The use of multiple observations near noon with a traditional sextant to determine a fix is common among celestial navigators. A recent invention is the fixed-angle ‘Bris sextant’ that comes with advantages, but imposes constraints due to its invariant nature. We propose a method by which both longitude and latitude can be fixed using only two sightings with such a device, each equidistant from the meridian. By modelling the solution space for the method, we explore some of the potential utility across geography and seasonal variation. Although this method was developed for use with a Bris fixed-angle sextant, it can also be conveniently used with a more traditional marine or level-bubble sextant. Because this method is computationally cumbersome, it is most convenient when used in a computer or tablet application, or with tables.

A simple scheme is presented for mapping the 2D probability density for an observer's position, defined by any number of lines of position (LOPs) on the surface of the Earth, assuming that the LOPs result from uncorrelated observations that have normally distributed errors. Although the mapping can be used to determine the position fix corresponding to the LOPs (which is consistent with other methods), its intended use is computing the total probability that the observer is located within (or outside) some specified area of interest, such as a zone of avoidance around a navigational hazard. Numerical experiments with areas where the average total interior probability is known, such as the triangles and polygons formed by nearly convergent LOPs, show that the method provides correct answers. The numerical experiments also revealed that theoretical probabilities associated with commonly used error ellipses are overstated for navigational solutions based on small numbers of LOPs.

We present a method of teaching the basics of celestial navigation, using the Javascript programming interface to Google Maps. This offers the teacher of celestial navigation many highly visual and flexible investigative and presentation options, all using the familiar Google Map system. In this paper, we discuss the programming route to Google Maps, and demonstrate an exercise of correcting the assumed position of a hypothetical lost ship, using sights of the Sun, suitable for classroom use. We also present an analysis of the relationship between the proximity of the assumed to the actual position of the ship, and the outcome of a sight-reduction procedure.

The basis of all recommended methods for obtaining position by using celestial bodies is the known altitude of the celestial body being observed. Accordingly, it is necessary to have a sextant, classic or with an artificial horizon, or some other device that can measure altitude. However, there is a way to obtain position using astronomical navigation without determining the altitude of a celestial body, and this method will be analysed in this paper. The introduced method requires only precise measurement of the azimuth, and is based on determining two positions close to the dead reckoning position and lying on the isoazimuthal curve, i.e. a curve of the same great circle azimuths of a celestial body. Furthermore, the model assumes that a part of this curve, between the selected positions, can be replaced by a straight line. Special attention will be given to the analysis of errors of the line of position for various azimuth errors and various dead reckoning (assumed) positions. It will also be shown how a modern Electronic Chart Display and Information System (ECDIS) can help in approximate position determination, knowing only the azimuths of celestial bodies.

The running fix or sight-run-sight fix is a classic problem in celestial navigation. Methods employed to obtain the fix traditionally involve advancing a Line of Position (LoP) taken at an earlier time and crossing it with one obtained later. Attempts to generalise the operation of advancing an LoP when the Earth's surface is represented by a plane to the case of the sphere have resulted in proposals that contain poorly constrained approximations or are otherwise fundamentally flawed. A simple rapidly-convergent iterative procedure to obtain a running fix is described that avoids the notion of advanced LoPs and is readily applicable to both the sphere and ellipsoid.

A Celestial Navigation System (CNS) is a feasible and economical autonomous navigation system for deep-space probes. Ephemeris errors have a great influence on the performance of CNSs during the Mars approach phase, but there are few research studies on this problem. In this paper, the analysis shows that the ephemeris error of Mars is slowly-varying, while the ephemeris error of Phobos and Deimos is periodical. The influence of the ephemeris errors of Mars and its satellites is analysed in relation to both the Sun-centred frame and the Mars-centred frame. The simulations show that the position error of a probe relative to the Sun caused by the Mars ephemeris error is almost equal to the ephemeris error itself, that the velocity error is affected slightly, and that the position and velocity relative to Mars are hardly affected. The navigation result of a Mars probe is also greatly affected by the quantities and periodicities of the ephemeris errors of Phobos and Deimos, especially that of Deimos.

This paper reports on innovative mathematical and analytical approaches to celestial navigation. Beyond the common-sense recommendation for automation of the calculations, it is necessary to consider the specific algorithms used. The advantages of these new algorithms become evident when navigational practice is extended beyond the usual twilight round of sights or the noon Sun line. But exploiting the full advantages of the algorithms would probably require new software. This leads us to consider the prospects of applying new information technologies to the task of taking celestial observations.

A direct method for obtaining the two possible positions derived from two sights using vector analysis instead of spherical trigonometry is presented. The geometry of the circle of equal altitude and of the two body fixes is analyzed, and the vector equation for simultaneous sights is constructed. The running fix problem is also treated.

K. H. Zevering has recently proposed a way to calculate the position of a moving observer, from the intersection of position circles, that differs from the methods hitherto accepted by mariners1. Zevering concludes, from discrepancies between his own results and that of other methods, that those others are in error. However, the discrepancies result from incorrect assumptions in his proposed method, which give rise to erroneous answers. My comments will be restricted to the simplest case of two observations only, and will assume that the Earth is, for our purposes, spherical.