On Some Problems of Flight Dynamics to the Moon

V. A. Egorov

INTRODUCTION

With the present development of rocket engineering, it is becoming realistic to attain velocities sufficient not only for creating an artificial satellite of the Earth, but also for flight to the Moon. However, up to now the literature \(^{1-5}\) has not provided a satisfactory solution to a number of important questions in the theory of flight to the Moon: the form and classification of trajectories on the passive segment, possible trajectories of a flyby of the Moon with return to the Earth, the possibility of periodic flybys of the Moon and the Earth, the question of the minimal initial velocities required to reach the Moon and to impact the Moon, and also the very important question of the influence of scatter in the initial data on the characteristics of various trajectories of flight to the Moon. This is explained by certain difficulties. Indeed, it can be shown that, in the simplest formulation, the problem reduces to the unsolved restricted circular three-body problem of mechanics.

In 1953–1955, at the Mathematical Institute of the Academy of Sciences of the USSR, a systematic investigation of the above-mentioned circle of questions was carried out, and numerical calculations were performed using a high-speed digital electronic computer. The principal results of this work are set forth in the present article *).

In § 1 the problem is reduced to the circular restricted three-body problem: the Earth, the Moon, and the rocket. The equations of this problem in a coordinate system rotating together with the line joining the centers of the Earth and the Moon possess an energy integral (Jacobi’s). This makes it possible, following Hill \(^{7}\), to apply an energy approach to the problem and to obtain an exact theoretical solution of the question of the minimal initial velocities required to reach the Moon (§ 2).

However, the actual determination of trajectories with minimal velocity by the method of numerical integration shows that these trajectories remain for a long time close to ellipses with a focus at the center of the Earth, and that, before reaching the Moon, the rocket must make a sufficiently large number of revolutions around the Earth (on the order of hundreds or more). Therefore the indicated trajectories are of no practical interest. It also turns out that the minimal velocities required to reach the Moon on the first revolution can be computed from the condition of impact on the Moon, completely neglecting its influence (§ 3).

In § 4 the impossibility is established of capture of the projectile by the Moon’s sphere of action for trajectories starting at the Earth and approaching the Moon on the first revolution (encounter trajectories).

*) Reported at the Mathematical Institute of the Academy of Sciences of the USSR in February 1956. A brief communication is given in note \(^{18}\).

The impossibility of capture for arbitrary trajectories in the circular restricted three-body problem can be proved only for sufficiently small ratios of the attracting masses (smaller than the ratio of the masses of the Moon and the Earth).

For approach trajectories it proves possible to neglect the influence of the Moon outside its sphere of action and the influence of the Earth inside the Moon’s sphere of action, so that the motion outside the Moon’s sphere of action may be regarded as taking place along a geocentric conic section, and the motion inside the sphere of action along a selenocentric conic section. This provides a simple, yet at the same time sufficiently accurate, approximate method for investigating trajectories and makes it possible, in particular, to show that motion relative to the Moon along approach trajectories always occurs with velocities greatly exceeding the parabolic selenocentric velocities (§ 5).

The indicated approximate method of considering approach trajectories by characteristic segments makes it possible to establish the principal regularities and to calculate the characteristics of the motion. In particular, it turns out that, for initial velocities exceeding the parabolic velocity by only 0.5 km/sec, the nature of the regularities already approaches the asymptotic one corresponding to infinitely large initial velocities.

In the case of motion in the plane of the Moon’s orbit, it is possible approximately to determine the geocentric velocities of exit from the Moon’s sphere of action, and to do so at once for all approach trajectories with a given initial-velocity vector. This makes it possible to analyze the evolution of the entire set of approach trajectories as the magnitude and direction of the initial velocity are varied (§ 6), and permits one to solve the principal problems of the dynamics of flight to the Moon in the plane of its orbit.

In solving the problem of impact on the Moon (§ 7), a method was developed for the numerical determination, on a digital machine, of initial data corresponding to impact at the center of the Moon with a prescribed accuracy, which made it possible to carry out a large-scale computation of impact trajectories. With the aid of the approximate method it was established that the deviation of a trajectory from the center of the Moon is a quadratic, not a linear, function of small errors in the initial data. The computations showed that, in terms of the required accuracies of the initial data, the problem of hitting the Moon without any correction on the passive segment is, apparently, technically realistic.

In the problem of circumnavigation of the Moon (§ 8), a method was also found for determining the initial data corresponding to return to the Earth. In this problem the deviation of trajectories from the center of the Earth upon return is, as in the problem of impact on the Moon, a quadratic function of errors in the initial data. It was found that, according to the character of approach to the Moon, flyby trajectories may be of two different classes. There also exist two classes of approach trajectories returning to the Earth but not corresponding to a flyby—direct-flight classes. For all classes it is possible to trace their emergence, evolution, and disappearance as the initial velocity is varied. It was also found that the influence of the dispersion of the initial data decreases as the distance of the flyby trajectory from the Moon increases; that in the problem of flying around the Moon at distances greater than the radius of the Moon’s sphere of action (66,000 km), the accuracies required for return to the Earth are no higher than in the problem of impact on the Moon with the same initial velocity. Therefore, carrying out a flyby of the Moon at a sufficiently large distance, with return to the Earth without correction of the passive segment, is, apparently, technically realistic.

The more attractive problem of a flyby with a grazing entry into the Earth’s atmosphere (§ 9) is solved analogously and has an analogous classification.

However, owing to the relatively small thickness of the atmosphere, the implementation of a shallow entry without an appropriate correction of the approach trajectory after the lunar flyby appears hardly realistic.

As for a planar periodic flyby of the Moon and the Earth along approach trajectories (§ 10), it is practically impossible. It turns out that there exists only one family of flyby periodic approach trajectories and an infinite set (two sequences) of families of impact trajectories. The flyby solutions correspond to hyperbolic initial velocities, are distant from the Earth by a distance of the order of 100,000 km and more, and prove to be unstable.

The problem of accelerating a rocket with the aid of the Moon without expenditure of fuel was also considered, for example, with the aim of flight to other planets (§ 11). This problem is of interest because the plane of the Moon’s orbit forms small angles with the planes of the planetary orbits. Trajectories corresponding to an acceleration close to the maximum possible (of the order of 1.5 km/sec) must pass rather close to the surface of the Moon. In consequence, obtaining an acceleration close to the maximum without a corresponding correction of the passive segment before approach is apparently unrealistic.

In addition to the impact, flyby, and impact-flyby trajectories that have been found, and also trajectories corresponding to one or another acceleration of the rocket after approach, there also exist trajectories corresponding to a greater or lesser deceleration of it (relative to the Earth). Other approach trajectories do not exist, and therefore the classification obtained for planar approach trajectories is complete.

In the conclusion of the article, possible directions are indicated for generalizing the method and the results; in particular, the possibility of generalizing them to problems of flight from the Earth to the outer planets of the solar system is indicated.

§ 1. EQUATIONS OF MOTION

For the purposes of the present work there is no need to take into account all the forces acting on the passive segment of the trajectory of a flight to the Moon; it is sufficient to take into account only the principal, basic forces that determine the motion.

In analyzing the acting forces in the problem under consideration, it is natural to assume that the flight trajectory begins outside the atmosphere at an altitude of the order of hundreds of kilometers and lies entirely within the sphere of action of the Earth with respect to the Sun. Here the sphere of action of the Earth is determined in the following way.^6 The ratio is considered of the force with which the Sun perturbs the geocentric motion of the projectile to the force of attraction of the Earth. Also considered is the ratio of the force with which the Earth perturbs the heliocentric motion of the projectile to the force of attraction of the Sun. The region of space around the Earth in which the first ratio is smaller than the second is called the sphere of action of the Earth*).

To determine the radius of the sphere of action of a small body \(m\) with respect to a large one \(M\), there is the formula \((6,\ \text{p.}\ 194)\)

\[ r_* = A \left( \frac{m}{M} \right)^{2/5} \tag{1,1} \]

(where \(A\) is the distance \(mM\)). In the case of the Earth and the Sun, formula (1,1) gives \(r_* \simeq 930\,000\) km, which is approximately 2.5 times greater than the mean distance from the Earth to the Moon, equal to

\[ a = 384\,400\ \text{km}. \]

*) Geocentric motion is motion in a translationally moving coordinate system with origin at the center of the Earth. Heliocentric motion is defined analogously.

Let us estimate the magnitude of the perturbing action of the Sun in relation to the attraction of the Earth. The ratio of the magnitude \(\Delta F'\) of the perturbation to the attraction \(F\) of the central body is maximal at the boundary of the sphere of influence and has the estimate \(^{6}\)

\[ \frac{\Delta F'}{F} < \left(16\frac{m}{M}\right)^{1/5}, \tag{1.2} \]

which gives, for the case of the Earth and the Sun, \(\frac{\Delta F'}{F}<0.138\). As the distance \(r\) from the Earth decreases, \(\frac{\Delta F'}{F}\) decreases as \(r^{3}\), so that at the distance of the Moon’s orbit we have
\[ \frac{\Delta F'}{F}=\frac{0.138}{(2.5)^3}<0.01, \]
and at smaller distances from the Earth the ratio of the perturbation due to the action of the Sun to the attraction of the Earth is still smaller. We shall suppose that the projectile does not go far beyond the limits of the Moon’s orbit. Then, in the problem under consideration, the influence of the Sun may be disregarded.

Turning to the estimate of the perturbing action of the planets, we note that the distances from the Earth to the planets are comparable with the distance to the Sun, while the masses of the planets are negligible in comparison with the mass of the Sun. Hence, the perturbing action of the planets may all the more be disregarded.

Finally, let us consider the influence of the Moon. The radius \(\rho_{*}\) of the Moon’s sphere of influence with respect to the Earth is small: \(m_{1}:m_{2}=81.45\); \(\rho_{*}=a(m_{2}:m_{1})^{1/5}=66\,000\) km. Although for the main part of the space under consideration the perturbing action of the Moon in comparison with the attraction of the Earth is negligible, in the Moon’s sphere of influence it plays the principal role and cannot be neglected, especially since precisely trajectories passing through the Moon’s sphere of influence are of interest. Thus, in the problem under consideration, only the attractions of the Earth and the Moon should be taken into account.

Since the oblateness of the Earth and the Moon changes the force of gravitation by less than \(1\%\), it may be neglected. The problem is reduced to the so-called restricted elliptic three-body problem: \(m_{1}\) is the Earth, \(m_{2}\) is the Moon, \(m_{0}\) is the projectile.

The problem can be simplified by observing that the Moon’s orbit is close to circular (its eccentricity is \(e=0.0549\)). It can be shown that the forces arising from the ellipticity in the equations of motion, written in the coordinate system \(x,y,z\), rotating together with the line \(m_{1}m_{2}\), amount to a fraction of order \(e\) of the centrifugal and Coriolis forces, i.e. only a few percent. Therefore, instead of the elliptic problem one may consider the circular restricted three-body problem. Its equations are:

\[ \ddot{x}=2\dot{y}+\frac{\partial U}{\partial x};\qquad \ddot{y}=-2\dot{x}+\frac{\partial U}{\partial y};\qquad \ddot{z}=\frac{\partial U}{\partial z}. \tag{1.3} \]

Here

\[ U=\frac{1}{2}(x^{2}+y^{2})+\frac{f m_{1}}{r}+\frac{f m_{2}}{\rho}, \tag{1.4} \]

where \(f\) is the gravitational constant, and \(r\) and \(\rho\) are the distances of the projectile \(m_{0}\) from \(m_{1}\) and \(m_{2}\), respectively; the \(x\)-axis constantly runs from \(m_{2}\) to \(m_{1}\), while the \(y\)-axis passes through the center of gravity of the system \(m_{1},m_{2}\) in the plane of the Moon’s orbit.

As units we take: for lengths, the distance \(a\) between the centers of the Earth and the Moon, and for time, \(\frac{T}{2\pi}\), where \(T\) is the sidereal month (the period of revolution of the masses \(m_{1}\) and \(m_{2}\)). We note that, by Kepler’s third law, in these units
\[ f(m_{1}+m_{2})=a^{3}\left(\frac{T}{2\pi}\right)^{-2}=1. \]

The circular restricted three-body problem, even in the planar case, is known not to have been solved in general form in mechanics. For certain particular cases concerning the motion of bodies of the solar system, methods exist for finding solutions. But the trajectories of one-time flights from the Earth to the Moon bear little resemblance to the ordinary orbits of bodies of the solar system, and the indicated methods are ineffective.

In the present work, in order to find possible types of flight trajectories to the Moon and to compute them approximately, an approximate method has been used. For a refined calculation of the parameters of motion and for determining the influence of the scatter of the initial data, numerical integration of the equations of motion (1.3) was applied with the aid of a high-speed digital computer.

For the theoretical solution of the question of the minimum velocities required to reach the Moon, and of the question of the possibility of capture by the Moon of a projectile from the Earth, an energy approach was used, analogous to that applied by Hill^7.

§ 2. THEORETICAL SOLUTION OF THE QUESTION OF MINIMUM VELOCITIES

To determine the minimum velocities required to reach the Moon, we shall make use of the fact that equations (1.3) have the well-known Jacobi integral^6

\[ \frac{1}{2}V^2 = U + h, \quad \text{where } h=\mathrm{const}. \tag{2.1} \]

Obviously, for fixed \(h\), the motion is bounded by the zero-velocity surfaces introduced by Hill^7:

\[ U=-h. \tag{2.2} \]

With the aid of an investigation of equation (2.2) it can be shown (^6 p. 108) that for large negative \(h\) the motion is possible only inside nonintersecting surfaces \(s'\) and \(s''\), close to spheres with centers \(m_1\) and \(m_2\), and also outside the surface \(s\), which encloses \(s'\) and \(s''\). The section of each of the surfaces \(s, s', s''\) by the plane \(xy\) is close to a circle. For the initial velocity \(V_0=0\), the surface \(s'\) passes through the initial point, and as \(V_0\) increases it increases, moving away from \(m_1\).

Since the passive segment begins at distances much smaller than the distance to the Moon, for small \(V_0\) the motion can occur only inside the surface \(s'\), and approach to the Moon is impossible. As the initial velocity \(V_0\) increases, the value of \(h\) increases, the surfaces \(s'\) and \(s''\) expand and approach one another, while the surface \(s\) contracts. For a certain \(v_0=v_0'\), the value of \(h\) reaches such a value \(h=h_1\) that the surfaces \(s'\) and \(s''\) have one common point \(L_1\) (Fig. 1), and for small \((h-h_1)\) they will be connected by a neck near the point \(L_1\). Thus penetration becomes possible of a trajectory that began near \(m_1\) into the region around \(m_2\), i.e. approach of the rocket to the Moon becomes possible. One half of the section by the plane \(xy\) of the surfaces \(s, s', s''\) corresponding to \(h=h_1\) (the curves \(s_1, s'_1, s''_1\)) is shown in Fig. 1.

With further increase of \(h\) (\(V_0\)), a critical value \(h_2>h_1\) is reached (\(V_0=V_0^{(2)}\)), corresponding to contact of the surfaces \(s\) and \(s''\) at the point \(L_2\) (the curve \(s_2\) in Fig. 1), so that departure of the rocket from the Earth to infinity through the neck near \(L_2\) becomes possible. Thus, the minimum velocity required to reach the Moon is equal to \(V_0^{(1)}\), and the minimum velocity required for departure to infinity is equal to \(V_0^{(2)}\).

Besides the critical values \(h_1\) and \(h_2\), there also exist critical values \(h_3>h_2\) and \(h_4>h_3\). The value \(h_3\) corresponds to the possibility of departure to infinity through the neck near \(L_3\) (see curve \(s_3\) in Fig. 1). The val-

tion \(h_4 < 0\) and corresponds to the disappearance of the zero-velocity curves in the \(xy\)-plane at the points \(L_4\) and \(L_5\), symmetric with respect to the \(x\)-axis, i.e. it corresponds to the possibility of the rocket escaping to infinity in any direction in the \(xy\)-plane. The disappearance of the surfaces bounding the spatial motion occurs at \(h = 0\).

Fig. 1. Section of the critical zero-velocity surfaces for the Earth–Moon system by the plane of the lunar orbit. At velocities slightly greater than the minimal one, penetration of the projectile to the Moon is possible only through the neck at the point \(L_1\).

The points \(L_i\) (the so-called libration points) are located in the plane of the Moon’s orbit and can be found as special points of the surfaces (2.2); the values of \(h\) are found from (2.1) for \(V = 0\) from the coordinates of the points \(L_i\), and the corresponding initial velocities \(V_0^{(i)}\) are found from (2.1), with \(h = h_i\), from the coordinates of the specified initial point.

Let us note that the magnitude of the critical initial velocity \(V_0^{(i)}\) is obtained the same, independently of its direction, although, obviously, it varies from point to point. However, from (1.4) and (2.1) it follows that on a sphere of small radius \(r\) the velocities \(V_0^{(i)}\) change little from point to point. It turns out that on the sphere corresponding to an altitude of 200 km, the changes in the velocities \(V_0^{(i)}\) will be of the order of \(5 \cdot 10^{-7}\dfrac{2\pi a}{T}\), so that in practice the velocities \(V_0^{(i)}\) do not depend on the position on the sphere. (Here, as in the calculation of the curves in Fig. 1, \(m_1 : m_2 = 81.45\) was adopted.)

The results of calculating the distances \(r_i\) and \(\rho_i\) of the libration points from the Earth and the Moon, the critical energies \(h_i\), and the critical velocities \(V_0^{(i)}\) in units of \(\dfrac{2\pi a}{T}\) and in km/sec are given in Table I. The initial altitude was taken equal—

Table I

	\(r_i\)	\(\rho_i\)	\(h_i\)	\(V_0^{(i)}\left(\dfrac{2\pi a}{T}\right)\)	\(V_0^{(i)}\left(\dfrac{\text{km}}{\text{sec}}\right)\)
\(L_1\)	0.8491539	0.1508461	−1.594067	10.60335	10.84890
\(L_2\)	1.1677237	0.1677237	−1.585991	10.60411	10.84968
\(L_3\)	0.9929263	1.9929263	−1.506062	10.61165	10.85738
\(L_4\)	1	1	−1.494001	10.61278	10.85854

of 200 km. Such an initial altitude was chosen because the trajectories calculated for it are valid also for large initial altitudes. Smaller initial altitudes cannot be taken because of the increasing influence of atmospheric resistance.

We see that the difference between the first and fourth critical velocities is less than 10 m/sec, while the difference of the first velocity from the second and of the third from the fourth is only of the order of 1 m/sec.

In kilometers, the distances of the points \(L_1\) and \(L_2\) from the Moon are respectively 58,000 km and 65,000 km, i.e., both these points lie inside the Moon’s sphere of action, rather close to its boundary.

Let us note that, in investigating the position of the libration points, an error was found in a theorem of M. Martin\(^9\), reviewed but not corrected by A. Markov\(^ {10}\). According to this theorem, if one sets \(m_2=\mu\), \(m_1=1-\mu\), then the inequalities
\(\rho_1(\mu)<\rho_2(\mu)\) for \(0<\mu<\mu^*\), \(\rho_1(\mu)=\rho_2(\mu)\) for \(\mu=\mu^*\), and \(\rho_1(\mu)>\rho_2(\mu)\) for \(\mu^*<\mu<1\) should hold. It was asserted that \(\mu^*>\dfrac{1}{2}\).

However, it can be shown that always \(\rho_1(\mu)<\rho_2(\mu)\), and their equality is impossible for \(0<\mu<1\). The error in M. Martin’s proof crept in when like terms were combined.

In this connection, the work of D. Rosenthal\(^8\), devoted to the numerical determination of the value \(\mu^*\), turns out to be without object.

§ 3. NUMERICAL DETERMINATION OF TRAJECTORIES, TRAJECTORIES OF MINIMUM VELOCITY

It is interesting to find out, at least in the planar problem, what the trajectories of minimum velocity are and whether it is possible to launch a projectile from the Earth, giving it an initial velocity slightly greater than the minimum, so that along an ascending trajectory it will rise to the libration point \(L_1\), pass through the neck with a small velocity, and then reach the Moon. This can be found out by computing trajectories with the aid of a computing machine by one of the methods of numerical integration.

Let us note that the right-hand sides of the equations of motion grow rapidly as the projectile approaches the Moon and become unbounded for exact hits at the center of the Moon. This can be avoided by means of a regularizing transformation of variables.

In the planar problem the most convenient is the well-known Thiele transformation\(^ {15}\). For its application the origin of coordinates is moved to the middle of the segment \(m_1m_2\), and as the unit of measurement one takes \(c=\dfrac{a}{2}\) instead of \(a\) (Fig. 2):

\[ \xi=c_1+2x;\qquad \eta=2y,\quad \text{where}\quad c_1=\frac{m_1-m_2}{m_1+m_2}=0.9757478. \tag{3,1} \]

The equations obtained are

\[ \ddot{\xi}=2\frac{d\eta}{dt}+\frac{\partial U'}{\partial \xi};\qquad \ddot{\eta}=-2\frac{d\xi}{dt}+\frac{\partial U'}{\partial \eta} \tag{3,2} \]

and the Jacobi integral

\[ V^2=2U'+H,\quad \text{where}\quad U'=\frac{1}{2}(\xi^2+\eta^2)-c_1\xi+\frac{8m_1}{r}+\frac{8m_2}{\rho},\quad H=\mathrm{const}. \tag{3,3} \]

The Thiele transformation consists in passing to new variables \(\tau\), \(U(\tau)\), and \(V(\tau)\) according to the formulas \(dt=r\rho\,d\tau\) and \(\zeta=\cos W\), where

\[ \zeta=\xi+i\eta,\quad \text{and}\quad W=U+iV. \tag{3,4} \]

As is not difficult to verify, this transformation indeed leads to equations with right-hand sides that are regular in the entire finite plane \(W\). In Tile variables, on the high-speed electronic digital computer BESM-17 all trajectories (about a thousand) were computed with 5–7 significant figures. A convenient means of checking the accuracy proved to be the Jacobi integral (3.3).

Let us give the results of the calculation of trajectories with minimum velocity. The first trajectory computed was that with the first critical velocity

Fig. 2. Relation of the geocentric coordinates \(\xi_1,\eta_1\) to the rotating \(\xi,\eta\). Initial direction of the axis \(\xi_1\) from \(m_1\) (the Moon) to \(m_2\) (the Earth).

\(V_0=V_0^{(1)}\), directed perpendicular to the initial geocentric radius \(\mathbf r_1\) in the direction of the Moon’s rotation. In this case the geocentric initial velocity of the projectile, as may be inferred from Fig. 2, turned out to be maximal. The trajectory began at an altitude of \(200\ \mathrm{km}\) with \(\varrho=a\). It turned out that this trajectory returns to the Earth, not reaching the critical curve \(s_1'\) by approximately \(30\,000\ \mathrm{km}\). A further 5 revolutions of this trajectory were traced over an interval exceeding a month, but the apogee distance practically did not change.

The computed trajectory in rotating coordinates is shown in Fig. 3 (the 6th revolution is not shown, since it intersects the first). The flight time here and below is marked along the curves in days *). In the non-rotating geocentric system \(m_1\xi_1\eta_1\) (the axis \(\xi_1\) is directed along the axis \(\xi\) at the instant \(t=0\); Fig. 2), the figure-eight loops in Fig. 3 correspond to curves encircling the Earth in the same direction and lying almost on one and the same ellipse with a focus at the center of the Earth (Fig. 4). Although an increase of the apogee radius is noticeable, it is small.

If, for some trajectory with initial velocity \(V_0=V_0^{(1)}\), the surface \(s_1'\) is sooner or later reached in accordance with § 2, then from the calculation presented it follows that this will occur only after a sufficiently large number of revolutions about \(m_1\).

In addition to the one considered, trajectories differing from it in the direction of the initial velocity were computed (Fig. 5). It turned out that, for the apogee distance of the first revolution of the trajectory, the choice of direction

*) The Earth, the Moon, and the trajectories in the figures with divisions along the axes are drawn to the corresponding scale.

of the initial velocity is not indifferent. Namely, from a comparison of the trajectories in Fig. 5 (for example, trajectories I and IV) it is seen that the apogee distance turns out to be the greater, the larger the magnitude of the initial geocentric velocity.

Let us note that the curves of type IV in Fig. 5, going around the Earth clockwise, are, in the geocentric coordinate system, very close, as are

Fig. 3. First revolutions of trajectories with the minimum critical initial velocity in rotating coordinates. The critical curve \(s'_1\) is not reached during the month. (The curves in the figures with divisions along the axes are drawn to scale).

the curves in Fig. 4, to ellipses with a focus at the center of the Earth, and these ellipses are traversed clockwise.

Facts analogous to those given for trajectories with initial velocity \(V_0 = V_0^{(1)}\) also hold for trajectories \(V_0 = V_0^{(i)}\), \(i = 2, 3, 4\). It turned out, in particular, that on the first revolution these trajectories not only do not reach the corresponding curves \(s_i\), but do not even get as far as the curve \(s'_i\) (Fig. 1).

Thus, the minimum velocities obtained in § 2 with the aid of Jacobi’s integral for reaching the Moon on the first revolution (which is of greatest interest) are entirely insufficient. It is not possible to take into account the condition of reaching the Moon on the first revolution by the method of § 2, and another approach to the problem must be sought.

From the fact that on the first revolution the motions with minimum velocities proved, in geocentric coordinates, to be very close to motions along the corresponding ellipses with a focus at the center of the Earth, it follows that for such motions on the first revolution the influence of the Moon is practically negligible. This suggests trying to find the mini-

small velocities for reaching the Moon on the first revolution approximately, completely neglecting the influence of the Moon.

Fig. 4. The preceding trajectory in geocentric coordinates. Noticeable change of the orbit under the action of lunar perturbations.

In this case the initial velocity \(V_1\) in the nonrotating geocentric system \(m_1 \xi_1 \eta_1\), and not the velocity in the system \(O\xi\eta\) (Fig. 2), will be significant. The minimum geocentric initial velocity, as is not difficult to understand, must simply correspond to reaching an apogee radius \(r_a\) equal to the distance to the Moon.

Let us determine the magnitude \(V_1\) of the minimum velocity, regarding as given its angle \(\alpha_1\) with the initial geocentric radius \(r_1\).

Using the condition \(r_a=a\), we obtain the expression for the semimajor axis of the minimum ellipse:

\[ 2a_\ni=\frac{a^2-r_1^2\sin^2\alpha_1}{a-r_1\sin^2\alpha_1}. \tag{3,5} \]

Using the geocentric energy integral, we find the minimum initial velocity \(V_1(r_1,\alpha_1)\):

\[ \left. \begin{aligned} V_1^2&=r_1V_{\mathrm p}^2(r_1)\left(\frac{1}{r_1}-\frac{1}{2a_\ni}\right);\\ V_{\mathrm p}^2(r_1)&=\frac{2fm_1}{r_1}, \end{aligned} \right\} \tag{3,6} \]

where \(V_{\mathrm p}\) is the parabolic velocity. For an altitude of \(200\) km and the vertical direction \(\alpha_1=0\) of the initial velocity, we have

\[ V_{\mathrm p}=10.99967\ \text{km/sec}, \]

and the magnitude of the minimum initial velocity is \(V_1=10.90525\ \text{km/sec}\). This

Fig. 5. Influence of a change in the direction of the initial velocity in the rotating coordinates \(O\xi\eta\) on the trajectory.

the value by approximately \(60\)—\(50\ \mathrm{m/sec}\) exceeds the geocentric velocities corresponding to the first—fourth critical velocity (respectively) at \(\alpha=0\).

From (3.5) and (3.6) it follows that, as the direction of the initial velocity changes from vertical to horizontal, i.e., as \(0<|\alpha_1|<\dfrac{\pi}{2}\) changes, the magnitude of the initial velocity increases monotonically. Consequently, the given value of the initial velocity corresponding to its vertical direction is the least of the minima. However, its difference from the value corresponding to the horizontal direction \(|\alpha_1|=\dfrac{\pi}{2}\) is small. For example, at an initial altitude of \(200\ \mathrm{km}\) this difference is only \(1.6\ \mathrm{m/sec}\).

The initial position is computed quite simply. From the magnitude \(V_1\) and the direction \(\alpha_1\) of the initial velocity, the parameters of the ellipse, the angular distance \(\Phi\), and the flight time between the initial and final radii are easily found. The initial position corresponding to reaching the Moon is found from the condition that the projectile meet the Moon at the advanced point.

Fig. 6. Calculation of hitting the Moon without taking its influence into account for \(\alpha_1>0\).

In the case of motion in the plane of the Moon’s orbit, the initial position is determined only by the angle \(\lambda\) of the initial radius with the axis \(\xi\) \((|\lambda|<\pi)\). Under the condition that at the initial instant the axis \(\xi\) coincides with the axis \(\xi_1\), we have (Fig. 6):

\[ \lambda=\pi \operatorname{sign}\alpha_1-\Phi_0,\qquad \Phi_0=\Phi-\varphi_{\mathrm{л}},\qquad \varphi_{\mathrm{л}}=\omega T_{1,2}, \tag{3.7} \]

where \(\omega\) is the angular velocity of the Moon, and \(T_{1,2}\) is the flight time to the advanced point, determined by the formulas of the theory of conic sections. For all angles here and below, the positive direction is taken to be counterclockwise.

The trajectories determined by equations (3.2) and by the initial data obtained from formulas (3.5)—(3.7) were found by numerical integration on a machine for the values \(\alpha_1=-\dfrac{\pi}{2},\,0,\,\dfrac{\pi}{2}\) and a number of nearby values. One of them is shown in Fig. 7. It turned out that the condition \(r_a=a\) gives the minimum velocity required to reach the center of the Moon with sufficient accuracy (to an accuracy of about \(0.02\ \mathrm{m/sec}\)), so that co-

corresponding initial data one can indeed calculate, completely neglecting the influence of the Moon.

The result obtained means, in particular, that the widespread opinion that, in order to reach the Moon, it is sufficient to reach the distance at which the attractions of the Earth and the Moon are equal, is incorrect.

Fig. 7. Difference of the impacting trajectory II from the corresponding ellipse I at the minimum initial velocity. It is seen that the impact can be calculated without taking into account the influence of the Moon.

This was also confirmed by calculations of the corresponding trajectories taking into account the attraction of the Moon for
\( \alpha_1 = -\dfrac{\pi}{2},\ 0,\ +\dfrac{\pi}{2} \) (see, for example, in Fig. 8 the trajectory with \( \alpha_1 = +\dfrac{\pi}{2} \)).

Let us note that even in the cases shown in Figs. 7 and 8, when the projectile and the Moon, before their encounter, go around the Earth in the same direction and the influence of the Moon is especially strong, the motions with allowance for the Moon’s influence (thick curves) and without allowance for it (thin curves), up to entry into the Moon’s sphere of action (the circle \( \rho = \rho_* \)), practically coincide. At velocities greater than the minimum, the influence of perturbations proves still smaller.

Now one can obtain a simple explanation of why, for an initial velocity in the system \(O\xi\eta\), slightly greater than the first critical velocity \(V_0^{(1)}\)

(§ 2), the projectile cannot, on the first revolution of the trajectory, rise to the libration point \(L_1\), pass through the neck with a very small velocity, and reach the Moon. Indeed, the geocentric area constant \(\delta(L_1)\), corresponding to relative rest at the point \(L_1\), exceeds the initial area constant \(\sigma(A_1)\) by at least a factor of 3.9\(^*\), and the perturbations due to the Moon cannot reduce this difference to zero within one revolution.

Fig. 8. Perturbation by the Moon of an ellipse reaching the point of equality of the Earth's and Moon's attractions.

Let us estimate, to order of magnitude, the number of revolutions required to reach the critical curve \(s'_1\) (Fig. 1) at the first critical initial velocity. Since for the trajectory in Fig. 2, for the initial value \(\delta(A_1)=71300\ \mathrm{km^2/sec}\), over 6 revolutions there was found to be \(\Delta\delta = 6810\ \mathrm{km^2/sec}\), one may expect that, in order to reach the curve \(s'_1\), the number of revolutions required will be of the order

\[ \frac{\sigma(L_1)-\sigma(A_1)}{\Delta\sigma}\cdot 6, \]

i.e., of the order of 200 revolutions.

Let us note that the difference between the geocentric energies of a projectile at relative rest at the point \(L_1\) and of a projectile having the critical—

\[ {}^*)\ \sigma(L_1)=\omega r_{L_1}^{2}\simeq 282\,000\ \mathrm{km^2/sec},\quad |\sigma(A_1)|\simeq r_1 V_0^{(1)}<72\,400\ \mathrm{km^2/sec}. \]

any initial velocity \(V_0^{(1)}\), negligible in comparison with its initial kinetic energy \(\frac12\bigl(V_0^{(1)}\bigr)^2\). Therefore the disturbing action of the Moon, for velocities \(V_0\) close to \(V_0^{(1)}\), is reduced essentially to a change in the geocentric sectorial velocity \(\sigma\).

§ 4. On the possibility of capture by the Moon of a projectile from the Earth

Among the trajectories computed in solving the problem of minimal initial velocities, there were no trajectories corresponding to capture. With the aid of Hopf’s results[^11] one may arrive at the conclusion that the set of projectile trajectories in phase space about which it cannot be said that they do not correspond to capture has Lebesgue measure zero. For solving problems of flight to the Moon it is important to know whether there exists at least one trajectory corresponding to capture or not. If it exists, then, although its exceptional initial data cannot be realized exactly, nevertheless, by realizing data sufficiently close to them, it would be possible to obtain trajectories making an arbitrarily large number of revolutions around the Moon before moving away from it. This is obviously of interest, for example, for creating an artificial satellite of the Moon without the aid of an engine.

However, it can be proved that capture by the Moon of a projectile arriving from the Earth is impossible on the first revolution of its trajectory for any initial data. This assertion is proved with the aid of V. G. Fesenkov’s idea[^12], consisting in transforming the Jacobi integral (2,1) before capture to the geocentric elements \(a_1, p_1, i_1\), after capture—to the selenocentric elements \(a_2, p_2, i_2\), eliminating the constant \(h\), and establishing a contradiction in the result.

The contradiction can be proved owing to the specific character of the initial data of trajectories beginning at the Earth and approaching the Moon on the first revolution (cf.[^12]). As in V. G. Fesenkov, it was assumed that capture occurs if, beginning at some moment, the trajectory does not leave the sphere within which the attraction of the Moon is stronger than the attraction of the Earth. The radius of this sphere is

\[ \rho_0 \simeq \left(\frac{m_2}{m_1}\right)^{1/2}=42700\ \text{km}. \]

If instead of this sphere one uses the sphere of action of the Moon

\[ \left(\rho=\rho_*=\left(\frac{m_2}{m_1}\right)^{2/5}=66000\ \text{km}\right), \]

the proof remains valid. We note that this proof can also be carried out for a smaller ratio of the attracting masses \(m_2:m_1\), provided only that the initial distance of the body \(m_0\) from the mass \(m_1\) is sufficiently small in comparison with the distance \(m_1m_0\).

Thus, a projectile launched from the Earth and on the first revolution entering the sphere of action of the Moon will necessarily leave it. This means that obtaining a permanent satellite of the Moon on the first revolution without the aid of an engine is impossible.

For the case of approach of the projectile to the Moon along an arbitrary trajectory, it is not possible to prove the impossibility of capture.

The impossibility of capture for arbitrary trajectories in the circular restricted three-body problem can be proved only for sufficiently small ratios of the attracting masses, smaller than the ratio of the mass of the Moon to the mass of the Earth. We present this proof. Put \(m_1+m_2=1\) and let \(m_2=\mu\). Then \(m_1=1-\mu\), \(\rho_0\simeq\sqrt{\mu}\).

According to § 2, if the body \(m_0\) approaches \(m_2\) on some revolution from the interior of the critical surface \(S'\), then \(h>h_1\), and if from the бес-

finiteness, then \(h>h_2\). In this case

\[ -h_i=U_i=\frac{\left[1-\mu+(-1)^i\rho_i\right]^2}{2} +\frac{1-\mu}{1+(-1)^i\rho_i} +\frac{\mu}{\rho_i},\quad i=1,2. \tag{4,1} \]

For small \(\mu\) we have\({}^{13}\)

\[ \rho_i=\left(\frac{\mu}{3}\right)^{1/3} +(-1)^i\left(\frac{\mu}{3}\right)^{2/3} -\frac{1}{9}\left(\frac{\mu}{3}\right)^{3/3}+\ldots,\quad \text{and} \]

\[ -2h_i=3+10\left(\frac{\mu}{3}\right)^{2/3} -\left[3+(-1)^i\frac{4}{9}\right]\mu \tag{4,2} \]

with accuracy up to \(\left(\frac{\mu}{3}\right)^{4/3}\).

Assuming that capture has occurred, and transforming (2,1) to \(m_2\)-centric elements, by virtue of the inequality \(-2h<-2h_1\) we obtain:

\[ \frac{\mu}{a_2}+2\sqrt{\mu p_2}\cos i_2 <10\left(\frac{\mu}{3}\right)^{2/3}+\mu F_1, \tag{4,3} \]

where \(F_1\) is finite. By virtue of \(a_2<\sqrt{\mu}\) we have

\[ \sqrt{\mu}-2\mu^{3/4} <10\left(\frac{\mu}{3}\right)^{2/3}+\mu F_1. \]

Since the lowest power of \(\mu\) on the left is smaller than on the right, for sufficiently small \(\mu\) the last inequality is contradictory, and capture is impossible. It is not difficult to show that the result is preserved if, instead of the sphere \(\rho<(\mu)^{1/2}\), one uses the sphere of action \(\rho<(\mu)^{2/5}\).

Result (4,3) corrects V. G. Fesenkov’s criterion of impossibility of capture\({}^{12}\) for small \(\mu\)

\[ \frac{1}{a_2}+2\sqrt{p_2}\cos i_2<\mu F, \tag{4,4} \]

where \(F\) is finite. Criterion (4,4) differs from (4,3) because in\({}^{12}\), in the proof, \(\mu\) is omitted in the left-hand sides of formulas of type (4,4). Moreover, in\({}^{12}\), in deriving criterion (4,4), the expressions \(h_i(\mu)\) were not considered, while Tisserand’s transformation\({}^{13}\) and the following assertion\({}^{12}\) (p. 45) were used:

“It is easy to see that, under all circumstances of the motion of the body \(m_0\) in which its approach to \(m_1\) is possible, the expression \(\frac{1}{a_1}+2\sqrt{p_1}\cos i_1\) must always be less than 3.” This assertion is incorrect. Transforming (2,1) to the elements \(a_1,\rho_1,i_1\) and using (4,2), one can obtain

\[ \frac{1}{a_1}+2\sqrt{p_1}\cos i_1 =3+10\left(\frac{\mu}{3}\right)^{2/3}+\mu\varphi, \]

(where \(\varphi\) is finite), which for sufficiently small \(\mu\) is greater than 3.

For \(\mu^{-1}=81.45\), inequality (4,3) is not yet contradictory, and for arbitrary trajectories arriving at the Moon the question of capture remains open, whereas by V. G. Fesenkov’s criterion (4,4) it follows that capture is impossible.

Regardless of whether capture is possible or not, the creation of a temporary satellite of the Moon without the aid of an engine is possible in principle. Indeed, if the initial velocity of the projectile exceeds the first critical velocity but is less than the second, i.e. if \(h_1<h<h_2\), then, according to § 2, the projectile will not be able to escape to infinity, but can pass through the neck near the libration point \(L_1\) toward the Moon (Fig. 1). Of course, beforehand,

according to § 3, it must make a sufficiently large number of revolutions around the Earth. Since the throat can be made arbitrarily narrow by sufficiently decreasing the initial velocity, the projectile, apparently, can move near the Moon for an arbitrarily long time before it passes back through the throat.

As for the practical realization of such an artificial satellite of the Moon, it is hardly possible for the following reasons. First, the difference between the first and second critical velocities is less than \(1\ \text{m/sec}\), and an intermediate velocity is difficult to realize. Second, before reaching the throat, the projectile may fall to the Earth; and third, the perturbations of the Sun, which have been neglected, can easily take \(h\) out of the range \(h_1<h<h_2\), i.e., lead to the closing of the throat or to the escape of the projectile to infinity.

§ 5. APPROXIMATE METHOD OF INVESTIGATION

We shall call approach trajectories those trajectories which begin near the Earth (body \(m_1\)) and, on the first revolution around the Earth, enter the sphere of action of the Moon (body \(m_2\)).

Since an approach trajectory, having entered the sphere of action, necessarily leaves it, the motion along an approach trajectory may be divided into 3 segments: motion toward the sphere of action, motion inside the sphere of action, and motion away from the sphere of action.

The perturbation by the Moon of geocentric motion toward the sphere of action, as we saw in § 3, is practically negligible. The same applies to the motion away from the sphere of action. Just as slight is the perturbation by the Earth of selenocentric motion within the sphere of action, as follows from the very definition of the sphere of action. Therefore, perturbations may everywhere be neglected in comparison with the attraction of the central body.

Neglecting perturbations, we find that the portions of the trajectory outside the sphere of action, referred to the geocentric system \(m_1\xi_1\eta_1\zeta_1\), are conic sections with focus \(m_1\), while the portion of the motion inside the sphere of action, referred to the selenocentric system \(m_2\xi_2\eta_2\zeta_2\), is a conic section with focus \(m_2\) (see Fig. 9a, corresponding to the case of motion in the plane of the Moon’s orbit).

On the segment from the initial point to the point of entry into the sphere of action (the entry point), the trajectory, depending on the geocentric initial velocity, may be an ellipse, a hyperbola, or a parabola. It is evident that, for an elliptic velocity greater than the minimum, entry of the rocket into the sphere of action is possible both on the ascending branch of the geocentric trajectory and on its descending branch; while for parabolic and hyperbolic velocities, entry is possible only on the ascending branch.

The calculation of the motion toward the sphere of action is carried out from the geocentric initial data: \(r_1,\lambda_1,V_1,\alpha_1\). At the entry point the geocentric coordinates and velocity \(V_2\) (the geocentric entry data) are transformed into the selenocentric system \(m_2\xi_2\eta_2\zeta_2\), and the selenocentric entry data are obtained (Fig. 9a).

It can be shown that the segment of an approach trajectory lying inside the Moon’s sphere of action is always, in selenocentric coordinates, a hyperbola. At the boundary of the sphere of action the selenocentric velocity \(V'_2\) exceeds the parabolic velocity at the boundary of the sphere of action,

\[ V'_{\text{p}}=\sqrt{\frac{2fm_3}{\rho_*}}\simeq 0.383\ \text{km/sec}, \]

by more than a factor of two.

This fact explains the impossibility of capture for approach trajectories. Moreover, owing to the hyperbolicity of the near-...

the boundary of the sphere of action pass very rapidly, and although the perturbations attain near the boundary of the sphere of action a magnitude of the order of 0.7 of the attraction of the central body, they still do not have time to have an appreciable effect on the approach trajectories.

Motion within the Moon’s sphere of action is determined by selenocentric integrals of energy and areas. At the point of exit from the sphere

Fig. 9a. Approximate calculation of the motion. Trajectory. Outside the Moon’s sphere of action, perturbations from the Moon are neglected, and inside—perturbations from the Earth. Within the sphere of action the selenocentric motion proceeds along a hyperbola.

of action, the selenocentric coordinates and the exit velocity \(\mathbf{V}'_{3}\) (selenocentric exit data) are recalculated (Fig. 9a) into geocentric exit data.

After exit from the Moon’s sphere of action, the motion is calculated from geocentric integrals of energy and areas with new values of the constants. The motion after exit from the sphere of action may be, with respect to the Earth, either ascending or descending.

Let us show, using the example of the trajectory shown in Fig. 9a, how the hodograph of the incoming and outgoing velocities is constructed. In the velocity hodograph let us take the direction of the geocentric radius at the moment of entry \(t_{2}\) as horizontal (Fig. 9b). From a certain point \(O\) draw the vector of the incoming geocentric velocity \(\mathbf{V}_{2}\), making an angle \(\alpha_{2}\) with the incoming geocentric radius, and from its end draw the vector \(-\mathbf{V}_{n}(t_{2})\), making with the transverse direction an angle \(\varphi_{2}\) equal to the angle of the incoming radius with the direction

Earth–Moon. The resultant vector \(\mathbf V'_2=\mathbf V_2-\mathbf V_{\mathrm L}(t_2)\) will be the entry selenocentric velocity.

Since the selenocentric motion takes place along a conic section, \(|\mathbf V'_3|=|\mathbf V'_2|\). Let \(\alpha\) be the angle through which the direction of the projectile’s selenocentric motion changes due to the Moon during the time it remains in the sphere of action. Draw from the end of the vector \(\mathbf V'_2\) the vector \(\mathbf V'_3\), forming an angle \(\alpha\) with \(\mathbf V'_2\), and from its end draw the vector \(+\mathbf V_{\mathrm L}(t_3)\), forming an angle \(\omega(t_3-t_2)=\omega T_{2,3}\) with the vector \(\mathbf V_{\mathrm L}(t_2)\). The resultant vector \(\mathbf V_3=\mathbf V'_3+\mathbf V_{\mathrm L}(t_3)\) is the vector of the exit geocentric velocity.

Fig. 9b. Velocity diagram at the points where the trajectory intersects the sphere of action.

§ 6. ANALYSIS OF APPROACH TRAJECTORIES

Using the method described in § 5, one can calculate and construct any approach trajectory. It is of interest to analyze, at least approximately, the totality of all possible motions along approach trajectories. For this purpose let us consider the characteristics of the set of approach trajectories at the beginning and at the end of each of the successive segments of motion. At the same time we shall carry out the corresponding constructions on the velocity diagram.

The segment of motion to the sphere of action is determined by the initial data. We shall regard the initial altitude as fixed and in the calculations take it equal to \(200\) km. The results will also be applicable for altitudes exceeding the one considered by a factor of two or even several times (for example, for the altitude of a satellite-station), since changes in the initial altitude are small in comparison with the initial geocentric radius. The angle \(\alpha_1\) of the initial velocity with this radius will vary in the range \(-90^\circ<\alpha_1<90^\circ\), and the initial velocity \(V_1\) itself will be the principal variable parameter.

Let us consider the set of possible values of the geocentric entry data. The indicated quantities are calculated from the initial quantities with the aid of the geocentric integrals of energy and areas. Consider the entry geocentric velocity \(V_2\) and its angle \(\alpha_2\) with the entry geocentric radius for the mean value of this radius \(r_2=a\), i.e., at the distance of the lunar orbit. Then, according to the energy integral, the entry velocity will depend only on the initial velocity, or, equivalently, on the excess \(\Delta V_1\) of the velocity over the parabolic velocity \(V_{\mathrm p}\). The function \(V_2(V_1)\) is monotonically increasing (Fig. 10). The angle \(\alpha_2\) depends not only on the initial velocity, but also on the initial angle \(\alpha_1\), and has its sign. The function \(\alpha_2(V_1)\) is monotonically decreasing and, for the initial angle \(\alpha_1=90^\circ\), i.e., for the horizontal direction of the initial velocity, is shown in Fig. 10. When the initial angle is decreased, the corresponding curve, still beginning at the ordinate \(90^\circ\), will pass below the one shown.

To determine the possible ranges of variation of the entry data when the entry radius changes, curves analogous to those considered were computed (Fig. 10),

\[ \widehat V_2(V_1),\ \widehat\alpha_2(V_1)\ \text{and}\ \widetilde V_2(V_1),\ \widetilde\alpha_2(V_1), \tag{6.1} \]

corresponding, respectively, to

\[ r_2=a+\rho_* \quad \text{and} \quad r_2=a-\rho_*, \]

where \(\rho_*\) is the radius of the sphere of action.

It is seen that, for initial velocities not close to the minimum ones, the changes in the magnitude and direction of the entering geocentric velocity within the interval \(|r-a|<\rho_*\) are small. For such initial velocities one may approximately assume that, for fixed \(V_1\) and \(\alpha_1\), the magnitude

Fig. 10. Change of the characteristics at the point of entry into the sphere of action as a function of the excess \(\Delta V_1\) of the initial velocity \(V_1\) over the parabolic velocity \(V_{\mathrm p}\). \(\check V_2\), \(\hat V_2\) and \(\check \alpha_2\), \(\hat \alpha_2\) are the limiting values of the entering geocentric velocity \(V_2\) and its angle \(\alpha_2\) with the entering geocentric radius. \(U^+\) and \(U^-\) are the values of the entering selenocentric velocity for angles of the initial velocity with the radius of \(\pm 90^\circ\).

and the direction of the entering geocentric velocity do not depend on the point of entry, i.e., on the trajectory, and have the mean values

\[ V_2(r_2)\cong V_2(a) \quad \text{and} \quad \alpha_2(r_2)\cong \alpha_2(a). \tag{6,2} \]

Here and in what follows we shall agree to consider only approach trajectories passing in the plane of the Moon’s orbit. Let the initial velocity \(V_1\) and its angle with the radius \(\alpha_1\) be fixed, and let the approach trajectory vary only through the angle \(\lambda\) of the initial radius with the Moon–Earth direction. Then, for initial velocities not close to the minimum ones, the vector \(\mathbf V_2\) on the velocity diagram (Figs. 9b and 14), according to (6,2), will be the same for all the trajectories under consideration.

The times of flight to the Moon’s orbit \(T_{1,2}(V_1)\) for the vertical and horizontal directions of the initial velocity are presented in Fig. 11. They turned out to be very close. This fact, as well as the abruptness of the changes in the curves considered in Figs. 10 and 11 at velocities close to the minimum ones, is explained by the elongated form of the conic sections corresponding to motion toward the sphere of action along approach trajectories (their eccentricity is close to 1).

With a sufficient increase in the initial velocity, the curves

\[ V_2(V_1),\quad \alpha_2(V_1)\quad \text{and}\quad T_{1,2}(V_1) \]

approach, respectively, the asymptotes

\[ V_2=V_1;\quad \alpha_2=\arcsin \frac{r_1\sin\alpha_1}{r_2};\quad T_{1,2}=0, \tag{6.3} \]

where \(r_1\) and \(r_2\) are the initial and entry radii. The asymptotic character of these curves is noticeable already at initial velocities only \(0.5\ \mathrm{km/sec}\) greater than parabolic (Figs. 10 and 11).

Let us proceed to an analysis of the motion in the sphere of action. The selenocentric entry data are determined from the geocentric ones. In this case allowance for the small angle \(\varphi_2\) (Figs. 9a and 9b) between the geocentric radii of the Moon and the entry point, as can be shown, has no fundamental

Fig. 11. Flight times to the Moon’s orbit. The vertical direction of the initial velocity \((\alpha_1=0)\) corresponds to the smallest flight times, the horizontal direction to the largest.

importance for the subsequent results. If this angle is neglected, then the velocity of the Moon at the instant of entry \(t_2\) will be orthogonal to the entry radius, while the magnitude and direction of the entry selenocentric velocity \(V'_2\) in the velocity plane will be the same for all the trajectories considered (Fig. 14). This means that the initial segments of the selenocentric trajectories in the sphere of action will be parallel.

If the entry point is characterized by the angle \(\psi_2\) of its selenocentric radius vector with the velocity of the Moon (Fig. 9a), then, as the angle \(\lambda\) is varied, entry will be possible over a range of angles \(\psi_2\) whose extent is close to \(180^\circ\). The curves of one half of this range go around the center of the Moon clockwise (Fig. 12, b), and the curves of the other half counterclockwise (Fig. 12, a).

For plane approach trajectories it can be shown that the neighborhood of the value \(|\psi_2|=180^\circ\) is a forbidden zone on the sphere of action, through which entry cannot occur, whatever the initial-

data. This means that the projectile cannot enter the sphere of action while overtaking the Moon in its orbital motion.

Recall that the incoming and outgoing selenocentric velocities are equal in magnitude. Their common magnitude \(U\) is a monotonically increasing function of the initial velocity. For initial angles of the velocity with the radius \(\alpha_1=\pm 90^\circ\) it is shown in Fig. 10 (the curves \(U^+\) and \(U^-\)). It can be shown that, as \(\alpha_1\) varies in the range \(-90^\circ<\alpha_1<+90^\circ\), the magnitude of the selenocentric velocity at the boundary of the sphere of action decreases monotonically from \(U^-\) to \(U^+\). The difference of the velocities \(U^- - U^+\) is small and rapidly tends to zero as the initial velocity increases. Thus, already at \(\Delta V_1=-0.5\ \mathrm{km/sec}\) the difference is only about \(3\%\) of the half-sum of these velocities. Note that, for arbitrary initial velocities, the selenocentric velocity at the boundary of the sphere of action exceeds the incoming geocentric velocity, i.e. \(U(V_1)>V_2(V_1)\) (Fig. 10).

Fig. 12. Selenocentric trajectories in the Moon’s sphere of action. The entry points occupy approximately half of the sphere of action. \(V'_2\) is the incoming selenocentric velocity.

The motion inside the sphere of action and the selenocentric outgoing data are calculated by means of the selenocentric integrals of energy and areas. In particular, one can calculate the change, due to the Moon, in the direction of the selenocentric motion, i.e. the angle \(\alpha\) between the incoming and outgoing selenocentric velocities. Since the trajectories under consideration have one and the same magnitude of incoming velocity, the angle \(\alpha\) is determined only by the distance \(d\) of the line of action of this velocity vector from the center of the Moon. We shall agree to assign to this distance the sign of \(\alpha\), i.e. the sign of the direction of passage around the Moon. Obviously, the Moon will change the direction of motion the more strongly—that is, the angle \(|\alpha|\) will be the greater—the closer the initial direction of motion is to the Moon. For various fixed values of the incoming selenocentric velocity, the curves \(\alpha(d)\) are shown in Fig. 13.

If it is assumed that, for the trajectories under consideration, the distance \(d\) varies within the limits \(-\rho_*<d<\rho_*\), then the angle \(\alpha(d)\) will take any value between \(-180^\circ\) and \(+180^\circ\), and the vectors of the outgoing selenocentric velocities on the velocity plane will completely fill a circle of radius \(U\) with center \(A'_2\) (the upper circle in Fig. 14).

Let us consider the change of velocities and trajectories in the sphere of action when the angle \(\lambda\) of the initial radius with the Moon–Earth direction is varied, i.e. when the portion of motion toward the sphere of action, as a whole, is rotated about the center of the Earth. Let the ascending trajectory first touch the sphere of action, passing around it counterclockwise (Fig. 12, \(a\)). We have

\[ d=\rho_*,\quad a=0\quad \text{and}\quad V'_3\big|_{a=0}=V'_2 \]

Fig. 13. Angle between the tangents to the hyperbola at the entry and exit points of the sphere of action, as a function of the distance \(d\) of the tangent to the trajectory at the entry point from the center of the Moon.
\(U\) is the magnitude of the velocity.

Fig. 14. Plan of exit velocities. The upper circle shows selenocentric velocities, the lower one geocentric velocities. Motion after approach to the Moon is possible in all directions. The velocities \(V_3'|_{\alpha=0}\) and \(V_3|_{\alpha=0}\) correspond to the trivial solution.

(Fig. 14). As \(\lambda\) increases, the trajectory will approach the Moon; \(d\) will decrease and \(\alpha\) will increase, i.e., the vector \(\mathbf V'_3\) will rotate counterclockwise. When \(d=0\), \(\alpha\) will reach \(180^\circ\) and impact at the center of the Moon will occur (Fig. 12, \(a\)). With a further increase of \(\lambda\), \(d<0\), and abruptly \(\alpha_1=-180^\circ\), since the direction of going around the center of the Moon changes (Fig. 12, \(b\)). However, the point of departure from the sphere of influence and the endpoint of the exit-velocity vector \(\mathbf V'_3\) traverse the corresponding circle continuously, and everywhere counterclockwise. When the vector \(\mathbf V'_3\) has gone around a full circle, \(\alpha\) again becomes \(0\), \(\left.\mathbf V'_3\right|_{\alpha=0}=\mathbf V'_2\) (Fig. 14), and the approach no longer takes place. Solutions with \(\mathbf V'_3=\mathbf V'_2\) may be called trivial.

For small \(d\), as is not difficult to show, the distance of the trajectory from the center of the Moon is of small order \(d^2\), and this holds for any selenocentric velocities on the boundary of the sphere of influence. For impact with the Moon (§ 7) this will be important.

Let us note that for small \(d\) the trajectory in the sphere of influence bends so sharply that, in appearance, it approaches the angle formed by the corresponding velocities \(\mathbf V'_2\) and \(\mathbf V'_3\) in the velocity plane (Fig. 14).

Also of interest are the dependences of \(|d|\), \(\alpha\), \(\Phi'\) on \(U\), corresponding to trajectories with a fixed distance from the center of the Moon. (Here \(\Phi'\) is the angle traversed by the selenospheric radius vector during motion inside the sphere of influence.) They are decreasing functions, and for trajectories close to the surface of the Moon they are presented in Fig. 15. We see that the values of \(d\) corresponding to grazing the Moon are small, less than \(5400\) km. The magnitude of the angle of rotation of the direction of motion by the Moon, \(\alpha\), exceeds \(90^\circ\) only for velocities \(U\) close to the minimum, and with increasing \(U\) approaches zero. In Fig. 14 the vectors corresponding to grazing trajectories are drawn with a dashed line. Obviously, approach trajectories that do not collide with the Moon can correspond only to vectors enclosed between the dashed ones.

Fig. 15. Characteristics of trajectories passing near the surface of the Moon: \(\Phi'\) is the selenocentric angle traversed by the projectile inside the sphere of influence, \(\alpha\) is the angle between the tangents to the hyperbola at the entry and exit points, \(d\) is the distance of the tangent from the center of the Moon, \(U\) is the entry selenocentric velocity.

Flight times inside the sphere of influence, \(T_{2,3}=t_3-t_2\), as functions of the distance \(|d|\) for fixed values of the selenocentric velocity \(U\) on the boundary of the sphere of influence, are presented in Fig. 16. It turns out that the maxima of the flight times are reached at

\[ d=-\frac{\rho^* V'_{\mathrm p}}{U\sqrt{2}}, \]

where \(V'_{\mathrm p}\) is the selenocentric parabolic velocity on the boundary of the sphere of influence. However, these maxima are weakly expressed, and for distances \(|d|\) not close to the radius of the sphere of influence one may regard \(T_{2,3}\) as constant and dependent only on \(U\).

Finally, let us consider the geocentric output data. Without dwelling on them in detail, we shall construct only the output geocentric

velocities \(\mathbf V_3(d)\) on the velocity plane. These velocities are obtained from the corresponding selenocentric velocities \(\mathbf V'_3(d)\) by adding the velocity vector of the Moon \(\mathbf V_{\text{л}}(t_3)\), which is rotated relative to \(\mathbf V_{\text{л}}(t_2)\) through the small angle \(\omega(t_3-t_2)<\omega(T_{2,3})_{\max}\) between the Earth–Moon directions at the moments of entry and exit (Fig. 14). It can be shown that allowance for this angle is not of fundamental importance for the consideration being carried out, and it may be neglected. Then the locus of the ends of the vectors of the geocentric outgoing velocities \(\mathbf V_3(d)\) issuing from the point \(A'_2\) will be a circle of radius \(U\), but with center at the point \(A_2\) (the lower circle in Fig. 14). Thus the outgoing-velocity plane is extremely simple.

Fig. 16. Flight time in the sphere of action. The rapid decrease of the flight time with increasing velocity \(U\) is noticeable.

Let us now consider approach on the descending branches with the same initial data \(r_1, V_1, \alpha_1\), taking the initial velocities \(V_1\) to be elliptic. Under the same assumptions as for the ascending branches, we obtain the vectors \(O'A_2=\mathbf V_2\) and \(O'A'_2=\mathbf V'_2\) (Fig. 14), symmetric to the former ones with respect to the line \(A_2A'_2\). Since to every vector \(\mathbf V'_3\) corresponding to the angle \(\alpha\) there corresponds a vector \(\mathbf V'_3\) for the ascending branches, symmetric with respect to \(A_2A'_2\), corresponding to the angle \(-\alpha\), the velocity plane will simply be symmetric to the plane for the ascending branches, and the circles of the ends of the vectors \(\mathbf V_3\) will coincide. Owing to this, both planes can be investigated simultaneously.

In both cases we see from Fig. 14 that the maximum value of the outgoing geocentric velocity is

\[ V_3^{(M)}=U+V_{\text{л}}, \]

and the minimum is

\[ V_3^{(m)}=U-V_{\text{л}}, \]

and that for \(U>V_{\text{л}}\) there exist outgoing geocentric velocities \(\mathbf V_3\) of any direction, whatever the magnitude and direction of the initial velocity may be. Since the magnitude \(U\) decreases monotonically, as the angle \(\alpha_1\) of the initial velocity with the radius increases, the velocity plane constructed for the angle \(\alpha_1=90^\circ\) will have somewhat smaller dimensions than the plane for the angle \(\alpha_1=-90^\circ\), while for intermediate angles all the characteristics of the velocity plane are intermediate.

Let us briefly follow the evolution of the velocity plane with change in the initial velocity \(V_1\), beginning with large values of the excess \(\Delta V_1\) of the initial velocity over the parabolic one. For hyperbolic velocities

convergence is, evidently, possible only on the ascending branch. For \(\Delta V_1 = 0.5\ \text{km/sec}\) we have (Fig. 10): \(|\alpha_2^{\pm}| = 3^\circ\), \(V_2 = 3.64\ \text{km/sec}\), \(U^{+} = 3.73\ \text{km/sec}\), \(U^{-} = 3.83\ \text{km/sec}\), where the sign “\(+\)” corresponds to the angle \(\alpha_1 = +90^\circ\), and “\(-\)” corresponds to \(\alpha_1 = -90^\circ\). Since \(U^{+}\) and \(U^{-}\) differ little, the velocity planes for all angles \(\alpha_1\) practically coincide. All exit geocentric velocities are strongly hyperbolic.

With a decrease in the initial velocity, all velocities decrease, and soon the minimal geocentric exit velocities become elliptic, although the entry velocities are still hyperbolic. This first occurs at \(\Delta V_1 = 0.149\ \text{km/sec}\) for the angle \(\alpha_1 = +90^\circ\). With a further decrease of the initial velocity, elliptic exit velocities also appear for negative angles \(\alpha_1\). For the angle \(\alpha_1 = -90^\circ\) they appear only at \(\Delta V_1 = 0.114\ \text{km/sec}\).

When the initial velocity \(V_1\) passes through the parabolic value, convergence on the descending branch becomes possible. For \(V_1 = V_{\mathrm{p}}\) we have (Fig. 14):

\[ V_2 = 1.44\ \text{km/sec}; \qquad |\alpha_2^{\pm}| = 7^\circ.5; \]

\[ U^{+} = 1.65\ \text{km/sec}; \]

\[ U^{-} = 1.87\ \text{km/sec}. \]

The region of elliptic geocentric exit velocities occupies, in the velocity plane, approximately one third of the corresponding circle (Fig. 14). For the angles \(\alpha_1 = \pm 90^\circ\) we have, respectively, maximal velocities \(V_3^{(m)+} = 2.67\ \text{km/sec}\); \(V_3^{(m)-} = 2.9\ \text{km/sec}\), and minimal velocities \(V_3^{(m)+} = 0.63\ \text{km/sec}\); \(V_3^{(m)-} = 0.85\ \text{km/sec}\).

Fig. 17. Plan of exit velocities in the case when the incoming selenocentric velocity \(V'_2\) is, in magnitude, equal to the lunar velocity \(V_{\mathrm{L}}\). Motion outside the sphere of influence proves possible only on one side of the direction of the exit geocentric radius \(m_1A_2\).

Finally, when, with a decrease of the initial velocity at the angle \(\alpha_1 = +90^\circ\), the selenocentric entry velocity decreases to the velocity of the Moon, a zero exit geocentric velocity appears (Fig. 17). We have:

\[ V_3^{(m)} = 0; \qquad V_3^{(m)} = 2 \cdot V_{\mathrm{L}} \simeq 2.05\ \text{km/sec}; \qquad V_2 \simeq 0.6\ \text{km/sec}; \qquad \alpha_2^{+} \simeq 18^\circ.4. \]

In Fig. 17 the sector of practically realizable selenocentric exit velocities (between the dotted lines) exceeds \(180^\circ\). The corresponding value is \(\Delta V_{1*} \simeq 0.077\ \text{km/sec}\).

Although the corresponding vector of the incoming geocentric velocity \(V_2\), as is seen from Fig. 10, can no longer be regarded as constant when the entry point is varied, it can be shown that taking into account its variability, as well as the variability of the vector of incoming selenocentric velocity, has no fundamental significance for \(\Delta V_1 > \Delta V_{1*}\). For \(\Delta V_1 < \Delta V_{1*}\), as is not difficult to verify, trajectories with \(\alpha_1 = \pm 90^\circ\) cannot reach distances \(d\) close to \(-\rho_*\), and there will not be sufficiently small angles \(\alpha < 0\). A special forbidden region appears in the plane of exit velocities. Therefore a special consideration is needed, which we shall not present here.

Let us only note that the indicated forbidden region expands as \(V_1'\) decreases, covering the entire plane, with the exception of the vector \(\mathbf V_3\big|_{\alpha=0}\), corresponding to the trivial solution \(\left(\mathbf V_3\big|_{\alpha=0}=A_2'O_1'\right.\) for descending branches; see Figs. 14 and 17), and that for \(U<V_l\) only motions to one side of the initial geocentric radius \(r_3\) are possible, and not to both, as before (Fig. 17).

For initial velocities close to the minimum, the incoming geocentric velocities \(V_2\) are small, and the motion proceeds approximately as if the sphere of action were approaching a rocket at rest in space.

The consideration of the motion after exit from the sphere of action is analogous to the consideration on the segment that preceded the approach, and will not be carried out here.

§ 7. THE PROBLEM OF HITTING THE MOON

In the problem of hitting the Moon it is not necessary to know the behavior of the solutions after approach to the Moon, and their classification is very simple. Let the initial geocentric radius \(r_1\), the velocity \(V_1\), and the angle \(\alpha_1\) of the velocity with the radius be specified, while the angle \(\lambda\) of the radius with the \(\xi\)-axis (Fig. 6) is chosen from the condition of hitting the Moon.

Obviously, for elliptic initial velocities the approach of the projectile to the Moon is possible both on the ascending (в) and on the descending (н) branch of its trajectory. Accordingly we have two classes of hits, \(P^{\mathrm{в}}\) and \(P^{\mathrm{н}}\). The limiting trajectories of each of the classes, i.e. the trajectories corresponding to \(\alpha_1=+\dfrac{\pi}{2}\) and \(\alpha_1=-\dfrac{\pi}{2}\), are schematically shown in Fig. 18.

Fig. 18. Classes of hitting trajectories: \(I\) — hitting on the ascending branch of the trajectory, \(II\) — on the descending branch.

According to the sign of the direction of revolution about the Earth, i.e. according to the sign of \(\alpha_1\), each class can be divided into two subclasses:

\[ P^{\mathrm{в}}\ \text{into}\ P^{\mathrm{в}+}\ \text{and}\ P^{\mathrm{в}-}; \qquad P^{\mathrm{н}}\ \text{into}\ P^{\mathrm{н}+}\ \text{and}\ P^{\mathrm{н}-}. \]

Here the trajectory separating the classes is purely radial with respect to the Earth. With monotonic variation of the angle \(\alpha_1\) in the interval

\[ -\frac{\pi}{2}<\alpha_1<+\frac{\pi}{2} \]

the trajectory changes monotonically within one and the same class between its limiting trajectories.

Let us trace the evolution of the solutions as the initial velocity decreases. For hyperbolic initial velocities, obviously, there exist only solutions of the class \(P^{\mathrm{в}}\). When the velocity passes through the parabolic value, there appear—

are descending branches, and immediately for all \(\alpha_1\) there appear solutions of class \(\Pi^{\mathrm n}\). With decreasing velocity the solutions of both classes corresponding to one and the same angle \(\alpha_1\) approach each other. Finally, when the initial velocity passes through the minimum value \(V_{1\min}(|\alpha_1|)\), the corresponding solutions of each class merge and disappear. The solutions with \(|\alpha_1|=\frac{\pi}{2}\) disappear first, so that for \(|\alpha_1|=\frac{\pi}{2}\) hitting the center of the Moon becomes impossible, although there exist angles \(|\alpha_1|<\frac{\pi}{2}\) for which it is still possible.

The value of \(\lambda\) corresponding to impact, as follows from § 3, can be found approximately entirely without taking into account the influence of the Moon, by formulas (3.7). For this purpose, beforehand, by the formulas of the theory of conic sections, the angle \(\Phi\) between the geocentric radii of the initial point and of the point of encounter is found, as well as the flight time \(T_{1,2}\) between these points. The functions \(\Phi(V_1)\), \(\lambda^+(V_1)\), \(\lambda^\circ(V_1)\), and \(\lambda^-(V_1)\) (shown in Fig. 19) and the function \(T_{1,2}(V_1)\) (in Fig. 11) have been computed for class \(\Pi^{\mathrm v}\) with an initial altitude of 200 km. The function \(\Phi(V_1)\) corresponds to the value of the angle \(\alpha_1=+\frac{\pi}{2}\), while the functions \(\lambda^+(V_1)\), \(\lambda^\circ(V_1)\), and \(\lambda^-(V_1)\) correspond, respectively, to the values \(\alpha_1=+\frac{\pi}{2}, 0, -\frac{\pi}{2}\). For excesses of the initial velocity over the parabolic velocity of \(0.5\ \mathrm{km/sec}\), the tendency of the curves toward asymptotes is already noticeable.

Fig. 19. The angle \(\lambda\) of the initial radius with the direction Moon–Earth, ensuring impact on the Moon, as a function of the initial velocity, in the case of vertical \((\alpha_1=0)\) and horizontal \((|\alpha_1|=90^\circ)\) firing directions. \(\Phi\) is the angular range.

Trajectories of class \(\Pi^{\mathrm v}\) corresponding to an initial altitude of 200 km, the angle \(\alpha_1=\frac{\pi}{2}\), and the approximate values \(\lambda^+(V_1)\) calculated above were found, taking into account the attraction of the Moon, by numerical integration of equations (3.2) on a digital machine. It turned out that the deviation \(\rho_m\) of the trajectory from the center of the Moon, arising as a result of neglecting the attraction of the Moon in determining \(\lambda\), decreases very rapidly as the initial velocity increases from the minimum. If at minimum initial velocities the miss distance \(\rho_m\) obtained is of the order of tens of kilometers, then as the velocity approaches the parabolic value \(\rho_m<1\ \mathrm{km}\).

To obtain an idea of the impact trajectories themselves, an accuracy of tens of kilometers would be sufficient; however, to clarify the effect of scatter in the initial data, an accuracy of impact on the center of the Moon of the order of a kilometer will be needed.

Trajectories hitting the center of the Moon with arbitrarily high accuracy can be obtained by the method of iteration to the value \(\rho_m=0\) with respect to the argument \(\lambda\). In this case it can be shown that the convergence will be of higher order if, as the function, one takes not \(\rho_m\) with the sign of the direction of motion around the Moon, but \(\sqrt{\rho_m}\) with the same sign.

This iterative process was programmed for a digital computer, which made it possible to carry out a mass calculation of impact trajectories. The results of calculating class \(I^{\mathrm{B}}\) trajectories for \(\alpha_1 = + \dfrac{\pi}{2}\) are presented in Fig. 20 in the \(O\xi\eta\) system, rotating together with the direction toward the Moon. The dependences \(T_{1,2}\), \(\Phi\), and \(\lambda\) on the initial velocity corresponding to these trajectories practically coincide with those given in Figs. 11 and 19.

Let us consider the question of the required accuracies of the initial data. We note that if, in determining the influence of small errors in the initial data, the action of the Moon is neglected, then the miss distance \(\rho_m\) will be a linear

Fig. 20. Ascending impact trajectories in rotating coordinates; \(T_{1,2}\) — flight time.

	\(\Delta V_1\left(\dfrac{\mathrm{km}}{\mathrm{sec}}\right)\)	\(T_{1,2}\) (days)
I	\(+0,48251\)	\(1,08386\)
II	\(+0,106094\)	\(1,62688\)
III	\(0\)	\(2,06981\)
IV	\(-0,057828\)	\(2,64816\)
V	\(-0,082828\)	\(3,33284\)
VI	\(-0,092828\)	\(4,73092\)

function of the errors. If, however, the influence of the Moon within its sphere of action is taken into account, then \(d\) (the distance from the center of the Moon to the line of action of the vector of the incoming selenocentric velocity) will be a linear function of the errors. And since, according to § 6, for small \(d\) the quantity \(\rho_m\) is proportional to \(d^2\), when the influence of the Moon is taken into account the miss distance turns out to be a quadratic function of the errors. This means that it is much easier to achieve a sufficiently accurate impact on the Moon than to ensure impact on a non-attracting point moving in the same way as the Moon.

Determining the miss distances \(\rho_m, \rho_v, \rho_\alpha, \ldots\), corresponding respectively to errors \(\delta V_1, \delta \alpha_1, \ldots\), by means of the approximate method (i.e., taking account of the Moon’s influence only within its sphere of action) is rather laborious. A more accurate and easier procedure is to determine the deviations \(\rho_m\) by directly computing, on the machine, trajectories close to the trajectory of a sufficiently accurate impact at the center of the Moon (the nominal trajectory).

Varying one of the nominal initial data \(x_i\) by a small amount \(\delta x_i\), and calculating the corresponding trajectory, we find its rea-

distance from the center of the Moon, \(\rho_i \simeq k_l(\delta x_i)^2\). The curves \(k_\alpha(V_1)\) and \(k_V(V_1)\) are shown in Fig. 21. The function \(k_\alpha(V_1)\) increases monotonically, while \(k_V(V_1)\) decreases monotonically, passing through zero. Accordingly, the maximum admissible error in the direction \(|\delta \alpha_1|_{R_\text{л}}\) (i.e., the error corresponding to a miss equal to the radius of the Moon, \(\rho_m=R_\text{л}\), when the remaining initial data are realized exactly) is a monotonically decreasing function of the initial

Fig. 21. Coefficients determining the deviation of the trajectory from the center of the Moon.

Fig. 22. Magnitude of the error corresponding to a deviation of the trajectory to the edge of the Moon. The optimal value of the initial velocity depends on the available accuracies.

velocity \(V_1\), while the maximum admissible error in the initial velocity \(|\delta V_1|_{R_\text{л}}\) has a maximum with respect to \(V_1\) (Fig. 22). The passage of the function \(k_V(V_1)\) through zero is explained by the fact that, for the corresponding initial velocity and for the angles \(\alpha_1=+\frac{\pi}{2}\), the displacement of the encounter point due to the change in the curvature of the trajectory is compensated by the displacement of the Moon due to the change in the flight time of the projectile to the encounter point.

The value of the maximum of the function \(|\Delta V_1|_{R_{\mathrm{L}}}\) in Fig. 22 can be found approximately by means of the formula \(|\Delta V|_{R_{\mathrm{L}}}=\sqrt[4]{R_{\mathrm{L}}/k_3}\), which follows, for \(k_V=0\), from the relation \(\rho_m=k_V(\Delta V_1)^2+k_4(\Delta V_1)^4+\cdots\) (in Fig. 22 the value of this maximum is plotted approximately).

From Fig. 22 we see that, for example, at velocities close to parabolic ones, \(|\delta \alpha_1|_{R_{\mathrm{L}}}\simeq 0^\circ.3\), and \(|\Delta V_1|_{R_{\mathrm{L}}}\simeq 50\ \mathrm{m/sec}\). The optimal initial velocity obviously depends on the ratio of the available accuracies. It is also seen that the influence of the spread \(\partial V_1\) is especially large at

Fig. 23. Change in the character of the impact trajectory for errors in the initial velocity close to the minimum. For \(H=200\ \mathrm{km}\), \(\alpha_1=75^\circ\), \(\Delta V_1=0.6919\ \mathrm{km/sec}\), impact occurs. For \(\Delta V=\Delta V_1\pm 2\ \mathrm{m/sec}\) there is no impact; the sign “\(+\)” corresponds to I, the sign “\(-\)” corresponds to II.

velocities close to the minimum. How strongly an error \(\Delta V_1=\pm 2\ \mathrm{m/sec}\) can change the trajectory of the projectile at initial velocities close to the minimum is shown in Fig. 23.

Let us note that at the angle \(\alpha_1=-\dfrac{\pi}{2}\) of the initial velocity with the radius, instead of compensation of the displacements of the trajectory and the Moon under errors in the initial velocity, addition of the displacements should occur. As a result, the maximum permissible errors \(|\delta V_1|_{R_{\mathrm{L}}}\) for \(\alpha_1=-\dfrac{\pi}{2}\) must be considerably smaller than for \(\alpha_1=+\dfrac{\pi}{2}\).

The influence of errors \(\delta r_1\) turned out to be practically negligible. For example, for an error \(\delta r_1=50\ \mathrm{km}\), for initial velocities \(V_1-V_{\mathrm{p}}=\)

\(= -0.092328;\ 0;\) and \(+0.106094\ \text{km/sec}\), respectively, the misses \(\rho_r = 56,\ 43\) and \(140\ \text{km}\) were obtained.

The errors \(\delta\lambda\) are connected chiefly with errors in the starting time. The maximum admissible errors in the angle \(\lambda\) are of the order of one degree; therefore the starting time must be maintained with an accuracy of the order of minutes.

The errors \(\delta \xi\) and \(\dot{\delta}\xi\) along the normal \(\zeta\) to the plane of the Moon’s orbit, as can be shown, will lead respectively to distances

\[ |d| \cong \frac{a|\zeta|}{r_1} \]

and

\[ |d| < \frac{a \dot{\xi}}{4V_1}. \]

For \(V_1 \cong V_{\text{II}}\) we have \(U^+ \cong 1.65\ \text{km/sec}\), and from Fig. 15 the admissible values are \(|d| < 3000\ \text{km}\).

It follows from this that, in any case, for \(\delta\xi < 50\ \text{km}\) or \(\dot{\delta}\xi < 50\ \text{m/sec}\), impact will still occur.

Thus, the admissible errors when only one of the initial data is perturbed will be: in velocity about \(50\ \text{m/sec}\), in its direction about \(0.5^\circ\), in the position of the initial point about \(50\ \text{km}\), and in the starting time of the order of minutes. It can be shown that the order of simultaneously admissible errors will be approximately the same.

This order will not change even when perturbations from the Sun and other factors not taken into account by equations (3.2) are included. Indeed, the perturbations are small and will enter into the exact equations of motion with small parameters (which in equations (3.2) were taken to be zero). The derivatives of the solutions with respect to the initial data, by which the required accuracies in the case under consideration are determined, will, as is not difficult to see, depend continuously on the small parameters. Hence these derivatives and the required accuracies will be of the same order as for zero values of the small parameters.

Let us note that the influence of dispersion upon impact with the Moon on the descending branches turns out to be \(2\)—\(5\) times greater than for impact on the ascending branches of the trajectory.

It follows from the results obtained that the influence of dispersion of the initial data on real trajectories of impact with the Moon is comparatively small, so that carrying out an impact on the Moon, apparently, is possible without correction of the trajectory on the passive segment.

§ 8. THE PROBLEM OF FLYING AROUND THE MOON

Hohmann\(^{14}\) already pointed out the possibility of a sufficiently distant flyby of the Moon.

In works\(^{4,5}\) the possibility of flying around the Moon along symmetric trajectories is shown. However, it is of interest to investigate all possible planar trajectories of a close flyby of the Moon, i.e., all flyby trajectories of approach, and also to determine the accuracies of the initial data required for their realization. To obtain flyby trajectories of approach we shall solve a more general problem—to find those trajectories of approach which return to a prescribed neighborhood of the Earth, namely to the sphere of a prescribed radius \(r_k \ll a\), where \(a\) is the distance to the Moon. This problem may be called the problem of return.

The set of planar solutions of the problem under consideration is four-parametric. The set of return trajectories tangent to the geocentric circle of prescribed radius \(r_k \ll a\) is three-parametric. The set of symmetric return trajectories is two-parametric. Using the reversibility of the motion, it is not difficult to show that symmetric trajectories must intersect the straight—

Earth–Moon line (axis of symmetry) at a right angle. Therefore they are completely characterized by the distance of the point of intersection from the center of the Moon and by the velocity at this point. This circumstance was used in works \(^{4,5}\) for the numerical determination of symmetric trajectories.

Let us proceed to the analysis of the return problem formulated above.

Just as in the problem of impact, trajectories passing through the center of the Moon were nominal, so in the return problem the nominal trajectories are those passing through the center of the Earth.

Obviously, the nominal trajectories are characterized by the fact that, after leaving the sphere of action, they have a zero geocentric constant of areas \(\sigma_\kappa\).

Let us note that the small angle \(\varphi_3\) between the departure geocentric radius (Fig. 9,a) and the Earth–Moon direction may be neglected, since it turns out that taking this angle into account has no fundamental significance. Then it is obvious that return trajectories can correspond in the velocity plane (Fig. 14) to only two vectors of the departure geocentric velocity \(V_3\), parallel to the straight line \(A_2m_1\) (Figs. 24 and 26). One of them, \(V_{3\mathrm{в}}\), corresponds to ascending motion after approach, and the other, \(V_{3\mathrm{н}}\), to descending motion, whatever the motion before approach may have been. Ascending motion, in contrast to descending motion, can be a solution of the problem only when \(V_3 < V_n(a)\), where \(V_n(a)=1.44\ \text{km/sec}\) is the geocentric parabolic velocity at the distance of the Moon’s orbit.

Fig. 24. Vectors of the velocity plane for the case of return of the trajectory to the center of the Earth with positive initial sectorial velocity.

Adopting the latter condition, let us first consider approach on the ascending branch, corresponding to a positive angle \(\alpha_1\) of the initial velocity with the radius.

We see that the departure selenocentric velocities \(V'_{3\mathrm{в}}\) and \(V'_{3\mathrm{н}}\) form, with the incoming selenocentric velocity \(V'_2\), the angles \(\alpha_{\mathrm{в}}<0\) and \(\alpha_{\mathrm{н}}>0\) (Fig. 24). Consequently, the center of the Moon is passed by the corresponding trajectories clockwise, so that flybys occur. We shall call these trajectories respectively \(C_{\mathrm{в}}^{\mathrm{в}+}\) and \(C_{\mathrm{н}}^{\mathrm{в}+}\). Here the sign is \(\operatorname{sign} a_1\) (the sign of the initial sectorial velocity \(\sigma_1\)); the upper letter characterizes the type of branch before approach, and the lower one the type after approach. The solution \(C_{\mathrm{н}}^{\mathrm{в}+}\) passes much closer to the center of the Moon than \(C_{\mathrm{в}}^{\mathrm{в}+}\), since \(|\alpha_{\mathrm{н}}|\gg|\alpha_{\mathrm{в}}|\). The flight times to approach for both solutions, obviously, must be approximately the same as for impacts corresponding to the same initial velocities. During approach, the geocentric sectorial velocity

should vary from the value \(\sigma_1\) to zero. The geocentric motion after the encounter is purely radial. The solutions considered are shown schematically in Fig. 25.

In the case \(a_1<0\) (Fig. 26), as before the angle \(\alpha_{\mathrm n}<0\) and corresponds to a flyby of type \(C_{\mathrm n}^{\mathrm v-}\) (Fig. 25). However, the angle \(\alpha_{\mathrm v}>0\), so that the center of the Moon is bypassed counterclockwise, and there is no flyby. We shall call such encounter trajectories reaching ones. If they are denoted by the letter \(D\), then the latter solution is denoted \(D_{\mathrm v}^{\mathrm v-}\) (Fig. 27).

Let us finally consider an encounter on the descending branch. The corresponding velocity planes, according to § 6, are symmetric with respect to the straight line \(A_2A'_2\) to the planes shown in Figs. 24 and 26, and, analogously to the preceding case, four solutions can be obtained: for \(a_1>0\), the solutions \(D_{\mathrm v}^{\mathrm n+}\) and \(D_{\mathrm n}^{\mathrm n+}\) (Fig. 27); for \(a_1<0\), the solutions \(D_{\mathrm v}^{\mathrm n-}\) (Fig. 27) and \(C_{\mathrm n}^{\mathrm n-}\) (Fig. 25). From the analysis of Figs. 24–26 it is seen that, according to the consequences of the encounter, the flyby solutions (Fig. 25) are divided into two classes:

I. A close encounter; the type of branch after it changes to the opposite one (class \(C_{\mathrm n}^{\mathrm v}\));

II. A weak encounter; the type of branch after it does not change. This class, unlike I, is divided into two subclasses \(C_{\mathrm v}^{\mathrm v+}\) and \(C_{\mathrm n}^{\mathrm n-}\), which do not pass one into the other under continuous variation of \(a_1\) and fixed initial velocity. Analogously, the reaching solutions are also divided into two classes: I \(D_{\mathrm v}^{\mathrm n}\); II \(D_{\mathrm v}^{\mathrm v-}\) and \(D_{\mathrm n}^{\mathrm n+}\) (Fig. 27). As \(\alpha_1\to0\), the flyby and reaching nominal solutions of the second classes degenerate into trivial ones, i.e. ones not corresponding to entry into the Moon’s sphere of action (an encounter).

Fig. 25. Classes of flyby nominal trajectories. The reduction of the sectorial velocity under the action of the Moon to zero is shown.

Fig. 26. Vectors of the velocity plane corresponding to return trajectories to the center of the Earth for negative initial sectorial velocity.

Let us note that the obtained system of nominal flyby and reaching encounter trajectories is complete.

Nominal trajectories, as in the impact problem, are found by the method of iterations with respect to some initial datum, as argument, for the value of the function \(y=0\). In this case \(y=\sqrt{r_m}\operatorname{sign}\sigma_k\), where \(r_m\) is the distance of the return branch from the center of the Earth. By this method, using numerical integration, flyby trajectories of all subclasses and some impact trajectories were computed on the machine, which confirmed the correctness of the approximate method.

Fig. 27. Classes of flyby nominal trajectories. The analogy with the flyby classes is visible.

validity of the approximate method. The iterations were carried out with respect to the argument \(\lambda\) for fixed initial data \(r_1=6571\) km, \(\Delta V_1=-0.07228\) km/sec and \(a_1=\pm \dfrac{\pi}{2}\), where \(\Delta V_1\) is the excess of the initial velocity above the parabolic one.

It turned out that to the class \(C_{\mathrm{n}}^{\mathrm{v}}\) there correspond flight times \(T_{1,k}=5\div10\) days, to the class \(D_{\mathrm{v}}^{\mathrm{n}}\)—\(15\div20\) days, and to the remaining classes—intermediate ones.

Let us consider the evolution of nominal solutions as the initial velocity \(V_1\) decreases. At sufficiently high velocities there exist only solutions of the class \(C_{\mathrm{n}}^{\mathrm{v}}\). At \(\Delta V_1=0.017\) km/sec a solution \(C_{\mathrm{v}}^{\mathrm{v}+}\) appears; for \(\Delta V_1<0\)—solutions \(D_{\mathrm{n}}^{\mathrm{n}+}\), \(D_{\mathrm{v}}^{\mathrm{n}+}\), and \(C_{\mathrm{n}}^{\mathrm{n}-}\). Finally, at \(\Delta V_1=-0.017\) km/sec there appear solutions \(D_{\mathrm{v}}^{\mathrm{v}-}\) and \(D_{\mathrm{v}}^{\mathrm{n}-}\). With further decrease of the initial velocity, pairwise approach takes place, and then merging and disappearance of solutions: first the solutions \(C_{\mathrm{n}}^{\mathrm{v}+}\) and \(C_{\mathrm{v}}^{\mathrm{v}+}\) disappear, then \(D_{\mathrm{v}}^{\mathrm{n}+}\) and \(D_{\mathrm{n}}^{\mathrm{n}+}\), then \(C_{\mathrm{n}}^{\mathrm{v}-}\) and \(C_{\mathrm{n}}^{\mathrm{n}-}\). The last to disappear are the solutions \(D_{\mathrm{v}}^{\mathrm{v}-}\) and \(D_{\mathrm{v}}^{\mathrm{n}-}\). At smaller initial velocities it is no longer possible to obtain \(\sigma_k=0\), although the trajectories may still reach the sphere of action.

Solutions of the second classes at all initial velocities pass outside the disk of the Moon, while solutions of the first classes do so only at initial velocities close to the minimum ones.

Let us consider the question of the influence of the scatter of the initial data. Since for nominal trajectories, in contrast to the others, the value of the distance from the center of the Earth \(r_m\) is a quadratic, and not a linear, function—

of small errors, they are preferable to trajectories close to them in terms of the required accuracy of realization of the initial data.

As in the impact problem, in the flyby problem the determining factor is the influence of the dispersion in the initial velocity \(V_1\) and its angle with the radius \(\alpha_1\).

The influence of dispersion in the problem under consideration depends not only on the character of the passage of the trajectory relative to the Earth, but also on the distance \(\rho_m\) of the trajectory from the center of the Moon. As \(\rho_m\) decreases, the influence of errors grows rapidly. Therefore, for the first classes it is stronger than for the second; moreover, errors in the direction of decreasing the distance \(\rho_m\) have a stronger effect than those in the direction of increasing it. For example, for a flyby of type \(C^{\mathrm{B}+}_{\mathrm{v}}\) with

\[ \alpha_1=\frac{\pi}{2} \]

and \(\Delta V_1=-0.07228\), we have \(\rho_m=12900\) km. For errors \(\delta V_1=-1\) m/sec and \(\delta\alpha_1=-0.01\) rad., and also \(\delta V_1=+10\) m/sec and \(\delta\alpha_1=+0.1\) rad., the trajectories still return to the Earth. However, for errors \(\delta V_1=-10\) m/sec and \(\delta\alpha_1=-0.1\) rad., the trajectories either collide with the Moon or pass around the Moon clockwise, so that there is no flyby. The realization of the considered and closer flybys with return to the Earth without correction on the passive segment is hardly possible.

As the minimum distance of the trajectory from the Moon increases, the accuracy requirements rapidly decrease.

For trivial solutions (passing from the Moon at distances \(\rho_m\sim\rho_*\)), i.e., sufficiently distant ones, of the Hohmann flyby type, the influence of dispersion is comparatively small, and the accuracies required for return to the Earth prove to be no higher than for impact on the Moon at the same initial velocities. The realization of such a flyby without correction on the passive segment is apparently technically feasible.

§ 9. SPECIAL PROBLEM OF A FLYBY OF THE MOON

By the special problem of a flyby of the Moon is meant the problem of seeking flybys in which the projectile returns into the Earth’s atmosphere at a shallow angle. Such flybys are of greatest interest, since they correspond to the easiest conditions of atmospheric entry. Obviously, for these flybys the minimum geocentric radius on the return segment \(r_m\) must be equal to the radius of the upper layers of the atmosphere \(R\).

It can be shown that, for initial velocities \(V_1<V_{\mathrm{p}}+0.5\) km/sec, trajectories approaching \(r_m=R\) at the point of departure from the sphere of influence possess the property

\[ |V_{3\tau}|\simeq \frac{R}{r_3} V_{\mathrm{p}}, \]

where \(V_{3\tau}\) is the component, normal to the exit radius \(r_3\), of the exit geocentric velocity \(V_3\) (Fig. 9), and \(V_{\mathrm{p}}\) is the parabolic velocity for the initial altitude. This property makes it possible, as in the preceding problem, to find all classes of solutions by means of the velocity diagram.

It turns out that to each pair of subclasses of solutions of the preceding problem there correspond, in the problem under consideration, two classes of solutions passing around the Moon in the same direction as the solutions of the preceding problem, but the Earth in different directions. Here, however, the classes need not be divided into subclasses, since their solutions vary continuously when the sign of the angle of the initial velocity with the radius \(\alpha_1\) passes through zero. Therefore they may be denoted without indicating the sign at the top: \(C^{\mathrm{B}}_{\mathrm{н}+}\) and \(C^{\mathrm{B}}_{\mathrm{н}-}\) for the first flyby class, and \(C^{\mathrm{B}}_{\mathrm{в}-}\) and \(C^{\mathrm{н}}_{\mathrm{н}+}\) for the second flyby class. Similarly, we have the impact types of solutions \(D^{\mathrm{н}}_{\mathrm{в}+}\), \(D^{\mathrm{н}}_{\mathrm{в}-}\), \(D^{\mathrm{в}}_{\mathrm{в}+}\), and \(D^{\mathrm{н}}_{\mathrm{н}-}\). The sign below indicates the direction of the Earth flyby.

When the angle $\alpha_1$ is a right angle, the second pairs of solutions may be expressed in trivial ones (not corresponding to an encounter with the Moon). These solutions are ellipses with a focus at the center of the Earth, touching the upper layers of the atmosphere at perigee. For the mean value $\alpha_1=0$, the flyby classes are shown in Fig. 28. The geocentric sectorial velocity at the beginning is zero, and the solutions begin radially. Then the perturbations of the Moon increase the magnitude of the sectorial velocity so much that the return branch touches the upper layers of the Earth’s atmosphere.

Fig. 28. Classes of trajectories returning to the Earth’s atmosphere on a half-turn.

The evolution of the solutions of the problem under consideration with a change in the initial velocity is traced analogously to the evolution of the solutions of the corresponding classes of the preceding problem.

Let us note that, in form, all encounter trajectories returning to the Earth are intermediate between the corresponding solutions of the problem under consideration with $\alpha_1=+\dfrac{\pi}{2}$ and $\alpha_1=-\dfrac{\pi}{2}$ and with the same initial radius and velocity. Let us also note that the symmetric solutions of the problem of a special return, as may be concluded from Fig. 28, can be contained only in the classes $C_{\text{н}+}^{\text{в}}, D_{\text{в}+}^{\text{н}}$ (corresponding to $\alpha_1=+\dfrac{\pi}{2}$) and $C_{\text{н}-}^{\text{в}}, D_{\text{в}-}^{\text{н}}$ (corresponding to $\alpha_1=-\dfrac{\pi}{2}$). By independently varying the magnitude of the initial radius and the initial velocity, we obtain, according to § 8, the set of all symmetric encounter trajectories (cf. 5).

The method for obtaining the solutions of the preceding problem is applicable to this problem as well, only the iterations are carried out not to the value $y=r_m=0$, but to the values $y=\pm\sqrt{R}$. However, the relative smallness of the thickness of the atmosphere leads to much more stringent requirements on the accuracy of the initial data than in the preceding problem.

For example, for a symmetric flyby with an excess of the initial velocity over the parabolic one $\Delta V_1=-0.083773$, with the initial angle of the velocity with the radius $\alpha_1=\dfrac{\pi}{2}$ and a flight time of $823\,600$ sec, the distance from the Moon turned out to be $\rho_m=27\,000$ km. Even such small errors in the initial data as $\delta V_1=0.2$ m/sec and $\delta\alpha_1=5\cdot10^{-3}$ rad. cause, upon return, changes in altitude of 160 and 190 km respectively, i.e., they are inadmissible (changes in altitude must not exceed tens of kilometers).

This example corresponds to a comparatively large value of $\rho_m$. Decreasing $\rho_m$ just as rapidly increases the influence of errors as in the preceding problem.

Since the required accuracies in realizing the initial data are extremely—

randomly large, the practical realization of a flyby of the Moon along a close-approach trajectory with a shallow entry into the Earth’s atmosphere without correction of the trajectory on the passive segment appears hardly realistic.

§ 10. THE PROBLEM OF A PERIODIC FLYBY OF THE MOON

In the present section we consider the question of whether it is possible to launch a projectile in the plane of the lunar orbit in such a way that, periodically flying around the Moon, it would return to the Earth and regularly provide information about the Moon, for example, about its far side. In this connection not just any periodic solutions are of interest, but solutions passing close to the Moon. We shall evidently solve the question posed if we find all plane periodic close-approach trajectories. In the literature such a problem, apparently, has not arisen.

We shall agree to regard as the terminal point of a period the point nearest to the center of the Earth. It is easy to see that the required trajectories must be closed not in geocentric coordinates \(m_1\xi_1\eta_1\), but in the coordinates \(0\xi\eta\), rotating together with the Earth–Moon line (see Fig. 2). Nevertheless, at the terminal point of a period the final geocentric values of the velocity \(V_k\), the radius \(r_k\), and the angle between them \(\alpha_k = \pm \dfrac{\pi}{2}\), respectively, must coincide with the initial values \(V_1\), \(r_1\), and \(\alpha_1\).

Using this, and also the reversibility of the motion, one can prove that the required solutions have an axis of symmetry in the coordinate systems \(0\xi\eta\) and \(m_1\xi_1\eta_1\). Therefore they can belong only to the symmetric solutions of the preceding problem, i.e. to solutions of the types \(C^{\mathrm{v}}_{\mathrm{n}+}\), \(D^{\mathrm{n}}_{\mathrm{v}+}\) with \(\alpha_1 = +\dfrac{\pi}{2}\) and \(C^{\mathrm{v}}_{\mathrm{n}-}\), \(D^{\mathrm{n}}_{\mathrm{v}-}\) with \(\alpha_1 = -\dfrac{\pi}{2}\). It can also be shown that the periodic solutions must belong to a one-parameter family with parameter \(r_1\), and that they begin on the \(\xi\)-axis, i.e. correspond to \(\lambda = 0\) or \(\lambda = \pi\), where \(\lambda\) is the angle of the initial radius with the Moon–Earth direction.

If \(\Phi_1\) is the angle between the geocentric radii of the initial point and of the point nearest to the Moon, then the condition that the angle \(\lambda\) for the initial point and the terminal point in the system \(0\xi\eta\) coincide, as is not difficult to understand, reduces to a condition imposed on the angular displacements of the projectile and of the Moon during the period \(T_{1,k}\), and has the form

\[ 2\Phi_1 - \omega T_{1,k} = \pm 2k\pi,\qquad k = 0, 1, 2,\ldots \tag{10.1} \]

Since, owing to the smallness of the sphere of action, \(\Phi_1 \cong \Phi\), where \(\Phi\) is the angle between the geocentric radii \(r = r_1\) and \(r = a\) of the conic section to which the initial segment of the trajectory belongs, we have approximately:

\[ 2\Phi - \omega T_{1,k} = \pm 2k\pi,\qquad k = 0, 1, 2,\ldots \tag{10.2} \]

With the aid of condition (10.2) one can find all periodic close-approach trajectories. It turns out that a periodic flyby of type \(C^{\mathrm{v}}_{\mathrm{n}+}\) is impossible, while to type \(C^{\mathrm{v}}_{\mathrm{n}-}\) there belongs only one family of periodic solutions, corresponding to \(2\Phi - \omega T_{1,k} = -2\pi\). The solutions of this family pass around the Earth and the Moon clockwise, i.e. correspond to \(\Phi < 0\) (Fig. 29).

Circumlunar periodic solutions are possible of both types \(D^{\mathrm{n}}_{\mathrm{B}+}\) and \(D^{\mathrm{n}}_{\mathrm{B}-}\), with type \(D^{\mathrm{n}}_{\mathrm{B}+}\) including two countable sets of families satisfying the condition \(2\Phi-\omega T_{1,k}=-2k\pi\), and respectively \(\lambda=\pi,\ k\geqslant 0\) and \(\lambda=0,\ k\geqslant 1\); while type \(D^{\mathrm{n}}_{\mathrm{B}-}\) includes the analogous two sets satisfying the condition \(2\Phi-\omega T_{1,k}=-2l\pi\), and respectively \(\lambda=\pi,\ l\geqslant 2\) and \(\lambda=0,\ l\geqslant 3\).

Fig. 29. Trajectories of periodic flyby of the Earth and the Moon. Trajectories passing above the surface of the Moon are separated from the Earth by a distance of 100,000 km or more.

N	\(r_m\) (km)	\(V_0\) \(\left(\dfrac{\text{km}}{\text{sec}}\right)\)	\(\rho_m\) (km)
I	6 571	11,1242	150
II	42 203	4,4008	825
III	82 824	3,1833	1500
IV	116 371	2,7410	2000

Solutions of type \(D^{\mathrm{n}}_{\mathrm{B}+}\) go around the Earth and the Moon counterclockwise, corresponding to \(\Phi>0\). In this case, during a half-period the Moon completes somewhat more than \(0.5;\ 1.5;\ldots,\ k+0.5;\ldots\) revolutions for the first countable set of families (with \(\lambda=\pi\)), and somewhat more than \(1,\ 2,\ldots,\ k,\ldots\) revolutions for the second countable set (with \(\lambda=0\)).

Solutions of the type under consideration with \(\lambda=\pi\) for \(k=0,1\) are shown schematically in Figs. 30 and 31, and the solution with \(\lambda=0\) for \(k=1\) is shown in Fig. 32.

Solutions of type \(D^{\mathrm{n}}_{\mathrm{B}-}\) go around the Earth clockwise, and the Moon counterclockwise, and correspond to \(\Phi<0\). In this case, during a half-period the Moon completes sever-

to less than \(0.5;\ 1.5;\ldots;\ l-2+\dfrac{1}{2};\ldots\) revolutions for a countable set of families with \(\lambda=\pi\), and to somewhat less than \(1;\ 2;\ldots,\ l-2,\ldots\) revolutions for a countable set with \(\lambda=0\). The solutions under consideration with \(\lambda=\pi\) for \(l=2,3\) are shown schematically in Figs. 33 and 34, and those with \(\lambda=0\) for \(l=3\) in Fig. 35.

Fig. 30. The simplest periodic flight trajectory of subclass \(D_+\) (direct revolution about the Earth, \(k=0,\lambda=\pi\)); the period is somewhat more than half a month.

Fig. 31. Flight periodic trajectory of subclass \(D'_{1+}\) \((k=1,\lambda=\pi)\). The period is somewhat more than \(1.5\) months.

Fig. 32. Flight periodic trajectory of subclass \(D'_{1+}\) \((k=1,\lambda=\pi)\). The period is somewhat more than a month.

Fig. 33. The simplest periodic flight trajectory of subclass \(D_-\) (retrograde revolution about the Earth, \(l=2,\lambda=\pi\)). The period is somewhat less than half a month.

Between solutions of the types \(D^{\mathrm n}_{\mathrm{в}+}\) and \(D^{\mathrm n}_{\mathrm{в}-}\) with identical \(\lambda\) one can establish a one-to-one correspondence by putting \(l=k+2\). Then it is not difficult to see that, as the parameter \(r_1\) decreases to zero, the corresponding solutions tend to one and the same solution, i.e., to coincidence. These limiting solutions for \(\lambda=\pi\) and \(k=0,1\) (i.e., \(l=2,3\)) are shown in Figs. 36 and 37, and for \(\lambda=0\) and \(k=1\) \((l=3)\) in Fig. 38. The limiting solutions correspond to angular displacements of the Moon over a half-period exactly equal to \(0.5;\ 1.5;\ldots;\ k+\dfrac{1}{2};\ldots\) and \(1,\ 2,\ 3,\ldots,\ k,\ldots\) revolutions, respectively.

Fig. 34. Pre-impact periodic trajectory of subclass \(D_{1-}\) (\(l=3,\ \lambda=\pi\)). The period is somewhat less than 1.5 months.

Fig. 35. Pre-impact periodic trajectory of subclass \(D'_{1-}\) (\(l=3,\ \lambda=0\)). The period is somewhat less than a month.

Fig. 36. Simplest periodic post-impact trajectory of ejection of class \(D\), separating the subclasses \(D_{+}\) and \(D_{-}\) (Figs. 30 and 33). Period 0.5 month.

Fig. 37. Periodic post-impact trajectory of ejection of class \(D_1\), separating the subclasses \(D_{1+}\) and \(D_{1-}\) (Figs. 31 and 34). Period 1.5 months.

Fig. 38. Periodic post-impact trajectory of ejection of class \(D'\), separating the subclasses \(D'_{1+}\) and \(D'_{1-}\) (Figs. 32 and 35). Period 1 month.

Thus, in reality, there are not four but two countable sets of flyby periodic families, corresponding to \(\lambda=0\) and \(\lambda=\pi\). We shall denote the family corresponding to \(k=0\) and \(\lambda=\pi\) by the letter \(D\), and the families corresponding to \(k>0\) and \(\lambda=\pi,\ 0\) by the letters \(D_k\) and \(D'_k\), respectively. These notations are also given in the captions to Figs. 30–38.

Comparing the results obtained with the known results of the work of Strömgren and the Copenhagen school1, obtained for the case of equality of the masses of the attracting bodies \(m_1=m_2\), one may observe that the flyby family (we denote it by the letter \(D\)) corresponds to Strömgren’s class “\(m\),” while to the families \(D,\ D_k\) there corresponds his class “\(c\).” The families \(D'_k\) have no analogues in Strömgren, and, conversely, he gives periodic orbits (the class “\(k\),” found by Lous2) that have no analogues in our case, although for \(m_1=m_2\) they are trajectories of close approach. Since the system \(C,\ D,\ D_k,\ D'_k\) of periodic close-approach trajectories, according to the method by which it is obtained, is complete for the mass ratio of the Earth and Moon

\[ \frac{m_1}{m_2}=81.45, \]

it follows that for \(m_1=m_2\) there may exist periodic solutions that disappear when

\[ \frac{m_1}{m_2} \]

is increased to 81.45.

It should be noted that the trajectories \(D_k,\ D'_k\) are not the simplest ones, i.e., they make no more than one revolution in the system \(O\xi\eta\) during a period. In the case of the Earth and Moon the families of the simplest periodic close-approach trajectories are only \(C\) and \(D\).

Fig. 39. Escape to infinity from an almost periodic flyby orbit. Orbits \(I\) and \(III\) are located between the images. \(H=76253\ \text{km}\), \(\alpha_1=-\dfrac{\pi}{2}\), \(\Delta V_0=0.085058\ \text{km/sec}\), \(\delta V_0<0.001\ \text{m/sec}\), \(I-I'\) — first revolution, \(III'\) — end of the third revolution, \(IV\) — fourth revolution.

Let us consider the family \(C\) as the only flyby one. It turns out that its solution concerning the atmosphere passes inside the Moon (Fig. 29), while the solutions passing over the surface of the Moon are more than \(94\,800\ \text{km}\) from the center of the Earth. All the solutions in Fig. 29 were found by a method analogous to Strömgren’s method, in the system \(O\xi\eta\), and \(V_0\) is the velocity in this system. These solutions proved to be unstable. For example, the solution with distances from the center of the Earth \(r_m=82\,800\ \text{km}\) and from the center of the Moon \(\rho_m=1500\ \text{km}\), with an error \(\delta V_0=0.001\ \text{m/sec}\), escapes to infinity already from the fourth revolution (Fig. 39). Thus a practically periodic flyby of the Moon at close distance is impossible.

Let us note that if, for a periodic solution, in the Jacobi integral (2.1) one had \(h<h_2\), then the solution, according to § 2, could not go off to infinity. However, in order that it might pass from the Earth to the Moon, it must be that \(h>h_1\). For close-approach trajectories, including periodic ones, according to § 3, \(h>h_2\). Consequently, periodic solutions satisfying the condition \(h_1<h<h_2\), if they exist, must pass sufficiently far from the Earth. Since the difference \(h_2-h_1\) is negligible, such solutions, even if they exist, cannot

be of practical significance—not only because of the scatter of the initial data, which makes it difficult to fall within the range \(h_1<h<h_2\), but also because the unaccounted-for perturbations of the Sun and others can easily take \(h\) out of the range \(h_1<h<h_2\).

§ 11. THE PROBLEM OF ACCELERATING OR BRAKING FLIGHT WITH THE AID OF THE MOON. CLASSIFICATION OF TRAJECTORIES

Let us consider the problem of the maximum acceleration of a space rocket with the aid of the Moon without using the thrust of the engine. From the velocity diagram (see Fig. 14) it is clear that the maximum outgoing geocentric velocity after the encounter is \(U_3=U+V_{\text{л}}\), where \(U\) is the magnitude of the outgoing selenocentric velocity, and \(V_{\text{л}}\) is the velocity of the Moon; however, it can be shown that the corresponding encounter trajectory, for any initial data, must pass inside the Moon.

It turns out that the practically realizable trajectory corresponding to the greatest acceleration \(\Delta V_{2,3}=V_3-V_2\) (where \(V_2\) is the incoming geocentric velocity) must pass at the surface of the Moon, going around it clockwise in the encounter on the descending branch (class \(\widetilde{C}^{\text{н}}\)) and counterclockwise in the encounter on the ascending branch (class \(\widetilde{D}^{\text{в}}\)). Class \(\widetilde{D}^{\text{в}}\) corresponds to those shown by the dashed vectors \(V_3\) in Figs. 14 and 17. In seeking accelerating solutions, the iterations were carried out for the function \(y=\sqrt{\rho_m}\), with the sign of the direction of going around the Moon set equal to \(+\sqrt{R_{\text{л}}}\) for solutions \(\widetilde{D}^{\text{в}}\) and to \(-\sqrt{R_{\text{л}}}\) for solutions \(\widetilde{C}^{\text{н}}\). Here \(\rho_m\) is the distance of the trajectory from the center of the Moon, \(R_{\text{л}}=1736.7\) km is the radius of the Moon. The limiting

Fig. 40. Classes of trajectories of maximum acceleration with the aid of the Moon. The analogy with the classes of impact trajectories is visible (Fig. 18).

solutions of each of the classes were found, i.e., the solutions with \(\alpha_1=+\dfrac{\pi}{2}\) and the solutions with \(\alpha_1=-\dfrac{\pi}{2}\). They are represented schematically in Fig. 40 in geocentric \((\xi_1,\eta_1)\) and rotating \((\xi,\eta)\) coordinates. According to the sign of the angle \(\alpha_1\), i.e.

according to the direction of revolution about the Earth, each class can be divided into two subclasses

\[ \widetilde{D}^{\,\mathrm{в}} \text{ into } \widetilde{D}^{\,\mathrm{в}+} \text{ and } \widetilde{D}^{\,\mathrm{в}-};\quad \widetilde{C}^{\,\mathrm{н}} \text{ into } \widetilde{C}^{\,\mathrm{н}+} \text{ and } \widetilde{C}^{\,\mathrm{н}-}. \]

Accelerating solutions pass around the Moon in such a way as to leave the sphere of action in a direction possibly closer to the direction of the Moon’s velocity.

The resulting outgoing geocentric velocity \(V_3\) after acceleration is always hyperbolic, independently of the initial velocity \(V_1\), so that after its encounter with the Moon the projectile escapes to infinity. However, the magnitude of the acceleration \(\Delta V_{2,3}\) depends on \(V_1\); it is maximal (of the order of \(1.5\ \text{km/sec}\)) for velocities \(V_1\) close to the minimum, and with increasing \(V_1\) decreases monotonically (to zero as \(V_1 \to \infty\)). The maximum velocity due to the monthly rotation of the Moon can be obtained in any direction in the plane of the Moon’s orbit. Since this plane makes small angles with the planes of the planets’ orbits, acceleration without expenditure of fuel can be used for interplanetary flight.

Turning to consideration of the influence of scatter in the initial data on accelerating solutions, let us note that it is greater than for impact solutions, since the miss \(\delta \rho_m\) is not a quadratic but a linear function of the errors. For example, for acceleration \(\widetilde{D}^{\,\mathrm{в}+}\) with an initial velocity less than parabolic by \(0.0723\ \text{km/sec}\) and \(\alpha_1 = 90^\circ\), errors \(\delta V_1 = 1\ \text{km/sec}\) and \(\delta \alpha_1 = 1^\circ\) cause, respectively, changes \((\delta \rho_m)_v = 120\ \text{km}\) and \((\delta \rho_m)_\alpha = 100\ \text{km}\).

Because of the danger of collision with the Moon owing to errors in the initial data, the computed trajectory must be raised above the surface of the Moon. Since raising the trajectory above the Moon reduces the gain in velocity, accelerations sufficiently close to the maximum can hardly be carried out without correcting the trajectory before its encounter with the Moon.

Let us note that solutions with \(\alpha_1 = +\dfrac{\pi}{2}\) are more advantageous than other accelerating solutions in that they make it possible, at launch, to use the component of the Earth’s diurnal rotation lying in the plane of the Moon’s orbit (as, incidentally, do the solutions of the problems considered earlier that correspond to the angle

\[ \alpha_1 = +\frac{\pi}{2} \]
).

It is easy to see that, because of the reversibility of the motion after replacing \(t\) by \(-t\), the solution of the problem of any acceleration always gives a solution of the problem of the same braking with the aid of the Moon, for motion along a trajectory mirror-symmetric to the accelerating one with respect to the \(\xi\)-axis. In this case the trajectory obtained from the most advantageous accelerating one proves to be the most advantageous in the sense of braking, since, on entering the atmosphere, it uses its diurnal rotation together with the Earth, reducing the velocity of the projectile relative to the Earth’s surface. Trajectories of maximum braking may be used, for example, in the return of a rocket from interplanetary flight.

As may be concluded from the velocity plan (Fig. 14), besides impact trajectories and trajectories returning to the Earth after encounter with the Moon, and also trajectories corresponding to one or another acceleration of the rocket as a result of the encounter, there may also exist trajectories corresponding to a greater or smaller deceleration of it (relative to the Earth), and no other encounter trajectories exist. Thus, the plan

velocities makes it possible to give a complete classification of planar flyby trajectories.

The deceleration trajectories in Fig. 14 correspond to the outgoing geocentric velocities which, with the velocity \(-V_{\text{л}}\), form an angle smaller than the incoming velocities corresponding to trajectories of the types \(C^{\text{в}}_{\text{н}+}\) and \(D^{\text{в}}_{\text{в}+}\); moreover, the maximum deceleration corresponds to the trajectory leaving the sphere of action in the direction opposite to the velocity of the Moon, independently of the entry conditions.

Let us note that, as is seen from Fig. 14, the trajectory of maximum deceleration corresponds to the maximum geocentric area constant \(\sigma_k\), and therefore this trajectory passes at the greatest distance from the center of the Earth in comparison with nearby trajectories.

With the aid of the velocity plane one can trace the evolution of the solutions as the angle \(\lambda\) or another initial datum is varied within the range corresponding to a flyby; however, we shall not dwell on this here.

Summarizing the problems considered above, one may draw the following conclusions. The high requirements on the accuracy of the initial data lead to the fact that, in order to realize a close flyby of the Moon, and especially a close flyby with subsequent entry into the Earth’s atmosphere, correction of the passive segment appears necessary. For impact trajectories to the Moon and trajectories of a distant flyby of the Moon, the accuracy requirements are sufficiently moderate; as a result, flight along these trajectories, apparently, can be carried out without correction on the passive segment, as soon as rockets attain velocities of the order of parabolic velocity.

REMARKS

1. The results obtained make it possible, for each concrete objective of a flight to the Moon, to choose planar trajectories of the most suitable forms and properties. When such a choice has been made, the method described can be used to find the initial data and the required accuracies of their realization. These initial data may be taken as a first approximation for an exact determination of the computed initial data with allowance for perturbations by the Sun and other secondary factors.

2. The corresponding exact equations should be integrated numerically, on high-speed computing machines. In doing so, for the numerical determination of the computed trajectory, the iterations may be carried out with the same functions and arguments and by the same methods that were used for finding the nominal trajectories in the problems set forth.

3. The solutions of concrete problems obtained in the present work should be of practical interest even in the case where their implementation is impossible without correcting the trajectory on the passive segment (with the aid of an auxiliary engine), since it is evident that they require minimal correction in comparison with other solutions.

4. It can be shown that the approximate method applied is valid not only for the mass ratio of the Moon and the Earth \((m_2:m_1=1:81.45)\), but also for arbitrarily small \(m_2:m_1\). Moreover, as \(m_2:m_1\) decreases to zero, its accuracy increases without bound. Consequently, it is applicable to the motion of a projectile in the gravitational field of the Sun \(m_1\) and a single outer planet \(m_2\). Flyby trajectories will be those which pass through the sphere of action of \(m_2\) with respect to \(m_1\).

Since the orbits of the planets are inclined to the Earth’s orbit at small angles, the method described is applicable to the investigation of interplanetary-flight trajectories. The principal part of such a trajectory will pass outside the Earth’s sphere of action with respect to the Sun. Initial

these data should be referred to the boundary of the Earth's sphere of action. Then the role of the initial radius \(r_1\) will be played by the initial geocentric radius or, approximately, by the radius of the Earth's orbit, and the role of the radius of the Moon's orbit \(a\) by the radius of the planet's orbit.

The influence of planets other than \(m_2\) may be disregarded, since the entry of the trajectory of approach to \(m_2\) into their spheres of action is unlikely.

Obviously, for trajectories of approach, the plane problems of hitting the planet \(m_2\), of flying around it with return to the Earth's orbit, of a special flyby (with tangential return to the Earth's orbit), and of acceleration with the aid of \(m_2\), for example for the purpose of flight to the orbit of a more distant planet, are meaningful.

If the ratio \(r_1 : a\) is small, as, for example, for Neptune, not only the method but also many results are applicable. In particular, the theoretically possible classes and forms of trajectories of flight to Neptune from the Earth's orbit will evidently be the same as for trajectories of flight from the Earth's surface or from its satellite to the Moon. If, however, the ratio \(r_1 : a\) is not small (as, for example, for Mars), a treatment generalizing that carried out in the present work is necessary.

1. The approximate method set forth in the paper is generalized without particular difficulty to spatial motion. In this case the characteristic segments of which the trajectory consists remain the same, although they are located in different planes. The regularities obtained on each of the segments are preserved. Only the formulas for recalculation at the points where the trajectory intersects the boundary of the sphere of action change.

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