Celestial Navigation

Measure the angle between the horizon and a heavenly body, and enter the required data in the plugin. Each measurement results in a Circle of Position on the sphere (COP, or Circle of Equal Altitude). Two or more observations result in intersecting COP’s from which a position fix can be obtained.

Measure the azimuth (bearing) of a heavenly body. This method is hopelessly inaccurate, especially on a small boat in high seas. However, it is interesting for demonstration purposes, and possibly - with accurate digital compasses - it may be a viable navigation method in the future.

Measure the angle between two heavenly bodies. The computer then attempts to determine clock error from this measurement, and the system time may be corrected.

Standard practice in navy and merchant navy with regard to celestial positioning is as follows (assuming no artificial horizon is available on the sextant):

Many books have been written about the art and science of celestial positioning (Resources below).

Document containing the test data used for the example below: Example Worksheet PDF (4 star observations).

Enter Type (Altitude, Azimuth, Lunar), Celestial Body, Limb, Measurement and Degrees of Certainty. Degrees of Certainty is the accuracy the navigator assumes for the observation. A larger value results in a larger line width for the Circle of Position on the chart.

Enter Date (based on GMT/UT) and Time in GMT/UT, Certainty and Shift. Note that entering a shift removes the calculated numerical fix. This is due to the computation method used, which presently does not allow to shift COP’s. However, a position fix can be established by visually observing the COP’s (which are graphically shifted) on the chart.

The date and time is populated using current computer time and time zone (verify your computer’s time), to match the Greenwich UTC Date & Time. Times for sights are entered in UTC. Sights are likely taken extemporaneously with time details, unique exact time for each sight must be entered separately, overriding the computer time & date.

When a DR Shift with values > 0 is entered the Circle of Position will shift accordingly, the “Fix” button will not compute and the Fix must be done manually.

The DR Shift is used to “advance” a sight to the time of last sight in a “group of sights” which have been taken at different times (usually 1-5 minutes apart), so that the fix can e more accurately determined.

Please see David Burch’s Videos: Methodology_with_Celestial_Navigation_pi below for a much clearer explanation.

Enter Transparency and Color you wish to use for the COP.

Enter Eye Height, Temperature, Pressure and Index Error.

Showing the input figures and some calculated results for the observation. Together with the calculated numerical position fix showed in the main window of the plugin, this can be used for comparison with results that are obtained by other calculation methods (traditional manual method using logarithms, traditional or direct computation methods as mentioned in Nautical Almanac, shortened methods using e.g.

A Circle of Position (COP) indicates all the positions on earth where a navigator may observe the same altitude of a heavenly body at a certain time. Using traditional methods, only the part of a COP the navigator is interested in is used, and replaced by a tangent line (LOP).

The MOB icon shows the initial DR position entered. The red circle indicates the intersection of the crossing red lines, the calculated position fix. Hover cursor over the crossing, right click and place a mark. If required, visually adjust this to get best latitude and longitude of the fix. In Sight Properties - Sight Tab, Degrees Certainty was set to 0.05.

The plugin is still under development and the computation methods used are innovative and based on vector, matrix and least squares methods. The author, Sean d’Epagnier, uses this innovative method to directly calculate a fix position. Only he knows the background and details.

General information on direct computation methods can be found on pages 277 to 285 of the Nautical Almanac 1994 (See Celestial Navigation Links below) and in the following articles:

Presently, the plugin is not capable of advancing COP’s to a common time. When a shift is entered, the calculated numerical position on the main window disappears. In this case, the fix can only be established by visual examination of the graphics on the screen (also see above) 3. Sight Properties - Date and Time Tab and 9. Four Circles of Position.

The data and formulae contained in the Nautical Almanac form a standard in itself. The plugin utilizes astronomical data from VOP87d (for the planets and indirectly for the sun), ELP2000/82 (for the moon) and contains Right Ascension (RA; star’s SHA = 360° - star’s RA) and Declination (Dec) data for the selected stars.

During development of the plugin, the calculated (intermediate) correction values for dip, refraction, horizontal parallax, parallax in altitude and semi diameter, as well as the calculated position fix, should be compared to values that result from other computation methods.

The astronomical data used in the plugin is more accurate than data taken from the Nautical Almanac. However, for navigation purposes the differences are generally not important. With regard to altitude reductions, so far test data indicates that the differences found in calculated observed altitude (Ho) are small. Measurement and reading errors made by the navigator will be larger. Using the present version, calculated fix positions can still differ from those calculated with traditional methods.

The plugin astronomical data are from Jean Meeus' Astronomical Algorithms Wikipedia and Sourceforge.

The largest error that can occur in GHA and declination of any body other than the Sun or Moon is less than 0.2'; it may reach 0.25' for the Sun and 0.3' for that of the Moon. In practice it may be expected that only one third of the values of GHA and declination will have errors larger than 0.05', and less than one tenth will have errors larger than 0.1'.

The errors in the altitude corrections are nominally in the same order (but the actual values of dip and refraction at low altitudes may differ considerably in extreme atmospheric conditions).

Depending on the type of sextant, the reading accuracy of the sextant can be 0.2', 0.1' or 10“. Measurement and reading errors made by the navigator will be larger.

This page allows you to obtain all the astronomical information necessary to plot navigational lines of position from observations of the altitudes of celestial bodies. Simply fill in the form below and click on the “Get data” button at the end of the form.

A table of data will be provided giving both almanac data and altitude corrections for each celestial body that is above the horizon at the place and time that you specify. Sea-level observations are assumed. Very useful for study, testing and comparisons.

by Ming-Cheng Tsou, Ph.D., National Kaohsiung Marine University, Taiwan POLISH MARITIME RESEARCH 3(75) 2012 Vol 19; pp. 53-59 10.2478/v10012-012-0031-5

ABSTRACT In this work, we employ a genetic algorithm, from the field of artificial intelligence, due to its superior search ability that mimics the natural process of biological evolution. Unique encodings and genetic operators designed in this study, in combination with the fix principle of celestial circles of equal altitude in celestial navigation, allow the rapid and direct attainment of accurate optimum vessel position. Test results indicate that this method has more flexibility, and avoids tedious and complicated computation and graphical procedures.

by Tien-Pen Hsu (1), Chih-Li Chen (2) and Jiang-Ren Chang (3)

(1) Institute of Civil Engineering, National Taiwan University (2) Institute of Merchant Marine, National Taiwan Ocean University (3) Institute of Systems Engineering and Naval Architecture, National Taiwan Ocean University THE JOURNAL OF NAVIGATION (2005), 58, 315–335. The Royal Institute of Navigation, doi: 10.1017/S0373463305003188, Printed in the United Kingdom

ABSTRACT In this paper, a simplified and direct computation method formulated by the fixed coordinate system and relative meridian concept in conjunction with vector algebra is developed to deal with the classical problems of celestial navigation. It is found that the proposed approach, the Simultaneous Equal-altitude Equation Method (SEEM), can directly calculate the Astronomical Vessel Position (AVP) without an additional graphical procedure. The SEEM is not only simpler than the matrix method but is also more straightforward than the Spherical Triangle Method (STM). Due to tedious computation procedures existing in the commonly used methods for determining the AVP, a set of optimal computation procedures for the STM is also suggested. In addition, aimed at drawbacks of the intercept method, an improved approach with a new computation procedure is also presented to plot the celestial line of position without the intercept. The improved approach with iteration scheme is used to solve the AVP and validate the SEEM successfully. Methods of solving AVP problems are also discussed in detail. Finally, a benchmark example is included to demonstrate these proposed methods.

by Stanley W. Gery Neptune Power Squadron, Huntington, New York, Received April 1996, Revised December 1996

ABSTRACT This work presents a direct method for obtaining the latitude and longitude of an observer from the observed altitudes of two celestial bodies. No assumed position or dead-reckoned position or plotting is required. Starting with the Greenwich hour angles, declinations, and observed altitudes of each pair, the latitude and longitude of the two points from which the observations must have been made are directly computed. The algorithm is presented in the paper, along with its derivation. Two different, inexpensive, programmable pocket electronic calculators were programmed to execute the algorithm, and they do it in under 30 s. The algorithm was also programmed to run on a personal computer to examine the effect of the precision of the calculations on the error in the results. The findings show that the use of eight decimal places in the trigonometric computations provides acceptable results.

by A. Ruiz Industrial engineer, Navigational Algorithms Journal of Maritime Research, Vol. VIII. No. 3, pp.51-58, 2011

ABSTRACT A direct method for obtaining the points of a circle of equal altitude using the vector analysis as an alternative to the spherical trigonometry is presented, and a solution where celestial navigation and Global Navigation Satellite Systems are complementary and coexist is proposed.

by Andrés Ruiz González Navigational Algorithms, San Sebastián website: Navigational Algorithms

ABSTRACT A direct method for obtaining the two possible positions derived from two sights using the vector analysis instead the spherical trigonometry is presented. The geometry of the circle of equal altitude and of the two body fixes is analyzed, and then the vector equation for simultaneous sights is constructed. Also the running fix problem is treated. Finally the C++ source code for the algorithm is provided in an easy implementation, susceptible for being translated to other common programming language

by George H. Kaplan, U.S. Naval Observatory

Determine Position & Motion of a Vessel

George Kaplan’s Website and other Celestial Navigation Algorithms

Current Directions in Navigation Technology - 2017 Mystic Seaport GPS Spoofing Issues, Response: Inertial Navigation and Celestial Navigation. eLoran under study.

ABSTRACT Although many mathematical approaches to the celestial fix problem have been published, all of them fundamentally assume a stationary observer. Since this situation seldom occurs in practice, methods have been developed that effectively remove the observer’s motion from the problem before a fix is determined. As an alternative, this paper presents a development of celestial navigation that incorporates a moving observer as part of its basic construction. This development allows recovery of the information on the vessel’s course and speed contained in the observations. Thus, it provides the means for determining, from a suitable ensemble of celestial observations, the values of all four parameters describing a vessel’s rhumb-line track across the earth: latitude and longitude at a specified time, course, and speed. In many cases, this technique will result in better fixes than traditional methods.

Rick Fisher August 2010 "www.cv.nrao.edu/~rfisher/Ephemerides/earth_rot.html" dead link * Wikipedia to replace above dead link * Two mysteries about wobbling earth

Abstract “By the standards of modern astrometry, the earth is quite a wobbly platform from which to observe the sky. The earth’s rotation rate is not uniform, its axis of rotation is not fixed in space, and even its shape and relative positions of its surface locations are not fixed. For the purposes of pointing a telescope to one-arcsecond accuracy, we need not worry about shape and surface feature changes, but changes in the orientation of the earth’s rotation axis are very important. ”

Discusses small errors in measurements and standards due to perturbations of the earth. 2/28/2017

Coordinate Systems for Direction John Thorstensen, Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755

This subject is fundamental to anyone who looks at the heavens; it is aesthetically and mathematically beautiful, and rich in history…

Copyright © 1997-2011 Henning Umland; PDF file can be found on this page on his web site:

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 March 15, 2019, First Published May 20th, 1997

Copyright © 1986, 1992 Bruce A. Bauer International Marine ISBN 0-07-005219-0

The Sextant Handbook is dedicated to the premise that electronic navigation devices, while too convenient to disregard, are too vulnerable to rely on exclusively. The book is designed to make beginner and expert alike conversant with this most beautiful and and functional of the navigator’s tools.

by Richard E. Rice and David Burch

This is an update of work done originally in 2012. We have used it in our classes but not published it. We revive it here with new examples and free apps for computation and experimentation with the solution. Details of the derivations are published in another format. The derivation applies to n LOPs with random and systematic variances. This example is three only, addressing the navigator’s famous “cocked hat” problem.

This free and open to the public, online course is made possible by The Blended & Online Learning Design (BOLD) Fellows Program and is hosted by Vanderbilt University. The BOLD Fellows program allows graduate student-faculty teams to create course materials in STEM subject areas rooted in good course design principles which benefit from the online content delivery.

This course serves to address the lack of widely-available instruction in astronavigation. Specifically targeted here are the steps of performing a sight reduction to obtain a terrestrial position using this technique. These steps are explicitly illustrated after a brief overview provides a solid context for their relevance. Difficult concepts such as plotting on a navigational chart and the complexities of using of navigational publications should be better served through this online content delivery.

Content created by: David D. Caudel, PhD. Candidate, Physics, Vanderbilt University

Stellarium is a free open source planetarium for your computer. It shows a realistic sky in 3D, just like what you see with the naked eye, binoculars or a telescope. It is being used in planetarium projectors. Just set your coordinates and go.

Shawn Cook has provided a comprehensive Spreadsheet for Calculations:

Mer Pass is Meridian passages of the Sun (LL) or The Noon Sight (RYA Astronavigation Chapter 5).

All my sights are NOT in the same time so you need to do “running fix”(maybe somebody can improve this plugin to have build in drawing “running fix”). For all 17 sights, I first calculated on paper during passage using Sight Reduction Tables Almanac for 2017 and to compere it, I do it again using Long Term Almanac 2000-2050 - Kolbe (which isgreat). Lastly I put my sight into plugin to check it and it looks OK. Same as my paper work (except Mer Pass).

What I would like to see as an option to this plugin is Meridian passage of the Sun. I used those sight as Sun LL in the plugin but it is NOT as precise as could be (Astro17 - I have on the paper 18°10’N [on GPS it was 18°10,6’N] - plugin draw circle in 18°12.9’N - the reason is that time of the Mer Pass of the Sun is very difficult to measure precisely).

Enter these stars and values Using “Find” and Altitude set for the Star and enter the Lat/Long above:

I suppose I should go up to the rooftop to use my sextant and learn how to take sights again. But that is not the purpose here. We want to check Celestial_Navigation_pi. So this an armchair method that I think may be ok using the “Find” Button. (Short Answer: I think the problem was the default setting of “Clock Offset: -10000 seconds”! This should be set at default=0 IMHO)

Here is a sample test

file that you can use if you would like. Remove the ”.doc“ please. You can rename your own sights.xml file for reuse later, and load this one….for Windows Users this file is in C:\ProgramData\opencpn\plugins\celestial_navigation.

7. Time Tab: Boston Time (UTC-5): Oct 10, 2017, 13:00 so UTC 10/10/17 18:00

8. DR Shift: Distance=0 Bearing=0

9. Config; Set color wanted.

10. Parameters; Eye height=2.0 meters; Temp 10 c.; Pressure=1010; Index Error 0 min. Click Set as Defaults.

11. Sight Tab: Type=Altitude; Celestial Body=Sun; Limb=Lower; then pick “Find“

12. Make sure to change Latitude: 42.35, Longitude to -71.1 (Would very much like to Right Click > Move Boat Lat/Long!)

13. Read Altitude of Sun on 10/10/17 UTC 1800 = 36.888867, Select “Done”

14. Enter “Degrees” 36.888867, make the Minutes 0. Hit OK.

16. Sight Tab:Then enter another Type= Altitude Celestial Body=Arcturus Limb=Lower, check that the Time, DR Shift, Config are the same. Hit Find.

17. Enter Lat=42.35 Long=-71.1 See Altitude of Arcturus UTC 10/10/17 18000 is 66.507224 Hit Done.

18. Enter Degrees=66.507224, make Minutes=0. Hit OK.

15. Arcturus Calculation Page (Printable)

19. Found Clock Offset= -10000 or something, set it at 0 then screwed around for awhile checking other things. Sights changed position, better… This was definitely a problem from earlier!

20. Sight Tab:Then enter another Type= Altitude Celestial Body=Kochab Limb=Lower, check that the Time, DR Shift, Config are the same. Hit Find.

21. Enter Lat=42.35 Long=-71.1 See Altitude of Kochab UTC 10/10/17 18000 is 58.133196 Hit Done.

22. Enter Degrees=58.133196, make Minutes=0. Hit OK.

23. Fix Then find Fix. The fix is 41 nm off. To many circles east to west.

24. Sight Tab:Pick “New” and set Celestial Body=Alkiaid. Check all Tabs set correctly. Pick “Find“

25. Enter Lat=42.35 Long=-71.1 See Altitude of Alkaid UTC 10/10/17 18000 is 79.501993 Hit Done.

26. Fix Hit Fix new red X draw and it is 31 nm away. Better.

27 Turn off the Sun as it is the worst sighting compared to the other 3 by clicking on the “eye”. Better.

28. Fix Hit Fix and new red X drawn and it is 8nm away.

sights.xml.rick2.doc File Please remove ".rick2.doc" for use.

29. Later added more sights and selected the 4 best ones and hit Fix and got about .6nm away.