## GRAVITY and Gravity

In Alfonso Cuarón‘s Oscar-winning 2013 movie GRAVITY, actress Sandra Bullock‘s Dr. Ryan Stone makes an emergency entrance into an abruptly abandoned International Space Station. This month, European Space Agency astronaut Samantha Cristoforetti recreated this scene onboard the space station, as in the photograph below, with the movie playing on the ISS Viewscreen.

In GRAVITY, “gravity” refers to both the Newtonian force that causes the astronauts and their spacecraft to endlessly free-fall around Earth and to the grave nature of their situation, a brilliant and stunning drama of adversity, courage, persistence, and triumph.

Astronaut Samantha Cristoforetti (bottom) enters a module of the real International Space Station as actress Sandra Bullock’s Dr. Ryan Stone enters the fictional station in the movie GRAVITY (top)

## Science, serendipity, and coincidence

As part of my science history project, the article “Science, serendipity, coincidence, and the Oregonator at the University of Oregon, 1969–1974” has been published in Chaos: An Interdisciplinary Journal of Nonlinear Science.

It’s especially exciting because it’s the Feature article in the Focus Issue, From Chemical Oscillations to Applications of Nonlinear Dynamics: Dedicated to Richard J. Field on the Occasion of his 80th Birthday.

After many months of email exchanges and virtual meetings, I met Bob Mazo for the first time in person this January (see blog entry ‘50 years later‘). Together with Dick Field, the sole living scientist of the Field-Kőrös-Noyes (FKN) mechanism, developed in the early 1970s we revisited the exciting start of a new field in physical chemistry. The FKN mechanism was the first complete reaction scheme to describe the behavior of the Belousov-Zhabotinsky (BZ) reaction, a nonlinear chemical reaction-diffusion system (more here). Shortly after, they developed a mathematical three-variable model to describe the BZ reaction’s nonlinear behavior – the Oregonator model. The name was a response to the Brusselator model, developed by Ilya Prigogine in Brussels in the 1950s and 1960s. For his work on non-equilibrium thermodynamics and dissipative structures, Ilya Prigogine was awarded the Nobel Prize in Chemistry in 1977.

Fun fact: The total author age is 225, the average author age is 75! This will be difficult to beat.

## Alien Suns Reversing in Exoplanet Skies

Not only can suns stand still in the sky, from some exoplanets their motion can apparently reverse! Wooster physics-math double majors Xinchen (Ariel) Xie ’21 and Hwan (Michelle) Bae ’19 and I just published an article elucidating these apparent solar reversals.

Michelle and I began studying the architecture of daylighting terrestrial buildings for energy efficiency as part of her senior thesis, but due to our mutual interests, we gravitated to extraterrestrial possibilities. After my return from a yearlong sabbatical, Ariel eagerly continued the work for her senior thesis. Despite a pandemic keeping us a planet apart and meeting via video conference at simultaneously very early and very late hours, we completed the project, one of the most beautiful in my 33 years at Wooster. During one of our early morning, late evening sessions, we mathematically derived the solar reversals condition for the important special case of zero obliquity or tilt.

In the first figure below, exoplanet spin-orbit ratio $\rho$ increases rightward, orbital eccentricity $e$ increases upward, and time $t$ increases outward. Red rods represent planetary observers, and coils represent apparent position and angle of their suns for 8 orbits. Yellow and cyan indicate apparent clockwise and counterclockwise motion, which signify reversals in coils with both colors. Our own solar system’s Mercury (Me) is just inside the reversal region. Red-white-blue background colors represent the product of the differences of the planet’s spin angular speed and its extreme orbital angular speeds at apoapsis and periapsis; red means no apparent solar reversals, blue means reversals, and the saturation indicates the reversal magnitude.

On Mercury, one (solar) day lasts two years, and once a year an equatorial observer witnesses a brief solar reversal, as in the second figure below, surely a special day for any future inhabitants. For a civilization inhabiting such a planet, we expect “reversal day” to be culturally significant.

Apparent solar motion for different spin-orbit ratios and orbital eccentricities; red background means no apparent solar reversals, blue means reversals, and saturation indicates the reversal magnitude.

Solar strobe plots. (a) Earth-like planet whose sun appears to rise in the east and set in the west every day; rainbow hues code time. (b) Mercury-like exoplanet whose sun appears to reverse its motion once a year near periapsis where it appears largest.

## Black Hole Above the Fold

When grocery shopping, I normally just glance at the newspapers in the newsstand. However, this morning, I was excited to see “above the fold” of the Wall Street Journal a large reproduction of the first image of the supermassive black hole Sagittarius A* at the center of our Milky Way galaxy!

The Event Horizon Telescope team combined signals from radio telescopes that span Earth to reconstruct the image using Very Long Baseline Interferometry. Famously, not even light can escape a black hole, but EHT can see the glow of compressed and ionized gas or plasma in its orbiting accretion disk and the “shadow” of its event horizon. A  conventional false-color black-orange-white palette makes visible the averaged radio-wave data.

The Wall Street Journal in a newsstand at my local grocery story unveiling “above the fold” the first photo of the back hole at the center of the Milky Way

## Slide Rule Examples

Slide rules were widely used in engineering, science, and mathematics until the early 1970s, including during the Gemini and Apollo space programs. Although rendered largely obsolete by the advent of inexpensive electronic calculators, their descendants continue to have specialized applications, such as backup flight computers.

Buzz Aldrin and a floating slide rule during Gemini 12 on 1966 November 13 (NASA).

The giant Taylor Bowl slide rule used to be used to teach the slide rule but today is the trophy for the annual Taylor Hall bowling tournament between the Physics and Math Clubs.

The giant Taylor Bowl slide rule trophy and a victorious Physics Club on 2019 April 28.

## Slide Rule Theory

Slide rules were the analog computers that ruled science and engineering for 400 years. Their brilliant innovation was using logarithms to convert multiplication and division to addition and subtraction,

$$\log xy = \log x + \log y$$

and

$$\log \frac{x}{y} = \log x - \log y.$$

Slide rules feature logarithmic scales that slide past each other. For straight slide rules, logarithms of the numbers are proportional to their distances along them. To multiply $2\times 3 = 6$, as below, slide the upper (blue) scale from 1 to 2 along the lower (red) scale, add the distance from 1 to 3 along the upper (blue) scale, and read the product from the lower (red) scale.

Logarithms of the numbers are proportional to their distances along the straight slide rule scales. Slide the upper (blue) scale to multiply numbers by adding these distances.

For circular slide rules, logarithms of the numbers are proportional to their distances around them. To multiply $3\times 7 = 21$, as below, rotate the outer (blue) scale from 1 to 3 around the inner (red)  scale, add the distance from 1 to 7 around the outer (blue) scale, and read the product from the inner (red) scale. Circular slide rules eliminate off-scale calculations because they naturally wrap around, with each wrap multiplying (or dividing) the numbers by 10.

Logarithms of the numbers are proportional to their distances around the circular slide rule scales. Rotate the outer (blue) scale to multiply numbers by adding these distances.

## Diffraction Limited

Yesterday, Webb optical telescope element manager Lee Feinberg said “We made the right telescope” while reporting that its focus has reached the $\theta \sim \lambda/D$ diffraction limit of 0.7 arcseconds at the infrared wavelength of 2 microns. (For comparison, from Earth, Luna subtends 31 arcminutes, which is about 1/2°.) Unlike the Hubble space telescope, whose primary mirror was very finely polished to the wrong curvature — but later corrected by additional optics — the Webb telescope optics will meet or exceed its design goals without any corrections. The only way to increase Webb’s resolution would be to increase its size.

The test image below shows a long exposure of a faint star. The radial lines are diffraction spikes. Confine light in one direction and it spreads in the perpendicular direction. Here, the six large spikes are from Webb’s 18 hexagonal mirror-segment edges and the one horizontal spike is from the vertical strut supporting the secondary mirror, which is visible in the Webb selfie. (The remaining two secondary mirror struts are parallel to mirror edges and their diffraction patterns combine with the mirror edge patterns.)

Due to Webb’s unprecedented resolution and sensitivity, many galaxies are also visible in this alignment image, whetting the appetite of Earth-bound astronomers!

Webb image of a faint test star, diffraction spikes perpendicular to mirror segment edges and support struts, pixel bleed due to the long exposure, and background galaxies. (NASA)

Webb selfie shows hexagonal mirror segments and three struts supporting the secondary mirror. (NASA)

## Shackleton’s Valiant Voyage

Although a child of the Apollo program, I was gripped by Alfred Lansing‘s 1962 book Shackleton’s Valiant Voyage, a great tale of endurance, leadership, and survival and an inspiring true story from the heroic age of Antarctic exploration. In the 1910s, shortly after Roald Amundsen and Robert Scott separately reached the south pole, Ernest Shackleton organized an expedition to cross Antartica from sea to sea via the pole. Unfortunately, over the course of nearly a year, the ice slowly caught, crushed, and sank the expedition’s ship, the Endurance. After camping on ice floes for months, Shackleton and his crew of 28 (including one stowaway) used lifeboats to reach a desolate island, from which Shackleton and 5 companions sailed 1330 km of stormy, icy ocean via dead reckoning to bring rescue. All of the crew survived.

Even as a child, the expedition seemed remote in time, but Frank Hurley meticulously documented it, and the cover image of Lansing’s book is an actual photograph of the Endurance trapped in the ice. Today, almost exactly 100 years after Shackleton’s death, news broke that a team of scientists and adventurers have found Endurance 3008 meters beneath the surface of the Weddell Sea. Protected by cold waters and the Antarctic Treaty, the wreck is upright, well preserved, and will not be disturbed.

Front cover of Lansing’s 1962 account of the great survival story of the crew of Endurance

2022 photo of the wreck of the Endurance upright on the seabed.
Image © Falklands Maritime Heritage Trust / National Geographic.

## Halo Orbit

The Webb telescope has fully deployed and arrived at its halo orbit about the second Earth-Sun Lagrange point. But how can it orbit an empty point in space?

In the accompanying animation, a star (red) and planet (cyan) orbit their common center of mass (cross). The inward force to pull a moving mass (orange) into a circular orbit of fixed period increases proportionally with distance, but at the distance of $L_2$, the combined gravitational pulls of the star and planet are sufficient. Shifting the mass upward increases its distances from the star and planet, which decreases their pull. Shifting the mass inward compensates for this radially (orange), but perpendicularly a force component (yellow) now pulls the moving mass into a circle about $L_2$ in a reference frame rotating with the planet (and the mass bobs up and down sinusoidally along its orbit in an inertial frame fixed to the distant stars.)

Star (red) and planet (cyan) gravitationally pull a mass (orange) into a halo orbit (yellow).
[You may need to click to see the animation.]

## 50 years later

After MANY months of not traveling, I scheduled a meeting with Robert (Bob) M. Mazo, Professor emeritus from the University of Oregon, now living outside Philadelphia.

In 1971/1972 he helped developing the key model to describe chemical reaction-diffusion systems. But, as he stated, he was “only the catalyst” and only accepted to be recognized in the acknowledgments. The now called FKN mechanics, named after the three authors Dick Field, Endre Kőrös, and Dick Noyes, was the first model to describe the necessary reactions to create temporal, spatial, and spatiotemporal patterns in the nonlinear chemical Belousov-Zhabotinsky reaction. It had been published between 1972 and 1974 in a series called Oscillations in chemical systems I, II, III, IV, and V.

I am honored to write a manuscript with him, now at age 91, and the sole survivor of the original FKN papers, Dick Field, for a special edition in Chaos this year to celebrate Dick Field’s 80th birthday last October. During our meeting, Dick Field joined us via Skype from Missoula, Montana. It was a wonderful time travel back five decades while they were exchanging anecdotes and talked about nearly forgotten stories.

At the end of our meeting he joked that people in 50 years will know pretty much exactly which year our selfie had been taken…

Robert (Bob) M. Mazo and Niklas Manz on 4 January 2022.

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