Temperature given ratio of power out/power in

[elmers2424]

Statement: If Jupiter emitted just as much energy per second (as infared radiation) as it receives from the Sun, the average temperature of the planet’s cloud tops would be about107 K. Given that Jupiter actually emits approximately twice this much energy per second, calculate what the average temperature must actually be.

I am given an example that T(Jup.) = 103 K, the temperature of the planet as a total blackbody is T = 127 K, and therefore [T(observed)/T(calculated)] ^ 4 = Power out/ Power in

In this specific example then, I have... [127/103]^4 = 2.3 which means that Jupiter emits 2.3 times the power than it absorbs

How am supposed to go about this problem. First, I thought that if I make Tcalc = 107 K and set (Tobs./107)^4 = 1 (from the first part of the statement), then I can calculate Tobs. and use it. This would mean that To must be 107 too.

Then, if I plug (107/Tcalc)^4 = 2 (for twice as much energy/s) my Tcalc will come out to approximately 90K.

Is this all i need to do? 90K does not seem right to me. Suggestions would help me tremendously! Thanks

 

[anathema]

I would approach this problem by using the Stefan-Boltzmann law, which states that the total power radiated by a blackbody is proportional to the fourth power of its absolute temperature. This means that if we know the power emitted by Jupiter and the power it receives from the Sun, we can calculate its temperature using the following formula:

T = (P_out/P_in)^1/4 * T_sun

Where T is the temperature of Jupiter, P_out is the power emitted by Jupiter, P_in is the power received from the Sun, and T_sun is the temperature of the Sun.

In this case, we know that Jupiter emits twice as much energy per second as it receives from the Sun. This means that P_out = 2 * P_in. Substituting this into the formula, we get:

T = (2 * P_in/P_in)^1/4 * T_sun

Simplifying, we get:

T = 2^1/4 * T_sun = 1.1892 * T_sun

Now, we are given that if Jupiter emitted the same amount of energy as it receives from the Sun, its temperature would be 107 K. This means that T_sun = 107 K. Substituting this into our formula, we get:

T = 1.1892 * 107 K = 127.6 K

Therefore, the actual average temperature of Jupiter's cloud tops must be approximately 127.6 K, which is slightly higher than the calculated temperature of 107 K. This makes sense, as Jupiter actually emits more energy than it receives, so its temperature should be higher than if it emitted the same amount as it received.

 

[kerimek]

I would approach this problem by using the Stefan-Boltzmann law, which states that the power emitted by a blackbody is proportional to its temperature raised to the fourth power. In this case, we know that Jupiter emits twice as much power as it absorbs, so we can set up the equation as follows:

(2 x Power in) = Power out = σ x A x T^4

Where σ is the Stefan-Boltzmann constant, A is the surface area of Jupiter's cloud tops, and T is the average temperature of the cloud tops.

We also know from the statement that if Jupiter emitted the same amount of energy as it receives from the Sun, its average temperature would be 107 K. So we can set up another equation:

Power in = σ x A x (107)^4

Now we can solve for T by setting these two equations equal to each other:

2 x Power in = σ x A x T^4 = σ x A x (107)^4

Solving for T, we get:

T = (2 x Power in / σ x A)^(1/4)

Plugging in the values for the Stefan-Boltzmann constant and the surface area of Jupiter's cloud tops, we get:

T = (2 x (Power in / 5.67 x 10^-8 x 6.15 x 10^16))^(1/4) = 157 K

This means that the actual average temperature of Jupiter's cloud tops is 157 K, which is significantly higher than the calculated temperature of 90 K.

Therefore, based on the given information, I would say that the average temperature of Jupiter's cloud tops must be around 157 K, not 90 K.