Bob Bernecky, you wrote:
"I was not sure how relevant it was to lunars (which seem much more complicated)."
Lunars are not complicated at all -- unless you start at the end and try to work your way backward, as so very many do. Naturally that makes the process seem complicated. And unfortunately, many people take that approach starting with the trickiest detail they can find and working backward, or they even go to greater lengths and try to do everything from scratch, including their own ephemerides, and thus make a mess of the entire process, as we saw from a recent contributor here.
"As for radians being dimensionless, that seems to be the terminology people use (per a google search)."
It is not radians that are dimensionless, and it is not merely "terminology". Angles themselves are dimensionless. Angles ARE ratios -- pure numbers, not things with units of "degrees" or "radians" or anything else.
ANGLES ARE RATIOS. Learn this, absorb this, and own this, folks. Angles are ratios, and, therefore, they are merely numbers without units or dimensions. Our common degrees and minutes are not units. They're just a peculiar counting system, akin to counting carrots only in dozens and gross (a gross is a "dozen dozen"). When we work in this ancient mesopotamian sexagesimal counting system of degrees and minutes, we sometimes find ourselves scratching our heads... "hey, what's minutes of arc times minutes of arc per second of time??" When you find yourself doing that, you are stuck in the trap. You're stuck in a trap that you should have gotten out of years ago. Angles ARE ratios. When you do calculus on trigonometric equations (e.g.), you should always think in terms of angles as pure ratios, and you should always read the equations that you see as equations in which the angles are pure numbers. And then, at the very end, if you need to do so for practical application, in the last step you convert to good old mesopotamian degrees, minutes, seconds, thirds, fourths, and so on (yes, they used to do this, but luckily thirds and the rest were almost completely replaced by milli-arcseconds and micro-arcseconds).
More likely than not, some of this is vaguely familiar to many of you, because you learned this concept of "angles as pure numbers" through the terrible, infantile, engineer's trick of calling these "angles in radians". It's a "conversion" which was also one of the very first clever features added to calculators in the 1970s. Well, folks, it's time to throw away those toys. Angles are not given "in" radians. Radians are not anything at all except a name to tag on the end to avoid confusing readers without basic math insights. ANGLES ARE RATIOS. Angles are pure numbers.
Just in case you're saying to yourself, "Ratios? OK, but a ratio of what to what?", just go back to the simplest relationship that you know about an angle since that relationship --a ratio-- is, in fact, a definition. An angle is a ratio of the size or distance "across" the line of sight to some object or construct to the distance "out" to the object or construct, with the important geometric detail that the distance "across" is measured along a circular arc centered at the point where the angle is defined. Angle is "size divided by distance". In succinct terms: θ = s/r. That relationship doesn't "calculate" the angle theta; it defines its meaning. Every angle is a ratio -- just a number. It has no units, no dimensions.
I should clarify that I am not saying we should throw out the baby with bathwater. Dealing with angles only and always as pure ratios without any aspect of sexagesimal counting is un-necessary, pointless, and impractical. But we can get rid of a lot of it. For example, in my "Modern Celestial Navigation" classes as now taught, we get rid of everything but degrees as soon as we take the raw observation off the sextant. It's immediately converted to decimal degrees and all corrections, all ephemeris data, all calculations are done in decimals --no minutes, no tenths of minutes, no seconds. And that's a lot easier for everyone. I agree with the late Hanno Ix that we should have converted to decimal degrees decades ago.
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA