Temperature of the Sun and
radiant exitance vs.
black-body temperature. Red arrows show that
5780 K black bodies have 501 nm peak and 63.3 MW/m2 radiant exitance. With his law, Stefan also determined the temperature of the
Sun's surface. He inferred from the data of
Jacques-Louis Soret (1827–1890) that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal
lamella (a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same
angular diameter as the Sun. Soret estimated the temperature of the lamella to be approximately 1900
°C to 2000 °C. Stefan surmised that 1/3 of the energy flux from the Sun is absorbed by the
Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5. Precise measurements of atmospheric
absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.574 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K. This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as were claimed. The lower value of 1800 °C was determined by
Claude Pouillet (1790–1868) in 1838 using the
Dulong–Petit law. Pouillet also took just half the value of the Sun's correct energy flux.
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. So: L = 4 \pi R^2 \sigma T^4 where is the
luminosity, is the Stefan–Boltzmann constant, is the stellar radius and is the
effective temperature. This formula can then be rearranged to calculate the temperature: T = \sqrt[4]{\frac{L}{4 \pi R^2 \sigma}} or alternatively the radius: R = \sqrt{\frac{L}{4 \pi \sigma T^4}} The same formulae can also be simplified to compute the parameters relative to the Sun: \begin{align} \frac{L}{L_\odot} &= \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4 \\[1ex] \frac{T}{T_\odot} &= \left(\frac{L}{L_\odot}\right)^{1/4} \left(\frac{R_\odot}{R}\right)^{1/2} \\[1ex] \frac{R}{R_\odot} &= \left(\frac{T_\odot}{T}\right)^2 \left(\frac{L}{L_\odot}\right)^{1/2} \end{align} where R_\odot is the
solar radius, and so forth. They can also be rewritten in terms of the surface area
A and radiant exitance M^{\circ}: \begin{align} L &= A M^{\circ} \\[1ex] M^{\circ} &= \frac{L}{A} \\[1ex] A &= \frac{L}{M^{\circ}} \end{align} where A = 4 \pi R^2 and M^{\circ} = \sigma T^{4}. 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.
Effective temperature of the Earth Similarly we can calculate the
effective temperature of the Earth
T⊕ by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation (Earth's own production of energy being small enough to be negligible). The
luminosity of the Sun,
L☉, is given by: L_\odot = 4\pi R_\odot^2 \sigma T_\odot^4 At Earth, this energy is passing through a sphere with a radius of
a0, the distance between the Earth and the Sun, and the
irradiance (received power per unit area) is given by E_\oplus = \frac{L_\odot}{4\pi a_0^2} The Earth has a radius of
R⊕, and therefore has a cross-section of \pi R_\oplus^2. The
radiant flux (i.e. solar power) absorbed by the Earth is thus given by: \Phi_\text{abs} = \pi R_\oplus^2 \times E_\oplus Because the Stefan–Boltzmann law uses a fourth power, it has a stabilizing effect on the exchange and the flux emitted by Earth tends to be equal to the flux absorbed, close to the steady state where: \begin{align} 4\pi R_\oplus^2 \sigma T_\oplus^4 &= \pi R_\oplus^2 \times E_\oplus \\ &= \pi R_\oplus^2 \times \frac{4\pi R_\odot^2\sigma T_\odot^4}{4\pi a_0^2} \\ \end{align}
T⊕ can then be found: \begin{align} T_\oplus^4 &= \frac{R_\odot^2 T_\odot^4}{4 a_0^2} \\ T_\oplus &= T_\odot \times \sqrt\frac{R_\odot}{2 a_0} \\ & = 5780 \; {\rm K} \times \sqrt{6.957 \times 10^{8} \; {\rm m} \over 2 \times 1.495\ 978\ 707 \times 10^{11} \; {\rm m} } \\ & \approx 279 \; {\rm K} \end{align} where
T☉ is the temperature of the Sun,
R☉ the radius of the Sun, and
a0 is the distance between the Earth and the Sun. This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere. The Earth has an
albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of
effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving . The above temperature is Earth's as seen from space, not ground temperature but an average over all emitting bodies of Earth from surface to high altitude. Because of the
greenhouse effect, the Earth's actual average surface temperature is about , which is higher than the effective temperature, and even higher than the temperature that a black body would have. In the above discussion, we have assumed that the whole surface of the earth is at one temperature. Another interesting question is to ask what the temperature of a blackbody surface on the earth would be assuming that it reaches equilibrium with the sunlight falling on it. This of course depends on the angle of the sun on the surface and on how much air the sunlight has gone through. When the sun is at the zenith and the surface is horizontal, the irradiance can be as high as 1120 W/m2. The Stefan–Boltzmann law then gives a temperature of T=\left(\frac{1120\text{ W/m}^2}\sigma\right)^{1/4}\approx 375\text{ K} or . (Above the atmosphere, the result is even higher: .) We can think of the earth's surface as "trying" to reach equilibrium temperature during the day, but being cooled by the atmosphere, and "trying" to reach equilibrium with starlight and possibly moonlight at night, but being warmed by the atmosphere. == Origination ==