Basic Astronomical Data for the Sun (BADS)
Eric Mamajek
last updated 22 October 2012 (list of updates at bottom)
_________________________________________________________
This is a list of fundamental values adopted for the Sun. I am not a
solar astronomer, but I occasionally need a solar value time and again
for my calculations. If you do not like the values that I have
adopted, feel free to email me and justify why I should adopt a
different value. If you note the timing of the updates at the bottom
of the page, I have been maintaining this file for 5 years now (since
May 2007)!
####################################################################
# Apparent V Magnitude: V = -26.74 (+-0.02) mag
# Absolute V Magnitude: M_V = 4.83 (+-0.02) mag
# Absolute B Magnitude: M_B = 5.485 (+-0.02) mag
# Bolometric Correction: BCv = -0.08 (+-0.02) mag
# Absolute Bolometric Magnitude: Mbol = 4.7554 (+-0.0004) mag
# Photospheric Radius: R = 695660 (+-100) km
# Solar Oblateness: f = (a-b)/a = 8 X 10^-6
# Spectral Type: SpT = G2 V
# Effective Temperature: Teff = 5771.8 +- 0.7 K
# Effective Temperature: log10(Teff/K) = 3.76131(+-0.00005) dex
# Solar Gravitational Constant (TDB) = 1.32712440041e20 m^3 s^-2
# Solar Gravitational Constant (SI,TCB) = 1.32712442099e20 m^3 s^-2
# Solar Mass: M = 1.98855(24) e30 kg
# Solar Bulk Density: rho = 1.4111(2) g cm^-3
# Moment of Inertia: I = 5.96e47 kg m^2 = 5.96e53 g cm^2
# Inertia Constant: k = 0.062 = I/MR^2
# Solar Surface Gravity: g = 27423.2 (+-7.9) cm/s^2 (ignoring rotation, oblateness)
# Solar Surface Gravity: g = 274.232 (+-0.079) m/s^2 (ignoring rotation, oblateness)
# Solar Surface Gravity: log(g) = 4.43812 (+-0.00013) dex [cgs] (ignoring rotation, oblateness)
# Astronomical Unit: AU = 149597870700 m (exact)
# Mean Earth-Sun Distance: <r> = 149618753000 m
# Total Solar Irradiance: S(@1AU) = 1360.8 (+-0.5) W/m^2
# Luminosity: L_bol = 3.8270 (+-0.0014) e33 erg/s
# Mt. Wilson S-value: <S_MW> = 0.1762
# Chromospheric Activity: <logR'HK> = -4.903 dex
# X-ray Luminosity (0.1-2.4 keV) L_X = 2.24e27 erg/s = 10^27.35 erg/s
# X-ray Surface Flux (0.1-2.4 keV) f_X = 36800 erg/s/cm^2
# X-ray/Bol. Ratio log(L_X/L_bol) = -6.24
# Age(Solar System) t = 4572 +- 4 Myr
# Protosolar Hydrogen Mass Fraction: Xo = 0.7028-0.7154
# Protosolar Helium Mass Fraction: Yo = 0.2703-0.2783
# Protosolar Metal Mass Fraction: Zo = 0.0142-0.0189
# Photospheric Metal Mass Fraction: Zs = 0.0134-0.0172
# Equatorial Rotation Period: P_eq = 24.47 days
# Equatorial Rotation Velocity: V_eq = 2.067 km/s
# Mean Rotation Period: <P> = 26.09 days
# Mean Solar Wind Mass Loss: <dM/dt> = 2e-14 Msun/yr ~ 1e12 g/s
# Median Daily Int'l Sunspot #: <ISN> = 40 (years 1818-2008)
# Solar (B-V) Color: (B-V) = 0.653 +- 0.003 [Johnson bands]
# Solar (U-B) Color: (U-B) = 0.158 +- 0.009 [Johnson bands]
# Solar (V-Rc) Color: (V-Rc) = 0.356 +- 0.003 [Johnson & Cousins bands]
# Solar (V-Ic) Color: (V-Ic) = 0.701 +- 0.003 [Johnson & Cousins bands]
# Solar (V-J) Color: (V-J) = 1.198 +- 0.005 [Johnson & 2MASS bands]
# Solar (V-H) Color: (V-H) = 1.484 +- 0.009 [Johnson & 2MASS bands]
# Solar (V-Ks) Color: (V-Ks) = 1.560 +- 0.008 [Johnson & 2MASS bands]
# Solar (J-H) Color: (J-H) = 0.286 +- 0.01 [2MASS bands]
# Solar (J-Ks) Color: (J-Ks) = 0.362 +- 0.01 [2MASS bands]
# Solar (H-Ks) Color: (H-Ks) = 0.076 +- 0.01 [2MASS bands]
# Solar (V-W1) Color: (V-W1) = 1.608 +- 0.008 [Johnson & WISE bands]
# Solar (V-W2) Color: (V-W2) = 1.563 +- 0.008 [Johnson & WISE bands]
# Solar (V-W3) Color: (V-W3) = 1.552 +- 0.009 [Johnson & WISE bands]
# Solar (V-W4) Color: (V-W4) = 1.604 +- 0.011 [Johnson & WISE bands]
####################################################################
I have not had time to properly include complete bibliographic
information for all of the references yet -- usually I list just the
author and a year. The correct references can be easily retrieved
using the author name and year from the Smithsonian/NASA ADS server
at: http://adsabs.harvard.edu/abstract_service.html. Uncertainties are
alternatively listed either showing parentheses (indicating the
uncertainty in the last digits) or using "+-" and a value indicating a
1-sigma uncertainty (as is commonly used in astronomy).
In addition to the data, I've included a discussion on a
parameterization of the luminosity evolution of the Sun from the ZAMS
through the end of its Main Sequence stage.
In addition to this table, I also recommend "Table 2. Astrophysical
Constants and Parameters" (edited by E. Bergren & D.E. Groom (LBNL))
from the 2011 Review of Particle Physics (Nakamura et al. (Particle
Data Group), J. Phys. G. 37, 075021 (2010).
(http://pdg.lbl.gov/2011/reviews/contents_sports.html)
What follows is some discussion on each of the constants. - EEM
####################################################
# Apparent V Magnitude: V(Sun) = -26.74 +- 0.02 mag
####################################################
This value varies slightly in the literature. Good discussions and
reviews are presented by Hayes (1985; 1985IAUS..111..225H) and
Bessell, Castelli, & Plez (1998, A&A 333, 231; Table A4). Note that
Bessell98 and Torres10 have pointed out instances of quoted
combinations of V, Mv, Mbol, and BC from some authors that are not
mutually consistent.
Vmag(Sun) Ref.
-26.70(1) Gallouet64
-26.70 Durrant81 (Landolt Bornstein vol VI/2A, p.82)
-26.705 Engelke10 (Rieke08 synthetic + Engelke08 zero reference)
-26.706 Engelke10 (Rieke08 synthetic + Vega from Rieke08)
-26.723 Engelke10 (ASUN model + Engelke08 zero reference)
-26.723 Engelke10 (Kurucz model + Engelke08 zero reference)
-26.724 Engelke10 (ASUN model + Vega from Rieke08)
-26.724 Engelke10 (Kurucz model + Vega from Rieke08)
-26.73(44) Stebbins & Kron (1957,ApJ,126,266) [original value, p.e.=0.03 quoted)]
-26.74 Allen76 (Astrophysical Quantities, 2nd ed.)
-26.74 Schmidt-Kaler82 (Landolt Bornstein, Num. Data..., Vol 2, p.451)
-26.740(44) Stebbins & Kron (EEM recalc with new Vmags, adopting p.e.=0.03)
-26.740 Casagrande06 (ATLAS9 ODFNEW w/Grevesse & Sauval abundances)
-26.742 Casagrande06 (Colina96 synthetic)
-26.743 Casagrande06 (Thuillier04 synthetic)
-26.744(15) Stebbens & Kron (1957; updated by Bessell98)
-26.746 Casagrande06 (Kurucz04 model R=100,0000 synthetic)
-26.75(2) Hayes85 (1985IAUS..111..225H, synthetic)
-26.75(6) Hayes85 (1985IAUS..111..225H, direct measurements N=3)
-26.75 Colina96 (synthetic)
-26.75 Cox00 (Allen's Astrophysical Quantities, 4th Ed., p.341)
-26.753 Casagrande06 (MARCS synthetic)
-26.76(3) Torres11 (adopted)
-26.76 Bessell+98 (A&A 333, 231) [adopted]
-26.760(44) Stebbins & Kron (EEM recalc with new Vmags, applying Hayes85 corr.)
-26.764 Stritzinger05 (PASP, 117, 810) (synthetic)
-26.77 Bessell+98 (A&A 333, 231)[SUN-OVER(ATLAS9, overshoot)]
-26.77 Bessell+98 (A&A 333, 231)[SUN-NOVER(ATLAS9, no overshoot)]
-26.77 Lang74 (Astrophysical Formulae, p. 562)
-26.78 Lang91 (Astrophysical Data: Planets and Stars, p.103)
-26.78 Allen63 (Astrophysical Quantities, 2nd ed.)
-26.81(5) Nikonova49 tranformed to V-mag by Martynov60
The listed values that are *not* from reviews seem to be distributed as:
<V(Sun)> = -26.74 +- 0.02 (rms)
(and one gets the same distribution when looking at only the post-1990 literature).
Note that Hayes (1985) applies a horizontal extinction correction to
the result from Stebbins & Kron (1957) [-26.73+-0.03 p.e.] to get
-26.75+-0.03. i.e. they subtract 0.02 mag.
The Engelke10 ASUN model is a Kurucz model for the Sun scaled assuming
solar constant of 1367 W m^-2. However, more recent work suggests
1360.8+-0.5 W m^-2 (Kopp & Lean 2011, Geop. Res. Let., 38, L01706).
This suggests that the Engelke10 Vmag values should be offset +0.00494 mag
(fainter). The Engelke10 values corrected to the Kopp & Lean (2011)
solar irradiance are:
-26.700 Engelke10 (Rieke08 synthetic + Engelke08 zero reference)
-26.701 Engelke10 (Rieke08 synthetic + Vega from Rieke08)
-26.718 Engelke10 (ASUN model + Engelke08 zero reference)
-26.718 Engelke10 (Kurucz model + Engelke08 zero reference)
-26.719 Engelke10 (ASUN model + Vega from Rieke08)
-26.719 Engelke10 (Kurucz model + Vega from Rieke08)
So all of the Engelke10 synthetic V magnitudes appear to be consistent
with V = -26.71+-0.01 mag. The effect of the shift on the previously
calculated mean and rms is negligible.
Two important papers from the mid-1980s on the subject appear to given
consistent values.
-26.75 +- 0.025 mag ; Neckel (1986; A&A 159, 175; synthetic photometry)
-26.75 mag ; Hayes (1985; IAU Symp. 111, 225; synthetic photometry)
-26.75 +- 0.06 mag ; Hayes (1985; IAU Symp. 111, 225; direct estimate)
The Hayes (1985) mean of published "direct" Vmag estimates comes from
Nikonova (1949; transformed to V-magnitude scale by Martynov 1960),
Stebbins & Kron (1957), and Gallouet (1964). Both the Hayes (1985)
and Neckel (1986) "synthetic" estimates assume that V=0.03 mag for
Vega.
Torres (2010; AJ, 140, 1158) adopted a consensus estimate of
-26.76+-0.03 mag based primarily on the Hayes (1985) paper. I would
argue that this value is too low - possibly weighed down by treating
the value of V=-26.81+-0.05 in an unweighted manner (it is the most
extreme of >20 values listed). All recent values (since 1995) are
between -26.70 and -26.76 -- none lower.
V(Sun) = -26.75 mag is favored by Hayes85, Neckel86, and adopted in
Cox00 in the modern edition of Allen's Astrophysical Quantities, while
Bessell98 and Torres10 advocate V(Sun) = -26.76 mag. However this
seems on the low side compared to the ensemble of published values.
I adopt a consensus value of V(Sun) = -26.74 +- 0.02 mag.
###################################################
# Absolute V Magnitude: Mv(Sun) = 4.83 +- 0.02 mag
###################################################
This value varies slightly in the literature. Here are various quoted
values summarized in Bessell, Castelli, & Plez (1998, A&A 333, 231;
Table A4).
Mv
4.79 Allen63 (Astrophysical Quantities, 2nd ed.)
4.79 Lang74 (Astrophysical Formulae, p. 562)
4.81 Bessell+98 (A&A 333, 231)
4.81(3) Torres10 (AJ 140, 1158) [assumes V = -26.76+-0.03]
4.82 Lang91 (Astrophysical Data: Planets and Stars, p.103)
4.82 Cox00 (Allen's Astrophysical Quantities, 4th Ed., p.341)
4.83 Allen76 (Astrophysical Quantities, 3rd ed.)
4.83 Schmidt-Kaler82 (Landolt Bornstein, Num. Data..., Vol 2, p.451)
4.87 Durrant81 (Landolt Bornstein vol VI/2A, p.82)
Bessell has found that some quotes combinations of V, Mv, Mbol, and BC
from some authors are not mutually consistent, a point iterated upon
in Section 4 of Torres (2010; AJ, 140, 1158).
Recent reviews by Bessell98 and Torres10 advocate Mv(Sun) = 4.81.
Note that the distance modulus for the Sun is constant, independent
of what particular value you adopt for the AU in physical units.
The distance modulus for 1 AU will always be (m-M) = -31.5721.
So the absolute magnitude of the Sun is set by the adopted Vmag for the
Sun. I have adopted Vmag(Sun) = -26.74 +- 0.02 mag as a consensus
value based on published estimates since 1995 (but statistically consistent
with older values).
Hence, I adopt Mv = (-26.74+-0.02) - (-31.5721)
Mv = 4.8321 +- 0.02
Mv = 4.83 +- 0.02 (rounded)
where the uncertainty comes from the rms in the estimated solar V magnitude.
####################################################
# Absolute B Magnitude: M_B(Sun) = 5.485 +- 0.02 mag
####################################################
I simply calculate this from the adopted values for Mv and B-V
(where the solar B-V comes from Ramirez et al. 2012):
M_B = M_V + (B-V) = (4.8321+-0.02) + (0.653+-0.003)
M_B = 5.4851 +- 0.020 mag
M_B = 5.485 +- 0.02 mag
##################################################################
# Bolometric Correction: BCv(Sun) = -0.08 +- 0.02 mag
# Absolute Bolometric Magnitude: Mbol(Sun) = 4.7554 +- 0.0004 mag
##################################################################
WARNING! Different authors have different ways of setting the
zero-point for their bolometric correction and bolometric magnitude
scales. I recommend reading Appendix D of Bessell, Castelli, & Plez
(1998; A&A 333, 231) for a more detailed discussion. If you mix and
match systems, you can systematically affect the stellar luminosities
that you calculate (which can affect the ages and masses you infer
from putting stars on theoretical isochrones; and possibly deny you
tenure!). Since the Bessell et al. 1998 paper was published, however,
two IAU commissions have agreed upon a zero point flux for the
bolometric magnitude scale (see below).
Kurucz (1979; ApJS, 40, 1) set the zero-point of his bolometric
correction scale for the object which had the minimum bolometric
correction in his suite of stellar models: a Teff=7000, log(g) = 1.0
model. On this system, the bolometric correction for the Sun (Teff ~
5780K) is BCv = -0.194. One sometimes sees authors use this scale.
Bessell, Castelli, & Plez (1998) adopt a consistent system where
V(Sun) = -26.76, and the solar bolometric magnitude is *defined* as
Mbol = 4.74. This gives a bolometric correction of BCv(Sun) = -0.07.
IAU Commissions 25 (Stellar Photometry and Polarimetry) and 36 (Theory
of Stellar Atmospheres) adopted a zero point in 1999 for the
bolometric luminosity scale, where M_bol = 0 corresponds to an
absolute bolometric luminosity of L = 3.055e28 W. From the text from
IAU Commission 36 attributed to Cram & Pallavicini, "This choice is
intended to be close to the most current practice, and its equivalent
to taking the value M_bol = 4.75 (C. Allen, "Astrophysical
Quantities") for the nominal bolometric luminosity adopted for the Sun
by international GONG project (L_Sun = 3.846e26 W)." The choice of
constant also dethrones the Sun (a variable, evolving, and
surprisingly poorly calibrated source of luminosity!) as the defining
body for the bolometric magnitude and luminosity scale.
Another thing that has occurred in recent years which affect
the value of absolute bolometric magnitude for the Sun is a
revision in the total solar irradiance (TSI; see section on TSI).
The TSI has recently been revised downwards by ~5 W m^-2 to
1360.8+-0.5 W/m^2 (Kopp & Lean 2011), taking advantage of recent
TSI measurements since 2003 with the SORCE/TIM radiometer, which
is absolutely calibrated to 0.035%.
Using the 1999 IAU zero point for the bolometric luminosity scale
(3.846e26 W), and the solar luminosity that I calculated using the
Kopp & Lean (2011) total solar irradiance (and adopting the 2009 IAU
value for the astronomical constant (AU = 149597870700 m) => L_Sun =
3.8270(+-0.0014)e33 erg/s), then the bolometric magnitude of the Sun
becomes:
M_bol(Sun) = 4.7554 +- 0.0004 mag
As I have adopted V = -26.74 (+-0.02) and Mv = 4.832 (+-0.02), then
the bolometric correction of the Sun will be defined as:
BC_V(Sun) = -0.077 +- 0.02 mag =>
BC_V(Sun) = -0.08 +- 0.02 mag (rounded)
where the uncertainty is dominated by the uncertainty in the solar
Vmag (+-0.02 mag). Both the updated TSI uncertainty (0.035%) and
bolometric magnitude scale luminosity zero-point (defined exactly)
contribute negligibly to the uncertainty.
Hence, if one adopts the IAU bolometric flux zero point constant, and
M_bol(Sun) = 4.7554 (+-0.0004), then one should make sure that one's
choice of bolometric correction relations as a function of stellar
Teff (and/or other variables) is calibrated to BC_V(Sun) for the solar
Teff and/or color: BC_V(Sun) = -0.08 +- 0.02 mag.
#######################################
# Radius: R(Sun) = 695660 (+-100) km
#######################################
Tables of published solar radii values (angular radii and/or physical
radii) are given by Gobasi+2000, Kuhn+2004 (ApJ, 613, 1241), and
Haberreiter+2008 (ApJ 675, L53).
For evolutionary models, one would define a star's radius to be where
the temperature equals the effective temperature, i.e. the Rosseland
mean opacity = 2/3. According to Haberreiter, Schmutz & Kosovichev
(2008, ApJ, 675, L53), this is approximately 14 km higher than the
radius at which the optical depth at 5000A equals unity. They also
estimate that the layer at which one measures an inflection point in
5000A light intensity at the solar limb is 333+-8 km/s higher than the
radius as measured by opacity_Rosseland = 2/3 (studies that measure
the solar diameter using e.g. solar meridian transits like Brown &
Christiansen-Dalsgaard 2008, take this model-dependent correction into
account). Haberreiter+2008 claim that the differences in the intensity
profile radii and seismic radii can be reconciled, and that both are
now consistent with 695660 km.
Haberreiter, Schmutz, & Kosovichev (2008) summarize
the solar radius literature in their abstract: "Two methods are used
to observationally determine the solar radius: One is the observation
of the intensity profile at the limb; the other one uses f-mode
frequencies to derive a "seismic" solar radius which is then
corrected to optical depth unity. The two methods are inconsistent and
lead to a difference in the solar radius of $(C!-(B0.3 Mm. Because of the
geometrical extension of the solar photosphere and the increased path
lengths of tangential rays the Sun appears to be larger to an observer
who measures the extent of the solar disk." They also compile a list
of published solar radii (their Table 2) and discuss the subtle and
systematic differences between measured and seismic radius
estimates. There are systematic differences at the hundreds of km*
level that likely occur due to differences in the techniques and what
level of the solar photosphere defines the solar radius.
(* = or tenths of a Mm level, to use the silly choice of obscure SI
units elected by some authors - probably the only use of megameter as
an SI unit that I have seen in the literature).
Published angular radii for the Sun (or inferred angular radius from a
radius published in km or Mm):
959".03 +-0".07 ; Golbasi+ 2000 (A&A 368, 1077)
959".28 +-0".15 ; Emilio, Kuhn, & Bush 2010 IAU 264, 21 (MDI/SOHO)
959".29 +-0".15 ; Kuhn+ 2004 (MDI)
959".321 +-0".024 ; Ribes91 ("The Sun in Time"; HAO data)
959".35 ; Antia98 (seismic) (as quoted in Haberreiter08)
959".41 +-0".01 ; Laclare+ 1996 (OCA astrolab)
959".44 +-0".08 ; Ribes91 ("The Sun in Time"; CERGA data)
959".52 +-0".03 ; Emilio & Leister 2005 MNRAS 361, 1005 (visual data)
959".53 +-0".06 ; Sofia+ 1994 (SDS)
959".57 +-0".04 ; Emilio & Leister 2005 (SP astrolab) [in Haberreiter08]
959".61 +-0".05 ; Emilio & Leister 2005 MNRAS 361, 1005 (CCD data)
959".62 +-0".03 ; Neckel+ 1995 (McMath ST)
959".63 +- ; Cox 2000 (Allen's Astrophys. Quan. 4th Ed; rad = 959".63)
959".63 +-0".10 ; Auwers 1891 AN 128, 361 (diam = 1919".26 +- 0".10)
959".63 +-0".01 ; Allen 1963 (Astrophys. Quan. 2nd Ed.; "circular to +-0".01)
959".63 +-0".01 ; Allen 1953 (Astrophys. Quan. 1st Ed.; "circular to +-0".01)
959".64 +-0".02 ; Chollet & Sinceac 1999 A&AS 139, 219
959".6795+-0".018 ; Brown&Christensen-Dalsgaard 1998 (ApJ 500,L195)
959".73 +-0".05 ; Wittmann 1997
960".12 +-0".09 ; Emilio+2012 ApJ 750 135 (MDI/SOHO, Mercury transit)
960".63 +-0".04 ; Whittmann 1997 (as quoted by Haberreiter+2008)
Surprisingly, Cox 2000 quotes the physical radius from Brown &
Christensen-Dalsgaard 1998, but not their angular radius. Cox 2000
quotes an oblateness as the semidiameter equator-pole difference as
0".0086 (see next section).
Djafer, Thuillier, & Sofia (2008; ApJ 676, 651) compare a few solar
diameter datasets and confirm that there are (unsurprisingly)
systematic differences between measurements from different
instruments. They conclude that once systematic effects are taken
account of (plausibly modeled by the authors), the Calern, SDS, and
MDI angular radii for the Sun are consistent within their quoted
errors. Their re-analysis of the three datasets give corrected
estimates of:
959".705 +- 0".150 (MDI data; Djafer+ 2008)
959".811 +- 0".075 (Calern data; Djafer+ 2008)
959".898 +- 0".091 (SDS data; Djafer+ 2008)
Djafer et al. do not estimate a mean value. Calculating an unweighted
mean of these three estimates gives:
959".805 +- 0".056 (mean)
Here are some recently quoted values for the photospheric solar radius
(by no means exhaustive).
Rsun(km)
695508+-26 Brown & Christensen-Dalsgaard 1998 (adopted by Cox 2000)
695660 Haberreiter+2008 (radius where T = Teff)
695680+-300 Schou+1997 (helioseismic)
695700 Goldberg in Kuiper51 ("The Sun", p. 18)
695740+-110 Kuhn02,Kuhn04,EmilioKuhnBush10 (SOHO MDI experiment)
695749+-241 Richards04(adopted mean of Brown98,Antia98,Schou97)
695780 Antia98 (seismic)
695830+-7 Laclare96 (OCA astrolab)
695917+-43 Sofia94 (SDS)
695946+-29 EmilioLesiter05 (SP astrolab)
695980+-70 Allen63 (Astrophysical Quantities, 2nd Ed.) - no reference
695980+-70 Lang74 (Astrophysical Formulae)
695982+-22 Neckel95 (McMath ST)
695990+-70 Allen73 (Astrophysical Quantities, 3rd Ed.)
696000+-100 Allen55 (Astrophysical Quantities, 1st Ed.)
696342+-65 Emilio+ 2012 (SOHO/MDI Mercury transit; ApJ 750, 135)
Inexplicably, the most commonly adopted value for the solar radius is
that of Allen (1973), for which Brown & Christensen-Dalsgaard (1998)
state "It is not clear how the value quoted by Allen (1973) was
obtained."
Haberreiter+2008 claim to reconcile the large range in published solar
radii by correcting the inflection point measurements to the radius of
the effective temperature, and their corrected inflection point radii
(corrected literature mean value 695568+-98 km) and seismic radii
(literature mean 695658+-140 km) appear to agree within 90+-171 km.
Harmanec & Prsa (2011) support use of the Brown &
Christensen-Dallsgard (1998) radius as the standard value.
The recent review on eclipsing binaries by Torres, Andersen, & Gimenez
(2010, ARA&A) adopts the new (exact) radius from Haberreiter et
al. 2008 for quoting the radii of stars. I agree that this is a good
idea, and adopt the Haberreiter+2008 solar radius (695660 km).
The uncertainty in the value is probably ~+-100 km (+-0.014%) based on
Table 3 of Haberreiter+2008, although it is probably safe to adopt it
as "exact" for the purposes of quoting stellar radii.
###############################
# Solar Oblateness = 8 x 10^-6
###############################
Oblateness is a measurement of the flattening of an object due to
rotation. The solar oblateness is very tiny.
Oblateness is usually quoted as f = (a-b)/a where a is the equatorial
radius and b is the polar radius. For the Sun, the oblateness is often
quoted as the angular difference between the equatorial and polar
radius (usually in arcseconds or milliarcseconds [mas]).
Fivian+ 2008 (Science, 322, 560) states "The surface rotation rate, ~2
kilometers per second at the equator, predicts an oblateness
(equator-pole radius difference) of 7.8 milli arcseconds, or ~0.001%."
There are claims of slight variations in the oblateness measured at
optical wavelengths which appear to be anti-correlated with solar
activity (Egidi+ 2005, Solar Physics, 235, 407).
oblateness =
4.3-10.3 X 10^-6 ; Egidi+2005, Solar Physics, 235, 407
11.5 +- 3.4 mas ; Rozelot Cool Stars 10, Vol. 154, 685
8.6 milliarcseconds ; Cox+2000 Allen's Astrophysical Quantities
8.01 +- 0.14 mas ; Fivian+2008
Fivian+2008 "The corrected oblateness of the nonmagnetic Sun is 8.01 ,A1(B
0.14 milli arcseconds, which is near the value expected from
rotation." For the solar radius adopted previously (959680
milliarcseconds), this translates to an oblateness of 8.35 X 10^-6 ~
1/120,000, or a difference in the polar and equatorial radii of ~6 km.
I am still investigating the literature on this topic, but it appears
that for all practical purposes the oblateness is negligible (~10^-5).
######################
# Spectral Type: G2V
######################
G2V ubiquitous
(except apparently the first edition of Allen's Astrophysical
Quantities from 1955, which listed G1V).
The integrated solar spectrum (as inferred from reflection spectra of
bodies like the moon, Uranus, Callisto, etc.) is *the* G2V standard or
"dagger type" of the MK system (Morgan & Keenan 1973; Houk 1988;
Garrison 1994).
The Sun's spectral type varies as a function of angle from the
limb, from roughly ~G0 near the center to ~K0 near the limb.
Morgan & Keenan (1939) listed the following spectral types
as a function of distance from center of the Sun. The spectral
types are on the "MW" system, which, given the authorship of the
paper, can be construed as "Morgan-Keenan" system.
Distance Spectral
from Type
Center (MW system) Teff
_____________________________
Center G1 5990K
0.750R G4 5720K
0.945R G9p^1 5070K
0.985R K0p^2 .....
(1) Spectral type determined from ratio Fe4045/Hdelta; the strong
metallic arc lines are weaker than in a G9 dwarf.
(2) Metallic arc lines and Ca+ are much weaker than in a K0 dwarf.
Spectral type determined from ratio Fe4045/Hdelta. The H-lines are
very weak.
I have been unable to find out why the Sun was called "G2" instead of
"G0" or "A1" or "obviously the best spectral type in the universe",
but rumor has it that God mentioned it to Aaron somewhere deep in the
passages of Leviticus.
###############################################
# Effective Temperature: Teff = 5771.8 +- 0.7 K
###############################################
5777 K Cox 2000 (Allen's Astrophysical Quantities, 4th Ed.)
5781 K Bessell et al. 1998 A&A 333, 231
The effective temperature can be calculated using the total solar
irradiance (TSI) value, solar radius, AU, and the Stefan-Boltzmann
constant. If one adopts the total solar irradiance from Kopp & Lean
(2011, Geop. Res. Let., 38, L01706), the IAU 2009 definition of the
astronomical constant, the solar radius from Haberreiter, Schmutz &
Kosovichev (2008, ApJ, 675, L53), and use the CODATA 2010 value for
the Stefan-Boltzmann constant sigma:
R = 695660(+-100) km [solar radius from Haberreiter+2008]
f = 1360.8(+-0.5) W/m^2 [TSI from Kopp & Lean 2011]
D = 149597870700 m [AU from IAU 2012 resolution; exact]
sigma_SB = 5.670373(+-0.000021) e-8 W/m^2/K^4 [SB constant from CODATA 2010]
Then one derives:
Teff = (f*D^2/(sigma_SB*R^2))^(1/4)
= 5771.777 +- 0.673 K
= 5771.8 +- 0.7 K
~ 5772 K
This is ~5-10K cooler than most previous estimates, but flows from the
slightly lower value for the total solar irradiance in the very recent
literature (e.g. Kopp & Lean 2011).
######################################################################
# Solar Mass: M(Sun) = 1.98855(24)x10^30 kg #
# Solar Gravitational Constant (TDB) = 1.32712440041e20 m^3 s^-2 #
# Solar Gravitational Constant (SI, TCB) = 1.32712442099e20 m^3 s^-2 #
######################################################################
Cox 2000 (Allen's Astrophysical Quantities, 4th Ed.) lists
Msun = 1.989e30 kg
The IAU's recommended dynamical constants for the solar system
(including the solar system) are listed at the website for the IAU
Working Group on Numerical Standards for Fundamental Astronomy (NSFA)
at:
http://maia.usno.navy.mil/NSFA/IAU2009_consts.html
Heliocentric Gravitational Constant (GMsun) values
(for the solar system barycentric reference frame - not "SI"):
1.32712438 e20 m^3 s^-2 (IAU 1976 constant)
1.32712440 e20 m^3 s^-2 (Cox 2000)
1.32712440018e20 m^3 s^-2 (DE405 ephemeris; Klioner 2005 astro-ph/0508292)
1.32712440041e20 m^3 s^-2 (DE423 ephemeris; see below)
1.32712440041e20 (+-1.0e10) m^3 s^-2 (IAU 2009 constant ; TDB-compatible)
DE423 ephemeris (31 Mar 2010; JPL website:
ftp://ssd.jpl.nasa.gov/pub/eph/planets/ascii/de423/header.423 ) lists
GMsun = 0.295912208285591100D-03 AU^3/day^2. Using the DE423 value for
the IAU and 86400 sec/day, I translate this to be GMsun =
1.32712440041e20 m^3 s^-2 (i.e. the IAU 2009 TDB-compatible constant).
To put these values in SI units, they must be corrected for the
difference in timescales between the solar system barycentric time and
terrestrial time. A review of the complex history and current system
of IAU-sanctioned time systems is beyond the scope of this document,
but I briefly review some of the material relevant to understanding
the differences in quoted masses.
Note that the ephemerides and their associated constants are quoted on
Barycentric Coordinate Time (TCB), and in a space-time coordinate
system called the Barycentric Celestial Reference System (BCRS). TCB
is the time coordinate ("clock") for the solar system barycenter,
running at a rate equal to the SI second. But TCB runs at a different
rate compared to Terrestrial Time (TT), and these can not be mixed up
in calculations without introducing systematic errors. In practice,
the TCB is realized through the IAU's (2006) definition of Barycentric
Dynamical Time (TDB), which follows the JPL ephemeris time argument in
JPL Development Ephemeris 405 (DE 405), and which is used in the
Astronomical Almanac. A 2006 IAU resolution (#3) defined TDB to be a
linear transformation of TCB, where 1 - d(TDB)/d(TCB) = 1.550519768e-8
(from IAU NSFA working group webpage
http://maia.usno.navy.mil/NSFA/CBE.html and
http://maia.usno.navy.mil/NSFA/IAU2009_consts.html). Many thanks to
Erik Bergren for notes on TCB vs. TT and its effects on astronomical
constants.
The "SI" versions of GM_Sun:
1.32712442076 e20 m^3 s^-2 (Kovalenvsky & Seidelmann 2004,SI)
1.3271244208 e20 m^3 s^-2 (DE405 ephemeris; Klioner 2005 astro-ph/0508292, SI)
1.327124420997e20 m^3 s^-2 (DE423 ephemeris; calc. Erik Bergen, priv. comm.))
1.32712442099 e20 (+-1.0e10) m^3 s^-2 (IAU 2009 constant ; TCB-compatible)
The CODATA 2010 value for Newtonian constant G is:
6.67384e-11 m^3 kg^-1 s^-2 (1.2e-4 relative uncertainty)
http://physics.nist.gov/cgi-bin/cuu/Value?bg|search_for=universal_in!
Adopting the CODATA 2010 value for G, and the IAU 2009 constant for GMsun,
I calculate:
M(Sun) = GMsun/G = 1.98855(24)x10^30 kg, where the uncertainty is
dominated by the uncertainty in G (~1.2e-4).
Note that while the product of G times the mass of the Sun ("GMsun")
is known to a precision of 1 part in 13 billion, the uncertainty in
the Newtonian gravitational constant is still about 1 part in
8,300. This translates into an uncertainty in our estimate of the mass
of the Sun to be a whopping 2.4x10^26 kg -- or *40 Earth masses*!
In rough terms:
Mass(Sun) = 1047.35 x Mass(Jupiter)
Mass(Sun) = 322,946 x Mass(Earth)
Mass(Sun) = 27,068,703 x Mass(Moon)
####################################
# Sun's Bulk Density = 1.411 g cm^-3
####################################
Adopting the following parameters:
Mass M = 1.98855e30 kg (updated April 2012 using IAU 2009 GM and CODATA 2010 G)
Equatorial radius = a = 695508 km
Polar radius = b = a(1-f) = 695502 km
Density = Mass/Volume = Mass/(4 pi a^2 b / 3)
Density = 1411.1 +- 0.2 kg/m^3
= 1.4111 +- 0.0002 g/cm^3
Where the uncertainty in the solar mass (relative error 1.2e-4;
dominated by the uncertainty in G) dominates the uncertainty in
density.
##################################################################
# Moment of Inertia I = 5.96e53 g cm^2 = 5.96e47 kg m^2 #
# Inertia Constant k = 0.062 #
##################################################################
The moment of inertia is calculated by integrating int(r^2 dm) from
the core to the surface. Moment of inertia is usually parameterized
by the form I = k M R^2.
Given our adopted solar mass (1.98842e30 kg) and radius (695508 km),
the product of (M R^2) = 9.61861e54 g cm^2 = 9.61861e47 kg m^2
Allen's Astrophysical Quantities quotes I = 5.7e53 g cm^2 (implying k = 0.059)
The value estimated for a 1 Msun solar metallicity star
from the Lyon models (assuming mixing length = pressure scale height)
is k = 0.062 (implying I = 5.96e53 g cm^2 = 5.96e46 kg m^2).
Here is a list of quoted k-values:
0.059 Allen's Astrophysical Quantities (Cox 2000)
0.06 Moons & Planets, 5th Edition, W.K. Hartmann (2005, p. 198)
0.062 Lyon models
I've adopted the k-value and MOI inferred from the Lyon models.
##############################################################
# Solar Surface Gravity: g = 27423.2 (+-7.9) cm/s^2
# g = 274.232 (+-0.079) m/s^2
# g = 27.9638 X g(Earth)
# log(g) = 4.43812 (+-0.00013) dex [cgs]
##############################################################
For now, I will simply calculate a "standard" value for the solar
surface gravity, which ignores solar rotation and oblateness. One can
simply estimate the Sun's "surface" gravity at the solar photosphere
as:
g = GMsun/Rsun^2
adopting the best values (from previous discussion) of:
GMsun = 1.32712442099e20 (+-1.0e10) m^3 s^-2 (IAU 2009 constant ; TCB-compatible)
Rsun = 695660 (+-100) km
"GM" values are usually quoted in MKS units, and the Sun's radius is
usually quoted in km. So of course, astronomers usually quote stellar
surface gravities in log10 of the surface gravity... in cgs
units. Life is not fair.
One derives a solar surface gravity of:
g = 27423.2 (+-7.9) cm/s^2
g = 274.232 (+-0.079) m/s^2
g = 27.9638 times Earth's "standard gravity" (assuming g(Earth) = 9.80665 m/s^2)
log(g) = 4.43812 (+-0.00013) dex [cgs]
This value will be slightly lower at the equator due to rotation (to
be calculated later), but this standard value should be very accurate
at the solar poles (at least to the degree to which we are ignoring solar
oblateness).
##############################################################
# Distance = Astronomical Unit (AU) = 149597870700 m (exact) #
# Mean Earth-Sun Distance <r> = 149618753000 m #
##############################################################
In August 2012, the IAU General Assembly adopted resolution B2, which
re-defined the astronomical unit "to be a conventional unit of length
equal to 149 597 870 700 m exactly". The IAU resolution also adopted
the symbol "au" to be used for the astronomical unit. As I find "au"
to be a silly symbol (as it can be confused with atomic units), I use
"AU" throughout.
The 2012 re-definition of the AU was a major change, as historically
the astronomical unit was defined with respect to an auxiliary
constant: the Gaussian gravitational constant. The astronomical unit
was previously defined as that length for which the heliocentric
gravitational constant (GM_Sun) is equal to (0.01720209895)^2
AU^3/d^2, where the mean sidereal motion of the Earth's orbit is
0.01720209895 radians per day. This Gaussian gravitational constant
(0.0172...) was originally defined by Simon Newcomb (1895) in "Tables
of the Motion of the Earth on its Axis and Around the Sun". This
constant remained at that value for over a century.
I've split discussion on the AU into five sections:
I: Pre-2009 values for the astronomical constant
II: The new IAU value for the astronomical constant (2009, 2012)
III: Notes on the astronomical constant
IV: The mean Earth-Sun distance (not the AU!)
V: AU in light seconds
Defining the AU as a constant was a good idea for a few reasons.
Really, we want to use the AU as a convenient yardstick with which to
quote distances on the scale of planetary orbits and separations
between stars (at least on sub-parsec scales). Unfortunately, the
actual semi-major axis of the Earth's orbit is changing all the time,
making its "true" value an inconvenient yardstick. The mass of the Sun
is constantly (but subtly and negligibly) decreasing due to nuclear
reactions (via photons and neutrinos) and the solar wind (via hot
coronal plasma escaping the solar system). The predicted change in the
AU due to these known mechanisms results in +0.3 meter/century
(Krasinsky & Brumberg 2004, Celestial Mechanics and Dynamical
Astronomy, 90, No. 3-4, p. 267). The semi-major axis of the Earth's
orbit already changes all the time due to perturbations of the other
planets, and indeed the Astronomical Almanac quotes "osculating"
orbital elements for the Earth's orbit which oscillate around the 1 AU
level. The semi-major axis of the Earth's orbit may also suffer from
long-term variations due to subtle interactions between the planets
and minor bodies, etc. Indeed, there is some evidence for a recent
trend where the Earth's semi-major axis is increasing at the ~7-15
meters/century level (Krasinsky & Brumberg 2004; Standich 2004). If
this trend is significant, a perusal of the literature suggests that
the bulk of the motion is not as yet satisfactorily explained. While
the variations in the Earth-Sun distance are a field of study, the
2012 IAU resolution sets the AU as a set number of meters, and allows
its usage as a yardstick independent of the vagaries of the behaviour
of the Earth's actual orbit around the Sun.
___
I: Pre-2009 values for the astronomical constant
Here is a list of some pre-2009 published values for the AU:
149597870000 m IAU 1976 constant (standard value)
149597870660 m IAU 1976 value used in preparing ephemerides (not clear why different)
149597870660 +- 2 m JPL DE118/LE118 (DE200/LE200), Seidelmann 1992
149597870691 m JPL DE403 (1995),
IAA's EPM2000,
IERS2003 values
149597870691 +- 3 m DE405 (1997)
149597870698 +- 2: m Standich (2004; IAU 196, p. 163; see below)
149597870696.0 +- 0.1 m EPM2004 (Pitjeva 2005)
149597870697 +- 1 m DE410
149597870700.8 +- 0.15 m DE414 (Standich 2006)
149597870699.6 +- 0.15 m DE421
149597870695.4 +- 0.1 m EPM2008 (Pitjeva 2008)
149597870699.22 +- 0.11 m INPOP2008 (Fienga et al. 2009)
149597870700 +- 3 m Pitjeva & Standich (2009; proposal to IAU
Working Group on Numerical Standards for
Fundamental Astronomy)
149597870699.626200 m DE423 (no uncertainty,
ftp://ssd.jpl.nasa.gov/pub/eph/planets/ascii/ 31 March 2010)
149597870700 (exact) m 2012 IAU General Assembly Resolution B2
According to the IAU NSFA (2009) website, "The value for au is
TDB-compatible. An accepted definition for the TCB-compatible value of
au is still under discussion." The TDB is appropriate for the solar
system barycentric reference frame, or practically equivalent to the
JPL ephemeris time argument T_eph as implemented in JPL ephemeris
DE405 (as used in the Astronomical Almanacs since 2003). TCB is
equivalent to the proper time measured by a clock at rest in the solar
system barycentric coordinate frame - i.e. not subject to
gravitational time dilation caused by the solar system's bodies. An
extensive discussion the IAU's time systems can be found in USNO
circular 179 by George Kaplan:
www.usno.navy.mil/USNO/astronomical-applications/publications/Circular_179.pdf
Table 2 of Fienga et al. 2009 summarizes recent results regarding
estimation of the astronomical unit and some other physical parameters
for planetary ephemerides. The Fienga et al. 2009 value is fitted by
adopting the GM_sun value from DE405. The paper can be found at:
http://www.imcce.fr/fr/presentation/equipes/ASD/inpop/
Pitjeva (2005) tabulates the recent ephemeris updates and what new data
was included in the new analyses.
Standish (2004, "Transits of Venus: New Views of the Solar System and
Galaxy, IAU 196, p. 163) reports that "The recent addition of the MGS
and Odyssey ranges tend to indicate a value for the au which is a
couple of meters shy of 149,597,870,700 m", and lists 149,597,870,698
with a ~2 meter uncertainty (in Q&A discussion after paper).
II: The new IAU value for the astronomical unit (2009, 2012)
Pitjeva & Standich (2009; Celestial Mechanics and Dynamical Astronomy,
103, 365) "proposed the... astronomical unit in meters obtained from
the ephemeris improvement processes at JPL in Pasadena and at IAA RAS
in St. Petersburg... AU = 149597870700(3) m."
On 13 Aug 2009, the XXVIIth General Assembly of the IAU at the meeting
in Rio de Janeiro, passed resolution B2, which adopted a set of
current best estimates for astronomical constants proposed by the IAU
Working Group on Numerical Standards for Fundamental Astronomy (NSFA
WG). The table of adopted constants is at:
http://maia.usno.navy.mil/NSFA/CBE.html
The 2009 IAU value for the astronomical constant was adopted directly
from Pitjeva & Standich (2009): a = 149597870700+-3 m.
At the 2012 IAU General Assembly in Beijing, the assembly passed
resolution B2, which decided to adopt the Pitjeva & Standich (2009
IAU) value as an exact number, and drop the uncertainty when using the
astronomical unit as a unit of length. Hence, the IAU definition of
the astronomical unit is now decoupled from the dreaded Gaussian
gravitational constant (see IAU 1976 constants) *and* the actual
semi-major axis of the Earth's orbit around the Sun.
III: Notes on the astronomical unit
Discussion on 2009 definition:
One finds the AU quoted in two conventions: based on barycentric
dynamical time (TDB) and "SI" (calculated for a hypothetical observer
measuring proper length and proper time at the solar system
barycenter). According to a footnote on the IAU NSFA Working Group
website: "The value for au is TDB-compatible. An accepted definition
for the TCB-compatible value of au is still under discussion."
The 2012 definition of the astronomical constant sets it as an integer
number of meters, independent of which reference frame it is used in.
IV: The mean Earth-Sun distance
Note that the AU is *not* the "mean" distance between the Earth and
Sun!
One must take into account the fact that a planet will spend a longer
portion of its orbit near aphelion and shorter time near perihelion.
The *mean distance* is then (Standish 2004, D. Williams 2003):
<r> = a(1+e^2/2)
Where a is the semi-major axis, and e is the mean eccentricity.
(e_Earth = 0.016708617; IAU 1976 value). The mean value of the
eccentricity from the osculating orbital elements from the 2012
Astronomical Almanac is e=0.016700. Using the IAU 1976 eccentricity
value, and the IAU 2012 value of the AU, I estimate:
<r> = 149618753000 m (with uncertainty in the last four digits)
V. Astronomical Unit in light-seconds
Adopting the IAU 2012 estimate for the AU: 149,597,870,700 m (exact), and
the speed of light c = 299,792,458 m/s (exact), one can easily
calculate the AU in light-seconds:
AU = 499.00478383615643451776122674345970153808593750 sec (exact)
AU = 499,004,783,836.15643451776122674345970153808593750 nanosec (exact)
All digits after the final zero are zero (not that anyone needs this
number to that precision!)
Commentary: I find the choice of "au" (as spelled out in the 2012 IAU
resolution B2) to be a silly choice to be used for the astronomical
unit. I say this because "au" also means "atomic units" in atomic
physics parlance, which isn't too far removed from the astronomical
literature. As far as I know, "AU" (capitalized) is fairly unique in
science, and is how I usually see the astronomical unit quoted in the
literature. "Au" is obviously the symbol for gold, and would not have
been a good choice.
############################################################
# Total Solar Irradiance: S(1AU) = 1360.8 (+-0.5) W/m^2 #
############################################################
Total solar irradiance is the power per unit area of energy coming
from the Sun's light (at all wavelengths) measured at the reference
distance 1 astronomical unit (AU). I have occasionally called "total
solar irradiance" (TSI) by some other names: the solar flux constant,
solar bolometric flux, etc. (where one assumes it is as measured at 1
AU). The values in the literature always refer to the power per unit
area measured at 1 AU, since the eccentricity of the Earth's orbit
produces a small annual amplitude for measurements taken from Earth or
Earth orbit.
<S(1AU)> W/m^2 = Total Solar Irradiance at 1 AU:
1360.6+-0.5 [400 yr mean] from Kopp & Lean, and Wang+2005 (see below)
1360.8+-0.5 [2008 minimum] Kopp & Lean (2011,Geop.Res.Let.,38,L01706)
1365.5 Brusa (1983, Publ. Phys.-Meteorol. Obs. Davos, No. 598)
1365.5 ["contemp. quiet Sun"] Wang+ (2005, ApJ, 625, 522)
1366.85 Mekaoui & Dewitte (2008, Solar Phys. 247, 203)
1367 Frohlich (1983, Publ. Phys.-Meteorol. Obs. Davos, No. 599)
1367.2976 Tobiska (2002, Adv. Space Res. 29, 1969)
1368.2 Willson (1982, The Symp. on the Solar Constant..., p.3)
1372.7 Hickey+ (1982, The Symp. on the Solar Constant..., p.10)
1365-1369 W/m^2 Cox 2000 (error in units)
Neckel (1986; A&A 159, 175) lists multiple published values.
There is a table of total solar irradiance data that is regularly
updated by Frohlich and Jean which can be downloaded from the NGDC at:
http://www.ngdc.noaa.gov/stp/solar/solarirrad.html#composite
From their dataset for the Sun from Nov. 1978-Oct. 2003,
I find that the total solar irradiance (TSI) has the
following statistics, based on 8405 measurements
(all in W/m^2):
median = 1365.922
mean = 1366.001 (unweighted)
mean = 1366.003 (Chauvenet clipped; N=8 data points clipped)
and "dispersion statistics":
68%CL = +0.705 -0.444 (+-0.574)
probit st.dev = +-0.552 (from probability plots)
I do not list the standard error or error in the true median, as the
uncertainties are no doubt dominated by the absolute calibration. Note
that they already normalize the data to 1AU, and the flux is
calibrated to the SARR: Space Absolute Radiometer Reference.
Frohlich et al. (2006, Nature 443, 161) lists an average solar
irradiance at solar minimum of 1365+-0.009 W/m^2, with the irradiances
from one solar minimum to another during 1978-2005 only varying in
their average minima by +-<0.09 W/m^2.
Kopp & Lean (2011) quote measurements from the Total Irradiance
Monitor (TIM) on NASA's Solar Radiation and Climate Experiment
(SOURCE), which has been measuring solar irradiance since 2003. The
TIM irradiances are believed to have absolute accuracy of +-0.035% or
approximately +-0.48
W/m^2. [http://lasp.colorado.edu/sorce/instruments/tim.htm]. The TIM
instrument was designed to measure TSI to absolute accuracy 100 parts
per mission (Kopp & Lawrence 2005, Solar Phys. 230, 91), and the TIM
instrument is calibrated against the NIST Primary Optical Watt Radiometer.
Note that their
irradiances are ~5 W/m^2 below previous values, and they claim that
previous TSI experiments had slightly higher values to the
effects of scattered light in the radiometers.
Kopp & Lean (2011) take the total solar irradiance reconstruction from
Wang, Lean, & Sheeley (2005) and tie the flux calibration to the
SORCE/TIM system. They list yearly estimated and measured TSI values
from 1610 through 2011 (402 values).
The 1610-2011 mean value is:
<S(1AU)> = 1360.6459 (sem +-0.0194, st.dev. +-0.3882) W/m^2
The 1978-2011 mean value is:
<S(1AU)> = 1361.2498 (sem +-0.0634, st.dev. +-0.3696) W/m^2
Kopp & Lean (2011) quote a TSI during the 2008 solar minimum -
which appears to define a more-or-less stable value for the quiet Sun:
<S(1AU)> = 1360.8 +- 0.5 W/m^2 [2008 solar minimum; "quiet Sun"]
where the 0.5 W/m^2 uncertainty is completely dominated by the
SORCE/TIM absolute calibration error of +-0.035%. As they show in
their Fig. 1 (which combines the TIM data with rescaled TSI
measurements from spacecraft since 1978), the TSI at solar minima has
been repeatably stable during the past 3 solar minima. The monthly
averaged TSI during recent solar maxima have been approximately 1.6
W/m^2 higher (~1362.4 W/m^2).
Note that the differences between the mean values measured
over solar cycles and the historic reconstruction are all
within 1sigma of the calibration uncertainty.
I adopt the Kopp & Lean (2011) value of 1360.8 +- 0.5 W/m^2.
##################################################
# Luminosity: Lsun = 3.8265 (+-0.0013) e33 erg/s #
##################################################
Lsun(erg/s)
3.826e33 Lang74 (Astrophysical Formulae)
3.827e33 calc. by EEM from Kopp & Lean (2011) TSI historical reconstruction
3.845e33 Cox 2000
3.846e33 GONG project value (IAU Comm. 36; Andersen, Trans.IAU,1999)
3.846e33 Harmanec & Prsa (2011)
3.86(+-0.03)e33 Allen55 (Astrophysical Quantities, 1st ed.)
The Kopp & Lean (2011) reconstruction of the Wang et al. (2005)
historical TSI data (calibrated to the SORCE/TIM TSI flux calibration)
is consistent with having a long-term (1610-2011) of:
<S(1AU)> = 1360.646 +-0.477 W/m^2
where the uncertainty is completely dominated by the claimed +-0.035%
flux calibration of SORCE/TIM.
Combining this long-term TSI value with the 2009 IAU value for the
astronomical unit (149597870700 m), I estimate:
L(Sun) = <S(1AU)>*4*pi*AU^2
= (3.8265 +- 0.0013) e33 erg/s
= (3.8265 +- 0.0013) e26 W
The luminosity in log10 cgs units is then:
log(L_Sun/(erg/s)) = 33.583
This is only 0.48% lower than that from Cox (2000), and 0.51% lower
than that from value adopted by the GONG project.
Note that Harmanec & Prsa (2011) propose Lsun = 3.846e33 erg/s (exact)
as their standard luminosity. But the primary justification for this
value was given to be agreement with the 1997 IAU bolometric magnitude
zero-point - i.e. not justified with quality measurements, but by
being in agreement with an out-dated value. But this luminosity value
is difficult to justify as the Kopp & Lean (2011) recalibration of the
TSI scale has made it obsolete.
#################################################
# Main Sequence Luminosity Evolution of the Sun #
#################################################
I have not seen a useful formula for the luminosity evolution
of the Sun during its main sequence phase. So I estimate one here.
The estimate of the luminosity vs. time comes from a 1 solar mass
model from the Yale-Yonsei evolutionary tracks, where I adopted
Z=0.0181 (their recommended solar value) and [alpha/Fe]=0.0.
Between an age of 45 Myr (when the Sun reached the Zero-age main
sequence, a luminosity minimum) and 11.1 Gyr (the end of the main
sequence stage), one can approximate the luminosity of the Sun as:
log10(L/Lsun,now) = a0 + a1*(t/Gyr) + a2*(t/Gyr)^2 + a3*(t/Gyr)^3
where the coefficients are:
a0 = -0.152212064
a1 = 0.0400317
a2 = -0.002721567
a3 = 2.745474E-4
Note that I rescaled the actual Y^2 track by ~0.01 dex so as to force 1
solar luminosity at age 4.567 Gyr.
Here are the inferred normalized solar luminosities at
some interesting points in the Sun's life:
L/Lsun(now) Time Notes
Runaway GH 1.410 3730 Myr future "Runaway Greenhouse limit" (Kasting+1993)
10% brighter 1.100 1210 Myr future "Water loss limit" (Kasting+1993)
Now 1.000 0 Myr ago This is now, now
K/T boundary 0.995 65 Myr ago Asteroid 1, Dinosaurs 0
P/T boundary 0.982 251 Myr ago The Great Dying
Cryogenian 0.947 ~750 Myr ago Snowball Earth epoch ~850-630 Myr ago?
First life 0.750 3850 Myr ago Evidence of life in Greenland rocks?
#####################################################
# Mt. Wilson S-value: S = 0.1762
# Chromospheric Activity Index logR'HK = -4.903 dex
#####################################################
The Mt. Wilson S-value is a bandpass ratio which measures the strength
of the Ca II H & K emission lines from the Sun. These emission lines
originate from the Sun's chromosphere - a hot plasma layer above the
Sun's photosphere. The Mt. Wilson S-value and its associated index
logR'HK (logarithm of the Ca H & K flux to the star's bolometric flux)
are indicators of stellar magnetic activity, related to the generation
and evolution of stellar magnetic fields.
Here is a list of published long-term average Mt. Wilson S-values
for the Sun:
S_MW
0.179 Baliunas+1995 (~1966-1993; cycle 20,21,22)
0.170 Hall+2007 (~1994-2006; cycle 23)
=>
0.1762 Baliunas+95 & Hall+07 year-weighted mean (1996-2006)***
In Mamajek & Hillenbrand (2008; ApJ 687, 1264; Table 1), we quoted a
mean chromospheric activity index of logR'HK = -4.905. This was based
off of adopting a mean Mt. Wilson S-value of S = 0.1762, solar B-V
color of 0.650, and using the equations of Noyes et al. (1984) to
convert S to logR'HK.
A more recent estimate (April 2012), adopting B-V = 0.645 and S_MW =
0.1762 (see above), would give logR'HK = -4.903.
Donahue (1998; Cool Stars, Stellar Systems, and the Sun, ASPC
Vol. 154) provides the following table of representative activity
levels (Mt. Wilson S-values; Ca H&K emission lines) for the Sun:
Epoch Smean Est. log
Age(Gyr) R'HK
Activity Maximum (Cycle 22) 0.205 2.5 -4.780
Mean Activity (Cycle 20-22) 0.182 3.5 -4.877
Mean Activity (Cycle 20) 0.171 4.5 -4.932
Activity Cycle Minimum 0.165 5 -4.966
Maunder Minimum 0.145 8 -5.102
The "estimated age" would be the age inferred from the Sun's Ca HK
activity index via the R'HK-to-age calibration in Donahue's thesis
(1993; NMSU; also listed in the 1998 Cool Stars conference
proceedings). I calculated the last column from Donahue's Mt. Wilson
S-values following Noyes et al. 1984 and assuming B-V(Sun)=0.65 (Cox
2000; median of 19 published values).
The estimated activity during the Maunder minimum (~1645-1715)
(logR'HK = -5.10) was estimated by Baliunas & Jastrow (1990; Nature
348, 520), which they quote as Mt. Wilson S-value ~ 0.145.
Radick et al. (1998; ApJS 118, 239; Sec. 3.2.2) combined data from the
Mt. Wilson survey and the NSO/Kitt Peak K-index data, and found that
the typical mean activity level for the Sun near solar minimum is S =
0.169. For adopted solar color (B-V=0.645 mag), this converts to
<logR'HK(solar minimum)> = -4.941.
Keil, Henry, & Fleck (1998; ASPC 140, 301) present NSO/Kitt Peak
K-index data on Solar cycles 21 and 22. I convert their K-index data
to logR'HK following the K-index-to-Mt. Wilson S-index conversion of
Radick et al. (1998), and the Mt. Wilson S-index to logR'HK conversion
of Noyes et al. (1984). The extrema measured during these two cycles
correlated with logR'HK = -4.804 (Cycle 22 peak) and -4.958 (Cycle 22
minimum).
Baliunas & Jastrow (1990; Nature 348, 520) says that the Mt Wilson
S-value for the Sun "ranges between 0.164 and 0.178 during the 11-year
sunspot cycle (cycle 20) and averages ~0.171 (Wilson 1978)." Using the
Noyes et al. 1984 conversion, and adopting B-V(Sun)=0.645, I estimate
these three S-values to correspond to logR'HK=-4.970, -4.894, and
-4.930, respectively. They define a "Maunder minimum" star as a star
"exhibiting prolonged low levels of magnetic activity." That is, it is
not defined by a given activity level, but by the very flat evolution
of the activity seen over time.
Livingston, Wallace, White, & Giampapa (2007; ApJ 657, 1137) present
~33 years of Ca II K 3933A K-index data integrated over the solar
disk. They present 1302 measurements of the K-index between 1974.82
and 2008.07. Based on their full disk measurements described in the
Appendix to their paper, and converting the K-indices to logR'HK
following Radick et al. (1998) and Noyes et al. (1984), and assuming
(B-V)Sun = 0.642, I find the following moments:
Measurements for 1A K-index [logR'HK in brackets]
Minimum: 0.082747 ; [-4.978]
Maximum: 0.107183 ; [-4.803]
Mean: 0.091993 ; [ -4.903]
St.Dev.: 0.004818 ; [+-0.036]
skew : 0.5229 (platykurtic)
+95%CL: 0.102362 [-4.832 ; median + 0.076]
+68%CL: 0.097365 [-4.865 ; median + 0.043]
Median : 0.091358 [-4.908 ]
-68%CL: 0.086900 [-4.942 ; median - 0.034]
-95%CL: 0.085389 [-4.955 ; median - 0.047]
If you force symmetric confidence intervals,
then the Sun's activity can be approximately
stated as:
logR'HK = -4.902 (+-0.039; 68%CL) (+-0.063; 95%CL)
######################################################################
# Solar X-ray Luminosity (0.1-2.4 keV): L_X = 10^27.35 (+-50%) erg/s
# Solar X-ray/Bolometric
# Luminosity Ratio: log(L_X/L_bol) = -6.24 +- 0.24 dex
# Solar X-ray Surface Flux (0.1-2.4 keV): f_X = 36800 erg/s/cm^2
# = 10^4.566 erg/s/cm^2
######################################################################
When comparing X-ray luminosity and X-ray/bolometric flux ratios, the
largest uniform database of X-ray data for nearby stars and members of
clusters and associations is the ROSAT All Sky Survey (Voges et
al. 1999) which covers the 0.1-2.4 keV bandpass. For this reason, here
I only discuss the Sun's X-ray luminosity in this bandpass.
Orlando, Peres, & Reale (2001; ApJ 560, 499) convert flux measurements
taken with the Yohkoh X-ray satellite between 1992-1996 to ROSAT X-ray
luminosity, and claim that the whole solar corona was consistent with
having an X-ray luminosity (0.1-2.4 keV) of 1e26-5e27 erg/s during the
four years of observations. This suggests log(L_X) = 26.0-27.7 erg/s,
or a mean log(L_X) = 26.85 erg/s. Note that Orlando et al. 2001 did
not attempt to estimate a mean X-ray luminosity averaged out over a
full solar activity cycle. From a look at the smoothed sunspot number
data: ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SUNSPOT_NUMBERS/SMOOTHED
it appears that most of the Orlando et al. data covered the bottom
half of the solar cycle (1993-1996). The midway between solar maximum
(~1989.6) and solar minimum (~1996.4) was roughly 1993.0. At this
point in the Orlando et al. data, the X-ray luminosity of the "whole
solar corona" was ~1e27 erg/s (their Figure 6). This is probably a
fair assessment of the "mean" solar X-ray luminosity from the Orlando
et al. data averaged out over a full activity cycle.
Judge, Solomon, & Ayres (2003; ApJ 593, 534) made an extensive study
of the solar X-ray emission with the SXP instrument on the SNOE
satellite. After accounting for the differences in sensitivities
between SNOE-SXP and ROSAT, and correcting for the fact that SNOE only
observed the Sun for a partial solar cycle (~1998-2000), Judge et
al. conclude "We find that the Sun's 0.1-2.4 keV luminosity lies
between 10^27.1 and 10^27.75 [erg/s] (measured over the time space of
the SNOE-SXP data) and between 10^26.8 and 10^27.9 [erg/s]
(extrapolated over a full activity cycle)." They claim an accuracy of
50% in their calibration between the SNOE-SSXP and ROSAT bandpasses.
From the discussion of Judge et al., I adopt a mean solar X-ray
luminosity (0.1-2.4 keV) of:
L_X = 2.24e27 erg/s (+-1.12e27 erg/s = 50% unc.)
log(L_X) = 27.35 (+0.18,-0.30; +-0.24; 1sigma) erg/s
Adopting a solar luminosity of 3.8416e33 erg/s (see above), and
assuming negligible error in the solar luminosity (correct to first
order given the huge error in the X-ray luminosity), I derive:
L_X/L_bol = 5.82e-7 (+-2.91e-7; 50%; 1sigma)
log(L_X/L_bol) = -6.24 (+0.18,-0.30; +-0.24; 1sigma)
f_X = 36800 erg/s/cm^2 = 10^4.566 erg/s/cm^2
Check: I used 157 stars with chromospheric activity log(R'HK) > -4.3
and X-ray luminosity data to measure the correlation between log(R'HK)
and log(L_X/L_bol), first demonstrated by Sterzik et al. 2007. I find:
log(L_X/L_bol) = 7.081 + 2.63075*log(R'HK), with rms scatter in
log(L_X/L_bol) of 0.24 dex. For the mean log(R'HK) estimated
previously (-4.905), I estimate log(L_X/L_bol) = -5.82 +- 0.24 dex.
This would translate into a solar X-ray luminosity of log(L_X) =
27.76+-0.24 erg/s. This is only 1.2 sigma off of the value derived
from Judge et al., given the 0.24 dex rms in the chromospheric-X-ray
fit, and the quoted error in Judge et al.'s X-ray luminosity. So we
have an independent check that the solar log(L_X/L_bol) value is
probably ~ -6, and log(L_X) ~ 27 erg/s.
#####################################
# Age(Solar System) = 4572 +- 4 Myr
#####################################
In nature, there appears to be a continuum of objects ranging
from cold molecular clouds to protostars to optically visible,
accreting T Tauri stars. Where to define t=0 for the "birth" of
a star or planetary system is a matter of taste or definition.
The radioactive isotopes in meteorites give us samples of the
first "rocks" to have accreted from the protosolar nebula,
orbiting the still accreting proto-Sun, and hence are our
best "clocks" for dating the age of the solar system and Sun.
In a well-cited paper, G.J. Wasserburg wrote an appendix on the age of
the Sun in the paper by Bahcall, Pinsonneault, & Wasserburg (1995,
Rev. Mod. Phys. 67, 781). He concludes that the meteoritic evidence is
consistent with an age of the sun between 4563 and 4576 Myr. The upper
bound comes from consideration of the decay of 26Al between the source
(supernova?) and injection into CAIs. The lower bound of this age
(4563 Myr) should be revised upward given the ages of the oldest CAIs
(4568 Myr).
Here are some recent, relevant ages from isotopic studies of meteorites:
>4569.5 +- 0.2 Myr ; Baker+(2005; Nature, 436, 1127)
4568.2 Myr ; Bouvier & Wadhwa (2010; Nature Geosci. 3, 637)
4568 +0.91-1.17 Myr ; Moynier+ (2007; ApJ 671, L181)
4567.2 +- 0.6 Myr ; Amelin+(2002; Science 297, 1678)
4567.18 +- 0.50 Myr ; Amelin+(2010; E&PSL 300, 343)
4567.11 +- 0.16 Myr ; Amelin+(2006; update to 2002; Lun.Pl.Sci.Conf. 37, 1970)
The Baker et al. study claims "the accretion of differentiated
planetesimals pre-dated that of undifferentiated planetesimals, and
reveals the minimum Solar System age to be 4.5695+-0.0002 billion
years." They find the basaltic angrite (read: igneous rock from a
large asteroid or protoplanet) is 4566.2 +- 0.1 Gyr old, suggesting
that there were large, differentiated planetary bodies with volcanism
by this time. From dating of a carbonaceous chrondrite, Moynier+2007
says "therefore the formation of the first solid igneous objects as
well as the accretion of the undifferentiated kilometer-sized
carbonaceous chondrite parent bodies must have been complete within
+0.91 to -1.17 Myr at 4568 Myr ago."
The age from Amelin et al. is from isotopic studies of Ca-Al-rich
inclusions (CAIs) in the chondrite Efremovka. CAIs are the oldest
known parts of meteorites, and are thought to be the most primitive
solids to have survived the protosolar nebula (the Sun likely accreted
>99% of the material that ever passed through the protosolar nebula
disk). Connelly et al. (2008; ApJ 675, L121) says "the currently most
precise and accurate estimate of the timing of primary CAI formation -
and consequently the age of the solar system - is that defined by the
E60 Efremovka CAI at 4567.11 +- 0.16 Myr (Amelin et al. 2002, 2006)."
The age from Bouvier & Wadhwa (2010) calculate a 207Pb-206Pb age for a CAI
in the meteorite NWA 2363.
Combining Wasserburg's upper limit (4576 Gyr) and the ages of the
oldest CAI (4568 Gyr), it would appear that a consensus age for the
Sun with conservative uncertainty bars would be:
4572 +- 4 Myr (+-0.09%)
At +1 sigma it is consistent with upper limit given by Wasserburg (in
Bahcall+ 1995) and at -1 sigma it is consistent with the ages of the
oldest CAIs (Amelin+ 2006). The Sun is currently unique among stars
for having an age that we estimate its age so precisely.
Note that the early Sun was powered predominantly by the release of
gravitational energy as it contracted to the main sequence. Based on
contemporary models (which agree well with the back-of-the-envelope
Kelvin-Helmholtz contraction timescale), the Sun likely did not reach
the main sequence for another ~40 Myr after its protostellar
phase. The transition in the Sun's dominant fuel source from
gravitational energy to proton-proton (PP) chain fusion probably had
negligible impact on the evolution of meteorites, so the timescales
from isotopic studies should *not* be confused with the timescale
since the Sun reached the "zero-age main sequence" or "the start of
main sequence behaviour" (e.g. Bahcall et al. 1995). In the author's
opinion, starting t=0 at the zero-age main sequence is a very bad,
silly, and confusing habit still adopted by some theorists. As it
appears that stars in clusters form within <few Myr of one another
(e.g. Hartmann+2001, Preibisch+2002), defining t=0 using the ZAMS
complicates cross-comparison of evolutionary tracks of different
masses (which have different pre-MS contraction times!). A more useful
metric for t=0 might be the "stellar birth-line" which corresponds to
the deteurium-burning sequence for young stars, and corresponds well
with the observed distributions of luminosities for accreting T Tauri
stars (Stahler 1983, 1988).
###########################################
# Photospheric and Protosolar Composition #
###########################################
There is a healthy debate on this right now, and the situation has yet
to be resolved.
To summarize, the fraction of mass of the Sun in the form of "metals"
(elements heavier than He; denoted "Z") is currently a matter of
debate, and is uncertain at the tens of percent level. Solar Z is
somewhere between Z(Sun) ~ 0.12-0.19, with the some
atmospheres/abundances experts favoring smaller values, and the
helioseismologists and various stellar interiors theorists favoring
higher values.
Before discussing the solar composition, it is worth noting that there
are multiple types of numbers quoted for the composition. The mass
fractions in H, He, and all elements heavier than He ("metals") are
labeled by the capitalized letters X, Y, and Z, respectively. They
are related by:
X + Y + Z = 1
Often (Z/X) ratios are quoted, so
X = (1 + Y)/(1 + (Z/X))
Y = 1 - Z - Z/(Z/X)
Z = (1 - Y)/(1 + 1/(Z/X))
In most contexts, these are subscripted with letters/symbols that
refer to (1) the composition of the modern-day solar convection zone
and photosphere (which can be probed with abundance analyses of the
stellar photosphere via spectroscopy, or through helioseismology;
usually unscripted or with subscript "s"), (2) the modern-day solar
bulk composition (which is not terribly useful as as the distribution
of X and Y vary greatly between the core and convection zone; I will
subscript these with a "b"), and (3) a theoretical "protosolar"
composition (subscript "0"), which is useful as it can provide the
starting point for producing stellar evolution models. Note that often
one sees the subscript "p" (for "primordial") when discussing the
abundance of helium from the Big Bang: "Yp".
Complicating matters, the convection zone (and by virtue of its
mixing, the photosphere) of the Sun has been subject to diffusion over
its lifetime, which results in the settling of He and metals to lower
depths in the Sun (relative to the lightest element H). Lodders (2010;
Principles and Perspectives in Cosmochemistry, Astrophysics and Space
Sci. Proc., p. 379) estimate that the Sun's convective zone has seen
its He abundance decrease 0.061 dex (15%) since the protostellar
phase, and for all elements heavier than He, a loss of 0.053 dex
(13%). Similarly, Grevesse et al. 2010 (Astrophy. Space Sci.)
and Asplund et al. 2010 (ARA&A, 47, 481)
estimate that the protosolar bulk composition was 0.05 dex (12%)
higher for He (Y) and 0.04 dex (10%) higher for metals (Z) compared to
the modern-day solar photosphere/convection zone. As stated in the
review by Asplund, Grevesse, Sauval, & Scott (2009, Annual Rev. of
Astro. & Astrophys., 47, 481), "With the exception of a
general $(C!-(B10% modification owing to diffusion and gravitational settling
and depletion of Li and possibly Be, today's photospheric abundances
are believed to reflect those at the birth of the Solar System."
Asplund, Grevesse, Sauval, & Scott 2009 (ARA&A 47, 481) summarizes the
past two decades of solar mass fractions in their Table 4. I reproduce
their table here, and include a few other recent entries to enhance
the historical completeness ("phot(modern)" is modern photospheric
values or constrained from the convective envelope using
helioseismology methods):
X Y Z Z/X type reference
0.7314 0.2485 0.0201 0.0274 phot(modern) Anders & Grevesse (1989)
0.7336 0.2485 0.0179 0.0244 phot(modern) Grevesse & Noels (1993)
0.7345 0.2485 0.0169 0.0231 phot(modern) Grevesse & Sauval (1998)
0.6937 0.2875 0.0188 0.0271 phot(modern) Demarque & Guenther (in Cox 2000)
0.7491 0.2377 0.0133 0.0177 phot(modern) Lodders (2003)
0.7389 0.2485 0.0126 0.0171 phot(modern) Basu & Antia (2004) [Z=0.0126 model]
0.7392 0.2485 0.0122 0.0165 phot(modern) Asplund, Grevesse, Sauval (2005)
... ... 0.0172 ... phot(modern) Antia & Basu (2006)
0.7390 0.2469 0.0141 0.0191 phot(modern) Lodders, Palme & Gail (2009)
0.7381 0.2485 0.0134 0.0181 phot(modern) Asplund, Grevesse, Sauval, & Scott (2009)
0.7380 0.2485 0.0134 0.0181 phot(modern) Grevesse, Asplund, Sauval, & Scott (2010)
0.7321 0.2832 0.0153 0.0209 phot(modern) Caffau et al. (2010)
0.7096 0.2691 0.0213 0.0301 protosolar Anders & Grevesse (1989)
0.7112 0.2697 0.0190 0.0268 protosolar Grevesse & Noels (1993)
0.7120 0.2701 0.0180 0.0253 protosolar Grevesse & Sauval (1998)
0.7111 0.2741 0.0149 0.0210 protosolar Lodders (2003)
0.7166 0.2704 0.0130 0.0181 protosolar Asplund, Grevesse & Sauval (2005)
0.7112 0.2735 0.0153 0.0215 protosolar Lodders, Palme & Gail (2009)
0.7154 0.2703 0.0142 0.0199 protosolar Asplund, Grevesse, Sauval, & Scott (2009)
0.7154 0.2703 0.0142 0.0199 protosolar Grevesse, Asplund, Sauval, & Scott (2010)
... 0.278 ... ... protosolar Serenelli & Basu (2010)
"Canonical" (high-Z) values:
An excellent recent review by Basu & Antia (2008, Physics Reports,
457, 217-283) states that "Seismic determinations of the solar
heavy-element abundances yield results that are consistent with the
older, higher values of the solar abundance, and hence no major
changes o the inputs to solar models are required to make
higher-metallicity models consistent with the helioseismic data."
A recent review by Lodders (2003; ApJ 591, 1220, Table 4) lowered
the solar metal fraction slightly:
Sun Sun
"Protosolar" Present
X = 0.7110+-0.0040 0.7491+-0.0030 Hydrogen mass fraction
Y = 0.2741+-0.0120 0.2377+-0.0030 Helium mass fraction
Z = 0.0149+-0.0015 0.0133+-0.0014 Metals mass fraction
Z/X = 0.0177 0.0178
Lodders (2003; Table 5) shows that estimates of Z/X have been declining
from 1984 through 2003, from values of Z/X ~ 0.027 in the mid-1980s to
Z/X ~ 0.018 in 2003.
"Heretical" (low-Z) values:
The "heretical" value is approximately Z = 0.012 (Grevesse, Asplund, &
Sauval 2007, Space Science Reviews 130, 105). Asplund, Grevesse, &
Sauval 2006 (Comm. in Asteroseismology 147, 76) report Z=0.0122 and
Z/X=0.0165. Asplund et al. (2009 ARA&A) and Grevesse et al. (2010;
Astrophys. Space Sci. 328, 179) now lists Z=0.0134.
The canonical value can be used to match the helioseismological sound
speed profile of the Sun with ease, however the lower Z would require
that there is some missing opacity source in the Sun. The groups that
first proposed the lower Z have claimed to have ruled out enhanced Ne
as the culprit (see Asplund et al. 2005 astro-ph/0510377 in response
to Drake & Testa 2005; Nature 436, 525). Stay tuned.
Asplund, Grevesse & Sauval (2006; Comm. in Asteroseismology 147, 76)
list the following "heretical" solar composition:
X = 0.7393, Y = 0.2485, Z = 0.0122, Z/X = 0.0165
The estimated protosolar abundances are:
Zo = 0.0132, Zo/Xo = 0.0185, (Yo = 0.2733, Xo = 0.7135)
The most recent quote from this same group (Grevesse, Asplund, Sauval,
Scott 2010, Astrophys. Space Sci.) for the photospheric abundances:
X = 0.7380, Y = 0.2485, Z = 0.0134, X/Z = 0.0181
and for the modern day bulk composition:
Xb = 0.7154, Yb = 0.2703, Zb = 0.0142.
Their estimated protosolar abundances are:
Xo = 0.7154, Yo = 0.2703, Zo = 0.0142
Conclusions on Solar Z:
Antia & Basu (2006) estimate Z = 0.0172+-0.002 from modeling of
helioseismological sound speed profiles for the solar interior. In
their review (Basu & Antia, 2008, Physics Reports, 457, 217-283), they
provide a comprehensive review of helioseismic constraints on the
solar abundances, and conclude that "if the GS98 [Grevesse & Sauval
1998, Space Sci. Rev. 85, 161] abundances are correct then the
currently known input physics is consistent with seismic data." While
the low solar metal fraction corroborates studies of CNO abundances in
nearby B-stars and ISM, it cannot account for the shape of the main
sequence turn-off for the ~4 Gyr-old cluster M67 as well as the high Z
models (Vandenberg et al. 2007, ApJ, 666, L105; and references
therein).
Remarkably, as seen in Table 4 of the review by Asplund et al.
2010 (ARA&A, 47, 481), the estimates of the protosolar Y (helium abundance)
have varied negligibly over the years, remaining very close to Yo = 0.270,
however the work by Basu & Antia (2008) suggests a slightly higher value
(Yo = 0.278).
Based on the Antia & Basu (2006, 2008) study and review, I would adopt
a photospheric solar Z = 0.0172 (X = 0.739, Y = 0.2438; see Fig. 1 of
Antia & Basu 2006).
For protosolar values, I would take the Antia & Basu modern-day solar
photosphere/ convection zone Z (0.0172), and correct it for diffusion
by 0.04 dex (Zo = 0.0189). Basu & Antia estimated the solar convection
zone helium fraction to be Y ~ 0.2485 from modeling helioseismology
data from GONG and MDI (most published estimates are in the Y =
0.24-0.25 range, Table 3 of Basu & Antia 2008). Accounting for
diffusion (0.05 dex), this suggests a protosolar value of Yo =
0.2783. Hence Xo = 1 - Yo - Zo = 0.7028, and hence
(Xo, Yo, Zo = 0.7028, 0.2783, 0.0189).
A note on "dY/dZ" - i.e. the slope of the helium and metals mass
fractions used in scaling stellar evolution models. One gets fairly
similar inferred dY/dZ slopes whether one adopts the protosolar
abundances scaled to the Basu & Antia results (Xo, Yo, Zo = 0.7028,
0.2783, 0.0189) or the recent Asplund, Grevesse, Sauval, & Scott
results (Xo, Yo, Zo = 0.7154, Yo = 0.2703, Zo = 0.0142). If one
adopts what I would consider the best recent estimate of the
primordial (Big Bang) helium abundance (Yp = 0.2486; Cyburt, Fields, &
Olive 2008; see summary of published values at:
http://www.pas.rochester.edu/~emamajek/memo_Yp.html ), then one would
estimate dY/dZ = 1.57 using the protosolar abundances estimated from
the Basu-Antia work, or dY/dZ = 1.528 using the protosolar abundances
from Asplund-Grevesse-Sauval-Scott. This is somewhat lower than the
median of the recent published estimates (see summary of dY/dZ values:
http://www.pas.rochester.edu/~emamajek/memo_dydz.html ), but not
statistically inconsistent with recent estimates from extragalactic
HII regions, eclipsing binaries, nearby K dwarfs, etc. (given their
large uncertainties). So I would adopt these new dY/dZ values based on
which protosolar abundances you adopt.
##################################################
# Equatorial Rotation Period: P_eq = 24.47 days
# Equatorial Rotation Velocity: V_eq = 2.067 km/s
##################################################
There is some useful discussion in Wikipedia on measuring solar rotation:
http://en.wikipedia.org/wiki/Solar_rotation
Solar rotation periods are either synodic (measured from Earth) or
sidereal (with respect to background stars). Synodic period simply
measures the interval it takes for a feature (spot?) to return to the
same position from the perspective of an observer on Earth. Obviously
the Earth is moving a considerable distance in its orbit during the
course of a solar rotation period (nearly a month), so while we
measure synodic periods from Earth-bound observations, one needs to
correct for the Earth's orbital motion to derive a sidereal period
(which is what an observer outside the solar system would measure, who
is not participating in orbital motion around the Sun).
R. Howard states in "Allen's Astrophysical Quantities" (2000,
Sec. 14.9, P. 363) that "The period of sidereal rotation adopted for
heliographic longitudes is 25.38 days". This is *not* the equatorial
rotation period, but appears to correspond to the sidereal solar
rotation period at an arbitrary latitude.
Snodgrass & Ulrich (1990, ApJ, 351, 309; http://adsabs.harvard.edu/abs/1990ApJ...351..309S )
report a *sidereal* solar rotation rate for the photosphere of:
omega(phi) [deg/day] = 14.71 - 2.39 sin^2(phi) - 1.78 sin^4(phi)
where phi is the heliocentric latitude in degrees. This translates
into an equatorial sidereal rotation period of 24.47 days. Their rate is "~2%
faster than the magnetic and sunspot rates and ~4% faster than Mount
Wilson spectroscopic rate".
One sometimes encounters the "Carrington Rotation" synodic period for
the Sun, which is defined to be 27.2753 days (see
http://en.wikipedia.org/wiki/Carrington_rotation ). The "Carrington
Rotation Number" defines the number of solar rotations since 9
November 1853. This synodic period translates to a sidereal period of
25.38 days. Note that this is not meant to correspond to an
equatorial velocity, but given the differential rotation of the Sun,
the Carrington Rotation period corresponds roughly to latitude 26
degrees, which is a typical latitude for sunspots.
Using the sidereal equatorial rotation period from Snodgrass & Ulrich (1990),
I calculate the solar equatorial rotation velocity:
V_eq(Sun) = 2*pi*Rsun/Per(equator)
where I adopt Rsun = 695660 (+-100) km.
Hence: V_eq(Sun) = 2.067 km/s.
##########################################
# Mean Rotation Period: <P> = 26.09 days
##########################################
This refers to the mean rotation period as inferred from searching for
periodicities in chromospheric activity measurements or due to
starspots (a rotation that can be more directly compared to periods
measured for other stars), *not* the equatorial rotation period. I've
adopted this period as it provides a useful comparison to rotation
periods for other Sun-like stars derived using chromospheric emission
(which is what is typically used for older, slower rotating stars like
the Sun).
Donahue, Saar, & Baliunas (1996; ApJ 466, 384) studied 19 seasons of
solar chromospheric activity (as measured with the Mt. Wilson S-index;
>= 30 days of observations each) over a 8-year period, and was able to
detect periodicity due to solar rotation in 8 seasons. The detected
periods range from 24.5 to 28.5 days, with a mean detected period of
26.09 days. Although individual periods within a given season can be
measured to tenths or hundreds of a day accuracy, from season to
season as the active regions vary by longitude, the measured period
can vary by rms ~ 10% for the Sun (+- ~2 days).
Another useful period is the "Carrington Rotation" synodic period for
the Sun, which is defined to be 27.2753 days (see
http://en.wikipedia.org/wiki/Carrington_rotation ). The "Carrington
Rotation Number" defines the number of solar rotations since 9
November 1853. This synodic period translates to a sidereal period of
25.38 days. The Carrington Rotation period corresponds roughly to
latitude 26 degrees, which is a typical latitude for sunspots.
##########################################################
# Mean Solar Wind Mass Loss Rate <dM/dt> = 2e-14 Msun/yr
##########################################################
The flow of charged particles escaping the Sun (the solar wind) has
been measured by many spacecraft. Here is a very brief summary.
Solar Wind Velocity:
Using data from the Ulysses spacecraft, which sampled the solar wind
at a wide range of heliographic latitudes (-80deg to +80 deg) in
1994-1995, Goldstein et al. (1996 A&A 316, 296; Figure 1) show that
the solar wind velocity varies as a function of heliographic latitude.
For heliographic latitudes of +-20-80 degrees, the solar wind velocity
was in the range of ~600-830 km/s (mean ~750 km/s; values are inferred
by-eye and ruler from their Fig. 1). There appears to be a sharp
discontinuity in solar wind velocities at plus and minus 20
deg. heliographic latitude. At latitudes below +-20 deg latitude
(i.e. where the ecliptic plane is), the solar wind velocity ranged
from ~320-700 km/s, with an approximate mean of ~460 km/s (again, by
eye from their Fig. 1).
Gosling et al. 1976 (Jrnl. Geophys. Res. 81, 5061) reports solar
wind velocity statistics for the period 1962-1974, presumably sampled
near the Earth, and hence at low heliographic latitudes. The median
solar wind velocity over this period was 408 km/s.
Data from the Voyager 2 probe sampled the solar wind between 1977 and
2008, between radii of 1 and 87.07 AU (as of 10/31/2008). The probe
reached the termination shock and entered the heliosheath at a
distance of 83.6 AU. During the period 1977.64-2007.64, the median
solar wind velocity was 432 km/s, and the mean was 439 km/s. Since its
pass of Neptune in 1989, Voyager 2 is heading towards a point in the
sky 47 degrees below the ecliptic plane. It appears to have been
sampling the denser, slower moving solar wind (that Ulysses detected
within 20 deg of the heliographic equator) throughout.
The OMNIWeb data (Goddard Space Flight Center:
http://omniweb.gsfc.nasa.gov/html/ow_data.html) reports best daily
solar wind density and velocity values in near-Earth space from a
variety of spacecraft measurements. For 11006 daily solar wind
velocity measurements between 1963 and 2003, I find a median velocity
of 419 km/s (68%CL +123-70 km/s) and mean of 441 km/s (stdev = 98
km/s). The minimum daily velocity was 212 km/s and the maximum was 923
km/s.
Solar Wind Density:
The proton densities measured by Ulysses as a function of heliographic
latitude show a discontinuity similar to that seen for velocities. The
mean proton densities at high heliographic latitude (>+-20 deg) were
typically 2-3 cm^-3 (range: ~1.5-4 cm^-3), while at low latitudes
(<+-20 deg) the densities were typically ~8 cm^-3 (approximate range:
~2-20 cm^-3).
Taking the Voyager 2 data and correcting for distance
(i.e. normalizing the densities to what one would find at 1 AU), it
detected a median proton density of 5.63 cm^-3 and a mean of 6.68
cm^-3 between 1977.64-2007.64. This agrees well with the Ulysses data
at low heliographic latitudes.
The OMNIWeb database reports 11006 daily solar wind density
measurements between 1963 and 2003. The median proton density is 6.10
cm^-3 (68%CL +5.5-2.8) and the mean is 7.39 cm^-3 (st.dev. =
4.85). The minimum daily value was 0.1 cm^-3 (2002 day 144, when Mdot
= 10^-15.30 Msun/yr) and the maximum was 60.3 cm^-3 (1964 day 16, when
Mdot = 10^-12.85 Msun/yr). Both the median and mean values are within
10% of the Voyager 2 numbers.
Calculation:
A nice derivation for estimating the solar mass loss from the
parameters for the solar wind is Example 11.2.1 (p. 374) of Carroll &
Ostlie's "An Introduction to Modern Astrophysics (Second Edition)"
(2007).
In the special case of the solar wind velocity and density being
independent of heliographic latitude (not true), and assuming that the
detected particles are all protons (not true), one can estimate the
mass loss due to solar wind as measured at some radial distance R (in
AU) from the Sun using the following derived formula:
dM/dt [Msun/yr] = 7.41e-18 Msun/yr * n[cm^-3] * V[km/s] * (R[AU]^2)
Where n is the proton density in cm^-3, and V is plasma velocity in
km/s.
Using the OMNIWeb mean solar wind parameters (<n>=7.39 cm^-3,
<V>=441 km/s, assume all protons) and this formula, I estimate:
dM/dt = ~2.4e-14 Msun/yr
The range of the mass loss rate inferred from the density and velocity
of the solar wind detected near Earth leads to mean mass loss
rates of 2.26e-14 Msun/yr (st.dev. 1.11e-14), and median mass loss
of 1.93e-14 Msun/yr (68%CL 1.08e-14).
From Goldstein et al.'s Ulysses data it appears that the product
density X velocity for the solar wind is approximately double at lower
latitudes (<20 deg) than at higher latitudes. Hence approximately
1/3rd of the heliographic latitudes and longitudes are emitting
protons at the rate we measured, while ~2/3rds has a product of n X V
that is roughly half. This suggests that a more realistic ~2nd-order
estimate might be:
dM/dt = ~1.6e-14 Msun/yr
Given the uncertainties, I would just adopt 2e-14 Msun/yr.
In cgs units:
(2e-14 Msun/yr)*(1 yr/3.1557e7 s)*(1.988e33 g/Msun) =>
dM/dt = ~1.3e12 g/s => 1e12 g/s
#############################################
# Mean Solar Wind Pressure
#############################################
The dynamical pressure due to the solar wind is nicely discussed at:
http://www.swpc.noaa.gov/SWN/sw_dials.html
The dynamical pressure in nanopascals can be estimated in terms of the
solar wind proton density (in cm^-3) and velocity in (km/s):
P = 1.6726e-6[nPa] * n[cm^-3] * V[km/s]^2
Using the mean densities and velocities from the OMNIWeb database
(see section on solar wind mass loss rate), i.e. <n> = 7.39 cm^-3,
<V> = 441 km/s, I find a mean solar wind pressure of
<P> = 2.40 nanoPascals
Using the previous formula along with 11006 solar wind measurements
from the OMNIWeb database, I estimate the median solar wind dynamical
pressure to be 1.92 nPa (68%CL +1.23-0.79) and the mean to be <P> =
2.20 nPa (st.dev. 1.33).
SUMMARY ON SOLAR WIND PARAMETER CORRELATIONS
So during the solar cycle variations, we see that the 1sigma variation
in proton density is +- ~66%, the velocity varies by +- ~22%, the
solar mass loss rate by +- ~49%, and the pressure varies by +- ~60%.
P vs. Mdot
There is a strong correlation between the solar wind pressure and mass
loss rate (Pearson r = 0.91 ; N=11006), for range 0.07-19.59 nPa.
Mass loss [Msun/yr] = P[nPa] * 1.0573e-14[Msun/yr/nPa]
P vs. V
There is a gentle correlation between the solar wind pressure and
velocity (Pearson r = 0.20 ; N=11006), for range 0.07-19.59
nPa. However there is significant rms about this relation (~ +-100
km/s).
V [km/s] = 408.2 km/s + P[nPa] * (14.82+-0.76 [km/s/nPa])
#############################################
# Median International Sunspot Number (ISN)
#############################################
The Solar Influences Data Analysis Center (SIDC) has a nice database
of historical sunspot number measurements:
http://sidc.oma.be/sunspot-data/
The SIDC lists a daily estimate of the International Sunspot Number
(ISN) going back to January 1818 (although not every day had a
measurement in the early data), with 66515 daily measurements between
8 Jan 1818 and 31 Dec 2008.
The moments of the ISN can be summarized as such:
median = 40
mean = 54
68% interval = 4 to 105
95% interval = 0 to 187
max = 355 (measured at year = 24 & 25 Dec 1957)
min = 0 (measured 10243 times, or 15.4% of daily observations)
################################################
# Solar Colors (B-V)o(Sun) = 0.653+-0.003 mag
# other colors listed below
################################################
The color of the Sun as measured in various combinations of bands, has
been estimated in many studies. I do not attempt to summarize all of
the measurements of all of the solar colors which have been estimated.
Instead, I discuss the solar B-V color in detail, and list several
other solar colors in optical/near-IR bands from two recent studies
which appear to have nailed the values to high precision (Casagrande
et al. 2012; ApJ, in press, arXiv:1209.6127, Ramirez et al. 2012; ApJ,
725, 5; arXiv:1204.0828). I advocate adopting the solar colors from
these two papers as they appear to be the best available.
Solar B-V:
The solar B-V color has been the source of some controversy over the
years, and quoted values have spanned a (relatively) large range of
values (from B-V=0.62 in Allen63, to B-V=0.686 in Tug & Schmidt-Kaler
1982).
Arguably, the most recent authoritative study of the solar B-V is by
Ramirez et al. (2012, ApJ, 752, 5) who derives the color 3 ways: (1)
comparing spectroscopically determined Teff, log(g), and [Fe/H] values
for 10 solar twins, (2) the same for 112 solar analogs, and (3)
comparing spectral-line-depth ratios and colors for solar twins. The
three analyses yield very similar solar B-V values of: 0.653+-0.005,
0.658+-0.014, and 0.653+-0.003. Although they do not list a weighted
mean for their 3 independent estimates, it would be <B-V(Sun)> =
0.653+-0.003. They adopted Teff=5777K in their analysis.
Here is sorted list of published *pre-2000* solar B-V values. It is
probably not exhaustive, but it should be fairly representative of the
quoted values.
B-V(Sun) reference
0.62 Allen63 (Astrophysical Quantities, 2nd ed.; from Epstein & Motz 1954)
0.628 Taylor98
0.629 Napiwotski93 calib. for T=5778K
0.63 Tayler94
0.63 Colina96
0.642 Cayrel96
0.648 Gray95 (+-0.006)
0.648 Porto de Mello & da Silva 1997 (+-0.006)
0.649 Colina96 (as cited by Bessell98)
0.650 Neckel86 (+-0.005)
0.651 Freil93 (+-0.008)
0.652 Cayrel96 (as cited by Bessell98, Table 6 analog)
0.656 Gray92 (+-0.005)
0.66 Wamsteker81
0.665 Hardorp80
0.667 Bessell98 ("Sun-Nover")
0.679 Bessell98 ("Sun-over")
0.68 Lang+92
0.686 Tug & Schmidt-Kaler 1982
Here are the values inferred from *post-2000* literature and
calibrations:
B-V(Sun) reference
0.617 Median for 223 G2V *s in Hipparcos (d<75pc), mostly Houk types
0.626 Sekiguchi00
0.631 Ramirez05 calib. for T=5778K
0.637 Vandenberg03 (p. 779)
0.641 Biazzo07 calib. for T=5778K
0.641 Casagrande10 calib. for T=5778
0.642 Holmberg+05 (+-0.016)
0.646 Engelke10 [Reike08 synthetic]
0.647 Median B-V for 48 G2V stars from Gray01/Gray03/Gray06.
0.649 Pasquini08 (+- 0.016)
0.650 Cox+2000
0.651 Casagrande06 (Table 4, "our temperature scale")
0.652 Engelke10 [Kurucz synthetic]
0.653 Ramirez12 (solar twins, +-0.005)
0.653 Ramirez12 (line depth ratios, +-0.003)
0.658 Ramirez12 (solar analogs, +-0.014)
0.661 Valenti05 calib. for Teff=5778K
* Solar twins
There are a few famous solar twins of note, so I mention their B-V
colors as a sanity check. This list is not exhaustive, but I believe
that these stars have been the most strongly argued to be similar in
parameters to our Sun.
18 Sco: The most famous solar twin is 18 Sco (HR 6060, HD 146233) was
noted as a solar twin by Porto de Mello & Da Silva (1997, ApJ, 482,
L89). They find Teff=5789K, log(g)=4.49, [Fe/H]=0.05. They find
abundances for HR 6060 within 1sigma of the solar values for 24
different elements (only Sc and V showed slightly excesses compared to
solar). The Hipparcos catalog lists B-V=0.652+-0.009 for HR 6060.
HIP 56948: HIP 56948 is a solar twin mentioned by Melendez & Ramirez
(2007; ApJ, 669, L89), Takeda & Tajitsu (2009, PASJ, 61, 471), and
Melendez et al. (2012; A&A, 543, A29). The 2012 paper notes that it
has Teff only 17+-7 K hotter than the Sun, log(g) higher by 0.02 dex,
[Fe/H] of +0.02+-0.01 dex, microturbulence velocity higher by
0.01+-0.01 km/s, and mass of 1.02+-0.02 Msun. Takeda & Tajitsu (2009)
stated "HIP 56948 most resembles the Sun in every respect, including
the Li abundance... and deserves the name of ``closest-ever solar
twin''. The Hipparcos catalog lists B-V = 0.647+-0.014, derived from
the Tycho photometry.
HD 44594: HD 44594 was considered "the most solar like dwarf found so
far in the neighborhood of the sun" by Cayrel de Strobel & Bentolila
(1989, A&A, 211, 324). The Hipparcos catalog lists B-V=0.657+-0.006.
So the truly noteworthy solar twins have B-V colors of: 0.652 (18
Sco), 0.647 (HIP 56948), 0.657 (HD 44594), i.e. these 3 stars alone
argue for a solar color of B-V = 0.652 +-0.005 (rms).
* B-V colors of Gray et al. G2V stars:
The nearby star survey of Gray et al. (2003,2006) lists a total of 44
G2V stars. Using Hipparcos B-V colors, I find a median B-V
0.648+-0.001 mag, and Chauvenet-clipped mean B-V = 0.641+-0.006 mag
(N=42, 2 clipped). The sample has Chauvenet mean Teff = 5766+-11 K,
so close to the solar value (Teff=5772K), and median [M/H] = -0.09.
* B-V colors of G2V stars from Houk and listed in Hipparcos:
Most of the stars classified as "G2V" in the Hipparcos catalog are
taken from the Michigan Spectral Survey catalogs published by N. Houk
and colleagues. For the 265 stars classified as "G2V" in the
Hipparcos catalog with parallaxes of >13.33 mas (distance < 75 pc;
i.e. probably within the Local Bubble with negligible reddening) and
parallax errors of <12.5%, the median and mean (regular, probit, and
Chauvenet-clipped) values of B-V *all* converge towards 0.620+-0.003
mag, with a standard deviation of ~0.03 mag. Hence, among the
Houk/Hipparcos G2Vs, the Sun appears to be ~1 sigma redder than the
typical G2V in the field (for (B-V)sun = 0.65, 83% of G2Vs are bluer,
17% are redder).
Hence, there appear to be subtle differences between what stars are
classified as G2V by Gray vs. those of Houk. This may be due to MK
"standards" which appeared to have changed by 1-2 subtypes over the
years as the MK system aged (notably eta Cas [G0V ~> F9V] and beta Com
[Morgan, Keenan, and Gray call G0V, but Houk calls G2V]. These subtle
changes among the standards may be responsible for the color offset
between the Houk/Hipparcos <B-V> for G2V stars, and that measured for
the Gray et al. G2V stars.
* Brian Skiff has written a memo with useful data
tables and references on "Near-Solar MK Standards and Photometric Standards of
Similar Color" at: ftp://ftp.lowell.edu/pub/bas/starcats/solar.list
* Summary:
Median values:
0.620 B-V for Houk G2V stars
0.641 B-V for Gray G2V stars
0.644 B-V for post-2000 literature values
0.651 B-V for pre-2000 literature values
0.652 B-V for 3 of the best solar twins (actually the mean)
The true median for the 33 B-V values listed (excluding the Ramirez et
al. 2012 values) plus that for the 3 solar analogs is
<B-V>=0.648+-0.003.
This agrees well with the best determined modern value from Ramirez et
al. (2012) of 0.653+-0.003.
The new Ramirez et al. (2012) value is very precise, but also
appears to be consistent (within ~+-0.01 mag), of the median of published
values in the literature.
The preponderance of evidence suggests that a solar B-V color bluer
than 0.64 or redder than 0.66 appears extremely very unlikely, despite
strong statements to the contrary (e.g. Sekiguchi & Fukugita 2000).
At this point, I would advocate adopting the new Ramirez et al. (2012)
value.
Other Colors:
Two recent papers (Casagrande et al. 2012; ApJ, in press,
arXiv:1209.6127, Ramirez et al. 2012; ApJ, 725, 5; arXiv:1204.0828)
have determined precise optical/near-IR colors for the Sun, through
comparing measured photometry for solar twins, and measuring precision
effective temperatures relative to the Sun using the line-depth ratio
(LDR) technique applied to high S/N, high resolution stellar spectra.
Three of the authors are on both studies (Ramirez, Casagrande,
Melendez), and they usually identical techniques, so I proceed
discussing them collectively. The temperatures of the solar twins
were also determined through excitation and ionization equilibrium
analysis, and two different implementations of the infrared flux
method (both using 2MASS photometry, but one using Tycho-2 photometry,
and the other using Johnson-Cousins photometry). The 4 techniques
gave mutually consistent means, however the LDR technique had smaller
uncertainties, and is the most model independent, and the other
techniques may suffer from small systematic Teff errors at the +-20 K
level, so the authors (Casagrande12) cite the LDR numbers as their
final values.
I refer the reader to those two studies, and cite their solar colors
here. The photometry system is 2MASS for J/H/Ks bands, WISE for W1/W2/W3/W4
bands, and Johnson-Cousins for UBVRcIc. On the right side are some of my
estimates from looking at color-color trends for field stars (not necessarily
only solar twins), evaluated at adopted solar color (B-V)=0.651. My (EEM) estimates
are of inferior quality (averaged out for field stars of a wide range of metallicities
and gravities), but (perhaps?) provide a useful sanity check.
(B-V) = 0.653 +- 0.003 ; Ramirez12
(U-B) = 0.158 +- 0.009 ; Ramirez12 [EEM: for (B-V)=0.651 => (U-B) = 0.135]
(V-Rc) = 0.356 +- 0.003 ; Ramirez12 [EEM: for (B-V)=0.651 => (V-Rc)= 0.363]
(V-Ic) = 0.701 +- 0.003 ; Ramirez12 [EEM: for (B-V)=0.651 => (V-I) = 0.714]
(V-J) = 1.198 +- 0.005 ; Casagrande12 [EEM: for (B-V)=0.651 => (V-J) = 1.201]
(V-H) = 1.484 +- 0.009 ; Casagrande12 [EEM: for (B-V)=0.651 => (V-H) = 1.494]
(V-Ks) = 1.560 +- 0.008 ; Casagrande12 [EEM: for (B-V)=0.651 => (V-Ks)= 1.567]
(J-H) = 0.286 ; Casagrande12 [EEM: for (B-V)=0.651 => (J-H) = 0.294]
(J-Ks) = 0.362 ; Casagrande12 [EEM: for (B-V)=0.651 => (J-Ks)= 0.366]
(H-Ks) = 0.076 ; Casagrande12 [EEM: for (B-V)=0.651 => (H-Ks)= 0.073]
(V-W1) = 1.608 +- 0.008 ; Casagrande12 [EEM: for (B-V)=0.651 => (V-W1)= 1.595]
(V-W2) = 1.563 +- 0.008 ; Casagrande12
(V-W3) = 1.552 +- 0.009 ; Casagrande12
(V-W4) = 1.604 +- 0.011 ; Casagrande12
Uncertainties for the (J-H), (J-Ks), and (J-Ks) colors were not
listed, but should be of similar order (i.e. +-0.01 mag). The
agreement between the finely done Ramirez12/Casagrande12 colors and my
overly-simplified estimates using color-color trends for dwarf stars
of a wide range of metallicities & gravities is fairly good. The
agreement is <=0.015 mag for (V-Rc), (V-I), (V-J), (V-H), (V-Ks),
(J-H), (J-Ks), (H-Ks), and (V-W1). Disagreement is 0.023 mag for
(U-B), however as U-band is fairly sensitive to metallicity effects,
this could be due to subtle offsets between the solar metallicity
and that of the Galactic disk stars that comprised the color-color
plots used for my own estimates.
The solar colors from Ramirez12 and Casagrande12 appear to be the best
available, and should be adopted.
#########################
# UPDATES & CORRECTIONS
#########################
29 May 2007: Fixed solar mass units correctly to kg & g, where appropriate.
2 Jun 2007: Added comments on solar spectral type from Morgan & Keenan (1939).
13 Jun 2007: Added Sun's B magnitude.
28 Aug 2007: Added brief discussion on B-V of G2Vs in the field.
28 Aug 2007: Added Solar GM value discussion and value.
28 Aug 2007: Updated AU value and discussion.
12 Sep 2007: Added logR'HK discussion on Maunder minimum.
23 Oct 2007: Added solar X-ray luminosity (Judge et al. 2003, Orlando et al. 2001).
12 Nov 2007: Added log of bolometric luminosity and age/yr.
29 Nov 2007: Added X-ray surface flux and defined ROSAT X-ray energy ranges (0.1-2.4 keV).
29 Nov 2007: Edited comments on age of solar system.
7 Dec 2007: Added discussion of solar B-V using Gray et al. 2003,2006 samples.
2 Feb 2008: Added equatorial rotation period and mean rotation period.
26 Mar 2008: Added discussion on solar abundances.
1 Apr 2008: Added calculations for statistics regarding logR'HK.
2 Apr 2008: Added discussion on temporal evolution of luminosity.
24 Apr 2008: Added discussion on Maunder minimum to chromospheric activity section.
1 Sep 2008: Changed author's affiliation to U. Rochester.
31 Oct 2008: Added discussion and estimate regarding the solar wind/mass loss.
1 Dec 2008: Added B. Skiff reference to solar-type MK stars & colors.
2 Dec 2008: Edited solar wind/mass loss discussion, Added reference.
2 Jan 2009: Added daily International Sunspot Number observations.
7 Jan 2009: Updated and reorganized discussion on solar Z value.
30 Mar 2009: Updated solar radius and age discussion (new age listed).
21 Apr 2009: Added discussion on moment of inertia.
15 Jun 2009: Updated astronomical unit to Pitjeva & Standich (2009) value.
23 Jun 2009: Corrected year in Bessell+ reference (thanks to M. Cushing).
1 Jul 2009: Added B-V estimate using relation from Valenti & Fischer (2005).
25 Aug 2009: Revised AU following 2009 IAU resolution.
4 Sep 2009: Revised solar bolometric magnitude to reflect 2009 IAU definition of AU.
4 Sep 2009: Thanks to Erik Bergren for pointing out the IAU Mbol zero point.
4 Sep 2009: Adopted Vmag(Sun) from Hayes85, Neckel86, Cox00.
4 Sep 2009: Revised bolometric correction (negligibly) due to revised Mv and Mbol.
10 Feb 2010: Added OMNI solar wind parameters and adopted those for solar mass loss.
29 Apr 2010: Added cgs estimate of solar mass loss rate.
4 Nov 2010: Updated discussion on solar abundances, included Asplund09 table, and dY/dZ.
9 Nov 2010: Added Lodders+2009 mass fractions to abundance discussion.
12 Nov 2010: Added Caffau+2010 reference on solar abundances, plus XYZ equations.
20 Nov 2010: Added discussion on DE423 JPL ephemeris on solar mass and AU discussion.
30 Dec 2010: Added discussion on TDB and "SI" estimates of GMsun (Thanks Erik Bergren)
18 Apr 2011: Added oblateness and bulk density estimates
31 May 2011: Added solar V magnitude estimates by Engelke10 and Reike08
21 Jun 2011: Updated discussion on solar B-V and adopted value
14 Apr 2012: Included estimate of AU in light-days
14 Apr 2012: Updated solar mass - using new G (CODATA 2010) and GMsun (IAU 2009)
14 Apr 2012: Update to chrom. activity discussion, revised logR'HK & S_MW
23 May 2012: Updated solar B-V value and discussion
24 May 2012: Updated solar irradiance and luminosity values and discussion (Kopp & Lean 2011)
24 May 2012: Updated bolometric magnitude value and discussion
24 May 2012: Minor revision to logR'HK due to revised B-V
25 May 2012: Updated solar V, Mv, BCv values and discussion based on new review of V values
8 Jun 2012: Added some values from Allen 1963 (Astrophysical Quantities, 2nd. Ed.)
9 Jun 2012: Adopted new solar radius from Haberreiter+2008 as standard.
11 Jun 2012: Adopted new Teff taking into account new TSI value.
28 Jun 2012: Major update to B-V discussion to reflect Ramirez+2012.
2 Sep 2012: Major update to AU (2012 IAU resolution B2) - AU is now exact.
2 Oct 2012: Added discussion of other solar colors, adopted Ramirez+2012 B-V, recalc. M_B.
22 Oct 2012: Added discussion on surface gravity (log(g)) and equatorial rotation velocity.
22 Oct 2012: Updated solar sidereal equatorial rotation period and added discussion.