Stefan-Boltzmann law
The Stefan-Boltzmann law, also known as Stefan's law, states that the
total energy radiated per unit surface area of a black body in unit time
(known variously as the black-body irradiance, energy flux density,
radiant flux, or the emissive power), j*, is directly proportional to the
fourth power of the black body's thermodynamic temperature T (also
called absolute temperature):
The irradiance j* has dimensions of power density (energy per time per
square distance), and the SI units of measure are joules per second per
square meter, or equivalently, watts per square meter. The SI unit for
absolute temperature T is the kelvin. ε is the emissivity of the blackbody;
if it is a perfect blackbody, ε = ١.
The constant of proportionality σ, called the Stefan-Boltzmann constant
or Stefan's constant, is non-fundamental in the sense that it derives from
other known constants of nature. The value of the constant is
where k is Boltzmann constant. Thus at ١٠٠ K the energy flux density is
۵٫۶٧ W/m٢, at ١٠٠٠ K ۵۶،٧٠٠ W/m٢, etc.
The Stefan-Boltzmann law is an example of a power law.
The law was discovered experimentally by Jožef Stefan (١٨٣۵-١٨٩٣) in
١٨٧٩ and derived theoretically, using thermodynamics, by Ludwig
Boltzmann (١٨۴۴-١٩٠۶) in ١٨٨۴. Boltzmann treated a certain ideal heat
engine with the light as a working matter instead of the gas. This law is
the only physical law of nature named after a Slovene physicist. The law
is valid only for ideal black objects, the perfect radiators, called black
bodies. Stefan published this law on March ٢٠ in the article Über die
Beziehung zwischen der Wärmestrahlung und der Temperatur (On the
relationship between thermal radiation and temperature) in the Bulletins
from the sessions of the Vienna Academy of Sciences.
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Derivation of the Stefan-Boltzmann law
Integration of intensity derivation
The Stefan-Boltzmann law can be easily derived by integrating the
emitted intensity from the surface of a black body given by Planck's law
of black body radiation over the half-sphere into which it is emitted, and
over all frequencies.
where Ω٠ is the half-sphere into which the radiation is emitted, and
I(ν,T)dν is the amount of energy emitted by a black body at temperature T
per unit surface per unit time per unit solid angle in the frequency range
[ν,ν + dν]. The cosine factor is included because the black body is a
perfect Lambertian radiator. Using dΩ= sin(θ) dθdφ and integrating
yields:
(See appendix for the solution of this integral)
Thermodynamic derivation
The fact that the energy density of the box containing radiation is
proportional to T۴ can be derived using thermodynamics. It follows from
classical electrodynamics that the radiation pressure P is related to the
internal energy density:
The total internal energy of the box containing radiation can thus be
written as:
Inserting this in the fundamental law of thermodynamics
yields the equation:
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We can now use this equation to derive a Maxwell relation. From the
above equation it can be seen that:
and
The symmetry of second derivatives of S w.r.t. P and V then implies:
Because the pressure is proportional to the internal energy density it
depends only on the temperature and not on the volume. In the derivative
on the r.h.s. the temperature is thus a constant. Evaluating the derivatives
gives the differential equation:
This implies that
Examples
Temperature of the Sun
With his law Stefan also determined the temperature of the Sun's surface.
He learned from the data of Charles Soret (١٨۵۴–١٩٠۴) that the energy
flux density from the Sun is ٢٩ times greater than the energy flux density
of a warmed metal lamella. A round lamella was placed at such a distance
from the measuring device that it would be seen at the same angle as the
Sun. Soret estimated the temperature of the lamella to be approximately
١٩٠٠ °C to ٢٠٠٠ °C. Stefan surmised that ⅓ of the energy flux from the
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Sun is absorbed by the Earth's atmosphere, so he took for the correct
Sun's energy flux a value ٣/٢ times greater, namely ٢٩ × ٣/٢ = ۴٣٫۵.
Precise measurements of atmospheric absorption were not made until
١٨٨٨ and ١٩٠۴. The temperature Stefan obtained was a median value of
previous ones, ١٩۵٠ °C and the absolute thermodynamic one ٢٢٠٠ K. As
٢٫۵٧۴ = ۴٣٫۵, it follows from the law that the temperature of the Sun is
٢٫۵٧ times greater than the temperature of a lamella, so Stefan got a value
of ۵۴٣٠ °C or ۵٧٠٠ K (modern value is ۵٧٨٠ K). This was the first
sensible value for the temperature of the Sun. Before this, values ranging
from as low as ١٨٠٠ °C to as high as ١٣،٠٠٠،٠٠٠ °C were claimed. The
lower value of ١٨٠٠ °C was determined by Claude Servais Mathias
Pouillet (١٧٩٠-١٨۶٨) in ١٨٣٨ using the Dulong-Petit law. Pouilett also
took just half the value of the Sun's correct energy flux. Perhaps this
result reminded Stefan that the Dulong-Petit law could break down at
large temperatures.
Temperature of stars
The temperature of stars other than the Sun can be approximated using a
similar means by treating the emitted energy as a black body
radiation.[١][٢] So:
where L is the luminosity, σ is the Stefan-Boltzmann constant, R is the
stellar radius and T is the effective temperature. This same formula can
be used to compute the approximate radius of a main sequence star
relative to the sun:
where
, is the solar radius, and so forth.
With the Stefan-Boltzmann law, astronomers can easily infer the radii of
stars. The law is also met in the thermodynamics of black holes in so
called Hawking radiation.
Temperature of the Earth
Similarly we can calculate the temperature of the Earth TE by equating the
energy received from the Sun and the energy transmitted by the Earth:
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where TS is the temperature of the Sun, rS the radius of the Sun and a٠
astronomical unit, giving ۶°C.
Summarizing: the surface of the Sun is ٢١ times as hot as that of the
Earth, therefore it emits ١٩٠،٠٠٠ times as much energy per square metre.
The distance from the Sun to the Earth is ٢١۵ times the radius of the Sun,
reducing the energy per square metre by a factor ۴۶،٠٠٠. Taking into
account that the cross-section of a sphere is ١/۴ of its surface area, we see
that there is equilibrium (٣۴٢ W per m٢ surface area, ١،٣٧٠ W per m٢
cross-sectional area).
This shows roughly why T ~ ٣٠٠ K is the temperature of our world. The
slightest change of the distance from the Sun might change the average
Earth's temperature.
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