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By Charles Rhodes, P. Eng., Ph.D.


The frequency dependent emissivity of a body is the ratio of the radiant energy per unit volume per unit frequency in that body to the radiant energy per unit volume per unit frequency in a theoretically ideal black body at that same frequency and temperature.

The radiant energy per unit volume in a solid sets the rate per unit area at which thermal radiant energy is emitted from the surface of that solid. Similarly the thermal infrared radiant energy per unit volume in the Earth's upper atmosphere sets the rate at which the Earth emits thermal infrared radiation into outer space.

For many practical purposes the frequency dependent emissivity Ftw is integrated over the Planck energy distribution to arrive at an effective frequency independent emissivity Ft. This procedure, although strictly speaking not mathematically correct, greatly simplifies the related mathematics and hence is widely used. The results usually give a qualitatively correct temperature response. However, in certain situations, such as when dealing with green house gases, global warming, thermal runaway, and water near its freezing point, the results can be very misleading if the underlying frequency dependence of emissivity is not properly taken into account.

When the earth is viewed from outer space in the thermal infrared region, what is observed are thermal infrared radiative emissions from molecular species that can readily interact with electromagnetic radiation at thermal infrared frequencies. The amplitude if these emissions is in part set by the Planck molecular energy distribution. Oxygen and nitrogen do not interact with photons in the thermal emission spectrum and hence are not seen. Water molecules, which interact with thermal infrared radiation photons right across the thermal emission spectrum, are seen. For wavenumbers less than 400 cm^-1 and greater than 1300 cm^-1 due to the low value of the Planck molecular energy distribution the thermal radiation emission is low and infrared absorption by water molecules obscures everything else.

For wavenumbers in the range 400 cm^-1 to 1300 cm^-1 the combination of a high value of the Planck molecular energy distribution and a low atmospheric water vapor concentration potentially permits transmission of thermal infrared photons. However, a cloud will prevent thermal infrared photons originating below it from reaching a space craft above the atmosphere. These photons will be absorbed by water molecules in the cloud and converted into atomic/molecular kinetic energy. Inside the cloud the molecules energized by infrared photon absorption exchange energy with N2, O2 and other H2O molecules and acquire the temperature of neighbouring other molecules the cloud. At locations in the cloud where the molecular temperature is below 273.15 K water droplets freeze and during the liquid-solid phase transition emit thermal infrared radiant energy. Some of this radiant energy escapes from the top of the cloud into space. Hence a space craft above the atmosphere looking in the thermal infrared spectrum sees only the top of a cloud and the radiation temperature that the space vehicle sees (~ 270 degrees K) is the temperature of the radiation photons emitted by the liquid-solid phase transition of water.

Microscopic ice particles which form at the top of a cloud fall through the cloud, melt through acquisition of either radiant or kinetic energy from other molecules and then diffuse to the top of the cloud where they again emit thermal infrared radiant energy.

From the perspective of Earth energy balance the nominal temperature of the infrared radiation emitted into space that cools the Earth to balance absorbed solar radiation is 270.0 K. However, the emitted thermal infrared radiation spectrum is modified by filtering by greenhouse gases in the upper atmosphere.

At a wavenumber of 669 cm^-1 carbon dioxide has a stronger absorption than water vapor. Hence, when the earth is viewed from a space vehicle through 669 cm^-1 bandpass filter the carbon dioxide in the upper atmosphere will make the atmosphere appear opaque.

At a wavenumber of 1054 cm^-1 infrared absorption by ozone in the upper atmosphere makes the atmosphere appear opaque.

For other wavenumbers in the range 400 cm^-1 to 1300 cm^-1 the space vehicle can see down to the cloud level.

On an exceptionally clear day, when there is no cloud, the remaining water vapor is almost transparent in the region 600 cm^-1 to 1200 cm^-1. Under these exceptional circumstances within the thermal infrared band the space vehicle can see almost down to ground level, except within the carbon dioxide and ozone absorption bands.

In high latitude regions where the air temperature at ground level is consistently below 270 K the clouds cannot emit infrared radiation via the liquid-solid phase transition of water because this phase transition does not occur. Hence in high latitude regions the cloud emissivity in the thermal infrared spectrum is less than in lower latitude regions. Hence there is less infrared energy emission per unit area from clouds, which causes a higher ground level temperature than would otherwise be the case. Instead at high latitudes infrared radiation originates from ground level or lower altitudes, where it is more sensitive to the atmospheric CO2 concentration. This effect contributes to the higher increase in atmospheric temperature with increasing atmospheric CO2 concentration observed in northern Canada as compared to temperature increases observed in lower latitude regions.

Recall that in the Radiation Physics section it was shown that when there is no net heat absorption the average emission temperature of an element of surface area of the Earth is given by:
Ta = (Ho dAc / dAs Cb)^.25[(1 - Fr) / Ft]^.25
= Te [(1 - Fr) / Ft]^.25

where emissivity Ft is measured at infrared radiation frequencies and albedo Fr is measured at solar radiation frequencies.

The infrared emission spectrum of the Earth has absorption bands due to the presence of greenhouse gases. These absorption bands cause Ft to be less than unity. The temperature increase due to Ft being less than unity is known as the greenhouse effect. However, this temperature increase is partially offset by a temperature decrease due to bond albedo Fr being greater than zero. The local value of the bond albedo Fr is subject to wide variations. However, the measured average value of Fr for the entire earth, obtained from Astronomical Measurements of the Earth's Reflectance, is stable at about Fr = 0.297.

On November 23, 1996 the following Infrared Thermal Emission Spectrum of the Earth was recorded by the Mars Global Surveyor Spacecraft at a distance from earth of 4,776,000 km. The subearth point was approximately 152 degrees West longitude, 18 degrees North latitude (in the Pacific Ocean near Hawaii). The Earth filled approximately 9.3% of the Thermal Emission Spectrometer field of view.

The collection of this data is described in detail in a paper titled: "Initial data from the Mars Global Surveyor thermal emission spectrometer experiment: Observations of the Earth", JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. E5, PAGES 10,875-10,880, MAY 25, 1997. The Thermal Emission Spectrometer apparatus used to obtain this data is described in detail in a paper titled: Mars Global Surveyor Thermal Emission Spectrometer experiment: Investigation description and surface science results", Journal of Geophysical Research, VOL. 106, NO. E10, Pages 23,823-23,871, OCTOBER 25, 2001.

The following references are applicable to the TES equipment on the Mars Global Surveyor space probe:

What IS TES?

TES Mission Highlights

The First TES Data!

Initial Data from the Mars Global Surveyor thermal emission spectrometer experiment:Observations of the Earth

Mars Global Surveyor Thermal Emission Spectrometer experiment: Investigation description and surface science results

The Thermal Emission Spectrometer was capable of either 5 cm^-1 or 10 cm^-1 spectral resolution. We are uncertain as to the setting used for collection of this data.

On the above thermal emission graph the red black body curve represents the temperature at the top of the cloud layer over the ocean (270 K). The blue black body curve (215K) represents the temperature high in the upper atmosphere of the Earth.

Note the strong absorption band for carbon dioxide in the region 575 cm^-1 to 775 cm^-1.
Note the ozone absorption band in the region 1000 cm^-1 to 1075 cm^-1.
Note the dominant water absorption bands from 200 to 475 cm^-1 and from 1260 to 1600 cm^-1. Note the water pass band from 1600 to 1650 cm^-1. Note that there may also be methane absorption in the range 1200 cm^-1 to 1400 cm^-1 and nitrous oxide absorption in the range 1200 cm^-1 to 1350 cm^-1.

Note that in order for a spacecraft borne thermal emission spectrometer to see the ground temperature it is necessary that there be no clouds in its field of view. Water vapor has a sufficient broad band absorption that clouds are opaque in this infrared spectal range. The 270 K radiative emission corresponds to the cloud temperature.

Note that the data shows that the high altitude water vapor concentration plays a key role in determining emissivity Ft. This high altitude water vapor concentration will increase or decrease with the average open water surface temperature.

Scaling amplitude data off an enlarged version of this plot with a transparent plastic rule having 1 mm divisions at 50 cm^-1 wavenumber intervals gives:
215 K
Column Totals:123.585163.47

E = estimate

The peak in the 270 K reference curve is at wavenumber = 525 cm^-1 where the scaled value is 9.540. At this peak:
Z = 2.8214
F(Z) = 1.4214
Thus the 270 K reference curve values are given by: Value(Z) = (9.540 / 1.4214) F(Z)
= 6.71169 F(Z)
This equation was used to calculate the 270 K reference curve values.

For this calculation:
dZ = (h dF) / (K T)
d(wave number) = d(F / C) = 50 cm^-1

dZ = [(h C) / (K T)] 50 cm^-1
= [(6.6256 X 10^-34 J-s X 2.99792 X 10^8 m / s) / (1.38054 X 10^-23 J / K X 270 K)] 50 cm^-1 X 100 cm / m
= 0.266442

The emissivity Ft of the whole earth on November 23, 1996 is given by:
Ft = Integral from Z = 0 to Z = infinity of:
{Pr(W) / Pro(W)} {15 / Pi^4}{Z^3 dZ / [exp(Z) - 1]}

This integral is evaluated using a BASIC programme to find:
Ft = 0.755492.

Temperature T = 270 K from the emission spectrum plot.

A quantity of interest is the difference in temperature between the cloud temperature Ta and the corresponding temperature Te that would pertain if there was no atmospheric absorption of infrared radiation. Recall that:
P = Cb Te^4 = Ft Cb Ta^4
Te = Ta (Ft)^0.25
Ta - Te = Ta(1 - (Ft)^0.25)
= 270 K (1 - (.755492)^0.25)
= 270 K (1 - .9323038281)
= 18.28 K
This temperature difference is the atmospheric warming due to the greenhouse effect as of November 23, 1996.

An issue that should not be forgotten is that normally greenhouse warming is largely offset by cooling due to the Earth's bond albedo (solar energy reflection from the atmosphere and the Earth's surface). Loss of bond albedo due to conversion of ice into water can cause a major increase in total warming.


This web page last updated November 10, 2015

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