Parameter
 Var.  Value  Unc. 
Apparent V magnitude  V  26.71 mag  0.02 
Absolute V magnitude  M_{V}  4.862 mag  0.02 
Absolute B magnitude  M_{B}  5.515 mag  0.02 
Vband Bolometric Correction  BC_{V}  0.107 mag  0.02 
Absolute Bolometric Mag. [IAU nominal Sun]
 M_{bol}  4.74 mag  ... 
Apparent Bolometric Mag. [IAU nominal Sun]
 m_{bol}  26.832 mag  ... 
Photospheric Radius
[IAU Nominal Solar Radius]
 R_{sun}  695660 km
[695700 km]
 100
[exact]

Photospheric Angular Radius  theta_{Sun}  959".176  0".138

Oblateness f = (ab)/a  f  8e6  ... 
Spectral Type  SpT  G2 V  exact 
Effective Temperature
[IAU Nominal Solar Effective Temperature]
 T_{eff}  5771.8 K
[5772 K]
 0.7
[exact]

Effective Temperature log_{10 [log10 of IAU Nominal Solar Effective Temperature]}  log(T_{eff})  3.76131 dex
[3.7613...]
 0.00005
[exact]

Gravitational Constant (TDB)  GM_{Sun}  1.32712440041e20 m^{3} s^{2}
 1.0e10 
Gravitational Constant (SI,TCB)
[IAU Nominal Solar Mass Parameter]
 GM_{Sun}  1.32712442099e20 m^{3} s^{2}
[1.3271244e20 m3 s2]
 1.0e10
[exact]

Mass  M_{Sun}  1.98855e30 kg  0.00024e30 
Bulk Density  rho_{Sun}  1.4111 g cm^{3}  0.0002 
Moment of Inertia  I_{Sun}  5.96e47 g cm^{2}  ... 
Inertia Constant (k = I/MR^{2})  ^{ k}
 0.062  ... 
Surface Gravity (cgs units)  g  27423.2 cm s^{2}  7.9 
Surface Gravity (MKS units)  g  274.232 m s^{2}  0.079 
Surface Gravity log_{10}(cgs)  log(g)  4.43812 dex  0.00013 
Escape Velocity  Vesc  617.69 km s^{1}  0.04 
Astronomical Unit [IAU]  AU  149,597,870.700 km  exact 
Mean Separation EarthMoon Barycenter and Sun (J2000)
 <r>  149,641,158 km  ... 
Tot. Solar Irradiance@1AU(MKS)
[IAU Nominal Total Solar Irradiance]
 S_{1AU}  1360.8 W m^{2 }[1361 W m^{2}]  0.5
[exact]

Tot. Solar Irradiance@1AU(cgs)  S_{1AU}  1360800 erg s^{1}cm^{2}  500 
Luminosity (cgs units)  L_{Sun}  3.8270e33 erg s^{1}  0.0014e33 
Luminosity (MKS units)
[IAU Nominal Solar Luminosity]
 L_{Sun}  3.8270e26 W
[3.828e26 W]
 0.0014e26
[exact]

Mean Chromospheric Activity (Mt. Wilson Svalue) (Average Solar Minimum, rms) (Average Solar Maximum, rms)  S_{MW}  0.1694
(0.1621) (0.177)
 0.0005
(0.0008) (0.003)

Mean Chromospheric Activity (R'_{HK} index) (Average Solar Minimum, rms) (Average Solar Maximum, rms)
 logR'_{HK}  4.9427
(4.9844) (4.905)
 0.0072 (0.0049) (0.016)

Mean Xray Luminosity (0.12.4keV; ROSAT PSPC band)
 L_{X}  2.24e27 erg s^{1}  50%: 
Mean Xray Surface Flux (0.12.4keV; ROSAT PSPC band)
 f_{X}  36800 erg s^{1} cm^{2}  50%: 
Xray/Bolometric Luminosity Ratio  log(L_{X}/ L_{bol})  6.24 dex  0.24: 
Age of Sun and Solar System  t_{Sun}  4567.30 Myr  0.16 
Protosolar Hydrogen (Z=1) Mass Fraction
 X_{o}  0.70280.7154  ... 
Protosolar Helium (Z=2) Mass Fraction  Y_{o}  0.27030.2783  ... 
Protosolar Metal (Z>=3) Mass Fraction
 Z_{o}  0.01420.0189  ... 
Photospheric Metal (Z>=3) Mass Fraction
 Z_{phot}  0.01340.0172  ... 
Equatorial Rotation Period  P_{eq}  24.47 day  ... 
Equatorial Rotation Velocity  v_{eq}  2.067 km s^{1}  ... 
Mean Rotation Period  <P>  26.09 day  ... 
Solar Wind Mass Loss  <dM/dt>  1.3e14 M_{sun} yr^{1}  ... 
Solar Wind Pressure @1AU  <P_{SW}>  1.77 nPa (median) 2.05 nPa (mean)  ... 
Daily International Sunspot Number (yrs:18182008)
 <ISN>  40 (median) 54 (mean)
 ... 
BV color (Johnson)  (BV)_{Sun}  0.653 mag  0.003 
UB color (Johnson)  (UB)_{Sun}  0.158 mag  0.009 
VR_{c} color (Johnson,Cousins)  (VR_{c})_{Sun}  0.356 mag  0.003 
VI_{c} color (Johnson,Cousins)  (VI_{c})_{Sun}  0.701 mag  0.003 
VJ color (Johnson,2MASS)  (VJ)_{Sun}  1.198 mag  0.005 
VH color (Johnson,2MASS)  (VH)_{Sun}  1.484 mag  0.009 
VK_{s} color (Johnson,2MASS)  (VK_{s})_{Sun}  1.560 mag  0.008 
JH color (2MASS)  (JH)_{Sun}  0.286 mag  0.01 
JK_{s} color (2MASS)  (JK_{s})_{Sun}  0.362 mag  0.01 
HK_{s} color (2MASS)  (HK_{s})_{Sun}  0.076 mag  0.01 
VW1 color (Johnson,WISE)  (VW1)_{Sun}  1.608 mag  0.008 
VW2 color (Johnson,WISE)  (VW2)_{Sun}  1.563 mag  0.008 
VW3 color (Johnson,WISE)  (VW3)_{Sun}  1.552 mag  0.009 
VW4 color (Johnson,WISE)  (VW4)_{Sun}  1.604 mag  0.011 
by color (Stromgren)  (by)_{Sun}  0.4105 mag  0.0015 
m1 color (Stromgren)  m1_{Sun}  0.2122 mag
 0.0018 
c1 color (Stromgren)  c1_{Sun}  0.3319 mag  0.0054 
Beta color (Stromgren)  Beta_{Sun}  2.5915 mag
 0.0024 
Distance to Center of Galaxy (Sun's Galactocentric Dist.)
 R_{O}  8.0 kpc  0.4 
Nearest Known Star to Sun (Proxima Centauri)
 d_{star}  1.3009 pc  0.0005 
Nearest Known Brown Dwarf to Sun (Luhman 16)  d_{BD}  2.0 pc  0.15 
Solar Wind Velocity @1AU  V_{SW(1AU)}  416 km s^{1} (median) 439 km s^{1} (mean)  ... 
Solar Wind Density @1AU  n_{SW(1AU)}  5.7 cm^{3} (median) 6.9 cm^{3} (mean)
 ... 
Interplanetary Magnetic Field (IMF) @1AU  B_{IMF}(1AU)  5.8 nT (median) 6.4 nT (mean)
 ... 
Coronal Temp. (emission measureweighted)
 T_{corona}  1.5 MK  ... 
SemiMajor Axis of EarthMoon Barycenter and Sun (J2000.0)
 a(J2000)  1.0000010176 AU 149,598,022.93 km
 ... 
   
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. As you can see, plenty of tedious
editing would be required for including links to each and every reference that I cite here 
time I unfortunately do not have at the moment. 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.
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 solar values listed in the table.  EEM
Solar Apparent V Magnitude:
V(Sun) = 26.71 + 0.02 mag
Note that I need to update this section  it has not been revisited over 20152016.
This value varies somewhat in the literature, at the hundreds of a magnitude
level. 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.
I've included a few new (unpublished) estimate:
Adopting equation #5 and coefficients from Table 1 of Boyajian+ (2014)
(angular diameter as function of color and metallicity), adopting solar colors from
Casagrande+2012, [Fe/H]=0.00, and angular radius 959.176" (calculated using Haberreiter+ 2008),
corresponds to solar V magnitudes of 26.753 (from BV), 26.754 (from VIc), 26.798
(from VRc), 26.665 (from VW3), 26.733 (from VW4). These 5 estimates
are consistent with Vmag = 26.747 (+0.021 sem, +0.035 stdev).
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)
26.71 Pecaut & Mamajek (2013) [adopted]
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)
26.730(44) Stebbins & Kron (1957,ApJ,126,266) [original value, p.e.=0.03 quoted)]
26.74 Allen76 (Astrophysical Quantities, 2nd ed.)
26.74 SchmidtKaler82 (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)
26.741 EEM calculated for 22 solar analogs using Casagrande10 bolometric fluxes
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)
26.747 EEM calculated using Boyajian14 relation, BV=0.653, Haberreiter+08 radius
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)[SUNOVER(ATLAS9, overshoot)]
26.77 Bessell+98 (A&A 333, 231)[SUNNOVER(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 Vmag 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 post1990 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. The Rieke08 Sun spectrum integrates to a solar
constant of 1366.7 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 ASUN Vmag values should be offset +0.00494 mag
(fainter), and the Engelke10 solar Vmags dependent on the Rieke08 synthetic
solar spectrum should be offset +0.00470 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 mid1980s 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 Vmagnitude 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.
When fitting synthetic atmosphere models (BTSettl & Kurucz) to the
solar colors from Ramirez et al. (2013), Pecaut & Mamajek (2013) we were able
to solve for selfconsistent solar Teffs (~5772K) and solar angular radius
when adopting V = 26.71 + 0.02 mag (consistent with the Engelke+2010 Vmag).
The fits were much worse (predicting incorrect solar radii) when the V magnitude
was allowed to vary to other commonly quoted values (e.g. 26.74, 26.76).
Following the analysis of Engelke+2010 and Pecaut & Mamajek (2013),
I adopt V(Sun) = 26.71 + 0.02 mag.
Solar Absolute V Magnitude:
Mv(Sun) = 4.862 + 0.02 mag
As V magnitude varies at the hundredths of a magnitude level in the literature,
so does the absolute V magnitude. Here are various quoted
values summarized in Bessell, Castelli, & Plez (1998, A&A 333, 231;
Table A4), along with some more recent estimates.
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 SchmidtKaler82 (Landolt Bornstein, Num. Data..., Vol 2, p.451)
4.862 Pecaut & Mamajek (2013)
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.
This is due to the reference distance for absolute magnitudes (10 parsecs)
being defined by an exact number of AU (in this case, 10*3600*180/pi).
The distance modulus for 1 AU will always be (mM) = 31.5721, so long
as the absolute magnitude scale is set using 10 pc as the reference distance.
So the absolute magnitude of the Sun is set by the adopted Vmag for the
Sun. Following the analysis in Engelke+2010 and Pecaut & Mamajek (2013),
I have adopted Vmag(Sun) = 26.71 + 0.02 mag.
Hence, I adopt Mv = (26.71+0.02)  (31.5721)
Mv = 4.862+0.02
where the uncertainty comes from the rms in the estimated solar V magnitude.
Solar Absolute B Magnitude:
M_B(Sun) = 5.515 + 0.02 mag
I simply calculate this from the adopted values for Mv and BV
(where the solar BV comes from Ramirez et al. 2012):
M_B = M_V + (BV) = (4.862+0.02) + (0.653+0.003)
M_B = 5.515 + 0.020 mag
Solar Bolometric Correction:
BCv(Sun) = 0.107 + 0.02 mag
Solar Absolute Bolometric Magnitude:
Mbol(Sun) = 4.74 mag
Note that I need to update the section on the solar bolometric correction 
it has not been revisited over 20152016. The discussion on the solar
bolometric absolute magnitude has been updated to reflect the IAU 2015 scale.
Post2015 discussion: In August 2015, the IAU General Assembly
adopted IAU 2015 Resolution B2, which defined the bolometric magnitude
scale once and for all. The zero point of the scale was set
such that the "nominal" Sun (with nominal solar luminosity 3.828e26 W)
corresponded to absolute bolometric magnitude Mbol(Sun) = 4.74.
The nominal total solar irradiance (1361 W/m2) corresponds to
apparent bolometric magnitude mbol(Sun) = 26.832. Note that this
bolometric scale replaces provisional IAU zeropoints introduced
in 1999 by IAU Commissions 25 and 36, which were not widely
advertised nor adopted by the astronomical community.
Pre2015 discussion for historical purposes: <obsolete>
WARNING! In pre2015 publications, different authors used different
zeropoints 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 introduce systematic errors in your stellar
luminosities. 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 their zeropoint 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. I
do not advocate it.
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
(3.055e28 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
Where the uncertainty is *solely* due to the uncertainty in the
adopted solar luminosity, which is propagated mainly from the
systematic uncertainty in the Kopp & Lean (2011) estimate of the total
solar irradiance.
Following Engelke+2010 and Pecaut & Mamajek (2013), I have
adopted V = 26.71 (+0.02) and Mv = 4.862 (+0.02), then
the bolometric correction of the Sun will be defined as:
BC_V(Sun) = 0.107 + 0.02 mag =>
BC_V(Sun) = 0.11 + 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 zeropoint (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) = 0.11 + 0.02 mag.
Note that while the best estimate of the solar V magnitude might
change in the future, so long as there is common usage of the IAU
Commissions 25 & 36 bolometric magnitude scale, then (1) the solar
bolometric magnitude will not change, but (2) the solar bolometric
correction might change.
</obsolete>
One can convert between stellar absolute bolometric magnitude M_{bol}
and stellar luminosity L_{star} on the IAU 2015 luminosity and bolometric
magnitude scale using the following formulae (recall 10^{7} erg/s = 1 W):
M_{bol}(star) = 2.5 log_{10}(L_{star}/L_{sun}) + M_{bol}(Sun)
M_{bol}(star) = 2.5 log_{10}(L_{star}/3.828e33 erg/s) + 4.74 mag
M_{bol}(star) = 2.5 log_{10}(L_{star}/3.0128e28 W)
L_{star} = L_{sun} * 10^(0.4(M_{bol}  M_{bol}(Sun))
L_{star} = (3.828e33 erg/s) * 10^(0.4(M_{bol}  4.74))
L_{star} = (3.0128e28 W) * 10^(0.4 M_{bol})
log_{10}(L_{star}/L_{sun}) = 0.4 (M_{bol}  M_{bol}(Sun))
log_{10}(L_{star}/L_{sun}) = 0.4 (M_{bol}  4.74)
Solar Apparent Bolometric Magnitude:
mbol(Sun) = 26.832 mag
Post2015 discussion: In August 2015, the IAU General Assembly
adopted IAU 2015 Resolution B2, which defined the bolometric magnitude
scale once and for all. The zero point of the scale was set
such that the "nominal" Sun (with nominal solar luminosity 3.828e26 W)
corresponded to absolute bolometric magnitude Mbol(Sun) = 4.74.
The nominal total solar irradiance (1361 W/m2) corresponds to
apparent bolometric magnitude mbol(Sun) = 26.832. Note that this
bolometric scale replaces provisional IAU zeropoints introduced
in 1999 by IAU Commissions 25 and 36, which were not widely
advertised nor adopted by the astronomical community.
Note that the distance modulus (the difference between the absolute
and apparent bolometric magnitudes) for distance = 1 AU will always be
(mM) = 31.5721, as a 2012 IAU resolution defined the astronomical unit
to be an exact number of meters. The IAU 2015 Resolution B2 values
for the nominal Sun (Mbol = 4.74, mbol = 26.832) reflect this.
Apparent bolometric magnitudes are useful as they be easily
calculated using apparent magnitudes and bolometric corrections,
and can be easily tied to a bolometric flux on SI scale (f_bol; in
either W/m^2 or erg/cm^2). On the IAU 2015 scale, these relations
apply:
f_bol = f_bol(Sun)*10^(0.4*(mbol  mbol(Sun))
f_bol = (1361 W/m^2)*10^(0.4*(mbol  26.832))
m_bol = m_bol(Sun)  2.5*log(f_bol/f_bol(Sun))
m_bol = 26.832  2.5*log(f_bol/1361 W/m^2)
Solar Radius:
R(Sun) = 695660 (+100) km
[IAU Nominal Solar Radius = 695,700 km (exact)]
IAU Value: The IAU had previously adopted a solar radius of 696000 km in its
1976 System of Astronomical Constants (presumably for e.g. eclipse
calculations). This value was repeated by the IAU Working Group
on Cartographic Coordinates and Rotational Elements (2009), reported in
Archinal et al. (2011, CMDA, 109, 101). The value was listed as a
comparison for the adopted planetary radii, and no effort was mentioned
to adopt a newer, more accurate value (the Working Group's focus
is specifically the poles of rotation and prime meridians of the
planets, satellites, minor planets, and comets). The literature
on the solar radius was reviewed extensively by the IAU Working
Group on Nominal Units for Stellar and Planetary Astronomy in drafting
of IAU 2015 Resolution B3 (Prsa et al. 2016), which was adopted
by the IAU General Assembly in 2015 in Honolulu. The resolution
adopted a nominal solar radius of 695,700 km for use when quoting
the radii of objects in units of the Sun's radius (now in units
of nominal solar radii). The purpose of adopting a nominal solar
radius as an exact number of meters (or km) was because the
radii of some eclipsing stars were becoming so accurately known
that their precision was degraded when quoting their radii
in terms of an observed solar radius (which also had the downside
that "which" solar radius was best was at the caprice of the authors).
Recent Summaries on observed estimates of the solar photospheric
radius: 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).
Details: The Sun obviously does not have a solid surface, so some
physical definition must be made to define a radius. For stellar
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 &
ChristiansenDalsgaard 2008, take this modeldependent 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 fmode
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 +0.3 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.
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".176 ; Haberretier+ 2008 (ApJ 675, L53) calc. by EEM as theta = arcsin(Rsun/AU)
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&ChristensenDalsgaard 1998 (ApJ 500,L195)
959".73 +0".05 ; Wittmann 1997
959".74 +0".008 ; Kubo 1993 (PASJ 45, 819; 1970 eclipse data)
959".84 +0".015 ; Kubo 1993 (PASJ 45, 819; 1973 eclipse data)
959".84 +0".012 ; Kubo 1993 (PASJ 45, 819; 1980 eclipse data)
959".86 ; Meftah+ 2014 (Solar Physics 289, 1; Picard/SODISM, 607.1 nm)
959".88 +0".008 ; Kubo 1993 (PASJ 45, 819; 1991 eclipse data, 461.5 nm)
959".89 +0".19 ; Hauchecorne+ 2014 (ApJ 783, 127; Picard/SODISM, 607.1 nm)
959".90 +0".06 ; Hauchecorne+ 2014 (ApJ 783, 127; HMI, 617.3 nm)
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 &
ChristensenDalsgaard 1998, but not their angular radius. Cox 2000
quotes an oblateness as the semidiameter equatorpole 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 reanalysis 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 & ChristensenDalsgaard 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.)
696072+6 Kubo93 (1970 eclipse data, calc. using angular diam.)
696144+11 Kubo93 (1973 eclipse data, calc. using angular diam.)
696144+9 Kubo93 (1980 eclipse data, calc. using angular diam.)
696156+145 Meftah14 (2012 Venus transit/PICARD; 2014SPIE.9143E..1RM)
696173+6 Kubo93 (1991 eclipse data, calc. using angular diam.)
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 & ChristensenDalsgaard (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.
The uncertainty in the value is probably ~+100 km (+0.014%; +0".138)
based on Table 3 of Haberreiter+2008. The Haberretier+2008 radius translates
to a solar angular radius of 959".176.
In 2015, the IAU adopted the nominal solar radius to be exactly 695,700 km.
This is the value that should be used for calculating radii of stars in
units of nominal solar radii, rather than the "observed" solar value (which
may vary at the +100 km level in the future due to more refined measurements).
See Prsa et al. (2016) for further discussion.
Solar Oblateness
f = (ab)/a = (8.5+0.7) 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 = (ab)/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
(equatorpole 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 anticorrelated with solar
activity (Egidi+ 2005, Solar Physics, 235, 407).
oblateness =
4.310.3 X 10^6 ; Egidi+2005, Solar Physics, 235, 407
8.5+0.7 X 10^6 ; Meftah+2014 2014SPIE.9143E..1RM
calculated using quoted 5.9+0.5 km equatorpole radius difference.
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.
Meftah+2014 quote the equatorpole radius to be 5.9+0.5 km,
and so adopting their equatorial radius (696,156 km) leads to f =
8.5+0.7 x 10^6, which I adopt.
Solar Spectral Type: G2V
G2 V ubiquitous
(except, apparently, the first edition of Allen's Astrophysical
Quantities from 1955, which inexplicably listed G1V).
A Brief History of the Sun as a G2 V Standard Star:
The integrated solar spectrum (as inferred from reflection spectra of
bodies like the moon, Uranus, Callisto, etc.) has been often
considered the standard spectrum defining G2V. "V" implies "dwarf" luminosity
class, as the Sun is clearly a main sequence star. The Sun was not mentioned
at all in the "MKK" atlas (Morgan, Keenan, Kellman 1943), however the Sun was
one of 3 G2V standard stars in Johnson & Morgan (1953) defining the "MK" system
(the other two were 16 Cyg A and HR 483, however these were later
classified as G1V and G1.5V standards by Keenan). The Sun's status was reiterated
in later papers by Morgan (Morgan & Hiltner 1965, Morgan et al. 1971),
and Morgan considered the Sun to be the G2V "dagger" standard defining
the "revised MK" (or "MK73") system in Morgan & Keenan (1973). Keenan later
reiterated the Sun as a G2V standard in his standard star
compilations: Keenan & McNeil (1976), Keenan & Pitts (1980), Keenan
(1983), Keenan & Yorka (1985), Keenan & Yorka (1988), Keenan & McNeil (1989).
Keenan & Pitts (1980) states "The MK system for stars with essentially solar
composition necessarily rests on several standard stars defining the zero point and
scale for both temperature types and luminosity classes..." and lists the Sun (G2 V)
as the only G dwarf among the standard stars.
Garrison (1994) considers the Sun to be the "anchor" standard
defining G2V in the MK system. More notes on G2V stars can be found
at: http://www.pas.rochester.edu/~emamajek/spt/G2V.txt
Note that the *resolved* spectrum over the visible solar photosphere
appears to vary 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 "MorganKeenan" 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 Hlines 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".
As the Sun is "the" G2 V standard star  and one that has anchored the
MK spectral classification system since Johnson & Morgan (1953),
by definition there is no uncertainty in its subtype.
Solar Effective Temperature:
Teff = 5771.8 + 0.7 K
[Nominal Solar Effective Temperature = 5772 K ]
5777 K Cox 2000 (Allen's Astrophysical Quantities, 4th Ed.)
5781 K Bessell et al. 1998 A&A 333, 231
5772.0 + 0.8 K Prsa+2016 (calculated value for Sun)
5772 K Prsa+2016/IAU Resolution 2015 B3 (nominal value)
The effective temperature can be calculated using the total solar
irradiance (TSI) value, solar radius, AU, and the StefanBoltzmann
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 StefanBoltzmann constant sigma:
R = 695660(+100) km [solar radius from Haberreiter+2008]
f = 1361(+0.5) W/m^2 [TSI in Prsa et al. 2016]
D = 149597870700 m [AU from IAU 2012 resolution; exact]
sigma_SB = 5.670367(+0.000013) e8 W/m^2/K^4 [SB constant from CODATA 2014]
Then one derives:
Teff = (f*D^2/(sigma_SB*R^2))^(1/4)
= 5772.0 + 0.8 K
~ 5772 K
This is ~510K cooler than most previous estimates, but flows from the
slightly lower value for the total solar irradiance in the very recent
literature (see discussion in Prsa et al. 2016 and IAU Resolution 2015 B3).
Solar Mass:
M(Sun) = 1.98855(24)x10^30 kg
Solar Gravitational Constant (TDB)
GMsun(TDB) = 1.32712440041e20 m^3 s^2
Solar Gravitational Constant (SI, TCB)
GMsun(SI,TCB) = 1.32712442099e20 m^3 s^2
[nominal solar mass parameter
GM^{N}_{Sun} = 1.3271244e20 m^3 s^2 exact]
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 astroph/0508292)
1.32712440041e20 m^3 s^2 (DE423 ephemeris; see below)
1.32712440041939400e20 m^3 s^2 (DE430/DE431 ephemerides; Folkner+ 2014)
1.32712440041e20 (+1.0e10) m^3 s^2 (IAU 2009 constant ; TDBcompatible)
DE423 ephemeris (31 Mar 2010; JPL website:
ftp://ssd.jpl.nasa.gov/pub/eph/planets/ascii/de423/header.423 ) lists
GMsun = 0.295912208285591100D03 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 TDBcompatible constant).
The DE430/DE431 JPL ephemeris value for GMsun is from
Folkner, Williams,Boggs, Park, & Kuchynka (2014; IPN Progress Report 42196).
Uncertainties are not provided, only best values taking into account a wide
range of observational material. This appears to be the most recent value, and
is consistent with the IAU 2009 best estimate at the 7e12 fractional error level.
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 IAUsanctioned 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.
Ephemerides and their associated constants are quoted on
Barycentric Coordinate Time (TCB), and in a spacetime 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) = 1.550519768e8
(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 astroph/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 ; TCBcompatible)
IAU 2015 Resolution B3 did not explicitly recommend a nominal
solar mass due to the large uncertainty in G, and the fact that the product
GM was known to much higher precision.
The CODATA 2014 value for Newtonian constant G is:
6.67408e11 m^3 kg^1 s^2 (1.2e4 relative uncertainty)
http://physics.nist.gov/cgibin/cuu/Value?bgsearch_for=universal_in!
Adopting the CODATA 2014 value for G, and the IAU 2015 nominal solar
mass parameter, Prsa et al. 2016 calculates:
M(Sun) = GM^{N}_{sun}/G = 1.988475(92)x10^30 kg, where the uncertainty is
completely dominated by the uncertainty in G.
Here is a table of the ratios of the Sun's mass to that of the planets and Pluto
(or the planets + their satellites) as listed on the JPL "Astrodynamic Constants" page:
http://ssd.jpl.nasa.gov/?constants
mass ratio: Sun/Mercury 

6023600. (± 250.) 
mass ratio: Sun/Venus 

408523.71 (± 0.06) 
mass ratio: Sun/(Earth+Moon) 

328900.56 (± 0.02) 
mass ratio: Sun/(Mars system) 

3098708. (± 9.) 
mass ratio: Sun/(Jupiter system) 

1047.3486 (± 0.0008) 
mass ratio: Sun/(Saturn system) 

3497.898 (± 0.018) 
mass ratio: Sun/(Uranus system) 

22902.98 (± 0.03) 
mass ratio: Sun/(Neptune system) 

19412.24 (± 0.04) 
mass ratio: Sun/(Pluto system) 

1.35 (± 0.07) x 10^{8}_{ } 
These ratios are really ratios of the mass parameters ("GM") for these
objects, so one can quote them to much higher precision than
G is known as the G's cancel.
Solar Bulk Density
rho(Sun,bulk) = 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(1f) = 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 1.2e4;
dominated by the uncertainty in G) dominates the uncertainty in
density.
Solar Moment of Inertia
I = 5.96e53 g cm^2 = 5.96e47 kg m^2
Solar 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 kvalues:
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 kvalue 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 discussions) of:
GMsun = 1.32712442099e20 (+1.0e10) m^3 s^2 (IAU 2009 constant ; TCBcompatible)
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).
A convenient formula for estimating log(g) from stellar mass, Teff, and
luminosity (variables that often appear in stellar evolutionary tracks) is:
log(g) = log_{10}(M/Msun)  log_{10}(L/Lsun) + 4*log_{10}(Teff)  10.6068
Solar Escape Velocity:
Vesc = 617.96 + 0.04 km/s
Here I calculate the escape velocity unrigorously, using the Newtonian definition,
and ignoring any nuisances that one may wish to impose (i.e. ignoring the solar
rotation, latitude of "launch site", atmospheric drag, relativistic effects, etc.).
I have only seen the solar escape velocity relevant to solar wind calculations, where
accuracy beyond 2 or 3 decimal places seems hardly warranted, so I do not make any effort
to calculate it more accurately than that. You can think of this estimate as a
nonrelativistic escape velocity appropriate for a spherical, nonrotating, atmosphereless
"Sun" of solar mass and radius:
Vesc = sqrt(2GMsun/Rsun)
adopting the best values (from previous discussions) of:
GMsun = 1.32712442099e20 (+1.0e10) m^3 s^2 (IAU 2009 constant ; TCBcompatible)
Rsun = 695660 (+100) km
There are slight differences between which definition of GMsun is appropriate
(due to choice of reference frame). For now, I adopt the TCBcompatible "SI" definition of
GMsun. The differences between the adopted GMsun (whether one uses a TDBcompatible or
TCBcompatible "SI") are at the 10^{10} level, and impact the escape velocity at the 10^{5} level.
This is negligible compared to the uncertainty in the solar radius, which impacts the escape
velocity at the 10^{4} level.
Vesc = 617692 + 44 m/s => 617.69 + 0.04 km/s
Astronomical Unit:
AU = 149,597,870.700 km (exact)
SemiMajor Axis for EarthMoon Barycenter and Sun:
a(J2000.0) = 149,598,023 km
Mean Separation between EarthMoon
Barycenter and Sun:
<r> = 1.00028020 AU
<r> = 149,639,787 km
There is often some confusion about what exactly the astronomical unit
(AU) is. A common answer, repeated on a NASA/JPL website
(http://neo.jpl.nasa.gov/glossary/au.html), is that the AU is:
"An Astronomical Unit is the mean distance between
the Earth and the Sun." Historically, as observationally defined,
this was the correct. It is now a precisely determined length
defined by the International Astronomical Union (IAU) in meters
(more on this below), but does NOT correspond exactly to either the
"mean distance between the Earth and the Sun" nor the "semimajor
axis of the Earth's orbit", but it is VERY close to these values
(differing only after a few decimal places). Part of the problem
is that we are dealing with 1) the EarthMoon barycenter, not just
Earth, and 2) the Earth's orbit is constantly being perturbed by the
other masses in the solar system (e.g. planets, asteroids),
so its orbital elements (i.e. semimajor axis, eccentricity, etc.)
are constantly changing ever so slightly ("osculating"). You can
estimate the semimajor axis of the EarthMoon barycenter's and Sun's
orbit at a particular time (e.g. J2000.0 = Julian Date 2451545.0).
So if you are explaining the AU to a student, simply calling it the
mean EarthSun distance is probably OK, but it is not entirely
correct. Many thanks to email discussions with Erik Bergren on
this topic.
The AU is now a precisely defined unit of length defined in meters.
In August 2012, the IAU General Assembly adopted resolution B2, which
redefined 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. I find "au"
to be confusing as it makes me think of atomic units. I've been writing
it as "AU" for so many years, it is difficult for me to change now,
so throughout I will be using "AU".
The 2012 redefinition 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 used by Simon Newcomb (1895) in "Tables
of the Motion of the Earth on its Axis and Around the Sun", and
was used as a constant to define the AU for over a century.
I've split discussion on the AU and the characteristics of the
Earth's orbit around the Sun into several sections:
I: 19762009 values for the astronomical constant
II: The new IAU value for the astronomical constant (2009, 2012)
III: AU in lightseconds
IV: SemiMajor Axis of Orbit of EarthMoon Barycenter and Sun
V: The mean EarthSun distance (not the AU!)
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 subparsec scales). Unfortunately, the
actual semimajor 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. 34, p. 267). The semimajor 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 semimajor axis of the Earth's orbit may also suffer from
longterm variations due to subtle interactions between the planets
and minor bodies, etc. While the variations in the EarthSun 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: 19762009 values for the astronomical constant
Here is a list of some pre2009 published values for the AU, all of
which depended on combining an adopted Gaussian gravitational constant
with the most uptodate estimate of GMsun:
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
Note that the 2012 IAU value is independent of rest frame.
According to the IAU NSFA (2009) website, "The [2009] value for au [was]
TDBcompatible. An accepted definition for the TCBcompatible 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/astronomicalapplications/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 unit of length in meters. Hence, the IAU definition of
the astronomical unit is now decoupled from the dreaded Gaussian
gravitational constant (see IAU 1976 constants) *and* the actual
semimajor axis of the Earth's orbit around the Sun. The 2012
definition of the astronomical constant sets it as an integer number
of meters, independent of which reference frame it is used in.
III. Astronomical Unit in lightseconds
Adopting the 2012 IAU length 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 lightseconds:
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!)
IV. SemiMajor Axis of Orbit of EarthMoon Barycenter and Sun
Due to perturbations on the orbit of the EarthMoon Barycenter (EMB) around
the Sun, its semimajor axis is usually varying subtly around 1 AU.
As the value is changing, if one is going to quote a value for the
semimajor axis of the EMB's solar orbit, one should either quote
an epoch (time) for which the estimate is accurate, or a range of
times over which the quoted semimajor axis is understood to be
an average value. Here are a few estimates from the literature
and calculated by myself from published data (note that some
older values use older estimates of the astronomical unit, and
may differ at the ~10th decimal place):
a(AU)
1.00000105726665 ; Bretagnon (1982) VSOP82 fit to DE200 (Table 5; J2000)
1.000001017788 ; Bretagnon (1982) VSOP82 fit to DE200 (Table 7 & 8; J2000)
1.00000011* ; Cox (2000) Allen's Astrophys. Quantities citing
Seidelmann (1992) Explanatory Suppl. to Astronomical Almanac (Tab 5.8.1;
Classical Keplerian elements at epoch J2000 (JED 2451545.0)
1.0000010176 ; Simon+ (2013) VSOP2013 solution to DE405
0.99999911 ; Recent (20082013) average of 56 epochs from osculating
orbital elements in Astronomical Almanac
(rms variations +0.00001070 AU)
1.00000102 ; average from present to 50 million years ago at 1kyr
intervals from Laskar+2011 "La2010" longterm solution
(rms variations +0.00001101 AU)
* = assumes AU = 149,597,870.66 km
For the rest, assume AU = 149,597,870.700 km.
The osculating orbital elements in the Astronomical Almanac
published between 2008 and 2013 show that over a 5 year period,
the rms variations in the semimajor axis of the EarthMoon
Barycenter orbit around the Sun  quoted at 40 day intervals 
are at the +0.000001070 AU = +1601 km level! Remarkably, if
one examines the Laskar+2011 reconstruction of the Earth's orbit
back 50 million years (the simulations goes back 250 million years,
but they only validate the past 50 Myr), one sees that the
rms variations in the semimajor axis are almost identical to that
seen over a recent 5 year span: +0.00001101 AU = +1647 km.
The Laskar+2011 reconstruction finds no longterm trend in the
migration of the semimajor axis. So whether you are talking
about months, years, or megayears, the semimajor axis
of the EarthMoon barycenter/Sun system is varying at the
10^5 level. These results illustrate the usefulness of adopting
a yardstick value for the AU that isn't tied to the shortterm
wanderings of the actual semimajor axis of the orbit of the
EarthMoon Barycenter about the Sun.
The Simon+ (2013) VSOP2013 value is the probably the best
"snapshot" value of the semimajor axis (EMB/Sun) at epoch 2000.0,
however the average value from the Laskar+2011 simulations
probably represents the best longterm average for the
semimajor axis.
V: The mean EarthSun distance
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 semimajor axis, and e is the eccentricity.
Here I adopt the eccentricity for the orbit of the EarthMoon barycenter
and Sun for epoch J2000 (e = 0.0167086298; Simons+ 2013 VSOP2013 value
for epoch J2000). Note that the eccentricity varies somewhat:
During 55 epochs between 2008 and 2013, the Astronomical Almanac
listed osculating orbital elements for the EMB/Sun orbit which
had mean eccentricity <e> = 0.016703 with rms scatter +0.000023.
In the longterm orbit simulations by Laskar+2010, they show
that over the past 50 million years, the eccentricity of the
orbit of the EarthMoon barycenter and Sun has range
from nearly 0 (e_min ~ 1.5e4) to as much as 0.067!
Using the Simons+2013 (VSOP2013) values for a and e for epoch J2000.0,
one derives a mean separation between the Sun and EarthMoon barycenter to be:
<r> = a(1+e^2/2) = 1.00028020 AU = 149,639,787 km
Looking at the longterm orbital evolution of the EMB/Sun orbit
by Laskar+2010, it appears that there may have been epochs during
the past 50 million years where the mean separation between
the EarthMoon barycenter and Sun may have been as large as:
r_max ~ a(1+e_max^2/2) ~ 1.0045 AU.
The highest estimated eccentricity during the past 50 million years
during the Laskar+2010 simulations was 27.355 million years ago.
However this would have only decreased the Earth's average equilibrium
over the course of its orbit by about 0.6 K.
Total Solar Irradiance:
S(1AU) = 1360.8 (+0.5) W/m^2
In their recent review on "Solar Irradiance Variability and Climate",
Solanski, Krivova, & Haigh (2013, ARA&A, 51, 311) define Total Solar
Irradiance (TSI) as "as the total power from the Sun impinging on a unit area
(perpendicular to the Sun’s rays) at 1AU (given in units of Wm−2). The TSI is the wavelength
integral over the solar spectral irradiance, or SSI (Wm−2nm−1)."
TSI may also be called the "solar flux constant", "solar bolometric flux", and even more
generically "the solar constant" (e.g. http://en.wikipedia.org/wiki/Solar_constant).
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.
Here is a table of TSI values, both published and calculated medians estimated
by the author from online datasets from recent experiments. Recent values are in bold.
<S(1AU)> W/m^2 = Total Solar Irradiance at 1 AU:
1360.6+0.5 Kopp & Lean, and Wang+2005 (see below)
1360.8+0.5 Kopp & Lean (2011, Geop.Res.Let.,38,L01706) [2008 minimum]
1360.8+0.4 SORCE/TIM TSI measurements [Feb 2003Nov 2014; 3990 measurements]
1360.9+0.4 Schmutz+ (2013; AIPC 1531, 624) PREMOS/Picard [2010 1st light]
1361.3+0.4 TCTE TSI measurements (PI Kopp) [Dec 2013Nov 2014; 172 measurements]
1362.1+2.4 Meftah+ (2014) SOVAP/Picard [20102013]
1364.5+1.4 Mekaoui+ (2011) DIARAD/SOVIN on ISS [3 days during June 2008]
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; ACRIM on SMM)
1372.7 Hickey+ (1982, The Symp. on the Solar Constant..., p.10; ERB/Nimbus 7)
13651369 Cox 2000 (error in units)
For astronomers that need to know the solar luminosity or absolute
bolometric magnitude in their calculations, one should note that there
has been a slight downward revision to the solar bolometric flux (TSI)
in recent years at the ~0.4% level. This is due to improved radiometric
calibration of new TSI detectors, and correction for
internal scatter on older instruments
(Kopp & Lean 2011; Geophys. Res. Letters 38, L01706).
A brief summary of the TSI historical record and recent experiments is
presented by Kopp (2014; J. Space Weather Space Clim. 4, A14). While many
pre2010 papers quote TSI values around ~13651370 W/m^{2}, 5 recent
experiments (SORCE/TIM, ACRIM3, PREMOS/Picard, SOVAP/Picard, TCTE/TIM) are all
producing TSI measurements near 1361 W/m^{2}. The TIM experiments
were calibrated at the TSI Radiometer Facility (TRF) to a
NISTcalibrated radiometric standard (operating at vacuum conditions)
which is tied to the World Radiometric Reference
(WRR) and SI radiometric scale at the ~0.35% level
(Fehlmann+ 2012; Metrologia 49, S34; Kopp 2014; Kopp & Lean 2011).
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 have estimated 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.
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 16102011 mean value is:
<S(1AU)> = 1360.65 (sem +0.02, st.dev. +0.39) W/m^2
The 19782011 mean value is:
<S(1AU)> = 1361.25 (sem +0.06, st.dev. +0.37) W/m^2
Note that the standard errors of the mean and standard deviations
quoted here are smaller than the calibration errors (+0.48 W/m^2).
Kopp & Lean (2011) quote a TSI during the 2008 solar minimum 
which appears to define a moreorless 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). As Kopp & Lean (2011) also demonstrate,
their longterm reconstructed solar constant history based on
Wang et al. (2005) has a mean similar to that of recent measurements
of the quiet Sun (within a few tenths of a W/m^2).
The differences between the mean values estimated
over solar cycles and the historic reconstruction are all
within 1sigma of the calibration uncertainty.
SORCE/TIM update: As of 26 Nov 2014, the SORCE/TIM TSI experiment
(http://lasp.colorado.edu/lisird/sorce/sorce_tsi/index.html)
has reported 3990 daily TSI measurements between 25 Feb 2003 and
19 Nov 2014. This range covers a full solar cycle from max to min to
max. The SORCE/TIM TSI measurements are consistent with a median TSI
of 1360.84 W/m^2 (+0.39 W/m^2 68.3%CL; standard error of the median
+0.007 W/m^2).
Kopp is PI of the new Total Irradiance Monitor (TIM) on the
TSI Calibration Transfer Experiment (TCTE) mission which launched
19 November 2013. The new TCTE/TIM measurements are being reported
at this website: http://lasp.colorado.edu/lisird/tcte/tcte_tsi/index.html.
As of 26 Nov 2014, the TCTE/TIM experiment reported 172 daily measurements of
TSI between 16 Dec 2013 and 15 Nov 2014. The values are consistent
with a median TSI of 1361.33 W/m^2 (+0.41 W/m^2 68.3%CL; standard error
of the median +0.03 W/m^2).
For calculations of solar luminosity and other related derived quantities,
I adopt the Kopp & Lean (2011) value of 1360.8 + 0.5 W/m^2.
Solar Luminosity:
Lsun = 3.8270 (+0.0014) e33 erg/s
Since this value is obviously related to the Total Solar Irradiance
(TSI) measurements, the reader should review that section.
Here is an incomplete compilation of published estimates of the solar
luminosity in 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.)
As mentioned in the section on TSI, the most recent TSI measurements
are ~0.4% lower than those widely published during the 1980s through
early 2000s. Newer instruments are all recovering TSI measurements of
~1361 W/m^{2}.
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 longterm (16102011) of:
<S(1AU)> = 1360.646 +0.477 W/m^2
where the uncertainty is completely dominated by the +0.035%
flux calibration of SORCE/TIM.
I have decided to estimate the luminosity using the "quiet Sun"
Total Solar Irradiance estimated by Kopp & Lean (2011)
(1360.8+0.5 W/m^{2}). The TSI appears to repeatably revert to this level
during solar minima whereas the TSI during solar maxima varies somewhat.
The "quiet Sun" TSI appears to be a benchmark value quoted in recent
studies (e.g. Solanski, Krivova, & Haigh 2013, ARA&A, 51, 311).
Combining this longterm TSI value with the 2009 IAU value for the
astronomical unit (149597870700 m), I estimate:
L(Sun) = <S(1AU)>*4*pi*AU^2
= (3.8270 + 0.0014) e33 erg/s
= (3.8270 + 0.0014) e26 W
The luminosity in log10 cgs units is then:
log(L_Sun/(erg/s)) = 33.58286
This is ~0.5% lower than that adopted by Cox (2000) and the GONG project.
See section on TSI measurements for discussion of the downward revisions
to TSI values.
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
zeropoint  i.e. not justified with quality measurements, but by
being in agreement with an outdated value. But this luminosity value
is difficult to justify as the recent total solar irradiance measurements
by multiple experiments over the past decade appear to be converging
towards a somewhat lower TSI (~1361 W/m^2).
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 1.0 solar mass
model from the YaleYonsei 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 Zeroage 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 = 2.745474E4
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 is a reconstruction of the Sun's history over the past 4,568 million
years, tagged at some major solar evolutionary stages or geological epochs
of significance. Geological timescales are set using the Geological
Society of America scale: http://www.geosociety.org/science/timescale/.
I have used a Bressen+2012 stellar evolutionary track for 1 Msun,
combined with the rotationactivity relations from Mamajek & Hillenbrand (2008)
[but imposing an observationallyconstrained Skumanich rotation law with
slope dlog(Period)/dlog(age) = 0.5263], the BV/Teff color relation from
Casagrande+2010, and Xray saturation level of Wright+2011.
tGya tGyr TeffK Lsun Rsun BV Per logRx logLx SpType Epoch
4.567 0.001 4513 2.076 2.357 1.171 <4 3.1 30.8 K3.2V 1 Myrold preMS Hayashi Track
4.563 0.005 4405 0.645 1.379 1.230 <4 3.1 30.3 K3.7V 5 Myrold preMS Hayashi Track
4.556 0.012 4510 0.455 1.105 1.172 <4 3.1 30.1 K3.2V End of Hayashi PreMS Stage
4.538 0.030 5676 0.930 0.997 0.684 <4 3.1 30.4 G4.2V End of Henyey PreMS Stage
4.523 0.045 5630 0.686 0.871 0.699 <4 3.3 30.1 G5.4V ZeroAge Main Sequence (ZAMS)
4.448 0.120 5645 0.707 0.879 0.694 3.8 3.9 29.5 G5.2V Pleiades Age
4.000 0.568 5660 0.734 0.891 0.690 8.7 4.8 28.6 G5.0V Archaen Eon/Eoarchean Era
3.918 0.650 5662 0.739 0.893 0.689 9.3 4.9 28.5 G4.9V Hyades Age
3.600 0.968 5672 0.756 0.900 0.686 11.5 5.1 28.3 G4.4V Paleoarchean Era
3.450 1.118 5676 0.764 0.904 0.684 12.4 5.2 28.2 G4.2V Barberton Greenstone Belt dacite
3.200 1.368 5684 0.777 0.909 0.681 13.8 5.4 28.1 G3.9V Mesoarchean Era
2.800 1.768 5696 0.800 0.918 0.678 15.8 5.5 27.9 G3.6V Neoarchean Era
2.500 2.068 5705 0.818 0.926 0.675 17.2 5.6 27.8 G3.4V Proterozoic Eon/Paleoproterozoic Era/Siderian period
2.300 2.268 5710 0.830 0.931 0.673 18.0 5.7 27.8 G3.2V Rhyacian Period
2.050 2.518 5717 0.846 0.937 0.670 19.1 5.8 27.7 G3.1V Orosirian Period
1.800 2.768 5725 0.862 0.944 0.668 20.0 5.8 27.6 G2.9V Stratherian Period
1.600 2.968 5730 0.875 0.949 0.666 20.8 5.9 27.6 G2.8V Mesoproterozoic Era/Calymmian Period
1.400 3.168 5736 0.889 0.955 0.664 21.5 5.9 27.6 G2.7V Ectasian Period
1.200 3.368 5742 0.904 0.961 0.663 22.2 6.0 27.5 G2.6V Stenian Period
1.000 3.568 5747 0.919 0.967 0.661 22.9 6.0 27.5 G2.5V Tonian Period
0.850 3.718 5751 0.930 0.971 0.660 23.4 6.0 27.5 G2.4V Cryogenian Period
0.635 3.933 5757 0.947 0.978 0.658 24.1 6.1 27.4 G2.3V Ediacaran Period
0.541 4.027 5759 0.954 0.981 0.657 24.4 6.1 27.4 G2.3V Cambrian Period
0.485 4.083 5760 0.959 0.983 0.657 24.6 6.1 27.4 G2.2V Ordovican Period
0.444 4.124 5761 0.962 0.985 0.656 24.7 6.1 27.4 G2.2V Silurian Period
0.419 4.149 5762 0.964 0.985 0.656 24.8 6.1 27.4 G2.2V Devonian Period
0.359 4.209 5764 0.969 0.987 0.656 25.0 6.1 27.4 G2.2V Carboniferous Period
0.299 4.269 5765 0.974 0.989 0.655 25.2 6.1 27.4 G2.1V Permian Period
0.252 4.316 5766 0.978 0.991 0.655 25.3 6.2 27.4 G2.1V Triassic Period
0.201 4.367 5767 0.983 0.993 0.654 25.5 6.2 27.4 G2.1V Jurassic Period
0.145 4.423 5768 0.987 0.995 0.654 25.7 6.2 27.4 G2.1V Cretaceous Period
0.066 4.502 5770 0.994 0.998 0.653 25.9 6.2 27.3 G2.0V Paleogene Period
0.023 4.545 5771 0.998 0.999 0.653 26.0 6.2 27.3 G2.0V Neogene Period
0.003 4.565 5772 1.000 1.000 0.653 26.1 6.2 27.3 G2.0V Quaternary Period
0.000 4.568 5772 1.000 1.000 0.653 26.1 6.2 27.3 G2.0V Anthropocene Period (Present)
tGya tGyr TeffK Lsun Rsun BV Per logRx logLx SpType Epoch
Much of this table has been published in the supplementary material
to Tarduno, Blackman, & Mamajek (2014; Physics of the Earth and Planetary
Interiors Review Articles, in press) and a PNG version of the published
table (including predicted solar wind mass loss rates) is posted here.
Mean Solar Mt. Wilson Svalue:
<S_MW> = 0.1694+0.0005
Mean Solar Chromospheric Activity Index
<logR'HK> = 4.9427+0.0072 dex
Quiet Sun Chromospheric Activity Index
logR'HK_min = 4.98
Active Sun Chromospheric Activity Index (Peak, varies by Cycle)
logR'HK_max = 4.93 (Cycle 15, 24) to 4.88 (Cycle 19)
The Mt. Wilson Svalue 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  the layer of plasma above the
Sun's photosphere but below the hotter corona. The Mt. Wilson Svalue
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.
See studies by Egeland+ 2016 and references therein.
Here is a list of published longterm average Mt. Wilson Svalues:
<S_MW>
0.179 Baliunas+1995 (~19661993; cycle 20,21,22)
0.170 Hall+2007 (~19942006; cycle 23)
0.1694 Egeland+2016 (cycle 1524 average)
This last estimate by Egeland+ is a recent construction of the Mt. Wilson
Svalue record going back to 1913 using several data sets and proxy indices,
and averaged over these cycles. The average minimum S_MW values over these
cycles 1524 is S_min = 0.1621+0.0008, and the average maximum S_MW value
over cycles 1424 is S_max = 0.177+0.001.
The Egeland+2016 value should be taken in preference to the mean
value quoted in Mamajek & Hillenbrand (2008; ApJ 687, 1264; Table 1), where
we quoted a mean chromospheric activity index of logR'HK = 4.905.
This was based off of adopting a mean Mt. Wilson Svalue of S = 0.1762,
solar BV color of 0.650, and using the equations of Noyes et al. (1984)
to convert S to logR'HK.
Donahue (1998; Cool Stars, Stellar Systems, and the Sun, ASPC
Vol. 154) provides the following table of representative activity
levels (Mt. Wilson Svalues; 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 2022) 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'HKtoage 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
Svalues following Noyes et al. 1984 and assuming BV(Sun)=0.65 (Cox
2000; median of 19 published values).
The estimated activity during the Maunder minimum (~16451715)
(logR'HK = 5.10) was estimated by Baliunas & Jastrow (1990; Nature
348, 520), which they quote as Mt. Wilson Svalue ~ 0.145. However
Egeland+2016 find that the typical minimum activity value
over the past nine cycles was logR'HK = 4.9844+0.0087.
These minima usually occurred when the Sun had no starspots.
So it is unclear why the Sun would have a significantly lower
activity level during the Maunder minimum compared to recent
solar minima (in which the Sun would exhibit long periods with no
sunspots). The "quiet Sun" appears to correspond to an activity
level of logR'HK ~ 4.98.
Estimates of the solar maximum and minimum activity levels over the
past 9 cycles (15 through 24) are compiled in Table 2 and 3 of Egeland+2016.
These represent smoothed values fit to the data over each cycle, and
represent the peak of a curve of activity index vs. time.
Solar minima over the past 9 cycles typically have logR'HK(min) = 4.9844
(+0.0087 measurement, +0.0049 rms scatter), while solar maxima have
logR'HK(max) = 4.905 (+0.008 measurement, +0.016 rms scatter). The most
active recent solar maxima were Cycle 19 (logR'HK(max)=4.879 and Cycle 22
(logR'HK(max)=4.884). The most recent solar maximum (Cycle 24; ~2015)
logR'HK(max)=4.931 was the weakest of the past 9 cycles.
Solar Xray Luminosity (0.12.4 keV):
L_X = 10^27.35 (+50%) erg/s
Solar Xray/Bolometric Luminosity Ratio:
log(L_X/L_bol) = 6.24 + 0.24 dex
Solar Xray Surface Flux (0.12.4 keV):
f_X = 36800 erg/s/cm^2
f_X = 10^4.566 erg/s/cm^2
Mean Coronal Temperature:
log(T_{corona}) = 1.5 MK
When comparing Xray luminosity and Xray/bolometric flux ratios, the
largest uniform database of Xray data for nearby stars and members of
clusters and associations is the ROSAT All Sky Survey (Voges et
al. 1999) which covers the 0.12.4 keV bandpass. For this reason, here
I only discuss the Sun's Xray luminosity in this bandpass.
Peres, Orlando, Reale, Rosner, & Hudson (2000; ApJ 528, 537) use solar
Yohkoh/SXT Xray data to estimate that in the ROSAT/PSPC band, the solar
Xray luminosity varies from ~2.7e26 erg/s at minimum to ~4.7e27 erg/s
at maximum. A later paper by a subset of the same authors (Orlando, Peres,
& Reale 2001; ApJ 560, 499) quoted that the Yohkoh/SXT data were
consistent with the Sun having an Xray luminosity (0.12.4 keV) of
1e265e27 erg/s during the four years of observations. This suggests
log(L_X) = 26.027.7 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 Xray 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 (19931996). 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 Xray luminosity of the "whole
solar corona" was ~1e27 erg/s (their Figure 6). This is probably a
fair assessment of the "mean" solar Xray 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 Xray emission with the SXP instrument on the SNOE
satellite. After accounting for the differences in sensitivities
between SNOESXP and ROSAT, and correcting for the fact that SNOE only
observed the Sun for a partial solar cycle (~19982000), Judge et
al. conclude "We find that the Sun's 0.12.4 keV luminosity lies
between 10^27.1 and 10^27.75 [erg/s] (measured over the time space of
the SNOESXP 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 SNOESSXP and ROSAT bandpasses.
Telleschi et al. (2005; ApJ 622, 653), in their large survey "Coronal Evolution
of the Sun in Time: Highresolution Xray Spectroscopy of Solar Analogs
with Different Ages", adopts a mean solar Xray luminosity in the ROSAT band
(0.12.4 keV) of Lx = 10^27.3 erg/s ~ 2e27 erg/s.
From the discussion of Judge et al., I adopt a mean solar Xray
luminosity (0.12.4 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 Xray luminosity), I derive:
L_X/L_bol = 5.82e7 (+2.91e7; 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 Xray 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 Xray 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 chromosphericXray
fit, and the quoted error in Judge et al.'s Xray 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.
Mean coronal temperature: The corona obviously consists of gas
of a wide range of temperatures. However an emissionmeasureweighted
coronal temperature (in megaKelvin = 10^{6} K = MK!) can be estimated through modeling
of Xray spectra. Emission measure (EM) is the product of the electron
density (n_{e}), hydrogen number density (n_{H}), and volume (V), or EM = n_{e}n_{H}V
(see e.g. Sec. 4.2 of Gudel 2007, Living Reviews in Solar Physics, vol. 4,
no. 3). Peres, Orlando, Reale, Rosner, & Hudson (2000; ApJ 528, 537)
estimate that the Sun's coronal EMweighted temperature varies between
~1 MK at solar minimum to ~2 MK at solar maximum, with an average value
of T_{corona} ~ 1.5 MK.
Evolution of mean coronal temperature: Telleschi et al. (2005; ApJ 622, 653)
estimate emission measureweighted mean coronal temperatures for 6 Sunlike
stars at different ages and a wide range of Xray luminosities. Combining the
estimates of Telleschi et al. [dropping the outlier point for Beta Com] with
the mean solar coronal temperature from Peres et al. (2000) and the adopted
Lx value mentioned above, one finds a narrow trend line:
logT_{corona} = 1.54 + 0.282*logL_{X} .
Age(Sun & Solar System)
t_{Sun} = 4567.30 + 0.16 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 (young, premain sequence stars). Defining
t=0 for the "birth" of a star or planetary system is subjective.
For theoretical models of stellar evolution, sometimes the depletion
of deuterium through fusion (the lowest temperature nuclear reaction)
is used to mark t=0 (the "birthline"), however for the Sun there would
be no practical way to connect this event in the Sun's early history
with the radioactive decay timescales inferred from meteoritic agedating.
The radioactive isotopes in meteorites give us samples of the first "rocks"
to have accreted from the protosolar nebula and crystallized,
orbiting the still accreting protoSun.
In a wellcited 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 ^{26}Al 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
(4567 Myr). However, given the meteoritic ages over the past two decades,
there seems to be no evidence for an age exceeding ~4570 Myr.
Here are some recent, relevant ages from isotopic studies of meteorites:
>4569.5 + 0.2 Myr ; Baker+(2005; Nature, 436, 1127)
4567.2 + 0.6 Myr ; Amelin+(2002; Science 297, 1678)
4567.11 + 0.16 Myr ; Amelin+(2006; update to 2002; Lun.Pl.Sci.Conf. 37, 1970)
4568 +0.911.17 Myr ; Moynier+ (2007; ApJ 671, L181)
4567.18 + 0.50 Myr ; Amelin+(2010; E&PSL 300, 343)
4568.2 Myr ; Bouvier & Wadhwa (2010; Nature Geosci. 3, 637)
4567.30 + 0.16 Myr ; Connelly+(2017; GeCoA 201, 345)
The Baker et al. study claims "the accretion of differentiated
planetesimals predated 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 kilometersized
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 CaAlrich
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 207Pb206Pb age for a CAI
in the meteorite NWA 2363.
The age from Connolly+(2017) state: "This paper reviews the theory and
methods behind this chronometer, offers criteria to critically evaluate
Pb–Pb ages and presents a summary of the current state and immediate future
of the chronometry of the early Solar System. We recognize that there is some
debate over the age of the Solar System, but conclude that an age of
4567.30 ± 0.16 Ma based on four CAIs dated individually by the same method
in two different laboratories is presently the best constrained published
value. We further conclude that nebular chondrules dated by the PbPb method
require that they formed contemporaneously with CAIs and continued to form
for at least ∼4 Myr, a conclusion that implies heterogeneous distribution
of the shortlived^{ 26}Al nuclide in the protoplanetary disk. Planetesimals
were already forming by ∼1 Myr after CAI formation, consistent with their
growth predominantly through the accretion of chondrules. Nebular
chondrule formation was completed by ∼5 Myr after CAI formation when
the impactgenerated Cba chondrules formed after the disk was cleared of
gas and dust." This is arguably the best recent value, and the one adopted.
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 backoftheenvelope
KelvinHelmholtz 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 protonproton (PP) chain fusion probably had
negligible impact on the crystallization of meteorites, so the timescales
from isotopic studies should *not* be confused with the timescale
since the Sun reached the "zeroage 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 zeroage main sequence is a very bad,
silly, and confusing habit still adopted by some theorists and some
planetary scientists. 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 crosscomparison of evolutionary tracks
of different masses (which have different preMS contraction times!).
A more useful metric for t=0 for stellar evolutionary tracks might be
the "stellar birthline" which corresponds to the deteuriumburning 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) ~ 0.120.19, 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 modernday 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 modernday 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 modernday 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)
0.7299 0.2547 0.0154 0.0211 phot(modern) Caffau et al. (2010)
0.7321 0.2526 0.0153 0.0209 phot(modern) Caffau et al. (2011)
0.7380 0.2485 0.0134 0.018 phot(modern) Grevesse, Asplund, Sauval, Scott (2012)
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)
0.7154 0.2703 0.0142 0.0199 protosolar Grevesse, Asplund, Sauval, Scott (2012)
0.72 0.264 0.016 0.0222 protosolar Gruberbauer et al. (2012)
"Canonical" (highZ) values:
An excellent recent review by Basu & Antia (2008, Physics Reports,
457, 217283) states that "Seismic determinations of the solar
heavyelement 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
highermetallicity 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 mid1980s to
Z/X ~ 0.018 in 2003.
"Heretical" (lowZ) 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 astroph/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
Some Closing Comments on Solar Composition:
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, 217283), 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 Bstars and ISM, it cannot account for the shape of the main
sequence turnoff for the ~4 Gyrold 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 modernday 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 =
0.240.25 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 BasuAntia work, or dY/dZ = 1.528 using the protosolar abundances
from AsplundGrevesseSauvalScott. 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.
A few notes on Fe abundance:
Grevesse98: 7.55+0.05 (photospheric),
Grevesse07: 7.45+0.05 (photospheric), 7.45+0.03 (meteoritic) http://adsabs.harvard.edu/abs/2007SSRv..130..105G
Asplund09: 7.50+0.04 (photospheric), 7.45+0.01 (meteoritic) http://adsabs.harvard.edu/abs/2009ARA%26A..47..481A
Grevesse10: 7.50+0.04 (photospheric), 7.45+0.01 (meteoritic) http://adsabs.harvard.edu/abs/2010Ap%26SS.328..179G
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 Earthbound 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 Sunlike 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 Sindex;
>= 30 days of observations each) over a 8year 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> = 1.3e14 Msun/yr
Median Solar Wind Density @1AU:
n = 5.7 cm^{3} Mean Solar Wind Density @1AU:
n = 6.9 cm^{3}
Median Solar Wind Velocity @1AU:
v = 416 km/s
Mean Solar Wind Velocity @1AU:
v = 439 km/s
The flow of charged particles escaping the Sun (the solar wind) has
been measured by many spacecraft. The solar wind mass loss rate calculation
obviously requires some discussion of the solar wind velocities and
particular densities.
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
19941995, 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 +2080 degrees, the solar wind velocity
was in the range of ~600830 km/s (mean ~750 km/s; values are inferred
byeye 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 ~320700 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 19621974, 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 1977.642007.64, 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 nearEarth space from a
variety of spacecraft measurements. Among 15,800 daily solar wind
velocity measurements at 1 AU between 1963 and 2014, I find a median
solar wind velocity of 416 km/s (68%CL +12772 km/s) and mean of 439
km/s (stdev = 100 km/s). The minimum daily velocity was 193 km/s and
the maximum was 1003 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 23 cm^3 (range: ~1.54 cm^3), while at low latitudes
(<+20 deg) the densities were typically ~8 cm^3 (approximate range:
~220 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 1977.642007.64. This agrees well with the Ulysses data
at low heliographic latitudes.
The OMNIWeb database (accessed 19 Apr 2014) reports 15121 daily solar wind
density measurements between 1963 and 2014. The median proton density is 5.7
cm^3 (68%CL +5.22.6) and the mean is 6.94 cm^3 (st.dev. =
4.60). Daily values ranged from 0.1 to 60.3 cm^3.
Calculation of Solar Mass Loss Rate:
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] = 7.41e18 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 daily solar wind velocities and number densities
with this formula, I estimate solar wind mass loss rates of:
median dM/dt = 1.80e14 Msun/yr = 10^13.745 Msun/yr
mean dM/dt = 1.94e14 Msun/yr = 10^13.713 Msun/yr
The mass loss rate has a lognormal dispersion of values, with rms scatter
in log(dM/dt) of 0.23 dex
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 ~2ndorder
estimate might be:
dM/dt ~ 1.3e14 Msun/yr
In cgs units:
(1.3e14 Msun/yr)*(1 yr/3.1557e7 s)*(1.988e33 g/Msun) =>
dM/dt ~ 8.2e11 g/s
Aarnio, Matt & Stassun (2010, ApJ, 760, 9) estimate that mass loss
via coronal mass ejections (CMEs) alone accounts for approximately
2.2e15 Msun/yr of the Sun's total mass loss (or ~17% of the average
solar wind mass loss derived above; they estimate ~10% in their paper).
Median Solar Wind Pressure @1AU:
P_{SW} = 1.77 nPa
Mean Solar Wind Pressure @1AU:
P_{SW} = 2.05 nPa
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 = 1.6726e6[nPa] * n[cm^3] * V[km/s]^2
Using 15,121 daily values for the solar wind density and velocity
measured at 1 AU from the OMNIWeb database (see section on solar wind
mass loss rate), I find the solar wind pressure at 1 AU to be:
median P_{SW@1AU} = 1.77 nanoPascals (+1.22,0.74; 68%CL)
mean P_{SW@1AU} = 2.05 nanoPascals (+1.27; st.dev.)
This is commensurate with the average solar wind pressure from Shue et al.
(1997; Journal of Geophysical Research, 102, A5, 9497):
P_{SW@1AU} = 1.915 nPa. With this pressure, the Earth's magnetospheric
"standoff" radius towards the Sun is approximately 10.1 Earth radii,
with typical excursions in the range ~812 Earth radii as the Zcomponent
(B_{Z}) of the interplanetary magnetic field (IMF) varies from ~12 to +15 nT
(average value 0.595 nT; Shue et al. 1997).
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 0.0719.59 nPa.
Mass loss [Msun/yr] = P[nPa] * 1.0573e14[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 0.0719.59
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 Interplanetary Magnetic Field:
<B(IMF)>@1AU = 5.8 nT
Mean Interplanetary Magnetic Field:
<B(IMF)>@1AU = 6.4 nT
Median Interplanetary Magnetic Field
Pressure <P(IMF)>@1AU = 0.013 nPa
The OmniWEB database (http://omniweb.gsfc.nasa.gov/; accessed 2014 Apr 19) lists
15,754 daily measurements of the interplanetary magnetic field measured between
1963 and 2014. The median IMF is 5.8 nT (+2.81.8 nT; 68%CL; +7.93.1 nT; 95%CL).
The (unclipped) mean IMF is 6.37 nT (+0.02 nT s.e.m.; +2.77 nT st.dev)
Steinhilber et al. 2010 (Journal of Geophysical Research: Space Physics,
Volume 115, Issue A1, CiteID A01104) report that a 9300 year reconstruction of
the interplanetary magnetic field (IMF) record based on radiogenic 10Be
measurements suggests that the IMF ranges from ~2 nT (during e.g. Maunder minimum)
to ~8 nT. Longterm periodicities with period ~2 kyr and amplitude ~0.75 nT
are seen in the record. They corroborate that the recent (4decade) IMF average
is 6.6 nT.
The solar wind magnetic field pressure can be calculated as:
P_{B} = B_{IMF}^{2}/8\pi
Entering a magnetic field strength in Gauss easily yields a magnetic
field pressure in cgs units of dynes/cm^{2}. For B_{IMF} = 5.8 nT = 5.8e5 G,
P_{B} = 1.3e10 dyne/cm^{2} = 0.013 nPa
This is only ~0.07% of the solar wind's dynamical pressure (~2 nPa).
Solar LargeScale Surface
Magnetic Field Strength:
B(surf,Sun) = 2 Gauss
Vidotto et al. (2014; MNRAS, in press; http://arxiv.org/abs/1404.2733) list
an estimate of the Sun's largescale magnetic field strength during:
solar minimum (1.89 Gauss; January 1982; when logLx/Lbol = 7.15) and
solar maximum (3.81 Gauss; March 1986; when logLx/Lbol = 5.91).
Smith, E.J. ("The Sun, solar wind, and magnetic field. I,"
Proceedings of the International School of Physics "Enrico Fermi" Course
CXLII, B. Coppi, A Ferrari and E. Sindoni (Eds.), IOS Press, 179, Amsterdam 2000)
estimates the Sun's polar field strength to be ~7 Gauss (7e4 Tesla).
Smith also mentions that the Ulysses results are consistent with the radial
component of the Sun's magnetic field (and hence magnetic flux)
being independent of latitude. This is not reflected at the Sun's surface
(where smaller scale magnetic fields are obviously observed).
Janardhan et al. (2010; http://arxiv.org/abs/1009.4299) plotted solar magnetic
field in the polar regions (latitudes 78 to 90; north & south) over a 3 decade
(~3 solar cycle) period, showing that the poles flip polarity every ~decade with
a quasisinusoidal amplitude ~+10 Gauss. Their Figure 3 shows that the absolute
magnitude of the polar magnetic field is indeed around ~7 Gauss averaged over
the solar cycles.
Median International Sunspot Number:
ISN = 40
Mean International Sunspot Number:
ISN = 54
The Solar Influences Data Analysis Center (SIDC) has a nice database
of historical sunspot number measurements:
http://sidc.oma.be/sunspotdata/
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:
(BV)o(Sun) = 0.653+0.003 mag
other JohnsonCousinsStromgren2MASSWISE
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 BV color in detail, and list several
other solar colors in optical/nearIR 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 BV:
The solar BV color has been the source of some controversy over the
years, and quoted values have spanned a (relatively) large range of
values (from BV=0.62 in Allen63, to BV=0.686 in Tug & SchmidtKaler
1982).
Arguably, the most recent authoritative study of the solar BV 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 spectrallinedepth ratios and colors for solar twins. The
three analyses yield very similar solar BV 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 <BV(Sun)> =
0.653+0.003. They adopted Teff=5777K in their analysis.
Here is sorted list of published *pre2000* solar BV values. It is
probably not exhaustive, but it should be fairly representative of the
quoted values.
BV(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 ("SunNover")
0.679 Bessell98 ("Sunover")
0.68 Lang+92
0.686 Tug & SchmidtKaler 1982
Here are the values inferred from *post2000* literature and
calibrations:
BV(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 [Rieke08 synthetic]
0.647 Median BV 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 BV
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 BV=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 ``closestever solar
twin''. The Hipparcos catalog lists BV = 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 BV=0.657+0.006.
So the truly noteworthy solar twins have BV 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 BV = 0.652 +0.005 (rms).
Casagrande et al. (2010, A&A, 512, 54) lists a top 10 list of closest
solar twins in their survey: HIP 30502, 36512,
41317, 44935, 44997, 55409, 56948, 64713, 77883, and 89650.
* BV 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 BV colors, I find a median BV
0.648+0.001 mag, and Chauvenetclipped mean BV = 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.
* BV 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
Chauvenetclipped) values of BV *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 (BV)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 12 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 <BV> 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 "NearSolar MK Standards and Photometric Standards of
Similar Color" at: ftp://ftp.lowell.edu/pub/bas/starcats/solar.list
* Summary:
Median values:
0.620 BV for Houk G2V stars
0.641 BV for Gray G2V stars
0.644 BV for post2000 literature values
0.651 BV for pre2000 literature values
0.652 BV for 3 of the best solar twins (actually the mean)
The true median for the 33 BV values listed (excluding the Ramirez et
al. 2012 values) plus that for the 3 solar analogs is
<BV>=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 BV 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/nearIR colors for the Sun, through
comparing measured photometry for solar twins, and measuring precision
effective temperatures relative to the Sun using the linedepth 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 Tycho2 photometry,
and the other using JohnsonCousins 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 JohnsonCousins for UBVRcIc. On the right side are some of my
estimates from looking at colorcolor trends for field stars (not necessarily
only solar twins), evaluated at adopted solar color (BV)=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.
(BV) = 0.653 + 0.003 ; Ramirez12
(UB) = 0.158 + 0.009 ; Ramirez12 [EEM: for (BV)=0.651 => (UB) = 0.135]
(VRc) = 0.356 + 0.003 ; Ramirez12 [EEM: for (BV)=0.651 => (VRc)= 0.363]
(VIc) = 0.701 + 0.003 ; Ramirez12 [EEM: for (BV)=0.651 => (VI) = 0.714]
(VJ) = 1.198 + 0.005 ; Casagrande12 [EEM: for (BV)=0.651 => (VJ) = 1.201]
(VH) = 1.484 + 0.009 ; Casagrande12 [EEM: for (BV)=0.651 => (VH) = 1.494]
(VKs) = 1.560 + 0.008 ; Casagrande12 [EEM: for (BV)=0.651 => (VKs)= 1.567]
(JH) = 0.286 ; Casagrande12 [EEM: for (BV)=0.651 => (JH) = 0.294]
(JKs) = 0.362 ; Casagrande12 [EEM: for (BV)=0.651 => (JKs)= 0.366]
(HKs) = 0.076 ; Casagrande12 [EEM: for (BV)=0.651 => (HKs)= 0.073]
(VW1) = 1.608 + 0.008 ; Casagrande12 [EEM: for (BV)=0.651 => (VW1)= 1.595]
(VW2) = 1.563 + 0.008 ; Casagrande12
(VW3) = 1.552 + 0.009 ; Casagrande12
(VW4) = 1.604 + 0.011 ; Casagrande12
Uncertainties for the (JH), (JKs), and (JKs) 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
overlysimplified estimates using colorcolor trends for dwarf stars
of a wide range of metallicities & gravities is fairly good. The
agreement is <=0.015 mag for (VRc), (VI), (VJ), (VH), (VKs),
(JH), (JKs), (HKs), and (VW1). Disagreement is 0.023 mag for
(UB), however as Uband 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 colorcolor
plots used for my own estimates.
Melendez et al. 2010 (A&A, 522, A98) recently estimated the Stromgren
colors for the Sun based on photometric studies of solar twins. They derive:
(by) = 0.4105 + 0.0015 ; Melendez+2010
m1 = 0.2212 + 0.0018 ; Melendez+2010
c1 = 0.3319 + 0.0054 ; Melendez+2010
Beta = 2.5915 + 0.0024 ; Melendez+2010
The solar colors from Melendez+2010, Ramirez+2012 and Casagrande+2012 appear
to be the best available, and should be adopted.
Distance to Center of Galaxy
(Sun's Galactocentric Distance):
R0 = 8.0 + 0.4 kpc
The literature discussing determinations of the distance to the center of the Galaxy R_{0}
(effectively marked by the radio source and massive black hole Sgr A*) is extensive, and
I will not (yet) review all of this literature. I refer the reader to an extensive and recent
literature review on the subject by Malkin (2013; Astronomy Reports 57, 128;
http://arxiv.org/abs/1301.7011). After reviewing 20 years of literature, encompassing 52 determinations
of the distance to the center of the Galaxy, Malkin (2013) finds no statistically significant
trend (or evidence of a "bandwagon effect"). My interpretation of Malkin's review is that the past two
decades of investigations have moreorless converged on a consensus estimate of 8.0 kpc and that the
1sigma uncertainty appears to be of order +0.4 kpc (5%).
Distance to Nearest Known Star to Sun:
d = 1.3009 + 0.0005 pc
The nearest known star to our solar system is the dim red dwarf Proxima Centauri (Alpha Centauri C),
which is the lowest mass member of the triple system Alpha Centauri. Benedict et al. (1999; AJ, 118, 1086;
http://adsabs.harvard.edu/abs/1999AJ....118.1086B) used the Hubble Space Telescope Fine Guidance Sensor 3
to calculate a trigonometric parallax to Proxima Centauri of 768.7+0.3 milliarcseconds. The corresponding
distance is 1/parallax = 1.3009+0.0005 pc. Given that a parsec is 206,264.8063 AU in length, the distance
to Proxima translates to 268300+100 AU, or approximately 40 trillion km.
Distance to Nearest Known Brown Dwarf to Sun:
d = 2.0 + 0.15 pc
The nearest known brown dwarf to our solar system is actually a pair of such substellar objects:
the recently discovered pair WISE J104915.57531906.1 (also known by its double star designation
Luhman 16). The pair was discovered in infrared images from the WISE mission by Kevin Luhman in early 2013
(Luhman 2013 ApJ, 767, 1), and he estimated a preliminary trigonometric parallax of 496+37 milliarcseconds
(distance 2.0+0.15 parsecs). The Luhman 16 system consists of a L8 and T2type brown dwarf, and
is the nearest known object outside the solar system after the Alpha Centauri triple system (1.3 pc)
and Barnard's Star (1.8 pc).
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 BV 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 Xray luminosity (Judge et al. 2003, Orlando et al. 2001).
12 Nov 2007: Added log of bolometric luminosity and age/yr.
29 Nov 2007: Added Xray surface flux and defined ROSAT Xray energy ranges (0.12.4 keV).
29 Nov 2007: Edited comments on age of solar system.
7 Dec 2007: Added discussion of solar BV 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 solartype 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 BV 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 Rieke08
21 Jun 2011: Updated discussion on solar BV and adopted value
14 Apr 2012: Included estimate of AU in lightdays
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 BV 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 BV
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 BV 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 BV, 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.
16 Nov 2012: Added solar photospheric angular radius estimate.
26 Dec 2012: Edited the discussion on solar bolometric magnitude and correction for clarification.
30 Dec 2012: Updated discusson on the Sun as the G2V standard star.
1 Jan 2013: Added solar escape velocity. Added list of Sun/planet mass ratios.
13 Jan 2013: Added solar apparent bolometric magnitude.
20 Jan 2013: Minor revisions to solar radius discussion, and inclusion of Kubo (1993) radii.
12 Feb 2013: Fixed text in discussion on bolometric magnitude scale (thanks to Trent Dupuy).
The previous text erroneously listed the IAU 1999 bolometric mag zero point as the
solar luminosity.
25 Mar 2013: Fixed Caffau+2010 solar abundances, and added Caffau+2011 abundances.
10 Jun 2013: Minor corrections to revised version of Engelke10 Vmag for Sun.
16 Jun 2013: List total solar irradiance in cgs units.
26 Jun 2013: Added Sun's galactocentric distance (8.0+0.4 kpc).
26 Jun 2013: Added distances to nearest star and brown dwarf to the Sun.
11 Sep 2013: Edited solar constant (S@1AU) section.
7 Feb 2014: Incorporated small revisions to the adopted solar apparent V and B magnitudes,
absolute V magnitude, and bolometric correction. These were motivated by analysis
in Pecaut & Mamajek (2013).
4 Mar 2014: Added Sun's Stromgren colors from Melendez+2010.
19 Apr 2014: Updated solar wind mass loss, velocity and density measured at 1AU based on new
analysis of OMNIweb solar wind data (19632014). Added measurement and discussion
on interplanetary magnetic field (IMF).
7 May 2014: Added discussion of emission measureweighted mean coronal temperature and
an estimate of the interplanetary magnetic field pressure.
11 May 2014: minor edits. List both median and mean in the primary table for some quantities related
to solar wind.
20 May 2014: updates to text on Total Solar Irradiance and solar luminosity values.
19 Jun 2014: Included reference to Tarduno, Blackman, & Mamajek (2014)  reconstruction of
history of solar parameters.
26 Nov 2014: Added TSI measurements from DIARAD/SOVIM and TCTE/TIM to TSI discussion.
27 Nov 2014: Added recent solar angular diameter measurements from Hauchecorne+ 2014.
24 Dec 2014: Included GMsun estimate from JPL ephemerides DE431/DE432 (Folkner+ 2014).
1 Jan 2015: Updated discussion on solar radius, repaired some syntax in this bizarre editing
environment that Google uses. It still does not look very good!
14 Jan 2015: Fixed values for semimajor axis and mean distance of EarthMoon Barycenter/Sun orbit.
22 Feb 2015: Added Meftah+2014 values for radius, TSI, and oblateness.
21 Nov 2016: Updated Chromospheric activity discussion to reflect new findings of Egeland+2016.
Updated table to include nominal IAU 2015 values in brackets.
15 Jun 2017: Edited section on age of Sun and solar system, including recent findings from Connelly+2017.
References citing this table:
Torres (2010) "On the Use of Empirical Bolometric Corrections for Stars"Mamajek (2012) "On the Age and Binarity of Fomalhaut"
BenJaffel & Ballester (2013) "Hubble Space Telescope detection of oxygen in the atmosphere of exoplanet HD 189733b"
de Freitas et al. (2013) "New Suns in the Cosmos"