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.

Posted in Physics | Leave a comment

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

 

Posted in Space Exploration | Leave a comment

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

Posted in Mathematics | Leave a comment

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

Posted in Adventure, Space Exploration | Leave a comment

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

Posted in Adventure, Space Exploration | Leave a comment

“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

Posted in Adventure, Space Exploration | Leave a comment

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 traversed by a moon orbiting a moon orbiting a moon orbiting ... a planet orbiting a star

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

Posted in Mathematics, Physics | Leave a comment

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.

Posted in Physics | Leave a comment

Black Hole Radii

I set the alarm for 8:55 AM. Brutal, but I wanted to watch live the National Science Foundation Event Horizon Telescope news conference. I was expecting the first image of a black hole, and the EHT team did not disappoint. But the black hole was not the Milky Way’s Sgr A*, but M87*, a thousand times further but a thousand times larger (billions rather than millions of solar masses).

For Astronomy Table lunch at Kitt’s Soup & Bread, I quickly created the graphic below to illustrate various radii of a mass M Schwarzschild black hole, a good approximation to this rotating Kerr black hole. The event horizon with reduced circumference R_s = 2 G M / c^2 is the point of no return, the causal disconnection from which even light can’t escape. The innermost stable circular orbit at 3R_s marks the inner edge of the accretion disk. Massless particles like photons can orbit even closer, at the 1.5R_s photon sphere, where you can see your back by looking straight ahead! The ring of light (mainly 1.3 mm synchrotron radiation) in the EHT photo comes from photons with impact parameter \sim 2.6R_s that just graze the photon sphere and can orbit multiple times before spiraling in or out.

Schwarzschild ("black shield") black hole with event horizon, unstable photon sphere, grazing photon sphere, and innermost stable circular orbit

Schwarzschild (“black shield”) black hole with event horizon, unstable photon sphere, grazing photon sphere, and innermost stable circular orbit

Posted in Astronomy, Physics, Space Exploration | Leave a comment

The Longest Day

December 22 is the longest day of the year, despite being near the northern hemisphere’s shortest daylight.

Earth’s sidereal day is the time to rotate 360° with respect to distant stars, about 23 hours and 56 minutes, and its solar day is the time between successive noons, about 24 hours. Earth’s obliquity (tilt) and revolution (orbit) require an extra rotation (spin) of about 4 minutes to go from noon to noon. Because Earth is coincidentally at perihelion (nearest sun and moving fastest) during the December solstice (maximum tilt toward sun), these effects combine to produce the longest solar day, about 24 hours and 30 seconds. (Clocks average this variation by recording mean solar time.)

In a discrete two-step approximation, the diagram illustrates the difference between solar and sidereal days for planets with 3 different tilts. From the first row to the second, the planets orbit the sun as they spin 360° (in the sense of the green right-handed arrow). From the second row to the third, the planets spin through extra angles (in the sense of the yellow right-handed arrow), so that the marked longitudinal planes again includes the sun. Tilted planets must spin further, taking extra time, to extend a sidereal day to a solar day.

Planet (yellow) orbits sun (green) near perihelion through solar and sidereal days (rows) for 3 planetary tilts (columns). Longitudinal square rotates 360° for a sidereal day and >360° for a solar day.

Planet (yellow) orbits sun (green) near perihelion through solar and sidereal days (rows) for 3 planetary tilts (columns). Longitudinal square rotates 360° for a sidereal day and >360° for a solar day.

Posted in Astronomy | Leave a comment