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Emission Temperature of Planets The emission temperature of a planet, Te, is the temperature with which it needs to

emit in order to achieve energy balance (assuming the average temperature is not decreasing c.f. Jovian planets). We equate the absorbed solar energy with the energy emitted by a blackbody.

–  Solar radiation absorbed = planetary radiation emitted –  Absorbed Solar Radiation = Sp π rp

2 × (1-αp)

αp is the planetary reflectivity or albedo. For Earth, αp is ~ 0.3. For Venus, αp is ~ 0.7 and so solar energy input per unit area is less despite being at 0.7 AU.

Emitted radiation = σ T4 4 π rp2

The 4 π rp2 accounts for the fact that emission occurs over the entire area of the

sphere. Equating the absorbed and emitted radiation: Sp π rp

2 × (1-αp) = σ Te4 4 π rp

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Te = [(Sp / 4)( 1-αp)/ σ]1/4

Figure Ruddiman 2001 Emission Temperature of Planets

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Emission Temperature of Earth Today:

Te = [(1367 W m-2 / 4) (1 – 0.3) / (5.67 × 10-8 W m-2 K-4)]¼ = 255 K = -18 C (O oF)

Obviously this is not the emission temperature of the surface! As we will see, the greenhouse effect warms the surface significantly above Te (fortunately). The problem is even more severe early in the solar evolution:

4 Billion years ago: Te = [(1000 W m-2 / 4) (1 – 0.3(?)) / (5.67 × 10-8 W m-2 K-4)]¼

= 235 K = -38 C (-36 oF)

Contrast with Jupiter: Jupiter is 7.8 × 108 km from the sun. Its albedo is 0.73.

SJupiter = 3.9 x 1026 W / (4 π dJupiter2) = 50 W m-2

Te = [(50 W m-2 / 4) (1 – 0.73) / (5.67 × 10-8 W m-2 K-4)]¼ = 88 K

Actual emission temperature of Jupiter is ~ 134 K. Most of the energy emitted by Jupiter today (> 80% - note T4 dependence) is associated with cooling from gravitational accretion.

Spectral Dependence of Black Body Radiation: The Planck Function

Understanding black body radiation is one of the great triumphs of 20th century physics and led to the discovery of quantum theory. In many ways, it was a more important development than relativity, since in BB radiation q.m. is manifest in the macroscopic world. Max Planck (1858-1947; http://wwwfml.mpib-tuebingen.mpg.de/max.htm) developed the theory of blackbody radiation leading to the discovery of quantum mechanics (for which he earned the Nobel Prize in physics in 1918). Planck postulated that the energy, E, of molecules is quantized and can undergo only discrete transitions that satisfy:

where n is integer, h is Planck's constant (the fundamental constant of quantum mechanics), and ν = c/λ is the frequency of the radiation emitted (or absorbed) in the transition. The radiation is therefore quantized in photons (not Planck's work) that carry energy in units of hν. From this postulate, Planck derived the relationship between the intensity radiation emitted by a blackbody at a given wavelength to its temperature (Planck's Law):

where kB is the Boltzmann constant (the fundamental constant of statistical mechanics, Eaverage = kBT, for molecules, light, everything) and c, is the speed of light in vacuum.

The integral of Bν (T) over all frequency (and over all angles in a hemisphere) is σ T4

Hotter objects have their peak emission at shorter wavelength (higher frequency)

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Is the area under these curves the energy emitted? Are the units correct?

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Greenhouse Effect Note that the solar radiance is almost completely separated spectrally from the thermal emission

from Earth. We now develop a very simple but illustrative model to illustrate how an atmosphere warms the surface of a planet.

Consider a slab atmosphere that is transparent (emissivity = 0) at all wavelengths < 2 µm (2000 nm). Now assume that this atmosphere is a black body at terrestrial wavelengths (emissivity = 1 at wavelengths > 2 µm):

Since the atmosphere absorbs all terrestrial radiation, the only energy emitted to space is from the atmosphere. The energy balance at the top of the atmosphere is then:

So π rp2 × (1-αp) = σ TA

4 4 π rp2

or (So/4) (1-αp) = σ TA4 = σ Te

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The surface temperature, however, is much warmer. Energy balance at the surface requires: σ TS

4 = 2 σ TA4 ⇒ σ TS

4 = 2 σ Te4

Surface

σ TA4

(OLR)

σ TS4

So/4 (1- αp)

σ TA4

The atmosphere does not inhibit the flow of energy to the surface, but augments the solar heating of the surface with its own emission (here equal to the solar heating). For this slab, isothermal, atmosphere of unit emissivity, the surface temperature is nearly 20% higher than Te (255 K ⇒ 303 K)

In this course, our aim is to understand in growing sophistication the feedbacks and forcing that control Earth's surface temperature. In fact, the isothermal blackbody atmosphere, is not really the extreme. Because of the thermal structure of the atmosphere (Chapter 1), the surface temperature can be forced substantially above (2)^0.25.

The simple model described above can be extended to an arbitrary number of 'layers' in our slab atmosphere:

Thus, the greater the 'optical depth' of the atmosphere, the warmer the surface.

σ TA4

Atmosphere (1)

Surface

σ TA(1)4

σ TS4

So/4 (1- αp) Atmosphere (2)

σ TA(2)4

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Global Radiative Flux Energy Balance for Earth In our model above, the themal coupling between the surface and atmosphere is weak. Only

radiation transfers energy and the isothemal atmosphere is substantially cooler than the surface. Clearly, our head is not cooler than our feet! The vertical transfer of heat in the atmosphere is one of the most important climate processes. Energy is transferred both radiatively and non-radiatively and both are critical for climate. The transmission of solar radiation to the surface (and the absorption there) and the degree of coupling between the warm surface and the atmosphere determines the strength of the greenhouse effect.

Figure 2.4 below (from Hartmann, Global Physical Climatology) illustrates the globally averaged movement of energy through Earth's atmosphere (the values are % of the top of atmosphere radiative flux (340 W m-2) . 50% of the insolation is transmitted and absorbed at the surface. The UV (3%) is significantly attenuated in the stratosphere by O2 and O3. In the troposphere (below ~10 km), ~20% is absorbed by water vapor (13%), clouds (3%), and CO2, O3, O2 (together a few percent).

Note the large thermal fluxes within the atmosphere, indicating the strength of the greenhouse effect. The main contributers to the absorption of long wavelength radiation in the atmosphere (and their mixing ratio) are:

H2O (variable < 3%), clouds (variable), CO2 (370 ppm) , O3 (0-7 ppm), N2O (315 ppb), CH4 (1800 ppb), ……

The gases that make up most of the atmosphere N2 (78%), O2 (21), Ar (1%) are essentially transparent (more later). Water vapor and clouds are by far the most important greenhouse drivers (> 80%) of the surface forcing. Only ~10% of the photons emitted at the surface pass through the atmosphere unimpeded. What makes a greenhouse gas a greenhouse gas, we will take up in much more detail in week 3.

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Examining Figure 2.4 we can begin to understand a number of things about Earth climate:

The very strong downward emission from the atmosphere is essential for maintaining the relatively small diurnal variations in surface temperature. Compare desert to continental interior (humidity/clouds).

We also can see a number of positive feedbacks:

Water is a critical component of the climate system. ⇑ temperature ⇒ ⇑ absolute humidity ⇒ ⇑ temperature

Planetary Reflectivity (Albedo)

Visible albedo of some surfaces:

•  The difference between the reflectivity of ice/snow and ocean/forest is an important climate feedback. Much more about this later.

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Latitude and Solar Insulation Up until now, our model of energy transport has been of the 'copper' planet of spherical geometry. In

this model (miraculously) the entire surface is at the same temperature and all regions get the same insolation (averaged over a day).

Let us first consider the input of solar radiation.

We define the solar zenith angle (SZA), θS, as the angle between the local normal and a line between a point on Earth's surface and the sun.

The SZA depends on latitude, season, and time of day. The season can be expressed in terms of the declination angle, δ, which is the latitude of the point on the surface of Earth directly under the sun at noon. (23.45 oN at summer solstice 23.45 oS at winter solstice). The hour angle, h, is defined as the longitude of the subsolar point relative to its position at noon. We these definitions, the cosine of SZA can be derived for any latitude (φ), season, and time of day:

cos(θs)= sin φ sin δ + cos φ cos δ cos h

From this, we can derive the radiant flux as a function of latitude and season. The albedo also is higher at high latitude (ice / higher inclination angle).

Solar Flux

Zenith

θS Shadow Area = A cos θS

Surface Area = A

Hartmann

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Hartmann