Appendix: Physical and Scientific Constants
Values are from CODATA 2022 (published May 2024 by NIST). After the 2019 SI redefinition, seven constants are exact by definition; many others become exact because they derive purely from those seven. Constants marked (exact) have zero measurement uncertainty.
Sources: NIST Fundamental Physical Constants, Frink units database, Wikipedia: List of physical constants.
Notation
299_792_458 m/s Int quantity — exact integer value with unit
8.314462618 J/mol·K Decimal quantity — exact non-integer value (arbitrary precision)
~6.674e-11 m³/kg·s² Float quantity — approximate/measured value (IEEE 754 double)The ~ prefix selects hardware Float, which supports e notation for scientific notation. Measured constants should use Float since experimental uncertainty dwarfs floating-point rounding error. Exact constants use Int or Decimal to preserve full precision.
Constants already available as built-in variables are shown with their variable name (e.g. c, ℎ, Nₐ).
1. Defining Constants of the SI
The 2019 SI redefinition fixed these seven constants to exact numerical values. All SI units derive from them.
| Constant | Symbol | Value | SI Unit | Tungsten |
|---|---|---|---|---|
| Speed of light in vacuum | c | 299 792 458 | m·s⁻¹ | c = 299_792_458 m/s |
| Planck constant | h | 6.626 070 15 × 10⁻³⁴ | J·Hz⁻¹ | ℎ = ~6.62607015e-34 J*s |
| Elementary charge | e | 1.602 176 634 × 10⁻¹⁹ | C | e₀ = ~1.602176634e-19 C |
| Boltzmann constant | k | 1.380 649 × 10⁻²³ | J·K⁻¹ | kB = ~1.380649e-23 J/K |
| Avogadro constant | N_A | 6.022 140 76 × 10²³ | mol⁻¹ | Nₐ = ~6.02214076e23 mol⁻¹ |
| Luminous efficacy of 540 THz radiation | K_cd | 683 | lm·W⁻¹ | 683 lm/W |
| Hyperfine transition frequency of ¹³³Cs | Δν_Cs | 9 192 631 770 | Hz | 9_192_631_770 Hz |
2. Universal Constants
| Constant | Symbol | Value | SI Unit | Rel. Uncertainty | Tungsten |
|---|---|---|---|---|---|
| Newtonian constant of gravitation | G | 6.674 30(15) × 10⁻¹¹ | m³·kg⁻¹·s⁻² | 2.2 × 10⁻⁵ | G = ~6.67430e-11 m³/kg·s² |
| Reduced Planck constant | ℏ | 1.054 571 817... × 10⁻³⁴ | J·s | exact (h/2π) | ℏ = ~1.054571817e-34 J*s |
| Planck mass | m_P | 2.176 434(24) × 10⁻⁸ | kg | 1.1 × 10⁻⁵ | ~2.176434e-8 kg |
| Planck length | l_P | 1.616 255(18) × 10⁻³⁵ | m | 1.1 × 10⁻⁵ | ~1.616255e-35 m |
| Planck time | t_P | 5.391 247(60) × 10⁻⁴⁴ | s | 1.1 × 10⁻⁵ | ~5.391247e-44 s |
| Planck temperature | T_P | 1.416 784(16) × 10³² | K | 1.1 × 10⁻⁵ | ~1.416784e32 K |
3. Electromagnetic Constants
| Constant | Symbol | Value | SI Unit | Rel. Uncertainty | Tungsten |
|---|---|---|---|---|---|
| Vacuum electric permittivity | ε₀ | 8.854 187 8188(14) × 10⁻¹² | F·m⁻¹ | 1.6 × 10⁻¹⁰ | ε₀ = ~8.8541878188e-12 F/m |
| Vacuum magnetic permeability | μ₀ | 1.256 637 061 27(20) × 10⁻⁶ | N·A⁻² | 1.6 × 10⁻¹⁰ | μ₀ = ~1.25663706127e-6 H/m |
| Characteristic impedance of vacuum | Z₀ | 376.730 313 412(59) | Ω | 1.6 × 10⁻¹⁰ | ~376.730313412 Ω |
| Fine-structure constant | α | 7.297 352 5643(11) × 10⁻³ | (dimensionless) | 1.5 × 10⁻¹⁰ | α = ~7.2973525643e-3 |
| Inverse fine-structure constant | α⁻¹ | 137.035 999 177(21) | (dimensionless) | 1.5 × 10⁻¹⁰ | ~137.035999177 |
| Josephson constant | K_J | 483 597.848 4... × 10⁹ | Hz·V⁻¹ | exact (2e/h) | ~4.835978484e14 Hz/V |
| Von Klitzing constant | R_K | 25 812.807 45... | Ω | exact (h/e²) | ~25812.80745 Ω |
| Bohr magneton | μ_B | 9.274 010 0657(29) × 10⁻²⁴ | J·T⁻¹ | 3.1 × 10⁻¹⁰ | ~9.2740100657e-24 J/T |
| Nuclear magneton | μ_N | 5.050 783 7393(16) × 10⁻²⁷ | J·T⁻¹ | 3.1 × 10⁻¹⁰ | ~5.0507837393e-27 J/T |
| Electron magnetic moment | μ_e | −9.284 764 7043(28) × 10⁻²⁴ | J·T⁻¹ | 3.0 × 10⁻¹⁰ | ~-9.2847647043e-24 J/T |
| Proton magnetic moment | μ_p | 1.410 606 797 36(60) × 10⁻²⁶ | J·T⁻¹ | 4.2 × 10⁻¹⁰ | ~1.41060679736e-26 J/T |
| Neutron magnetic moment | μ_n | −9.662 3653(23) × 10⁻²⁷ | J·T⁻¹ | 2.4 × 10⁻⁷ | ~-9.6623653e-27 J/T |
| Muon magnetic moment | μ_μ | −4.490 448 30(10) × 10⁻²⁶ | J·T⁻¹ | 2.2 × 10⁻⁸ | ~-4.49044830e-26 J/T |
4. Atomic and Nuclear Constants
| Constant | Symbol | Value | SI Unit | Rel. Uncertainty | Tungsten |
|---|---|---|---|---|---|
| Electron mass | m_e | 9.109 383 7139(28) × 10⁻³¹ | kg | 3.1 × 10⁻¹⁰ | mₑ = ~9.1093837139e-31 kg |
| Proton mass | m_p | 1.672 621 925 95(52) × 10⁻²⁷ | kg | 3.1 × 10⁻¹⁰ | mₚ = ~1.67262192595e-27 kg |
| Neutron mass | m_n | 1.674 927 500 56(85) × 10⁻²⁷ | kg | 5.1 × 10⁻¹⁰ | ~1.67492750056e-27 kg |
| Muon mass | m_μ | 1.883 531 627(42) × 10⁻²⁸ | kg | 2.2 × 10⁻⁸ | ~1.883531627e-28 kg |
| Tau mass | m_τ | 3.167 54(21) × 10⁻²⁷ | kg | 6.8 × 10⁻⁵ | ~3.16754e-27 kg |
| Atomic mass constant | m_u | 1.660 539 068 92(52) × 10⁻²⁷ | kg | 3.1 × 10⁻¹⁰ | ~1.66053906892e-27 kg |
| Proton-electron mass ratio | m_p/m_e | 1 836.152 673 426(32) | (dimensionless) | 1.7 × 10⁻¹¹ | ~1836.152673426 |
| Bohr radius | a₀ | 5.291 772 105 44(82) × 10⁻¹¹ | m | 1.6 × 10⁻¹⁰ | a₀ = ~5.29177210544e-11 m |
| Rydberg constant | R_∞ | 10 973 731.568 157(12) | m⁻¹ | 1.1 × 10⁻¹² | ~10973731.568157 m⁻¹ |
| Hartree energy | E_h | 4.359 744 722 2(60) × 10⁻¹⁸ | J | 1.4 × 10⁻¹⁰ | ~4.3597447222e-18 J |
| Classical electron radius | r_e | 2.817 940 3205(13) × 10⁻¹⁵ | m | 4.7 × 10⁻¹⁰ | ~2.8179403205e-15 m |
| Compton wavelength (electron) | λ_C | 2.426 310 238 67(73) × 10⁻¹² | m | 3.0 × 10⁻¹⁰ | ~2.42631023867e-12 m |
| Compton wavelength (proton) | λ_C,p | 1.321 409 855 39(40) × 10⁻¹⁵ | m | 3.1 × 10⁻¹⁰ | ~1.32140985539e-15 m |
| Compton wavelength (neutron) | λ_C,n | 1.319 590 905 82(75) × 10⁻¹⁵ | m | 5.7 × 10⁻¹⁰ | ~1.31959090582e-15 m |
| Thomson cross section | σ_e | 6.652 458 7051(62) × 10⁻²⁹ | m² | 9.3 × 10⁻¹⁰ | ~6.6524587051e-29 m² |
5. Thermodynamic and Physico-Chemical Constants
| Constant | Symbol | Value | SI Unit | Rel. Uncertainty | Tungsten |
|---|---|---|---|---|---|
| Molar gas constant | R | 8.314 462 618... | J·mol⁻¹·K⁻¹ | exact (N_A · k) | R = 8.314462618 J/mol·K |
| Faraday constant | F | 96 485.332 12... | C·mol⁻¹ | exact (N_A · e) | 𝐹 = 96_485.33212 C/mol |
| Stefan-Boltzmann constant | σ | 5.670 374 419... × 10⁻⁸ | W·m⁻²·K⁻⁴ | exact (derived) | σ = ~5.670374419e-8 W/m²·K⁴ |
| Wien displacement law constant | b | 2.897 771 955... × 10⁻³ | m·K | exact (derived) | ~2.897771955e-3 m·K |
| First radiation constant | c₁ | 3.741 771 852... × 10⁻¹⁶ | W·m² | exact (derived) | ~3.741771852e-16 W·m² |
| Second radiation constant | c₂ | 1.438 776 877... × 10⁻² | m·K | exact (derived) | ~1.438776877e-2 m·K |
| Loschmidt constant (273.15 K, 101.325 kPa) | n₀ | 2.686 780 111... × 10²⁵ | m⁻³ | exact (derived) | ~2.686780111e25 m⁻³ |
| Molar volume of ideal gas (STP) | V_m | 22.413 969 54... × 10⁻³ | m³·mol⁻¹ | exact (derived) | 0.02241396954 m³/mol |
6. Adopted and Conventional Constants
| Constant | Symbol | Value | SI Unit | Tungsten |
|---|---|---|---|---|
| Standard acceleration of gravity | g_n | 9.806 65 | m·s⁻² | g₀ = 9.80665 m/s² |
| Standard atmosphere | atm | 101 325 | Pa | 101_325 Pa or 1 atm |
| Standard-state pressure | p° | 100 000 | Pa | 100_000 Pa |
| Molar mass of carbon-12 | M(¹²C) | 12 × 10⁻³ | kg·mol⁻¹ | 0.012 kg/mol |
| Conventional Josephson constant | K_J-90 | 483 597.9 × 10⁹ | Hz·V⁻¹ | ~483597.9e9 Hz/V |
| Conventional von Klitzing constant | R_K-90 | 25 812.807 | Ω | 25_812.807 Ω |
7. Astronomical Constants
| Constant | Symbol | Value | SI Unit | Tungsten |
|---|---|---|---|---|
| Astronomical unit | au | 149 597 870 700 | m | 149_597_870_700 m or 1 au |
| Light-year | ly | 9.460 730 472 5808 × 10¹⁵ | m | 1 ly |
| Parsec | pc | 3.085 677 581 4914 × 10¹⁶ | m | 1 pc |
| Solar mass | M_⊙ | 1.988 92 × 10³⁰ | kg | 1 solarmass |
| Earth mass | M_⊕ | 5.9722 × 10²⁴ | kg | 1 earthmass |
| Jupiter mass | M_J | 1.8986 × 10²⁷ | kg | 1 jupitermass |
| Moon mass | M_◊ | 7.342 × 10²² | kg | 1 moonmass |
| Solar radius | R_⊙ | 6.96 × 10⁸ | m | 1 solarradius |
| Earth radius (mean) | R_⊕ | 6.371 × 10⁶ | m | 1 earthradius |
| Hubble constant | H₀ | ~67.4(5) | km·s⁻¹·Mpc⁻¹ | ~67.4 km/s·Mpc |
| CMB temperature | T_CMB | 2.725 48(57) | K | ~2.72548 K |
8. Constants Implicit in the Unit System
These constants are already encoded in the unit conversion factors of built-in units. Writing 1 eV automatically carries the correct SI value.
| Unit | Implicit Constant | SI Factor | Tungsten |
|---|---|---|---|
eV (electronvolt) |
elementary charge | 1.602 176 634 × 10⁻¹⁹ J | 1 eV |
Da (dalton) |
atomic mass constant | 1.660 539 066 60 × 10⁻²⁷ kg | 1 Da |
atm (atmosphere) |
standard atmosphere | 101 325 Pa | 1 atm |
cal (calorie) |
thermochemical calorie | 4.1868 J | 1 cal |
u (atomic mass unit) |
same as dalton | 1.660 539 066 60 × 10⁻²⁷ kg | 1 u |
Notes
Exact vs Measured
The 2019 SI redefinition made seven constants exact by definition (Section 1). Constants derived purely from those seven (e.g. R = N_A · k, F = N_A · e, σ = 2π⁵k⁴/15h³c²) are also exact. The gravitational constant G remains the least-precisely known fundamental constant, with a relative uncertainty of 2.2 × 10⁻⁵.
Numeric Type Semantics
Tungsten's three numeric types map naturally to the epistemological status of constants:
Int exact integer 299_792_458 m/s (speed of light)
Decimal exact non-integer 9.80665 m/s² (standard gravity)
Float measured / approximate ~6.674e-11 m³/kg·s² (gravitational constant)Built-in Variables
The interpreter pre-defines 17 physical constants as global variables:
c ℎ ℏ G g₀ Nₐ kB e₀ R ε₀ μ₀ σ α mₑ mₚ a₀ 𝐹Plus 6 mathematical constants: π, τ, ϕ/φ, ℯ, ℇ, ∞.
Note: 𝐹 (U+1D439, Mathematical Italic Capital F) is used for the Faraday constant to disambiguate from F (Farad unit). Local variables like c = 1 will shadow the built-in — this is by design.
Values: CODATA 2022 — physics.nist.gov/cuu/Constants Frink: frinklang.org/frinkdata/units.txt