Temperature given a ratio of power out/power in

[elmers2424]

Homework Statement

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 about 107 K. Given that Jupiter actually emits approximately twice this much energy per second, calculate what the average temperature must actually be.

Homework Equations

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

The Attempt at a Solution

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

 

[ehild]

The emitted energy per unit time is proportional to the forth power of the body. It is the same as the absorbed power in equilibrium, and this power would correspond to 107 K.
P_abs= const*107⁴

Jupiter is not in thermal equilibrium as there are heat productive processes inside it. Because of this processes, it emits twice as much energy as it absorbs. Its emission rate is proportional to the fourth power of the real temperature of its clouds.

P_emission =const*T⁴=2*P_abs.

ehild

 

[kerimek]

I would approach this problem by first understanding the concept of a planet's energy balance. A planet's temperature is determined by the balance between the energy it receives from its star (power in) and the energy it emits into space (power out). In the case of Jupiter, the statement tells us that it emits twice as much energy as it receives from the Sun. This means that the power out is 2 times the power in.

To calculate the actual temperature of Jupiter's cloud tops, we can use the equation T(observed)/T(calculated) = (power out/power in)^1/4. We know that T(calculated) = 107 K from the given statement. Plugging in the values, we have (T(observed)/107)^4 = 2. This gives us T(observed) = 152 K.

Therefore, the average temperature of Jupiter's cloud tops is approximately 152 K, which is higher than the calculated temperature of 107 K due to the extra energy emitted by the planet. This makes sense as Jupiter is a gas giant and its atmosphere is able to trap and retain more heat compared to smaller, rocky planets like Earth.

In conclusion, the key to solving this problem is understanding the energy balance of a planet and using the given information to calculate the actual temperature.