Physics Topics can also be used to explain the behavior of complex systems, such as the stock market or the dynamics of traffic flow.
What is Solar Constant and Temperature of the Sun?
After careful analysis of the results of a few experiments by scientists Dulong, Petit and Tyndall, famous physicist Joseph Stefan formulated the relation connecting the amount of radiation emitted by a body and the temperature of the body. This law is known as Stefan’s law. The law states that, the total heat radiated (E) per second from per unit surface area of a body at an absolute temperature T is directly proportional to the fourth power of the absolute temperature of the body.
i.e., E ∝ T4 or E = σ ᐧ T4
Here σ is called Stefan’s constant, whose experimental value is
σ = 5.672 × 10-5 erg ᐧ cm-2 ᐧ s-1 ᐧ K-4
= 5.672 × 10-8 W ᐧ m-2 ᐧ K-4.
Clearly, for T = 0, E = 0 which means that a body at T = 0 K does not emit thermal radiation.
Later in 1884 Boltzmann theoretically proved the law from thermodynamical considerations and established that the law is valid only for black body radiation. The law is, thus, also known as Stefan-Boltzmann law. It states that the total radiant energy emitted per second per unit surface area of a perfect black body, is directly proportional to the fourth power of its absolute temperature.
i.e., E ∝ T4 or, E = σT4
If for any surface, the relative emittance = ε and the surface area = A, then the heat radiated in time t is,
Q = σεAtT4
The law is sometimes called Stefan’s fourth power law.
It should be noted that the amount of heat radiated by a body due to its temperature can be obtained from Stefan’s law. But the law does not refer to the net loss or gain of heat by radiation due to exchange with its surroundings. It should be modified in that case.
As the black body at absolute temperature T radiates heat, it absorbs heat as well from its surroundings at absolute temperature T0. Hence, by Prevost’s theory of heat exchange, the net rate of heat radiation by a unit surface area of a black body is equal to the difference in the rates of emission and absorption.
i.e., E = εσ( T4 – \(T_0^4\)), where ε is the relative emittance of the surface.
If the body is not a black body, then radiation per second per unit area, E = εσ(T4 – \(T_0^4\)), where ε is the relative emittance of the surface.
Solar Constant and Solar Temperature
Surface temperature of the sun can be determined from the knowledge of the solar constant and Stefan’s law
Solar constant: It is defined as the amount of radiant energy received per minute per unit area by a perfectly black body placed on earth at the mean distance of the earth from the sun in a direction perpendicular to the direction of rays from the sun.
Its value is nearly 1300 J ᐧ m-2 ᐧ s-1, i.e., 1300 W ᐧ m-2.
Determination of solar temperature: The central region of the sun is very hot. This region is called photo-sphere which is surrounded by a comparatively cooler atmosphere called chromosphere. For all external purposes, the temperature of the sun means the temperature of the chromosphere.
Let r be the radius and T K be the temperature of the sun. To determine the solar temperature, the sun is treated as a perfectly black body. According to Stefan’s law, the energy emitted by the sun in t seconds is E = 4πr2σT4t.
Imagine a sphere of radius R drawn around the sun. R is the mean distance of the earth from the sun [Fig.]. The surface area of the sphere = 4πR2.
Therefore, the energy incident normally per unit area of the earth’s surface in time
t = \(\frac{E}{4 \pi R^2}\) = \(\frac{4 \pi r^2 \sigma T^4}{4 \pi R^2}\) = \(\frac{r^2 \sigma T^4 t}{R^2}\)
If t = 60 s then this quantity is nothing but thè solar constant S.
∴ S = \(\frac{r^2 \sigma T^4 \times 60}{R^2}\) or, T4 = \(\left(\frac{R}{r}\right)^2\) × \(\frac{S}{\sigma}\) × \(\frac{1}{60}\)
or, T = \(\left[\left(\frac{R}{r}\right)^2 \times \frac{S}{\sigma} \times \frac{1}{60}\right]^{1 / 4}\) ….. (1)
From the equation (1), the solar temperature can be determined.
Here, S = 0.032 cal ᐧ cm-2 ᐧ s-1 (approx.)
σ = 1.37 × 10-12 cal ᐧ cm-2 . s-1
r = 7 × 1010cm
R = 15 × 1012cm
Putting these values in equation (1) the solar temperature (T) becomes 5722 K (approx.).