Physicist Portraits Vol. 3 Pixel Sticker Sheet

10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Physicist Portraits Vol. 3. Every character below has its own sticker page with the full physics panel.

Satyendra Nath Bose

Bose counted photons as truly identical particles and recovered Planck's law. The resulting Bose–Einstein distribution gives the average number of particles in a state of given energy, and it allows many bosons to pile into the same state.

n̄(ε) = 1/(e^((ε−μ)/(k_BT)) − 1)

̄n: mean occupation of a single-particle state; ε: state energy (J); μ: chemical potential (J); k_B: Boltzmann constant (J/K); T: temperature (K)

Cecilia Payne-Gaposchkin

Saha's equation gives how much of each element is ionized in a hot gas at a given temperature and density. Cecilia Payne used it to correct for ionization in stellar spectra, and showed that stars are made mostly of hydrogen and helium.

n_i+1n_e/n_i = (2/Λ³)(g_i+1/g_i) exp(−χ_i/(k_BT))

nᵢ, nᵢ₊₁: number densities of ionisation stages i and i+1 (m⁻³); nₑ: electron number density (m⁻³); gᵢ: statistical weight of stage i; χᵢ: ionisation energy from stage i (J); k_B: Boltzmann constant (J/K); T: temperature (K)

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_☉)

Enrico Fermi

The Fermi–Dirac distribution gives the chance that a state of energy E is occupied by a fermion. Deep below the chemical potential states are full, far above it they are empty, and exactly at it the chance is one half.

f(E) = 1/(e^((E−μ)/(kT)) + 1)

f(E): Fermi–Dirac occupation probability; E: single-particle energy (J); μ: chemical potential (J); k: Boltzmann constant (J/K); T: temperature (K)

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)

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

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)

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)

Ole Rømer

Io's eclipses behind Jupiter ran later when Earth was farther from Jupiter, because the light had farther to travel. Rømer used that delay to show that light has a finite speed, crossing the width of Earth's orbit in a matter of minutes.

Δt = Δd/c

Δ t: extra light travel time (s); Δ d: extra distance light must travel (m); c: speed of light in vacuum (m/s); AU: astronomical unit, about the mean Earth–Sun distance (exactly 149 597 870 700 m since 2012) (m)

Ludwig Boltzmann

Boltzmann's formula says entropy measures the number of microscopic arrangements W that look the same from outside. Disordered states have overwhelmingly more arrangements than ordered ones, so systems drift toward disorder. Since 2019 the constant k_B has an exact defined value.

S = k_B ln W

S: entropy (J/K); k_B: Boltzmann constant (J/K); W: number of microstates consistent with the macrostate; ln: natural logarithm (written 'log' on the tomb)