N. N. Goryachev.

Secular Perturbations of Ceres by Eight Influential Planets.

Dedicated to TII.

The present article of mine should be regarded as a brief preliminary communication on the results of the work I have carried out on the problem of computing the secular perturbations of Ceres; this work is a link in the larger work of Prof. M. F. Subbotin on the creation of a theory of the motion of Ceres; by agreement with M. F. Subbotin, I have taken upon myself this link in the theory of Ceres.

The creation of as perfect as possible a theory of the motion of the minor planets is a task of current importance at the present stage in the development of astrometry; it makes it possible to determine more accurately the position of so fundamental a point of the sky as the point of Aries (the vernal equinox), without resorting to comparatively less precise observations of the Sun, but replacing the Sun by another member of the solar system—a minor planet, observations of which, as a luminous point, are more precise.

The question of finding the secular perturbations of the planets has, like every scientific question, its own history; the founder of this enterprise is the great Gauss, who in 1818 published his famous memoir under the title: “Determination of the attraction which a planet exerts on a given point with a given position of that point, if its mass were distributed continuously over its entire orbit according to the law of direct proportionality to the time which the planet takes to traverse any infinitely small segment of the orbit” (a paraphrase of the Latin title of this memoir).

The point is that the perturbing function, as is known from celestial mechanics, consists of two parts: 1) the force function of the direct attraction by the influential planet; 2) the transfer, if one may so express it, of the force function of the influence of the perturbing planet on the Sun; this second part of the perturbing function, as is known, gives no secular perturbations, but only periodic ones, so that the question reduces to taking into account the influence of only the first part.

In that case Gauss’s idea becomes clear; if one confines oneself only to secular perturbations of the first order, i.e. if, in computing them, one does not take into account the perturbations experienced by the perturber itself (regarding its orbit as constant), then the following may be said: to a given position of the perturbed planet in its orbit, during its successive revolutions around the Sun, there will correspond, in the case of incommensurability of the periods of revolution of the perturbing and perturbed planets (and such a case always occurs in our solar system), an innumerable continuous set of positions of the perturber, the density of these positions on the orbit being nonuniform: where the perturber moves faster, there will be fewer such positions in that region, and where it moves more slowly, there will be more; for the exact justification of this see Tisserand, Traité de mécanique céleste, Vol. 1, pp. 434–436.

Without entering, for lack of space, into a detailed exposition of the solution of the problem of the attraction of the indicated infinitely thin nonhomogeneous ellip-

of an elliptic ring upon an external point, excellently expounded by Halphen in his Traité des fonctions elliptiques (Vol. II, pp. 310–328), I shall indicate only the principal stages of this solution and dwell on the errors committed by Halphen in his extremely ingenious analysis.

Halphen considers a cone of the 2nd order, whose vertex is the perturbed planet and whose directrix is the orbit of the perturbing body. Taking the perturbed planet as the origin of coordinates and orienting the rectangular Cartesian coordinate axes arbitrarily, Halphen convinces himself that the projections on the coordinate axes of the force of attraction of the elliptic ring of the perturbing body’s orbit (with the above-mentioned law of density distribution in it) at the origin will be partial derivatives with respect to \(x_0;\ y_0;\ z_0\) of the polynomial

\[ \Phi=\frac{1}{h}\left\{P_x x_0^2+Q_y y_0^2+R_z z_0^2+2Q_x x_0 y_0+2R_y y_0 z_0+2P_z z_0 x_0\right\}; \]

here \(h\) is the distance of the perturbed body from the plane of the influential planet’s orbit, and \(P_x;\ Q_y\ldots P_z\) are numbers depending only on the form of the cone.

In the special case when the coordinate axes are along the principal axes of the cone, then \(Q_x=0;\ R_y=0;\ P_z=0\), and the polynomial \(\Phi\) has a “canonical” form; this canonical form is Halphen’s starting point: he takes the equation of the cone in the form

\[ \frac{x^2}{p}+\frac{y^2}{q}+\frac{z^2}{r}=0 \]

(\(p\) and \(q\) are negative, \(r\) positive), expresses the current coordinates \(x;\ y;\ z\) through two auxiliary parameters \(s\) and \(\rho\) by the formulae:

\[ x^2=-\frac{p(s-p)}{(p-q)(p-r)}\,\rho^2;\quad y^2=-\frac{q(s-q)}{(q-r)(q-p)}\,\rho^2;\quad z^2=-\frac{r(s-r)}{(r-p)(r-q)}\,\rho^2, \]

forms for this case the coefficients \(P_x;\ Q_y;\ R_z\) and convinces himself that

\[ dP_x=\frac{C}{2S}(r-q)(s-p)\,ds;\quad dQ_y=\frac{C}{2S}(p-r)(s-q)\,ds;\quad dR_z=\frac{C}{2S}(q-p)(s-r)\,ds, \]

where

\[ C=\frac{\sqrt{pqr}}{(p-q)(q-r)(r-p)};\quad S=\sqrt{4(s-p)(s-q)(s-r)}. \]

Now, if one introduces the Weierstrass elliptic function \(\rho(u)\) under the condition

\[ \frac{s-p}{\rho(u)-e_\alpha} = \frac{s-q}{\rho(u)-e_\beta} = \frac{s-p}{\rho(u)-e_\gamma} =1, \]

then it turns out that

\[ S=\rho'(u);\quad S\,du=ds,\quad \text{and therefore}\quad h\Phi=\left[\eta-\frac{p+q+r}{3}\,\omega\right]M+\omega N, \]

where

\[ M=C\left[(q-r)x_0^2+(r-p)y_0^2+(p-q)z_0^2\right]; \]

\[ N=C\left[p(q-r)x_0^2+q(r-p)y_0^2+r(p-q)z_0^2\right]; \]

here \(\eta\) and \(\omega\) are known quantities from the theory of elliptic functions; \(u\) in the present case varies so that \(u-\omega'\) is real.

The extreme theoretical elegance and the simplicity of practical application of Halphen’s method (the latter will be evident from what follows) to the computation of secular perturbations are due to the fact that he undertook to express the function

\(\Phi\) by means of the following fundamental polynomials or, as mathematicians say, “forms” (omitting the subscripts in \(x_0; y_0; z_0\)):

\[ \Psi=x^2+y^2+z^2;\quad f=\frac{x^2}{p}+\frac{y^2}{q}+\frac{z^2}{r};\quad \varphi=px^2+qy^2+rz^2; \]

the first of these is the square of the distance of the point from the origin, the second is the left-hand side of the equation of our cone resting on the orbit of the perturbing body, the third is the left-hand side of the equation of the “inverse” cone—in the same sense in which the gyration ellipsoid is inverse to the ellipsoid of inertia in mechanics.

Without giving, for lack of space, Halphen’s elegant computations, I shall indicate the final result:

\[ h\Phi=\frac{1}{g_2}\sqrt[4]{\frac{4k_3^2}{g_2}} \left[ \frac{1}{3}(k_1\psi-3\varphi)\Psi(\xi) -\frac{144g_3}{g_2^3}\,\mu\,\Psi'(\xi) \right]; \]

here \(k_1=p+q+r;\ k_2=pq+qr+rp;\ k_3=pqr;\ \mu=(k_1^2k_2-2k_2^2-3k_1k_3)\Psi+\)
\[ +2k_3(3k_2-k_1^2)f+(9k_3-k_1k_2)\varphi, \]
and \(g_2,g_3\) are invariants of the function \(\rho(u)\); as for \(\Psi(\xi)\), then \(\Psi(\xi)=\omega\sqrt[4]{4g_2}\), finally \(\xi=27g_3^2:g_2^3\);

Further,

\[ \Psi(\xi)=2AF\left(\frac{1}{12};\frac{5}{12};\frac{1}{2};\xi\right) - BF\left(\frac{7}{12};\frac{11}{12};\frac{3}{2};\xi\right) \sqrt{\frac{\xi}{3}}, \]

where

\[ A=\int_0^1\frac{dx}{\sqrt{1-x^4}}=1.311028777146\ldots;\quad B=\frac{\pi}{4A}=0.59907011737\ldots; \]

for \(\sqrt{\xi/3}\) the same sign must be taken as that of \(k_3g_3\), and \(F(\alpha;\beta;\gamma;x)\) is the ordinary Gauss hypergeometric series; here Halphen has a strange error, which is repeated in both volumes of his Traité: he has

\[ \Psi(\xi)=2AF\left(\frac{1}{12};\frac{1}{12};\frac{1}{2};\xi\right) - BF\left(\frac{7}{12};\frac{7}{12};\frac{3}{2};\xi\right) \sqrt{\frac{\xi}{3}}. \]

This error had not been noticed by anyone before me; incidentally, another error of Halphen’s: he writes in vol. 2 (p. 318) that if \(u\) varies from \(\omega'+a\) to \(\omega'+a+2\omega\), then the point will traverse a complete circuit of the cone; this, of course, is incorrect: the point will then go around only one half of the circuit of the cone, so that, if one computes the secular perturbations according to Halphen, they will all come out half as small as they should. How Halphen, a first-rate mathematician, could have made such errors I do not comprehend!! Is this not the reason why, before me, no one had applied Halphen’s method in practice? The second error also had not been noticed by anyone before me. If \(k_3g_3\) is positive and \(\xi\) is between \(\frac12\) and 1, then it is more convenient for computation to use

\[ \Psi(\xi)=\frac{\pi}{\sqrt[4]{3}}\, F\left(\frac{1}{12};\frac{5}{12};1;1-\xi\right) \]

When the form

\[ \Phi=a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{31}zx \]

has been composed, then the projections of the force of attraction onto the positive directions of the chosen coordinate axes are

\[ \frac{\partial \Phi}{\partial x}=2(a_{11}x+a_{12}y+a_{13}z) \]

\[ \frac{\partial \Phi}{\partial y}=2(a_{21}x+a_{22}y+a_{23}z) \]

\[ \frac{\partial \Phi}{\partial z}=2(a_{31}x+a_{32}y+a_{33}z) \]

The approach to the practical application of Halphen’s method is as follows: from the given orbital elements of the perturbed and the influential planet, the orbit of the perturbed planet must first of all be referred to the orbit of the influential one, so that the orbit of the influential planet will play the role of the ecliptic, and the perihelion of its orbit—the role of the vernal equinox point; more simply, in these special “coordinates” one can easily, by solving a single spherical triangle, find: 1) the longitude of the ascending node of the perturbed planet \((\Omega)\); 2) the inclination of its orbit \((I)\); 3) the distance of its perihelion from the ascending node \((\Phi)\); thus, for example, for Jupiter and Ceres, at the epoch 1850.0, it is found that \(\Omega=66^\circ26'08''.12\); \(\Phi=70^\circ11'36''.24\); \(I=9^\circ22'52''.26\); all the data here are on the heliocentric sphere.

After finding \(\Omega\), \(\Phi\), \(I\) for the perturbed planet, several (I took 12) positions are taken on the orbit, equally spaced in eccentric anomaly, and for values \(0^\circ, 30^\circ, 60^\circ\ldots\) of the eccentric anomaly the true anomalies \(w\) are computed by the known formulas, and after this, by the likewise known formulas

\[ x=r[\cos\Omega\cos(\Phi+w)-\sin\Omega\sin(\Phi+w)\cos I] \]

\[ y=r[\sin\Omega\cos(\Phi+w)+\cos\Omega\sin(\Phi+w)\cos I] \]

\[ z=r\sin(\Phi+w)\sin I \]

one finds the heliocentric coordinates of the perturbed planet; the axis \(x'ov\)—toward the perihelion of the perturbing planet, the axis \(y\)—in the plane of its orbit at \(90^\circ\) from the axis \(x'ov\) in the direction of motion, and the axis \(z\)—perpendicular to the plane of its orbit to the north.

The origin of coordinates is the Sun.

Then one finds the coordinates \(\alpha,\beta,\gamma\) of the perturbed planet, as if the origin of coordinates were transferred to the center of the orbit of the perturbing planet, while the axes were left in the same directions:

\[ \alpha=x+ae;\quad \beta=y;\quad \gamma=z; \]

at the same time, for each of the selected positions of the perturbed planet we find

\[ \cos\alpha=\frac{x}{r};\quad \cos\beta=\frac{y}{r};\quad \cos\gamma=\frac{z}{r}; \]

\[ \cos\alpha'=-\cos\Omega\sin(\Phi+w)-\sin\Omega\cos(\Phi+w)\cos I; \]

\[ \cos\beta'=-\sin\Omega\sin(\Phi+w)+\cos\Omega\cos(\Phi+w)\cos I;\quad \cos\gamma'=\cos(\Phi+w)\sin I; \]

\[ \cos\alpha''=+\sin\Omega\sin I;\quad \cos\beta''=-\cos\Omega\sin I;\quad \cos\gamma''=\cos I. \]

Here \(\alpha,\beta,\gamma\) are the angles of the radius vector of the perturbed planet with the adopted coordinate axes; \(\alpha',\beta',\gamma'\) are the angles of the perpendicular to the radius vector in the plane of the orbit of the perturbed planet, in the direction of its motion; and \(\alpha'',\beta'',\gamma''\) are the angles of the perpendicular to the plane of the perturbed orbit, toward the north, with

by the same coordinate axes. In short, the 9 angles \(\alpha_i,\ \beta_i,\ \gamma_i\) \((i=1;\ 2;\ 3)\) characterize the direction of the axes, the so-called Hansen axes.

At this point the process of trigonometric computations ends; what follows is arithmetical calculation, which undoubtedly makes Halphen’s method very convenient when using calculating machines.

After this, quite simple considerations of analytic geometry show that, in the coordinate axes adopted by us (along the principal axes of the orbit of the disturbing body, with the origin at the center of the ellipse), the equation of the cone projecting this orbit from the perturbed planet will be (and Halphen himself points this out)

\[ \frac{(\alpha z-\gamma x)^2}{a^2}+\frac{(\beta z-\gamma y)^2}{b^2}=(z-\gamma)^2, \]

and after a parallel translation of the axes to the perturbed planet,

\[ f=\frac{\zeta^2}{\gamma^2}-\frac{(\alpha \zeta-\gamma \xi)^2}{a^2\gamma^2}-\frac{(\beta \zeta-\gamma \nu)^2}{b^2\gamma^2}=0 \]

Then

\[ k_1=p+q+r=\alpha^2+\beta^2+\gamma^2-a^2-b^2 \quad;\quad g_2=\frac{4}{3}(k_1^2-3k_2); \]

\[ k_2=pq+qr+rp=a^2b^2-\alpha^2b^2-\beta^2a^2-b^2\gamma^2-a^2\gamma^2; \quad g_3=\frac{4}{27}(27k_3-9k_1k_2+2k_1^3) \]

\[ k_3=pqr=-a^2b^2\gamma^2 \quad;\quad 27g_3^2:g_2^3=\xi; \]

After this, using the preceding formulas, we compute the sums of the series \(\Psi(\xi)\) and \(\Psi'(\xi)\), and then form Halphen’s form \(\Phi\) according to the scheme

\[ a_{11}'=\frac{1}{9}k_1(9k_1k_2-27k_3)-2k_2^2+\frac{3g_2}{2}\frac{k_3}{a^2} -(9k_3-k_1k_2)(a^2-\alpha^2); \]

\[ a_{22}'=\frac{1}{9}k_1(9k_1k_2-27k_3)-2k_2^2+\frac{3g_2}{2}\frac{k_3}{b^2} -(9k_3-k_1k_2)(b^2-\beta^2); \]

\[ a_{33}'=\frac{1}{9}k_1(9k_1k_2-27k_3)-2k_2^2+\frac{3g_2}{2}(a^2b^2+\beta^2a^2-\alpha^2b^2) -(9k_3-k_1k_2)(-\gamma^2); \]

\[ a_{12}'= -(9k_3-k_1k_2)(-\alpha\beta); \]

\[ a_{13}'= +\frac{3g_2}{2}k_3\left(-\frac{\alpha}{\gamma a^2}\right) -(9k_3-k_1k_2)(-\alpha\gamma); \]

\[ a_{23}'= +\frac{3g_2}{2}k_3\left(-\frac{\beta}{\gamma b^2}\right) -(9k_3-k_1k_2)(-\beta\gamma); \]

\[ a_{11}''=-\frac{1}{3}k_1-(a^2-\alpha^2); \qquad A=\frac{\sqrt{2.144\,g_3}}{\pi g_2\sqrt[4]{g_2}}\,\Psi'(\xi) \]

\[ a_{22}''=-\frac{1}{3}k_1-(b^2-\beta^2); \]

\[ a_{33}''=-\frac{1}{3}k_1+\gamma^2 \quad;\quad B=-\frac{\sqrt{2}}{\pi g_2\sqrt[4]{g_2}}\,\Psi(\xi) \]

\[ a_{12}''=+\alpha\beta \quad;\quad a_{ik}=a'_{ik}A+a''_{ik}B \]

\[ a_{13}''=+\alpha\gamma \quad;\quad (i,\ k=1;\ 2;\ 3). \]

\[ a_{33}''=+\beta\gamma \quad; \]

As a result of projecting the perturbing force onto the coordinate axes directed along the principal axes of the orbit of the perturbing planet, we obtain

\[ P_x=2(a_{11}x_0+a_{12}y_0+a_{13}z_0);\quad P_y=2(a_{21}x_0+a_{22}y_0+a_{23}z_0); \]

\[ P_z=2(a_{31}x_0+a_{32}y_0+a_{33}z_0), \]

where \(x_0=-x;\ y_0=-y;\ z_0=-z\), and \(x;\ y;\ z\) have been computed earlier. Finally, \(S_0;\ T_0;\ W_0\) (the projections of the perturbing force onto Hansen’s axes) will be

\[ S_0=P_x\cos\alpha+P_y\cos\beta+P_z\cos\gamma;\quad T_0=P_x\cos\alpha' + P_y\cos\beta' + P_z\cos\gamma'; \]

\[ W_0=P_x\cos\alpha''+P_y\cos\beta''+P_z\cos\gamma'', \]

and the secular perturbations themselves are computed by the usual formulas (7), p. 435, Tisserand, Traité de mécanique céleste, Vol. I, and by formulas (23) (see ibid., p. 441).

An important check on the computations is provided by the well-known property of secular perturbations, namely

\[ \left(\frac{da}{dt}\right)_{0,0}=0. \]

The deviation of this quantity from zero, obtained in the computational process, indicates the number of correct decimal places in the perturbations. For Jupiter and Saturn the computations are usually carried out with 8-place logarithms, and for the other planets with 7- or 6-place logarithms.

It is not without interest to note what type of hypergeometric series I had to deal with in the computations; the type

\[ 2AF\left(\frac{1}{12};\ \frac{5}{12};\ \frac{1}{2};\ \xi\right) - BF\left(\frac{7}{12};\ \frac{11}{12};\ \frac{3}{2};\ \xi\right) \]

occurred only for Mars at eccentric anomalies \(270^\circ;\ 300^\circ;\ 330^\circ;\ 0^\circ;\ 30^\circ;\ 60^\circ\) and \(90^\circ\), and for Jupiter at \(120^\circ;\ 150^\circ;\ 180^\circ;\ 210^\circ;\ 240^\circ;\ 270^\circ\) and \(300^\circ\); in all other cases I had to deal with the simpler type

\[ \frac{\pi}{\sqrt[4]{3}}\, F\left(\frac{1}{12};\ \frac{5}{12};\ 1;\ 1-\xi\right); \]

here is the table of the secular perturbations of Ceres by the influential planets (per Julian year):

In conclusion, a list of the principal literature on secular perturbations:

1. Gauss. Determinatio attractionis quam in punctum quodvis positionis data exerceret planeta, si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispersita (Werke, III, p. 331).

2. Nicolai. Neue Berechnung der Säcularänderungen der Erdbahn (Astronomisches Jahrbuch, p. 224, 1820).

1. Clausen. Alia Solutio problematis a celeberrimo Gauss in opere: „Determinatio attractionis…“ tractati (Journ. de Crelle, t. VI. 1830).

2. Adams. On the orbit of the november meteors (Monthly Notices, t. XXVII).

3. Bour. Thèse de Doctorat. 1855.

4. Seeliger. Ueber das von Gauss herrührende Theorem die Säcularstörungen betreffend (Astron. Nachr., t. XCIV, 1879).

5. G. W. Hill. On Gauss’s method of computing secular perturbations. Collected Works, Mem. No. 37 and No. 69.

6. O. Callandreau. Détermination des perturbations d’une planète par les méthodes de M. Gylden. Application a (103) Héra. Annales de l’observatone de Paris, t. XVI.

7. O. Callandreau. Calcul des variations séculaires des éléments des orbites (Ann. de l’obs. de Paris, t. XVIII).

8. Halphen. Traité des fonctions elliptiques, t. II, p. 310—328.

9. L. Arndt. Recherches sur le calcul des forces perturbatrices dans la théorie des perturbations séculaires. Neuchâtel. 1906.

10. R. T. A. Innes. Computation of secular perturbations (Monthly Notices. 1907. May).

A critical evaluation of all these methods in comparison with the application, carried out by me here for the first time, of Halphen’s method cannot be given, for lack of space.

University. Astronomical Cabinet.
April 14, 1935.