This category introduces celestial concepts and how they relate to celestial navigation. The motion of celestial bodies and celestial coordinates are discussed, as well as the use of the Air Almanac. The celestial horizon is explained and detail is given on how to determine observed altitude, true azimuth, and how to find a celestial fix.
Introduction to Celestial Concepts
Celestial navigation is a universal aid to dead reckoning (DR). Because it is available worldwide and is independent of electronic equipment, it is a very reliable method of fixing the position of the aircraft. It cannot be jammed and emanates no signals. Each celestial observation yields one line of position (LOP). In the daytime, when the sun may be the only visible celestial body, a single LOP may be all you can get. At night, when numerous bodies are available, LOPs obtained observing two or more bodies may be crossed to determine a fix.
It is impossible to predict, in so many miles, the accuracy of a celestial fix. Celestial accuracy depends on the navigator’s skill, the type and condition of the equipment, and the weather. With the increase in aircraft speed and range, celestial navigation is very demanding. Fixes must be plotted and used as quickly as possible.
You do not have to be an astronomer or mathematician to establish a celestial LOP. Your ability to use a sextant is a matter of practice, and specially designed celestial tables have reduced the computations to simple arithmetic.
Although you do not need to understand astronomy in detail to establish an accurate celestial position, celestial work and celestial LOPs mean more if you understand the basics of celestial astronomy. Celestial astronomy includes the navigational bodies in the universe and their relative motions. Although there are an infinite number of heavenly bodies, celestial navigation utilizes only 63 of them: 57 stars, the moon, the sun, Venus, Jupiter, Mars, and Saturn.
Assumptions
We make certain assumptions to simplify celestial navigation. These assumptions help you obtain accurate LOPs without a detailed knowledge of celestial astronomy. However, celestial positioning is more than extracting numbers from various books. A working knowledge of celestial concepts helps you crosscheck your computations.
First, assume the earth is a perfect sphere. That puts every point on the earth’s surface equidistant from the center, forming the terrestrial sphere. Next, assume the terrestrial sphere is the center of an infinite universe. Finally, assume all other bodies, except the moon, are an infinite distance from the terrestrial sphere. Imagine them on the inside surface of an enormous concentric sphere, the celestial sphere. If the stars, planets, and sun are infinitely distant from the earth’s center, then the earth’s surface (or aircraft’s altitude) is approximately the center of the universe.
Ptolemy proposed the celestial concept of the universe in AD 127. He said the earth is the center of the universe, and all bodies rotate about the earth from east to west. In the relatively short periods of time involved with celestial positioning, you can assume that all bodies on the celestial sphere rotate at the same rate. In actuality, over months or years, the planets move among the stars at varying rates.
Establishing an artificial celestial sphere with an infinite radius simplifies computations for three celestial spheres has a corresponding point on the terrestrial sphere and; conversely, every point on the terrestrial sphere has a corresponding point on the celestial sphere.
Second, the celestial sphere’s infinite radius dwarfs variations in the observer’s location. An infinite radius means all light rays from the celestial body arrive parallel, so the angle is the same whether viewed at the earth’s center, on the surface, or at the aircraft’s altitude.
Third, the relationships are valid for all bodies on the celestial sphere. Because the moon is relatively close to the earth, it must be treated differently. With certain corrections, the moon still provides an accurate LOP. This is addressed in a later category.
Figure 8-1. Celestial points and subpoints on earth have the same relationship.
Because the celestial sphere and terrestrial sphere are concentric, each sphere contains an equator, two poles, meridians, and parallels of latitude or declination. The observer on earth has a corresponding point directly overhead on the celestial sphere called the zenith. A celestial body has a corresponding point on the terrestrial sphere directly below it called the subpoint or geographic position. At the subpoint, the light rays from the body are perpendicular to the earth’s surface. [Figures 8-1 and 8-2]
Figure 8-2. Elements of the celestial sphere.
Consistent with the celestial assumptions, the earth and the celestial meridians do not rotate. All bodies on the celestial sphere rotate 15° per hour past the celestial meridians. The moon moves at approximately 14.5° per hour.
Motion of Celestial Bodies
All the celestial bodies have two types of motion: absolute and apparent. Apparent motion is important to navigators. Apparent motion is the motion of one celestial body as perceived by an observer on another moving celestial body. Since apparent motion is relative, it is essential to establish the reference point for that motion. For example, the apparent motion of Venus would be different if observed from the earth or the sun.
Apparent Motion
The earth’s rotation and revolution causes the apparent motion of the celestial bodies. Rotation causes celestial bodies to appear to rise in the east, climb to a maximum height, then set in the west. All bodies appear to move along a diurnal circle, approximately parallel to the plane of the equator.
The apparent effect of rotation varies with the observer’s latitude. At the equator, the bodies appear to rise and set perpendicular to the horizon. Each body is above the horizon for approximately 12 hours each day. At the North and South Poles, a different phenomenon occurs. The same group of stars is continually above the horizon; they neither rise nor set, but move on a plane parallel to the equator. This characteristic explains the periods of extended daylight, twilight, and darkness at higher latitudes. The remainder of the earth is a combination of these two extremes; some bodies rise and set, while others continually remain above the horizon.
The greater the northerly declination (Dec) of a body, the higher it appears in the sky to an observer at the North Pole. Polaris, with a Dec of almost 90°, appears overhead. Bodies with southern Dec are not visible from the North Pole.
A circumpolar body appears to revolve about the pole and never set. If the angular distance of the body from the elevated pole is less than the observer’s latitude, the body is circumpolar. For example, the Dec of Dubhe is 62° N. Therefore, it is located at an angle of 90°– 62° from the North Pole, or 28°. So, an observer located above 28° N views Dubhe as circumpolar. Although Figure 8-3 uses the North Pole, the same characteristics can be observed from the South Pole.
Figure 8-3. Some bodies are circumpolar.
If the earth stopped rotating, the effect of the earth’s revolution on the apparent motion of celestial bodies would be obvious. The sun would appear to circle around the earth once each year. It would cover 360° in 365 days or move eastward at slightly less than 1 degree per day. The stars would move at the same rate. That is why different constellations are visible at different times of the year. Every evening, the same star appears to rise 4 minutes earlier.
After half a year, when the earth reached the opposite extreme of its orbit, its dark side would be turned in the opposite direction in space, facing a new field of stars. Hence, an observer at the equator would see an entirely different sky at midnight in June, than the one that appeared at midnight in December. In fact, the stars seen at midnight in June are those that were above the horizon at midday in December.
Figure 8-4. Seasonal changes of earth’s position. [click image to enlarge]
Seasons
The annual variation of the sun’s declination and the consequent change of the seasons are caused by the revolution of the earth. [Figure 8-4] If the celestial equator coincided with the ecliptic, the sun would always be overhead at the equator, and its Dec would always be zero. However, the earth’s axis is inclined about 66.5° to the plane of the earth’s orbit, and the plane of the equator is inclined about 23.5°. Throughout the year, the axis points in the same direction. That is, the axis of the earth in one part of the orbit is parallel to the axis of the earth in any other part of the orbit. [Figure 8-5]
Figure 8-5. Ecliptic with solstices and equinoxes.
In June, the North Pole is inclined toward the sun so that the sun is at a maximum distance from the plane of the equator. About June 22, at the solstice, the sun has its greatest northern Dec.
The solstice brings the long days of summer, while in the Southern Hemisphere, the days are shortest. This is the beginning of summer for the Northern Hemisphere and of winter for the Southern Hemisphere. Six months later, the axis is still pointing in the same direction; but, since the earth is at the opposite side of its orbit and the sun, the North Pole is inclined away from the Sun. At the winter solstice, about December 21, the sun has its greatest southern Dec. Days are shortest in the Northern Hemisphere, and winter is beginning. Halfway between the two solstices, the axis of the earth is inclined neither toward nor away from the sun, and the sun is on the plane of the equator. These positions correspond to the beginning spring and fall.
Celestial Coordinates
Celestial bodies and the observer’s zenith may be positioned on the celestial sphere using a coordinate system similar to that of the earth. Terrestrial lines of latitude correspond to celestial parallels of Dec. Lines of longitude establish the celestial meridians.
The observer’s celestial meridian is a great circle containing the zenith, the nadir, and the celestial poles. [Figure 8-2] A line extended from the observer’s zenith, through the center of the earth, intersects the celestial sphere at the observer’s nadir, the point on the celestial sphere directly beneath the observer’s position. The poles divide the celestial meridians into upper and lower branches. The upper branch contains the observer’s zenith. The lower branch contains the nadir.
Figure 8-2. Elements of the celestial sphere.
A second great circle on the celestial sphere is the hour circle. The hour circle contains the celestial body and the celestial poles. Unlike celestial meridians, which remain stationary, hour circles rotate 15° per hour. Hour circles also contain upper and lower branches. The upper branch contains the body. Again, the moon’s hour circle moves at a different rate. The subpoint is the point on the earth’s surface directly beneath the celestial body.
You can locate any body on the celestial sphere relative to the celestial equator and the Greenwich meridian using Dec and Greenwich hour angle.
Declination (Dec)
Dec is the angular distance a celestial body is north or south of the celestial equator measured along the hour circle. It ranges from 0° to 90° and corresponds to latitude.
Greenwich Hour Angle (GHA)
GHA is the angular distance measured westward from the Greenwich celestial meridian to the upper branch of the hour circle. It has a range of 0° to 360°. The Air Almanac lists the GHA and the Dec of the sun, moon, four planets, and Aries. The subpoint’s latitude matches its Dec, and its longitude correlates to its GHA, but not exactly. GHA is always measured westward from the Greenwich celestial meridian, and longitude is measured in the shortest direction from the Greenwich meridian to the observer’s meridian.
The following are examples of converting a body’s celestial coordinates to its subpoint’s terrestrial coordinates. If the GHA is less than 180°, then the subpoint is in the Western Hemisphere and GHA equals longitude. When the GHA is greater than 180°, the subpoint is in the Eastern Hemisphere and longitude equals 360° GHA. Again, Dec and latitude are equal. [Figure 8-6]
Figure 8-6. Declination of a body corresponds to a parallel of latitude.
You will use two other hour angles in celestial navigation in addition to GHA, local hour angle (LHA), and sidereal hour angle (SHA). [Figure 8-7] LHA is the angular distance from the observer’s celestial meridian clockwise to the hour circle. LHA is computed by applying the local longitude to the GHA of the body. In the Western Hemisphere, LHA equals GHA – W Long, and in the Eastern Hemisphere, LHA equals GHA + E Long. [Figure 8-8] When the LHA is 0, the body’s hour circle and the upper branch of the observer’s celestial meridian are collocated, and the body is in transit. If the LHA is 180, the hour circle is coincident with the lower branch of the observer’s celestial meridian. SHA is used with the first point of Aries.
The first point of Aries is the point where the sun appears to cross the celestial equator from south to north on the vernal equinox or first day of spring. Though not absolutely stationary relative to the stars, Aries moves so slowly that we consider it fixed on the celestial equator for as long as a year. The SHA is the angular measurement from the hour circle of Aries to the star’s hour circle. [Figure 8-9] Aries and the stars move together so the SHA remains constant for a year.
Figure 8-7. Greenwich hour angle.Figure 8-8. Local hour angle.Figure 8-9. Sidereal hour angle.
Use of the Air Almanac
Although the Air Almanac contains astronomical amounts of data, most of it is devoted to tabulating the GHA of Aries and the GHA and Dec of the sun, moon, and the three navigational planets most favorably located for observation. Enter the daily pages with Greenwich date and GMT to extract the GHA and Dec of a celestial body.
Finding GHA and Dec
The GHA is listed for 10-minute intervals on each daily sheet. If the observation time is listed, read the GHA and Dec directly under the proper column opposite the time.
For example, find the sun’s GHA and Dec at GMT 0540 on 11 August 1995. [Figure 8-10] The GHA is 263°–41′ and Dec is N 15°–24′. (Extractions of GHA and Dec are to the nearest whole minute.) To convert these values to the subpoint’s geographical coordinates, latitude is North 15°–24′. When GHA is greater than 180°, subtract it from 360° to get east longitude. The subpoint’s longitude in this example is (360°–00′ minus 263°–41′) East 96°–19′.
Figure 8-10. Daily page from Air Almanac—11 August 1995. [click image to enlarge]When you do not observe at a 10-minute interval, use the time immediately before the observation time. Then, use the Interpolation of GHA table on the inside front cover of the Air Almanac or the back of the star chart and add the increment to the GHA. [Figure 8-11]
Figure 8-11. Interpolation of Greenwich hour angle, Air Almanac.
For example, on 11 August 1995, you observe the sun at 1012 GMT. Enter Figure 8-10 to find the GHA listed for 1010 (331°–1 1′). Since the observation was 2 minutes after the listed time, enter the Interpolation of GHA table and find the correction listed for 2 minutes of time (30′). [Figure 8-11] Add this correction to the listed GHA to determine the sun’s exact GHA at 1012 (331°–41′). The Dec for the same time is N 15°–21′. Thus, at the time of the observation, the subpoint of the sun is at latitude 15°–21′ N, longitude (360°–00′ minus 331°–41′) 028°–19′ E.
For example, at 0124 GMT on 11 August 1995, you observe Altair. To find the GHA and Dec, look at the extracts from the tables in Figures 8-11 and 8-12.
Figure 8-12. Sidereal hour angle obtained from table. [click image to enlarge]You can find the GHA and Dec of a planet in almost the same way as the sun. Because the planet’s Dec change slowly, they are recorded only at hourly intervals. Use the Dec listed for the entire hour. For example, to find the GHA and Dec of Jupiter at 1109 GMT, 11 August 1995, enter the correct daily page for the time of 1100 GMT. [Figure 8-10] The GHA is 240°–35′ and the Dec is S20°–40′. Enter the Interpolation of GHA table under sun, etc., to get the adjustment for 9 minutes of time, 2°–15′. [Figure 8-11] Therefore, GHA is 242°–50′. Jupiter’s subpoint is at latitude 20°–40′ S, longitude (360°–00′ minus 242° –50′) 1 17°–10′ E.
If you need to find an accurate GHA and Dec without the Air Almanac, you can find the procedures and applicable tables in Publication No. 249, Volume 1 for Aries or Volume 2 or 3 for the sun.
Finding GHA and Dec of Moon
The moon moves across the sky at a different rate than other celestial bodies. In the Interpolation of GHA table, the intervals for the moon are listed in the right column where the values for the sun, Aries, and the planets are in the left column.
The interpolation of GHA table is a critical table and the increment is opposite the interval in which the difference of GMT occurs. If the difference (for example, 06′-3 1″ for the moon) is an exact tabular value, take the upper, or right, of the two possible increments (that is, 1°–34′). The up, or right, rule applies to all critical tables.
For example, at 1136 GMT on 11 August 1995, you observe the moon. The following information is from the Air Almanac [Figures 8-10 and 8-11]:
GHA of moon at 1130 GMT 163° 16′
GHA correction for 6 minutes 1° 27′
GHA 164° 43′ Dec S8° 00′
Thus, at 1136Z, the moon’s subpoint is located at S 8°–00′, longitude 164°–43’W.
Finding GHA and Dec of a Star
The stars and the first point of Aries remain fixed in their relative positions in space, so the gas of the stars and Aries change at the same rate. Rather than list the GHA and Dec of every star throughout the day, the Air Almanac lists the GHA of Aries at 10-minute intervals and gives the sidereal hour angle (SHA) of the stars. The GHA of a star for any time can be found by adding the GHA of Aries and the SHA of the star. The GHA of a star is used to precomp any star that falls within 29° (Dec) of the equator using Volume 2 or Volume 3.
The table, STARS, is inside the front cover of the almanac and on the back of the star chart. This table lists navigational stars and the following information for each star: the number corresponding to the sky diagram in the back, the name, the magnitude or relative brightness, the SHA, the Dec, whether used in Publication No. 249, and stars that can be used with Dec tables. NOTE: If you need a higher degree of accuracy, the SHA and Dec of the stars are listed to tenths of a degree in the Air Almanac’s appendix. Thus, the subpoint of Altair is 08°–52′ N 042°–24′ W.
All the celestial concepts and assumptions you have learned may help you obtain a cestial LOP. A celestial LOP is simply a circle plotted with the center at the subpoint and a radius equal to the distance from the observer to the subpoint. To accurately compute this distance and the direction to the subpoint of the body, you must initially position the subpoint and then measure the angular displacement of the body above the horizon. GHA and Dec position the body, and the sextant measures the height above the horizon. A basic knowledge of celestial theory and LOPs will help you appreciate celestial navigation and detect errors. The next section explains how angular displacement is measured.
Celestial Horizon
You use a sextant to measure a body’s angular displacement above the horizon. The celestial horizon is a plane passing through the earth’s center perpendicular to the zenith-nadir axis. The visual horizon approximates this plane at the earth’s surface.
Figure 8-13 depicts the zenith-nadir axis and the celestial horizon. The angular displacement you see through a sextant is the height observed (Ho). Ho is measured along the vertical circle above the horizon. The vertical circle is a great circle containing the zenith, nadir, and celestial body. The body’s altitude is the same whether measured at the earth’s surface from an artificial horizon or at the center of the earth from the celestial horizon, because these horizons are parallel and the light rays from the body are essentially parallel. Figure 8-14 shows that the infinite celestial sphere makes the difference in angle for light rays arriving at different points on the earth infinitesimal.
Figure 8-13. Celestial horizon is 90° from observer Zenith and Nadir.Figure 8-14. Parallel lines make equal angles with parallel planes.
The angle between light rays is called parallax. In Figure 8-15, parallax is shown at its maximum; that is, when the observer and the subpoint are separated by 90°. Since the earth’s radius is tiny compared to the infinite distance to the stars, the angle p is very small. For the sun, angle p is a negligible 9 seconds of arc or 0.15 nautical miles (NM). Observed altitudes from either the artificial or celestial horizon are practically the same.
Figure 8-15. Parallax.
The bubble in a sextant or artificial horizon is most used by navigators. As in a carpenter’s level, a bubble indicates the apparent vertical and horizontal. With the bubble, the navigator can level the sextant and establish an artificial horizon parallel to the plane of the celestial horizon. Figure 8-16 shows that the plane of the artificial (bubble) horizon and the plane of the celestial horizon are parallel and separated by the earth’s radius. Compared to the vast distances of space, the radius of the earth is inconsequential. Thus, the artificial horizon and the celestial horizon are nearly identical.
Figure 8-16. The two planes are parallel.
Observed Altitude
The distance of the observer from the subpoint and a body’s Ho are related. [Figure 8-17] When the body is directly overhead, the Ho is 90°, and the subpoint and the observer’s position are collocated. When the Ho is 0°, the body is on the horizon and the subpoint is 90° (5,400 NM) from the observer’s position. (See Figure 8-18, where C is the center of the earth, AB is the observer’s horizon, and S is the subpoint of the body.) Since the sum of the angles in a triangle equals 180°, the angle OX is equal to 180° – (Ho + P). The sum of the angles on a straight line equals 180°, so angle OXC is equal to Ho + P. The horizon AB being tangent to the earth at O is perpendicular to OC, a radius of the earth. Thus, angle OCX equals 90° (Ho + P). The preceding discussion showed that angle P is negligible, so this angle becomes 90° – Ho. The arc on the surface subtended by the angle OCX at the center of the earth is arc OS. This arc then is equal to 90° – Ho.
Figure 8-17. Measure altitude from celestial horizon along vertical circle.Figure 8-18. Finding observed altitude.
The distance from the subpoint to the observer is the zenith distance or co-alt and is computed using the astronomical triangle described in Chapter 9. Basically, the zenith distance equals 90° minus the Ho. [Figure 8-19] The figures are then converted to NM by multiplying the number of degrees by 60 and adding in the odd minutes of arc. Zenith distance is the radius of the circle which becomes the celestial LOP. This circle is called the circle of equal altitude [Figure 8-20], as anyone located on it views an identical Ho. Now that you can determine the distance to the subpoint, you must next find the direction.
Figure 8-19. Co-altitude and Zenith distance.Figure 8-20. Constructing a circle of equal altitude.
True Azimuth (Zn)
The direction to a body from an observer is called Zn. A celestial body’s Zn is the true bearing (TB) to its subpoint. The Zn is the angle measured at the observer’s position from true north (TN) clockwise through 360° to the great circle arc joining the observer’s position with subpoint. [Figure 8-21] If you could measure the Zn when you measure its altitude, you could have a fix. Unfortunately, there is no instrument in the aircraft that measures Zn accurately enough. Except in the case of a very high body (85–90°), if you observe a body with a Ho of 40° and you mismeasure the Zn by 1°, the fix will be 50 NM off.
Figure 8-21. Relationship of true azimuth to an observer.
Celestial Fix
Since you cannot normally fix off a single body, you usually need to cross two or more LOPs. The fix position is the intersection of the LOPs. When two celestial LOPs are plotted, they intersect at two points, only one of which can be your position. In practice, these two intersections usually are so far apart that DR removes all doubt as to which is correct.
