Rayleigh-Jeans law can be used and this leads to a simple relationship between the
brightness temperature and the actual temperature
T b ¼ εT
ð2:22Þ
In the more general case, the emissivity can be thought of as the ratio of the
thermal radiation from a surface to the radiation from an ideal black surface at the
same temperature as given by the Stefan–Boltzmann law. The emitted flux (radiant
exitance), R BB , of a black-body is
R BB ¼ σT
4
ð2:23Þ
in units of [W m
À2 ]. So that
ε ¼
R obs
R BB
ð2:24Þ
where R obs is the observed radiant exitance.
It is here important to note that the brightness temperature is not a constant over
all wavelengths because, in the general case, the emitted power from a surface within
a wavelength interval is proportional to the emission coefficient at that wavelength
and only in the specific case of a black-body is this independent of wavelength and
equal to one. This is a consequence of Kirchhoff’s law which can be loosely defined
as saying that if an arbitrary body in thermodynamic equilibrium, is emitting and
absorbing thermal radiation, the emissivity is equal to the absorptivity. However, the
absorptivity of a surface is not constant with wavelength—objects have colour and
thus the spectral dependence of emissivity is a property of the material and its grain
size as has been shown by laboratory investigations.
The integrated emissivity over all wavelengths is required for the energy balance
calculation and is therefore an integral over wavelength weighted by a Planck
function, B λ (T), at the temperature of the surface such that
ε ¼
R 1
0 B λ T
ð Þ 1 À A H λ
ð Þ
ð
Þdλ
R 1
0 B λ T
ð Þ dλ
:
ð2:25Þ
where the wavelength dependence of emissivity has been replaced by 1-absorptivity
using the hemispherical albedo. We can now see that the spectral radiance from a
real surface is not necessarily distributed according to a Planck function at the
temperature of that surface. It is this property of the brightness temperature concept
that can lead to significant confusion.
A further complexity arises from surface roughness. The effect of surface roughness on the sizes of nuclei measured from Earth-based telescopes is probably small,
particularly if the measurements are made close to zero phase angle. However, the
significance is much greater for resolved surfaces at intermediate phase angles where
32
2 The Nucleus
brightness temperature and the actual temperature
T b ¼ εT
ð2:22Þ
In the more general case, the emissivity can be thought of as the ratio of the
thermal radiation from a surface to the radiation from an ideal black surface at the
same temperature as given by the Stefan–Boltzmann law. The emitted flux (radiant
exitance), R BB , of a black-body is
R BB ¼ σT
4
ð2:23Þ
in units of [W m
À2 ]. So that
ε ¼
R obs
R BB
ð2:24Þ
where R obs is the observed radiant exitance.
It is here important to note that the brightness temperature is not a constant over
all wavelengths because, in the general case, the emitted power from a surface within
a wavelength interval is proportional to the emission coefficient at that wavelength
and only in the specific case of a black-body is this independent of wavelength and
equal to one. This is a consequence of Kirchhoff’s law which can be loosely defined
as saying that if an arbitrary body in thermodynamic equilibrium, is emitting and
absorbing thermal radiation, the emissivity is equal to the absorptivity. However, the
absorptivity of a surface is not constant with wavelength—objects have colour and
thus the spectral dependence of emissivity is a property of the material and its grain
size as has been shown by laboratory investigations.
The integrated emissivity over all wavelengths is required for the energy balance
calculation and is therefore an integral over wavelength weighted by a Planck
function, B λ (T), at the temperature of the surface such that
ε ¼
R 1
0 B λ T
ð Þ 1 À A H λ
ð Þ
ð
Þdλ
R 1
0 B λ T
ð Þ dλ
:
ð2:25Þ
where the wavelength dependence of emissivity has been replaced by 1-absorptivity
using the hemispherical albedo. We can now see that the spectral radiance from a
real surface is not necessarily distributed according to a Planck function at the
temperature of that surface. It is this property of the brightness temperature concept
that can lead to significant confusion.
A further complexity arises from surface roughness. The effect of surface roughness on the sizes of nuclei measured from Earth-based telescopes is probably small,
particularly if the measurements are made close to zero phase angle. However, the
significance is much greater for resolved surfaces at intermediate phase angles where
32
2 The Nucleus