Losing Betelgeuse

At my computer Tuesday evening, I receive a message from a university physics chat that is both thrilling and chilling: LIGO + Virgo report a “burst” gravitational wave event, possibly due to a core-collapse supernova (or a binary collision where one object is in the hypothetical “mass gap” between black holes and neutron stars). The burst event is in Orion — near Betelgeuse.

Betelgeuse is a red supergiant star evolving rapidly to an expected supernova. It has dimmed dramatically in recent months, and I’ve seen estimates of thousands to hundreds of thousands of years until it explodes catastrophically in a surge of neutrinos, but 10^{4\pm1} years is so soon astronomically. (In physics, the uncertainty is in the mantissa, but in astrophysics, the uncertainty is in the exponent.)

While a Betelgeuse supernova would irrevocably scar my favorite constellation, it would be the most dramatic astronomical event of my lifetime, outshining Earth’s moon for months before dimming to dark. For a minute or two I seriously contemplate losing Betelgeuse — and gaining a spectacular naked-eye supernova. I frantically search online for more information. Betelgeuse seems safe, but at least one astronomer walks outside to visually check. My breathing returns to normal. More time to prepare the next generation neutrino detectors.

Enjoy Orion while you can, because sometime soon, Betelgeuse is gonna blow.

A Betelgeuse supernova would irrevocably damage my favorite constellation, but it would be the most spectacular astronomical event of my lifetime

A Betelgeuse supernova would irrevocably damage my favorite constellation, but it would be the most spectacular astronomical event of my lifetime

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Continental Bridge

I remember looking at a classroom map of Earth and thinking the continents seem like puzzle pieces, especially north and south America in the west and Europe and Africa in the east. I mentally fit them together. Later I learned about continental drift and plate tectonics, driven by gravity and mantle convection, with newborn crust at the Mid Atlantic Ridge recording reversals of Earth’s magnetic field in its cooling magma.

Last week I was in Iceland for the winter solstice in search of aurora. Came for astronomy; stayed for geology.

Iceland is a hot spot, probably the expression of an entrenched mantle plume, straddling the boundary between the North American and Eurasian tectonic plates. In southwest Iceland, on the lava-scarred Reykjanes peninsula, the Mid Atlantic Ridge is above ground. As the plates drift apart, several centimeters each year, a rift valley widens. Although the rift itself is several kilometers across, at Sandvik a small footbridge over a fissure in the rift symbolizes the connection between the continents.

Fissure in the rift between continents, Reykjanes peninsula, southwest Iceland, December 2019[

Fissure in the rift between continents, Reykjanes peninsula, southwest Iceland, December 2019

Symbolic bridge between North America and Eurasia

Symbolic bridge between North America and Eurasia

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Table of Nuclides

As of 2019, we have identified or synthesized 118 distinct elements with Z protons, but about 2900 distinct nuclides with N neutrons (where atom is to element as nucleus is to nuclide).

The start of my version of the table of nuclides is below, where number of protons Z increases toward 2 o’clock, number of neutrons N increases toward 11 o’clock, and atomic mass A = Z + N increases toward 12 o’clock on average (because more neutrons than protons are needed to bind large nuclei). Rainbow colors code lifetimes t from short (violet) to long (red). For example, the heavy hydrogens are very short lived. The whole chart is a very tall 880 KB PDF table of nuclides. Enjoy!

Start of table of atoms. Rainbow colors code lifetimes, violet (short) to red (long). Number of protons increases toward 2 o'clock, number of neutrons increases toward 11 o'clock.

Start of table of atoms. Rainbow colors code lifetimes, violet (short) to red (long). Number of protons increases toward 2 o’clock, number of neutrons increases toward 11 o’clock.

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Intrepid-Surveyor

Fifty years ago, Apollo 12 landed within sight of another spacecraft, a dramatic demonstration of pinpoint landing capability. While Dick Gordon orbited Luna in the command module Yankee Clipper, Pete Conrad and Al Bean left the lunar module Intrepid and walked over to the robotic Surveyor, which had landed over two years earlier. They retrieved parts of Surveyor and returned them to Earth for engineering analysis. Bean’s photograph of Conrad at Surveyor with Intrepid on the horizon is a spoce exploration icon. Recently, the Lunar Reconnaissance Orbiter photographed the landing site and revealed Surveyor and Intrepid’s descent stage connected by dark tracks in the lunar regolith left by the astronauts.

Al Bean photographed Pete Conrad at the Surveyor 3 spacecraft with the lunar module Intrepid on the horizon, November 20, 1969

Al Bean photographed Pete Conrad at the Surveyor 3 spacecraft with the lunar module Intrepid on the horizon, November 20, 1969

2011 Lunar Reconnaissance Orbiter photograph of the Apollo 12 landing site including the astronauts' tracks from two moonwalks

2011 Lunar Reconnaissance Orbiter photograph of the Apollo 12 landing site including the astronauts’ tracks from two moonwalks

 

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Relaxing Fermat

In 1637, while reading a copy of Diophantus’s Arithmetica, Pierre de Fermat famously scribbled

“Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”

which roughly translates to

“It is impossible to separate a cube into two cubes, or a quartic into two quartics, or in general, a power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

In modern notation, the equation

x^n + y^n = z^n

has no positive integer solutions for exponents n > 2. Although Fermat did leave a proof for the case n = 4, 358 years past before Andrew Wiles published his proof of the general case in 1995.

Relaxing Fermat’s constraints to allow non-integers greatly expands the number of solutions. The looping animation shows all solutions for 1 \le n < \infty and 1 \le \{x,y,z\} \le 11. All points on the arcs

y_{z,n}[x] = (z^n - x^n)^{1/n}

are solutions, and red dots indicate integer solutions. Watch the the famous Pythagorean triple \{3,4,5\} flash by for n = 2. Integer solutions are visibly harder for large finite n. Many more solutions exist for n < 1.

Points along arcs are solutions to the generalized Fermat equation; red points are integer solutions

Points along arcs are solutions to the generalized Fermat equation; red points are integer solutions

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Stainless Steel Starship

Welders in a Texas swamp have built a starship. But don’t bet against SpaceX.

Starship is a prototype upper stage for a next-generation, fully reusable, two-stage-to-orbit launch vehicle designed to enable the human exploration of the solar system and the colonization of Mars. It’s made from stainless steel. (A little carbon converts iron to hard steel; a little chromium converts steel to corrosion-resistant stainless steel.) At cryogenic temperatures, grade 301 stainless steel has higher strength-to-weight and strength-to-cost ratios than carbon fiber reinforced polymer, and it has a higher melting temperature.

Starship will dissipate orbital energy by entering a planetary atmosphere like a sky diver, belly first, its fore and aft fins rapidly moving to control its descent prior to a tail-first rocket-powered landing. Strong electric motors powered by Tesla batteries will flap the fins.

Starship is powered by next-generation Raptor engines, the first full-flow staged combustion rocket engines to fly. In these efficient closed cycle engines, no propellant is wasted: all the oxidizer (and some fuel) power the oxidizer turbopump and all the fuel (and some oxidizer) power the fuel turbopump, which together pump gaseous oxidizer and fuel into the combustion chamber to combust and thrust. The oxidizer is liquid oxygen; the fuel is liquid methane, the primary component of natural gas, which can be manufactured from the martian (or terrestrial) atmosphere.

SpaceX Stainless Steel Starship Prototype

SpaceX Stainless Steel Starship Prototype

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After the Moonwalk

Iconic is Neil Armstrong’s photograph of Buzz Aldrin during the first moon walk, with Armstrong reflected in Aldrin’s visor. Much less well-known is this pair of photographs taken just after the moon walk. To my eyes, Armstrong seems exhausted but happy; Aldrin seems satisfied … and over his shoulder, almost casually, is a window, and outside the window is the lunar horizon, with its stunning airless black sky at day! Fifty years later, I still imagine A & A trying to catch a few zzzs … in hammocks … in their home … on the moon.

After the moonwalk, Monday, July 21, 1969

Neil Armstrong and Buzz Aldrin in the LEM after the first moonwalk, Monday, July 21, 1969

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“Contact Light”

Our TV is broken, so Aunt Nora invites us to her apartment. (Aunt Nora isn’t really our aunt, but she introduced our parents to each other, so that’s what we call her.) My brother Jim and I lie on the floor close to the TV, while the adults sit on the couch. We watch NBC not CBS, so we miss Cronkite’s commentary. The late afternoon video is a simple animation; the famous 16-mm film — only later synchronized with the audio — would return to Earth with the astronauts 4 days later.

The tension is palpable. The cartoon lander reaches the surface at the expected time, but Aldrin’s monotone readouts continue. Absence of video heightens the audio. Mission control radios “60 seconds” of fuel remaining. Then “30 seconds”. I hold my breath. At last, Aldrin reports “Contact light” — we have touched the moon — followed by Armstrong’s famous, “Houston … Tranquility Base here, the Eagle has landed”. Of the landing site, my mother observes, “They’ve already named it”.

No one wants to cook, so we go to McDonald’s for dinner. As we drive, I see a small shop with photos of the three astronauts in its window. The streets are still. The world seems stopped.

10:56:15 PM EDT, Sunday, July 20, 1969

10:56:15 PM EDT, Sunday, July 20, 1969

As Collins orbits the moon solo, Armstrong and Aldrin forgo a scheduled sleep period, moving forward the moonwalk. Finally video — live from the surface of the moon —  shows a LEM landing leg, first inverted but quickly rectified. Armstrong describes the surface as “almost like a powder”. Again I hold my breath, a lump in my throat. “Okay. I’m going to step off the LEM now … that’s one small step for [a] man, one giant leap for mankind.” I don’t hear the indefinite article, but I immediately grasp the meaning. Armstrong reads the plaque, “We came in peace for all mankind”. Aldrin practices locomotion, which “would get rather tiring”. The president phones. We leave Aunt Nora’s as the astronauts prepare to return to the LEM.

The next morning I sit on my living room floor reading two newspapers: The New York Times, with its simple banner headline “Men Walk on Moon” — which still brings me tears of joy, triumph, and wonder as I write this 50 years later — and the local newspaper, with its astonished “Now Do You Believe!”.

My Monday morning newspapers, July 21, 1969

My Monday morning newspapers, July 21, 1969

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Wooster Epicycles

A vector is the sum of its components, a mechanical vibration is a combination of its normal mode motions, a quantum state is a superposition of its eigenstates, and any “nice” function is a Fourier sum of real or complex sinusoids, e^{i \varphi} = \cos \varphi + i \sin \varphi.

The animation below traces the Wooster W in epicycles of 100 circles-moving-on-circles in the complex plane. Algebraically, the trace is a complex discrete Fourier series \sum c_n e^{i n \omega t} =\sum r_n e^{i (n\omega t + \theta_n)}, where r_n are the circle radii, \theta_n are carefully chosen phase shifts, \omega is the fundamental angular frequency, and t is time.

Using Fourier analysis, any reasonable path can be traversed by a moon orbiting a moon orbiting a moon orbiting ... a planet orbiting a star

Using Fourier analysis, any reasonable path can be traversed by a moon orbiting a moon orbiting a moon orbiting … a planet orbiting a star

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Redefining SI

Today the SI (Système international d’unités) base units are redefined. The following are now exact. Memorize these numbers!

Cs-133 transition frequency constant Δν_{\text{Cs}} = 9\,192\,631\,770~\text{s}^{−1} defines the second.

Then light speed constant c = 299\,792\,458~\text{m}\cdot\text{s}^{−1} defines the meter.

Then Planck’s constant h = 6.626\,070\,15\times 10^{−34}~\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{−1} defines the kilogram.

Then electron charge constant e = 1.602\,176\,634\times 10^{−19}~\text{A}\cdot{\text{s}} defines the Ampere.

Then Boltzmann’s constant k = 1.380\,649\times 10^{−23}~\text{kg}\cdot\text{m}^{2}\cdot\text{K}^{−1}⋅\text{s}^{−2} defines the Kelvin.

And Avogadro’s constant N_{\text{A}} = 6.022\,140\,76\times 10^{23}~\text{mol}^{−1} defines the mole.

And luminous efficacy constant K_{\text{cd}} = 683~\text{cd}\cdot\text{sr}\cdot\text{s}^{3}\cdot\text{kg}^{−1}\cdot\text{m}^{−2} defines the candela.

(Where “sr” is the steradian or square radian, the 3D analogue of the 2D radian.) Discussion continues about the mole and the candela, including whether they should even be base units. The new definitions break the relationship between the C-12 mass, the dalton, the kilogram, and Avogadro’s constant, and the candela is arguably a photo-biological quantity.

I wish my phone number were 919-263-1770.

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