CelNav

A Program for Calculating Lines of Position from Altitudes of Celestial Bodies

By Ron Baker

April 2009

Background

Prior to the early 1800s, the “lunar distance” method was sometimes used to attempt to find longitude at sea. This method had the advantage of not requiring an accurate time source. But it is mathematically complex, and had to be performed separately from the daily latitude sights made at noon. By the mid 1800s, precise and reliable chronometers were in regular use. But determining longitude with these accurate time pieces also depended on obtaining the precise time of local apparent noon, which is not as easy as one might think. During this time, improvements in accuracy and completeness appeared in the Nautical Almanac, published annually by the Astronomer Royal of England. With these new resources, St Hilaire (and others) developed the ingenious “intercept method”, which is still the basis for celestial navigation today. This new method fixes the current position in both latitude and longitude simultaneously. Although the method has remained basically unchanged for more than 150 years, new tools have appeared which significantly impact how the method is practiced.

The basic steps for the intercept method can be stated briefly. A marine sextant is used to measure the altitude of a celestial body above the horizon at a precise time. Corrections for atmospheric refraction and parallax are applied to the measurement to obtain the body’s true observed altitude. Given the geographic position of the body at the time of the observation, the body’s altitude and true azimuth are calculated using trigonometry. The calculated altitude is then compared with the observed altitude measured with a sextant. With those measurements and calculations, a line of position (LOP) is plotted on a chart. The observer’s true position appears on the chart at the intersection of 2 or more LOPs.

Why CelNav?

Sophisticated celestial navigation programs based on the intercept method are readily available today for calculators and laptop computers. These programs are very efficient, and completely eliminate the need for performing the calculations by hand. Most of these full featured programs are designed for practical use while underway, and are particularly useful to the experienced navigator. But the programs handle the mathematics so efficiently that an overall perspective of the process can be hard to keep. One of the best is StarPilot designed by David Burch of the StarPath School of Navigation. This program runs on laptop computers and hand held calculators. It manages all aspects of celestial navigation including the use of a native perpetual calendar, automatic calculations of sight reductions, celestial fix calculations, and dead reckoning updates. But for efficiency reasons, many of the individual steps in the process are intentionally hidden.

The manual approach, on the other hand, provides the opportunity to investigate details, and can be carried out with nothing more than a sextant, a watch, the Nautical Almanacc, sight reduction tables, paper/pencil, and a work form. This approach relies on the Nautical Almanac for the predicted geographical position of selected celestial bodies in time. The calculated altitude and true azimuth of the body are determined by referencing special sight reduction tables. Although somewhat intimidating at first glance, these voluminous tables are simply pre-determined trigonometric solutions for all possible spherical triangles. From a

2 mathematical standpoint, the calculation of the body’s altitude and true azimuth is perhaps the most interesting part of the whole process. But to find the answers in the sight reduction tables (as you might look up a number in a phone book) does not promote an understanding of the overall geometry. Why not perform the trigonometry directly, and gain an understanding of the big picture?

The rest of this document provides some general background about celestial navigation. It also describes a specific approach that is something of a hybrid between the completely manual methods used for many decades, and the automation provided by today’s full featured programs. It’s true that an observer can easily find the required geographic positions of celestial bodies from various internet sources. But due to the Nautical Almanac’s unique appeal, CelNav was intentionally designed to require manual input from this annual publication. The altitude correction tables appearing in the Nautical Almanac, however, have been dropped in favor of calculating the refraction and parallax corrections empirically. The traditional sight reduction tables used for looking up the calculated altitude and true azimuth are not needed as spherical trigonometry provides the solution for the navigational triangle directly. CelNav was designed to perform the mathematics quickly, but in a way that does not hide the details. It was designed to run on the TI-89 calculator, and using such a device is convenient due to its mobility. But an excel spreadsheet version is described in the appendix, and is perhaps even more transparent in that the formulas can be viewed within the worksheet cells. Although this approach does not provide the overall completeness of a full featured program for navigation at sea, it can be used as a tool to explore the fascinating subject of celestial navigation.

Celestial Sphere & Apparent Motion

The celestial sphere can be imagined as a sphere of infinite radius with the earth at the center. The celestial equator is the circle where the plane which contains the earth’s equator is projected on the celestial sphere. The celestial poles are the points on the celestial sphere where a line which contains the earth’s poles is projected on the celestial sphere. Just as locations on the earth’s surface are defined by the terrestrial coordinates latitude (LAT) and longitude (LON), celestial bodies have a specific address on the celestial sphere defined by their celestial coordinates. In astronomy, these coordinates are right ascension (RA) and declination (DEC). The celestial equator divides the celestial sphere into the northern and southern half. A celestial body’s DEC is the arc angle in degrees north or south of the celestial equator. A body’s RA is the arc distance in arc hours east of the first point of Aries (FPA). The FPA is the point where the ascending ecliptic intersects the celestial equator. This point is named for Aries because it was located in that constellation when discovered 2000 years ago. Drifting westward due to precession, the FPA is currently located in the constellation Pisces. In roughly 600 years from now it will move into the constellation Aquarius. The RA at the FPA is defined to be 0 Hr. RA increases eastward through an entire 360 degrees along the equator from 0 to 24 Hr. In celestial navigation, RA is replaced by the sidereal hour angle (SHA). SHA is the arc distance in degrees a celestial body is located west of the FPA.

SHA  0  (RA *15)

3 To an observer, the celestial sphere appears to move from east to west around the sky through time due to the daily rotation of the earth. The positions of the stars appear to move with it, but their own proper motions cause slight changes over large time scales. In comparison to the stars, the positions of the moon and planets on the celestial sphere change much more rapidly due to their own respective orbital motions. The sun’s position also varies due to the earth’s orbital motion. And the address of each celestial object changes slowly through time due to precession.

Nautical Almanac & Geographic Position

A line drawn from a celestial body’s address on the celestial sphere to the earth center intersects the earth surface at the body’s geographic position (GP). A celestial body will appear at the zenith (Z) to an observer located at the body’s GP. The body’s Greenwich hour angle (GHA) is the distance in degrees west of the prime meridian at a particular time. A body’s GP is defined by its GHA and DEC.

The Nautical Almanac is published annually by the U.S. Naval Observatory in collaboration with the HM Nautical Almanac Office in the United Kingdom. It is a publication of great practical and historical significance appearing continuously in various forms since 1766. The main purpose of the almanac is to provide the hourly GP of the sun, the moon, and the planets Venus, Mars, Jupiter, and Saturn by listing their GHA and DEC for each hour throughout the year. In addition, the GHA of the FPA is listed on the daily pages. The SHA is listed for each of the 57 navigational stars for each day of the year. The GHA for stars is calculated by adding the GHA of the FPA to the SHA of the star. Each star’s DEC is listed for each day of the year. These navigational stars provide good coverage for all areas on the celestial sphere, but are not necessarily the brightest stars in the sky. The familiar 1st magnitude stars are included: Arcturus, Vega, Rigel, Deneb, Sirius, Fomalhaut, and all the rest. But also included are many dimmer stars with names that are not quite so familiar: Alioth, Elnath, Hamal, Sabik, Nunki, Zubenelgenubi, and others.

Navigational Triangle

The navigational triangle (figure 1) is a spherical triangle defined by the Geographic Position of the celestial body (GP), the Assumed Position of the observer (AP), and the North Pole (N), each located at a vertex of the triangle. Each of the 3 vertex angles are formed by 2 adjacent sides. In addition, each of the 3 sides of the triangle subtend an angle formed by 2 rays extending through the respective vertices from the earth center (EC). The triangle is often imagined to be 3 points on the earth’s surface. But it can also be considered by projecting the rays which pass through the vertices onto the celestial sphere.

A line drawn from the earth center to the celestial body intersects the earth’s surface at GP. This point, a vertex of the navigational triangle, is defined at a particular point in time by the celestial body’s Greenwich hour angle (GHA) and declination (DEC). The observer’s Assumed Position (AP) is defined by longitude (LON) and latitude (LAT), which mark the observer’s current position on the surface of the earth determined by dead reckoning. Normally the AP is within a reasonable distance of the true position. The North Pole (N) is

4 the point where a line drawn from the earth center to the north celestial pole intersects the earth’s surface. The local hour angle (LHA) is the celestial body’s angular distance west of the LON, and must be calculated by subtracting LON from GHA.

For each celestial observation, a spherical triangle is formed with the 3 vertices (GP, AP, N). These vertices, combined with the earth center (EC), create 3 vertex angles and 3 sides. We are particularly interested in the 2 sides (90-DEC) and (90-LAT), and the vertex angle (LHA).

 side (90 - DEC) subtends angle ∠GP/EC/N  side (90 - LAT) subtends angle ∠AP/EC/N  vertex angle (LHA) is ∠GP/N/AP

These 3 angles will allow us to calculate 2 important angles to be used later. We will apply the spherical law of cosines to make the calculations. First we must calculate side z (see figure 1) which subtends angle ∠GP/EC/AP. This angle represents the zenith distance of the body observed at AP. Using the given angles, the law of cosines can be expressed as follows, from which the side z is calculated:

 cos(z)  cos(90  Lat)*cos(90  Dec)  sin(90  Lat)*sin(90  Dec)*cos(LHA)

Once side z has been calculated, we use it as a known side. Now we can solve for Az which is the vertex angle ∠N/AP/GP. But to do that, first examine the law of cosines equation with the calculated side z substituted for side 90-DEC, and vertex angle Az substituted for vertex angle LHA:

 cos(90  Dec)  cos(90  Lat)*cos(z)  sin(90  Lat)*sin(z)*cos(Az)

Careful comparison of these 2 equations will show that the various terms are in the same relative orientation, and that each equation is a valid representation of the law of cosines. Figure 1 will help with that comparison. Note, for example, that the position of the LHA in the first equation is replaced with the position of Az in the second equation, as if the triangle has been rotated clockwise. Of course we are not calculating side 90-DEC since it is known. But we do need Az, which will require us to rearrange the terms in the equation. The specific examples later on will demonstrate this. See in particular the section labeled “Screen #6 – calculated altitude & true azimuth (Hc & Zn)”.

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Figure 1. Navigational Triangle. The 3 vertices (GP, AP, and N) define the spherical triangle, and are used to calculate the side z which subtends the angle GP/EC/AP and the vertex angle Az. This figure is based on graphics by Henning Umland (see references).

Sight Reduction

The altitude of a body measured with a sextant (Hs) must be corrected for various factors before it can be compared to the calculated altitude (Hc). First, Hs must be corrected because the visible horizon observed in the sextant always appears lower than the celestial horizon due to the combined effect of the sextant’s height above the water (usually limited to a few feet) and the atmospheric refraction of the horizon. This partially corrected altitude (Ha) is with respect to the celestial horizon. The sextant altitude must also be corrected with respect to the true position of the celestial body. The first step is to correct for atmospheric refraction. The position of the celestial body always appears higher than its true position due to the refraction of the body’s light. Next, in certain circumstances, a correction must be made for parallax. The parallax correction must be made for the moon, and should be made for the sun and the 2 planets Venus and Mars when those planets are less than 1 astronomical unit distant. Finally, when observing the upper or lower limb of the moon or sun, a correction must be made to account for the body’s semi diameter. After all corrections, we

6 arrive at the observed altitude (Ho). This fully corrected altitude is with respect to the celestial horizon and the true position of the celestial body.

The calculated altitude (Hc) of a celestial body is subtracted from the observed altitude (Ho). The difference (a-int), and the calculated true azimuth (Zn), are used to plot a single line of position (LOP) on a chart. A celestial fix is the point on the chart where 2 or more LOPs intersect.

Examples

CelNav was initially designed to run on the TI-89 hand held calculator. The intercept method can be demonstrated by moving through the program in steps. Screen shots will be displayed at each step, along with additional comments about the procedures. We carry through the examples with 2 celestial observations, one of the moon and one of the star Spica. The observations are assumed to have taken place during evening twilight on 6/5/08 off the coast of the Bahamas. The specific celestial bodies were selected to demonstrate how the program handles various factors that are unique to each body. See details in the appendix regarding an excel spreadsheet version of CelNav.

Although only 2 observations are shown in the examples, it is best to observe 3 different celestial bodies if possible. That will allow 3 lines of position to be plotted on a chart, which will yield higher accuracy in the final fix. If multiple sights of the same body are made, then they should be averaged and used as a single sight. It is always better to take multiple sights of just 2 or 3 celestial bodies, rather than take one sight each of 5 or 6 different bodies. The celestial bodies should be selected so that the true azimuth of each is roughly separated by 120 degrees.

The procedure for using a sextant to measure the altitude of a celestial body involves several steps. At the instant the object is observed to be on the horizon through the sextant, read the watch. Record the date and time in a field notebook, along with the celestial body name, sextant altitude, index correction, assumed position, and height of eye (optionally temperature and barometric pressure can also be recorded). The notebook is also a good place to record details about the weather during the sight session, and miscellaneous notes such as the type of horizon used.

Celnav requires the input data to be in a special format. The format follows the convention used in StarPilot (see references), and facilitates the data entry into the program. The date entries must be in the format (yyyy.mmdd), time (hh.mmss), and angles (ddd.mmm). For example, the date & time 2008.0605 & 20.0657 represents June 5, 2008 at 8:06:57 pm. The arc angle 152.431 represents 152 deg, 43.1 min. All dates, times, and angles are converted internally to decimal dates, hours, and angles for use in the calculations.

7 Screen #1 – Watch Time (WT)

From the field notebook, enter the watch date, time, & error (if any) at the respective prompts. The data must be entered in the format shown, but that is the easiest way to enter the data. Watch time must be entered in 24-hour format. Knowing the accurate time to the second is critical for good results. A digital watch will perform like a good chronometer (constant error). It’s not necessary for the watch to display the actual time as long as the watch error is known. Watch error should normally not exceed a few seconds, and should be constant.

Moon Spica

8 Screen #2 – Greenwich Mean Time (GMT)

The watch zone adjustment is needed if the watch does not display GMT directly. When the watch displays time from a time zone other the Greenwich time zone, the time difference must be known and entered here. For zones west of the prime meridian the adjustment is positive, for zones to the east the adjustment is negative. Daylight savings time must also be taken into consideration. For example, if the watch displays Eastern Daylight Time, the value +4 should be entered. From the entered watch date, watch time, watch error, and watch zone, the GMT date and time is calculated and displayed. Note in the example the +4 hour adjustment for the watch zone resulted in the GMT date to move ahead 1 day. This is common for twilight sights in the summer for assumed positions west of the prime meridian. The GMT date displayed this way is a useful reminder to select the correct date when using the Nautical Almanac.

Moon Spica

9 Screen #3 – Greenwich Hour Angle (GHA)

We need to know the celestial body’s angular distance west of the Greenwich meridian at the precise time it is observed in the sextant. The angles are found in the Nautical Almanac for each hour of each day. The Daily Page for June 6 from the 2008 Nautical Almanac is reproduced as figures 3 & 4. There are 3 days on each page. In our examples, June 6 is the first of the 3 days. The hours throughout the day appear in the first column. The GHA for the FPA appears in figure 3, and the GHA for the moon appears in figure 4. Celnav asks for the GHA for the hour prior to the actual observation time (GHA-b) and also for the hour following it (GHA-e). With this data, the program will estimate the actual GHA for the body at the precise time of observation. The interpolation is linear due to the regular nature of the earth’s rotation. The label A/P/S/M is a reminder to select Aries/Planet/Sun/Moon.

Due to space limitations, the GHA for each of the 57 navigations stars is not shown in the Nautical Almanac. Instead, the GHA of the First Point of Aries is tabulated for each hour. To obtain the GHA for one of the stars (in this example Spica), add the GHA of the First Point of Aries to Spica’s Sideral hour angle (SHA). The SHA (along with declination) defines the position occupied by each star on the celestial sphere. SHA is similar to Right Ascension (RA) used by astronomers. The units for SHA are in arc degrees west of the First Point of Aries, while RA is measured east in hours starting also from the First Point in Aries (currently in Pisces).

Moon Spica

10 Screen #4 – Declination (DEC)

The declination (DEC) of the celestial body at the time of the observation is obtained from the Nautical Almanac. In our examples, the declination values can be found in figure 3 & 4.

We use the same linear method for interpolating the body’s declination at the precise time of observation. Theoretically, this is not a valid assumption. In the case of the sun, the rate of change in declination varies depending on the season. During the spring and autumn the sun’s declination changes faster than in summer and winter. The rate of change in the moon’s declination varies even more. But the variances are small considering that the range of time is only 1 hour. So the precision in the calculated declination appears to be adequate for our purposes.

Declinations north of the celestial equator are positive. If the body is south of the celestial equator, the declination is negative. The declination of the celestial body allows us to calculate one of the required sides of the navigational triangle. The side, measured in arc degrees, is 90 degrees minus the declination, with a minimum of 0 and maximum of 180.

Moon Spica

11 Screen #5 – Assumed Position (AP) and Local Hour Angle (LHA)

The latitude and longitude of the observer’s assumed position must be entered, which results in another of the navigational triangle’s vertices. If the Assumed Position (AP) is north of the equator, then the latitude is positive. If the AP is south of the equator, then the latitude is negative. The latitude of the AP allows us to calculate another of the required sides of the navigational triangle. This side, also measured in arc degrees, is 90 degrees minus the latitude, with a minimum of 0 and maximum of 180.

The LHA is measured in arc degrees and is the difference between the body’s GHA and the observer’s AP longitude. It is also the angle at the N vertex of the navigational triangle. Traditionally, longitude is measured east (positive) and west (negative) from the Greenwich meridian in arc degrees up to 180. As a result, before calculating the body’s GP, we must convert the AP longitude to the GHA convention. (For example, if AP longitude is positive (eastern hemisphere), then the corresponding AP hour circle is 360 minus the AP longitude. But if AP longitude is negative (western hemisphere), then the corresponding AP hour circle is 0 minus the AP longitude. Celnav will make this conversion automatically.)

The range in arc degrees for the LHA is 0 to 360. If the body’s GHA is less than the AP longitude (converted to the GHA convention), then 360 must be added to the body’s GHA before the LHA can be calculated.

Moon Spica

12 Screen #6 – Calculated Altitude & True Azimuth (Hc & Zn)

Given the 2 sides (90-DEC) and (90-LAT), and the vertex angle (LHA), we can calculate the altitude (Hc) and true azimuth (Zn) using the Law of Cosines:

Altitude (Hc):  arc angle ∠GP/EC/AP = z (zenith distance) z  arccos(cos(90  Lat)*cos(90  Dec)  sin(90  Lat)*sin(90  Dec)*cos(LHA))  Hc = 90 – z

True Azimuth (Zn):  vertex angle ∠N/AP/GP = Az (cos(90  Dec)  cos(90  Lat))*cos(z) Az  arccos sin(90  Lat)*sin(z)

 If LHA ≥ 180, then Zn = Az If LHA < 180, then Zn = 360 – Az

Moon Spica

With Hc and Zn now calculated, CelNav leads us through the corrections we need to make to the sextant altitude (Hs).

13 Screen #7 – Sextant Height (Hs)

A careful measurement with a sextant will provide the measured altitude (Hs) of a celestial body with a precision of up to 0.1 arcmin under good conditions. The most accurate measurements are made of bodies no higher above the water than 70 degrees or so. When the body is “brought down” to the horizon in the sextant telescope, it is good to rock the instrument back and forth slowly. The body will appear to move in an arc, and the bottom of the arc can be aligned accurately with the horizon. However, if the celestial body is close to the zenith, the arc observed while rocking the sextant will be more like a line and will be difficult to align with the horizon.

Several sights of the same body made over a period of several minutes can be averaged to create a single sight for reduction. To average several sights, a scatter plot can be made with the precise time of the sight on the X-axis and Hs on the Y-axis. Using linear regression (conveniently built into Exel), a trend line can be fitted to the points. Then select any time between the first and last observation. Find the Hs on the trend line corresponding with the selected time. The use of the average Hs will likely produce more accurate results than any one of the actual observations.

Even the best metal sextants need to be calibrated before each session. This can easily be done by observing the horizon. Ideally, such an observation would show 0.0 for the altitude. However, the reading will often show a small difference, such as 1.2 arcmin. This index correction is of no significance but must be used to adjust any measured altitudes.

Moon Spica

14 Screen #8 – Horizon corrections (HOR)

The sea horizon is formed by the line where the sky and water meet. The distance to the horizon depends on the height of the eye, and can be calculated using the following formula. Distance is in nautical miles, and Height of Eye is in feet.

Dist =1.14* Sqrt(Ht Eye)

A true sky/water horizon exists even in the case where land can be seen a good way off. If the land is farther away than the calculated horizon distance, then the Celnav sky/water horizon (option 1) should be used.

The Dip correction is a combination of two factors. First, the Height of Eye causes the horizon to appear slightly lower than it really is. Second, the atmosphere refracts the light from the horizon causing the horizon to appear higher than it really is. The correction for Dip is solved empirically with the following formula. The height if eye is in feet, and Dip is expressed in arcmin.

DIP  0.97 *Sqrt(HtEye)

Celnav can also be used when land is closer than the calculated sea horizon (option 2). This option uses the Dip Short formula for calculating Dip. This option is especially useful for practice. The artificial horizon (option 3) can also be used for practice, but some type of substitute horizon must be available. At this point we have corrected the measured altitude with respect to the celestial horizon. This corrected altitude is referred to as Ha.

Moon Spica

15 Screen #9 – Atmospheric Refraction (Ro)

As the light from a celestial body passes through the atmosphere, it is bent. And the lower the altitude, the more it is bent. If the body is lower than 15 degrees above the horizon, the correction for atmospheric refraction is large and also unstable. It is best to observe bodies higher than 15 degrees.

Refraction (Ro) will cause the body to appear to be higher than it really is, and the amount must be calculated and subtracted from Ha. The correction for atmospheric refraction is mostly a function of the body’s Ha, although temperature and barometric pressure will also slightly affect the value. The following formula is precise for altitudes greater than 15 degrees. CelNav will automatically use a different formula which is better suited for altitudes less than 15 degrees. Ro is expressed in arcmin.

Ro  0.97127 *tan(90  Ha)  0.00137 *tan3 (90  Ha)

Moon Spica

16 Screen #10 – Parallax corrections (HP & PA)

Parallax will cause the celestial body to appear lower than its true altitude. For all bodies except for the moon (which is very close), the parallax correction is small. The parallax correction for the sun is only 0.15 arcmin. For objects farther away than the sun, parallax is ignored. This is true certainly for the stars, but also for Jupiter and Saturn since those planets are never closer to earth than 1 AU. Venus and Mars are interesting since those planets are sometimes closer to earth than 1 AU and sometimes farther away. The Nautical Almanac contains the parallax adjustments for Venus and Mars in a special table called “Altitude Correction Table”, and those values should be entered into CelNav when appropriate.

For the moon, the parallax correction is always significant. The altitude (Ha) of a body is with respect to the earth center. But the observer is on the earth surface, roughly 4000 miles away from the earth center. When the moon is on the horizon, the arc angle (as viewed from the moon) between the earth center and earth surface is fairly large. This angle is called Horizontal Parallax (HP). As the moon rises in altitude, this angle gets smaller. Theoretically, at the zenith (never the case from our location in NE Ohio) the angle is zero and parallax ceases to be a factor. Parallax in Altitude (PA) is a function of the celestial body’s HP and the body’s altitude Ha.

The HP for the moon is listed on the daily pages of the Nautical Almanac (see figure 4). PA is calculated with the following formula, expressed in arcmin. This formula applies to a perfect sphere, but the earth is slightly oblate. The actual formula used in Celnav includes a term which accounts for the oblateness of the earth.

PA  HP*cos(Ha)

Moon Spica

17 Screen #11 – Semi Diameter (SD)

When stars are observed, their light is considered to be a point source. This is not truly the case with the planets, but the disks are small enough that we can treat the light as being a point source, with Venus being a possible exception.

However, the sun and the moon have large disks. The GP for the sun and moon listed in the Nautical Almanac is based on the center of the sun and moon. It is not practical to try to align the center of either the sun or moon with the horizon with the sextant. Instead, we align the upper or lower limb of the bodies with the horizon. A correction for the body’s Semi Diameter must be made to account for the difference between the limb and the center of the body. As the distances changes between the bodies throughout the year, the Semi Diameter for both the sun and the moon found on the daily pages of the Nautical Almanac also changes. The Semi Diameter of the sun and moon appears at the bottom of the daily pages of the Nautical Almanac (figure 4).

When the apparent altitude with respect to the celestial horizon (Ha) is corrected for Atmospheric Refraction, and for Parallax and Semi Diameter (when appropriate), we arrive at the fully corrected observed altitude (Ho).

Moon

18 Screen #12 – Intercept & True Azimuth (a-int & Zn)

To plot a line of position on the chart, we need to know in which direction the body’s GP lies from the observer’s assumed position (AP). The calculated true azimuth (Zn) provides this, and was previously calculated (Screen #6) . We also need to know the difference between the calculated altitude (Hc) and the observed altitude (Ho). That difference is the altitude intercept (a-int). The intercept represents the distance the observer’s true position is from the assumed position. The distance can be either be in the direction of the true azimuth (Toward) or in the opposite direction (Away).

Moon Spica

19 Plotting a celestial fix (figure 2)

The true azimuth (Zn) and the altitude intercept (a-int) are used to plot a line of position (LOP) on the chart. The line of position (LOP) is perpendicular to the true azimuth line, and is place on the chart at a distance from the assumed position (AP) equal to the altitude intercept (a-int). The true position of the observer is located at the intersection of 2 or more LOPs. In this example, the LOP derived from the moon sight is in green, and from the Spica sight in blue. If the recommended 3 LOPs are plotted, normally a very small triangle is formed near the intersection of 2 of the LOPs. In that case, the true position of the observer is located at the center of the small triangle.

Figure 2. The LOP for the moon and Spica sights are drawn using a-int and Zn. The celestial fix is the point where the lines of position intersect.

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Figure 3. Daily Page for June 5 from the 2008 Nautical Almanac showing the hourly GHA for the FPA. The hourly GHA and Dec angles are also list for the planets, and the SHA and Dec is listed for the 57 navigational stars.

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Figure 4. Continuation of the Daily Page for June 5 from the 2008 Nautical Almanac showing the hourly GHA and Dec for the sun and moon. The Horizontal Parallax (HP) for the moon is listed hourly. The Semi Diameter for the sun and moon appear at the bottom of the page.

22 Comment

There are many factors that combine to create the fascination with celestial navigation. The method is filled with history and practical utility, and it draws from many disciplines including astronomy, mathematics, celestial mechanics, geography, history, navigation, and horology. The author believes that the practice of celestial navigation provides a sense of connection with the cosmos in a way that using a GPS never will.

References:

 US Naval Observatory & United Kingdom Hydrographic Office (2007), “The 2008 Nautical Alamanc”.

 Umland, H. (1997-2008), “A Short Guide to Celestial Navigation.” http://www.celnav.de/

 Burch, D. (2002), “Celestial Navigation, A Home Study Course”. The Starpath School of Navigation.

 Sorbel, D. (2003), “Longitude”.

23 Appendix: Screen prints from the CelNav spreadsheet version

Moon – Relates to the calculator screens #1 to #6

24 Moon – Relates to the calculator screens #7 to #12

25 Spica – Relates to the calculator screen #1 to #6

26 Spica – Relates to the calculator screens #7 to #12

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