Venus’s Supercritical Ocean

The pressure and temperature near the surface of Venus are so high that its carbon dioxide atmosphere is a global ocean of a remarkable state of matter, a supercritical fluid, which fills any container like a gas but is as dense as a liquid.

I created a carbon dioxide pressure versus temperature phase diagram using Mathematica and its curated computable data. Phase transitions separate single-phase regions. Moving along the boiling-condensing curve from the triple point, the liquid and gas densities converge at the critical point, beyond which carbon dioxide can transition between liquid and gas without boiling or condensing! I added points representing the near-surface atmospheres of Earth and Venus, with the latter being in the supercritical region above both the critical temperature and pressure.

Only the Soviet Union‘s Venera spacecraft have landed on Venus’s alien surface, and only between 1975 and 1982. Their cameras provided us our first and so far only glimpses of Venus from beneath its supercritical “ocean”.

Carbon dioxide pressure versus temperature phase diagram, created in Mathematica. Carbon dioxide is a gas near Earth’s surface (blue dot) but is a supercritical fluid near Venus’s surface (white dot).

Carbon dioxide pressure versus temperature phase diagram, created in Mathematica. Carbon dioxide is a gas near Earth’s surface (blue dot) but is a supercritical fluid near Venus’s surface (white dot).

At the bottom of a supercritical “ocean”, the surface of Venus (top) reconstructed by Don Mitchell based on Venera 14 panoramas (bottom) processed by Ted Stryk.

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Percy & Ginny

A chill went through the spaceflight community last week as NASA reported that it had lost contact with the Ingenuity Mars helicopter. Delivered to Mars underneath the Perseverance rover and intended as a 5-flight 30-sol tech demo, it had vastly exceeded expectations, including the first successful powered flight off Earth, a true Wright brothers moment, and more than two hours cumulative flying time in 72 flights over 1000 sols.

Although Perseverance (nicknamed Percy) was able to reestablish communication with Ingenuity (nicknamed Ginny), NASA reports today that one of Ginny’s rotors is broken, and the first Mars helicopter will not fly again. Like the Apollo 11 lunar lander Eagle, Ginny carries a swatch from the original Wright flyer, and just last month the Smithsonian’s National Air and Space Museum added one of Ginny’s prototypes to its collection.

Flying Ingenuity helicopter photographs grounded Perseverance rover (left) and vice versa (right) -- on Mars! (NASA photos.)

Flying Ingenuity helicopter photographs grounded Perseverance rover (left) and vice versa (right)
— on Mars! (NASA photos.)

Ginny and Percy selfie. (NASA photo.)

Ginny and Percy selfie. (NASA photo.)

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Summer of ’19

Due to the pandemic, the summer of 2019 was regrettably and unexpectedly my last Wooster summer research program, but the team was amazing. Niklas Manz and I obtained Sherman-Fairchild funding to work with Margaret McGuire ’20, Yang (Fish) Yu ’21, and Chase Fuller ’19 to computationally study reaction-diffusion phenomena.  All three of their projects have now been published, with them as lead authors. Fish was lead on Disruption and Recovery, published in Physica A; Margaret was lead on Geographic Tongue, published in Chaos; and Chase was lead on Diffusion Diodes, published today in the International Journal of Unconventional Computing.

Summer 2019 Sherman-Fairchild scholars take a break in the lovely Taylor 211 computational physics lab

Summer 2019 Sherman-Fairchild scholars take a break in the lovely Taylor 211 computational physics lab.

With important later contributions by co-author Daniel Cohen-Cobos ’23, Chase used the Tyson-Fife model of the Belousov-Zhabotinsky reaction to numerically investigate the propagation of reaction-diffusion waves through narrow, quasi-one-dimensional channels. He created “soft” obstacles where the inhibitor’s diffusion was larger than the activator’s diffusion, so the system exhibited unidirectional or one-way propagation – a diffusion diode. Furthermore, he discovered a nonlinear compensation relationship between higher activator diffusion (causing increased wave speed) and light illumination (causing decreased wave speed) that enables normal propagation. This effect should facilitate the creation of very energy efficient on-off switches for chemical computation circuits, where small changes in light levels cause the diffusion diodes to pass or block waves.

Schematic time-space plots of periodic reaction diffusion waves with variable diffusion $D_v/D_u$ and illumination $\phi$ illustrate light switches. Leftward (red) and rightward (blue) waves are independent experiments. Left: Faint light increase creates a diode. Middle: Strong light pulse destroys the diode and creates a 2-way barrier. Right: Faint light decrease destroys a diode and creates a 2-way opening.

Schematic time-space plots of periodic reaction diffusion waves with variable spatial diffusion and temporal illumination illustrate light switches. Leftward (red) and rightward (blue) waves are independent experiments. Left: Faint light increase creates a diode. Middle: Strong light pulse destroys the diode and creates a 2-way barrier. Right: Faint light decrease destroys a diode and creates a 2-way opening.

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Chemistry Does General Relativity

I hired Kiyomi from Hawai’i for our NSF REU summer program in spring 2020 amidst fears of the pandemic that eventually postponed the program two years. When she finally arrived in summer 2022, I had already retired from Wooster, where my last year was completely remote, classes via Teams, but several of those included Daniel. Despite the pandemic, I was fortunate to collaborate with both Daniel and Kiyomi — as well as colleagues in Chemical Physics, Physics, Astrophysics, and Computer Science — on an article just published in Frontiers in Physics. Based on Daniel’s senior thesis and Kiyomi’s summer research, the article is the very-Wooster, very-interdisciplinary Chemistry Does General Relativity: Reaction-Diffusion Waves Can Model Gravitational Lensing.

Gravitational lensing is a general relativistic (GR) phenomenon where a massive object redirects light, deflecting, magnifying, and sometimes multiplying its source. In the research, we used the chemistry of Belousov-Zhabotinsky (BΖ) reaction-diffusion (RD) waves to model this astronomical effect in a table-top experiment. We began by experimentally passing BΖ RD waves through non-planar quasi-two-dimensional molds. We next reproduced the waveforms in computer simulations of planar RD waves with variable diffusion. We then varied the diffusion parameter so the effective wave speed of planar waves matched the GR predictions for light deflection near a massive object. We thereby recovered Einstein’s famous light deflection formula, as summarized by the figure below.

Article summary. (a) General relativity predicts the observed gravitational deflection of light near stars with mass M . The deflection angle α depends on the impact parameter b, the perpendicular distance between the initial ray and the star’s center. (b) Experimental reaction-diffusion (RD) over spherical cap obstacles vets our (c) simulated RD over a plane with variable diffusion. (d) Planar RD with variable diffusion can match the effective light speed near a star or black hole and (e) well approximate the famous angle deflection relation α ~ M / b.

Article summary. (a) General relativity predicts the observed gravitational deflection of light near stars with mass M . The deflection angle α depends on the impact parameter b, the perpendicular distance between the initial ray and the star’s center. (b) Experimental reaction-diffusion (RD) over spherical cap obstacles vets our (c) simulated RD over a plane with variable diffusion. (d) Planar RD with variable diffusion can match the effective light speed near a star or black hole and (e) well approximate the famous angle deflection relation α ~ M / b.

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Magic Scroll

When I bought my new house, I knew I would soon need to replace its heat pump, which was almost 20 years old. Earlier this month, with my old pump laboring under a cold snap, I upgraded to a new version, which boasts a history of elegant inventions.

Powered by electricity, heat pumps circulate a low-boiling-point hydrofluorocarbon (HFC) refrigerant between two heat exchangers, one inside the house and one outside it, separated by a compressor and an expansion valve, cycling the refrigerant between liquid and gas phases. When the refrigerant flows one way, it evaporates in the inside heat exchanger, absorbing energy and cooling the interior (while it condenses in the outside heat exchanger, liberating energy). Alternately, when the refrigerant flows the other way, it condenses in the inside heat exchanger, liberating energy and heating the interior (while it evaporates in the outside heat exchanger, absorbing energy).

The heart of my heat pump is its scroll compressor, invented by Léon Creux over a century ago, where one interleaved scroll orbits another, channeling and squeezing the injected fuel, as in the 2D animation below. What a wonderful and unexpected use of spiral curves! With less moving parts than a traditional reciprocating compressor, a scroll compressor can be more efficient, smoother, quieter, and more reliable.

Scroll compressor intakes low pressure fluid (left), smoothly compresses it, and exhausts high pressure fluid (center). Rainbow colors suggest pressures, increasing from red to violet. (You may need to click to see the animation.)

Scroll compressor intakes low pressure fluid (left), smoothly compresses it, and exhausts high pressure fluid (center). Rainbow colors suggest pressures, increasing from red to violet. (You may need to click to see the animation.)

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All Engine(s) Running

I asked Siri to wake me at 7:15 this morning so I could watch SpaceX’s second Integrated Flight Test of Super Heavy Starship, the biggest and most powerful rocket ever built. Unfortunately, my house suffered a rare power outage an hour or two earlier, so I found myself lying in bed watching the coverage on my iPhone.

The sunrise launch was truly spectacular, a major improvement over the first test earlier this year, when multiple engines failed at launch. This morning, for the first time in any test, all 33 Raptor methalox engines ignited together and ran throughout the boost phase! With a thrill and a shiver, I recalled the excited “Voice of Apollo” Jack King announcing a good start to Apollo 11 with inspiring words that have reverberated throughout my life, “All engine(s) running” (without the “s”).

All engines running! The largest and most powerful rocket ever made under full power, 2023 November 18. (Photo credit: SpaceX.)

All engines running! The largest and most powerful rocket ever made under full power, 2023 November 18. (Photo credit: SpaceX.)

Sunrise, Boca Chica, Texas, 2023 November 18. (Photo credit: SpaceX.)

Sunrise, Boca Chica, Texas, 2023 November 18. (Photo credit: SpaceX.)

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Diversity Improves Machine Learning

For the last two years, the Nonlinear Artificial Intelligence Lab and I have labored to incorporate diversity in machine learning. Diversity conveys advantages in nature, yet homogeneous neurons typically comprise the layers of artificial neural networks. In software, we constructed neural networks from neurons that learn their own activation functions (relating inputs to outputs), quickly diversify, and subsequently outperform their homogeneous counterparts on image classification and nonlinear regression tasks. Sub-networks instantiate the neurons, which meta-learn especially efficient sets of nonlinear responses.

Our examples included conventional neural networks classifying digits and forecasting a van der Pol oscillator and physics-informed Hamiltonian neural networks learning Hénon-Heiles stellar orbits

As a final real-world example, I video recorded my wall-hanging pendulum clock, ticking beside me as I write this. Engineered to be nearly Hamiltonian, and assembled with the help of a friend, the pendulum’s Graham escapement periodically interrupts the fall of its weight as gravity compensates dissipation. Using software, we tracked the ends of its compound pendulum, and extracted its angles and angular velocities at equally spaced times. We then trained a Hamiltonian neural network to forecast its phase space orbit, as summarized by the figure below. Once again, meta-learning produced especially potent neuronal activation functions that worked best when mixed.

Meta-learning 2 activations for forecasting a real pendulum clock engineered to be almost Hamiltonian. Left: Falling weight (not shown) drives a wall-hanging pendulum clock. Center: State space flow from video data is nearly elliptical. Right: Box plots summarize distribution of neural network mean-square-error validation loss, starting from 50 random initializations of weights and biases, for a fully connected neural networks of sine neurons (blue), type-1 neurons (yellow), type-2 neurons (orange), and a mix of type 1 and type 2 neurons (red).

Meta-learning 2 neuronal activations for forecasting a real pendulum clock engineered to be almost Hamiltonian. Left: Falling weight (not shown) drives a wall-hanging pendulum clock. Center: State space flow from video data is nearly elliptical. Right: Box plots summarize distribution of neural network mean-square-errors for fully connected neural networks of sine neurons (blue), type-1 neurons (yellow), type-2 neurons (orange), and a mix of type 1 and type 2 neurons (red).

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Neural network does quantum mechanics

A particle confined to an impassable box is a paradigmatic and exactly solvable one-dimensional quantum system modeled by an infinite square well potential. Working with Bill Ditto, Elliott Holliday and I recently explored some of its infinitely many generalizations to two dimensions, including particles confined to regions that exhibit integrable, ergodic, or chaotic classical billiard dynamics, using physics-informed neural networks. In particular, we generalized an unsupervised learning algorithm to find the particles’ eigenvalues and eigenfunctions, even in cases where the eigenvalues are degenerate. During training, the neural network adjusts its weights and biases, one of which is the energy eigenvalue, so that its output approximately solves the stationary Schrödinger equation with normalized and mutually orthogonal eigenfunctions.

2D classical billiard orbits (left) and corresponding 2nd quantum eigenfunctions (right)

2D classical billiard orbits (left) and corresponding 2nd quantum eigenfunctions (right)

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The Ringed Planets

When I was a kid, Saturn was the ringed planet. But today, we know that all of the outer planets have rings. The James Webb Space Telescope has now imaged each of them in infrared revealing their distinctive structures, including Jupiter‘s very faint ring (located by the arrow and dashed curve). The planet images below are roughly to scale, 11:9:4:4, where Earth is 1.

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Simplest Chaos

The motion of one of the simplest dynamical systems, a torqued, damped, nonlinear pendulum, can be infinitely complicated.

Consider a simple pendulum of length l and mass m rigidly connected to an axle of radius r wrapped by a rope that hangs down one side with a mass M climbing up and down it, as in the attached animation.

If the climber’s height varies sinusoidally (relative to the axle)

x = x_0  + \frac{a}{\omega^2} \sin \omega t,

then its acceleration also varies sinusoidally

\ddot x = - a \sin \omega t,

so the total force on the climber

M\ddot x = \sum_\text{down} f = Mg-T

implies upper rope tension

T = Mg + ma \sin \omega t,

where 0 < a < g. If the axle and rope have negligible inertia, then the total torque on the axle

m l^2 \ddot \theta = \sum_\text{CCW}\tau = - mgl \sin\theta + rT - \gamma \dot\theta,

where \gamma is the axle viscosity. The full motion equation

m l^2 \ddot \theta = - mgl \sin\theta + rMg + rma \sin \omega t - \gamma \dot\theta

reduces to

\ddot \theta = - \sin\theta + 0.7155 + 0.4 \sin 0.25 t - 0.75 \dot\theta

for parameters that describe the animation’s chaotic motion.

Green mass sinusoidally climbs up and down brown rope torquing blue pendulum into chaotic motion.

Green mass sinusoidally climbs up and down brown rope torquing blue pendulum into chaotic motion.
Upper rope tension is largest with climber lowest and smallest with climber highest.

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