Class 6- Equilibrium Temperature

Where and when was this picture taken?

Equilibrium Temperature

 

Equilibrium means no change with time.

Knowing that the total output of sunlight and that light decreases as 1/distance2, we can calculate the amount of sunlight that should be hitting a square meter of each planet. The ALBEDO (A) of a planet is the reflectivity of a planet. Therefore, the total amount of sunlight absorbed the by the planet per square meter is

Power absorbed/area = (1-A) (Solar Flux@Earth) / (distance)2

where distance from the Sun is in AU. In equilibrium, we expect

ENERGY IN = ENERGY OUT

The energy emitted per square meter is described by the Stefan-Boltzmann law for thermal emission:

Power/Area = sigma x T4

where T = Temperature of the radiating surface and sigma = the Stefan-Boltzmann constant = 5.67 x 10-8 W m-2 K-1

Equating these two, and allowing for the fact that objects receive an area piR2 of sunlight but emit from all 4piR2,

(1-A) (Solar Flux@Earth) / (distance)2 = 4x sigma x T4

Solving for T and "normalizing" to the Earth (at 1 AU), we get

T4 = [1368 W m-2 / 4sigma] (1-A)/a2

where a is the distance from the Sun in AU. Taking the 4th root we get....

T = {1368 W m / (4sigma )}1/4 (1-A)1/4 /a1/2

evaluating the constant in the curly brackets {} we get a handy-dandy formula

Tequilibrium = 280K [(1-A)/a2]1/4

OR

Tequilibrium = 280K [1-A]1/4/a1/2

 

(if you compare with some textbooks - e.g. Hartmann's page 297 - you will see we are ignoring emissivity - a small correction generally, as it is inside the 4th root and the 4th root of a number close to 1 is very close to 1).

What happens when this equilibrium temperature is compared with the TRUE temperature (measured by looking at the IR spectrum and measuring the wavelength of maximum emission and using Wein's Law)?

This figure (from Hubbard's chapter in The New Solar System) shows that the giant planets tend to emit more energy than they receive - all except Uranus where the internal heat source (red arrow) is negligible. Hartmann quotes these ratios:

  Jupiter Saturn Uranus Neptune
Heat Emitted / Sunlight Absorbed 2.5 2.3 ~1.1 2.7

This table tells us that all of the giant planets except Uranus emits about 21/2 times the amount of solar energy absorbed. What's the story with Uranus? Why does it emit so much less energy? This is a MAJOR issue of planetary science.

But we digress.... For the GIANT PLANETS the issue is an internal heat source. For the TERRESTRIAL PLANETS we need to think about the GREENHOUSE EFFECT (why not internal heat sources for TPs?)

More quantitatively...

See the Solar System Collaborative for an interactive.(go to Greenhouse effect module).

More on the details of radiative transfer when we look at Earth-Venus-Mars in more detail.

Scattering

Rayleigh scattering - by molecules (particularly water) in the visible part of the spectrum where the wavelength is comparable to the scale of the molecule. Scattering is inversely proportional to l4

Mie scattering - by particulates - aerosols - many, many molecules. The wavelength is less than the scale of the particles. The scattering function of wavelength and angle is complicated.

The Brown Cloud over Denver - January 23, 2005

The main cause of the Brown Cloud is scattering by particulates - aerosols - dirt in the atmosphere - that preferentially scatters at the red end of the spectrum. There is also absorption of blue light by NO2 - nitrogen dioxide - a pollutant from vehicles.

Why is the sky blue? Why are sunsets red?

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Left - alpen glow - all the blue light is scattered away, only the red light gets through. Right - sunset - again, blue scattered away leaving only red to get through - but also scattering by particles, preferentially at red wavelengths.

High scattering - by larger particles - across the spectrum (white). Note that farther ridges are whiter - more scattered light between observer and object.

Titan vs. Earth

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View of Sky on Other Planets

No atmosphere - no scattered sunlight.

Mars sky line at sunset.

Below are Titan - something between reality and artist rendering....from creative amateurs

Here are two more colorful versions - color completely invented...

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The rest we have to make up. Here are some old artistic renderings of Mercury, Venus, Earth and Mars.