Pursuit of Orbit

Bill Guier sees an equation as paragraphs of lucid prose. There are nuances and implications stemming from the relationship that the equation establishes between different aspects of the physical world. To George Weiffenbach, numbers, their intrinsic values and relationship to one another, are like a film. How they change tells him how the physical events that they represent are unfolding. These ways of viewing the world are quite usual for physicists, and the one is typical of theoretician, the other of the experimentalist.

Each understood something of the others outlook and could, to some extent, see from the other’s viewpoint. The combination proved in­valuable.

Some textbooks still describe the miss-distance method that Guier and Weiffenbach initially adopted as the basis for Doppler tracking. It is not the technique that APL developed. Had Guier and Weiffenbach stuck to determining miss distances, they would not have had tracking data that were accurate enough for a good orbital determination, because one of the values needed in the calculation is satellite transmitter frequency. Sputnik’s oscillator, though loud and clear, was varying slightly from its nominal value in an unknown way. Those at the IGY who had discounted Doppler as a means of obtaining tracking data had done so precisely because they were concerned about the stability of the oscillators that generated the radio signals. Oscillator stability would be of concern, too, in the method that Guier and Weiffenbach developed, but their approach offered a way to tackle the problem.

On Tuesday morning, October 8, Guier and Weiffenbach were still thinking in terms of miss distance. And they were conceptualizing the problem. As is usual when creating a mathematical model of a physical phe­nomenon, they simplified the problem. In this way, they would find out what general principles were at play before turning to the inherent com­plexities of this particular case. They assumed that the satellite was moving in a straight line, just like our train. Since they were looking only for the range at the point of closest approach, the simplification was reasonable.

Guier and Weiffenbach knew all but one of the values needed to calculate miss distance, and that value—the range rate when the Doppler
shift was zero—was available experimentally from the slope of the curve as it passed through T = 0 in the graphs of frequency against time. It is the steepest part of the curve.

Guier recalls watching colleagues plotting graphs, fitting plastic splines to them and measuring the slope, then looping string around the lab to represent an orbit and marking the miss distances with rulers. It was for fun as much as anything, though such a physical representation of the orbit does give a conceptual idea of what is going on in three dimensions.

To two people who saw what Guier and Weiffenbach saw in equa­tions and numbers, the ruler and string approach was not the way forward. They determined the maximum slope from tables of frequency and time using a mechanical calculator. Soon, they switched to the lab’s new Umvac ПОЗА, APL’s first fully digital electronic computer. With its combination of vacuum tubes and transistors, the ПОЗА could perform several thou­sand additions per second, zipping through the tables of time and fre­quency far more rapidly than was possible with a mechanical calculator. It was an advance on APL’s partly analog machine—a type of computer that performed arithmetic operations by converting numbers to some physical analogue, say length or voltage.

By the late 1950s, university departments were slowly converting to digital machines, but not all scientists and engineers had, as yet, seen the value of computers. So, even though their project was not sanctioned offi­cially, Guier and Weiffenbach had no problem getting time on the computer, something that would not have been the case even a few months later.

When Guier and Weiffenbach first turned to the Univac, some of their colleagues asked, Why do you need a computer to calculate the slope of the curve? Given that they intended only a rough calculation, the ques­tion was reasonable. Guier, however, had worked with computers at Los Alamos. Though those machines were mechanical and rattled and banged around once Guier had fed in the numbers and instructed the machine to add or multiply, he had been won over by them. To him, the 1103A was a luxury, and he was happy to find a problem for the new machine.

It was no easy job, however, to program those early computers. A task that would take an hour to code in a higher-level language such as FOR­TRAN might take eight hours to code in assembly language, where each mathematical operation had to be coded for separately. If two numbers were to be multiplied together, the code needed to tell the computer in which memory locations to find each number, in what location to carry out the multiplication, and where to store the result.

Guier wrote instructions for those writing the code. As the orbital determination grew more elaborate, the number of code writers increased, and it not always possible to tell what each programmer had done. Guier started to insist that code be accompanied by notes describing why a par­ticular approach had been taken, and, though it looked like a tedious task, he learned assembly language himself in order to provide some continuity in the software effort (of course, this all took place before the word soft­ware had been invented). Thus APL, like others at the time, started to for­malize software development.

Today s computers can translate high-level languages, which are rich in symbolic notation, into the low-level assembly language that the machine “understands.” Computers then did not have the memory to hold the pro­grams needed for such a translation. To make matters worse, code writers often had to go back to the basic binary words of zeros and ones, which the limited symbology of assembly language represents, in order to find errors.

Despite what today would seem to be daunting limitations, the Uni- vac ПОЗА was a godsend to Guier’s and Weiffenbach s research.

As the two sought values for the miss distance, they soon recognized the ambiguities inherent in the approach. In the case of the train, it is easy to see that the way the frequency varies depends on how close the listener is to the track. But if you have only a tape recording, how do you know whether that recording was made two yards to the east or two yards to the west of the track? In the two-dimensional case, you are stuck with the ambiguity. In the case of the satellite, the earth is rotating beneath its orbit, and the east-west symmetry is broken.

But there are other ambiguities. Consider the train again; a recorder two yards west of a north-south track will record the same frequency shift if it maintains its westward distance but moves a mile due north. Fortu­nately, satellites inhabit our three-dimensional world. Rather than moving in a straight line, the satellite’s path with relationship to the lab might be a shallow or pronounced arc, near to the lab or like a distant rainbow, low or high on the horizon, and the arcs could be at many different angles. For each arc, the relative motion between the lab and the satellite, and thus the Doppler shift, was different.

Guier and Weiffenbach began to recognize the richness of the situa­tion, its complexity. The miss distance soon ceased to look like a particu­larly interesting value to calculate. They were beginning to suspect that their Doppler curves might contain a lot more information about satellite motion than was immediately apparent.

So they decided to explore, to find out how much information the data contained. To do this they needed to know how the Doppler data changed for different orbits. But only one satellite was aloft. Guier turned to the computer and created a general mathematical description of an orbit that would then generate the theoretical Doppler shift associated with a hypothetical satellite’s motion.

Guier pulled together the geometry, trigonometry, and algebra required to describe a generalized ellipse in space and the relationships between the ellipse, the lab, and the center of the earth. He drew up flow charts showing how his mathematical description of this physical phenom­enon could be turned into a computer program.

The model assumed the earth to be spherical, which it is not, and orbits to be circular, which they are not, though Sputnik’s orbit was nearly so. The idea, again, was to deal first with the simplest possible situation in order to clarify the underlying principles. This construction of a general­ized mathematical representation is akin to preparing a canvas for an oil painting. Guier’s and Weiffenbach’s painting would prove to be ambitious.

They began to generate theoretical Doppler curves. Of course, they were not creating curves but churning out lists of Doppler-shifted fre­quencies and times for the computer to work through, looking for ways in which the Doppler data varied as the orbital path varied. These numbers were octal rather than decimal, so instead of a counting scheme based on the digits 0 to 9, the digits run only from 0 to 7.

Guier and Weiffenbach pored over these numbers, and as necessary turned to their mechanical calculator to convert octal to decimal. They decided to make their model more realistic and so included mathematics describing the earth’s oblateness (a bulging at the equator caused by the centrifugal force resulting from the earth’s rotation). Oblateness causes orbits to precess because the Earth’s gravitational field is not uniform, as it would be around a homogeneous sphere. Inclusion of a term for oblate­ness was the first brush stroke on the newly prepared canvas.

In 1957, geophysicists thought they had a good understanding of the earth’s shape and structure and thus of its gravitational field. They turned out to be very wrong. Observations of satellites in near-Earth orbit showed all kinds of deviations from Keplerian orbits, each one rep­resenting some variation in the gravitational field, which in turn was a result of inhomogeneities in the earth. The field of satellite geodesy was established to tease a new understanding of the earth from these observa­tions of satellite motion. In 1957, the revolution that satellite geodesy pre­cipitated in our understanding of the earth had not yet begun, and the mathematical model that Guier devised to generate orbits was extremely rudimentary compared with what is possible today. In the years following Sputnik’s launch, observations of satellite motion would lead to many more brush strokes on Guier’s mathematical canvas, each representing a newly understood aspect of the Earth’s structure and its associated gravi­tational field.

Using their basic model in October 1957, Guier and Weiffenbach generated more theoretical orbits. They changed imaginary perigees, then the inclinations, ascending nodes, and eccentricity, and they observed what impact these changes in the Keplerian elements had on the theoretical Doppler curves generated by the computer. By this stage, they had com­pletely lost interest in calculating miss distances.

Given the multiplicity of tasks and the extent to which they exchanged ideas, it is hard now for Guier and Weiffenbach to remember who did what when, but they agree that at some point in those first days, Guier recognized that the detail of the Doppler curve must change in a unique way, depending on the geometry of the satellite pass. This was the conceptual breakthrough that others failed to see or, if they saw it, did not use: that each ballistic trajectory of the satellite generates a unique Doppler shift. They asked, Could they find the Keplerian elements that would generate theoretical Doppler curves that matched experimental observations of a real pass? If they could—bingo! The Keplerian elements fed into the theo­retical model would be the elements defining Sputnik’s orbit. They could test their findings by predicting the time that the real satellite would be at a given position if it had the Keplerian elements they had found by this method.

They knew enough from their observations and knowledge of physics to make rough initial guesses about Sputniks Keplerian elements. They fed these to the computer, generating theoretical lists of Doppler shifts. They watched the output, the reams of octal numbers, looking at how closely the theoretical values matched the observations for a particular pass.

Their project was still not sanctioned officially, but other members of the research center knew that something potentially interesting was afoot.

Their boss, Frank McClure, always insightful if not always diplomatic, was watching—at a discreet distance. From time to time, the trademark pipe of Ralph Gibson, APL’s director, appeared stealthily around a corner.

At some time in the first few weeks, when things happened in a less orderly fashion than this account suggests, Guier heard that Charles Bit- terli, one of APL’s few computer programmers, was writing an algorithm for a standard statistical tool, known as least squares fitting, that would be very useful in Guier’s and Weiffenbach’s research. Guier went in search of Bitterli.

Bitterli, who needed a real-life test problem, was happy to oblige by making his work available. An algorithm is a step-by-step set of instruc­tions for carrying out a computational process. When translated into a computer program, it becomes a standard tool to be pulled out when needed. Today, thousands exist in software libraries. Back then the task of building such libraries had scarcely begun.

Guier’s and Weiffenbach’s problem was that vast numbers of theoret­ical Doppler curves could be generated by varying the Keplerian elements. If it were not for the fact that the physics ruled out some combinations, the number of options would have been impossibly large. Even with the limits that physics imposed, there were many possibilities.

Each theoretical curve had to be checked point by point against observations of Sputnik’s Doppler curves. One theoretical curve might at first look like the best match, but then a comparison point by point could well show that it was not. Further, once they found the best possible match, they needed some quantitative way of assessing how good that fit was, and thus what the errors in the Keplerian elements probably were. So an algorithm for least-squares curve fitting, a statistical technique for find­ing the curve that best represents a set of experimental data, was very desirable to Guier and Weiffenbach. Bitterli’s program, which was for linear equations only (those whose graph is a straight line), had to be generalized for nonlinear equations relating many parameters. It was now that it became apparent that new methods would have to be learned and that specialists in numerical analysis would be needed.

Bitterli’s algorithm worked well, and the comparison of theoretical and observed curves speeded up. Even so, and despite the fact that their intuition was working full time as they made informed guesses about the input values of the Keplerian elements, they couldn’t generate a match that was close enough to be of any help in defining Sputnik’s orbit. Yet, theoret­ically, the method should work.

What next? Check everything; examine the setup for recording the Doppler shifts—connection by connection. Review the raw data, the cal­culations. Weiffenbach combed through everything he could think of. He concluded that the data were good. Guier reviewed the theory, the equa­tions, and the computer programs, which by now could have covered enough rolls of wallpaper to decorate a small room. They discussed the problem with one another and with colleagues, listened to critiques, and incorporated suggestions that seemed apt.

They continued to feed the computer the initial conditions that rep­resented their best guess about the satellite’s orbit. Then the computer would run through the calculations to produce the Doppler shift associ­ated with those elements. The process was iterative, with small changes being made to the starting conditions and fresh Doppler curves generated for each set of conditions. Sometimes, it looked as though they had found as good a fit as they were going to find. Listening still to their intuition, they were convinced they could find a set of Keplerian elements that would generate a theoretical Doppler curve even closer to the experimen­tal data. The new Keplerian elements would sometimes make the curve first grow away from a fit, but then as small changes were made, the theo­retical curve would grow closer than it had been before. The process was a little like finding yourself in a valley, only to climb through trees and find a deeper valley beyond. But still, the match was not good enough.

They grew despondent, questioned themselves, and searched some more. They noticed that from one experimental curve to the next, the fre­quency of the transmitter varied. Sputnik’s oscillator was not stable. Exactly as those planning tracking for the IGY had feared, transmission frequency could not be treated as a constant. But Guier and Weiffenbach needed to know the transmitter frequency in order to generate theoretical Doppler curves.

Now they started varying the value of the transmitter frequency. The same principle applied as when varying the Keplerian elements: if they could find a fit between theory and experiment, then the theoretical val­ues of transmitter frequency as well as of the Keplerian elements must be the values of the actual setup. The task was now way beyond what would have been possible without the recently installed Univac.

The whole process was barreling along when Sputnik I suddenly stopped transmitting. Guier and Weiffenbach had data to work with retro­spectively, much of which still had to be reduced. In practice therefore, Sputnik’s, silence made little difference, but psychologically it was disap­pointing to Guier. The pair were also beginning to suspect that an aspect of the physical world that they had thought could be ignored—the iono­sphere—was, in fact, significant. They were right. But to solve the problem they needed another satellite transmitting two frequencies.

Once again the Soviets obliged. Sputnik II was launched November 3, 1957. It transmitted at twenty and forty megahertz. With these two fre­quencies they could make the correction necessary to solve their problem. Theory and experiment started to come together.

Shortly after the launch of Sputnik II, they did it—found a theoreti­cal curve that gave them Sputnik’s orbit. It was nowhere near as accurate as later orbital determinations would be, but they had a method to determine all the Keplerian elements and the transmitter frequency from data col­lected during one satellite pass over one ground station. Guier and Weif­fenbach were delighted. It was the first time, recalls Guier, that they jumped for joy. Henry Elliott, who with Harry Zinc had joined Guier and Weiffenbach shortly after their work started, recalls being ecstatic. Bitterli, who went on to code far more complicated programs than a least-squares algorithm, came to see this work as one of the most important accom­plishments of his professional career.

Guier and Weiffenbach were now on the edge of new ground, sur­veying terrain that was full of quagmires and briars. The basic computa­tional and statistical methods existed for a novel method of orbital deter­mination and prediction, one that did not rely, as did every other, on measurements of angle. The same computational and statistical techniques would be applied, in an inverted manner, to satellite navigation.

Guier and Weiffenbach suspected that their method was correct to within two or three miles. McClure suggested checking their predictions against orbits determined by other groups. They compared their results with those from Jodrell Bank and the Royal Aircraft Establishment and found good agreement. Later they checked their orbital predictions for Sputnik II and the first American satellite, Explorer I, with Minitrack’s results. Again, the methods gave comparable accuracies.

The insight that each orbital trajectory had a unique Doppler shift associated with it was not new physics; rather, it was an inevitable conse­quence of the Doppler phenomenon in the dynamic, three-dimensional world. Nevertheless, Guier and Weiffenbach were the first to recognize that consequence and its implications for orbital determination.

More importantly, they got the process to work in the real world, where very little behaves ideally. If they had stopped to think about the task too much, they might have concluded that the method would never work to any useful degree of accuracy. If they had known more than they did initially about orbital mechanics, the ionosphere, or the discussions at the IGY, they might not have started their work as they did and thus not developed their techniques.

Some people who heard of their research found it hard to believe that all of the Keplerian elements and the transmitter frequency could be found by fitting a theoretical curve to one experimental set of data. But the trajectory of every satellite pass, part of a definable orbit, has a unique Doppler shift. And in three dimensions the ambiguities of two dimensions are eliminated. The method, to use their terminology, worked because of its exquisite sensitivity to the range rate. It was an excellent example of applied physics, and the danger is that it might sound trivial in the telling, particularly a telling that leaves out the mathematics. To fall back on scien­tific cliche, the task was nontrivial.