Energy Transport
Energy can be transmitted by:
1. Conduction
2. Radiation
3. Convection
Chapter 3 – Energy Balance
and Temperature
p
Astro 9601
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Topics to be covered
One mechanism usually dominates
In solids, conduction dominates
In space and tenuous gases, radiation
dominates
Convection is important in atmospheres (and
liquid interiors)
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Temperature
• Energy Balance and Temperature (3.1) - All
• Conduction (3.2.1), Radiation (3.2.2 and 3.2.2.1)
• Convection (3.2.3), Hydrostatic Equilibrium
(3.2.3.1), First Law of Thermodynamics (3.2.3.2)
and
d Adi
Adiabatic
b ti L
Lapse rate
t (3
(3.2.3.3)
2 3 3)
– All to be discussed in lecture notes with Ch. 4 (where
it makes sense!)
• The temperature of an object is
proportional to the average translational
kinetic energy of its molecules.
• Note that one object can have many
temperatures
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Radiation and Planetary Science
Blackbody - Introduction
• All solar system bodies are illuminated by
the sun
• Balance between solar radiation received
(plus any
(p
y internal energy)
gy) and that emitted
defines temperature
• Blackbody – a hypothetical (idealized) body that
– Absorbs all incident radiation (hence the term “black”)
– Emits the maximum possible radiant energy in all
wavelength bands in all directions
– No
N radiation
di ti iis reflected
fl t d
– ultimately equilibrium is reached which
defines T
• Temperature of bodies critical to behaviour
of atmospheres, surfaces and interiors
All bodies with temperatures above absolute zero emit
radiation
Max Planck
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• The amount of radiation emitted by a
blackbody is uniquely determined by its
temperature (Planck’s law):
Classical Limit (small f, large λ)
The black body specific intensity or brightness
is defined (following discovery by Max Planck
in 1900) as either
Bλ (T ) =
2hc 2
λ
5
1
e
hc / λkT
− 1 or
Bν (T ) =
In the limit of small f:
2ν 2 k BT
c2
Rayleigh-Jeans
2hν 3
1
c 2 e hν / kT − 1
≈
where c=2.99x1010 cm/s, h=6.57x20-27 erg s,
k=1.38x10-16 erg/s. Using cgs units (λ in
Angstroms) we have
Bλ (T ) =
Bv (T ) ≈
1
λ4
1.19 x1027 λ5
8
e1.44 x10
/ λT
−1
Max Planck
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This equation doesn’t involve Planck’s constant – was originally derived from purely
classical considerations. Classical physics predicts the so-called ultraviolet catastrophe
– an infinite amount of energy being radiated at high frequencies or short wavelengths
(derived from the equipartition theorem).
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• At the other extreme for high f (or for short
wavelengths), Planck’s law simplifies to Wiens
Law:
• Blackbody radiation is • Units are J m-2 Hz-1 s-1
isotropic; the radiance
ster-1 (erg cm-2 Hz-1 s-1
is independent of
ster-1)
direction
• Recall 107 ergs = 1 J
Bv (T ) ≈
Bλ ≅
2 hν 3
1
Bν (T ) = 2 hν / kT
c e
−1
hν
2 hν 3 − k B T
e
c2
2hc 2
⎡
⎛ − hc ⎞ ⎤
⎟⎟ ⎥
⎝ λk T ⎠ ⎦
λ5 ⎢exp ⎜⎜
⎣
Max Planck
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The Wien displacement law
• Characteristic shape for blackbody
radiation plotted using Planck’s law
• Using Planck’s law and differentiating to find the peak (ie. solve ∂B/
∂λ=0) , one can find the wavelength of peak emission for a
blackbody at temperature T:
λm =
2897 (μm K )
T
known as the Wien displacement law. This law
makes possible the estimate of the temperature
of a radiation source from knowledge of its
emission spectrum.
Sharp short wavelength cutoff, steep rise to the maximum, gentle
dropoff toward longer wavelengths – often can use limiting
expressions at high f (Wien Law) or low f (Rayleigh-Jeans Law) 9
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The Wien displacement law
Albedos
• When the sun illuminates an object, some of the radiation
is absorbed, and some scattered.
• The albedo (ratio of reflected and scattered intensity to
incident intensity) varies with wavelength. Aν is the
monochromatic albedo.
• The luminosity observed depends on the geometry,
specifically the phase angle.
• Consequence:
– solar radiation (due to the temperature of the sun) is
concentrated in the visible and near-IR parts of the
spectrum
–p
planetary
a e a y radiation
ad a o a
and
d that
a o
of their
e a
atmospheres
osp e es is
s
largely confined to the IR
Earth
(normalized)
Object
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The Wien displacement law
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Albedos
• The geometric albedo is
the ratio of the flux reflected
head-on (back to the sun)
to the incident flux
• The bond albedo is the
ratio of the total flux
reflected to the incident. It
incorporates an integral
over phase angle
• Note the lack of overlap…
that allows separation of the radiative transfer
problems of the earth and of the sun
Sun
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A0 =
F (ϕ = 0)
Fincident
Ab = A0 q ph
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The Stefan-Boltzmann law
• If we integrate Planck’s law just above the surface of an
object and over all frequencies, we find:
∞
∞
0
0
Marley et al. (1999)
F (T ) ≡ ∫ Fν dν = π ∫ Bν (T )dν
F = σ T4
where F is the flux (power/unit area)
which is known as the Stefan-Boltzmann law
• F = Flux, (power/unit area), T = Temp. in Kelvin,
σ = 5.67 x 10-8 W/m2K4 (conductivity)
• For non-ideal black body, F = σT4ε
Josef Stefan
• where ε = emissivity < 1.
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Phase Function:
φ=
I (ϕ )
I (0)
Sudarsky et al. (2005)
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Equilibrium temperature
Fin = (1 − Ab )
Fout = 4πR 2εσT 4
Lsun
πR 2
4πr 2
We can calculate the equilibrium temperature by setting
q
to each other.
the two equal
Eros from NEAR
1/ 4
⎛ F (1 − Ab ) ⎞
Teq = ⎜ sun
⎟
2
4εσ ⎠
⎝ r
The temperature depends on the distance to the sun, but
not on the size of the object.
Muinonen et al. (2002)
Equilibrium temperature
• The sunlit hemisphere of a planet absorbs
Cross-sectional area
radiation:
L
of planet
Fin = (1 − Ab ) sun2 πR 2
Area over which solar
4πr
radiation is spread at
distance r from sun
• If the planet rotates rapidly
rapidly, its temperature is
uniform. In that case, it emits radiation:
Fout = 4πR 2εσT 4
We can calculate the equilibrium temperature by setting
the two equal to each other.
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Planetary Temperatures
Mercury
Venus
Earth
Moon
Mars
Teq
446 K
238
263
277
222
Teff
446 K
238
263
277
222
Jupiter
Saturn
Uranus
Neptune
113
83
60
48
124
95
59
59
Tsurf
100 – 725 K
733
288
277
215
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Thermal radiation is reflected in all directions (slow
rotator) so as seen at the Earth the thermal radiation
received is:
Albedos in the solar system
Rocky surfaces:
Icy bodies
Gaseous planets:
The Moon:
Venus:
0.1 – 0.2
0.2 – 0.7
~0.3
0.07
0.75
Thus the ratio of visible to thermal radiation is:
Therefore if we can simultaneously measure the 2
2
thermal and visible flux we can 2directly measure the
visible (and hence thermal) albedos.
We can measure the visual albedo by
comparing the reflected and emitted radiation.
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Reflected visible light
Heat Conduction
Av=0.20
• Conduction is the transport of energy by collisions
between particles. Conduction is important in the
upper atmosphere, where the mean free path is long
and collisions are important.
• Sunlight heats many surfaces during the day. The
energy is transported downwards from the surface.
• The rate of flow of heat is known at the heat
flux, Q.
• Q depends on the temperature gradient,
or
and the thermal conductivity KT.
• KT is a measure of the material’s ability to
conduct heat.
Av=0.05
Units of KT:
erg s-1 cm-1 K-1 or
J s-1 m-1 K-1
IR emission
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Solar radiation flux falling on an asteroid surface
per square meter:
Total reflected visible luminosity of the asteroid
is given by:
Energy not reflected is absorbed and then re-emitted
at IR wavelengths:
Conduction as diffusion
The energy that goes into a volume element per unit time is:
How much does this heat up the material?
Combining this with
We get:
Assume asteroid is at opposition with the
Earth and reflects visible radiation
uniformly over its sunlit hemisphere (2π
2
steradians).
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Visible radiation detected at the Earth is
then:
or
where
This is known as the diffusion equation
d
Compare to the wave equation:
which has oscillating solutions.
2
The diffusion equation has exponentially spreading solutions.
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t
t
t
t
t
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Thermal diffusion coefficients
CP (J/kgK)
Water 4200
Iron
450
Stone 700
ρ (kg/m3)
1000
7800
3000
KT (W/mK)
2.18
80
2-7
Typical Near-Earth Asteroid rotation period ~ 104 sec
Longest known asteroid rotation period ~ 107 sec
For Mars/Moon Z ~ 5 cm
Kd (m2/s)
5.5 x 10-7
2.3 x 10-5
2.3 x 10-5
Z ~ 10 cm
Z ~ 10 m
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