Measuring the Solar System

Thousands of years ago, ancient astronomers like Aristarchus and Eratosthenes combined careful observations with simple mathematics to measure the solar system, especially the diameters D of Earth, Luna (Earth’s moon), Sol (Earth’s star, the sun), and the radii r of their orbits. You too can do this, but it helps to observe an eclipse or two.

Step 1: Diameter of Earth

Measure how shadow lengths vary with latitude. No need to pace the distance between Alexandria and Syene, just use your favorite map software! Assume Sol is far from Earth (and check in Step 3), so Sol’s rays are nearly parallel.

Step 2:  Distance to Luna

Measure the duration of a lunar eclipse.

Step 3:  Distance to Sol

Measure the angle  between Sol and Luna at first quarter moon, when Luna appears to be ahead of Earth in its orbit, like a signpost to a car on a road. This is the most difficult step, as the angle is nearly but not quite ninety degrees, but the result is the astronomical unit. (Alternately, measure the ratio of time Luna is crescent to gibbous.)

Step 4:  Diameter of Luna

Measure the time for Luna to enter Earth’s shadow during a lunar eclipse. Consistent with Step 3, again assume Earth is far from Sol, so Earth’s shadow is nearly uniform.

Step 5:  Diameter of Sol

Note that Sol and Luna have about the same apparent angular size (both subtending about half a degree). This is most spectacularly evident during a solar eclipse, where Luna just barely covers Sol — if you were ever fortunate enough to experience a solar eclipse.

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Wooster’s Time Crystals

Saturday, March 8, 2008. A heavy snow, one of the heaviest I remember, shuts down the city of Wooster. Streets are undriveable, so I walk to Taylor Hall, getting snow in my boots.

Taylor is deserted, as the College has begun Spring Break, but like yesterday, Kelly and I work all afternoon and evening in the Physics Shop, me with wet socks. We’re leaving tomorrow, weather permitting, for the New Orleans American Physical Society meetingwhere Kelly will present her senior thesis research, but with just hours until our departure, and just days until her I.S. is due, the apparatus is still not working.

Todd brings us sandwiches, which we eat for dinner in the empty Reading Room. Then, just after dinner, back in the shop, the apparatus works for the first time, just as in our computer simulations. But simulation is one thing, reality is another! With no time to celebrate, we video record the dynamics, and move upstairs to the intro physics lab. On one computer, Kelly steps through the video frame-by-frame recording the motion, while at another, I work on the APS poster, which we output with the Taylor large-format printer shortly before midnight.

Next morning is sunny and white, a winter wonderland. The roads are plowed but still snowy. We cautiously drive to the airport and fly to New Orleans via Houston. Kelly’s poster presentation goes well, but I have a fever, sore throat, and stuffy nose. I blame the wet socks.


 Group photo of Wooster Physics at the March 2008 APS meeting

Wooster Physics at the March 2008 APS meeting in New Orleans. Kelly is on the right, and I am next to her. Todd is on the left.


Kelly’s work spawned three publications in refereed journals, two in the Physical Review and one in Chaos, involving many other undergraduates, as we gradually refined the apparatus over the next decade. The final version is on a shelf in my living room beside me as I write this 16 years later.

We designed and constructed a mechanical array of bistable pendulums coupled one-way in a topological ring, so each element affected the next one, but not vice versa, simultaneously violating Newton’s third law of action and reaction, momentum conservation, and energy conservation. Nevertheless, we achieved this by powering the device with a constantly flowing fluid, first water and later air. Each element directed the fluid flow on the next element to rotate it from one stable state to the other.

In this way, solitary waves or solitons of see-sawing elements propagated in one direction along the array, each soliton undoing what the previous one had done. Periodic boundaries enabled solitons to annihilate pairwise in arrays with an even number of elements, but solitons propagated indefinitely in arrays with an odd number of elements, where the oddness frustrated pairing and forbade a “ground state” of alternating elements. The frequency of motion depended continuously on the fluid speed and discretely on the number of elements.

We called them one-way arrays, although today we’d probably call them time crystals, as they are not only periodic in space, but periodic in time, for as long as the constant fluid flow continues.


3D printed final design after a decade of development. Wind blows down, solitons move right. Deflector of one element selectively shields the wing of the next, causing wind torque to rotate the next’s tail to the opposite stopping rod. Periodic boundary element, shown in motion, is split into two pieces connected coaxially via rod and (cyan) gears.

3D printed final design after a decade of development. Wind blows down, solitons move right. Deflector of one element selectively shields the wing of the next, causing wind torque to rotate the next’s tail to the opposite (red) stopping rod. Periodic boundary element, shown in motion, is split into two (cyan) pieces connected coaxially via (gray) rod and (cyan) gears.


REFERENCES (* indicates undergraduate coauthor)

“Experimental observation of soliton propagation and annihilation in a hydromechanical array of one-way coupled oscillators”, J. F. Lindner, K. M. Patton*, P. M. Odenthal*, J. C. Gallagher*, B. J. Breen, Physical Review E, volume 78, pages 066604(1-5) (2008)

“Electronic and mechanical realizations of one-way coupling in one and two dimensions”, B. J. Breen, A. B. Doud*, J. R. Grimm*, A. H. Tanasse*, S. J. Tanasse*, J F. Lindner, K. J. Maxted*, Physical Review E, volume 83, pages 037601(1-4) (2011)

“A wind-powered one-way bistable medium with parity effects”, T. Rosenberger*, G. Schattgen*, M. King-Smith*, P. Shrestha*, K. J. Maxted*, J. F. Lindner, Chaos: An Interdisciplinary Journal of Nonlinear Science, volume 27, pages 023114(1-5) (2017)

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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”.


CO2

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).


Venus

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 spatial diffusion and temporal illumination illustrate light switches.

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.

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 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, smoothly compresses it, and exhausts high pressure fluid.

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.

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.

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 neuronal activations for forecasting a real pendulum clock engineered to be almost Hamiltonian.

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 and corresponding 2nd quantum eigenfunctions

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

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