Cosmic Extremes Pixel Sticker Sheet

11 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Cosmic Extremes. Every character below has its own sticker page with the full physics panel.

Stephen Hawking

Hawking found that black holes radiate like warm bodies, with a temperature inversely proportional to their mass. Bekenstein's entropy is proportional to the area of the horizon, not its volume, which links gravity, thermodynamics and quantum theory.

T_H = ħc³/(8πGMk_B)

T_H: Hawking temperature (K); M: black-hole mass (kg); S_(BH): black-hole entropy (J/K); A: horizon area (m²); ħ, c, G, k_B: fundamental constants (J·s, m/s, m³·kg⁻¹·s⁻², J/K)

Subrahmanyan Chandrasekhar

A white dwarf is held up by electron degeneracy pressure. Chandrasekhar showed that once the electrons become relativistic, this pressure can only support a limited mass, the Chandrasekhar limit. Above that, collapse cannot be stopped.

M_Ch ≈ (5.73/μ_e²) M_☉

M_(Ch): Chandrasekhar mass limit (idealised: cold, non-rotating, ideal electron gas) (kg); ω₃⁰: constant from the Lane–Emden n = 3 solution (about 2.018); ħ: reduced Planck constant (J·s); c: speed of light (m/s); G: gravitational constant (m³·kg⁻¹·s⁻²); μₑ: mean molecular weight per electron

Fritz Zwicky

In a gravitationally bound cluster, the galaxies' speeds are set by the total mass. Zwicky used their velocity spread to estimate the Coma cluster's mass and found it far larger than the light suggested.

σ² ∼ GM/R

σ: line-of-sight velocity dispersion of galaxies (km/s); G: gravitational constant (pc M_☉⁻¹ (km/s)²); M: gravitating mass within radius R (M_☉); R: characteristic cluster radius (pc); M_(vir): virial mass implied by \sigma and R (M_☉)

Vera Rubin

If most of a galaxy's mass were in its bright centre, stars far out would orbit more slowly. Vera Rubin found the speeds stay roughly constant far out, which means the enclosed mass keeps growing with radius: unseen matter.

v(r)² = GM(r)/r

v(r): circular orbital speed (m/s); M(r): enclosed mass (kg); r: radius (m); G: gravitational constant (m³·kg⁻¹·s⁻²)

Jocelyn Bell Burnell

A pulsar's very regular pulses mean it must be extremely dense to spin that fast without flying apart, which points to a neutron star. Radio pulses at lower frequencies arrive later after passing through interstellar electrons, and that delay measures how many electrons lie along the way.

ρ̄ ≳ 3π/(GP²)

ρ̄: minimum mean density for a body spinning with period P not to fly apart (kg/m³); G: gravitational constant (m³·kg⁻¹·s⁻²); P: rotation period (s); Δ t: arrival-time delay between frequencies f₁ < f₂ (s); DM: dispersion measure, ∫ nₑ dl (free electrons per area along the line of sight) (m⁻²); f₁, f₂: observing frequencies (Hz)

Relicta

The cosmic microwave background is light left over from when the universe became transparent. It has a nearly perfect blackbody spectrum, and its temperature has fallen in proportion to the stretching of space, to about 2.7 K today.

T(z) = T₀(1 + z)

T(z): temperature of the background radiation at redshift z (K); z: redshift (last scattering: about 1090); T₀: temperature today (Fixsen 2009: 2.72548 ± 0.00057 K) (K); B_ν: blackbody spectral radiance per unit frequency (W·m⁻²·Hz⁻¹·sr⁻¹); ν: frequency (Hz); h: Planck constant (J·s)

Primordix

In the first minutes after the Big Bang, the ratio of neutrons to protons froze out at a small fixed value. Nearly all those neutrons ended up in helium-4, which predicts the fraction of ordinary mass that came out as helium, matching observations.

Y_p ≃ 2(n/p)/(1 + n/p)

Y_p: primordial helium-4 fraction by mass (observed 0.245 ± 0.003); n/p: neutron-to-proton ratio; Q: rest-energy difference between neutron and proton (MeV); k_B: Boltzmann constant (eV/K); T_f: temperature when the weak reactions freeze out (thermal energy about 0.7–0.8 MeV) (K); m_n, m_p: neutron and proton masses (kg)

Skyrider

In 1912 Victor Hess carried electroscopes up in balloons to 5 350 m (Phys. Z. 13, 1084). Their discharge rate, a measure of ionising radiation, first dipped slightly, then climbed steeply with height. The radiation was coming from above, not from the ground. A flight during a near-total solar eclipse and at night showed no drop, so the Sun was not the source. He shared the 1936 Nobel Prize for cosmic rays.

ionisation rate rises with altitude

altitude: height above sea level, up to 5 350 m in 1912; ionisation rate: ions formed per cm³ per second in the electroscope

Mirrorix

LIGO measures gravitational waves by comparing the lengths of two 4 km arms with laser light. At the end of each arm hangs a test mass: a 40 kg fused-silica mirror 34 cm across, hung from a four-stage pendulum to isolate it from ground vibration (Advanced LIGO, Class. Quantum Grav. 32, 074001, 2015). On 14 September 2015 a wave from two merging black holes, GW150914, stretched and squeezed the arms with a peak strain of 1.0 × 10⁻²¹ (Phys. Rev. Lett. 116, 061102, 2016).

h ≈ ΔL/L

h: gravitational-wave strain, dimensionless; ΔL: change in the difference of the arm lengths (m); L: arm length, 4 km

Muonaut

Muons made high in the atmosphere live only about two microseconds, too short to cross the atmosphere at face value. Moving near the speed of light, their clocks run slow by the factor γ, so many reach the ground; in the muon's frame, the atmosphere is shorter instead.

Δt = γτ, L = L₀/γ

Δ t: muon lifetime as timed by clocks on the ground (s); τ: mean lifetime in the muon's own rest frame (PDG 2024: 2.1969811 μs; the half-life is shorter, 1.52 μs) (s); γ: Lorentz factor; L₀: thickness of the atmosphere in the ground frame (m); L: the same thickness measured in the muon's frame (m); v: muon speed (m/s)

Homestaka

Raymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.

measured 2.56 ± 0.16 ± 0.16 SNU vs predicted 7.6⁺¹·³_−1.1 SNU

SNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)