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Re: Azimuth Angle and the Time Diagrams
From: Frank Reed
Date: 2015 Nov 28, 10:54 -0800

Steve Bryant, you wrote:
"Why is it that the azimuth angle, Zn, for a celestial body, in this case the sun, is not descernable on the time diagram? I'm speculating it might have something to do with measurements that can be represented wth two dimentional drawings and the three dimentional nature of the Zn; but, that is only a guess. I still can't conceptionalize any diffrence in the application."

Fundamentally, it's as Henry Halboth explained, but perhaps I can elaborate...

Hour angles and longitudes, the elements of your "time diagram," are angles measured about the Earth's axis. You can think of these either as viewed from above the Earth's pole as angles that wrap around a central point at the pole, or projected out onto the sky, and then they are angles centered at the celestial pole (near the North Star, in the northern hemisphere). By contrast, azimuth is an angle measured about the axis of the local vertical. You can think of this as an angle measured on the ground centered on the observer, or projected on the sky as an angle centered on the zenith. Since these angles are not, in general, measured about the same axis (equivalently centered on the same point on the sky), they cannot be displayed on the same simple polar diagram.

But wait. Sometimes azimuth and local hour angle have a simple relationship. As we approach the poles, the two axes (earth's axis and local vertical axis) become nearly parallel, and in the limit, when we reach the pole, they coalesce (at either pole, your local vertical is the same thing as the earth's axis). This has the interesting consequence that azimuth and local hour angle are essentially the same thing, apart from a fixed offset value, when we are close to one of the poles. And in fact, even a great distance from the poles, in higher latitudes, azimuth and hour angle are roughly the same in many cases. This has led to certain "rules of thumb" that are approximately valid in high latitudes, but completely wrong in general terms. There's a centuries-old rule in English maritime lore relating the time of the high tide to the Moon's compass bearing or azimuth. But we know, in fact, that the general rule should be based on the Moon's local hour angle. There's not much error in England because there is a rough match between the Moon's hour angle (again, apart from a constant offset value) and its azimuth in high latitudes, and in this case, England is a "high enough" latitude to make this work.

At the equator, there is a near complete disconnect between azimuth and local hour angle because there the two axes are perpendicular: the local vertical is everywhere perpendicular to the earth's axis for points on the equator. The simplest example of this is the case of the Sun's motion on the equinox at the equator. All morning long, its LHA changes by the usual 15 degrees per hour, but its azimuth is fixed near 90 degrees.

To see the simple connection between hour angle and azimuth near the pole, go to the USNO Celestial Navigation data page here and for your assumed position enter 88° 00' N and 180° 00' W (the longitude doesn't matter much but it makes it easier to see what's going on). When you calculate the data, you will discover that the GHA values and the Zn values are nearly identical. This even leads to a simple technique of navigation when near the pole: use the pole itself as the AP and then the declination of any body is identical to the Hc while the GHA serves as a surrogate for Zn. While azimuth (Zn) is formally undefined at the pole, this is really a mathematical curiosity rather than a real problem. In mathematical terms we would imagine a "final approach" to the pole "in the limit" as latitude approaches 90° along the meridian of 180° longitude since then GHA and Zn are identical on that path, in the limit. At the pole, replace Zn by GHA, and we're done. See how simple that is? Sight reduction near the pole is as easy as this: Hc=Dec, Zn=GHA for any body at any time. So as long as we can deal with the usual issues of long intercepts, anywhere within a few hundred miles of the pole, we can immediately compare measured (corrected) altitudes with the body's declimation and then get the distance of our LOP from the pole. The "azimuth" of the intercept from the pole is simply the GHA. Plot and you're done. By the way, does anyone recall: was this methodology a Weems invention?

Frank Reed
Conanicut Island USA

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