The Physics behind “Hidden Figures”

The Physics behind “Hidden Figures”

• Math plays a starring role in the movie "Hidden Figures,"
• Adapted from a book, the movie has to do with three three African-American women working as "human computers" for NASA. The film's standout math whiz is Katherine Goble Johnson.
• Johnson is shown trying to solve equations for the trajectory of astronaut John Glenn's space capsule. They're stumped until Johnson hits upon a solution: "Euler's Method,"
• How she used  Euler's Method was very helpful to sending the astronauts into orbit.
• First off, Euler's Method is indeed pretty old, if not exactly ancient. It was developed by Leonhard Euler a prolific Swiss mathematician who lived 1707-1783.
• He was a mathematician at Carnegie Mellon University
• What has come to be known as Euler's Method is just a tiny fraction of his legendary contributions.

• The method tackles what many people may not realize is a common challenge in math  often the equations just can't be solved exactly.
• When that happens, mathematicians must figure out ways to approximate the answers for specific situations.
• Euler's method is one such technique applied to what is called a differential equation. These equations often show up, among many other places, in physics problems that describe the path of a moving object subject to changing forces.
• When a capsule is flying through space, gravity is constantly tugging at it.
• How hard gravity pulls is related to distance, so as the spacecraft gets nearer to or farther away from Earth, the forces on it also change.

• One way to visualize the meaning of the equations could be as ocean currents
• As you travel through the water, the currents change direction and speed
• If you're planning to navigate from a remote island on a raft, you'd want to determine exactly how you'd float through the water, which is a pathway made up of infinitely many points which is analogous to an exact solution to a differential equation.
• Sometimes it's impossible to get an ideal solution.
• Instead, as you drift along, you could measure the current at regular time intervals.
• By knowing your starting point and assuming the current stays roughly constant between readings, you could plot an approximate trajectory.
• This process of calculating solutions at discrete points and connecting them is essentially Euler's Method
• The method works best when the points are close together and when the solution changes slowly and smoothly, because errors can accumulate at each step of the process.
• Approximating a pathway made up of infinitely many points, by linking together a finite number of calculations, is an example of something called a numerical approach in mathematics.

• Euler's Method works numerically
• Euler's Method is one of the simplest of many numerical methods that now exist for solving differential equations.
• Rudy Horne, a mathematician at Morehouse College in Atlanta, was the math advisor to the movie, and it was he who suggested Euler's Method for the key blackboard scene.
• The scene focuses on how to get John Glenn's capsule back to Earth, and Horne says NASA had derived a set of differential equations in the late 1950s to describe the re-entry.
• Euler's Method is one way to solve the equations

• The work for solving these coupled differential equations was done by the whole team of researchers at NASA and possibly in part by Katherine Johnson
• Katherine Johnson began work at the National Advisory Committee for Aeronautics, the predecessor to NASA, in 1953.
• She analyzed test flight data and helped calculate the trajectory of the first American manned space flight, Alan Shepard 1961 trip.
• In 1960, she became the first woman in the Space Flight Division to co author a published technical report.
• In the report Johnson and engineer Ted Skopinski work through some of the key calculations needed to make sure an orbiting space capsule passes over a specified latitude and longitude on the Earth.
• Such calculations were essential so that the Navy could be at the right place to rescue the astronaut once he splashed down in the ocean
• Johnson and Skopinski calculations draw upon multiple branches of math and require numerical methods.
• Complicating the picture is the fact that the Earth is not a perfect sphere, as assumed in idealized orbital mechanics calculations, but bulges slightly in the middle, like a squashed ball, which causes the capsule's orbit to shift slightly over time.
• But the full extent of the math that Katherine Johnson and the other women and men of NASA used to send astronauts safely into space and back is much richer and deeper than any one approach.

Jared Blatt

Period G