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Celestial Navigation Fundamentals

Angles, angles, angles...

Angles are the most common type of number used by the celestial navigator. The position of the celestial bodies and points on the surface of the earth may be described by angles. The sextant is an instrument that measures angles. Angles are usually measured in degrees, minutes and seconds. The complete circumference has 360 degrees (360°). One degree is equivalent to 60 minutes. The seconds of arc are not used in the celestial navigation, since the angle measurement instrument - the sextant - is not precise enough to measure them. The smallest unit of angle used by navigators is the tenth of minute. Recently, the popularization of GPS devices added the 1/100 of minute.

The nautical mile (=1852 m) is a unit conveniently selected to simplify the conversions between angles and distances. One nautical mile corresponds to an arc of one minute on the surface of earth. Angles and distances on the surface of earth are, therefore, equivalent. One exception is the minute of longitude, equivalent to one mile only near the Earth Equator. Another important equivalence is between time and degrees of longitude. Since the earth goes one complete turn (360°) in 24 hours, each hour corresponds to 15° of longitude. Or 900 Nautical miles (NM).

The Earth and the Celestial Sphere

Fig. 1 - The Earth and the Celestial Sphere

Imagine that the Earth is the center of the universe and that around the Earth there is a larger sphere, centered in the same point, in which the stars are fixed, as if they were painted in its internal surface. This other ball we call the Celestial Sphere.

Fig. 2 - Earth coordinate system

To specify a position on the surface of Earth we use a system of coordinates that consists of two angles: latitude and longitude. Latitude is the angle measured from the Equator in direction North-South. Longitude is the angle in the Pole between the Meridian of Greenwich and that of the considered position (fig. 2).

Fig. 3- Celestial Coordinate System

A similar system is used for the Celestial Sphere. The angle analogous to the latitude in the celestial sphere we call declination. The declination is measured in the plane North-South, from the Celestial Equator. The analog to the longitude is named Right Ascension or RA.

Like the longitude, the Right Ascension is measured from an arbitrary Meridian: the Vernal Equinox Point (a.k.a. first point of Aries).

Apparent movement of the stars

The stars have nearly fixed positions in the Celestial Sphere. The Sun, Moon and planets move around during the year, but their movement is slow when compared to the apparent movement due to the rotation of the Earth. So let's consider for now that the celestial objects ( stars, planets, Sun and Moon) are fixed in the Celestial Sphere.

Using the Earth-at-the-center-of-the-universe model, imagine that the Earth is stopped and the celestial sphere is turning around it, completing a turn every 24 hours. You should not be confused by this idea: it's exactly what you observe if you seat and watch the night sky long enough.

The Earth's and Celestial Sphere's axes of rotation are in the same line. Both equators are, therefore, in the same plane (see fig. 1).

The stars, fixed to the celestial sphere, turn around the earth. The celestial sphere poles, being in the axis of rotation, remain fixed in the sky. So, a star located near a celestial pole will appear to be stationary in the sky. That's the case of Polaris, a star that is near the North Celestial Pole (its declination is 89°05' N). It's always in the north direction, a wonderful fact known by every navigator. Unfortunately there's no corresponding bright star near the South Celestial Pole.

Finding the Earth position by observing the stars

Now consider a line connecting the center of a star and the center of the Earth. The point where this line crosses the surface of the Earth we call Geographical Position of this star (or GP). An observer positioned in the GP of a star will see it directly in the vertical, above the head.

Since stars move with the celestial sphere, their GPs also move on the surface of the Earth. And they are fast. The Sun's GP, for example, travels a mile every four seconds. The GPs of other stars, closer to the celestial poles, move more slowly. The GP of Polaris moves very slowly, since it's very close to the North Pole.

Fig. 4- Geographical Position of a Star

Because both Earth and Celestial equators are in the same plane, the latitude of the GP is equal to the declination of the star. The longitude of the GP is known as Greenwich Hour Angle - or GHA - in a reference to the correspondence between hours and longitude.

We can determine, using a Nautical Almanac, the GP of a star (it's GHA and declination) in any moment of time. But we must know the exact time of the observation. As we have seen, 4 seconds may correspond to one mile in the GP of a star. This shows the importance of having a watch with the correct time for the celestial navigation. The Beagle - ship of Charles Darwin's travel in 1830 - carried 22 chronometers on board when she went around the globe in a geographic survey.

Another important point is the Zenith. The Zenith is the point in the celestial sphere located in the vertical, over the head of the navigator. The line that connects the Zenith and the center of the Earth crosses the surface in the position of the navigator, the one we want to find. So, we have the following correspondence between points:

Surface of Earth:

Geographical Position of a star
Position of the navigator

Celestial Sphere:

Center of the star
Zenith

In the figure 5, the GP of the star is represented by X and the Zenith by Z. The distance XZ, from the GP of the star to the point Z of the navigator is called Zenith Distance. This distance, as we have seen, can be expressed in miles or degrees, since it's an arc on the surface of the Earth.

The angle that XZ makes with the True North (i.e. the "bearing" of the star) is called Azimuth ( Az ) (fig. 6).

Fig. 5 - GP of a star and the Zenith

Fig. 6 - Azimuth of a star

Fig.7 - Altitude and Zenith distance of a star

The stars are at a great distance from the earth and so the light rays coming from them that reach the Earth are parallel.

Therefore, as illustrated in the figure 7, we may say that the distance XZ (as an angle) is equal to the angle that the navigator observes between the star and the vertical. This is important.

The distance XZ, measured as an angle, is equal to the angle that the navigator observes between the star and the vertical.

However, it's difficult to determine the Zenith distance with precision, since it's difficult to find the vertical direction in a rocking boat. It's a lot easier to measure the angle between the star and the horizon. This important angle for the celestial navigator is called altitude (H) of the star. The altitude of a star is taken with the sextant held in the vertical plane, measuring the angle between the horizon and the star. In the fig. 7, we can see that the zenith distance equals 90° less the altitude of the star.

We have seen how to determine the zenith distance of a star using the sextant. The Zenith Distance and the GP of a star, however, are not enough to determine our position. With this data we can only say that our position is in a big circle, with the center in the GP of the star and radius equal to the Zenith Distance. This is known as the Circle of Position.

Fig.8 - Circle of Position

Figure 8 shows a Circle of Position. Point X is the GP of the star.

Any observer located on this circle will see the star at the same altitude, but with different Azimuths. In the example of the figure, suppose the navigator observes the star with an altitude of 65°. As we have seen, the Zenith Distance is 90°-H, or 25°. To determine this distance in miles, we multiply by 60, since one degree is equal to 60 nautical miles (NM). So, the Zenith Distance in the example - the radius of our circle - is 1500 NM.

If we just could determinate the exact direction where the GP of the star is - it's Azimuth - that would establish where in the circle we are. How about using a compass? Unfortunately, the compass is not precise enough for celestial navigation. One error of just 3°, common when reading a compass, corresponds to 78 miles of error in our example! Not an acceptable error.

The way to find our position is to draw two or more circles - for two or more celestial bodies - and see where they intercept each other. But drawing these big circles would require really big charts! We work around this problem by making a guess at our position. No matter how lost we are, we can always make a guess. Using this assumed position we can calculate expected altitude for the star at a given time (using the Nautical Almanac).

This Calculated Altitude can then be compared with the Observed Altitude (the actual altitude, measured with the sextant). The difference is the error of our assumed position (also known as Delta) in the direction of the star. The Delta can be towards the star or away from it.

Fig. 9 - Line of Position

Because a Geographical Position of a star is normally thousands of miles from our position, the circle of position is very large and the small piece that interests us - the one near our position - may be considered a straight line, orthogonal to the Azimuth of the star. This line is called the Line of Position or LOP (fig. 9).

We managed, from the measured altitude of a star at a certain time and our assumed position, to draw a line of position. We know that our actual position is somewhere along this line. To determine this point we can draw another line, for another star. The point were they intercept each other is our position - or our Astronomical Position.

Fig. 10 - Triangle formed by the intersection of three lines of Altitude

Normally, the navigator should repeat this procedure for yet another star, just to be sure. Since measurements are affected by minor imprecisions, the three lines will probably not intercept in a single point, resulting in a small triangle. Our position is probably in some point of this triangle (fig. 10). The smaller the triangle, the better. We usually assume that our Astronomical Position is in the center of the triangle.

In figure 10 above, we can see how three circles of position determine 3 Lines of Position r1, r2 and r3.

In traditional celestial navigation the determination of a Line of Position involves the computation of the GP of the star (GHA and declination) using the Nautical Almanac and the solution of the Position Triangle PXZ, formed by the terrestrial pole (P), the GP of the star (X) and the assumed position of the navigator (Z) (see fig.11).

Fig. 11 Triangle of Position PXZ

This solution, using tables, yields the Calculated Altitude and the Azimuth of the star. The difference, in minutes of degree, between the calculated altitude and the altitude of the star measured with the sextant is the distance between the line of position and our assumed position - the error Delta of our estimate. This can be away or towards the star.

Using Navigator software, the GP of a star and the triangle of Position are solved by the computer using formulas. All you will have to do is enter the sextant reading (date, time and altitude), name of the star and the assumed position (latitude and Longitude).

Determination of the Astronomical Position

It's not necessary to draw the lines of position when using Navigator software. But let's see how this is done using pencil and paper:

Plot your assumed position.
Using a parallel ruler, draw a line passing on the assumed position, in the direction of the Azimuth of the star.
Over this line, measure the error Delta of the estimate - in the direction of the star or contrary to it - according to the sign of the Delta.
Draw the line of position, orthogonal to the Azimuth, at this point.

Detailed Nautical Charts are usually only available for places near the shore. When in high seas, we normally don't have charts with the adequate scale to plot our position. Special plotting paper is used instead.

When navigating using Navigator software, the computer determines the altitude lines interceptions and calculates the astronomical position. A simplified map is drawn, showing the parallels, meridians, lines of altitude and the astronomical position.

The sextant

The sextant is an instrument that measures angles. Fig 12 shows a schematic sextant. The eyepiece is aligned to the small mirror, which is fixed in the frame of the instrument. This mirror is half transparent. By the transparent part, the navigator can see the horizon directly. The small mirror also partially reflects the image from the big mirror, where you see the star. The big mirror is mobile and turns with the arm of the sextant. Doing that, we change the angle between the two mirrors. The altitude of the star is measured in the scale. There is a drum to make the fine adjustments. Whole degrees are read in the scale and the minutes in the drum.

Fig. 12 - The Sextant

The sextant has two sets of filters (or shades) to eliminate the excess of light, especially when observing the Sun. The use of two or more filters in front of the big mirror is necessary when observing the Sun. Serious eye injuries will result from observing the Sun without filters, even for a brief period.

When looking through the eyepiece and adjusting the sextant, you will see something like figure 13, to the left. Sextant readings must be made with the sextant in the vertical position.

Fig. 13 - Image of the Sun in the sextant

Inclining (rocking about the axis of the eyepiece) the adjusted instrument slightly, the image of the celestial body describes a small arc that touches the horizon in a point near the center of the mirror. In this situation, the angle is ready to be read in the instrument scale.

Altitude corrections

But before we can use this apparent reading in our calculations, some corrections must be made, in order to obtain the true observed altitude. These corrections are: 1) the height of the eye, 2) semi diameter of the body (only for Sun and Moon), 3) instrumental error, 4) atmospheric refraction and 5) parallax (only for the Moon).

Since most of these corrections depend only on the selected celestial object and altitude, they are performed automatically by Navigator software. The only information you will have to provide to the program are the height of the eye (a.k.a. Dip) and the instrumental error. The application of these corrections to the instrumental altitude gives the corrected altitude, the one used in calculations.

An observer located in a high place will see a star with an altitude bigger than other at sea level, in the same location. This error is called height of the eye (Dip).

Fig. 14 - Error due to the height of the eye (Dip)

Fig. 15 - eyepiece image of the sextant index error

Fig. 16 - Sign of the instrumental error

Fig. 17 - Semi diameter correction

The sextant index error (IE) is due to a small misalignment of the scale of the sextant (the "zero" of the instrument). To read the index error, adjust the scale to 0°00.0' and point towards the horizon. In fig. 15 left we can see this error. Turn the drum until the horizon forms a single line (fig. 15 right). Then you can read the index error.

The index error can be positive or negative, as shown in fig. 16. The index correction has opposite signal (i.e. must be subtracted from altitude if positive and vice-versa).

Since the navigator is not in the Earth's center, but in its surface, the apparent object position is below the true geocentric position.

Parallax is only meaningful for the Moon. Other objects are so far, their parallax is very small.

Nautical Almanac data is tabulated for the centers of the celestial objects. For the Sun and Moon, however, it's easier to measure the altitude of the lower part of the body, as illustrated in fig. 17. This is known as the lower limb. Of course a correction must be applied in order to obtain the altitude of the center of the body. This correction is called semi diameter. Sometimes, the upper limb is also used.

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