Lecture 1: The Sun
The Earth’s climate system is fundamentally driven by the Sun. It is by far the largest
energy source for the Earth’s climate system.
The Sun emits most of its radiation to space from a layer called the photosphere.
For the purpose of this class, we will define the temperature of the photosphere as
Tp=5780 K (Emission temperature and not actual temperature), with the radius of the
photosphere given as rp=6.96x108 m.
Figure 1.1 shows the Sun with the photosphere indicated in yellow. The photosphere is
where darker sunspots and brighter areas called faculae are present. We will discuss these
features and their impacts on the Sun’s energy output.
To understand emission of radiation from the Sun, we first need to understand some basic
concepts in radiative transfer.
Basic Quantities of Radiative Transfer
We want to calculate the net emission of radiation from a unit area of a “blackbody”,
where a blackbody is defined as an object that:
1) Absorbs all incident radiation
2) Has maximum possible emissivity over all wavelengths and directions
Blackbodies emit isotropically. In other words, emission is equally probable in all
directions.
Before we calculate blackbody emission, we need to define a quantity called intensity or
radiance:
Intensity=If (,)
If (,) gives the energy flux per unit area per frequency per unit solid angle (steradian)
Intensity units: Joules m-2 s-1 (s-1)-1 steradian-1
Figure 1.2 shows the geometry we will be using, where is the zenith angle, and is the
azimuth angle. Solid angle, given in steradians, is an area element on a unit sphere
defined by:
d(steradian) = dw = sinq dq dj
Where the factor sin q accounts for the convergence of the meridians at 0 zenith angle.
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We are interested in the total amount of radiation that passes through an increment of
area (dA). If we integrate intensity over all angles of the upper hemisphere, we get a
quantity called the spectral flux density:
Spectral Flux Density = Ff = ò I f (q ,j )cosq dw =
2 p p /2
ò òI
0
f
(q ,j )cosq sinq dq dj
0
In the integration, the factor cos  accounts for the reduction in energy flux per unit area
when the radiation is not normal to the surface. Energy is effectively spread over a larger
area. We will see this factor again below.
Ff units: Joules m-2
Ff gives the energy flux as a function of frequency. Note that this flux can alternately be
expressed as a function of wavelength using the phase speed of electromagnetic radiation:
c*  f  , whereis expressed in meters and f is expressed in s-1. c*=3x108 m s-1 is the
speed of light.
If we integrate the spectral flux density over all frequencies, we obtain the energy flux
density:

Energy flux density= F   F f df
0
F units: Joules s-1 m-2, otherwise expressed as Watts per m2.
F = The flux of energy passing through a unit area in one direction.
The energy flux density in units of W m-2 is the most widely used quantity for
characterizing the energy flow through the Earth’s climate system. Most energy
calculations throughout the course will be expressed in these units.
Back to blackbody emission and the Sun.
Blackbody Emission
We can think of the photosphere of the Sun as emitting like a blackbody.
The intensity for blackbody emission is given as:
I f (T ) = B f (T ) =
2hf 3
1
, which is called Planck’s Law.
2
hf
c * (exp( ) -1)
kT
2
where
k= Boltzmann’s constant=1.37x10-23 J K-1
h =Planck’s constant=6.625x10-34 J s
c*=speed of light=3x108 m s-1
Bf (T) units: J s-1 m-2 (s-1)-1 steradian-1.
Remember that blackbody emission is “isotropic”, and not dependent on angle. Thus, no
angle dependence is apparent in Bf (T) and this simplifies the integration over angle.
Now, integrate blackbody emission over all angles to get the spectral flux density:
Ff,, blackbody=Bf (T),
with units of W m-2 (s-1)-1.
Using this relationship, the frequency (or wavelength) of maximum emission can be
found as a function of temperature:
Wien’s Law:  max 
2897
, wheremax is in units of 10-6 m, and T is in deg K.
T
You will derive this expression in Homework 1.
If we examine the wavelengths of maximum emission of the Sun (Tp=5780 K) and the
Earth (Te=255 K), this provides a nice demonstration of Wien’s law.
Figure 1.3: Emission curves for the Sun and Earth from which the wavelengths of
maximum emission of the Sun and Earth can be inferred
Note that the Sun’s emission peaks in the visible, as determined by Wien’s Law. Earth’s
and Sun’s emission spectra are almost distinct, which is crucially important to the climate
system as will be discussed below.
Now, we can integrate the blackbody spectral flux density over all frequencies to get the
energy flux passing through a unit area:
¥
¥
0
0
Fblackbody = ò p B f (T )df = p ò
2hf 3
1
df
2
c * (exp( hf ) -1)
kT
Integration of this quantity (by parts a few times), produces possibly the most
fundamental equation of climate dynamics, called the Stephan-Boltzmann Law.
The Stephan-Boltzmann Law:
3
Fblackbody  T 4 ,
where s =
2p 5 k 4
= 5.67x10-8 W m-2 k-4, =Stephan-Boltzmann constant.
2 3
15c * h
Fblackbody units: W m-2.
The Stephan-Boltzmann Law is extremely important in that it gives the net upward
energy emission per unit area by a blackbody.
We will continually use the Stephan-Boltzmann equation to describe energy fluxes in the
Earth’s climate system (and you will likely be seeing it in your sleep).
Now, back to the Sun.
Figure 1.4: Again shows the Solar disk.
We can assume that emission from the Sun’s photosphere occurs like a blackbody.
Recall that for the Sun, Tp=5780 K, rp=6.96x108 m.
Using the Stephan-Boltzmann law for blackbody emission, the solar energy flux from the
photosphere per unit area is given by:
F p  T p
4
Fp=(5.67x10-8)(5780K)4
Fp=6.33x107 W m-2.
We can calculate the total rate of energy emission from the Sun, the Luminosity (Lo), by
integrating the solar flux density over the entire surface area of the photosphere.
Area of photosphere=4rp2
Thus, the luminosity is given by:
Lo=4rp2 Fp
Lo=3.85x1026 J s-1 (or Watts) = Rate of energy emission from the Sun.
We will get back to how constant or variability the solar luminosity may actually be.
First though, let’s consider the impact of this solar energy emission at the radius of the
Earth’s orbit.
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Solar Flux at a Distance
Let us first note that the Earth’s orbit around the Sun is not circular, but instead is
elliptical. Thus, there is an annual cycle in the Sun-Earth distance.
Figure 1.5: Shows the elliptical nature of Earth’s orbit.
The Earth is closest to Sun (perihelion) in NH Winter and furthest from Sun (aphelion)
in NH summer.
We assume that at a substantial distance from the Sun that solar rays are effectively
coming from a single direction, and are thus parallel.
At a distance d from the Sun, we can calculate the solar flux passing through a square
meter of area as follows:
Sd 
Lo
4d 2
d
Sun
For now, let’s just ignore the eccentricity of the Earth’s orbit and assume that we can
calculate the flux density at the radius of the Earth’s orbit using an average radius d ,
where the overbar represents a time mean .
d  d  1.5 1011 m = mean radius of Earth’s orbit.
At the radius d ,
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Sd = So =
Lo
4p d
2
=1363Wm-2 = Solar Constant
This gives the flux density of solar radiation at the mean radius of Earth’s orbit, a
quantity we call the solar constant. The exact value of the solar constant is actually
difficult to measure from satellite, and hence our best estimate of this value has changed
over time. Climate papers through the 1990s commonly use values around 1367-8 W m-2.
But recent satellite measurements have suggested a much lower value. Kopp and Lean
(2010, GRL) estimated a value of 1361 W m-2 from the SORCE mission. Previous higher
estimates of the solar constant are thought to be contaminated by instrument perfections
including scattering of light into absorbing cavity by edge imperfections in the aperture.
The most updated measurements derived from the Total Irradiance Monitor (TIM) are
derived from a setup shown on the left side of Figure 1.6 that limits scattering, whereas
previous satellite measurements were taken with apertures that have more scattering.
Recent measurements from the SORCE mission provided courtesy of Greg Kopp show
this lower value for solar irradiance and how it has varied over the last decade (Figure
1.7). This plot also shows hints of decade-timescale variability in the solar irradiance that
we will discuss next. Regardless, given the uncertainty in these measurements, we will
assume a value of 1363 W m-2 throughout this class. The exact value for the long-term
average of the solar “constant” is of limited practical importance for determining Earth’s
climate, although efforts to measure the balance (or degree of imbalance) of top of
atmosphere radiation depend on how accurately we can measure the solar constant.
How “constant” is So ?
Although we define a quantity called “solar constant”, the Sun’s output reaching the
Earth varies in reality. We will explore some of this variability here.
1) We know that d changes during the course of an annual cycle. Thus, Sd changes during
the course of a year. The magnitude of this variation is about 7% from our winter (~1412
W m-2) to our summer (~1321 W m-2), and will become more apparent when we discuss
the season cycle of insolation later. This variation in Earth-Sun distance over the annual
cycle will also prove important for glacial cycles associated with Earth’s orbital
parameters, a topic to be explored in the second half of the course.
2) We also know that the Sun has 11-year sunspot cycles due to quasi-regular fluctuations
in the Sun’s magnetic field, as will be described below.
3) The amplitude and length of these 11-year sunspot cycles can vary on timescales of
centuries, which can produce longer-term variability in solar output.
Cool sunspots are accompanied by larger, hotter patches called faculae. Both are
associated with disruptions in the Sun’s magnetic field. Faculae cover a larger area of the
Sun’s photosphere than sunspots, which raises the average temperature of the
photosphere and leads to increased solar luminosity.
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Figure 1.8 once again shows the full disk of the Sun, on which both sunspots and faculae
can be noticed.
Figure 1.9 from from Wallace and Hobbs (2006) details the 11-year cycle in sunspots.
Periods of greater sunspot activity are characterized by increased solar luminosity. This
figure shows the percent area of each zonal band of the sun that is covered by Sunspots,
as well as the total fractional area of the photosphere covered by sunspots.
Retrievals of the solar constant from satellite show how the solar constant varies as a
function of time. It is also striking the systematic biases among the instruments that
sometimes make calculation of the solar constant difficult.
Figure 1.10: Shows these direct retrievals from various satellite sources, calibrated to
match the output from one particular recent estimate from the TIM instrument we discuss
in Figure 1.6 above. This plot is from the Intergovernmental Panel on Climate Change’s
2013 report (IPCC2013).
The solar constant recently has varied by about 1.6 W m-2 , or about 0.12%, over an 11year solar cycle (Kopp and Lean 2010). Note that we would have to divide by 4 (to
account for the ratio of the Earth disk to its surface area) and account for albedo to get the
actual global-averaged flux perturbation per meter squared of the Earth’s surface. Higher
frequency fluctuations on timescales of days to weeks occur on the order of 4-5 W m-2
(Figure 1.7). This short timescale variability appears highest during the maxima in the
11-year solar cycle.
Figure 1.11 shows the reconstruction of the solar constant back to 1600 using the
methods of Lean (2000) and Wang et al. (2005) from IPCC2007. The knowledge gained
from direct measurements, coupled with knowledge of past sunspot cycles and/or other
proxies for solar activity, can lead to (very uncertain) retrievals of trends and variability
in the solar constant. Traditional methods such as Lean (2000) typically used relatively
simple methods linking sunspot activity over the last 500 years to solar irradiance, such
as tying irradiance to the length or amplitude of the solar cycle. Recent, more
sophisticated methods (e.g. Wang et al. 2005), have modeled solar magnetic flux
variations as they relate to sunspots, and have thus developed more accurate estimates of
irradiance variations over the centuries. Upward trends in solar irradiance in Wang et al.
(2005) are only about 0.27 times that of Lean (2000). The purple curves agree, at least in
the magnitude of the solar constant changes since 1750, with more recent estimates from
IPCC Assessment report 5 (IPCC2013).
Some evidence exists that the Maunder Minimum in sunspot activity in the 1600s,
associated with a relatively cold Europe, might have been due to the relatively low solar
luminosity then. But this is not at all certain.
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Examining the variability and trends in solar constant, and how they compare to other
climate forcings since the mid 18th Century, it is a good approximation to assume that the
solar luminosity is constant (for modern climate studies at least).
Figure 1.12, showing the IPCC2013 global mean radiative forcings since 1750, indicates
that the global average climate forcing due to increase solar activity since 1750 is modest
compared to other radiative forcing such as increases in greenhouse gases. Note that
volcanic forcing is ignored in this plot. The anthropogenic forcing estimate has gone up
by about 0.5 W m-2 since IPCC2007 due to both increases in the greenhouse gas forcing,
and reduced estimates for the importance of aerosol-cloud interactions.
Some researchers try very hard to find an unknown positive feedback in the climate
system that links recent global mean temperature increases to the relatively modest
increase in solar constant. One such example is the impact of relatively modest solar
variations on cloud nucleation. See the recent paper by Pierce and Adams (2009) that
provides a nice rebuttal to this latter view.
For the majority of this class, we will assume that solar output is constant, such that:
So=1363 W m-2.
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