's face when they see that for themselves, with the Sun two constellations away from where their beloved charts say it is, is priceless.

It's actually a common belief in astrology that, like the first point of Aries, the "houses" were named for the constellations they were in when they were discovered. Don't get me wrong, I don't buy into astrology.

I just noticed the text you added to your previous post. In it, you stated that...

That's not exactly right. While it is true that if a body is directly overhead, at the observer's zenith, you don't have to worry about refraction...this is a rare occurrence. It only happens when the declination of the body exactly matches one's latitude. Any other time, the altitude of a body on the local meridian will be equal to 90° minus the declination plus or minus the observer's latitude (depending on whether they are in the same or different hemispheres.)

You also wrote:

I'd be happy to. But it'll have to wait. I need to get some rest before work. :)

Also, while I was able to see the screenshot from before when you linked to it, I still cannot see any of the images you are posting in-line. I'm not sure if it's a problem on my end or what.

Alright...back home from work and I've gotten some more rest. Now, for the noon sight:

The noon sight is rarely, if ever used to find longitude. The reason is that, in order to accurately find longitude at noon using a sextant (and chronometer), one needs to determine the exact time at which the Sun transits the local meridian. When the Sun is on the local meridian, it is at its highest altitude for the observer's location. A sextant measures the altitude of heavenly bodies...so that should be easy, right? Not so much. Right around noon, the Sun seems to "hang" at the same altitude for up to several minutes.

Take a look at this graph. (You should be able to zoom it if it is not clear.) The blue line shows the altitude of the Sun every fifteen minutes on July 17th, 2017 at 45°N, 0°E. The red line (which uses the right-hand vertical axis) shows the rate of change in altitude in arc minutes per minute. Notice what happens at local noon: the rate of change drops to nearly zero. That makes it very difficult to determine exactly when the Sun has reached culmination.

There is a "trick" which can be used to try and find the time of local noon: double altitudes. The navigator measures the altitude of the Sun some number of minutes before noon and notes the time. The sextant is left at whatever reading was taken at that time. Then, the navigator waits until after noon, when the Sun drops to the exact same altitude again, and notes the time. splitting the difference between these times should give the time of local noon...if you're stationary...and not on a pitching and rolling ship...and make perfect observations. But, even then, it's tough to get an accurate time.

For the sake of explanation, let's say you did get an accurate time. How does that tell you your longitude? Well, your chronometer would be set to GMT (Greenwich Mean Time or UT [Universal Time], essentially the same thing). Let's say you found that local noon occurred at 14:32:17 GMT on July 17th, 2017. Now, you know that noon in Greenwich occurred at 12:00:00 GMT...or did it? Not necessarily. You have to consider something called the "equation of time". Because the Earth speeds up and slows down in its yearly orbit around the Sun, the Sun appears to race ahead or lag behind "mean" time (the time that a regular clock keeps). On this date, the equation of time is about -6m13s. IOW, the Sun is six minutes and thirteen seconds behind where the mean Sun would be, and noon would occur later. (This information can be found in the Nautical Almanac [see the bottom of page 2 in this example].)

So, Greenwich noon occurred at 12:06:13 GMT. If we subtract that from the time of our local noon, we get 14:32:17-12:06:13 = 2:26:04. Local noon occurred two hours, twenty-six minutes and four seconds after Greenwich noon. Since we know that the Earth rotates 360° every 24 hours, we know that the Sun appears to move 15° through the sky each hour. So, if we multiply 2:26:04 by 15, we get: 2:26:04∙15 = 36°31'00"...this is our longitude. And since local noon occurred after Greenwich noon, we know our longitude is west: 36°31'00"W.

However, as we have learned, the noon sight is not typically used to find longitude. What it is used for is to find latitude. So, how is this done? Well, in this case, it's kind of handy that the sun "hangs" at the same altitude for several minutes. Because that's what we're after - the maximum altitude. Let's say that, after correcting for index error, height of eye, refraction and semi-diameter, we get a maximum altitude of 66°05.5' on July 17th, 2017 at 14:32:17 UT. We subtract the altitude from 90° to get the "zenith distance" - the distance the Sun is from being directly overhead. (This is equal to the distance we are from the Sun's GP.) 90°-66°05.5' = 23°54.5'. Now, we look in our almanac and find, after interpolation, that the Sun has a declination of N21°05.5'. Since the Sun's declination is in the same hemisphere as our DR, we add the declination to the zenith distance to find our latitude: 23°54.5'+21°05.5' = N45°00.0'.

Finding longitude was usually done by "time sight". But that is another subject altogether.

That raises the question of how accurately can you measure the altitude of the Sun, an extended object, on the pitching deck of a submarine? I bet you couldn't duplicate two sightings within 10' of arc. I wouldn't be a bit surprised at an error envelope of a half degree. That would make finding longitude with a noon sight impossible.

It still seems to me the altitude of the north pole is the best way to establish latitude. If you're on land that can be done with incredible precision using a polar alignment scope. Alternately, you could use Polaris itself when the position angle of the pole is parallel with the hofizon. That happens twice a day.

The altitude of the Sun is always going to be a problem. The Sun isn't a point, its a disk. You have to find and align on the exact center of the disk.

Typically, a series of sights are taken around noon. The navigator can choose to use the highest measured altitude directly, or can plot all of the observations onto a graph and rough a curve through them. The top of the curve can then be used as the culmination altitude.

Included in the Nautical Almanac (and also in some sight reduction tables) is a table for correcting the offset of Polaris from the elevated pole. The navigator simply calculates LHA Aries and enters the table to find the needed correction for the time at which the sight was taken. An accurate latitude can be determined fairly easily this way. A corrected azimuth can also be obtained for checking a compass.

It's practically impossible to accurately shoot the center of the Sun or Moon with a sextant. That's why a correction for semi-diameter is included in the calculations. The navigator shoots the lower limb (typically, but sometimes the upper limb for the Moon) and corrects the altitude for half of the diameter to find the altitude of the center. The SD of the Sun is listed at the bottom of each daily page, but the correction is included in the altitude correction tables for both Sun and Moon.