Program Manager, Blavatnik Awards for Young Scientists (Physical Sciences and Engineering) | New York Academy of Sciences
The movie gives a shout-out to Euler's Method -- a centuries-old math technique. Did it really help send astronauts into orbit?
Courtesy of 20th Century FoxCulture
Friday, February 24, 2017 - 15:30
(Inside Science) -- Math plays a starring role in the movie "Hidden Figures," which is nominated for three Oscars, including Best Picture, at this weekend's Academy Awards.
Adapted from a book of the same name by Margot Lee Shetterly, the movie chronicles the grit and ultimate triumphs of three African-American women working as "human computers" for NASA in the segregated south during the space race. The film's standout math whiz is Katherine Goble Johnson. During a pivotal scene, Johnson and a team of white, male engineers are staring at a blackboard, 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," she says. "That's ancient," says one of the engineers incredulously. "Yes. But it works," she counters. "It works numerically."
The scene -- and indeed the timing of key details in the film -- is clearly dramatized for Hollywood, but it raises the questions: What exactly is Euler's Method? And did Katherine Johnson actually use it to help send astronauts into orbit in the 1960s?An "ancient" approximation technique
First off, Euler's Method is indeed pretty old, if not exactly ancient. It was developed by Leonhard Euler (pronounced oy-ler), a prolific Swiss mathematician who lived 1707-1783. "He was one of the greatest in history," said Po-Shen Loh, a mathematician at Carnegie Mellon University in Pittsburgh. 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. For example, 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, Loh said. 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 -- analogous to an exact solution to a differential equation.
Sometimes, however, it's impossible to get this comprehensive, 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, Loh said.
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. That's what Johnson's character in the movie means when she says Euler's Method works "numerically."
Euler's Method is one of the simplest of many numerical methods that now exist for solving differential equations.Euler meets Glenn?
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.
“[Director Theodore Melfi] really liked that concept,” Horne said in an interview with NPR station KGOU. “I didn't expect him to, but he actually put that into the script.”
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, Horne said, which is why he proposed it for the film.
"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," Horne wrote in an email to Inside Science.
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's 1961 trip. In 1960, she became the first woman in the Space Flight Division to coauthor 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, and are discussed in the movie "Hidden Figures" when Johnson's character is first allowed to attend a daily briefing.
Johnson and Skopinski's 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.
"It's really a heroic calculation," said Steve Strogatz, a mathematician at Cornell University in Ithaca, New York, after reading the technical paper. "John Glenn isn't the only hero in this."
In the movie, highlighting Euler's Method works as a dramatic plot device. 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.Filed under
Authorized news sources may reproduce our content. Find out more about how that works. © 2016 American Institute of PhysicsAuthor Bio & Story Archive Catherine Meyers
Catherine Meyers is a senior science writer at AIP.
Tiny Mass, Huge Costs.Physics
Friday, February 24, 2017 - 10:15
(Inside Science) -- “In October 1993 the U.S. Congress terminated the Superconducting Super Collider (SSC) with an estimated price tag that went above 10 billion dollars. The SSC was designed to do a pretty wide range of physics goals, but at the very core of it was to find the Higgs boson. Ever since, this project has been referred to as ‘water shed’ in the history of science, in the history of physics especially, and that the funding for high energy physics [has been] either flat or declining ever since,” said Michael Riordan, adjunct professor of physics, University of California, Santa Cruz.
Riordan also stated, “One day director emeritus of Stanford Linear Accelerator Center (SLAC), Wolfgang Panofsky, best known as "Pief", came into my office and said, ‘You know I think this was a bridge too far.’ I always often puzzled with what that meant, and finally resolved it years later when I realized that this was the title of a book about a disastrous allied campaign during World War II. But that really keyed my thinking about this project.”
According to Riordan, “the SSC was designed to understand, why do particles have mass? The theories generally had mass-less particles in them and they had a high degree of symmetry. You had to break that symmetry down by giving particles mass, and there was a mechanism called the Higgs mechanism that did this in a very natural way, but it required the existence of a particle called the Higgs boson, and nobody could really specify the mass it had to have.”
“There was a large range of masses, ranging from about the mass of a proton, up to about a thousand times the mass of a proton, in which it might occur. So successive particle colliders and accelerators went looking for this. The SSC was built to be sure that they could either find the Higgs boson or find some other phenomena responsible for why particles have mass,” stated Riordan.
“The SSC was understood that it would have to be a multibillion dollar project. Everything the community had done before that had been hundreds of millions of dollars -- that had been evaluated in 1990 dollars -- this one was clearly going to be over 10 billion dollars. When you get to those kinds of numbers you have to bring in partners. In this case it was the Texas politico's members of the military industrial complex. To them it was a high-tech jobs project, and that was the image that they projected in Congress. And that is like the kiss of death, especially when you have a multibillion dollar project that is growing seemingly out of control -- pressing on many other projects, not just science projects, but also water projects, in various districts throughout the nation. Was this project too big for the U.S. high energy physics community to embrace?” said Riordan.
“In general, I think it was. The SSC was indeed a bridge too far,” concluded Riordan.Filed under
Authorized news sources may reproduce our content. Find out more about how that works. © 2016 American Institute of PhysicsAuthor Bio & Story Archive Jason Socrates Bardi
Jason Socrates Bardi is the News Director of the American Institute of Physics and a longtime science writer.
Scientists reveal plans for future experiments to study the faint remnants left behind by the Big Bang.
Southern polar lights over the Amundsen-Scott South Pole Station, with BICEP2 telescope visible in the foreground.Image credits:
Dr. Keith Vanderlinde, NSFSpace
Thursday, February 23, 2017 - 13:45
(Inside Science) -- What exactly happened in the beginning of the universe?
Scientists are trying to answer this ultimate question by deploying more telescopes to help in their quest. The global collaborative project known as Cosmic Microwave Background-Stage 4 aims to construct new ground- and space-based telescopes, as well as expand the capabilities of existing ones. It will be the continuation of the ongoing CMB-Stage 3 project to study the remnants of our distant cosmic past.
Adding new telescopes will allow scientists to obtain higher resolution data of the cosmic microwave background, which is the oldest observable light in the universe and contains tiny hints of what happened at the very beginning.Peeking behind the curtain
The cosmic microwave background was first measured in 1964 by Arno Penzias and Robert Wilson by accident. The detection of this background radiation confirmed the Big Bang theory as the correct explanation of our universe's origin, beating its main rival, the Steady State theory, which posited that matter is created continuously throughout the universe over an infinite time scale. In contrast, the Big Bang theory describes an initial creation event, the remnants of which can be detected today. Penzias and Wilson were awarded a share of the Nobel Prize in physics in 1978 for their discovery.
The cosmic microwave background, as measured by telescopes that see in the microwave spectrum, provides a backdrop to our visible universe. Everywhere you look in the sky, if you can see microwaves, you would see the background radiation from 13.8 billion years ago.
However, those microwaves do not mark the beginning of the universe, but only the beginning of free photons, which can travel long distances and someday reach a telescope. Before the curtain of the early universe lifted, the universe was opaque and photons couldn't travel freely. In fact, according to our current understanding, this first light occurred about 380,000 years after the Big Bang. This was long before any stars or galaxies formed, but also long after the initial inflation of our universe.
But if we cannot see past the cosmic curtain of the early universe, how can we ever figure out what happened during the first 380,000 years?
"The idea is that the universe leaves imprints on [the background] in different ways, and we can look for those imprints, and infer what the universe was doing," said Mark Devlin, an astrophysicist from University of Pennsylvania in Philadelphia. He is also a member of the Conceptual Design Team of the CMB-S4 project.
One of the main motivations for the CMB-S4 project is to better understand the initial rapid expansion of the universe, which occurred within the first instant of the birth of our universe. This period of lightning-fast growth, known as the inflationary epoch, was responsible for creating a huge portion of our universe, and is estimated to have lasted less than one quadrillionth of a quadrillionth of a second. It is truly the "big" and "bang" part of the Big Bang.
"The physics of inflation is still unknown," said Cora Dvorkin, an astrophysicist from Harvard University in Cambridge, Massachusetts. She spoke about the CMB-S4 project at a meeting of the American Physical Society, held in Washington, D.C. this January. "We can shed light on the physics of inflation by using CMB observations."
Timeline of the universe, from the Big Bang to present time. (NASA/Public Domain)More eyes on the sky
The cosmic microwave background emits microwaves with energy that correspond to a temperature of around 2.7 degrees Kelvin. In comparison, the "imprints" mentioned by Devlin correspond to fluctuations around 0.0001 K, approximately a difference of one part in 100,000. Looking for these signals is like looking for fingerprints on the roof of a football stadium.
The detailed, all-sky picture of the cosmic microwave background. The color map covers a range of 0.0002 Kelvin. (NASA and WMAP Science Team/ Public Domain)
"To get to the sensitivity levels that are required to find either inflationary signal, or just say that's not findable, requires about a factor of ten increase in the current capabilities of all of [the current telescopes] combined," Devlin said.
In other words -- the effort requires more data. The data can come from either building more telescopes or making the current ones bigger, or both.
"The detectors are sensitive as they ever could be. The only way to get more sensitivity is to have more detectors on the sky," Devlin said.
Those additional capabilities will take time and money to build. According to the project's website, the CMB-S4 will be a U.S.-led effort with participation from international partners. Per the request of potential U.S. funding agencies, the scientific community is preparing a report on the technical and financial requirements of the project, which will be reviewed by a federal advisory committee this August, before being formally transmitted to the agencies for their approval.
Cosmic inflation is not the only thing CMB-S4 is set to study. A 221-page "Science Book" detailing the main objectives of the project is now available online. "We expect to learn more about dark matter, dark energy, neutrinos, [in addition to] inflation," Dvorkin said.Filed under
Authorized news sources may reproduce our content. Find out more about how that works. © 2016 American Institute of PhysicsAuthor Bio & Story Archive Yuen Yiu
Yuen Yiu covers the Physics beat for Inside Science. He's a Ph.D. physicist and fluent in Cantonese and Mandarin. Follow Yuen on Twitter: @fromyiutoyou.
Aerosol pollution may have suppressed global warming and increased Arctic sea ice from 1950-1975.
Thursday, February 23, 2017 - 10:00
(Inside Science) -- For 40 years, heat trapped by human-caused greenhouse gas pollution has been eating away at Arctic sea ice. But for a brief span in the 20th century, Arctic sea ice was actually increasing -- and that may be our fault too, according to a new study.
The researchers used historical measurements to learn how the ice had changed, then modeled climate variables to untangle the causes. They found that between 1950 and 1975, certain types of air pollution counteracted the effects of greenhouse gases that hold heat in the Earth's atmosphere. The balance of these gases at that time cooled the Earth and helped sea ice to grow. The findings emphasize the fragility of the Arctic sea ice system, with humanity's changing pollution habits buffeting the region first one way, then the other.
"I think what it's telling us is that humans had an impact on the Arctic much earlier than had been thought," said John Fyfe, a senior scientist at Environment and Climate Change Canada in Victoria, British Columbia and one of the authors of the study, which appeared today in Geophysical Research Letters.
When people burn fossil fuels, they produce a range of different pollutants that can "force" the climate to react in different ways. Greenhouse gases -- like carbon dioxide -- trap heat and warm the planet, while fine particles suspended in droplets, known collectively as aerosols, usually cool the planet by reflecting the sun's energy away.
"It's a tug-o-war between these two forcings as to what the climate is actually going to do," said Fyfe.
In the mid-20th century, coal-burning power plants and other sources released huge amounts of sulfur dioxide, which then formed toxic sulfate aerosols in the atmosphere. Past research has suggested that the cooling influence of these sulfate aerosols helped counteract warming from greenhouse gases, said Fyfe. But around 1970, the U.S. Clean Air Act and other air quality regulations started curbing sulfate aerosol production. Sulfate aerosols degrade relatively quickly, so people soon enjoyed air that was healthier to breathe. Meanwhile, production of greenhouse gases -- which linger in the atmosphere much longer than sulfate aerosols -- has continued, causing average global temperatures to rise.
The new study examined what this meant for Arctic sea ice. Unlike land ice, floating sea ice doesn't affect sea levels when it melts and freezes. But it does play an important role in the global climate and ocean currents, acting somewhat like the radiator and water pump in a car, said Andrew Mahoney, a sea ice geophysicist at the University of Alaska in Fairbanks who was not involved in the study. Sea ice is also crucial to Arctic ecosystems, supporting nutritious algae that other organisms feed on, he said. The ice releases algae as it melts, so changes to the yearly freezing and thawing cycles could throw the food web out of whack.
"The growth and melt of sea ice is like a pulse in the Arctic that's timed to all of the other species that live there," said Mahoney.
Researchers have documented rapid declines in Arctic sea ice since the late 1970s, when the satellite record begins. Earlier data are scarce, but past research has suggested that Arctic sea ice increased between about 1950 and 1975. Fyfe and his colleagues confirmed these findings using data from several historical sources, including maps that once helped Russian ships navigate icy northern waters. The data show that Arctic ice went through an expansion as dramatic as its subsequent decline, said Fyfe.
Next, the researchers used a mathematical model of Earth's climate to simulate different scenarios, playing out what would have happened without particular pollutants or natural phenomena. The simulations indicated that about a third of the sea ice increase was due to natural processes, while two-thirds was due to aerosol pollution from human, or anthropogenic, sources.
"Arctic sea ice was increasing -- that's clear and unequivocal now during this time period," said Fyfe. "And through these model experiments, we've been able to determine that the increase was due to anthropogenic aerosols."
Not everyone is convinced about the role of aerosols, however. Vladimir Semenov, a climate scientist at the Institute of Atmospheric Physics and the Institute of Geography in Moscow, who was not involved in the study, attributes the period of ice growth mostly to natural variability, although he said aerosols probably also play some role.
"It's an interesting result in the sense that the model showed that such a response [of aerosols on sea ice] is possible," said Semenov. "But to make sure whether it's just an outlier, or really some reliable physical result, you have to check other models."
Mahoney agrees that the new study needs to be replicated with other climate models. But, he said, the results are plausible, and they demonstrate the profound effects of our species' actions.
"I think it suggests that the climate may be more sensitive than we realize to human activities," he said. "When we try to do an environmental accounting of the impacts of our activities, we need to be very broad in our thinking about what to include."Filed under
Nala Rogers is a staff writer and editor at Inside Science, where she covers the Earth and Creature beats. She has a bachelor’s degree in biology from the University of Utah and a graduate certificate in science communication from U.C. Santa Cruz. Before joining Inside Science, she wrote for diverse outlets including Science, Nature, the San Jose Mercury News, and Scientific American. In her spare time she likes to explore wilderness.