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Previously I explained that the celestial poles are over the earth's poles and the earth's equator is directly beneath the celestial equator. This system of celestial coordinates remains the same no matter where you are on the earth. Remember that the celestial sphere has a similar coordinate system to the earth's latitude and longitude. Recall also that the zenith is always directly over your head on the celestial sphere, while the nadir is directly below your feet on the opposite side of the celestial sphere. The celestial horizon is a plane passing through the earth's center and perpendicular to the observer's zenithnadir axis. At sea, the celestial horizon is parallel to the observer's horizon, or the visible horizon from your eye to where the sky meets the ocean. The celestial horizon is dependent upon the position of the observer and moves with him as he changes positions. These two systems of coordinates are intertwined to solve the celestial triangle.
The visible horizon is separated from the celestial horizon by the radius of the earth and the two are parallel. In comparison to the vast distances to the stars, the radius of the earth is immeasurably small. Since the celestial and the visible horizons are parallel, the angular displacement of any celestial body above the celestial horizon, as measured from both the center and the surface of the earth, will be almost equal. The small difference in this angle is called parallax and except for the sun, moon, and two nearest planets, it is not even considered.
"At sea, the celestial horizon is parallel to the observer's horizon, or the visible horizon from your eye to where the sky meets the ocean." 
Stars will appear as a point of light in the sextant, while the sun and moon will appear considerably larger. By custom, we usually observe either the lower or upper edge of the sun and moon. This is a more accurate method than trying to estimate the center of the body, especially when the moon is not full. This correction is called semidiameter and is used to correct the angle observed or the upper or lower edge to the center of the sun, moon, or planet.
As light rays from the celestial bodies leave the vacuum of space and enter the earth's atmosphere they are bent, and the body will appear higher than its actual position. This same phenomenon occurs when we look at a pencil in a glass of water. The different density of air and water bends the light rays and creates the distortion. Refraction correction is applied to all celestial observations and varies with the altitude of the body above the horizon. When a body is near the horizon the light rays must penetrate a greater density of air than when it is near our zenith and the refraction correction is larger.
If the eye of the observer were at the surface of the earth, he would be looking directly along the visible horizon. However, on a boat we are always at some point above the surface of the ocean and the sensible horizon will not be parallel to the visible horizon. The difference between them is called Dip (D) also sometimes called the height of eye correction.
Parallax, Dip, and refraction corrections are used to correct the sextant observation from the sensible horizon to the celestial horizon. The semidiameter correction converts the sextant measurement from the edge of the body to its center. These corrections are listed in the Nautical Almanac and the actual math will be covered in future articles.
A celestial LOP is actually a circle on the earth's surface plotted from the body's subpoint with a radius equal to the distance from the observer to that subpoint. To accurately compute this distance, the navigator must measure the angular position of the observed body above the celestial horizon. A sextant is used to measure this angle and the subpoint, as explained in the previous article, is determined by extracting the GHA and declination of the body from the Nautical Almanac. Remember that GHA and declination can be plotted as latitude and longitude on the earth and this geographical position is called the subpoint. The angle of the body above the celestial horizon is called the observed altitude (Ho). The altitude measured with the sextant is always from the observer's sensible horizon. The corrections mentioned earlier must be applied to arrive at an altitude measured from the celestial horizon to the center of the body.
The angle from the celestial horizon, through the body, to the observer's zenith is always 90 degrees. There is a definite relationship between the Ho of a body and the distance of the observer from the body's subpoint. The angular distance from the body to the observer's zenith is called the coaltitude and, of course, is equal to 90 degrees minus the observed altitude (Ho). This coaltitude is also called the zenith distance and when multiplied by 60 will give you the distance in nautical miles from the subpoint to the observer. When the body is directly overhead (at the observer's zenith) the Ho is 90 degrees and the subpoint is equal to the observer's location since the distance is 0 miles (CoAlt = 90 minus 90 times 60 = 0). When the Ho is 0 degrees, the body is on the observer's horizon and the distance from the subpoint to the observer is 90 degrees, or 5,400 nm (90 minus 0 times 60). Zenith distance is used for the radius of a circle which, when plotted, becomes the celestial LOP. This circle is called the circle of equal altitude, because anyone located on it and viewing the body at the same time will have an identical Ho.
The next consideration is the direction from the observer to the body's subpoint. This direction is called the true azimuth (Zn). The Zn of a celestial body corresponds exactly to the true bearing of an object located at the subpoint. Here is the formal definition: The Zn of a celestial body is the angle measured at the observer's position, from true north clockwise to the great circle arc joining the observer's position with the subpoint.
If the true azimuth of a body were measured at the same time as the Ho, a fix could be established. This would be just like taking a fix from a radar range and bearing on a known return. Unfortunately there is no instrument that can measure the Zn to the required accuracy. For example, if a body is observed with an Ho of 40 degrees and the Zn is off by onetenth of a degree, the resultant fix would be in error by five nautical miles. This accuracy is not too bad in the middle of the ocean and I know many celestial navigators who would be pleased with this accuracy. But onetenth of a degree would be a very small measurement error if the error were one degree the resultant fix would be off by 50 miles. This amount of error is not acceptable under any circumstances except perhaps for emergency lifeboat navigation. Except for a noonday fix using three LOPs on the sun, a fix is not usually determined from a single body. Normally, LOPs from two or more bodies must be crossed to get a fix.
I think you would agree that using a chart with a scale small enough to plot the subpoints of the celestial bodies, and then drawing circular LOPs with thousands of miles of radii would be unwieldy and not very accurate. For this reason, the intercept method of plotting celestial LOPs has evolved. Before I get into the technical details, here is a simple explanation of the intercept method: instead of plotting from the subpoint, we plot from the DR position. After you take a sextant measurement of a body's altitude, you calculate what the actual altitude and true azimuth of that body are for that DR position and time by using the Nautical Almanac and other special navigational tables. This altitude is called the computed altitude (Hc). Now, if you were actually at that DR position, the altitude you measured (Ho) would be exactly the same as the computed altitude (Hc). If there is any difference between the two altitudes, then you are not at that DR position.
This difference between these altitudes is called the intercept (a). The intercept, in minutes of arc, is equal to nautical miles and an LOP is plotted from that position, either away from or toward the subpoint, along the true azimuth for the intercept distance perpendicular to the ZN. If the observer's altitude (Ho) is greater than the calculated altitude (Hc), then the intercept is plotted from the DR toward the subpoint. Conversely, if the Hc is greater than the Ho, the LOP is plotted away from the subpoint. Remember, the closer you are to the body's subpoint, the greater the vertical angle will be. Another way of looking at this is by considering two observers measuring the altitude of a lighthouse. The observer closer to the lighthouse will measure a greater angle than the one farther away. Conversely, the one who measures the smaller vertical angle is farther away from the lighthouse than the one who measures a larger angle. If they were both at the same distance from the lighthouse, then they would measure the same angle. Plotting the LOP perpendicular to the ZN is the same as plotting a line tangent to the circle of equal altitude. Drawing the LOP as a straight line will not induce any errors, since any short segment of arc on a circle with a radius measured in the thousands of miles will approximate a straight line.
So there it is. Almost everything you need to know about celestial theory to navigate by the stars. One of the areas left is learning how to use a sextant and correcting the sextant altitude (Hs) to arrive at the observed altitude (Ho). The parallax, semidiameter, dip, and refraction corrections, along with any sextant index errors, are applied to the Hs to arrive at the Ho.
The other area is learning to use the Nautical Almanac and navigational tables to calculate Hc and Zn and we'll cover this topic in a future article.
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