Physics Constants

In 2019 the SI2019 standardization was completed, based on the 7 physics dimensions specific to the Metric system. That is actually an inadequate and insufficient unit system standard, as it is mathematically impossible to unify all historical units with that standard. In 2020, Michael Reed set out to work around that impossibility with a new project called UnitSystems.jl, which ended up completely solving the problem with a brand new 11 dimensional Unified System of Quantities (USQ) for physics.

The following is a new set of formulas for fundamental physics constants: DOI PDF 2020-2025

\[\alpha = \frac{\lambda e^2}{4\pi\varepsilon_0\hbar c} = \frac{\lambda c\mu_0 (e\alpha_L)^2}{4\pi\hbar} = \frac{e^2k_e}{\hbar c} = \frac{\lambda e^2}{2\mu_0ch} = \frac{\lambda c\mu_0\alpha_L^2}{2R_K} = \frac{e^2Z_0}{2h}\]

There exists a deep relationship between the fundamental constants, which also makes them very suitable as a basis for UnitSystem dimensional analysis. All of the formulas on this page are part of the Test suite to guarantee their universal correctness.

\[\mu_{eu} = \frac{m_e}{m_u}, \qquad \mu_{pu} = \frac{m_p}{m_u}, \qquad \mu_{pe} = \frac{m_p}{m_e}, \qquad \alpha_\text{inv} = \frac{1}{\alpha}, \qquad \alpha_G = \left(\frac{m_e}{m_P}\right)^2\]

MeasureSystems.Universe — Constant 
μₑᵤ, μₚᵤ, μₚₑ, αinv, αG, ΩΛ

Physical measured dimensionless Coupling values with uncertainty are the electron to proton mass ratio μₑᵤ, proton to atomic mass ratio μₚᵤ, proton to electron mass ratio μₚₑ, inverted fine structure constant αinv, and the gravitaional coupling constant αG.

julia> μₑᵤ # electronunit(Universe)
μₑᵤ = 0.000548579909065(16)

julia> μₚᵤ # protonunit(Universe)
μₚᵤ = 1.007276466621(53)

julia> μₚₑ # protonelectron(Universe)
μₑᵤ⁻¹μₚᵤ = 1836.15267343(11)

julia> αinv # 1/finestructure(Universe)
α⁻¹ = 137.035999084(21)

julia> αG # coupling(Universe)
𝘩²𝘤⁻²R∞²α⁻⁴mP⁻²2² = 1.751810(39) × 10⁻⁴⁵

julia> ΩΛ # darkenergydensity(Universe)
ΩΛ = 0.6889(56)

Relativistic Constants

\[c = \frac1{\alpha_L\sqrt{\mu_0\varepsilon_0}} = \frac{1}{\alpha}\sqrt{E_h\frac{g_0}{m_e}} = \frac{g_0\hbar\alpha}{m_e r_e} = \frac{e^2k_e}{\hbar\alpha} = \frac{m_e^2G}{\hbar\alpha_G}\]

MeasureSystems.lightspeed — Constant 
lightspeed(U::UnitSystem) = 𝟏/sqrt(vacuumpermeability(U)*vacuumpermittivity(U))/lorentz(U)
speed : [LT⁻¹], [LT⁻¹], [LT⁻¹], [LT⁻¹], [LT⁻¹]
LT⁻¹ [𝘤] Unified

Speed of light in a vacuum 𝘤 for massless particles (m⋅s⁻¹ or ft⋅s⁻¹).

julia> lightspeed(Metric) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] Metric

julia> lightspeed(English) # ft⋅s⁻¹
𝘤⋅ft⁻¹ = 9.835710564304461×10⁸ [ft⋅s⁻¹] English

julia> lightspeed(IAU) # au⋅D⁻¹
𝘤⋅au⁻¹2⁷3³5² = 173.1446326742(35) [au⋅D⁻¹] IAU☉

\[h = 2\pi\hbar = \frac{2e\alpha_L}{K_J} = \frac{8\alpha}{\lambda c\mu_0K_J^2} = \frac{4\alpha_L^2}{K_J^2R_K}\]

MeasureSystems.planck — Constant 
planck(U::UnitSystem) = turn(x)*planckreduced(x)
action : [FLT], [FLT], [ML²T⁻¹], [ML²T⁻¹], [ML²T⁻¹]
FLT⋅(τ = 6.283185307179586) [ħ⋅ϕ] Unified

Planck constant 𝘩 is energy per electromagnetic frequency (J⋅s or ft⋅lb⋅s).

julia> planck(SI2019) # J⋅s
𝘩 = 6.62607015×10⁻³⁴ [J⋅s] SI2019

julia> planck(SI2019)*lightspeed(SI2019) # J⋅m
𝘩⋅𝘤 = 1.9864458571489286×10⁻²⁵ [J⋅m] SI2019

julia> planck(CODATA) # J⋅s
RK⁻¹KJ⁻²2² = 6.626070039(82) × 10⁻³⁴ [J⋅s] CODATA

julia> planck(Conventional) # J⋅s
RK90⁻¹KJ90⁻²2² = 6.626068854361324×10⁻³⁴ [J⋅s] Conventional

julia> planck(SI2019)/elementarycharge(SI2019) # eV⋅s
𝘩⋅𝘦⁻¹ = 4.135667696923859×10⁻¹⁵ [Wb] SI2019

julia> planck(SI2019)*lightspeed(SI2019)/elementarycharge(SI2019) # eV⋅m
𝘩⋅𝘤⋅𝘦⁻¹ = 1.2398419843320026×10⁻⁶ [V⋅m] SI2019

julia> planck(British) # ft⋅lb⋅s
𝘩⋅g₀⁻¹ft⁻¹lb⁻¹ = 4.887138541095932×10⁻³⁴ [lb⋅ft⋅s] British

\[\hbar = \frac{h}{2\pi} = \frac{e\alpha_L}{\pi K_J} = \frac{4\alpha}{\pi\lambda c\mu_0K_J^2} = \frac{2\alpha_L}{\pi K_J^2R_K}\]

MeasureSystems.planckreduced — Constant 
planckreduced(U::UnitSystem) = planck(x)/turn(x)
angularmomentum : [FLTA⁻¹], [FLT], [ML²T⁻¹], [ML²T⁻¹], [ML²T⁻¹]
FLTA⁻¹ [ħ] Unified

Reduced Planck constant ħ is a Planck per radian (J⋅s⋅rad⁻¹ or ft⋅lb⋅s⋅rad⁻¹).

julia> planckreduced(SI2019) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹ = 1.0545718176461565×10⁻³⁴ [J⋅s] SI2019

julia> planckreduced(SI2019)*lightspeed(SI2019) # J⋅m⋅rad⁻¹
𝘩⋅𝘤⋅τ⁻¹ = 3.1615267734966903×10⁻²⁶ [J⋅m] SI2019

julia> planckreduced(CODATA) # J⋅s⋅rad⁻¹
RK⁻¹KJ⁻²τ⁻¹2² = 1.054571800(13) × 10⁻³⁴ [J⋅s] CODATA

julia> planckreduced(Conventional) # J⋅s⋅rad⁻¹
RK90⁻¹KJ90⁻²τ⁻¹2² = 1.0545716114388567×10⁻³⁴ [J⋅s] Conventional

julia> planckreduced(SI2019)/elementarycharge(SI2019) # eV⋅s⋅rad⁻¹
𝘩⋅𝘦⁻¹τ⁻¹ = 6.582119569509067×10⁻¹⁶ [Wb] SI2019

julia> planckreduced(SI2019)*lightspeed(SI2019)/elementarycharge(SI2019) # eV⋅m⋅rad⁻¹
𝘩⋅𝘤⋅𝘦⁻¹τ⁻¹ = 1.973269804593025×10⁻⁷ [V⋅m] SI2019

julia> planckreduced(British) # ft⋅lb⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ft⁻¹lb⁻¹τ⁻¹ = 7.778122563903315×10⁻³⁵ [lb⋅ft⋅s] British

\[m_P = \sqrt{\frac{\hbar c}{G}} = \frac1k\sqrt{\hbar c\frac{m_\odot}{\text{au}^3}} =\frac{m_e}{\sqrt{\alpha_G}} = \frac{2R_\infty hg_0}{c\alpha^2\sqrt{\alpha_G}}\]

MeasureSystems.planckmass — Constant 
planckmass(U::UnitSystem) = electronmass(U)/sqrt(coupling(U))
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(𝘩⁻¹𝘤⋅R∞⁻¹α²mP⋅2⁻¹ = 2.389222(26) × 10²²) [mₑ] Unified

Planck mass factor mP from the gravitational coupling constant αG (kg or slugs).

juila> planckmass(Metric)*lightspeed(Metric)^2/elementarycharge(Metric) # eV⋅𝘤⁻²
𝘩⁻¹ᐟ²𝘤⁵ᐟ²α⁻¹ᐟ²mP⋅τ¹ᐟ²2⁻⁷ᐟ²5⁻⁷ᐟ² = 1.220890(13) × 10²⁸ [V] Metric

juila> planckmass(Metric) # kg
mP = 2.176434(24) × 10⁻⁸ [kg] Metric

juila> planckmass(Metric)/dalton(Metric) # Da
𝘩⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅mP⋅2⁻¹ = 1.310679(14) × 10¹⁹ [𝟙] Metric

juila> planckmass(Metric)*lightspeed(Metric)^2/elementarycharge(Metric)/sqrt(𝟐^2*τ) # eV⋅𝘤⁻²
𝘩⁻¹ᐟ²𝘤⁵ᐟ²α⁻¹ᐟ²mP⋅2⁻⁹ᐟ²5⁻⁷ᐟ² = 2.435323(27) × 10²⁷ [V] Metric

julia> planckmass(PlanckGauss) # mP
𝟏 = 1.0 [mP] PlanckGauss

\[k = \frac{1}{m_P}\sqrt{\hbar c\frac{m_\odot}{\text{au}^3}} = \frac{1}{m_e}\sqrt{\hbar c\alpha_G\frac{m_\odot}{\text{au}^3}} = \sqrt{G\frac{m_\odot}{\text{au}^3}} = c^2\sqrt{\frac{\kappa m_\odot}{8\pi\text{au}^3}}\]

MeasureSystems.gaussgravitation — Constant 
gaussgravitation(U::UnitSystem) = sqrt(gravitation(U)*solarmass(U)/astronomicalunit(U)^3)
angularfrequency : [T⁻¹A], [T⁻¹], [T⁻¹], [T⁻¹], [T⁻¹]
T⁻¹A⋅(𝘤⁻¹R∞⁻¹α²kG⋅2⁻¹⁵3⁻⁷5⁻⁵ = 2.56456351221(79) × 10⁻²⁸) [ħ⁻¹𝘤²mₑ⋅g₀⁻¹] Unified

Gaussian gravitational constant k of Newton's laws (Hz or rad⋅D⁻¹).

julia> gaussgravitation(Engineering)
kG⋅τ⋅2⁻¹⁴3⁻⁷5⁻⁵ = 1.990983676471466×10⁻⁷ [s⁻¹rad] Engineering

julia> gaussgravitation(MetricGradian)
kG⋅2⁻¹⁰3⁻⁷5⁻³ = 1.2674995749028348×10⁻⁵ [s⁻¹gon] MetricGradian

julia> gaussgravitation(MetricDegree)
kG⋅2⁻¹¹3⁻⁵5⁻⁴ = 1.1407496174125516×10⁻⁵ [s⁻¹deg] MetricDegree

julia> gaussgravitation(MetricArcminute)
kG⋅2⁻⁹3⁻⁴5⁻³ = 0.0006844497704475308 [s⁻¹amin] MetricArcminute

julia> gaussgravitation(MetricArcsecond)
kG⋅2⁻⁷3⁻³5⁻² = 0.041066986226851857 [s⁻¹asec] MetricArcsecond

juila> gaussgravitation(MPH)
kG⋅τ⋅2⁻¹⁰3⁻⁵5⁻³ = 0.0007167541235297278 [h⁻¹] MPH

julia> gaussgravitation(IAU)
kG⋅τ⋅2⁻⁷3⁻⁴5⁻³ = 0.017202098964713464 [D⁻¹] IAU☉

\[G = k^2\frac{\text{au}^3}{m_\odot} = \frac{\hbar c}{m_P^2} = \frac{\hbar c\alpha_G}{m_e^2} = \frac{c^3\alpha^4\alpha_G}{8\pi g_0^2 R_\infty^2 h} = \frac{\kappa c^4}{8\pi}\]

MeasureSystems.gravitation — Constant 
gravitation(U::UnitSystem) = lightspeed(U)*planckreduced(U)/planckmass(U)^2
nonstandard : [FM⁻²L²], [F⁻¹L⁴T⁻⁴], [M⁻¹L³T⁻²], [M⁻¹L³T⁻²], [M⁻¹L³T⁻²]
FM⁻²L²⋅(𝘩²𝘤⁻²R∞²α⁻⁴mP⁻²2² = 1.751810(39) × 10⁻⁴⁵) [ħ⋅𝘤⋅mₑ⁻²ϕ] Unified

Universal gravitational constant G of Newton's law (m³⋅kg⁻¹⋅s⁻² or ft³⋅slug⁻¹⋅s⁻²).

juila> gravitation(Metric) # m³⋅kg⁻¹⋅s⁻²
𝘩⋅𝘤⋅mP⁻²τ⁻¹ = 6.67430(15) × 10⁻¹¹ [kg⁻¹m³s⁻²] Metric

julia> gravitation(English) # ft³⋅lbm⁻¹⋅s⁻²
𝘩⋅𝘤⋅g₀⁻¹ft⁻²lb⋅mP⁻²τ⁻¹ = 3.322929(73) × 10⁻¹¹ [lbf⋅lbm⁻²ft²] English

julia> gravitation(PlanckGauss)
𝟏 = 1.0 [mP⁻²] PlanckGauss

\[\kappa = \frac{8\pi G}{c^4} = \frac{8\pi k^2\text{au}^3}{c^4m_\odot} = \frac{8\pi\hbar}{c^3m_P^2} = \frac{8\pi\hbar\alpha_G}{c^3m_e^2} = \frac{\alpha^4\alpha_G}{g_0^2R_\infty^2 h c}\]

MeasureSystems.einstein — Constant 
einstein(U::UnitSystem) = 𝟐^2*τ*gravitation(U)/lightspeed(U)^4
nonstandard : [FM⁻²L⁻²T⁴], [F⁻¹], [M⁻¹L⁻¹T²], [M⁻¹L⁻¹T²], [M⁻¹L⁻¹T²]
FM⁻²L⁻²T⁴⋅(𝘩²𝘤⁻²R∞²α⁻⁴mP⁻²τ⋅2⁴ = 4.402779(97) × 10⁻⁴⁴) [ħ⋅𝘤⁻³mₑ⁻²ϕ] Unified

Einstein's gravitational constant from the Einstein field equations (s⋅²⋅m⁻¹⋅kg⁻¹).

julia> einstein(Metric) # s²⋅m⁻¹⋅kg⁻¹
𝘩⋅𝘤⁻³mP⁻²2² = 2.076648(46) × 10⁻⁴³ [N⁻¹] Metric

julia> einstein(IAU) # day²⋅au⁻¹⋅M☉⁻¹
𝘤⁻⁴au⁴kG²τ³2⁻⁴⁰3⁻²⁰5⁻¹⁴ = 8.27497346775(66) × 10⁻¹² [M☉⁻¹au⁻¹D²] IAU☉

Atomic & Nuclear Constants

\[m_u = \frac{M_u}{N_A} = \frac{m_e}{\mu_{eu}} = \frac{m_p}{\mu_{pu}} = \frac{2R_\infty hg_0}{\mu_{eu}c\alpha^2} = \frac{m_P}{\mu_{eu}}\sqrt{\alpha_G}\]

MeasureSystems.dalton — Constant 
dalton(U::UnitSystem) = molarmass(U)/avogadro(U)
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(μₑᵤ⁻¹ = 1822.888486209(53)) [mₑ] Unified

Atomic mass unit Da of 1/12 of the C₁₂ carbon-12 atom's mass (kg or slugs).

julia> dalton(Metric) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2 = 1.66053906660(51) × 10⁻²⁷ [kg] Metric

julia> dalton(Hartree) # mₑ
μₑᵤ⁻¹ = 1822.888486209(53) [𝟙] Hartree

julia> dalton(QCD) # mₚ
μₚᵤ⁻¹ = 0.992776097862(52) [mₚ] QCD

julia> dalton(Metric)*lightspeed(Metric)^2 # J
𝘩⋅𝘤⋅R∞⋅α⁻²μₑᵤ⁻¹2 = 1.49241808560(46) × 10⁻¹⁰ [J] Metric

julia> dalton(SI2019)*lightspeed(SI2019)^2/elementarycharge(SI2019) # eV⋅𝘤⁻²
𝘩⋅𝘤⋅𝘦⁻¹R∞⋅α⁻²μₑᵤ⁻¹2 = 9.3149410242(29) × 10⁸ [V] SI2019

julia> dalton(British) # lb
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ft⋅lb⁻¹2 = 1.13783069118(35) × 10⁻²⁸ [slug] British

\[m_p = \mu_{pu} m_u = \mu_{pu}\frac{M_u}{N_A} = \mu_{pe}m_e = \mu_{pe}\frac{2R_\infty hg_0}{c\alpha^2} = m_P\mu_{pe}\sqrt{\alpha_G}\]

MeasureSystems.protonmass — Constant 
protonmass(U::UnitSystem) = protonunit(U)*dalton(U)
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(μₑᵤ⁻¹μₚᵤ = 1836.15267343(11)) [mₑ] Unified

Proton mass mₚ of subatomic particle with +𝘦 elementary charge (kg or mass).

julia> protonmass(Metric) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg] Metric

julia> protonmass(SI2019)*lightspeed(SI2019)^2/elementarycharge(SI2019) # eV⋅𝘤⁻²
𝘩⋅𝘤⋅𝘦⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 9.3827208816(29) × 10⁸ [V] SI2019

julia> protonmass(Metric)/dalton(Metric) # Da
μₚᵤ = 1.007276466621(53) [𝟙] Metric

julia> protonmass(Hartree) # mₑ
μₑᵤ⁻¹μₚᵤ = 1836.15267343(11) [𝟙] Hartree

julia> protonmass(QCD) # mₚ
𝟏 = 1.0 [mₚ] QCD

\[m_e = \mu_{eu}m_u = \mu_{eu}\frac{M_u}{N_A} = \frac{m_p}{\mu_{pe}} = \frac{2R_\infty h g_0}{c\alpha^2} = m_P\sqrt{\alpha_G}\]

MeasureSystems.electronmass — Constant 
electronmass(U::UnitSystem) = protonmass(U)/protonelectron(U) # αinv^2*R∞*2𝘩/𝘤
mass : [M], [FL⁻¹T²], [M], [M], [M]
M [mₑ] Unified

Electron rest mass mₑ of subatomic particle with -𝘦 elementary charge (kg or slugs).

julia> electronmass(Metric) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] Metric

julia> electronmass(CODATA) # kg
𝘤⁻¹R∞⋅α⁻²RK⁻¹KJ⁻²2³ = 9.10938355(11) × 10⁻³¹ [kg] CODATA

julia> electronmass(Conventional) # kg
𝘤⁻¹R∞⋅α⁻²RK90⁻¹KJ90⁻²2³ = 9.1093819203(28) × 10⁻³¹ [kg] Conventional

julia> electronmass(International) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²Ωᵢₜ⋅Vᵢₜ⁻²2 = 9.1078806534(28) × 10⁻³¹ [kg] International

julia> electronmass(Metric)/dalton(Metric) # Da
μₑᵤ = 0.000548579909065(16) [𝟙] Metric

julia> electronmass(QCD) # mₚ
μₑᵤ⋅μₚᵤ⁻¹ = 0.000544617021487(33) [mₚ] QCD

julia> electronmass(Hartree) # mₑ
𝟏 = 1.0 [𝟙] Hartree

julia> electronmass(Metric)*lightspeed(Metric)^2 # J
𝘩⋅𝘤⋅R∞⋅α⁻²2 = 8.1871057769(25) × 10⁻¹⁴ [J] Metric

julia> electronmass(SI2019)*lightspeed(SI2019)^2/elementarycharge(SI2019) # eV⋅𝘤⁻²
𝘩⋅𝘤⋅𝘦⁻¹R∞⋅α⁻²2 = 510998.95000(16) [V] SI2019

julia> electronmass(English) # lb
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹2 = 2.00827533796(62) × 10⁻³⁰ [lbm] English

\[E_h = \frac{m_e}{g_0}(c\alpha)^2 = \frac{\hbar c\alpha}{a_0} = \frac{g_0\hbar^2}{m_ea_0^2} = 2R_\infty hc = \frac{m_P}{g_0}\sqrt{\alpha_G}(c\alpha)^2\]

MeasureSystems.hartree — Constant 
hartree(U::UnitSystem) = electronmass(U)/gravity(U)*(lightspeed(U)*finestructure(U))^2
energy : [FL], [FL], [ML²T⁻²], [ML²T⁻²], [ML²T⁻²]
FL⋅(α² = 5.3251354520(16) × 10⁻⁵) [𝘤²mₑ⋅g₀⁻¹] Unified

Hartree electric potential energy Eₕ of the hydrogen atom at ground state is 2R∞*𝘩*𝘤 (J).

julia> hartree(SI2019)/elementarycharge(SI2019) # eV
𝘩⋅𝘤⋅𝘦⁻¹R∞⋅2 = 27.211386245989(52) [V] SI2019

julia> hartree(Metric) # J
𝘩⋅𝘤⋅R∞⋅2 = 4.3597447222072(83) × 10⁻¹⁸ [J] Metric

julia> hartree(CGS) # erg
𝘩⋅𝘤⋅R∞⋅2⁸5⁷ = 4.3597447222072(83) × 10⁻¹¹ [erg] Gauss

julia> hartree(Metric)*avogadro(Metric)/kilo # kJ⋅mol⁻¹
𝘤²α²μₑᵤ⋅2⁻⁶5⁻⁶ = 2625.49964038(81) [J⋅mol⁻¹] Metric

julia> hartree(Metric)*avogadro(Metric)/kilocalorie(Metric) # kcal⋅mol⁻¹
𝘤²α²μₑᵤ⋅Ωᵢₜ⋅Vᵢₜ⁻²2⁻⁸3⁻²5⁻⁷43 = 627.09920344(19) [mol⁻¹] Metric

julia> 𝟐*rydberg(CGS) # Eₕ/𝘩/𝘤/100 cm⁻¹
R∞⋅2⁻¹5⁻² = 219474.63136320(42) [cm⁻¹] Gauss

julia> hartree(Metric)/planck(Metric) # Hz
𝘤⋅R∞⋅2 = 6.579683920502(13) × 10¹⁵ [Hz] Metric

julia> hartree(Metric)/boltzmann(Metric) # K
kB⁻¹NA⁻¹𝘤²α²μₑᵤ⋅2⁻³5⁻³ = 315775.024913(97) [K] Metric

In a Gaussian unit system where 4π*ε₀ == 1 the Hartree energy is 𝘦^2/a₀.

\[R_\infty = \frac{E_h}{2hc} = \frac{m_e c\alpha^2}{2hg_0} = \frac{\alpha}{4\pi a_0} = \frac{m_e r_e c}{2ha_0g_0} = \frac{\alpha^2m_ec}{4\pi\hbar g_0} = \frac{m_Pc\alpha^2\sqrt{\alpha_G}}{2hg_0}\]

MeasureSystems.rydberg — Constant 
rydberg(U::UnitSystem) = hartree(U)/2planck(U)/lightspeed(U) # Eₕ/2𝘩/𝘤
wavenumber : [L⁻¹], [L⁻¹], [L⁻¹], [L⁻¹], [L⁻¹]
L⁻¹⋅(α²τ⁻¹2⁻¹ = 4.2376081491(13) × 10⁻⁶) [ħ⁻¹𝘤⋅mₑ⋅ϕ⁻¹g₀⁻¹] Unified

Rydberg constant R∞ is lowest energy photon capable of ionizing atom at ground state (m⁻¹).

julia> rydberg(Metric) # m⁻¹
R∞ = 1.0973731568160(21) × 10⁷ [m⁻¹] Metric

The Rydberg constant for hydrogen RH is R∞*mₚ/(mₑ+mₚ) (m⁻¹).

julia> rydberg(Metric)*protonmass(Metric)/(electronmass(Metric)+protonmass(Metric)) # m⁻¹
𝘩⋅𝘤⁻¹R∞²α⁻²μₑᵤ⁻¹μₚᵤ⋅2⋅5.9753831112(19) × 10²⁶ = 1.09677583403(48) × 10⁷ [m⁻¹] Metric

Rydberg unit of photon energy Ry is 𝘩*𝘤*R∞ or Eₕ/2 (J).

julia> hartree(Metric)/2 # J
𝘩⋅𝘤⋅R∞ = 2.1798723611036(42) × 10⁻¹⁸ [J] Metric

julia> hartree(SI2019)/𝟐/elementarycharge(SI2019) # eV
𝘩⋅𝘤⋅𝘦⁻¹R∞ = 13.605693122994(26) [V] SI2019

Rydberg photon frequency 𝘤*R∞ or Eₕ/2𝘩 (Hz).

julia> lightspeed(Metric)*rydberg(Metric) # Hz
𝘤⋅R∞ = 3.2898419602509(63) × 10¹⁵ [Hz] Metric

Rydberg wavelength 1/R∞ (m).

julia> 𝟏/rydberg(Metric) # m
R∞⁻¹ = 9.112670505824(17) × 10⁻⁸ [m] Metric

julia> 𝟏/rydberg(Metric)/τ # m⋅rad⁻¹
R∞⁻¹τ⁻¹ = 1.4503265557696(28) × 10⁻⁸ [m] Metric

Precision measurements of the Rydberg constants are within a relative standard uncertainty of under 2 parts in 10¹², and is chosen to constrain values of other physical constants.

\[a_0 = \frac{g_0\hbar}{m_ec\alpha} = \frac{g_0\hbar^2}{k_e m_ee^2} = \frac{r_e}{\alpha^2} = \frac{\alpha}{4\pi R_\infty}\]

MeasureSystems.bohr — Constant 
bohr(U::UnitSystem) = planckreduced(U)*gravity(U)/electronmass(U)/lightspeed(U)/finestructure(U)
angularlength : [LA⁻¹], [L], [L], [L], [L]
LA⁻¹⋅(α⁻¹ = 137.035999084(21)) [ħ⋅𝘤⁻¹mₑ⁻¹g₀] Unified

Bohr radius of the hydrogen atom in its ground state a₀ (m).

julia> bohr(Metric) # m
R∞⁻¹α⋅τ⁻¹2⁻¹ = 5.29177210902(81) × 10⁻¹¹ [m] Metric

julia> bohr(IPS) # in
R∞⁻¹α⋅ft⁻¹τ⁻¹2⋅3 = 2.08337484607(32) × 10⁻⁹ [in] IPS

julia> bohr(Hartree) # a₀
𝟏 = 1.0 [a₀] Hartree

\[r_e = g_0\frac{\hbar\alpha}{m_ec} = \alpha^2a_0 = g_0\frac{e^2 k_e}{m_ec^2} = \frac{2hR_\infty g_0a_0}{m_ec} = \frac{\alpha^3}{4\pi R_\infty}\]

MeasureSystems.electronradius — Constant 
electronradius(U::UnitSystem) = finestructure(U)*planckreduced(U)*gravity(U)/electronmass(U)/lightspeed(U)
angularlength : [LA⁻¹], [L], [L], [L], [L]
LA⁻¹⋅(α = 0.0072973525693(11)) [ħ⋅𝘤⁻¹mₑ⁻¹g₀] Unified

Classical electron radius or Lorentz radius or Thomson scattering length (m).

julia> electronradius(Metric) # m
R∞⁻¹α³τ⁻¹2⁻¹ = 2.8179403262(13) × 10⁻¹⁵ [m] Metric

julia> electronradius(CODATA) # m
R∞⁻¹α³τ⁻¹2⁻¹ = 2.8179403262(13) × 10⁻¹⁵ [m] CODATA

julia> electronradius(Conventional) # m
R∞⁻¹α³τ⁻¹2⁻¹ = 2.8179403262(13) × 10⁻¹⁵ [m] Conventional

julia> electronradius(Hartree) # a₀
α² = 5.3251354520(16) × 10⁻⁵ [a₀] Hartree

\[\Delta\nu_{\text{Cs}} = \Delta\tilde\nu_{\text{Cs}}c = \frac{\Delta\omega_{\text{Cs}}}{2\pi} = \frac{c}{\Delta\lambda_{\text{Cs}}} = \frac{\Delta E_{\text{Cs}}}{h}\]

MeasureSystems.hyperfine — Constant 
hyperfine(U::UnitSystem) = frequency(ΔνCs = 9.19263177×10⁹,U)
frequency : [T⁻¹], [T⁻¹], [T⁻¹], [T⁻¹], [T⁻¹]
T⁻¹⋅(𝘤⁻¹ΔνCs⋅R∞⁻¹α²τ⁻¹2⁻¹ = 1.18409248138(36) × 10⁻¹¹) [ħ⁻¹𝘤²mₑ⋅ϕ⁻¹g₀⁻¹] Unified

Unperturbed groundstate hyperfine transition frequency ΔνCs of caesium-133 atom (Hz).

julia> hyperfine(Metric) # Hz
ΔνCs = 9.19263177×10⁹ [Hz] Metric

Thermodynamic Constants

\[M_u = m_uN_A = N_A\frac{m_e}{\mu_{eu}} = N_A\frac{m_p}{\mu_{pu}} = N_A\frac{2R_\infty hg_0}{\mu_{eu}c\alpha^2}\]

MeasureSystems.molarmass — Constant 
molarmass(U::UnitSystem) = avogadro(U)*electronmass(U)/electronunit(U)
molarmass : [MN⁻¹], [FL⁻¹T²N⁻¹], [MN⁻¹], [MN⁻¹], [MN⁻¹]
MN⁻¹ [Mᵤ] Unified

Molar mass constant Mᵤ is the ratio of the molarmass and relativemass of a chemical.

julia> molarmass(CGS) # g⋅mol⁻¹
𝟏 = 1.0 [g⋅mol⁻¹] Gauss

julia> molarmass(Metric) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] Metric

julia> molarmass(SI2019) # kg⋅mol⁻¹
NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2 = 0.00099999999966(31) [kg⋅mol⁻¹] SI2019

julia> molarmass(International) # kg⋅mol⁻¹
Ωᵢₜ⋅Vᵢₜ⁻²2⁻³5⁻³ = 0.0009998350000179567 [kg⋅mol⁻¹] International

\[N_A = \frac{R_u}{k_B} = \frac{M_u}{m_u} = M_u\frac{\mu_{eu}}{m_e} = M_u\frac{\mu_{eu}c\alpha^2}{2R_\infty h g_0}\]

MeasureSystems.avogadro — Constant 
avogadro(U::UnitSystem) = molargas(x)/boltzmann(x) # Mᵤ/dalton(x)
nonstandard : [N⁻¹], [N⁻¹], [N⁻¹], [N⁻¹], [N⁻¹]
N⁻¹⋅(μₑᵤ = 0.000548579909065(16)) [mₑ⁻¹Mᵤ] Unified

Avogadro NA is molarmass(x)/dalton(x) number of atoms in a 12 g sample of C₁₂.

julia> avogadro(SI2019) # mol⁻¹
NA = 6.02214076×10²³ [mol⁻¹] SI2019

julia> avogadro(Metric) # mol⁻¹
𝘩⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅2⁻⁴5⁻³ = 6.0221407621(19) × 10²³ [mol⁻¹] Metric

julia> avogadro(CODATA) # mol⁻¹
𝘤⋅R∞⁻¹α²μₑᵤ⋅RK⋅KJ²2⁻⁶5⁻³ = 6.022140863(75) × 10²³ [mol⁻¹] CODATA

julia> avogadro(Conventional) # mol⁻¹
𝘤⋅R∞⁻¹α²μₑᵤ⋅RK90⋅KJ90²2⁻⁶5⁻³ = 6.0221419396(19) × 10²³ [mol⁻¹] Conventional

julia> avogadro(English) # lb-mol⁻¹
𝘩⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅lb⋅2⁻¹ = 2.73159710074(84) × 10²⁶ [lb-mol⁻¹] English

julia> avogadro(British) # slug-mol⁻¹
𝘩⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅g₀⋅ft⁻¹lb⋅2⁻¹ = 8.7886537756(27) × 10²⁷ [slug-mol⁻¹] British

\[k_B = \frac{R_u}{N_A} = m_u\frac{R_u}{M_u} = \frac{m_e R_u}{\mu_{eu}M_u} = \frac{2R_uR_\infty h g_0}{M_u \mu_{eu}c\alpha^2}\]

MeasureSystems.boltzmann — Constant 
boltzmann(U::UnitSystem) = molargas(x)/avogadro(x)
entropy : [FLΘ⁻¹], [FLΘ⁻¹], [ML²T⁻²Θ⁻¹], [ML²T⁻²Θ⁻¹], [ML²T⁻²Θ⁻¹]
FLΘ⁻¹ [kB] Unified

Boltzmann constant kB is the entropy amount of a unit number microstate permutation.

julia> boltzmann(SI2019) # J⋅K⁻¹
kB = 1.380649×10⁻²³ [J⋅K⁻¹] SI2019

julia> boltzmann(Metric) # J⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³ = 1.38064899953(43) × 10⁻²³ [J⋅K⁻¹] Metric

julia> boltzmann(SI2019)/elementarycharge(SI2019) # eV⋅K⁻¹
kB⋅𝘦⁻¹ = 8.617333262145179×10⁻⁵ [V⋅K⁻¹] SI2019

julia> boltzmann(SI2019)/planck(SI2019) # Hz⋅K⁻¹
kB⋅𝘩⁻¹ = 2.0836619123327576×10¹⁰ [Hz⋅K⁻¹] SI2019

julia> boltzmann(CGS) # erg⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2¹¹5¹⁰ = 1.38064899953(43) × 10⁻¹⁶ [erg⋅K⁻¹] Gauss

julia> boltzmann(SI2019)/calorie(SI2019) # calᵢₜ⋅K⁻¹
kB⋅Ωᵢₜ⋅Vᵢₜ⁻²2⁻²3⁻²5⁻¹43 = 3.2976728498006145×10⁻²⁴ [K⁻¹] SI2019

julia> boltzmann(SI2019)*°R/calorie(SI2019) # calᵢₜ⋅°R⁻¹
kB⋅Ωᵢₜ⋅Vᵢₜ⁻²2⁻²3⁻⁴43 = 1.832040472111452×10⁻²⁴ [K⁻¹] SI2019

julia> boltzmann(Brtish) # ft⋅lb⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ft⁻¹lb⁻¹2⁴3⁻²5⁴ = 5.6573024638(17) × 10⁻²⁴ [lb⋅ft⋅°R⁻¹] British

julia> boltzmann(SI2019)/planck(SI2019)/lightspeed(SI2019) # m⁻¹⋅K⁻¹
kB⋅𝘩⁻¹𝘤⁻¹ = 69.50348004861274 [m⁻¹K⁻¹] SI2019

julia> avogadro(SI2019)*boltzmann(SI2019)/calorie(SI2019) # calᵢₜ⋅mol⁻¹⋅K⁻¹
kB⋅NA⋅Ωᵢₜ⋅Vᵢₜ⁻²2⁻²3⁻²5⁻¹43 = 1.9859050081929637 [K⁻¹mol⁻¹] SI2019

julia> dB(boltzmann(SI2019)) # dB(W⋅K⁻¹⋅Hz⁻¹)
dB(kB) = -228.59916717321767 [dB(kg⋅m²s⁻²K⁻¹)] SI2019

\[R_u = k_B N_A = k_B\frac{M_u}{m_u} = k_BM_u\frac{\mu_{eu}}{m_e} = k_BM_u\frac{\mu_{eu}c\alpha^2}{2hR_\infty g_0}\]

MeasureSystems.molargas — Constant 
molargas(U::UnitSystem) = boltzmann(x)*avogadro(x)
molarentropy : [FLΘ⁻¹N⁻¹], [FLΘ⁻¹N⁻¹], [ML²T⁻²Θ⁻¹N⁻¹], [ML²T⁻²Θ⁻¹N⁻¹], [ML²T⁻²Θ⁻¹N⁻¹]
FLΘ⁻¹N⁻¹⋅(μₑᵤ = 0.000548579909065(16)) [kB⋅mₑ⁻¹Mᵤ] Unified

Universal gas constant Rᵤ is factored into specific gasconstant(x)*molarmass(x) values.

julia> molargas(SI2019) # J⋅K⁻¹⋅mol⁻¹
kB⋅NA = 8.31446261815324 [J⋅K⁻¹mol⁻¹] SI2019

julia> molargas(English)/𝟐^4/𝟑^2 # psi⋅ft³⋅°R⁻¹⋅lb-mol⁻¹
kB⋅NA⋅g₀⁻¹ft⁻¹2⁻¹3⁻⁴5⁴ = 10.731577089016287 [lbf⋅ft⋅°R⁻¹lb-mol⁻¹] English

julia> molargas(English)/atmosphere(English) # atm⋅ft³⋅R⁻¹⋅lb-mol⁻¹
kB⋅NA⋅ft⁻³lb⋅atm⁻¹2³3⁻²5⁴ = 0.7302405072952731 [ft³°R⁻¹lb-mol⁻¹] English

julia> molargas(English)/thermalunit(English) # BTU⋅°R⁻¹⋅lb-mol⁻¹
kB⋅NA⋅Ωᵢₜ⋅Vᵢₜ⁻²2⁻²3⁻²5⁻¹43 = 1.9859050081929637 [°R⁻¹lb-mol⁻¹] English

julia> molargas(Metric)/atmosphere(Metric) # atm⋅m³⋅K⁻¹⋅mol⁻¹
kB⋅NA⋅atm⁻¹ = 8.205736608095969×10⁻⁵ [m³K⁻¹mol⁻¹] Metric

julia> molargas(Metric)/torr(Metric) # m³⋅torr⋅K⁻¹⋅mol⁻¹
kB⋅NA⋅atm⁻¹2³5⋅19 = 0.062363598221529364 [m³K⁻¹mol⁻¹] Metric

julia> molargas(English)/torr(English) # ft³⋅torr⋅°R⁻¹⋅lb-mol⁻¹
kB⋅NA⋅ft⁻³lb⋅atm⁻¹2⁶3⁻²5⁵19 = 554.9827855444075 [ft³°R⁻¹lb-mol⁻¹] English

julia> molargas(CGS) # erg⋅K⁻¹⋅mol⁻¹
kB⋅NA⋅2⁷5⁷ = 8.31446261815324×10⁷ [erg⋅K⁻¹mol⁻¹] Gauss

julia> molargas(English) # ft⋅lb⋅°R⁻¹⋅lb-mol⁻¹
kB⋅NA⋅g₀⁻¹ft⁻¹2³3⁻²5⁴ = 1545.3471008183453 [lbf⋅ft⋅°R⁻¹lb-mol⁻¹] English

julia> molargas(British) # ft⋅lb⋅°R⁻¹⋅slug-mol⁻¹
kB⋅NA⋅ft⁻²2³3⁻²5⁴ = 49720.07265826846 [lb⋅ft⋅°R⁻¹slug-mol⁻¹] British

julia> molargas(SI1976) # J⋅K⁻¹⋅mol⁻¹ (US1976 Standard Atmosphere)
8.31432 = 8.31432 [kg⋅m²s⁻²K⁻¹mol⁻¹] SI1976

\[\frac{p_0}{k_B T_0} = \frac{N_Ap_0}{R_uT_0} = \frac{\mu_{eu}M_up_0}{m_e R_u T_0} = \frac{M_u \mu_{eu}c\alpha^2p_0}{2R_uR_\infty hg_0 T_0}\]

MeasureSystems.loschmidt — Function 
loschmidt(U::UnitSystem) = atmosphere(U)/boltzmann(U)/temperature(T₀,SI2019,U)
nonstandard : [L⁻³], [L⁻³], [L⁻³], [L⁻³], [L⁻³]
L⁻³⋅(kB⁻¹R∞⁻³α⁶T₀⁻¹atm⋅τ⁻³2⁻³ = 1.5471467610(14) × 10⁻¹²) [ħ⁻³𝘤³mₑ³ϕ⁻³g₀⁻³] Unified

Number of molecules (number density) of an ideal gas in a unit volume (m⁻³ or ft⁻³).

julia> loschmidt(SI2019) # m⁻³
kB⁻¹T₀⁻¹atm = 2.686780111798444×10²⁵ [m⁻³] SI2019

julia> loschmidt(Metric,atm,T₀) # m⁻³
kB⁻¹NA⁻¹𝘩⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅T₀⁻¹atm⋅2⁻⁴5⁻³ = 2.68678011272(83) × 10²⁵ [m⁻³] Metric

julia> loschmidt(Conventional,atm,T₀) # m⁻³
kB⁻¹NA⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅T₀⁻¹atm⋅RK90⋅KJ90²2⁻⁶5⁻³ = 2.68678063809(83) × 10²⁵ [m⁻³] Conventional

julia> loschmidt(CODATA,atm,T₀) # m⁻³
𝘤⋅R∞⁻¹α²μₑᵤ⋅T₀⁻¹atm⋅RK⋅KJ²Rᵤ2014⁻¹2⁻⁶5⁻³ = 2.6867811(16) × 10²⁵ [m⁻³] CODATA

julia> loschmidt(SI1976,atm,T₀) # m⁻³
𝘩⁻¹𝘤⋅R∞⁻¹α²μₑᵤ⋅T₀⁻¹atm⋅2⁻⁴5⁻³/8.31432 = 2.68682619991(83) × 10²⁵ [m⁻³] SI1976

julia> loschmidt(English) # ft⁻³
kB⁻¹ft³T₀⁻¹atm = 7.608114025223316×10²³ [ft⁻³] English

julia> loschmidt(IAU) # au⁻³
kB⁻¹au³T₀⁻¹atm = 8.99514898792(54) × 10⁵⁸ [au⁻³] IAU☉

\[\frac{S_0}{R_u} = log\left(\frac{\hbar^3}{p_0}\sqrt{\left(\frac{m_u}{2\pi g_0}\right)^3 \left(k_BT_0\right)^5}\right)+\frac{5}{2} = log\left(\frac{m_u^4}{p_0}\left(\frac{\hbar}{\sqrt{2\pi g_0}}\right)^3\sqrt{\frac{R_uT_0}{M_u}}^5\right)+\frac{5}{2}\]

MeasureSystems.sackurtetrode — Function 
sackurtetrode(U::UnitSystem,P=atm,T=𝟏,m=Da) = log(kB*T/P*sqrt(m*kB*T/τ/ħ^2)^3)+5/2
dimensionless : [𝟙], [𝟙], [𝟙], [𝟙], [𝟙]
log(FL⁻²Θ⁻⁵ᐟ²A³ᐟ²⋅(μₑᵤ⁻³ᐟ²atm⁻¹τ⁻³ᐟ²exp(2⁻¹5) = 0.594141574194(26)))

Ideal gas entropy density for pressure P, temperature T, atomic mass m (dimensionless).

julia> sackurtetrode(Metric)
log(kB⁵ᐟ²NA⁵ᐟ²𝘩⋅𝘤⁻⁴R∞⁴α⁻⁸μₑᵤ⁻⁴atm⁻¹τ³ᐟ²2²³ᐟ²5¹⁵ᐟ²⋅12.182493960703473) = -1.1648705244 ± 1.2e-9

julia> sackurtetrode(SI2019)
log(kB⁵ᐟ²NA⁵ᐟ²𝘩⋅𝘤⁻⁴R∞⁴α⁻⁸μₑᵤ⁻⁴atm⁻¹τ³ᐟ²2²³ᐟ²5¹⁵ᐟ²⋅12.182493960703473) = -1.1648705244 ± 1.2e-9

julia> sackurtetrode(Conventional)
log(kB⁵ᐟ²NA⁵ᐟ²𝘩⋅𝘤⁻⁴R∞⁴α⁻⁸μₑᵤ⁻⁴atm⁻¹τ³ᐟ²2²³ᐟ²5¹⁵ᐟ²⋅12.182493960703473) = -1.1648705244 ± 1.2e-9

julia> sackurtetrode(CODATA)
log(kB⁵ᐟ²NA⁵ᐟ²𝘩⋅𝘤⁻⁴R∞⁴α⁻⁸μₑᵤ⁻⁴atm⁻¹τ³ᐟ²2²³ᐟ²5¹⁵ᐟ²⋅12.182493960703473) = -1.1648705244 ± 1.2e-9

julia> sackurtetrode(SI2019,𝟏𝟎^5)
log(kB⁵ᐟ²NA⁵ᐟ²𝘩⋅𝘤⁻⁴R∞⁴α⁻⁸μₑᵤ⁻⁴τ³ᐟ²2¹³ᐟ²5⁵ᐟ²⋅12.182493960703473) = -1.1517075379 ± 1.2e-9

\[\frac{180 R_uV_{it}^2}{43 k_BN_A\Omega_{it}} = \frac{180 k_BM_uV_{it}^2}{43 R_um_u\Omega_{it}} = \frac{90 k_BM_u\mu_{eu}c\alpha^2V_{it}^2}{43 hg_0R_uR_\infty\Omega_{it}}\]

MeasureSystems.mechanicalheat — Function 
mechanicalheat(U::UnitSystem) = molargas(U)/molargas(Metric)*calorie(Metric)
energy : [FL], [FL], [ML²T⁻²], [ML²T⁻²], [ML²T⁻²]

Heat to raise 1 mass unit of water by 1 temperature unit, or kB⋅NA⋅Ωᵢₜ⋅Vᵢₜ⁻²2⁻²3⁻²5⁻¹43 = 1.9859050081929637 mechanicalheat per molaramount per temperature units (J or ft⋅lb).

julia> mechanicalheat(Metric) # J
Ωᵢₜ⁻¹Vᵢₜ²2²3²5⋅43⁻¹ = 4.186737323211057 [J] Metric

julia> mechanicalheat(English) # ft⋅lb
g₀⁻¹ft⁻¹Ωᵢₜ⁻¹Vᵢₜ²2⁵5⁵43⁻¹ = 778.1576129990752 [lbf⋅ft] English

julia> mechanicalheat(British) # ft⋅lb
ft⁻²Ωᵢₜ⁻¹Vᵢₜ²2⁵5⁵43⁻¹ = 25036.480825188257 [lb⋅ft] British

\[\sigma = \frac{2\pi^5 k_B^4}{15h^3c^2} = \frac{\pi^2 k_B^4}{60\hbar^3c^2} = \frac{32\pi^5 h}{15c^6\alpha^8} \left(\frac{g_0R_uR_\infty}{\mu_{eu}M_u}\right)^4\]

MeasureSystems.stefan — Constant 
stefan(U::UnitSystem) = τ^5/𝟐^4*boltzmann(U)^4/(𝟑*𝟓*planck(U)^3*lightspeed(U)^2)
nonstandard : [FL⁻¹T⁻¹Θ⁻⁴], [FL⁻¹T⁻¹Θ⁻⁴], [MT⁻³Θ⁻⁴], [MT⁻³Θ⁻⁴], [MT⁻³Θ⁻⁴]
FL⁻¹T⁻¹Θ⁻⁴⋅(τ²2⁻⁴3⁻¹5⁻¹ = 0.16449340668482262) [kB⁴ħ⁻³𝘤⁻²ϕ⁻³] Unified

Stefan-Boltzmann proportionality σ of black body radiation (W⋅m⁻²⋅K⁻⁴ or ?⋅ft⁻²⋅°R⁻⁴).

julia> stefan(SI2019) # W⋅m⁻²⋅K⁻⁴
kB⁴𝘩⁻³𝘤⁻²τ⁵2⁻⁴3⁻¹5⁻¹ = 5.670374419184431×10⁻⁸ [W⋅m⁻²K⁻⁴] SI2019

julia> stefan(Metric) # W⋅m⁻²⋅K⁻⁴
kB⁴NA⁴𝘩⋅𝘤⁻⁶R∞⁴α⁻⁸μₑᵤ⁻⁴τ⁵2¹²3⁻¹5¹¹ = 5.6703744114(70) × 10⁻⁸ [W⋅m⁻²K⁻⁴] Metric

julia> stefan(Conventional) # W⋅m⁻²⋅K⁻⁴
kB⁴NA⁴𝘤⁻⁶R∞⁴α⁻⁸μₑᵤ⁻⁴RK90⁻¹KJ90⁻²τ⁵2¹⁴3⁻¹5¹¹ = 5.6703733026(70) × 10⁻⁸ [W⋅m⁻²K⁻⁴] Conventional

julia> stefan(CODATA) # W⋅m⁻²⋅K⁻⁴
𝘤⁻⁶R∞⁴α⁻⁸μₑᵤ⁻⁴RK⁻¹KJ⁻²Rᵤ2014⁴τ⁵2¹⁴3⁻¹5¹¹ = 5.670367(13) × 10⁻⁸ [W⋅m⁻²K⁻⁴] CODATA

julia> stefan(Metric)*day(Metric)/(calorie(Metric)*100^2) # cal⋅cm⁻²⋅day⁻¹⋅K⁻⁴
kB⁴NA⁴𝘩⋅𝘤⁻⁶R∞⁴α⁻⁸μₑᵤ⁻⁴Ωᵢₜ⋅Vᵢₜ⁻²τ⁵2¹⁷5¹²43 = 0.0011701721683(14) [m⁻²K⁻⁴] Metric

julia> stefan(English) # lb⋅s⁻¹⋅ft⁻³⋅°R⁻⁴
kB⁴NA⁴𝘩⋅𝘤⁻⁶R∞⁴α⁻⁸μₑᵤ⁻⁴g₀⁻¹ft⋅lb⁻¹τ⁵2¹²3⁻⁹5¹⁵ = 3.7012656963(46) × 10⁻¹⁰ [lbf⋅ft⁻¹s⁻¹°R⁻⁴] English

\[a = 4\frac{\sigma}{c} = \frac{8\pi^5 k_B^4}{15h^3c^3} = \frac{\pi^2 k_B^4}{15\hbar^3c^3} = \frac{2^7\pi^5 h}{15c^7\alpha^8} \left(\frac{g_0R_uR_\infty}{\mu_{eu}M_u}\right)^4\]

MeasureSystems.radiationdensity — Constant 
radiationdensity(U::UnitSystem) = 𝟐^2*stefan(U)/lightspeed(U)
nonstandard : [FL⁻²Θ⁻⁴], [FL⁻²Θ⁻⁴], [ML⁻¹T⁻²Θ⁻⁴], [ML⁻¹T⁻²Θ⁻⁴], [ML⁻¹T⁻²Θ⁻⁴]
FL⁻²Θ⁻⁴⋅(τ²2⁻²3⁻¹5⁻¹ = 0.6579736267392905) [kB⁴ħ⁻³𝘤⁻³ϕ⁻³] Unified

Raditation density constant of black body radiation (J⋅m⁻³⋅K⁻⁴ or lb⋅ft⁻²⋅°R⁻⁴).

julia> radiationdensity(Metric) # J⋅m⁻³⋅K⁻⁴
kB⁴NA⁴𝘩⋅𝘤⁻⁷R∞⁴α⁻⁸μₑᵤ⁻⁴τ⁵2¹⁴3⁻¹5¹¹ = 7.5657332399(93) × 10⁻¹⁶ [J⋅m⁻³K⁻⁴] Metric

julia> radiationdensity(SI2019) # J⋅m⁻³⋅K⁻⁴
kB⁴𝘩⁻³𝘤⁻³τ⁵2⁻²3⁻¹5⁻¹ = 7.565733250280007×10⁻¹⁶ [J⋅m⁻³K⁻⁴] SI2019

julia> radiationdensity(Conventional) # J⋅m⁻³⋅K⁻⁴
kB⁴NA⁴𝘤⁻⁷R∞⁴α⁻⁸μₑᵤ⁻⁴RK90⁻¹KJ90⁻²τ⁵2¹⁶3⁻¹5¹¹ = 7.5657317605(93) × 10⁻¹⁶ [J⋅m⁻³K⁻⁴] Conventional

julia> radiationdensity(CODATA) # J⋅m⁻³⋅K⁻⁴
𝘤⁻⁷R∞⁴α⁻⁸μₑᵤ⁻⁴RK⁻¹KJ⁻²Rᵤ2014⁴τ⁵2¹⁶3⁻¹5¹¹ = 7.565723(17) × 10⁻¹⁶ [J⋅m⁻³K⁻⁴] CODATA

julia> radiationdensity(International) # J⋅m⁻³⋅K⁻⁴
kB⁴NA⁴𝘩⋅𝘤⁻⁷R∞⁴α⁻⁸μₑᵤ⁻⁴Ωᵢₜ⋅Vᵢₜ⁻²τ⁵2¹⁴3⁻¹5¹¹ = 7.5644848940(93) × 10⁻¹⁶ [J⋅m⁻³K⁻⁴] International

\[b = \frac{hc/k_B}{5+W_0(-5 e^{-5})} = \frac{hcM_u/(m_uR_u)}{5+W_0(-5 e^{-5})} = \frac{M_u \mu_{eu}c^2\alpha^2/(2R_uR_\infty g_0)}{5+W_0(-5 e^{-5})}\]

MeasureSystems.wienwavelength — Constant 
wienwavelength(U::UnitSystem) = planck(U)*lightspeed(U)/boltzmann(U)/(𝟓+W₀(-𝟓*exp(-𝟓)))
nonstandard : [LΘ], [LΘ], [LΘ], [LΘ], [LΘ]
LΘ/4.965114231744276 [kB⁻¹ħ⋅𝘤⋅ϕ] Unified

Wien wavelength displacement law constant based on Lambert W₀ evaluation (m⋅K or ft⋅°R).

julia> wienwavelength(Metric) # m⋅K
kB⁻¹NA⁻¹𝘤²R∞⁻¹α²μₑᵤ⋅2⁻⁴5⁻³/4.965114231744276 = 0.00289777195618(89) [m⋅K] Metric

julia> wienwavelength(SI2019) # m⋅K
kB⁻¹𝘩⋅𝘤/4.965114231744276 = 0.0028977719551851727 [m⋅K] SI2019

julia> wienwavelength(Conventional) # m⋅K
kB⁻¹NA⁻¹𝘤²R∞⁻¹α²μₑᵤ⋅2⁻⁴5⁻³/4.965114231744276 = 0.00289777195618(89) [m⋅K] Conventional

julia> wienwavelength(CODATA) # m⋅K
𝘤²R∞⁻¹α²μₑᵤ⋅Rᵤ2014⁻¹2⁻⁴5⁻³/4.965114231744276 = 0.0028977729(17) [m⋅K] CODATA

julia> wienwavelength(English) # ft⋅°R
kB⁻¹NA⁻¹𝘤²R∞⁻¹α²μₑᵤ⋅ft⁻¹2⁻⁴3²5⁻⁴/4.965114231744276 = 0.0171128265129(53) [ft⋅°R] English

\[\frac{3+W_0(-3 e^{-3})}{h/k_B} = \frac{3+W_0(-3 e^{-3})}{hM_u/(m_uR_u)} = \frac{3+W_0(-3 e^{-3})}{M_u \mu_{eu}c\alpha^2/(2R_uR_\infty g_0)}\]

MeasureSystems.wienfrequency — Constant 
wienfrequency(U::UnitSystem) = (𝟑+W₀(-𝟑*exp(-𝟑)))*boltzmann(U)/planck(U)
nonstandard : [T⁻¹Θ⁻¹], [T⁻¹Θ⁻¹], [T⁻¹Θ⁻¹], [T⁻¹Θ⁻¹], [T⁻¹Θ⁻¹]
T⁻¹Θ⁻¹⋅2.8214393721220787 [kB⋅ħ⁻¹ϕ⁻¹] Unified

Wien frequency radiation law constant based on Lambert W₀ evaluation (Hz⋅K⁻¹).

julia> wienfrequency(Metric) # Hz⋅K⁻¹
kB⋅NA⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³⋅2.8214393721220787 = 5.8789257556(18) × 10¹⁰ [Hz⋅K⁻¹] Metric

julia> wienfrequency(SI2019) # Hz⋅K⁻¹
kB⋅𝘩⁻¹⋅2.8214393721220787 = 5.8789257576468254×10¹⁰ [Hz⋅K⁻¹] SI2019

julia> wienfrequency(Conventional) # Hz⋅K⁻¹
kB⋅NA⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³⋅2.8214393721220787 = 5.8789257556(18) × 10¹⁰ [Hz⋅K⁻¹] Conventional

julia> wienfrequency(CODATA) # Hz⋅K⁻¹
𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹Rᵤ2014⋅2⁴5³⋅2.8214393721220787 = 5.8789238(34) × 10¹⁰ [Hz⋅K⁻¹] CODATA

julia> wienfrequency(English) # Hz⋅°R⁻¹
kB⋅NA⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴3⁻²5⁴⋅2.8214393721220787 = 3.2660698642(10) × 10¹⁰ [Hz⋅°R⁻¹] English

\[K_{\text{cd}} = \frac{I_v}{\int_0^\infty \bar{y}(\lambda)\cdot\frac{dI_e}{d\lambda}d\lambda}, \qquad \bar{y}\left(\frac{c}{540\times 10^{12}}\right)\cdot I_e = 1\]

MeasureSystems.luminousefficacy — Constant 
luminousefficacy(U::UnitSystem) = Kcd*power(U)
luminousefficacy(U::UnitSystem{𝟏}) = 𝟏
luminousefficacy : [F⁻¹L⁻¹TJ], [F⁻¹L⁻¹TJ], [M⁻¹L⁻²T³J], [M⁻¹L⁻²T³J], [M⁻¹L⁻²T³J]
F⁻¹L⁻¹TJ [Kcd] Unified

Luminous efficacy of monochromatic radiation Kcd of frequency 540 THz (lm⋅W⁻¹).

julia> luminousefficacy(Metric) # lm⋅W⁻¹
Kcd = 683.01969009009 [lm⋅W⁻¹] Metric

julia> luminousefficacy(CODATA) # lm⋅W⁻¹
𝘩⋅Kcd⋅RK⋅KJ²2⁻² = 683.0197015(85) [lm⋅W⁻¹] CODATA

julia> luminousefficacy(Conventional) # lm⋅W⁻¹
𝘩⋅Kcd⋅RK90⋅KJ90²2⁻² = 683.0198236454071 [lm⋅W⁻¹] Conventional

julia> luminousefficacy(International) # lm⋅W⁻¹
Kcd⋅Ωᵢₜ⁻¹Vᵢₜ² = 683.1324069249656 [lm⋅W⁻¹] International

julia> luminousefficacy(British) # lm⋅s³⋅slug⋅ft⁻²
Kcd⋅g₀⋅ft⋅lb = 926.0503548878947 [lb⁻¹ft⁻¹s⋅lm] British

Electromagnetic Constants

\[\lambda = \frac{4\pi\alpha_B}{\mu_0\alpha_L} = 4\pi k_e\varepsilon_0 = Z_0\varepsilon_0c\]

MeasureSystems.rationalization — Constant 
rationalization(U::UnitSystem) = spat(U)*biotsavart(U)/vacuumpermeability(U)/lorentz(U)
demagnetizingfactor : [R], [𝟙], [𝟙], [𝟙], [𝟙]
R [λ] Unified

Constant of magnetization and polarization density or spat(U)*coulomb(U)*permittivity(U).

julia> rationalization(Metric)
𝟏 = 1.0 [𝟙] Metric

julia> rationalization(Gauss)
τ⋅2 = 12.566370614359172 [𝟙] Gauss

\[\alpha_L = \frac{1}{c\sqrt{\mu_0\varepsilon_0}} = \frac{\alpha_B}{\mu_0\varepsilon_0k_e} = \frac{4\pi \alpha_B}{\lambda\mu_0} = \frac{k_m}{\alpha_B}\]

MeasureSystems.lorentz — Constant 
lorentz(U::UnitSystem) = spat(U)*biotsavart(U)/vacuumpermeability(U)/rationalization(U)
nonstandard : [C⁻¹], [𝟙], [𝟙], [𝟙], [𝟙]
C⁻¹ [αL] Unified

Electromagnetic proportionality constant αL for the Lorentz's law force (dimensionless).

julia> lorentz(Metric)
𝟏 = 1.0 [𝟙] Metric

julia> lorentz(LorentzHeaviside)
𝘤⁻¹2⁻²5⁻² = 3.335640951981521×10⁻¹¹ [cm⁻¹s] LorentzHeaviside

julia> lorentz(Gauss)
𝘤⁻¹2⁻²5⁻² = 3.335640951981521×10⁻¹¹ [cm⁻¹s] Gauss

\[\alpha_B = \mu_0\alpha_L\frac{\lambda}{4\pi} = \alpha_L\mu_0\varepsilon_0k_e = \frac{k_m}{\alpha_L} = \frac{k_e}{c}\sqrt{\mu_0\varepsilon_0}\]

MeasureSystems.biotsavart — Constant 
biotsavart(U::UnitSystem) = vacuumpermeability(U)*lorentz(U)*rationalization(U)/𝟐/τ
nonstandard : [FT²Q⁻²C], [FT²Q⁻²], [MLQ⁻²], [𝟙], [L⁻²T²]
FT²Q⁻²C⋅(τ⁻¹2⁻¹ = 0.07957747154594767) [μ₀⋅λ⋅αL] Unified

Magnetostatic proportionality constant αB for the Biot-Savart's law (H/m).

julia> biotsavart(Metric) # H⋅m⁻¹
2⁻⁷5⁻⁷ = 1.0×10⁻⁷ [H⋅m⁻¹] Metric

julia> biotsavart(CODATA) # H⋅m⁻¹
𝘤⁻¹α⋅RK⋅τ⁻¹ = 1.00000000040(28) × 10⁻⁷ [H⋅m⁻¹] CODATA

julia> biotsavart(SI2019) # H⋅m⁻¹
𝘩⋅𝘤⁻¹𝘦⁻²α⋅τ⁻¹ = 1.00000000055(15) × 10⁻⁷ [H⋅m⁻¹] SI2019

julia> biotsavart(Conventional) # H⋅m⁻¹
𝘤⁻¹α⋅RK90⋅τ⁻¹ = 9.9999998275(15) × 10⁻⁸ [H⋅m⁻¹] Conventional

julia> biotsavart(International) # H⋅m⁻¹
Ωᵢₜ⁻¹2⁻⁷5⁻⁷ = 9.995052449037726×10⁻⁸ [H⋅m⁻¹] International

julia> biotsavart(InternationalMean) # H⋅m⁻¹
2⁻⁷5⁻⁷/1.00049 = 9.995102399824085×10⁻⁸ [H⋅m⁻¹] InternationalMean

julia> biotsavart(EMU) # abH⋅cm⁻¹
𝟏 = 1.0 [𝟙] EMU

julia> biotsavart(ESU) # statH⋅cm⁻¹
𝘤⁻²2⁻⁴5⁻⁴ = 1.1126500560536184×10⁻²¹ [cm⁻²s²] ESU

julia> biotsavart(Gauss) # abH⋅cm⁻¹
𝘤⁻¹2⁻²5⁻² = 3.335640951981521×10⁻¹¹ [cm⁻¹s] Gauss

julia> biotsavart(HLU) # hlH⋅cm⁻¹
𝘤⁻¹τ⁻¹2⁻³5⁻² = 2.654418729438073×10⁻¹² [cm⁻¹s] LorentzHeaviside

\[Z_0 = \mu_0\lambda c\alpha_L^2 = \frac{\lambda}{\varepsilon_0 c} = \lambda\alpha_L\sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{2h\alpha}{e^2} = 2R_K\alpha\]

MeasureSystems.vacuumimpedance — Constant 
vacuumimpedance(U::UnitSystem) = vacuumpermeability(U)*lightspeed(U)*rationalization(U)*lorentz(U)^2
resistance : [FLTQ⁻²], [FLTQ⁻²], [ML²T⁻¹Q⁻²], [LT⁻¹], [L⁻¹T]
FLTQ⁻² [𝘤⋅μ₀⋅λ⋅αL²] Unified

Vacuum impedance of free space Z₀ is magnitude ratio of electric to magnetic field (Ω).

julia> vacuumimpedance(Metric) # Ω
𝘤⋅τ⋅2⁻⁶5⁻⁷ = 376.7303134617706 [Ω] Metric

julia> vacuumimpedance(Conventional) # Ω
α⋅RK90⋅2 = 376.730306964(58) [Ω] Conventional

julia> vacuumimpedance(CODATA) # Ω
α⋅RK⋅2 = 376.73031361(10) [Ω] CODATA

julia> vacuumimpedance(SI2019) # Ω
𝘩⋅𝘦⁻²α⋅2 = 376.730313667(58) [Ω] SI2019

julia> vacuumimpedance(International) # Ω
𝘤⋅Ωᵢₜ⁻¹τ⋅2⁻⁶5⁻⁷ = 376.5439242192821 [Ω] International

julia> vacuumimpedance(InternationalMean) # Ω
𝘤⋅τ⋅2⁻⁶5⁻⁷/1.00049 = 376.5458060168223 [Ω] InternationalMean

julia> 120π # 3e8*μ₀ # Ω
376.99111843077515

julia> vacuumimpedance(EMU) # abΩ
𝘤⋅τ⋅2³5² = 3.767303134617706×10¹¹ [cm⋅s⁻¹] EMU

julia> vacuumimpedance(ESU) # statΩ
𝘤⁻¹τ⋅2⁻¹5⁻² = 4.1916900439033643×10⁻¹⁰ [cm⁻¹s] ESU

julia> vacuumimpedance(HLU) # hlΩ
𝘤⁻¹2⁻²5⁻² = 3.335640951981521×10⁻¹¹ [cm⁻¹s] LorentzHeaviside

julia> vacuumimpedance(IPS) # in⋅lb⋅s⋅C⁻²
𝘤⋅g₀⁻¹ft⁻¹lb⁻¹τ⋅2⁻⁴3⋅5⁻⁷ = 3334.344236337137 [lb⋅in⋅s⋅C⁻²] IPS

\[\mu_0 = \frac{1}{\varepsilon_0 (c\alpha_L)^2} = \frac{4\pi k_e}{\lambda (c\alpha_L)^2} = \frac{2h\alpha}{\lambda c(e\alpha_L)^2} = \frac{2R_K\alpha}{\lambda c\alpha_L^2}\]

MeasureSystems.vacuumpermeability — Constant 
vacuumpermeability(U::UnitSystem) = 𝟏/vacuumpermittivity(U)/(lightspeed(U)*lorentz(U))^2
permeability : [FT²Q⁻²R⁻¹C²], [FT²Q⁻²], [MLQ⁻²], [𝟙], [L⁻²T²]
FT²Q⁻²R⁻¹C² [μ₀] Unified

Magnetic permeability in a classical vacuum defined as μ₀ in SI units (H⋅m⁻¹, kg⋅m²⋅C⁻²).

julia> vacuumpermeability(Metric) # H⋅m⁻¹
τ⋅2⁻⁶5⁻⁷ = 1.2566370614359173×10⁻⁶ [H⋅m⁻¹] Metric

julia> vacuumpermeability(Conventional) # H⋅m⁻¹
𝘤⁻¹α⋅RK90⋅2 = 1.25663703976(19) × 10⁻⁶ [H⋅m⁻¹] Conventional

julia> vacuumpermeability(CODATA) # H⋅m⁻¹
𝘤⁻¹α⋅RK⋅2 = 1.25663706194(35) × 10⁻⁶ [H⋅m⁻¹] CODATA

julia> vacuumpermeability(SI2019) # H⋅m⁻¹
𝘩⋅𝘤⁻¹𝘦⁻²α⋅2 = 1.25663706212(19) × 10⁻⁶ [H⋅m⁻¹] SI2019

julia> vacuumpermeability(International) # H⋅m⁻¹
Ωᵢₜ⁻¹τ⋅2⁻⁶5⁻⁷ = 1.2560153338456637×10⁻⁶ [H⋅m⁻¹] International

julia> vacuumpermeability(EMU) # abH⋅cm⁻¹
𝟏 = 1.0 [𝟙] EMU

julia> vacuumpermeability(ESU) # statH⋅cm⁻¹
𝘤⁻²2⁻⁴5⁻⁴ = 1.1126500560536184×10⁻²¹ [cm⁻²s²] ESU

\[\varepsilon_0 = \frac{1}{\mu_0(c\alpha_L)^2} = \frac{\lambda}{4\pi k_e} = \frac{\lambda e^2}{2\alpha hc} = \frac{\lambda}{2R_K\alpha c}\]

MeasureSystems.vacuumpermittivity — Constant 
vacuumpermittivity(U::UnitSystem) = 𝟏/vacuumpermeability(U)/(lightspeed(U)*lorentz(U))^2
permittivity : [F⁻¹L⁻²Q²R], [F⁻¹L⁻²Q²], [M⁻¹L⁻³T²Q²], [L⁻²T²], [𝟙]
F⁻¹L⁻²Q²R [𝘤⁻²μ₀⁻¹αL⁻²] Unified

Dielectric permittivity constant ε₀ of a classical vacuum (C²⋅N⁻¹⋅m⁻²).

julia> vacuumpermittivity(Metric) # F⋅m⁻¹
𝘤⁻²τ⁻¹2⁶5⁷ = 8.854187817620389×10⁻¹² [F⋅m⁻¹] Metric

julia> vacuumpermittivity(Conventional) # F⋅m⁻¹
𝘤⁻¹α⁻¹RK90⁻¹2⁻¹ = 8.8541879703(14) × 10⁻¹² [F⋅m⁻¹] Conventional

julia> vacuumpermittivity(CODATA) # F⋅m⁻¹
𝘤⁻¹α⁻¹RK⁻¹2⁻¹ = 8.8541878141(24) × 10⁻¹² [F⋅m⁻¹] CODATA

julia> vacuumpermittivity(SI2019) # F⋅m⁻¹
𝘩⁻¹𝘤⁻¹𝘦²α⁻¹2⁻¹ = 8.8541878128(14) × 10⁻¹² [F⋅m⁻¹] SI2019

julia> vacuumpermittivity(International) # F⋅m⁻¹
𝘤⁻²Ωᵢₜ⋅τ⁻¹2⁶5⁷ = 8.85857064059011×10⁻¹² [F⋅m⁻¹] International

julia> vacuumpermittivity(EMU) # abF⋅cm⁻¹
𝘤⁻²2⁻⁴5⁻⁴ = 1.1126500560536184×10⁻²¹ [cm⁻²s²] EMU

julia> vacuumpermittivity(ESU) # statF⋅cm⁻¹
𝟏 = 1.0 [𝟙] ESU

julia> vacuumpermittivity(SI2019)/elementarycharge(SI2019) # 𝘦²⋅eV⁻¹⋅m⁻¹
𝘩⁻¹𝘤⁻¹𝘦⋅α⁻¹2⁻¹ = 5.52634935805(85) × 10⁷ [kg⁻¹m⁻³s²C] SI2019

\[k_e = \frac{\lambda}{4\pi\varepsilon_0} = \frac{\mu_0\lambda (c\alpha_L)^2}{4\pi} = \frac{\alpha \hbar c}{e^2} = \frac{R_K\alpha c}{2\pi} = \frac{\alpha_B}{\alpha_L\mu_0\varepsilon_0} = k_mc^2\]

MeasureSystems.electrostatic — Constant 
electrostatic(U::UnitSystem) = rationalization(U)/𝟐/τ/vacuumpermittivity(U)
nonstandard : [FL²Q⁻²], [FL²Q⁻²], [ML³T⁻²Q⁻²], [L²T⁻²], [𝟙]
FL²Q⁻²⋅(τ⁻¹2⁻¹ = 0.07957747154594767) [𝘤²μ₀⋅λ⋅αL²] Unified

Electrostatic proportionality constant kₑ for the Coulomb's law force (N⋅m²⋅C⁻²).

julia> electrostatic(Metric) # N⋅m²⋅C⁻²
𝘤²2⁻⁷5⁻⁷ = 8.987551787368176×10⁹ [m⋅F⁻¹] Metric

julia> electrostatic(CODATA) # N·m²⋅C⁻²
𝘤⋅α⋅RK⋅τ⁻¹ = 8.9875517909(25) × 10⁹ [m⋅F⁻¹] CODATA

julia> electrostatic(SI2019) # N·m²⋅C⁻²
𝘩⋅𝘤⋅𝘦⁻²α⋅τ⁻¹ = 8.9875517923(14) × 10⁹ [m⋅F⁻¹] SI2019

julia> electrostatic(Conventional) # N·m²⋅C⁻²
𝘤⋅α⋅RK90⋅τ⁻¹ = 8.9875516323(14) × 10⁹ [m⋅F⁻¹] Conventional

julia> electrostatic(International) # N·m²⋅C⁻²
𝘤²Ωᵢₜ⁻¹2⁻⁷5⁻⁷ = 8.983105150318768×10⁹ [m⋅F⁻¹] International

julia> electrostatic(EMU) # dyn⋅cm²⋅abC⁻²
𝘤²2⁴5⁴ = 8.987551787368175×10²⁰ [erg⋅g⁻¹] EMU

julia> electrostatic(ESU) # dyn⋅cm²⋅statC⁻²
𝟏 = 1.0 [𝟙] ESU

julia> electrostatic(HLU) # dyn⋅cm²⋅hlC⁻²
τ⁻¹2⁻¹ = 0.07957747154594767 [𝟙] LorentzHeaviside

\[k_m = \alpha_L\alpha_B = \mu_0\alpha_L^2\frac{\lambda}{4\pi} = \frac{k_e}{c^2} = \frac{\alpha \hbar}{ce^2} = \frac{R_K\alpha}{2\pi c}\]

MeasureSystems.magnetostatic — Constant 
magnetostatic(U::UnitSystem) = lorentz(U)*biotsavart(U) # electrostatic(U)/lightspeed(U)^2
nonstandard : [FT²Q⁻²], [FT²Q⁻²], [MLQ⁻²], [𝟙], [L⁻²T²]
FT²Q⁻²⋅(τ⁻¹2⁻¹ = 0.07957747154594767) [μ₀⋅λ⋅αL²] Unified

Magnetic proportionality constant kₘ for the Ampere's law force (N·s²⋅C⁻²).

julia> magnetostatic(Metric) # H⋅m⁻¹
2⁻⁷5⁻⁷ = 1.0×10⁻⁷ [H⋅m⁻¹] Metric

julia> magnetostatic(CODATA) # H⋅m⁻¹
𝘤⁻¹α⋅RK⋅τ⁻¹ = 1.00000000040(28) × 10⁻⁷ [H⋅m⁻¹] CODATA

julia> magnetostatic(SI2019) # H⋅m⁻¹
𝘩⋅𝘤⁻¹𝘦⁻²α⋅τ⁻¹ = 1.00000000055(15) × 10⁻⁷ [H⋅m⁻¹] SI2019

julia> magnetostatic(Conventional) # H⋅m⁻¹
𝘤⁻¹α⋅RK90⋅τ⁻¹ = 9.9999998275(15) × 10⁻⁸ [H⋅m⁻¹] Conventional

julia> magnetostatic(International) # H⋅m⁻¹
Ωᵢₜ⁻¹2⁻⁷5⁻⁷ = 9.995052449037726×10⁻⁸ [H⋅m⁻¹] International

julia> magnetostatic(EMU) # abH⋅m⁻¹
𝟏 = 1.0 [𝟙] EMU

julia> magnetostatic(ESU) # statH⋅m⁻¹
𝘤⁻²2⁻⁴5⁻⁴ = 1.1126500560536184×10⁻²¹ [cm⁻²s²] ESU

julia> magnetostatic(HLU) # hlH⋅m⁻¹
𝘤⁻²τ⁻¹2⁻⁵5⁻⁴ = 8.85418781762039×10⁻²³ [cm⁻²s²] LorentzHeaviside

\[e = \sqrt{\frac{2h\alpha}{Z_0}} = \frac{2\alpha_L}{K_JR_K} = \sqrt{\frac{h}{R_K}} = \frac{hK_J}{2\alpha_L} = \frac{F}{N_A}\]

MeasureSystems.elementarycharge — Constant 
elementarycharge(U::UnitSystem) = √(𝟐*planck(U)*finestructure(U)/vacuumimpedance(U))
charge : [Q], [Q], [Q], [M¹ᐟ²L¹ᐟ²], [M¹ᐟ²L³ᐟ²T⁻¹]
Q⋅(α¹ᐟ²τ¹ᐟ²2¹ᐟ² = 0.302822120872(23)) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] Unified

Quantized elementary charge 𝘦 of a proton or electron 2/(klitzing(U)*josephson(U)) (C).

julia> elementarycharge(SI2019) # C
𝘦 = 1.602176634×10⁻¹⁹ [C] SI2019

julia> elementarycharge(Metric) # C
𝘩¹ᐟ²𝘤⁻¹ᐟ²α¹ᐟ²τ⁻¹ᐟ²2⁷ᐟ²5⁷ᐟ² = 1.60217663444(12) × 10⁻¹⁹ [C] Metric

julia> elementarycharge(CODATA) # C
RK⁻¹KJ⁻¹2 = 1.6021766207(99) × 10⁻¹⁹ [C] CODATA

julia> elementarycharge(Conventional) # C
RK90⁻¹KJ90⁻¹2 = 1.602176491612271×10⁻¹⁹ [C] Conventional

julia> elementarycharge(International) # C
𝘩¹ᐟ²𝘤⁻¹ᐟ²α¹ᐟ²Ωᵢₜ⋅Vᵢₜ⁻¹τ⁻¹ᐟ²2⁷ᐟ²5⁷ᐟ² = 1.60244090637(12) × 10⁻¹⁹ [C] International

julia> elementarycharge(EMU) # abC
𝘩¹ᐟ²𝘤⁻¹ᐟ²α¹ᐟ²τ⁻¹ᐟ²2⁵ᐟ²5⁵ᐟ² = 1.60217663444(12) × 10⁻²⁰ [g¹ᐟ²cm¹ᐟ²] EMU

julia> elementarycharge(ESU) # statC
𝘩¹ᐟ²𝘤¹ᐟ²α¹ᐟ²τ⁻¹ᐟ²2⁹ᐟ²5⁹ᐟ² = 4.80320471388(37) × 10⁻¹⁰ [g¹ᐟ²cm³ᐟ²s⁻¹] ESU

julia> elementarycharge(Hartree) # 𝘦
𝟏 = 1.0 [𝘦] Hartree

\[F = eN_A = N_A\sqrt{\frac{2h\alpha}{Z_0}} = \frac{2N_A\alpha_L}{K_JR_K} = N_A\sqrt{\frac{h}{R_K}} = \frac{hK_JN_A}{2\alpha_L}\]

MeasureSystems.faraday — Constant 
faraday(U::UnitSystem) = elementarycharge(U)*avogadro(U)
nonstandard : [QN⁻¹], [QN⁻¹], [QN⁻¹], [M¹ᐟ²L¹ᐟ²N⁻¹], [M¹ᐟ²L³ᐟ²T⁻¹N⁻¹]
QN⁻¹⋅(α¹ᐟ²μₑᵤ⋅τ¹ᐟ²2¹ᐟ² = 0.000166122131531(14)) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²mₑ⁻¹Mᵤ⋅ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] Unified

Electric charge per mole of electrons 𝔉 based on elementary charge (C⋅mol⁻¹).

julia> faraday(SI2019) # C⋅mol⁻¹
NA⋅𝘦 = 96485.33212331001 [C⋅mol⁻¹] SI2019

julia> faraday(Metric) # C⋅mol⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅τ⁻¹ᐟ²2⁻¹ᐟ²5¹ᐟ² = 96485.332183(37) [C⋅mol⁻¹] Metric

julia> faraday(CODATA) # C⋅mol⁻¹
𝘤⋅R∞⁻¹α²μₑᵤ⋅KJ⋅2⁻⁵5⁻³ = 96485.33297(60) [C⋅mol⁻¹] CODATA

julia> faraday(Conventional) # C⋅mol⁻¹
𝘤⋅R∞⁻¹α²μₑᵤ⋅KJ90⋅2⁻⁵5⁻³ = 96485.342448(30) [C⋅mol⁻¹] Conventional

julia> faraday(International) # C⋅mol⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅Ωᵢₜ⋅Vᵢₜ⁻¹τ⁻¹ᐟ²2⁻¹ᐟ²5¹ᐟ² = 96501.247011(37) [C⋅mol⁻¹] International

julia> faraday(InternationalMean) # C⋅mol⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅τ⁻¹ᐟ²2⁻¹ᐟ²5¹ᐟ²⋅1.0001499490173342 = 96499.800064(37) [C⋅mol⁻¹] InternationalMean

julia> faraday(EMU) # abC⋅mol⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅τ⁻¹ᐟ²2⁻³ᐟ²5⁻¹ᐟ² = 9648.5332183(37) [g¹ᐟ²cm¹ᐟ²mol⁻¹] EMU

julia> faraday(ESU) # statC⋅mol⁻¹
𝘩⁻¹ᐟ²𝘤³ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅τ⁻¹ᐟ²2¹ᐟ²5³ᐟ² = 2.8925574896(11) × 10¹⁴ [g¹ᐟ²cm³ᐟ²s⁻¹mol⁻¹] ESU

julia> faraday(Metric)/kilocalorie(Metric) # kcal⋅(V-g-e)⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅Ωᵢₜ⋅Vᵢₜ⁻²τ⁻¹ᐟ²2⁻¹¹ᐟ²3⁻²5⁻⁷ᐟ²43 = 23.0454706695(89) [kg⁻¹m⁻²s²C⋅mol⁻¹] Metric

julia> faraday(Metric)/3600 # A⋅h⋅mol⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²μₑᵤ⋅τ⁻¹ᐟ²2⁻⁹ᐟ²3⁻²5⁻³ᐟ² = 26.801481162(10) [C⋅mol⁻¹] Metric

\[G_0 = \frac{2e^2}{h} = \frac{4\alpha}{Z_0} = \frac{2}{R_K} = \frac{hK_J^2}{2\alpha_L^2} = \frac{2F^2}{hN_A^2}\]

MeasureSystems.conductancequantum — Constant 
conductancequantum(U::UnitSystem) = 𝟐*elementarycharge(U)^2/planck(U) # 2/klitzing(U)
conductance : [F⁻¹L⁻¹T⁻¹Q²], [F⁻¹L⁻¹T⁻¹Q²], [M⁻¹L⁻²TQ²], [L⁻¹T], [LT⁻¹]
F⁻¹L⁻¹T⁻¹Q²⋅(α⋅2² = 0.0291894102771(45)) [𝘤⁻¹μ₀⁻¹λ⁻¹αL⁻²] Unified

Conductance quantum G₀ is a quantized unit of electrical conductance (S).

julia> conductancequantum(SI2019) # S
𝘩⁻¹𝘦²2 = 7.748091729863649×10⁻⁵ [S] SI2019

julia> conductancequantum(Metric) # S
𝘤⁻¹α⋅τ⁻¹2⁸5⁷ = 7.7480917341(12) × 10⁻⁵ [S] Metric

julia> conductancequantum(Conventional) # S
RK90⁻¹2 = 7.74809186773062×10⁻⁵ [S] Conventional

julia> conductancequantum(CODATA) # S
RK⁻¹2 = 7.7480917310(18) × 10⁻⁵ [S] CODATA

julia> conductancequantum(International) # S
𝘤⁻¹α⋅Ωᵢₜ⋅τ⁻¹2⁸5⁷ = 7.7519270395(12) × 10⁻⁵ [S] International

julia> conductancequantum(InternationalMean) # S
𝘤⁻¹α⋅τ⁻¹2⁸5⁷⋅1.00049 = 7.7518882990(12) × 10⁻⁵ [S] InternationalMean

julia> conductancequantum(EMU) # abS
𝘤⁻¹α⋅τ⁻¹2⁻¹5⁻² = 7.7480917341(12) × 10⁻¹⁴ [cm⁻¹s] EMU

julia> conductancequantum(ESU) # statS
𝘤⋅α⋅τ⁻¹2³5² = 6.9636375713(11) × 10⁷ [cm⋅s⁻¹] ESU

\[R_K = \frac{h}{e^2} = \frac{Z_0}{2\alpha} = \frac{2}{G_0} = \frac{4\alpha_L^2}{hK_J^2} = h\frac{N_A^2}{F^2}\]

MeasureSystems.klitzing — Constant 
klitzing(U::UnitSystem) = planck(U)/elementarycharge(U)^2
resistance : [FLTQ⁻²], [FLTQ⁻²], [ML²T⁻¹Q⁻²], [LT⁻¹], [L⁻¹T]
FLTQ⁻²⋅(α⁻¹2⁻¹ = 68.517999542(10)) [𝘤⋅μ₀⋅λ⋅αL²] Unified

Quantized Hall resistance RK (Ω).

julia> klitzing(SI2019) # Ω
𝘩⋅𝘦⁻² = 25812.80745930451 [Ω] SI2019

julia> klitzing(Metric) # Ω
𝘤⋅α⁻¹τ⋅2⁻⁷5⁻⁷ = 25812.8074452(40) [Ω] Metric

julia> klitzing(Conventional) # Ω
RK90 = 25812.807 [Ω] Conventional

julia> klitzing(International) # Ω
𝘤⋅α⁻¹Ωᵢₜ⁻¹τ⋅2⁻⁷5⁻⁷ = 25800.036427200(40) [Ω] International

julia> klitzing(CODATA) # Ω
RK = 25812.8074555(59) [Ω] CODATA

julia> klitzing(EMU) # abΩ
𝘤⋅α⁻¹τ⋅2²5² = 2.58128074452(40) × 10¹³ [cm⋅s⁻¹] EMU

julia> klitzing(ESU) # statΩ
𝘤⁻¹α⁻¹τ⋅2⁻²5⁻² = 2.87206216508(44) × 10⁻⁸ [cm⁻¹s] ESU

\[K_J = \frac{2e\alpha_L}{h} = \alpha_L\sqrt{\frac{8\alpha}{hZ_0}} = \alpha_L\sqrt{\frac{4}{hR_K}} = \frac{1}{\Phi_0} = \frac{2F\alpha_L}{hN_A}\]

MeasureSystems.josephson — Constant 
josephson(U::UnitSystem) = 𝟐*elementarycharge(U)*lorentz(U)/planck(U)
nonstandard : [F⁻¹L⁻¹T⁻¹QC⁻¹], [F⁻¹L⁻¹T⁻¹Q], [M⁻¹L⁻²TQ], [M⁻¹ᐟ²L⁻³ᐟ²T], [M⁻¹ᐟ²L⁻¹ᐟ²]
F⁻¹L⁻¹T⁻¹QC⁻¹⋅(α¹ᐟ²τ⁻¹ᐟ²2³ᐟ² = 0.0963912748286(74)) [ħ⁻¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ⁻¹ᐟ²λ⁻¹ᐟ²] Unified

Josephson constant KJ relating potential difference to irradiation frequency (Hz⋅V⁻¹).

julia> josephson(SI2019) # Hz⋅V⁻¹
𝘩⁻¹𝘦⋅2 = 4.8359784841698356×10¹⁴ [Hz⋅V⁻¹] SI2019

julia> josephson(Metric) # Hz⋅V⁻¹
𝘩⁻¹ᐟ²𝘤⁻¹ᐟ²α¹ᐟ²τ⁻¹ᐟ²2⁹ᐟ²5⁷ᐟ² = 4.83597848549(37) × 10¹⁴ [Hz⋅V⁻¹] Metric

julia> josephson(Conventional) # Hz⋅V⁻¹
KJ90 = 4.835979×10¹⁴ [Hz⋅V⁻¹] Conventional

julia> josephson(CODATA) # Hz⋅V⁻¹
KJ = 4.835978525(30) × 10¹⁴ [Hz⋅V⁻¹] CODATA

julia> josephson(International) # Hz⋅V⁻¹
𝘩⁻¹ᐟ²𝘤⁻¹ᐟ²α¹ᐟ²Vᵢₜ⋅τ⁻¹ᐟ²2⁹ᐟ²5⁷ᐟ² = 4.83757435839(37) × 10¹⁴ [Hz⋅V⁻¹] International

julia> josephson(EMU) # Hz⋅abV⁻¹
𝘩⁻¹ᐟ²𝘤⁻¹ᐟ²α¹ᐟ²τ⁻¹ᐟ²2⁻⁷ᐟ²5⁻⁹ᐟ² = 4.83597848549(37) × 10⁶ [g⁻¹ᐟ²cm⁻³ᐟ²s] EMU

julia> josephson(ESU) # Hz⋅statV⁻¹
𝘩⁻¹ᐟ²𝘤¹ᐟ²α¹ᐟ²τ⁻¹ᐟ²2⁻³ᐟ²5⁻⁵ᐟ² = 1.44978987700(11) × 10¹⁷ [g⁻¹ᐟ²cm⁻¹ᐟ²] ESU

\[\Phi_0 = \frac{h}{2e\alpha_L} = \frac{1}{\alpha_L}\sqrt{\frac{hZ_0}{8\alpha}} = \frac{1}{\alpha_L}\sqrt{\frac{hR_K}{4}} = \frac{1}{K_J} = \frac{hN_A}{2F\alpha_L}\]

MeasureSystems.magneticfluxquantum — Constant 
magneticfluxquantum(U::UnitSystem) = planck(U)/𝟐/elementarycharge(U)/lorentz(U)
magneticflux : [FLTQ⁻¹C], [FLTQ⁻¹], [ML²T⁻¹Q⁻¹], [M¹ᐟ²L³ᐟ²T⁻¹], [M¹ᐟ²L¹ᐟ²]
FLTQ⁻¹C⋅(α⁻¹ᐟ²τ¹ᐟ²2⁻³ᐟ² = 10.374382969600(79)) [ħ¹ᐟ²𝘤¹ᐟ²μ₀¹ᐟ²ϕ¹ᐟ²λ¹ᐟ²] Unified

Magnetic flux quantum Φ₀ is 𝟏/josephson(U) (Wb).

julia> magneticfluxquantum(SI2019) # Wb
𝘩⋅𝘦⁻¹2⁻¹ = 2.0678338484619295×10⁻¹⁵ [Wb] SI2019

julia> magneticfluxquantum(Metric) # Wb
𝘩¹ᐟ²𝘤¹ᐟ²α⁻¹ᐟ²τ¹ᐟ²2⁻⁹ᐟ²5⁻⁷ᐟ² = 2.06783384790(16) × 10⁻¹⁵ [Wb] Metric

julia> magneticfluxquantum(Conventional) # Wb
KJ90⁻¹ = 2.0678336278962334×10⁻¹⁵ [Wb] Conventional

julia> magneticfluxquantum(International) # Wb
𝘩¹ᐟ²𝘤¹ᐟ²α⁻¹ᐟ²Vᵢₜ⁻¹τ¹ᐟ²2⁻⁹ᐟ²5⁻⁷ᐟ² = 2.06715168784(16) × 10⁻¹⁵ [Wb] International

julia> magneticfluxquantum(InternationalMean) # Wb
𝘩¹ᐟ²𝘤¹ᐟ²α⁻¹ᐟ²τ¹ᐟ²2⁻⁹ᐟ²5⁻⁷ᐟ²/1.00034 = 2.06713102335(16) × 10⁻¹⁵ [Wb] InternationalMean

julia> magneticfluxquantum(EMU) # Mx
𝘩¹ᐟ²𝘤¹ᐟ²α⁻¹ᐟ²τ¹ᐟ²2⁷ᐟ²5⁹ᐟ² = 2.06783384790(16) × 10⁻⁷ [Mx] EMU

julia> magneticfluxquantum(ESU) # statWb
𝘩¹ᐟ²𝘤⁻¹ᐟ²α⁻¹ᐟ²τ¹ᐟ²2³ᐟ²5⁵ᐟ² = 6.89755126494(53) × 10⁻¹⁸ [g¹ᐟ²cm¹ᐟ²] ESU

\[\mu_B = \frac{e\hbar\alpha_L}{2m_e} = \frac{\hbar\alpha_L^2}{m_eK_JR_K} = \frac{h^2K_J}{8\pi m_e} = \frac{\alpha_L\hbar F}{2m_e N_A} = \frac{ec\alpha^2\alpha_L}{8\pi g_0R_\infty}\]

MeasureSystems.magneton — Constant 
magneton(U::UnitSystem) = elementarycharge(U)*planckreduced(U)*lorentz(U)/2electronmass(U)
nonstandard : [FM⁻¹LTQA⁻¹C⁻¹], [L²T⁻¹Q], [L²T⁻¹Q], [M¹ᐟ²L⁵ᐟ²T⁻¹], [M¹ᐟ²L⁷ᐟ²T⁻²]
FM⁻¹LTQA⁻¹C⁻¹⋅(α¹ᐟ²τ¹ᐟ²2⁻¹ᐟ² = 0.151411060436(12)) [ħ³ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²mₑ⁻¹ϕ¹ᐟ²λ⁻¹ᐟ²] Unified

Bohr magneton μB natural unit for expressing magnetic moment of electron (J⋅T⁻¹).

julia> magneton(SI2019) # J⋅T⁻¹
𝘤⋅𝘦⋅R∞⁻¹α²τ⁻¹2⁻² = 9.2740100783(28) × 10⁻²⁴ [J⋅T⁻¹] SI2019

julia> magneton(Metric) # J⋅T⁻¹
𝘩¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²τ⁻³ᐟ²2³ᐟ²5⁷ᐟ² = 9.2740100808(36) × 10⁻²⁴ [J⋅T⁻¹] Metric

julia> magneton(CODATA) # J⋅T⁻¹
𝘤⋅R∞⁻¹α²RK⁻¹KJ⁻¹τ⁻¹2⁻¹ = 9.274010001(58) × 10⁻²⁴ [J⋅T⁻¹] CODATA

julia> magneton(Conventional) # J⋅T⁻¹
𝘤⋅R∞⁻¹α²RK90⁻¹KJ90⁻¹τ⁻¹2⁻¹ = 9.2740092541(28) × 10⁻²⁴ [J⋅T⁻¹] Conventional

julia> magneton(International) # J⋅T⁻¹
𝘩¹ᐟ²𝘤¹ᐟ²R∞⁻¹α⁵ᐟ²Ωᵢₜ⋅Vᵢₜ⁻¹τ⁻³ᐟ²2³ᐟ²5⁷ᐟ² = 9.2755397877(36) × 10⁻²⁴ [J⋅T⁻¹] International

julia> magneton(ESU) # statA⋅cm²
𝘩¹ᐟ²𝘤³ᐟ²R∞⁻¹α⁵ᐟ²τ⁻³ᐟ²2¹³ᐟ²5¹⁷ᐟ² = 2.7802782776(11) × 10⁻¹⁰ [g¹ᐟ²cm⁷ᐟ²s⁻²] ESU

julia> magneton(SI2019)/elementarycharge(SI2019) # eV⋅T⁻¹
𝘤⋅R∞⁻¹α²τ⁻¹2⁻² = 5.7883818060(18) × 10⁻⁵ [m²s⁻¹] SI2019

julia> magneton(Hartree) # 𝘤⋅ħ⋅mₑ⁻¹
2⁻¹ = 0.5 [𝘦] Hartree

Astronomical Constants

MeasureSystems.eddington — Constant 
eddington(U::UnitSystem) = mass(𝟏,U,Cosmological)
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(𝘩⁻²𝘤³R∞⁻¹α²ΩΛ⁻¹ᐟ²H0⁻¹au⋅mP²τ⁻¹ᐟ²2⁸3⁷ᐟ²5⁶ = 2.804(21) × 10⁸²) [mₑ] Unified

Approximate number of protons in the Universe as estimated by Eddington (kg or lb).

julia> 𝟐^2^2^3/α # mₚ
α⁻¹2²⁵⁶ = 1.58676846347(24) × 10⁷⁹

julia> eddington(QCD) # mₚ
𝘩⁻²𝘤³R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹ΩΛ⁻¹ᐟ²H0⁻¹au⋅mP²τ⁻¹ᐟ²2⁸3⁷ᐟ²5⁶ = 1.527(11) × 10⁷⁹ [mₚ] QCD

julia> eddington(Metric) # kg
𝘩⁻¹𝘤²ΩΛ⁻¹ᐟ²H0⁻¹au⋅mP²τ⁻¹ᐟ²2⁹3⁷ᐟ²5⁶ = 2.555(19) × 10⁵² [kg] Metric

julia> eddington(IAU) # M☉
𝘤³ΩΛ⁻¹ᐟ²H0⁻¹au⁻²kG⁻²τ⁻⁷ᐟ²2³⁷3³⁵ᐟ²5¹⁶ = 1.2847(95) × 10²² [M☉] IAU☉

julia> eddington(Cosmological)
𝟏 = 1.0 [M] Cosmological
MeasureSystems.solarmass — Constant 
solarmass(U::UnitSystem) = mass(𝘩⁻¹𝘤⁻¹au³kG²mP²τ³2⁻²⁸3⁻¹⁴5⁻¹⁰ = 1.988409(44) × 10³⁰,U)
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(𝘩⁻²R∞⁻¹α²au³kG²mP²τ³2⁻²⁹3⁻¹⁴5⁻¹⁰ = 2.182814(48) × 10⁶⁰) [mₑ] Unified

Solar mass estimated from gravitational constant estimates (kg or slug).

julia> solarmass(Metric) # kg
𝘩⁻¹𝘤⁻¹au³kG²mP²τ³2⁻²⁸3⁻¹⁴5⁻¹⁰ = 1.988409(44) × 10³⁰ [kg] Metric

julia> solarmass(British) # slug
𝘩⁻¹𝘤⁻¹g₀⁻¹au³ft⋅lb⁻¹kG²mP²τ³2⁻²⁸3⁻¹⁴5⁻¹⁰ = 1.362493(30) × 10²⁹ [slug] British

julia> solarmass(English) # lb
𝘩⁻¹𝘤⁻¹au³lb⁻¹kG²mP²τ³2⁻²⁸3⁻¹⁴5⁻¹⁰ = 4.383692(97) × 10³⁰ [lbm] English

julia> solarmass(IAUE) # ME
au³kG²GME⁻¹τ²2⁻²⁸3⁻¹⁴5⁻¹⁰ = 332946.04409(67) [ME] IAUE

julia> solarmass(IAUJ) # MJ
au³kG²GMJ⁻¹τ²2⁻²⁸3⁻¹⁴5⁻¹⁰ = 1047.565484(74) [MJ] IAUJ

julia> solarmass(QCD) # mₚ
𝘩⁻²R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹au³kG²mP²τ³2⁻²⁹3⁻¹⁴5⁻¹⁰ = 1.188798(26) × 10⁵⁷ [mₚ] QCD

julia> solarmass(Metric)/dalton(Metric) # Da
𝘩⁻²R∞⁻¹α²μₑᵤ⋅au³kG²mP²τ³2⁻²⁹3⁻¹⁴5⁻¹⁰ = 1.197448(26) × 10⁵⁷ [𝟙] Metric
MeasureSystems.jupitermass — Constant 
jupitermass(U::UnitSystem) = mass(𝘩⁻¹𝘤⁻¹mP²GMJ⋅τ = 1.898124(42) × 10²⁷,U)
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(𝘩⁻²R∞⁻¹α²mP²GMJ⋅τ⋅2⁻¹ = 2.083702(46) × 10⁵⁷) [mₑ] Unified

Jupiter mass estimated from gravitational constant estimates (kg or slug).

julia> jupitermass(Metric) # kg
𝘩⁻¹𝘤⁻¹mP²GMJ⋅τ = 1.898124(42) × 10²⁷ [kg] Metric

julia> jupitermass(British) # slug
𝘩⁻¹𝘤⁻¹g₀⁻¹ft⋅lb⁻¹mP²GMJ⋅τ = 1.300628(29) × 10²⁶ [slug] British

julia> jupitermass(English) # lb
𝘩⁻¹𝘤⁻¹lb⁻¹mP²GMJ⋅τ = 4.184647(92) × 10²⁷ [lbm] English

julia> jupitermass(IAU) # M☉
au⁻³kG⁻²GMJ⋅τ⁻²2²⁸3¹⁴5¹⁰ = 0.000954594262(68) [M☉] IAU☉

julia> jupitermass(IAUE) # ME
GME⁻¹GMJ = 317.828383(23) [ME] IAUE

julia> jupitermass(QCD) # mₚ
𝘩⁻²R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹mP²GMJ⋅τ⋅2⁻¹ = 1.134820(25) × 10⁵⁴ [mₚ] QCD

julia> jupitermass(Metric)/dalton(Metric) # Da
𝘩⁻²R∞⁻¹α²μₑᵤ⋅mP²GMJ⋅τ⋅2⁻¹ = 1.143077(25) × 10⁵⁴ [𝟙] Metric
MeasureSystems.earthmass — Constant 
earthmass(U::UnitSystem) = mass(𝘩⁻¹𝘤⁻¹mP²GME⋅τ = 5.97217(13) × 10²⁴,U)
mass : [M], [FL⁻¹T²], [M], [M], [M]
M⋅(𝘩⁻²R∞⁻¹α²mP²GME⋅τ⋅2⁻¹ = 6.55606(14) × 10⁵⁴) [mₑ] Unified

Earth mass estimated from gravitational constant estimates (kg or slug).

julia> earthmass(Metric) # kg
𝘩⁻¹𝘤⁻¹mP²GME⋅τ = 5.97217(13) × 10²⁴ [kg] Metric

julia> earthmass(British) # slug
𝘩⁻¹𝘤⁻¹g₀⁻¹ft⋅lb⁻¹mP²GME⋅τ = 4.092234(90) × 10²³ [slug] British

julia> earthmass(English) # lb
𝘩⁻¹𝘤⁻¹lb⁻¹mP²GME⋅τ = 1.316637(29) × 10²⁵ [lbm] English

julia> earthmass(IAU) # M☉
au⁻³kG⁻²GME⋅τ⁻²2²⁸3¹⁴5¹⁰ = 3.0034896577(60) × 10⁻⁶ [M☉] IAU☉

julia> earthmass(IAUJ) # MJ
GME⋅GMJ⁻¹ = 0.00314635210(22) [MJ] IAUJ

julia> earthmass(QCD) # mₚ
𝘩⁻²R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹mP²GME⋅τ⋅2⁻¹ = 3.570542(79) × 10⁵¹ [mₚ] QCD

julia> earthmass(Metric)/dalton(Metric) # Da
𝘩⁻²R∞⁻¹α²μₑᵤ⋅mP²GME⋅τ⋅2⁻¹ = 3.596523(79) × 10⁵¹ [𝟙] Metric
MeasureSystems.lunarmass — Constant 
lunarmass(U::UnitSystem) = earthmass(U)/μE☾
mass : [M], [FL⁻¹T²], [M], [M], [M]
M/81.300568 ± 3.0e-6 [mₑ] Unified

Lunar mass estimated from μE☾ Earth-Moon mass ratio (kg or slug).

julia> lunarmass(Metric) # kg
𝘩⁻¹𝘤⁻¹mP²GME⋅τ/81.3005680(30) = 7.34579(16) × 10²² [kg] Metric

julia> lunarmass(British) # slug
𝘩⁻¹𝘤⁻¹g₀⁻¹ft⋅lb⁻¹mP²GME⋅τ/81.3005680(30) = 5.03346(11) × 10²¹ [slug] British

julia> lunarmass(English) # lb
𝘩⁻¹𝘤⁻¹lb⁻¹mP²GME⋅τ/81.3005680(30) = 1.619469(36) × 10²³ [lbm] English

julia> lunarmass(IAU) # M☉
au⁻³kG⁻²GME⋅τ⁻²2²⁸3¹⁴5¹⁰/81.3005680(30) = 3.69430341(14) × 10⁻⁸ [M☉] IAU☉

julia> lunarmass(IAUE) # ME
𝟏/81.3005680(30) = 0.01230003707(45) [ME] IAUE

julia> lunarmass(IAUJ) # MJ
GME⋅GMJ⁻¹/81.3005680(30) = 3.87002474(31) × 10⁻⁵ [MJ] IAUJ

julia> lunarmass(QCD) # mₚ
𝘩⁻²R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹mP²GME⋅τ⋅2⁻¹/81.3005680(30) = 4.391780(97) × 10⁴⁹ [mₚ] QCD

julia> lunarmass(Metric)/dalton(Metric) # Da
𝘩⁻²R∞⁻¹α²μₑᵤ⋅mP²GME⋅τ⋅2⁻¹/81.3005680(30) = 4.423736(98) × 10⁴⁹ [𝟙] Metric
MeasureSystems.gravity — Constant 
gravity(U::UnitSystem) = # mass*acceleration/force
gravityforce : [F⁻¹MLT⁻²], [𝟙], [𝟙], [𝟙], [𝟙]
F⁻¹MLT⁻² [g₀] Unified

Gravitational force reference used in technical engineering units (kg⋅m⋅N⁻¹⋅s⁻²).

julia> gravity(Metric)
𝟏 = 1.0 [𝟙] Metric

julia> gravity(Engineering) # m⋅kg⋅N⁻¹⋅s⁻²
g₀ = 9.80665 [kgf⁻¹kg⋅m⋅s⁻²] Engineering

julia> gravity(English) # ft⋅lbm⋅lbf⁻¹⋅s⁻²
g₀⋅ft⁻¹ = 32.17404855643044 [lbf⁻¹lbm⋅ft⋅s⁻²] English
MeasureSystems.earthradius — Constant 
earthradius(U::UnitSystem) = sqrt(earthmass(U)*gravitation(U)/gforce(U))
length : [L], [L], [L], [L], [L]
L⋅(R∞⋅α⁻²g₀⁻¹ᐟ²GME¹ᐟ²τ⋅2 = 1.6509810466(17) × 10¹⁹) [ħ⋅𝘤⁻¹mₑ⁻¹ϕ⋅g₀] Unified

Approximate length of standard Earth two-body radius consistent with units (m or ft).

julia> earthradius(KKH) # km
g₀⁻¹ᐟ²GME¹ᐟ²2⁻³5⁻³ = 6375.4163237(64) [km] KKH

julia> earthradius(Nautical) # nm
τ⁻¹2⁵3³5² = 3437.7467707849396 [nm] Nautical

julia> earthradius(IAU) # au
g₀⁻¹ᐟ²au⁻¹GME¹ᐟ² = 4.2617025856(43) × 10⁻⁵ [au] IAU☉
MeasureSystems.greatcircle — Constant 
greatcircle(U::UnitSystem) = τ*earthradius(U)
length : [L], [L], [L], [L], [L]
L⋅(R∞⋅α⁻²g₀⁻¹ᐟ²GME¹ᐟ²τ²2 = 1.0373419854(11) × 10²⁰) [ħ⋅𝘤⁻¹mₑ⁻¹ϕ⋅g₀] Unified

Approximate length of standard Earth two-body circle consistent with units (m or ft).

julia> greatcircle(KKH) # km
g₀⁻¹ᐟ²GME¹ᐟ²τ⋅2⁻³5⁻³ = 40057.922172(40) [km] KKH

julia> greatcircle(Nautical) # nm
2⁵3³5² = 21600.0 [nm] Nautical

julia> greatcircle(IAU) # au
g₀⁻¹ᐟ²au⁻¹GME¹ᐟ²τ = 0.00026777067070(27) [au] IAU☉
MeasureSystems.gaussianmonth — Constant 
gaussianmonth(U::UnitSystem) = τ*sqrt(lunardistance(U)^3/earthmass(U)/gravitation(U))
time : [T], [T], [T], [T], [T]
T⋅1.6987431854323947e6 [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Orbit time defined by lunardistance and earthmass for neglible mass satellite (s).

julia> gaussianmonth(Metric) # s
GME⁻¹ᐟ²τ⋅2⁹ᐟ²3⁹ᐟ²5⁹ᐟ²⋅1.6987431854323947×10⁶ = 2.3718343493(24) × 10⁶ [s] Metric

julia> gaussianmonth(MPH) # h
GME⁻¹ᐟ²τ⋅2¹ᐟ²3⁵ᐟ²5⁵ᐟ²⋅1.6987431854323947×10⁶ = 658.84287479(66) [h] MPH

julia> gaussianmonth(IAU) # D
GME⁻¹ᐟ²τ⋅2⁻⁵ᐟ²3³ᐟ²5⁵ᐟ²⋅1.6987431854323947×10⁶ = 27.451786450(28) [D] IAU☉
MeasureSystems.siderealmonth — Constant 
siderealmonth(U::UnitSystem) = gaussianmonth(U)/√(𝟏+lunarmass(IAU))
time : [T], [T], [T], [T], [T]
T⋅1.68839128266e6 ± 0.00038 [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Orbit time defined by standard lunardistance and the Earth-Moon system mass (s).

julia> siderealmonth(Metric) # s
GME⁻¹ᐟ²τ⋅2⁹ᐟ²3⁹ᐟ²5⁹ᐟ²⋅1.68839128266(38) × 10⁶ = 2.3573807233(24) × 10⁶ [s] Metric

julia> siderealmonth(MPH) # h
GME⁻¹ᐟ²τ⋅2¹ᐟ²3⁵ᐟ²5⁵ᐟ²⋅1.68839128266(38) × 10⁶ = 654.82797870(67) [h] MPH

julia> siderealmonth(IAU) # D
GME⁻¹ᐟ²τ⋅2⁻⁵ᐟ²3³ᐟ²5⁵ᐟ²⋅1.68839128266(38) × 10⁶ = 27.284499112(28) [D] IAU☉
MeasureSystems.synodicmonth — Constant 
synodicmonth(U::UnitSystem) = 𝟏/(𝟏/siderealmonth(U)-𝟏/siderealyear(U))
time : [T], [T], [T], [T], [T]
T⋅29.487179323 ± 3.3e-8 [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Orbit time defined by siderealmonth and siderealyear of Sun-Earth-Moon system (s).

julia> synodicmonth(Metric) # s
2⁷3³5²⋅29.487179323(33) = 2.5476922935(28) × 10⁶ [s] Metric

julia> synodicmonth(MPH) # h
2³3⋅29.487179323(33) = 707.69230376(79) [h] MPH

julia> synodicmonth(IAU) # D
29.487179323(33) = 29.487179323(33) [D] IAU☉
MeasureSystems.gaussianyear — Constant 
gaussianyear(U::UnitSystem) = turn(U)/gaussgravitation(U)
time : [T], [T], [T], [T], [T]
T⋅(𝘤⋅R∞⋅α⁻²kG⁻¹τ⋅2¹⁵3⁷5⁵ = 2.45000183355(75) × 10²⁸) [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Orbit time defined by gaussgravitation constant kG for neglible mass satellite (s).

julia> gaussianyear(Metric) # s
kG⁻¹2¹⁴3⁷5⁵ = 3.155819598840209×10⁷ [s] Metric

julia> gaussianyear(MPH) # h
kG⁻¹2¹⁰3⁵5³ = 8766.165552333914 [h] MPH

julia> gaussianyear(IAU) # D
kG⁻¹2⁷3⁴5³ = 365.2568980139131 [D] IAU☉
MeasureSystems.siderealyear — Constant 
siderealyear(U::UnitSystem) = gaussianyear(U)/√(𝟏+earthmass(IAU)+lunarmass(IAU))
time : [T], [T], [T], [T], [T]
T/1.0000015202151904 ± 3.1e-15 [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Orbit time defined by gaussgravitation constant kG and Earth-Moon system mass (s).

julia> siderealyear(Metric) # s
kG⁻¹2¹⁴3⁷5⁵/1.0000015202151904(31) = 3.1558148013226100(98) × 10⁷ [s] Metric

julia> siderealyear(MPH) # h
kG⁻¹2¹⁰3⁵5³/1.0000015202151904(31) = 8766.152225896140(27) [h] MPH

julia> siderealyear(IAU) # D
kG⁻¹2⁷3⁴5³/1.0000015202151904(31) = 365.2563427456725(11) [D] IAU☉
MeasureSystems.jovianyear — Constant 
jovianyear(U::UnitSystem) = τ*day(U)*√(jupiterdistance(U)^3/solarmass(U)/gravitation(U))/√(𝟏+jupitermass(IAU))
time : [T], [T], [T], [T], [T]
T⋅1.321238687229e8 ± 0.0045 [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Orbit time defined by jupiterdistance and the Sun-Jupiter system mass (s).

julia> jovianyear(Metric) # s
au⁻³ᐟ²kG⁻¹2²³3¹⁷ᐟ²5¹⁴⋅1.321238687229(45) × 10⁸ = 3.74444292140(17) × 10⁸ [s] Metric

julia> jovianyear(MPH) # h
au⁻³ᐟ²kG⁻¹2¹⁹3¹³ᐟ²5¹²⋅1.321238687229(45) × 10⁸ = 104012.3033722(47) [h] MPH

julia> jovianyear(IAU) # D
au⁻³ᐟ²kG⁻¹2¹⁶3¹¹ᐟ²5¹²⋅1.321238687229(45) × 10⁸ = 4333.84597384(20) [D] IAU☉
MeasureSystems.radarmile — Constant 
radarmile(U::UnitSystem) = 𝟐*nauticalmile(U)/lightspeed(U)
time : [T], [T], [T], [T], [T]
T⋅(R∞⋅α⁻²g₀⁻¹ᐟ²GME¹ᐟ²τ²2⁻³3⁻³5⁻² = 9.605018384(10) × 10¹⁵) [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] Unified

Unit of time delay from a two-way nauticalmile radar return (s).

julia> radarmile(Metric)
𝘤⁻¹g₀⁻¹ᐟ²GME¹ᐟ²τ⋅2⁻⁴3⁻³5⁻² = 1.2372115338(12) × 10⁻⁵ [s] Metric
MeasureSystems.hubble — Constant 
hubble(U::UnitSystem) = time(U,Hubble)
frequency : [T⁻¹], [T⁻¹], [T⁻¹], [T⁻¹], [T⁻¹]
T⁻¹⋅(𝘤⁻¹R∞⁻¹α²H0⋅au⁻¹2⁻¹¹3⁻⁴5⁻⁶ = 2.824(18) × 10⁻³⁹) [ħ⁻¹𝘤²mₑ⋅ϕ⁻¹g₀⁻¹] Unified

Hubble universe expansion frequency parameter.

julia> hubble(Metric)
H0⋅au⁻¹τ⋅2⁻¹⁰3⁻⁴5⁻⁶ = 2.193(14) × 10⁻¹⁸ [Hz] Metric

julia> hubble(Hubble)
𝟏 = 1.0 [T⁻¹] Hubble

julia> hubble(Cosmological)
ΩΛ⁻¹ᐟ²τ¹ᐟ²2⋅3⁻¹ᐟ² = 3.487(14) [T⁻¹] Cosmological

julia> 𝟏/hubble(Metric)/year(Metric)
H0⁻¹aⱼ⁻¹au⋅τ⁻¹2³3⋅5⁴ = 1.4452(90) × 10¹⁰ [𝟙] Metric
MeasureSystems.cosmological — Constant 
cosmological(U::UnitSystem) = 𝟑*darkenergydensity(U)*(hubble(U)/lightspeed(U))^2
fuelefficiency : [L⁻²], [L⁻²], [L⁻²], [L⁻²], [L⁻²]
L⁻²⋅(𝘤⁻²R∞⁻²α⁴ΩΛ⋅H0²au⁻²2⁻²²3⁻⁷5⁻¹² = 1.649(24) × 10⁻⁷⁷) [ħ⁻²𝘤²mₑ²ϕ⁻²g₀⁻²] Unified

Cosmological constant from Einstein's controversial theory expanded on by Hubble.

julia> cosmological(Metric)
𝘤⁻²ΩΛ⋅H0²au⁻²τ²2⁻²⁰3⁻⁷5⁻¹² = 1.106(16) × 10⁻⁵² [m⁻²] Metric

julia> cosmological(Hubble)
ΩΛ⋅3 = 2.067(17) [T⁻²] Hubble

julia> cosmological(Cosmological)
τ⋅2² = 25.132741228718345 [T⁻²] Cosmological

Constants Index

Wolfram plagiarism timeline

Timeline of UnitSystems.jl registration and Wolfram Research plagiarism:

  • 2019: The SI2019 standard is formalized with a primitive SI only unit-system based on 7 physics dimensions (massive collaboration).
  • 2020: Registered DOI 10.5281/zenodo.7145479, UnitSystems.jl
  • 2021: Discused with Ted Corcovilos about what the unsolved and nuanced issues are with defining physics units, which I then solved by independently creating the never before seen 11 dimensional Unified System of Quantities (USQ) for physics, which was standardized in detail and completely handcrafted by myself alone.
  • 2021: Wolfram Research invited me to their Summer School, where everyone was hinting at the fact I would be hired there.
  • 2022: Wolfram Research interviewed and then hired me, with an explicit interest in my UnitSystems.jl work from lead developers. They requested that I present them my independently discovered UnitSystems.jl results in the Wolfram Language to make a comparison with their existing system. While I was shortly an employee at Wolfram, I indeed directly handed them my newly discovered Unified System of Quantities. My work was already independently complete and prepared ready to incorporate into their stack. They acknowledged that their system was old and outdated compared to mine, as they only implemented a Metric and Imperial unit system, and neither of these was up to the standard of my UnitSystems.jl standard. However, they told me that I would not be allowed to work on this project further because they didn't want to upgrade their systems. Instead, they did the software equivalent of placing me in a backroom shed to mop the floors. After 6 months they ended the contract and it turned out they lied to me on the job application about what my role would be (they said I would be part of the core team with Jonathan Gorard, but this was a blatant lie).
  • 2023-2025: Wolfram keeps inviting me to their Winter/Summer schools to help mentor people, but I declined because I am too busy making progress in my research (why directly help mentor my competitors, who made it clear that they don't want to actually support my work); their use of social environments feels predatory.
  • 2025: Wolfram bribes Memes of Destruction at Wolfram Summer School to take my fully prepared work and use it to boost the Wolfram brand on social media, presenting my completed project with AI generated text as if it was Wolfram's idea, without crediting that I was the person who directly handed them the completed project years earlier (but without AI generated text they added).

Did Wolfam think that they can pluck low hanging fruits from my garden to build their brand on social media? My only goal here is to show that these low hanging fruit Wolfram plucked, these fruits came from my public garden and were not grown or developed by them from scratch, it's my solo-project.

Academic institutions should be direclty investing in my research instead of funding and enabling Wolfram Research to systematically gangstalk me with an army of employees. I can feel the presence of Wolfram looking over my shoulder and monitoring my every step. There seems to be an entire economy of people being paid to monitor and surveil me, while I struggle to survive with my resources. Stephen Wolfram never seemed to care about earning my respect. Every time I interacted with him, he was only focused on talking about himself and that was the only topic.

It's fascinating to me how unaware Stephen Wolfram is of the fact that people perceive him as textbook specimen of ultra-narcissism. This is because he is entirely surrounded by people with a salary depending on how much they inflate Stephen Wolfram's ego, which completely divorces these people from the reality of doing actual scientific research. Wolfram's premise seems to be that they can use gangstalking to target open source developers like me to data-mine our work, enabled by funding granted from academic institutions who don't check for Wolfram's plagiarism violations.

Combining the ultra-narcissism of Wolfram with the economic incentive to target open source developers with gangstalking by an army of employees, it becomes highly uncomfortable knowing that these people are incentivized to gangstalk me for the rest of my life with smear campaigns and so on.

I urge academic institutions to quit enabling and sponsoring Stephen Wolfram's systematic gangstalking of individuals like me. He shouldn't be rewarded for plucking fruit from my public gardens, which I handcrafted. % by myself. Wolfram's goal seems to be taking the fruit of my work in a cowardly and uncollaborative way. Wolfram does not acknowledge that my science research is what's boosting their brand in the social media campaign funding Memes of Destruction.

Julia Computing are no better stewards, they are also unehtical people, but at least their product is open source and therefore a solid foundation. My work on UnitSystems.jl and the entire process of creating the new 11 dimensional Unified System of Quantities (USQ) was all done entirely in public on GitHub and each release registered with several scientific websites. This is only one of my side projects, the mainstream of my research is my differential geometric algebra software development, Grassmann.jl and Cartan.jl, and various related work at the cutting edge of science, making me a bigger target for Wolfram's gangstalking. Wolfram is now constantly being observed in attempting to keep up with my research by systematically gangstalking me in a hush-hush way, not acknowledging me. With shady business practices, I have to wonder what other fraud is being commited.

It appears that Wolfram tends to resort to plagiarism of other people's works by data mining other people's creativity through employment, ghostwriters, summer schools, shady business practices, identity theft, bribes.

The incentive behind this systematic gangstalking appears to be this: instead of working with me directly, they all wish to ostracize, isolate, and erase me. Their eventual goal is replacing me and then retroactively claiming credit for my past achievements to boost their brands. Ironically, the temptation (to incorrectly eat the fruit of my labor like this) will be their downfall, as this choice is accompanied by firm evidence of plagiarism. Plagiarism is considered a violation of academic standards by the academic institutions funding Wolfram Research. My projects are effectively ego-traps, which will trigger the downfall of an ultra-narcissit ego if incorrectly consumed. I know the academic institutions don't acknowledge me either, so all I can do is to permanently add the Wolfram plagiarism disclaimer to the original sources.

Having a quick 0-60 speed in pathological lying is not necessarily a sign of high intelligence in long term thinking. Rather, it's an indicator of a complete lack of long term thinking, demonstrating optimization toward the short term illusions of success, which falls apart upon any scrutiny.

If Wolfram does not want to be perceived as confirmed plagiarist, then Wolfram must acknowledge Michael Reed as the original creator of the new Unified System of Quantities (USQ), which is the underlying foundation for the completed research project I handed them (and they padded with AI generated text). Wolfram is well known for the claims that LLMs will replace writing code and text, so we have to assume the foundations of their work rests in AI generated text, on top of my presented complete project foundation. The LLMs and AI models all know about my UnitSystems.jl work and mine was the only reference work in existence which completed this type of work. Therefore, if using AI or LLM generated text to manipulate my unique project, this is effectively transforming the original source data which was ingested from my work using my own knowledge embedded in the LLMs. Wolfram is regurgitating the fruits of my labor without acknowledging that I directly handed this to them as a completed project.

Memes of Destruction self proclaims to not be an expert on the topic and publicly discloses the paid sponsorship from Wolfram for the social media campaign, at least this is some transparency.

– Michael Reed's audience reaction to Wolfram's plagiarism

This preface was written in 2025, the UnitSystems.jl Appendix has been documented on my website and registered as Julia package since 2020.

Core UnitSystems.jl} standard was last updated in 2022, while Similitude.jl and MeasureSystems.jl have received minor software design updates since.