Example: the calculations needed for space missions. Let me give you a quick idea of some of those. To start, we have to figure out how fast a spacecraft must travel to escape the Earth’s gravity and head off into space. How powerful must its engines be to accelerate it to that speed? How much fuel will they need? Once in space, what’s the best path to the craft’s destination, whether the Moon or Mars or some even farther world? How do we account for the destination’s own path through space? How do we use Earth’s gravity, and that of other planets en route, as a slingshot to spur the craft on its way?
Those just scratch the surface. There are plenty more where they came from. Today, powerful computers—the electronic kind—can do these calculations so swiftly, we forget that once upon a time, human computers had to do them.
This week, one of the best known such computers died at 101. That’s Katherine Johnson of the National Aeronautics and Space Administration (Nasa).
It’s a pity, if not very surprising, that it took a 2017 Hollywood film (Hidden Figures) for the wider world to come to know of Johnson—and indeed, others like her. After all, she was a critical part of plenty of Nasa’s missions: the first American in space, Alan Shephard’s 1961 flight; John Glenn’s orbit of the Earth in 1962; the Apollo lunar missions including Apollo 11’s Moon landing in 1969; and later even the Space Shuttle. But especially in Nasa’s early days, their human computers worked in the shadows, never really part of the agency’s public image. Somehow the calculations just got done, and the flights just happened, and few outside Nasa thought to ask, “but how did you get all this right?"
So as my own tribute to a fine mind, let me give you a flavour of some of the calculations above. Let’s start with easily the most straightforward of them, the escape velocity from the Earth.
The way to understand this is, first, to remember those hoary old terms, potential and kinetic energy. A rocket at rest has a certain inherent potential energy that it converts into kinetic energy when it starts moving. (Not just rockets: All of us do this energy conversion when we go from being at rest to moving). This involves the mass of the Earth (M), the mass of the rocket (m), the size of the Earth (its radius r), and Newton’s gravitational constant, G, which is a measure of how strong gravity really is (actually pretty weak): the rocket’s potential energy is GMm/r. In effect, this tells us how powerfully gravity is acting on the rocket.
When the rocket starts moving, its kinetic energy is a function of its mass m and its velocity v: mv2/2. (For here and now, trust me about these energy formulae). If you think about it, we want the rocket to be moving fast enough that its kinetic energy equals the potential energy inherent in gravity’s hold on it. Because that’s when it will be able to break that hold.
Thus we equate the two energies: GMm/r = mv2/2. Some manipulation later, we get v = sqrt (2GM/r). That’s the escape velocity.
Note that this velocity is dependent solely on the mass and size of the Earth. Stick in figures for each of those and for Newton’s constant G, and we find that v is just over 11km/second. That is, a rocket would have to accelerate to that speed to break free of Earth’s clutches. Naturally, the same reasoning applies to other planets as well. If our spacecraft lands on Jupiter, say, it may be pretty hard to get it off. For it would have to ramp up to nearly 60km/s to get away.
Here’s is an early step in planning the exploration of space: If we know that escape velocity, we know how powerful the rocket’s engine’s will have to be.
But as I said, this is definitely the most straightforward, the easiest, of the calculations space travel needs. Katherine Johnson’s work went far beyond that.
For example, think of a spacecraft that’s orbiting the Earth. We want it to return, and to a specific spot on the planet. (Makes sense: No use rushing all over the world chasing it down). At what point in its orbit will it have to begin its descent to Earth? Given its orbital velocity and direction, how will it need to be positioned when it shuts off its engine? How do you take into account the “oblateness" of the Earth, meaning that it is a slightly flattened sphere? All this is important to determine, for a safe return of the craft.
In 1960, Johnson co-authored a Nasa report called Determination of Azimuth Angle at Burnout for Placing a Satellite Over a Selected Earth Position (go.nasa.gov/2Vr2Ty5). In it, she gathered together various measures she needed for the calculations. Two of them: the Earth’s radius in feet (20,902,260) and the speed at which the Earth is rotating in degrees per minute (0.25068). Plenty of intricate calculations followed, all done essentially by hand, and eventually Johnson’s report showed how to answer these fundamental questions about spaceflight. Today, all of us have easy access to computational knowledge and resources that Nasa could not even dream of in the early 1960s, and it would still be an interesting challenge to tackle what Johnson addressed in that report. Hers was a remarkable feat of reasoning and mathematics, and especially remarkable given the era in which she wrote that report.
Two years later, Nasa had IBM computers in place that had calculated and were going to control the trajectories of John Glenn’s orbit of the earth. It’s a measure of the respect the early astronauts had for Johnson and her colleagues that Glenn asked Nasa to “get the girl". He wanted her to verify the same calculations just as she had done all along—yes, essentially by hand. “If she says they’re good," said Glenn, “then I’m ready to go." Of course, Johnson did say they were good and Glenn did go.
Still later, there was the Apollo mission. While Apollo 11’s lunar module (LM) was on the surface of the Moon, its command module (CM) was orbiting above, waiting for the LM to lift off and then link up so Armstrong, Aldrin and Collins could come home. And the intricate calculations for the delicate business of linking a rising LM to the orbiting CM, what of those? Katherine Johnson did them, and forever saw it as her greatest contribution to space exploration.
The movie Hidden Figures pays tribute to Johnson and her equally remarkable African-American colleagues at Nasa, celebrating not just their mathematical ability but their civil rights struggles too. After all, the US in the late 1950s and early 1960s was not an easy place to be black, nor yet black and talented. When I went to see the film in 2017, I had a Johnson story on my mind, from a friend in Washington I’ll call S. In the early 1960s, S’s father was a fresh graduate, hired as one of the first programmers of Nasa’s new electronic computers from IBM. When he and his new wife moved into town, one of Nasa’s earlier computers invited them to join their church choir. In 1965, just before S was born, it was this same computer who got the choir together to give a gift to S’s parents. And while they knew this computer was a mathematician, it was only when they saw Hidden Figures in 2017 that S’s parents fully appreciated the role she had played in Nasa’s journey.
That computer was, of course, Katherine Johnson. “I liked what I was doing," she says in a tribute Nasa released after her death (bit.ly/3aaBAwm). “I liked work. I liked stars, and the stories we were telling."
Go well, Katherine Johnson. They are some fantastic stories you told.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun