With the advent of "fellow-travelers" about the sun and their portent of things to come, it is appropriate that we who have long looked to the heavenly bodies to find our way, look now to the problem of finding our way among these heavenly bodies.
Interplanetary navigation will differ markedly from terrestrial navigation in several respects. First, we must fix our position in terms of three co-ordinates rather than two. Second, "earthly" velocity is not dependent upon the navigator's position on earth, but in space, velocity depends primarily on a vehicle's distance from the Sun, with its strong gravitational pull. Third, if we travel from New York to Southampton, we may reasonably expect Southampton to have zero velocity with respect to New York; not so when traveling, say, from Earth to Mars. Fourth, the magnitude of the velocities, distances, and times of flight are of a much higher order in space than they are close to the Earth.
Let us assume that in the not-too-distant future we may want to guide an interplanetary vehicle from "here" to "there." Let us further assume that the vehicle is unmanned, and that we cannot communicate with it. Additionally, we may consider ourselves severely restricted in the expenditure of precious fuel.
Since we have established a rather difficult problem at best, let us ease the strain somewhat by allowing ourselves to begin and end our journey not necessarily at a given point on the surface of a given planet, but rather, in orbit around a given planet.
If we are to provide our vehicle with sufficient intelligence to accomplish its mission, a transit between two bodies of the solar system, we must assess the problem in the following terms:
1. With respect to what does the vehicle compare its actual location and motion to its desired location and motion?
2. What action will the vehicle undergo in its environment?
3. How will the vehicle establish comparison data between actual and desired motion?
4. Once such comparison is made, how will the vehicle use its knowledge to assure arrival?
Let us first establish a reference framework within which the vehicle may locate itself and its target.
Co-ordinate System Considerations
Celestial bodies may be located by means of any of several different co-ordinate systems (Figure 1).
In determining the position of our vehicle, the familiar Earth-centered Horizon and Equator Systems will have to be discarded as being too provincial space-wise. The Galactic Systems are inept since we are concerned with only a limited portion of the Galaxy; they will deserve further consideration when we are ready to venture outside the Solar System. The Heliocentric Ecliptic System, however, offers three powerful advantages:
1. The reference zero latitude plane includes the Earth and Sun at all times.
2. All other planets' orbits lie close to this plane.
3. Tables of planetary positions are available with their data based on this system.
If we are to use a co-ordinate system involving the Sun and the Ecliptic, the plane of the Earth's motion about the Sun, we should look at the various motions this system is experiencing. The Sun moves at 12.2 miles per second in the direction of the constellation Hercules. This motion is with respect to the "nearer" stars. Then these "nearer" stars, together with the Sun, appear to be traveling at about 170 miles per second toward the constellation Cygnus. Also, the Solar System revolves about the center of the Galaxy once in approximately 2,200,000 years. In addition, our Galaxy is a member of the so-called Local Group including the Andromeda Galaxy, the Large and Small Magellanic Clouds, and several small clusters, which all rotate about a common center of mass located somewhere between our Galaxy and the Andromeda Galaxy. These motions of the Solar System show that the Sun is probably undergoing intricate and complex acceleration. The scale of distances involved, however, is such that during the time of flight of our vehicle within the solar system the effect of these ill-defined accelerations is neither appreciable nor measurable. Having settled upon a co-ordinate system in which to measure our vehicle's travel, let us now examine its proposed path.
The Trajectory
Within the Heliocentric Ecliptic co-ordinate system, the path of a planet or vehicle in orbit about the Sun can be completely described by seven elements, insofar as the body is influenced by the Sun's gravitational field (Figure 2, next page). These are:
1. Semi-major axis- a
2. Eccentricity- e (1 – b2/a2)
3. lnclination- i
4. Longitude of ascending node- ?
5. Longitude of perihelion- ?
6. Period- T (not illustrated)
7. Epoch- t (not illustrated)
Elements 1 and 2 describe the elliptical orbit. Elements 3, 4, and 5 orient the path with respect to the Ecliptic. Elements 6 and 7 fix the position of the body with respect to real and relative time.
The sum of the accelerations of our vehicle due to the gravitational pull of bodies other than the Sun, Earth, Moon, and target planet under the worst possible conditions is only about one millionth of one foot per second per second. If all the planets were lined up in the exact positions required to produce this worst possible effect on our vehicle, the error induced during a typical flight would be approximately 30,000 miles, not a large navigational error in space. This unlikely planetary arrangement would occur about once every 500,000 years. At a given time, the effects would be much more likely to cancel, rather than reinforce, each other. Thus we need consider the gravitational fields of the Sun, Earth, Moon, and the target planet only.
Our vehicle's flight paths may be grouped into three classifications: (1) the powered trajectory, (2) the unpowered trajectory, and (3) the combination trajectory. The term "powered trajectory" implies continuous thrust with either acceleration or deceleration used during the entire flight. Also implied is the ability to apply this thrust in any direction so that the vehicle can change course to a desired trajectory. The powered trajectory would in this application require nearly unlimited thrust duration and fuel. Guidance requirements would be simplified to the extent that time of flight would be less, so that the vehicle would be exposed to possible error for a shorter time. Since constant corrections to the vehicle course would be introduced, the initial accuracy requirements would be low.
The unpowered trajectory implies the application of a large initial thrust for a small fraction of the total flight time. The result is a hyperbolic path with respect to the Earth during the application of thrust. This hyperbolic trajectory merges with, and becomes, an elliptical path with respect to the Sun, and the vehicle would then be in free flight, acted upon by the gravitational attraction of the Sun with slight perturbations caused by the planets. A trajectory of this sort allows simplification of the in-flight guidance and computer section. Its great disadvantage lies in the extreme accuracy required in the initial course and speed of the vehicle, since uncorrected errors early in a 200,000,000-mile flight would result in an unacceptably large terminal error.
The combination trajectory, containing thrust periods and no-thrust periods, permits slightly less accuracy in the initial vector, but requires fairly accurate mid-course position determination. Ideally, our vehicle would require no corrective action in its transit. We will consider, however, that during the transit it becomes apparent that corrective action is required. A new trajectory is then computed, and proper thrust is applied to fly this new path.
One might reasonably wonder at this point why we do not force our vehicle to return to the originally computed trajectory. Aside from the fact that this would waste fuel, we are faced with the situation that once our vehicle deviates from its planned trajectory, being unable to speed up appreciably, it cannot easily return to a given point on the trajectory at the originally planned time. Since the target planet would be in some other position at the later time of arrival of our vehicle back on the original trajectory, a new trajectory would have to be computed in any case.
Position Determination
The following possible methods may be applied to vehicle position determination:
1. Radar or radio ranging
2. Measurement of incident versus reflected solar radiation
3. Integrating accelerometer
4. Optical star tracker
Aside from unacceptable power requirements, radar or radio ranging has the obvious disadvantages of bad bearing resolution and large time delay- 18 minutes for a target 100,000,000 miles distant. Incident and reflected solar radiation vary too slightly to provide an accurate ranging method. For example, in the vicinity of the Earth, the variation in this radiation is approximately one erg per square inch second for a. change in range of one mile from the Sun. A ranging device capable of measuring this variation would have to be extremely sensitive. The use of integrating accelerometers would require an instrument sensitive to accelerations as small as one ten-millionth of a foot per second per second- an extremely accurate instrument. The optical star tracker, with a maximum error of five seconds of arc on any given measurement, may not be impossible to build. It appears that a measurement system consisting of integrating accelerometers monitored by a star tracker system would provide sufficiently accurate data.
Given, then, a dead reckoning position and an observed position, both with reference to the Sun at the center of our co-ordinate system, our vehicle can compare the two and compute a new trajectory on the basis of the observed position.
To find the velocity vector of our vehicle, it is necessary to add the force vector of the vehicle itself to the force vectors at the vehicle's position imposed by the gravitational fields of the Sun, Earth, Moon, and target planet, and integrate this value to get vehicle velocity. The mathematics involved is based on two of Newton's basic equations, force equals mass times acceleration, and the gravitational force between two bodies is proportional to the product of the masses of the two bodies divided by the square of the distance between them. In addition, the directions of the Sun, Earth, Moon, and target planet from the vehicle must be computed. By this method, here oversimplified to be sure, the behavior of our vehicle with respect to the Sun, at the center of our co-ordinate system, can be determined.
Instrumentation
The instrumentation of any guidance and control system for interplanetary navigation will be extremely complex. Here is a proposed system for accomplishing such controls as appear required for the successful carrying out of our vehicle's mission. We must assume the system contains a miniature low power digital computer, can remember tables of position for celestial bodies in an ephemeris memory and has a basic time generating mechanism.
The functional responsibilities of guidance and control are segmented into three basic categories:
(1) The Tracking System- maintains lines-of-sight upon prescribed bodies and provides tracking line data there from to the Computing System.
(2) The Computing System- performs the necessary computations to the tracking lines to obtain signals proportional to the corrections required to the vehicle velocity vector to determine proper trajectory.
(3) The Thrust System- provides vehicle orientation and corrective thrust in accordance with the received signals.
The Tracking System would consist of one, two, or three star trackers rigidly mounted to a platform isolated from the vehicle's rotational motion by gimbals. Gimbal drive motion activated by error signals generated in the star trackers would maintain the platform in its selected attitude. A gyro package mounted on the platform would assist in maintaining the platform orientation in space. Three additional tracker units mounted on the platform would maintain lines-of-sight on the Sun, the Earth, and target planet.
Within the Computing System, a tracking line computer would receive the angular measurements of the tracking section together with data from the ephemeris memory and time generator and would determine an observed position of the vehicle with respect to the Sun. From discrete measurements, all the elements of the trajectory may be determined.
A prediction computer would establish continuously, as a function of time, the position of the vehicle relative to a given body. This is a predicted position in the sense that it does not stem directly from observation, but rather represents what should occur if no external influence altered the vehicular motion. If no corrective trajectories were envisaged, all the outputs could be stored prior to launch. Since corrections are anticipated, however, the computer must predict vehicle position during the entire time of flight without reference to any source of information from outside the vehicle itself. Essentially this is the continuous solution of the force-equals-mass-times-acceleration equation.
A comparator and correction computer would compare predicted vehicle position with observed position and, when necessary, compute a new trajectory and incremental change in velocity required to fly this trajectory. This computer would signal the Thrust System the required orientation and thrust.
The Thrust System would be essentially a modification of the familiar aircraft control loop, altered to include a thrust device rather than wing control surfaces. Several means of directional control are available:
(1) Movable vanes in the jet stream
(2) Small individual variable thrust jets
(3) Universally mounted main engine
(4) Gimbal-mounted thrust chambers.
The Thrust System would be activated by the Correction System, as would the controlled metering of propellant to insure a velocity change of proper magnitude. The resultant change in motion of the vehicle would be sensed by accelerometers for comparison by the tracking system as apparent changes in position of the bodies being tracked.
In summary, it is evident that there is nothing radically new or thought-provoking in this approach to the problem of interplanetary navigation. A great deal of study has gone into this problem over many years. Successful navigation to the planets will require tremendous expenditures of effort, but the rewards may well compensate for it. And beyond the planets, of course, there are always the stars.