CELESTIAL MECHANICS

If you wonder why the astronauts don’t whip out their trusty slide rules and quickly compute a new orbit when something goes a little wrong—try this basic course in “Celestial Mechanics and why it drives people nuts”. There is a solution to the three body problem…only it can’t be worked out. To catch up to a ship ahead of you in orbit, you must slow down! Roland E. Burns is a NASA orbital mechanics mathematician and knows the frustrations of which he speaks.

BY ROWLAND E. BURNS

In one of the more perceptive science-fiction stories of recent years Isaac Asimov describes a planet which is associated with a system of six stars. This story, “Nightfall,” deals with the collapse of a civilization which occurs once every two thousand and forty-nine years when darkness descends upon a planet which otherwise lives always in light. The fundamental point of this story is the profound interaction between a civilization and the local celestial mechanics of a star system. The psychological, social, and even religious aspects of Asimov’s imaginary culture are shown to follow in large measure from celestial mechanics.

Most people would agree that the variations of the seasons, length of the year, weather, and various other manifestations of the geometry of the Earth’s path about the sun have had similar effects on our everyday mode of thought. It seems to be less well-known that much of our philosophy, most of macroscopic physics, and all but the most recent of our mathematics have proceeded from the same source.

Specifically, applied mathematics has followed what is generally known as an analytical bent. By this we mean that the end product in the study of a mathematical problem is a formula which relates the variables of the problem, not a numerical answer. It is only quite recently that the advent of large computers has produced the field of numerical methods as being a field which is respectable in its own right. The historical inertia of analytical mathematics is so strong that most new college graduates go forth into the real world with a belief that mathematical problems of the real world can be solved analytically.

The basic impetus behind the surge of analytical mathematics lies in the fortunate complexity of the two-body problem. The two-body problem was first formulated mathematically by Newton after discovery of the inverse square law of gravitation. By way of definition, the two-body problem is the mutual motion of two material objects which are either point masses—a mathematical fiction—or perfect spheres—no less a fiction—which attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This problem was sufficiently difficult that the invention of much of mathematics was necessary to describe the motion, yet sufficiently simple that there was hope for such a solution. Thus, the “fortunate complexity.”

In order to better understand the mechanics of a two-body problem, let us imagine the following thought experiment: Imagine, if you will, that you are located in a universe which is quite empty except for a massive perfectly spherical perfectly homogeneous planet and a smaller projectile which fits the same criterion. You, as a massless ghost, are equipped with an equally massless gun and a quantity of massless powder. In the course of the experiment you are to load the projectile—or bullet—into the gun and fire it with varying amounts of powder. The bullet is to be fired while standing—better, lying—on the surface of the planet.

Suppose that the first powder charge is quite small. The bullet will travel a short distance and, due to the attraction of the planet, impact the surface. (Keep in mind that no air exists in the make-believe universe, so no air drag will slow the bullet.) If the bullet is now reloaded into the gun with a larger powder charge, it will travel farther before impact. After each shot, as the bullet speeds away faster and faster, the curvature of the trajectory of the bullet comes closer and closer to matching the curvature of the spherical planet. Finally, at a critical powder loading, the curvature of the trajectory of the bullet exactly matches the curvature of the planet. Impact will never occur; an orbit has been established.

This orbit will be a perfect circle under the idealized conditions that have been postulated here. But, as most people know, other types of orbits are possible. Even under the conditions stated above we can obtain not only circles but ellipses, parabolas, and hyperbolas as well. It is interesting to consider these geometric figures from two points of view. The first point of view is purely geometric and was known long before Kepler discovered the laws of planetary motion. The second point of view is dynamic and dates from Newton.

To proceed with the geometric point of view, we shall temporarily disregard our ideal universe and instead consider an idealized ice cream cone with perfectly smooth sides. In order to proceed with the geometrical discussion it is first necessary to make a few definitions about a cone. The tip of the cone is simply the point and any line through the tip which stays in the surface of the cone is called a generator of the cone. A line which passes through the tip of the cone and bisects the angle at the top of the cone is called the axis of the cone. These quantities are shown in Figure 1.

Figure 1

THE CONIC SECTIONS AND DESCRIPTORS

If we now take a sharp knife and cut through the cone perpendicular to the axis of the cone, the perimeter of either half of the resulting figures are circles. If we had chosen to cut the cone at some angle other than in a plane perpendicular to the axis of the cone, other figures would have resulted. For example, if we choose to cut the cone at some angle such that the knife blade will emerge from the other side of the cone, an ellipse will result as the perimeter of either of the two pieces that are formed. If the angle of cut had been chosen in such a way that the plane of the cut exactly paralleled the generator of the cone on the opposite side of the cone, the resultant figures would have had perimeters in the shape of parabolas. If the angle of cut is even steeper than that used to shape a parabola, the resultant pieces will have hyperbolas for perimeters.

It is most interesting to note that a considerable degree of freedom exists in the choice of the angle used to generate either the ellipse or the hyperbola. In the case of the circle, or parabola, no such choice exists; one angle, and only one angle, will do the job. This situation has an analogy in the dynamic case which will be pointed out below.

Figure 1 illustrates the various conic sections which result from the cone cutting procedures just discussed.

Having dispensed with the geometric definitions of the conic sections let us proceed to the dynamic aspects of conic orbits by performing additional experiments on the previously discussed imaginary planet.

In the initial set of experiments it might have seemed reasonable to record the amount of powder charge which was used and then record the length of time which was required for the projectile to return to the starting point. For the first experiments no real correlation would have been observed since the planet interfered with the projectile and it never returned. Finally, a critical powder load did return the bullet to the firing site and it could be argued that a further increase in powder loading should be expected to decrease the time until return of the bullet just as a car traveling at 70 mph could be expected to circle the globe in less time than one traveling at 50 mph.

‘Taint so.

Consider the energy that the bullet possesses when it is fired. Energy is divided, as usual, into two parts. The first part of the energy is called kinetic energy and is measured by the velocity of the bullet. The second part of the energy is called potential energy and is measured by the distance of the bullet from the center of the planet. In the case of the circular orbit which was first generated, the kinetic and potential energies were individually constant since a circular orbit maintains a constant height above the surface of a planet. Once we gain more velocity than the precise value which a circular orbit demands—by increasing the amount of the powder charge—only the sum of the kinetic and potential energies are constant. The bullet has excess kinetic energy and it begins to immediately convert it into potential energy. Directly opposite the launch site, after traveling through a central angle of 180° the bullet passes through the point farthest from the planetary center and begins to descend. It is now converting potential energy, bought at the expense of kinetic energy, back into kinetic energy. This process continues through all time and we have established a periodic elliptic orbit.

One of the most important points of the preceding discussion is that the velocity which the bullet has at the firing point is not retained throughout the orbit. The bullet is fastest at the point nearest the planet and slowest at the point farthest from the planet. Furthermore, it is geometrically obvious that the bullet must traverse a longer path length while in the elliptic orbit than it did in the circular orbit. The combined effects of the lowered speed and longer path length result in a longer time of return with increased initial velocity. This effect is seen to be more pronounced as we continue to increase the initial velocity. The sum total is the paradoxical result that the faster we initially fire the bullet, the longer it will take until the bullet returns!

Just as there was a critical slicing angle that we could not exceed and still obtain an ellipse in the geometric case discussed above, there is a critical powder charge that we cannot exceed and still obtain an ellipse in the dynamic case. If the initial velocity becomes too large we move from the realm of the ellipse to the realm of the parabola. In the case of the ellipse it was pointed out that the velocity at the highest point decreases as the initial velocity increases. It is possible to define a parabola as a figure that has a zero velocity at the highest point … but in this case the highest point will be an infinite distance from the center of the planet. At a critical powder loading the time before the bullet returns becomes infinite since the arc length suddenly becomes infinite and, at the apex of the trajectory, the velocity becomes zero.

If the value of the powder charge is increased beyond the value required for a parabolic orbit, a hyperbolic orbit will exist. The hyperbola, like the parabola, is not a closed figure and the bullet will never return, of course.

The velocity at infinity is not zero in the case of a hyperbola, however. In this case even at an infinite distance the velocity still has some nonzero value. (Hyperbolic orbits are often used for interplanetary probes leaving Earth and are sometimes categorized by their “hyperbolic excess velocity.” This measures the amount of velocity which the probe would have at infinity, even though it could never arrive at that point.)

It was earlier mentioned that there was an interesting correlation between the exact slicing angle required in the case of generating the circle and parabola in the cone slicing exercise and the dynamic description of our orbits. That correlation is now clear. In the case of these two figures an exact value of the powder charge is required but in the case of the ellipse and hyperbola there is a range of powder charges that will still produce the latter two figures.

Figure 2 illustrates that origin of various conic sections from our imagined experiment.

Figure 2

CONIC ORBITS, P=Planet

With a rather few exceptions which will be specifically mentioned, the remainder of this article will be concerned with the ellipse. This is because ellipses are closed figures which account for most real orbits. The open conic sections—the parabola and hyperbola—could be covered but tend to muddy the water by consideration of each case. The circles are special cases of ellipses and most of the comments made about ellipses will apply to that case.

A few definitions about ellipses are in order. One of the more important points is that of the focus. Each ellipse has two foci and one of the more common definitions of the ellipse comes from the property that the sum of the distances from the two foci to any point on the perimeter of the ellipse is a constant.

The attracting planet for any orbit is always located at one focus of the orbital ellipse—which always leaves one empty focus. It does not matter which focus is occupied by the attracting planet, but once we chose such a focus we must keep the planet there.

Another term that is frequently encountered is that of apopoint and peripoint. Apopoint refers to the point of the orbit farthest from the attracting center and peripoint to that point of the orbit closest to the planet. These words are usually encountered with a suffix which additionally indicates a particular planet such as perigee for the Earth, perilune for the Moon, et cetera. The introduction of the generalized suffix-point seems defensible on the grounds of clarity. Furthermore, when satellites are established in orbit about Venus the term periveneral would surely be misinterperted by all but the most dedicated astronomers.

The semi-major axis of the ellipse is one half the sum of the peripoint and apopoint distances. This measures the length of the ellipse along the longest axis of the ellipse. The eccentricity of the ellipse is a measure of the flattening of the ellipse. An ellipse with an eccentricity of zero is perfectly round, i.e., a circle. An ellipse with an eccentricity of 1 is perfectly flat, i.e., a straight line.

Thus far we have considered nothing but the geometrical shape of the orbits which result from a two-body problem, specifically the ellipse. It remains yet to describe how this orbit is located in space by standard astronomical specifications. This description may be given in a number of ways and various workers have had various descriptors which they personally preferred. The set given here is the so-called classical set and has the advantage of being intuitive. The following discussion is illustrated by Figure 3.

Imagine a point fixed anywhere in space. About this point we wish to explicitly define an orbiting .body which we assume to be in an elliptic orbit. The first two descriptors which shall be assumed given are the eccentricity of the ellipse and the semi-major axis. Place one focus of this ellipse at the center of the attracting planet. If this is pictured as a geometric construction it is apparent that the ellipse still has a considerable degree of freedom to flop freely about the fixed focal point. We now proceed to specify other parameters to remove this freedom.

Imagine that the planet at one focus of the ellipse is rotating about some fixed axis in space. This defines the equator of the planet and the plane in which the equator lies is convenient as a reference. If we now fix the angle between the orbital plane and the equatorial plane of the planet—the angle is called the orbital inclination—the orbital ellipse has far less freedom to “flop.” The inclination is the third of the classical orbital elements and has equal standing with the eccentricity and semi-major axis.

Thus far we have not defined the top and bottom of the planet. Define these arbitrarily—imagine that we simply point an “x” at one pole of the planet and call that the top. There are now two points at which the satellite moving in the elliptic path will pierce the equatorial plane of the planet. Call the point at which the satellite moves from below the equatorial plane to above the equatorial plane the ascending node and the point at which the satellite moves from above the equatorial plane to below the equatorial plane the descending node. A line drawn between these two nodal points is called the line of nodes. The line of nodes is not, in itself, an orbital element. A moment’s reflection upon our construction to date shows that we have not yet specified the direction in space where this line of nodes must lie. This direction is specified by an angle between the line of nodes and a fixed direction in space such as the vernal equinox. The angle, called the longitude of the eccentric node, is another orbital element. *

(* The standard symbol in the literature for the longitude of the ascending node is [omega]. This symbol is a printer’s nightmare since it is so rarely used. A few years ago one of the sets of the proceedings of the International Astronautical Congress were delayed over an argument as to whether or not the capital Greek omega could be substituted for this symbol.)

The orbital plane is now almost completely oriented. One final degree of freedom that must yet be eliminated is that the orbit can turn about an axis through the focus and perpendicular to the plane of the orbit. To remove the final degree of freedom we specify the angle between the line of nodes and a line drawn from the peripoint to the center of the attracting planet. This angle is called the argument of the peripoint and is the fifth orbital element. Once these five quantities are specified the orbit is uniquely oriented in space.

The usual problem is not just to locate the orbit in space but rather to locate the exact position of a satellite, or planet, in the orbit. In order to locate the exact position of the satellite in the orbit at any given time we must know where it was at some time in the past. For this reason the final orbital element is the time of peripoint passage. Once these quantities are all specified—the semi-major axis, the eccentricity, the orbital inclination, the longitude of the ascending node, the argument of the peripoint, and the time of peripoint passage—then the position of the satellite in the two-body problem is known for all time.

It was earlier mentioned that the classical orbital elements which we have just described are one of the favorite ways of describing an orbit, but they are not the only way. One alternative to this set is simply to specify the three components of position and three components of velocity, again six quantities as before. Each of these two sets of elements have advantages and disadvantages. The classical elements can experience singularities. For example, the argument of the peripoint is not defined if the orbit is circular and thus has no peripoint. The position and velocity designation do not experience singularities but have the disadvantage that they give no intuitive Feel for the size and shape of the orbit. Very often this latter designation is useful for computers.

Since the orbit has now been described in general, it is possible to specialize the discussion to two very specific orbits which are of interest. These orbits are the equatorial orbit and the polar orbit. We begin with the equatorial orbit. In both cases we shall further restrict the discussion to circular orbits to simplify the arguments.

At the outset of the discussion it should be remembered that the rotational velocity of the Earth is unrelated to the mass of the Earth. The mass of the Earth and the altitude of the orbit determine the period of the satellite orbit and the fact that the Earth rotates in twenty-four hours in no way depends upon either the mass of the Earth or, of course, the altitude of the satellite orbit. It is thus a pure accident that low orbit satellites—say on the order of 100 miles altitude above the surface of the Earth—complete an orbit in less time than it takes the Earth to rotate about the polar axis. If we now recall that orbits of higher altitude produce longer periods for the satellites which lie in them; it becomes apparent that a satellite with an altitude of 1,000 miles will have a period closer to the rotation period of the Earth than one in orbit at 100 miles. This slowing of the period of the orbit shows that eventually we shall reach an altitude such that the time required for the satellite to circle the Earth is exactly the same as the time that is required for the Earth to rotate about the polar axis. This altitude is 22,300 miles.

It should be noted that a satellite period of twenty-four hours does not depend upon whether or not the orbit is equatorial. It is perfectly possible to launch a satellite into an orbit having an inclination of, say, 30° which has a twenty-four hour period. The sub-satellite point will, however, trace a line on the Earth which is bounded between the north and south latitudes which are numerically equal to the inclination of the orbit. Thus, a satellite is a twenty-four hour period altitude and an inclination of 30° will trace a line from thirty degrees north latitude to thirty degrees south latitude along a line which is perpendicular to the equator. In order to establish a satellite which appears to hang motionlessly in the sky it is now only necessary to reduce the inclination to zero—i.e., establish an equatorial orbit.

One final point about geostationary orbits should be made. It was mentioned in the general description of orbits that the center of the attracting planet must lie at the focus of the orbital ellipse—or the center of a circular orbit. It is this reason that makes it impossible to establish an orbit which lies in a plane which is parallel to the equator. Such a plane is exactly what is required to establish a stationary satellite over a nonequatorial site such as Chicago, for example. In passing it can be noted that an equatorial orbit at an altitude of 22,300 miles has such a large “look angle” over the surface of the Earth that even if a satellite could be established in a fixed position over Chicago it would accomplish virtually nothing that could not be accomplished from a presently existing equatorial stationary satellite—unless observation of the polar regions is of prime importance.

The second type of orbit which we will discuss in some detail is that of a polar orbit. A polar orbit, by definition, passes over both poles of a planet, be it Earth or some other planet. Polar orbits are usually used for observation such as close studies of cloud cover and are important because every point of the planet is subject to close surveillance. To understand this point we begin with the assumption that it is easier to observe something when you are close to it than when you are farther away. In other words we want satellite orbits of low altitude. As in the equatorial case a low-orbit satellite means a relatively short period as well as a small look angle. Since the orbit passes over both poles it will cover a narrow strip from the north pole to the south pole then back to the north pole. But during the time that this satellite has completed an orbit am returned to the north pole the Earth has turned about the polar axis a few degrees. The satellite then observes new strip of terrain on the next orbit. This continues until the entire surface of the Earth has been observed.

It is interesting to contrast the terrain view from a low-altitude equatorial satellite with that of a low-altitude polar satellite. If we consider at equatorial satellite starting at some specific point over the equator such as the Galapagos Islands—it is amazing how few well-known places one finds on the equator—then on the first orbit the satellite will photograph a narrow band about the equator. By the time that we have returned to our starting point over the Galapagos Earth will have rotated about the polar axis … but this simply changes the time that we pas; the starting point. The terrain viewed on the next orbit will be exactly the same as that passed over on the first orbit.

Thus far we have only established, intuitively, the fact that an orbit can exist, the forms that orbits can take, the parameters used to describe the orbits, and two special cases of orbits. This, in simplest terms, is the area which was of interest to the classical astronomers who were interested in natural bodies such as stars, planets, comets, and natural satellites. The advent of artificial satellites has introduced the notion of orbit modification which was never even considered in the classical literature.

There are many reasons why it is desirable—and even necessary—to modify satellite orbits. Following a launch, the entire Apollo mission is basically a question of orbit modification as is the establishment of an equatorial twenty-four hour satellite. Before proceeding to the actual techniques of orbit modification insofar is flight mechanics is concerned, it is well to remember that the basic tool used in accomplishing such modifications is the rocket motor. A number of different types of motors have been used for orbit modification depending upon the circumstances. The case of “station keeping”—i.e., making sure that small perturbative forces do not destroy an established orbit—is usually handled by small jets of compressed gas. The ion engines, which produce very low thrusts for very long periods of time, have been proposed for some special applications. In this article we shall be concerned with a third type of rocket motor … the so-called high-thrust chemical engines which produce spectacularly large orbit changes in very short periods of time.

The fact that these engines produce very large changes in velocity M very short times gives rise to a rather common mathematical approximation known as impulsive orbit transfer. The amount of propellant used in changing the velocity between two orbits is usually not given directly as a measure of how “expensive” the transfer is, since the pounds of propellant necessary to effect the transfer depends upon the type of propellant that is used. Instead, the velocity difference between the orbits is specified and this number, known as “delta V” is independent of the type of rocket used.

It is important at this point to carefully define a word which is often used in a nontechnical sense. We wish to distinguish between the terms “velocity” and “speed” even though they are often used interchangeably in common speech. Technically the two words are quite different. Velocity is a vector, that is, it is specified by both a magnitude and a direction. Speed is a scalar and can be completely specified by a magnitude. Thus we could say that an automobile has a speed of seventy miles per hour but to say that it has a velocity of seventy miles per hour makes no sense. A direction must be added to this second statement and made to read something like “the auto had a velocity of seventy miles per hour heading due north.” It is easy to see that two autos which have the same constant velocity can never collide, but two with the same speed could easily destroy each other. It is intuitively apparent that to turn the velocity of an automobile through some given angle requires an expenditure of fuel even though the speed may be the same before and after the turn. A similar case holds in the case of orbital mechanics and just because a satellite has the same speed before and after a maneuver in no way indicates the delta V which has been expended in the transfer.

Having dispensed with preliminary definitions we are now in a position to discuss the fundamental problem of orbital transfer mechanics. We are given an initial set of orbit elements, a desired set of orbit elements, and a rocket engine capable of delivering a certain delta V—which measures the fuel on board. The delta V may be added in any desired number of impulses as long as the total does not exceed the given reserve. We must find the position in space where the rocket engine is to be fired and the direction of firing at each firing point in order to transfer from the initial set of orbit elements to the final set of orbit elements … but we must perform the entire set of maneuvers in such a way that a minimum amount of fuel is expended.

This problem may not appear difficult at first sight but appearances can be deceiving. It is so difficult that no fully general solution has ever appeared. A few special cases have been solved … and these may be used as the basis for educated guesses in more complicated cases. The guess must then be checked on a computer for verification. A point to be kept in mind is that the problem stated above is simply the mathematical bare bones of a realistic orbit transfer problem.

In the real case additional effects must be recognized. For example, air drag can severely limit results which are predicted theoretically, and such areas as the van Allen belts must be strictly avoided. An additional complication can come from psychological requirements such as the chase vehicle in a rendezvous situation wishing to keep the target within sight. Mathematics also has the annoying habit of believing literally the problem which is assigned to it. If we design a mathematical system to predict a transfer between orbits which expends the minimum amount of fuel, it may well show us that this minimum occurs using an infinitely long time for the transfer. This possibility often crops up as a part of the solution and it can be expected that both the astronauts and their families might complain. To guard against such nonsense is not simple, however, and it very often becomes a matter of engineering judgment as to just how close to the true minimum fuel expenditure we wish to come.

For the time being we shall exclude such real world effects since we must crawl before we walk. To this end we return to our imaginary planet with ideal properties and the equally ideal bullet. This time we shall add the possibility of impulsive orbit change to the bullet in orbit in order to discuss orbit transfer. At first there will be no consideration of attempts to reach a second vehicle, an operation known as rendezvous. In other words we wish only to achieve the orbit which is different from the initial orbit without consideration of when we achieve the orbit, or where in the orbit we end up.

Before we decide the actual firing direction which will be required to modify one or all of the given orbital elements, it is well to ask which direction of thrust will modify each of the elements individually. It would be nice if each orbital element could be changed by firing in a separate direction and the firing would leave all other elements unchanged but nature is not nearly this kind.

In order to answer the question of “which firing direction changes which orbital element” we must first define reference directions in space. These directions, a set of coordinate axes shown in Figure 4, all originate at the center of our imaginary planet. The first direction that we shall choose is along the radius vector drawn from the center of the planet to the satellite. The second direction is drawn from the center of the planet in a direction which is perpendicular to the orbital plane. The third direction lies in the plane of the orbit plane and is perpendicular to the first two directions. We shall refer to these directions as reference directions (1), (2), and (3).

The scorecard below will now help to keep up with the game.

From the table we see that firing along the first and third directions will modify the same orbital elements but two orbital elements are uniquely modified by firing along the second reference line. Thus, if an inclination change is desired, we must fire along the second reference line since firing along the first and third reference lines has no effect on this quantity. However, it should be noted that if we do fire in this way the longitude of the ascending node and the argument of peripoint will be simultaneously modified. Of course the reference directions chosen here are quite arbitrary and for any other choice of axes a new table, such as that shown here, must be constructed. Some sets of axes would be more convenient for some problems and much of the work of the orbit transfer problem is the isolation of a judicious choice of axes.

In order to examine an actual minimum fuel orbit transfer we shall consider the first such problem ever solved. This solution was due to a German engineer named Walter Hohmann and it occurred long before the hardware for a rocket launching ever existed. The transfer maneuver bears his name, the Hohmann transfer. This is shown in Figure 5. The problem is to transfer from a circular orbit about a planet to a second coplanar circular orbit at a higher altitude and minimize the fuel expended in the process. Hohmann showed that the best maneuver that could transfer a vehicle between these two orbits starts by firing along reference line (3), i.e., tangent to the lower circle. This transfers the vehicle to an elliptical orbit which begins coasting upward toward the second circle. The magnitude of the first impulse is chosen to be exactly the value required to produce an ellipse with an apopoint exactly equal to the radius of the second circle. At the time the apopoint distance is achieved a second impulse is added which raises the velocity at that point of the ellipse to a value necessary to achieve circular velocity. This is the most elementary of the two impulse transfers.

This solution, first published in the early part of this century, was believed to cover all cases of the outlined problem until quite recently. During the late 1950s numerical calculations indicated that the Hohmann transfer is the true minimum fuel expenditure transfer only if certain other conditions are satisfied. This restriction can be summarized as follows; if the ratio of the radii of the final circular orbit to the initial circular orbit exceeds a value of approximately 15.6, then the Hohmann transfer no longer provides the least fuel expenditure. Once the ratio of the radii exceeds this value it is cheaper, fuelwise, to first kick the spacecraft to a distance infinitely far from the attracting center along a parabolic orbit, modify the zero velocity at infinity by adding an impulse of zero magnitude, coast back from infinity along a second parabola and add a third impulse at the required circularization altitude. It is interesting to note that a fuel savings results from adding a third impulse even though the results are wildly impractical. In practice the modified Hohmann transfer can be used by coasting to some distance above the target orbit and adding a second impulse of nonzero magnitude then drifting down to the required altitude and adding a third impulse.

Even this modified scheme will beat the original Hohmann transfer if the ratio of the radii exceeds the above given value. These modified Hohmaim transfers are shown in Figures 6 and 7.

The Hohmann transfers just described are examples of transfers between nonintersecting orbits. It is fairly apparent that transfers between such orbits must necessarily involve two impulses since position does not change during an impulse. If two orbits do intersect—that is, they have at least one point in common—then it is equally obvious that a transfer between them could be accomplished with a single impulse. It is not obvious—and indeed untrue—that a single impulse is always the best way to transfer between intersecting orbits. Whether one impulse, or multiple impulses, can best be used to accomplish an orbital transfer is a very difficult question and individual cases must be examined.

Another case of orbit transfer which is of much importance is that of change of inclination. The inclination is one of the most difficult orbital parameters to modify since the satellite in orbit exhibits some of the characteristics of a huge gyroscope with an equally large angular momentum. The change of inclination involves the rotation of the angular momentum through an angular displacement … and that costs in fuel consumption. The problem of inclination modification is so severe that, if the cost of transporting fuel to the satellite is counted, it is sometimes cheaper to land a payload and relaunch it into the desired new inclination rather than attempt to modify the inclination directly.

Several important mission constraints can be derived from the fact that orbital inclination is so expensive to modify—where expense, as usual, is measured in the coin of the realm, fuel. One of the direct results is that it is impossible to launch reasonably large satellites much out of the plane of the Earth’s motion about the sun with today’s rockets. The reason for this is that the satellite carries with it much of the velocity of Earth’s motion about the sun and this, in effect, fixes the inclination of the orbit to be the same as the inclination of Earth. To launch a probe to some other planet which travels much out of the plane of motion of Earth about the sun—such as Pluto—would be prohibitively expensive—unless we caught the planet while it was at the line of nodes.

One real life case in which inclination modification is absolutely essential is the case of launching an equatorial satellite. The reason that an inclination change is necessary in this case depends upon a quirk of geography and a bit of the mechanics of rocket launching. The quirk of geography is that the United States does not extend southward to the equator. For reasons which are both political and logistic it is desirable to have the rocket-launching site located on home territory. If the United States actually did cross the equator, it would have been important to establish a spaceport at such a point for at least two reasons. One such reason is that the rotational velocity of Earth is a maximum at the equator and a rocket vehicle at liftoff would already have this velocity as a contribution toward the required orbital velocity. But a second—and far more important—reason is that it is possible to launch a satellite into any desired inclination from an equatorial launch site without subsequent orbit modification. (The amount of payload will vary with the required inclination. It will be largest for a due east launch and smallest for a due west launch since in the former case the satellite makes full use of the rotational velocity and in the latter case actually must fight the Earth rotational velocity.)

To see why nonequatorial launch sites cannot efficiently launch equatorial satellites, we consider a specific site such as Cape Kennedy. Kennedy lies at approximately 30° north latitude. If the equatorial orbit is to be directly launched from Kennedy with orbit transfer, then the cutoff point of the rocket engines must necessarily be at a point on the equator. However, this entails traversing an arc of approximately two thousand miles of distance—without yet even worrying about achieving orbital velocity. Such a launch is obviously prohibitive. Thus with our quirk of geography we are left with the highly expensive problem of inclination modification to achieve the necessary equatorial orbits.

The saving grace comes from the accident that the twenty-four hour satellite orbit is so high-22,300 miles. At this large an altitude the velocity in the orbit is quite low and the velocity vector which must be added to the satellite velocity in order to turn the orbit is small.

This trick of modification of inclination at high altitudes is a special case of a general result from orbit transfer mechanics. It has been shown that if the inclination change is large it is often cheaper to kick the vehicle infinitely far from the attracting center along a parabolic trajectory. At the apopoint, since the velocity is zero, it is possible to rotate the velocity vector by an impulse of zero magnitude, swoop back from infinity and recircularize the orbit. The real life case of the twenty-four hour orbit represents an approximation to this idealized maneuver. It is interesting to compare this case with that of the three impulse Hohmann transfer described earlier.

Two other classes of orbital maneuvers which are more stringent in their requirements than the case of simply matching a desired orbit are interception and rendezvous. Interception is defined as having to match position of one spacecraft with another with no particular worry about a velocity match. Rendezvous involves the matching of both position and velocity and is the most difficult of the orbit transfer problems.

The case of rendezvous has achieved considerable notoriety due to the Apollo program and it is well to examine one case of it in detail. We must assume a specific initial orbit configuration for the pair of vehicles. For convenience we shall label them “chase” and “target” and assume that they are initially in circular orbits with chase in a lower orbit than is target. During the maneuver target will be assumed passive and chase will do all the thrusting. Suppose that chase is in an orbit which is more than ten miles below target. The first maneuver is for chase to perform a Hohmann transfer from whatever circular orbit he initially occupies to an orbit which is again circular but ten miles below target. Since chase is still below target he is gaining on the higher vehicle—a lower altitude involves a greater velocity. Thus the angular separation of the vehicles will gradually diminish until chase passes below target and then the whole process of catching up starts all over again.

At a critical angular separation between the vehicles chase again fires his thruster and moves into a very special elliptic orbit. This ellipse has the property that the apopoint of the ellipse lies as far above target’s orbit as the peripoint of the ellipse lies below the circular orbit that chase has just left: Furthermore the ellipse has the property that the angle traversed between injection into the ellipse and the final rendezvous point should equal one hundred and forty degrees—due to line of sight considerations.

This ellipse should bring chase into the near vicinity of target and a final thrust is used to circularize chase’s orbit. This portion of the maneuvering is usually referred to as the “gross” rendezvous. From this point onward seat-of-the-pants flying takes over for docking—if you can count anything involving radar, computers, et cetera, as seat-of-the-pants. The point to make is that within close proximity vehicles will behave in the intuitively obvious manner with respect to each other, just as an astronaut does not need orbital mechanics to pull in a pencil floating about the vehicle cabin.

The reason that intuitive targeting does not apply in the case of gross rendezvous, but does apply in the case of docking, is often a point of confusion to the layman. The reason behind this situation goes back ultimately to the definition of a vector. Suppose that two vehicles are located in the same circular orbit but are separated from each other by an angle of, say, 90°. The gravitational force acting on the vehicles is of equal magnitude in either case—but gravity is a force which means that it is a vector quantity. Since the vehicles are so widely separated the force of gravity, which acts upon one of the vehicles, is not the same as the force of gravity, which acts upon the other. As the two vehicles close upon one another the force of gravity upon the two vehicles becomes almost identical and close-range intuitive targeting can take over.

The situation might be thought of as being analogous to the case where a friend stands at the axis of a rapidly spinning merry-go-round and you wish to walk from the rim to meet him. While you are at the rim a force acts upon you which does not act upon him. Although this force decreases as you approach him you must still account for the force until you actually arrive at the axis of rotation. It is apparent in this case that your path from the rim to your friend will not be a straight line as seen from a third man standing on stationary ground. Similarly, the path of rendezvous between two vehicles will not be a straight line.

Thus far it has been continually emphasized that the discussion applies strictly to idealized cases in an imaginary universe. It seems unfair to leave even this brief discussion of celestial mechanics without giving consideration to some of the real world effects that continually confront workers in astronautics even though the resultant whirlwind tour may appear to be a bit of a hodgepodge. Specifically, we shall look briefly at air drag effects, multi-body problems, and nonspherical planetary effects.

To begin with air drag. It is known that the actual atmosphere of the Earth, for example, extends many hundreds of miles into space and is usually the cause of the demise of most satellites. Air drag is a basically different type of force than is gravity. Gravitational forces are known as conservative forces since energy is conserved by them. This is a generalization of the earlier comment that a satellite in a noncircular orbit continually trades energy between the kinetic and potential forms. Air drag is not a conservative force but rather a dissipative force. Energy lost to air drag cannot be recovered and ends up as increased entropy.

The problems of celestial mechanics which involve only gravity are attractive from a theoretical point of view since the only necessary tool the investigator must possess is ingenuity. Problems involving air drag are far more complex and must be largely empirical, at least at our present state of knowledge. The approximate equations which yield pressure, temperature, and density as functions of altitude are not in the least aesthetic to a theoretician. Due to the complexity of the system almost any meaningful calculation involving drag must resort to numerical analysis rather than hope for an analytic solution.

One of the results of air drag which is sometimes referenced in the literature is the so-called “satellite paradox.” The phenomenon is really rather easily explained and is hardly deserving of the title “paradox.” The original observation was that as a satellite lost energy due to air drag it attained increased velocity. The velocity increase, as pointed out several times, comes from the fact that the radius of the orbit is decreasing which results in a shorter period. If the total energy of the satellite—kinetic plus potential—is considered it will be seen that the total energy decreases due to the drag.

Another aspect of the real world that has not been considered as yet is the influence of other massive bodies on the two-body problem. Examples of this case are quite easy to construct. For example as a satellite moves in orbit about the Earth the gravity attraction of the Moon, the Sun, Altair, and, in short, every other body of the universe perturbs the motion away from the ideal two-body case. M a second idealization to the physical case the three-body problem was posed. This problem is to describe the motion of three massive bodies—in an otherwise empty universe—which attract each other according to Newton’s law of universal gravitation. Virtually every great mathematician from Newton on has made contributions to this problem but it was not solved until Sundmann gave the solution in 1907—a fact which is apparently unknown to most people.

To understand the difficulty involved in the solution of the three problems, let us begin by recalling that Newton’s second law of motion—which is the basis of all classical mechanics—relates the acceleration experienced by a body to the mass of the body and the forces acting upon it. In the present case the forces axe simply those of gravity and at first sight the three-body problem may not appear more difficult than the two-body problem. Now the question arises as to what is meant by a solution. The desired end point of the analysis is an equation, or set of equations, which allow us to predict the position and velocity of the body once the time is specified. Very often much of the necessary information comes from what are called conservation laws and a number of conservation laws are known for the three-body problem.

The first such conservation law tells us that the energy of the system is conserved since the forces are conservative. Furthermore the angular momentum of the system about the center of mass must remain constant. The linear momentum of the center of mass must, likewise, remain constant and this final fact additionally allows us to predict where in space the center of mass will lie at any given time. Since there is one scalar conservation law—energy—and three vectorial conservation laws, we have a total of ten pieces of information that can be of value. But the total required is eighteen since we have three bodies, each of which must be specified by three velocity components and three position coordinates. Thus a total of eight additional pieces of information is necessary to solve the three-body problem.

Things began to look fairly hopeless with respect to the possibility of obtaining a solution when an investigator named Bruns showed that no more purely algebraic relationships—which would be additional laws of conservation—could ever be obtained. This theorem relegated many of the ongoing studies concerning the three-body problem to that limbo occupied by efforts to trisect the angle and square the circle.

Enter Sundmann. Sundmann in no way negated the theorem of Bruns, but rather circumvented it by using nonalgebraic functions. In order to solve the problem it was necessary for Sundmann to introduce certain functions that were very exotic and could be described only as the sum of an infinite number of terms—a common practice in mathematics. In order to use such a series it is absolutely necessary that the series approach some limit value as more and more terms are added; that is, it must converge. Sundmann investigated the convergence of his series and demonstrated that they must converge—but he did not investigate how fast they converged. This was not an error on his part since there are no usable tests in mathematics that yield the rate of convergence of a series, but it was fatal nonetheless. Numerical calculations involving the Sundmann solution showed that the rate at which the series converged was so slow that no practical importance could ever be attached to the solution.

It is tempting at this point to consider the feeling that Sundmann had when the numerical analysts pointed out the fatal flaw. He had obtained the solution to a problem which stopped Newton, Gauss, Euler, and Poincare only to have it pulled from him at the last instant. Let us hope that he took comfort in the solution, be it practical or impractical.

Long before the work of Sundmann other attacks on the three-body problem were in progress. The most notable of these is the restricted problem of three bodies. This was originally introduced as a simplification of the three-body problem but turned out to be a brand-new can of worms. The basic ground rules of the problem are to assume that two of the bodies are quite large and the third is extremely small. The two large bodies are required to move about their common center of mass in circular orbit while the third wanders about under their gravitational attraction. The problem is to predict the position and velocity of the very small body as a function of time.

Since the third body was so small that it did not affect the motion of the two larger bodies the law of mutual gravitational attraction was violated. Physics revenge from this affront is that none of the quantities conserved in the three-body problem, such as energy and momentum, are conserved. Over the years it has become apparent that the restricted problem of three bodies is at least as difficult as the full blown three-body problem; recently it was demonstrated that the Sundmann solution is completely inapplicable to the restricted case.

Euler, probably one of the greatest of the great mathematicians, was determined to find a problem more complex than the two-body problem that could be solved. To this end he further simplified the restricted problem by requiring that the two massive bodies no longer rotated about each other but rather were fixed in space. This problem, of course, bears the title of “Elder’s problem of two fixed centers” and it is solvable in terms of only slightly exotic functions.

Since the restricted problem is made to order for Moon probes it seemed that Euler’s problem would be a good working approximation to a Moon probe motion but this was not found to be the case by early workers in the astronautics industry. It has been found that, if we wish to launch a probe toward the Moon, it is a better approximation to completely disregard the Moon and use the two-body approximation than it is to assume that the Moon is on station but motionless. This result can be predicted once it is known, hindsight being 20/20.

To justify the case let us begin with the observation that the units and dimensions that mankind has developed are purely arbitrary; in problems of celestial mechanics it is often convenient to use units and dimensions which are inherent to the problem. To apply this to the motion of the Moon probe we begin by measuring distance in units of the distance from the Earth to the Moon, and time in units of the time that is required for the Moon to make one complete circuit about the Earth—that is the lunar month.

The third applicable quantity that is available to us is the unit of mass and we choose this unit such that the sum of the masses of the Earth and the Moon are equal to I.

In the system of units which we have chosen the problem is now simplified. The time that it takes the Moon to complete one orbit about the Earth is 1, which is not a small quantity. On the other hand Earth possesses 80/81 of the mass the Earth-Moon system while the Moon possesses only 1/81 of combined mass. It seems apparent that we can disregard the small quantity 1/81 and make only a slight error.

Another major problem area in the calculation of celestial mechanics is that the attracting planets are not spheres. Before describing this further it is well to keep things in perspective by pointing out that the Earth is much closer to being a sphere than is the average bowling ball. But even these small deviations from a spherical shape are enough to cause easily observable perturbations in the orbit of a satellite.

Fortunately there Earth’s equator is quite circular so we must be more concerned with deviations from sphericity along lines of latitude than we are along lines of longitude. (This is not the case with lunar satellites since the equator of the Moon is not close to circular.) Before the advent of geodetic satellites not much was known about the actual shape of the Earth. The shape was written in terms of an infinite series of terms and, since the Earth obviously has a finite gravitational field, it was assumed that each succeeding term would be much smaller than the preceding term in order that the sum of the series would be finite. Measurements from satellites have produced the disturbing result that the terms are not growing smaller. The resulting dilemma, known as the King-Hele catastrophe, has not been resolved.

The major contributor to the mathematical description of the Earth’s gravitational field is the zeroth term of the series mentioned above. This term simply tells us that as a first guess Earth is a sphere. The second term—which is much smaller than the zeroth—distorts the Earth to an oblate spheroid—which is flattened at the poles. It is of interest to note that the first two terms of this series are exactly equivalent to the field observed if we study Euler’s problem of two fixed centers but choose the distance between the two bodies to be an imaginary, number! This problem was first solved several hundred years ago and then rediscovered in the 1930s—a case which is not uncommon in celestial mechanics. Almost any modern worker in celestial mechanics today at one time or another will make a discovery which he later finds is a special case of a problem treated by a long dead genius.

Another case of this sort is the pear shaped Earth which made a splash in the newspapers a while back. The pear shaped Earth is treated by the old master, Tisserand, in a multi-volume work entitled “Mecanique Celeste,” published in 1889.

The treatment of additional perturbations such as radiation pressure, outgassing of vehicles, et cetera, could be extended almost indefinitely. The interesting topic of the motion of a rocket vehicle when the burning time is not short—as it was in our impulsive approximation—is so extensive that it includes normal celestial mechanics as a special case and cannot be considered here.

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