A PRELIMINARY ESTIMATION
OF THE ACCURACY
OF THE INNER PLANET'S CO-ORDINATES
B y D. K. K U L I K O V ( t ) .
RESUME. — L'auteur £tudie successivement les effets d'erreurs dans les
d^veloppements analytiques des theories des planetes, ou dans les
valeurs adoptees pour les masses, pour les elements et pour le temps des
£ph6m6rides, sur la precision des coordonn£es des planetes inf£rieures
calcutees a partir des d£veloppements et des constantes de Newcomb
(61£ments de Ross pour Mars). La precision est de cinq d6cimales.
La source principale d'erreurs sur les positions des planetes est l'impr^cision des Elements. On pourrait ameliorer considerablement les ephem^rides en introduisant des corrections appropri^es.
ABSTRACT. — In the construction of analytical theories of planetary
motion, the effect is considered of errors in the adopted values of masses,
orbital elements and ephemeris time upon the accuracy of computed
co-ordinates of the inner planets in accordance with Newcomb's expan­
sions and constants (with Ross's elements for Mars). These are accurate
to five decimal places. The errors in the orbital elements are the prin­
cipal source of inaccuracy in the positions of the planets. The precision
of the ephemerides can be considerably improved by introducing appro­
priate corrections.
ZUSAMMENFASSUNG. — Der Einfluss von Fehlern bei der Aufstellung
analytischer Theorien der Planetenbewegung und in den angenommenen
Werten fiir die Massen, die Bahnelemente u n d die Ephemeridenzeit
auf die Genauigkeit der berechneten Koordinaten der inneren Planeten
nach den Ausdrucken und mit den Konstanten von Newcomb (mit den
Marselementen von Ross) wird untersucht. Die Genauigkeit betragt
funf Dezimalen. Die Fehler in den Bahnelementen sind die h a u p t sachliche Ursache fiir die Ungenauigkeit der Planetenorter. Die Genauig­
keit der Ephemeriden kann betrachtlich gesteigert werden, wenn man
geeignete Korrektionen einfuhrt.
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i4o
D. K. KULIKOV.
Pe3K)Me. — ABTOP nocJieaoBaTejibHO H3y*iaeT 3<|><j>eKTbi norpeuiHOCTeft B
aHaJiHTHqecKHX pa3jio>KeHHHX B TeopHHX njiaHeT, HJIH B npHHHTbix
3HaneHHHx Mace H 3JieMeHT0B, HJIH eme B 3<J>eMepHjjHOM BpeMeHH, Ha
TOHHOCTb
KOOpAHHaT
BHyTpeHHHX
njiaHeT,
BblHHCJieHHblX
n0JIb3VHCb
pa3Jio>KeHHHMH H nocTOHHHbiMH HbK>K0M6a (HJIH Mapca, ajieMeHTbi
Pocca). OHH TOHHW 30 nHTH aecflTHHHbix 3HanoB. IIorpeiHHOCTH B op6HTajibHbix 3JieMeHTax HBJIHIOTCH rjiaBHOH npn^HHOH HeTOHHOCTH nojio>KeHHH njiaHeT. 3<j)eMepHabi MoryT 6biTb 3Ha*iHTejibH0 yTOHHeHbi, npnMeHHH HajjJioKamne nonpaBKH.
The National Almanacs, since i960, give the ephemerides of the Moon,
the outer planets and the day numbers with an additional decimal figure.
Apparent places of stars are computed more precisely. However the
accuracy of ephemerides of the inner planets has remained unchanged.
Besides it is to be noted, that more than a hundred years have elapsed
since Newcomb had worked out his theory of motion of the inner planets,
and the accumulated observational material remains unused in the
computation of ephemerides; whereas the demands for precision of the
astronomical ephemerides rise continuously. At the present time astro­
nomical almanacs are used not only in studies of the motion of the
bodies of the solar system, and in discussions of different kinds of astro­
nomical observations, but also in computations of the paths of space
rockets.
In accordance with the need for higher precision in astronomical
ephemerides by both theory and practice, especially in experimental
astronomy, it may be of interest to make an at least provisional esti­
mate of the accuracy of the ephemerides of the inner planets, based on
Newcomb's theory ([1], [2]), which are at present published in astro­
nomical almanacs.
The principal sources of errors in the determination of planetary
co-ordinates are the following :
1.
2.
3.
4.
Errors
Errors
Errors
Errors
in
in
in
in
the
the
the
the
analytical theories;
adopted masses of the planets;
determination of orbital elements;
determination of ephemeris time.
We shall discuss the effect of the enumerated sources of errors on the
heliocentric position of the planets, resolving the vector of error into
three components — along the radius-vector (r), in the normal to the
orbital plane (s), and in that direction perpendicular to the radiusvector, which lies in the orbital plane (r)r It is convenient to express
the vector of error in astronomical units of length (a. u.).
Errors in the analytical theories of planetary motion arise from
unaccounted or incorrectly computed terms in Newcomb's expansions.
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ACCURACY OF THE INNER PLANET'S CO-ORDINATES.
I/,I
These errors may be detected by a comparison of Newcomb's expansions,
representing an approximate solution of differential equations, with an
accurate solution of these equations, which can be readily obtained
within a given number of decimals by numerical integration.
In order that the co-ordinates computed directly from Newcomb's
expansions might be comparable with those obtained by numerical
integration, the initial conditions of motion and the parameters of the
differential equations must be consistent with the constants adopted by
Newcomb in his analytical theories of the motion of the planets. Besides,
care must be taken that the accumulation of errors in the process of
numerical integration should not affect the result within the stated
limits of accuracy.
To make the initial conditions of motion agree, we used the formulae
of Eckert-Brouwer in the form
(i;
A/- = C x / ^ / • C
1
+(r-^rjC
5
+ I Hr -+- K/v) C,-,,
where
II = :
K —
(/■ -+- i> \ /•/•
1
1
pa' n
The integration of the equations of motion for the period from 1961
to 1965 has been performed by Cowell's modificated method of quadra­
tures ([3], [4]), taking account of mutual perturbations, and pertur­
bations from Jupiter, Saturn and Uranus. An estimate of the accu­
mulation of errors in the numerical integration shows that these errors
are practically zero.
Thus, the discrepancies between the integrated values of the co-ordi­
nates and Newcomb's are the result of a deficiency in the analytical
theories of motion of the inner planets. The maximum discrepancies
between integrated values of the co-ordinates and Newcomb's in the
interval 1961-1965 are shown in the first line of table IV.
It may be seen from this table that the defect of the theory is espe­
cially great in the case of Mars.
A similar estimate of accuracy of the new theory of Mars by
Glemence ([5]-[7]) yields the following results :
io 8 I A r I < 22,
10s I
AT
I < 26,
io 8 | \s | < 10.
Now let us consider.the influence of errors in the values of masses
adopted by Newcomb. First these errors will cause an error in the semimajor axis and secondly they will affect the perturbations in longitude
and in the radius-vector. Perturbations in latitude are very small,
owing to the small mutual inclinations of the orbits, and are beyond the
limits of accuracy.
Symposium L\ A. I., n° 21.
10
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i/,2
D. K. KUL1K0V.
Periodic perturbations in longitude and in the radius-vector by planet i
are of the form
A /•/ = miS
a<k cos D,/,.
A-
where n\i is the mass of the perturbing planet.
Therefore we may write
|Am;|
A,-^~»
in
(2)
AT.'.
a( i - h 6M > B, I
i
I
i
i
where
On the basis of Newcomb's expansions we easily find the values of A,
and B/. They are shown in table I.
TABLE
Mercury.
1.
2.
U.
A.
5.
G.
7.
(i
Planets
lO A,-.
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
17.7
2.3
0.0
8.1
0.0
-
1.
Venus.
B
JO'B,.
I0 A,-.
r>7. o
IJ.O
0.0
36.<S
3.9
-
Earth.
10'B,.
o.<S
{">.•>.
1.6
9.0
o.\
-
3.8
176. j
10.7
3o.">
\>..\
:
10'' A,-.
o.i
'J1. 3
10.0
3/j.'>
1.9
-
Mars.
10''B,.
10" A,
o.3
8 *>. 3
-33. o
7>.6
4-8
-
10'B,.
o.o
r>. *>
iS">.S
<>.<>
71.7
'12.3.7
fifty. 8
20.0
o.">
72.4.8
27.6
1 .0
To estimate the errors in the values of planetary masses adopted by
Newcomb, the most reliable values of masses obtained by different
authors from independent determinations are summarized in table II.
TABLE
Author.
Year.
II.
\/m.
Prob. error.
Method of determination.
From the motion of the
Earth's perihelion
F r o m t h e motion of Eros
(1926-1945)
Mercury.
1. Glemence [8]
19I9
6 \oo 000
rt:20o 000
2. Rabe [9]
1960
6120000
— 43 000
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ACCURACY OF THE INNER PLANET'S CO -0RD1NATES.
TABLE II
Author
Year.
i43
{continued).
\/m.
Proh. error.
Method of determination.
: *-V>oooo
From secular perturbations
of Mercury and Earth
F r o m the motion of Venus
(1750-1949)
F r o m t h e motion of comet
Encke-Backlund ( 1 8 9 8 -
Mercury.
\\. Brouwer ([10], [I 1|)
19"><) (> iSoooo
A. Duncombe [12J
i9r>(i
5 970 000
f /j5o 000
5. Makover, Bochan [Bl|. . .
i960
5980000
-M70000
1911)
Weighted mean
Arithmetic mean
:
(> r>7 000
(> 190000
4° 000
- : 70000
Venus.
1. Fotheringham f 14|
i\y>.i\
.<
| ><> 'i'»S
:'■■ 7>.'i
2. Spencer Jones [1o|
i9'*>
4°{ 7<>n
^<)()
.*>. Fotheringham [1(>]
19T)
{08000
1 900
i. Morgan, Scott [17|
M)'*9
{07000
">oo
i). Clemence [18]
19IJ
{09 3oo
1 ioo
(>. Clemence [8]
19^9
{08 1 ~>o
7. Broilwer ([10, [ H ] i
i9)()
{08000
800
S. Babe [9]
M)M>
i<>8 (>i">
'■>■»*
{08 1 >o
{07 ">7o
1 3o
3:>o
Weighted mean
A r i t h m e t i c mean
Earth
>oo
From t h e molion of inner
planets
From the motion of t h e
Earth (i83f>-i()23)
From perturbations of the
obliquity of t h e ecliptic
F r o m t h e motion of t h e
Earth (1900-1937)
From the motion of Mer­
cury (1705-1937)
F r o m t h e motion of the
perihelia of Earth and
Mercury
From secular perturbations
of Mercury and Earth
From t h e motion of Eros
(i9«>J)-i9{">)
-h Moon.
\. Noteboom [19]
19> 1
3:>8 3 7 o
zh
<>9
2. Bosch [20]
199.7
327 9r)o
71:
'.>.oo
3. W i t t [21]
1933
328 390
:T:
IO3
A. Brouwer [11]
1930
3>8 \\-j
Z. Rabe( [9] : [22])
1954
3:>8 ', \(\
(\~j
:-
43
From t h e motion of Eros
(1893-1914)
F r o m t h e weighted mean
values of parallaxes
F r o m t h e motion of Eros
(1893-1931)
F r o m the relation of De Sit­
ter with />o = 8".7984
From t h e motion of Eros
(i9:>(>-i94r))
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ACCURACY OF THE INNER PUNET'S CO-ORDINATES.
TABLE II
Author.
(continued).
Method of determination.
Prol>. error.
l/m.
Year.
i45
Saturn.
2. Gaillot [35]
<
.
1913
3 499-9
rh
I.18
3. Bosch [20]
1927
3 49^
rh
3
4. Hertz ]3(>]
1953
3 497 M
±:
0.27
5. Jeffreys [37f
1954
3 494-8
" :
2.0
6. Clemence [38]
19O0
3 499-7
Weighted mean
Arithmetic mean
3 498.85
3 498.38
0.4
:. :
-j:
From the motion of Jupi­
ter (1750-1907)
Weighted mean from satel­
lites
From ihe motion of Jupi­
ter (1884-1948)
From the motion of satel­
lites (1924-1937)
From the motion of Jupiter
0.20
1.12
Uranus.
1. Hill [i!>]
1898
23 2*9
:-' ;
«9
2. Bosch [20]
1927
22 53o
rh
5o
3. De Sitter [2<»|
i. Clemence [8]
0. Harris [11]
1938
1949
1950
22750
22 800
r::
200
:
3(K)
Weighted mean
Arithmetic mean
()
22 934
22929
::
6
22 851
:■:
78
From the motion of Saturn
(1751-1888)
From the motion of satel­
lites
Weighted mean
Weighted mean
From the motion of satel­
lites
Rounded-off probable values of the masses of major planets, finally
adopted by us, as well as the relative errors of Newcomb's values of
these masses are shown in table III.
TABLE 111.
Probable values of
Planets.
Mercury
Venus..
Earth -f- Moon
Mars
Jupiter
Saturn
Uranus
l/m.
6 i5o 000
zh 5o 000
407 85o
-J- 25o
328 5oo
~:
40
3 089 000
zh 3 000
1 047.40 zh
0.02
3498.6 zh
o.5
22 900
zh
5o
Newcomb's values of
l/m.
Relative errors.
10s
0 000 000
2.5oo=bo.833
4o8 0 0 0
0 . 0 3 7 rh 0 . 0 6 1
329 390
3 093 5oo
1 047.355
3 5oi.6
22869
0.270 z h o . o i 3
0 . i 4 5 zh 0 . 0 9 7
0 . 0 0 4 zh 0 . 0 0 2
0 . 0 8 6 zh 0 . 0 1 4
0 . 1 3 6 zh 0 . 2 1 9
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ACCURACY OF THE IxNNER PLANET'S CO-ORDINATES.
TABLE H
Author.
Year.
i/45
(continued).
\/ni.
Prob. error.
Method of determination.
Saturn.
191]
3 {99.9
r
\\. Bosch \-i)\
19 >7
3 ;«)()
3
i. Hertz ]3<>|
19V3
3 497.f> j
o.>/j
.3. Jeffreys [37|
upj
3 49J.S
>.o
(>. (llenience |->S|
io,<io
3 499.7
±. Gaillot [3.3]
.,
,s ,s
Weighted mean
Arithmetic mean
3 i9 - >
3 jyN.3N
-1,s
(
>- i
From t h e motion of Jupi­
ter (i-5o-i907)
W e i g h t e d mean from satel­
lites
F r o m t h e motion of Jupi­
t e r (1884-1948)
From t h e motion of satel­
lites (199.4-1937)
From the motion of Jupiter
o. ><
.>
1 . r.>
Uranus.
1. H i l l \~i\)\
189N
2 . P>osch \~H)\
1 «)->7
•.'.•■» V)(>
.3. De Sitter \~H\\
■i. Glenience [S]
19)8
19{9
■>■>. 7 " > o
•>•> 8<><>
.3. Harris [1 l|
1 <) "><>
:>•>. 9 3 j
Weighted mean
Arithmetic mean
>3 ■>: )<>
•>•>. 9>.<j
:>.->. <s*n
«S()
H)
><>o
3<><>
<>
From the motion of Saturn
(1751-1888)
From t h e motion of satel­
lites
W e i g h t e d mean
W e i g h t e d mean
From the motion of satel­
lites
(>
7N
Rounclecl-oiT probable values of the masses of major planets, finally
adopted by us, as well as the relative errors of Ne\vcomb\s values of
these masses are shown in table III.
TABLE I I I .
Probable values of
Planets.
Mercury
\ enus
Eartli-f- Moon
Mars
Jupiter
Saturn
I ranus
\/m.
(> i5o ooo
1078m
3->,8 ;M>O
3 089 000
IOI7.|0
3 {98.0
>:i 900
:
:
Ncweomb's values of
\lm.
3o ooo
'>.:"><>.
{0
3 000
o.o>
o.~>
5<>
(> ooo ooo
408000
3:>9 3 9 0
3 093 5oo
10.17. 3 V>
3ooi.()
±:>. 8 O 9
Relative errors.
I \ 1
10-— \ •
| m |
:>. 3oo : ; o.833
0.0J7
o.ofu
o.:>7o : : o . o i 3
o. 1 \3 -_-'■- 0.097
o.oo4 : : <>-oo>
o.o8(>
0.01 \
o . i3<>
o.:>K)
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D. K. KULIKOV.
146
According to relation (2) and the numerical values listed in tables I
and III, we may easily find the maximum errors in the position of inner
planets resulting from errors in the values of the masses.
They are represented in the second line of table IV. Large errors
in the position of Mars are due mainly to an error in the Earth's mass.
The effect of errors in the masses upon the semi-major axes, according
to Kepler's third law, does not exceed some units of the ninth decimal
place, i. e. it amounts practically to zero.
The following relations were used to estimate the effect of errors
in the elements of the planetary orbits upon the accuracy of heliocentric
positions of the planets [39] :
1 1 1
/
TT
I A r I ^ a\/(e&L
I AT I ,y a (i -+- e)[±L
I A.v I :_< a(\
1
•)-
—1
— cAm)--h
- I A II i
(Ac)'- -+- -~ a ( 1 -+- e)
5
-+- :> \J(e A L — e Aw)* -4- (1 -+- e) ( A e )-'].
-+- e) ^(Myt-h
(sin* ±Q)-:
AL = AL (I + TA/>.
Corrections to the orbital elements in formulae (3) can be found only
from a discussion of series of planetary observations.
During 1940-1950 discussions of observations of the Sun (1700-19 H ) ,
Mercury (1762-1937) and Venus (1700-1949) were made by Morgan [40],
Clemence [18] and Duncombe [12] respectively as a result of which correc­
tions were obtained to Newcomb's values of the orbital elements.
We shall make use of the new theory of the motion of Mars by
Clemence [6] for an estimate of the accuracy of the orbital elements
of this planet.
Thus we get the following table of corrections to Newcomb's values
of the elements with their probable errors.
Mercury.
// + o.o«S
//
A L» = + 0.'j.'i
Venus.
A n = -4- 0.08 r;T ° • * 4
AL»
In
A e = — 0. 4o zh 0. o4
\e
e Anr = - ( - o . i 9 z b o . o 5 - t - o 2 1 T
sin;'AQ=
o . o o ± o . o 6 + o. 19 T
A/ = + o . o ( ) ± o . o 7
e ATU
sin iAl>
li
-+- o . l o ± o.o(')
-h
— 0.1:'. ± o . o 3 + u.oi T
-h o. o 1 ± o. o4 — o. t>4 T
-h o.:>.i ± o . o i -h 0.02 T
-+- o. 08 -+z o • o'i — o. 02 T
Mars.
Earth.
AL 0 = -f- o.o3 ± o.o5
A/i= — o.iizho.16
A e = — o. 12 db o. o 1
eAnr = — o . 0 7 z f z 0 . o i — 0 . 0 8 T
o . Vi ~z 0 . 1 8
AL 0
A/?
Ae*
— 5. 5o z t o. 11
+ i 3 . 2 ' > ± o . 14
-+- o. 18 zb o. 06
e\m
sin i\Q
M
-+■ o . 30 zb o . o4
— o.iozbo.08
-+-
O.OJ
zb 0 . 0 7
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13/J
TOTAL
rh
TOTAL ERROR
1 "*
I'I
-~_ 1
^
Planet, masses
Orbital elem
ERROR
91
<>
Ephemeris time
•>.
Planet, masses
Orbital elem
/j i
A/-.
Analyt. theory
S o u r c e s of e r r o r .
rz
x)
ii*
><)•
AT.
Mercury.
zr> 1
18
1
errors
Fortuitous
o
<)«
1 *)
errors
3S
Systematic
IV.
A.v.
A/.
lot)
<s<>
111
«X">
■'■-.
'S
o
(in 0.0000 oooi a. u . ) .
rj
■17
l >
8
(in o.oooo oooi a. 11. 1
AT
Venus.
TABLE
1o
X)
")(S
?.()"{
iSS
AT.
Eiirth.
o
o
0
13
o
A.V
"j(i
i7<)
>\\\
A,.
;
:i8o
1 So
1 '>■
Sio
3')
z
^
S
o
»■<
:" 8*9
9
—
:«<)
17<S
-
C/5
o
o
O
i
^
«- £
» g
<)<>
IS
A.v.
o
c°
!>-*•^
d
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D. K. KULIKOV.
118
The time T is reckoned in Julian centuries from 1900 Jan. o Greenwich
mean noon. The correction An applies to the mean motion in a Julian
century. Corrections to the mean longitude and mean motion of the
Earth are taken according to Clemence [18]; they agree well with the
correction to the mean longitude for the epoch 19^0 found by Kulikov [41]
from a discussion of observations of the Sun (1925-1953).
An estimate of errors in the planetary co-ordinates for the epoch 1960,
resulting from an inaccuracy of the elements, has been made on the
basis of the above data and relations (3) ('). These estimates are shown
in the third line of table IV, the probable errors of the estimates being
found by means of formulae [39] :
a\
e-i;t-+-
y
' i >
c-i-n-¥- z;.-i
— £-.
9 n-
\ ( 1 -+- ■>■ ej- zf - h 4 r11% -+- /, ( i -+- e)ff..
a(i-hc)
« ( ! - + - e) V V + >in-izr1:
s f . = sf (i H-
T-zfn
where 3/ is the probable error of element /.
The fourth line of table IV contains the maximum errors in the posi­
tion of the planets, resulting from an error in ephemeris time. At present,
ephemeris time is known only approximately, and we may assume that
the error does not exceed 2 s.
The total effect of all the above sources of error is shown in the fifth
line of table IV.
Thus the provisional estimation of accuracy of planetary positions
computed strictly in accordance with Xewcomb's theory on the one hand,
and with Newcomb's theory with the corrections of the elements on the
other, may be presented as follows :
Mcrcurj
Theory and elements
of Newcomb.
I o s I A /" ; • '.' I ') 1
s
K) j
AT
I -' \ 1 \
10 s j A.v ; ■:
10 s I A/^ 4> r i : * »
10 s I A T ; • " 119 -I- :H>
I >
;
JO
u
10 s I A 5 , <
:
j(i
10 s I A r j < i'>.- 1 : ")()
38 -
10 s I A r I < 43(i :
Theory of Newcomb.
Elements of ("lemonee.
f> : : - ,>i
Venus.
Theory of Newcomb.
Elements of Duncombe.
Theory and eleme; ts
of Newcomb.
10 s j \ r I <
98 : -
18
10 s j A T j < \->> --■-
J;
1 0 s I A A- I < 109 =•: 1")
io
8
I A r j < \\-
r :
(V>
8
io
|Arj<
10 s |
AT
48 ~-
18
1 < 1 - 4 ™- 5 7
10 s j A .9 j <
:>o zh
ir>
^io I A r ; < i8-.> ±
(v>
8
(') T h e e s t i m a t i o n s for M a r s a r e o b t a i n e d b y d i r e c t c o m p a r i s o n of t h e c o - o r d i n a t e s
from C l e m e n c e ' s [7] t h e o r y w i t h t h o s e from N e w c o m b [1],
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ACCURACY OF THE INNER PLANET'S CO-ORDINATES.
IVJ
Earth.
Theory of Newcomb.
Elements of Morgan,
Clemence, Duncombe.
Theory an d eleinenls
of Newcomb.
10 s | A/- i < \\ \
10s |
AT
10 s | A.v
: • •>(i'i
:
10 s 1 A T i • '
~n)
1 ); ;
10 s | \ T \ ■: O.Sf)
io8 ! \ r '
11
•
O
■ C)()
31 -J-: 11
- 5 : - : - 5()
IO s I A S j • ^
I 3 ; ■■;
10 s I A r
<S:> : ; (io
:
(>
Mars.
Theory of N e w c o m b .
Elements of R o s s .
Theory and elements
of Clemence.
io« j A/-' < 77 :-!- 4<>
i<>« | A H ■ ; 1 ; D ±
i<i
10 s | AT
• ; N i o : : l<So
i o N ! A T I ;; ■>(>(> - ± 1 8 0
10s ! A.v ; - ; 1 78 - ■-: 89
I(js j A \ j
IO ; : ; 8()
10 s | A/•; ■. <)">7
10s I A / ; ;
>77 : ■ :>o()
; :>o(>
It may be seen from these data, that the heliocentric planetary co-ordi­
nates computed in accordance with Newcomb's expansions and constants
(with Ross's elements for Mars) are accurate to only five decimal places.
However we must bear in mind, that this estimate of accuracy of
planetary co-ordinates is considerably overstated. It may be seen from
table IV, that the main source of errors in the planetary positions is the
inaccuracy of the orbital elements. Introducing appropriate correc­
tions into the elements we can essentially increase the precision of the
ephemerides of Mercury, Venus and Earth. As to Mars, the new theory
of its motion by Clemence ([5] to [7]) must be used to improve the accu­
racy of its ephemeris.
However the most efficient solution of this problem of a higher accuracy
of ephemerides would be a complete revision of the theories of motion
of the inner planets.
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