In order to describe a flight path in space, the trajectory experts simply need to know two things: where the spacecraft is and how fast it is going at any particular time in the flight. But before they can express these two concepts they must have some frame of reference against which to measure them.
If I were throwing a stone across my back garden and wished to define its path -assuming I had access to the necessary measuring equipment - I might be able to state that 1 second into its flight, the stone was 4 metres from my neighbour's fence behind me, 3 metres above the lawn and 2 metres from my house wall. For the same moment in time, I could also analyse the stone's speed, stating how fast it was moving away from the fence, its speed away from or towards the lawn, and how fast it was moving with respect to my house. In total, for that moment in time, I would have six numbers that would not only define the stone's position and speed in three dimensions, but could also be applied to Newton's laws of motion to predict the stone's continuing journey.
Describing a spacecraft's trajectory is exactly analogous. Its position and speed at a given time are expressed in three dimensions with respect to some reference or sense of which way is up. Position is expressed as three coordinates; each plotted along the x, y and z axes of the current reference. Likewise, speed is resolved to three velocities, a definition of speed where the direction of motion is taken into account; and again, they are plotted along the x, y and z axes. This set of six numbers is collectively known as the state vector. Computers use the state vector as a starting point in their calculations and extrapolate the flight path forward from that point to predict where the spacecraft will be at any time in the future, given the gravitational fields of any bodies in the solar system that would significantly affect its flight through space. Since the lives of the Apollo crews depended upon the accuracy of a spacecraft's state vector, a lot of effort was expended in refining it.
In the early days, around the time Apollo became a mission to the Moon as opposed to being a generic, advanced spacecraft whose role had not been defined, managers expected that the state vector would be determined solely by the crew. Apollo had become a part of the Cold War, a grab for prestige by the United States at a time when they and the Soviet Union stared at each other, each with nuclear weapons in hand, waiting for the other to blink. There were serious concerns that the Soviets might try to interfere with the Apollo flights, perhaps by jamming radio transmissions, therefore it was decided that the guidance and navigation system should be completely autonomous. Once dispatched to the Moon, the crew should be able to navigate, conduct their mission, and return home entirely without assistance from the ground. This philosophy drove the design of the spacecraft's guidance system by the Instrumentation Laboratory of the Massachusetts Institute of Technology. However, by the time Apollo was ready to fly, a lot had changed.
In the mid-1960s, NASA had begun to send probes to the Moon that were either deliberately crash-landed (the Ranger probes), soft-landed (the Surveyors) or sent into orbit (the Lunar Orbiters obviously). These were mainly for reconnaissance purposes in support of Apollo, with some scientific gain coming off the back of them. With time, the people running these missions became increasingly adept at controlling their spacecraft from the ground. Moreover, techniques to accurately determine the state vector from Earth using radio tracking were refined to levels of exquisite accuracy. Also, although Cold War rivalry was still real, it had become less belligerent. NASA decided that ground-based techniques would be the prime means of determining the state vector. The crew would still make their own separate determination, but only as a backup to be used in case of emergencies.
Keeping track of a spacecraft's motion far away from Earth is a wondrous application of human ingenuity and knowledge. It requires a blend of heavy engineering to build and control huge dish antennae, allied to the subtle, precise reception and measurement of vanishingly weak radio signals.
Two techniques were used for Apollo tracking and both of these were cleverly interwoven into the same radio signals that carried all the communication between the spacecraft and Earth: voice, television and telemetry from the spacecraft's systems and data uploads from the ground station on Earth to the spacecraft's computer. Because so many functions were brought together into one S-band radio signal, the system was known as Unified S-band. Its tracking capability could determine range and radial velocity - that is, the distance to the spacecraft and how fast it was approaching, or receding from, the antenna system on Earth.
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