The rate at which radiant energy from the sun reaches the...

The rate at which radiant energy from the sun reaches the earth's upper atmosphere is about 1.50 kW/m$^2$. The distance from the earth to the sun is $1.50 \times 10{^1}{^1} m$, and the radius of the sun is $6.96 \times 10{^8} m$. (a) What is the rate of radiation of energy per unit area from the sun's surface? (b) If the sun radiates as an ideal blackbody, what is the temperature of its surface?

Transcript

00:01 Okay, first of all, let me summarize the current problem.

00:04 So we imagine the earth and the sun.

00:06 This is the sun and this is the earth.

00:09 The sun is radiating with some unknown power per unit area, hs.

00:15 And when the sun's radiation reaches the earth, the power per unit area is estimated to be 1 .5 kilowatt per meter squared.

00:27 We know that the distance between the earth and the sun is 1 .5 times 10 to the 11 meters and the radius of the sun is 6 .96 times 10 to 80 meters and we need to find the power per unit area of the sun at the surface of the sun and so in order to do this we need to recall that the power the total power is given by the power per unit area multiplied by the area of the sun so the total power radiated by the sun is equal to the power per unit area at the surface of the sun multiplied by the surface area of the sun.

01:06 The surface area of the sun is equal to 4 pi times the radius of the sun squared.

01:15 So this is the total power that is radiated by the sun and this total power is, can be reasonably assumed to be a constant when it reaches the earth.

01:27 So assuming that there is no power, you know, um, um, being absorbed from the from the from the from the you know on the trajectory between the sun on the trajectory between the sun and the earth then the total power of the sun when it reaches the the the sphere that the earth is rotating on will be the same as the total power at the surface of the sun and so this is equal to p at the position of the earth so so, on the other hand, pe, the total power at the position of the earth is equal to he multiplied by the surface area of this whole sphere right here.

02:13 So this is, in turn, is equal to 4 pi times the distance between the earth and the sun squared.

02:22 And this is equal to ps.

02:24 And this is equal to hs multiplied by 4 pi rs squared.

02:33 And so from here we can rearrange the equation to find that the power, the total power per unit area at the surface of the sun is equal to the power per unit area at the position of the earth multiplied by the distance between the earth squared divided by the radius of the sun.

02:52 And then and so all we have to do now is to plug in these numbers so this is equal to 1 .5 kilowatt per meter square multiplied by 1 .5 times 10 to the 11 divided by 6 .96 times 10 to the 8th square...

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