As navigators, we depend on some fundamental concepts to work our trade. Locating a celestial body, for example, is done using the coordinate system of declination and Greenwich hour angle (GHA).
Declination, in its simplest form, tells us the location of a heavenly body, north or south of the celestial equator. Declination is similar to latitude in two ways: 1) It can vary from 0° at the celestial equator to 90° at the celestial poles, and 2) it is always labeled either north or south. However, declination is much more interesting than just a way of measuring angular distance from the celestial equator. Knowledge of declination can also provide us with an understanding of how the navigational objects in our solar system move through the skies.
To get a handle on declination we must first review the apparent and absolute motions within our solar system. All celestial bodies are in motion. And since our celestial body, Earth, is also in motion, our observations of other body’s movement can properly be termed apparent motion. This apparent motion is a combination of two principal motions of celestial bodies: rotation and revolution. Rotation is the spinning motion of a body about its axis and revolution is the motion of a body in its orbit around another body. Earth’s rotation is the most significant cause of apparent motion. This motion causes bodies to appear to rise in the east, climb in the sky to a maximum altitude, and then set in the west.
However, declination also has a role to play in the apparent motion of various bodies. The best example of this is the sun. The sun’s declination is defined by the fact that the earth’s axis of rotation is not perpendicular to its orbit. It is this inclination that causes the sun’s seasonal change of height in the skyhigh in the summer and low in the winter for the northern hemisphere. This angle of inclination, measured from the celestial equator is approximately 23° 27′, and is referred to as the obliquity of ecliptic (the ecliptic is the sun’s annual path across the sky). Due to this obliquity, the sun’s declination varies from 23° 27′ N to 23° 27′ S during the course of the year.
It is the sun’s declination, combined with an observer’s latitude, that determines the sun’s daily height in the sky. The closer the observer’s latitude to the sun’s declination, the higher the body will appear in the sky. A position directly under the sun would indicate a latitude equal to the sun’s declination.
Declination is tabulated for the sun (as well as the four navigational planets, moon, and 57 navigational stars) in the Nautical Almanac. The values of declination are determined using formulas (algorithms) developed after years of observations by the U.S. Naval Observatory and other astronomical organizations. The sun’s declination algorithm is: (sin e sin l )where e is the obliquity of the ecliptic and l is ecliptic longitude.
Obliquity of ecliptic, as mentioned earlier, is the earth’s inclination to the celestial equator. Ecliptic longitude is the sun’s longitude adjusted for the two somewhat arcane factors of aberration and anomaly. These are irregularities in Earth’s orbit that are described by mathematical formulas. Aberration is an apparent shift in light rays coming from celestial bodies due to the fact that Earth is clipping along in its orbit at 18.5 miles per second. This apparent shift is in the direction of Earth’s motion. Anomaly is the term used to describe the time necessary for Earth to make one revolution around the sun. The ellipse of Earth’s orbit is slowly rotating through space, so Earth’s revolution, with respect to the sun, is not the same as its revolution with respect to the background stars. An anomalistic year is approximately five minutes longer than a sidereal year, a sidereal year being the time necessary for a body to move through 360° as measured against the stars.
Using the declination formula, a graph can be constructed to represent the sun’s changing declination over the course of a year. This curve, resembling a sine wave, shows a consistent monthly change in declination from January through May with a slowing in June followed by a steady change from July through November with a slowing in December. The peaks of declination in June and December are referred to as solstices (sun standing still) and can be thought of, in analogy, as similar to the stand of a tide at high and low water. (This slowing of motion is only in the north/south direction and not a slowing of the daily apparent revolution of the sun around Earth.) The sun’s declination, then, is never greater than 23.5° N or 23.5° Sit is always somewhere between these two values.
Points of zero declination, on March 20 and September 23, are called equinoxes (equal night and day). On the equinoxes the sun’s declination is 0°, and it rises due east and sets due west. Days and nights are approximately equal in length at every latitude. There is no slowing of declination change at the equinoxes.
Examining the tabulated declination values for the sun in the Nautical Almanac for June 21, 1994the day on which the Sun experiences its highest declination in the Northern Hemispherewe observe the declination remaining unchanged for 20 hours, hovering at N 23° 26.3′. Again, six months later, on December 21, and into December 22, we see declination remaining at S 23° 26.3′ for 14 hours. During the remainder of the year, the Sun’s declination changes at an hourly rate ranging from 0.1′ per hour, just before and after the solstices, to a maximum of 1.0′ per hour in March/April and September/October. The northern limit of the Sun’s declination, N 23° 26.3′, is at a line known as the Tropic of Cancer and the southern limit, S 23° 26.3′, as the Tropic of Capricorn. These names come from the constellations which the sun entered at the solstices more than 2,000 years ago.
Earth’s axis is undergoing two types of motion: precession and nutation. Precession results from the sun, moon, and planet’s gravitational forces pulling down on the tilted Earth so that its axis of rotation is perpendicular to the plane of its orbit. It takes approximately 25,800 years to complete one of these precessionary cycles. Precession has carried Earth’s axis 30° from where it was 2,000 years ago.
Nutation is a short-term “wobble” in Earth’s axis, caused predominately by the gravitational forces of the moon. Nutation accounts for three or four wobbles over the course of a year.
Earth’s orbit around the Sun is elliptical and, therefore, the distance between the sun and Earth varies between 91,300,000 miles and 94,500,000 miles. Earth’s closest approach to the sun (perihelion) is in early January with the farthest point (aphelion) being in early June.
Kepler’s orbital laws
The laws explaining elliptical orbits were deduced by Johannes Kepler in the 1600s and are now known as Kepler’s laws. Of interest to us in the understanding of declination is Kepler’s second law which states, “The straight line joining the sun and a planet (earth) sweeps over equal areas in equal intervals of time.” Thus, Earth travels faster when nearest the sun.
Since Earth is nearest the sun during the northern hemisphere winter and farthest during the summer, it is the declination of the sun to Earth, not the proximity of Earth and sun that accounts for the Earth’s change in temperature and, thus, the seasons. Following Kepler’s second law, Earth’s variation in speed around the sun causes Earth’s astronomical winter to be seven days shorter than summer. The sun’s altitude in the sky and the length of time above the horizon, both a function of declination, are the factors which determine the difference in temperature between winter and summer.
Additionally, because Earth is a sphere, there is not a linear relationship between the sun’s declination, an observer’s latitude, and the sun’s height in the sky. Our view of the sun as Earth revolves is defined by the mathematics of great circles.
Here is an example: on June 21, 1994, with the sun’s declination at its highest value of N 23° 26.3′, an observer at 45° north latitude would observe the sun rising at a true bearing of 055° T, while an observer at the equator would observe the sun rise at a bearing of 067° T. These bearings, far to the north of east, are due to the sun’s bearing being along a great circle path (see diagram on page 71).
When one takes a sight of the sun and wishes to do a sight reduction, one of the items that one needs is declination. To find the sun’s declination at a given time, we enter the daily pages of the Nautical Almanac. Each set of daily pages lists hourly declinations for the sun for a three-day period. Since one will rarely, if ever, take a sight exactly on the hour, these hourly amounts of declination must be adjusted for the exact time the sight was taken. This is done by employing an interpolation factor. (The interpolation factor is based on the amount of declination change between 1200 and 1300 GMT on the middle of the three days listed.)
This interpolation factor is recorded at the bottom of the declination column, designated by the letter “d”. We use this “d” value, the average hourly change in declination, to enter the increment and corrections section of the Nautical Almanac and extract the appropriate correction to be added or subtracted to the hourly declination.
The meaning of little d
According to Dr. Paul Janiczek of the U.S. Naval Observatory, the average hourly change in declination was designated “d” many years ago, and its origins are arcane and not precisely recorded. “You could say the little d stands for departure from the mean value of declination, if you like,” Dr. Janiczek said.
Let’s look at an example: say we shot the sun at 16:41:19 GMT on October 22, 1994. As part of our sight reduction process we need to determine the sun’s declination at 16:41:19 GMT. First we look up the declination at 1600 GMT on October 22the value is S 11° 07.6′. The “d” value, or average hourly change of declination over the three day period, at the bottom of the page is 0.9′. If we note the declination entries for 1200 and 1300 GMT on 23 October we see they are S 11° 25.1′ and S 11° 26.0′ respectively, confirming the 0.9′ difference. This 0.9′ amount is now used as the entering argument in the increments and corrections tables in the back of the Nautical Almanac.
We go to the page for 41 minutes, the increment of an hour for our shot, and enter the v/d correction columns. We travel down the first column, locate the value of 0.9′ and extract the correction which is located horizontally to the right in the Corr column. For 0.9′ the correction is 0.6′. Logically this makes sense, the hourly change in the sun’s declination is 0.9′ and since the sun has progressed 41/60 into the next hour the change in declination is 41/60 x 0.9′, which equals .6′. This amount, 0.6′, is added to the 1600 GMT tabulated value of declination because declination is increasing in value and will continue to increase until December 21st.
Use of an average hourly change of declination over a three-day period does not cause a significant degradation in accuracy. For example, at the extreme ends of this three-day periodmidnight on the 22nd and midnight on the 24th the hourly change is 0.9′ and 0.8′ respectively.
The subtle, yet daily change in the Sun’s declination accounts for our changes in daylight and seasonal temperature. Declination also provides one of the key pieces of information necessary for a navigator to determine their position.