SIMULTANEOUS DETERMINATIONS
OF LONGITUDE AND LATITUDE
IN AFRICA WITH THE PRISMATIC ASTROLABE
b y G u n th e r Z i m m e r m a n n
In s titu t fü r A ngew andte Geodäsie, F r a n k fu r t/M .
P a p e r s u b m itte d by the G overnm ent o f the F e de ra l R e p u b lic o f G e rm a n y
to the S econd U n ite d N atio ns R e g io n a l C arto g rap hic Conference for A fric a ,
T unis (T u n is ia ) 12-2i Septem ber Í966.
In th is paper the a u th o r describes a stro n o m ica l d e te rm in atio n s of
control p o in ts w h ic h he carried o u t some years ago in connection w ith
technical aid projects in L iberia. The purpose of the w ork w as to establish
by m eans of a stro no m ica l observations the hom ogeneous systems of control
p oin ts requ ire d for the ph o to g ra m m e tric p re p a ra tio n of large-scale p la n n in g
m aps of the regions n o t yet surveyed.
The execution o f this w o rk revealed th a t the observation a n d e v a lu a ­
tio n procedures, as w ell as the in s tru m e n ts , involved certain re q u ire m e n ts
w h ic h are fa irly ty pical for astro no m ic controls in the regions of developing
countries th a t are either n o t yet surveyed or else isolated geodetically.
For this reason some fu rth e r details on th is subject w ill be given.
The fo llo w in g circum stances characterise the s itu a tio n :
(1) The astro n o m ica l observations h a d to be plan n e d , prepared, per­
fo rm ed a n d evaluated w ith in a relatively short period of tim e , i.e. w ith in
the in te rval betw een the ph o to g ra m m e tic flig h t over the area to be m a p p e d
a n d the a e ro tria n g u latio n .
(2) D u e to the d iffic u lt o perational c o n d itio n s (i.e. the flig h t to the
w ork site, the use o f bad roads d u r in g field w o rk , a n d the fre qu e nt tra n s p o rt
of e q u ip m e n t b y carriers) the in s tru m e n ts s h o u ld be lig h t, s m a ll a n d
sturdy, b u t they sho uld nevertheless guarantee very accurate a stro n o m ic a l
d e te rm in atio n s of position,
(3) As the p la n n in g , p re p ara tio n a nd c o m p u ta tio n of the observations
h a d to be pe rfo rm e d far fro m the a ctu al site o f w o rk — n a m e ly in
G e rm a n y — it w as necessary to determ ine, d u r in g the stay on each station ,
w hether or n o t the observations o f the stars really supplied the re q u ire d
accuracy for the coordinates. T his check h a d not o n ly to be rigorous, b u t
also so easy th a t it could be pe rfo rm e d q u ic k ly u n d e r the d iffic u lt
o pe ratio n al c o nd itio ns in v irg in forests u n d e r tropical ra in a n d w ith o u t
special a u x ilia r y m eans, a n d by a person still inexperienced in a stro n o m ic a l
c o m p u ta tio n s.
These re q u ire m e n ts led to the decision th a t longitudes a n d la titu d e s
of the co ntro l p o in ts sh o u ld be s im u lta n e o u sly determ ined by m e ans of a
p ris m a tic astrolabe by the m e th o d of e qual altitud e s of stars. The
observations, how ever, co uld n o t be evaluated b y the u s u a l direct positionlin e m e th o d because it was im p rac tica b le to com pute the in d iv id u a l
position-lines o n the spot w ith the re q u ire d accuracy, due to lac k of
tim e a n d the necessary a u x ilia r y m eans. T hus we h a d recourse to the
in d ire c t m e th o d ; i.e. we h a d to pre-calculate the tim es of tra n s it o f selected
stars at the k n o w n ze n ith d istance o f the astrolabe used, u s in g the
a p p ro x im a te values o f the s ta tio n ’s coordinates. B y u sing this m e th o d it
was possible to process the observations in the field w ith sim ple a u x ilia ry
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The special econom y of the observation a n d e v alu atio n procedures,
as developed a n d tested by the a u th o r, m a y be fo u n d in the fact th a t for
all the stars in questio n the exact ephem erides of the tra n s it tim es at the
in s tr u m e n t’s z e n ith distance o f 30°00,40" have been com puted in advance
by a n electronic c o m p u te r for each observation station. T hus the e v a lu a ­
tio n could be based on the differences betw een the c o m pu ted tim es
— referred to the *Greenw ich m e rid ia n — a n d the tim es d ete rm in e d by
c hro no m ete r a n d stop w atch.
The ephem erides h a d been co m p u te d at 10-day intervals, so th a t the
tra n s it tim es co rrespo n d ing to the a ppro x im ate values o f lo ng itude , la titu d e
a n d ze n ith d istance c o uld be q u ic k ly d ete rm in e d by in te rp o la tio n for the
v ario u s observation days, w h ich , n a tu ra lly , were not k n o w n in advance.
T h u s the rig o ro u s e v alu atio n , u p to the observation equations, c o u ld be
p e rfo rm e d in the fie ld itself a n d provid e d a p p ro x im ate values o f clock
correction a n d of latitu d e . The setting up of the n o rm a l e q u atio n s, the
d e te rm in a tio n o f the corrections in la titu d e a n d lo ng itude , as w ell as the
c o m p u ta tio n o f errors, were a ll pe rform e d after the te rm in a tio n of fie ld
w ork . It is w o r th m e n tio n in g th a t the pre-calculation of an ephem eris
by a Z use Z 23 electronic c o m p u te r (in c lu d in g the p rin tin g ) took a p p ro x i­
m a te ly one h o u r per station, w hereas the e v alu atio n of the observations
re s u ltin g in the fin a l coordinate valu e s，a n d p e rform e d after the re tu rn
fro m A frica, took a p p ro x im a te ly 40 m in u te s per station.
T w o exam ples w ill serve to dem onstrate the scope of the w o rk
connected w ith the a fo re m e n tio n e d procedure. The first exam ple concerns
the p h o to g ra m m e tic p re p ara tio n s for p ro d u c in g p la n n in g m aps at scale of
1/10 000 fo r the e x p lo ita tio n o f an extensive deposit of iro n ore situate d
a p p ro x im a te ly 80 k m no rth east of M o nro via, th e L ib e ria n capital.
F o r the execution o f this pro je ct six a stro n o m ica l stations，a m e an
reciprocal d istance of a p p ro x im ate ly 4.5 k m apart, have been established
in order to fix the area o f m in in g ge og rap hically a n d to co ntro l the
a e ro tria n g u la tio n . D u e to the b e g in n in g of the ra in y season the w eather
c o nd itio ns were extrem ely b ad, since each evening the dew-point w as
alre ad y reached tw o h o u rs after n ig h tfa ll at the latest. The astro n o m ica l
observations h a d therefore to be carried out w ith in these few evening h o u rs
before a heavy g ro u n d
cloudiness.
fog set in ,
follow ed
shortly
after by
com plete
A lth o u g h the stay in L ib e ria totalled 40 days, the m easurem ents co uld
o n ly be m ade d u r in g 13 nig hts, w ith an usable observation tim e of 35 hours,
in the course o f w h ic h 394 star transits were registered by m eans o f a
W ild A strolabe w ith a m e rc u ry horizon. T he results of the aerotriangulatio n disclosed th a t, w ith an average o f 66 stars per station, a m e a n accuracy
of ± 15 m for the coordinates w as obtained. F o r fu rth e r details see the
recent p u b lic a tio n m entio ne d in reference [1].
The second exam ple concerns the p ro d u c tio n of 11 sheets of the
T o po g ra ph ic M a p o f L iberia, at a scale of 1 /20 000，w h ic h were p u b lish e d
in 1963 by the R e p u b lic of L ib e ria in cooperation w ith the Federal R e p u b lic
of G erm any, as a basis for the to w n p la n n in g of M onro via. I n th is case
seven a stro n o m ic a l stations were established w h ic h (together w ith the
n in e baselines w hose directions were d e te rm in e d by astro n o m ica l m easure­
m ents) served as a basis for o rie n ta tio n a n d scale checking o f a n aerial
b lock tria n g u la tio n . T his stay in L ib e ria lasted o n ly 20 days o u t of w h ic h
16 n ig h ts co uld be used for m easurem ents, w ith an usable observation
tim e of 44 h o u rs, in the course of w h ic h 853 star transits were registered
b y m e an s of a Zeiss Ni2 Astrolabe. W it h an average of 122 stars per
sta tio n a m e an accuracy of ± 9 m for the coordinates w as obtained.
I n the fo llo w in g table the characteristic d a ta for b o th projects are
listed by w ay o f com parison.
Y ear ..........................................................................
P r o j e c t .....................................................................
1961
Bong-M ining
1962/63
T ow n
p la n n in g
of M onrovia
M ap scale ............................................................
M ean distance between control stations (k m )
1/10 000
4.5
1/20 000
18
Type o f astrolabe ...............................................
N u m b e r of observation nights ....................
N u m b e r of observation h o u r s .........................
N um be rs of stars ...............................................
W ild-T 3
14
34.9
376
Zeiss-Ni2
16
44.1
853
N u m b e r of stars per h o u r .............................
M ean n u m b e r of stars per s t a t i o n ................
M ean accuracy of coordinates ( m ) ................
10.8
66
土 15
19.4
122
土9
In c onclu sio n it sho u ld be m e n tio n e d th a t the dispersion of the evening
values o f lo n g itu d e a n d latitud e , co m pared w ith the m e an values re su ltin g
fro m the w hole a d ju s tm e n t, is of exactly th a t order of m a g n itu d e w h ic h
s h o u ld be theoretically expected. As the in s tru m e n ts h a d to be m o u n te d
a n d d e m o u n te d before a n d after each observation it m a y be concluded
th a t the results do not c o n tain a n y noticeable errors due to the tran sp o rt
o f in s tru m e n ts .
REFERENCE
[1 ] Z im m erm ann , G. : A stronom isch-geodätische A rbeiten in L ib eria. D t.
Geod, K o m m is s io n , V e rö ffe n tlic h u n g R . B, H. 117，F r a n k fu r t/M ,
1965.