# Katherine Johnson, NASA mathematician and scientist

Katherine Johnson, now a retired NASA research mathematician, was no stranger to geometry. She was the one who would calculate the trajectory of Alan Shepard's 1961 trip into space. This was America's first big step to putting a man on the moon.

Johnson commented that the early trajectory of the Mercury program space launches would be a parabola, and it was easy to predict where it would be at any point. Early in the Mercury program, scientists wanted the capsule to come down at a particular place on Earth after re-entry, and to do this they were trying to compute when it should launch. Johnson said, “’Let me do it. You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off.' That was my forte.”

A typical Mercury Project launch and trajectory which Johnson would use to calculate when to take off in order to splash down in a desired area of the ocean. (Image courtesy of NASA)

The motion of any object which is under gravity’s influence, can be determined by using the acceleration of gravity, the objects launch speed, and launch angle if air friction is negligible. One can separate the horizontal and vertical motions and describe both by using the general motion equations for constant acceleration. (Image courtesy of Georgia State University)

Johnson knew that the initial vector components of velocity would be used in the general motion equations. The diagram above shows trajectories with the same launch speed but with differing launch angles. It can be seen that the 60o and 30o trajectories have the identical range, as will any two pairs of launches at complementary angles. Launching at 45o will give the object maximum range.

Subsequent flights became more complicated, with more variables involving the position and rotation of the Earth and the moon for orbiting. By the time John Glenn was to go up to orbit the Earth, NASA had gone to computers.

Realizing that subsequent Mercury flights and beyond would become increasingly more complex, Johnson acknowledged that they could do much more, so much faster on the computer. As the scientists went to computers, they also asked in parallel for Johnson to manually check and see if the computer trajectory they had calculated was correct. She did check it and it was correct.

Johnson’s, amazing mind, also verified early electronic computer calculations, which she did by using her mathematical genius, for John Glenn’s 1962 launch to orbit as well as the 1969 Apollo trajectory to the moon.

The “Celestial Training Device” shown here in front of Katherine Johnson, was a globe-within-a-globe, which was basically a celestial training device that showed the key coordinates for navigation in Earth orbit in space. (Image courtesy of NASA)

Here is how the Celestial Training device worked for astronauts like John Glenn: Astronauts needed to be able to identify points in space, so they used an angular coordinate system centered on our home planet called celestial coordinates1 which they used at the beginning with the Mercury Program in the early 60s. The device’s key components were declination, the space-analog to latitude, and right ascension, which is a form of astro-longitude. Declination is defined as degrees north or south of the celestial equator, the projection of our Earthly equator into space. Right ascension is roughly defined as an equivalent to longitude, but with a few unique differences. Instead of degrees east or west, right ascension is typically measured in hours, minutes, and seconds (where an hour is 15o s of apparent sky motion as the Earth rotates). The zero-point of the Vernal Equinox is only slightly more sensible than longitude’s arbitrary Prime Meridian: it’s the intersection of the ecliptic and celestial equator at the moment of the spring equinox.

Katherine Johnson has been awarded the Presidential Medal of Freedom, which is the nation's highest civilian honor, by President Obama on Nov. 24, 2015. On May 5, 2016, she returned to NASA Langley, on the 55th anniversary of Alan Shepard's historic flight, to attend a ceremony where a \$30 million, 40,000-square-foot Computational Research Facility was named in her honor. As part of the event, Johnson also received a Silver Snoopy award from Leland Melvin, an astronaut and former NASA associate administrator for education. Often called the astronaut’s award, the Silver Snoopy goes to people who have made outstanding contributions to flight safety and mission success. (Courtesy of NASA)

References

1 What are Celestial Coordinates?, Alan MacRobert, Sky & Telescope, 2006

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