Unitates Planckianae

Formula:Res in cosmologia physica et in physica particularum elementarium sunt systema Formula:Creanda a Maximiliano Planck physico Germanico anno 1899 propositum (undevicensimo anno ab unitatibus Stoneyanis creatis), sic excogitatum ut ex quinque tantum quantitatibus constantibus primariis dependeret: [[celeritas lucis|Formula:Mvar/c]], [[constans gravitatis|Formula:Mvar/G]], [[constans Planckiana minuta|Formula:Mvar/hbar]], [[constans Boltzmanniana|Formula:Mvar/k B]] et [[constans Coulombiana|Formula:Mvar/k e]].

Unitates

Ad systema mensurae Planckianum definiendum, hae quantitates constantes adhibentur:

• Formula:Mvar/c, celeritas lucis in vacuo (Formula:Dimensiones)
• Formula:Mvar/G, constans gravitatis (Formula:Dimensiones)
• Formula:Mvar/hbar, constans Planckiana minuta (Formula:Dimensiones)
• Formula:Mvar/k e, constans Coulombiana (Formula:Dimensiones)
• Formula:Mvar/k B, constans Boltzmanniana (Formula:Dimensiones)

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

Nomen	Dimensio	Aequatio	Mensura ad SI versa
Longitudo Planckiana	longitudo (L)	$l_{\text{P}}=\sqrt{\frac{\hslash G}{c^3}}$	1.616255 × 10−35 m
Massa Planckiana	massa (M)	$m_{\text{P}}=\sqrt{\frac{\hslash c}{G}}$	2.176434 × 10−8 kg
Tempus Planckianum	tempus (T)	$t_{\text{P}}=\sqrt{\frac{\hslash G}{c^5}}$	5.391247 × 10−44 s
Onus electricum Planckianum	onus electricum (Q)	$q_{\text{P}}=\sqrt{\frac{\hslash c}{k_{\text{e}}}}=\sqrt{4\pi {\epsilon }_0\hslash c}=\frac{e}{\sqrt{\alpha }}$	1.875545956 × 10−18 C
Temperatura Planckiana	temperatura (Θ)	$T_{\text{P}}=\sqrt{\frac{\hslash c^5}{G{k}_{\text{B}}^2}}$	1.416784 × 1032 K

Forma originalis systematis a Maximiliano Planck proposita constantem Planckianam (Formula:Mvar) ad vicem constantis Planckianae minutae (Formula:Mvar/hbar) adhibebat et constante Coulombiana carebat. Idcirco unitates originales erant factoris $\sqrt{2\pi }\approx 2.5066$ maiores hodiernis unitatibus et sine ulla unitate oneris electrici.

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

$\begin{array}{cc}c& =1l_{\text{P}}\cdot t_{\text{P}}^{-1}\\ G& =1l_{\text{P}}^3\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}^{-1}\\ \hslash & =1l_{\text{P}}^3\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}^{-1}\\ k_{\text{B}}& =1l_{\text{P}}^2\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}\cdot T_{\text{P}}^{-1}\\ k_{\text{e}}& =1l_{\text{P}}^3\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}\cdot q_{\text{P}}^{-2}\end{array}$

Proportio inter unitates Planckianae et unitates Stoneyanas est $\sqrt{1/\alpha }\approx 11.7$. Idcirco: $l_{\text{S}}=\sqrt{\alpha }{l}_{\text{P}};$ $m_{\text{S}}=\sqrt{\alpha }{m}_{\text{P}};$ $t_{\text{S}}=\sqrt{\alpha }{t}_{\text{P}};$ $q_{\text{S}}=\sqrt{\alpha }{q}_{\text{P}}.$

Ad vicem constantis gravitatis (Formula:Mvar/G) et constans Coulombiana (Formula:Mvar/k e), nonnulli auctores, rationem Oliverii Heaviside sequentes, praeferunt Formula:Math et Formula:Math uni aequare (Formula:Math). Talis emendatio, “rationalizatio” appellata, aliquando in physica altarum energiarum adhibetur et hoc modo unitates Planckianas transformat:

$\begin{array}{c}{l}_{\text{P}}^{\prime }=\sqrt{\frac{4\pi G\hslash }{c^3}}\approx 5.7295\times 10^{-35}\text{m}\\ m_{\text{P}}^{\prime }=\sqrt{\frac{\hslash c}{4\pi G}}\approx 6.1396\times 10^{-9}\text{kg}\\ t_{\text{P}}^{\prime }=\sqrt{\frac{4\pi G\hslash }{c^5}}\approx 1.9111\times 10^{-43}\text{s}\\ q_{\text{P}}^{\prime }=\sqrt{\frac{\hslash c}{4\pi k_e}}\approx 5.2908\times 10^{-19}\text{C}\\ T_{\text{P}}^{\prime }=\sqrt{\frac{\hslash c^5}{4\pi G{k}_{\text{B}}^2}}\approx 3.9967\times 10^{31}\text{K}\end{array}$

Notae

• Longitudo Planckiana
• Massa Planckiana
• Celeritas lucis
• Constans Planckiana
• Constans Coulombiana
• Constans gravitatis
• Onus electricum Planckianum
• Physica particularum elementarium
• Radius Schwarzschildianus
• Systema internationale unitatum
• Temperatura Planckiana
• Tempus Planckianum
• Unitates Stoneyanae