Physics Constants

The following are fundamental constants of physics: DOI

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

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 [ħ⋅𝘤⁻¹mₑ⁻¹ϕ⋅g₀] 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⁻¹𝘤⁴mₑ³Kcd⋅ϕ⁻¹g₀⁻²] 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 [mₑ] 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

\[m_P = \sqrt{\frac{\hbar c}{G}} = \frac{m_e}{\sqrt{\alpha_G}} = \frac{2R_\infty hg_0}{c\alpha^2\sqrt{\alpha_G}}\]

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) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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

\[k = \frac{\sqrt{\hbar c}}{m_P} = \frac{\sqrt{\hbar c\alpha_G}}{m_e} = \frac{\alpha^2}{2g_0R_\infty}\sqrt{\frac{c^3\alpha_G}{2\pi h}} = c^2\sqrt{\frac{\kappa}{8\pi}}\]

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) [ħ⁵ᐟ²𝘤⁻⁵ᐟ²μ₀⁻¹ᐟ²mₑ⁻³ϕ⁵ᐟ²λ⁻¹ᐟ²g₀²] 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☉

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

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⁻¹𝘤⁴μ₀⋅mₑ²Kcd⋅ϕ⁻¹λ⋅αL²g₀⁻¹] 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

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

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⁻⁵𝘤⁸μ₀⋅mₑ⁶Kcd⋅ϕ⁻⁵λ⋅αL²g₀⁻⁵] 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)) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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)) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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 [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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) [ħ⁵ᐟ²𝘤⁻⁵ᐟ²μ₀⁻¹ᐟ²mₑ⁻²ϕ⁵ᐟ²λ⁻¹ᐟ²g₀²] 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) [ħ⁵ᐟ²𝘤⁻¹¹ᐟ²μ₀⁻¹ᐟ²mₑ⁻⁴Kcd⁻¹ϕ⁵ᐟ²λ⁻¹ᐟ²g₀³] 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⁻³ᐟ²𝘤³ᐟ²μ₀¹ᐟ²mₑ²ϕ⁻³ᐟ²λ¹ᐟ²g₀⁻¹] 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⁻³ᐟ²𝘤³ᐟ²μ₀¹ᐟ²mₑ²ϕ⁻³ᐟ²λ¹ᐟ²g₀⁻¹] 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) [ħ⁷ᐟ²𝘤⁻¹³ᐟ²μ₀⁻¹ᐟ²mₑ⁻⁵Kcd⁻¹ϕ⁷ᐟ²λ⁻¹ᐟ²g₀⁴] 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 [kB⁻¹𝘤²mₑ⋅g₀⁻¹] 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)) [kB⁻¹ħ⁻¹ᐟ²𝘤⁵ᐟ²μ₀¹ᐟ²mₑ⋅ϕ⁻¹ᐟ²λ¹ᐟ²αL⋅g₀⁻¹] 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 [ħ⁻¹𝘤³mₑ²ϕ⁻¹g₀⁻²] 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⁻¹ħ⁻³ᐟ²𝘤¹¹ᐟ²μ₀¹ᐟ²mₑ³ϕ⁻³ᐟ²λ¹ᐟ²αL⋅g₀⁻³] 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) [ħ¹⁵ᐟ²𝘤⁻³³ᐟ²μ₀⁻³ᐟ²mₑ⁻¹²Kcd⁻³ϕ¹⁵ᐟ²λ⁻³ᐟ²g₀⁹] 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) [ħ⁻³𝘤²mₑ⋅Kcd⁻³ϕ⁻³g₀⁻⁴] 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) [ħ⁻⁴𝘤³mₑ²Kcd⁻³ϕ⁻⁴g₀⁻⁵] 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 [ħ⋅Kcd⋅ϕ⋅g₀] 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 [𝘤⁻¹mₑ⁻¹Kcd⁻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 [mₑ⋅Mᵤ⁻¹] 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 [ϕ] 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 [λ] 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) [ħ⋅𝘤⁻²mₑ⁻¹ϕ²λ⋅g₀] 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 [ħ²𝘤⁻³mₑ⁻²ϕ³λ²g₀²] 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 [ħ⋅𝘤⁻²mₑ⁻¹ϕ⋅g₀] 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 [ħ⁻³𝘤⁴mₑ³ϕ⁻³λ⁻²g₀⁻³] 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) [ħ³𝘤⁻⁴mₑ⁻³ϕ⁴λ²g₀³] 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) [ħ⋅𝘤⁻²mₑ⁻¹ϕ²λ²g₀] 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)) [ħ⁻³ᐟ²𝘤⁷ᐟ²mₑ⁵ᐟ²Kcd¹ᐟ²ϕ⁻²λ⁻¹g₀⁻²] 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)) [kB⁻¹ħ⁻²𝘤⁶μ₀¹ᐟ²mₑ⁷ᐟ²Kcd¹ᐟ²ϕ⁻⁵ᐟ²λ⁻¹ᐟ²αL⋅g₀⁻³] 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)) [ħ⁻²𝘤³mₑ²ϕ⁻³λ⁻²g₀⁻²] 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)) [ħ²𝘤⁻³mₑ⁻²ϕ³λ²g₀²] 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⁻¹ᐟ²mₑ⁻¹ᐟ²Kcd⁻¹ᐟ²ϕ⁻¹] 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)) [ħ¹ᐟ²𝘤¹ᐟ²mₑ¹ᐟ²Kcd¹ᐟ²ϕ] 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)) [ħ⁻²𝘤⁴μ₀¹ᐟ²mₑ⁷ᐟ²Kcd¹ᐟ²ϕ⁻⁵ᐟ²λ¹ᐟ²αL⋅g₀⁻²] 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) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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) [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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 [ħ¹ᐟ²𝘤⁻¹ᐟ²μ₀⁻¹ᐟ²ϕ¹ᐟ²λ⁻¹ᐟ²αL⁻¹] 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 [αL⁻¹] 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⁻⁵ᐟ²𝘤¹¹ᐟ²μ₀¹ᐟ²mₑ⁴Kcd⋅ϕ⁻⁵ᐟ²λ¹ᐟ²g₀⁻³] 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⁻⁵ᐟ²𝘤¹¹ᐟ²μ₀¹ᐟ²mₑ⁴Kcd⋅ϕ⁻⁵ᐟ²λ¹ᐟ²g₀⁻³] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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⁻⁷ᐟ²𝘤¹³ᐟ²μ₀¹ᐟ²mₑ⁵Kcd⋅ϕ⁻⁷ᐟ²λ¹ᐟ²g₀⁻⁴] 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) [ħ⁷ᐟ²𝘤⁻¹³ᐟ²μ₀⁻¹ᐟ²mₑ⁻⁵Kcd⁻¹ϕ⁷ᐟ²λ⁻¹ᐟ²g₀⁴] 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) [ħ⁵𝘤⁻¹¹μ₀⁻¹mₑ⁻⁸Kcd⁻²ϕ⁵λ⁻¹g₀⁶] 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