Search: a003676 -id:a003676
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A003678
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Decimal expansion of the SI unit c (speed of light in vacuum), c = 299792458 meters/second.
(Formerly M1912)
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+10
76
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OFFSET
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9,1
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COMMENTS
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Since 1983, the speed of light has been defined to be exactly 299792458 m/s. - Ron Marcinski (ronmarcinski(AT)hotmail.com), Apr 18 2002
General context: Within current metrological systems (SI + IAU definitions), several physics constants have been "assigned" immutable values. They thus became metrological reference points, no longer subject to experimental assessment. These should not be confused with "conventional" values of some empirical quantities (such as Josephson's constant) used in applied metrology, but not assigned, and therefore subject to possible future revisions.
Assigned metrological constants [before the inception of the 2019 SI, which introduced some changes, see below for references] and some of their combinations that appear in the OEIS include the speed of light (this sequence); magnetic permeability of vacuum (A019694); electric permittivity of vacuum (A081799); characteristic impedance of vacuum (A213610); standard gravity acceleration (A072915), standard atmosphere (A213611), Julian year (A213612), Gregorian year (A213613) and the light-year (A213614), all in basic SI units.
(End)
Prime factors of this number are 2^1, 7^1, 73^1, 293339^1. - John W. Nicholson, Jun 15 2014
c is also the speed of gravity. - Omar E. Pol, Jun 23 2017
In the 2019 SI system of units (see the second BIPM link, and A322415) one of the seven defining constants is c = 299792458 m/s. - Wolfdieter Lang, Feb 12 2019 [corrected by Ivan Panchenko, May 20 2019]
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REFERENCES
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CRC Handbook for Chemistry and Physics, 75th edition, (1994-1995), Page 1-1.
H. J. Fischbeck and K. Fischbeck, Formulas. Facts and Constants, Springer-Verlag, NY, 2nd ed., 1987.
R. F. Fox and T. P. Hill, An exact value for Avogadro's number, American Scientist, 95 (No. 2, 2007), 104-107.
K. R. Lang, Astrophysical Data: Planets and Stars, Springer-Verlag, NY, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
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c = 299792458 m/s (equals 299792.458 km/s).
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MATHEMATICA
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CROSSREFS
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More assigned constants: A003676 (h), A230458 (Δν_{Cs}), A081823 (e), A322578 (N_A), A070063 (k), A021687 (1/K_{cd}); A182999 (c^2), A183000 (c^3), A183001 (c^4), A019694, A081799, A213610, A072915, A213611, A213612, A213613, A213614.
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approved
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A120070
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Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.
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+10
49
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3, 8, 5, 15, 12, 7, 24, 21, 16, 9, 35, 32, 27, 20, 11, 48, 45, 40, 33, 24, 13, 63, 60, 55, 48, 39, 28, 15, 80, 77, 72, 65, 56, 45, 32, 17, 99, 96, 91, 84, 75, 64, 51, 36, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21
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OFFSET
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2,1
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COMMENTS
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The rationals r(m,n):=a(m,n)/(m^2*n^2), for m-1 >= n, else 0, are used to compute the frequencies of the spectral lines of the H-atom according to quantum theory: nu(m,n) = r(m,n)*c*R' with c*R'=3.287*10^15 s^(-1) an approximation for the Rydberg frequency. R' indicates, that the correction factor 1/(1+m_e/m_p), approximately 0.9995, with the masses for the electron and proton, has been used for the Rydberg constant R_infinity. c:=299792458 m/s is, per definition, the velocity of light in vacuo (see A003678).
In order to compute the wave length of the spectral lines approximately one uses the reciprocal rationals: lambda(m,n):= c/nu(m,n) = (1/r(m,n))*91.1961 nm. 1 nm = 10^{-9} m. For the corresponding energies one uses approximately E(m,n) = r(m,n)*13.599 eV (electron Volts).
The author was inspired by Dewdney's book to compile this table and related ones.
For the approximate frequencies, energies and wavelengths of the first members of the Lyman (n=1, m>=2), Balmer (n=2, m>=3), Paschen (n=3, m>=4), Brackett (n=4, m>=5) and Pfund (n=5, m>=6) series see the W. Lang link under A120072.
This triangle also has an interpretation related to particle spin. For proper offset such that T(0,0) = 3, then, where h-bar = h/(2*Pi) = A003676/A019692 (= The Dirac constant, also known as Planck's reduced constant) and Spin(n/2) = h-bar/2*sqrt(n(n+2)), it follows that: h-bar/2*sqrt(T(r,k)) = h-bar/2*sqrt(T(r,0) - T(k-1,0)) = sqrt((Spin((r+1)/2))^2 - (Spin(k/2))^2). For example, for r = k = 4, then h-bar/2*sqrt(11) = h-bar/2*sqrt(T(4,4)) = h-bar/2*sqrt(T(4,0) - T(3,0)) = sqrt(h-bar^2/4*T(4,0) - h-bar^2/4*T(3,0)) = sqrt(h-bar^2/4*35 - h-bar^2/4*24) = sqrt((Spin((4+1)/2))^2 - (Spin(4/2))^2); 35 = 5*(5+2) & 24 = 4*(4+2). - Raphie Frank, Dec 30 2012
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REFERENCES
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A. K. Dewdney, Reise in das Innere der Mathematik, Birkhäuser, Basel, 2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley & Sons, N.Y., 1999.
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FORMULA
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a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - Raphie Frank, Dec 27 2012
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EXAMPLE
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Triangle begins
[ 3];
[ 8, 5];
[15, 12, 7];
[24, 21, 16, 9];
[35, 32, 27, 20, 11];
...
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MATHEMATICA
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ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)
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PROG
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(PARI) nmax=400; a=vector(1+nmax*(nmax-1)\2); idx=1; for(n=2, nmax, for(k=1, n-1, a[idx]=n*n-k*k; idx++)) \\ Stanislav Sykora, Feb 17 2014
(PARI) T(n, k)=n^2-k^2;
for (n=1, 10, for(k=1, n-1, print1(T(n, k), ", ")));
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CROSSREFS
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Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).
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approved
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A081823
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Decimal expansion of the elementary charge e in coulombs (C).
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+10
36
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OFFSET
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-18,2
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COMMENTS
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e is exactly 1.602176634*10^(-19) Coulomb according to the SI definition (effective on May 20, 2019). - Ralf Steiner, Dec 29 2018
See the second BIPM link with the CGPM resolutions and A322415. The elementary charge e is one of the seven units of the 2019 SI system. - Wolfdieter Lang, Feb 12 2019
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e = 1.602176634*10^(-19) Coulomb, [C = A s].
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EXAMPLE
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electron volt, eV = 1.602176634*10^(-19) J.
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CROSSREFS
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EXTENSIONS
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Updated to conform with CODATA 2010 recommended value by Raphie Frank, Dec 29 2012
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approved
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A070058
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Decimal expansion of Newton's gravitational constant in SI units.
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+10
29
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-10,1
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COMMENTS
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Also called "Newtonian constant of gravitation".
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REFERENCES
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CRC Handbook of Chemistry and Physics, 75th edition, (1994-1995) Page 1-1
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LINKS
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FORMULA
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EXAMPLE
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6.674 30(15) * 10^(-11) m^3 kg^(-1) s^(-2) according to CODATA 2018 (digits in parentheses represent standard uncertainty).
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CROSSREFS
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KEYWORD
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AUTHOR
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Ron Marcinski (ronmarcinski(AT)hotmail.com), Apr 18 2002
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EXTENSIONS
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Updated by Omar E. Pol, Sep 01 2009, Sep 01 2012, Nov 14 2016
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STATUS
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approved
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A070063
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Decimal expansion of the Boltzmann constant k in the 2019 SI system in units J/K.
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+10
23
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OFFSET
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-22,2
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COMMENTS
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The exact Boltzmann constant is one of the seven units in the 2019 system of units. See the BIMP link with the CGPM resolutuions from November 2018 which will become effective May 20 2019. See also A322415. - Wolfdieter Lang, Feb 12 2019
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REFERENCES
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CRC Handbook for Chemistry and Physics, 75th edition, (1994-1995), Page 1-1.
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LINKS
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BIPM, CGPM-2018 [See the link ``Resolutions of the CGPM" there.]
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FORMULA
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k = 1.380649×10^{-23} J/K. Joule J = kg m^2 s^(-2).
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CROSSREFS
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KEYWORD
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AUTHOR
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Ron Marcinski (ronmarcinski(AT)hotmail.com), Apr 18 2002
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EXTENSIONS
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Updated to conform with CODATA 2010 value by Ivan Panchenko, Jan 27 2015
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STATUS
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approved
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A081799
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Decimal expansion of the electric permittivity of vacuum in SI units as defined before 20 May 2019.
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+10
16
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8, 8, 5, 4, 1, 8, 7, 8, 1, 7, 6, 2, 0, 3, 8, 9, 8, 5, 0, 5, 3, 6, 5, 6, 3, 0, 3, 1, 7, 1, 0, 7, 5, 0, 2, 6, 0, 6, 0, 8, 3, 7, 0, 1, 6, 6, 5, 9, 9, 4, 4, 9, 8, 0, 8, 1, 0, 2, 4, 1, 7, 1, 5, 2, 4, 0, 5, 3, 9, 5, 0, 9, 5, 4, 5, 9, 9, 8, 2, 1, 1, 4, 2, 8, 5, 2, 8, 9, 1, 6, 0, 7, 1, 8, 2, 0, 0, 8, 9, 3, 2, 8, 6, 7
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OFFSET
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-11,1
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COMMENTS
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Denoted epsilon_0, this is a function of assigned metrological constants. Regarding these, see A003678 for general context notes, references and links. - Stanislav Sykora, Jun 16 2012
Also called electric constant. - Omar E. Pol, Oct 17 2013
The new value as defined after 20 May 2019 is (8.8541878128 +- 0.0000000013) * 10^(-12) F/m. - Jianing Song, Mar 13 2021
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FORMULA
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Explicitely: 1/([magnetic_permeability of vacuum]*[speed_of_light in vacuum]^2). - Stanislav Sykora, Jun 16 2012
Also 625000/(22468879468420441*Pi). - Omar E. Pol, Oct 17 2013
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EXAMPLE
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8.854187817...*10^-12 F m^-1.
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MAPLE
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A322415
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Decimal expansion of h * Δν_{Cs} / c^2 in units of kg in the 2019 SI system of units.
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+10
16
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6, 7, 7, 7, 2, 6, 5, 3, 1, 2, 3, 1, 2, 0, 6, 7, 6, 1, 0, 6, 1, 7, 2, 7, 0, 2, 3, 4, 6, 5, 8, 4, 3, 5, 5, 3, 9, 4, 4, 6, 4, 5, 5, 7, 0, 6, 2, 4, 2, 3, 2, 3, 9, 8, 0, 4, 2, 6, 6, 2, 2, 4, 5, 0, 7, 0, 9, 9, 8, 6, 1, 7, 6, 7, 8, 8
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OFFSET
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-40,1
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COMMENTS
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The seven defining constants of the SI system (May 20 2019) are (see the BIPM link):
Δν_{Cs} = Δν(133Cs)_{hfs} = 9192631770 s^(-1)
c = 299792458 m/s
h = 6.62607015*10^{-34} J s [kg m^2 s^(-1)]
e = 1.602176634*10^{-19} C [A s]
k = 1.380649*10^{-23} J/K [kg m^2 s^(-2) K^(-1)]
N_A = 6.02214076*10{23} mol^(-1)
K_{cd} = 683 lm/W [cd sr s^3 kg^(-1) m^(-2)]
In brackets are the units given in s, m, kg, A, K, and cd*sr. The symbol sr stands for steradian (or sterad).
K_{cd} is the luminous efficacy of monochromatic radiation of frequency 540*10^12 Hz, with Hz = 1/s.
The given combination of three constants defines the kg from the Planck constant h after s and m have been expressed in terms of Delta nu_{Cs} and c.
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FORMULA
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Equals 0.677726531231206761061727023465843553944645570624232398042662245... * 10^{-40} kg.
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CROSSREFS
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Cf. A230458 (Δν_{Cs}), A003678 (c) A003676 (h), A081823 (e), A070063 (k), A248510 (R_K), A248508 (K_{J}), A322579 (m), A322580 (kg), A324031 (A), A324032 (K), A324033 (lm), A324034 (J).
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approved
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A322578
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Decimal expansion of the Avogadro constant N_A in the 2019 SI system in units mol^(-1).
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+10
14
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OFFSET
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24,1
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COMMENTS
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The Avogadro comstant N_A is one of the seven units of the 2019 SI system of units. See the BIMP link with the CGPM resolutions which become effective om May 20 2019. See also A322415.
This is also the conversion factor of the old SI unit mol into the reciprocal 2010 SI unit 1/N_A.
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LINKS
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BIPM, CGPM-2018 [See the link "Resolutions of the CGPM" there.]
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FORMULA
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N_A = 6.02214076×10^23 mol^(-1) = 2^17*5^15*563*267413 mol^(-1).
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KEYWORD
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approved
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A230458
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Decimal expansion of Δν_{Cs} in unit s^{-1}; one of the seven units of the 2019 SI system.
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+10
13
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OFFSET
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10,1
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COMMENTS
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The second s is defined to be exactly 9192631770 periods of the resonance frequency of the hyperfine transition of the cesium 133 atom.
Prime factors of this number are 2^1, 3^2, 5^1, 7^2, 47^1, 44351^1. - John W. Nicholson, Jun 15 2014
The value was fixed in 1968 by the 13th CGPM in terms of periods of the radiation due to the transition between the two hyperfine levels of the ground state of 133Cs atoms. - Stanislav Sykora, Mar 30 2015
In the 2019 SI system of units (see the BIPM link, and A322415) one of the seven units is Δν_{Cs} = Δν(^{133}Cs)_{hfs} = 9192631770 s^{-1}. - Wolfdieter Lang, Feb 12 2019
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FORMULA
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Δν_{Cs} = 9192631770 s^{-1}.
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EXAMPLE
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c = 9192631770 m/s.
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MATHEMATICA
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approved
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A254181
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Decimal expansion of reduced Planck constant (in joule seconds).
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+10
12
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1, 0, 5, 4, 5, 7, 1, 8, 1, 7, 6, 4, 6, 1, 5, 6, 3, 9, 1, 2, 6, 2, 4, 2, 8, 0, 0, 3, 3, 0, 2, 2, 8, 0, 7, 4, 4, 7, 2, 2, 8, 2, 6, 3, 3, 0, 0, 2, 0, 4, 1, 3, 1, 2, 2, 4, 2, 1, 9, 2, 3, 4, 7, 0, 5, 9, 8, 4, 3, 5, 9, 1, 2, 7, 3, 4, 7, 3, 9, 0, 6, 2, 4, 9, 8, 5, 3
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OFFSET
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-33,3
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COMMENTS
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1.054 571 817... * 10^(-34) J s
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KEYWORD
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STATUS
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approved
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