S I(RI FTE R Nr.90

DET KONGELIGE INDUSTRI-, HANDVERK­

OG SKIPSFARTSDEPARTEMENT

NORSK POLARINSTITUTT

(Formerly Norges Svalbard- og Ishavs-undersokelser)

S I(RI FTE R

Nr.90

ASTRONOMICAL

OBSERVATIONS ON HOPEN

BY

HANS HENlE

OSLO

I KOMMISJON HOS JACOB DYBWAD 1948

NORSK POLARINSTITUTT

Observatoriegaten 1, Oslo

SKRIFTER

Nr. 1. Ho EL, A., The Norwegian Svalbard Expeditions 1906-1926. 1929. Kr. 10,00.

2. RAVN, J. P. J., On the Mollusca of the Tertiary of Spitsbergen. 1922. Kr. 1,60.

" 3. WERENSKIOLD, W. and 1. OFTEDAL, A burning Coal Seam at Mt. Pyramide, Spitsbergen. 1922. Kr. 1,20.

" 4. WOLLEBJEK, A., The Spitsbergen Reindeer. 1926. Kr. 10,00.

5. LYNGE, B., Lichens from Spitsbergen. 1924. Kr.2,50.

" 6. HOEL, A., The Coal Deposits and Coal Mining of Svalbard. 1925. Kr. 10,00.

[Out of print.]: I

7. DAHL, K., Contributions to the Biology of the Spitsbergen Char. 1926. Kr. 1,00.

" 8. HOL TEDAHL, 0., Notes on the Geology of Northwestern Spitsbergen. 1926. Kr.5,50.

" 9. LYNGE, B., Lichens from Bear Island (BjfiJrnfiJya). 1926. Kr. 5,80.

" 10. IVERsEN, T., Hopen (Hope Island), Svalbard. 1926. Kr. 7,50.

" 11. QUENSTEDT, W., Mollusken a. d. Redbay- u. Greyhookschichten Spitzb. 1926. Kr.8,50.

Nos. 1-11: Vo!. I. From Nr. 12 the papers are only numbered consecutively.

Nr. 12. STENSIO, E. A:SON, The Downtonian and Devonian Vertebrates of Spitsbergen.

Part l. Cephalaspidae. A. Text, and B. Plates. 1927. Kr.60,OO.

" 13. LIND, J., The Micromycetes of Svalbard. 1928. Kr. 6,00.

" 14. KJJER, R. and J. E. FJELDSTAD, Tidal Observations in the Arctic. 1934. Kr.6,OO.

" 15. HORN, G. and A. K. ORVIN, Geology of Bear Island. 1928. Kr. 15,00.

" 16. JELSTRUP, H. S., Determinations astronomiques. 1928. Kr. 2,OO.

17. HORN, G., Beitriige zur Kenntnis der Kohle von Svalbard. 1928. Kr.5,50

" 18. HOEL, A. und A. K. ORVIN, Das Festungsprofil auf Spitzbergen. Karbon-Kreide.

I. Vermessungsresultate. 1937. Kr.4,OO.

" 19. FREBOLD, H., Das Festungsprofil aUf Spitzbergen. Jura und Kreide. 11. Die Strati­

graphie. 1928. Kr.3,00.

» 20. FREBOLD, H., Oberer Lias und unteres Callovien in Spitzbergen. 1929. Kr. 2,50. 21. FREBOLD, H., Ammoniten aus dem Valanginien von Spitzbergen. 1929. Kr.4,OO.

" 22. HEINTZ, A., Die Downtonischen und Devonischen Vertebra ten von Spitzbergen.

Il. Acanthaspida. 1929. Kr. 15,00.

" 23. HEINTZ, A., Die Downtonischen und Devonischen Vertebraten von Spitzbergen.

Ill. Acanthaspida. - Nachtrag. 1929. Kr.3,OO.

" 24. HERITSCH, F., Eine Caninia aus dem Karbon des De Geer-Berges. 1929. Kr. 3,50.

" 25. ABS, 0., Untersuchungen iiber die Erniihrung der Bewohner von Barentsburg, Svalbard. 1929. Kr. 5,00.

" 26. FREBOLD, H., Untersuchungen iiber die Fauna, die Stratigraphie und Paliio­

geographie der Trias Spitzbergens. 1929. Kr. 6,00.

" 27. THoR, S. , Beitriige zur Kenntnis der invertebraten Fauna von Svalbard. 1930. Kr.18,00.

28. FREBOLD, H., Die Altersstellung des Fischhorizontes, des Grippianiveaus und des unteren Saurierhorizontes in Spitzbergen. 1930. kr.4,00.

" 29. HORN, G., Franz Jose! Land. Nat. Hist., Discovery, Expl., and Hunting. 1930. Kr. 5,00.

30. ORVIN, A. K., Beitriige zur Kenntnis des Oberdevons Ost-Gronlands. HEINTZ, A .•

Oberdevonische Fischreste aus Ost-Gronland. 1930. Kr.4,00.

" 31. FREBOLD, H., Verbr. und Ausb. des Mesozoikums in Spitzbergen. 1930. Kr. 17,00.

" 32. ABS, 0., Ober Epidemien von unspezijischen Katarrhen der Luftwege auf Svalbard.

1930. Kr. 2,00.

" 33. KlJER, J.. Ctenaspis, a New Genus of Cyathaspidian Fishes. 1930. Kr. 1,00.

" 34. TOLMATCHEW, A., Die Gattung Cerastium in der Flora von Spitzbergen. 1930. Kr. 1,00.

" 35. SOKOLOV, D. und W. BODYLEVSKY, Jura- und Kreidefaunen von Spitzb. 1931. Kr.15,00.

" 36. SMEDAL, G., Acquisition of Sovereignty over Polar Areas. 1931. Kr. 10,00.

DET KONGELIGE INDUSTRI-, HANDVERK­

OG SKIPSFARTSDEPARTEMENT

NORSK POLARINSTITUTT

(Formerly Norges Svalbard- og Ishavs-undersokelser)

S I(RI FTE R

Nr.90

ASTRONOMICAL

OBSERVATIONS ON HOPEN

BY

HANS HENlE

OSLO

I KOMMISjON HOS jACOB DYBWAD 1948

A. W. B[�0GGERS BOKTRYKKER[ Ais

Contents.

Page Introductory. .. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Determination of Geographical Longitude ...... 9 Determination of Geographical Latitude .. . . . . .. 20 Determination of Azimuth. . . . . . . . . . . . . .. 26 Determination of the Magnetic Declination ... 30

Introductory.

1. T h e i s I a n d H o p e n in the Arctic Ocean has been described in "Skrifter om Svalbard og Ishavet" Nr. 10 by Thor Iversen. In this publication the author gives a survey of the various maps which in the course of time have been made of the Arctic Ocean. The position of Hopen changes from one map to the other and all these positions have been collected on a general survey map, showing in all 8 different positions. The latest of these positions has been laid down by Thor I versen based on two series of observations of the sun's altitude made in August 1924. The observations of altitude have been calculated by B. Caspersen and discussed by W. Werenskiold in a supplementary note.

Werenskiold finds a geographical latitude of 76° 34' and a geographical longitude of 25° 6', 3 east of Greenwich, and adds that the longitude is well determined, but there is still some uncertainty as to the latitude.

These co-ordinates apply to Husdalen in the southern part of the island.

As Norges Svalbard- og Ishavs-unders0kelser have started the surveying of Hopen, and as there is a greater demand for a more accurate position of the island after the establishment of a meteorological station in connection with a wireless station, it became necessary to determine its position once more. On the request of the leader of the Svalbard Office, Dr. Anders K. Orvin, the author of the present treatise, Dr. Hans Henie, undertook to make the necessary observations and calculations in order to carry out a sufficiently exact determination of a point on which the survey of the island could be based, and which could be used for laying down the exact position of the island on the map.

2. The a s t r on o m ica 1s t a t i on. OnJuly 10th, 1947 the MiS" Minna", chartered by N orges Svalbard- og Ishavs-unders0kelser, landed a party of topographers on the east coast of Hopen under the leadership of T. Askheim as well as H. Henie with his instruments for astronomical observations. The expedition stayed on the island until the 12th September. When selecting the site for the astronomical station one had to consider the visibility to Koefoedodden on the south coast, where a concrete foundation for a trigonometrical signal had been built previously. This foundation should now serve as a meridian mark.

-6-

One also had to consider that the station was situated so that it could be directly connected to the geodetic net, and finally it had to be in the vicinity of the wireless station at Husdalen, where the chronometer and the instruments were kept, and were a room had been placed at the disposal of the observer.

A point abt. 55 m to the south of the southern most building of the wireless station was selected. The point is on the same marine terrace as the meteorological station, and here, where the ground was firm and dry, we dug down into the frozen ground where the foundation for the observation pillar was erected. The pillar is of concrete, abt. 80 cm above the surface of the ground, square 50 by 50 cm and provided with a copper bolt fastened in the centre of the pillar. Three steel plates with groves for the foot screws of the instrument were also placed in the concrete on the top of the pillar. Around the pillar we built a hut propped up by drift wood. At a distance of 18 m west of the pillar a copper bolt was imbedded in a concrete block at the ground level for greater safety. The area was surveyed by the topographer Askheim.

3. I n s t r u m e n t s. The instruments used were lent by Norges Geografiske Oppmaling, Oslo. They consisted of an old universal instrument, the so-called Olsen small microscope-theodolite, a magnetic declinometer and a sidereal time chronometer. The universal instrument has a horizontal circle with microscopes and is read by estimation to one second. Its vertical circle, fitted with verniers, is read to 10". The telescope has a broken axis and by means of. simple mechanism it can be reversed in the bearings. It has a filament, specially designed for trigonometrical observations. There is only one horizontal thread, whereas the vertical thread in the centre is double, and has on each side two threads, so that the filament consists of 6 threads in all for the observation of transits. The instrument has no ocular-micrometer. It is provided with a striding level to be mounted on the horizontal axis, and another level attached to the verniers of the vertical circle. The value of one division of the striding level is 4".

The chronometer which is adjusted for sidereal time is an ordinary chronometer of the type used on board ships. It has been made by Aug. Michelet, Oslo.

A description of the declinometer will be given in the chapter:

Magnetic Declination.

4. T h e p r o g r a m m e for t h e 0 b s e r v a t i 0 n s . The four problems we had to solve were:

1. Determination of the geographical longitude by means of star transits in the meridian and the time signals by wireless.

2. Determination of the latitude by means of star transits in the prime vertical.

-7-

The meteorological station in Husdalen, Hopen.

Astronomical station to the right.

Th. Askheim pho!. 29/8 1947.

Dr. H. Henie in the astronomical station.

Th. Askheim pho!. 29/8 1947.

-8-

3. Determination of azimuth of an object near the meridian by observations of the Polar Star.

4. Determination of the magnetic declination by observations of the sun.

In order to carry through this programme on the island Hopen we had, when planning the work, to consider the difficulties which are always connected with the observation of stars in the Arctic as well as the special difficulties we were likely to encounter on Hopen. These difficulties are due to the bright nights and the climate.

Only few stars can be observed at full daylight and with a small instrument. Even when the sun is low, one cannot count on observing stars of smaller magnitude than 2.5 m, and when the sun is near the meridian even the brightest stars will be invisible against the bright sky. Thus there will always be a comparatively long interval between the observations of stars, and it will be almost impossible to carry through a continuous series of observations. Provided one could count on a cloudless sky for some length of time it would, however, be feasible to follow some sort of a plan, but when a cloudless sky is seldom seen, the chances of making satisfactory observations is even further reduced. The only thing to do under such conditions is to keep the instrument in constant readiness, and as soon as the weather allows observations to be made, one must try to observe as many stars a possible.

The prevailing wind on Hopen comes from the northeast and always brings a heavily clouded sky, generally accompanied by fog and drizzle. A clearing of the sky can only be expected when the wind is north or north-west, especially by strong wind, and a wind from these directions never lasts long. On the east coast of the island there is also another obstacle to cope with, viz. the fog in the afternoon on clear days. Currents of air are coming down the slopes in the west, and as soon as the sun is low in the sky, they bring thick fog, which soon covers land and sea. Even on clear days when conditions seemed favourable for observations, the air appeared to be hazy, and the sky, seen through the ocular, presented itself absolutely white to the eye, so that no stars could be discerned. The unfavourable weather conditions in summer on Hopen are illustrated by the fact that during the 8 weeks we stayed on the island we had only eighteen hours allowing us to make our observations, but in the course of this short span of time we carried through the programme for the observations.

When preparing for the observations on Hopen we tried to make such arrangements that everything was ready for making observations at any time of the day. For this purpose pertinent data for stars were compiled in two tables, one pertaining to culminating stars and containing the meridian altitudes for 80 culminations, and one pertaining to stars in the prime vertical and containing the computed time and

- 9 -

altitude at transit across the centre thread and the outer side thread of the filament, also for 80 passages. In addition we prepared a table for stars with great transit altitude and a transit time near to � U rsae Minoris. This last table was intended for observations according to the Bessel Method, i. e. observation of the same star in east and west and reversing the instrument between the transits.

Determination of Geographical Longitude.

5. O b s e rva t i o n s o f t r a n s i t s i n t h e m e r i d i a n were taken between 6.00 p. m. on August 22nd and 1. 30 a. m. on August 23rd, G. M. T. At noon on August 22nd there was a northwest breeze, partly clouded with drifting clouds and fog. In the afternoon the sky gradually became clear. A thin veil extinguished the weak stars at daylight. These observations were made in the course of the evening.

and the night.

T a b l e I.

Transit Observations zn the Meridian 22nd-23rd August.

e ...

Cl. Lyra -y Cygn

CL Cygn

Cl. Ceph et U M -y Ceph Cl. Andr -y Cass (3 Andr

i ... . i ...

ei ...

aj ...

ei ...

0 . . .. . . . i ...^....

0 . . . ..

I I

E E E E E E W

I�

II

m s m s m

52 53,5 53 39,0 38 2.0 38 48,4 57 9,5 57 59,6 33 45,0 35 1,5 36 22 52,6 21 37,6 50 5,5 52 46,8 55 26 15,1 25 34,2 14 46,0

1

1

27 24,6 26 41,1

III

I

I

s

9,5 12,8

h m s m s m s In s

16 54 21,0 55 2,0 55 43.5 18 39 31,2 40 13,0 40 55,0 18 58 46,0 59 31,0 60 17,4 19 36 13,0 36 16,5 37 21,4 38 31,4 21 20 26,0 19 18,7 18 10,0 21 55 18,9 55 25,0

22 24 56,7

1 ¹

¹

1

26,5 III

I

The distance between the two threads - III and IV - of the double centrethread is so small that transits across both threads can only be measured for slow stars. For stars with greater velocity we observed the position of the star in the centre between III and IV. Observing the y Cephei we reversed the instrument in order to obtain the transit across two threads with the ocular in a different position. However, we missed the star. Observing y Cassiopeia we did not find the star till it was close upon thread V, so that we are not quite certain about the exact time. Observations were discontinued at 1. 30 a. m., when the Polaris became visible in the field, and the azimuth was determined by the aid of this star.

The transit observations were made under fairly favourable conditions. The noise from the breakers was rather disturbing. Both eye- and ear-observations were carried out with precision and composure.

-- la- T a b l e 2.

w E b

" Lyrae . .. . ... . . 33,5 3.2 +36,7 -0,12

3,9 34,6 -38.5

Y Cygni ............. 34,3 3,5 +37,8 +0,10

2,8 33,5 -36,3

" Cygni ........... 5,0 35.8 .. 40,8 -0,31

33,4 2,8 +36.2

" Cephei ......^{...... 34,8 3,9 +38,7 -0,27}

5,9 36,8 -·42,7

rr. U. Maj ............ 35,9 4,9 +40,8 -0,01 y Cephei ....^..... ... 5,0 36,0 -41,0

rr. Androm ........ . .. 35,9 3,1 +39,0 +0,09 Y Cassio ... . .. . .. . . 2,4 35.3 ·37,7

� Androm ........... 36,5 3,9 +40,4 +0,01

3,8 36,4 -40,2

The readings of the striding level are shown in table 2.

6. T he i n t e r v a l s o f t h e f i l a m e n t. The values of the thread intervals previously used for the Olsen small microscope-theodolite appeared to be of no avail. As it turned out to be infeasible to make special observations in order to determine the intervals, these had to be determined by the transit observations at hand. The values in table 3 have been computed from the differences between the centre of the filament, and the side thread, 9, expressed by the equation

i = 9 cos ()

Table 3.

Thread log i

l

to the upper culmin- 68,53 1,83588 ation, ocular east.

33,06 1,51930 31,79 1,50229

VI 64,20 1,80754

The two central threads are not shown in this table; we used the mean of the transits of these threads. The values previously used in 1923 are about one half of those which were determined in 1947.

Based on the above values the transit observations have been reduced to the centre of the filament with the result shown in table 4.

� 1 1 �

Bj 0 r 11 -

HOPEN Husdulen

'-+0 60 8� < 1°10 _____ , I 'Colltour" ,"ntervo I 2 )I)

Th. Askheim, 1947

- 12- T a b l e 4.

IF!

\

l

^\

¹

^I

I ex Lyrae

.... \

\

I'

1

\

\

1

1 \

\

\

9 � Androm ... W 2,4 23 ²⁶ 0,6 ^0,6 0,5 ^0,4 0,4 23 ^{26 0,54} +⁰,⁰1 9 Co-ordinates of the stars by culmination Hopen, Nautical Almanac 1947, are shown in table 5.

T a b l e 5.

I I "'1

^\

I

\

1

8 y Cassiop ... 0 53 31,94

1

9 � Androm .....^..^{. I 6 47,39}

Decl.

38044' ^10"

40 5 ²⁰ 45 5 34 62 21 46 1 17 57 41 S.P.

77 20 12 28 48 0 60 25 48 35 2^{0 30}

7. Co r r e c t i o n of c h r 0 n o m e t e r according to Greenwich Sidereal Time (G. S. T. ) was determined by means of wireless signals. The intention was to use the signals from Bureau International de I'Heure, at Pontoise. These signals are sent over one long-wave and the short­

wave bands, according to the new ONOGO-system and the rhythmic system. We did not bring any receiver sets with us to Hopen. We had in advance made inquiries whether the meteorological station disposed of the necessary wireless sets, and we were told that they had sets for all wave lengths. It appeared, however, that the wireless station on Hopen was out of order. The aggregate had been broken down and there was very little electricity left for the receivers. Not till the new crew arrived at the station with new machinery did the station come into normal operation.

The receiver on the station was unable to take the time signal in the long wave band. The short wave, 40. 38 m, gave the best results, but conditions were not favourable for reception. The signals were as a rule disturbed by strong telegraphic signals on a neighbouring frequency .

- 13--

The rhythmic signals could only seldom be discerned without great disturbances. The registration of the ONOGO-signals was somewhat better, and the correction of the chronometer was, therefore, based on these signals. Before and after the star observations on August 21 st and 23rd the correction of the chronometer was based on the entire ONO GO-signals. Before the departure, during the voyage -and our stay on the island, the chronometer was continually checked by the time signals from Oslo.

The correction and the rate of the chronometer on the days of observation were checked by series of 10 ONO GO-signals and gave this result:

Aug. 21st 19h 59m Os Aug. 23rd 19h 59m OS G. M. T.

17 56 10.44 18 4 3.55 G. S. T.

Chronometer 17 54 39.74 18 2 31.29 Correction + 1 m 308.70 + Im 328.26

Chronometer correction Aug. 21st 17h 56m 10S = Im 308.70

» 2 3rd 18 4 3 ⁼ I 32 .26 rate in 48h 7m 538 ⁼ 1 s.56

48h. 1314

rate per hour = 0.s03241 (8.51066)

As the first star transit was observed on Aug. 22nd 16h 54m G. S. T.

we have chosen as a starting point for the correction: Aug. 22nd 16h Om which gives us the following equation for the chronometer on the days of the observations:

Correction

=

T

h m s

1 16 54 21 2 18 39 3 2 3 18 58 47 4 19 36 13 5 21 20 27 6 21 55 18 7 22 24 57 89

1

T a b l e 6 .

I,

h

0,906 2,659 2,979 3,604 5,341 5.922 6,416 7,208 7,450

0,03 0,09 0,10 0,10 0,17 0,19 0, 21 0,23 0,24 31,42

COff.

m

+ 1 31,45 51 52 5 2 59 61 63 65 66

As a verification of the chronometer correction as shown in this time table we may mention the value of the correction derived from the

-- 14-

signals from Oslo. As we have been unable to obtain any information as to the accuracy one may ascribe to the Oslo signals, they have been left out of consideration in the following calculations. However, they give us a determination of time entirely independent of the ONOGO-signals, and may, therefore, serve as a verification and a control that no important mistakes. have been made in the above computation of the correction.

Signals from Oslo Aug. ²¹ st 911 55m 51.s75 G. S. T.

» 23rd ¹⁰ 3 44. 86 4811 7m53.811

Chronometer

9h 54m 21.s 2 10 2 ¹2^. 6

Correction I ^{m 30.s55}

32. 2⁶

----

Rate ^{I. 71} Computed correction Aug. 22nd ^{1611 Om} Oslo I^{m 31.60}

ON OG O Im ^{31.4 2} Difference ^0.s18 The result corresponds to the accuracy with which an isolated signal can be discerned. The accuracy of the chronometer correction has thus been confirmed.

8. T h e c o n.st a n t s o f t h e i n s t r u m e n t . The collimation constant was checked by pointing the instrument towards the meridian mark at the Koefoedodden. As the instrument is not fitted with an ocular micrometer the constant could not be determined, but its value was reduced as much as possible. The collective discussion of the observations showed that it was c ^{= Os.537.}

The inclination of the horizontal axis, b, was by each observation determined by means of the striding level. The value of one division of the level is ^4/1. It was difficult to use the level as it must be absolutely vertical to give a reliable result. The pivots of the axis were examined, but no inequality in the pivots was found.

Our plan was to orientate the instrument in the meridian by corresponding solar altitudes, considering the uncertainty as to the exact position of the island Hopen. But as the sun was visible only at very short intervals the instrument had to be orientated from single observations of the sun. On August 6th and ¹¹ th we had glimpses of a clear sky between drifting clouds and we observed a passage of Vega and also made an uncompleted observation of Polaris. They both showed that the instrument had to be turned slightly to the west.

The later complete transit observations showed that the instrument was now definitely to the west of the meridian, so that the azimuth constant was rather great. For the reduction of the observations we used a value a ^{= -748.5} which through the n ormal equations got a correction of ^{-08.41 O.} But even if the azimuth is considerable it is well within the allowed limit. This value was later con firmed by a

direct azimuth determination by means of Polaris.

- 15 -

9. C al c u l a t i o n o f t h e e qu a t i o n s . The calculation of the transit observations are made according to the usual method with corrections for the instrument constants. The transit time, corrected in accordance with the correction and the rate of the chronometer and the aberration which amounts to

- 0.80049. sec 8 is dessignated by T.

The reduction equation is:

oc�T+�T+a sin (9-8). sec 8+b cos ('.;,-8). sec 8+c sec8 . ,

where the last term changes its sign according to ocular east or west.

We give the equation this form

where we put

and obtain

oc=T + � T + A.a + B.b + C.c

�T=&+�D a=ao+�a

oc=T +It+ A.ao + B.b + �t}+ A. �a + C.c We can then calculate

T + A.ao + B.b =t whence A. �a + C.C + It-(oc-t) ⁼⁰

Putting &-(oc-t)=w we get this equation of condition for each passage

A. �a + C.c + �{} + w ^=0.

This forms the basis for the n ormal equation.

- 16 -

T a b l e 7 . Transits in the Meridian

I

1

1

76° 30' 0" ... 37 45 50 36 24 40 31 24 26 14 9 14

A 0,785 0,776 0,738 0,527

�I

₁

Observation time ... 16h 54m 21,s22 18h39m 31,838 18h58m 46, s30 19h 36m 12,886 Chronometer correction + 1 31, 45 I ^{31, 5 1} I ^{31, 52} 1 31, 52

Aberration ... ... ^- 1 ^{- 1 -} I ^- 1

T 16 55 52, 66

1

1

I

-74s,5 Aao -58, 49

1

1

1

I.

2.

3.

4.

5.

6.

7.

8.

9.

Bb - 0, 12 + 0, 10 -- 0, 37 - 0, 56

t 16 54 54, 05

1

1

1

a 18 35 10, 38 20 2O 21, 66 20 39 39, 68 21 17 a -t S- 1 I ^{40 16, 3340 16}

1

1

1

S--(a -t)=w -0,833

1

1

1

0,616 0.602 0,545 0,207 1,994 0,004 0,712 0,3 15 0,651

Equations of condition

I . 0,785 :3. a + 1,282 c +:3. f}-0,33 ⁼ 0 2. 0,776 � a + 1,307 c i- 6&- 0,48=0 3. 0,738�a + 1,4 17 c +�{}'- 1,23=0 4. 0,527 �a + 2, 156 c + �{}- I,IO ⁼ 0

Ta b l e 8 .

10. The 'normal equations.

Aw CC

°1

1,006 0,785 -0,259 1,644 1,282 1,014 0,776 -0.373 1,708 1,307 1,046 0,738 -0,908 2.008 1,4 17 1,136 0,527 -0,580 4,6i8 2,156 -3,012 1,412 1,737 4,550 -2, 133 -0,306 -0.067 0. 196 20,8 12 4,562 -0,963 0,844 0.675 1,302 -1, 14 1 -1,137 0,561 0,550 4, 105 - 2,026 -0,989 0,807 0,565 1,503 - 1,226

- 0,423 - 0,627

- 1,743

- 2,372

- 2,642 -13,32 1 - 0,913 - 1,986 - 0,858

4, 54 21, 64 17, 10

_ 1,s 10

-0,33 -0,48 - 1,23 -1,10 + 1,23 - 2,92 +0,80 +0,98 +0,70 Sum

1

1

1

1

1

1

1

1

, t

t

- 17-

22nd and 23rd August 1947 T a b l e 7.

5 6 7 8 9

" Urs.Maj. SP e y Cephei e

I

I

1

1170 57' 42"

77°

-- 41 27 42 -0 50

1

4

4 1

1,412 -0,067 0^,84

4

-1,598 4,56 1 0,768

1,947

-2,133 4,562 1,141 2,026 1,226

-O,sOI -0, 501 +0,s09 T

0,s09

21 h20m 26,s 66 21h 55m 18, s20 22h 24m 56, s?8 23h 12m 26 s,90 23h26m 0,554

+

-- I + 21 20 23 ° I ⁴⁰

31,

1

I ^{- 2 -} I - ^{1 -}

58, 26

I

79

1 ^{+4, 96}

₊

⁹⁸

⁷⁹ ₁₇

₊

13, 08

1

⁴

⁷

37

14, 77 1

+ 1, 523

I

+0,s98

5. 1,4 12 �a -2, 133 c + �{}+ 1,23=0 6. -0,067 � a + 4,562 c + �{}-2,92 ⁼ 0 7. 0,844 � a- 1,¹4 1 c + �{} + 0,80 =0 8. 0,56 1 �a -2,026 c + �{} + 0,98 ⁼ 0 9. 0,807 � a-^I,226 c +�{} + 0,70=0

The equations are

5,7 17 � a - 2,2 05 c + 6,383.6. {} + 1,603 ⁼ 0 - 2,205 � a + 42,280 c + 4, 198.6.{} - 24,867 ⁼ 0 6,383 � a + 4, 198 c + 9,000 � {} ^- 2,350 ⁼ 0 4 1,430 c + 6,660 � {} -24,249 ⁼⁼ 0 6,660 c + 1,874.6. {} - 4,14 0 ⁼ 0 0,803 � .{} - 0,242 ⁼ 0 The values sought are then

�{} ⁼ 0,30 1

c ⁼ 0,537

� a ⁼ -0,410

1 ⁴⁰

31,

I 32, 19 0, 11 0,01 32, 09 47, 39 15, 30

+ O,sJO

- 18 -

1 1. T h e g e o g r a p h i c a l l o n g i t u d e. Substituting the values of a, c, and b.(} in the original equations of condition, we find the residuals v as shown in table 9.

A.a .. . . ... . ... ^. C.c .... ... . Sum ... ^.

,,�t ... . Diff . ... . .. ... .

&+il& ... ^.

T a b l e 9.

I ² I ³ I ⁴ I ^{5 6} 1 ^{7 8}

I ⁹

I-O,321-0,321� 0,301-0,221-0,581 +0,031

0,351-0,231 � 0,33 1+0,691+0,701+0,761+1,161-1,151+2,451-0,611-1,091-0,66

1 ^,0,37 16,33 16,48 17,23 17,10 14,77 18,92 15,20 15,02 15,30 1 ^+0,38 1 ^+0,46 1 ^+0,94 1 ^-1,73 1 ^+2,48 1 ^-0,96 1 ^-1,32 1 ^--0,99 1 ^15,96 16,30 16,30 16,30 16.30 16,30 16,30 16,30 16,30 16,30 1 ^16,10 1 ^16,77 1 ^16,16 1 ^16,50 1 ^16,44 1 ^16,16 1 ^16,34 1 ^16,29

1+ 0,341

0,201-0,471

0,141-0,201-0,141

0,141-0,041

0,01

Thus we find the probable error as shown in table 10.

T a b l e 1 0.

v

+0,34 +0,20 --0,47 +0,14 --0, 14 -0,20 +0,14 -0,04

+0,01

[vv]

vv

0,1156 0,0400 0,2209 0,0196 0,0400 0,0196 0,0196 0,0016 0,0001 0,4770

As we have 9 equations with 3 unknown quantities, the probable error at the computation of the chronometer correction according to the local sidereal time of Hopen will be:

1/ [^{v v} 1

e = ± 0.67 r (9 ^{_ 3) . P COS? where p = 0.803}

e = ± 0.8049

The factor cos rp has been included to have the probable error ot the geographical longitude expressed in seconds of the great circle in the same way as the probable error for the latitude.

The values computed above give this result:

[} ^-� 1 ^h 40m 16 s, 00 b.{} ⁼ + 0 , 30

.:1 T ⁼ 1 ^h 40m 168,30 ± 0.s049

t

t

t

r I

I

V E

-19-

R

GiE

I

I Bb"

I

'\ -

-- �--- ----\-^{--- -- �8 •}

EDG[OYA

o

\

er o I--

IL _{- -- -}

AM'R"�;;l T

\ _\,

\ __ �76

- 20--

In the longitudinal difference between Greenwich and Hopen there will also be included an error depending on the accuracy of the registration of the Greenwich sidereal time. We have no exact figures here so it can only be estimated. As all time signals are registered by the ear only without any technical facilities one may, to be on the safe side, reckon with an average error of 0. s2. The total probable error in the longitudinal difference will then be:

The final result will then be:

The longitudinal difference Greenwich-Hopen 1947 1 h 40m 16. s30 ± 0.s058 (2504' 4."5 ± 0."87)

Determination of Geographical Latitude.

12. O b s e rv a t i o n s in t h e p ri m e v e r t i c a l were made on August 23rd between 17hOO and 23hOO G. M. T., under the same conditions as for the meridian observations. The work had to be discontinued as the sky became more and more cloudy as the night advanced. Gradually the entire sky was covered and we had fog and rain.

The preliminary work for the observations in the prime vertical also planned a special observation of � U rsae Minoris at its passage both east and west with a reversal of the instrument. The declination for � Ursae Minoris is 74° 22' and it is 8° from zenith in the prime vertical, and there is no star of any avail, which comes nearer. The declination for y U rsae Minoris is 72° and it passes at 12° from zenith, but it was so feeble that it was difficult to discern it. As late in the summer as the end of August, however, it was impossible to make observation of the passage to the east, as the sun was so near that the sky was too bright. The passage to the west was observed. Th e work, therefore, had to be limited to observations of the stars which we could discern. The transits of four stars were observed, one zenith star to the west and three stars to the east.

Transit observations in the prime vertical on August 23rd between 17hOO and 23hOO G. M. T.

- 2 1- Ta b l e 1 1.

Ocular N. 11

I

I

IIVI V

m ^s m ^s m ^s h m ^s m ^s 0 Urs.Min w . . . 22 5 17 51 13 58 15 13 49 13 40 l-r Androm e .^.....^. 9 25 19 1 0 11 ,5 o Cassiop e . . ^..^... 17 21,0 18 40,8 19 19 55,0 :x Persei e ... ... 43 49,5 44 47,9 20 45 42,5

The co-ordinates of the stars on August 23rd.

Tab l e 12.

Nautical Almanac.

Dec!.

I. � Urs. Min .. ^. 2. r Androm ... . 3. 0 Cassiop ... .

4. ^et. Persei .... .

1

I

I

1

30,"3 38, 1 37, 3 26, 2

m s

10 56,S 21 7,0 46 35,4

VI

m s

1 1 43,0 22 20,5 47 29,2

In order to examine the influence upon the computed latitude of

a possible uncertainty about the declination of the star or about the time observed, the equation referring to the prime vertical is being differentiated

which gives

tg cp ⁼ tg 0 ^. sec t

cos ^{Q •}

d ^Q . ⁼ cos 0 cos z ^{' .} d 0 + SIO ^{Q '} . tg z . dt

If this formula is applied to the stars observed we find the following values giving information about the applicability of the observation method:

Urs.Min .... ^. Androm .... ^. Cassiop .... ^. :x Persei ... ^.

T a b l e 13.

An uncertainty of 0," 1 in the decli­

nation gives an uncertaintv in ^the

latitude' of:

0,"11 0, 22 0, 19 0, 22

An uncertainty of O,s 1 in time gives an uncertainty in the latitude of:

0,"2\

1, 53 0, 75 1, 15

-22-

The first part of the differential formula shows two things. Firstly, because of the factor cos cp an uncertainty in the declination of the star will be less appreciable in polar regions than on lower latitudes. The figures show that an error in the declination of a star like y Androm, despite its distance from zenith of more than 53°, will not be more than doubled. If, therefore, we keep to stars whose exact declination is known, there is no reason to leave out stars passing the prime vertical even at a considerable distance from zenith.

The second part clearly shows the advantage of using zenith stars, but the figures also show that even other stars can be used safely.

There is also another fact speaking in favour of using stars at a greater distance from zenith, namely the fact that in polar regions, such as the Svalbard (Spitsbergen) region, zenith stars will always have a great declination. The result of this again will be that zenith stars pass very slowly across the field of the ocular. A star like � Ursae Minoris, passing 8° from zenith, takes more than 17 minutes from the first to the last thread. This slow passage across the threads of the filament makes the time observation less exact. This is especially felt when the threads are not particularly fine, as was the case with the instrument which we used. For stars with a smaller declination and a swifter movement the time observations are more reliable, a fact speaking in favour of using such stars.

Also another circumstance complicates the making of exact time observations in the prime vertical, viz. the oblique passage of the stars through the field. This is especially noticeable for stars with a great declination. A star like � U rsae Minoris falls off as much as 40' during its passage in the west.

Both these circumstances make it desireable to have a chronograph·

to hand for observations in the prime vertical.

13. Re d u c t i o n o f t h e o b s e rvations i n t h e p r i m e v e r tical.

The chronometer correction was on August 23rd 18,h068 + I m 32, s26 with a rate ^=.c 0,s0324 per hour.

The correction for passages through the centre of the filament is:

rate correction

I, ISh 13,m8 18,h07 - 2,1¹74 -0,s09 Im 32,s 17

2, 19 10, 2 + ^I, 10 + 0,04 30

3, 19 19, 9 -I- 1, 29 + 0, 04 30

4, 20 45, 7 + 2, 69 + 0, 09 35

As the transit through all threads of the filament was not achieved for all stars, a simplified, summary system of reduction cannot be applied.

We shall, therefore, have to reduce each individual transit to the centre of the filament. The transit is reduced according to the formula: