[utglyvmo] dark matter cloud

what would we observe if the earth, or the solar system, passed though a dense cloud of dark matter?

assume WIMP.  we would see not the dark matter directly but only indirect effects.  hypothesize that orbits will change.  will the sun gravitationally capture dark matter and increase in mass, increasing its fusion rate?

do dense clouds of dark matter exist?  how dense might they be?

update: a low-surface-brightness galaxy could be considered a galaxy-sized dense cloud of dark matter.  but I don't think their average dark matter density is that much higher than that of the Milky Way.

previously on dark matter in the solar system.

[ygnpgfnt] potential energy of a compressible ball

model a spherical solid as a lattice of atoms connected by springs.  let the solid's own gravity compress itself.  how much energy gets stored in the compressed springs?

the earth's nickel-iron core is denser than nickel-iron at room temperature, denser than nickel-iron meteorites.

aside: does this mean that destroying a planet requires less than its gravitational binding energy?  harness this internal potential energy.  but this requires more thought: gravitational potential energy balances spring energy.

to model the lattice of springs, we need spring potential energy to approach infinity as distance between atoms approaches zero.  Hooke's Law is not the right model for this.

aside: does it matter what lattice (e.g., cubic, rhombic dodecahedral) of springs?

we also need potential energy to head to infinity as distance increases to infinity so that atoms do not convert to gas and evaporate in our model.  Hooke's Law could work for this direction.

U = x^2/2 + 1/x.  first term is Hooke's Law; second is electrostatic repulsion.  the leading coefficient of 1/2 places the local minimum at x=1.

alternatively, a U-shaped function that has local minimum at x=1 and is infinite at x=0 and x=2: U = 1/x - 1/(x-2) = 2/(x*(2-x)).  the two expressions after U= are the two most straightforward ways of constructing a function with the two vertical asymptotes.  it is neat that they yield the same result.

consider general relativity: the potential energy of compressed strings exerts gravity, bends spacetime.

[xflxiqkv] a level of stupidity achievable only by a god

e.g., accidentally set the number of dimensions in the universe to not an integer.

...to not a real number.

previously, gods stupidly at war.

[kzrvwmdz] random oscillating field

model a real scalar field oscillating in space and time as a sum of cosines:

f(t,x,y,z) = sum over i of a[i]*cos(b[i] + c[i]*t + d[i]*x + e[i]*y + f[i]*z)

note that constant offset DC can be represented by b[0] = c[0] = d[0] = e[0] = 0.

does this capture all typical shapes?  I think it still works even if you do a rotation or translation of axes.

can it model spherical waves f(t,r)?

can it model dipole radiation?  dipole radiation is a time-varying vector not scalar field, so model it is 3 scalar fields, one for each coordinate.  or, model just its magnitude.

3D or 4D Fourier transform might be better.  3D Fourier is sum of spherical harmonics (I think): how can they be made to vary over time?

oscillating vector fields?  tensor fields (in flat space)?  complex scalar fields?  Fourier series with complex exponentials probably gets you complex scalar fields.

[lavnkjop] easy 21cm radio astronomy

is the 21 cm (1420 MHz) hydrogen line the easiest radio astronomy "target"?  not really a target if observing the hydrogen radiation from "everywhere" with an omnidirectional (dipole) antenna.  I have seen such a simple antenna and software-defined radio successfully detect 21 cm radiation.  it was fun to see a peak right at 1420 MHz.  does 21 cm signal strength vary significantly by antenna orientation and what part of the sky (probably the center of the Milky Way) is overhead?  such space and time variance could make observing 21 cm with simple antenna more interesting than just a peak.

an even easier astronomical target to observe in radio frequencies might be the sun.  I think the sun emits broadband noise, not a clear peak.  by measuring intensity, how easy is it to detect active regions rotating into view, or track the sunspot cycle through radio?  for the former, you need to keep your radio calibrated through months; for the latter, through decades.

is 21 cm the easiest astronomical radio thing to observe not also observable in visible light?  it is fun to see the invisible.  (previously: easiest visible-light astronomy with binoculars.)

while not astronomical, lightning can be detected with radio, even from great distance.  the necessary equipment to triangulate lighting strikes is more complicated than a simple dipole antenna but seems gettable by amateurs.

is 21 cm the longest astronomically interesting intrinsic wavelength of electromagnetic radiation?

(the extrinsic effect of cosmological redshift stretches wavelength.  at cosmological distance z=9, 21 cm radiation becomes 21 * (1 + z) = 210 cm.)

the 21 cm line is the hyperfine electron transition of ground-state hydrogen.  I think hyperfine transitions in other elements have shorter wavelengths.  previously, cesium, 3 cm.

electron energy levels become closely spaced near ionization, so transitions between adjacent high energy levels correspond to longer wavelengths.  but do we care about radiation from transitions near but not quite at ionization?  do they happen often enough to observe?  what about hyperfine transitions of excited hydrogen?  the electron probably unexcites much faster (more frequently) than it spin-flips.

[sggdczyg] private moon

order the objects in the solar system by decreasing mass.  what is the mass of object number N, where N is human population (currently 8.2 billion)?

at what N do things become messy, for example, the precise definition of the extent of an object affects its place in line (e.g., atmospheres, aggregates of rocks and ice so loosely gravitationally bound that there might be gaps between them)?

sort similarly the objects in the Milky Way galaxy.  currently, everyone gets a star, provocatively presuming that we are space-faring and the only ones space-faring.

inspired by Gaila's Moon.

previously, your patch of the sky.

[mktjcnja] pretty Thomson solutions

solutions to the Thomson problem up to 12 points (excluding 11) have nice symmetry.  (though I feel that the solution for 1 point also lacks pretty symmetry.)

2 . digon . vertex types: 1

3 . equilateral triangle . vertex types: 1

4 . regular tetrahedron . vertex types: 1

5 . triangular dipyramid . vertex types: 2

6 . regular octahedron . vertex types: 1

7 . pentagonal dipyramid . vertex types: 2

8 . square antiprism . vertex types: 1

9 . triaugmented triangular prism . vertex types: 2

10 . gyroelongated square dipyramid . vertex types: 2

12 . regular icosahedron . vertex types: 1

"vertex types" refers to vertex transitivity.  we do not care about face or edge types of the convex hull polyhedron because the Thomson problem is about points (vertices).

https://www.mathpages.com/home/kmath005/kmath005.htm describes some (locally) minimum energy configurations.

maybe we care about the "balanced" entries in this Wikipedia table in which magnitude(sum(r[i])) = 0 (precisely), where r[i] are the positions of the charges as vectors.  the table has no citation, but Laszlo Hars has replicated energy results for small solutions.

assuming we trust the Wikipedia table, the balanced sequence continues 2 3 4 5 6 7 8 9 10 12 14 15 16 17 18 20 22 23 24 27 28 29 30 32 which does not have an OEIS number.

there is also the sequence 2 3 4 6 8 12 24 in which the solution has only one vertex type.  24 is snub cube.  there might be some more entries between 12 and 24, but probably not.  the sequence probably ends at 24.

not all icosahedral geodesic spheres appear in the minimum energy table.  GP(2,0) = 42, GP(3,0) = 92, GP(4,0) = 162 are the first few missing.  clearly GP(n,0) seems to not be preferred.  which geodesic spheres are more likely to be minimum energy solutions?  this is likely similar to how cube is not minimum energy for 8 points, instead square antiprism. do icosahedral geodesic spheres continue to appear as minimum energy configurations, or is there a final geodesic sphere which is a solution?

[rcpodzpi] Kelvin vs. Celsius

0 K = -273.15 C
10^-12 K = -273.149999999999 C
10^-11 K = -273.14999999999 C
10^-10 K = -273.1499999999 C
10^-9 K = -273.149999999 C
10^-8 K = -273.14999999 C
10^-7 K = -273.1499999 C
10^-6 K = -273.149999 C
10^-5 K = -273.14999 C
10^-4 K = -273.1499 C
10^-3 K = -273.149 C
10^-2 K = -273.14 C
10^-1 K = -273.05 C
10^0 K = -272.15 C
10^1 K = -263.15 C
10^2 K = -173.15 C
10^3 K = 726.85 C
10^4 K = 9726.85 C
10^5 K = 99726.85 C
10^6 K = 999726.85 C
10^7 K = 9999726.85 C
10^8 K = 99999726.85 C
10^9 K = 999999726.85 C
10^10 K = 9999999726.85 C
10^11 K = 99999999726.85 C
10^12 K = 999999999726.85 C

Celsius values are not weird (not containing lots of nines) only between about 0.01 K and 1000 K, roughly the temperatures relevant to the human experience.  but if you want a temperature scale actually designed for human experience, Fahrenheit is generally better.

science often works with extremely low or extremely high temperatures.  as illustrated above especially for low temperatures, Kelvin is better than Celsius.  for high temperatures, at what threshold is it acceptable to say that the temperature in Kelvin is the same as temperature in Celsius?

use Kelvin for science, Fahrenheit for humans.  Celsius is useful only for discussing liquid or potentially liquid water.

[zbkseudn] 12 faces of the sky

project a dodecahedron onto the celestial sphere, then present each of the 12 faces as star chart images for a monthly calendar.  the images have no correlation with what is visible in the night sky during that month.  (people in urban areas can't see much in the night sky anyway.)

both regular dodecahedron and rhombic dodecahedron would work.  unspecified: map projection of each spherical polygonal face onto flat paper.  regular dodecahedron induces less distortion, but maybe a projected rhombus fits more nicely on a rectangular calendar page.  (but why not pentagonal calendar pages?)

also possible: cube with each face cut in half (pyritohedron) can yield nice rectangular regions.  tetrahedron with each face cut into 3 can yield kites.  (other shapes possible depending on how you cut.)

optimize the orientation of the dodecahedron so that every month has a good collection of labeled interesting objects.  also try to avoid cutting constellations and any other large objects (can't think of any others) awkwardly.  for the regular dodecahedron, a face centered on each of the celestial poles is also desirable.

annotate edges with the name of the month whose map lies beyond the edge.  overlay grid of celestial coordinates (right ascension, declination).

would it be better if the views overlapped?  spherical caps yield circular segments of the sky.  what is the minimum size spherical cap that covers a pentagonal face of a regular dodecahedron (this should be easy)?  how thick should the overlaps be?  maybe variable to avoid cutting constellations awkwardly.

alternative to a star chart: cosmic microwave background.

also possible: various other spherical astronomical bodies (including earth) whose surfaces we've fully mapped.  but these seem less interesting, not looking out at infinity but looking in at something finite.

for a weekly or daily calendar, find equal(ish)-area compact(ish) subdivisions of a sphere into the appropriate number of pieces.  Goldberg polyhedra are pretty.  there exist Goldberg polyhedra of 42 and 362 faces.

[iuggtvmq] Death Star radio telescope

when not obliterating planets, which was most of the time, the Death Star emitter dish was used as a radio telescope by astronomers on board the space station.  may they rest in peace.

inspired by the Arecibo radio dish which could both transmit and receive.

the whole Death Star needs to turn to face its dish toward an observation (or obliteration) target, and it needs to stably keep facing exactly that direction for the duration of the observation.  presumably gyroscopes.  however, any movement of mass inside the space station, e.g., a person walking, will induce via conservation of angular momentum a rotation of the station.  (future work: calculate how much rotation.)  this seems difficult to deal with.  maybe the dish is mechanically isolated from the station: it floats a short distance from the superstructure.

or, the Force keeps the station in line with its target.  the Emperor and Darth Vader are keen astronomers.  this was the original purpose for the space station: politics forced them to add a military capability in order to get it built, much like how the Hubble Space Telescope succeeded becoming approved and built because research and development for its design was reused a dozen times over for U.S. spy satellites.

[aqyyoppn] relativistic bullet

let

(relativistic kinetic energy) = (relativistic mass-energy) - (rest mass-energy)

(rest mass-energy) = m * c^2

let resq = "remaining speed squared" = c^2 - v^2, having units of speed^2.

aside: lorentz = sqrt(c^2/resq) is unitless.  we prefer to work with resq rather than lorentz because with resq, some mistakes can be caught by dimensional analysis.

aside: for very small v, resq has problems with numerical precision.  for very high v, lorentz has problems with numerical precision.

(relativistic mass-energy) = sqrt(c^2/resq) * m * c^2

(relativistic kinetic energy) = (sqrt(c^2/resq)-1) * m * c^2

assume a bullet has mass 100 grain = 6.49 gram.  (aside: the fact that "in" in grain looks like "m" in gram in some fonts is unfortunate.) (another aside: dram is another unit of mass used in firearms, 1/256 lb.  dram sounds similar to gram.  1 dram = 1.77 gram.  "if you are facing a charging rhino, be sure your shotgun shell is loaded with at least ___ [unintelligible]rams of black powder in order to stop it.  but not too much or the gun will explode in your face.")

(rest mass-energy of 100 grain bullet) = 5.82e14 J

at typical bullet speed 0.0000031622777 c = sqrt(1e-11) c = 3110 ft/s = 2121 mph = 948 m/s : (relativistic kinetic energy) = 2911.9 J

0.01 c : (relativistic kinetic energy) = 2.9121e10 J (3162x speed ~= 10,000,000x energy, matching non-relativistic approximation)

0.1 c : 2.93e12 J (10x speed ~= 100x energy, matching non-relativistic approximation)

0.5 c : 9.01e13 J (50x speed ~= 3094x energy, not 2500x by non-relativistic approximation)

0.866 c = sqrt(0.75) c : 5.82e14 J (kinetic energy equals rest mass-energy)

0.9 c : 7.54e14 J

0.99 c : 3.546e15 J

expressing energy in terms of tons of TNT:

(rest mass-energy) = 5.82e14 J = 139 kiloton tnt

typical bullet speed 0.0000031622777 c : (relativistic kinetic energy) = 696 nanoton tnt = 9.74 grain tnt (which unsurprisingly is in the same order of magnitude as typical amounts of black powder in gun rounds)

0.01 c : 6.96 ton tnt

0.1 c : 701 ton tnt

0.5 c : 21.53 kiloton tnt

0.866 c : 139 kiloton tnt

0.9 c : 180 kiloton tnt

0.99 c : 848 kiloton tnt

inspiration: if a SpaceX Starship (approximately 10 kiloton tnt) were to explode on the launchpad, and a significant fraction of its chemical energy were freakishly transferred into a single bullet-sized piece of debris, how far would be safe distance to view the launch and not get hurt by debris?

answer, assuming no air resistance: there is no safe distance.  inverting the equations above, a bullet with relativistic kinetic energy 10 kiloton tnt has speed 0.36 c.  5 kiloton tnt is 0.26 c, because maybe we care about conservation of momentum.  in comparison, speed of low earth orbit is 0.000026 c, and earth escape velocity is 0.000037 c.  a bullet in orbit can hit anywhere on earth; a bullet escaping earth can hit anywhere in the universe.

the SpaceX Starship is of course trying to deliver payloads much more massive than a bullet into orbit or to infinity.  this fact alone is enough to conclude there is no safe distance.

[erlrqhiz] hydrogen as a dipole antenna

if we hypothesize that the 21 cm hydrogen line is produced by a half-wave dipole emitter, then this predicts that the hydrogen atom is 10.5 cm wide.   the actual diameter of the hydrogen atom is about 1e-8 cm, so the error factor between theory and experiment is about 1 billion.

we can go further, seeking greater error.  the 21 cm hydrogen radiation has energy (c*h / (21.106114054160 cm)) = 941.17082 rontojoule = 5.8743262 micro eV = 0.068168725 boltzmann degK per photon.  ascribing that energy change to a change in distance between the hydrogen's electron and proton, the distance change (obtained by solving the quadratic of the Coulomb force) is -(k_C*e^2 / (c*h / (21.106114054160 cm))) + sqrt((k_C*e^2 / (c*h / (21.106114054160 cm)))^2 + bohrradius^2) = 5.7462715e-16 cm via the "units" program.  the error factor compared to the predicted (21.106114054160 cm)/2 dipole antenna is 1.8365051e+16.

these are bad predictions, but not even remotely close to the 1e+50 or 1e+120 error factor of the vacuum energy density, the "worst prediction of all science".

[tnhftoso] biodegradable One Ring

now that Peter Jackson has established that New Zealand is Middle Earth, go on a real-life quest to dispose of a replica One Ring into some New Zealand volcano, of which there are conveniently many.

for example, construct a One Ring out of sodium (or other group 1 element) and drop it into Lake Taupo, which is not just a volcano but a supervolcano, the earth's most recently erupting one at that.  the destruction of the One Ring in water will be fiery and explosive.

or, laser engrave the One Ring inscription onto a bagel, perhaps an Everything Bagel.  (incidentally, the bagel in Everything Everywhere All at Once behaved like a black hole.  was it a deliberate reference to the Kerr ring singularity?)  let it decompose at a dormant or extinct volcano.

[ugvaecin] convenient volcanic land bridge

when has a volcano's lava flow into water conveniently built a bridge over a gap that humans wanted to bridge?

there must be evidence of regular human crossings (over water) before the eruption.  there must now be a road across the land bridge.

Sakurajima, Japan is one possible example.  but the road is there (in part) to visit the volcano famous for the 1914 eruption that created the land bridge.  is (was) there demand for travel to the peninsula (island) apart from that?

[lzkxzkgm] the sproingly-oinglies

a coined word in the style of walkie-talkie: sproingly-oingly.  not sure exactly what it should mean.  maybe chaotic motion of a system with many springs, or nonlinear springs.

[phnkdtec] fun astronomy with a small telescope

the Moon has texture.  Venus has phases.  Jupiter has moons.  Saturn has rings.  there are many stars, much more than 7, in the Pleiades.  there are many, many stars in the Milky Way.  the Sun has spots.

all of these were discovered soon after the invention of the telescope: the early history of the telescope is practically useful for deciding what astronomical objects are the most exciting to observe with a small modern telescope or binoculars.  Galileo has already tried looking at everything for you.

(Galileo went blind observing the sun directly with a telescope.  don't be like Galileo: to see sunspots without repeating his mistake, project the telescopic image of the sun onto a surface.)

[tgjnjvqe] human cubic volume

volume of average human = 62 liter = 0.062 m^3 = 2.2 ft^3 = 16 gallon

blend.  (but how much of the internal volume of a human is gas?)

slurry would fill a cube with side length 40 cm = 1.3 ft.

100 square foot room with 9 ft ceiling fits 411 people.

all of humanity fits in a cube with side length less than 1 km.  16 billion people = 1 cubic kilometer.  67 billion people = 1 cubic mile.

perhaps someday humans gain shapeshifting ability (like Odo), pack themselves into a temperature-controlled Borg Cube, then head for the stars.

[fojocgym] no music of the sphere

prove that spherically symmetric redistributions of mass do not emit gravitational waves.   (you are smarter than Einstein.)

[bvvsscqe] irony of fear about vaccines rewriting your DNA

it is deeply ironic that one of the fears antivaxxers have about COVID-19 vaccination is the fear that mRNA vaccines will rewrite your DNA.  it is ironic because COVID-19 the disease will absolutely definitely rewrite your DNA.  because that's what viruses do: they rewrite your DNA, reprogramming your own cells to become virus factories.  viruses are very good at rewriting your DNA.

if it turns out that children and future descendants of people infected with COVID-19 have a higher rate of birth defects or autism or homosexuality or whatever, no biologist will be very surprised, because that's what viruses do: they rewrite your DNA.  most of the time, the COVID-19 virus (SARS-CoV-2) infects and rewrites the DNA of cells in your respiratory system, but everything is connected inside your body, so it is completely plausible that the virus might manage to infect a gamete (sperm or egg cell) and thereby rewrite the DNA of your yet-to-be-conceived children and all your future descendants.

the human genome is absolutely littered with the DNA of viruses which successfully rewrote the DNA of some human ancestor.  most of those virus sequences have been rendered inert by evolution (virus rewrites the human DNA of something critical, baby is stillborn, that virus sequence then fails to propagate through human DNA), but it is an open research question how much virus DNA is still active, how much of what makes us human is actually the result of ancestral viruses which rewrote our DNA.  previously.

if you want to protect the sanctity of your DNA, get vaccinated.

(counterargument: presume that the government (secretly puppeteered of course by the Illuminati, Rothschilds, Freemasons, Lizard People, etc.) and COVID-19 are both equally competent at rewriting your DNA if they are allowed to introduce a foreign substance into your body.  in choosing whether to get vaccinated or let the disease put virus into you, you are trying to choose the lesser evil.  the government has both a documented history of being evil and plainly visible incentives to continue to be evil in order to maintain power.)

[cvqnvaxd] flying droplets on a ocean planet

model an incompressible liquid under the influence of its own gravity.  the completely static solution is a sphere, and there is probably a rotating hydrostatic solution the shape of an oblate ellipsoid.  exactly which ellipsoidoid (because might not exactly be an ellipsoid)?

what if general relativity?

if the fluid is allowed to move, then things likely become insane due to Navier-Stokes.  turbulence might cause droplets to break off and be ejected faster than escape velocity.