Knowing (fixing) the geographic location of your vessel to the highest degree of precision remains one of the most important tasks of a mariner. Aside from DGPS technology which produces reliable fixes accurate to within a few yards, the most reliable fix comes from visual observation of the position of known objects relative to your vessel.
Among the simplest ways to do this is to take bearings or ranges to charted objects. You draw a line on the chart corresponding to the observation, and you know that the vessel was somewhere on that line when the observation was taken. We call this line a line of position (LOP). The point where two such LOPs meet marks your position and is called a fix. It is rare for more than two the LOPs to meet at a single point, and the resulting open triangle or box indicates the accuracy of the fix. Two components are required to determine any reliable fix: 1) knowing the exact position of the object you are observing, and 2) having a means for measuring its position relative to yours. The numerous ways of converting the observation to a LOP and subsequently plotting a fix are all variations on this simple theme.
Celestial navigation most closely corresponds to using LOPs derived from ranges to known objects. Celestial navigation applies the basic idea of determining a fix in a fairly straightforward way. In celestial navigation the object from which you obtain a LOP is a celestial body. Because the earth rotates on its axis and revolves around the sun, and because the solar system moves through space relative to the stars, celestial navigation requires the application of three fundamental principles in order to derive the two components for a reliable fix. It may appear complicated at first, yet once understood you should discover simplicity in its elegance and an appreciation for the extensive work done long ago by others.
Principle One: The the first and most important principle is really quite simple: at any given instant you can draw a line from the center of the earth to any celestial body and that line pierces the surface of the earth. We call that point where the line pierces the surface the geographic position (GP) of the celestial body, and the Nautical Almanac provides the means of fixing it with an accuracy of one second of time. (It also fixes the GP to within about one second of great circle arc or about 33 yards.) The geographic position satisfies the first component of the fix (knowing the exact position of the object you are observing) and is the observed object from which we base our LOP.
Unlike in terrestrial navigation where the object from which we take a bearing remains stationary, the geographic position of a celestial body continually moves since the earth is in constantly spinning on its axis as well as moving through space. The majority of the work done in celestial navigation involves establishing the geographic position, and it should be easy to see why. The accuracy of the geographic position is the most important determining factor affecting the accuracy of the resulting fix. Fortunately, the Nautical Almanac contains the means for determining the geographic position for over fifty easy-to-observe celestial bodies at any given time. Therefore, recording the exact time of the observation is imperative to accurately determining the position of the basis of our LOP. We’ll deal with this in detail later.
Once the geographic position is determined it should be easy to see the importance of the other two principles of obtaining a LOP by celestial observation. Remember that celestial LOPs resemble those of taking ranges to known objects. For terrestrial range-based LOPs you usually take a pair of dividers and put one end on the object and swing an arc near where you believe you dead reckoning position is. When exclusively using ranges you seldom think about the bearing to the object, but you should remember that once your position is fixed you could go back and plot bearings to each object you took a range from. The second principle deals with range and the third deals with bearing. (The calculations of the final two principles rely on spherical trigonometry and are also used in great circle sailings calculations. They are dealt with in detail in Volume II of Bowditch and we will examine them later. Remember that, at this point, you do not know our own exact position. The “known position” is one you arbitrarily choose to make our sight reduction calculations easier. You will learn more about that below.)
Principle Two: The second principle is that from any known position away from the body's geographic position we can calculate the angle that the celestial body would appear above the horizon from that known position. We call this angle the computed altitude, and it is abbreviated Hc in the Sight Reduction Tables (H.O. 229). It should be easy to accept that the altitude changes if you move toward or away from the geographic position and the altitude remains the same if you move sideways such that your distance to the GP is constant. If you continued moving sideways in this manner you would trace a circle with the GP as its center. This is known as the circle of equal altitude with a radius that equals the distance to the GP.
Principle Three: The third and final principle is that we can calculate the bearing from any known position on the earth to the celestial body’s geographic position. We call this the azimuth, and it is abbreviated Z (The HO 229 tables give the azimuth as Zn which must be corrected to Z according to the instructions found at the top and bottom of each page.) in the Sight Reduction Tables. The azimuth remains the same if you move on a line directly toward or away from the geographic position, and the azimuth changes if you deviate sideways along the circle of equal altitude. If you continued moving along the azimuth line it would scribe a great circle completely around the earth, therefore each minute of arc along the azimuth line is equal to one nautical mile.
If you find these principles difficult to comprehend, try this exercise or a variation of it. Go outside and find some object above you and some distance away like a street light, church steeple, tree top, etc. First imagine the spot on the ground directly underneath that object; that spot corresponds to the geographic position. Next imagine taking a compass bearing to the object and imagine calling it the azimuth. Now move a few steps directly toward or away from the object and see that the bearing remains constant. Move a few steps sideways and see how the bearing changes. Finally, stand in one place and imagine measuring the angle of altitude to the top of the object. Now move a several steps closer and several steps further away from the object and see how the angle changes. As you move closer the angle gets larger until you get directly underneath when the angle would be 90°; as you move further away the angle decreases. Lastly move in a circle sideways such that the angle remains the same and you will experience the angle of equal altitude.
“OK,” you say; “but what good does this do me in learning how it all works? In other words, how do I use these principles to get a LOP from a celestial body?”
Let's go back to the components of a fix. The first component, knowing the exact location of the object from which you are taking measurements, is satisfied by the GP. But the second component, having the means to measure the GP relative to your position, is complicated by the scale of the distances involved. The solution is to use assumed positions for each observation because the sight reduction tables let you know what the observation would have been had you been if you were in that location. In essence what you are saying is, “I don't know my exact location when I take my sextant observations, but I know that if I were at a the assumed position I would have observed the body at this altitude (Hc) and this azimuth (Z). Therefore, I can compare my sextant observation to what I would have observed had I been at the assumed position to see whether to draw my LOP (the segment of the circle of equal altitude that contains my position) closer to the GP or further away from it.” The process for making the comparison between observation and assumed position is referred to as sight reduction and will be covered in detail next