Influence of chronometer error uncertainties on the Longitude of Shackleton's vessel, Endurance

Abstract

In 1915 while the Imperial Trans-Antarctic Expedition's vessel Endurance was icebound in the Weddell Sea, lunar occultation timings were carried out in order to rate the chronometers and thereby find longitude. The original observations have been re-analysed using modern lunar ephemerides and catalogues of star positions. The times derived in this way are found to differ by an average of 20 s from those obtained during the expedition using positions given from the Nautical Almanac and introduces an additional offset of the true positions to the east of those recorded in the log.

1. Introduction

The navigational procedures employed by Captain Frank Worsley on the 1914 Imperial Trans-Antarctic Expedition led by Sir Ernest Shackleton have been studied in detail (Bergman and Stuart, 2018, 2019a, 2019b; Bergman et al., 2018) by examining his original log and workbook housed at the Canterbury Museum, Christchurch, New Zealand (Worsley, 1916b). The majority of these publications were principally concerned with expounding the navigational techniques used in practice and replicating the calculations performed in the workbook. The astronomical data in the analyses were taken from the Nautical Almanacs of the period.

Particularly complex and challenging is the reduction of the occultation timings used to rate the chronometers when the expedition's vessel Endurance was trapped in the Weddell Sea ice during the southern winter of 1915 (Bergman and Stuart, 2018). It was found that both Worsley and expedition physicist Reginald James carried out the necessary calculations accurately and consistently.

One of us (DLM) discovered a paper in the archives of the Scott Polar Research Institute (James, 1919) which describes occultation timing reductions carried out by A. D. Crommelin of the Greenwich Observatory based on corrections to the moon's position from observational data. The occultation times deduced in this manner were found to occur on average 22 s earlier than those obtained from tables in the 1915 Nautical Almanac, meaning that the chronometers were running around 22 s faster than the chronometer errors recorded in the expedition's logs. This displaces the true positions by 5 ⋅ 5′ in longitude to the east and has implications for the location of the wreck of Endurance that sank on 21 November 1915 at a position reported as 68°39′30^″S 52°26′30^″W.

In this paper, the occultation timings are reanalysed using modern ephemerides of the moon (Folkner et al., 2014) and star positions taken from the Hipparcos catalogue (Perryman et al., 2009). The results are broadly consistent with Crommelin's but produce a much better quality fit to the actual observations recorded in Worsley's workbook.

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, 2019b, 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, 1998, 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, 1861) 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., 2018, p. 31). It is usually referred to in the log as Chronometer X. Several other chronometers were given similar designations, see Bergman and Stuart (2018, 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, 2018 p. 86., 2019b, 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 (1905). Close takes the method from Notes for Travellers (Godwin-Austen et al., 1883) which introduced some potential errors of interpretation arising from the somewhat ambiguous description that Raper (1840) provides. However, these potential errors do not affect Worsley's calculations. A complete description of the observations has been given by Bergman and Stuart (2018). 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 (1905, p. 190) states

To get the full benefit from the accuracy of the method, it is necessary to obtain from some fixed observatory the observed declination and right ascension of the moon during the night in question, so as to correct the co-ordinates given (by prediction) in the ‘Nautical Almanac’; differences even in the second place of decimals of seconds in these quantities appreciably affect the result of the calculation.

James (1919) describes corrections deduced by Crommelin, see also Dyson and Crommelin (1923), 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., 2014). The occulted star positions were taken from the Hipparcos catalogue (Perryman et al., 2009), 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, 2016) 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., 2001). 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 (1916b) and James (1919). 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 (1919) 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 (1919) 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 (1998, 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, 2018, pp. 87–88; Bergman et al., 2018, p. 54). From these observations, it was determined that Mount Percy was ‘23′W of Chron:’ (Worsley, 1916b, 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. (2018). 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

5.1 Effects of chronometer error

The foregoing information can be used to estimate ranges by which the true longitude differs from that given in the log. Each 1 s of UCE fast (slow) moves the true position 0 ⋅ 25′ of longitude or at the latitude at which Endurance sank to 0 ⋅ 09NM east (west) of the position given in the log.

For 27 October 1915, the day that Endurance was abandoned, Worsley (Worsley, 1916b, p. 39) gives a CE as 2^m38^s fast. However taking the sum of the values for ‘Worsley CE’ and ‘Worsley UCE’ in Table A1 and regressing them in time predicts a true CE for the Mercer 5229, Chronometer X, of 3^m6 ⋅ 8^s fast on that date and suggests that the working chronometer was actually fast by an additional 28 ⋅ 8 s. On that date, the expedition's true position would then be 7 ⋅ 2′ of longitude east of where the log entries put it.

Worsley may have begun using the Smith 192/262 immediately after Endurance was abandoned or, alternatively, since it makes its first appearance on 2 November, the CE for the Mercer 5229 may have been carried forward at CR of 0 ⋅ 2 s/day, gaining until this date by which time the UCE would be 29 ⋅ 8 s fast. Exactly what transpired, how the role of working chronometer was transferred from the Mercer to the Smith and what errors may have been introduced in the process is unknown.

If the Smith was the working chronometer from 27 October 1915, then its UCE on that date would be 28 ⋅ 9 s fast compared with Worsley's reckoning. Taking the observation of Mount Percy on 24 March 1916, for which the UCE was 2^m16^s slow, and performing simple linear interpolation gives a UCE on 22 November of zero. In this scenario, the positions as stated in the log are correct.

If the Mercer was the working chronometer until 2 November, the UCE inherited by the Smith on that date would be 29 ⋅ 8 s fast. If it were to keep perfect time, then true positions would be 7 ⋅ 5′ (2 ⋅ 7 NM) east of those given in the log. If it began to drift immediately at a rate determined by the time sight from Elephant Island taken on 24 April 1916 and for which the UCE was 2^m25^s slow, then the UCE on 22 November would be 9 ⋅ 7 s fast. This corresponds to the true position of the sinking being 2 ⋅ 4′ (0 ⋅ 9NM) of longitude east of that given in the log.

5.2 Effects of ice drift

Endurance sank at around 5 p.m. local time on 21 November 1915, but sights to fix the position could not be taken until the following day. As noted elsewhere (Bergman and Stuart, 2019a), Shackleton's journal entries as well as those from other expedition members indicate that the wind blew with a persistent southerly component during that period. The position of the sinking reported in the log is consistent with the noon position of Ocean Camp on 22 November plus an offset of 1 ⋅ 2NM S 33° E (1′S 1′45^″E) to account for the distance and bearing of Endurance from Ocean Camp at the time of the sinking and is consistent with accounts by Worsley and others. On 10 November, Worsley (1916a) recorded, ‘ … our camp is roughly 2 m NW of the position of the ship … ’ In his diary entry of 21 November, Orde-Lees (1916) wrote ‘ … there was our poor ship a mile and a half away breathing her last’. If indeed the reported position of the wreck is simply based on the noon position as determined the following day then the influence of the wind and ice drift in the intervening 19 hour period needs to be considered. There is very little quantitative guidance available here.

On 21 November, Shackleton (1915) reported, ‘ … a SSE fair wind all day’ and ‘The wind veered later to the west, and the sun came out at 9 p.m.’ (Shackleton, 1920. Ch. V, P.98). For the same day Orde-Lees (1916) writes ‘wind S. to S.W. increasing about 6pm’.

As noted previously (Bergman and Stuart, 2019a, p. 263), the difference in longitude that Worsley obtained in the a.m. and p.m. time sights on 22 November show a roughly north-westerly drift with a westward component of 0 ⋅ 9NM in 6^h47^m, giving an average speed of 0 ⋅ 13 knots. If maintained over 19 hours, that would push the true position of the wreck a further 2 ⋅ 6 NM east of the noon position given in the log. However, since the a.m. and p.m. time sights were not made when the sun was on the prime vertical, the longitude difference has some dependency on the latitudes that were assumed in the sight reduction and is, therefore, open to question. Moreover, the westward drift on 22 November suggests an easterly component to the wind and a reversal of direction since the evening before. Which direction predominated over the period is unknown.

6. Conclusions

The occultation timings made over the winter of 1915 by the Imperial Trans-Antarctic Expedition in order to rate their chronometers and thereby fix longitude have been re-analysed using modern lunar ephemerides and star positions. This shows that previous estimates of chronometer errors based on positions from the 1915 Nautical Almanac are too slow by an average of 20 s. It was noted elsewhere (Bergman and Stuart, 2019a) that Captain Frank Worsley had an apparent tendency to underestimate how slow the Smith 192/262 chronometer was running in the period following the sinking of Endurance, which would favour a wreck site to west of the position in the log. The new correction examined here more than compensates for that bias and pushes the wreck site back toward the east. As previously noted, a westerly component of ice drift would tend to offset the position still further to the east.

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 (2019a) and the scenarios for longitude discussed in Section 5.