# Stefan-Boltzman Law Proof with Regards to Solar Radiation

• walterwhite
In summary, the unit area intensity of radiation from the Sun at the photosphere is 6.33*107 W/m2 and can be checked using the Stefan-Boltzmann Law, assuming the Sun is a blackbody emitter with a surface temperature of 5777K. The Stefan-Boltzmann Law involves an integral that can be solved using Planck's Law, but it is a difficult integral that involves zeta functions. However, the Stefan-Boltzmann constant already includes the integral, making it unnecessary to solve the integral.
walterwhite

## Homework Statement

The unit area intensity of radiation from the Sun at the photosphere is 6.33*107 W/m2.

a) Check this value using the calculus of the Stefan-Boltzman Law, assuming the Sun is a blackbody emitter ($\epsilon$ = 1) with a surface temperature of 5777K.

## Homework Equations

Stefan-Boltzmann Law:
$$E_b = \int_0^∞ \! E_{\lambda,b} \, \mathrm{d} \lambda = σT^4$$
Planck's Law:
$$E_{\lambda, b} = {\frac{2\pi hc^2}{\lambda^5[e^{(hc/\lambda kT)}-1]}}$$

where:

$c = 2.998*10^{14} \ \mu m \ s^{-1}$
$h = 6.626 * 10^{-34} \ J \ s$
$k = 1.381 * 10^{-23} \ J/K$

## The Attempt at a Solution

I'm not sure how to solve the integral of the Stefan-Boltzmann Law. I know I can substitute $E_{\lambda, b}$ from Planck's law into the Stefan-Boltzmann law, but I have no idea how to integrate it then. Integration by substitution fails here and I have to prove using calculus, that the sun's $E_b$ is equal to 6.33*107 W/m2. Thanks.

It's not an elementary integral, as you've probably guessed. It involves zeta functions and stuff like that. You can reduce it to a dimensionless integral which contains the hard part. See http://en.wikipedia.org/wiki/Stefan–Boltzmann_law But I'm not sure you need to do that. Doesn't it just ask you to use the Stefan-Boltzann constant? You can look that up. Do they really mean you should derive it from Planck's Law?

Last edited:
Thanks for the reply. It's certainly not. The Stefan-Boltzmann constant comes from the integration, right? No, it doesn't say I need to derive it from Planck's law. However, I don't see how to integrate it without the substitution.

walterwhite said:
Thanks for the reply. It's certainly not. The Stefan-Boltzmann constant comes from the integration, right? No, it doesn't say I need to derive it from Planck's law. However, I don't see how to integrate it without the substitution.

I don't think you need to do any integration. The Stefan-Boltzmann constant already includes the integral. Just put numbers in. It's a hard integral. Involves functions that aren't elementary. Stuff beyond just substitution and integration by parts.

Alright thanks. I'll give it a shot and see if I can come up with anything.

## What is the Stefan-Boltzman Law?

The Stefan-Boltzman Law is a physical law that describes the relationship between the temperature of an object and the amount of thermal radiation it emits. It states that the total energy emitted per unit time by a blackbody is proportional to the fourth power of its absolute temperature.

## How does the Stefan-Boltzman Law relate to solar radiation?

The Stefan-Boltzman Law is used to calculate the amount of solar radiation emitted by the Sun. By knowing the temperature of the Sun's surface, which is approximately 5800 Kelvin, we can use the law to determine the amount of solar radiation reaching the Earth's surface.

## What is the proof for the Stefan-Boltzman Law?

The proof for the Stefan-Boltzman Law is based on theoretical and experimental evidence. It was first derived by Josef Stefan in 1879 and later refined by Ludwig Boltzman in 1884. The law has been tested and confirmed through various experiments and observations, making it a well-established principle in physics.

## How is the Stefan-Boltzman Law used in climate science?

The Stefan-Boltzman Law is an important tool in climate science as it helps scientists understand the Earth's energy balance. By knowing the amount of solar radiation reaching the Earth and the amount of energy emitted by the Earth, they can determine how much of the Sun's energy is trapped in the Earth's atmosphere, affecting global temperatures and climate patterns.

## Are there any limitations to the Stefan-Boltzman Law?

While the Stefan-Boltzman Law is a useful tool for understanding thermal radiation, it is based on the assumption that the object emitting the radiation is a perfect blackbody. In reality, most objects are not perfect blackbodies, so the law may not be accurate in all situations. Additionally, the law does not take into account factors such as the Earth's atmosphere, which can absorb and reflect some of the incoming solar radiation.

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