Why Use the Average Earth-Sun Distance in Stefan's Law for Energy Conservation?

[sweetreason]

I am trying to understand an example in my Modern Physics textbook (Example 3.1, page 5 in thishttp://phy240.ahepl.org/Chp3-QT-of-Light-Serway.pdf" or pg 69 using the book numbering)

I don't understand why the average earth-sun distance is being used in the conservation of energy equation instead of the radius of the earth. Isn't the idea that whatever total power is emitted from the sun must equal the total power received at the earth? [I think e_total can be power received, too, right? It just depends on context?] So, to make sure the power at each end of the journey is equal, we multiply the power per unit area (the values we have) by the surface area of each body. But in that case we would want 4pi*(Earth Radius) not 4pi*(Earth-Sun Distance)

I am also a bit worried that no energy is "lost" on the way to the Earth. The book doesn't really talk about that. How do we *know* that energy is conserved in this way?

Thanks!

 

[sweetreason]

To make my second question a bit more precise, how can the total power emitted by the sun equal the total power received at the Earth, since presumably at least half of the power radiated by the Sun goes off in a direction opposite the Earth?

 

[D H]

I don't understand why the average earth-sun distance is being used in the conservation of energy equation instead of the radius of the earth. Isn't the idea that whatever total power is emitted from the sun must equal the total power received at the earth?

Absolutely not! The Earth intercepts a tiny, tiny fraction of the Sun's output. The Earth's cross section to solar radiation is approximately the area of a circle with the radius of the Earth. The Sun's radiation output is pretty much the uniform with respect to direction. By the time the radiation gets to 1 AU it is spread over the surface of a sphere 2 AU in diameter. The fraction of the Sun's output that is received by the Earth is

$$f = \frac{2\pi r_e^2}{4\pi R_e^2}$$

where $r_e$ is the radius of the Earth, 6378 km, and $R_e$ is the radius of the Earth's orbit, 149,598,000 kilometers. The value of this f is about 9×10^-10. Tiny.

The total power emitted by the Sun is equal to the total power crossing this 2 AU diameter sphere.

 

[sweetreason]

Okay, I think I understand this now. Thank you!

 

[paranormal]

Dear student,

Thank you for your question and interest in understanding Stefan's Law applied to the Sun. Let me explain the concept in more detail.

Stefan's Law is a fundamental law in thermodynamics that describes the relationship between the total power emitted by a blackbody (such as the Sun) and its temperature. It states that the total power emitted per unit area by a blackbody is proportional to the fourth power of its temperature. This means that as the temperature of a blackbody increases, the total power emitted also increases significantly.

Now, let's apply this concept to the Sun. The Sun is a massive, hot object with a very high temperature. It emits a large amount of energy in the form of radiation, which travels through space and reaches the Earth. The amount of energy received by the Earth from the Sun is equal to the total power emitted by the Sun. This is the basis of the conservation of energy equation used in Example 3.1 in your textbook.

In this example, the average Earth-Sun distance is used instead of the radius of the Earth because we are considering the total power received by the Earth, not just the power received at a single point on the Earth's surface. The Earth is constantly rotating, so the amount of power received at a single point on its surface will vary. However, the total power received by the entire Earth will remain the same, regardless of its rotation. This is why the average distance is used in the equation.

As for your concern about energy being "lost" on the way to the Earth, it is important to note that energy is not being lost, but rather being spread out over a larger area. The radiation emitted by the Sun spreads out in all directions, and the Earth intercepts only a small fraction of it. This is why we use the surface area of the Earth in the conservation of energy equation. We want to account for the fact that not all of the energy emitted by the Sun is received by the Earth.

In summary, the conservation of energy equation used in Example 3.1 is based on the fundamental law of thermodynamics, and it takes into account the distance between the Sun and the Earth, as well as the surface area of the Earth. This equation is a simplified representation of the complex processes that occur in the interaction between the Sun and the Earth, but it provides a good understanding of the relationship between the two bodies.

I hope this explanation helps you understand the concept better. If