2. Astrophysical Constants and Parameters1

2. Astrophysical Constants and Parameters
Table 2.1: Revised August 2023 by D.E. Groom (LBNL) and D. Scott (U. of British Columbia). The figures in parentheses after somevalues give the 1-σ uncertainties in the last digit(s). Physical constants are from Ref. [1]. While every effort has been made to obtainthe most accurate current values of the listed quantities, this table does not represent a critical review or adjustment of the constants,
and is not intended as a primary reference. The values and uncertainties for the cosmological parameters depend on the exact data sets,priors, and basis parameters used in the fit. Many of the derived parameters reported in this table have non-Gaussian likelihoods andparameters may be highly correlated, so care must be taken in propagating errors. Unless otherwise specified, cosmological parameters
are derived from a 6-parameter ΛCDM cosmology fit to Planck cosmic microwave background 2018 temperature (TT) + polarization(TE,EE+lowE) + lensing data [2]. For more information see [3] and the original papers.
Quantity
Newtonian constant of gravitation
Planck mass

Symbol, equation
GN
p
MP =
~c/GN

p

Value
Reference, footnote
6.674 30(15) × 10−11 m3 kg−1 s−2[1]1.220 890(14) × 1019 GeV/c2 = 2.176 434(24) × 10−8 kg[1]

1.616 255(18) × 10−35 m[1]
31 556 925.1 s = 365.242 189 days
[4]
31 558 149.8 s ≈ π × 107 s[4]
23h 56m 04s.090 53[4]
149 597 870 700 m
exact [5]3.085 677 581 49 . . . × 1016 m = 3.261 56 . . . ly
exact [6]0.306 601 . . . pc = 0.946 073 . . . × 1016 m[7](π/180)2 sr = 3.046 17 . . . × 10−4 sr[8]2.953 250 1 km
[9]
1.988 41(4) × 1030 kg[10]
6.957 × 108 m
exact [11]−21361 W mexact [11, 12]5772 K
exact [11]26
3.828 × 10 W
exact [11, 13]8.870 056 mm[9]5.972 17(13) × 1024 kg[10]
6.3781 × 106 m
exact [11]3.097 972 µ−2 MP3 /m2H = 1.433 77(6) (µ/2)−2 M[14, 15]
1.257 065 179 8(12) × 1031 (M/M ) W[16, 17]4= 3.283 869 330 8(31) × 10 (M/M ) L
jansky (flux density)Jy
10−26 W m−2 Hz−1
definition
luminosity conversion
f0
3.0128 × 1028 × 10−0.4 MBol W
exact [18](MBol = absolute bolometric magnitude = bolometric magnitude at 10 pc)
flux conversion
F
2.518 021 002 × 10−8 × 10−0.4 mBol W m−2
exact [18](mBol = apparent bolometric magnitude)−1−2
ABsolute monochromatic magnitudeAB−2.5 log10 fν − 56.10 (for fν in W m Hz )[19]
= −2.5 log10 fν + 8.90 (for fν in Jy)
Solar distance from Galactic center
R0
8.275 ± 0.009(stat.) ± 0.033(sys.) kpc[20]
local circular angular velocity
Θ0 /R028.2(10) km s−1 kpc−1[21]−1circular velocity at R0
v0 or Θ0233.4 ± 0.3(stat.) ± 8.0(sys.) km s
[22]
local escape velocity from the Galaxyv esc
477 km s−1 < v esc < 502 km s−1 (68%)[23]
local disk density
ρ disk6.6(9) × 10−24 g cm−3 = 3.7(5) GeV/c2 cm−3
[24]
local dark matter density
ρχ
canonical value 0.3 GeV/c2 cm−3 within factor 2–3[25]
present-day CMB temperature
T0
2.7255(6) K[26, 27]
present-day CMB dipole amplituded3.3621(10) mK[26, 28]
Solar velocity with respect to CMBv
369.82(11) km s−1 towards (l, b) = (264.021(11)◦ , 48.253(5)◦ )[28]
Local Group velocity with respect to CMB vLG
620(15) km s−1 towards (l, b) = (271.9(20)◦ , 29.6(14)◦ )
[28]
3
−3
number density of CMB photonsnγ
410.73(27) (T /2.7255 K) cm
[29]
4
−34
−3
−3
density of CMB photons
ργ
4.645(4) (T /2.7255 K) × 10
g cm
≈ 0.260 eV cm
[29]
3
−3
entropy density/Boltzmann constants/k
2 891.2 (T /2.7255 K) cm
[29]
−1
−1
−1
present-day Hubble expansion rate
H0
100 h km s Mpc= h × (9.777 752 Gyr)[30]
scaling factor for Hubble expansion rateh0.674(5) from CMB anisotropies (Planck)[2]
or 0.730(10) from the distance ladder (SH0ES)[31, 32]
26−1
26
Hubble length
c/H00.925 0629 × 10 h m = 1.372(11) or 1.267(18) × 10 m
[2, 31]
scaling for cosmological constantc2 /3H02
2.85247 × 1051 h−2 m2 = 6.29(10) or 5.35(15) × 1051 m2[2, 31]
2−29
2−3
critical density of the Universe
ρcrit = 3H0 /8πGN
1.878 34(4) × 10
h g cm
= 1.053 672(24) × 10−5 h2 (GeV/c2 ) cm−3
= 2.77536627 × 1011 h2 M Mpc−3
baryon-to-photon ratio (from BBN)
η = nb /nγ
6.04(12) × 10−10[33, 34]
number density of baryons
nb2.515(17) × 10−7 cm−3 (from CMB)[2, 35]2.48(5) × 10−7 cm−3 (from BBN, η × nγ )[3, 33]
CMB radiation density of the UniverseΩγ = ργ /ρcrit2.473 × 10−5 (T /2.7255 K)4 h−2 =5.38(15) × 10−5[29]
primordial helium fraction
Yp
0.2448(33)[36]- - - Planck 2018 6-parameter fit to flat ΛCDM cosmology - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ‡ 0.02237(15) h−2 = † 0.0493(6)baryon density of the UniverseΩb = ρb /ρcrit[2, 3, 26]‡ 0.1200(12) h−2 = † 0.265(7)cold dark matter density of the UniverseΩc = ρc /ρcrit[2, 3, 26]‡ 1.04092(31)100 × approximation to r∗ /DA
100 × θMC[2, 3, 26]‡ 0.054(7)reionization optical depthτ[2, 3, 26]‡ 3.044(14)ln(power prim. curv. pert.) (k0 = 0.05 Mpc−1 )ln(1010 ∆2R )[2, 3, 26]‡ 0.965(4)scalar spectral index
ns[2, 3, 26]† 0.315(7)pressureless matter density parameterΩm = Ωc + Ωb
[2, 3]† 0.685(7)dark energy density parameterΩΛ[2, 3]† 5.83(16) × 10−30 g cm−3energy density of dark energyρΛ[2]† 1.088(30) × 10−56 cm−2
cosmological constantΛ[2]† 0.811(6)fluctuation amplitude at 8 h−1 Mpc scale
σ8[2, 3]

Planck lengthlP =~GN /c3
tropical year (equinox to equinox, 2020)
yrsidereal year (period of Earth around Sun relative to stars)
mean sidereal day (Earth rotation period relative to stars)
astronomical unit
auparsec (1 au/1 arcsec)
pc
light year (deprecated unit)lysolid angle
deg2
Schwarzschild radius of the Sun
2GN M /c2
Solar mass
Mnominal Solar equatorial radiusRnominal Solar constantSnominal Solar photosphere temperatureTnominal Solar luminosity
L
Schwarzschild radius of the Earth
2GN M⊕ /c2
Earth mass
M⊕nominal Earth equatorial radiusR⊕
Chandrasekhar mass
MCh
Eddington luminosityLEd2

2. Astrophysical Constants and Parameters

Quantity
Symbol, equation.
amplitude of matter fluctuations
σ8 (Ωm /0.3)0.5
redshift of matter-radiation equality
zeq
age at matter-radiation equality
teqredshift at which optical depth equals unityz∗
comoving size of sound horizon at z∗r∗
age when optical depth equals unityt∗redshift at half reionizationzi
age at half reionizationti
redshift when acceleration was zerozq
age when acceleration was zerotq
age of the Universe todayt0
effective number of neutrinos
Neff
sum of neutrino masses
Σmν−2
neutrino density of the UniverseΩν = h Σmν /93.14 eV
curvatureΩKrunning spectral index, k0 = 0.05 Mpc−1
dns /d ln ktensor-to-scalar perturbation ratio
r0.05 = T /S
dark energy equation of state parameter
w

Value

Reference,footnote[2, 3]† 3402(26)[2, 37]† 51.0(8) kyr[2, 38]† 1089.92(25)[2]† 144.43(26) Mpc[2, 39]† 371.8(10) kyr[2, 38]† 7.7(7)[2, 40]† 690(90) Myr[2]† 0.656(18)[2, 38]† 7.70(10) Gyr[2]† 13.797(23) Gyr[2]] 2.99(17) (CMB + BAO)
[2, 41, 42]] < 0.12 eV (95%, CMB + BAO); ≥ 0.06 eV (mixing)[2, 42–44]] < 0.003 (95%, CMB + BAO); ≥ 0.0012 (mixing)[2, 43, 44]] 0.0007(19) (CMB + BAO)
[2]
] −0.004(7)[2]< 0.036 (95%, k0 = 0.05 Mpc−1 )[2, 45–48]] −1.028(31) (CMB + BAO +SNe)[2, 49]† 0.832(13)‡ Parameter in 6-parameter ΛCDM fit; † derived parameter in 6-parameter ΛCDM fit; ] extended model parameter, Planck + BAO data [2].

References[1] CODATA recommended 2018 values of the fundamental physical constants:
physics.nist.gov/cuu/Constants/index.html.

[25] J. I. Read, J. Phys. G41, 063101 (2014), [arXiv:1404.1938]; A. M.
Green, J. Phys. G44, 8, 084001 (2017), [arXiv:1703.10102]; the conclusion is ρlocal= 0.39 ± 0.03 GeV cm−3 .
DM[3] O. Lahav and A. R. Liddle, “The Cosmological Parameters,” Sec. 25in this Review.

[26] D. Scott and G. F. Smoot, “Cosmic Microwave Background,” Sec. 29
in this Review.
[27] D. J. Fixsen, Astrophys. J. 707, 916 (2009), [arXiv:0911.1955].
[28] Planck Collab. 2018 Results I, Astron. Astrophys. 641, A1 (2020),[arXiv:1807.06205].

[4] The Astronomical Almanac for the year 2020.

[29] nγ =[2] Planck Collab. 2018 Results VI, Astron. Astrophys. 641, A6 (2020),[arXiv:1807.06209].

[5] The astronomical unit of length (au) in meters is re-defined (IAU
XXVIII General Assembly 2012, Resolution B2) to be a conventional
unit of length in agreement with the value adopted in IAU XXVII GA
2009, Resolution B2. It is to be used with all time scales.

[30]
[31]

[6] The distance at which 1 au subtends 1 arc sec:
π/648 000.

[32]

1 au divided by[7] From c ×(tropical year)/pc, c ×(tropical year), see IAU XVI GA 1976,
Recommendations; T. Lederle, Astron. Gesellschaft 48, 59 (1980).
[8] The number of square degrees on a sphere is 3602 /π = 41 259.9 . . . .
[9] Observationally determined mass parameter GN M × 2/c2 [1] for either
the Sun or the Earth, using the nominal values GM = 1.327 124 4 ×1020 m3 s−2 and GM⊕ = 3.986 004×1014 m3 s−2 [11]; the calligraphic
symbol indicates the recommended nominal value. The combination
GN M is known much more precisely than either GN or M individually. The digits are truncated here at the point where one would need to
distinguish between Barycentic Coordinate Time (TCB) and Barycentric Dynamical Time (TDB).

[33]
[34]
[35]

[36]
[37]

[10] GN M ÷ GN [1].
[11] IAU XXIX GA 2015 Resolution B3; A. Prša et al., Astron. J. 152, 2,
41 (2016), [arXiv:1605.09788].
[12] See also G. Kopp and J. L. Lean, Geophys. Res. Lett. 38, L01706(2011), who give (1360.8 ± 0.6) W m−2 ; see paper for caveats andother measurements.
[13] 4π (1 au)2 × S , assuming isotropic irradiance.

[38]
[39]
[40]

[41]

[14] S. Chandrasekhar, Astrophys. J. 74, 81 (1931).
[15] This value assumes an ideal Fermi gas, using a numerical constantfrom the Lane-Emden equation [50], and with µ the average molecularweight per electron, defined relative to the mass of the single-protonhydrogen atom.

[42]
[43]

[16] A. S. Eddington, Mon. Not. Roy. Astron. Soc. 77, 16 (1916).
[17] The maximum luminosity assuming pure electron scattering for theoutward force arising from radiation pressure: 4πGN M mp c/σT .
[18] IAU XXIX GA 2015, Resolution
arXiv:1510.06262 (2015).

B2;

E.

E.

Mamajek

et al.

[44][19] J. B. Oke and J. E. Gunn, Astrophys. J. 266, 713 (1983).

[45][20] GRAVITY Collaboration,[arXiv:2101.12098].

[46]

Astron. Astrophys. 647,

A59 (2021),[21] A.-C. Eilers et al., Astrophys. J. 871, 1, 120 (2019), [arXiv:1810.09466];
this is representative of other published values and the systematic uncertainty here is an estimate.

[47][22] Product of angular velocity and R0 , derived from Ref. [21] with anestimate of the systematic uncertainty.

[48][23] L. Necib and T.
[arXiv:2102.02211].

(2022),[49][24] C. F. McKee, A. Parravano and D. J. Hollenbach, Astrophys. J. 814, 1,
13 (2015), [arXiv:1509.05334]; this is representative of other publishedestimates.

[50]

Lin,

Astrophys.

J.

926,

2,

189

2ζ(3)
π2

kT
~c

3

2

; ργ = π kT
2
15 c

kT
~c

3

2

; s/k= 2·43·π
11·45

kT
~c

3

;

kT /~c = 11.90 235(T /2.7255)/cm.
Conversion using length of sidereal year.
A. G. Riess et al., Astrophys. J. Lett. 934, 1, L7 (2022),[arXiv:2112.04510].
Distance-ladder estimates of H0 tend to give higher values than derived
from the CMB; Lahav and A. R. Liddle, “The Cosmological Parameters,” Sec. 25 in this Review.
T.-H. Yeh et al., J. Cosmol. Astropart. Phys. 2022, 10, 046 (2022),[arXiv:2207.13133].
B. D. Fields, P. Molaro and S. Sarkar, “Big-Bang Nucleosynthesis,”Sec. 24 in this Review.
nb depends only upon the measured Ωb h2 , the average baryon
mass at the present epoch (G. Steigman, J. Cosmol. Astropart.
Phys., 0610, 016 (2006), [astro-ph/0606206]), and GN : nb =(Ωb h2 )(h−2 ρcrit )/(0.93711 GeV/c2 per baryon).
E. Aver et al., Mon. Not. Roy. Astron. Soc. 510, 1, 373 (2022),[arXiv:2109.00178].
Here “radiation” includes three species of light neutrinos as well as
photons.
D. Scott, A. Narimani and D. N. Page, Phys. Canada 70, 258 (2014),[arXiv:1309.2381].
D. H. Weinberg and M. White, “Dark Energy,” Sec. 28 in this Review.
Planck Collab. Interm. Results XLVI, Astron. Astrophys. 596, A107
(2016), [arXiv:1605.02985]; the range can be extended by ∆z ≈ 1,
depending on the reionization model.
Summary Tables in this Review list Nν = 2.984(8) (Standard Modelfits to LEP-SLC data). Because neutrinos are not completely decoupled
at e± annihilation, the effective number of massless neutrino species
is 3.044, rather than 3.
J. Lesgourgues and L. Verde, “Neutrinos in Cosmology,” Sec. 26 in thisReview.
The sum is over all neutrino mass eigenstates, the lower limit following
from neutrino mixing results reported in this Review combined with theassumptions that there are three light neutrinos and that the lightest
neutrino is substantially less massive than the others.
Astrophysical determinations of Σmν , reported in the Full Listings ofthis Review under “Sum of the neutrino masses,” range from < 0.17 eV
to < 2.3 eV in papers published since 2003.
P. A. R. Ade et al. (BICEP2, Keck Array), Phys. Rev. Lett. 121, 221301(2018), [arXiv:1810.05216].
M. Tristram et al., Astron. Astrophys. 647, A128 (2021),[arXiv:2010.01139].
BICEP/Keck Collaboration, Phys. Rev. Lett. 127, 151301 (2021),[arXiv:2110.00483].
Planck data alone give r < 0.056, using all power spectra; the currently
tightest constraint comes from BICEP/Keck data, using WMAP and
Planck to remove foregrounds.
This constraint uses BAO and SNe data, as described in Ref. [2]; seediscussion in D. H. Weinberg and M. White, “Dark Energy,” Sec. 28 inthis Review.
G. P. Horedt, Astrophys. Space Sci. 126, 2, 357 (1986).