The UnitSystem

Physical unit system constants (Metric, English, Natural, etc...)

The UnitSystem

By default, UnitSystems provides a modern unified re-interpretation of various historical unit systems which were previously incompatible. In order to make each UnitSystem consistently compatible with each other, a few convenience assumptions are made. Specifically, it is assumed that all default modern unit systems share the same common Universe of dimensionless constants, although this can be optionally changed. Therefore, the philosophy is to characterize differences among UnitSystem instances by means of dimensional constants. As a result, all the defaults are ideal modern variants of these historical unit systems based on a common underlying Universe, which are completely consistent and compatible with each other. These default UnitSystem values are to be taken as a newly defined mutually-compatible recommended standard, verified to be consistent and coherent.

Metric SI Unit Systems

In the Systeme International d'Unites (the SI units) the UnitSystem constants are derived from the most accurate possible physical measurements and a few exactly defined constants. Exact values are the avogadro number, boltzmann constant, planck constant, lightspeed definition, and elementary charge definition.

Construction of UnitSystem instances based on specifying the the constants molarmass, the vacuumpermeability, and the molargas along with some other options is facilitated by MetricSystem. This construction helps characterize the differences between

MeasureSystems.Metric — Constant

Metric = MetricSystem(milli,𝟐*τ/𝟏𝟎^7)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Standard Metric system based on exact molarmass and vacuumpermeability.

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

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

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

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

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

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

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

MeasureSystems.SI2019 — Constant

SI2019 = MetricSystem(Mᵤ,μ₀)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Systeme International d'Unites based on approximate molarmass and vacuumpermeability.

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

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

julia> lightspeed(SI2019) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] SI2019

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

julia> electronmass(SI2019) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] SI2019

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

julia> luminousefficacy(SI2019) # lm⋅W⁻¹
Kcd = 683.01969009009 [lm⋅W⁻¹] SI2019

MeasureSystems.SI1976 — Constant

SI1976 = MetricSystem(milli,𝟐*τ/𝟏𝟎^7,8.31432)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Reference UnitSystem with universal gas constant of 8.31432 from 1976 standard atmosphere.

julia> boltzmann(SI1976) # J⋅K⁻¹
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³⋅8.31432 = 1.38062531722(43) × 10⁻²³ [kg⋅m²s⁻²K⁻¹] SI1976

julia> planckreduced(SI1976) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹ = 1.0545718176461565×10⁻³⁴ [kg⋅m²s⁻¹] SI1976

julia> lightspeed(SI1976) # m⋅s⁻¹
𝘤 = 2.99792458×10⁸ [m⋅s⁻¹] SI1976

julia> vacuumpermeability(SI1976) # H⋅m⁻¹
τ⋅2⁻⁶5⁻⁷ = 1.2566370614359173×10⁻⁶ [kg⋅m⋅C⁻²] SI1976

julia> electronmass(SI1976) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg] SI1976

julia> molarmass(SI1976) # kg⋅mol⁻¹
2⁻³5⁻³ = 0.001 [kg⋅mol⁻¹] SI1976

julia> luminousefficacy(SI1976) # lm⋅W⁻¹
Kcd = 683.01969009009 [kg⁻¹m⁻²s³lm] SI1976

MeasureSystems.Engineering — Constant

Engineering = MetricSystem(milli,𝟐*τ/𝟏𝟎^7,Rᵤ,g₀)
F=F, M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=A, R=𝟙, C=𝟙

Standard Metric Engineering system based on kilogram and kilopond (kilogram-force) units.

julia> boltzmann(Engineering) # kgf⋅m⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹2⁴5³ = 1.40787016925(43) × 10⁻²⁴ [kgf⋅m⋅K⁻¹] Engineering

julia> planckreduced(Engineering) # kgf⋅m⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹τ⁻¹ = 1.0753639802033891×10⁻³⁵ [kgf⋅m⋅s⋅rad⁻¹] Engineering

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

julia> vacuumpermeability(Engineering) # kgf⋅s²⋅C⁻²
g₀⁻¹τ⋅2⁻⁶5⁻⁷ = 1.2814131853751459×10⁻⁷ [kgf⋅s²C⁻²] Engineering

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

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

julia> luminousefficacy(Engineering) # lm⋅s⋅m⁻¹⋅kgf⁻¹
Kcd⋅g₀ = 6698.135043821981 [kgf⁻¹m⁻¹s⋅lm] Engineering

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

Additional Metric variants with angle scaling include MetricTurn, MetricSpatian, MetricGradian, MetricDegree, MetricArcminute, MetricArcsecond.

Historically, the josephson and klitzing constants have been used to define Conventional and CODATA variants.

MeasureSystems.Conventional — Constant

Conventional = ConventionalSystem(RK1990,KJ2014)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Conventional electronic UnitSystem with 1990 tuned josephson and klitzing constants.

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

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

julia> boltzmann(Conventional) # J⋅K⁻¹
kB⋅NA⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹RK90⁻¹KJ90⁻²2⁶5³ = 1.38064872956(43) × 10⁻²³ [J⋅K⁻¹] Conventional

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

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

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

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

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

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

MeasureSystems.CODATA — Constant

CODATA = ConventionalSystem(RK2014,KJ2014,Rᵤ2014)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Reference UnitSystem based on Committee on Data of the International Science Council.

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

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

julia> boltzmann(CODATA) # J⋅K⁻¹
𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹RK⁻¹KJ⁻²Rᵤ2014⋅2⁶5³ = 1.38064851(80) × 10⁻²³ [J⋅K⁻¹] CODATA

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

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

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

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

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

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

Originally, the practical units where specified by resistance and electricpotential.

MeasureSystems.International — Constant

International = ElectricSystem(Metric,Ωᵢₜ = 1.000495,Vᵢₜ = 1.00033)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

International UnitSystem with United States measurements of Ωᵢₜ and Vᵢₜ.

julia> resistance(International,Metric) # Ω⋅Ω⁻¹
Ωᵢₜ = 1.000495 [kg⋅m⋅s⁻²C⁻²]/[kg⋅m⋅s⁻²C⁻²] International -> Metric

julia> electricpotential(International,Metric) # V⋅V⁻¹
Vᵢₜ = 1.00033 [V⋅m⁻¹]/[V⋅m⁻¹] International -> Metric

julia> boltzmann(International) # J⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹Ωᵢₜ⋅Vᵢₜ⁻²2⁴5³ = 1.38042119247(42) × 10⁻²³ [J⋅K⁻¹] International

julia> planckreduced(International) # J⋅s⋅rad⁻¹
𝘩⋅Ωᵢₜ⋅Vᵢₜ⁻²τ⁻¹ = 1.0543978133151816×10⁻³⁴ [J⋅s] International

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

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

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

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

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

MeasureSystems.InternationalMean — Constant

InternationalMean = ElectricSystem(Metric,1.00049,1.00034)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

International UnitSystem with mean measurements of Ωᵢₜ and Vᵢₜ.

julia> resistance(InternationalMean,Metric) # Ω⋅Ω⁻¹
1.00049 = 1.00049 [kg⋅m⋅s⁻²C⁻²]/[kg⋅m⋅s⁻²C⁻²] InternationalMean -> Metric

julia> electricpotential(InternationalMean,Metric) # V⋅V⁻¹
1.00034 = 1.00034 [V⋅m⁻¹]/[V⋅m⁻¹] InternationalMean -> Metric

julia> boltzmann(InternationalMean) # J⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁴5³/1.0001900224889804 = 1.38038669501(42) × 10⁻²³ [J⋅K⁻¹] InternationalMean

julia> planckreduced(InternationalMean) # J⋅s⋅rad⁻¹
𝘩⋅τ⁻¹/1.0001900224889804 = 1.0543714633563797×10⁻³⁴ [J⋅s] InternationalMean

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

julia> vacuumpermeability(InternationalMean) # H⋅m⁻¹
τ⋅2⁻⁶5⁻⁷/1.00049 = 1.2560216108466024×10⁻⁶ [H⋅m⁻¹] InternationalMean

julia> electronmass(InternationalMean) # kg
𝘩⋅𝘤⁻¹R∞⋅α⁻²2/1.0001900224889804 = 9.1076530427(28) × 10⁻³¹ [kg] InternationalMean

julia> molarmass(InternationalMean) # kg⋅mol⁻¹
2⁻³5⁻³/1.0001900224889804 = 0.0009998100136127059 [kg⋅mol⁻¹] InternationalMean

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

Electromagnetic CGS Systems

Alternatives to the SI unit system are the centimetre-gram-second variants, where the constants are rescaled with centi meter and milli kilogram units along with introduction of additional rationalization and lorentz constants or electromagnetic units.

MeasureSystems.EMU — Constant

EMU = GaussSystem(Metric,𝟏,𝟐*τ)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L¹ᐟ², Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Centimetre-gram-second UnitSystem variant based on EMU (non-rationalized).

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

julia> planckreduced(EMU) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] EMU

julia> lightspeed(EMU) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] EMU

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

julia> electronmass(EMU) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] EMU

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

julia> luminousefficacy(EMU) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.8301969009009×10⁻⁵ [lm⋅s⋅erg⁻¹] EMU

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

MeasureSystems.ESU — Constant

ESU = GaussSystem(Metric,(𝟏𝟎*𝘤)^-2,𝟐*τ)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L³ᐟ²T⁻¹, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Centimetre-gram-second UnitSystem variant based on ESU (non-rationalized).

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

julia> planckreduced(ESU) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] ESU

julia> lightspeed(ESU) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] ESU

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

julia> electronmass(ESU) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] ESU

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

julia> luminousefficacy(ESU) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.8301969009009×10⁻⁵ [lm⋅s⋅erg⁻¹] ESU

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

MeasureSystems.Gauss — Constant

Gauss = GaussSystem(Metric,𝟏,𝟐*τ,𝟏𝟎^-2/𝘤)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L³ᐟ²T⁻¹, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=LT⁻¹

Centimetre-gram-second UnitSystem variant CGS (Gauss-Lorentz, non-rationalized).

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

julia> planckreduced(Gauss) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] Gauss

julia> lightspeed(Gauss) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] Gauss

julia> vacuumpermeability(Gauss) # statH⋅cm⁻¹
𝟏 = 1.0 [𝟙] Gauss

julia> electronmass(Gauss) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] Gauss

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

julia> luminousefficacy(Gauss) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.8301969009009×10⁻⁵ [lm⋅s⋅erg⁻¹] Gauss

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

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

MeasureSystems.LorentzHeaviside — Constant

LorentzHeaviside = GaussSystem(Metric,𝟏,𝟏,centi/𝘤)
F=MLT⁻², M=M, L=L, T=T, Q=M¹ᐟ²L³ᐟ²T⁻¹, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=LT⁻¹

Centimetre-gram-second UnitSystem variant HLU (Heaviside-Lorentz, rationalized).

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

julia> planckreduced(LorentzHeaviside) # erg⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁷5⁷ = 1.0545718176461565×10⁻²⁷ [erg⋅s] LorentzHeaviside

julia> lightspeed(LorentzHeaviside) # cm⋅s⁻¹
𝘤⋅2²5² = 2.99792458×10¹⁰ [cm⋅s⁻¹] LorentzHeaviside

julia> vacuumpermeability(HLU) # hlH⋅cm⁻¹
𝟏 = 1.0 [𝟙] LorentzHeaviside

julia> electronmass(LorentzHeaviside) # g
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁴5³ = 9.1093837016(28) × 10⁻²⁸ [g] LorentzHeaviside

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

julia> luminousefficacy(LorentzHeaviside) # lm⋅s⋅erg⁻¹
Kcd⋅2⁻⁷5⁻⁷ = 6.8301969009009×10⁻⁵ [lm⋅s⋅erg⁻¹] LorentzHeaviside

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

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

There are multiple choices of elctromagnetic units for these variants based on electromagnetic units, electrostatic units, Gaussian non-rationalized units, and Lorentz-Heaviside rationalized units. Note that CGS is an alias for the Gauss system.

Modified (Entropy) Unit Systems

Most other un-natural unit systems are derived from the construction above by rescaling time, length, mass, temperature, and gravity; which results in modified entropy constants:

MeasureSystems.Gravitational — Constant

Gravitational = EntropySystem(Metric,𝟏,𝟏,g₀)
F=F, M=FL⁻¹T², L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Standard Gravitational system based on hyl and kilopond units.

julia> boltzmann(Gravitational) # kgf⋅m⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹2⁴5³ = 1.40787016925(43) × 10⁻²⁴ [kgf⋅m⋅K⁻¹] Gravitational

julia> planckreduced(Gravitational) # kgf⋅m⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹τ⁻¹ = 1.0753639802033891×10⁻³⁵ [kgf⋅m⋅s] Gravitational

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

julia> vacuumpermeability(Gravitational) # H⋅m⁻¹
g₀⁻¹τ⋅2⁻⁶5⁻⁷ = 1.2814131853751459×10⁻⁷ [kgf⋅s²C⁻²] Gravitational

julia> electronmass(Gravitational) # hyl
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀⁻¹2 = 9.2889862507(28) × 10⁻³² [hyl] Gravitational

julia> molarmass(Gravitational) # hyl⋅mol⁻¹
g₀⁻¹2⁻³5⁻³ = 0.00010197162129779284 [kgf⋅m⁻¹s²mol⁻¹] Gravitational

julia> luminousefficacy(Gravitational) # lm⋅s⋅m⁻¹⋅kgf⁻¹
Kcd⋅g₀ = 6698.135043821981 [kgf⁻¹m⁻¹s⋅lm] Gravitational

MeasureSystems.MTS — Constant

MTS = EntropySystem(SI2019,𝟏,𝟏,kilo)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Metre-tonne-second UnitSystem variant of Metric system.

julia> boltzmann(MTS) # kJ⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2 = 1.38064899953(43) × 10⁻²⁶ [t⋅m²s⁻²K⁻¹] MTS

julia> planckreduced(MTS) # kJ⋅s⋅rad⁻¹
𝘩⋅τ⁻¹2⁻³5⁻³ = 1.0545718176461566×10⁻³⁷ [t⋅m²s⁻¹] MTS

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

julia> vacuumpermeability(MTS) # kH⋅m⁻¹
τ⋅2⁻⁹5⁻¹⁰ = 1.2566370614359174×10⁻⁹ [t⋅m⋅C⁻²] MTS

julia> electronmass(MTS) # t
𝘩⋅𝘤⁻¹R∞⋅α⁻²2⁻²5⁻³ = 9.1093837016(28) × 10⁻³⁴ [t] MTS

julia> molarmass(MTS) # t⋅mol⁻¹
2⁻⁶5⁻⁶ = 1.0×10⁻⁶ [t⋅mol⁻¹] MTS

julia> luminousefficacy(MTS) # lm⋅kW⁻¹
Kcd⋅2³5³ = 683019.6900900899 [t⁻¹m⁻²s³lm] MTS

MeasureSystems.KKH — Constant

KKH = EntropySystem(Metric,HOUR,kilo,𝟏)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Kilometer-kilogram-hour UnitSystem variant of Metric system.

julia> boltzmann(KKH) # kg⋅km²⋅h⁻²⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹2⁶3⁴5 = 1.78932110338(55) × 10⁻²² [kg⋅km²h⁻²K⁻¹] KKH

julia> planckreduced(KKH) # kg⋅km²⋅h⁻¹
𝘩⋅τ⁻¹2⁻²3²5⁻⁴ = 3.7964585435261634×10⁻³⁷ [kg⋅km²h⁻¹] KKH

julia> lightspeed(KKH) # km⋅hr⁻¹
𝘤⋅2⋅3²5⁻¹ = 1.0792528488×10⁹ [km⋅h⁻¹] KKH

julia> vacuumpermeability(KKH) # kg⋅km⋅C⁻²
τ⋅2⁻⁹5⁻¹⁰ = 1.2566370614359174×10⁻⁹ [kg⋅km⋅C⁻²] KKH

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

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

julia> luminousefficacy(KKH) # lm⋅h³⋅kg⁻¹⋅km⁻²
Kcd⋅2⁻⁶3⁻⁶ = 0.014639482383618185 [kg⁻¹km⁻²h³lm] KKH

MeasureSystems.MPH — Constant

MPH = EntropySystem(FPS,HOUR,mi,𝟏)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Mile-pound-hour specification based on FPS absolute UnitSystem.

julia> boltzmann(MPH) # lbf⋅mi²⋅hr⁻²⋅F⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹ft⁻²lb⁻¹2²5⁶11⁻² = 8.4615956484(26) × 10⁻²³ [lb⋅mi²h⁻²°R⁻¹] MPH

julia> planckreduced(MPH) # lbf⋅mi²⋅hr⁻¹⋅rad⁻¹
𝘩⋅ft⁻²lb⁻¹τ⁻¹2⁻⁶11⁻² = 3.2315817800735083×10⁻³⁷ [lb⋅mi²h⁻¹] MPH

julia> lightspeed(MPH) # mi⋅hr⁻¹
𝘤⋅ft⁻¹2⁻¹3⋅5⋅11⁻¹ = 6.706166293843951×10⁸ [mi⋅h⁻¹] MPH

julia> vacuumpermeability(MPH) # lbm⋅mi⋅C⁻²
ft⁻¹lb⁻¹τ⋅2⁻¹¹3⁻¹5⁻⁸11⁻¹ = 1.7214532710813804×10⁻⁹ [lb⋅mi⋅C⁻²] MPH

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

julia> molarmass(MPH) # lbm⋅lb-mol⁻¹
𝟏 = 1.0 [lb⋅lb-mol⁻¹] MPH

julia> luminousefficacy(MPH) # lm⋅h³⋅lb⁻¹⋅mi⁻²
Kcd⋅ft²lb⋅2⁻²3⁻⁴5⁻⁴11² = 0.017198446999173198 [lb⁻¹mi⁻²h³lm] MPH

MeasureSystems.Nautical — Constant

Nautical = EntropySystem(Metric,HOUR,nm,em^3,𝟏,τ*𝟑^3/𝟐^10/𝟓^12,milli)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Nautical miles, kilo-earthgram, hour specification based on Meridian definition.

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

julia> boltzmann(Nautical) # keg⋅nm²⋅hr⁻²⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁵2⁴⁹3¹⁰5³² = 5.180046618(26) × 10⁻²³ [keg⋅nm²h⁻²K⁻¹] Nautical

julia> planckreduced(Nautical) # keg⋅nm²⋅hr⁻¹⋅rad⁻¹
𝘩⋅g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁶2⁴¹3⁸5²⁷ = 1.0990666907(55) × 10⁻³⁷ [keg⋅nm²h⁻¹] Nautical

julia> lightspeed(Nautical) # nm⋅hr⁻¹
𝘤⋅g₀¹ᐟ²GME⁻¹ᐟ²τ⁻¹2⁹3⁵5⁴ = 5.8195383759(58) × 10⁸ [nm⋅h⁻¹] Nautical

julia> vacuumpermeability(Nautical) # keg⋅nm⋅eC⁻²
τ⋅2⁻¹⁰3³5⁻¹² = 6.785840131753954×10⁻¹⁰ [keg⋅nm⋅eC⁻²] Nautical

julia> electronmass(Nautical) # keg
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀³ᐟ²GME⁻³ᐟ²τ⁻³2²⁸5²¹ = 9.069925385(27) × 10⁻³¹ [keg] Nautical

julia> molarmass(Nautical) # keg⋅eg-mol⁻¹
2⁻³5⁻³ = 0.001 [keg⋅eg-mol⁻¹] Nautical

julia> luminousefficacy(Nautical) # lm⋅h³⋅keg⁻¹⋅nm⁻²
Kcd⋅g₀⁻⁵ᐟ²GME⁵ᐟ²τ⁵2⁻⁴⁹3⁻¹²5⁻³¹ = 0.05056853095(25) [keg⁻¹nm⁻²h³lm] Nautical

MeasureSystems.Meridian — Constant

Meridian = EntropySystem(Metric,𝟏,em,em^3,𝟏,τ/𝟐^6/𝟓^7,milli)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Modern ideal Meridian system defined by France's original earthmeter definition.

julia> greatcircle(Meridian) # em
2⁹5⁷ = 4.0×10⁷ [em] Meridian

julia> boltzmann(Meridian) # eJ⋅K⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁵2⁴⁹5³⁸ = 1.3706960050(69) × 10⁻²³ [eJ⋅K⁻¹] Meridian

julia> planckreduced(Meridian) # eJ⋅s⋅rad⁻¹
𝘩⋅g₀⁵ᐟ²GME⁻⁵ᐟ²τ⁻⁶2⁴⁵5³⁵ = 1.0469694890(53) × 10⁻³⁴ [eJ⋅s] Meridian

julia> lightspeed(Meridian) # em⋅s⁻¹
𝘤⋅g₀¹ᐟ²GME⁻¹ᐟ²τ⁻¹2⁹5⁷ = 2.9935896996(3) × 10⁸ [em⋅s⁻¹] Meridian

julia> vacuumpermeability(Meridian) # kegf⋅s²⋅eC⁻²
τ⋅2⁻⁶5⁻⁷ = 1.2566370614359173×10⁻⁶ [eH⋅em⁻¹] Meridian

julia> electronmass(Meridian) # keg
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀³ᐟ²GME⁻³ᐟ²τ⁻³2²⁸5²¹ = 9.069925385(27) × 10⁻³¹ [keg] Meridian

julia> molarmass(Meridian) # keg⋅eg-mol⁻¹
2⁻³5⁻³ = 0.001 [keg⋅eg-mol⁻¹] Meridian

julia> luminousefficacy(Meridian) # lm⋅W⁻¹
Kcd⋅g₀⁻⁵ᐟ²GME⁵ᐟ²τ⁵2⁻⁴⁵5⁻³⁵ = 687.9792808(35) [lm⋅eW⁻¹] Meridian

Foot-Pound-Second-Rankine

In Britain and the United States an English system of engineering units was commonly used.

MeasureSystems.FPS — Constant

FPS = RankineSystem(Metric,ft,lb)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Absolute English UnitSystem based on the foot, pound, second, and poundal.

julia> boltzmann(FPS) # ft⋅pdl⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹ft⁻²lb⁻¹2⁴3⁻²5⁴ = 1.82018324169(56) × 10⁻²² [lb⋅ft²s⁻²°R⁻¹] FPS

julia> planckreduced(FPS) # ft⋅pdl⋅s⋅rad⁻¹
𝘩⋅ft⁻²lb⁻¹τ⁻¹ = 2.5025369304889247×10⁻³³ [lb⋅ft²s⁻¹] FPS

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

julia> vacuumpermeability(FPS) # lb⋅ft⋅C⁻²
ft⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 9.089273271309688×10⁻⁶ [lb⋅ft⋅C⁻²] FPS

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

julia> molarmass(FPS) # lb⋅lb-mol⁻¹
𝟏 = 1.0 [lb⋅lb-mol⁻¹] FPS

julia> luminousefficacy(FPS) # lm⋅s³⋅lb⁻¹⋅ft⁻²
Kcd⋅ft²lb = 28.78252493663283 [lb⁻¹ft⁻²s³lm] FPS

MeasureSystems.IPS — Constant

IPS = RankineSystem(Metric,ft/𝟐^2/𝟑,lb*g₀*𝟐^2*𝟑/ft)
F=F, M=FL⁻¹T², L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

British Gravitational UnitSystem historically used in the United States of America.

julia> boltzmann(IPS) # in⋅lb⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ft⁻¹lb⁻¹2⁶3⁻¹5⁴ = 6.7887629566(21) × 10⁻²³ [lb⋅in⋅°R⁻¹] IPS

julia> planckreduced(IPS) # in⋅lb⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ft⁻¹lb⁻¹τ⁻¹2²3 = 9.333747076683978×10⁻³⁴ [lb⋅in⋅s] IPS

julia> lightspeed(IPS) # in⋅s⁻¹
𝘤⋅ft⁻¹2²3 = 1.1802852677165354×10¹⁰ [in⋅s⁻¹] IPS

julia> vacuumpermeability(IPS) # slinch⋅in⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.825032496413345×10⁻⁷ [lb⋅s²C⁻²] IPS

julia> electronmass(IPS) # slinch
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀⁻¹ft⋅lb⁻¹2⁻¹3⁻¹ = 5.2015921425(16) × 10⁻³³ [slinch] IPS

julia> molarmass(IPS) # slinch⋅slinch-mol⁻¹
𝟏 = 1.0 [lb⋅in⁻¹s²slinch-mol⁻¹] IPS

julia> luminousefficacy(IPS) # lm⋅s⋅in⁻¹⋅lb⁻¹
Kcd⋅g₀⋅ft⋅lb⋅2⁻²3⁻¹ = 77.17086290732456 [lb⁻¹in⁻¹s⋅lm] IPS

MeasureSystems.British — Constant

British = RankineSystem(Metric,ft,slug)
F=F, M=FL⁻¹T², L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

British Gravitational UnitSystem historically used by Britain and United States.

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

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

julia> vacuumpermeability(British) # slug⋅ft⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.825032496413345×10⁻⁷ [lb⋅s²C⁻²] British

julia> electronmass(British) # slugs
𝘩⋅𝘤⁻¹R∞⋅α⁻²g₀⁻¹ft⋅lb⁻¹2 = 6.2419105710(19) × 10⁻³² [slug] British

julia> molarmass(British) # slug⋅slug-mol⁻¹
𝟏 = 1.0 [lb⋅ft⁻¹s²slug-mol⁻¹] British

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

MeasureSystems.English — Constant

English = RankineSystem(Metric,ft,lb,g₀/ft)
F=F, M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=A, R=𝟙, C=𝟙

English Engineering UnitSystem historically used in the United States of America.

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

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

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

julia> vacuumpermeability(English) # lbm⋅ft⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.825032496413345×10⁻⁷ [lbf⋅s²C⁻²] English

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

julia> molarmass(English) # lbm⋅lb-mol⁻¹
𝟏 = 1.0 [lbm⋅lb-mol⁻¹] English

julia> luminousefficacy(English) # lm⋅s⋅ft⁻¹⋅lbf⁻¹
Kcd⋅g₀⋅ft⋅lb = 926.0503548878947 [lbf⁻¹ft⁻¹s⋅lm] English

julia> gravity(English) # lbm⋅ft⋅lbf⁻¹⋅s⁻²
g₀⋅ft⁻¹ = 32.17404855643044 [lbf⁻¹lbm⋅ft⋅s⁻²] English

MeasureSystems.Survey — Constant

Survey = RankineSystem(Metric,ftUS,lb,g₀/ftUS)
F=F, M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=A, R=𝟙, C=𝟙

English Engineering UnitSystem based on the geophysical US survey foot (1200/3937).

julia> boltzmann(Survey) # ftUS⋅lbf⋅°R⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹g₀⁻¹ftUS⁻¹lb⁻¹2⁴3⁻²5⁴ = 5.6572911492(17) × 10⁻²⁴ [lbf⋅ft⋅°R⁻¹] Survey

julia> planckreduced(Survey) # ftUS⋅lbf⋅s⋅rad⁻¹
𝘩⋅g₀⁻¹ftUS⁻¹lb⁻¹τ⁻¹ = 7.77810700765819×10⁻³⁵ [lbf⋅ft⋅s⋅rad⁻¹] Survey

julia> lightspeed(Survey) # ftUS⋅s⁻¹
𝘤⋅ftUS⁻¹ = 9.835690892883334×10⁸ [ft⋅s⁻¹] Survey

julia> vacuumpermeability(Survey) # lbm⋅ftUS⋅C⁻²
g₀⁻¹lb⁻¹τ⋅2⁻⁶5⁻⁷ = 2.825032496413345×10⁻⁷ [lbf⋅s²C⁻²] Survey

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

julia> molarmass(Survey) # lbm⋅lb-mol⁻¹
𝟏 = 1.0 [lbm⋅lb-mol⁻¹] Survey

julia> luminousefficacy(Survey) # lm⋅s⋅ft⁻¹⋅lbf⁻¹
Kcd⋅g₀⋅ftUS⋅lb = 926.0522069923087 [lbf⁻¹ft⁻¹s⋅lm] Survey

julia> gravity(Survey) # lbm⋅ftUS⋅lbf⁻¹⋅s⁻²
g₀⋅ftUS⁻¹ = 32.17398420833334 [lbf⁻¹lbm⋅ft⋅s⁻²] Survey

An impractical yet humorous unit system is the FFF specification.

MeasureSystems.FFF — Constant

FFF = EntropySystem(Metric,𝟐*𝟕*DAY,fur,𝟐*𝟑^2*𝟓*lb,°R,0,𝟏)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Furlong–firkin–fortnight FFF is a humorous UnitSystem based on unusal impractical units.

julia> boltzmann(FFF) # fir⋅fur²⋅ftn⁻²⋅F⁻¹
kB⋅NA⋅𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹ft⁻²lb⁻¹2¹⁵5⁵7²11⁻² = 6.7931043720(21) × 10⁻¹⁸ [fir⋅fur²ftn⁻²°R⁻¹] FFF

julia> planckreduced(FFF) # fir⋅fur²⋅ftn⁻¹⋅rad⁻¹
𝘩⋅ft⁻²lb⁻¹τ⁻¹2³3⁻¹5⁻¹7⋅11⁻² = 7.721326066522302×10⁻³⁵ [fir⋅fur²ftn⁻¹] FFF

julia> lightspeed(FFF) # fur⋅ftn⁻¹
𝘤⋅ft⁻¹2⁶3²5⋅7⋅11⁻¹ = 1.8026174997852542×10¹² [fur⋅ftn⁻¹] FFF

julia> vacuumpermeability(FFF) # fir⋅fur⋅Inf⁻²
𝟏/Inf = 0.0 [fir⋅fur⋅Inf⁻²] FFF

julia> electronmass(FFF) # fir
𝘩⋅𝘤⁻¹R∞⋅α⁻²lb⁻¹3⁻²5⁻¹ = 2.23141704217(68) × 10⁻³² [fir] FFF

julia> molarmass(FFF) # fir⋅fir-mol⁻¹
𝟏 = 1.0 [fir⋅fir-mol⁻¹] FFF

julia> luminousefficacy(FFF) # lm⋅ftn³⋅fir⁻¹⋅fur⁻²
Kcd⋅ft²lb⋅2⁻¹⁹3⁻⁵5⁻³7⁻³11² = 6.375788993269436×10⁻¹⁰ [fir⁻¹fur⁻²ftn³lm] FFF

Astronomical Unit Systems

The International Astronomical Union (IAU) units are based on the solar mass, distance from the sun to the earth, and the length of a terrestrial day.

MeasureSystems.IAU — Constant

IAU☉ = EntropySystem(Metric,DAY,au,GM☉/G)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Solar UnitSystem defined by International Astronomical Union and solarmass.

julia> boltzmann(IAU) # M⊙⋅au²⋅D⁻²⋅K⁻¹
kB⋅NA⋅𝘩²R∞⋅α⁻²μₑᵤ⁻¹au⁻⁵kG⁻²mP⁻²τ⁻³2⁴⁶3²⁰5¹⁷ = 2.316083(51) × 10⁻⁶⁶ [M☉⋅au²D⁻²K⁻¹] IAU☉

julia> planckreduced(IAU) # M⊙⋅au²⋅D⁻¹⋅rad⁻¹
𝘩²𝘤⋅au⁻⁵kG⁻²mP⁻²τ⁻⁴2³⁵3¹⁷5¹² = 2.047544(45) × 10⁻⁸² [M☉⋅au²D⁻¹] IAU☉

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

julia> vacuumpermeability(IAU) # M⊙⋅au²⋅C⁻²
𝘩⋅𝘤⋅au⁻⁴kG⁻²mP⁻²τ⁻²2²²3¹⁴5³ = 4.224533(93) × 10⁻⁴⁸ [M☉⋅au⋅C⁻²] IAU☉

julia> electronmass(IAU) # M⊙
𝘩²R∞⋅α⁻²au⁻³kG⁻²mP⁻²τ⁻³2²⁹3¹⁴5¹⁰ = 4.58124(10) × 10⁻⁶¹ [M☉] IAU☉

julia> molarmass(IAU) # M☉⋅mol⁻¹
𝘩⋅𝘤⋅au⁻³kG⁻²mP⁻²τ⁻³2²⁵3¹⁴5⁷ = 5.02915(11) × 10⁻³⁴ [M☉⋅mol⁻¹] IAU☉

julia> luminousefficacy(IAU) # lm⋅D³⋅M☉⁻¹⋅au⁻²
𝘩⁻¹𝘤⁻¹Kcd⋅au⁵kG²mP²τ³2⁻⁴⁹3⁻²³5⁻¹⁶ = 4.71247(10) × 10⁴⁰ [M☉⁻¹au⁻²D³lm] IAU☉

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

MeasureSystems.IAUE — Constant

IAUE = EntropySystem(Metric,DAY,LD,GME/G)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Astronomical (Earth) UnitSystem defined by lunardistance around the earthmass.

julia> boltzmann(IAUE) # ME⋅LD²⋅D⁻²⋅K⁻¹
kB⋅NA⋅𝘩²R∞⋅α⁻²μₑᵤ⁻¹mP⁻²GME⁻¹τ⁻¹2¹²5/202692169 = 1.167923(26) × 10⁻⁵⁵ [ME⋅LD²D⁻²K⁻¹] IAUE

julia> planckreduced(IAUE) # ME⋅LD²⋅D⁻¹⋅rad⁻¹
𝘩²𝘤⋅mP⁻²GME⁻¹τ⁻²2⋅3⁻³5⁻⁴/202692169 = 1.032508(23) × 10⁻⁷¹ [ME⋅LD²D⁻¹] IAUE

julia> lightspeed(IAUE) # LD⋅D⁻¹
𝘤⋅2⁴5⁻¹/14237 = 67383.2876027253 [LD⋅D⁻¹] IAUE

julia> vacuumpermeability(IAUE) # ME⋅LD²⋅C⁻²
𝘩⋅𝘤⋅mP⁻²GME⁻¹2⁻⁹3⁻³5⁻¹⁰/14237 = 5.47389(12) × 10⁻⁴⁰ [ME⋅LD⋅C⁻²] IAUE

julia> electronmass(IAUE) # ME
𝘩²R∞⋅α⁻²mP⁻²GME⁻¹τ⁻¹2 = 1.525306(34) × 10⁻⁵⁵ [ME] IAUE

julia> molarmass(IAUE) # ME⋅mol⁻¹
𝘩⋅𝘤⋅mP⁻²GME⁻¹τ⁻¹2⁻³5⁻³ = 1.674434(37) × 10⁻²⁸ [ME⋅mol⁻¹] IAUE

julia> luminousefficacy(IAUE) # lm⋅D³⋅ME⁻¹⋅LD⁻²
𝘩⁻¹𝘤⁻¹Kcd⋅mP²GME⋅τ⋅2⁻¹⁵3⁻³⋅202692169 = 9.34520(21) × 10²⁹ [ME⁻¹LD⁻²D³lm] IAUE

julia> turn(IAU)/gaussianmonth(IAU) # D⁻¹
GME¹ᐟ²2⁵ᐟ²3⁻³ᐟ²5⁻⁵ᐟ²/1.6987431854323947×10⁶ = 0.22888074402(23) [D⁻¹] IAU☉

MeasureSystems.IAUJ — Constant

IAUJ = EntropySystem(Metric,DAY,JD,GMJ/G)
F=MLT⁻², M=M, L=L, T=T, Q=Q, Θ=Θ, N=N, J=J, A=𝟙, R=𝟙, C=𝟙

Astronomical (Jupiter) UnitSystem defined by jupiterdistance around the solarmass.

julia> boltzmann(IAUJ) # MJ⋅JD²⋅D⁻²⋅K⁻¹
kB⋅NA⋅𝘩²R∞⋅α⁻²μₑᵤ⁻¹mP⁻²GMJ⁻¹τ⁻¹2⁶3⁴5⁻⁵/67336617049 = 8.95968(20) × 10⁻⁶⁵ [MJ⋅JD²D⁻²K⁻¹] IAUJ

julia> planckreduced(IAUJ) # MJ⋅JD²⋅D⁻¹⋅rad⁻¹
𝘩²𝘤⋅mP⁻²GMJ⁻¹τ⁻²2⁻⁵3⋅5⁻¹⁰/67336617049 = 7.92084(17) × 10⁻⁸¹ [MJ⋅JD²D⁻¹] IAUJ

julia> lightspeed(IAUJ) # JD⋅D⁻¹
𝘤⋅2⋅3²5⁻⁴/259493 = 33.272661653300865 [JD⋅D⁻¹] IAUJ

julia> vacuumpermeability(IAUJ) # MJ⋅JD²⋅C⁻²
𝘩⋅𝘤⋅mP⁻²GMJ⁻¹2⁻¹²3⁻¹5⁻¹³/259493 = 8.50430(19) × 10⁻⁴⁶ [MJ⋅JD⋅C⁻²] IAUJ

julia> electronmass(IAUJ) # MJ
𝘩²R∞⋅α⁻²mP⁻²GMJ⁻¹τ⁻¹2 = 4.79915(11) × 10⁻⁵⁸ [MJ] IAUJ

julia> molarmass(IAUJ) # MJ⋅mol⁻¹
𝘩⋅𝘤⋅mP⁻²GMJ⁻¹τ⁻¹2⁻³5⁻³ = 5.26836(12) × 10⁻³¹ [MJ⋅mol⁻¹] IAUJ

julia> luminousefficacy(IAUJ) # lm⋅D³⋅MJ⁻¹⋅JD⁻²
𝘩⁻¹𝘤⁻¹Kcd⋅mP²GMJ⋅τ⋅2⁻⁹3⁻⁷5⁶⋅67336617049 = 1.218177(27) × 10³⁹ [MJ⁻¹JD⁻²D³lm] IAUJ

julia> sqrt(gravitation(IAUJ)*solarmass(IAUJ)/jupiterdistance(IAUJ)^3) # D⁻¹
au³ᐟ²kG⋅τ⋅2⁻¹⁶3⁻¹¹ᐟ²5⁻¹²/1.3218691602384917×10⁸ = 0.001449102839405(44) [D⁻¹] IAUJ

MeasureSystems.Hubble — Constant

Hubble = AstronomicalSystem(Metric,th,𝘤*th,mₑ)
F=T⁻¹, M=𝟙, L=T, T, Q, Θ=𝟙, N=𝟙, J=T⁻¹, A=𝟙, R=𝟙, C=𝟙

Hubble UnitSystem defined by hubble parameter.

julia> boltzmann(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> planckreduced(Hubble)
𝘤⁻¹R∞⁻¹α²H0⋅au⁻¹2⁻¹¹3⁻⁴5⁻⁶ = 2.824(18) × 10⁻³⁹ [T] Hubble

julia> lightspeed(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> vacuumpermeability(Hubble)
τ⋅2 = 12.566370614359172 [TQ⁻²] Hubble

julia> electronmass(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> molarmass(Hubble)
𝟏 = 1.0 [𝟙] Hubble

julia> luminousefficacy(Hubble)
𝟏 = 1.0 [𝟙] Hubble

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

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

MeasureSystems.Cosmological — Constant

Cosmological = AstronomicalSystem(Metric,lc/𝘤,lc,mc)
F=MT⁻¹, M, L=T, T, Q, Θ=M, N=M, J, A=𝟙, R=𝟙, C=𝟙

Cosmological scale UnitSystem defined by darkenergydensity.

julia> boltzmann(Cosmological)
𝟏 = 1.0 [𝟙] Cosmological

julia> planckreduced(Cosmological)
𝘩²𝘤⁻⁴ΩΛ⋅H0²au⁻²mP⁻²2⁻²⁰3⁻⁷5⁻¹² = 2.888(43) × 10⁻¹²² [MT] Cosmological

julia> lightspeed(Cosmological)
𝟏 = 1.0 [𝟙] Cosmological

julia> vacuumpermeability(Cosmological)
τ⋅2 = 12.566370614359172 [MTQ⁻²] Cosmological

julia> electronmass(Cosmological)
𝘩²𝘤⁻³R∞⋅α⁻²ΩΛ¹ᐟ²H0⋅au⁻¹mP⁻²τ¹ᐟ²2⁻⁸3⁻⁷ᐟ²5⁻⁶ = 3.566(26) × 10⁻⁸³ [M] Cosmological

julia> molarmass(Cosmological)
𝟏 = 1.0 [𝟙] Cosmological

julia> luminousefficacy(Cosmological)
𝟏 = 1.0 [M⁻¹TJ] Cosmological

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

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

MeasureSystems.CosmologicalQuantum — Constant

CosmologicalQuantum = AstronomicalSystem(Metric,tcq,lcq,mcq)
F=M², M, L=M⁻¹, T=M⁻¹, Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Cosmological quantum scale UnitSystem defined by darkenergydensity.

julia> boltzmann(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> planckreduced(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> lightspeed(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> vacuumpermeability(CosmologicalQuantum)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁼²] CosmologicalQuantum

julia> electronmass(CosmologicalQuantum)
𝘩¹ᐟ²R∞⋅α⁻²ΩΛ⁻¹ᐟ⁴H0⁻¹ᐟ²au¹ᐟ²mP⁻¹ᐟ²τ¹ᐟ⁴2¹³ᐟ²3⁷ᐟ⁴5³ = 2.2733(84) × 10⁸ [M] CosmologicalQuantum

julia> molarmass(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

julia> luminousefficacy(CosmologicalQuantum)
𝟏 = 1.0 [𝟙] CosmologicalQuantum

Natural Unit Systems

With the introduction of the planckmass a set of natural atomic unit systems can be derived in terms of the gravitational coupling constant.

\[\alpha_G = \left(\frac{m_e}{m_P}\right)^2, \qquad \tilde k_B = 1, \qquad (\tilde M_u = 1, \quad \tilde \lambda = 1, \quad \tilde \alpha_L = 1)\]

julia> αG # (mₑ/mP)^2
𝘩²𝘤⁻²mP⁻²R∞²α⁻⁴2² = 1.75181e-45 ± 3.9e-50

Some of the notable variants include

Planck       ::UnitSystem{1,1,1,1,√(4π*αG)}
PlanckGauss  ::UnitSystem{1,1,1,4π,√αG}
Stoney       ::UnitSystem{1,αinv,1,4π,√(αG*αinv)}
Hartree      ::UnitSystem{1,1,αinv,4π/αinv^2,1}
Rydberg      ::UnitSystem{1,1,2αinv,π/αinv^2,1/2}
Schrodinger  ::UnitSystem{1,1,αinv,4π/αinv^2,√(αG*αinv)}
Electronic   ::UnitSystem{1,αinv,1,4π,1}
Natural      ::UnitSystem{1,1,1,1,1}
NaturalGauss ::UnitSystem{1,1,1,4π,1}
QCD          ::UnitSystem{1,1,1,1,1/μₚₑ}
QCDGauss     ::UnitSystem{1,1,1,4π,1/μₚₑ}
QCDoriginal  ::UnitSystem{1,1,1,4π/αinv,1/μₚₑ}

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 1, \qquad \tilde m_e = \sqrt{4\pi \alpha_G}\]

MeasureSystems.Planck — Constant

Planck = UnitSystem(𝟏,𝟏,𝟏,𝟏,√(𝟐*τ*αG))
F=M², M, L=M⁻¹, T=M⁻¹, Q=𝟙, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Planck UnitSystem with the electronmass value √(4π*αG) using gravitational coupling.

julia> boltzmann(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> planckreduced(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> lightspeed(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> vacuumpermeability(Planck)
𝟏 = 1.0 [𝟙] Planck

julia> electronmass(Planck)
𝘩⋅𝘤⁻¹R∞⋅α⁻²mP⁻¹τ¹ᐟ²2³ᐟ² = 1.483708(16) × 10⁻²² [M] Planck

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 4\pi, \qquad \tilde m_e = \sqrt{4\pi \alpha_G}\]

MeasureSystems.PlanckGauss — Constant

PlanckGauss = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ,√αG)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Planck (Gauss) UnitSystem with permeability of 4π and electronmass coupling √αG.

julia> boltzmann(PlanckGauss)
𝟏 = 1.0 [𝟙] PlanckGauss

julia> planckreduced(PlanckGauss)
𝟏 = 1.0 [𝟙] PlanckGauss

julia> lightspeed(PlanckGauss)
𝟏 = 1.0 [𝟙] PlanckGauss

julia> vacuumpermeability(PlanckGauss)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁻²] PlanckGauss

julia> electronmass(PlanckGauss)
𝘩⋅𝘤⁻¹R∞⋅α⁻²mP⁻¹2 = 4.185463(46) × 10⁻²³ [mP] PlanckGauss

The well known PlanckGauss values for length, time, mass, and temperature are:

julia> length(PlanckGauss,SI2019) # ℓP
𝘩⋅𝘤⁻¹mP⁻¹τ⁻¹ = 1.616255(18) × 10⁻³⁵ [m]/[mP⁻¹] PlanckGauss -> SI2019

julia> time(PlanckGauss,SI2019) # tP
𝘩⋅𝘤⁻²mP⁻¹τ⁻¹ = 5.391247(59) × 10⁻⁴⁴ [s]/[mP⁻¹] PlanckGauss -> SI2019

julia> mass(PlanckGauss,SI2019) # mP
mP = 2.176434(24) × 10⁻⁸ [kg]/[mP] PlanckGauss -> SI2019

julia> temperature(PlanckGauss,SI2019) # TP
kB⁻¹𝘤²mP = 1.416784(16) × 10³² [K]/[mP] PlanckGauss -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = \frac1\alpha, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 4\pi, \qquad \tilde m_e = \sqrt{\frac{\alpha_G}{\alpha}}\]

MeasureSystems.Stoney — Constant

Stoney = UnitSystem(𝟏,𝟏/α,𝟏,𝟐*τ,√(αG/α))
F=MT⁻¹, M, L=T, T, Q, Θ=M, N=M, J, A=𝟙, R=𝟙, C=𝟙

Stoney UnitSystem with permeability of 4π and electronmass coupling √(αG/α).

julia> boltzmann(Stoney)
𝟏 = 1.0 [𝟙] Stoney

julia> planckreduced(Stoney)
α⁻¹ = 137.035999084(21) [MT] Stoney

julia> lightspeed(Stoney)
𝟏 = 1.0 [𝟙] Stoney

julia> vacuumpermeability(Stoney)
τ⋅2 = 12.566370614359172 [MTQ⁻²] Stoney

julia> electronmass(Stoney)
𝘩⋅𝘤⁻¹R∞⋅α⁻⁵ᐟ²mP⁻¹2 = 4.899602(54) × 10⁻²² [M] Stoney

The well known Stoney values for length, time, mass, and charge are:

julia> length(Stoney,SI2019) # lS
𝘩⋅𝘤⁻¹α¹ᐟ²mP⁻¹τ⁻¹ = 1.380679(15) × 10⁻³⁶ [m]/[T] Stoney -> SI2019

julia> time(Stoney,SI2019) # tS
𝘩⋅𝘤⁻²α¹ᐟ²mP⁻¹τ⁻¹ = 4.605448(51) × 10⁻⁴⁵ [s]/[T] Stoney -> SI2019

julia> mass(Stoney,SI2019) # mS
α¹ᐟ²mP = 1.859209(21) × 10⁻⁹ [kg]/[M] Stoney -> SI2019

julia> charge(Stoney,SI2019) # qS
𝘦 = 1.602176634×10⁻¹⁹ [C]/[𝘦] Stoney -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = \frac1\alpha, \qquad \tilde \mu_0 = 4\pi\alpha^2, \qquad \tilde m_e = 1\]

MeasureSystems.Hartree — Constant

Hartree = UnitSystem(𝟏,𝟏,𝟏/α,𝟐*τ*α^2,𝟏)
F=L⁻³, M=𝟙, L=L, T=L², Q=Q, Θ=L⁻², N=𝟙, J=L⁻⁴, A=𝟙, R=𝟙, C=𝟙

Hartree atomic UnitSystem based on bohr radius and elementarycharge scale.

julia> boltzmann(Hartree)
𝟏 = 1.0 [𝟙] Hartree

julia> planckreduced(Hartree)
𝟏 = 1.0 [𝟙] Hartree

julia> lightspeed(Hartree)
α⁻¹ = 137.035999084(21) [a₀⁻¹] Hartree

julia> vacuumpermeability(Hartree)
α²τ⋅2 = 0.00066917625662(21) [a₀⋅𝘦⁻²] Hartree

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

The well known Hartree atomic unit values for length, time, mass, and charge are:

julia> length(Hartree,SI2019) # lA
R∞⁻¹α⋅τ⁻¹2⁻¹ = 5.29177210902(81) × 10⁻¹¹ [m]/[a₀] Hartree -> SI2019

julia> time(Hartree,SI2019) # tA
𝘤⁻¹R∞⁻¹τ⁻¹2⁻¹ = 2.4188843265857(46) × 10⁻¹⁷ [s]/[a₀²] Hartree -> SI2019

julia> mass(Hartree,SI2019) # mA
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg]/[𝟙] Hartree -> SI2019

julia> charge(Hartree,SI2019) # qA
𝘦 = 1.602176634×10⁻¹⁹ [C]/[𝘦] Hartree -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = \frac2\alpha, \qquad \tilde \mu_0 = \pi\alpha^2, \qquad \tilde m_e = \frac{1}{2}\]

MeasureSystems.Rydberg — Constant

Rydberg = UnitSystem(𝟏,𝟏,𝟐/α,τ/𝟐*α^2,𝟏/𝟐)
F=MLT⁻², M, L, T, Q, Θ=T⁻¹, N=M, J=T², A=𝟙, R=𝟙, C=𝟙

Rydberg UnitSystem with lightspeed of 𝟐/α and permeability of π*α^2.

julia> boltzmann(Rydberg)
𝟏 = 1.0 [ML²T⁻¹] Rydberg

julia> planckreduced(Rydberg)
𝟏 = 1.0 [ML²T⁻¹] Rydberg

julia> lightspeed(Rydberg)
α⁻¹2 = 274.071998168(42) [LT⁻¹] Rydberg

julia> vacuumpermeability(Rydberg)
α²τ⋅2⁻¹ = 0.000167294064155(51) [MLQ⁻²] Rydberg

julia> electronmass(Rydberg)
2⁻¹ = 0.5 [M] Rydberg

The well known Rydberg atomic unit values for length, time, mass, and charge are:

julia> length(Rydberg,SI2019) # lR
R∞⁻¹α⋅τ⁻¹2⁻¹ = 5.29177210902(81) × 10⁻¹¹ [m]/[a₀] Rydberg -> SI2019

julia> time(Rydberg,SI2019) # tR
𝘤⁻¹R∞⁻¹τ⁻¹ = 4.8377686531713(93) × 10⁻¹⁷ [s]/[T] Rydberg -> SI2019

julia> mass(Rydberg,SI2019) # mR
𝘩⋅𝘤⁻¹R∞⋅α⁻²2² = 1.82187674031(56) × 10⁻³⁰ [kg]/[M] Rydberg -> SI2019

julia> charge(Rydberg,SI2019) # qR
𝘦⋅2⁻¹ᐟ² = 1.1329099625600371×10⁻¹⁹ [C]/[Q] Rydberg -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = \frac1\alpha, \qquad \tilde \mu_0 = 4\pi\alpha^2, \qquad \tilde m_e = \sqrt{\frac{\alpha_G}{\alpha}}\]

MeasureSystems.Schrodinger — Constant

Schrodinger = UnitSystem(𝟏,𝟏,𝟏/α,𝟐*τ*α^2,√(αG/α))
F=MLT⁻², M, L, T, Q, Θ=T⁻¹, N=M, J=T², A=𝟙, R=𝟙, C=𝟙

Schrodinger UnitSystem with permeability of 4π/αinv^2 and electronmass of √(αG*αinv).

julia> boltzmann(Schrodinger)
𝟏 = 1.0 [ML²T⁻¹] Schrodinger

julia> planckreduced(Schrodinger)
𝟏 = 1.0 [ML²T⁻¹] Schrodinger

julia> lightspeed(Schrodinger)
α⁻¹ = 137.035999084(21) [LT⁻¹] Schrodinger

julia> vacuumpermeability(Schrodinger)
α²τ⋅2 = 0.00066917625662(21) [MLQ⁻²] Schrodinger

julia> electronmass(Schrodinger)
𝘩⋅𝘤⁻¹R∞⋅α⁻⁵ᐟ²mP⁻¹2 = 4.899602(54) × 10⁻²² [M] Schrodinger

\[\tilde k_B = 1, \qquad \tilde \hbar = \frac1\alpha, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 4\pi, \qquad \tilde m_e = 1\]

MeasureSystems.Electronic — Constant

Electronic = UnitSystem(𝟏,𝟏/α,𝟏,𝟐*τ,𝟏)
F=T⁻¹, M=𝟙, L=T, T, Q, Θ=𝟙, N=𝟙, J=T⁻¹, A=𝟙, R=𝟙, C=𝟙

Electronic UnitSystem with planckreduced of 1/α and permeability of 4π.

julia> boltzmann(Electronic)
𝟏 = 1.0 [𝟙] Electronic

julia> planckreduced(Electronic)
α⁻¹ = 137.035999084(21) [T] Electronic

julia> lightspeed(Electronic)
𝟏 = 1.0 [𝟙] Electronic

julia> vacuumpermeability(Electronic)
τ⋅2 = 12.566370614359172 [TQ⁻²] Electronic

julia> electronmass(Electronic)
𝟏 = 1.0 [𝟙] Electronic

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 1, \qquad \tilde m_e = 1\]

MeasureSystems.Natural — Constant

Natural = UnitSystem(𝟏,𝟏,𝟏,𝟏,𝟏)
F=𝟙, M=𝟙, L=𝟙, T=𝟙, Q=𝟙, Θ=𝟙, N=𝟙, J=𝟙, A=𝟙, R=𝟙, C=𝟙

Natural UnitSystem with all primary constants having unit value.

julia> boltzmann(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> planckreduced(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> lightspeed(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> vacuumpermeability(Natural)
𝟏 = 1.0 [𝟙] Natural

julia> electronmass(Natural)
𝟏 = 1.0 [𝟙] Natural

The well known Natural values for length, time, mass, and charge are:

julia> length(Natural,SI2019)
R∞⁻¹α²τ⁻¹2⁻¹ = 3.8615926796(12) × 10⁻¹³ [m]/[𝟙] Natural -> SI2019

julia> time(Natural,SI2019)
𝘤⁻¹R∞⁻¹α²τ⁻¹2⁻¹ = 1.28808866819(39) × 10⁻²¹ [s]/[𝟙] Natural -> SI2019

julia> mass(Natural,SI2019)
𝘩⋅𝘤⁻¹R∞⋅α⁻²2 = 9.1093837016(28) × 10⁻³¹ [kg]/[𝟙] Natural -> SI2019

julia> charge(Natural,SI2019)
𝘦⋅α⁻¹ᐟ²τ⁻¹ᐟ²2⁻¹ᐟ² = 5.29081768990(41) × 10⁻¹⁹ [C]/[𝟙] Natural -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 4\pi, \qquad \tilde m_e = 1\]

MeasureSystems.NaturalGauss — Constant

NaturalGauss = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ,𝟏)
F=𝟙, M=𝟙, L=𝟙, T=𝟙, Q=Q, Θ=𝟙, N=𝟙, J=𝟙, A=𝟙, R=𝟙, C=𝟙

Natural (Gauss) UnitSystem with the Gaussian permeability value of 4π.

julia> boltzmann(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

julia> planckreduced(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

julia> lightspeed(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

julia> vacuumpermeability(NaturalGauss)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁻²] NaturalGauss

julia> electronmass(NaturalGauss)
𝟏 = 1.0 [𝟙] NaturalGauss

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 1, \qquad \tilde m_e = \frac1{\mu_{pe}} = \frac{m_e}{m_p}\]

MeasureSystems.QCD — Constant

QCD = UnitSystem(𝟏,𝟏,𝟏,𝟏,𝟏/μₚₑ)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=𝟙, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Qunatum chromodynamics UnitSystem based on the protonmass scale.

julia> boltzmann(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> planckreduced(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> lightspeed(QCD)
𝟏 = 1.0 [𝟙] QCD

julia> vacuumpermeability(QCD)
𝟏 = 1.0 [𝟙] QCD

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

The well known QCD values for length, time, mass, and charge are:

julia> length(QCD,SI2019) # lQCD
R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 2.10308910335(66) × 10⁻¹⁶ [m]/[mₚ⁻¹] QCD -> SI2019

julia> time(QCD,SI2019) # tQCD
𝘤⁻¹R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 7.0151501388(22) × 10⁻²⁵ [s]/[mₚ⁻¹] QCD -> SI2019

julia> mass(QCD,SI2019) # mQCD
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg]/[mₚ] QCD -> SI2019

julia> charge(QCD,SI2019) # qQCD
𝘦⋅α⁻¹ᐟ²τ⁻¹ᐟ²2⁻¹ᐟ² = 5.29081768990(41) × 10⁻¹⁹ [C]/[𝟙] QCD -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 4\pi, \qquad \tilde m_e = \frac1{\mu_{pe}} = \frac{m_e}{m_p}\]

MeasureSystems.QCDGauss — Constant

QCDGauss = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ,𝟏/μₚₑ)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Qunatum chromodynamics (Gauss) UnitSystem based on the protonmass scale.

julia> boltzmann(QCDGauss)
𝟏 = 1.0 [𝟙] QCDGauss

julia> planckreduced(QCDGauss)
𝟏 = 1.0 [𝟙] QCDGauss

julia> lightspeed(QCDGauss)
𝟏 = 1.0 [𝟙] QCDGauss

julia> vacuumpermeability(QCDGauss)
τ⋅2 = 12.566370614359172 [𝘦ₙ⁻²] QCDGauss

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

The well known QCDGauss values for length, time, mass, and charge are:

julia> length(QCDGauss,SI2019) # lQCD
R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 2.10308910335(66) × 10⁻¹⁶ [m]/[mₚ⁻¹] QCDGauss -> SI2019

julia> time(QCDGauss,SI2019) # tQCD
𝘤⁻¹R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 7.0151501388(22) × 10⁻²⁵ [s]/[mₚ⁻¹] QCDGauss -> SI2019

julia> mass(QCDGauss,SI2019) # mQCD
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg]/[mₚ] QCDGauss -> SI2019

julia> charge(QCDGauss,SI2019) # qQCD
𝘦⋅α⁻¹ᐟ² = 1.87554603778(14) × 10⁻¹⁸ [C]/[𝘦ₙ] QCDGauss -> SI2019

\[\tilde k_B = 1, \qquad \tilde \hbar = 1, \qquad \tilde c = 1, \qquad \tilde \mu_0 = 4\pi\alpha, \qquad \tilde m_e = \frac1{\mu_{pe}} = \frac{m_e}{m_p}\]

MeasureSystems.QCDoriginal — Constant

QCDoriginal = UnitSystem(𝟏,𝟏,𝟏,𝟐*τ*α,𝟏/μₚₑ)
F=M², M=M, L=M⁻¹, T=M⁻¹, Q=Q, Θ=M, N=M, J=M², A=𝟙, R=𝟙, C=𝟙

Qunatum chromodynamics (original) UnitSystem scaled by protonmass and elementarycharge.

julia> boltzmann(QCDoriginal)
𝟏 = 1.0 [𝟙] QCDoriginal

julia> planckreduced(QCDoriginal)
𝟏 = 1.0 [𝟙] QCDoriginal

julia> lightspeed(QCDoriginal)
𝟏 = 1.0 [𝟙] QCDoriginal

julia> vacuumpermeability(QCDoriginal)
α⋅τ⋅2 = 0.091701236889(14) [𝘦⁻²] QCDoriginal

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

The well known QCDoriginal values for length, time, mass, and charge are:

julia> length(QCDoriginal,SI2019) # lQCD
R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 2.10308910335(66) × 10⁻¹⁶ [m]/[mₚ⁻¹] QCDoriginal -> SI2019

julia> time(QCDoriginal,SI2019) # tQCD
𝘤⁻¹R∞⁻¹α²μₑᵤ⋅μₚᵤ⁻¹τ⁻¹2⁻¹ = 7.0151501388(22) × 10⁻²⁵ [s]/[mₚ⁻¹] QCDoriginal -> SI2019

julia> mass(QCDoriginal,SI2019) # mQCD
𝘩⋅𝘤⁻¹R∞⋅α⁻²μₑᵤ⁻¹μₚᵤ⋅2 = 1.67262192369(52) × 10⁻²⁷ [kg]/[mₚ] QCDoriginal -> SI2019

julia> charge(QCDoriginal,SI2019) # qQCD
𝘦 = 1.602176634×10⁻¹⁹ [C]/[𝘦] QCDoriginal -> SI2019