Fontem paginae Unitates Planckianae inspicere

{{Prospectus unitatum Planckianarum}}

{{Res|Unitates Planckianae}} in [[cosmologia physica]] et in [[physica particularum elementarium]] sunt [[systema]] {{Creanda|en|Natural units|d|Q3962243|unitates naturales|unitatum naturalium}} a [[Maximilianus Planck|Maximiliano Planck]] [[physicus|physico]] [[Germania|Germanico]] anno [[1899]] propositum<ref name="planck-1899">{{cite journal
	| last = Planck
	| first = Maximilianus
	| author-link = Maximilianus Planck
	| year = [[1899]]
	| title = Über irreversible Strahlungsvorgänge
	| journal = Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin
	| volume = 5
	| pages = 440–480
	| url = https://www.biodiversitylibrary.org/item/93034#page/498/mode/1up
	| language = de
	| access-date = 23 May 2020
	| archive-date = 17 November 2020
	| archive-url = https://web.archive.org/web/20201117200137/https://www.biodiversitylibrary.org/item/93034#page/498/mode/1up
	| url-status = live
}}</ref> (undevicensimo anno ab [[unitates Stoneyanae|unitatibus Stoneyanis]] creatis), sic excogitatum ut ex quinque tantum [[quantitas constans|quantitatibus constantibus]] primariis dependeret: [[celeritas lucis|{{mvar/c}}]], [[constans gravitatis|{{mvar/G}}]], [[constans Planckiana minuta|{{mvar/hbar}}]], [[constans Boltzmanniana|{{mvar/k_B}}]] et [[constans Coulombiana|{{mvar/k_e}}]].<ref>Forma originalis systematis a Maximiliano Planck proposita [[constans Planckiana|constantem Planckianam]] ({{mvar|h}}) ad vicem constantis Planckianae minutae ({{mvar/hbar}}) adhibebat et constante Coulombiana carebat. Vide {{caput|Unitates}}.</ref>

{{Folium memoratum|

{{Barbarice|lingua=de|... 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.

}}

{{Norma dextra|― [[Maximilianus Planck]], "Über irreversible Strahlungsvorgänge"<ref name="planck-1899" />}}

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

* {{mvar/c}}, [[celeritas lucis]] in vacuo ({{Dimensiones|1|0|-1}})
* {{mvar/G}}, [[constans gravitatis]] ({{Dimensiones|3|-1|-2}})
* {{mvar/hbar}}, [[constans Planckiana minuta]] ({{Dimensiones|2|1|-1}})
* {{mvar/k_e}}, [[constans Coulombiana]] ({{Dimensiones|3|1|-2|-2}})
* {{mvar/k_B}}, [[constans Boltzmanniana]] ({{Dimensiones|2|1|-2|0|-1}})

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

{| class="wikitable" style="margin-left: auto; margin-right: auto;"
|-
! Nomen
! Dimensio
! Aequatio
! Mensura ad [[Systema internationale unitatum|SI]] versa
|- style="text-align:left;"
| [[Longitudo Planckiana]]
| [[longitudo]] (L)
| <math>l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}</math>
| 1.616255&thinsp;×&thinsp;10<sup>−35</sup> [[metrum|m]]<ref>{{cite web
	| url = https://physics.nist.gov/cgi-bin/cuu/Value?plkl
	| title = 2018 CODATA Value: Planck length
	| date = [[20 Maii]] [[2019]]
	| work = The NIST Reference on Constants, Units, and Uncertainty
	| publisher = [[National Institute of Standards and Technology|NIST]]
	| access-date = 2019-05-20
}}</ref>
|-
| [[Massa Planckiana]]
| [[massa]] (M)
| <math>m_\text{P} = \sqrt{\frac{\hbar c}{G}}</math>
| 2.176434&thinsp;×&thinsp;10<sup>−8</sup> [[chiliogramma|kg]]<ref>{{cite web
	| url = https://physics.nist.gov/cgi-bin/cuu/Value?plkm
	| title = 2018 CODATA Value: Planck mass
	| date = [[20 Maii]] [[2019]]
	| work = The NIST Reference on Constants, Units, and Uncertainty
	| publisher = [[National Institute of Standards and Technology|NIST]]
	| access-date = 2019-05-20
}}</ref>
|-
| [[Tempus Planckianum]]
| [[tempus]] (T)
| <math>t_\text{P} = \sqrt{\frac{\hbar G}{c^5}}</math>
| 5.391247&thinsp;×&thinsp;10<sup>−44</sup> [[secundum|s]]<ref>{{cite web
	| url = https://physics.nist.gov/cgi-bin/cuu/Value?plkt
	| title = 2018 CODATA Value: Planck time
	| date = [[20 Maii]] [[2019]]
	| work = The NIST Reference on Constants, Units, and Uncertainty
	| publisher = [[National Institute of Standards and Technology|NIST]]
	| access-date = 2019-05-20
}}</ref>
|-
| [[Onus electricum Planckianum]]
| [[onus electricum]] (Q)
| <math>q_\text{P} = \sqrt{\frac{\hbar c}{k_\text{e}}} = \sqrt{4 \pi \varepsilon_0 \hbar c} = \frac{e}{\sqrt{\alpha}} </math>
| 1.875545956&thinsp;×&thinsp;10<sup>−18</sup> [[coulomb (unitas)|C]]<ref name="qp">Pars numeralis e {{math|{{mvar|q<sub style{{=}}"font-style: normal;">P</sub>}} {{=}} {{ofrac|{{mvar/phye}}|√{{mvar/phyalpha}}}}}} ducitur – {{cite web
	| url = https://physics.nist.gov/cgi-bin/cuu/Value?e
	| title = 2018 CODATA Value: elementary charge
	| date = [[20 Maii]] [[2019]]
	| work = The NIST Reference on Constants, Units, and Uncertainty
	| publisher = [[National Institute of Standards and Technology|NIST]]
	| access-date = 2019-05-20
}}</ref>
|-
| [[Temperatura Planckiana]]
| [[temperatura]] (Θ)
| <math>T_\text{P} = \sqrt{\frac{\hbar c^5}{G k_\text{B}^2}}</math>
| 1.416784&thinsp;×&thinsp;10<sup>32</sup> [[kelvinus (unitas)|K]]<ref>{{cite web
	| url = https://physics.nist.gov/cgi-bin/cuu/Value?plktmp
	| title = 2018 CODATA Value: Planck temperature
	| date = [[20 Maii]] [[2019]]
	| work = The NIST Reference on Constants, Units, and Uncertainty
	| publisher = [[National Institute of Standards and Technology|NIST]]
	| access-date = 2019-05-20
}}</ref>
|}

Forma originalis systematis a Maximiliano Planck proposita [[constans Planckiana|constantem Planckianam]] ({{mvar|h}}) ad vicem constantis Planckianae minutae ({{mvar/hbar}}) adhibebat et constante Coulombiana carebat.<ref name="planck-1899" /> Idcirco unitates originales erant factoris <math>\sqrt{2 \pi} \approx 2.5066</math> maiores hodiernis unitatibus et sine ulla unitate oneris electrici.

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

: <math>\begin{align}
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{align}</math>

Proportio inter unitates Planckianae et unitates Stoneyanas est <math>\sqrt{{1}/{\alpha}} \approx 11.7</math>.<ref>{{Citation
	| last1 = Duff
	| first1 = M. J.
	| author-link1 = Michael Duff (physicist)
	| last2 = Okun
	| first2 = L. B.
	| author-link2 = Lev Okun
	| last3 = Veneziano
	| first3 = G.
	| author-link3 = Gabriele Veneziano
	| date = [[3 Martii]] [[2002]]
	| title = Trialogue on the number of fundamental constants
	| journal = Journal of High Energy Physics
	| volume = 2002
	| issue = 3
	| page = 3
	| doi = 10.1088/1126-6708/2002/03/023
	| arxiv = physics/0110060
	| bibcode = 2002JHEP...03..023D
	| s2cid = 15806354
	| issn = 1029-8479
}}</ref> Idcirco: <math>l_\text{S} = \sqrt{\alpha}\,l_\text{P};</math> <math>m_\text{S} = \sqrt{\alpha}\,m_\text{P};</math> <math>t_\text{S} = \sqrt{\alpha}\,t_\text{P};</math> <math>q_\text{S} = \sqrt{\alpha}\,q_\text{P}.</math>

Ad vicem constantis gravitatis ({{mvar/G}}) et constans Coulombiana ({{mvar/k_e}}), nonnulli auctores, rationem [[Oliverius Heaviside|Oliverii Heaviside]] sequentes, praeferunt {{math|4{{mvar/pi}}{{mvar/G}}}} et {{math|4{{mvar/pi}}{{mvar/k_e}}}} uni aequare ({{math|1={{mvar/c}} = {{mvar/hbar}} = 4{{mvar/pi}}{{mvar/G}} = 4{{mvar/pi}}{{mvar/k_e}} = {{mvar/k_B}} = 1}}).<ref>{{cite journal
	| title = Kaluza-Klein Monopole
	| journal = [[Physical Review Letters|Phys. Rev. Lett.]]
	| volume = 51
	| issue = 2
	| pages = 87–90
	| year = 1983
	| last = Sorkin
	| first = Raphaël
	| doi = 10.1103/PhysRevLett.51.87
	| bibcode = 1983PhRvL..51...87S
}}</ref> Talis emendatio, “rationalizatio” appellata, aliquando in [[Physica particularum elementarium|physica altarum energiarum]] adhibetur<ref>{{cite book 
	| last = Rañada 
	| first = Antonius F. 
	| chapter = A Model of Topological Quantization of the Electromagnetic Field 
	| title = Fundamental Problems in Quantum Physics 
	| date = [[31 Octobris]] [[1995]]
	| editor = M. Ferrero 
	| editor2 = Alwyn van der Merwe 
	| chapter-url = https://books.google.com/books?id=1wUFoP7HC38C&pg=PA271 
	| page = 271 
	| publisher = Springer 
	| isbn = 9780792336709 
	| access-date = 16 January 2018 
	| archive-date = 1 September 2020 
	| archive-url = https://web.archive.org/web/20200901153435/https://books.google.com/books?id=1wUFoP7HC38C&pg=PA271 
	| url-status = live
}}</ref> et hoc modo unitates Planckianas transformat:

: <math>\begin{align}
l_{\text{P}}^{\prime}=\sqrt{\frac{4\pi G\hbar}{c^{3}}}\approx5.7295\times10^{-35}\,\text{m}\\
m_{\text{P}}^{\prime}=\sqrt{\frac{\hbar c}{4\pi G}}\approx6.1396\times10^{-9}\,\text{kg}\\
t_{\text{P}}^{\prime}=\sqrt{\frac{4\pi G\hbar}{c^{5}}}\approx1.9111\times10^{-43}\,\text{s}\\
q_{\text{P}}^{\prime}=\sqrt{\frac{\hbar c}{4\pi k_{e}}}\approx5.2908\times10^{-19}\,\text{C}\\
T_{\text{P}}^{\prime}=\sqrt{\frac{\hbar c^{5}}{4\pi Gk_{\text{B}}^{2}}}\approx3.9967\times10^{31}\,\text{K}
\end{align}
</math>

== Notae ==
<references />

{{NexInt}}
* [[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]]

{{Ast}}

{{Unitates Planckianae}}

[[Categoria:Mensura]]
[[Categoria:Notiones in physica]]
[[Categoria:Physica]]
[[Categoria:Quantitates constantes physicae]]
[[Categoria:Unitates naturales]]
[[Categoria:Unitates physicae]]
{{Myrias|Physica}}