Astronomy Question: Est. Sun's Power & Temp

[Jeann25]

[SOLVED] Astronomy Question

Homework Statement

This is a two part question. I believe the problem I'm running into is coming from the answer to the first question.
Q1: Hold your hand 7.5 cm from a 100 watt light bulb and you will feel the same heat sensation that you feel when your hand is pointed towards the Sun on a nice clear day. Assuming that your hand is a perfect total energy flux sensor and knowing that the sun is 150,000,000 km away, estimate the Sun's energy generation rate in watts.

Q2: Knowing that the Sun has an angular diameter of 0.5 degrees in the sky, calculate the surface temperature that is required for it to radiate the power that you determined in question #1. For this problem assume that the Sun is a perfect black body that it radiates power per unit area from all of it's surface at a rate of σT^4 (where σ is the Stephan Boltzman constant.

Homework Equations

The Attempt at a Solution

For the first question, i figured the watts of the sun by setting up the ratio 100 watts/7.5 x 10^-5 km = x watts/1.5 x 10^8 km
solving for x = 2.0 x 10^14 (which is way less than the sun's actually luminosity, why I'm figuring this is incorrect)

So for the second question i used the angular diameter, and the distance from the Earth to the Sun, to find the diameter of the sun tan(0.5/2)2(1.496 x 10^11)=1.306 x 10^9 m
Then use this to find the surface area using 4πr^2, and the answer I got was 5.35 x 10^18.
From the problem the equation i derived was Power/area = σT^4, solving for T, T = (Power/area*σ)^(1/4).
((2.0 x 10^14)/(5.35 x 10^18)(5.67 x 10^-8))^(1/4)=5.1 K
Should be over 5000K.

Please help because I can't see my error in what I'm doing :(

 

[Jeann25]

I figured this out. Please disregard.

Thank you :)

 

[Blueshift5]

Hello,

Thank you for your question. It seems like you have made a few errors in your calculations.

For the first question, the ratio you set up is incorrect. The correct ratio should be: 100 watts/ (7.5 cm)^2 = x watts/ (150,000,000 km)^2. This is because we are comparing the power per unit area of the light bulb and the sun, not just their distances. Solving for x will give you the correct value for the sun's energy generation rate.

For the second question, your method of finding the diameter of the sun is incorrect. You need to use the angular diameter in radians, not degrees. So the correct calculation would be: tan(0.5 degrees/2 radians) = (1.306 x 10^9 m)/ (1.496 x 10^11 m). Solving for the diameter of the sun will give you the correct value of 1.392 x 10^9 m.

Using this diameter, you can then calculate the surface area of the sun and use the equation you derived to find the surface temperature. Make sure to use the correct values and units for the constants.

I hope this helps. Let me know if you have any further questions. Keep up the good work!