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.

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.

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.

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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 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)

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

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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

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.

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

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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 animated sequence of inertial space diagrams, 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 (solid white) 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.]

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

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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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Merlin & Raptor

The turbopump is the heart of most liquid-fueled rocket engines.

Gas generator engines tap off and burn a little propellant to drive a turbine, which turns a centrifugal pump, which rapidly pushes the fuel and oxidizer to the combustion chamber, after the cryogenic fuel first circulates around the rocket nozzle to cool it. The combustion generates supersonic exhaust out the converging-diverging nozzle that pushes the rocket. (The nozzle pushes the exhaust, and the exhaust pushes the nozzle).

The Merlin is an open-cycle engine because the fuel-rich flow of its gas generator is dumped overboard and not fully burned in the combustion chamber. The Raptor is a closed-cycle and full-flow staged-combustion engine as all of its propellants flow through the pre-burners (aka gas generators) to fully burn in the combustion chamber in the optimal ratio and exhaust out the nozzle for maximum efficiency.

Open and closed cycle gas generator rocket engines are driven by turbopumbs

Open and closed cycle gas generator rocket engines are driven by turbopumbs

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Astronomy Christmas Gift

I awoke early this Christmas morning to watch the successful launch of the James Webb Space Telescope. I remember the genesis of the telescope a quarter of a century ago when it was called the Next Generation Space Telescope. (The telescope’s name is controversial, and I would have preferred an astronomer’s name.) Its development has been prolonged and difficult, as the graphs below attest, but like its predecessor, the Hubble Space Telescope, I am confident of its revolutionary importance — if the upcoming weeks of deployment and months of commissioning succeed.

Development of the unprecedented Webb telescope has been long and challenging

Development of the unprecedented Webb telescope has been long and challenging

Hubble is a warm visible light telescope, with some near infrared and ultraviolet capability, in a low 540-kilometer Earth orbit. Webb will be a cold infrared telescope, with some red and orange capability, in a halo orbit about the second Earth-Sun Lagrange point 1.5 million kilometers anti-sunward from Earth.

(L_2 is an unstable equilibrium point: in an inertial reference frame, solar and terrestrial gravity combine to pull a mass into a solar orbit with a larger radius but the same period as Earth’s, so the mass appears to hover almost a million miles above Earth’s center; in a frame rotating with Earth about Sun, the combined gravitational force inward balances the centrifugal pseudo-force outward.)

Webb is optimized to study the early universe. Light from the first galaxies is both fainter due to distance and redder due to the expansion of the universe, so a larger, colder, more remote telescope is desirable. By contrast, Hubble is warm enough that its own infrared emission blinds it to this light.

During Webb’s long gestation, exoplanet science dramatically bloomed. Exoplanets are easier to detect in infrared light where the contrast between the glow of a star and its exoplanets is less. Consequently, the study of exoplanets and their atmosphere will be another Webb focus.

First though, Webb must survive six months of terror as it unfolds its giant sunshield and segmented mirror through hundreds of single points of failure.

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Burning Plasma

In August I received an urgent email from my brother with the title “Fusion”. The National Ignition Facility (NIF) had created a burning plasma — a star on Earth — a major milestone on the long road to controlled nuclear fusion.

A plasma is an ionized gas, but in this context “burning” does not mean chemically combining with oxygen. Rather it refers to the nuclear reaction

{}^2_1 \mathrm{H}^{+} + {}^3_1 \mathrm{H}^{+} \rightarrow {}^4_2 \mathrm{He}^{++} + {}^1_0 \mathrm{n} + 18~\text{MeV},

where hydrogen isotopes deuterium and tritium combine to form helium. Nuclear reactions produce millions of times the energy of chemical reactions, but they are hard to initiate as hydrogen ions repel one another electromagnetically unless they are close enough to allow the short-ranged nuclear force to convert mass energy into kinetic energy.

NIF uses the world’s most powerful laser, capable (briefly) of almost 1000 times the power output of the United States’ electrical grid. The ultraviolet laser pulse heats a small gold cavity or hohlraum, which converts the incident radiation into x-rays, which causes an enclosed deuterium-tritium pellet to implode, which increases the pellet’s temperature and density to initiate fusion.

After missing its 2012 ignition goal, and steadfastly fine-tuning the experimental design since then, the August shot generated much more energy than was delivered to the pellet (but slightly less energy than was delivered to the hohlraum). NIF director Mark Herrmann reported that, “Everyone has a spring in their step”. To me, controlled nuclear fusion, like humans on Mars, has always seemed 20 years away; with progress at NIF (and elsewhere), today it seems closer.

 

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Part Science, Part Art, Part Luck

Launched just last month, Lucy will be the first spacecraft to visit Jupiter’s trojan asteroids, rocky swarms that orbit about 60 degrees ahead and behind Jupiter in its orbit. Hal Levison, Lucy’s Principal Investigator, has described Lucy’s complicated trajectory, which includes an Earth gravity-assist and visits to both trojan swarms, as “part science, part art, and part luck”.

Named after the Lucy fossil that elucidates hominid history, Lucy will illuminate solar system history as the trojans are thought to be pristine remnants of planetary formation. The Lucy fossil itself was named after The Beatles‘ song “Lucy in the Sky with Diamonds” and, indeed, Lucy’s thermal emission spectrometer includes a large diamond beam splitter. During its 12-year primary mission, Lucy will visit seven trojans, including a near-twin binary pair, and one tiny main-belt asteroid named after Donald Johanson, the discover of the Lucy fossil.

Lucy's trajectory involves an Earth (green) flyby and visits to the trojan asteroids leading and lagging Jupiter (brown) in its orbit. (NASA image.)

Lucy’s trajectory involves an Earth (green) flyby and visits to the trojan asteroids leading and lagging Jupiter (brown) in its orbit. (NASA)

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4D Unknot

In four dimensions, you can’t tie your shoelaces — because 4D knots don’t work. Any 1D curve in 4D space can be continuously deformed to the unit circle, which is an unknot.

The looping animation below demonstrates how to undo a trefoil knot in 4D, where rainbow colors code the 4th dimension. The animation pauses when curve segments appear to intersect, but the segments’ different colors reveal their separation in the fourth dimension.

2D is too small to allow complicated neural circuits, and 4D is too large to enable knots; perhaps unsurprisingly, we find ourselves in a 3D world.

Unknotting a trefoil knot in 4D, where rainbow colors code the 4th dimension

Unknotting a trefoil knot in 4D, where rainbow colors code the 4th dimension

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