IFR by the Sun and Stars
I achieved a degree of fascination and competence with celestial navigation over 50 years ago as an Air Force navigator and recently have found that some of my flying friends have expressed an interest in the basics of this "lost" art, because ... well, just because.
Here's a primer for any of you who might be interested as well. There is, perhaps, even a tidbit that will help in your modern-day IFR flying. But let's put that aside for just now.
How to Shoot a Star
Imagine a celestial body—the sun, moon or a prominent star—is exactly over your head at 90° elevation. Granted, it's out there in space, but from a flat-world point of view it's exactly where you are because it's right over your head. So we say its "subpoint," the point on the earth directly beneath the object, is zero miles from where you are.
Now imagine a celestial body that is on the horizon, or 0° elevation. With a lot of simplifying assumptions, its subpoint is 90° x 60 miles/degree from where you are (and we're talking nautical miles for this whole discussion). That's 5400 miles from your position. So we could draw a line on our chart offset 5400 miles from that subpoint and perpendicular to the direction to that subpoint. This is a line of position (LOP), and you must be located somewhere on that line. Actually, you are on a circle around the subpoint with a radius of 5400, but short segments of that circle can be treated as a straight line.
Of course, the devil is in the details, and to put the above basics into a system for practical application takes an organized library of books of tables much more boring than the FAR/AIM. But, similarly, one doesn't have to read them at one sitting; one just picks the part needed and uses it.
These celestial tables are based on one grand, simplifying assumption: All celestial bodies are located on the celestial sphere that rotates around a stationary Earth. With apologies to Copernicus, this assumption is convenient and works well for the stars because they are pretty much fixed on the celestial sphere. The Sun, Moon, and planets wander through this nice, ordered environment and thus require special treatment, but the tables help out with that nicely. (The word planet comes from the Greek for "wandering star.")
The Celestial LOP
To locate a body on the celestial sphere, one uses a coordinate system analogous to that used on the Earth. "Latitude" north and south from the celestial equator is measured in degrees and is called declination. The analog of longitude is the Greenwich Hour Angle, or GHA, and it is measured westward from a particular spot on the celestial sphere, arbitrarily selected, called the "first point of Aries." Think of Aries as the analog of the prime meridian or Greenwich meridian.
But, since the celestial sphere is moving westward at 15° per hour, the big trick in celestial navigation is knowing your time. For a particular date and time, the Air Almanac gives, in tabular form, the GHA of Aries for stars and also individual GHAs for the Sun, Moon, and selected planets. But what you really want to know is where the celestial body is with respect to your position. For this, you need the Local Hour Angle (LHA), which is simply the GHA minus your longitude (for the Western Hemisphere).
Now, you can go into the Sight Reduction Tables for Air Navigation for an assumed latitude and LHA, which is based on your estimated position, and read the expected height (Hc) and bearing (Zn) for the body you are going to shoot. You don't have to be real close with your estimated position, but you need to start with an assumed point on the globe for this to work. That's why dead reckoning is king for celestial navigation. I'd argue that it should be for all IFR pilots, but that's another matter.
The celestial shot is taken using an aircraft sextant that has a bubble level to establish the horizon. This is different from a marine sextant that uses the visible horizon. The scene through the sextant eyepiece has vertical and horizontal cross hairs. The bubble is kept centered by slightly tilting the sextant left and right and fore and aft. The body is kept centered by rotating the sextant so the vertical hair is on the body and by using your right thumb on an external knob to change the observed height and keep the body on the horizontal hair.
The sextant has a two-minute timer. When the time runs out, a shutter drops down and blocks the view of the body. By realigning indices on the timer, you then read the average "observed height" (Ho) directly from the altitude counter.
The LOP is then plotted by calculating a distance from the assumed position. If the Ho is greater than the Hc, the distance in miles is plotted toward the body, in the direction of the Zn. If the Ho is less than Hc, the distance is plotted away from the body. Once you have that distance relative to your assumed position, you can draw a line perpendicular to the Zn on the chart. This is an LOP and your real position is somewhere on that line—assuming no errors.
When I was flying the North Atlantic with the Air Force's Military Air Transport Service (MATS) (today's MAC) in the late '50s and early '60s, there were still large areas of these oceans that were without any electronic navigation aids. We navigators would pride ourselves in being able to shoot and plot a three-star fix in about 15 minutes. We would pre-compute the shots so the Hc and Zn for each would be listed on our shot form.
For a three-star fix, you would have your shot times four minutes apart because the LHA would change an even one degree each four minutes (remember, the Earth rotates 15° per hour) and this "simplified" the computations.
Assuming a fix time of 0100Z, the first shot would be timed for 0052Z and the second for 0056Z. Since we were getting a two-minute average, the first shot would start at 0051 and end at 0053. The second shot would start at 0055, so that gave us two minutes to hop down from the sextant, finish computing and plotting the first LOP, set the sextant up for the second shot, and get our eyeball on the next star, ready to start the timer.
We would do the second and third star the same way. By 0105Z we could have the fix plotted and, by comparing our current position with our last known fix, provide an updated wind, true course and ground speed, as well as a new compass heading, to the pilot.
Walk the Line
From time to time, I reflect with empathy on the task that Fred Noonan had on his ill-fated flight with Amelia Earhart to Howland Island in 1937. Fred was well-experienced, having established most of the Pan Am's China Clipper seaplane routes across the Pacific. Noonan and Earhart left Lae, New Guinea, in late morning local time for the roughly 2200-mile, 20-hour flight, with a planned arrival at Howland after sunrise, about 8 a.m. local time. That gave Noonan all night for three-star navigation and a morning arrival to find the little island.
Based on Earhart's last few voice transmissions, "We are on the line 157 337 ...," Noonan was using a navigation technique called celestial landfall. Celestial landfall requires intentional off-setting the track to the left or right of the intended course, so that when the craft gets abeam the destination the navigator knows which way to turn.
The approach to Howland was quite literally the textbook reason for the landfall technique. With only a single celestial body to shoot—the Sun—Noonan had good information to determine his groundspeed but nothing for course guidance, because their true course was about 082° and the Sun's Zn was 067°. If that's not clear, think about it this way: They were flying toward the rising sun, so Noonan could calculate the distance toward the sun with good precision and know how fast they were moving along the course. Deviation north or south of the planned course, however, would be tough to calculate, as their LOP would run 157° to 337°.
I would guess that Noonan altered course to put them perhaps 60 miles north of their flight-planned true course. About an hour prior to his ETA, he would have taken a sun shot that gave him the 157°-337° LOP. Using his latest computed ground speed, he would have slid that LOP forward on the chart to where it would run through Howland.
Then he would have taken up a course that was perpendicular to the LOP, in this case 067°, and would have computed a dead reckoning position and time for intercepting the LOP. When they reached that time, they would have turned right to about 157° to fly down the LOP to Howland. That turning point, the dead-reckoning position, may have been on the order of 60 miles or about 30 minutes flying time from Howland.
During those 30 minutes there might have been a bit of conflict in crew imperatives. My experience with the South Pacific is that it is common to have wide stretches of scattered-to-broken, fair-weather cumulus clouds with bases as low as 1000 feet and tops somewhere around 3000 feet. Noonan would have wanted to stay on top to keep shooting the Sun to make sure the LOP continued to fall through Howland. Earhart would have wanted to drop down to see the island. Unfortunately, each of those little clouds has a shadow that looks just like a flat island.
Back to the Future
For today's IFR pilot, a lot of things have to go wrong for you to need to fall back on an E-6B, dead reckoning, and a terrestrial landfall. And few of us really know how to use these techniques anymore. We still use lines of position, in a sense, because GPS is based on a similar set of principles.
But there is still real value in knowing your estimated position as both a point on the chart and a time hack. It matters because your GPS can lose its mind or its satellites. It matters because it's good tonic against confusion on an instrument approach. It matters because the simple act of bothering with such details promotes a self-discipline that works its way into all your flying.
And perhaps you'll find some amusement in a connection with the navigators of yore who used the original GPS: Gienah, Procyon, and Sirius. Some things are fun to know, just because.