2. Background

In the navigational procedures that Worsley followed, longitude was found by means of a time sight. Using a sextant or theodolite, the altitude of a celestial body, normally the sun, is measured when it lies well off the meridian. After a calculation that incorporates the observer's estimated latitude, Local Mean Time (LMT) is determined. Longitude is the difference between LMT and Greenwich Mean Time (GMT) as read from a chronometer. Ideally, the time sight should be made when the body being observed lies on the prime vertical, due east or west, as then the longitude obtained is independent of estimated latitude and any uncertainties therein. It is known (Bergman and Stuart, Reference Bergman and Stuart2019b, p. 17) that this was the standard procedure adhered to while Endurance was underway from Portsmouth, England, to Buenos Aires, Argentina, in 1914. The procedure was relaxed while at Ocean Camp on the Weddell Sea ice. Around the time of the sinking, time sights were taken at about 9 a.m. local time, which may represent a convenient interval in the camp's daily routine. On 22 November 1915, the day following the sinking of Endurance, the morning time sight was taken at 8:52 a.m. local time. The sun would have been at an altitude of 34°. On the prime vertical, the sun's altitude was 21°, which would be sufficient to allow a time sight to be made.

Management of the chronometers was a crucial task aboard any ship. Endurance set out carrying 24 (Worsley, Reference Worsley1998, p. 101). Ideally, they would be kept in a temperature-controlled environment and be wound at the same time each day so as to use the same portion of the spring. Treatises (Shadwell, Reference Shadwell1861) were written on the methods and procedures that should be followed. From 18 March until 27 October 1915 when Endurance was abandoned, the principal working chronometer used for navigation was a boxed chronometer by Thomas Mercer with serial number 5229. Now in the collection at the National Maritime Museum in Greenwich (Object Id: ZAA0029), it is believed to have been carried on the James Caird on the famous voyage from Elephant Island to South Georgia in 1916 (Bergman et al., Reference Bergman, Huxtable, Morris and Stuart2018, p. 31). It is usually referred to in the log as Chronometer X. Several other chronometers were given similar designations, see Bergman and Stuart (Reference Bergman and Stuart2018, Figure 8). Sometime between 28 October and 2 November 1915, the role of working chronometer passed to the Smith watch with serial number 192/262, now in the collection of the Scott Polar Research Institute of Cambridge University (Reference number: N: 999a).

The time on chronometers was rarely reset. Instead, careful records were kept of their individual chronometer error (CE) fast or slow of GMT and chronometer rate (CR) gaining or losing in seconds per day. In this paper, the term unaccounted-for chronometer error (UCE) will be adopted to refer to the difference between the true CE and the CE as given in the log.

The CE is generally found by making a time sight from a location of known longitude or by comparison to some authoritative standard time source. When a fixed observing platform was available GMT, and hence the CE, could be determined from the timing of lunar occultations, meridian passages of moon-culminating stars or eclipses of Jupiter's moons. It is known that the chronometers were rated on 24 October 1914 while Endurance was in Buenos Aires (Bergman and Stuart, Reference Bergman and Stuart2018 p. 86., Reference Bergman and Stuart2019b, p. 13). Upon rating, the UCE will be zero.

3. Occultations

The observer's latitude and local mean time (LMT) can be obtained by observation without precise knowledge of GMT, and from them the GMT at immersion or emersion of an occulted star can be computed. Between 24 June and 15 September 1915, Worsley and physicist Reginald James timed 10 occultations and independently reduced their observations using Raper's method as set out by Close (Reference Close1905). Close takes the method from Notes for Travellers (Godwin-Austen et al., Reference Godwin-Austen, Laughton and Freshfield1883) which introduced some potential errors of interpretation arising from the somewhat ambiguous description that Raper (Reference Raper1840) provides. However, these potential errors do not affect Worsley's calculations. A complete description of the observations has been given by Bergman and Stuart (Reference Bergman and Stuart2018). It was found that Worsley and James correctly carried out the reductions based on the positions of the moon and stars as tabulated in the Nautical Almanac (1915), where they are quoted to 0 ⋅ 01^s in right ascension and 0 ⋅ 1^″ in declination.

The level of precision to which positions are given in the Nautical Almanac does not reflect the accuracy to which they could be computed in advance. Close (Reference Close1905, p. 190) states

James (Reference James1919) describes corrections deduced by Crommelin, see also Dyson and Crommelin (Reference Dyson and Crommelin1923), aimed at improving the CEs obtained from the occultations.

The CEs for the Mercer 5229 have been calculated using the Jet Propulsion Laboratory's DE430 model for the motion of the moon (Folkner et al., Reference Folkner, Williams, Boggs, Park and Kuchynka2014). The occulted star positions were taken from the Hipparcos catalogue (Perryman et al., Reference Perryman2009), which gave an average difference of 1 ⋅ 3^″ in apparent positions obtained from the Nautical Almanac. The maximum difference was 3 ⋅ 2^″ for the star B.D. −17°4053 (HIP 69792). Calculations were performed using the Skyfield software package (Rhodes, Reference Rhodes2016) and are summarised in the Appendix in Tables A1 and A2. A comparison of the positions of the moon given in the Nautical Almanac with those obtained with Skyfield on the hour closest to the occultation finds that the former consistently lag by 0 ⋅ 9^s in right ascension on average and have a root-mean-square difference in declination of 1 ⋅ 7^″.

There is a complication with regard to the occultations of the star A Ophiuchi (HIP 84405) observed on 23 July and 15 September. It is a binary system consisting of two nearly identical components that at the time of observation were separated by 4 ⋅ 24^″, with the B star at position angle (J2000 ⋅ 0) of 184 ⋅ 1° relative to the A star (Hartkopf et al., Reference Hartkopf, Mason and Worley2001). The B star is the reference component in the Hipparcos catalogue. To reduce the occultations, the position and proper motion of the system's centre of mass (CM) was found by averaging those of the two components. The apparent positions of the A and B stars at immersion were offset from the position of the CM based on the separation and position angle.

In relation to occultations, the Nautical Almanac (1915) refers to ‘A Ophiuchi (1st star)’. The precise meaning of this term is ambiguous. From the position angle, it can be seen that the A star and the B star have very nearly the same right ascension and the one that is occulted first depends on where on the lunar limb the disappearance occurs, and that in turn depends on the observer's latitude. For the occultation of 23 July, the A star disappeared first, with the B star following 0 ⋅ 15^s later. However on 15 September, the B star was the first to be occulted and the A star disappeared 4 ⋅ 25^s afterwards. It is assumed that Worsley recorded the time of the first immersion.

The time calculated for an occultation depends somewhat on the assumed value of the constant k. This is the ratio of the moon's radius to the equatorial radius of the Earth but is adjusted in an attempt to account for observational effects, such as irradiance. In these calculations, k = 0 ⋅ 272550 is used as recommended in the Nautical Almanac (1915, p. 641). The calculations take the observer's latitude and LMT of immersion as inputs. Both of these quantities are determined from observation with a sextant or theodolite. The point on the moon's limb where the star will disappear is not known to any better accuracy than the uncertainties in the observer's position on the Earth, and often much less than that. There is, therefore, little justification for attempting to include additional small effects, such as the impact of irregularities in the moon's limb.

Table A1 gives the latitude and local mean time of immersion for the 10 occultations observed on the expedition. The Universal Time (UT1) deduced from these observations is given. This modern timescale is 12 h ahead of GMT as it was recorded in the log. The CEs listed come from Worsley (Reference Worsley1916b) and James (Reference James1919). The UCEs represent the corrections that must be added to the CEs in order to be consistent with the DE430 model and Hipparcos catalogue. They range around an average value of 20^s. The UCEs computed by Crommelin are deduced from James (Reference James1919) and have an average of 22^s.

Table A2 gives the observer's latitude and longitude at the time of the occultation based on the UT1 listed in Table A1. Also given are the apparent topocentric positions for the equator of date of the moon and occulted stars along with the moon's semidiameter.

As seen in Figure 1, Worsley computed CEs for the Mercer 5229 chronometer from the occultations. Those of 24 June and 16 August are averaged over the multiple stars observed. From these, the chronometer rate (CR) over the intervals between occultations is obtained. The occultation of 15 September is omitted. It gives a CR of gaining 3 ⋅ 9 s/day from 13 September, which is far larger than the other values, and Worsley apparently rejected it. The average CR for the three intervals between listed occultations is gaining 0 ⋅ 3 s/day, which is extrapolated backwards to estimate the CE on 18 March. However, perhaps motivated by the smaller-than-average CR of gaining 0⋅ 1s/day obtained for 13 September, Worsley adopts a CR of gaining 0 ⋅ 2 s/day from 17 September until Endurance was abandoned on 27 October.

Figure 1.

Chronometer errors deduced by Worsley for the Mercer 5229 (Chronometer X) from the occultations. The occultation of 15 September has been rejected. (Canterbury Museum 2001. 177.1–pg 73.)

The time of immersion recorded for each occultation will be subject to random errors and consequently the CEs derived from them will inherit some random variation. A standard least-squares linear regression can be performed on the CEs and corrected CEs (CE + UCE) in Table A1 against the tabulated UT1. This is done to extract the best estimate for the two parameters, CE at some specified time and the CR. If the underlying error statistics are Gaussian or binomial, then the least-squares regression represents the maximum likelihood fit to the observational data. More generally and under much weaker conditions, the Gauss–Markov theorem states that the least-squares estimator has the lowest variance of any estimator constructed from linear combinations of the observations.

Linear regression is consistent with the navigational practice of using the two parameters to extract, the CE at specified time and the CR. It is performed by finding the slope, m, and intercept, c, that minimises the sum of the squares of residuals, $\sum\limits_i {r_i^2}$, for ${r_i} = m{T_i} + c - ({\textrm{C}{\textrm{E}_i} + \textrm{UC}{\textrm{E}_i}} )$. T_i is the occultation date and time in some convenient time scale, and CE_i and UCE_i represent the corresponding entries in Table A1. As a figure of merit, the square of the correlation coefficient

(1)\begin{equation}{R^2} = 1 - \frac{{\sum\limits_i {r_i^2} }}{{\sum\nolimits_i {{{({\textrm{C}{\textrm{E}_i} + \textrm{UC}{\textrm{E}_i} - \overline {\textrm{CE} + \textrm{UCE}} } )}^2}} }} ,\end{equation}

is quoted. Here, $\overline {\textrm{CE} + \textrm{UCE}}$ is the average of the total corrected chronometer errors. In general, R ² gives the percentage the total variance in the data that is explained by the model − in this case a straight line. The closer that R ² is to 100%, the better the fit.

When the procedure was performed manually, as would have been the case in the early part of the 20th century, the CE and CR would be most easily be extracted by drawing a line by eye through a set of plotted points. This has intuitive appeal and allows potential outliers to be visually identified and excluded if appropriate. CEs and corrected CEs are plotted in Figure 2 along with the lines of least-squares best fit.

Figure 2.

Plots of the CEs as originally obtained by Worsley (top), after correction modern positions for the moon and stars (middle) and as corrected by James and Crommelin (bottom). The horizontal axis shows days prior to GMT noon on 27 October 1915. The solid marker at right is the CE at that time as predicted by the least-squares linear regression

Table 1 lists the results obtained from the linear regression. James (Reference James1919) suggests that Crommelin's UCE for the star B.A.C. 5253 on 24 June seems to be wrong and it is, therefore, omitted from the regression. The fit yields an estimate of the CR. The CE is given at GMT noon on 27 October 1915. This was the day that Endurance was abandoned and after which time the Mercer 5229 chronometer could not be depended on to perform reliably. R ² for Worsley and James's initial CEs indicates a fair fit. Correction by the DE430/Hipparcos UCEs produces an excellent fit with R ² = 99 ⋅ 2% and gives a CR of gaining 0 ⋅ 366 s/day. The fit residuals are plotted in Figure 3. For clarity, James's original values are not included because they lie very close to those of Worsley. Although the average of the UCEs derived from modern calculations is close to the average for Crommelin's, the former produces a considerably better fit.

Figure 3.

Residuals to the linear least-squares best fit of chronometer errors and corrected chronometer errors. James's CEs fall very close to Worsley's and are not plotted

Table 1.

Chronometer error (CE), chronometer rate (CR) and R ² obtained by a linear least-squares best fit to the CEs and corrected CEs (CE + UCE) in Table A1 omitting Crommelin's C.E. for BAC 5253. The CE is given as of GMT noon on 27 October 1915 when Endurance was abandoned

The foregoing section dealt with the offset of positions versus true positions arising from systematic errors in the tabulated values of the moon's position in the Nautical Almanac. In order to estimate the final position of the wreck of Endurance, the influence of other factors must be incorporated, and these are discussed next.

4. Connecting the Mercer and Smith chronometers

As previously noted, to function reliably, chronometers should be kept in a temperature-controlled environment. After Endurance was abandoned and the expedition took up residence in Ocean Camp on the ice floe, this would not have been possible for a boxed chronometer like the Mercer 5229. The log entry for 2 November shows that the role of working chronometer had passed to the Smith 192/262 with a CE of ‘40^m26^s slow’. As it takes the form of a large pocket watch, it could be worn close to the body to maintain it at a constant temperature. Worsley (Reference Worsley1998, p. 191) mentions, ‘I carried … the chronometer … slung around my neck by lampwick, inside my sweater, to keep it warm’. Similar watches were apparently placed in the care of Hudson (192/232) and Wild (192/231). The exact procedures followed remain uncertain, however the Smith 192/262 would have inherited the UCE from the Mercer 5229 when it became the working chronometer. On 7 November, the Smith 192/262 is recorded as having a CE ‘fast 3^m28 ⋅ 5^s Cor’, indicating that it had been reset.

Little information is available to pin down the behaviour of the UCE for the Smith chronometer between 2 and 22 November. Long-term averages can be obtained from sights taken in March and April 1916, but these may not reliably reflect short-term variations that could have occurred in the period.

Mount Percy on Joinville Island was sighted on 24 March 1916 from Patience Camp on the Weddell Sea pack ice, and an attempt was made to rate the chronometers by triangulation on its changing bearing over a 3-day period (Bergman and Stuart, Reference Bergman and Stuart2018, pp. 87–88; Bergman et al., Reference Bergman, Huxtable, Morris and Stuart2018, p. 54). From these observations, it was determined that Mount Percy was ‘23′W of Chron:’ (Worsley, Reference Worsley1916b, p. 80). Worsley did not have sufficient confidence in the observations to allow the CE to be adjusted based on them. In fact, Mount Percy lies at 55°49′W or 11′ further west than the position that Worsley had for it. This implies a UCE of 2^m16^s slow.

On 24 April 1916 immediately prior to departure from Elephant Island for South Georgia, Worsley was able to take a time sight from Cape Wild (now Point Wild). A detailed analysis can be found in Bergman et al. (Reference Bergman, Huxtable, Morris and Stuart2018). Geographical constraints mean that the location where the sight was taken is quite well determined at 61°6′S 54°51′30^″W giving a UCE of 2^m25^s slow.

5. Scenarios and unknowns

7. Postscript

After this paper was submitted, the Endurance22 expedition announced that they had located the wreck of Endurance on 5 March 2022. The position was later given as 68°44^′21^″S 52°19^′47^″W which is 4·9NM South, 2·4NM East (5·4NM total distance) from the log position and is fully consistent with the prediction for latitude given in Bergman and Stuart (Reference Bergman and Stuart2019a) and the scenarios for longitude discussed in Section 5.