Unitates Planckianae in cosmologia physica et in physica particularum elementarium sunt systema unitatum naturalium(en)(d) a Maximiliano Planck physico Germanico anno 1899 propositum (undevicensimo anno ab unitatibus Stoneyanis creatis), sic excogitatum ut ex quinque tantum quantitatibus constantibus primariis dependeret: c, G, ℏ, kB et ke.

> *... die Möglichkeit gegeben ist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und aussermenschliche Culturen nothwendig behalten und welche daher als »natürliche Maasseinheiten« bezeichnet werden können.*  
>   
> ... possibile est quasdam unitates longitudinis, massae, temporis temperaturaeque sic definire ut e specialibus corporibus vel materie non dependeant, ut semper et per omnes civitates, extraterrestribus vel haud humanis inclusis, significationes suas servent, ut igitur "unitates naturales mensurae" appellari possint.– Maximilianus Planck, "Über irreversible Strahlungsvorgänge"

Unitates
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Ad systema mensurae Planckianum definiendum, hae quantitates constantes adhibentur:

* c, celeritas lucis in vacuo
* G, constans gravitatis
* ℏ, constans Planckiana minuta
* ke, constans Coulombiana
* kB, constans Boltzmanniana

Unitates Planckianae emergunt numerum unius parti numerali talium quantitatum constantium imponendo et systema quinque aequationum linearium solvendo.

| Nomen | Dimensio | Aequatio | Mensura ad SI versa |
| --- | --- | --- | --- |
| Longitudo Planckiana | longitudo (L) | {\displaystyle l\_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}} | 1.616255 × 10−35 m |
| Massa Planckiana | massa (M) | {\displaystyle m\_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}} | 2.176434 × 10−8 kg |
| Tempus Planckianum | tempus (T) | {\displaystyle t\_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}} | 5.391247 × 10−44 s |
| Onus electricum Planckianum | onus electricum (Q) | {\displaystyle q\_{\text{P}}={\sqrt {\frac {\hbar c}{k\_{\text{e}}}}}={\sqrt {4\pi \varepsilon \_{0}\hbar c}}={\frac {e}{\sqrt {\alpha }}}} | 1.875545956 × 10−18 C |
| Temperatura Planckiana | temperatura (Θ) | {\displaystyle T\_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk\_{\text{B}}^{2}}}}} | 1.416784 × 1032 K |

Forma originalis systematis a Maximiliano Planck proposita constantem Planckianam (h) in loco constantis Planckianae minutae (ℏ) adhibebat et constante Coulombiana carebat. Idcirco unitates originales erant factoris {\displaystyle {\sqrt {2\pi }}\approx 2.5066} maiores hodiernis unitatibus et sine ulla unitate oneris electrici.

Constantes quae systema Planckianum definiunt, cum per ipsas unitates Planckianas exprimantur, partem numeralem habent unius:

{\displaystyle {\begin{aligned}c&=1\ l\_{\text{P}}\cdot t\_{\text{P}}^{-1}\\G&=1\ l\_{\text{P}}^{3}\cdot t\_{\text{P}}^{-2}\cdot m\_{\text{P}}^{-1}\\\hbar &=1\ l\_{\text{P}}^{3}\cdot t\_{\text{P}}^{-2}\cdot m\_{\text{P}}^{-1}\\k\_{\text{B}}&=1\ l\_{\text{P}}^{2}\cdot t\_{\text{P}}^{-2}\cdot m\_{\text{P}}\cdot T\_{\text{P}}^{-1}\\k\_{\text{e}}&=1\ l\_{\text{P}}^{3}\cdot t\_{\text{P}}^{-2}\cdot m\_{\text{P}}\cdot q\_{\text{P}}^{-2}\end{aligned}}}
Proportio inter unitates Planckianae et unitates Stoneyanas est {\displaystyle {\sqrt {{1}/{\alpha }}}\approx 11.7}. Idcirco: {\displaystyle l\_{\text{S}}={\sqrt {\alpha }}\,l\_{\text{P}};} {\displaystyle m\_{\text{S}}={\sqrt {\alpha }}\,m\_{\text{P}};} {\displaystyle t\_{\text{S}}={\sqrt {\alpha }}\,t\_{\text{P}};} {\displaystyle q\_{\text{S}}={\sqrt {\alpha }}\,q\_{\text{P}}.}

Notae
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1. 1 2 3  Planck, Maximilianus (1899). "Über irreversible Strahlungsvorgänge". *Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin* **5**: 440–480
2. ↑  Forma originalis systematis a Maximiliano Planck proposita constantem Planckianam (h) in loco constantis Planckianae minutae (ℏ) adhibebat et constante Coulombiana carebat. Vide § Unitates.
3. ↑  "2018 CODATA Value: Planck length". *The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 Maii 2019
4. ↑  "2018 CODATA Value: Planck mass". *The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 Maii 2019
5. ↑  "2018 CODATA Value: Planck time". *The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 Maii 2019
6. ↑  Pars numeralis e qP = e⁄√α ducitur – "2018 CODATA Value: elementary charge". *The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 Maii 2019
7. ↑  "2018 CODATA Value: Planck temperature". *The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 Maii 2019
8. ↑  Duff, M. J.; Okun, L. B.; Veneziano, G. (3 Martii 2002), "Trialogue on the number of fundamental constants", *Journal of High Energy Physics* **2002** (3): 3, arXiv:physics/0110060

Nexus interni
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* Longitudo Planckiana
* Massa Planckiana
* Celeritas lucis
* Constans Planckiana
* Constans Coulombiana
* Constans gravitatis
* Onus electricum Planckianum
* Radius Schwarzschildianus
* Systema internationale unitatum
* Temperatura Planckiana
* Tempus Planckianum
* Unitates Stoneyanae

Unitates Planckianae

|  |  |
| --- | --- |
| Mensurae Planckianae primariae | Tempus Planckianum · Longitudo Planckiana · Massa Planckiana · Onus electricum Planckianum · Temperatura Planckiana |
| Mensurae Planckianae secundariae | Energia Planckiana · Vis Planckiana · Potentia Planckiana · Densitas Planckiana · Frequentia angularis Planckiana · Pressio Planckiana · Fluxus Planckianus · Differentia potentialis Planckianae · Impedantia Planckiana |