### 3. TRAJECTORIES AND ORBITS

A. FUNDAMENTAL TYPES OF TRAJECTORIES AND ORBITS

The terms trajectory and orbit both refer to the path of a body in space. Trajectory is commonly used in connection with projectiles and is often associated with paths of limited extent, i. e., paths having clearly identified initial and end points. Orbit is commonly used in connection with natural bodies (planets, moons, etc.) and is often associated with paths that are more or less indefinitely extended or of a repetitive character, like the orbit of the Moon around the Earth. In discussions of space flight, both terms are used, with the choice usually dependent upon the nature of the flight path. Thus we speak of trajectories from the Earth to Moon, and of satellite orbits around the Earth.

The basic types of paths in space are determined by the gravitational-attraction properties of concentrated masses of material and the laws of motion discovered by Newton.

Virtually all major members of the solar system are approximately spherical in shape; and a spherical body will produce a force of attraction precisely like that of a single mass point located at the center of the body. Therefore, the fundamental problem is that of motion under the gravitational influence of a mass concentrated at a point.

Two general and several special types of paths are possible under the gravitational influence of a point mass, The two main types are illustrated in figure 1.

Figure 1a is an elliptical orbit-the familiar artificial Earth satellite kind of orbit. Figure 1b is a hyperbolic orbit-the kind that will characterize the start of an interplanetary flight. The elliptical orbit is closed on itself and would be traversed repetitively. The hyperbolic orbit is open, extending to infinity. Separating these two cases is a special one-the parabolic orbit-similar in general appearance to the hyperbolic. The parabolic orbit is the Borderline case between open and closed orbits and therefore identifies the border line condition between space vehicles that are tied to paths (elliptical) in the general vicinity of their parent planet and those that can take up paths (hyperbolic) extending to regions remote from their parent planet. For any of these orbits the vehicle's velocity will be greatest at the point of nearest approach to the parent body, and it will be progressively less at more remote points.

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B. ESCAPE VELOCITY

The type of path that will be taken up by an unpowered space vehicle starting at a given location will depend upon its velocity. It will take up an open-ended path if its velocity equals or exceeds escape velocity; escape velocity is, by definition, that velocity required at a given location to establish a parabolic orbit. Velocities greater than escape velocity result in hyperbolic orbits. Lower velocities result in closed elliptical orbits-the vehicle is tied to the neighborhood of the planet.

Since it essentially separates "local" from "long distance" flights, escape velocity is clearly a primary astronautical parameter. The exact value of this velocity is dependent upon two factors: (a) The mass of the parent planet and (b) the distance from the center of the planet to the space vehicle. Escape velocity increases as the square root of the planet's mass, and decreases as the square root of the distance from the planet's center. The speeds required for escape directly from the surfaces of various bodies of interest are listed in table 1. These escape velocity requirements are a measure of the difficulty of departure from these bodies.

ASTRONAUTICS AND ITS APPLICATIONS 21

TABLE 1.-Surface escape velocity
Planet Feet per
second
Mercury
13.600
Venus
33,600
Earth
36,700
Moon
7,800
Mars
16,700
Asteroid Eros
~50
Jupiter
197,000

The projection speed required to escape directly from the Earth's surface is about 36,700 feet per second. If a vehicle takes up unpowered flight (end of rocket propulsion) at an altitude of, say, 300 miles, it requires the somewhat lesser speed of 35,400 feet per second to escape into interplanetary space. This reduction in required velocity has, of course, been obtained at the expense of the energy expended in lifting the vehicle to an altitude of 300 miles.

C. SATELLITE ORBITS

The elliptical orbits generated by velocities below escape velocity are the type followed by artificial satellites, as well as by all the planets and moons of the solar system.

The period of the satellite-the time required to make one full circuit-is dependent upon the mass of the parent body and the distance across the orbit at its greatest width (the length of the major axis). The period is less if the parent body is more massive-the Earth's Moon moves more slowly than similarly placed moons of Jupiter. The period gets longer as the length of the major axis increases-the period of the Moon, with a major axis of about 500,000 miles, is much longer than those of the first artificial satellites, with major axes of about 9,000 miles.

The velocity required to establish a satellite at an altitude of a few hundred miles above the Earth is about 25,000 feet per second. This required orbital velocity is less at greater altitudes. At the distance of the Moon it is only about 3,300 feet per second.

D. LUNAR FLIGHT

The gravitational attraction of the Moon affords some assistance to a vehicle for an Earth-Moon flight. However, the Moon is so far removed that this assistance is only enough to reduce the required launching velocity slightly below escape velocity.

E. INTERPLANETARY FLIGHT

To execute a flight to one of the other planets, a vehicle must first escape from the Earth. Achievement of escape velocity, however, is only part of the problem; other factors must be considered, particularly the Sun's gravitational field and the motion of the Earth about the Sun.

Before launching, the vehicle is at the Earth's distance from the Sun, moving with the Earth's speed around the sun-about 100,000 feet per second. Launching at greater than Earth escape velocity

22 ASTRONAUTICS AND ITS APPLICATIONS

results in the vehicle taking up an independent orbit around the Sun at a velocity somewhat different from that of the Earth. If it is fired in the same direction as the Earth's orbital motion, it will have an independent velocity around the Sun greater than that of the Earth. It will then take up an orbit like A, figure 2 which moves farther from the Sun than the Earth's orbit; the vehicle could, if properly launched, reach the outer planets Mars, Jupiter, and so forth, The minimum launch velocities required to reach these planets are given in table 2.

Fig. 2 - Interplanetary trajectories

If the vehicle is launched "backward," or against the Earth's velocity, it will assume an independent velocity less than that of the Earth and move on an orbit like B, figure 2, so that it could reach the inner planets Venus and Mercury.

To reach the more distant portions of the solar system requires that the vehicle take up a velocity, relative to the Sun, that is considerably greater than that of the Earth. A large launch velocity is required to produce this excess (after a good deal of it has been absorbed by the Earth's gravitational field). On the other hand, to travel in close to the Sun requires that the vehicle take up a velocity, relative to the Sun, that is considerably less than that of the Earth. A large launch velocity is this time required to cancel out the component of vehicle velocity due to the Earth's motion, and again much of the launch velocity is absorbed by the Earth's gravitational field. Thus, as seen from table 2, it is almost as hard to propel a vehicle in to Mercury as it is to propel it out to Jupiter.

ASTRONAUTICS AND ITS APPLICATIONS 23

Planet Minimum
Launching
velocity
(feet per
second)
Transit time
Mercury
44,000 110 days
Venus
38,000 150 days
Mars
38,000 260 days
Jupiter
46,000 2.7 years
Saturn
49,000 6 years
Uranus
51,000 16 years
Neptune
52,000 31 years
Pluto
53,000 40 years

The velocities in table 2 are minimum requirements, and lead to flight times as shown. Higher velocities will reduce flight times.

F. ESCAPE FROM THE SOLAR SYSTEM

If a vehicle reaches escape velocity with respect to the sun it will leave the solar system entirely and take up a trajectory in interstellar space. Starting from the surface of the Earth, a launch velocity of about 54,000 feet per second will lead to escape from the solar system The course of the vehicle will be a parabola, with the Sun at its focus; until eons later lt is deflected by some star or other body.

The flight capabilities that become available as the total velocity potential of a ballistic rocket vehicle increases are illustrated in summary form in figure 3.

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Fig. 3 - Velocity requirements for ballistic missile and space flight

ASTRONAUTICS AND ITS APPLICATIONS 25

G. POWERED TRAJECTORIES IN SPACE

Once a vehicle is in space, moving at high velocity, say in a satellite orbit, it requires no further propulsion to stay aloft. Its flight path can, however, be very appreciably influenced and great increases in velocity imparted by very small forces acting over long periods of time. The fact that useful results can be derived from small thrusts in space-thrusts that would be entirely insignificant on the earth- leads to interest in unique propulsion systems based on electrical accelerator principles. One kind of application of particular interest involves use of heavy conventional propulsion systems to develop orbital velocity (say, 25,000 feet per second) and then to build up the remaining 12,000 feet per second to reach escape velocity by a low-thrust electrical system.1 2

H. VELOCITIES NEAR THAT OF LIGHT

As the velocity of a space vehicle nears that of light (not likely to be achieved in the foreseeable future), the effects of relativity theory enter into the situation. Of particular interest is the so-called "time dilatation" effect predicted by this theory-and supported by experimental evidence in the physics of high-speed particles.

Briefly, the predicted effect is as follows: Consider 2 men, A and B, of identical age, say 20 years old. A will remain at home on the Earth, and B will undertake a voyage in space at a speed very near that of light and eventually return to Earth. The total duration of the voyage will be different, as measured by the two men, the exact amount of the difference depending upon how close B's vehicle approached the speed of light.

As an example, suppose B took a round trip to the vicinity of a nearby star at a speed very near that of light (about 186,000 miles per second). It would appear to A that the trip took, say, 45 years- he would be 65 years old when his friend returned. To B, however, the trip might appear to take about 10 years, including a year or so for acceleration to flight speed and deceleration for the return landing-he would be 30 when he returned.

Different values of vehicle speed will lead to widely different time disparities. By approaching ever closer to the speed of light, B could take more extended trips that would last millions of years in earthtime, but still appear to him to take only a few years,

1 Boden, R. H., The Ion Rocket Engine, North American Aviation, Rocketdyne Division, Rept No. R-645, August 26, 1957.

2 Stuhlinger, E., Space Travel With Electric Propulsion Systems, Army Ballistic Missile Agency, November 11, 1958.

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Achievement of near-light velocities would require stupendous amounts of propulsion energy-nothing less than complete conversion of matter into usable energy will do,3-5

In addition to the fact that no presently foreseeable propulsion scheme will deliver the required quantities of energy there are also problems of a very severe and uncertain nature concerning environment, The traveler at speeds approaching that of light will find himself immersed in a grossly altered natural environment and will also face the problems of carrying with him a source of extremely intense radiation-in whatever form his propulsion system may take,

3 Gamow G., One, Two, Three Infinity, Mentor MS 97, Paperbound Books, c. 1947.

4 Von Handel, P. F., and H. Knothe, Relativistic Treatment of Rocket Kinematics and Propulsion, Air Force Missile Development Center, Rept. No. AFMDC TR 58-3, January 1958

5 Haskins, J. R., Notes on Asymmetrical Aging In Space Travel, Quarterly Research Review No. 18, Army Rocket and guided Missile Agency, Rept. No. 2A18, September 1,1958.