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Chapter 3 – Energy Balance and Temperature

Astro 9601

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Topics to be covered• Energy Balance and Temperature (3.1) - All• Conduction (3 2 1) Radiation (3 2 2 and 3 2 2 1)• 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 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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Radiation and Planetary Science

• All solar system bodies are illuminated by the sunthe sun

• Balance between solar radiation received (plus any internal energy) and that emitted defines temperature – ultimately equilibrium is reached which

defines Tdefines T• Temperature of bodies critical to behaviour

of atmospheres, surfaces and interiors

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Energy Transport

Energy can be transmitted by:1 Conduction1. Conduction2. Radiation3. Convection

One mechanism usually dominatesIn solids, conduction dominates

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,In space and tenuous gases, radiation

dominatesConvection is important in atmospheres (and

liquid interiors)

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Temperature

• The temperature of an object isti l t th t l ti lproportional to the average translational

kinetic energy of its molecules. • Note that one object can have many

temperatures

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Blackbody - Introduction• Blackbody – a hypothetical (idealized) body that

– Absorbs all incident radiation (hence the term “black”)– Absorbs all incident radiation (hence the term black )– 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

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Max Planck6

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• 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 in 1900) as either

112)( /5

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−= kThce

hcTB λλ λ

in 1900) as either

or

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

112)( /2

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−= kThec

hTB ννν

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Max Planck

11019.1)(

/1044.1

527

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−=

TxexTB

λλλ

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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)direction

112)( /2

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−= kThec

hTB ννν

)• Recall 107 ergs = 1 J

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• Characteristic shape for blackbody radiation plotted using Planck’s law

Sharp short wavelength cutoff, steep rise to the maximum, gentledropoff toward longer wavelengths – often can use limiting expressions at high f (Wien Law) or low f (Rayleigh-Jeans Law) 9

Classical Limit (small f, large λ)

In the limit of small f:2

22)(c

TkTB Bv

ν≈

41λ

≈

Rayleigh-Jeans

This equation doesn’t involve Planck’s constant – was originally derived from purelyclassical 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). 10

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• At the other extreme for high f (or for short wavelengths), Planck’s law simplifies to Wiens Law:

Tkh

BehTBν

ν −

≈32)(

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −≅

Tkhc

hcB

λλ

λ

exp

2

5

2

v ec

TB ≈ 2)(

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Max Planck11

The Wien displacement 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:

( )T

Kμm2897=mλ

known as the Wien displacement law. This law makes possible the estimate of the temperature p pof a radiation source from knowledge of its emission spectrum.

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The Wien displacement law• Consequence:

– solar radiation (due to the temperature of the sun) is– 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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The Wien displacement law

• Note the lack of overlap…

that allows separation of the radiative transferproblems of the earth and of the sun

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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:

4TF σ=where F is the flux (power/unit area) which is known as the Stefan-Boltzmann law

∫∫∞∞

=≡00

)()( νπν νν dTBdFTF

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Josef Stefan15

• 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• where ε = emissivity < 1.

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.

Earth

16Sun

Object

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Albedos

• The geometric albedo is the ratio of the flux reflected

FA )0(0

==

ϕthe 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

incidentFA0

qAA =reflected to the incident. It incorporates an integral over phase angle

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phb qAA 0=

Marley et al. (1999)

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Phase Function:

)0()(

II ϕφ =

Sudarsky et al. (2005) 19

Eros from NEAR

Muinonen et al. (2002)20

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Equilibrium temperature• The sunlit hemisphere of a planet absorbs

radiation:2)1( RLAF sun π−=

Cross-sectional area of planet

• If the planet rotates rapidly, its temperature is uniform. In that case, it emits radiation:

24)1( R

rAF bin π

π=

Area over which solar radiation is spread at distance r from sun

424 TRF εσπ=

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4 TRFout εσπ=

We can calculate the equilibrium temperature by setting the two equal to each other.

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Equilibrium temperature

22)1( RLAF sun

bin π−= 424 TRFout εσπ=24)(

rbin π out

We can calculate the equilibrium temperature by setting the two equal to each other.

4/1

2 4)1(⎟⎠⎞

⎜⎝⎛ −

=εσ

bsuneq

Ar

FT

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⎠⎝The temperature depends on the distance to the sun, but not on the size of the object.

Planetary TemperaturesTeq Teff Tsurf

M 446 K 446 K 100 725 KMercury 446 K 446 K 100 – 725 KVenus 238 238 733Earth 263 263 288Moon 277 277 277Mars 222 222 215

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Jupiter 113 124Saturn 83 95Uranus 60 59Neptune 48 59

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Albedos in the solar system

Rocky surfaces: 0.1 – 0.2Icy bodies 0.2 – 0.7Gaseous planets: ~0.3The Moon: 0.07Venus: 0.75

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We can measure the visual albedo by comparing the reflected and emitted radiation.

Reflected visible light

Av=0.20

Av=0.05

IR emission26

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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:

Assume asteroid is at opposition with the Earth and reflects visible radiation

Energy not reflected is absorbed and then re-emitted at IR wavelengths:

dEarth and reflects visible radiation uniformly over its sunlit hemisphere (2πsteradians).Visible radiation detected at the Earth is then:

2

2 2

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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:

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Therefore if we can simultaneously measure the thermal and visible flux we can directly measure the visible (and hence thermal) albedos.

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2 2

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Heat Conduction• Conduction is the transport of energy by collisions

between particles. Conduction is important in the upper atmosphere, where the mean free path is long pp p , p gand 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

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Units of KT:erg s-1 cm-1 K-1 or J s-1 m-1 K-1

conduct heat.

The energy that goes into a volume element per unit time is:

How much does this heat up the material?

Conduction as diffusion

How much does this heat up the material?

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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t

tt

tt

t

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Thermal diffusion coefficientsCP (J/kgK) ρ (kg/m3) KT (W/mK) Kd (m2/s)

Water 4200 1000 2 18 5 5 x 10-7

Typical Near-Earth Asteroid rotation period ~ 104 sec Z ~ 10 cm

Water 4200 1000 2.18 5.5 x 10 7

Iron 450 7800 80 2.3 x 10-5

Stone 700 3000 2 - 7 2.3 x 10-5

yp pLongest known asteroid rotation period ~ 107 sec Z ~ 10 mFor Mars/Moon Z ~ 5 cm

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