Question

Calculate the greenhouse-free equilibrium temperature of Jupiter. Assume it has an albedo of 0.3. a) What is the equilibrium temperature assuming there is no internal heat? b) We know that the amount of internal heat is twice as much as the solar irradiation reaching the outer atmosphere of Jupiter (before some fraction being reflected back). Calculate the equilibrium temperature in this scenario. (Hint: use the slides that calculated the energy balance for a planet with solar irradiation). c) What is the peak wavelength of Jupiter's infrared radiation in the latter scenario?

Answer 1

a) for this problem we assume no internal heat produced in Jupiter.

We have albedo a=0.3

To solve this problem,we use the energy balance for a planet with no internal heat. So the total heat irradiated by the planet is solely because of the solar insolation(on presence of an albedo of 0.3)

So we can write an expression as follows,

.................(1)

7P115 (D – Ι)οΙΙ 4πησΤ –

Where, T= surface temperature of Jupiter=?

r=radius of Jupiter(we won't bother with it because it cancels out)

$\sigma$= the Stefan-Boltzmann constant=5.67×10^-8watt/m^{​​​​​​2} k^{​​​​​​4}

Lo=luminosity of sun in watts=3.846×10²⁶ watts

In this expression, the LHS corresponds to the total heat irradiated by the Jupiter (with no internal heat) and the RHS corresponds to the solar irradiance at Jupiter (with albedo).

We can solve the equation (1) for temperature T and get,

::14 = 4 Lo(1-a) 167d20

3.846 x 1020 (1 -0.3) :14 = = 16 * 3.14 x (7.785 x 1011)2 x 5.67 x 10-8

:.14 = 155942028.98K4

::T = 111.74K

So the equilibrium temperature of Jupiter assuming no internal heat is 111.74°K

b)

In this problem,we have been given the information that the internal heat of Jupiter is twice as much as the solar irradiance reaching the outer atmosphere of Jupiter before some of it is reflected back. So for this case we don't consider the albedo. So we can write an expression as follows

.............(2)

47roT^ = 2(ArʻLO T=24902

The LHS of equation (2) corresponds to the total heat irradiated by Jupiter (considering the internal heat) and the RHS corresponds to the double the solar irradiance at Jupiter (without albedo)

So if we solve equation (2) for T we get,

Lo 87do

...1' = 2 3.846 x 1026 8x (7.785 x 1011)2 x 5.67 x 10-8

:14 = 1.39 x 10°K4

::T = 193K

So the equilibrium temperature of Jupiter with presence of internal heat is 193°K

c)

At this temperature Jupiter will emit a peak wavelength that can be given by the Wien's displacement law

2.898 x 10-3 peak = 2.898 x 10-3 -= 1.50 x 10 193 m = 0.15um

So at this temperature Jupiter will emit a peak wavelength of

​​​​​​​

0.15рт