Astro 9601
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• Energy Balance and Temperature (3.1) - All
• Convection (3.2.3), Hydrostatic Equilibrium
(3.2.3.1), First Law of Thermodynamics (3.2.3.2) and Adiabatic Lapse rate (3.2.3.3)
– All to be discussed in lecture notes with Ch. 4 (where it makes sense!)
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• All solar system bodies are illuminated by the sun
• Balance between solar radiation received
(plus any internal energy) and that emitted defines temperature
– ultimately equilibrium is reached which defines T
• Temperature of bodies critical to behaviour of atmospheres, surfaces and interiors
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Energy can be transmitted by:
2. Radiation
3. Convection
One mechanism usually dominates
In space and tenuous gases, radiation dominates
Convection is important in atmospheres (and liquid interiors)
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• The temperature of an object is ti l t th t l ti l kinetic energy of its molecules.
• Note that one object can have many temperatures
• Blackbody – a hypothetical (idealized) body that
– Emits the maximum possible radiant energy in all wavelength bands in all directions
– No radiation is reflected
All bodies with temperatures above absolute zero emit radiation
Max Planck http://home.wanadoo.nl/paulschils/07.02.html
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3

• The amount of radiation emitted by a blackbody is uniquely determined by its temperature ( Planck’s law ):
The black body specific intensity or brightness is defined (following discovery by Max Planck
B
λ
( T ) =
2 hc
2
λ 5
1 e hc
/ λ kT
− 1 or B
ν
( T ) =
2 h
ν 3 c
2 e h
ν
1
/ kT
− 1 where c=2.99x10
10 cm/s, h=6.57x20
-27 erg s, k=1.38x10
-16 erg/s. Using cgs units ( λ in
Angstroms) we have
B
λ
(
T
) =
1 .
19 x
10 27 e
1 .
44 x
10 8 / λ
T
λ 5
− 1
Max Planck http://home.wanadoo.nl/paulschils/07.02.html
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• Blackbody radiation is isotropic ; the radiance is independent of di ti
• Units are J m -2 Hz -1 s -1 ster -1 (erg cm -2 Hz -1 s -1 ster -1 ) )
• Recall 10 7 ergs = 1 J
B
ν
(
T
) =
2 h
ν 3 c
2
1 e h
ν / kT − 1 http://www.tpub.com/content/neets/14182/css/14182_179.htm
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4

• Characteristic shape for blackbody radiation plotted using Planck’s law
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)
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Classical Limit (small f, large
λ
)
In the limit of small f:
B v
(
T
) ≈
2 ν 2 c
2 k
B
T
Rayleigh-Jeans
≈
1
λ 4
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:
B v
( T ) ≈
2 h
ν 3 − c
2 h
ν k
B
T
B
λ
≅
λ 5
2 hc
2
⎡
⎢ exp
⎛ −
⎜⎜
λ k hc
T
⎞
⎟⎟
⎤
⎥
Max Planck http://home.wanadoo.nl/paulschils/07.02.html
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• 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:
λ m
=
2897
(
μ m K
)
T known as the Wien displacement law. This law of a radiation source from knowledge of its emission spectrum.
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• Consequence:
– solar radiation (due to the temperature of the sun) is concentrated in the visible and near-IR parts of the spectrum
– planetary radiation and that of their atmospheres is largely confined to the IR
(normalized)
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• Note the lack of overlap… that allows separation of the radiative transfer problems of the earth and of the sun
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• If we integrate Planck’s law just above the surface of an object and over all frequencies, we find:
F ( T ) ≡ ∫
0
∞
F
ν d
ν = π ∫
0
∞
B
ν
( T ) d
ν
F
=
T
4 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/m 2 K 4 (conductivity)
• For non-ideal black body, F = σ T 4 ε
• where ε = emissivity < 1.
Josef Stefan http://home.wanadoo.nl/paulschils/07.02.html
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• 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. monochromatic albedo.
A
ν is the
• The luminosity observed depends on the geometry, specifically the phase angle.
Earth
Object
Sun
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8

• The geometric albedo is head-on (back to the sun) to the incident flux
• The bond albedo is the ratio of the total flux incorporates an integral over phase angle
=
F ( ϕ = 0 )
F incident
A b
=
A
0 ph
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Marley et al. (1999)
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Sudarsky et al. (2005)
Phase Function:
φ =
I ( ϕ )
I
( 0 )
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Eros from NEAR
Muinonen et al. (2002)
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• The sunlit hemisphere of a planet absorbs radiation: in
( ( 1 −
A b
) )
L sun
4 π r
2
2
Cross-sectional area of planet
Area over which solar radiation is spread at distance r from sun
• If the planet rotates rapidly, its temperature is uniform. In that case, it emits radiation:
F out
2 εσ 4
We can calculate the equilibrium temperature by setting the two equal to each other.
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F
= ( ( 1 −
A
) )
L sun
4 π r
π
R
2 F
= 4 π
R
2 εσ
T
4
We can calculate the equilibrium temperature by setting the two equal to each other.
T eq
=
⎝
F sun r
2
( 1 −
A b
4 εσ
)
⎟
⎠
⎞
1 / 4
The temperature depends on the distance to the sun, but not on the size of the object.
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T eff
T surf
T eq
M 446 K
Venus
Earth
Moon
Mars
238
263
277
222
238
263
277
222
733
288
277
215
Jupiter
Saturn
113
83
Uranus 60
Neptune 48
124
95
59
59
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Rocky surfaces:
Icy bodies
Gaseous planets:
0.1 – 0.2
0.2 – 0.7
~0.3
The Moon: 0.07
Venus: 0.75
We can measure the visual albedo by comparing the reflected and emitted radiation.
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Reflected visible light
A v
=0.20
A v
=0.05
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:
Assume asteroid is at opposition with the d uniformly over its sunlit hemisphere (2 π
2 steradians).
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Visible radiation detected at the Earth is then:
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Thermal radiation is reflected in all directions (slow rotator) so as seen at the Earth the thermal radiation received is:
Thus the ratio of visible to thermal radiation is:
Therefore if we can simultaneously measure the
2 2 thermal and visible flux we can directly measure the visible (and hence thermal) albedos.
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• Conduction is the transport of energy by collisions between particles. Conduction is important in the 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 K
T
K
T conduct heat.
. is a measure of the material’s ability to
Units of
K
T
: erg s -1 cm -1 K -1 or
J s -1 m -1 K -1
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The energy that goes into a volume element per unit time is:
Combining this with
We get: or where
This is known as the diffusion equation
Compare to the wave equation: which has oscillating solutions.
The diffusion equation has exponentially spreading solutions.
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31 t tt t t t
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C
P
(J/kgK) ρ (kg/m 3 ) K
T
(W/mK) K d
(m 2 /s)
5 5 x 10 -7
Iron 450
Stone 700
7800
3000
80
2 - 7
2.3 x 10 -5
2.3 x 10 -5 p 4 sec Z ~ 10 cm
Longest known asteroid rotation period ~ 10 7 sec Z ~ 10 m
For Mars/Moon Z ~ 5 cm
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