THEORY OF THE DETERMINATION OF THE ELEMENTS OF A PARABOLIC ORBIT FROM TWO OBSERVATIONS OF APPARENT POSITION, AND ONE OF THE MOTION IN THE LINE OF SIGHT.

By F. R. MOULTON.

WITH the great light-gathering power of modern telescopes, and the many improvements in the construction and methods of application of the spectroscope, it may be expected that measurements of relative motion in the line of sight will be made with a very high degree of precision. If such measurements of relative motion. can be secured for a body moving around the Sun, together with the necessary number of apparent positions, it is evident that these data may be made the basis for the determination of the elements of its orbit. The purpose of this paper is to set forth the theory of the computation of the elements, supposing that the orbit is a parabola, and that at two epochs observations of the apparent position have been made, and that at the second epoch the motion in the line of sight has been observed.

SI. FORMULATION OF THE PROBLEM.

Let A, B, and p represent the longitude, latitude, and distance respectively of the comet, taking the center of the Earth as origin. Let the derivatives of these quantities with respect to the time be represented by the same letters with primes. Suppose the observations are made at the two epochs, t, and t.

The observed quantities A,, A, B1, B2, and p2 depend upon the elements of the unknown orbit, hence we may write the following equations :

These five equations involve the five elements as unknowns, and when solved, furnish the solution to the problem. They are, however, transcendental, and so exceedingly involved that a direct solution of them would present almost insurmountable difficulties. This suggests the alternative of first determining intermediate quantities from which the elements can be found. As such, it may be mentioned that one absolute position with reference to the Sun, and the velocity and direction of motion, are sufficient for the unique determination of the elements. If one desired to determine these intermediate quantities it would only be necessary to find one geocentric distance and the velocity and direction of motion. That is, there would be but four unknowns involved, instead of the five elements, and the problem would be reduced to the solution of four simultaneous equations.

As another set of intermediate quantities, the six coördinates defining two positions of the comet with respect to the Sun, and the interval of time required for the comet to move from one to the other, might be used. The interval of time is known, and the only unknowns in the two positions are the two geocentric distances. Therefore, λ, P may be considered as functions of the two unknowns, P, and P. In this case then, which is clearly the simplest possible, it is only necessary to find and solve two independent equations involving P1, P2 and known quantities.

Neglecting perturbations, the work must be consistent with and involve the following fundamental theorems of parabolic motion.

Theorem I. The motion of the comet is in a plane passing through the center of the Sun.

Theorem II. The areas swept over by the radius vector from the Sun are proportional to the intervals of time in which they are described.

Theorem III. The motion is in a parabola with the Sun at its

focus.

It will be supposed that the effects of the Earth's motions upon the observed velocities in the line of sight have been

eliminated, so that p represents the velocity in the direction of the Earth with respect to the Sun. The formulae for accomplishing this were given in the March number of the ASTROPHYSICAL JOURNAL, by Dr. Frank Schlesinger.

2. THE TWO INDEPENDENT EQUATIONS INVOLVING P1 AND PÅ As

UNKNOWNS.

1

2

Let x, y, and z represent the rectangular heliocentric coördinates, and r, I, and b the polar. Denote by s the chord joining the extremities of the radii r, and r,. Let k represent the Gaussian constant. Then Euler's equation is

6 k(t ̧ — t ̧) = (r, +r, + s)3 = (r1 +r, − s)}.

(2)

The upper sign is to be used if the heliocentric motion in the interval of time t2--t, is less than 180°, which will henceforth be supposed to be the case.

x

It will now be shown that (2) can be expressed in terms of P1 and P, as unknowns. r,,,, and s depend upon x1, 1, 1, 2, P2 12, 2, which are equal respectively to the differences of the corresponding geocentric coördinates of the comet and the Sun. The geocentric cöordinates of the Sun are given in the Nautical Almanacs, and the geocentric coördinates of the comet involve as unknowns p1 and p2 alone. Therefore Euler's formula is one equation of the type sought.

As a consequence of Theorem I we may write the following equations:

The constants A, B, and C, depend upon the position of the plane of the orbit, with respect to the plane of the ecliptic and the vernal equinox.. Eliminating them we have,

The expansion of this determinant may be written in the three

(5)

Z1 (X, Y1⁄2 − Y ̧ x')+≈2 (y, x' − x, y) +z; (x1 ÿ, −y1x1)=0.

As these equations are written they are not independent, but if by some new means the parentheses can be determined, they become independent and can be used for the elimination of any two unknowns. Let it be supposed for the moment that the parentheses have been computed. Then x1,y1, 1, 2, y2, and 22 depend upon p, and p2 alone as unknowns, as was seen in the discussion of Euler's equation. x, y, and z depend upon the direction of motion and velocity of the comet at the time t. They may be expressed in terms of p, and two other quantities. Eliminating these two unknown components of velocity, we have from (5) the desired relation between p1 and p2, which may be

written,

1

F (P1, P2)=0.

ρι

(6)

Therefore (2) and (6) are the equations from which p1 and P may be found.

Consider the parentheses in the first equation of (5). By the law of areas (y, z,y) is twice the projection, upon the yz plane, of the area described by the radius vector in a unit of time. Denoting the angles between the plane of the orbit and the three fundamental planes by i,,, i, and i respectively, we have,

xy

(7) where p is the parameter of the parabola. (y, z, -12) is twice 22 12) the projection, upon the yz plane, of the area between the radii Γι and r2' and the chord joining their extremities. Then we have,'

'OPPOLZER'S Bahnbestimmung der Kometen und Planeten, 1, p. 99, or WATSON'S Theoretical Astronomy, p. 176.

I

It remains to consider the second parenthesis of 51.

The development for 2, and 1, are obtained by writing in place of y, z, and 1, respectively. Substituting these expressions in the parenthesis ..-.-, we fad.

7.

From the fundamental equations of acceleration 2.7.

identically zero. It is found by differentiating

As a consequence of the law of areas this reduces to

There is the quest on whether in any case the terms of higher