标签: trajectories

  Fly me to the moon Let me play among the stars Let me see what spring is like On a Jupiter and Mars. —Bart Howard, 1954 (jazz standard sung by Frank Sinatra, among others)   We have a new topic this month, and it has absolutely nothing—well, almost nothing—to do with the topics we've been discussing lately—software development, testing, and the like. On the other hand, it certainly has to do with embedded systems and software. Systems that tend to be rather remote. Like 240,000 miles remote.   If you happened to see my column (" Calculating trajectories for Apollo program "), where I reminisced about my days with NASA, you know that I was deeply involved in the planning stages of Project Apollo. During that time, I spent a lot of energy learning neat concepts, making interesting relationships, and picking up skills that make designing lunar missions easier. One of the big disappointments of my young life is that, after Apollo was canceled in 1972, we've never had the curiosity or motivation to go back. So all those neat concepts have pretty much lain fallow.   What you probably don't know is that now, after more than 40 years, I'm finally getting another chance to go back to the Moon, at least vicariously and in spirit. Since last year, I've been working with one of the groups competing for the Google Lunar X-Prize competition. ( www.googlelunarxprize.org/ ). The competition invites non-governmental teams to soft-land a robotic rover on the Moon, and have it carry out certain tasks, which include relaying back real-time TV images, roving over at least 500m, and surviving the cold, two-week long lunar night. The first two teams to do this will share a $30 million prize.   I'm working with a group called Part Time Scientists ( www.part-time-scientists.com/ ), which is based in Germany. This effort has been a very exciting one for me. After all these years, I had pretty much abandoned all hope of working on things Lunar again, but here I am doing just that. What I'm doing for the X-Prize effort is pretty much the same as I did before: designing the trajectories. Managing Editor Susan Rambo and I thought you might be interested in how it's done.   1959 all over again Years before President Kennedy made his famous "Before the decade is out" speech, lots of people were saying, "Let's go to the Moon." The obvious response, to anyone with the slightest spirit of adventure, was to say, "Yeah, let's!" closely followed by, "How do we do that?" In 1959, we didn't have a very good answer.   Most of the work I did for NASA was to design the trajectory that gets people to the Moon and back. At the time, space was truly a new frontier for us. We knew virtually nothing about the physics of the problem, we didn't have very good reference books to study, and we didn't have time to go get degrees in astrodynamics. One of the NASA engineers, who had studies astrodynamics, created a handwritten "Cliff Notes" sort of document, and we all studied Xeroxed copies of it.   My first attempts to get a satisfactory lunar trajectory were pretty pitiful, but I learned. Early on, I decided that I wanted more than just to get a final, satisfactory trajectory. I wanted to understand, at a very deep level, the principles involved.   In the beginning I had more questions than answers, but from time to time I'd get little glimmers of insight. Whenever I thought about the problem a little more, I'd have one more little glimmer. Now, after more than 40 years of thinking, I believe I finally understand what's going on to that deeper level. I'd like to share that understanding with you.   It's not easy The movie Apollo 13 makes the task of getting to the Moon seem pretty easy. You point the nose of the spacecraft at the Moon, light off that big J-2 rocket motor on the Saturn upper stage, and off you go. Simple.   The reality is not simple at all. You can't just aim at the Moon, because there's the small matter of an Earth and a Moon and their gravitational fields. Before you lit the candle, you were orbiting close to the Earth. Afterwards, the spacecraft still wants to orbit the Earth, so your path is going to be curved in weird and wonderful ways.   Take a look at figure 1 , which depicts the now-familiar "Figure 8" circumlunar trajectory. This trajectory, which goes out, swings around the Moon, and returns to the Earth, attracted a lot of my attention in those early days, 1 where the spacecraft goes out to the Moon, swings around it, and returns to the Earth.     Figure 1: The circumlunar figure-8 trajectory.   The circumlunar trajectory became central to the Apollo missions, whether they landed on the Moon or not, and for good reason: We didn't just want to send people to the Moon. We wanted to get them back. The nice part about the circumlunar trajectory is that you're coasting in free-fall all the way. Once the booster has done its job, executing what was called the Trans-Lunar Injection (TLI) burn, the spacecraft can coast all the way to the Moon and back. If you've executed that burn perfectly, the spacecraft will coast out to the Moon, swing around it, and return to a safe reentry at Earth, with no more rocket burns required. For obvious reasons, we called it the Free Return trajectory.   Of course, nobody had any illusions that the Free Return was going to be truly free, because the TLI is not going to be perfect Every time you fire a rocket motor, you risk getting it slightly wrong. and any error at the beginning of the trajectory translates into a bigger error at the end. What's more, the circumlunar trajectory is particularly sensitive to errors in its initial conditions. Even so, using the Free Return trajectory, the astronauts had at least a fighting chance of getting home alive, if something bad happened. And if they could still apply some midcourse corrections, as Jim Lovell and his crew did during Apollo 13, the fighting chance gets a whole lot more optimistic.   Ironically, the first Apollo mission not to start with a Free Return trajectory was Apollo 13. After the explosion in the liquid oxygen tank, they had to apply a burn, using the LEM motor, to get back onto a Free Return path.   Since the days of Sputnik, we've become used to seeing orbits depicted as circular or elliptical. Not surprisingly, the physics problem of the motion of two gravitating bodies—Sun and planet, Earth and Moon, or Earth and spacecraft—is called the two-body problem. Way back in 1609, Johannes Kepler conjectured, based on Tycho Brahe's meticulous observations of Mars, that the planets moved in elliptical orbits about the Sun.   Sixty years later, Isaac Newton solved the two-body problem, and confirmed Kepler's laws. He did it, by the way, by first inventing his three laws of motion, his law of gravity, and calculus. He was 22. So, Isaac, how was your summer vacation?   Newton found that the general solution of the two-body problem is not always an ellipse. It can be any conic section , which includes the circle, ellipse, parabola, and hyperbola.  

with NASA, you know that I was deeply involved in the planning stages of Project Apollo. During that time, I spent a lot of energy learning neat concepts, making interesting relationships, and picking up skills that make designing lunar missions easier. One of the big disappointments of my young life is that, after Apollo was canceled in 1972, we've never had the curiosity or motivation to go back. So all those neat concepts have pretty much lain fallow.   What you probably don't know is that now, after more than 40 years, I'm finally getting another chance to go back to the Moon, at least vicariously and in spirit. Since last year, I've been working with one of the groups competing for the Google Lunar X-Prize competition. ( www.googlelunarxprize.org/ ). The competition invites non-governmental teams to soft-land a robotic rover on the Moon, and have it carry out certain tasks, which include relaying back real-time TV images, roving over at least 500m, and surviving the cold, two-week long lunar night. The first two teams to do this will share a $30 million prize.   I'm working with a group called Part Time Scientists ( www.part-time-scientists.com/ ), which is based in Germany. This effort has been a very exciting one for me. After all these years, I had pretty much abandoned all hope of working on things Lunar again, but here I am doing just that. What I'm doing for the X-Prize effort is pretty much the same as I did before: designing the trajectories. Managing Editor Susan Rambo and I thought you might be interested in how it's done.   1959 all over again Years before President Kennedy made his famous "Before the decade is out" speech, lots of people were saying, "Let's go to the Moon." The obvious response, to anyone with the slightest spirit of adventure, was to say, "Yeah, let's!" closely followed by, "How do we do that?" In 1959, we didn't have a very good answer.   Most of the work I did for NASA was to design the trajectory that gets people to the Moon and back. At the time, space was truly a new frontier for us. We knew virtually nothing about the physics of the problem, we didn't have very good reference books to study, and we didn't have time to go get degrees in astrodynamics. One of the NASA engineers, who had studies astrodynamics, created a handwritten "Cliff Notes" sort of document, and we all studied Xeroxed copies of it.   My first attempts to get a satisfactory lunar trajectory were pretty pitiful, but I learned. Early on, I decided that I wanted more than just to get a final, satisfactory trajectory. I wanted to understand, at a very deep level, the principles involved.   In the beginning I had more questions than answers, but from time to time I'd get little glimmers of insight. Whenever I thought about the problem a little more, I'd have one more little glimmer. Now, after more than 40 years of thinking, I believe I finally understand what's going on to that deeper level. I'd like to share that understanding with you.   It's not easy The movie Apollo 13 makes the task of getting to the Moon seem pretty easy. You point the nose of the spacecraft at the Moon, light off that big J-2 rocket motor on the Saturn upper stage, and off you go. Simple.   The reality is not simple at all. You can't just aim at the Moon, because there's the small matter of an Earth and a Moon and their gravitational fields. Before you lit the candle, you were orbiting close to the Earth. Afterwards, the spacecraft still wants to orbit the Earth, so your path is going to be curved in weird and wonderful ways.   Take a look at figure 1 , which depicts the now-familiar "Figure 8" circumlunar trajectory. This trajectory, which goes out, swings around the Moon, and returns to the Earth, attracted a lot of my attention in those early days, 1 where the spacecraft goes out to the Moon, swings around it, and returns to the Earth.     Figure 1: The circumlunar figure-8 trajectory.   The circumlunar trajectory became central to the Apollo missions, whether they landed on the Moon or not, and for good reason: We didn't just want to send people to the Moon. We wanted to get them back. The nice part about the circumlunar trajectory is that you're coasting in free-fall all the way. Once the booster has done its job, executing what was called the Trans-Lunar Injection (TLI) burn, the spacecraft can coast all the way to the Moon and back. If you've executed that burn perfectly, the spacecraft will coast out to the Moon, swing around it, and return to a safe reentry at Earth, with no more rocket burns required. For obvious reasons, we called it the Free Return trajectory.   Of course, nobody had any illusions that the Free Return was going to be truly free, because the TLI is not going to be perfect Every time you fire a rocket motor, you risk getting it slightly wrong. and any error at the beginning of the trajectory translates into a bigger error at the end. What's more, the circumlunar trajectory is particularly sensitive to errors in its initial conditions. Even so, using the Free Return trajectory, the astronauts had at least a fighting chance of getting home alive, if something bad happened. And if they could still apply some midcourse corrections, as Jim Lovell and his crew did during Apollo 13, the fighting chance gets a whole lot more optimistic.   Ironically, the first Apollo mission not to start with a Free Return trajectory was Apollo 13. After the explosion in the liquid oxygen tank, they had to apply a burn, using the LEM motor, to get back onto a Free Return path.   Since the days of Sputnik, we've become used to seeing orbits depicted as circular or elliptical. Not surprisingly, the physics problem of the motion of two gravitating bodies—Sun and planet, Earth and Moon, or Earth and spacecraft—is called the two-body problem. Way back in 1609, Johannes Kepler conjectured, based on Tycho Brahe's meticulous observations of Mars, that the planets moved in elliptical orbits about the Sun.   Sixty years later, Isaac Newton solved the two-body problem, and confirmed Kepler's laws. He did it, by the way, by first inventing his three laws of motion, his law of gravity, and calculus. He was 22. So, Isaac, how was your summer vacation?   Newton found that the general solution of the two-body problem is not always an ellipse. It can be any conic section , which includes the circle, ellipse, parabola, and hyperbola.    

All of my work on Apollo came in a frenetic four-year period, from 1959 through 1963. It was in 1959 that I began work for the Theoretical Mechanics Division at NASA, at Langley Research Center. This was just shortly after NASA was formed. Shortly after I arrived there, a paper came out of the think tank, Rand Inc., describing a class of lunar trajectories called free-return, circumlunar trajectories—the now-familiar figure-8 paths. It was immediately obvious that this class of trajectories was the only reasonable way to go to the Moon and back. We began studying them intensely, using first a two-dimensional simulation of the restricted three-body problem, and later a 3-D, exact simulation. In those days, we didn't have spreadsheet programs to draw graphs for us; we had to draw them ourselves. As low man on the TMD totem pole, I got elected to run parametric studies on the computer and plot the results. That task worked in my favor, though, because I gained an understanding of the physics of the problem and the relationship between parameters that I don't think I would have gotten, otherwise. I wasn't content to just make runs and plot curves; I wanted to UNDERSTAND what was going on, and I think that put us ahead of the Rand guys.     "After lifting off from the lunar surface, the lunar module made its rendezvous with the command module. The Eagle docked with Columbia, and the lunar samples were brought aboard. The LM was left behind in lunar orbit while the three astronauts returned in the command module to the blue planet in the background. (NASA photo ID AS11-44-6642)." -- From Apollo 11 mission, July 20, 1969; photo courtesy of NASA from   30th anniversary site I pretty much designed the parametric studies. Our group, the Lunar Trajectory Group, was small. Our group leader, Bill Michael, gave me the assignment, and he and I talked daily. But he never had to tell me, Ok, run this trajectory ... now run that one. I was the one making the day-to-day decisions. Bill designed the computer program but neither of us built it. In those days, things were still done "closed shop," and someone from the computer division wrote the code. But I did what would now be called desk-checking, checking the code (in IBM 702 assembler) to make sure it was right. Later, I did a sensitivity study, plotting the sensitivity of final to initial conditions. Nowadays, we'd call that a state transition matrix, but we didn't know that term, at the time. My boss and I published a paper in 1961, which was the second paper published on circumlunar trajectories. We also developed quite a number of rules of thumb, approximations, and "patched conic" methods that allowed us to study circumlunar trajectories without spending tons of money for computer time. Steering At the time, we weren't thinking of Project Apollo. In fact, I had never heard of it. We had plans to send a "Brownie" camera around the Moon before 1965, using NASA's solid-fuel Scout research vehicle. When Scout's projected payload at the Moon went from a few hundred pounds, through zero and negative, those plans were abandoned. However, the effort left us more than ready for Apollo when it came along. Next, I began studying the problem of steering the spacecraft, i.e., midcourse corrections. We were all pretty dismayed by the great sensitivity of the circumlunar trajectory to errors in initial conditions, and we knew the accuracy of the Scout boost guidance was orders of magnitude too low. Midcourse guidance would be essential. To my knowledge, my work on that topic was one of the earliest done, though I suspect the fellows from MIT, like Richard Battin, were studying the same problem. In fact, it was Battin's seminal work, seeking a midcourse correction scheme for Apollo, that helped make the Kalman filter practical. During this time, I also discovered a family of trajectories that were relatively insensitive to injection errors. These trajectories had a nominally vertical reentry. You see, for orbits as highly elliptical as translunar orbit, the perigee is almost totally a function of the angular momentum. The requirement that we return with essentially the same angular momentum as the original orbit implies that the passage past the Moon should not alter that momentum. To achieve this result requires very tight guidance. The required angular momentum, translated to velocity at the distance of the Moon, works out to be around 600 f/s. This should not be altered if we expect to achieve the kind of grazing entry required for a manned mission. Suppose, however, that we design an orbit that leaves the Moon with zero velocity. Then it should reenter the Earth, with a vertical entry—fatal for astronauts, but not for instruments. Even if our errors impart as much as 600 f/s of unplanned velocity, we will still enter the Earth's atmosphere and land, somewhere. If you don't care where, this peculiar class of trajectories allows one to fly to the Moon and back, even with the crude guidance of the old Scout. Of course, the mission never flew, so the trajectory is now nothing more than a footnote to space history. But an interesting one, nevertheless. In 1961 I moved to General Electric in Philadelphia, where they had landed a study contract for Apollo and were also planning to bid on the hardware contract. I was responsible for all the generation of nominal trajectories for those efforts. I did similar stuff for lunar missions that never flew or flew with radically altered plans; the names of projects Surveyor and Prospector come to mind. I think I did some of my very best work at GE. Generating a lunar trajectory is not easy. The problem is a two-point boundary-value problem, complicated by the fact that both points (the launch and return sites) are fixed on a rotating Earth, and we have the "minor" midpoint constraint that the trajectory come somewhere near the Moon. We quickly learned that simply trying to guess at suitable launch conditions was a hopeless venture. Therefore a large part of my energies went into building quite a number of patched-conic approximations, programs to solve the complicated spherical trig, front-end and back-end processors, and "wrappers" for GE's N-Body program. My crowning achievement was a fully automated program that required only the barest minimum of inputs, such as the coordinates of the launch and landing points, plus a few other things such as year of departure, lunar miss distance, and lighting conditions at both the Moon and Earth reentry. The program would then seek out the optimal, minimum-energy trajectory that would meet the constraints. I was also asked to study the problem of returning from the Moon to the Earth. Thanks to my experience with approximate methods, I knew exactly how to do this: Give the spacecraft a tangential velocity of about 600 f/s, relative to the Moon. From that desired end, it's fairly easy to figure out what sort of launch one needs at the Moon's surface. We generated quite a number of trajectories, both approximate and exact, that effected the Moon-to-Earth transfer. As far as I know, they were the first such studies ever done. During this same period, I was developing a method for computing nearly parabolic orbits. The classical two-body theory upon which all approximate methods are based has three distinct solutions, depending upon whether the orbit is elliptic, parabolic, or hyperbolic. The equations for the elliptic and hyperbolic cases all go singular for an eccentricity of one. Unfortunately, that's exactly the kind of orbits involved in translunar missions. It drove both the computers and their programmers crazy, trying to solve problems so close to a singularity. I developed a set of power series solutions based on Herrick's "Unified Two-Body Theory." Though Herrick's original formulation was unsatisfactory (his functions required two arguments), it was a rather short step to replace them by new functions that used only one argument. Between 1961 and 1963, I published quite a number of papers on these functions, most of them internal to GE and widely distributed within NASA. I was also busily putting them to work in trajectory generators. Unfortunately for me, I never managed to get credit for them; you'll find them today in astrodynamics texts, called the Lemon-Battin functions. In 1963 I spent nine months at GE's Daytona Beach facility, studying the problem of how to abort from the lunar mission. At first glance, you might think nothing much can be done; once zooming toward the moon, you're pretty much committed to continue. However, out near the Moon, the velocity relative to the Earth is rather low, and you have tons of available fuel for maneuvering, thanks to the need to enter and exit lunar orbit. Therefore, it's theoretically possible during some portions of the mission to simply turn around and head back for Earth. In other phases, you can lower the trip time quite a bit, by accelerating either toward the Moon (to hasten the swingby) or toward the Earth. In my studies, I discovered yet another class of returns, in which you make no attempt to slow down, but simply deflect the velocity downwards enough to graze the Earth's atmosphere. Aerodynamic braking does the rest. All these abort modes ended up designed into the missions and programmed into the Apollo flight computers. Two of them—accelerating toward the Moon on the way out, and toward the Earth on the way back—were used to reduce the life-critical flight time during Apollo 13. If you watch the movie of the same name, you'll hear discussions of Fast Return and Free Return trajectories, both of which I had a hand in designing. The computers Most of our work was done on the IBM 70x series, from 702 through 7094. Later in the Apollo era, we had Univac 1108s and 1110s, and the CDC 6600. None of them, of course, could hold a candle to the modern PC with Pentium processor. I think their cycles times were all around a microsecond, which is equivalent to a 1MHz clock speed; not even up to the smallest microprocessor of today. Memory was via magnetic cores, which were even slower. The IBM's had a maximum of 32K RAM (but 36-bit words, not bytes). Nevertheless, the later processors, notably the Univacs and CDCs, were no slouch. Both had hardware floating point and long word lengths (CDC's words were 60 bits long), so one could do some serious number crunching. Despite their low performance, on paper, we got a lot of work out of those old machines. For those used to waiting 15 seconds for a Windows spreadsheet to even load, it's difficult to imagine how much work a 1MHz computer can do when it's running full tilt in machine language, not encumbered by bloated software, interpreted languages, and a GUI interface. I'm old enough to remember, and look back on those days wistfully.  

All of my work on Apollo came in a frenetic four-year period, from 1959 through 1963. It was in 1959 that I began work for the Theoretical Mechanics Division at NASA, at Langley Research Center. This was just shortly after NASA was formed. Shortly after I arrived there, a paper came out of the think tank, Rand Inc., describing a class of lunar trajectories called free-return, circumlunar trajectories—the now-familiar figure-8 paths. It was immediately obvious that this class of trajectories was the only reasonable way to go to the Moon and back. We began studying them intensely, using first a two-dimensional simulation of the restricted three-body problem, and later a 3-D, exact simulation. In those days, we didn't have spreadsheet programs to draw graphs for us; we had to draw them ourselves. As low man on the TMD totem pole, I got elected to run parametric studies on the computer and plot the results. That task worked in my favor, though, because I gained an understanding of the physics of the problem and the relationship between parameters that I don't think I would have gotten, otherwise. I wasn't content to just make runs and plot curves; I wanted to UNDERSTAND what was going on, and I think that put us ahead of the Rand guys.   "After lifting off from the lunar surface, the lunar module made its rendezvous with the command module. The Eagle docked with Columbia, and the lunar samples were brought aboard. The LM was left behind in lunar orbit while the three astronauts returned in the command module to the blue planet in the background. (NASA photo ID AS11-44-6642)." -- From Apollo 11 mission, July 20, 1969; photo courtesy of NASA from   30th anniversary site I pretty much designed the parametric studies. Our group, the Lunar Trajectory Group, was small. Our group leader, Bill Michael, gave me the assignment, and he and I talked daily. But he never had to tell me, Ok, run this trajectory ... now run that one. I was the one making the day-to-day decisions. Bill designed the computer program but neither of us built it. In those days, things were still done "closed shop," and someone from the computer division wrote the code. But I did what would now be called desk-checking, checking the code (in IBM 702 assembler) to make sure it was right. Later, I did a sensitivity study, plotting the sensitivity of final to initial conditions. Nowadays, we'd call that a state transition matrix, but we didn't know that term, at the time. My boss and I published a paper in 1961, which was the second paper published on circumlunar trajectories. We also developed quite a number of rules of thumb, approximations, and "patched conic" methods that allowed us to study circumlunar trajectories without spending tons of money for computer time. Steering At the time, we weren't thinking of Project Apollo. In fact, I had never heard of it. We had plans to send a "Brownie" camera around the Moon before 1965, using NASA's solid-fuel Scout research vehicle. When Scout's projected payload at the Moon went from a few hundred pounds, through zero and negative, those plans were abandoned. However, the effort left us more than ready for Apollo when it came along. Next, I began studying the problem of steering the spacecraft, i.e., midcourse corrections. We were all pretty dismayed by the great sensitivity of the circumlunar trajectory to errors in initial conditions, and we knew the accuracy of the Scout boost guidance was orders of magnitude too low. Midcourse guidance would be essential. To my knowledge, my work on that topic was one of the earliest done, though I suspect the fellows from MIT, like Richard Battin, were studying the same problem. In fact, it was Battin's seminal work, seeking a midcourse correction scheme for Apollo, that helped make the Kalman filter practical. During this time, I also discovered a family of trajectories that were relatively insensitive to injection errors. These trajectories had a nominally vertical reentry. You see, for orbits as highly elliptical as translunar orbit, the perigee is almost totally a function of the angular momentum. The requirement that we return with essentially the same angular momentum as the original orbit implies that the passage past the Moon should not alter that momentum. To achieve this result requires very tight guidance. The required angular momentum, translated to velocity at the distance of the Moon, works out to be around 600 f/s. This should not be altered if we expect to achieve the kind of grazing entry required for a manned mission. Suppose, however, that we design an orbit that leaves the Moon with zero velocity. Then it should reenter the Earth, with a vertical entry—fatal for astronauts, but not for instruments. Even if our errors impart as much as 600 f/s of unplanned velocity, we will still enter the Earth's atmosphere and land, somewhere. If you don't care where, this peculiar class of trajectories allows one to fly to the Moon and back, even with the crude guidance of the old Scout. Of course, the mission never flew, so the trajectory is now nothing more than a footnote to space history. But an interesting one, nevertheless. In 1961 I moved to General Electric in Philadelphia, where they had landed a study contract for Apollo and were also planning to bid on the hardware contract. I was responsible for all the generation of nominal trajectories for those efforts. I did similar stuff for lunar missions that never flew or flew with radically altered plans; the names of projects Surveyor and Prospector come to mind. I think I did some of my very best work at GE. Generating a lunar trajectory is not easy. The problem is a two-point boundary-value problem, complicated by the fact that both points (the launch and return sites) are fixed on a rotating Earth, and we have the "minor" midpoint constraint that the trajectory come somewhere near the Moon. We quickly learned that simply trying to guess at suitable launch conditions was a hopeless venture. Therefore a large part of my energies went into building quite a number of patched-conic approximations, programs to solve the complicated spherical trig, front-end and back-end processors, and "wrappers" for GE's N-Body program. My crowning achievement was a fully automated program that required only the barest minimum of inputs, such as the coordinates of the launch and landing points, plus a few other things such as year of departure, lunar miss distance, and lighting conditions at both the Moon and Earth reentry. The program would then seek out the optimal, minimum-energy trajectory that would meet the constraints. I was also asked to study the problem of returning from the Moon to the Earth. Thanks to my experience with approximate methods, I knew exactly how to do this: Give the spacecraft a tangential velocity of about 600 f/s, relative to the Moon. From that desired end, it's fairly easy to figure out what sort of launch one needs at the Moon's surface. We generated quite a number of trajectories, both approximate and exact, that effected the Moon-to-Earth transfer. As far as I know, they were the first such studies ever done. During this same period, I was developing a method for computing nearly parabolic orbits. The classical two-body theory upon which all approximate methods are based has three distinct solutions, depending upon whether the orbit is elliptic, parabolic, or hyperbolic. The equations for the elliptic and hyperbolic cases all go singular for an eccentricity of one. Unfortunately, that's exactly the kind of orbits involved in translunar missions. It drove both the computers and their programmers crazy, trying to solve problems so close to a singularity. I developed a set of power series solutions based on Herrick's "Unified Two-Body Theory." Though Herrick's original formulation was unsatisfactory (his functions required two arguments), it was a rather short step to replace them by new functions that used only one argument. Between 1961 and 1963, I published quite a number of papers on these functions, most of them internal to GE and widely distributed within NASA. I was also busily putting them to work in trajectory generators. Unfortunately for me, I never managed to get credit for them; you'll find them today in astrodynamics texts, called the Lemon-Battin functions. In 1963 I spent nine months at GE's Daytona Beach facility, studying the problem of how to abort from the lunar mission. At first glance, you might think nothing much can be done; once zooming toward the moon, you're pretty much committed to continue. However, out near the Moon, the velocity relative to the Earth is rather low, and you have tons of available fuel for maneuvering, thanks to the need to enter and exit lunar orbit. Therefore, it's theoretically possible during some portions of the mission to simply turn around and head back for Earth. In other phases, you can lower the trip time quite a bit, by accelerating either toward the Moon (to hasten the swingby) or toward the Earth. In my studies, I discovered yet another class of returns, in which you make no attempt to slow down, but simply deflect the velocity downwards enough to graze the Earth's atmosphere. Aerodynamic braking does the rest. All these abort modes ended up designed into the missions and programmed into the Apollo flight computers. Two of them—accelerating toward the Moon on the way out, and toward the Earth on the way back—were used to reduce the life-critical flight time during Apollo 13. If you watch the movie of the same name, you'll hear discussions of Fast Return and Free Return trajectories, both of which I had a hand in designing. The computers Most of our work was done on the IBM 70x series, from 702 through 7094. Later in the Apollo era, we had Univac 1108s and 1110s, and the CDC 6600. None of them, of course, could hold a candle to the modern PC with Pentium processor. I think their cycles times were all around a microsecond, which is equivalent to a 1MHz clock speed; not even up to the smallest microprocessor of today. Memory was via magnetic cores, which were even slower. The IBM's had a maximum of 32K RAM (but 36-bit words, not bytes). Nevertheless, the later processors, notably the Univacs and CDCs, were no slouch. Both had hardware floating point and long word lengths (CDC's words were 60 bits long), so one could do some serious number crunching. Despite their low performance, on paper, we got a lot of work out of those old machines. For those used to waiting 15 seconds for a Windows spreadsheet to even load, it's difficult to imagine how much work a 1MHz computer can do when it's running full tilt in machine language, not encumbered by bloated software, interpreted languages, and a GUI interface. I'm old enough to remember, and look back on those days wistfully.