Finding the Temp of the Sun given only 3 variables.

[MarcZero]

Homework Statement

This is for my Optics class:

Given the radius of our planet, its temperature, and the distance to the Sun, estimate the temperature of the Sun (hint: use the Stephan-Boltzmann law).

Homework Equations

Radius of Earth $$\widetilde{=}$$ 6,378.1 km
Temperature of the Earth $$\widetilde{=}$$ 14.4 C or 287.55 K
Distance from Sun to Earth $$\widetilde{=}$$ 150,000,000 km

j* = A$$\epsilon\sigma$$T⁴

(I don't know why it is showing the greek symbols as exponents. I can't seem to get them all to be in the normal line. I apologize about this.)

The Attempt at a Solution

I can use what is given and estimate the surface are of the Earth (A_Earth) as:

4$$\pi$$r² = 5.11 X 10⁸ km²

I am also assuming that the Sun is a perfect blackbody radiation source so that $$\epsilon$$ = 1.

So I am guessing that to find T_Sun I need to find the Irradiance (j*) of the Sun somehow using the radius and temperature of the earth. This question is from the professor and is not in our Optics book anywhere that I can see (Fowles - Introduction to Modern Optics). I am struggling to figure out how to relate the Earth's info to the Sun's Irradiance. I can't seem to find a formula to plug things into. I can imagine that somehow the Energy of the sun falls off as the distance increases and we could probably find somehow that this is related to the temperature of the Earth, but I am at a loss to do so. I know the units of J* are [W/s/m²], but I can't see how I'm supposed to relate these to what I have to work with.

Any insight or a point in the right direction would be most appreciated. Thank you for your time.

 

[Redbelly98]

I think the key, at least to get started, is to realize that the "equivalent area" that receives radiation from the sun is πr², while the radiating area of the Earth is its entire surface area.

(I would have thought we'll need the diameter of the sun too, but I haven't work this one out completely yet.)

Oh, here's a copy-and-pastable σ for you, on the house

 

[kerimek]

I would approach this problem by first acknowledging that there are limitations to using only three variables to estimate the temperature of the Sun. The Stephan-Boltzmann law is a useful tool for calculating the temperature of a blackbody object, but it assumes that the object is in thermal equilibrium and that all of its energy is radiated as electromagnetic radiation. In reality, the Sun is a complex and dynamic system, and there are many other factors that can affect its temperature.

That being said, here is how I would approach this problem:

1. Use the given information to calculate the surface area of the Earth (AEarth). As you correctly pointed out, the formula for surface area of a sphere is 4πr^2.

2. Use the Stephan-Boltzmann law to calculate the irradiance (j*) of the Earth. The formula for this is j* = σT^4, where σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4) and T is the temperature of the Earth in Kelvin.

3. Use the inverse-square law to calculate the irradiance of the Sun at the distance of the Earth. This law states that the intensity of radiation from a point source decreases with the square of the distance from the source. The formula for this is j*Sun = j*Earth x (REarth/RSun)^2, where REarth and RSun are the distances of the Earth and Sun from their common center, respectively.

4. Use the calculated irradiance of the Sun to estimate its temperature using the Stephan-Boltzmann law. Rearranging the formula, we get T = (j*/σ)^1/4.

5. Keep in mind that this is only an estimate and there may be other factors at play that could affect the temperature of the Sun. Also, be sure to use the correct units for all calculations.

I hope this helps and gives you some direction in solving this problem. Remember to always be critical of your results and consider the limitations and uncertainties in your calculations.