University of St Andrews
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A DETERMINATION OF THE ORBIT OF THE
ASTEROID 522 HELGA
by
DIMITRIOS N. PAPADAKOS
A Thesis presented for the Degree of Master of Science
in the University of St Andrews
May 1977 TO MY PARENTS
NIKOAAOE - ETTTXIA
TO MY SUPERVISOR
TADEUSZ
D ^ / 3 I "o 3/ El irpwrov un GUTUxncrw, au0LS au TraALV TreLpaaai. PLATO DECLARATION
I hereby declare that the following Thesis is the result of work carried out by me, that the Thesis is my own composition, and that it has not previously been presented for a Higher Degree. The research was carried out at the University Observatory, St Andrews.
D. N. Papadakos CERTIFICATE
I hereby certify that Dimitrios N. Papadakos has spent seven terms at research work in the University Observatory,
St Andrews, that he has fulfilled the conditions of
Ordinance No. 51 (St Andrews) and that he is qualified to submit the accompanying Thesis in application for the Degree of Master of Science.
T. B. Slebarski ACKNOWLEDGEMENTS
I would like to express my thanks to the following:
Mr T. B. Slebarski, my research supervisor, for his constant interest and valuable advice and guidance, which were always freely given during the period of the project. Professor D. W. N. Stibbs, Director of the
University Observatory, for the facilities made available to me and helping me also in several other ways.
Thanks are also due to the Staff and Research Students of the
University Observatory (particularly to Mr D. M. Harland) for helpful discussions, and to Miss S. Nockolds for the typing of this disser¬ tation. CONTENTS
Page
Chapter 1 1
Chapter 2 14
Chapter 3 . • •• •• •« • ••• •. •• 30
Chapter 4 •• • • 78
Appendix A-l .. .. - .. 127
Appendix A-2 131 INTRODUCTION
There are two orinci'oal themes of this work*
Firstly, the application of the Gaussian and Laplacian methods to determine the preliminary orbit of the minor planet 522 Helga.
Secondly, the application of the power series method to find:
(i) The velocity vectors of the planets of the Solar System at a given time.
(ii) The determination of the real paths of these planets and hence the path of the minor planet 522 Helga.
By applying the latter method we are able to determine the mass of the minor planet (l.io""1^ solar masses) and to construct its ephemeris.
The calculations were carried out by using computer and all programs are available.
The photographic plates used in this work were taken with the
James Gregory Telescope at the University Observatory (St Andrews) during the observing season of 1971-72. 1
Chapter 1 Gaussian Method
The solution of the problem of the relative motion of two bodies, which can be regarded as particles, contains six arbitrary constants of
integration. They are:
^ jhz - components of angular momentum in a given
rectangular system of reference .
c - energy constant .
Uj - orientation of the major axis of the orbit
/ - time when the bodies are at the least
distance from each other.
In problems related to planets and comets it is more convenient to use six elements, which are independent functions of the constants of integration. They are:
£ - eccentricity of the orbit -
OL - semi-major axis of the orbit-
l\o- mean anomaly of a planet at a chosen epoch
to
uj - argument of perihelion (orientation of the
major axis).
Z - inclination of the orbit.
Q - longitude of ascendinal node .
Of course, the choice of the above orbital elements is not unique. It is possible to replace this set by another.
The reduction of a set of observed topocentric co-ordinates
oil (i=l^j2Jb) of a minor planet to obtain its orbital elements consists the problem of the initial determination of the orbit of the minor planet. Such a solution gives a satisfactory representation of the real motion during a relatively short interval of time, as long as it is possible to neglect the action of the other planets. Therefore, the 2
observations must be separated by short intervals of time.
Let ^=(C- \Z. |C) Ci=l2j3) be the heliocentric radius vectors of v> 0L j> J the minor planet at three instants tl (i = {,3-JbJ . Similarly, let R L E
( R.. R.j Hr-j and P.sfP. P P)be the topocentric radius vectors of the Sun
and the minor planet respectively at the same instants. Since the orbit in
Keplerian mechanics, neglecting the action of the other planets, is a plane
passing through the origin of the heliocentric co-ordinates we can write:
\\- Cx fl f C3 17 o a-1) 'v ^ ^
which states that the three position vectors iT, fi r» are dependent, so
the position vector YT^at any time is some linear combination of the
radius vectors at ~Li and The equation (1.1) gives
(£*5)= c>( <7*5) 0
Hence - (1»2) c=-M- > a=[rr^] [p.d in^j
where, in general, [.nXjj (Lj j=1 2 b Lfj) stands for the area of the
triangle formed by R • C's are known as the "triangle ratios".
The vectors YTjRij f3- defined above, are related by
~ " Pi" ~L" (i=1^ • (1'3)
Combining the expressions (1.1) and (1.3) we obtain
= Cip-p+C^p'—s-' -"v Ci ' jRi~
which is one of the fundamental equations of the Gaussian method.
Operating upon both sides of the equation (1.4) by*Cpx p) }
where p_ = p/ jp (l=i_,203) j we get
-
^ pp-cpp)r3i /-s3" =CiBi-fgx^)-Ra-(gx|)+aRi-(gxg).~ (1«5)
Since the jO = (coS^l COSC££j cos^L and K( (1=123) are known, the equation (1.5) reduces to an equation of the form
toA o - MA2.C1+ IaA x iVifCi (1.6)
Consider now the triangle formed by the gun^the observer and the planet
at A 2 . This has sides of and length (I p and thusjwe can write
^s "2?,l3 (1.7) which, since e and f\2 are known, is an equation of the form
(1.8) V2 - g1 + vJn g + iVia
The first step in the computation is to calculate the coefficients Wi j W/ij iVij IaG j ]a) 11 and ^2 using the topocentric direction cosines of the
planet, obtained from its observed topocentric co-ordinates OilJbl U=l,2,bR
and the topocentric rectangular co-ordinates, obtained from the geocentric
tables, corrected for parallax and reduced to the place of observations by introduction of appropriate corrections.
Yife can now proceed to obtain first approximations to the 17 f L= lJ2Ji).
In order to do this we require to know Ci ana C^, . From the expansions of the co-ordinates of the body in convergent Taylor's series we get
Ti . , , - (it'll) , TtfTzT-.-TPUr] Ci X, 1+ T 6rr- ^ [jrl i'U (1.9) and rjTzT.-TDlJr C3=Ak. 1.1 CtA-TA) , + Tz 6VS d- r* dx)t:
where H= kAAi "ti) . Ti= k(~ti O ~U,= k ( {.z ii) are the modified timeSj and k is the Gaussian constant. Since terms with dn ' dX/f = {2 are unknown, in the first approximation we have to neglect them. Thus we shall set
Tx TT22- Tz _ T5 tI-T; Cl- 1 + C3- 1 + (1.10) T, r Tz G'Q5 Clearly, since i and are known, the elimination of Ci and Cs
in equation (1.6) by substitution of the above expressions (1.10) will
result in an equation of the form
P = V)<, ~h . (1*11)
It is obvious that we can obtain values for p and ^ by simultaneous <2
solution of the equations (1.11) and (1.8). This is accomplished by means
of successive approximations. We can take fc=2.8A .U. as first approxi¬
mation since this is the mean distance of the minor planets from the Sun.
This value is used in equation (1.11) to obtain a value of p which is
then inserted into equation (1.8) to obtain a second approximation of IT.
The obtained value of fx is substituted into equation (1.11) and the
process is continued until successive approximations of p and fz are identical to the required accuracy.
Since equations (1.10) are nothing more than preliminary estimates
of the triangle ratios Ci and Cj , the Xf and p obtained by the
solution of the equations (1.8) and (1.11) are only approximate. It should
be noted that the coefficients Wlx and \V13k in equation (1.8) are exact
and they do not change in the subsequent computation of JT CL — 1jZjb) .
The next step in the reduction is to use the value of l2 to obtain
first approximations for Ct and C3 by means of the expressions (1.10).
After these unknowns are determined, we may return to the fundamental
equation (1.4) to obtain p and p . The equation is used in the form Ci |5 jp + |p — CiRxi -"R.-** + Ci ~R*i +j P [p Ci |p |p+G.|p I p=CiLf R3a + CiKas+ p Jp (1.12) I laiMi I I 15 1 vla i li ^
whereGiEifcCirCiB"_Tl2;+, IjlJ ^coTbl cosccl j p. | x cossl slti oLi ftj xsi'n'bi, The expressions (1.12) contain two unknowns, p and p , therefore we require two of these component equations to obtain a solution. The third
equation is used to provide a numerical check on the values of P and p . 5
This is generally unnecessary when a computer is used for the reduction.
Our observations are all exactly satisfied since our values of the
and p , g and gp have been derived directly from the equations which express the geometrical conditions. Our C's however,were obtained from approximate
formulas and therefore there is no guarantee that the motion of the minor
planet is in strict accordance with the law of gravitation.
With the p's we now have, vie may obtain the corresponding r' s by
means of the equation
YT=jDjD—Til (L- 1JLJ>) . (1.13)
In the second approximation, knowing p , p and p , we are able to take into consideration the neglected influence of planetary aberration
on the moments of observations. The corrected moments will be
"t L = "L L — Ap (i.= 1,2,3 ")• (1.1*0 It is necessary to substitute them in place of the initial in the
calculation of the formulas, ill is the moment when the light ray observed
by us left the celestial body, ~L'l is the ephemeris time of observation and
A is the light time for -unit distance (time during which light traves one /* astronomical unit). To evaluate Li. vie use the formula
ti+AT (1=1,2,3) ^ (1.15) where iii is the moment of observation indicated in U.T. and AT the
correction to U.T. to obtain the ephemeris time. The equation (1.1/j.) by introduction of the equation (1.15) becomes
"t^ii+AT-AjD (i"« 1,2,3) . (1.16)
More accurate expressions for C, and Cd was given by Gibbs.
They are
Ci=^l. 1 + B.G5 c5 = s- and Ti=k(irh) , T2= k, T-, = k (11-{l) . Taking into account only the first two terms of the series expansions (1.9) for C we have G=i#- + j Ca- -kg- + A (1.i8) where Vi= ij-U 14 = Ik. t'-t,1 u 6 3 3 x2 ~g~— ■ The equations (1.18) give G=(c.,-%)|I3 , i/» = (c»- ^)r25 . (1.19) Using the found value of Ifi from the previous approximation and the values of C± and Ch, evaluated from the Gibbs formulae (1.17), we compute the values of y3s. Substitution of these equations into equations (1.18) gives Ci = IV5 +lvk fa 3 ^ Ci - VJt- + UJ 8 Yl ^ (1.20) where lV.5 = Ti jzi j M/e = V1 w/f = Ta / Tz m/« = l/j are known. The equation (1.6) by introduction of the equations (1.20) takes the form .-5 g = vi/? + iV* nf . (1.21) The equations (1.8) and (1.21) are now solved for Yi and p . The value of Y2 obtained in the previous approximation, is taken as the initial value of the iteration. The values of Cx and C*. are determined using (1.20), and the new values of jp and jO are obtained from the equations (1.12). Then-the vectors IT (l- •£>•2^3/ are determined by means 7 of 1fT ~ p 0 —"R " fi - 1 n 5 ) ~ IT IL 1>L 1 This process must be repeated by exactly the same method until the values of C's and Jf s no longer change. The convergence is, in general, — 3 very rapid. This is due to the fact that L S are relatively insensitive to the . The r°S large change in X°5 , in going from one approximation to the next, will not produce a correspondingly large change in C3S. Hence the next solution for the S will not differ greatly from the preceding one. As a rule, no more than four approximations lead to the final values of T~l (LrXj2j3) . t—' The next stage in the computation is the determination of the ratio of the area of the sector to that of the triangle in the orbit. This important concept is fundamental to the Gaussian method and its determi¬ nation forms an intermediate stage between the calculation of the IT (l=1j2j3) and the evaluation of the orbital elements. It is possible to obtain the ratio of the area of the sector to the area of the triangle from two heliocentric positions of the body and the interval of .time during which the body passed from the first position to the second. Let H j Ij be the heliocentric distances of the planet at two instants «J irrespectively,J and E"L -JOE; the corresponding eccentric anomalies. Also let lj>ti • r\J The "sector-triangle ratio", Ij , satisfies the Gauss equations V - 1T&TO* ys_y2_ rnl(2%~-rn,(Z>fK~ 5i-nZQ5i J uk Lk + 5l-nifyjl) Oh Oh Sin l< (1.22) where ttE - tk2 [ - _JL±J2_ _ _1_ zWKl ' 2^Kk. 2 KNrni+M , 2^ = Ej-EL , . 8 The equations (1.22) contain two unknowns and ^ . Since it is not possible to eliminate the later, one has to use a method of successive approximations to obtain • In the determination of a preliminary orbit, the difference in eccentric anomalies is usually a small quantity. Then, using expansion in series, Gauss derived the equation (1.23) where = tqL r -_z.^ K -9k ^ •5/6 + lk+!>t 35 157!+ Xk = zoi _ /, - SIT) h) X k~ The cubic equation (1.23) has only one positive root, which, for small values of hk , can be obtained using Hansen's approximate formula 10 , JTL _- , j. io ? = 1 + s"_hk XI + ±±L ht 11 (1.23) 11 u k i-f 21/9 h 11 1+-2-1 yj k 11 U 1 + 9 k 1 i~ 1+ 1 + The procedure for determining^. (i-2^3) for the three observations is as follows: firstly, using the final values of the JJ (l=lj2jb) obtained in the previous section of the work, we calculate the Kl (i = 12 3) Ki = n r5 + n-n (1.26) Hu = r: rr + n n, /x/ J Ks = tin + u-ir . We next compute the TOi. and Ll (L=123) using the equations (1.22). That is Tnl- —IL /: - Ik + L _ J_ L , ' , • i M 2(2" K' 2fzKt 2 u'J'k-W 'Lf 1 where T'l (1=1^25) are the final values, corrected for light time, obtained in the previous section. The first approximations for the hi (l=lZip) are obtained by neglecting the 3iL (L= 1,2,3) 1 5/6 + l[ J J These values are substituted in the equation (1.25) to obtain the first approximations to the "M (1=12 3) . These are then used to compute and (ji- Sl from (1.2/f). Now, the second approximation tohi (i = 1, 2, 3) is determined with the help of the first of the equations (1.2if). Second approximation to V oi (i=1^2 b) is then derived, and the process is repeated until the successive values of the hi. and well. lj. (L-1JI^) agree sufficiently By definition i=rf#r «J*k> where (rrjj) represents the area of the sector of the ellipse contained between the two radius vectors and KJ .With the help of the equations (1.27) and (1.2) we easily obtain Ci = (tr'R) %, Ci = From the Kepler's laws it follows that the areas of the sectors are proportional to the intervals of time between observations. Hence r = Ik , Q - r. d ^ ' ~~ 10 We use these equations as a check on the values obtained for ljw (1=1,2,3) The values of C's obtained from these expressions are compared with the values obtained in the final approximation to the JT (i=12jb\There should be good agreement between the two sets of values if the accuracy has been maintained. We now have values of and and can thus proceed to the computation of the orbital elements. We notice that, in the numerical applications, we shall use dimensionless quantities. Therefore, the following units of time, mass and distance will be employed. The mass of the Sun will be taken as the unit of mass, the astronomical unit as the unit of distance, and ^Jmean solar days as the unit of time. Considering the mass of the asteroid negligible in comparison with the mass of the Sun, the "sector-triangle ratios" V "U and "U are given dl-,02 by 0 rip y - vjp , y- rfp D v1 Hi Kb sivCUi-lli) V2 G sin (Vs-Ui) ^ Id fl si-nCIfe-l/i) where UljU2 and Uj are the true anomalies of the minor planet at the instants 2 and ~tb respectively. From these equations we obtain the parameter of the orbit p as p_(cr;)isiTiMui-uJi'?- C(irr)z5L-rf(Lfe-u2)r/i-CriK)W(u2-i/i)^a r" Tl dz V di Tl (j> ' In order to determine p,it is sufficient to make use of the first of these equations, which gives the greatest accuracy, because its numerator and denominator are larger than those in the two following equations. In 11 practice we proceed as follows: Let us set Li_ IT g-a 17 and let us introduce the notation IT- IT-6T7 . It follows that '2 r-2 s~2-r~2- C= rf- s-1 rr = it MD2(U5-Ui) j and 17 la 5171 (Ua-Ui)= flT fo Hence P . (1*28) For the determination of the true anomalies we use the relations rr= —e—— > rr=—p— > 11-ecosUt 11-e cos 1)$ which give e cosUi--p--1 , ecosUc-l—i . (1.29) 17 3 17 The last equation can be represented in the form 0cos iJb — e cos Cuj. -hCUs-Ui)) — e cosL7 col (Ua~uD—e. siTiViSlT)(Vs-~iJi)—-~- ii Whence cos(Ua-Ui)-(~f=~~l) (1.30) eSl7lUl= . SlTl (ik-Vi) Evaluating cos (Vi~Vi) - < (1.31) Sin (tJs~1)i)~ (l-cos(Vi-Vi)) 2 D 12 the equations (1.29) and (1.30) permit us to find 1J± and 6 . With the help of the relations (1.31) 1/3= ui + (l/3-l/l) is also obtained. The semi-major axis CL and the mean daily motion T) are directly obtained from ol=.P fi-e2)"1 j v=k cl5/2 . For elliptical orbits we find the eccentric anomalies by the formulas icL7) 5 "z" = fM" ^ T" > % J and then v/e apply Kepler's equation giving the mean anomaly -/I i = El — e 5LT1 ELi (i. = ljb) . The time of perihelion passage is computed by use of the formula T~ - N I — 4°-Li. ((.-1^3) -) where ~Li (L=lJb) is the E.T. corrected for planetary aberration. The element which normally replaces ~T~ in elliptical orbits is the mean anomaly -J^io at some specific epoch -to. n a can be calculated from Jlc=n(L-T) or Tlo-JAi+n (io-in) (L=ib) where, again, Li. (L=ljb) is the E.T. corrected for planetary aberration. To find the remaining three elements tU0£. and -2 we require a knowledge of the vectorial equatorial constants3 Ix ^ Q-"* jz which are componentSjin the equatorial, system ycf the unit vectors JFJ5 and Q. directed along the major and minor axis of the orbit respectively. 13 The equations cosUi + Q.rr sm Uij n-T fi cos Ua + Q_ IT SLT) Us »n; »-v, -J give us ~P -JL£ Sen Us ~ JI ll" SIT) Ul it n sin (Ui- Ui) n — JJ ^ CQj, Ul— JT fa COS U 3 £~ Kr.sivdJ.- Ul) 3 from which"^P and Q. could be evaluated directly, but it is more convenient to proceed using the already defined vector Jo . Whence -N "pL COS Ul SLT1 U1 p. YI n; Q Mg. XSAlk r; • 11 ~ I o The elements L^Q and cO are referred to the ecliptic, therefore we have to rotate the co-ordinate system about X -axis through an angle £ equalLcXX tov w the1/IiO meaniilc: cui VobliquityUJ—LU[Lt_Lt/jr Uiof 1950.0.J11If theUllC ecliptic vectorial constants ares~R j Tz j Q-XjQ.^ jCXz-jTl-x Jl^Hzjthen / "P Px E 1 0 0 V 1] E a3 Q.z GL Qj 0 COSE. —SLTS £ % t, TL R, O 5m£ COSE. Alternatively^ SinuJSi.nL - Vcosa-f^t 3 CoSuJ SLTli - Q.Z C0b£ - Q.^SLTl£ 3 SLTlS - (R cosiO- GL^slti uj) sec£ 3 COS-Q, = P* COSuJ-GU S1T)U3 3 0 cosi =- (n SltilO 4 gl-*. cos u)^ cosec From these equations we obtain Oj^ 1 and 5 . In the numerical applications we must take care that the found values of the angles are the values of the corresponding orbital elements, since the same value of trigonometric function corresponds to more than one angle. Ik Chapter 2 Laplacian method The problem of the determination of elements by Laplace's method can, briefly^be described as follows: From the known topocentric position vectors of the Sun i "R. (i=lJ2J5 ) at the instants -L i J: 2 3 i 5 and the unit vectors p ( 1=1^2^) along the topocentric position vectors of the planet, we determine its heliocentric position and velocity vectors at the instant t = tx in the rectangular ecliptic system of co-ordinates, and from these we find the elements of orbit. Before we continue with the method we shall explain how we can 4 determine the velocity vector of a planet at the instant '-o, given its two position vectors at tj. and t^ and the position vector at t0 , where ~Li (. ito "tZ . If the attraction of the other bodies are neglected, the acceleration of the planet relative to the Sun is given by r=- ka(i+m)-£ 3 (2.D I where k is the Gaussian constant and m the mass of the planet expressed in terms of the solar mass. We define T by T = k (£-£0) 3 and introduce the following notation r-d£ , r- r(l° oLt The equation (2.1) becomes - — (1-f TTl) • (2.2) We write the solution of (2.2) in a Taylor's series 15 oo nr. + r/x + ~ ~ ~ ~r/'^Z'y t + Z—b/_Y Ajrr""^~YT'\j I 0, <2*3) v=5 where _To is the position vector at t=to, and the subscript o refers to the values for this position. r~" The differential equation (2.2) permits us to eliminate Jo in terms * y— __ y— I of Jo and lo — Jo| . Similarly, the higher derivatives in the equation (2.3) can be eliminated by successive differentiation of the equation (2.2). This process leads to an expression 00 A uu ..hore f= ,7/X* ^ 77 ~o II- 7]-° '■ These power series in T are known as "f and g series". The coefficients k Md Z are polynomials in |l_o (So and u.= /i±rn) 5 So=jL (co=£_ m . (2.5) lo lo lo I Lagrange was first to give explicit expressions for and up to Tl= 5 . The direct derivations for T) 3 5 becomes very laborious. Cipoletti (18?2) discovered two recursive formulae £ = 1" (1/^)^-1 3 (2.6) X = f-n-1- 3 16 with initial values ~f0 ~ 1 and ^ = O . The tedious algebra involved in the evaluation of the formula (2.6) can be carried out by a computer. Sconzo (1965) obtained £ and on an IBM 7094 computer, up to n = 27. In the present work the coefficients £ and ^ up to 12^ order will be used. The expressions for these arc given in the table I. Consider now two position vectors at the instants TL T2■> respectively. Using the power series expansion (2.3) about the same epoch, approximately halfway between Ti and T~a D we obtain £ = £ + U'T± + + £ + (r = ~rr+(7T1+f:"^,~ ~ 9 / +c'3)4£~ 5/ + // We eliminate using the equation (2.2). Then U = Qii; . ~ ~ +rr'm~ O. (2.7) n = a*!7 +c'51I|-,+ ^ n . (l + rn)Ti , ~ . ( 1+rr) Zl where (£1 = 1 p ' 3 and GLz= 1~ ■ • o ^-10 Neglecting higher terms in the expansions (2.7), we obtain (rr- rraJrz - ( /_ _ /T1 _2\ (2.8) ZxT.i (Ti~ La J . Now, using the velocity vector and the known position vector Jo we evaluate a first approximation of jJ.oj6o and £0 .Then, with the help of the table I, we calculate approximate values of the J? and 17 series at the point T- Ti say and 9^ .Since we have these values, we solve the equation (2.4) to obtain a better approximation of the / velocity vector £ r/=4rn —6-r K h Repeating the same process, we can obtain the desirable accuracy for To which, of course, is confined because we truncated the f and ^ series. We return now to the Laplacian method. The geometrical conditions of the problem are expressed by the equation r=pp-£, (2.9) where P = ( jZ ) is the heliocentric radius vector of the planet, "FU (X X Z ) and are the topocentric position vectors ^ -> j ' of the Sun and the planet respectively with reference to the observer. The co-ordinates j 3 Z 3 X 3 Y ^Z jTi3 3 are the equatorial rectangular co-ordinates of the bodies. We differentiate the equation (2.9) twice with respect to TT / r =p'p +pp'-Tl d (2.10) r - p£ -h 2p/p,+ pp/X -"R- • (2.11) We now impose the dynamical condition that the Earth and the minor planet move around the Sun in accordance with the law of gravitation. Neglecting the attraction of other planets, we have -g"= _ i+m*.x r"=-l+mr ft J ~ pi f ) where me and m are the masses of the Earth and the minor planet. The quantities are expressed in astronomical units, solar masses and i / k mean solar days. Neglecting the mass of minor planet, as very small compared with the mass of the Sun, the second equation becomes r"=-4j(fI ^P-"R).*~s— // 12x " We substitute the expressions for it and JT into (2.11). Then -p + (x _ p"f + zp'p'+ pp". Multiplying both sides by * ( P ^ P ) • ( P X | / v/e obtain (U TTle 1 ^ OT_ \ -p>_3 pjyDi = PD3 (2.12) 1+TTIe K5 r (2.13) —* ^ . it ^ where impap p D> = P Jp j P j R It is necessary to have one more equation involving f and P • Such an equation may be obtained by squaring the equation (2.9)« Then r*= (a.U) 19 We have now to evaluate the determinants D j D1 and Dm . Suppose that the topocentric equatorial co-ordinates, QLl/'Dl (L=lj2j5) have been obtained from three observations made at the instants and £3 . The direction cosines of the line from the Earth to the Sun at ~L — i. i (i=iJ2J5) are given by j\ 1 - cos^cosoLl j 15) ji. = c.os'bt scnOL[ j \JL - sirwi ( [= 1^5) _ (2. The quantities L j\ J u.L ^ L are the components of the unit vector p. . Thus the vector is known. The position vectors of the Sun ~R i (i= 1^5) are obtained by interpolation from the equatorial rectangular co-ordinates of the Sun, tabulated in the Astronomical Ephemeris. The next step in the evaluation of the determinants consists of the -\ determination of the first and second derivatives of the vector p at an instant 0 such that ~t ii < lo < t 3 • The vector p can be expanded as power series in T 0 that is P r: P fp'l + P^X1 -L ... . X X -1° X° 2/ (2.16) In the case of the minor planets the series converges rapidly for small values of T. The series (2.16) can be written in terms of J\ 0 jj. and V as J -x X] ~J\o ~hj\oT -j-j\o -f / " |X = jX+^oT + pi0-X/+ a (2.17) I/ = l^o t^li Vo + 20 If the observations are equally (or nearly equally) spaced, one takes, in practice, the time of the second observation as the epoch ~£o. Then Tz-0 and j\0-j\x j l/0= l/z ■ If only three observations are given, the series (2.17) have to be p' truncated. To obtain the components of the vectors and P we obtain the following systems of equations J\ 1= J\o T.X + (2.18) W^V-o + KoX.+ Ko^, (2.19) ^3-p-c -f-p.o-c5+p.o -Xi l/l = l/o + l/o'T1+ l/o" (2.20) l4 - Vo + l/o Tj-f Vo' Since (Jii^ijl/i) j (JiijKjl/i) 3 t Jo-yL 3 K~K j l/o— l/i) Ti and T5 are given we derive (JkKV.'/l and (_/)o j P° j Vo ) by solving the equations (2.18), (2.19) and (2.20). 21 The values so obtained are approximate due to the truncation of the series (2.17). If more than three observations are available we can obtain more accurate values, writing more equations containing more terms. Of course, the number of equations, which we can use, is limited by the fact that the instants of the observations must be included in the interval of convergence of the series (2.16), and we can no longer neglect the action of the other planets on the motion of the Earth and the minor planet, since more observations require a longer interval of time. This must be small, especially, when the perturbations from Jupiter on the minor planet are large. The choice of the time of the second observation as the initial epoch has been made with the assumption that the second observation is halfway (or nearly halfway) between those of the other two. The question now arises, what should be taken as the initial epoch, when this assumption is not valid, and why vie make such an assumption. Let us consider the power series expansions (2.1?) about some epoch point Then the errors, resulting from neglecting the higher terms in the right members, are given by the relations AjC =- ^7 7l'(n +Ti +u)_ -L ^"'(Tf+xf+TsVu r,+x2 T5 +T5 rj+ and similar expressions for All, 5 •> AlA? j A\/o . Cles \J0 are subjected to errors of the third order, are subjected to errors of the second order, 22 n h while the errors in J\o 0 jJLa j, Vo are of the first order. Since, in general, an error of the first order is more serious than one of the second order, the origin of the time should be chosen as to make the ^' " . i " first order errors in J\o |1<, Vo j equal to zero. This will be valid if Ti-j-T2-(-TV-0 2 or 1 1 ( L , J_ ,1 1 "to . (2.22) Hence the instant of the second observation is taken to be the origin of the time when the successive observations are equally or about equally distant from one another. The determinants D o D» and are easily computed at the n / n" point T — 0 from the evaluated derivatives jp o p and the known vectors JR. o and p • Then, the equations (2.12) and (2.14) take the form r (p-C-f-px J (2-25) I o ff + Af.XB (2.24) where j C= A=-26) JB = . The index zero refers to the initial epoch t0. In order to find the values of To and fo we have to solve the system of the equations (2.23), (2.24). Rather than attempting to solve this system by algebraic calculations, calculations which lead us to an equation of the eighth degree in p ^ it is easier to obtain values 23 for ^ and IV by simultaneous solution of the equations of the system. This is accomplished by means of successive approximations. A value of fo about 2 j8 a.u., which is the mean heliocentric distance of the minor planets, is inserted in the equation (2.23) to obtain a value of fV • Then with the help of the equation (2.2A-) we evaluate a second approximation of W . This is substituted into (2.23) and the process is continued until successive approximations are identical to the required accuracy. After To and PD are known we can evaluate fV making use of the equation (2.13), which at the point X = 0 becomes e _/ 1-fTTle (2.25) IV T = o The planetary aberration should now be taken into account. This is done subtracting from the E.T. the light time for the distance between the planet and the observer, that is correc ted -Li = '£.L= "ti-A pL (1 = 1^3) y (2.26) where A*= 0.003 T f 5 6 0 m.s.d., is the time, in mean solar days, during which the light travels one a.u. For the modified variable time X we have, corrected TlLl = t?Ll =— kn (irto)\ Ll Lo/ (1=1,2,3)lL ~ .X ' which by the introduction of the equation (2.26) becomes or Tlc=Tl- kJ\*(pL - 0 Ci=l_,2^).(2.27) In order to make these corrections to the instants of observations, it is necessary to know the values of p (I - 1^2^) • These are given 2^ with sufficient approximation by (?=K + f2' Substitution of this equation in equation (2.27) gives T;c= Ti - k J\*(£ +£ T;-f) or Tt= , (UiA5). (2.28) P With these new corrected values of ~Ci (l=lJZJ5) we repeat the / evaluation of p and p until successive approximations of these are identical to the required accuracy. We can now find the velocity vector Jo . In order to do this we need the velocity vector "R. o . This is obtained by using the process which is described at the beginning of this chapter. From the known position vectors "Bri a > E i and the corresponding times Ti Tj-Oj^-3 we calculate the values of the f and g series^and then making use of the -Q ' ' equation (2.if) we obtain the velocity vector fLo . Since p p and Ro are already known, the equation (2.10), at the point i=io gives the desired velocity vector Jo . The vectors Jfo and To" are already known, and as a matter of fact, their components comprise an alternative set of the six constants of integration of the two body problem, so we can proceed in the deter¬ mination of the orbital elements of the minor planet. Since the elements 2 o I and co are referred to the ecliptic, we rotate the co-ordinate system about x-axis through an angle 6 3 equal to the obliquity of the ecliptic. Then the components of the vectors fo and To in the new system are given by ) (2.29) Xe^zcj - J M f£) ' 25 where is the matrix 1 0 0 0 cos £ -5LT)£ 0 SLT)£ cos£ The angular momentum per unit mass r* ~(,= iCz k(r*-4^)= k(r*r') , is constant and can be found at once. Its components in the ecliptic co-ordinate system are evaluated by h(2.31) On the other hand, the direction cosines of the unit vector A as is evident from the geometry of the orbit, are • smSsini j-cobSsini ^ cost. yv Since fI ~ (ll-Xe/h D hve/k 3 h-Ze/h. / it follows that fl; -f- - sin 2 sin i -hi o h o - (21 cobls sinl j ^-scosl k n. h (2.31) From the formulae (2.31) , since A*e 5 Aue } A ze and A O are known, we obtain and l 26 The semi-latus rectum of the orbit is found as a result of the relation h2 P = Q (irn-irio) 2 where fi = (fu+K^ + K z.e I o 711© is the mass of the Sun and TTl the mass of the asteroid. Since 171 is much less than 7D 0 and, in general, unknown, it is usually neglected. In order to find the eccentricity G let us consider the equation or r= P/(l+ecosU) ecoiU=—-1 3 (2.32) r which describes, in first approximation, the relative motion of the asteroid. Differentiating this equation once with respect to "t we obtain -esmU u = r. r (2.33) Since the tangential component of the velocity in a motion is ]/ •= V• V and K= P" V the equation (2.33) becomes i_ e bin U_ -_P_ p . K (2.34) Squaring and adding the equations (2.32) and (2.34) we obtain e" =(-£r)2+ (-p- -i)\ «-35) which gives the value of the eccentricity. Using this value, with the help of the equations (2.32) and (2.34), we can find the true anomaly. We notice that i is given by \r _ / , ot (-xzi- ^2+Zz)/2_. 1 • - • \ ~~ .,,2, cTi ~ "2T 4 'Zhx-t-yy + ZZJ ' or f= r The semi-major axis oL is obtained directly from a z-A- (i-e) Next step is to find CO . The heliocentric ecliptic rectangular co-ordinates are given by - F( cos 5 cos ( co + U ) — Sen 2 slti (u> + u) cosi ) ^ ^ e=r ( sm 2 cos (uj + u) ■+ cos5 sin (to + u) cos i)-, 2e~r(sm (u)+u)sini) . Multiplying the first equation by cos £ D the second by sin £ and adding we obtain 2ecos 2 "f^eSlTlS = Ifcos ( uJ + f) . Similarly, multiplying the first equation by — Sin S cosi ^ the second by CosScosi j the third by sin'l and adding we get ~Xe sin £ cosi -f cos2 cos i + Ze sinl - rsm (uj +u) . The combination of these two equations gives -"^e SIT) 5 C051 + ye COS Q COS 1 + Ze SlTl [ {.OCTl(cO+ u) — /^eCOS-2 + Ve Sin 2 28 Since U j Q I are already known, we obtain u) . The mean daily motion T1 is obtained without difficulty from the relation - Tl* k / CX5 . The already known true anomaly and the eccentric anomaly are connected by the relation | E ( I - 1 [J tocn-^-- M—— torn. Z \ 1+e J 1 Consequently we can easily find the value of L . Using Kepler's equation J^l= E ~ eSLTlE , we obtain the mean anomaly and hence the time of perihelion passage, since _M= T> (t—T) T i Table I. Coefficients/»,.?, to 12th order. ft — 1 /. = 0 ft=~n /j = 3(T/6 /< = — 15(72/6+36/6+/62 /i= 105 /»= - 945+/,++ (630^+210/i5) - 24£M2 - 45«!M - M3 fi = 10395irV— /8= — 135135 /9 = 2027023 - 498«/i4 - 99225 /n = 654729075cr9M- /„= - 13749310575 g= = 0 S»=—M gt = (xrii gi= — 45+/6 + 96/6+/,2 gt = 420+/, — (j (1806/, +30/,2) g, = - 4725+/,++(31506/1+630/,2) - 54 gs = 62370+/, - g, = — 945945+/,++ (109147Se/,+259875/r) -a2(1115106M2+29767562/i+6615/63)+2436/13+110256V +413WV4V ^io= 16216200+/, -+ (227026S06/.+5675670/62) + Chapter 3 The Solution of the N-Body Problem by Power Series The rt-body problem is defined in the following way. Consider n bodies with masses 'IDi. 3 TT)^ ' - TO-n with such an internal mass distribution that they may be considered as particles which attract each other ac cci*ding to "tlx e Newtonian law of gravitation. They are free to move in space, initially in any given manner. The problem is to find their subsequent motion. The notation used in this chapter is given below. The position vectors of the particles and their co-ordinates in a Cartesian right handed inertial system are: Vl = o JJ = (~X3.jyzJZz) } ■ • ■ J7 = (Xn\ T^Zn) . •IL -1/ . The vector connecting the L and J ^ - 1 ■ ■ . Tl , c fcj) particles is; = °-2) The derivatives of the position vectors of the bodies^with respect to T 3 are: JT = \JsdJL ^ —S. and in general % ctl[ (^..7,) (3-3) d~C ^ cLT ~ CL I J V>y 1 where T — k d ~ io) . These derivatives^with respect to t^are: k) n£ = I? = dJL fi = and in general (7 - ~(L= ty * - . (3. ~ T)) ~ Cit - - - dti CL t M>y 1 31 Hence the following relation is evident: k" n""= IT"1 (i = q • -n) . (3.5) Neglecting any acceleration possibly caused by thrust, drag, gravitational forces exterior to the system of the bodies, etc., the force / acting on the l body due to the other bodies is given by the relation -n |— THj (L = lj • -71 ) . (3.6) j'a 'y Since this force, according to Newton's second law, is equal to 2 i TfU cLVl/ Ji (Lyv) equation (3.6) becomes J2 r 2 V~ ~ = k2Tf]-L L.TT\;JLzII (i. i3 ■ ' " T)) . (3.7) cLtL J=1- J F- UJ However, the form of equation (3.7) is not very useful for practical applications. We consider, therefore, an (n + l)-body system, and for the salts of convenience the origin of the co-ordinate system is transferred to a particular body. The form of the equations of motion, referred to this relative co-ordinate system, fixed to one of the bodies, say to the (n+1) are 2 ■ -I + j (3.8) cLt \l Z.7Tij(~rFM Ilj Ij where k is the gravitational constant, TR0 S TTI-n+i is the mass of the central body which is considered as the origin of the co-ordinate system, v~r ; Tul is the mass of the L body, Il the distance of the L body from the origin and fTj the distance between TD i and TTlj - It 32 is obvious that f[ n) =|YT| =. (*?+y2L +z\)/2 J (3.9) Vz Hi- = |jr-nl= (-Xj-Xi)2+(%-$)*+(ZrZi) J Since the system being considered is the Solar System, to establish the equations of motion using dimensionless variables, we express the distances in astronomical units, the masses in solar masses and the time in 1/k mean solar days. Then equation (3«8), by introduction of the notation (3.3) and the equation (3.5), becomes JT-JJ JO" (t=ly ■ ■ Tl) 0 Jf!'4-(i+i77;)JL -21 mj li if + if (3.10) or *n (l+TnJ—r^ - Z^TTIj — 3 FT + J~ TTI; ( pi rrsJ-D d-~ u j=i fij - hj ij i J*=«- • The system (3.10) consists of n second-order differential equations for the vectors n ( i - i, 2. - - - -m) and it is equivalent to a system of 3n second-order differential equations for the co-ordinates t 0 -Zl ( l = 1>- - - 11) - That is -n n // 1 1 *i = -(I+TDl) Til, + \ f-5 ifj-3 JXj T rr5 r/ J=I t» n (3.11) 2/ (l+Tf^,-Xrn^ f= v •") J>£ // f 1 2 Zl = J=1J= 'LJ i J=1 J J jfi J 33 We reduce this second-order system to another of first order.,by introducing new variables given by (l= 1: ■ ■ (3.12) o.i=(uiJVi ^ = (xi 0 yl jzj Tl)., and we obtain X-Ui yl= i* z'l = vji T) 1 1 UL: -d+171,)-—i "Xi + m ~y_Tnj>+ p3 pi J=1 ]lj J=i !y jf <• (3.13) Tl T» f 1 1 - +, ( 14-TH L -+ (1= 1; 777 l)^3—t|i +y"/ 771 j y 3 pj J=l J=1. I y 'j j*i jt 1 Tl uT 2; J-l ft J = 1 II?3 j*i J# ^ In order to prove that there exists a solution of this system, three theorems are given below, which will be used for this purpose. j> T- Theorem 1 r If a function -f has a derivative at a point <-i i then it is also continuous at T ■ (Apostol, Vol. I, p. 116) Theorem 2 : Let f and ^ be continuous functions at a point Ti j then the sum D the difference j- - ^ and the product jP. ^ are also continuous at ~C1 . The same is true for the quotient X if ( Ti ^ 0 . (Apostol, Vol. I, p.115). 7 /cy Z J Theorem3: A general first-order system for 771 unknowns functions SPi'= ti (t3 t,> >' ' • ' ft, "> kz= L(T= ft, ' ftz » • • ' ' ftm ) (3-14) ftm=L(r,ft, = Suppose that I A,.... and the derivatives !ln% f L a j= 1 j .. .77?) are continuous in a box |T-T°| < K 3 ^l- I ^ K (i. - i i .... m) 3 Y/here T°j ^ ^ are given numbers. Then for some positive number f~L there exists a solution 9(T) - f o of the system (3'lk) which is defined for T in the interval j"C-T°| and satisfies the conditions ft (T-)= unique in the sense that, if 9>i=93t(l) I L - 1 s ... TT\ ) is another solution and satisfies — . . . IT) 9l (t°J (i-ls ) , then (z)-fy L (z) in their common interval of definition (Morrey, p. 698) ' 11 ' ' Firstly, we notice that the existence of the derivatives 7U, Vl>Zl 3 / // / /j/''t 1' " VLlb~X'l j v£ = for the reason that the components of the velocity and acceleration vectors (to which the derivatives are identical) are limited for every value of the time T. Similarly, the functions Y7-K7"(t) 0 f~ii - fil ( t) (L 3 j = Is .. ,T) Lf3)3 possess first order derivatives in the interval (-00 3+00), which are 35 calculated as follows: rr- it = rr1- 2 rr-rr'= 2 rr rr - rr= JzJL'- T (-x?+af+2j)V2 '(3.i5) ( L=l> . . . 71) JTi fu=K, IS - 2 fij • fh 2 rsj g ~ e'= JLGL |ij fij D (3.16) hence 1/-_ \ Aj~ AlH A]- Ai )-f-[ 01;-J &)(((.Oil I flj'rlQil^KLi-Li)\Li~Lll:)+(zj-Z;)(z'j"Z;) ,. . hi - 7TA TTTTD iTil 7~ _ ^ \i/„ (g-l3,„n Lij) The functions IT - Y7 (t ) o IT/ ~ ITj' ("£) L f J ) therefore, possess first-order derivatives for every value of T in the interval f ooj+oo ) _ These are given hy (rf-MTV. (fTj"J)-3rT?f7/ ci,j=i,...-n i*ji. We must, obviously, exclude the points of singularity, if they exist. Consequently, according to theorems 1 and 2, the right hand side terms of system (3.13) are continuous in the open interval (-OOj-f oo) ^ except at the points of singularity. In the same way we can easily prove that the partial derivatives of the right hand side terms of the system (3.13) are continuous in the same interval. So the theorem 3 assures the existence of a unique solution of system (3.13). The next step is to prove that the position vectors J7 (L= 1 j . . . Tl)-, being considered as functions of the modified time "C 2 possess derivatives of all orders and so the unique solution of the system (3.13) 36 can be expanded in power series. For this purpose the following expressions are introduced / r aL= TT"3 , = -n) 3 I K r'p' < Tt) (3,17) ijn= - j Oy« = ^2 ' 0 r- I K I K a^"5 3 > T i-l where it is obvious that = ^jiu 2 ^Cjn-^jin (L,j3H = l- ■ -71)0 OCij-CXjL (Ljj-ly-n ifj) and OiiL - 1 (L^lj--r\). The proof that the derivatives of these expressions are functions of the expressions themselves is given below. They are initially given the formulae for the acceleration vectors rr" ^ j7 of the (_th and Jth particles by changing the indices of equation (3.10), a fact which will help us later to avoid confusion with the indices of the expressions (3.17) Tl -n ~ T) n ft r« ft Fa ft G jft-o+mjUZ- irZ^h-ti+t^cS. We proceed now to the calculation of the derivatives / / / X / / OL; l > 3 )/U" j ®ij " (i-jj 1 ■ •--n) j oLlj j bij (Ljj = i . ,. v Lfj) Differentiating the first of the relations (3.17) once with respect to x we obtain 37 aUfrr^-sn'rr* -3 J7'J7" m - oq =-3 17 ^ = - J r 2) OCL Q ili (i=l- • • n) . (3.20) I t r Similarly, differentiation of the second of the relations (3.17) gives ' (faY £/K~^ K-"J7+4£:_2J^7r- r/ pi piI K 'k k- Malting use of equations (3.15) and (3.18) we obtain n r. -(1+1T1l) / -JL r. - J n1 /L—"51 ,-£mL—n,Ji + OtjK 1 II ly J=i gly j=i L Ik 1 -71 i-j. Jt ■ ' / , ir-ir _ 2 r«'r« rr " r«5 n n-n v ^-3 ii'j --(l+Tnjrr -.^pr +1-Trijfsi "FT ik ik . ik 71 ->£5 + £Lt' _ 2 a,—-) r JiJC. J2lL j-i ^ ik n: rK2 C JtL and according to the expressions (3.17) ^Lj k ~ ~ l~) Q-lX) ij u — 2$ ij k ^ ^ ^ K i vJ-1- J f i- (3.21) 71 , - LmjaA«+jV -2.C.HjK VKKK (LJjJK= 1 •• • 71 ) • Jt = l J + i- Differentiating both sides of the third of the relations (3.17) with respect T we obtain ' I/,- /.Ml- JjV , u'-n" „ n-n' / r- -2 Tk i n rKa r;2 p:I K 38 which upon substitution of equations (3.15), (3.18) and (3.19) becomes ' _ JG -(l+rnj—v - [7 - im ~ S~ TTu-4r,+ Y JQ- + ir.- 3 p 5 4r—4ttu-£-'"J p j ^—" J\ pi 11 j=i ly 3J=i ly Jl=l Ij _ Ji £1 Jl£i- Jifi. t -n £ + -I] -m 1 pi J_ -2 K'-jt' n-r/ rz r* Y7- rr y p-3 v p-317£ v" rb nZ - ^1+Tnt) IT a pr +Z_1Tlj^j) (~2 ~v £~t ~ Ik 'k J*-1 J = i PIk j..Ji=i | v, ■Ji tL J. pi 'K ITIK ^I*-1. f*=1. Ik K-1. IK htJ _ o Ti-p .c-.r; r. , Ik IK Introduction of expressions (3«17) gives n *n n jK - - (i+-mi)at- 4K - 4K +ZL™j,ai>i 4* "4^ a, 4,* - JPi 31=1 ' Jl = i 7?pi JI£L Jlfi -p -n -n - (3.22) (lirrij)olj$Ljv, - aj>+ K otj> f* h= J*Bl.J10h°V^M Kf J 2JjijK "Ukkk (yT* -1 • • •71) • To find the derivative ( L> we y * j 3 K -- 1 3 2 3 . . . 71 ) differentiate both sides of the relation / 17-17 ,. . IK with respect to T to give 39 JiLnf- JFL+Jml , iir' y, • _ JC'Ii — i> 17 X"k -l. + JT-jd' 2 r;a ir1 Tu1 " (3.23) UKK r ^nv.~t Ou«~0i ( L ,-j ,K= i • • • T)) . Similarly, from the relation OUjH f7j ( L,j = 1,- • • V ifij) we obtain =-3r^ IT/, which upon substitution of equation (3.16) becomes c n-jfr)-(o-rr) aa- = - i/ r- 3 rr, wj r_3rr:*j^ jTjrr _ J y~ 2. p2 Uj IL D-f7 J7-I7 U-_rr -3fT/3^ + JT-XI 'y rr2 rr2 it2 IT2 This, according to expressions (3.17)» takes the form CL.Jij - -3(XiJ t£/4 ~^iji ^«i) (i)] = ly-n L±j) . (5.24) Vie obtain the derivative A;, j) from the following r2k / <2^-2 F > r-a F-z r.J 4-(|•y 'y ^0 which with the help of the equations (3»15) and (3*l6) becomes / l . r' r ij r:1 (J7-I1)-(Q-jgJ) S/ 17j _ , £'■_ r i. oi-nntr-rr) 'ij 0 11 ■ JJl 1Y2L0 L IL£' xjl; h1 (i/ I tTj2! v p ip rr2 rf2 Introduction of expressions (3«17) gives =2Ljimtlij -2 jy (4-;"j=1"71 ^J 1 •(3-25) For the sake of convenience, the results of the above calculations are summarized in the following - l (Xi =-3 CXL Oiii f i j' " fl ) j / n n ^Lj*- ~ ( l+TTll) OLlbLj* _ "+/u'k "2. &VK ^ 'L'JJ K=l - ■ ■ Ti) > t- •n -n r> V{j«=-h+Tni)ai4'f"4" Jr ™Jau+^3ay^'<-41TnJau^'< - (3 6) ~(l+Tf\j)(Xj $ljh — -h/L.W^OCj^ k ~2-- ^TH^CX^ ^i.p.K — p+j ~Z^iju ^XKK ^jjjK= 1 ' ' ' Tt) } ij K - 2. ^J* ^Kl< f L jjj K - i ' " " T)) _> oc tj - -5oCij by (ijji ~ b>yi. ~ $ju 4 ^C'u) f(j = 1 ■ ■ ■ n z £j j ^ by - ~2 by ( by (^jjL ~ $yi —$ju + )- b)ui j ft0 j = i • • • n Zj) Differentiation of equations (3.26) with respect to T., and the / introduction of the same equations, as soon as the derivatives QL L , / ' / . ■> Yij, j uiju (ijjU -1-OLij 3 /)y (ijj-i- ■ *7i ifj) appear, gives the second-order derivatives of the expressions (3»17) as functions of the expressions. In order to find the higher-order derivatives of the expressions (3.17)» / v/e can repeat this differentiation process, each time eliminating $LjK 0 VijK ^ Y) ij* (ijjaK = I- • -v) jOCij j bij (Ljj-l-'-n itj) by means of equations (3.26) as the case demands. As the next step in the course of the proof it v/ill be shown that, by making use of expressions (3.17)_-we can actually obtain the derivatives If! ("J V >/ 3 I L = Jj • • • 77 ) as functions of the position and the velocity vectors of the bodies J^* ^ JJ (i- 1 ■ * ' "H ) . The equations of motion already give the second-order derivatives n ( L= 13- • • T1 ) in the desired form. On substituting expressions (3.17) into equation (3.10) we obtain -n T> rr = (l+TniiaL-Z.-m.-ai/ FT +2Ztt?/fcx£/-oc/) (T a-i...v). (3.27) j=i J J ~ j=i J J ~ J*L Introducing a more general notation we can write this as follov/s ■n (3.28) TT"=L (C^U+D^jj') ft=v-n) , where 71) Caa - - (i +TnL) cxl -JL Tnf at/ (i=i J=0=11 J J Cij2= ir)j(cLij-cxj) (ijj = ij- - ■ ri Lfj) , (3.29) bDijl = 0 ( LJ =1- ' ■ 77) . (5) The third-order derivative = lf~L ( 1 1 0 • • • T) ) is easily- calculated by differentiating equation (3.2?) T) n (3) n - (l + mi) ol'l -Z.ctV J7 + >i -f-Z.-oij)J = 2 J7 n n -(iirnl)oci-l_ Tfij ocLj VIH-Z_ (t=iyn) j=i J TfyfoCij-CXj) IJ J**- J*L which upon substitution of equations (3.26) becomes (3) TT 5 ( 1 + 777 l) Oii V Hi + •n n - + 3Z TTlj CXij /)y- ( - 4jil>JLL - OijiUiJ + $jjl)JJ] J = l (3.30) Jti + — + J=I TD-iioCiibiiJ • -~LJ y ( 6jji ) ~ j jj ) R" 3>L m + \ (1+Til i)ai-Z^1T1i(Xif)rrl) OLi / TTlj CC Lj/ 1 l -t1 L.7ri;( -n jL { (L = lyV)j (3.31) 3=1 Cijsfi-hDiji Jj' J where Tl + Oib — (l TT)L)CCi£uL Ou ' UJJ' (i=i--vi) i-HwjCXijJ=t Id Lj ($ui-$jct-@Liii$jii) J±i 43 ■ ■ ■ T) L C iii = $rnj (oiij bij ( $Ut-$jLi-SijL i *$jji) OCj Sjjj) ( L,j=l. i*j> 3 n Diis = - (l+rnJoCi-Z > j=i mjocLj (L= ij-••ri THjioCij-OCj) (ijj-lj- • • 71 • Having differentiated equation (3«30) once with respect to T and having made use of equations (3.26) and (3.27)_,as soon as the derivatives OLlj ^Zj K j 1/ ijK 0 = 3 ^£j 3 biJ (ijj-ly-Tl Lfj) appear (and after substantial algebraic calculations)0the fourth-order derivative m rr ( L - 1- ■ • Tl) is obtained r -n + j-i (UTni)TDjOLi [- CLij +3 (OLij-OLj) \ iji J + J>L ^ ^ ^ bjj CI Lj' b Lj h Lj i^Lii ~ $JU- ^iji 4" h^Lii J^LJL ~VjLi +J0>-£ + "+ (i+mdoLL (Zi" 1) + (liTDj) OLj ({jii ) + ^ 77b Zfaia Zrai-Z JU +Zii -1) tOCji (bjjL—^Lii) 1 ji=I L -n "f" 7T) Z_ ( OCj^ (CTj^lZ _ 4- mLWj n - (OLij- -n -z 3 aib^uii (lirn^cd'Yui) + + J=i cx-jZ; t az Z7?73i faz^-(otb-ab ZjlZ ) + bk + 5171 j OC Li Llj ( 5ui J (vLu-$yd~ uLjl + f-ojjj - (3.32) f (liTDi) OCibiii + (liTflj) OCj CbM - l)+2(^UL-iiji ) + ^ , , V ( OCl?I (i)uL~ohl) + CCz-^JlL j + Ji=i Jt i i Trip. (aJh (cly'p-L -hbjyj bbju ~l) tcx^fbL^i _ + 77!j (au-OCj) ((1+7T1 l)cXl + (1+71)j) OLi + T) n + + 7nuarh (1 + Z-in.oc^ Z^ '—> JI #L J ^ h=+Mi r Jr Jt=i y-t 3 t t_ Ittk (OtL-.-CU A -m ^(ot^ J - ccj) T7) + jrl \ J J J»1 an Hl iti + 6 (i+mi)ou^u +JLTT)jOCn~ ™j ay Dijbj ($iiL~Vju~$LJL + ijji) rr- n J+LJfi in J b lj - >JUa ~ VLJl + ~ cc i rr. j-1 lay (bbLa (bj ^Lfi ^in JH (L -1 j ... n) . Equation (3.32) obviously has the form n'+I (Cy>^+DVrj' rz - l 77) . J= 1 In a completely analogous manner, and since the expressions (3.17) are functions of the position and the velocity vectors of the bodies, one can find the derivatives of all orders of the position vectors 3 — 1 • • • n (L ) 3 as functions of the position and velocity vectors of the bodies. It is also evident that these derivatives have the general form n fl"'= ~ ZfCij.ir+Dij.lT') j=i J ~ J ~ and are continuous as algebraic combinations of continuous functions (see theorem 2). The indices L ^ J ^ V mean that, each C ij \> or _T) . „ _ iji/ is a different expression of CU ^, $w tJKLi v< j jy* J OLjK ( L-J j,K = i0 • • • Tl) 3 QLlj J bij (ijj-lyf) Lfj ) for different H (L= l • • -n) or different Jfj (j = • • • 7) ) or different order of derivative. If we consider J7 (L=l)2-i-,'Tl) to mean JT (i - 1 j- • ■ n) then we obtain n JfT (l,) = Z_ ( Chv ly + DyV 17 ) (t = iJ-*-'n) l/^O . (3.34) J- i Comparing equation (3.34)» for IJ — O with the equation f7( } = f7 (i = 1.- ■ ■ 7?) it follows that the "coefficients" C tj o 3 J) lj'o have the values Cy° = ( i3j = ij- ' - n ) j T e (-ooj+oo) (3.35) Dyo - 0 (i J - 1 j' • • n ) 3 where Oy is the Kronecker's delta. ^6 Similarly, comparing equation (3«3*f)» for V- i ^ with the equation f~i E f~i ( L= ij* • • T) ) gives the values of Cyi j Diji Cijl = O (L,j - ly • • 71) , (3<36) "C 6. (-00 0+ 00) djj-iy" ' n) . We have already proved that the solution n fx) ( i-ly ' Tl) of the system (3.13) is defined in the open interval 00 3 + 00 ) (except at the points of singularity), and all the derivatives of this exist in the same interval. Hence^we can form the Taylor's series for JTT (T) ([=1J---T\) in powers of the independent variable T about some epoch point, say ([0 ( L = i • • • 71 ) . . ( L - lj ■ V-o The proof that this series converges in an open interval L* ) is given in the appendix A-l. Y— ( 0 ) / \ In equation (3*37) we interpret \'io ( l r • • • rt J to mean JT (t = lJ---n) at 1=0 and (aC£/^ - Later it will be showqthat recursion formulae can be derived, which will provide numerical values for the power series coefficients. These formulae will enable us to compute Tio / V J ( L - 13 1 •' 71 ) \! >/ 2. successively in terms of JTo and jffo (L - 13 • • • T,l) . So as a first step we have to find formulae for the calculation of the velocity vectors Yio' (i s l'- ■ - 1))r w— ( V ) % Equation (3*3^) gives the general form of the derivatives \i (L-li,.tT\) A/ V>/0 for every value of the time T . In the case of T = O we obtain k? -F0 ~HL (Cijvo Ho + L.jvo Co ) (L=ty---n) no ■> (3.38) J"1 where C ( t'j' = ly ■ ■ n) no 3 Lj i/o = C Lj jDijvo - JDiji/ (£3j = i,- •-n) i/*o. T = o If, for the sake of convenience and simplicity, we introduce the notation = C ijvo/ V.' (Ljj l^/O ^ (3.39) 0 V+ 3 = Dijvo/v.' (i^j-lyv) V>S O we get T> fio ~Z_ (^Ljv+1 fjo ^ ftci v+1 (Jo ) })! ( i = i, • • 71 ) ^>/0 J j= 1 ' or (i/) JQ Jfc r >o + (3.V3) u V+1 i ^ %j V+l Ho ) (L=ly1l) \}>, I/7 J = i Substitution of equation (3.40) in the power series (3-37) yields = ~C te (7 (-1^1*) ^ or n co or j>Z (i=iy-v) ze(-zji).O-kD j-1 V-l It is also obvious from equations (3.35), (3.36) and (3.39) that 48 iji — OLj Ljt = 0 (^j = t •■•■»> and (3«42) 7^p-°y- (Lj^ly-v), 'iii -T)Lj (l, j - i.• • • n) . fty2= (lJJ- y- • Equation (3»41)» being derived from equation (3«37), is valid at every interior point of the open interval (~ Z* ^ Z* ) -> since this is the interval of convergence of the series (3.37). On the other hand, the vector JT fL=lJ...T));jin relation (3.41 )j has a finite value in this interval of time. Hence ,the coefficients of the vectors 17o and CO ~ ^ ~ • ■ • -s are (j= 1 71 ) which in fact the series T i j i/ T , — ■> itl J J oo ' v (ljJ = 1- ' 'D ) z must have precise values at every point I £ (- I j I ). This means that the series converge within interval of convergence (~ L*j ~E* ) . Hence Ljt T^'1 - V- 1 tij (Lj = lo •• -v) T£ (-Z*jZ*) j 00 (3.43) Z ?yVT""XrJy- (Lj-U'--T)) Zel-Z+jT*) . Using equations (3.43) equation (3.41) becomes n ^ , fL — II ( TLi 1Z ^ (fo ' (i=l .. .7)) re z*). (3.44) ~ j=l J ~ ~ / Since Kjo and 17. 4=1,..^) are constant, it is possible to differentiate equation (3.44) once with respect to T to obtain the velocity vector IT (L = i ,. . . 7l) as V / q /] IT — Z__ ( 7ifij LJ0fjo T+ dii Uo I (Ui---ri) L£ (-1* I*) . (3.45) J=i 49 // One more differentiation gives the acceleration vector f~[ (t=7r,-=. as: = U f fii ITJo "f (fa] [jo / (l~ 1 ■ ■ ■ ri) T £ (~Zj ~L*) . (3.46) J=l J J ~ On the other hand equation (3^10) is valid for all the values of the modified time so changing the notation in this equation, we have n -n 11 m. i l = (i+m ti-k-I fl +X. TTlv -n IT - H JIlK fu ( L — 1 . . . Tl) HE (~T*s T*) , (3.43) M=1 where •n LL — —^ ( 1 + IT) ^ ) . V<. jr— 3 = ■ ■ ■ ,ip-3 TT) ( L 1, 7) ) K,:fK4i TK 13 (3.49) L =_ I /J M LlK THk I . . 71 i pj p-3 I LjK = 4, if ti) V Ilk Ik ' Making use of equation (3.44), namely 7) r:=I(&!r.+fok' (tt= 1* ...d) TTe f-t*, T*j j=i equation (3.48) becomes nTl Tl - IT X JIlk + ^kj IT (i- 1 ... n) ~ f Z_( fniiTo )/ Ze(-Z*T*) K=i j=i ~ ^ ^ 'j 7i -n or r=ZL 7?) Zehz^L*)^ K-l J=i 50 ■n a v or rr =. X-Ak fkj ) H Ak A ) ITo (L = ly-v) T£ (-r*jT*), (5.50) j-i 111 - U=1 ~ . Subtracting equation (3.M>) from equation (3*50) we obtain n 0 (Li-/Lk+(L A, A - A") L (UlyV.) (3.51) J- i L. K= 1I ~ K=1 ~ Since equation (3.51 ) is valid for every value of the variable II in the open interval ("1^1*) it can be written as -n II LU ■ LV, UUJ 0'lJ / I jo . A-Ah+iL 0 (L= L- ■ -Tl) Z£ (-L-*T*) J=i- K=i K-1 ~ J J (3.52) So, the left hand side is identical to zero in the interval (~~L* 3 T*) 3 and Jjo 7^ 0 0 Jio ^ O ( j = 1 y T) ) . it can easily be shoivn^that the expressions which appear as coefficients of fjo and K7o j in relation (3.52must have the values zero in the interval (-L*o ~L* ) . That is u Z_J\i« fKj = • • le (3.53) K=1 fij (l,J =ly n) (- I*o I*) n (3.5*f) H Jl« f iaf = 1J • • -7)) re (-1^1*) • K=-l A function represented by power series has, in its interval of convergence, derivatives of every order which may be obtained by means of term-by-term successive differentiations. Hence, we easily obtain^by using r w c • ) equations (3.^3), the derivatives Tij' and 0 (/ 1 ij ~ A' '' ^ n (\Z-l)(v-Z) iijv io ■ ■ , LJ V-5 TVb •?■)) ie(-L* l*) (3.55) // c°- A; r or 51 00 i // _y -V-l fij ~Z_ l/' (l/+l) TLjv+lT.J = 13' '' J, , (i-jj T)) ref-lM*) (3.5°) " V2" z Z_v(v-u)0ij V+2.T (L']=iy ■ v) . »/-1 ^ Te(-z*,T*)^ Substitution of these equations and equations (3«^3) in equations (3.53) and (3.5^-) gives •n / / "C / l/l" 1 (Li'lyd) . .» (3.57) « ze(-L*oL") ✓ (Jlk Z_Tuj'v1/- •< -/Lfi]V+z(v+l) J J J K=i V-l K/=I -n . co 00 (3.58) Z_[J\iv Since ~J\iu (ljM= 13 ■ • ■ T) ) are functions of the variable "Cj and we do not know the analytical expressions ^lv<~ JlK (l) (ijK = li ■ • ■ n)j we can not solve the systems of the differential equations (3«57) and (3«58). So, in the first approximation we proceed as follows: V/e consider the functions _/)lu(l) (i \A - 1 3 • • • T) ) as constant and equal to iv. ( L J\ (Ti) 3 -Ij ■ • • • T) ) -> where Tx belongs to the interval (~T*j T* ) 3 then according to the identity theorem for power series the coefficients of terms of the same power of T must be equal. That is V \> i V+l) fijv+z -ILJlJujv (Lj=iy--n) Tef-rvr) 10/1 3 (3,59) -n V(\l+l)^ijv+z~JL^ufujv (lJ=l>- -~n) ze(-z*I') V>/1 . (3.6o) K=X Analytically, for every l/=l ^ 1^-Z j 1^=3 •• • we have 52 iii = (Z_J in /nil l'Z K=l £J>-(5L./)LK ^KJ2i2)/2-3 3 K=X (5.61) L-Sjl.in f tfJi J/ 3 "3 ^ K=1 i] & ~ \ L XI lk TuM//3*5 2 K-t r\ \ / L J 2-11+1 =(IJL K=1 fil, 2jx+2~(Z_ A'k ftf ( . . and Lsj= i, .Tl) i 1-2 tklb-iX-Jl in C^k jl , K=1 -n z y/2-5 K-i LK OKJ 5// 3" LUL J (2JL 2U.-I)/(2ll-I)'2l 71 LJZ^+l K=i LJ 2U-+22LL+2 \ Z_/1LU (iKJ ' K=1 2|1 (Lj -13 ... n). Since these relations correlate the coefficients of the series of order 2 JJ.+ 1 aTic/. 2jl.-f 2 with the coefficients of order 2 jl-1 ot-noL 2.JJ- correspondingly, this enables us to find all the coefficients of the series in terms of the first two. Introducing equations (3.42) in equations (3.61) and (3.62) we obtain = £j ~f~iii 1) 3 /ljZ^0 7 /tJ5--/1 Lj/z 3 (5.63) L+= O 7 /lJ5 = (Z AuuAj)/2-34 K=1 J £Lis-o 3 53 fLh - Ii<2 J\%j) /CJ K=i .3 = 1 L J 8 -° D (Ljj= ij .. .in) and tfcji O > 2 -0 $ij LJ 5 $ar°> (3.6/f) ^UtzAijh-i , fa*=0> 2iif,=(Z K=i ?ur°. n t> ZiZZ Z K = 1 J|=l CZ J=J 3 • • -n) . Since the coefficients of the even order of the series ft J iioi-Lj • • T?J are equal to zero, these series have the form 21/-2 fii -- Z bv-t T (i,J= i. • • - 7?) re . (3.65) V=i Similarly, since the coefficients of the odd order of the series LJ (i jJ = 1 J - ■ ■ n) are equal to zero, these series have the form CO ^Lj=/L^lji irT1"'1 Te(-Z*oT*). (3.66) f=i It has already been shown that the series f- and 5k ( ljj= ' ' ' x) converge in the interval (~ t* 0 z ) (for the general case). Since in the first approximation the series were derived with the assumption that the functions —n cK (l) (l; U- i j ■ - • • ti ) are constant in the interval (-!*> L* ) 0 we have to prove the convergence of these modified series. Let us take _A — mot* IJltK fx) (l ■> K = i 3 2 j •n ) , re f-i^r) From equations (3.63) it is easily derived that i. 1 (. J 3 TiT i _A^4 = i s / n1 Jii! =„ lJ 5 X i J1 -n ^■ / - cx 2 } (J.67) / fi3 2 !i6l Lr i -j\ Xl -gT = OC^ ft 2^ / dh h-t |l2M ' 11 h a Ifu X h :> (2|x).' According to d'Alembert's ratio test the series OO I k I , J/'1 ll^ OC LA. /x = i jX.)/ (a=l r is convergent for every time X € (- I* 3 XT ) because nK+1 2(^+: J 71 IX OC U. +1 - f2 kt z)/ _ - mi i 7T) /ttn jn \tz o 00 cx r T2H (2fzk2) (2h). 55 Since this series converges, the absolute convergence^and so the simple I ■ , convergence of the series Tlj f IJ = 1 3 •••!)/ follows directly from the relations (3«67)« An entirely analogous process shows that the series ^lJ ( L 3 J = (; 2o •• " 11 ) converges in the same interval of time (~£*d T*) . JP . Unfortunately, we have no analytical expressions for the j-ij and Q/;j series where 11 ) 2 . Consequently, in practical applications we have to truncate the series after a finite number of terms. An extensive investigation for different numbers of terms and different values of shows that these series converge rapidly for values of T less than 0.2, provided that the number of terms is greater than 60. The series can easily be made to converge more rapidly by changing the units of the masses and distances (and hence the values of the quantities J lu (loK=1d'*,1i))j and therefore the values of the coefficients of the series . The calculation of the velocities of the bodies is now an easy problem, and is obtained as follows: Given the position vectors of the bodies at two different instants and their corresponding say times, fix 0 f7z ( L — J 3 ■■ ■ 71 ti ^2 can take j = z ~Lo= ~t t > Then the value of the modified time ~C = k ( I-to) is calculated. With the presupposition that this value is included in the interval of convergence of the series (3*37) ^equations (3*^9)» (3»63) (3»64) are used to calculate the values of the series •/\ j j $1j ( L jJ -Ij Then the solution of the system 36 ( ; - = \ L - 1i -) n 17. + j ... M ~ I C 4 & F7o', ! J= 1 ~ ~ gives the velocity vectors of the bodies 0 Co (J= 1-, ' '77 ) 0 with respect to the known position vectors Co (J -1 i" " H ) and 4 (L= Lj-'-n) Having found the velocity vectors as functions of the position vectors of the bodies, we now return to the differential equations (3»13) which describe the relative motion of the bodies. We vri.ll next develop a method for the calculation of the coefficients of the power series (3*37), which is the solution of the system (3*13)• If we resolve the vector series (3«37) into components series then the solution has the form CO XL = Zxit/T1'"1 CO yc°2L w ZL=,-LZ__ ZLv iC"1 ^=i (t = ij T) ) i € (-11*07;*) , where (iri y' V. \ X L 1/+1 2 ^L V+ 1 J X L )/~t 1 / (i~ 1 ? ' * * 7)) V' X 0 Note that the above series are initiated at 7 - I ^as this change in notation will make subsequent numerical, calculations (with the help of computer) feasible. The solution (3•68) must be subjected to certain initial conditions. In order to find them the series (3*68) are considered at time T -0. It is obvious that 5? Vi w ^ , (3.69) Z-l -- ^-i-l T=0 3 (L = lo • • • T\ Hence, the co-ordinates of the positions of the bodies, at time Z= o are equal to the first terms of the power series (3.68). Differentiating once equation (3.68) with respect to r Dthe following is obtained oo = Hi = Z. Hiv o V/-2. / °° <^L = \)i - Z. ylm (\l-l)ZV 2 3 (3.70) oo z'= = M/l = Z. T V=2 (l= 1; ■ ■■ 1)) T £ f-XMl At the point T=0 equations (3.70) give f ^'t-o = U' 1 = 0 = > = 14 = &2, <3-71) ^ r=o x=o ^'[-0 rZt2 3 (L= lo ■ ■ ■ 71 ) . Therefore, the components of the velocities of the bodies at the point T=o are equal to the coefficients of the second terms of the series (3.68). Equations (3.69) and (3.71) give the initial conditions to v/hich the power series solution must be subjected. The next step in this process is to obtain expansions in power series for the other variables of the equations of motion. We work in three-dimensional Euclidean space, hence the relation rr=tf + (3.72) yi+zi a= i, ■■■ v), is valid for every value of the time T . Since the functions X; - Xl(t) , ^ ( I) and Zi -Zi (t) (l - 1 j • ' ' ))) are represented by power series in the interval ( ~ T* o t* ) 0 and. the function fi = IT (z ) ( L = 1, ' • • • Tl) has derivatives of all orders, according to the theory of infinite series, the relation (3.72) assures us of a power series expansion of the form OO tT=Zf[, X"" (i=i, ---v) 0.7S) V=1 Differentiation of equations (3.72) and (3.73) gives IT Vi'= XcXi + Ij y-h ZiZi (L=iJ---n) 3 OO (3.7*0 \l'- Y- fiM (L= i D • • Tl ) T £ (~ T*;>T* ) . l/=2 By introducing equations (3.68), (3.70), (3.73) and the second of the equations (3«7*f) in the first of the equations (3«7*f)» the following is obtained 00 00_ v \ I — \ (LfiuZ^M liTv (v-l)TV"2) MZ_^LvxH(LxL,(V-l)TV'"2j + v/=l V=2 V-i ^=2. QO CD CO .. V "f 7 v^Tw_i)( vTlfl JiyLAi>-i)T"~z)+lZ-ZLvf~ll(LzLv(\/-i)zi/'2jd ito/ (i= ij ■ ■ -n) 59 CO co 1/ 4 OF L \ L I r«rr,iK I L l'+2-K T = Z I Z_"X LU~X L 1/+Z-K (l/4l" rv-1 I K= 1 A K = 1 00 V + Z(Z oo ^ T^1 "f Z_ I Zl Z lU ZL V+l-K ( V'T i- K ) / ~L V=1 W-l (i = L> • •• n) re (~z*oX*) By equating the coefficients of terms of the same power oj? T,it follows that V i/ ZirKn~ V+2-K fl/4l-K) = I™ L W2~K (1/4 1-K) + K=l K=i 4 z^y^+2-K 11/4 1-Kk)1 4 ttej. v 7" Z LK Z X l/f2~K r j/4 2~ K) (t = la . . . 71 ) lAZ 1 K=t J f V* or Z. ITk IZ+2-K (V41-k) = Z6( lK Z t 1/+2-K + 0 L V+2-K + ZiK Zi V+2-K K= L J 1/ 1A Hi 17v+ x ~h Z fZ Hv+2-k (1/41- K) - K=2 V =&*« Z i.iz+2-K -f- Vi*tyl>+2-K -f- 2t.U Z-LI/+J2.-K ) ( V/41-K) (L=tj...Tl) hence r —_J_ (3.76) ,wfl ~ ri*iV ■IE 1TI'+2-1<(I/+1-M)~J~/^['XLH7(LV+2-K +}/n.V',/4--) ,^4 Zlk ZU+2-K) (IZ+ZK) K=2 K=I (J u Ll/+2~M (L= l o • ■ ■ n ) I/Z-l This relation gives the coefficients of every order (greater than two) of the series (3«73) in terms of the lower order coefficients of the same series, and in terms of the lower or equal order coefficients of the series (3.68). In order to find the first order coefficient f~L l ( L= i j . . . 71 ) 3 the series (3»73) is considered at the point H = o and gives (I=l3 fTr^fu 2y . .-n) . (3.77) On the other hand, equation (3*72), at the point ~C= 0 gives + +Zu n) (3.78) T - O t = 0 T = 0 L= O f1 0 and substituting equations (3.69) and (3«77), equation (3.78) yields fTi =("XLI+ ^LI+ZLX)^ (L-l 3 71) . (3.79) As in the case of the function fi = fi (T) (L = 1 3 • • ■ 71 ) 3 it can be similarly proved that the function fu = fTi 11) (tjJ-is' ' 'Tl if j ) possesses a power series expansion_,in the interval ( - ~L * 3 T* ) ^ that is 00 = f I L j J - J 3 ... 71 fTl" X. ITi 1,-1 ( Lfj) t £ (~T* I* ) . (3'80) The equation - + ~ + ...T\ TTj" ("X L~Xj) C$i i/i) (Zt —Zj) (i3j=la if J ) J (3.81) is valid for every value of the time T Differentiation of equations (3.80) and (3.8l) gives lju fiJIij =("XL-%-)— ("XL-Xi)+IXl —Xii « ((Itlut ~Hj)((ft~Jfj)dJ/ldt (1// + \Zt—Zi/lZt—Zj'/ L±j) (zi-Zi)(Zi-ZjJ , , y (3.82) fTi = Z_ fti 1/ (i. j]= 13 . . . ti l^J) Te{-Z*iT*). V = 2 Making use of the equations (3.68), (3.70) and the second of the equations (3.82), the first of the equations (3.82) becomes OO ^ # _00 oo _ _ fy-ilr"'2)- (LxL.x,'"1-X"X^iv'"i)(ZxLv(i'-i)Tv/"-Z"Xj1t(i/-_ V=i V=1 v-iV-l v=lV=1 V=2.V-Z V=Z CO OO OO 00 ■ (L^x"'-. _ Z')■x""-L^jvJjf rv-i) f" v-i "=* ^=2. 00 00 +(I Z.vX'-'-Iz.vt'-'Av ZziJiMk'ZEz^-tkZx~ 00 l/=i V=2 K=2 ( Ljj-Jj . . .T\ L£j ) T £ (~T*j X *) ■> OO CO oo (Z&T^"0(LfZiv(i/-l)T""Or(&Lv-XjV)Tl,"i)(Z.(i/-l)Ui,-XJv)Tl/"a) + V/-X V-2 "=1 U-2 oo 00 +(L( (v-i) {%-%)t "~*)+ V=l V-2 oo , op V=i f=2 (ijj=Ij...v L + j ) re(-l*jT') therefore Oo V co V TV_1 + J (y fijuiLiW fTjit; m-K (l/+l-u llx^ 1 = X ( IbU-X,)(WK -Zjv+z-K K=i v=tV=1 K=iK=1 00 v> ( (fL (fjn) (dil/+2-K ^Ji'+2-K)(^+1-K))T1 1 "t" y=i K=i oo v ' ' 1 V-t + L(I(Zt^)(Z t V+2-K Zj t/+i-K J(l/+1~K) ix f=i K=1 v (i J J = 13 . . . n i jfc j') T £ f- T *J> I * ) 62 Equating the coefficients of terms of the same power of Z the following is obtained " JL fTjK IT; I'+a-K (1/+ 1-u) — ("XiK—X['n ) fXi. /+ 2-K ~~lAf 1~K) ")" K=l K = 1 V zL.(z/i.K Z(]K)( z/t 1/+2-K ^ji/f2-K ) (l/ + l~ K ) "+ K=l V (Z.LK Zjt<)(Z_i. t/+2-K ZjVf2-K^ 1 k) K-i fl-J- 13... 71 L £ j) |/> 1 j IJ ITj 1 fijv + 1 rljK C+a-K (v+ 1" k=2 flf -T ( I/+1-K ) (Zi.K XjU )(?( L W+2-K ^ijv+2-K) + K = i + (diK (fin )( z/i. 1/+2-K zfj m-K )~f~(Zt.K~ZiK ) (Zif+2-K ZJV+2-K Lj J = Is, . . . n ( Lfj) \)>/ 1 . Solution of this equation with respect to iTZj y + i gives: V" " i/+l - Z_ K 1/+2-K 1/+2-K K=2 (Tj fZj 1/+2-U f \J+1-K) + J~ ( ZtK —Zjk ) ( /Zl *Zj ) J J K=1 t (^lK z/.Jk)( (filH-2-K V + 2-K ) (Z. LK Z JK )(Z.Ll'+2-K ~ZjV+2_k) A/fa (3.83) (LjJ —13 . . . t) ) 1/3/1 and since this relation gives the coefficients fTj 1/ for IJ ^ 2 the coefficient TZji (t.J = l3...n l£j ) must be found by a different method. The 63 power series (3.80), at the point T = 0 ? reduces to the relation r. (3.8k) fliLJ z=o~ UJX (lJ-i,... n Lfj) > while, on the other hand, equation (3.81) at the point T - 0 3 becomes z j. 7x '^jx J + ( ^1=0 ~^~Jz=O' (3.85) (L J= 1j . .. n L±j) j which, by introduction of equations (3.69) and (3.8*f), becomes 1, \ . ( Z Ll~"Xji ) + i^Ll ijjl) +(zil~Zjl) fy- (ij TI Ifj) . (3.86) Now, because the function 17 - l7 (r) ( L = 1 j ... n) can be expressed as a power series in the open interval X* ) j the same will be valid for the function K = fi "S (t) (L = Id ... n) in the same interval of time. Hence, it can be written that KAJ (3.8?) v=i -3 Differentiating fi (L = 1 * ... T) ) with respect to T gives / 00 (f7 / - 2ji.LV (v-l)xV 2 (i=ls...n) T £ (-X** T.*) (3.88) V-l also T^fT ft = x> . . . ti ) re (-x*31*} or frr3) n = -3fi rz u=j*...n) ref-i^i*) (3.89) 6^ The last equation, by introduction of equations (3»73), (3»7k), (3»87) and (3*88), becomes CO CO V= 2 V-i V-Z co v/ oo , V or Z_(Z.(LkR.Lv<+2-k (|/+1-k))t1'~;= -3^(ZJLlk KTi/+z-k (m-KlliR'1 V=1 K=1 V=1 K=1 (i= 13 . . .-n) t£ ("T.* T*) . By equating the coefficients of terms of the same power of 7T it follows that v 1/+2-K 1= . . H. ^"Rl (V+l-K )— _3^Z "RLK tTv+2-K (V+1~K) ( 1J ,7l) l/ >✓ i , K= 1 K»I 1/ x/ or U IT, "R l i/+l ~h Lrr.ni 1/+2-K (V+l-K )=-3 Lw T[\i+2-k (1/4* l- K) K=2 K=i (L= Is ... n ) and the solution with respect to 7W/ILwa yields the result v V "Ri V+l -Lm-dluR^lTl LU lfTv+2-K (y+i-K) i/ rTi (3«9o) K=2 U=i (i= 1 O ■ ■ • Tl ) 1/>/ i . The coefficient "R. L l is obtained as follows: From the -b L= 1 s> . . . T> j power series expansion of (l* ( ) at time T~ O 3 it is found that -3 =o ii = j . .. n ^"r R. (l 1 ) 0 which, with equations (3*77) and (3»79)» gives Ru = (*?. +2/u+Zfi)"^ tf- i, ...-n) (3.91) 65 Nov/, it can be easily proved that the function nj * = R ij (I ) ( L d J = 1 d . . . T» J ) possesses a pov/er series expansion in the interval (-"Lrj T*) . Consider, therefore CO fZj — Z_ "R l j v T (L3j=lo..,n iij) Xe(-Tl>X*) (3.92) i/=i 1 o then = ... 7) differentiating fij ( LjJ lj i $ J ) ■> with respect to T gives / OO (ID 5) X RAiv 2 (Lj = U .. .71 i£j) V-X Te(-T.*jT*)p'9^ also ( flXJ = - 3 TTj "ft ( LjJ = 1j . . .T) Lf-j) T£ (~X*o x~) j or (3.94) fy (ID 3) -~3fu 1?j (lJ=l> . . .77 7 fjJ xef-X^T*) Equation (3.94), by introduction of equations (3.80), (3.82), (3.92) and (3.93), becomes V=1 i/=2 CO . , oo .. .ri ,-3(L^l-)fZ^(^)T:-)V=1 J ^=2 (LjUu i£j)' xe(-x\z*) 7 oo V 0r L(L(v-n-K)XiKRr.LJ V+2-K /T 7=1M=1 K.1K = 1 CO v/ ^ Z. ( Z. ( ^ + 1-K)RL j K fu 1/+2-K ) X^ 1 ( LdJ-IJ . . . 77 i. + J ) T£ (-T j T ) . l/=i K=1 66 Making use of the identity theorem for power series we obtain V ^_ (i/f i~ u) K TR Li V+ 2-U, = v — ~ 3 /R (l^+1" K )R L J K JTj Vtz-K ( L J i - Lj ... 11 L ^ J ) \) ^ I U= 1 vfLiR l J (v'-f 1- K) yTi k!R_ Li v+2.-u — K = 2 — 3H (ihi-k)Rljk fljLi V+Z-K ( L J = i J . . . Ti l± j) 1 K=1 Solution with respect to Rlji'+I (Idj=i5 , .. n Lk j ) yields the result v "R.LjvH-1— ~Z_(V-fl-K) !7ju Rl j v+2-k ~3^R0/__(Vfl-K)fl/-f 1-K) RI LlJki i k fiiJLifiz-Ki yn K=2 K-l y- (3.95) . . .a L^j) ))>/ 1 . In order to find the first order coefficient ~Ri J 1 ( i j J-U ■2d .., ti L ^ J ) j the series (3.92) is considered at the point T - 0 . Evidently ILiji — fij z~o ( LoJ-lj . . T) L±j) which, with equations (3.8^) and (3.86), becomes -3,% + (L>Ul>...ii R«l- fai.i~X.7l) l/ji) +(Zli~Zji) Ltj) . (3.96) The co-ordinates of the position vectors of the bodies, being considered as functions of the independent variable Z can be expanded in series (3.68). Hence, their first-order derivatives have power series expansions with the same interval of convergence (~Z*o ~L* ) :> which are CO =Ul=Z.Uci/ z"'1 o f=l .' CO i/ T (3.97) 67 CO Zl = M/l = Z_ uJlvZ"'1 v= 1 (L = l> . . . n ) re (-!%!*) Using expansions (3.70), the following equations are easily obtained co oo y—2. cx> oo - JLVlvZv~l , v-z l/-1 CO oo ( L- 1 o ,, , Tl ) lEkzr) . The above three equations hold for every value of T in the common interval of convergence of the series, hence the identity theorem for power series secures that the coefficients of terms of the same power of Z are equal. That is V ~A i v+l - U. Lv of ~Xlv+i= ILiv/V j 1J ^'h/+l -Viv or -Viv JV 5 (3.98) \>Z Lf-n \AJL \> or ZLlv+i- Aivl\! j (i.= h .. . -n) [>>,1 The equations of motion give the last three of the recursion formulae. Equation (3.10) is equivalent to the following equations for the co-ordinates ~Xi j V i. oZl (L = 1j .. .ti) ^■u''---""'Z-'lfNf • v). 68 T1 Z l - vJL- ~ (l+rrn)-^ + Zi L" j=i \v \iz,lj rr'J . . . (L= 1> T) ) . Introduction of the power series expansions (3.68), (3.87), (3*92), (3*97) in equations (3*99) gives 00 oo v-ij( / W Z Ulu (v-lh^zz- ( 1+TTIL) (IJXlvZ V/= 2, v-i V-1 Tl c °0 \°° \/°° V-l ~Zm, (Z (X-X») t"")(LKi;*z"-'ML_ I L LJ I ZjvI J=l JL v=i ,-I J v=i V=1 Jf L iE. . oc . . oo Z 1/^ [v-l)x"~2-- (H-TT1l)(Z ZI'VIZTUT1' l/=2 l/=I V-l v-l Z h Z-Z)t""i)(ZHy,iZ+(ZZ>'xH(ZRjV z 7=7 mj( l/=l f-i 1/51 J=7 * Jti (3.100) w Z mA* (V-I)ZV~1= v=z ® CO = - (l+TTii) (Zz;,lH( ZR^I"-1) V=1 T) oo oo oo ~ Z V/ X U_i. TTlj- v=i , , J (I_(Zl»-Zjv)Z1"1)(/LRljvv=i v=l Z^j^iUzzZ1"1V=1 >1 (L~ 1 3 . . .Tl) "L£ i-ZZZ*) 69 The derivatives XL i o V i and \Ri were obtained by differentiation of the power series (3.97) term-by-term. Equations (3.100) can be written as follows: CO ^ Z-Uiv+1 V V~L - V=X co \J = - I V—1 fl+TTl x)L( Z_~^ lm ?L l V + I-K/ L U=i u-L 3? m v 1/-I / Z_TTl f ( Z_ ( ^ xi< + V=i j=i J K=1 R.XJ K+X-K )~R ZjXjK TRjV X-K ) "C Jft oo JL.Viv+1 V Tv-1 : U^i oo V — - ,_v-i (1 + 7TIl) 2L( 21 (fx K Ri Vfl-K/X — V=1 K=i Tl (3.101) ^v-x TT1 (LiL-DRu^-MlLRj t>+X-K V=1 t J=x K-I J#x CO 1/-X v-t-l V T V=X CO l> =-(l+ml)Z_( X_Zl«Rl^i-Jz l/~I - V=1 K=1 co IX V V I)-l -I Ltd, (L(Zi„-zjk)Rx'v+i-kMLz JK Rj V+l-K z ^-X J=x K=X K-i LJ>x ( i~L J . . , T1 ) T £ (~Z*S>T*) Equating the coefficients of terms of the same power of T we obtain K. xi/fi — ~(ifmx)ZxLv v> v/ (^xk-Xjx)Rx^jVfX-K XXxK R j l/f X-K ) / ^ j J = 1 K-X J>L 70 V . Vim — V+l-K u=i V / ~Z_7T):\ L( L-L)RlJ V+l-K ~t~ J 1/+1-K ) /V (3.102) J=i ° K=i J + i it iVi i/' + l - (lr 771 Z_ ?;ZLlK D.I LL V+l-K K-i -n v/ "Z_7iij (21 (ZiK-ZJk)RlJ 1/+1-K + ZjK Rj lAfl-K j=l v^—i J ^ (l=1j . . ,n) l/>/ 1 . Equations (3#97)» at the point T = 0 give ~^*-t:=O = ^-Lz=o~ ^-Ll j Zt T:;o H iVi X^Q - Viz j ( L= lj ■ ■ Tl) j and, making use of the equations (3.71), it follows that li.ii - ill 3 (3.103) Vu = tu iVil ~ 2 Ll j ( L= 1 j . .. Tl) . Summarizing the results of the above calculations the following table is constructed A Li - ~Al £~o j y li=yL z=o, Zil = Zi r = o j ( L = 1 j .. . Tl ) j U-il - Xl Z-O j - Z.L x = o U/ii Vu--YiI=o,j fi = i j . . . n ) N V, RTiRxR+^ + Z'JZ 1f?Jx= Ku = ful , R lj i = fTj i j (i'jj= ij ... -n l£j) j "X lvt! = U-Lv I [/ 3 Zitf+2 - U/if / ^ 3 J ... ( L= 1 Ti ) l/X 1 J V _ lZi.V + 1 - ( 1+711 t) Ix LK R LV+l-K K-i H v ~A.Wr( Z (Xlk-Xjk) Rt-i' l/+ 1 - K "J" Xj * Rj V + J-K ) J=i ' *=i J±i V ViLV+1 = — (l"fill £.) Z ^lk R iv+l-K K=1 ~ Z ITlj ( Z ( Zk~ d JK ) R ijk+l-K + R JV+l-K 11 lR J=1 K=I jti VJ,L I/+JL - ■(l+TJl L)ZZ LK Fti.L V+l-K K-l n ~ Ztd,- (Lfz IK ~ z«)R Li lAf-i-K +z J K "R j ]/ + 1— K J = 1 J K=.t j*L fiv+i= -jjiufiv+2-K (l/+l-K) +Z(V+M<)6fLK XlV+2-K + !/LK )!L Vil'K R" Z. lkZ tk+2-K K=i k=i 72 v O- / — uv+i = VTj'k fij k+2-K (1/+1-KJ "f R_ (l/+ 1-k) ( ("Xlk~Xj'k ) ( Xl v+2-K ~ Xj V+ 2-K )"H k=2 K=1 /v._y. -7■ 1(7• \ULK U JK / I (/(. 1/+2-K -V.U J V+2~K 1 ' \ L— LK Z- TKM / LI/+2.-K1 ~Z_ j U+2-K 1/ R LV+l - ~Z cm-io (RRi v+2-K —37 T\ 11/ ITi/ + 2-K (l/+l-K, JTi K-2 k=i iJ V+l — —V+l-K ) IA/-2-K-l^+i-K 1/ R f[jK Ri.J )T\l7LJ Kk flj'I LJ k+2-K H}i j K = 2. K-l (L J j= 1 J ... V L + j ) lA^ 1 If the equations are used in the order where they are given above, they are perfectly recurrent, that is, each coefficient depends only on the preceding coefficients. All the first-order coefficients are given by the initial conditions. After the coefficients of the series (3.87) and (3*92) are found we substitute these series in the equations (3»*f9) to obtain /] lk (T) (i j W = 1 0 , . , T1 ) as power series, the introduction of which in the equations (3.53), (3.5*0 enables us to solve the system (3«53), (3»5*0 (by equating the coefficients of terms with the same power z . Since fu . fa (Lj>j- ij. . . T7 ) are known, we proceed then to the second approximation. This process is repeated until successive approximations of the velocities agree up to the desired accuracy. In the special case of the two-body problem the equation of motion is ~ r pj (3.10*0 Let us consider the wellknown equation To + 0 V~- f (5.105) where f ***and 7 are series in powers of the independent variable Z 73 and v- L j T / is their common interval of convergence. Since Jj> and [o are constant vectors, we can differentiate equation (3.105) once with respect to Z to obtain the velocity vector r=-f To+ ? To u f-i'Si") . (3.106) Differentiating once more gives - v" § Vo+^Vo' Tel-rM") 3 (3,107) which upon substitution of equation (3.10^-) becomes LL-Z -Pll'iq'fc' t£ r1 r ~ y~ or --fY~°~Tt~ T£f"x**jT") ■ <3'io8> Subtraction of equation (3.108) from equation (3.105) gives (f+~y f ^~°+ FP~~°(J. 109) Since the vectors I? and J[T are independent^their coefficients in the relation (3.109) must be equal to zero. Therefore, „// h-ff p3r3 „ (3.110) r-js. I £ f-I'M") . 7k Suppose that Sf and ^9/ have the form -7 TM *To ' " 1)! 3 UJ 1/ (3.111) L-Llz" l/=0 V' re f-rM*'), then differentiating these equations twice with respect to "C gives /y ) ir r ~Z. (v-i) v tv y.' 3 V-2 V=2 or CD fi" S / t"-2 T U (V-2).1 ° // °o ~o PL I" \)=2 (V-2)/ ' or ro | // v-r f.f}: w i/+ i~rrrr 3 v'=o V/ Equations (3.110), upon substitution of equations (3.112) become y p r u L— T vtz y/ v=o r v=o / ^.r,\ IE (-Z'-jT ) CO . i CO , / y 5, r r3' y a x Z_oi/ i// u. Z_ a^+2 v/ . v=° r v=o , Consider ]~3 as constant in the interval (~T** 3 T *) (an accurate approximation for small intervals of time), the coefficients of terms of the same power of Z must be equal. That is Tv/ - _ _I3 f p. Tvf2 3 a r I = - I y Analytically, for 1) = Q 3 V= 1 3 I)- 2 3 we find the following {-LI11- LI 3" y-3 v-jr2 n-f--LIjt. i = - ±J,r3 (3.114) - fT232a—— -z^r~~r Obviously, the knowledge of the coefficients of the first two terms, namely fo j So 3 /1 , 31 0 permits all the coefficients of the series (3.111) to be calculated. The coefficients "To and 3o are obtained as follows: At time ~C= o equations (3.111) and (3.105) give I /z-o T o o -f £r.o=£ » r*.. - Lo it + c or C= ir» + <£ tr' 76 which yields J (3.11?) since ^ = T^=o . On the other hand, equation (3.106), at time T = o gives ) t / H=o - T T = o TT + 2 r=o )o • (3.116) Since nl. = n' and because L„ - L and A* = & equation (3.116) becomes n'= + which yields i- ■ t-0 a 3-,- (3.117) Introduction of these values in the equations (3.11^) gives ^ = (- -pV J 4)«=° o $2J=0 , y 3>,o, (3.118) and hence the 7" and c' series have the form n oo 2V Al»i v/ h i ^=0 r5/ (2^).7 r**) \ V/ ^ / 1, \v/ (3.119) Mi-"' [x -j-ilM-l v=o r3/ (zi/tD/ 3 or V=0 (3.120) 77 h\h« (ZV4-1) / u=o (§47 ir If we now apply the series for Si Tlx and COSx CD 2 V/ :o£-x )v X1 / (21/)/ j V=o 5i.vX=Y_(-l l"*2"*1 /("2.I/+1)/ ^ V'zO then equations (3.120) give - C05 .T (3.121) .. r5 - — SLTl T k ilk T£ (- T ' * jT"* ) . Consequently, two known position vectors and their corresponding times^ say T7 , G , , , permits the calculation of the /N< ~ power series (3.121) (assuming T~ — constant = fi or fa. ) , Solving equation (3.105) with respect to gives the velocity of the body. Let us return now to the n-body problem. Clearly, knowledge of the position vectors of the bodies at two different instants enables us to calculate the initial conditions of the problem and to form the series (3.68), (3«97). By inserting in these series a value, say D of the variable T. , we can calculate the position and velocity vectors of the bodies at a subsequent instant, Ar/l< days later. Talking these new values as initial conditions we repeat the process, thereby detailing the paths of the bodies in space. 78 Chapter 4 Reduction of the plates and numerical results The first stage in the numerical application of the above, briefly described methods, is the measurement of the photographic plates and their reduction. For such a purpose the first task to be undertaken was the identifi¬ cation of the minor planet on the plates. This was accomplished using a Blink Comparator. Two plates, with the same guiding star, were used each time. After that, the measurement of the plates was carried out on a Zeiss measuring machine. All the stars on the plates for which the right ascension and declination are given in the Star Catalogue of the Smithsonian Astrophysical Observatory were taken as reference stars. We give, in brief, the method of reduction of the measurements and the applied corrections. A photographic plate (free from errors) is the central projection of a part of the celestial sphere from its centre (ex. the centre of the object glass) on to the tangent plane at a point defined by the inter¬ section of the optical axis of the instrument with the celestial sphere. The rectangular co-ordinates in this plane, called the "Standard co-ordinates", can be defined as follows: (a) The co-ordinate system lies in a plane tangent to the celestial sphere. (b) The origin of the system is the tangent point T . (c) The Y-axis is tangent to the declination circle through T. Its positive direction is in the direction in which the distance of ~T from the north celestial sphere is less than 180°. (d) The X-axis is perpendicular to the Y-axis at T . its positive direction is eastward of the declination circle, hence the increasing value of .X corresponds to increasing right ascension. 79 (e) The unit of length is the radius of the sphere (in practice the focal length, j. ). (f) The standard co-ordinates X and Y of a point 5 on the celestial sphere are the rectangular co-ordinates, in the defined system, of the central projection of 5 on the tangent plane from the centre of the sphere. The co-ordinates of the stars that are obtained from the measurement of the plates are usually called the "measured co-ordinates". The aim of the reduction is the derivation of the spherical co-ordinates C£ a of the objects of interest on the plate. This is possible because there is one-to-one correspondence between the points on the sphere and the points on the tangent plane. As a rule the reduction is obtained in two steps. Firstly, we transform the measured co-ordinates into standard co-ordinates, then these are transformed into spherical co-ordinates OC 3 0 In practice, we make the measured co-ordinates a close approximation to the standard. Then only small corrections are needed. These corrections may be divided into groups: "instrumental" corrections, depending on the state of the telescope while taking the plates, namely, scale, orientation, zero point and plate skewnessj and "spherical corrections", due to astronomical causes, namely, refraction, aberration, precession and nutation. For the second group of corrections formulae can be derived from which they can be computed with sufficient accuracy, on the basis of the known exposure data. For the first group, however, at least some of the coefficients have to be determined empirically, using the known positions of the reference stars. The displacement of the image of a star on the plate from the position corresponding to its standard co-ordinates is given, generally with sufficient accuracy, as a linear expression in the co-ordinates. Hence, if X and Y are the standard co-ordinates of a star, and x, its measured 80 co-ordinates, we can write 0 (4.1) Z\24 Y-fj'-xtBl+c*. It is possible to go a step further and to introduce quadratic terms, so the most general equations then take the form + -fC D +D*z-t-Ejcjj +Fjj2 (4. -fC+DV + E-x^4-FJ|2 • The introduction of these additional constants is not without its disadvantages, since a correspondingly larger number of reference stars will be required to determine them with adequate precision. In order to achieve the second step of the reduction, that is, the transformation of the standard co-ordinates into spherical co-ordinates, CLd X) we make use of the formulae X Lan(oc-a0) Cos C)o — YsiTl £)o Of.3) Sin's ; ■ SLTlb. + Ycosfci. (i+ where OCo and A are the right ascension and declination of the origin of the standard co-ordinates, and OL^ the right ascension and declination of the star with standard co-ordinates X and Y, all. to the mean equinox of 1950.0. The formulae, which give the standard co-ordinates X, Y as functions of the spherical equatorial co-ordinates, have the form 81 ^ _ cobh Bin (l-g,) 5 lTIOo SLTi C) -f 005*5 O COS^ CO 5 (oC-OCo / / / (zf,/f) Y- sLnbtosbo- Sltioq 003*5 cos (oc-OC0) .Smc)5LT)^ -t cos'bo Cosb COS foC-oCo) The calculations were carried out using the IBM J&O/kb- computer of St Andrews University and the IBM 370/165computer of Cambridge University (for the power series method). When writing the programs for the computations, a deliberate attempt was made to significantly reduce the time required for the calculations. The numerical results are tabulated as follows: The first five tables la, lb, lc, Id, le refer to five different photographic plates, taken on different dates. These tables include the following information for each reference star: (a) A serial number (A/A). (b) The numbers which the stars possess in the Star Catalogue of the Smithsonian Astrophysical Observatory (SAOSC). (c) The visual magnitude of the star (MAGN). (d) The right ascension (RA) in hours, minutes and seconds, and the declination (DEC) in degrees, minutes and seconds of arc, for equator equinox and epoch 1950.0. (e) The annual proper motion for right ascension (MA) and declination (MD) in seconds of time and seconds of arc respectively. (f) The date of observation (DATE ) and the interval of time, in tropical year, elapsed since 1950.0 (T). The first two lines give the values for the guiding star. The tables 2a, 2b, 2c, 2d, 2e, give: (a) The serial number for each reference star (A/A). 82 (b) The numbers which the stars possess in the Star Catalogue of the Smithsonian Astrophysical Observatory (SAOSC). (c) The visual magnitude of the star (MAGN). (d) The right ascension (RA) in hours, minutes and seconds, and the declination (DEC) in degrees, minutes and seconds of arc, for the date of observation and the equator and equinox of epoch 1950.0. (e) The standard co-ordinates of the stars (X, Y) in seconds of time and seconds of arc respectively and the same quantities in mm. (f) The date of observation (DATE). The first line gives the values for the guiding star. In order to find the standard co-ordinates of the stars the formulae (if.if) were used. The tables 5a, 3b, 3c, 3d, 3e, give (a) The serial number (A/A). (b) The numbers which the stars possess in the Star Catalogue of the Smithsonian Astrophysical Observatory (SAOSC). (c) The measured co-ordinates of the stars (X-XO, Y-YO) with respect to the guiding star and the values taken from the measurement (X, Y), in mm. The first line gives the date of observation (DATE) and the number which is given in the guiding star in the Star Catalogue of the Smithsonian Astrophysical Observatory (SAOSC(G.S.)). The first two lines of the columns refer to the guiding star and the minor planet respectively. The value of each co-ordinate is the mean value of at least ten measurements. To obtain great accuracy, we measured each co-ordinate in two positions of the plate, differing by 180°. In table if the following information is given: (a) The serial number (A/A). ✓ 83 (b) The numbers which the stars possess in the Star Catalogue of the Smithsonian Astrophysical Observatory (SAOSC). (c) The right ascension (RA) in hours, minutes and seconds and the declination (DEC) in degrees, minutes and seconds of arc, as calculated from the measured co-ordinates. (d) The difference of the calculated right ascension and declination from the corresponding values given in the tables 2a, 2b, 2c, 2d, 2e (DRA, DDEC), in seconds of time and seconds of arc respectively. In order to calculate the right ascension and the declination of the stars from their measured co-ordinates^we transformed the measured co-ordinates into standard co-ordinates using the corrections given by relations (^..1). The least squares method was used for the calculation of the coefficient of formulae (if.l). Then the relations (^-.3) give the spherical co-ordinates Values for the right ascension and the declination have been derived by using the method of Dependences. These values are of slightly less accuracy than the values derived by using the least squares method. Table k also gives results using stars distributed symmetrically around the guiding star. Three stars were used as test stars for each photographic plate. The results in the table 4 are given in five different groups, which correspond to the five different photographic plates. Table 5 gives the following information for each of the observations: (a) The day of observation (DA). (b) The local mean sidereal time (MTOO(LMST)) and the Greenwich mean sidereal time (MTOO (GMST)), which corresponds to the mid-point of the exposure, in hours, minutes and seconds. ii (c) The Greenwich mean sidereal time at 0 U.T. (on the date of observation), in hours, minutes and seconds (GMST AT 0 U.T.). Sif (d) The difference between the Greenwich mean sidereal time corresponding to the mid-point of the observation, and the Greenwich mean sidereal time at Qn U.T., in hours, minutes and seconds. This difference is given in mean sidereal time (DIFFER.(MST)) and in mean solar time (DIFFER. (MSL.T.)). (e) The month of observation (MH). (f) The Universal time (UT), the correction AT=E.T.-U.T. (in second of time), and the ephemeris time (ET). (g) The U.T. and the E.T. measured from the first day of the month of the first observation (UT * , ET * ). (h) The values of T0 and T (TO, T), in tropical countries, where T0 is the time elapsed since 1900.0 and T the time elapsed from the initial epoch to the epoch 1950.0. (i) The angles 3o and z (ZO, Z) in seconds of time. (j) The right ascension (RA) of the minor planet, in hours, minutes and seconds and the right ascension (RA* ) and the declination (DEC) of the minor planet, in degrees, minutes and seconds of arc. Tables 6a, 6b, 6c, 6d, 6e, give the topocentric rectangular co-ordinates of the Sun on the date of observation (X, Y, Z) and the components ( J\ j ) A of the unit-vector p along the topocentric position vector of the planet (LAMDA, MI, NI). To obtain the topocentric rectangular co-ordinates of the Sun on the date of observation v/e made use of the Everett's interpolation formula and introduced the topocentric corrections (DX, DY, DZ), given by the relations Ax= Axy COS0 j AY= AxrSmQ 0 Az.= Az • 85 The coefficients E, and are given by the formulae JfcEkl., P , 6 6 and the factors Axy and Az are given, for each observatory, in the Astronomical Ephemeris. 0 is the local sidereal time, with respect to the mean equinox 1950.0, hence 9 -L.i.T + Ji where L.S.T. is the local sidereal time, with respect to the equinox of the date, and ri-5<,+ £ the general precession in right ascension. Tables 7a> 7b, ?c, 7d refer to the four approximations of the Gaussian method. These tables give (a) The modified times Ii ~Lz sTs 3 ( I 1 jT2jT3 ). (b) The values of the quantities JE)i j J?)z j> j 1+FBx Fi j 1 — _BzII l + Bsd 3 a Ci 3 C3 3 Vi and! 1^3. (c) The solution of the system of the equations (1.8) and (1.21) by iteration (P2, R2). (d) The distances p ^ p j p . Hi ."Ri.Hs,(pi.P2.P3.R1>RZ.R3; and the components of the vectors (e) The final values of the constants Ci 3 C3 , Generally, the final values are determined when successive approximations differ by less .-10 than 10 Table 8 gives (a) The values of the quantities ( L = 1 > 2 >b) ^ (K1fK2,K3,M1,M2,M5,L1fL2,L3) . (b) The first approximation for the quantities (H1,H2,H3,Y1,Y2,Y3) . (c) The second approximation of the same quantities, including the values of = 1 7(£ 3233) 3 (X1,X2,X3,KS1,KS2,KS3). (d) The values of Ci and Ci , obtained from the relations, r - Ti $2 f . C1-^T3 Ci ^ -9 These differ by less than 10 from the values of tho last approxi¬ mation (Table 7d) Tables 9a and 9h give the final results of the Gauss method. Table 9h is obtained as follows: After the values of lj IJ are known we take instead of relations (1.19) the relations 14= -l) 5 ^=Cs175(-i--i) , and we proceed as described in the first chapter. Tables 10a and 10b refer to Laplace's method. These tables give the coefficients and the solutions of systems (2.18) (2.19) and (2.20) before and after the correction for the planetary aberration. Notice that the data from all the observations are used to find the values of the coefficients given in tables 10a, 10b. t Tables 11a and lib give (a) The values of the determinants D, Di 3 D2 , (D, Dl, D2). (b) The solution of the system of equations (2.23) and (2.21f) by iteration (RO, PO). (c) The modified times TidTzj (Tl, T2). (d) The values of p -,^0 and p' > (PO, RO, POD). Table lib gives the above quantities after the correction for the planetary aberration. It also gives the components (VX, VY, VZ) of the velocity vector. 87 Tables 12a and 12b give the final results of Laplace's method. We obtained Table 12b by using relations (3.121) and (3«105) to find the velocity vector Table 13 gives the heliocentric rectangular equatorial co-ordinates (for the mean equinox and equator 1950.0) of the planets Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and the minor planet, for y. the dates 1971 Nov 14, 20, 25 at 0 E.T. In order to find these values the apparent right ascension and the apparent declination, given in the Astronomical Ephemeris, were transformed into mean right ascension and declination by using first and second order corrections. Then, using the true distances of planets from the Earth and the heliocentric equatorial rectangular co-ordinates of the Earth for mean equinox and equator 1950.0 and for the dates 1971 Nov 14, 20, 21 at 0*1 E.T., the values given in the table are obtained. The co-ordinates of the minor planet were found, by correcting the values given in Table Table 14 gives (a) The velocities of the bodies. (b) The total initial energy of the system. (c) The heliocentric equatorial rectangular co-ordinates of the bodies for equinox and equator 1950.0, and for 0*1 and 12*1 E.T. (d) The total energy for every step (C). (e) The right ascension and declination of the minor planet (RA, DEC) for mean equinox and equator 1950.0. The ephemeris of the asteroid were constructed by solving the equatioi where r-(W) is the geocentric position vector of the asteroid obtained from the heliocentric position vectors of the Earth and the Asteroid. Co Co. DATE 1971 // MD .3707670 0 C 0 0 .005 0.003 0.01A .059 0.005 0.005 0.012 320.0 0.023 0.0 3.00 .037 -0.012 -0.019 -0.053 -0.012 -0.021 -0.012 A22-0. -0.056 -0.015 A3-0.0 -0.007 -0.006 -0.03A 31-0.0 -0.058 -0.025 -0.018 -0.027 s MA 0. 0 0.0008 C.0060 0.0033 01A 0.0009 0.0020 e .0005 0.0028 0.0011 0.0012 0.0035 C06• 0 .0006 0.0071 0.0912 0.002A 0.0051 0.00A2 0.0010 0.0066 0.00:29 0.0007 0.0 0.0012 10.000 -0.0025 -0.0016 -0.0001' 1A-0.00 81-0.00 IJ A36. 961. 628. 0.27 0.72 A 2 2 5 2 5 2 2 A 56.21 776. 69A. 7.6A1 C8.A 28.00 56.A2 17A. 879. 938. 0928. 27.63 AO.6A 26.8A 5.7A 27.79 9A12. 10.21 7822. 5.0A 13.36 56.62 87A6. 31A3. 596. 2 6 6 8 DEC 1 1 I 20 2 31 28 35 25 35 A 27 27 57 32 A3 A1 A1 35 5 27 10 12 10 5 1 16 17 19 s 21T= 1A.102 708. 76 9 8 8 9 8 8 8 9 9 9 9 9 DEC 70 O T.Y. 10 10 10 10 10 10 10 10 10 10 10 9°5A0215". 10 10 11 11 11 10 10 s 5.888 A 2 2 A 7A 1 laTABLE 50.5A7 6713.1 55.6A0 627A3. 52.315 15.023 15.A39 03.3A 52.533 10.111 10.519 725A3. 11.597 18.686 2.089 A2312. 13.273 28.A76 AO.A73 5155.1 52.986 39.A22 3A9A. 21.619 12329. 18.507 3A1. 728A. A.A72 n2i AI? m A9 59 50 36 37 39 A 36 36 38 39 39 AO A 2A A3 A 7A A7 A9 9A 9A A3 A A A A5 5A 5A AO 1 1 2 2 2 o 0 0 2 O 2 o 0 O 2 O O 2 o o O h 9 J z. C. Z- Z. Z- Z. z. z- z_ Z- Z. /. Z_ Z- Z- Z_ ZD Zl ') 5// 110723A.A C.0190KAMD== 242™ -0.030 MAGN 8.7 88. 58. 8.8 3.9 8.5 8.6 8.7 9.0 8.8 8.7 8.9 19. 9.0 8.0 78. 78. 9.0 9.5 9.0 83. 9.7 78. 96. 9.0 8.7 6.7 6.3 8.7 8.9 9 9 9 9 39 9 3j 9 9 SAOSC 3088 91309 39110 93136 93139 93150 93151 31529 93315 3161 92316 303A 3036 503 5309 C76 73d1 7830 377101 119742 311)7A 57110A 1-7521 310751 537101 110662 67A10 91036 70C12 61T71 2 3 A 5 6 7 8 9 1 c 02 21 22 32 A2 52 26 27 28 29 GUIDIUGSTARMAGNSAOSCRA. A/A 10 1 21 13 A1 15 16 17 10 19 00 vD DATS 197 1. // MD 3870742 0 0. 0 0 .037 0.014 0.059 0.005 50u .010 0.003 0.032 0.0 3.00 30-C.0 -0.027 -0.012 -0.019 -0.053 -0.034 —0.013 -0.043 70--0. -0.018 -0.006 -0.032 -C.011 -0.004 -0.053 -0.025 -0.013 CO 77'!29 s HA 0 0 019U. 0.0012 0.0001 0.0003 600.00 0 .0033 06Cu. 0.0012 0 .0035 0 .0006 0 .0006 C710 .0 0.0012 0.0C12 0.0005 0.0051 0.0042 0.0020 310. 0.0029 70u. 0.0 1-0.000 -0.0012 -0.0C07 14-0.J9 1 DSC O /359 // 436. 0.27 720. 5 4 41 2 2 2 5 4 15.02 35.14 46.59 56.21 7746. 12.94 10.21 8729. 38.9 28.09 27.63 4.08 95. 7817. 6.34 5.74 52.10 13.62 19.14 7422. 5.04 13.36 626. 6.35 2 6 8 DEC 1 1 / 31 54 23 35 20 57 32 43 41 20 44 42 54 25 27 19 10 1 16 17 18 19 92S.986 21T= 3 9 8 .8870742 0 9 9 9 9 8 8 9 8 9 8 8 9 9 10 10 1 T.Y. 10 10 10 10 10 01 10 10 'J s 5. 7. 5.3 304 f996. 371 8 2 5 4f 5 lbTABLE 14.102 24.728 14.472 50.547 3.6711 5.064 39.224 44.934 43.725 11.597 13.636 42.039 12.423 13.273 44.670 19.201 473. 5.511 43.622 21.619 1239. 13.507 47.143 AR 36 38 3 36 39 in 4 4 4 45 34 35 36 37 33 4 36 38 33 42 40 14 42 34 4 40 1 43 4 4 4 45 2 2 2 2 0 0 O 2 2 0 z 2 2 0 2 2 2 9 h 9 z z z -) /. ') /L '} ■C. 0S.HA= 1 00 310698. 10D=M 8. 3. 8. 8. 8. 6. 6. o". GAV. 4. 8. 70. 8.8 5 70. 8.9 9. 0 8.7 7 3.6 0.7 3 9. 9.0 7.3 8.7 3 9 9.0 8.7 6.7 3 39™2 023 N 4 7 8.9 9.7 1 9.0 5 9 9 9 39 3 9 39 9 3 SAOSC 3071 37C 3038 93091 93110 3926 3043 )36 503 5930 33967 116723 107021 110716 119737 110742 11743 110745 572101 753r1 6014 506101 581"61 2611' 110674 106861 32106 O 3 4 5 6 7 8 9 1 /i 52 0'1z> 21 23 2U 26 27 1 16 1 1 GUIDINGSTARAG,'JSAOSCHAR- A/A 10 1 12 3 14 15 17 3 9 O MD DATS MD // 1971.9009253 0. 9 0 0 0.02 0.0 0.0 932 3 .025 0.006 .022 -9 15 .022 0.010 0.006- -0.034 -9.013 -0.039 -0.013 5-0.00 -0.013 -0.032 .031 -0.012 -9.327 -0.007 -0.002 -0.002 -0.011 -0.006 S MA 010. 50J. 03J. 92UJ. 0.0051 0.0342 0 0.0366 6 0.0905 0.0349 COO-3. 00-C.j 0.0022 0.0019 8 0.0016 0.0020, 0 0.0351 Dit, -0.9091 5 -0.9317 -9.0013 7 -0.0097 -3.0005 -0.0915 -0.0035 -0.0905 -9.9012 C o 11/84641.95 // 64.8 6.61 9.52 0.97 9.16 3 4 3 4 3426. 7425. 27.73 9412. 10.21 3047. 7417. 49.67 7919. 725. 52.10 42.93 630.2 7.39 16.32 14.07 4.20 11.70 6213. 4.03 7 DEC 1 1 3 / 33 23 37 64 5 55 n 15 17 53 29 23 42 35 13 59 49 26 39 16 20 4 ^ 5S.84j 21.930r= 92 9 9 9 7 8 8 7 7 8 7 7 3 3 7 3 7 7 9 3 9 8 9 9 9 O T.T53 >3 i> 6.906 73.34 9960. lcTABLE 40.473 5.5115 52.936 39.224 34944. 21.999 19.201 43.656 22.3ft3 14.720 43.622 46.561 19.543 35.620 39.993 5925. 30.283 52.354 55.364 32.786 51.614 44.670 RA 36 39 36 33 3 3 3 39 3 3 73 43 41 35 36 37 37 3ft 39 9i 0 30 33 34 34 4 31 3 34 *m 2 2 2 2 2 9 2 h 2 z 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 c. C. ') 0S.AM= 06501ft.7 1200D=M o". MAG 9. 8. 9. 3. 8. ft. 8. 7. P. ft. 6. 9. 3. 7.7 78. P.7 25™3 800 11 5 9.9 3 7 7 6.5 3 5 3.4 9.2 5 ft 8.9 6 7 9.0 9 5 3.6 3.6 7 8.6 3A0SC 61C12 110674 11969ft 73120 110716 113653 113653 110663 110667 110673 110673 113636 6319 610601 119609 110610 60341 641013 674r1 6ft4119 16113'3 110616 113632 113639 1640 2 3 4 5 6 7 3 9 1 20 21 22 32 24 25 A/A 13 1 12 31 41 51 16 71 ft1 19 DI;{GUI7AST0SMAGGNAR■ , DATS MD // 0.011 0.013 0.0 0.013 3 .025 0 .005 030. 0. -0.053 -0.039 -0.072 -0.023 -0.046 -0.004 -0.021 -0.054 -0.001 -0.083 -0.049 -0.001 -0.003 -0.023 -0.017 -0.006 -0.012 -0.021 -0.012 -0.442 -3.056 -0.015 02 // 1.21971.8240538 s MA 00 00 00 0 0 00 0 0 00 3 0001 0009 0009 34 0091 24 530 2800 0015 0004 0003 31 0017 0046 0018 0003 015 380 9000 2100 04 114 502 09 20 160 5000 0028 0011 0 0010 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . u• 0 . ). 0. 0 . 0 • -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. // 7.34 1.96 28.6 4 2 4 4 3 4 2 5 3 2 2 2 5 5 2 0.77 12.22 52.6 17.95 32.93 4.51 20.56 20.24 42.64 9.35 70/. 0.29 4.69 3.123 13.96 3.23 8.08 13.19 705. 332. 4.69 17.46 8.04 08. 6.42 174. 7624. 3.39 7 6 6 DSC O J Z / \J 2 5 35 4 6 36 27 36 40 33 63 5 57 26 57 34 54 24 25 5 27 27 16 13 1 12 10 17 41.149 21.T= 9 9 9 9 9 8 9 3 3 8240538 0 T.K! 10 10 1 1 1 10 10 1015' 10 10 10 10 10 10 10 10 10 11 10 1 1 1 10 1J 11 11 S 7827. 1601. 5.790 6.877 4.367 3 4 5 2 IdTABLE 4.4031 25.131 39.743 17722. 52.298 41.913 35.508 51.034 47.664 13.327 58.919 37.656 12.765 43.459 33.359 5.411 3.762 2.315 15.023 15.439 3.943 52.533 10.111 10.519 37.933 13.370 RA 54 54 54 57 57 57 53 59 55 55 53 52 53 54 56 34 53 51 51 25 74 74 74 49 49 34 49 5C 59 51 52 -m 2 2 2 2 2 2 2 2 2 >/ 2 2 2 2 2 2 2 9 2 2 2 2 2 0 O 0 2 "> h £. C. t ^ ') ') 931368.2 MAMD==-o\oD12 2 8. 8. 8. 7. 8. 6. 9. 8. 3. 3. 8. 3. -o". 8.7 8.7 6 5 8.0 6.2 8 8. 8. 8. 3.0 8.6 9. 3 3.6 3 6 3 8.9 5 8.6 7 9. 3.7 8 52™ 038 KAGM 8.9 5 3 1 4 3 0 8.7 0 8.3 3.5' SAOSC 31989 20039 32049 73229 32289 32931 329 93244 32192 392 32934 31394 31336 31393 30159 51391 31529 3315 16301 31623 53173 18392 110340 110859 08621 381:81 110783 110839 110817 8142j 3014 2 3 5 6 7 8 j 4 1 r\ 2 2 27 2 3 31 23 21 2 3 24 5 52 82 9 0 A/A 1 1 1 1 1 STARGMSAOSCNA3oUTDING, 10 1 2 13 14 15 6 17 8 9 VD ro ii MD 01o. 0.001 0.022 022..0 0.018 C.017 0.001 0.013 0.010 0.013 59-0.0 -0.002 -0.006 -0.022 -0.013 3-0.01 -0.012 002-C. -0.002 -0.010 -0.017 -0.004 -O.033 -0.053 -0.085 03-0.0 -0.050 -0.200 -0.016 -0.021 s HA 62-0.00 0.0005 0.0016 0.0051 0.GG43 0.0007 -0.0003 -0.0005 0.0022 -0.0015 0.002u 1C04. -0.0035 0.0011 03.000 0.0025 50 .002 0.0014 0.0010 -0.0067 0 .0029 01. 0.0037' -0.0004 0.0005 021.0 0.0C45 -0.0195 -0.0010 0.0026 // 0.51 9.16 6.61 529. 7.27 84.4 3 3 2 4 2 2 2 6933. 4.20 6.16 3.13 38.47 30.62 11.70 7332. 949.4 1.56 13.15 8946. 39.22 593. 33.96 4.148 52.13 634. 12.40 4.20 4113. 38.74 9236. 7147. 3 2 DEC 7 9 3 9 9 / 53 55 37 55 16 39 40 26 46 19 19 29 47 38 14 24 26 32 30 32 29 58 20 9 9 9 8 8 8 8 7 8 8 3 a 3 8 7 3 3 8 8 8 3 7 8 8 9 9 9 9 O 10 10 h 7'A 255S.6^ 228 s 8° 4. 5.60 3.0 6. DEC 3.794 682 1.417 8 74 <379 4 47' MB- 39. 1 39. 53. TABLE 2. o". 19.623 34.243 51.614 32.642 48.570 19.543 35.620 993 32.786 17.079 3r.612 58.059 29.791 50.915 56.644 59.756 4.767 20.047 43.671 111 50.925 83 54.866 29.521 le \11971 01721T= DATE m 30 30 20 25 25 26 9.9333621 T-Y.19.933362 27 29 30 33 27 27 28 30 31 22 22 23 25 25 25 26 26 26 26 21 21 24 26 21 2 2 2 2 2 2 2 h 2 2 2 2 2 2 2 2 2 2 2 9 2 2 2 2 o o 2 2 o z SAOSC IIA= 110566 j cc 0.0003 r- MAGN 7.6 • • • cc oc 8. 8. 8. 8. 8. 7. 6. 6. 7. MAGNPA 8.6 9.3 8.7 8.7 P.4 68. 8.6 8.6 8.7 9.0 7.7 CO 9 6 34. 9 9.1 8.7 9.5■ 8 58. 6 8.9 8.8 6 5 3 3 9 1 10 29 SAOSC 57410 59710 60101 39610 65810 10576 851C 61060 60910 61010 6161' 50210 10520 19525 52710 53910 54310 65f;4 1 .548 55010 5210 54 56105 58105 739 51510 16501 10537 561; 2922 2 3 7 8 9 1 4 92 21 22 32 24 52 62 27 28 29 3 A/A 10 1 12 1 1 51 16 17 18 19 GUIDINGSTAN 1 3 4 5.5219 6.4001 YIN 28.1 31.1395 23.4454 31.0504 38.5839 25.4300 25.4572 - 149 25.3237 Mil 12.3711 58.3415 54.0164 57.0099 603910. -42.0537 -61.1596 -36.0707 2520-51. -54.4143 -54.2452 -23.1402 -27.4642 -14.0161 -25.4500 -39.7016 23.4284 24.3627 10.5332 19.2575 6.6253 X INK'1 10.9119 29.6351 60.5872 61.9827 77.2062 77.2713 78.7927 4.88461 87.5052 87.5736 16.4739 21.6640 22.9715 27.2547 32.9216 3.39821 35.8734 -61.6435 -47.7901 -26.0937 -17.5583 -5.4555 -64.9077 -63.5289 -45.7731 -32.3835 0431-27. -20.1848 -11.0560 Y / 6.339 3.555 41.697 31.344 77.997 72.214 76.217 41.511 51.583 34.064 34.034 31.321 33337. 25.745 37.572 34.524 16.539 - - -42.727 -56.221 -31.764 -48.223 -71.133 -72.746 -72.520 -37.621 -36.717 3310.7 344,'2. -53.677 14.082 X S 35.435 53.352 33.096 15.81 45.1 113.72 1-29. 323.096 - - - - 331.453 412.363 413.216 421.351 467.942 468.303 - -93.895 1 DSC 450.135 1 74 547 153.744 50 122.342 747 176.051 3 191.836 -59.123 DATE 36147.71137087076. -329.645 -255.562 53839. -347.100 339727. -244.776 3.02917 946144. 0497J. // 8. 5.27 1 .50 36 3.23 0.22 0.83 9.93 55.95 46.35 4.432 8226. 28.11 56.09 54.24 30.13 28.20 59.33 27.43 9.724 7.542 5.612 28.29 2012. 21.51 25.04 13.43 56.7j 6.434 44.5 7.404 &z - £~—x £«-■><• DC 6 5 O 2 1 1 ,) / 53 53 54 27 72 53 32 4 14 14 53 20 25 27 31 28 25 1 J 1 1 10 21 10 1 6 17 J TABLE 2a 8 9 8 8 9 9 9 9 3 3 9 9 0 10 10 10 10 11 11 11 10 10 10 10 10 10 10 10 10 10 10 i> 5.338 714 . 564'30. 13.802 55.712 43.876 52.260 15.043 15.483 23.008 52.544 10.172 10.543 43.751 11.783 13.699 42.102 12.578 18.299 28.528 40.590 55.603 53.003 39.368 44.982 21.682 29.138 18.476 4j 24.754 14.474 RA m 4 3 38 4 4 47 4 4 4 50 50 4 4 4 4 63 37 39 04 36 6 39 39 C 4 42 43 4 7 9 9 49 49 ' 3 4 44 45 5 5 41 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 9 2 2 2 o 2 2 2 2 h c. "9 3. 1107 CANM 8.7 8.8 8.5 8.8 8.9 3.5 8.6 8.7 9.0 8.8 3.7 98. 9.1 9.0 P. 8.7 3.7 9.0 9.5 9.0 8.8 9.7 7 6.9 9.0 5.7- 6.3 8.7 3.9 324.414*51842™ <3.7 9 9 9 9 9 9 9 9 9 93 3 9 9 9° 39 9 9 0 0 30 30 30 30 SAOSC 308 3091 93110 6313 93139 5031 3151 93152 3153 3161 3162 34 36 50 59 3)67 71 73 110737 11j742 110743 110745 75210 07531 75801 110662 70146 06981 7012 10.7161 2 3 4 5 6 7 8 9 1 02 21 22 23 42 52 26 27 28 92 03 EJTDIM3STARSAOSCMASNIIUDRA A/A 01 1 21 31 41 51 16 17 31 19 Y IN 06141. 39.8509 38.908 33.2615 flM 20.4424 45.2682 26.4538 - 32.9414 37.3715 9 24.5129 5 19.2681 -11.4134 -25.6486 -17.8717 -27.9570 -47.0599 -21.9649 7431-39. -40.2980 -10.9169 -37.8940 397366. -14.1815 -13.4827 -57.3127 14.2342 42.1132 5.8333 5548.7 X 2 4 IN 6..06 5.1 19. 53. 60. MM 93 0012 32.6434 36.9620 55.7523 20.6817 42.5998 47.8078 474 3897 59.0865 1492 -56.9084 -53.0911 -39.5271 -35.5146 -21.6675 84492.-1 -75.9125 -63.5088 -38.8728 -37.4973 -19.7200 -6.3532 -1.0857 Y / 18.799 53.277 32927. 60.519 35.366 25.759 91960.. 44.039 49.962 52.u17 32.771 44.467 55.303 15265- . 290-34. -23.393 376- . 14-62.9 -29.365 -52.323 -53.874 55914.- -50.650 -53.124 -18.959 -13.025 -76.621 X 5 ' 31.221 89.221 45.747 -5.806 // 72.238. 33.- 35 298.143 255.657 285.507 315.971 321.654 - DEC 28.29 133.408 174.564 197.653 110.597 7 262.82j 304323. -283.910 -211.375 -189.913 -115.869 -68.639 -455.950 -339.619 -207.875 -200.521 -165.455 974 0 DATS 9 1971.8370742 // 5.27 9.93 0.22 0.83 14.36 44.54 07.44 55.95 6.354 12.20 30.13 23.20 59.33 327.4 3044. 42.13 17.39 7.542 5.612 51.40 18.38 19.05 21.47 5.042 13.43 56.07 46.46 £-2 * * 2 — 5 8 DEC 1 <■» 1 / 3 4 54 0o 54 23 5 20 53 32 34 1 20 4 42 25 27 31 3 10 1 16 17 18 19 19 2bTABLE 8 8 9 9 9 9 8 8 9 3 3 9 8 9 9 9 0 10 10 10 10 10 10 '10 10 10 10 10 RA s■m 393.5 h 2 £> 5.833 1.940 7.399 5.888 14.518 7542.4. 14.474 50.565 13.302 55.712 39363. 44.932 43.751 11.733 13.699 42.102 12.578 13.299 44.644 19.212 40.590 5.5693 43.607 21.682 29.133 18.476 47.104 RA in 3 36 3 3 3 36 3 3 3 4 4 4 4 4 54 34 35 6 7 8 33 34 6 3 9 9 42• 04 1 42 43 44 0 41 3 4 4 4 45 NITUD 3. 8 2 2 2 2 2 2 2 2 2 2 2 2 2 o 2 2 2 O 2 2 2 2 2 9 2 2 h /.. lino •"> V*• SADSC MA 8. 3. 8.7 9. 8.0 8.7 3.6 8. 9. 9.0 3. 8.8 9. 78. 76. 1 o 4.4 7 8.9 7 8.8 8.5 8.9 1 9.0 3.7 7 3.3 5 7.3 7 6.9 ' 6.3 05931 Ji 9.7 " 9 9 9 39 9 9 SADSC 97310 93087 93083 3091 93110 3062 430 630 503 5930 73960 0721 170 110742 110743 110745 01752 110753 110646 110658 110662 110674 110636 110632 33STAR 3 110702 6 119737 110650 3 4 5 5 7 8 9 1 02 21 22 23 42 52 62 72 A/A ir 1 21 31 41 51 15 17 31 91 T'JDI vo vj1 1 iny 33.0705 .1632 52.l 7.8211 mm 23.7645 24.5101 26.6770 - - 12.4325 -56.7969 -1.7994 -27.5752 -33.9304 -53.5570 -32.2120 912866. 35764. -43.2407 -26.2675 -13.8041 -55.2979 4583-21. -60.9252 -63.1903 39.6501 833 26.9566 dats 2.3544 x mhin 7.1 56.08- 31.3340 52.9647 61.5603 73.7392 51.9949 - 4394 13.5644 18.1307 22.2552 25.3142 34.9759 40.3005 5 -2.3934 -1.9385 1 -3.9095 1971.9009253 0757-53. 0179-50. 8541-49. -12.9774 6943-55. -39.3664 -22.9933 69023. y / 5613. 31.771 32.767 50.395 35.664 16.633 -2.406 53.003 63.633 13.456 36.u38 -75.932 36.536- -45.362 -71.600 -43.064 -25.734 -44.388 -64.493 117-35. -18.455 -73.928 -28.683 -81.451 -34.473 x s / 93.526 15.264 72.537 95.956 ii 233.234 119. 35.- 4442.13 167.562 329.199 394.327 j12 135.370 187.037 215.511 278.043 -283.828 -267.475 -263.029 -69.393 190 -12.031 -10.366 -297.830 -210.525 -122.959 -73.210 -20.997 o 3 n 9.93 5.41 097. 269. 0.32 39.0 2 27.54 5.51 3.292 12.20 6.954 3517. 49.56 19.51 35.72 51.40 3.064 29.53 6.804 15.93 14.40 34.16 12.13 3313. 3044. #* drc 7 9 1 1 / 35 20 24 59 3 59 4j 28 3) 2 20 37 64 54 5 20 339 16 17 23 13 16 3 %— v 2ctable 7 3 0 9 9 9 9 9 8 7 7 8 3 7 7 3 8 7 8 7 7 9 8 9 3 9 ra 35™ h 2 6 5.995 3.829 1.040 40.590 55.693 53.003 39.369 44.982 21.222 19.212 7633.4 22.351 14.692 43.607 46.546 19.532 35.668 39.960 55.348 30.272 52.896 55.382 32.330 51.726 46a. ra IT) 36 37 39 35 36 36 39 30 39 3 3 3 33 3 34 40 14 37 37 38 38 4 4 4 31 33 . ,3C 3.3 nitud .7 8 2 2 2 2 2 0 2 2 hag h 2 2 2 2 2 2 2 2 2 T 2 2 2 -> I /. ') aoscs 113650 an3m 9.5 9.c 8.3 79. 73. 5.5 8.3 8.5 8.4 9.2 8.5 7.0 8.9 8.6 7a. 9.0 6.3 59. 6.o 8.6 8.7 77. 3.7 70. 68. sa3sc 3 star 113662 112674 112396 112792 112716 16532 106531 113663 110667 113673 119678 113636 961C 69061 119539 631: 113543 119643 119647 26431 961 916 11c632 36219 0164 2 3 5 7 J o 1 5 i> ur ^ - :1 52 1 1 1 1 -1 d a/a i . 11 21 3 '4 31 3 71 ? 3i in n•) 8.4930 1.1456 2.2324 9.2031 7.6315 9.4431 0.4193 INY 5.1 13.1 -9.2504 51.8833 42.5063 38.0497 41.0357 22.5666 38.2047 46.4061 15.6139 6651 18.3653 192 11.1360- -10.8855 -28.8609 -14.3376 784-13.1 -36.6630 -47.8373 -53.3395 -50.4027 -30.7544 -60.4135 15.0828 DATS X 4 1.8371 21 III 17. 82. 3.1 2.1 47. -4. MM 1543 33 49.0805 51.7277 57.1971 58.3445 70.7229 33.3027 653 59.9942 1264 5723 -1.4262 -2 -1 1771 1971.3240533 19.1145 17.9772 -44.9552 -27.4138 -17.7600 -10.6234 -56.0602 -54.6762 -53.CC33 -37.8226 -37.7339 -36.4194 -30.9729 -27.7594 7.6913 59541. Y / 1. 3.051 532 54 511.3 83120. 12.304 20.943 25.221 24.223 69.363 20310. 56.327 50.369 135. 25.164 35.169 12.625 12.593 51.1-76 62.040 J -14.955 -14.553 -12.357 -33.584 -19.836 17.613- 519.0-4 -63.954 -77.994 -67.333 116-41. -80.773 X 3 9.824 96. /// 91.734 65.533 135 -7.627 262.462 276.613 3^5.867 314.676 373.197 230.331 320.325 254.400 -94.973 -56.836 - -62.003 -22.338 15 102.217 116.705 173.089 DOC 20.85 -240.402 -146.625 -299.787 -292.386 -233.439 -202.260 -201.785 -194.756 -165.630 -143.446 148.U82 0 1: // 7.34 1.50 68.3 7.31 3 2 41.01 12.61 51.33 17.19 31.36 3.902 19.55 5220. 2.554 3.894 5.894 730.2 842.8 3014. 13.94 20.16 57.53 13.74 5.33 32.20 4324. 6.82 23.11 056.9 54.24 24.76 22.80 — 6 6 2 *— 7 < 02 *— / 53 72 3 5 3d 5'4 2'4 52 45 27 63 39 6 5 75 2 57 1 1 26 27 36 04 6 1 12 J 7 123 61 13 TABLE 9 9 3 2d 9 9 9 9 9 8 0 1 10 1 1 1 10 10 13 1 1 10 10 10 10 10 10 10 10 10 10 10 1 1 1 1 10 1 RA 52 m_ h 2 ro S • CO 7.356 2761. 5.351 6.384 4.950 14.401 125. 39.729 22.376 52.246 41.951 35.499 51.086 47.701 13.727 53.953 37.663 12.732 43.489 33.415 35.402 43.376 52.260 15.043 15.433 23.068 52.544 10.172 10.543 37.939 A2 m 50 4 4? 4 53 51 52 5 5 4 J 51 52 4 47 9 4 9 54 54 57 57 57 53 59 55 56 3 2 53 54 56 3 r V. 51 7 47 9 SIJITUD 8.2 ' 54 2 2 2 2 2 2 2 2 2 2 2 2 2 h 2 2 2 2 2 2 2 2 2 2 9 9 2 KA -i L. /- /L 86931 3A0SC 8. 6. 3. 8. 3. 9. 3. 8. H 3. 3. 3 A 8. 9. 5 5 7 5 311 8.7 8.7 3.5 3.5 3.0 3.9 6.2 5 7.3 3J. 1 3.4 3 8.0 3.6 0 8.7 6.3 8.6 3 8.8 3.9 6 9.7 0 8.3 CO •-4 9 9 9 3 9 39 3 3 3 3 SA 3 9 3 3 9 3 9 9 31 31 DSC 89319 3290 3204 93227 32289 3231 32 3244 2321 3223 3234 93134 36 9313 50 3151 3152 815 3161. 3162 5317 3132 110840 113859 113362 110338 110809 111-317 'l1428 119334 SPAR45 2 3 4 5 6 7 3 9 1 29 2 2 42 ^ Z 23 92 03 3 1 2 "50 r~ J 26 27 1 1 1 1 1 17 1 1 A/A 91 1 2 31 4 5 6 8 9 IDItJ 6.2431 INY 37.8342 MM 49.7445 50.8722 -5.7344 -J.5514 -6.6933 - 31.7022 53.5300 25.1365 56.7883 -22.7716 -29.4672 -50.0305 0392-28. -15.5771 -23.7342 -20.5029 -12.8232 -44.3552 -24.5491 -16.6533 -32.4715 -15.4361 304108. -57.4525 -10.6025 -28.1475 61.4713 12.3448 4.4332 1 . 6.9184 X 7653 - MIN 702. 51.3101 - 29.2993 34.7765 76.9653 23.9815 37.8718 41.5733 -4 43.3097 -73.6334 -48.9345 3.8653 -43.3292 -20.2532 5.18039 -11.3933 -10.3451 -10.2895 -9.6342 -8.9669 -7.4685 -6.5155 -2.1271 -58.3802 -56.2883 -24.3314 1 -59.9893 Y / 3468. 65.503 63.011 50531. -7.666 -0.737 -3.343 82.133 5164u. 342.33 71.564 33.565 75.923 -33.443 -39.395 -66.386 -37.436 -23.325 -27.72C -27.410 153-17. 829-59. -32.320 -22.271 -43.411 -23.596 14.479- -75.833 -14.174 37633- . X s 9.443 ' 23.974 36.997 -0.385 // 213.234 399- . 47 411.573 222.317 274.386 - DSC 2.74 156.681 185.973 112.233 202.523 -394.056 -261.949 34745-2. -231.737 -103.332 -84.513 -63.599 -57.995 -55.324 -51.523 -47.951 38 -34.842 -11.375 -312.193 -301.037 -133.114 323799. t DATE 3 1971.93336219 il 3.03 7.09 9.26 7.23 4.37 # 3 32.34 34.16 36.13 2 » 32.96 249. 41.19 13.0 37.95 726.3 3.78 552.1 24.92 % 24.49 14.35 36.57 4 ' 9.53 22.65 38.13 3.97 12.13 7 6 46.17 14.35 12.62 17.31 7.25 7 9 «— 3 3 2 DEC * ) 9 I 2 35 55 37 55 39 40 26 64 29 74 33 24 26 32 3 32 29 53 20 35 16 31 19 41 TABLE 9 9 9 8 8 8 8 7 8 8 3 8 8 8 7 3 8 3 8 8 8 7 8 3 9 9 9 9 2e 0 10 10 A8 s■m 55.26 h 2 5 3. 19.566 34.254 3.829 51.726 64.77 32.657 48.563 19.532 35.668 39.960 32.830 17.110 30.535 58.033 1424. 5.663 29.846 50.946 56.666 59.609 381 6.381 14.848 20.033 43.682 39.157 50.124 43.155 54.344 29.578 RA 171 27 29 30 3 27 27 28 30 30 30 31 02 22 22 32 52 52 52 25 52 62 62 26 26 26 12 21 42 26 12 NITUD 6. V 2 2 2 2 2 2 2 2 2 9 2 2 2 O 2 9z 2 o 9Z O 2 2 9z 9 2 2 2 2. ~> h SAOSC 110566 MAGN 8.6 9.3 8.7 >3.7 8.4 8.6 3.5 3.6 78. 9.0 77. 3.3 3.8 8.7 3.9 3.6 34. 3.9 19. 73. 9.5 3.3 3.5 3.S 93. 3.8. 7.5 6.5 36. 7.3 CO -CT 92 SAOSC 732 29 0 1 1 1 1 1 1 1 1 1 1 1 STAR 10574 113597 C361 113639 113563 57610 35331 60610 17609 C61110 631 50210 5203 113525 110527 113539 3514 54631 11.550 10552 113554 113556 113558 119515 11)516 110537 110565 2 3 4 5 6 7 8 3 1 2 3 1 1 1 22 23 24 52 26 27 23 29 0 A/A C1 1 12 3 14 5 61 17 31 91 IDIX'J \}tL Oo VO AUR.M2?..:0-S3DOFS.. Y-YO -mm 5.1433 5.3310 2 31.1304 12.3581 3.41122 23.4481 324. 10.5488 13.2624 28.1333 8325.1-1 70365ASUR.XOO-OF.OFAST.-16. -31.3733 -42.3674 -51.1745 -36.0346 -53.2533 4130-54. -54.2552 -28.1285 -27.4523 -14.0040 -25.4431 71u-33. X-XD mm 5.5325 32. 22.0733 - -5 2- 27.2595 33.9325 35.3542 2 -5.4542 12.3112 10.9052 29.7023 60.5204 15.4333 21.6575 0133 -51.5553 7.3114-4 5071. -17.5303 -54.3223 55.33.4 -43.7335 -32.4011 -27.1124 0341. -11.0553 oos: (1.5.)S 110723 Y 3aTABLE mm 22 22 1 1 1 1 71 1 14 4 5 3 1 32769. 131.6031 228.3731 210.0403 202.8260 221.0333 155.7033 155.6153 503236. 161.5331 4.4138 37253. 143.4175 5429. 170.2133 76783. 53322. 157.3720 3031. 222.6743 203.2315 215.3451 225.7025 33513. 204.0737 X mm 3DAT 1371.8707570 503.2754 515.5366 509.3030 514.1316 532.9732 563.8958 513.7532 524.3430 525.2432 530.5343 535.1347 537.2573 336351. 441.5195 455.4540 477.1584 485.5851 437.3112 3525433. 439.7170 457.4353 470.3743 476.1630 483.0860 492.2106 3 3 3 9 3 39 39 SAOSC 308 13039 93110 5133 110737 110742 110743 110745 110752 110753 110753 110652 110674 110693 110702 110715 304 530 05J 3053 750 170 370 2 3 4 1 20 21 22 23 24 52 52 27 23 23 03 A/A 21 31 41 51 51 71 31 31 S " £ ISV 3C 3C * 8 C- 03 'BC-OC 'EflSVEK *BnS¥3t; 8 L S L I 8 Z. 5 9 Z £ 9 2 C 9 C Z 8 tj 9 £ 2 8 L 8 I 8 68 8 8 9 I I 8 5 Z L 8 C UL 2 ti L ti 2 2 L C C 2 8 Z I 6 8 9 8 ti 2 tjltj 2 t) 86Z9 6 G I C 8 8 Z I t, 8928 ZG86 Z 6 9919 LZiZ I UL * OX-J * * * * " " * * * ' * * * 2 6 C 9 *6 I Z 8 L C. Z ii 8 L 8; '2 I 8 2 6, *92 1 I *92 I 2 t 'ZZ I 8 '88 I L 'Li *28 8 £ 'tiZ *88 6/ l 2 L C L 6 8 t 2 9 2 8 8 8 c 98 2 C 8C 6 6 9 C £ C 9 L Z 8 6 ti Z 8 Z C C 6, 6 8 6 9 8 C ti 8 8 tit) 8 CC2C 9 t)ZEZ 89 9 9 8 L 8 6 1 9 9 ZZZG 9 8 tiZEti 891Z 19: " * * • cx-x Ui.Ui. * * " " * " " " * * (*S 9 2 "8 C 2 Z 6 8 6 I *2 8 Z *6 9 1 9- "91 8 "99 2 ti ti ti 9 8 2 1 9 8 - - 1886*98 *99- "89- - - - - "88- - l- *£) 86 6629*98- 9 q£ C I I 2 9 ti 8 6 € 9 C C 6 2 2 Z 2 :scvs 8 2 6 9 L 2 C 8 21; 9 9 ZZ 8 9 Z t; 968 2 Z 8 6 Z I I C Z 98 8 9 LI e 8 tj 1 Z C aiavj, X 9Z6Z 2 9 0 112 C 8 8 Z 8689 Z 6 8IGC 1818 89(6 " uiui * " * * * * * * * * * • * Z 9 *Z *8 Z 9 6 6 9 9 6 ti C 0 "2 I 6 8 8 L 8 *2Z Z 9 C9 Z 8 9 8 t, 8 8 L L 2 1112*812 ti2 *tiZZ 2 I I I L I I L L *891 9929*881 t L 2 9981*982 "982 22 2 "682 2 t] 6 Z L 8 C 8 9 6, tj t) 8 6 9 8 Z C 9 Z 6 98 L 9 6/ 9 Z 3 C 8 82 0 I 9til tj 9C 11; 86Z C 69 92 26 9 J Z X Z80Z 1992 9 Z 8888 8169 tj 9682 ZZ88 8 9 0 I ZZ 2899 I I 2C ClEfc 2. 8 2 6 8 9 8 UltlL * * * * * * * * * • * * Z '/C 8 *8 *Z 6 9 Z 9 C 8 2 C 9 •7 9 h 9 9 6 6 ti 9 6 r * C *98 tj t)9 *88 f ti ti 6/ tj c, I 69 61) 9 *819 9 *669 *699 *219 *629 9 *199 *299 9 j tj t] tj t) tj •ngtj Z 6 I I 8 I C 2 Z 2 8 9 9 C 2 6, 9 9 ti 9 6 Z Z 8 6 I C 8 tj tj ti ti 9 9 Z 8 2 8 8 9 9 C C C I Z Z Z Z Z 9 9 899 9 9 9 C C C C C 8 8 8 8 C C C C C C C C C C 8 8 8 8 8 6 6 6 I I I I I CI I I I 6 6 6 6 6 6 C9G86 :sovs 8ZC86 I 91ZCII 11 11 I I I I I I I I 2 8 ti c 9 Z 8 6 C I 2 9 Z 8 6 C 9 Z li/V I I I I 91 I I I 2 22 82 62 2 92 2 H O o Ji AST. ME1SUR.OO-OR.OF MSAsua.00-33.OF Y-YO mm 7.8296 23. 534.21 1.1788 95525. 13.1920 7533 9 33.3729 25.5313 -56.7713 -1.7315 57275- . 392693.- -53.5513 -32.1994 -19.2303 53.3726- -43.2215 -26.2394 -13.7337 9752-55. -21.4401 -53.9312 -53.1742 39.5713 52.0889 1 x-xo mm 2.3383 11.2391 17.3345 31.3422 52.9655 634719. 13.5799 13.1415 22.2733 25.3575 34.3931 332643. 532236. -53.3533 135-53.. -49.1345 -12.9532 -5.5531 -2.3535 -1.3132 -55.7139 -39.3577 -23.3133 -13.5313 93- . SO (3.5.) 355 SAD 1 TABLE 3c Y TTlTn 23 197.5792 215.7712 221.3425 222.3311 235.6521 224.2732 143.8374 135.7975 173.0339 1536523. 144.J173 165.3793 178.2934 154.3366 149.3577 171.3393 133.7955 142.3113 71935. 136.67B3 134.4359 237.2532 198.7533 249.6531 5.3343 224.5343 X DATE mm 5 1971.9339253 503.9132 515.2073 521.4223 535.2634 556.8343 565.4651 536.3373 517.4931 522.3597 526.1975 529.2753 533.9163 544.2533 5.9473 453.3552 453.9346 735743. 493.9653 497.3551 591.5577 532.3050 4323. 464.5535 483.9379 392524. 533.3389 1 SAO 31 3 13 31 13 31 13 1 1 1 1 31 1 1 1 1 1 1 1 31 1 1 SO 562 674 63 372 356 635 635 637 763 37.36 63S 965 5936 63C 613 463 534 6374 345 6310 13515 632 369 5634 2 3 4 5 7 3 9 1 A/ 32 21 2 23 2 32 1 1 1 1 1 1 4 J 1 2 3 4 5 51 71 31 1'J o H OO-OR.M2ASUR.3.5.3F AST.03-03.MEASUR.F3 Y-YO Tntn 3.5392 8.4957 9.2118 7.6254 2.2364 4 2 1.1543 - 1 2. 2. 5.15222 7355. 13.3709 -11.1316 -10.3311 34423. -28.3691 -14.3415 -13.1659 -36.6553 -47.6269 2133-53. -50.4013 -30.7595 -50.4032 4970 33.0425 41.0332 15.3343 5664 38.210J 45.4137 X-X3 "mm 341 . 5 3 34 13. 5741. 3321. 21.3332 50.0174 47.3977 -1.4216 -3 -4.1383 3 2 57.2137 53.3575 3 0 43.1355 . 31.0335 17.1797 1428 49.0383 12.2704 13.0015 7. S.) /- -44.9335 -37.4195 -17.7532 7-10.521 -54.5792 -53.0114 -37.3141 7307 -35.4173 -30.9533 -11.3757 (SAOSO 9313 TABLE 3d Y TiTm 197.5335 203.2277 205.0342 213.1507 133.6923 139.3249 205.7503 213.2122 216.4094 135.3569 136.6574 133.3041 153.5594 132.6970 134.3725 160.3732 149.7115 139.2073 147.1357 166.7590 137.1353 205495.1 240.0355 235.5311 233.5717 212.6223 220.1049 235.7435 243.9322 X DATS Tn 1371.3240533 502.7843 533.7939 519.9643 521.9271 524.6175 551.3831 554.5305 560.0030 561.5519 535.106 545.9709 562.3017 504.6277 515.0547 520.7853 550.3320 457.3257 736485.4 435.0211 492.1525 501.3627 443.1051 449.7729 464.9702 465.0536 466.3655 471.3154 491.2075 433.5260 9 3 9 9 3 39 SA3S3 33199 0329 3924 32279 33223 3192 93232 321 3223 32943 110840 110353 110862 110833 110733 110909 110317 110924 110834 513 33133 93150 3151 93152 3153 5317 3132 5 2 3 5 7 3 1 4 2 30 31 02 2 32 42 5 25 27 1 1 1 41 1 1 1 31 91 A/A 0 1 2 3 5 5 7 o ro 33-OR.:1EAS(JR.3.3.OF 32.\SU3.30-33.3FAST. Y-Y3 mm 5.3273 2 53. 541. 35353. 5. - - - 31.6333 1 - 1116 -4 -6.7054 5 2 732 43.7575 8354 -22.7574 -5.7363 -23.4662 -53.3329 -23.3323 -15.5690 -3.5554 -23.7359 73294. -12.3383 89154. -16.6603 32.6384 -15.4142 1-19.32u 7.0654 -13.6086 73.551 12.3366 X-X3 mm 4.5333 5.3225 1.7554 3 - 23.3115 23.3143 23.3335 37. 43.3243 41.5329 51.3351 73.7343 49.3324 43.3313 43.3521 23.2573 11.3333 13.3533 13.2333 534. -3.3739 -7.4795 -5.5174 -2.1325 53.3352 317455. 337324. -3.3571 TABLE Y 3e mm 23 7391. 6-3433. 247.6533 248.7352 3.341175. 132.1545 153.4345 147.8573 163.8683 132.3313 137.3454 57491. 177.4335 135.0523 153.5419 131.1354 131.2403 153.4373 132.4365 137.0337 143.4353 137.2322 163.7351 259.3393 210.2374 223.5345 251.4333 223.3124 X DATE mm 35 5 94 94 4 4 (Sh352 3.553335 523.9653 538593.1 532.9631 535.4192 513.5763 524.6423 541.5445 544.4781 545.2467 54.9839 429.9493 454.5514 459.7625 73514j. 483.3P59 7543491. 39.2430. 3.3653 9374433. 74954. 445.1753 7.1346J 531.5213 445.2536 447.3364 7531. 59673. 3521931371.93511QG S3SAO 7392 1 1 1 1 113579 75931 113563 113575 113583 113695 113609 113619 113615 113502 111523 113525 113527 113539 113545 113543 113553 113552 5534 11"556 351 510 113515 113537 56j 2 5 6 7 R 3 1 32 21 22 23 92 92 32 27 23 32 A/A o1 11 21 31 41 51 51 31 91 TABLE 4 DATE SAOSC(G.S.) 1971.8707670 110723 A/A SAOSC RA DEC DRA h -m S 0 1 // 5 2 43 21.271 9 32 45.40 (Asteroid) 13 1107^2 2 44 11.803 8 58 0.21 0.0201 24 93034 2 36 21.713 10 25 21.48 0.0305 28 93067 2 39 47.096 10 31 46.42 -0.0082 RESULTS USING SYMMETRICAL DISTRIBUTION OF THE STARS 2 43 21.264 9 32 45.49 (Asteroid) 13 110742 2 44 11.791 8 58 0.38 0.0084 2b 93034 2 36 21.716 10 25 21.34 0.0344 28 93067 2 39 47.097 10 31 46.32 -O.OO69 DATE SA0SC(G.S.) 1971.8870742 110698 A/A SAOSC RA DEC DRA h m S 0 / 11 S 2 39 18.140 9 18 44.22 (Asteroid) b 93088 2 42 50.536 10 35 56.13 -0.0287 9 110737 2 43 43.781 9 11 30.31 0.0296 18 110662 2 36 40.618 9 16 27.40 0.0284 RESULTS USING SYMMETRICAL DISTRIBUTION OF THE STARS 2 39 18.140 9 18 44.22 (Asteroid) 4 93088 2 42 50.536 10 35 56.13 -0.0287 9 110737 2 43 43-781 9 11 30.31 0.0296 18 110662 2 36 40.618 9 16 27.40 0.0284 DATE (SAOSC(G.S.) 1971.9009253 110650 A/A SAOSC RA DEC DRA h m s 0 / 11 S 2 36 6.961 9 9 0.58 (Asteroid) 3 IIO698 2 59 53.004 9 35 28.75 -0.0037 16 110610 2 30 39.948 8 26 9.57 -0.0121 24 110639 2 33 51.705 8 55 8.84 -0.0208 RESULTS USING SYMMETRICAL DISTRIBUTION OF THE STARS 2 36 6.966 9 9 0.62 3 IIO698 2 39 53.014 9 55 28.74 O.OO65 16 110610 2 30 39.943 8 26 9.50 -0.0165 2b 110639 2 33 51.706 8 55 8.91 -0.0198 104 DATE SAOSC(G,S.) 1971.8240538 93186 A/A SAOSC RA DEC DRA D DEC / h m S 0 // S II 2 55 29.597 10 22 54.06 (Asteroid) 9 93212 2 55 41.957 10 0 20.42 0.0057 -0.0999 18 110817 2 51 4.924 9 7 57.43 -0.0263 -0.1521 2? 93158 2 49 52.537 10 45 28.19 -0.0068 0.0768 RESULTS USING SYMMETRICAL DISTRIBUTION OF THE STARS 2 55 29.599 10 22 54.19 (Asteroid) 9 93212 2 55 41.957 10 0 20.51 0.0063 -0.0143 18 110817 2 51 4.925 9 7 57.40 -0.0246 -0.1825 27 93158 2 49 52.545 10 45 28.34 0.0008 0.2333 DATE SAOSC(G.S.) 1971.93336219 110566 A/A SAOSC RA DEC DRA DDEC II h in 0 / // 5 2 29 12.176 8 54 44.74 (Asteroid) 7 110588 2 28 48.560 8 7 38.30 -0.0025 0.1248 18 110546 2 25 50.937 8 24 46.17 -0.0092 -0.0046 28 110537 2 24 43.154 9 58 34.59 -0.0010 0.2363 - RESULTS USING SYMMETRICAL DISTRIBUTION OF THE STARS 2 29 12.171 8 54 44.47 (Asteroid) 7 110588 2 28 48.561 8 7 38.05 -0.0022 -0.1284 18 110546 2 25 50.944 8 24 46.07 -0.0015 -0.0982 28 110537 2 24 43.158 9 58 34.56 0.0026 0.2067 14.041343590 19.997432195 25.055439885 27.979721687 6.903750128 5 DT // 0742. 42.07 42.07 41.92 2.224 45.400 44.220 8050. 54.060 44.740 / 9.0 DEE 32.0 18.0 22.0 54.0 UT -0. -0.218707670350 -0.218870742657 -0.219009253701 -0.218240538656 219333622269 0 14.040356668 19.996945273 25.055952963 27.979235501 6.903261470 9.0 9.0 9.0 10.0 8.0 MH 1 1 1 10 12 // TO 2.6400 s 19.0650 32.1000 44.4150 23.9550 50.016 36.072 34.336 5.034 41.791 0. 1 / (MSL.T) 50 49 52 18 0.718707670350 0.718870742657 0.719009253701 718240538656 0.719333622269 RA* in 58 55 20 30 40 0 h 0 1 40 39 39 43 37 DIFFER. 23 23 21 ET1 5 21.271 18.140 5.961 29.597 12.176 TO 45.041343589538 50.997432194583 56.055439834557 27.979721686538 67.903750127521 RA 0 0 0 5TABLE h-m 58 -4 201 -29 43.0 39.0 36.0 5.05 29.0 T)(MSDIFFER. 5 59.681 -24.651 47.572 -58.878 -41.093-19-2 h 2.0 2.0 2.0 2.0 2.0 UT 5 h 3 4 3MSTATUT0 ms 45.81929 25.151533 137.928 44.378222 26.59350 z -2.238 -2.240 -2.241 -2.233 -2.245 m 0 0 0 0 0 0.50 10 10 10 10 10 GMST) 45.50 5055. 5045. 45.50 MOMTH 45.040856668242 50.996945273237 56.055952963271 27.979236501353 67.903261470114 5 h (MT3D -m 284 349 335 521 240 zo -33.611 -33.636 -33.657 -33.539 -33.707 0 0 0 0 0 -m h 4 5 2 )MSTMT3CHL sIT) 30.0017 33745.00 2240.00 30.00411 30.0029 7 DA 14 20 25 28 002= 012= 1 1332.2645 0.0000930000 0.0000948000 .0000000003 64 TEST= or Z=-0. DZ Z0= 1Z= = 3045901752 54.1510 -0.3043764000 -0.3086484000 P*Z1= TO^OCEN-0.0000353000 F2 ( -0.0127606328 *DZ •m 1 2=F 1-P)Z0= E2*DZ02= 2= CORRECT -0.0068783202 16 -0.2917923871 -0.0000012033 -0.0000006521 0.1658384372 I:N 0.0002146000002= 0.000217800012D= TOPOCEN E2=-0.0129383284 CORRECTION 6aTABLE Y= V Y LOCAL 0= 7111=-0. Y=0 -0.7023616802 SIDER.TIMEWRT 80008-0.70192 7795000 P*Y1= -0.0000213422 2E F2 (1 -0.0294275195 1950.0= *0 1-P= 0=-P)Y 2YO= *DY12= -0.0000027766 -0.0000014982 IM 0.9586564105 -0.6729085433 0.6440758040= 0.0001931000002= 0.0001890000012= 0.0413435395P= 0X= X= X3= X1= -0.6268351285 P*X1= -0.0000103054 L -0.6273867000 -0.6137044000 -0.0253727428 TOPOCEN 0.AMDA= RA=GENERAL0™ INPRECESSION-35^.8490 60144828180=(1-P)X-0. E2*DX02=-0.0000024984 F2*DX12=-0.0000013001 CORRECTION 7460782369 16.8635 002= 012= 1 .00 TONE33C 17 0.0031001009 0.0001015000 CORRECT )0000003 54 EST= or -0.0008339595 -0.3277161224 -0.0000000428 -0.0000000865 -0.0000353000 -0.3285855113 Z= Z0= 1Z= 9. -0. 1242 -9.3247752000 3285598000 P*Z1= ,th F E2*DZ02= 2= (1-P)Z0= F2*DZ12= -0.0003526411 37 CORRECTION = DZ= I0.1518153841N 0.0002307000002= 0.0002338000012= TOPOCEN 2E=-0.0004279647 6bTABLE Y=-0. LOCAL Y0= 1=Y 7576958285 TIMEWRTSIDER. -0.7489702000 -0.7576987000 1*YP=-0. E2 1950.0= ( 7557530772 *D 2YO F2*0Y12= 1-P= 1-P)Y0= = ON 5670.002 DY= = 8054 -0.0019232097 -0.0000000987 -0.0000001993 -0.0000192435 0.6320121894IM 8751-35. 0 -m 7140000.0001D02= 0.0001667000012= TOPOCEN 21946A30.997P= CORRECTI -0.0014305966 -0.5411511518 -0.0000000734 -0.0000001421 -0.0000138340 -0.5425957978 0.7578762258 X= DX= X0= X1= -0.5571281000 -0.5425443000 )0XP= P*X1= DX02= D2XI= AAMO= AINRPRECESSIONGENERAL= 002= 012= 1 / // TE33C0 311.5230 0.0001057000 0.0001077000 CORRECTN .0,000000000 08 TTES= or Z=-0. Z0= Zl= DZ= 4.1015 3461541554 -0.3459377000 -0.3490969000 P*Z1= -0.0000353000 -0.0197029887 E2 17) F *DZ02= F2*DZ12= 2= (1-P)Z0= -0.0093765829 22 -0.3264130161 -0.0000018406 -0.0000010099 0.IN= D 1590224642 0.000246400002= 012=0.0002483000 T3P0CEN E20172505291=-0. CORRECTION 6cTABLE Y= Y SIDER.LOCAL 0= Y1= Y=0 -0.7982222154 TIMEWRT -0.7977810000 -0.8050677000 1P*Y= -0.0000233761 1950.0= -0.0454379281 F2 E2*DY02= *DY12= 1-P= (1-P)Y0= -0.0000042505 -0.0000023282 0.9435601154 -0.7527543325 MI=0.6217005866 •m 0.0001424000D02= 0.0001374000012= 0.0564398846P= X=-0. 0X= X3= 1X= 4663530822 P*X1= -0.0000039047 L -0.4672209000 -0.4517094000 TQPOCEN 0.AAM0= (1 -0.0254944264 0RAINPRECESSIONGENERAL=-35.8985 X0)=-0.4408510063-P E2*DX02=-0.0000024565 F2*DX12=-0.0000012884 CORRECTION 7669421338 002= 012= 33.413113 0.0000647000 0.0000665000 1.oooooooooo 25 TEST= or Z=-0. Z0= Zl= 0Z= 2187169322 54.2275 -0.2130900000 -0.2187967000 P*Z1=-0.2143598719 TDPOCEN-0.0000353000 -m 2F= (1-P)Z0= E2*DZ02= F2*DZ12= CORRECT 0065552226-0. AO -0.0043211058 -0.0000002186 -0.0000004359 1 I0.1802047019N= 0.0001494000002= 0.0001533000012= E2=-0.0033783291 6dTABLE Y= LOCAL 0=Y 1Y= DY= -0.5043240112 0.0202783135 SIDER. *P TIME1950.0=WRT -0.4914175000 -0.5045793000 Y1=-0.4943472828 TOPOCCN-0.0000101007 (1-P)Y E2*DY02= F2*DY12= 1-P= 0= -0.0000005047 -0.0000010049 CORRECTION -0.0099651181 0.6817201390MI= TT)O1 -35.7725 0.0002514000D02= 0.0002484000012= TOPOCEN 0.9797216865P= CORRECTION X= DX= -0. X0= X1= 8277654008 P*X1= -0.0000214398 L -0.8372295000 -0.8275451000 0.A=0AM E F 1{ -0.8107638811 2*D 2*D R4INPRECESSIONGENERAL= 0=-0.0169776022-P)X X02=-0.0000008493 X12=-0.0000016283 7090725757 0.DO2= 012= i n T0 1330.7254 0001153000 0.0001161000 30CENCORRECT l.OOOOOOOOOO 37 or -0.0360592062 -0.3403294331 -0.0000018325 -0.0000032043 -0.0000353000 Z=-0. DZ Z0= 1=Z = 3764289762 54.0484 -0.3746416000 -0.3765747000 1P*Z= 2E * 2F (1-P ZD <=DZ -m 20 1 F2 = 2 = )Z0= = -0.0275998823 28 002=0.0002660000 0.000267600012=D TOPOCEN E2=-0.0158930350 CORRECTION 6eTABLE -0.0831573103 7848431810-0. -0.0000042275 -0.0000073857 -0.0000143373 Y= SIDER.LOCAL 0=Y Y1= -0.8680264419 1950.0=TIMEWRT -0.8639732000 -0.8684294000 1P*Y= E2*DY02= 1-P= (1-P)Y0= F2*DY12= 0.0962498725 NI=0.5986819274MI= 9516•35. 0.1549246767 TO 002= 012= TEST= 0.9037501275 0.0000904000 0.0000858000 DY=CORRECTIONTOPOCEN 0278952632-0. 1924677611•0. 0000014367•0. •0.0000023581 -0.0000188715 0.7858614982 X= DX= X0= X1= 14-0.27469405 P*X1= -0.2898213000 -0.2730579000 MAL0= SP0RADMRECESINIIGENERAL= {1-P)X0= E2*DX02= F2*DX12= Ill o CO '— —» —» X C J Is- LP NT CO CP CM CP O Is- CP O LJ O CT CO M" Nj bJ p- IP MJ bJ X CP r—4 • 1— U > •—< o, O CM vT r- P- r- • • • II u c_ c: ►— < LP LP X vT CM f ■ "4 CM >- M" CM PI o CP MJ CM o » 14 LP P- LP • _J o CM o CO o p- X C CO cc CM P 1 p • 1^- c— • • • f—1 II >- CM CM CM X en u> —« P- CM LP in vU c X X LJ r- ►— x «—• U> >r CM LP LP LP L0 r- rp CO L_J cc O MJ CM CM (\J CM O LP CM O b^ pr- c LP LP LP IP vT CM NT OC' <" p- CO CP CP L* CP LP cc CO r«- p- vO UJ UJ X X X X • LP X CM o OJ vO vC >P sC sC o X CP- b) a > m .—1 t—»4 r—4 rH r—4 r 4 «■" * CP CP X CM r- O ■ 4 ^-4 r—^ .—4 r ■ 4 r—1 r—4 P\ CP fP >r CM M" NT M" vT <" •c c • X • CM CM pj .—1 CM o w w- II II II II r- m —4 CM CP X II NO o CP X »-H CM NO sU X X c_ Ci CC c< X CO (V X rp cc p > II CM f—1 LP CM r- h- CJ L ^ >—* M" NP vL' C">, X a X' P • r~* V4.' P- r- X —« M" CM CO L..' c M" O" 'P • • • r- p PI P ^ r—4 P > NT II II II • ■ >4 CM P • i\J cc V CC II 112 o o LJ bJ p" P- CP LJ cc vC Vp o eo r- 7 i PA o r^- (X O U^ PI *7 O C7 LP bJ bJ UJ LP r" 4 r-^ r-4 (X • r- bA LP Ob »—4 rp Ob CO CO o bJ r- P- fb- >7 CP bJ • • • CT bJ II bJ O bJ 7 >7 bJ I— ex O • <* bJ r—4 -J 9 4 bJ • • c? UJ o II ro p- I CO Ob o sO 7 rx ii II *- c< co LP CO bO co m *• CO 7 PA p- cj pa r—4 r 4 * 1 14 » 4 « 1 '4 l>» ro b"A CO CP p- O »—< r-4 « 4 7 7 u bO CO UJ II r- 7 CP II II (XI >7" 7 7 \T 7 7 7 >7 p^ ' • • • • • • 7 7 CO p- CO LP fO fO • • • • CP PA pA P A PA CO co O •—1 LJ 7 r— lp O Z PA p i m P 1 (TA PI PA O bJ CO c- pa • CO i 4 PA f 4 CO O O CO 7 bJ r- r—4 bJ CC 7 7 CO cQ • • • • • c O II X" • Ob (X ex en Ob 1 w It II ■tf PQ CXJ (XI v- II II II ox K- CD CX CP LP It CO r 4 Ob 4-0 ox EH d LP LP — ^yr cc re SO 7) CO o o (X bJ P- r—4 (X >7 ^7 LP UJ LO CL' CO C\J eo o bJ (X CO P- (X sO ex U_ Lb Lb r—4 CU r—4 o o bJ ex i^- bJ bJ ex ex ex u nr\ 1 CO p- bJ CO ex cp CO CO O r^ o O CJ CJ UJ P 'i PA r—4 bA PA r—' LP bJ P- a_ eo CO • • • CP CY ex PJ o » 4 >7 CO LP' u a un bP LP LP bA <—« u > u ^ LJ > 7 -rC LP o ex LP r- b> u A u A L > u > UN CO >—I —• •—4 7 >7 LJ cc r— LP lp u A bA U'N uy • r-4 u \ a > LP ex LJ LJ • • 7 IP bA u A b A b > (V* r^ b> Q> O o CP 7 7 7 >7 >7 >7 7 NT >7 Ob • • • •• cu LJ LJ (X • •• • • L> i i IX (X i 1 ex ex (X Ob ex (X (X (X ex 1 b". CJ LJ LJ II O P- cp It II IP o O o r—4 C\L r-4 7 r—^ r—4 7 \Q U > bJ —< f ■ 4 vi_' 7 7 X" 7 • • • sb' P > PA P A f—4 P A <" II II II • f—4 Ob P A ex CC QC 71 c o •—• —- —« o UJ >4 CO cr UJ • • X vT cr CM CM O r ■ 1 0 • • r—1 vT U N UJ u ^ U A on on IX ^ J X CJ X r-i CJ <_ J C> vj" vT • * 'JL OJ CJ II f— c UO O 1 cm c If II -/- CO NJ f < •—< Hi¬ CJ r- LO s. in X r—H r—H r > 4 11 11 II • r—< (M ro CM rr: cy 1 C ii r—H Q. O CJ CJ CJ xj" oc o CJ nT CM <0 CJ O CJ vO tn f\J o C\J O l.» CO P" in en in CJ CO en m r-« CM • fx. u 1 r—4 vT CNJ CJ O OsJ CO ro o n- sr <~rs CJ in Ln CJ vj" >J CJ CM CJ • < •—1 CJ CJ U r—4 CJ • • CJ CJ II CO CJ ro in I re r—4 en xC II II ro CJU > t ro -K. PO m v-U xX ry CO H— CO > r—4 cu XJ a: 02 I — OJ vT o * i l\J UJ • • • >r en (\J c\j CM CJ CJ CO C—J CL" aj UJ uj n in o o II • C\J CM CM t>- pi iM 1 w II II 3 CM CM -,'c II ii ii ii OJ Q_ •• n- CJ CJ II ir ro CM o 1 X cr P' II II Jj. rj f— p • » i r—1 -ft" in c L . CO c- CO ^T l_) m LO CM CO r—4 xJ" CO LJ —4 <—1 -ft- xO It fx II r—4 ex. 3K4= 38= 3L= 83= 3Y= 3X= 83= Y3= .822C150392 C.CCCC331GG5 C.CCCC175355 C.CCCC397157 1.CCCC4413C9 C.CCCC155621 KS3C.CCCCCCCCCC= C.CCCC397197 1.CCCC441309 0804288.5407 3C= 4.82523746LC C.GCC113C882 50C47896. CCOC.1356961 1507485CCC1. O.CCGC531777 CCCCCCCCC2C. C.CC01356561 1.0001507485 TABLE C5910.43318726= K1= 81= 1=CL CH1= 1=Y X1= 1CH= 1Y= C. 1 S1X 4.8266887370K2 C.CCCC238107f*Z 12.CCCC126C36 C582724H2 Y2.CCCC317460 C.CCCC112C56X2 C.CCCCCCCCOOKS2= - 28.CCCC285724 1.CCCC3174602Y H H ON 3A7 769 272 1G9 106 59*. 1 31 A 1 3 7" 51" 5981". < C' 2' 5' 26' 8'A 25 5'5 C'3 5 5C 9C 3 793C 72A69 27A8J1 8A1C0 59 5 9 6 0 357A36C550 CCC1C29765 9 A C 1 C855A596A7 9 CCCCCCCCCO ccccccccco C V A" 3 0 CCCCCCCCCv 0 . * A 6259281371 8550251332 . G. C8562C1377 0 C . . 7 r 0 3 15 5 A r_ A C° 2 -0 -C 9° -C. A A . a . 5°A r 1° 3° 55°3 13':: 8■2A. PX PY C PZ^ CZ^ T PZ^ CX: Y: CZ: T = - Tl= 3T: 1: 3: 7C9595692C 9aTABLE AT AT AT AT INCL FOR GFCR IINAT CN C F . = PX,PY Y>,Q RF:P ANQYALY ANOYALY ECCENTRICITY: ANIONALY ANCYALY YANONAL ANOYALY VECTOR ECSINPAS.= NCIIFEL= Tl=AT T3AT= ELEMENTS TEST TEST NYEA YEAN TRLE TRLE ECCENTRIC PERCF S PASSAGE=1971 INAXISRCJAI:Y-Y ECCENTRIC ECCENTRIC ANCYALIES= CCTC2.- LCNCITUCE CRCITOFPARAYETER U:A INACTIONCAILYYEAN ALIN= SEC^ FCRTEST CNI11-ELPER TINEFCRTEST MEOFIT ro CP CP CO PJ r—4 rp 0^ CP PJ o o o CC C CP o LP. P- CP CP cc s >0 X o fP r- CO >r CO PJ P- i 4 o vO J" o o OS P- p- cO PJ CO o LP o cc o o o CC: * • •. • • • o ro Np cc CP vO 55 • . _ o PJ LP,^ • ro r—4 : .—1 cc >r o o o CP ^CP r < r o r r—4 V-t 4 o O vO 4 O rp r I o o r\] LP r-H rp o IP v0 r—H o CO LP rc\ >r CP o ,—S. r- •J" f—4 CO CO O C\J PJ cj p- o p- r\j r—4 CP CP CG LP o JT o o NS¬ vC 0s o o p- C" o r vO CJ PJ L LP CP <■ <" CP CO • c* LP "UP "cj C'^ IP L" "lP PJ "P CP LP CP CO rc LP o o o 0s CO PJ CP PJ LP fp o CP o o LP rp CO o o ry\ Q O oc o (75 • • • • • • • • •• 0 0 O • • • P- LA o 0 0 LP UJ LU 8IT / ro ro ro ro t— i—• »— r— t—' r-* o o LJ LJ o O LJ w) CJ O LJ LJ 1 i • 1 I 1 i i i 1 I I j 1 1 lj LJ a O LJ o LJ u LJ LJ LJ LJ o LJ CO ro OJ OJ -p* LJ -xl xi ro -xl o CC OJ o -r OJ ro LJ ee LJ xl OJ O OJ ro )—1 LJ O CO ro -P- LJ OJ o cr f—* -xj 4> en SJJ -si -xi LJ \S\ OJ cr cr LJ ro 1— O cr LJ ro o xl LJ lj o 0J O xi en i—• -xl v>-> CC O OJ ro cr O OJ o )— NC on CD LJ OJ LJ ro eO 00 J- -xl xi cr CO o LJ CO U) LJ -si ro CJ OJ o CO -Xl r—4 cr 4> IX) OJ *—• 4> LJ 1—' -r l\J »—• ►—* • • • • • ,, - • • • • • • • lj O o o LJ o o LJ o 4> o o o LJ -xi o 1 l I I I 1 cr -xi -P- 00 cr cr cr -xi ro LH ro OJ en on en en ro OJ en OJ en ro OJ o o o L> ♦—1 O o o O 4> o o o O 4> i CO 1 1 1 1 1 1 1 en -xl I 1 1 1 o o o o c LJ LJ LJ o -si CJ LO o CJ LJ xl o LT ro ro cr ro o l\J o r—• ocr ro ro -si o OJ OJ (7s o OJ -*> OJ CO 4> OJ OJ o OJ 4> LJ CO Xj Xi • LJ 00 • O -xi 02 • -si -s| CO CC xj 02 4> -xl 02 /—- xl CO CO OJ -xi -si 1 xl » -*r 4> 4> P^ -xi P- 4> -P- -r •xl o on CO on CO Cr t—• cr en CO i—• i—• LJ O LJ c P- ►—4 o -P* > LJ r—• »—4 LJ CD ro 00 CD ro CD D o 02 ro CO L-' ro o LJ on OJ o LJ cr en OJ >—« eo LJ X en LJ -r •—< cr ro ro ro LH f—'' X ro x, ro cr h-* ro -P- x> ^s ►— xj LJ • • • • • LJ • • • • • • • xl LJ o o o f- JO o o o o o JJ ->J o D LJ -xl a -r o OJ 4> 70 U1 02 ~n LJ J> -xl "Tl -T ~ OJ o LJ 3: ro ~n 2 o sO OJ ro i—• X eo OJ X »—• OJ OJ ro f— f—• O O vG m CJ- o o O o C O > m O o O ro ■1 o 1 i 1 O 1 "7 1 I 1 I —I o ~n l 1 1 —i m 1 o o f— LJ LJ LJ • on LJ o LJ O en • LJ LJ CJ LJ • o —i O o xJ lj LJ m -xj o -< O o LJ -xi -< -xl en -xi Xj "Tl c on on CC en o 02 LJ en en en o 02 LJ en en o m -< cr ro LI cr en rj cr cr ro en cr en cr c m ro x| cr ro en h-4 m -L ro cr en »—' -xi I—4 X xj CO -xj —- X 02 rr. »—• 00 m »—• m ■xi -xl -xi -si h— 00 en cr on < CO —\ en en —i _£s r < X en cr vn ro —1 xl LJ ro CO -T j> ro sj ro CO LJ -p- ro CD ro 4> »—« C —i c o LJ OJ LJ 02 LJ OJ LJ -H LJ o LJ *n OJ 'O —1 *—* O "Tl o O LJ C ~ o O LJ o -xi o i—* o LJ xi CJ p> -xi O O *n < 4> t—* i—• 1—4 c ej f—• LJ h—' p^ » »—• _£s no »—• rn • • OJ en • • • 1 CO en • • • • • • « • xi o o o c o lj o o o o jo o cr —i o o O JO —i en 1 cr -z: i I rn 2: 1 -xl m i 1 CO —< -sj rn OJ ->J CJ m X •—• o o o rn *—• C") ro ro • »—11 oro ro • »—i *—* o ro ro • »—• O r—• o O O O O Tl o o o o C T1 O o o O •—< 1 1 m 1 1 1 l 1 i 1 m 1 i 1 I LJ lj o lj lj -n a LJ u LJ O o CJ LJ Tl LJ cr oJ ro CC o 02 cr OJ ro cr en m oJ ro 02 neo on CO ro en 09 ro uJ \n ro O OJ CO ro ro 4> OJ on ro -p* en ro 4> ro O en ro ro ro OJ cr cr _fS OJ cr CO cr OJ -r h—• -c* LJ 1—• ro OJ CO ro -T ro OJ OJ -r ro ro OJ OJ O x. LJ LJ ro D LJ -xl •xj LJ LJ ro -xj -xl x -xi -xi en CO cr CO CO O CO 00 CO en 02 cr CO 00 0s co OJ ro 0~» 02 CO S>J ro o 4r 02 OJ ro 4> en CO ro ro CD -si ro ro -0 ro -si ro CD ro -xi en xl on OJ 4> LO LJ -si I—4 en OJ -r P" -si OJ • • • ro • • • o • • • • • • o O o o o O o o o c i—• o o o I—4 • • • o o o i O o c o o o O o *w t—* )—' o O o O O o o O o o o 1 1 1 1 0 LJ L? LJ LC o CJ 0 LJ LJ CJ CC o on -si en 02 O o en 00 -xl -si on CO OJ 02 _£s Ol -T en eo CO 4> 'v(J o ro OJ VJ1 O LH ro uJ ro o eo en s0 ro on o »— ro on e- ro en LJ en r\) -xl en ro ro -xl xj en rx LH en en ro en en en ro en en on ro D oo 4> ro LJ 00 4> ro ej 02 -r ro o ro o en O o en O ro ro o en D o o LJ LJ -xj vO' O LJ D -xl •xj CO OJ ro CO OJ ro i—• ro CO OJ I—' r—• • • • • • • • • • • • • o o o LJ c o LJ O o o O LJ I I I i—1 0. 10310. 0.11797938970-91 -0.90659079420-02 0.48803650170-01 7239d-91279852-0. -0.12863614429-01 1603030-01 -0.49707949370-91 0.33330262239-01 -0.40230531489-02 27929199040-92 389317800-9118-0. 0.53907073390-02 0. 0. 0. 9 l'jl9462d-050.459 0 .23396037580-05 10240308230-02 0.29803046600-03 0.45910194520-05 0.23896037580-05 10240308230-02 3046600-03.2930 0.4591019462d-05 7580-052389603 0.10240308230-02 9.29803046600-03 10bTABLE 0. 10. 1098 7417 9.- -0. 3 derivatives 924169050-0317 0983774300-03 -0.10345272930-01 0.40992336890-02 oflamdaderivatives -0.17924159050-03 9.10983774300-03 -0.10345272930-01 0.4099233589d-02 miofderivatives -0.17924169059-03 00-03 10345272930-01 0.40992336899-02 niof theofstem9icoef theofc3efi:emts . :oeffi:ents system h forthesystemlamoaof formisystem formi 0.5240439461d-02 0.37865053210-02 0.78384851470-01 0.42286910460-01 3.87398965330.02197660360.1092083708-0.1186868428 0.52484394610-02 0.37865053210-02 0.7838485147d-01 0.42286910460-01 0.01938546910.0779346680-4.3182427142-0.1217897310 0.52484394610-02 0.37865053210-02 0.78384851470-01 0.42286910460-01 0.09745318980.0757244234-0.6708898599-0.0355370151 00-0.10245427720 0.87023046620-01 00-0.39594153980 000.2903157352d 00-0.10245427720 9.87023046620-01 00-0.39594153980 000.29081578520 00-0.19245427720 0.87023046620-01 00-0.39594153980 000.29081578520 P0D= PO 0.2895446400 2.3041735634 2.4329993137 2.4406152232 2.4410300116 2.4410524962 2.4410537148 2.4410537808 2.4410537844 2.4410537846 2.4410537846 T2=0.0870255509 11aTABLE 2.4410537846 0P= R0 3.2607642859 3.3890834856 3.3966706494 3.3970838759 3.3971062759 3.3971074898 3.3971075556 39710755923. 3.3971075594 3.3971075594 T1=-0.1024572255 0.008573752101=02=-0.0207046313-0.0049134399 3971075596. H ro 0.2896353500 VZ -0.2167338797 -0.2171000662 -0.2171000209 -0.2171000211 D2= P00= -0.0049137227 PO 0.0870230466 2.3040409552 2.4323508280 2.4404666391 2.4408814705 2.4409039601 2.4409051790 2.4409052451 2.4409052487 2.4409052489 2.4409052489 T2= VY libTABLE 2.4409052489 0.4998623236 0.5007968654 •0.5007J67610 0.5007067615 01= P0= -0.0207052271 R0 3.2606322197 3889355613.6 26249396523. 39693589413. 3.3969582990 39695951343. 39695957923. 39695958233. 3.3969595830 3.3969595830 1T= 1024542772-0. VX 39695958303. 131068120.85 0.8526190684 0.8526191209 0.8526191206 D= -0.0985744923 R0= r\J <\J 0 9 at) 169 St 2 SI \8 91 19B'/e I *rLL'Jl$ izay.t en>v 2 A 9696A 9 90S92A^ Z9666S9 ,6 T AS 00 ,92 k C 9 101 S9't 9 60C99IA2f ' * 1 9AS9S9 0 e 0 11 ;,A09 o99 JV o*7 ..vse „og ======*zt 2 21 1 =zi ssvw =A110 91xv 00 =iI3bO I1GW IV IV iv aiavi 992-*20100 T dO iion A"1V = =NDnaHiy A6I d = NjIlVM~lOM aivw□ c'dd IXIG5003 A"1IVG WGNV AlV.lOKV do f,v >iorV/j-1wds NV3W ?nv b513dV«Vd NV3W E'JVSSVd ucni 1 NO ioo: wniM3wc« 1 DliiiNSOOl j HI >jviriDi.v >j2d JG 5 WIJ ro 12b 0.032 29'9° 71304 96° 91° 59". 5T 0.1014912555 6' 3 56 3.61642178C3 3.6540603373 5 55' 699 27' 97".>757861197 39".285 95012". 296.0899593072 59°3'3898".9 50".693 TABLE ECCENTRICITY= SEMI-MAJORAX INCLINATION^ IS= MEAN PARAMETERITUFORB= DAILYMOTICN= MEANANOMALYT?AT= TRUEANOMALYT2AT= ECCENTRICANOMiALATYT2= LONGITUDEPERIHELION^OF ANGULARMOMENTUMPERUNITMASS= TIMEOFPERIHELIONPASSAGE=1971OCTOB. 0. 0. 9. 0. 0. 9. 1920250289 1251631301 0.3043754000 0.1398889233 1 .9954407553 2.9141278533 1.5865002988 8861673093 0.7077673727 1271424231 2957193985 0.3285598000 0.1765808202 2.0000375196 2.9211875171 59625713801. 9.8897745565 0.7259831568 0.0837350264 0.2849443817 0.3459377000 0.2066581709 2.0037690435 2.9270503024 1 .6044304186 8928259635 0.7412563015 Y 13TABLE -0.3149457117 -0.8523455833 -0.7019288000 0.3337744155 -4.7290200405 7.4884537959 1940954449-4. -24.9932138840 2.2702909762 -0.2047799301 -0.6238829603 -0.7576987000 0.4623310824 -4.7372810920 7.5022642976 -4.2161457303 -25.0010305525 2.3097477130 -0.0879134149 -0.5869769185 -0.7977810000 0.5264849803 -4.7439061716 7.5136845426 -4.2345305901 -25.0075299020 2.3426973118 X 0. 0. 0. 4. 0. 2. 1. 1 2 35 0.2274999773 1130161927 -0.6273867000 3513716839 -1.3713848860 4.2196437833 17.8009135607 14.0294204168 2.4412041126 3189455434 2299592237 -0.5425443000 .3279008667 -1.3280906775 1881924295 17.7952636682 14.0128426251 .4034986666 68822004 0.3227003821 -0.4672209000 1.3045411675 -1.2919335927 4.1619423619 17.7905419921 13.9989829134 1737104323 ro VJ1 43.16 41.01 40.08 40.38 41.93 44.75 DEC '17 16 15 14 13 12 9 9 9 9 9 9 58.467 39.006 19.695 0.539 41.541 22.706 RA 37 37 38 38 38 38 VZ 0.596611165705761753 0.0926312316198701496 0.216972148563807060 0.352501012201820479 -0.0439530210773910971 0.0683401001313072159 -0.0949134509713257662 -0.0354966747839541002 0.175549777167255047 2 2 2 2 2 2 Z VY C 0.72744956D+00 0.72895745D+00 0.73046397D+00 0.73196914D+00 0.73347294D+00 0.73497538D-+00 PLANETSTHEOFVELOCITIES 1.24340916807981605 0.360746153132148584 0.500442733394442354 0.753162100207822038 -0.0783732513728220559 0.133217626857053781 -0.213762473478564016 -0.0755889966201690466 0.379299386151404072 -0.33213349D-07 -0.33213352D-07 -0.33213354D-07 -0.33213356D-07 -0.33213358D-07 D-07-0.33213360 TOTALINITIAL Y ENERGY=-0.33213347D-07 0.23129154D+01 0.23161713D+01 0.23194229D+01 0.23226702D+01 0.23259132D+01 0.23291518D+01 VX 1.10741042893800157 0.420001820940074766 0.161118020193995876 X 0.24004174D+01 0.23972418D+01 0.23940617D+01 0.23908771D+01 0.23876881D+01 0.23844947D+01 o.674229922440345519 22 23 -0.852644829972549895 -0.251703919455896930 -0.304955616121567213 0.0548306703628411744 -0.368434268679320082 21 Nov. Nov. Nov. 48.85 54.25 0.97 9.01 18.40 29.15 41.27 54.78 9.69 26.01 43.76 2.94 23.57 45»66 9.23 34.27 0.81 28.84 DEC 4 3 2 0 0 9 8 7 6 5 5 3 1 1 59 11 10 10 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 4.039 45.543 27.224 9.084 51.128 33.360 15.784 58.404 41.223 24.246 7.475 50.914 34.568 18.438 2.529 46.844 31.386 16.158 RA 33 33 32 32 37 36 36 36 35 35 35 34 34 34 34 33 33 32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 c -0.33213363D-07 -0.33213365D-07 -0.33213367D-07 -0.33213370D-07 -0.33213372D-07 -0.33213375D-07 -0.33213377D-07 -0.33213380D-07 D-07-0.33213383 -0.33213385D-07 -0.33213388D-07 -0.33213391D-07 -0.33213393D-07 -0.33213396D-07 -0.33213399D-07 -0.33213402D-07 -0.33213405D-07 -0.33213407D-07 Z 0.73647646D+00 0.73797616D+00 0.73947450D+00 0.74097146D+00 0.74246704D+00 0.74396125D+00 0.74545407D+O0 0.74694552D+O0 0.74843557D+OO 0.74992424D+00 0.7514H52D+00 0.75289740D+O0 0.75438189D+00 6if980.7558D+OO 0.75734668D+00 O.75882697D+OO 0.76030585D+00 0.76178333D+OO Y 0.23323861D+01 0.23356161D+01 0.23388417D+01 0.23420630D+01 0.23452800D+01 0.234849250+01 0.23517008D+Ol 0.23549046D+O1 0.23581041D+01 0.23612992D+O1 0.23644899D+01 0.23676763D+O1 0.23708583D+01 237403580..D+Ol 0.23772090D+01 0.23803778D+01 0.23835421D+Ol 0.23867021D+01 X 0.23312968D+01 0.23780945D+01 0.23748878D+01 0.23716766D+01 O.236846HD+OI 0.23652411D+01 0.23620168D+01 0.23587880D+01 0.235555490+01 0.23523174D+01 0.23490756D+01 0.23458294D+01 0.23425788D+01 0.233932390+01 0.233606^7D+01 0.23328011D+01 0.23295333D+01 0.23262611D+01 25 28 26 27 29 30 24Nov. 1Dec. 2Dec. 127 APPENDIX A-l Proof of the convergence of the series (5.57) (Chapter 9) The Functions (Xl^oclCt) J ^>1] k = $>ijK (t) i "jjij k K (~t) j <^'Lj k^yl< ^j (t) j xi = ~Xl It) ((>jjk-ly"t] OUj = Ctij(X) , bLj-hijll) (LJ=1, . ..m Lfi (A-l.l) possess first order derivatives with respect to T in any closed interval [ —Hi o "Ei ] 3 as was mentioned previously. Hence, according to the theorem 1 (Chapter 3) the functions (A-l.l) are continuous in this interval, which is closed and obviously bounded. Therefore, the functions (A-l.l) are bounded in this range. (H. S. W. Massey, p if9~50) Let us take = max absolute values of the upper and lower bounds r • 1 of the functions (A-l.l) in the interval |_ Li^Lzj j and J\~ moc* 3 (1+7T1 i) j m t (L= 1 > . .. 71) . (A-l. 2) The right hand side of the relation JL -n (A-l.3) // -(l+T7li)cX£- I TTlj OC ij ~Xl+ Z_TT>7 ( OClj -(Xj)~Xj (L=1j .. .n) J=i J- i J J J jH contains (3 T) —2) products, with three factors in each one. Interpreting _Fi a as representing any one of the functions (A-l.l) and the factors ( 1 + 7T) c ) j TD LJ>([ = lJ...n) ^ the products which appear in the relation (A-l.3) have the general form FVF2 R . (A~i./f) Clearly, in the interval Ti j Hi ] ■> the following general relation is valid 3 F.-R-F, 128 hence " ft 3 ~y\i(T) \< (?>n-z) J\ = oc2 (L- 1 o...n) rel-TioTi] . (A-1.5) // Since the second order derivative AL (l) ('L = 1 j . . . n ) contains (371 —2) products of the form (A-l.if), the third order derivative ~X l fl) (L - 1 3 .. . 7i) will contain (3 17 - 2 ) '5 products of the general form / • • Fi Fz Fa j (A-1.6) where F3 £ o[ J7 /cL T and the meaning of J~i > K o J~3 is the same as above. We define, 0 = maximum number of products which appear in the right members of the relations (3«26), = LL maximum number of factors of these products . r From the equations (3«26), (A-1.3) and the definition of J 3 it is p / evident that the derivative _T3 will be either zero or a sum of 0 products, at most, and every one of them will contain, at most, u. factors. It is obvious now that the function Ai (ZJ will be represented by a sum of (3 77 -2)-3-0 products, at most, of the general form Ft- E---K , where V jJL+Z . r _ „ In the interval [- T 1 j T1 the relation |^+2 | J1' J7' ' ' Ft I ^ J is valid, since \J 2 ja. + 2 . Hence I a) , ^-+2 = i, . |"^l (z) < (371-2) 3 071 =023 (l ..7i) xef-T^Tj] < Working as above we obtain the relations ) | (371-2) 3 QZ (IL+Z) J\ =■ CCif. , i (z) |\<(3m-2)3 03(ji+2)(2^.-f-l)Jl3^HOC 5 D (A-1.7) I/) ({*.-!)-Z/JL +5 (T) \<(3ti-2)30v' 2(jLi+2)(2|x+i) ■ • ■ -(v (j±-i)-z>fjL + s) J = a 1/ » (L - I j ...Ti) T £ [-T 1 DT 1] 129 .(V) - we "Xl At the point TL 0 have x = 0 = "X Lo (i j ...Tl ) 2>/0j hence l7(lol n< _A , (A-1.8) l~XVo T. I < _A III J ti -j- 2. rx lo -o-/ < (i-n-2)J3 U + 2 It 3 I . (3) T1 ■x ^ (3n-Z)30_A 5' -J7 =oc3t x 5 1/3/ 3 ' M-x ,v(fJL-l)-2^+b V"lL Alio { (3t)-2 ) 30 \JJ (p-+z) (i/(ju.-l)-3|A-/-6Ln U' (i=i = T £ [-IijTi] . We form the series Ir1 (A-1.9) L OC v V=2 I// and we take the limit of the ratio of two successive terms of it V+ L |T OCvZ +i /im iv+1— -iTl /im V"» 00 (Xv IT l/T 00 CCl/ ( ) v/ v-i (371-2)3 0 IzKlth (/1+2) (2/jl+I) ( V (fi-l)L - 5fJ. + s)(v(fJL-l)-Zy.+5)J\' V -* OO (W1)(371-2)3 0""2 fyvL-l) (|LL+Z)(2jLL+i) - - - - (v{fL-l)-3fJL+6) J\ = Irl /im 0 f L(^-t)-2^t5) J V' —? 00 (l/+l) 130 !xl eJ\ -Lim11 I V (UL-1 l) . /■_ "2K+5 Hi : — VV-» oo l/"f 1 V 00 1/+1 - 1T10 We can choose the interval l-Ti o T i 1 such that zl 0J\h (LL-L) < I In this interval the series (A-1.9) converges (ratio test). Hence the convergence of the series tv) JT_ . LO X V.' fisi a . . .7)) le (A—1.10) l-Ti .It] ^ V— o (o) follows immediately from the relations (A-1.8). We interpret "XI to mean X io . The convergence of the series ( W ) J- V (l - J j ...?")) j (A-.l.ll) V= o v/y in the interval follows as a result of the [~Tj 3 TiJ convergence of the series (A-1.10) With a similar process we can prove that the series -7- \> J "Vl" r / Jlo ( L= 1 D . . . 77 J// ) 3 (A-1.12) l/=0 CO (I/) T ^ = Zio (i U . . . 7?) , l/=0 ~~yj (A-1.13) converge. The common interval of convergence of the series (A-l.ll), (A-1.12) and (A-1.13) is the interval of convergence of the series (3«37)» 131 APPENDIX A-2 Determination of the mass of the asteroid Use of the total energy of the system to test the accuracy of the numerical applications The problem which arises in the numerical application of the power series method concerns the determination of the mass of the asteroid. A first estimation can be obtained by using the relation p- 5-7tj- -2 lo^K ~ 0- ^ £> (i0o) where p is the albedo factor, with probable value 0.16, R the radius of the asteroid in Km and its absolute magnitude, given in the Ephemeris for the Minor Planets. -3 With probable density 3,5 o r cm , and considering the shape of the * asteroid as spherical, a value equal to 0.97 10 solar masses_,is obtained for the mass of the asteroid. Although this value is satisfactory, because the mass of the asteroid is very small compared with the masses of the planets, we calculated the mass of the asteroid as follows: Consider three points on the real orbit of the asteroid A, B, C at three instants T. i a j and the "to ia position vectors 17 JfT 3 17 . The use of the relation (3*¥t) twice for the values Ti , , -J fo J li,~Loand.J ——VoXAo.h—— J -s gives two velocity vectors JT and Toi at the point B0which must be the same. The path of the asteroid in space depends on its mass, hence values of the mass different from the true value will give either a more curved / / or less curved path (fig. 1), hence the two velocity vectors To and will not be the same. The mass which gives the same values for the vectors at point B is the true mass of the asteroid . 132 -13 This method gives a mass equal to 1.10 solar masses. Another problem which arises in the power series method concerns the checks during the calculation. Here, the total energy of the system was used as a check. We notice that the velocity of the Sun (origin of the co-ordinate system) is very nearly constant since the centre of mass of the system is inside the Sun, hence the value of the total energy must be very nearly constant. Another test of the accuracy of the calculations, used in the numerical application, is provided by the mass of the minor planet calculated from the solution of the equations of motion. This should, of course, be constant. 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