A Short Guide to Celestial Navigation
Copyright © 1997-2015 Henning Umland
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Revised June 5th, 2015
First Published May 20th, 1997 .;56B
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-02 ,>66 +<4A:6;@3@9<; /946;?6 Much is due to those who first broke the way to knowledge, and left only to their successors the task of smoothing it.
Samuel Johnson
Preface
Why should anybody still practice celestial navigation in the era of electronics and GPS? One might as well ask why some photographers still develop black-and-white photos in their darkroom instead of using a digital camera. The answer would be the same: because it is a noble art, and because it is rewarding. No doubt, a GPS navigator is a powerful tool, but using it becomes routine very soon. In contrast, celestial navigation is an intellectual challenge. Finding your geographic position by means of astronomical observations requires knowledge, skillfulness, and critical judgement. In other words, you have to use your brains. Everyone who ever reduced a sight knows the thrill I am talking about. The way is the goal.
It took centuries and generations of navigators, astronomers, geographers, mathematicians, and instrument makers to develop the art and science of celestial navigation to its present level, and the knowledge thus accumulated is a treasure that should be preserved. Moreover, celestial navigation gives us an insight into scientific thinking and creativeness in the pre-electronic age. Last but not least, celestial navigation may be a highly appreciated alternative if a GPS receiver happens to fail.
When I read my first book on navigation many years ago, the chapter on celestial navigation with its fascinating diagrams and formulas immediately caught my particular interest although I was a little intimidated by its complexity at first. As I became more advanced, I realized that celestial navigation is not nearly as difficult as it seems to be at first glance. Studying the literature, I found that many books, although packed with information, are more confusing than enlightening, probably because most of them have been written by experts and for experts. On the other hand, many publications written for beginners are designed like cookbooks, i. e., they contain step-by-step instructions but avoid much of the theory. In my opinion, one can not really comprehend celestial navigation and enjoy the beauty of it without knowing the mathematical background.
Since nothing really complied with my needs, I decided to write a compact manual for my personal use which had to include the most important definitions, formulas, diagrams, and procedures. As time went by, the project gained its own momentum, the book grew in size, and I started wondering if it might not be of interest to others as well. I contacted a few scientific publishing houses, but they informed me politely that they considered my work as dispensable (“Who is going to read this!”). I had forgotten that scientific publishing houses are run by marketing people, not by scientists. Around the same time, I became interested in the internet, and I quickly found that it is the ideal medium to share one's knowledge with others. Consequently, I set up my own web site to present my book to the public.
The style of my work may differ from other books on this subject. This is probably due to my different perspective. When I started the project, I was a newcomer to the world of navigation, but I had a background in natural sciences and in scientific writing. From the very beginning, it has been my goal to provide accurate information in a structured and comprehensible form. The reader may judge whether this attempt has been successful.
More people than I expected are interested in celestial navigation, and I would like to thank my readers for their encouraging comments and suggestions. However, due to the increasing volume of correspondence, I am no longer able to answer individual questions or to provide individual support. Unfortunately, I have still a few other things to do, e. g., working for a living. Nonetheless, I keep working on this publication at leisure, and I am still grateful for suggestions and error reports.
This publication is released under the terms and conditions of the GNU Free Documentation License. A copy of the latter is included.
June 5th, 2015
Henning Umland
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Traditionally, most formulas are based upon altitudes since these are easily measured using the visible sea horizon as a natural reference line. Direct measurement of the zenith distance requires an instrument with an artificial horizon, e. g., a pendulum or spirit level indicating the local direction of gravity (perpendicular to the plane of the sensible horizon) since a reference point in the sky does not exist. Instruments A marine sextant consists of a system of two mirrors and a telescope mounted on a sector-shaped metal frame (usually brass or aluminium alloy). A schematic illustration of the optical components is given in Fig. 2-1. The horizon glass is a half-silvered mirror whose plane is perpendicular to the plane of the frame. The index mirror, the plane of which is also perpendicular to the frame, is mounted on the so-called index arm rotatable on a pivot perpendicular to the frame. The optical axis of the telescope is parallel to the frame and passes obliquely through the horizon glass. During an observation, the instrument frame is held in an upright position, and the visible sea horizon is sighted through the telescope and horizon glass. A light ray originating from the observed body is first reflected by the index mirror and then by the back surface of the horizon glass before entering the telescope. By slowly rotating the index mirror on the pivot the superimposed image of the body is aligned with the image of the horizon line. The corresponding altitude, which is twice the angle formed by the planes of horizon glass and index mirror, can be read from the graduated limb, the lower, arc-shaped part of the sextant frame (Fig. 2-2). Detailed information on design, usage, and maintenance of sextants is given in [3] (see appendix). Fig. 2-2 On land, where the horizon is too irregular to be used as a reference line, altitudes have to be measured by means of instruments with an artificial horizon. 2-1 A bubble attachment is a special sextant telescope containing an internal artificial horizon in the form of a small spirit level the bubble of which (replacing the visible horizon) is superimposed on the image of the celestial body. Bubble attachments are expensive (almost the price of a sextant) and not very accurate because they require the sextant to be held absolutely still during an observation, which is rather difficult to manage. A sextant equipped with a bubble attachment is referred to as a bubble sextant. Special bubble sextants were used for air navigation before electronic navigation systems became standard equipment. On land, a pan filled with water or, preferably, a more viscous liquid, e. g., glycerol, can be utilized as an external artificial horizon. As a result of gravity, the surface of the liquid forms a perfectly horizontal mirror unless distorted by movements or wind. The vertical angular distance between a body and its mirror image, measured with a marine sextant, is twice the altitude of the body. This very accurate method is the perfect choice for exercising celestial navigation in a backyard. Fig. 2-3 shows a professional form of an external artificial horizon. It consists of a horizontal mirror (polished black glass) attached to a metal frame which is supported by three leg screws. Prior to an observation, the screws have to be adjusted with the aid of one or two detachable high-precision spirit levels until the mirror is exactly horizontal in every direction. Fig. 2-3 Fig. 2-4 A theodolite (Fig. 2-4) is basically a telescopic sight which can be rotated about a vertical and a horizontal axis. The angle of elevation (altitude) is read from the graduated vertical circle, the horizontal direction is read from the horizontal circle. The specimen shown above has vernier scales and is accurate to approx. 1'. 2-2 The vertical axis of the instrument is aligned with the direction of gravity by means of a spirit level (artificial horizon) before starting the observations. Theodolites are primarily used for surveying, but they are excellent navigation instruments as well. Some models can resolve angles smaller than 0.1' which is not achieved even with the best sextants. A theodolite is mounted on a tripod which has to rest on solid ground. Therefore, it is restricted to land navigation. Mechanical theodolites traditionally measure zenith distances. Electronic models can optionally measure altitudes. Some theodolites measure angles in the unit gon instead of degree (400 gon = 360°). Before viewing the sun through an optical instrument, a proper shade glass must be inserted, otherwise the retina might suffer permanent damage! The sextant shown in Fig. 2-2 has two sets of shade glasses attached to the frame (one for each optical path). Altitude corrections Any altitude measured with a sextant or theodolite contains errors. Altitude corrections are necessary to eliminate systematic altitude errors AND to reduce the topocentric altitude of a body to the geocentric altitude (chapter 1). Altitude corrections do NOT remove random observation errors. Index error (IE) A sextant or theodolite may display a constant error (index error, IE) which has to be subtracted from every reading before the latter can be used for further calculations. The error is positive if the angle displayed by the instrument is greater than the actual angle and negative if the displayed angle is smaller. Errors which vary with the displayed angle require the use of an individual correction table if the error can not be eliminted by overhauling the instrument. 1st correction: H 1 = Hs− IE The sextant altitude, Hs, is the altitude as indicated by the sextant before any corrections have been applied. When using an external artificial horizon, H1 (not Hs!) has to be divided by two. A theodolite measuring the zenith distance, z, requires the following formula to obtain H1: H 1 = 90° − z− IE Dip of horizon If the earth's surface were an infinite plane, visible and sensible horizon would be identical. In reality, the visible sea horizon appears several arcminutes below the sensible horizon which is the result of two contrary effects, the curvature of the earth's surface and atmospheric refraction. The geometrical horizon is a flat cone formed by an infinite number of straight lines tangent to the earth and converging at the observer's eye. Since atmospheric refraction bends light rays passing along the earth's surface toward the earth, all points on the geometric horizon appear to be elevated, and thus form the visible horizon. Visible and geometrical horizon would be the same if the earth had no atmosphere (Fig. 2-5). The vertical angular distance of the sensible horizon from the visible horizon is called dip (of horizon) and is a function of the height of eye, HE, the vertical distance of the observer's eye from the sea surface (the distance between sensible and geoidal horizon): Dip[' ] ≈ 1.76⋅ HE [m] ≈ 0.97⋅ HE [ ft] 2-3 The above formula is empirical and includes the effects of the curvature of the earth's surface and of atmospheric refraction*. *At sea, the dip of horizon can be obtained directly by measuring the angular distance between the visible horizon in front of the observer and the visible horizon behind the observer through the zenith. Subtracting 180° from the angle thus measured and dividing the resulting angle by two yields the dip of horizon. This very accurate method can not be accomplished with a sextant but requires a special instrument (prismatic reflecting circle) which is able to measure angles greater than 180°. The correction for the dip of horizon has to be omitted (Dip = 0) if any kind of an artificial horizon is used since the latter is solely controlled by gravity and thus indicates the plane of the sensible horizon (perpendicular to the vector of gravity). 2nd correction: H 2 = H 1 −Dip The altitude obtained after applying corrections for index error and dip is also referred to as apparent altitude, Ha. Ha = H 2 Atmospheric refraction A light ray coming from a celestial body is slightly deflected toward the earth when passing obliquely through the atmosphere. This phenomenon is called refraction, and occurs always when light enters matter of different density at an angle smaller than 90°. Since the eye is not able to detect the curvature of the light ray, the body appears to be on a straight line tangent to the light ray at the observer's eye, and thus appears to be higher in the sky. R is the vertical angular distance between apparent and true position of the body measured at the observer's eye (Fig. 2-6). Atmospheric refraction is a function of Ha (= H2). Atmospheric standard refraction, R0, is 0' at 90° altitude and increases progressively to approx. 34' as the apparent altitude approaches 0°: Ha [°] 0 1 2 5 10 20 30 40 50 60 70 80 90 R0 ['] ~34 ~24 ~18 9.9 5.3 2.6 1.7 1.2 0.8 0.6 0.4 0.2 0.0 There are several formulas to calculate R0. Smart's formula yields accurate results from 15° through 90° apparent altitude [2,9]: 0.96469 0.00113 R0['] = − 3 tan H2[°] tan H2[°] The coefficients shown here are not exactly those given by Smart but have been modified to match the results (within ±0.0003') obtained with Saastamoinen's highly accurate formula (conditions: 10°C, 1010 hPa, 75% relative air humidity) [10]. For the purpose of celestial navigation, Smart's formula is still accurate enough at 10° apparent altitude. For altitudes between 0° and 15°, the following formula is recommended [10]. H2 is measured in degrees: 2 34.1334.197⋅H 2 0.00428⋅H 2 R0 [' ] = 2 10.505⋅H 20.0845⋅H 2 2-4 A rather simple refraction formula including the whole range of altitudes from 0° through 90° was found by Bennett: 1 R ['] = 0 7.31 tan H [°] + ( 2 H [°] + 4.4 ) 2 Bennett's formula is sufficiently accurate for marine navigation. The maximum systematic error is smaller than 0.1' [2]. Atmospheric refraction is influenced by atmospheric pressure, air temperature, and, to a lesser degree, by relative air humidity. Therefore the standard refraction, R0, has to be multiplied with a correction factor, f, to obtain the actual refraction for a given combination of pressure and temperature (the influence of air humidity is usually ignored). p[hPa] 283.15 p[in . Hg] 510 f = ⋅ = ⋅ 1010 273.15+T [° C] 29.83 460+T [° F] P is the atmospheric pressure and T the air temperature at the observer's position. Standard conditions (f = 1) are 1010 hPa (29.83 in Hg) and 10°C (50°F, 283.15 K). The correction for pressure and temperature is sometimes omitted (f = 1) since the resulting error is often tolerable. The common refraction formulas refer to a fictitious standard atmosphere with an average density gradient. The actual refraction may differ significantly from the calculated one if anomalous atmospheric conditions are present (temperature inversion, mirage effects, etc.). The influence of atmospheric anomalies increases strongly with decreasing altitude. Particularly refraction below ca. 5° apparent altitude may become erratic, and calculated or tabulated values in this range should not be blindly relied on. It should be mentioned that the dip of horizon, too, is influenced by atmospheric refraction and may become unpredictable under certain meteorological conditions. 3rd correction: H 3 = H 2 − f ⋅R0 ≈ H 2 −R0 H3 represents the topocentric altitude of the body, the altitude with respect to the sensible horizon. Parallax The trigonometric calculations of celestial navigation are based upon geocentric altitudes. Fig. 2-7 illustrates that the geocentric altitude of an object, H4, is always greater than the topocentric altitude, H3. The difference H4-H3 is called parallax in altitude, P. P decreases as the distance between object and earth increases. Accordingly, the effect is greatest when observing the moon since the latter is the object nearest to the earth. On the other hand, P is too small to be measured when observing fixed stars (see chapter 1, Fig. 1-4). Theoretically, the observed parallax refers to the sensible, not to the geoidal horizon. However, since the height of eye is by several magnitudes smaller than the radius of the earth, the resulting error is usually not significant. The parallax (in altitude) of a body being on the geoidal horizon is called horizontal parallax, HP (Fig. 2-7). The horizontal parallax of the sun is approx. 0.15'. Current values for the HP of the moon (approx. 1°) and the navigational planets are given in the Nautical Almanac [12] and similar publications, e.g., [13]*. *Tabulated values for HP refer to the equatorial radius of the earth (equatorial horizontal parallax, see chapter 9). 2-5 P is a function of topocentric altitude and horizontal parallax of a body.* It has to be added to H3. P = arcsin sin HP⋅cos H 3 ≈ HP ⋅cos H 3 4th correction: H 4 = H 3 P An additional correction for the oblateness of the earth is recommended (P, see p. 2-8). H4 represents the geocentric altitude of the body, the altitude with respect to the celestial horizon. Semidiameter When observing sun or moon with a marine sextant or theodolite, it is not possible to locate the center of the body precisely. It is therefore common practice to measure the altitude of the upper or lower limb of the body and add or subtract the apparent semidiameter, SD. The latter is the angular distance of the respective limb from the center of the body (Fig. 2-8). We have to correct for the geocentric SD, the SD measured by a fictitious observer at the center the earth, because H4 is measured at the center the earth (see Fig. 2-4)**. The geocentric semidiameters of sun and moon are given on the daily pages of the Nautical Almanac [12]. The geocentric SD of a body can be calculated from its tabulated horizontal parallax. This is of particular interest when observing the moon. r Moon SDgeocentric = arcsin k ⋅sin HP ≈ k ⋅ HP k Moon = = 0.2725 r Earth The factor k is the ratio of the radius of the respective body to the equatorial radius of the earth (rEarth = 6378 km). **Note that Fig. 2-8 shows the topocentric semidiameter. Although the semidiameters of the navigational planets are not quite negligible (the SD of Venus can increase to 0.5'), the apparent centers of these bodies are usually observed, and no correction for SD is applied. With a strong telescope, however, the limbs of the brightest planets can be observed. In this case the correction for semidiameter should be applied. Semidiameters of stars are much too small to be measured (SD = 0). 5th correction: H 5 = H 4 ± SD geocentric (lower limb: add SD, upper limb: subtract SD) When using a bubble sextant, we observe the center of the body and skip the correction for semidiameter. The altitude obtained after applying the above corrections is called observed altitude, Ho: Ho = H 5 Ho represents the geocentric altitude of the center of the body. *To be exact, the parallax formula shown above is rigorous for the observation of the center of a body only. When observing the lower or upper limb, there is a small error caused by the curvature of the body's surface which is usually negligible. The rigorous formula for the observation of either of the limbs is: Lower limb: P = arcsin [ sin HP ⋅cos H 3 k ] − arcsink ⋅sin HP ≈ HP ⋅cos H 3 Upper limb: P = arcsin [sin HP⋅cos H 3 − k ] arcsin k ⋅sin HP ≈ HP⋅cos H 3 2-6 Combined corrections for semidiameter and parallax H3 can be reduced to the observed altitude in one step. The following formula includes the corrections for semidiameter and parallax in altitude: Ho = H 3 arcsin [ sin HP ⋅cos H 3 ± k ] ≈ H 3 HP ⋅cos H 3 ± k (lower limb: add k, upper limb: subtract k) Alternative procedure for semidiameter and parallax Correcting for semidiameter before correcting for parallax is also possible. In this case, however, we have to calculate with the topocentric semidiameter, the semidiameter of the respective body as seen from the observer's position on the surface of the earth (see Fig. 2-8). With the exception of the moon, the body nearest to the earth, there is no significant difference between topocentric and geocentric semidiameter. The topocentric SD of the moon is only marginally greater than the geocentric SD when the moon is on the sensible (geoidal) horizon but increases measurably as the altitude increases because of the decreasing distance between observer and moon. The distance is smallest (decreased by about the radius of the earth) when the moon is in the zenith. As a result, the topocentric SD of the moon being in the zenith is approximately 0.3' greater than the geocentric SD. This phenomenon is called augmentation (Fig. 2-9). The rigorous formula for the topocentric (augmented) semidiameter of the moon is: k SDtopocentric = arctan 1 2 − cos H ± k − sin H [ sin2 HP 3 ] 3 (observation of lower limb: add k, observation of upper limb: subtract k) The approximate topocentric semidiameter of the moon can be calculated with a simpler formula given by Meeus [2]. It refers to the center of the moon but is still accurate enough for the purpose of navigation (error < 1'') when applied to the altitude of the upper or lower limb, respectively: SDtopocentric ≈ k ⋅ HP⋅1 sin HP ⋅sin H 3 A similar formula was proposed by Stark [14]: k ⋅ HP SDtopocentric ≈ 1 − sin HP ⋅sin H 3 2-7 Thus, the alternative fourth correction is: 4th correction (alt.): H 4,alt. = H 3 ± SD topocentric (lower limb: add SD, upper limb: subtract SD) H4,alt represents the topocentric altitude of the center of the moon. Using the parallax formula explained above, we calculate Palt from H4,alt: Palt = arcsin sin HP ⋅cos H 4,alt ≈ HP ⋅cos H 4, alt Thus, the alternative fifth correction is: 5th correction (alternative): H 5, alt = H 4,alt Palt Ho = H 5, alt Since the geocentric SD is easier to calculate than the topocentric SD, it is usually more convenient to correct for the semidiameter in the last place or, better, to use the combined correction for parallax and semidiameter unless one has to know the augmented SD of the moon for special reasons. The topocentric semidiameter of the moon can also be calculated from the observed altitude (the geocentric altitude of the center of the moon), Ho: k SDtopocentric = arcsin 1 sin Ho 1 − 2⋅ sin 2 HP sin HP Instead of Ho, the computed altitude, Hc, can be used (see chapter 4). Correction for the oblateness of the earth The above formulas for parallax and semidiameter are rigorous for spherical bodies. In fact, the earth is not a sphere but rather resembles an oblate spheroid, a sphere flattened at the poles (chapter 9). In most cases, the navigator will not notice the difference. However, when observing the moon, the flattening of the earth may cause a small but measurable error (up to ±0.2') in the parallax, depending on the observer's position. Therefore, a small correction, ΔP, should be added to P if higher precision is required [12]. When using the combined formula for semidiameter and parallax, ΔP is added to Ho. 2 1 P ≈ f ⋅HP ⋅ sin2⋅Lat ⋅cos Az ⋅sin H − sin Lat ⋅cos H * f = [ N ] 298.257 * Replace H with H3 or H4,alt, respectively. Pimproved = P P Lat is the observer's estimated latitude (chapter 4). AzN, the true azimuth of the moon, is either measured with a compass (compass bearing) or calculated using the azimuth formulas given in chapter 4. Phase correction (Venus and Mars) Since Venus and Mars show phases similar to the moon, their apparent center may differ somewhat from the actual center. The coordinates of both planets tabulated in the Nautical Almanac [12] include the phase correction, i. e., they refer to the apparent center. The phase correction for Jupiter and Saturn is too small to be significant. In contrast, coordinates calculated with Interactive Computer Ephemeris refer to the actual center. In this case, the upper or lower limb of the respective planet should be observed if the magnification of the telescope is sufficient, and the correction for semidiameter should be applied. 2-8 Altitude correction tables The Nautical Almanac provides sextant altitude correction tables for sun, planets, stars (pages A2 – A4), and the moon (pages xxxiv – xxxv), which can be used instead of the above formulas if small errors (< 1') are tolerable (among other things, the tables cause additional rounding errors). Other corrections Sextants with an artificial horizon can exhibit additional errors caused by acceleration forces acting on the bubble or pendulum and preventing it from aligning itself with the direction of gravity. Such acceleration forces can be accidental (vessel movements) or systematic (coriolis force). The coriolis force is important to air navigation (high speed!) and requires a special correction formula. In the vicinity of mountains, ore deposits, and other local irregularities of the earth's crust, the vector of gravity may slightly differ from the normal to the reference ellipsoid, resulting in altitude errors that are difficult to predict (local deflection of the vertical, chapter 9). Thus, the astronomical position of an observer (resulting from astronomical observations) may be slightly different from his geographic (geodetic) position with respect to a reference ellipsoid (GPS position). The difference is usually small at sea and may be ignored there. On land, particularly in the vicinity of mountain ranges, position errors of up to 50 arcseconds (Alps) or even 100 arcseconds (Himalaya) have been found. For the purpose of surveying there are maps containing local corrections for latitude and longitude, depending on the respective reference ellipsoid. 2-9 Chapter 3 Geographic Position and Time Geographic and Astronomical Terms In celestial navigation, the earth is regarded as a sphere. This is an approximation only, but the errors caused by the flattening of the earth are usually negligible (chapter 9). A circle on the surface of the earth whose plane passes through the earth's center is called a great circle. A great circle has the greatest possible diameter of all circles on the surface of the earth. Any circle on the surface of the earth whose plane does not pass through the earth's center is a small circle. The equator is the only great circle whose plane is perpendicular to the polar axis, the axis of rotation. Further, the equator is the only parallel of latitude being a great circle. Any other parallel of latitude is a small circle whose plane is parallel to the plane of the equator. A meridian is a great circle going through the geographic poles, the points where the polar axis intersects the earth's surface. The upper branch of a meridian is the half from pole to pole passing through a given point, for example the observer's position. The lower branch is the opposite half. The meridian passing through the observer's position is called local meridian. The Greenwich meridian, the meridian passing through the center of the transit instrument at the Royal Greenwich Observatory, was adopted as the prime meridian at the International Meridian Conference in 1884. Its upper branch (0°) is the reference for measuring longitudes (0°...+180° to the east and 0°...–180° to the west), its lower branch (±180°) is the basis for the International Dateline (Fig. 3-1). Each point of the earth's surface has an imaginary counterpart on the surface of the celestial sphere obtained by central projection. The projected image of the observer's position, for example, is the zenith. Accordingly, there are two celestial poles, the celestial equator, celestial meridians, etc. The local celestial meridian is also a vertical circle, a great circle on the celestial sphere passing through the observer's zenith and nadir. Further, the local celestial meridian, passing throught the celestial poles, marks the north point and the south point on the horizon. The vertical circle perpendicular to the local meridian, called prime vertical, marks the west point and east point on the horizon. The Equatorial System of Coordinates The geographic position of a celestial body, GP, is defined by the equatorial system of coordinates, a spherical coordinate system the origin of which is at the center of the earth (Fig. 3-2). The Greenwich hour angle of a body, GHA, is the angular distance of the upper branch of the meridian passing through GP from the upper branch of the Greenwich meridian (Lon = 0°), measured westward from 0° through 360°. The meridian going through GP (as well as its projection on the celestial sphere) is called hour circle. The Declination, Dec, is the angular distance of GP from the plane of the equator, measured northward through +90° or southward through –90°. GHA and Dec are geocentric coordinates (measured at the center of the earth). 3-1 Although widely used, the term „geographic position“ is misleading when applied to a celestial body since it actually describes a geocentric position in this case (see chapter 9). GHA and Dec are equivalent to geocentric longitude and latitude of a position with the exception that longitudes are measured westward from the Greenwich meridian through −180° and eastward through +180°. Since the Greenwich meridian rotates with the earth from west to east, whereas each hour circle remains linked with the almost stationary position of the respective body in the sky, the Greenwich hour angles of all celestial bodies increase by approx. 15° per hour (360° in 24 hours). In contrast to stars (15° 2.46' /h), the GHA's of sun, moon, and planets increase at slightly different (and variable) rates. This is caused by the revolution of the planets (including the earth) around the sun and by the revolution of the moon around the earth, resulting in additional apparent motions of these bodies in the sky. For several applications it is useful to measure the angular distance between the hour circle of a celestial body and the hour circle of a reference point in the sky instead of the Greenwich meridian because the angle thus obtained is independent of the earth's rotation. The sidereal hour angle, SHA, of a given body is the angular distance of its hour circle (upper branch) from the hour circle (upper branch) of the first point of Aries (also called vernal equinox, see below), measured westward from 0° through 360°. Thus, the GHA of a body is the sum of its sidereal hour angle and the GHA of the first point of Aries, GHAAries: GHA = SHA + GHA Aries (If the resulting GHA is greater than 360°, subtract 360°.) The angular distance of a celestial body eastward from the hour circle of the vernal equinox, measured in time units (24h = 360°), is called right ascension, RA. The latter is mostly used by astronomers whereas navigators prefer sidereal hour angles. SHA [° ] RA [h] = 24 h − ⇔ SHA [°] = 360° − 15⋅ RA [h] 15 Fig. 3-3 illustrates the various hour angles on the plane of the equator as seen from the celestial north pole (time diagram). Declinations are not affected by the rotation of the earth. The declinations of sun and planets change primarily due to the obliquity of the ecliptic, the inclination of the earth's equator to the ecliptic. The latter is the mean orbital plane of the earth and forms a great circle on the celestial sphere. The declination of the sun, for example, varies periodically between ca. +23.5° (summer solstice) and ca. -23.5° (winter solstice) as shown in Fig.3-4. The two points on the celestial sphere where the great circles of ecliptic and celestial equator intersect are called equinoxes. The term equinox is also used for the times at which the apparent sun, moving westward along the ecliptic during the course of a year, crosses the celestial equator, approximately on March 21 and on September 23*. There is a vernal equinox (first point of Aries, vernal point) and an autumnal equinox. The former is the reference point for measuring sidereal hour angles (Fig. 3-5). At the instant of an equinox (Dec ≈ 0°), day and night have roughly (!) the same length (12 h), regardless of the observer's position (Lat. aequae noctes = equal nights). *To be more precise, the equinoxes are defined as the instants at which the ecliptic longitude (λ) of the apparent sun is either 0° (vernal equinox) or 180° (autumnal equinox) [10]. The actual declination of the sun at such an instant may slightly differ from 0° since the earth is not always exactly on the mean orbital plane (perturbations). 3-2 The declinations of the planets and the moon are also influenced by the inclinations of their own orbits to the ecliptic. The plane of the moon's orbit, for example, is inclined to the ecliptic by approx. 5° and makes a tumbling movement with a period of 18.6 years (Saros cycle). As a result, the declination of the moon varies between approx. -28.5° and +28.5° at the beginning and at the end of the Saros cycle, and between approx. -18.5° and +18.5° in the middle of the Saros cycle. Further, sidereal hour angles and declinations of all bodies change slowly due to the influence of the precession of the earth's polar axis. Precession is a slow, tumbling movement of the polar axis along the surface of an imaginary double cone. One revolution lasts about 26000 years (Platonic year). As a result, the equinoxes move westward along the celestial equator at a rate of approx. 50'' per year. Thus, the sidereal hour angle of each star decreases at about the same rate. In addition, there is a small elliptical oscillation of the polar axis with a period of 18.6 years (compare with the Saros cycle), called nutation, which causes the equinoxes to travel along the celestial equator at a periodically changing rate. Thus we have to distinguish between the ficticious mean equinox of date and the true equinox of date (see time measurement). Accordingly, the declination of each body oscillates. The same applies to the rate of change of the sidereal hour angle and right ascension of each body. Even stars are not fixed in space but move individually, resulting in a slow drift of their equatorial coordinates (proper motion). Finally, the apparent positions of bodies are influenced by other factors, e. g., the finite speed of light (light time, aberration), and annual parallax, the latter being caused by the earth orbiting around the sun [16]. The accurate prediction of geographic positions of celestial bodies requires complicated algorithms. Formulas for the calculation of low-precision ephemerides of the sun (accurate enough for celestial navigation) are given in chapter 15. Time Measurement in Navigation and Astronomy Since the Greenwich hour angle of any celestial body changes rapidly, celestial navigation requires accurate time measurement, and the instant of each observation should be measured to the second if possible. This is usually done by means of a chronometer and a stopwatch (chapter 17). The effects of time errors are dicussed in chapter 16. On the other hand, the earth's rotation with respect to celestial bodies provides an important basis for astronomical time measurement. Coordinates tabulated in the Nautical Almanac refer to Universal Time, UT. UT has replaced Greenwich Mean Time, GMT, the traditional basis for civil time keeping. Conceptually, UT (like GMT) is the hour angle of the fictitious mean sun, expressed in hours, with respect to the lower branch of the Greenwich meridian (mean solar time, Fig. 3-6). 3-3 UT is calculated using the following formula: GHA [°] UT [h] = MeanSun 12 15 (If UT is greater than 24 h, subtract 24 hours.) By definition, the GHA of the mean sun increases by exactly 15° per hour, completing a 360° cycle in 24 hours. The unit for UT is 1 solar day, the time interval between two consecutive meridian transits of the mean sun. The rate of change of the GHA of the apparent (observable) sun varies periodically and is sometimes slightly greater, sometimes slightly smaller than 15°/h during the course of a year. This behavior is caused by the eccentricity of the earth's orbit and by the obliquity of the ecliptic. The time measured by the hour angle of the apparent sun with respect to the lower branch of the Greenwich meridian is called Greenwich Apparent Time, GAT. A sundial located on the Greenwich meridian would indicate GAT. The difference between GAT and UT at a given instant is called equation of time, EoT: EoT = GAT − UT EoT varies periodically between approx. −14 and +17 minutes (Fig. 3-7). Predicted values for EoT for each day of the year (at 0:00 and 12:00 UT) are given in the Nautical Almanac (grey background indicates negative EoT). EoT is needed when calculating times of sunrise and sunset, or determining a noon longitude (chapter 6). Formulas for the calculation of EoT are given in chapter 15. The hour angle of the mean sun with respect to the lower branch of the local meridian (the upper branch going through the observer's position) is called Local Mean Time, LMT. LMT and UT are linked through the following formula: Lon [° ] LMT [h] = UT [h] 15 The instant of the mean sun passing through the upper branch of the local meridian is called Local Mean Noon, LMN. A zone time is the local mean time with respect to a longitude being a multiple of ±15°. Thus, zone times differ by an integer number of hours. In the US, for example, Eastern Standard Time (UT−5h) is LMT at −75° longitude, Pacific Standard Time (UT−8h) is LMT at −120° longitude. Central European Time (UT+1h) is LMT at +15° longitude. The hour angle of the apparent sun with respect to the lower branch of the local meridian is called Local Apparent Time, LAT: Lon[°] LAT [h] = GAT [h] 15 The instant of the apparent sun crossing the upper branch of the local meridian is called Local Apparent Noon, LAN. Time measurement by the earth's rotation does not necessarily require the sun as the reference point in the sky. 3-4 Greenwich Apparent Sidereal Time, GAST, is a time scale based upon the Greenwich hour angle (upper branch) of the true vernal equinox of date, GHAAries (see Fig. 3-3). GHA [° ] GAST [h] = Aries 15 The values for GHAAries tabulated in the Nautical Almanac refer to the true equinox of date. GAST can be measured by the Greenwich meridian transit of a chosen star since GAST and the right ascension of the observed star are numerically equal at the moment of meridian transit. The Greenwich hour angle (measured in hours) of the imaginary mean vernal equinox of date (travelling along the celestial equator at a constant rate) is called Greenwich Mean Sidereal Time, GMST. The difference between GAST and GMST at a given instant is called equation of the equinoxes, EQ, or nutation in right ascension. EQ can be predicted precisely. It varies within approx. ±1s. EQ = GAST − GMST Due to the earth's revolution around the sun, a mean sidereal day (the time interval between two consecutive meridian transits of the mean equinox) is slightly shorter than a mean solar day: 24 h Mean Sidereal Time = 23h56m 4.090524s Mean Solar Time Since the exact instant of the sun's meridian transit is difficult to find, UT is calculated from GMST, to which it is linked through a formula [10]. Today, sidereal time (and thus, UT) is obtained by observation of extragalactical radio sources (quasars). Quasars can be regarded as fixed to the imaginary celestial sphere since they do not exhibit any measurable proper motion. Their apparent motions are measured through Very Long Baseline Interferometry (VLBI). This technology, which involves a global network of observation stations, produces much more accurate results than the observation of meridian transits. By analogy with LMT and LAT, there is a Local Mean Sidereal Time, LMST, and a Local Apparent Sidereal Time, LAST: Lon[°] Lon[° ] LMST [h] = GMST [h] LAST [h] = GAST [h] 15 15 Solar time and sidereal time are both linked to the earth's rotation. The earth's rotating speed, however, decreases slowly (tidal friction) and, moreover, fluctuates in an unpredictable manner due to random movements of matter within the earth's body (magma) and on the surface (water, air). Therefore, neither of the two time scales is strictly linear. Many astronomical applications, however, require a linear time scale. One example is the calculation of ephemerides since the motions of celestial bodies in space are independent of the earth's rotation. International Atomic Time, TAI, is the most accurate time standard presently available. It is obtained by statistical analysis of data supplied by a world-wide network of atomic clocks. Among others, two important time scales are derived from TAI: Civil life is mostly determined by Coordinated Universal Time, UTC, which is the basis for time signals broadcast by radio stations, e. g., WWV or WWVH. UTC is controlled by TAI. Due to the variable rotating speed of the earth, UT tends to drift away from UTC. This is undesirable since the cycle of day and night is linked to UT. Therefore, UTC is synchronized to UT, if necessary, by inserting (or omitting) leap seconds at certain times (June 30 and December 31) in order to avoid that the difference, ΔUT, exceeds the specified maximum value of ±0.9 s. UT = UTC UT UTC = TAI − N N is the cumulative number of leap seconds inserted since 1972 (N = 34 in 2011.0). Due to the occasional leap seconds, UTC is not a continuous time scale. Predicted values for ΔUT are published by the IERS Rapid Service [15] on a weekly basis (IERS Bulletin A). Note that UT is a synonym for UT1 (UT1-UTC = DUT1 = ΔUT). The IERS also announces the insertion (or omission) of leap seconds in advance (IERS Bulletins A + C). Terrestrial Time, TT (formerly called Terrestrial Dynamical Time, TDT), is another time scale derived from TAI: TT = TAI 32.184s TT has replaced Ephemeris Time, ET. The offset of 32.184 s with respect to TAI is necessary to ensure a seamless continuation of ET. TT is used in astronomy (calculation of ephemerides) and space flight. The difference between TT and UT is called ΔT: T = TT − UT 3-5 At the beginning of the year 2015, ΔT was +67.6s. ΔT is of some interest to the navigator since computer almanacs require TT (TDT) as time argument (programs using UT calculate on the basis of interpolated or extrapolated ΔT values). A precise long-term prediction of ΔT is not possible. Therefore, computer almanacs using only UT as time argument may become less accurate in the long term. ΔT values for the presence and near future can be calculated with the following formula: T = 32.184s TAI − UTC − UT1 − UTC Current values and short-term predictions for UT1-UTC and for TAI-UTC (cumulative number of leap seconds) are published in the IERS Bulletin A. At present (2011) there is an ongoing discussion among scientists about the usefulness of leap seconds. Computer operating systems (mostly running on UTC), for example, would require a continuous time scale to create error-free time stamps for files at any instant. Another argument often heard is that the drift of UT with respect to atomic time is so slow that its effect on civil life will not become obvious in the foreseeable future. This argument cannot be denied since no one can tell if civilization as we know it today will still exist in, say, 3000 years. Up to now, no decision has been made. The practice of celestial navigation would not be affected by a possible abolishment of leap seconds since the calculation of ephemerides (RA, Dec) is based upon TT whereas the Greenwich hour angle of a celestial body is a function of its right ascension and sidereal time (the latter being determined by the earth's rotation). The GMT Problem The term GMT has become ambigous since it is often used as a synonym for UTC now. Moreover, astronomers used to reckon GMT from the upper branch of the Greenwich meridian until 1925 (the time thus obtained is sometimes called Greenwich Mean Astronomical Time, GMAT). Therefore, the term GMT should be avoided in scientific publications, except when used in a historical context. The Nautical Almanac Predicted values for GHA and Dec of sun, moon and the navigational planets are tabulated for each integer hour (UT) of a calendar year on the daily pages of the Nautical Almanac, N.A., and similar publications [12, 13]. GHAAries is tabulated in the same manner. Listing GHA and Dec of all 57 fixed stars used in navigation for each integer hour of the year would require too much space in a book. Therefore, sidereal hour angles are tabulated instead of Greenwich hour angles. Since declinations and sidereal hour angles of stars change only slowly, tabulated values for periods of 3 days are accurate enough for celestial navigation. GHA is obtained by adding the SHA of the respective star to the current value of GHAAries. GHA and Dec for each second of the year are obtained using the interpolation tables at the end of the N.A. (printed on tinted paper), as explained in the following directions: 1. We note the exact time of observation (UT), determined with a chronometer and a stopwatch. If UT is not available, we can use UTC. The resulting error is tolerable in most cases. 2. We look up the day of observation in the N.A. (two pages cover a period of three days). 3. We go to the nearest integer hour preceding the time of observation and note GHA and Dec of the observed body. In case of a fixed star, we form the sum of GHA Aries and the SHA of the star, and note the tabulated declination. When observing planets, we note the v and d factors given at the bottom of the appropriate column. For the moon, we take v and d for the nearest integer hour preceding the time of observation. The quantity v is necessary to apply an additional correction to the following interpolation of the GHA of moon and planets. It is not required for stars. The sun does not require a v factor since the correction has been incorporated in the tabulated values for the sun's GHA. The quantity d, which is negligible for stars, is the rate of change of Dec, measured in arcminutes per hour. It is needed for the interpolation of Dec. The sign of d is critical! 3-6 4. We look up the minute of observation in the interpolation tables (1 page for each 2 minutes of the hour), go to the second of observation, and note the increment from the respective column. We enter one of the three columns to the right of the increment columns with the v and d factors and note the corresponding corr(ection) values (v-corr and d-corr). The sign of d-corr depends on the trend of declination at the time of observation. It is positive if Dec at the integer hour following the observation is greater than Dec at the integer hour preceding the observation. Otherwise it is negative. v -corr is negative for Venus. Otherwise, it is always positive. 5. We form the sum of Dec and d-corr (if applicable). 6. We form the sum of GHA (or GHA Aries and SHA in case of a star), increment, and v-corr (if applicable). SHA values tabulated in the Nautical Almanac refer to the true vernal equinox of date. Interactive Computer Ephemeris Interactive Computer Ephemeris, ICE, is a computer almanac developed by the U.S. Naval Observatory (successor of the Floppy Almanac) in the 1980s. ICE is FREEWARE (no longer supported by USNO), compact, easy to use, and provides a vast quantity of accurate astronomical data for a time span of almost 250 (!) years. In spite of the archaic design (DOS program), ICE is still a useful tool for navigators and astronomers. Among many other features, ICE calculates GHA and Dec for a given body and time as well as altitude and azimuth of the body for an assumed position (see chapter 4) and, moreover, sextant altitude corrections. Since the navigation data are as accurate as those tabulated in the Nautical Almanac (approx. 0.1'), the program makes an adequate alternative, although a printed almanac (and sight reduction tables) should be kept as a backup in case of a computer failure. The following instructions refer to the final version (0.51). Only program features relevant to navigation are explained. 1. Installation Copy the program files to a chosen directory on the hard drive, floppy disk, USB stick, or similar storage device. ICE.EXE is the executable program file. 2. Getting Started DOS users: Change to the program directory and enter "ice" or "ICE ". Windows users can run ICE in a DOS box. Linux users: Install the DOS emulator "DOSBox". Copy the ICE files to a directory of your choice in your personal folder. ICE is started through the command “dosbox“, followed by a blank space and the path to the program file: dosbox /home/ After the program has started, the main menu appears. Use the function keys F1 to F10 to navigate through the submenus. The program is more or less self-explanatory. Go to the submenu INITIAL VALUES (F1). Follow the directions on the screen to enter date and time of observation (F1), assumed latitude (F2), assumed longitude (F3), and your local time zone (F6). Assumed latitude and longitude define your assumed position. Use the correct data format, as shown on the screen (decimal format for latitude and longitude). After entering the above data, press F7 to accept the values displayed. To change the default values permanently, edit the file ice.dft with a text editor (after making a backup copy) and make the desired changes. Do not change the data format. The numbers have to be in columns 21-40. An output file can be created to store calculated data. Go to the submenu FILE OUTPUT (F2) and enter a chosen file name, e.g., OUTPUT.TXT. 3-7 3. Calculation of Navigational Data From the main menu, go to the submenu NAVIGATION (F7). Enter the name of the body. The program displays GHA and Dec of the body, GHA and Dec of the sun (if visible), and GHA of the vernal equinox for the date and time (UT) stored in INITIAL VALUES. Hc (computed altitude) and Zn (azimuth) mark the apparent position of the body as observed from the assumed position. Approximate altitude corrections (refraction, SD, PA), based upon Hc, are also displayed (for lower limb of body). The semidiameter of the moon includes augmentation. The coordinates calculated for Venus and Mars do not include the phase correction. Therefore, the upper or lower limb (if visible) should be observed. ∆T is TT(TDT)-UT, the predicted difference between terrestrial time and UT for the given date. The ∆T value for 2011.0 predicted by ICE is 74.1s, the actual value is 66.3s (see below) which demonstrates that the extrapolation algorithm used by ICE is outdated. Horizontal parallax and semidiameter of a body can be extracted from the submenu POSITIONS (F3). Choose APPARENT GEOCENTRIC POSITIONS (F1) and enter the name of the body (sun, moon, planets). The last column shows the distance of the center of the body from the center of the earth, measured in astronomical units (1 AU = 149.6 . 106 km). HP and SD are calculated as follows: r [km] r [km] HP = arcsin E SD = arcsin B distance[ km] distance[km] rE is the equatorial radius of the earth (6378 km). rB is the radius of the respective body (Sun: 696260 km, Moon: 1378 km, Venus: 6052 km, Mars: 3397 km, Jupiter: 71398 km, Saturn: 60268 km). The apparent geocentric positions refer to TT (TDT), but the difference between TT and UT has no significant effect on HP and SD. To calculate the times of rising and setting of a body, go to the submenu RISE & SET TIMES (F6) and enter the name of the body. The columns on the right display the time of rising, meridian transit, and setting for the assumed location (UT+xh, according to the time zone specified). The increasing error of ∆T values predicted by ICE may lead to reduced precision when calculating navigation data in the future. The coordinates of the moon are particularly sensitive to errors of ∆T. Unfortunately, ICE has no option for editing and modifying the internal ∆T algorithm. The high-precision part of ICE, however, is not affected since TT (TDT) is the time argument. To circumvent the ∆T problem, extract GHA and Dec using the following procedure: 1. Compute GAST using SIDEREAL TIME (F5). The time argument is UT. 2. Edit date and time at INITIAL VALUES (F1). Now, the time argument is TT (UT+∆T). Compute RA and Dec using POSITIONS (F3) and APPARENT GEOCENTRIC POSITIONS (F1). 3. Use the following formula to calculate GHA from GAST and RA (RA refers to the true vernal equinox of date): GHA[°] = 15⋅GAST [h] 24h − RA[h] Add or subtract 360° if necessary. 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GHA B" VY cos D = sin Dec M sin Dec B cos Dec M cos Dec Bcos [15 RA M [h ! RA B [h " 8 QZ [PM OMVKMU[YQK S\UIY LQZ[IUKM& CPMZM NVYT\SIZ KIU JM \ZML [V ZM[ \W VUMcZ V^U [IJSM ^Q[P [PM IQL VN [PM ?I\[QKIS 5STIUIK VY IU` KVTW\[MY ISTIUIK QN I S\UIY LQZ[IUKM [IJSM QZ UV[ I]IQSIJSM& "5STVZ[ I KMU[\Y` IN[MY [PM VYQOQUIS =\UIY 8QZ[IUKM CIJSMZ ^MYM LYVWWML$ B[M]MU EMWZ[MY YMZ\TML [PM [YILQ[QVU& ;QZ [IJSMZ IYM WYMZMU[S` *((,! I]IQSIJSM [PYV\OP [PM QU[MYUM[ G),H& *951>8;6 @75 9A;1> 48?@1;35 6MNVYM I S\UIY LQZ[IUKM TMIZ\YML J` [PM VJZMY]MY KIU JM KVTWIYML ^Q[P [IJ\SI[ML ]IS\MZ$ Q[ PIZ [V JM YML\KML [V [PM KVYYMZWVULQUO OMVKMU[YQK IUOSM J` KSMIYQUO Q[ NYVT [PM MNNMK[Z VN YMNYIK[QVU IUL WIYISSI_& CPQZ MZZMU[QIS WYVKMZZ QZ KISSML 3951>8;6 @75 9A;1> 48?@1;35 & ?\TMYV\Z WYVKML\YMZ PI]M JMMU LM]MSVWML$ ITVUO [PMT YQOVYV\Z IUL dX\QKRe TM[PVLZ& /%) :QO& /%) ZPV^Z [PM WVZQ[QVUZ VN [PM TVVU$ >$ IUL I YMNMYMUKM JVL`$ 6$ QU [PM KVVYLQUI[M Z`Z[MT VN [PM PVYQaVU& EM LMUV[M [PM IWWIYMU[ WVZQ[QVUZ VN [PM KMU[MYZ VN [PM TVVU IUL [PM YMNMYMUKM JVL` J` >IWW IUL 6IWW $ YMZWMK[Q]MS`& F QZ [PM aMUQ[P& CPM ZQLM 8IWW VN [PM ZWPMYQKIS [YQIUOSM 6IWW %F%> IWW QZ [PM IWWIYMU[ S\UIY LQZ[IUKM& CPM IS[Q[\LMZ VN >IWW IUL 6IWW VJ[IQUML IN[MY IWWS`QUO [PM KVYYMK[QVUZ NVY QULM_ MYYVY$ LQW$ IUL ZMTQLQITM[MY! 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Az B CPM WVZQ[QVU VN MIKP JVL` QZ ZPQN[ML ISVUO Q[Z ]MY[QKIS KQYKSM J` I[TVZWPMYQK YMNYIK[QVU IUL WIYISSI_ QU IS[Q[\LM& 5N[MY KVYYMK[QUO ;>IWW IUL;6IWW NVY JV[P MNNMK[Z$ ^M VJ[IQU [PM OMVKMU[YQK WVZQ[QVUZ > IUL 6& EM LMUV[M [PM IS[Q[\LM VN > J` ;> IUL [PM IS[Q[\LM VN 6 J` ;6&;> QZ IS^I`Z OYMI[MY [PIU ;>IWW JMKI\ZM [PM WIYISSI_ VN [PM TVVU QZ IS^I`Z OYMI[MY [PIU YMNYIK[QVU& CPM IUOSM QZ UMQ[PMY INNMK[ML J` YMNYIK[QVU UVY J` [PM WIYISSI_ QU IS[Q[\LM2 Az M = Az Mapp Az B = Az Bapp CPM ZQLM 8 VN [PM ZWPMYQKIS [YQIUOSM 6%F%> QZ [PM \URUV^U OMVKMU[YQK S\UIY LQZ[IUKM& 5WWS`QUO [PM KVZQUM NVYT\SI [V [PM ZWPMYQKIS [YQIUOSM NVYTML J` [PM aMUQ[P IUL [PM IWWIYMU[ WVZQ[QVUZ$ ^M OM[2 cos Dapp = sin H Mapp sin H Bapp cos H Mapp cos H Bapp cos # cos D ! sin H sin H cos # = app Mapp Bapp cos H Mapp cos H Bapp AMWMI[QUO [PM WYVKML\YM ^Q[P [PM ZWPMYQKIS [YQIUOSM NVYTML J` [PM aMUQ[P IUL [PM OMVKMU[YQK WVZQ[QVUZ$ ^M OM[2 cos D = sin H M sin H B cos H M cos H Bcos # cos D ! sin H sin H cos # = M B cos H M cos H B /%* BQUKM QZ KVUZ[IU[$ ^M KIU KVTJQUM JV[P IaQT\[P NVYT\SIZ2 cos D ! sin H sin H cos D ! sin H sin H M B = app Mapp Bapp cos H M cos H B cos H Mapp cos H Bapp CP\Z$ ^M PI]M MSQTQUI[ML [PM \URUV^U IUOSM & ?V^$ ^M Z\J[YIK[ \UQ[` NYVT JV[P ZQLMZ VN [PM MX\I[QVU2 cos D ! sin H sin H cos D ! sin H sin H M B ! 1 = app Mapp Bapp ! 1 cos H M cos H B cos H Mapp cos H Bapp cos D ! sin H sin H cos H cos H cos D ! sin H sin H cos H cos H M B ! M B = app Mapp Bapp ! Mapp Bapp cos H M cos H B cos H M cos H B cos H Mapp cos H Bapp cos H Mapp cos H Bapp cos D ! sin H sin H ! cos H cos H cos D ! sin H sin H ! cos H cos H M B M B = app Mapp Bapp Mapp Bapp cos H M cos H B cos H Mapp cos H Bapp DZQUO [PM ILLQ[QVU NVYT\SI NVY KVZQUMZ$ ^M PI]M2 cos D ! cos H ! H " cos D ! cos H ! H " M B = app Mapp Bapp cos H M cos H B cos H Mapp cos H Bapp BVS]QUO NVY KVZ 8$ ^M VJ[IQU D2-1)./-'30 NVYT\SI NVY KSMIYQUO [PM S\UIY LQZ[IUKM2 cos H M cos H B cos D = [cos D app ! cos H Mapp ! H Bapp " cos H M ! H B" cos H Mapp cos H Bapp 5LLQUO \UQ[` [V JV[P ZQLMZ VN [PM MX\I[QVU QUZ[MIL VN Z\J[YIK[QUO Q[$ SMILZ [V $.2-(30 NVYT\SI2 cos H M cos H B cos D = [ cos Dapp cos H Mapp H Bapp " ! cos H M H B" cos H Mapp cos H Bapp /><354A>5 8MYQ]QUO DC NYVT I S\UIY LQZ[IUKM KVTWYQZMZ [PM NVSSV^QUO Z[MWZ2 . EM TMIZ\YM [PM IS[Q[\LM VN [PM \WWMY VY SV^MY SQTJ VN [PM TVVU$ ^PQKPM]MY QZ ]QZQJSM$ IUL UV[M [PM [QTM VN [PM VJZMY]I[QVU QULQKI[ML J` V\Y ^I[KP$ EC) =>IWW & EM IWWS` [PM KVYYMK[QVUZ NVY QULM_ MYYVY IUL LQW QN UMKMZZIY`! IUL OM[ [PM IWWIYMU[ IS[Q[\LM VN [PM SQTJ$ ;) =>IWW & EM YMWMI[ [PM WYVKML\YM ^Q[P [PM YMNMYMUKM JVL` IUL VJ[IQU [PM ^I[KP [QTM EC) 6IWW IUL [PM IS[Q[\LM ;) 6IWW & !. EM TMIZ\YM [PM IUO\SIY LQZ[IUKM JM[^MMU [PM SQTJ VN [PM TVVU IUL [PM YMNMYMUKM JVL`$ 8=IWW $ IUL UV[M [PM KVYYMZWVULQUO ^I[KP [QTM$ EC 8& CPM IUOSM 8=IWW PIZ [V JM TMIZ\YML ^Q[P [PM OYMI[MZ[ WVZZQJSM WYMKQZQVU& <[ QZ YMKVTTMULML [V TMIZ\YM I NM^ 8=IWW ]IS\MZ IUL [PMQY KVYYMZWVULQUO EC 8 ]IS\MZ QU YIWQL Z\KKMZZQVU IUL KISK\SI[M [PM YMZWMK[Q]M I]MYIOM ]IS\M& EPMU [PM TVVU QZ ISTVZ[ N\SS$ Q[ QZ UV[ X\Q[M MIZ` [V LQZ[QUO\QZP [PM SQTJ VN [PM TVVU NYVT [PM [MYTQUI[VY ZPILV^ SQUM!& /%+ ". EM TMIZ\YM [PM IS[Q[\LMZ VN JV[P JVLQMZ IOIQU$ IZ LMZKYQJML IJV]M& EM LMUV[M [PMT J` ;* =>IWW IUL ;* 6IWW $ IUL UV[M [PM KVYYMZWVULQUO ^I[KP [QTMZ VN VJZMY]I[QVU$ EC* =>IWW IUL EC* 6IWW & #. BQUKM [PM VJZMY]I[QVUZ IYM VUS` I NM^ TQU\[MZ IWIY[$ ^M KIU KISK\SI[M [PM IS[Q[\LM VN [PM YMZWMK[Q]M JVL` I[ [PM TVTMU[ VN [PM S\UIY LQZ[IUKM VJZMY]I[QVU J` SQUMIY QU[MYWVSI[QVU2 WT D ! WT1 LMapp H LMapp = H1 LMapp H2 LMapp ! H1 LMapp " WT2 LMapp ! WT1 LMapp WT D ! WT1 Bapp H Bapp = H1 Bapp H2 Bapp ! H1 Bapp " WT2 Bapp ! WT1 Bapp $. EM KVYYMK[ [PM IS[Q[\LM VN [PM TVVU IUL [PM IUO\SIY LQZ[IUKM 8=IWW NVY [PM I\OTMU[ML ZMTQLQITM[MY VN [PM TVVU$ B8 I\O & CPM SI[[MY KIU JM KISK\SI[ML LQYMK[S` NYVT [PM IS[Q[\LM VN [PM \WWMY VY SV^MY SQTJ VN [PM TVVU2 k tan SD aug = k = 0.2725 1 ! cos H ! k"2 ! sin H sin 2 HP LMapp LMapp [$ M upper limb: cos H LMapp ! k lower limb: cos H LMapp k CPM IS[Q[\LM KVYYMK[QVU QZ2 Lower limb: H Mapp = H LMapp SD aug Upper limb: H Mapp = H LMapp ! SD aug CPM Y\SMZ NVY [PM S\UIY LQZ[IUKM KVYYMK[QVU IYM2 Limb of moon towards reference body: Dapp = DLapp SD aug Limb of moon away from reference body: Dapp = DLapp ! SD aug CPM IJV]M WYVKML\YM QZ IU IWWYV_QTI[QVU ZQUKM [PM I\OTMU[ML ZMTQLQITM[MY QZ I N\UK[QVU VN [PM IS[Q[\LM KVYYMK[ML NVY YMNYIK[QVU& BQUKM YMNYIK[QVU QZ I ZTISS X\IU[Q[` IUL ZQUKM [PM [V[IS I\OTMU[I[QVU JM[^MMU (f IUL 1(f IS[Q[\LM QZ VUS` IWWYV_& (&+c$ [PM YMZ\S[QUO MYYVY QZ ]MY` ZTISS IUL TI` JM QOUVYML& CPM Z\U$ ^PMU KPVZMU IZ YMNMYMUKM JVL`$ YMX\QYMZ [PM ZITM KVYYMK[QVUZ NVY ZMTQLQITM[MY& BQUKM [PM Z\U LVMZ UV[ ZPV^ I TMIZ\YIJSM I\OTMU[I[QVU$ ^M KIU \ZM [PM OMVKMU[YQK ZMTQLQITM[MY [IJ\SI[ML QU [PM ?I\[QKIS 5STIUIK VY KISK\SI[ML ^Q[P I KVTW\[MY WYVOYIT& %. EM KVYYMK[ JV[P IS[Q[\LMZ$ ; >IWW IUL ; 6IWW $ NVY I[TVZWPMYQK YMNYIK[QVU$ A& p[mbar 283 0.97127 0.00137 R [' = ! i = Mapp , Bapp H % 10 ° i 1010 T [°C 273 tan H 3 i i tan H i " AQ QZ Z\J[YIK[ML NYVT [PM YMZWMK[Q]M IS[Q[\LM& CPM YMNYIK[QVU NVYT\SI QZ VUS` IKK\YI[M NVY IS[Q[\LMZ IJV]M IWWYV_& )(f& =V^MY IS[Q[\LMZ ZPV\SL JM I]VQLML IU`^I` ZQUKM YMNYIK[QVU TI` JMKVTM MYYI[QK IUL ZQUKM [PM IWWIYMU[ LQZR VN [PM TVVU IUL Z\U! IZZ\TMZ IU V]IS ZPIWM KI\ZML J` IU QUKYMIZQUO LQNNMYMUKM QU YMNYIK[QVU NVY \WWMY IUL SV^MY SQTJ& CPQZ LQZ[VY[QVU ^V\SL INNMK[ [PM ZMTQLQITM[MY ^Q[P YMZWMK[ [V [PM YMNMYMUKM JVL` QU I KVTWSQKI[ML ^I`& /%, &. EM KVYYMK[ [PM IS[Q[\LMZ NVY [PM WIYISSI_ QU IS[Q[\LM2 sin P M = sin HP M cos H Mapp ! RMapp " sin PB = sin HP Bcos H Bapp ! RBapp " EM IWWS` [PM IS[Q[\LM KVYYMK[QVUZ IZ NVSSV^Z2 H M = H Mapp ! RMapp PM H B = H Bapp ! RBapp PB CPM KVYYMK[QVU NVY WIYISSI_ QZ UV[ IWWSQML [V [PM IS[Q[\LM VN I NQ_ML Z[IY ;@6 4 (!& '. 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D2 " T 2 = T 1 3h T 3 = T 2 3h D1 & D2 & D3 or D1 % D2 % D3 8 TI` PI]M IU` ]IS\M JM[^MMU 8 ) IUL 8 +& CPMYM T\Z[ UV[ JM I TQUQT\T VY TI_QT\T VN 8 QU [PM [QTM QU[MY]IS GC )$ C+H& CPQZ WYVJSMT LVMZ UV[ VKK\Y ^Q[P I WYVWMYS` KPVZMU JVL` PI]QUO I Z\Q[IJSM YI[M VN KPIUOM VN 8& ?MIY I TQUQT\T VY TI_QT\T VN 8$ 8' C ^V\SL JM ]MY` ZTISS$ IUL [PM VJZMY]I[QVU ^V\SL JM MYYI[QK IU`^I`& 5N[MY NQULQUO C 8$ ^M KIU KISK\SI[M [PM ^I[KP MYYVY$ C& ' T = WT D ! T D C QZ [PM LQNNMYMUKM JM[^MMU V\Y ^I[KP [QTM I[ [PM TVTMU[ VN VJZMY]I[QVU$ EC 8$ IUL [PM [QTM NV\UL J` QU[MYWVSI[QVU$ C8& B\J[YIK[QUO [PM ^I[KP MYYVY NYVT [PM ^I[KP [QTM$ EC$ YMZ\S[Z QU DC& UT = WT !' T /%- -:=> CPM WYVKML\YMZ LMZKYQJML ZV NIY YMNMY [V I ZWPMYQKIS MIY[P& CPM KVYYMK[QVUZ YMX\QYM RUV^SMLOM VN [PM VJZMY]MYcZ SI[Q[\LM$ =I[$ [PM [Y\M IaQT\[P VN [PM TVVU$ 5a >$ IUL [PM [Y\M IaQT\[P VN [PM YMNMYMUKM JVL`$ 5a 6& BQUKM [PM KVYYMK[QVUZ IYM ZTISS$ [PM [PYMM ]IS\MZ LV UV[ UMML [V JM ]MY` IKK\YI[M& 9YYVYZ VN I NM^ LMOYMMZ IYM [VSMYIJSM& @IYISSI_ QU IS[Q[\LM2 CPQZ KVYYMK[QVU QZ IWWSQML [V [PM WIYISSI_ QU IS[Q[\LM IUL QZ \ZML [V KISK\SI[M ;> ^Q[P PQOPMY WYMKQZQVU JMNVYM KSMIYQUO [PM S\UIY LQZ[IUKM& 2 ' PM ( f HP M [sin 2 Lat "cos Az M sin H Mapp ! sin Lat cos H Mapp 1 N QZ [PM NSI[[MUQUO VN [PM MIY[P2 f = 298.257 PM , improved = P M ' PM H M = H mapp ! 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(4;0, )0<, *?45423< %070:,5 $87/4<487; 18: +4;4-454<@ 9UX ZNK VRGTTOTM UL UHYKX\GZOUTY# OZ OY [YKL[R ZU QTU] ZNK ZOSKY J[XOTM ]NOIN G IKXZGOT HUJ_ OY GHU\K ZNK NUXO`UT GY ]KRR GY ZNK ZOSKY UL Y[TXOYK# Y[TYKZ# GTJ Z]OROMNZ% 4 HUJ_ IGT HK GR]G_Y GHU\K ZNK NUXO`UT# GR]G_Y HKRU] ZNK NUXO`UT# UX GHU\K ZNK NUXO`UT J[XOTM G VGXZ UL ZNK JG_# JKVKTJOTM UT ZNK UHYKX\KX'Y RGZOZ[JK GTJ ZNK JKIROTGZOUT UL ZNK HUJ_% 4 HUJ_ OY .4:.=6985,: GR]G_Y GHU\K ZNK IKRKYZOGR NUXO`UT! ]NKT ZNK `KTOZN JOYZGTIK OY YSGRRKX ZNGT /&a GZ ZNK SUSKTZ UL 58?0: SKXOJOGT VGYYGMK# O% K%# ]NKT ZNK HUJ_ OY UT ZNK RU]KX HXGTIN UL ZNK RUIGR SKXOJOGT "&% !- # !% DNOY OY ZNK IGYK OL Lat Dec 90 ° 4 HUJ_ OY IUTZOT[GRR_ HKRU] ZNK IKRKYZOGR NUXO`UT ]NKT ZNK `KTOZN JOYZGTIK OY MXKGZKX ZNGT /&a GZ ZNK OTYZGTZ UL =990: SKXOJOGT VGYYGMK "&% !- $ !% DNOY OY ZNK IGYK OL Lat ! Dec 90 ° 4 IKRKYZOGR HUJ_ HKOTM UT ZNK YGSK NKSOYVNKXK GY ZNK UHYKX\KX OY KOZNKX YUSKZOSKY GHU\K ZNK NUXO`UT UX IOXI[SVURGX% 4 HUJ_ HKOTM UT ZNK UVVUYOZK NKSOYVNKXK OY KOZNKX YUSKZOSKY GHU\K ZNK NUXO`UT UX VKXSGTKTZR_ OT\OYOHRK# H[Z TK\KX IOXI[SVURGX% DNK Y[T VXU\OJKY G MUUJ K^GSVRK UL NU] ZNK \OYOHOROZ_ UL G HUJ_ OY GLLKIZKJ H_ RGZOZ[JK GTJ JKIROTGZOUT% 4Z ZNK ZOSK UL ZNK Y[SSKX YURYZOIK 7KI 2 "()%+a!# ZNK Y[T OY IOXI[SVURGX ZU GT UHYKX\KX HKOTM TUXZN UL ZNK ,:.<4. .4:.50 >GZ 3 ",,%+a!% 4Z ZNK YGSK ZOSK# ZNK Y[T XKSGOTY HKRU] ZNK IKRKYZOGR NUXO`UT GRR JG_ OL ZNK UHYKX\KX OY YU[ZN UL ZNK ,7<,:.<4. .4:.50 >GZ 1 −,,%+a!% 4Z ZNK ZOSKY UL ZNK KW[OTU^KY 7KI 2 &a!# ZNK Y[T OY IOXI[SVURGX UTR_ GZ ZNK VURKY% 4Z ZNK ZOSK UL ZNK ]OTZKX YURYZOIK 7KI 2 $()%+a!# ZNK Y[T OY IOXI[SVURGX YU[ZN UL ZNK GTZGXIZOI IOXIRK GTJ OT\OYOHRK TUXZN UL ZNK GXIZOI IOXIRK% (4;0 ,7/ )0< DNK K\KTZY UL XOYK GTJ YKZ IGT HK [YKJ ZU JKZKXSOTK RGZOZ[JK# RUTMOZ[JK# UX ZOSK% @TK YNU[RJ TUZ K^VKIZ \KX_ GII[XGZK XKY[RZY# NU]K\KX# YOTIK GZSUYVNKXOI XKLXGIZOUT SG_ HK KXXGZOI OL ZNK HUJ_ OY UT UX TKGX ZNK NUXO`UT% .$' DNK MKUSKZXOI XOYK UX YKZ UL G HUJ_ UII[XY ]NKT ZNK IKTZKX UL ZNK HUJ_ VGYYKY ZNXU[MN ZNK IKRKYZOGR NUXO`UT ; 2 &a!% 7[K ZU ZNK OTLR[KTIK UL GZSUYVNKXOI XKLXGIZOUT# GRR HUJOKY K^IKVZ ZNK SUUT GVVKGX GHU\K ZNK \OYOHRK GTJ YKTYOHRK NUXO`UT GZ ZNOY OTYZGTZ% DNK SUUT OY TUZ \OYOHRK GZ ZNK SUSKTZ UL NKX MKUSKZXOI XOYK UX YKZ YOTIK ZNK JKVXKYYOTM KLLKIZ UL ZNK NUXO`UTZGR VGXGRRG^ ∼'a! OY MXKGZKX ZNGT ZNK KRK\GZOTM KLLKIZ UL GZSUYVNKXOI XKLXGIZOUT% DNK GVVXU^OSGZK GVVGXKTZ GRZOZ[JKY XKLKXXOTM ZU ZNK YKTYOHRK NUXO`UT! GZ ZNK SUSKTZ UL ZNK GYZXUTUSOIGR XOYK UX YKZ GXK0 C[T RU]KX ROSH!0 '+' CZGXY0 (/' ARGTKZY0 (/' − ;A FNKT SKGY[XOTM ZNKYK GRZOZ[JKY ]OZN XKLKXKTIK ZU ZNK YKG NUXO`UT# ]K NG\K ZU GJJ ZNK JOV UL NUXO`UT INGVZKX (! ZU ZNK GHU\K \GR[KY% 9UX K^GSVRK# ZNK GRZOZ[JK UL ZNK RU]KX ROSH UL ZNK XOYOTM UX YKZZOTM Y[T OY GVVXU^% (&' OL ZNK NKOMNZ UL K_K OY .S% FK HKMOT ]OZN ZNK ]KRR$QTU]T GRZOZ[JK LUXS[RG INGVZKX *!% sin H = sin Lat "sin Dec cos Lat "cos Dec "cos t = 0 sin Lat "sin Dec cos t = ! cos Lat "cos Dec CUR\OTM ZNK KW[GZOUT LUX ZNK SKXOJOGT GTMRK# Z# ]K MKZ 0 t = arccos #! tan Lat "tan Dec $ DNK KW[GZOUT NGY TU YUR[ZOUT OL ZNK GXM[SKTZ UL ZNK OT\KXYK IUYOTK OY YSGRRKX ZNGT −' UX MXKGZKX ZNGT '% 7[K ZU ZNK GSHOM[OZ_ UL ZNK GXIIUY L[TIZOUT# ZNK KW[GZOUT NGY Z]U YUR[ZOUTY# UTK LUX XOYK GTJ UTK LUX YKZ% 9UX ZNK IGRI[RGZOUTY HKRU]# ]K NG\K ZU UHYKX\K ZNK LURRU]OTM X[RKY0 Lon = !GHA # t :;4 OY ZNK :XKKT]OIN NU[X GTMRK UL ZNK HUJ_ GZ ZNK SUSKTZ UL XOYK UX YKZ% DNK YOMT UL Z NGY ZU HK UHYKX\KJ IGXKL[RR_ YKK GHU\K!% =TU]OTM U[X VUYOZOUT# ]K IGT IGRI[RGZK ZNK ZOSKY UL Y[TXOYK GTJ Y[TYKZ0 Lon [° t [° UT = 12 ! EoT ! # Sunrise , Sunset 15 15 DNK ZOSKY UL Y[TXOYK GTJ Y[TYKZ UHZGOTKJ ]OZN ZNK GHU\K LUXS[RG GXK TUZ W[OZK GII[XGZK YOTIK 7KI GTJ 8UD GXK \GXOGHRK% COTIK ]K JU TUZ QTU] ZNK K^GIZ ZOSK UL XOYK UX YKZ GZ ZNK HKMOTTOTM# ]K NG\K ZU [YK KYZOSGZKJ \GR[KY LUX 7KI GTJ 8UD OTOZOGRR_% DNK ZOSK UL XOYK UX YKZ IGT HK OSVXU\KJ H_ OZKXGZOUT XKVKGZOTM ZNK IGRI[RGZOUTY ]OZN 7KI GTJ 8UD GZ ZNK IGRI[RGZKJ ZOSK UL XOYK UX YKZ!% 9[XZNKX# ZNK ZOSKY ZN[Y IGRI[RGZKJ GXK OTLR[KTIKJ H_ ZNK OXXKM[RGXOZOKY UL GZSUYVNKXOI XKLXGIZOUT TKGX ZNK NUXO`UT% DNKXKLUXK# G ZOSK KXXUX UL ±( SOT[ZKY OY TUZ [T[Y[GR% 4IIUXJOTMR_# ]K IGT IGRI[RGZK U[X RUTMOZ[JK LXUS ZNK ZOSK UL Y[TXOYK UX Y[TYKZ OL ]K QTU] U[X RGZOZ[JK0 15 12 Lon [° = "# ! GMT Sunrise , Sunset ! EoT $ # t [° .$( 4MGOT# ZNOY OY TUZ G \KX_ VXKIOYK SKZNUJ# GTJ GT KXXUX UL YK\KXGR GXISOT[ZKY OT RUTMOZ[JK OY TUZ [TROQKR_% =TU]OTM U[X RUTMOZ[JK# ]K GXK GHRK ZU JKZKXSOTK U[X GVVXU^OSGZK RGZOZ[JK LXUS ZNK ZOSK UL Y[TXOYK UX Y[TYKZ0 15 12 t [° = Lon [° ! "# ! GMT Sunrise, Sunset ! EoT $ cos t Lat = arctan ! # tan Dec $ &7 7,>42,<487, :4;0 ,7/ ;0< ,:0 /01470/ ,; <30 68607<; ?307 <30 =990: 546- 81 , -8/@ 4; 87 <30 >4;4-50 38:4A87 DNKYK K\KTZY IGT HK UHYKX\KJ ]OZNU[Z G YK^ZGTZ% ?U]# ]K NG\K ZU ZGQK OTZU GIIU[TZ ZNK KLLKIZY UL XKLXGIZOUT# NUXO`UTZGR VGXGRRG^# JOV# GTJ YKSOJOGSKZKX% DNKYK W[GTZOZOKY JKZKXSOTK ZNK GRZOZ[JK ;U! UL G HUJ_ ]OZN XKYVKIZ ZU ZNK IKRKYZOGR NUXO`UT GZ ZNK OTYZGTZ UL ZNK \OYOHRK XOYK UX YKZ% sin Ho ! sin Lat "sin Dec t = arccos cos Lat "cos Dec Ho = HP ! SD ! RH ! Dip 5_ JKLOTOZOUT# ZNK YZGTJGXJ XKLXGIZOUT LUX G HUJ_ HKOTM UT ZNK YKTYOHRK NUXO`UT# B;# OY )*' OT XKGROZ_ OZ OY Y[HPKIZ ZU XGTJUS \GXOGZOUTY!% FNKT UHYKX\OTM ZNK [VVKX ROSH UL ZNK Y[T# ]K MKZ0 Ho = 0.15 ' ! 16 ' ! 34 ' ! Dip %!50 ' ! Dip ;U OY TKMGZO\K% #A46=<3 ,7/ #6954<=/0 DNK G`OS[ZN GTMRK UL G XOYOTM UX YKZZOTM HUJ_ OY IGRI[RGZKJ ]OZN ZNK G`OS[ZN LUXS[RG YKK INGVZKX *!0 sin Dec ! sin H "sin Lat Az = arccos cos H "cos Lat FOZN ;2&# ]K MKZ0 sin Dec Az = arccos cos Lat 4` OY "/&a XOYK! GTJ −/&a YKZ! OL ZNK JKIROTGZOUT UL ZNK HUJ_ OY `KXU# XKMGXJRKYY UL ZNK UHYKX\KX'Y RGZOZ[JK% 4IIUXJOTMR_# ZNK Y[T XOYKY OT ZNK KGYZ GTJ YKZY OT ZNK ]KYZ GZ ZNK ZOSKY UL ZNK KW[OTU^KY MKUSKZXOI XOYK GTJ YKZ!% − FOZN ; IKTZKX 2 +&' [VVKX ROSH UL ZNK Y[T UT ZNK YKTYOHRK NUXO`UT!# ]K NG\K0 sin Dec 0.0145 "sin Lat Az = arccos 0.9999 "cos Lat DNK ZX[K G`OS[ZN UL ZNK XOYOTM UX YKZZOTM HUJ_ OY0 if 0 Az = Az t & N !360 °! Az if t 0 " DNK G`OS[ZN UL G HUJ_ GZ ZNK SUSKTZ UL XOYK UX YKZ IGT HK [YKJ ZU LOTJ ZNK SGMTKZOI JKIROTGZOUT GZ ZNK UHYKX\KX'Y VUYOZOUT IUSVGXK ]OZN INGVZKX ')!% .$) DNK NUXO`UTZGR GTM[RGX JOYZGTIK UL G XOYOTM UX YKZZOTM HUJ_ LXUS ZNK KGYZ XOYOTM! UX ]KYZ YKZZOTM! VUOTZ UT ZNK NUXO`UT OY IGRRKJ ,6954<=/0 GTJ IGT HK IGRI[RGZKJ LXUS ZNK G`OS[ZN% 4T GSVROZ[JK UL 8*+a?# LUX OTYZGTIK# SKGTY ZNGZ ZNK HUJ_ XOYKY *+a TUXZN UL ZNK KGYZ VUOTZ UT ZNK NUXO`UT% *?45423< 4Z YKG# Z]OROMNZ OY OSVUXZGTZ LUX ZNK UHYKX\GZOUT UL YZGXY GTJ VRGTKZY YOTIK OZ OY ZNK UTR_ ZOSK ]NKT ZNKYK HUJOKY ,7/ ZNK NUXO`UT GXK \OYOHRK% 5_ JKLOTOZOUT# ZNKXK GXK ZNXKK QOTJY UL Z]OROMNZ% DNK GRZOZ[JK# ;# XKLKXY ZU ZNK IKTZKX UL ZNK Y[T GTJ ZNK IKRKYZOGR NUXO`UT GTJ SGXQY ZNK HKMOTTOTM SUXTOTM! GTJ ZNK KTJ K\KTOTM! UL ZNK XKYVKIZO\K Z]OROMNZ% $4>45 ',=<4.,5 #;<:87864.,5 DNK SKXOJOGT GTMRK LUX ZNK Y[T GZ −,a GRZOZ[JK IKTZKX! 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( %; * ANK LUXS[RG LUX ZNK VGXGRRG^ OT G`OS[ZN OY GRYU GII[XGZK ZU G LXGIZOUT UL GT GXIYKIUTJ% :Z HKIUSKY RKYY GII[XGZK GY ZNK GRZOZ[JK GVVXUGINKY 0'c% >HYKX\OTM HUJOKY ]OZN Y[IN GRZOZ[JKY# NU]K\KX# OY JOLLOI[RZ GTJ [Y[GRR_ G\UOJKJ% -97=@=CA ;5B8=4! 7UX K\KT SUXK GII[XGZK XKY[RZY# ]K [YK ZNK B=>=35 ANK ZUVUIKTZXOI KW[GZUXOGR IUUXJOTGZKY UL G IKRKYZOGR HUJ_ GXK UHZGOTKJ LXUS ZNK MKUIKTZXOI UTKY ZNXU[MN IUUXJOTGZK ZXGTYLUXSGZOUT% ANK MO\KT W[GTZOZOKY GXK1 8KUMXGVNOI RGZOZ[JK UL ZNK UHYKX\KX ;GZ 8KUIKTZXOI SKXOJOGT GTMRK Z 8KUIKTZXOI JKIROTGZOUT 5KI 6W[GZUXOGR NUXO`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r Lat ' = arctan p !tan Lat = arctan [ 1 f 2 !tan Lat [ re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sin A1 sin A2 sin A3 = = sin s1 sin s2 sin s3 '.B :3 0:>692> 3:= >612># cos s1 = cos s2 cos s3 sin s2 sin s3cos A1 cos s2 = cos s1 cos s3 sin s1 sin s3 cos A2 cos s3 = cos s1 cos s2 sin s1 sin s2cos A3 '.B :3 0:>692> 3:= .9472># cos A1 = cos A2cos A3 sin A2 sin A3 cos s1 cos A2 = cos A1cos A3 sin A1sin A3 cos s2 cos A3 = cos A1cos A2 sin A1sin A2 cos s3 &%#& '.B :3 >692> .91 0:>692># F F F F F F '.B :3 0:?.9429?># F F F F F F (.;62='> .9.7:462># s s s s cos 1 2 sin 1 2 A1 A2 A3 2 A1 A2 A3 2 tan tan = tan tan = 2 2 s s 2 2 s s cos 1 2 sin 1 2 2 2 s s A A s s A A tan 1 2 cos 1 2 tan 1 2 sin 1 2 2 2 2 2 = = s A A s A A tan 3 cos 1 2 tan 3 sin 1 2 2 2 2 2 &.@>>' 3:=8@7.># A A s s A A s s sin 1 2 cos 1 2 cos 1 2 cos 1 2 2 2 2 2 A = s A = s cos 3 cos 3 sin 3 cos 3 2 2 2 2 A A s s A A s s sin 1 2 sin 1 2 cos 1 2 sin 1 2 2 2 2 2 A = s A = s cos 3 sin 3 sin 3 sin 3 2 2 2 2 1;8F8 9BE@H?4F 4A7 BG;8EF 78E *.=?60@7.=7C ?52 7.B :3 0:>692> 3:= >612> 6> :3 69?2=2>? ?: ?52 9.A64.?:= &%#' -52 +645? ,;52=60.7 -=6.9472 0B?I 38 4EE4A:8 G;8 F<78F 9BE@ +66BE7 sin s1 = tan s2tan !90 ° A2 " = cos !90 ° A1"cos !90 ° s3 " sin s2 = tan !90 ° A1"tan s1 = cos !90 ° s3 "cos !90 ° A2 " sin !90 ° A1 " = tan !90 ° s3"tan s2 = cos !90 ° A2 "cos s1 sin 90 ° s tan 90 ° A tan 90 ° A cos s cos s ! 3 " = ! 2 " ! 1" = 1 2 sin !90 ° A2 " = tan s1tan !90 ° s3" = cos s2 cos !90 ° A1" -A 4 F<@C?8E 9BE@" G;8F8 8DH4G sin s1 = tan s2 cot A2 = sin A1 sin s3 sin s2 = cot A1 tan s1 = sin s3 sin A2 cos A1 = cot s3 tan s2 = sin A2 cos s1 cos s cot A cot A cos s cos s 3 = 2 1 = 1 2 cos A2 = tan s1 cot s3 = cos s2 sin A1 1;8E8 4E8 F8I8E4? 4CC?<64G &%#( %1*7;.9 ## )1. (*=20*;265*3 )92*503. CNK 5*=20*;265*3 5*<;2,*3! ;92*503. OY ZNK [Y[GRR_! UHROW[K YVNKXOIGR ZXOGTMRK LUXSKJ H_ ZNK TUXZN VURK"@>" ZNK UHYKX\KX'Y GYY[SKJ VUYOZOUT" 4@" GTJ ZNK MKUMXGVNOI VUYOZOUT UL ZNK IKRKYZOGR UHPKIZ" 8@ %)(! "" " !$ $33 ,64465 :201; 9.-<,;265 796,.-<9.: *9. +*:.- <765 ;1. 5*=20*;265*3 ;92*503." ENKT [YOTM ZNK OTZKXIKVZ SKZNUJ" ZNK RGZOZ[JK UL ZNK GYY[SKJ VUYOZOUT" CNK LOXYZ YZKV OY IGRI[RGZOTM ZNK YOJK ` UL ZNK TG\OMGZOUTGR ZXOGTMRK H_ [YOTM ZNK RG] UL IUYOTKY LUX YOJKY0 cos z = cos 90 ° Lat AP !cos 90 ° Dec " sin 90 ° Lat AP !sin 90 ° Dec !cos t BOTIK IUY /&b#^! KW[GRY YOT ^ GTJ \OIK \KXYG" ZNK KW[GZOUT IGT HK ]XOZZKT OT G YOSVRKX LUXS0 cos z = sin Lat AP !sin Dec " cos Lat AP !cos Dec !cos t CNK YOJK ` OY TUZ UTR_ ZNK 09.*; ,29,3. -2:;*5,. HKZ]KKT 4@ GTJ 8@ H[Z GRYU ZNK @.52;1 -2:;*5,. UL ZNK IKRKYZOGR UHPKIZ GTJ ZNK XGJO[Y UL ZNK ,29,3. 6/ .8<*3 *3;2;<-. YKK INGVZKX '!$ B[HYZOZ[ZOTM ZNK GRZOZ[JK 9 LUX `" ]K MKZ0 sin H = sin Lat AP !sin Dec " cos Lat AP !cos Dec !cos t BUR\OTM ZNK KW[GZOUT LUX 9 RKGJY ZU ZNK GRZOZ[JK LUXS[RG QTU]T LXUS INGVZKX *0 H = arcsin sin Lat AP !sin Dec " cos Lat AP !cos Dec !cos t CNK GRZOZ[JK ZN[Y UHZGOTKJ LUX G MO\KT VUYOZOUT OY IGRRKJ ,647<;.- *3;2;<-. " ', $ ''#' CNK G`OS[ZN GTMRK UL ZNK UHYKX\KJ HUJ_ OY GRYU IGRI[RGZKJ H_ SKGTY UL ZNK RG] UL IUYOTKY LUX YOJKY0 cos 90 ° Dec = cos 90 ° Lat AP !cos z " sin 90 ° Lat AP !sin z!cos Az sin Dec = sin Lat AP !cos z " cos Lat AP !sin z!cos Az DYOTM ZNK IUSV[ZKJ GRZOZ[JK OTYZKGJ UL ZNK `KTOZN JOYZGTIK XKY[RZY OT ZNK LURRU]OTM KW[GZOUT0 sin Dec = sin Lat AP !sin Hc " cos Lat AP !cos Hc !cos Az BUR\OTM ZNK KW[GZOUT LUX 4` LOTGRR_ _OKRJY ZNK G`OS[ZN LUXS[RG LXUS INGVZKX *0 sin Dec sin Lat !sin Hc Az = arccos AP cos Lat AP !cos Hc CNK GXIIUY L[TIZOUT XKZ[XTY GTMRKY HKZ]KKT &b GTJ '.&b$ CNKXKLUXK" ZNK XKY[RZOTM G`OS[ZN GTMRK OY TUZ TKIKYYGXOR_ OJKTZOIGR ]OZN ZNK ZX[K G`OS[ZN" 4` > &b$$$ ),&b" SKGY[XKJ IRUIQ]OYK LXUS ZX[K TUXZN! IUSSUTR_ [YKJ OT TG\OMGZOUT$ :T GRR IGYKY ]NKXK Z OY TKMGZO\K 8@ KGYZ UL 4@! " 4` > KW[GRY 4`$ ?ZNKX]OYK Z VUYOZO\K" 8@ ]KYZ]GXJ LXUS 4@ GY YNU]T OT %)(! "" " !" 4` > OY UHZGOTKJ H_ Y[HZXGIZOTM 4` LXUS ),&b$ ENKT ZNK SKXOJOGT GTMRK" Z" UX ZNK RUIGR NU[X GTMRK" <94! OY ZNK W[GTZOZ_ ZU HK IGRI[RGZKJ ZOSK YOMNZ" &.*+','- SKZNUJ!" 6KI" 4MGOT" ZNK RG] UL IUYOTKY LUX YOJKY OY GVVROKJ0 cos z = cos 90 ° Lat AP !cos 90 ° Dec " sin 90 ° Lat AP !sin 90 ° Dec !cos t sin H = sin Lat AP !sin Dec " cos Lat AP !cos Dec !cos t sin H sin Lat !sin Dec cos t = AP cos Lat AP !cos Dec sin H sin Lat !sin Dec t = arccos AP cos Lat AP !cos Dec CNK UHZGOTKJ SKXOJOGT GTMRK" Z UX <94!" OY ZNKT [YKJ GY JKYIXOHKJ OT INGVZKX * GTJ INGVZKX +$ ENKT UHYKX\OTM G IKRKYZOGR HUJ_ GZ ZNK ZOSK UL SKXOJOGT VGYYGMK K$ M$" LUX JKZKXSOTOTM UTK'Y RGZOZ[JK!" ZNK RUIGR NU[X GTMRK OY `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` UX 9I! GTJ 4`$ ; OY ZNK MKUMXGVNOI RGZOZ[JK UL F$ 5UZN ZXOGTMRKY GXK YUR\KJ [YOTM (*72.9': 9<3.: 6/ ,29,<3*9 7*9;: YKK INGVZKX /!$ %)(! "" $ ORR[YZXGZKY ZNK IUXXKYVUTJOTM IOXI[RGX JOGMXGSY0 4IIUXJOTM ZU >GVOKX'Y X[RKY" 9I GTJ 4` GXK IGRI[RGZKJ H_ SKGTY UL ZNK LURRU]OTM LUXS[RGY0 sin R = sin t !cos Dec # R = arcsin sin t !cos Dec sin Dec sin Dec sin Dec = cos R!sin K # sin K = # K = arcsin cos R cos R B[HYZOZ[ZK '.&b −; LUX ; OT ZNK LURRU]OTM KW[GZOUT OL Z 3 /&b UX /&b 1 <94 1 (-&b!$ sin Hc = cos R !cos K Lat AP # Hc = arcsin [ cos R!cos K Lat AP sin R sin R sin R = cos Hc !sin Az # sin Az = # Az = arcsin cos Hc cos Hc 7UX L[XZNKX IGRI[RGZOUTY" Y[HYZOZ[ZK '.&b −4` LUX 4` OL ; GTJ ''#) CU UHZGOT ZNK ZX[K G`OS[ZN" 4` > &b$$$ ),&b!" ZNK LURRU]OTM X[RKY NG\K ZU HK GVVROKJ0 Az if Lat AP $ 0 N AND t % 0 180 °% LHA % 360 ° Az N = 360 ° Az if Lat AP $ 0 N AND t $ 0 0°% LHA % 180 ° ° Az Lat S ! 180 " if AP % 0 " CNK JO\OJKJ TG\OMGZOUTGR ZXOGTMRK OY UL IUTYOJKXGHRK OSVUXZGTIK YOTIK OZ LUXSY ZNK ZNKUXKZOIGR HGIQMXU[TJ LUX G T[SHKX UL :201; 9.-<,;265 ;*+3.: " K$M$" ZNK 4MKZUT CGHRKY YKK HKRU]!$ :Z OY GRYU [YKJ LUX 09.*; ,29,3. 5*=20*;265 INGVZKX '(!$ DYOTM ZNK YKIGTZ GTJ IUYKIGTZ L[TIZOUT YKI ^ 2 '%IUY ^" IYI ^ 2 '%YOT ^!" ]K IGT ]XOZK ZNK KW[GZOUTY LUX ZNK JO\OJKJ TG\OMGZOUTGR ZXOGTMRK OT ZNK LURRU]OTM LUXS0 csc R = csc t !sec Dec csc Dec csc K = sec R B[HYZOZ[ZK '.&b −; LUX ; OT ZNK LURRU]OTM KW[GZOUT OL Z 3 /&b0 csc Hc = sec R!sec K Lat csc R csc Az = sec Hc B[HYZOZ[ZK '.&b −4` LUX 4` OL ; GTJ :T RUMGXOZNSOI LUXS" ZNKYK KW[GZOUTY GXK YZGZKJ GY0 log csc R = log csc t " log sec Dec log csc K = log csc Dec log sec R log csc Hc = log sec R " log sec K Lat log csc Az = log csc R log sec Hc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` HYL PUKPZWLUZPISL ILJH\ZL [OL` HYL ULJLZZHY` [V JHSJ\SH[L KPZ[HUJL HUK KPYLJ[PVU JV\YZL! MYVT [OL HGBFK G? =>H:IKLI> # 3# [V [OL HGBFK G? :IIBM:D # 4# HZ ^LSS HZ [V JHSJ\SH[L [OL WVZP[PVU VM 4 MYVT [OL WVZP[PVU VM 3 PM JV\YZL HUK KPZ[HUJL HYL RUV^U% COL KIL> 8ALE; 4BF> 6:MB@:KBGF 3 IALE; DBF> # HSZV JHSSLK DGOG=IGE> # PZ H SPUL VU [OL Z\YMHJL VM [OL LHY[O PU[LYZLJ[PUN HSS TLYPKPHUZ H[ H JVUZ[HU[ HUNSL# 5% CO\Z# H YO\TI SPUL PZ YLWYLZLU[LK I` H Z[YHPNO[ SPUL VU H =LYJHY[VY JOHY[ ZLL JOHW[LY (*! ^OPJO THRLZ MGP:@> HD:FFBF@ X\P[L ZPTWSL% ?U H NSVIL# H YO\TI SPUL MVYTZ H ZWOLYPJHS ZWPYHS L_[LUKPUN MYVT WVSL [V WVSL \USLZZ P[ PZ PKLU[PJHS ^P[O H TLYPKPHU 5 2 'b VY (/'b! VY H WHYHSSLS VM SH[P[\KL 5 2 0'b VY ).'b!% 3 ]LZZLS Z[LLYPUN H JVUZ[HU[ JV\YZL [YH]LSZ HSVUN H YO\TI SPUL# WYV]PKLK [OLYL PZ UV KYPM[% AO\TI SPUL JV\YZL# 5# HUK KPZ[HUJL# K# HYL JHSJ\SH[LK HZ ZOV^U ILSV^% 7PYZ[# ^L PTHNPUL [YH]LSPUN [OL PUMPUP[LZPTHS KPZ[HUJL K_ MYVT [OL WVPU[ VM KLWHY[\YL# 3# [V [OL WVPU[ VM HYYP]HS# 4% ?\Y JV\YZL PZ 5 #%$ !"-! !1 COL KPZ[HUJL# K_# PZ [OL ]LJ[VY Z\T VM H UVY[O$ZV\[O JVTWVULU[# K d Lat 1 = d Lon cos Lat tan C :M [OL KPZ[HUJL IL[^LLU 3 KLMPULK I` Lat B Lon B d Lat 1 = d Lon cos Lat tan C Lat A Lon A Lat B " Lat A " Lon B $ Lon A ln tan ! $ ln tan ! = [ 2 4 # [ 2 4 # tan C ()$( Lon B $ Lon A tan C = Lat B " tan ! 2 4 # ln Lat A " tan ! 2 4 # =LHZ\YPUN HUNSLZ PU KLNYLLZ HUK ZVS]PUN MVY 5# ^L NL[1 Lon B $ Lon A C = arctan Lat B tan ! 45 ° 2 # ln Lat A tan ! 45 ° 2 # 9A> K>IE 4GF .#4GF - A:J KG ;> BF KA> I:F@> ;>KN>>F Q&*%R K:F= !&*%R$ 3? BK BJ GLKJB=> KABJ I:F@>" N> A:M> KG :== GI JL;KI: 9A> :I C if Lat B & Lat A AND Lon B & Lon A C % 360 °! C if Lat B & Lat A AND Lon B ' Lon A ° C Lat Lat !180 ! if B ' A " CV MPUK [OL [V[HS SLUN[O VM [OL YO\TI SPUL [YHJR# ^L JHSJ\SH[L [OL PUMPUP[LZPTHS KPZ[HUJL K_1 d Lat dx = cos C COL [V[HS SLUN[O K PZ MV\UK [OYV\NO PU[LNYH[PVU1 Lat B 1 Lat B $ Lat A d = d Lat = cos C cos C Lat A 7PUHSS`# ^L NL[1 40031.6 Lat B $ Lat A Lat B $ Lat A d [km = d [nm =60 360 cos C cos C :M IV[O WVZP[PVUZ OH]L [OL ZHTL SH[P[\KL# [OL KPZ[HUJL JHU FGK IL JHSJ\SH[LK \ZPUN [OL HIV]L MVYT\SHZ% :U [OPZ JHZL# [OL MVSSV^PUN MVYT\SHZ HWWS` 5 PZ LP[OLY 0'b VY ).'b!1 40031.6 d [km = Lon $ Lon #cos Lat d[nm =60 Lon $ Lon #cos Lat 360 B A B A ()$) 2I>:K /BI 2I>:K d Lat Lat Lat Lat Lon AB = arccos sin Asin B ! cos Acos Bcos ( AB # ( Lon AB = Lon B $ Lon A >VY[OLYU SH[P[\KL HUK LHZ[LYU SVUNP[\KL HYL WVZP[P]L# ZV\[OLYU SH[P[\KL HUK ^LZ[LYU SVUNP[\KL ULNH[P]L% 3 NYLH[ JPYJSL KPZ[HUJL OHZ [OL KPTLUZPVU VM HU HUNSL% CV TLHZ\YL P[ PU KPZ[HUJL \UP[Z# ^L T\S[PWS` P[ I` +''*(%-&*-' KPZ[HUJL PU RT! VY I` -' KPZ[HUJL PU UT!% sin Lat B $ sin Lat Acos d AB CA = arccos cos Lat Asin d AB :M [OL [LYT ZPU :U #%$ !"-" # 53 PZ [OL BFBKB:D @I>:K COLVYL[PJHSS`# ^L OH]L [V HKQ\Z[ [OL JV\YZL JVU[PU\HSS`% COPZ PZ WVZZPISL ^P[O [OL HPK VM UH]PNH[PVU JVTW\[LYZ HUK H\[VWPSV[Z% :M Z\JO TLHUZ HYL UV[ H]HPSHISL# ^L OH]L [V JHSJ\SH[L HU \WKH[LK JV\YZL H[ JLY[HPU PU[LY]HSZ ZLL ILSV^!% 8YLH[ JPYJSL UH]PNH[PVU YLX\PYLZ TVYL JHYLM\S ]V`HNL WSHUUPUN [OHU YO\TI SPUL UH]PNH[PVU% ?U H =LYJH[VY JOHY[ ZLL JOHW[LY (*!# H NYLH[ JPYJSL [YHJR HWWLHYZ HZ H SPUL ILU[ [V^HYKZ [OL LX\H[VY% 3Z H YLZ\S[# [OL UH]PNH[VY TH` ULLK TVYL PUMVYTH[PVU HIV\[ [OL PU[LUKLK NYLH[ JPYJSL [YHJR PU VYKLY [V ]LYPM` PM P[ SLHKZ [OYV\NO UH]PNHISL HYLHZ% FP[O [OL L_JLW[PVU VM [OL LX\H[VY# L]LY` NYLH[ JPYJSL OHZ [^V M>IKB<>J # [OL WVPU[Z MHY[OLZ[ MYVT [OL LX\H[VY% COL ]LY[PJLZ OH]L [OL ZHTL HIZVS\[L ]HS\L VM SH[P[\KL ^P[O VWWVZP[L ZPNU! I\[ HYL (/'b HWHY[ PU SVUNP[\KL% 3[ LHJO M>IK>O HSZV JHSSLK :H>O !# [OL NYLH[ JPYJSL PZ [HUNLU[ [V H WHYHSSLS VM SH[P[\KL# HUK 5 PZ LP[OLY 0'b VY ).'b JVZ 5 2 '!% CO\Z# ^L OH]L H YPNO[ ZWOLYPJHS [YPHUNSL MVYTLK I` [OL UVY[O WVSL# @ ># [OL ]LY[L_#E# HUK [OL WVPU[ VM KLWHY[\YL# 3 7PN% ()$*!1 ()$* CV KLYP]L [OL MVYT\SHZ ULLKLK MVY [OL MVSSV^PUN JHSJ\SH[PVUZ# ^L \ZL >HWPLY'Z Y\SLZ VM JPYJ\SHY WHY[Z 7PN% ()$+!% COL YPNO[ HUNSL PZ H[ [OL IV[[VT VM [OL JPYJ\SHY KPHNYHT% COL MP]L WHY[Z HYL HYYHUNLK JSVJR^PZL% 7PYZ[# ^L ULLK [OL SH[P[\KL VM [OL ]LY[L_# cos Lat V = sin CAcos Lat A BVS]PUN MVY COL HIZVS\[L ]HS\L VM ZPU 53 PZ \ZLK [V THRL Z\YL [OH[ ZUN _! PZ [OL JB@FLE M\UJ[PVU1 $1 if x'0 sgn x# = 0 if x=0 !!1 if x&0 " :M E PZ SVJH[LK IL[^LLU 3 HUK 4 SPRL ZOV^U PU 7PN% ()$*!# V\Y SH[P[\KL WHZZLZ [OYV\NO HU L_[YLT\T H[ [OL PUZ[HU[ ^L YLHJO E% COPZ KVLZ UV[ OHWWLU PM 4 PZ IL[^LLU 3 HUK E% ()$+ ;UV^PUN tan Lat A cos ( Lon AV = ( Lon AV = ( Lon V $ ( Lon A tan Lat V BVS]PUN MVY COL SVUNP[\KL VM E PZ tan Lat Lon Lon C A V = A ! sgn sin A#arccos tan Lat V 3KK VY Z\I[YHJ[ *-'b PM ULJLZZHY`%! COL [LYT ZUN ZPU 5 3! PU [OL HIV]L MVYT\SH WYV]PKLZ HU H\[VTH[PJ JVYYLJ[PVU MVY [OL ZPNU VM ;UV^PUN [OL WVZP[PVU VM E KLMPULK I` tan Lat X = cos ( Lon XV tan Lat V ( Lon XV = Lon V $ Lon X Lat Lon Lat X = arctan cos ( XV tan V # 7\Y[OLY# ^L JHU JHSJ\SH[L [OL JV\YZL H[ [OL WVPU[ G1 cos C X = sin ( Lon XV sin Lat V arccos sin ( Lon sin Lat if sin C & 0 C = XV V # A X Lon Lat ° C ! arccos sin ( XV sin V # ! 180 if sin A' 0 " 3S[LYUH[P]LS`# 5 G JHU IL JHSJ\SH[LK MYVT [OL VISPX\L ZWOLYPJHS [YPHUNSL MVYTLK I` G# @ ># HUK 4% COL HIV]L MVYT\SHZ LUHISL \Z [V LZ[HISPZO Z\P[HIS` ZWHJLK N:PHGBFKJ VU [OL NYLH[ JPYJSL HUK JVUULJ[ [OLT I` Z[YHPNO[ SPULZ VU [OL =LYJH[VY JOHY[% COL ZLYPLZ VM D>@J [O\Z VI[HPULK# LHJO VUL ILPUN H YO\TI SPUL [YHJR# PZ H WYHJ[PJHS HWWYV_PTH[PVU VM [OL PU[LUKLK NYLH[ JPYJSL [YHJR% 7\Y[OLY# ^L HYL UV^ HISL [V ZLL ILMVYLOHUK PM [OLYL HYL VIZ[HJSLZ PU V\Y ^H`% 5>:F D:KBKL=> 4LJH\ZL VM [OLPY ZPTWSPJP[`# [OL E>:F D:KBKL=> MVYT\SHZ HYL VM[LU \ZLK PU L]LY`KH` UH]PNH[PVU% =LHU SH[P[\KL PZ H NVVK HWWYV_PTH[PVU MVY YO\TI SPUL UH]PNH[PVU MVY ZOVY[ HUK TLKP\T KPZ[HUJLZ IL[^LLU 3 HUK 4% COL TL[OVK PZ SLZZ Z\P[HISL MVY WVSHY YLNPVUZ JVU]LYNLUJL VM TLYPKPHUZ!% 5V\YZL1 Lon B$ Lon A Lat A ! Lat B C = arctan cos Lat Lat = M Lat Lat M B$ A # 2 COL [Y\L JV\YZL PZ VI[HPULK I` HWWS`PUN [OL ZHTL Y\SLZ [V 5 HZ [V [OL YO\TI SPUL JV\YZL ZLL HIV]L!% 6PZ[HUJL1 40031.6 Lat B $ Lat A Lat B $ Lat A d [km = d [nm = 60 360 cos C cos C ()$, :M 5 2 0'b VY 5 2 ).'b# ^L OH]L \ZL [OL MVSSV^PUN MVYT\SHZ1 40031.6 d [km = Lon $ Lon cos Lat d [nm = 60 Lon $ Lon cos Lat 360 B A# B A# 0>:= 8> 0>:= I> BPUJL H 6A@ PZ VUS` HU HWWYV_PTH[L WVZP[PVU K\L [V [OL PUMS\LUJL VM KYPM[# L[J%!# [OL TLHU SH[P[\KL TL[OVK ZLL HIV]L! WYV]PKLZ Z\MMPJPLU[ HJJ\YHJ`% ?U SHUK# KLHK YLJRVUPUN PZ VM SPTP[LK \ZL ZPUJL P[ PZ \Z\HSS` UV[ WVZZPISL [V Z[LLY H JVUZ[HU[ JV\YZL HWHY[ MYVT KYP]PUN PU SHYNL# LU[PYLS` MSH[ HYLHZ SPRL# L%N%# ZHS[ MSH[Z!% 3[ ZLH# [OL 6A@ PZ ULLKLK [V JOVVZL H Z\P[HISL 3@ MVY [OL PU[LYJLW[ TL[OVK% :M JLSLZ[PHS VIZLY]H[PVUZ HYL UV[ WVZZPISL HUK LSLJ[YVUPJ UH]PNH[PVU HPKZ UV[ H]HPSHISL# KLHK YLJRVUPUN TH` IL [OL VUS` ^H` VM RLLWPUN [YHJR VM VUL'Z WVZP[PVU% 3WHY[ MYVT [OL ]LY` ZPTWSL NYHWOPJ ZVS\[PVU# [OLYL HYL [^V MVYT\SHZ MVY [OL JHSJ\SH[PVU VM [OL 6A@% 5HSJ\SH[PVU VM UL^ SH[P[\KL1 360 d[nm cos C Lat [° = Lat [° ! d [ km cos C Lat [° = Lat [° ! B A 40031.6 B A 60 5HSJ\SH[PVU VM UL^ SVUNP[\KL1 360 d[km sin C d[ nm sin C Lon B[° = Lon A[° ! Lon B [° = Lon A[° ! 40031.6 cos Lat M 60 cos Lat M :M [OL YLZ\S[PUN SVUNP[\KL PZ NYLH[LY [OHU "(/'b# ^L Z\I[YHJ[ *-'b% :M P[ PZ ZTHSSLY [OHU $(/'b# ^L HKK *-'b% :M V\Y TV]LTLU[ PZ JVTWVZLK VM ZL]LYHS JVTWVULU[Z PUJS\KPUN KYPM[# L[J%!# ^L OH]L [V YLWSHJL [OL [LYTZ KJVZ 5 HUK KZPU 5 ^P[O d C d C * i cos i and * i sin i # YLZWLJ[P]LS`% ()$- ".'59+7 ! ".'798 '3* %1499/3- &.++98 $+7)'947 ".'798 4N ORDER TO CONSTRUCT A 7ERCATOR CHART" WE HAVE TO REMEMBER HOW THE GRID PRINTED ON A GLOBE LOOKS$ /T THE EQUATOR" AN AREA OF" E$ G$" ( BY ( DEGREES LOOKS ALMOST LIKE A SQUARE" BUT IT APPEARS AS A NARROW TRAPEZOID WHEN WE PLACE IT NEAR ONE OF THE POLES$ >HILE THE DISTANCE BETWEEN TWO ADJACENT PARALLELS OF LATITUDE IS CONSTANT" THE DISTANCE BETWEEN TWO MERIDIANS BECOMES PROGRESSIVELY SMALLER AS THE LATITUDE INCREASES BECAUSE THE MERIDIANS CONVERGE TO THE POLES$ /N AREA WITH THE INFINITESIMAL DIMENSIONS D6AT AND D6ON WOULD APPEAR AS AN OBLONG WITH THE DIMENSIONS DX AND DY ON OUR GLOBE %'& !#-! !- dx = c' d Lon cos Lat dy = c' d Lat DX CONTAINS THE FACTOR COS 6AT SINCE THE CIRCUMFERENCE OF A PARALLEL OF LATITUDE IS PROPORTIONAL TO COS 6AT$ =HE CONSTANT C' IS THE SCALE FACTOR OF THE GLOBE MEASURED IN" E$ G$" MM%[!$ d Lat dy = c cos Lat =HE NEW CONSTANT C IS THE SCALE FACTOR OF THE CHART$ 8OW" DX REMAINS CONSTANT PARALLEL MERIDIANS!" BUT DY IS A FUNCTION OF THE LATITUDE AT WHICH OUR SMALL OBLONG IS LOCATED$ =O OBTAIN THE SMALLEST DISTANCE FROM ANY POINT WITH THE LATITUDE 6AT : TO THE EQUATOR" WE INTEGRATE- Y Lat P d Lat Lat P " Y = dy = c = c ln tan ! cos Lat 2 4 # 0 0 @ IS THE DISTANCE OF THE RESPECTIVE PARALLEL OF LATITUDE FROM THE EQUATOR$ 4N THE ABOVE EQUATION" ANGLES ARE GIVEN IN CIRCULAR MEASURE RADIANS!$ 4F WE MEASURE ANGLES IN DEGREES" THE EQUATION IS STATED AS- Lat [° Y = c ln tan P ! 45 ° 2 # ')#' =HE DISTANCE OF ANY POINT FROM THE 3REENWICH MERIDIAN 6ON . &[! VARIES PROPORTIONALLY WITH THE LONGITUDE OF THE POINT" 6ON :$ ? IS THE DISTANCE OF THE RESPECTIVE MERIDIAN FROM THE 3REENWICH MERIDIAN- Lon P X = dx = c Lon P 0 %'& !#-" SHOWS AN EXAMPLE OF THE RESULTING -7'9/):1+ '&[ SPACING!$ >HILE MERIDIANS OF LONGITUDE APPEAR AS EQUALLY SPACED VERTICAL LINES" PARALLELS OF LATITUDE ARE HORIZONTAL LINES DRAWN FARTHER APART AS THE LATITUDE INCREASES$ @ WOULD BE INFINITE AT ,&[ LATITUDE$ 7ERCATOR CHARTS HAVE THE DISADVANTAGE THAT GEOMETRIC DISTORTIONS INCREASE AS THE DISTANCE FROM THE EQUATOR INCREASES$ =HE 7ERCATOR PROJECTION IS THEREFORE NOT SUITABLE FOR POLAR REGIONS$ / CIRCLE OF EQUAL ALTITUDE" FOR EXAMPLE" WOULD APPEAR AS A DISTORTED ELLIPSE AT HIGHER LATITUDES$ /REAS NEAR THE POLES" E$ G$" 3REENLAND" APPEAR MUCH GREATER ON A 7ERCATOR MAP THAN ON A GLOBE$ 4T IS OFTEN SAID THAT A 7ERCATOR CHART IS OBTAINED BY PROJECTING EACH POINT OF THE SURFACE OF A GLOBE FROM THE CENTER OF THE GLOBE TO THE INNER SURFACE OF A HOLLOW CYLINDER TANGENT TO THE GLOBE AT THE EQUATOR$ =HIS IS ONLY A ROUGH APPROXIMATION$ /S A RESULT OF SUCH A PURELY GEOMETRICAL! PROJECTION" @ WOULD BE PROPORTIONAL TO TAN 6AT" AND CONFORMALITY WOULD NOT BE ACHIEVED$ %1499/3- &.++98 4F WE MAGNIFY A SMALL PART OF A 7ERCATOR CHART" E$ G$" AN AREA OF )&' LATITUDE BY *&' LONGITUDE" WE WILL NOTICE THAT THE SPACING BETWEEN THE PARALLELS OF LATITUDE NOW SEEMS TO BE ALMOST CONSTANT$ /N APPROXIMATED 7ERCATOR GRID OF SUCH A SMALL AREA CAN BE CONSTRUCTED BY DRAWING EQUALLY SPACED HORIZONTAL LINES" REPRESENTING THE PARALLELS OF LATITUDE" AND EQUALLY SPACED VERTICAL LINES" REPRESENTING THE MERIDIANS$ =HE SPACING OF THE PARALLELS OF LATITUDE" Y" DEFINES THE SCALE OF OUR CHART" E$ G$" +MM%NM$ =HE SPACING OF THE MERIDIANS" X" IS A FUNCTION OF THE MEAN LATITUDE" 6AT 7- Lat ! Lat $ x = $ ycos Lat Lat = min max M M 2 / SHEET OF OTHERWISE BLANK PAPER WITH SUCH A SIMPLIFIED 7ERCATOR GRID IS CALLED A 82'11 '7+' 51499/3- 8.++9 ANDISA VERY USEFUL TOOL FOR PLOTTING LINES OF POSITION$ ')#( 4F A CALCULATOR OR TRIGONOMETRIC TABLE IS NOT AVAILABLE" THE MERIDIAN LINES CAN BE CONSTRUCTED WITH THE GRAPHIC METHOD SHOWN IN %'& !#-# - >E TAKE A SHEET OF BLANK PAPER AND DRAW THE REQUIRED NUMBER OF EQUALLY SPACED HORIZONTAL LINES PARALLELS OF LATITUDE!$ / SPACING OF ) # '& MM PER NAUTICAL MILE IS RECOMMENDED FOR MOST APPLICATIONS$ >E DRAW AN AUXILIARY LINE INTERSECTING THE PARALLELS OF LATITUDE AT AN ANGLE NUMERICALLY EQUAL TO THE MEAN LATITUDE$ =HEN WE MARK THE MAP SCALE DEFINED BY THE SPACING OF THE PARALLELS! PERIODICALLY ON THIS LINE" AND DRAW THE MERIDIAN LINES THROUGH THE POINTS THUS LOCATED$ 1OMPASSES CAN BE USED TO TRANSFER THE MAP SCALE FROM THE CHOSEN MERIDIAN TO THE AUXILIARY LINE$ #34243/) ".'798 2OR GREAT CIRCLE NAVIGATION" THE -34243/) 5740+)9/43 OFFERS THE ADVANTAGE THAT '3< GREAT CIRCLE APPEARS AS A STRAIGHT LINE$ ;HUMB LINES" HOWEVER" ARE CURVED$ / GNOMONIC CHART IS OBTAINED BY PROJECTING EACH POINT ON THE EARTH'S SURFACE FROM THE EARTH'S CENTER TO A PLANE TANGENT TO THE SURFACE$ =HERE ARE THREE TYPES OF GNOMONIC PROJECTION- 4F THE PLANE OF PROJECTION IS TANGENT TO THE EARTH AT ONE OF THE POLES 541'7 -34243/) ).'79 !" THE MERIDIANS APPEAR AS STRAIGHT LINES RADIATING FROM THE POLE$ =HE PARALLELS OF LATITUDE APPEAR AS CONCENTRIC CIRCLES$ =HE SPACING OF THE LATTER INCREASES RAPIDLY AS THE POLAR DISTANCE INCREASES$ 4F THE POINT OF TANGENCY IS ON THE EQUATOR +6:'947/'1 -34243/) ).'79 !" THE MERIDIANS APPEAR AS STRAIGHT LINES PARALLEL TO EACH OTHER$ =HEIR SPACING INCREASES RAPIDLY AS THEIR DISTANCE FROM THE POINT OF TANGENCY INCREASES$ =HE EQUATOR APPEARS AS A STRAIGHT LINE PERPENDICULAR TO THE MERIDIANS$ /LL OTHER PARALLELS OF LATITUDE SMALL CIRCLES! ARE LINES CURVED TOWARD THE RESPECTIVE POLE$ =HEIR CURVATURE INCREASES WITH INCREASING LATITUDE$ 4N ALL OTHER CASES 4(1/6:+ -34243/) ).'79 !" THE MERIDIANS APPEAR AS STRAIGHT LINES CONVERGING AT THE +1+;'9+* 541+ $ =HE EQUATOR APPEARS AS A STRAIGHT LINE PERPENDICULAR TO THE CENTRAL MERIDIAN THE MERIDIAN GOING THROUGH THE POINT OF TANGENCY!$ :ARALLELS OF LATITUDE ARE LINES CURVED TOWARD THE POLES$ ')#) %'& !#-$ SHOWS AN EXAMPLE OF AN OBLIQUE GNOMONIC CHART$ $&% !"-# / GNOMONIC CHART IS A USEFUL GRAPHIC TOOL FOR LONG#DISTANCE VOYAGE PLANNING$ =HE INTENDED GREAT CIRCLE TRACK IS PLOTTED AS A STRAIGHT LINE FROM / TO 0$ 9BSTACLES" IF EXISTING" BECOME VISIBLE AT ONCE$ =HE COORDINATES OF THE CHOSEN WAYPOINTS PREFERABLY THOSE LYING ON MERIDIAN LINES! 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