Tutorial 2: Thermal Emission from Planetary Surfaces

Spring 2003

As before, in this tutorial we'll go quickly through some pretty simple stuff, but its important to get these concepts straight so that we're all speaking the same language when it comes to lectures, labs and homework.

In 1800, Sir William Herschel, the royal astronomer to the King of England, conducted an experiment to study the heating effects of sunlight. He used a prism to separate light into the colors of the spectrum and used a thermometer to measure the temperature in each color. As he moved from the violet to the red region, the temperature increased; however, when he placed the thermometer in the region just beyond the color red, the temperature continued to increase, even when no light was visible to the naked eye. William Herschel had just discovered the portion of the electromagnetic spectrum known as the infrared.

The term infrared spans a huge range of wavelengths. In this tutorial we're going to be talking about the thermal infrared. By thermal infrared we really mean heat as the 10 micron region corresponds to the peak of a 300 K blackbody, which is a nice cozy summer day for most of us.

We'll start off with the planck equation in both the wavelength and frequency forms. The units of B are J s^-1 m^-2 ster^-1 hz^-1 or m^-1. (We use S.I. units exclusively in these tutorials). This quantity is known by several names: Radiance, Emittance, Brightness (or more correctly specific brightness), surface brightness or specific intensity. This is a measure of power flowing from 1 square meter in some direction with a spread of propagation vectors of 1 ster-radian. Since no real surface emits as a blackbody, B is usually modified by a factor known as the emissivity which varies between 0 and 1. Just as there are many kinds of albedo there are many kinds of emissivity. However for our needs this introduces unnecessary complications.

The great thing about these equations is that they depend only on temperature. A quick note on nomenclature though. Terms like radiance, flux, irradiance and brightness are often brandied about in annoyingly imprecise ways as if they all mean the same thing [some of them do and some don't]. In an effort to set the record straight we'll try and provide exact definitions.

Radiance: (also known as: specific brightness, emittance,surface brightness or specific intensity) is given by the Plank equations above. When a spacecraft measures the luminosity (just energy per second) of a surface it is necessary to convert this to radiance, i.e. correct the quantity so that it comes from 1 m², divide by the bandwidth so that it is now per hertz, and divide by the collecting area of the instrument (in ster-radians) so that it is now per ster-radian. From the radiance we can derive things like surface temperature or emissivity in that band.
Intensity: (also known as brightness) This is simply the radiance integrated over all wavelengths or frequencies. It still contains a per solid angle term so the amount of energy you can collect depends on the amount of the sky your detector covers as seen by the source. The units are J s^-1 m^-2 ster^-1. Note the intensity does not vary with distance from the source. The amount of energy one receives changes with distance because the solid angle subtended by the detector as seen from the source changes. However the intensity remains constant.
Irradiance: (also known as specific flux) This is the radiance integrated over solid angle. It is usually used to describe collimated or near-collimated beams such as sunlight on a planets surface. The units are J s^-1 m^-2 hz^-1 or m^-1. This is still a function of wavelength / frequency.
Flux: This is the radiance integrated over all solid angles and over all wavelengths/frequencies. It is a measure of the total energy coming from 1 m² of a surface. Its not so easy to measure this quantity you could measure the radiance but then you'd have to make some assumption about how the surface emits over all other emission angles and over all other wavelengths. The units are J s^-1 m^-2.
Luminosity: This is the final quantity, it is the radiance integrated over all solid angles, all wavelengths and over the surface area of the whole body. The temperature may vary from surface area to surface area but for things like the sun you could assume its pretty constant. The units are J s^-1.

Example: A spacecraft orbiting 400 Km above the planet's surface has a far infrared instrument with a circular collecting area 50cm across. It observes a patch of the surface 3Km by 3Km at 50 microns through a filter 1 micron wide. The instrument records a luminosity of 9x10^-6 Watts. What is the radiance and hence the temperature of that patch of ground ?

The area observed is 3000x3000 meters, or 9x10⁶ m².
The area of the collector is pi*0.25² m². It is at a distance of 400 Km from the surface so the solid angle it subtends is pi*0.25² / 400000², which is 1.23x10^-12 ster-radians.
The bandwidth is 1 micron or 10^-6 meters.

We can divide the luminosity recorded by the spacecraft by these three quantities to derive the radiance. This gives us a radiance of 203721.14 J s^-1 m^-2 ster^-1 m^-1. We can rearrange the Planck formula as shown below and use it to get the temperature.

So the answer is 273 K, some major assumptions that went into this are : (1) The surface is assumed to emit isotropically, (2) The emissivity is assumed to be 1 (more on this later), (3) The atmosphere is assumed to play no role in the thermal infrared. The last assumption is the most serious. However accounting for the atmosphere would be a course in itself so we'll ignore it for now.

We saw in the example that you can't recover the emissivity and the temperature from a single measurement. One way to break the connection is to take a spectrum which resolves the black body peak. The wavelength of peak emission is directly connected to the temperature through the Wien Displacement Law, quoted below. The derivation basically follows the steps of taking the derivative of the Plank formula with respect to wavelength and setting it equal to zero to find the maximum. The resulting equation cannot be solved analytically using standard special functions of mathematical physics, but can be solved in terms of Lambert's W-Function.

If the surface emits isotropically then we can derive and expression for the flux emitted solely in terms of the temperature. Remember the flux is just the radiance integrated over solid angle and wavelength. For a patch of flat ground emitting into a hemisphere you may be tempted into thinking that the solid angle is simply half a sphere or 2 pi ster-radians. This would be true for a point. However because the emission is coming from a finite area (1 m² in this case) the effective solid angle is just pi due to geometrical effects. After multiplying by the solid angle we have to integrate over all wavelengths. The result is given by the Stefan-Boltzmann Law, given below.

Here sigma is the Stefan-Boltzmann constant, 5.67x10^-8 J s^-1 k^-4. Knowing the total flux emitted from a surface will be important when we consider the thermal state of the subsurface and heat balance of surfaces later in the course. As with the Plank law, the emissivity factor must be taken into account to find the flux from any real surface.

The emissivity may vary with wavelength in the same way as albedo does (see tutorial one). Below is shown a thermal spectrum of quartz along with its ideal black body spectra.

If we were to divide the real spectra by the hypothetical blackbody then we would recover the emissivity as a function of wavelength (spectral emissivity ). This can (and is) done for planetary surfaces. Different rock types show different emission spectra. Below are some examples. As mentioned before removing the effect of the atmosphere (especially in the case of Mars which has a lot of suspended dust) is a formidable challenge and is a permanent thorn in the side of workers in this field.

Questions:

(back to top)

(1a) Suppose you fly a mission to Mercury, and you go into polar orbit so you can sample both the day and night sides which have temperatures of ~700 and ~100 Kelvin respectively. For the orbit and instrument described in the example what is the range of powers that you would expect to record? Assume an albedo of 0.16 and an emissivity of 0.84 and a Lambert surface.

(1b) At what wavelengths does the intensity of reflected and emitted light (day and night sides) peak?

(1c) For observations on the day-side what wavelengths would you have to observe at to insure that what you are looking at is dominated by emitted rather than reflected radiation (by at least a factor of 10)? You will need to use also the methods described in tutorial one.

(2) In the emissivity spectra shown above, are the clay minerals more similar to the silicate or to carbonate rocks? Why ?, think about the chemical bonds involved.

- Shane Byrne -

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