Thursday, May 8, 2008

Celestial Coordinates and the Nautical Almanac

Imagine a clear plastic balloon covering the earth's surface. On it we shall trace the equator, the poles and the Greenwich meridian. Now, if we expand this balloon until it imaginarily touches the Sky, a view from inside this balloon shows these tracings against the sphere of stars. There is now a celestial equator directly above the earth's equator, a celestial north and south pole directly above the corresponding earth poles, and a celestial Green­wich meridian directly above the earth's prime meridian. A celestial body's coordinates can be defined exactly as ones on the earth except now we are using the expanded tracings. Sky coordinates are named differently. Sky latitude is called declination (Dec.) and is labeled north or south depending toward which pole the body is from the equator. Sky longitude is called Greenwich Hour Angle (GHA) and differs from earth longitude in that it is measured always westward from the Greenwich meridian, 0° to 360°.

By establishing the GHA and Dec of a celestial body for a particular instant of time, we have found the corresponding latitude and longitude of the geographical position of that body. Since the Sun, Moon and planets change their GHA and Dec constantly as they seem to move through the sky, these coordinates are listed in the Nautical Almanac for each hour of each day of the year with additional corrections in the yellow pages in the back of the Almanac for each minute and second of the hour. Listing the positions of these six bodies for each hour of the year is not a particularly space consuming process. Parts of two pages will cover a three day period as you can see by using the daily pages of the Nautical Almanac. However, there are fifty-seven navigational stars that may be easily observed during twilights and many more fainter stars which might also be picked up during this period. Therefore, to conserve space for these stars which in effect are stationary with respect to the celestial sphere, we can use another type of longitude called Sidereal Hour Angle (SHA). This is also an angular measurement along the equator but the starting point is not the Greenwich meridian.

Another arbitrary point has been chosen as the basis for SHA called the vernal equinox. Also called the first point of Aries, it is the position the Sun occupies on the first day of spring when the Sun changes from a southern declination to a northern one. At this transition point, the Sun is on the equator. The vernal equinox is designated by the sign for Aries. We can then establish a permanent longitude of stars with respect to the celestial sphere and are only left with the problem of relating this longitude to the Greenwich meridian which changes its position on the sphere as the earth rotates. To solve this problem, we need to find the GHA of the vernal equinox. Add to this the SHA of the star and you have the GHA of the star.

In southern latitudes, the celestial equator will cross the northern section of the meridian.
The vernal equinox can be roughly positioned on the equator by finding the constellation of Pegasus (if it is visible at the time you observe), taking the left side of the square as one unit of length and proceeding down from the lower left star in the square one more unit. The vernal equinox will be slightly to the right of where you arrived.

Our sight reduction forms are fairly self explanatory as to where to enter GRA, Dec, and SHA. We go into the Nautical Almanac with Greenwich time and date and pick out the value for the hour of GMT. In the case of a star, GHA is chosen out of the Aries column. At the bottom of the planets and Sun column and in separate columns of their own in the Moon section we find a small-type number designated "v" or "d" (the Sun has no "v" and the stars have no "v" or "d"). The "v" is an extra correction for additional longitude move­ment of the celestial body and the "d" is an extra correction for additional declination movement of the body. Planets, Sun and Moon move on their own in orbits around the Sun and earth and this change of position in the sky is accounted for in the "v" and "d" corrections. The "v" correction will always be positive except in the case of Venus where it may be negative. In this case, the "v" will be marked with a negative sign. The "d" correction can be either positive or negative depending on the trend of the declination. Look at the declination column of the body in question at the time in question and see if it is increasing or decreasing. This will establish the sign, increasing being positive, decreasing being negative.

These "v" and "d" numbers are only reference numbers for the celestial body's movement in one hour. If we took a sextant sight at thirty minutes after the hour, the actual correction would be 30/60 or 1/2 of the reference numbers. To save our doing the arithmetic, we enter the yellow pages in the back of the Nautical Almanac with the minute and second of our sight, pullout a value for the earth's movement in columns entitled Sun-Planets, Aries and Moon, and then go the the right hand side of the minute box and find the "v" or "d" values. The corresponding number to these values is the correction factor, to be added or subtracted as the sign of the "v" or "d".