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a planet's surface temperature, surface temperature

comes the end of our trilogy on the orbital temperature. In the first part

, we calculate the solar energy that Earth receives, and the second part

, which would estimate the temperature of the Earth if the Sun was not

calculate In this third part (last !) the equilibrium temperature. We have already determined that every second the earth receives $ 1.74 \\ times 10 ^ {17} $ J. What does the Earth with this energy? In short, it is heated. And we know that a hot body radiates energy according to the Stefan-Boltzmann (radiated energy increases with the fourth power of the temperature!). Suppose that the Earth has a temperature $ T_ \\ Oplus $ and behaves as a black body (a more reasonable assumption at this level), then the power radiated into space (usually using the letter $ L $ per light) is:

\\ [L_ {emit} = \\ left (4 \\ pi R_ \\ Oplus ^ 2 \\ right) \\ sigma T_ \\ Oplus ^ 4 \\]

where $ \\ sigma $ is the constant Stefan-Boltzmann and $ \\ left (4 \\ pi R_ \\ Oplus ^ 2 \\ right) $ is the Earth's surface (here we consider the total surface of the Earth and not only its cross section as in the case of the energy absorbed .)

The equilibrium occurs when the energy absorbed is equal to the radiated energy. If there is an imbalance, the parameters are adjusted so as to reach a new condition. Suppose for example that for some reason increases the solar luminosity $ L_ \\ odot $. In this case, the energy that Earth receives will be greater and this leads to an increase in temperature, which implies an increase in the radiated energy. This equilibrium condition is therefore stable. Then:

\\ [E_ {abs} = E_ {emit} \\]

remember the expression for $ E_ {abs} $ obtained in the first part

:

\\ [E_ {abs} = \\ frac {L_ \\ odot t} {4} \\ left (\\ frac {R_ \\ Oplus} {R_ {UA} \\ right) ^ 2} \\]

and since brightness is energy per unit time,

\\ [L_ {abs} = \\ frac {L_ \\ odot} {4} \\ left (\\ frac {R_ \\ Oplus} {R_ {UA} \\ right) ^ 2} \\]

Therefore, our equilibrium condition occurs when:

\\ [L_ {abs} = L_ {emit} \\]

\\ [ \\ frac {L_ \\ odot} {4} \\ left (\\ frac {R_ \\ Oplus} {R_ {UA} \\ right) ^ 2} = \\ Left (4 \\ pi R_ \\ Oplus ^ 2 \\ right) \\ sigma T_ \\ Oplus ^ 4 \\]

and then clearing the temperature $ T_ \\ Oplus $ we have:

\\ [T_ \\ Oplus = \\ sqrt [4] {\\ frac {L_ \\ odot} {16 \\ pi \\ sigma} \\ frac {1} {R_ {UA} ^ 2}} \\]

Before numbers forward some conclusions:

The temperature is independent of the size of the planet (note that $ R_ \\ $ Oplus canceled)

increases with the fourth root of the solar luminosity, and decreases with square root distance. The equation is valid for any star-planet system, and is used inter alia to determine the "

habitable zone", defined as the orbital region in which the temperature causes water to remain liquid (between 0 ° C and 100 ° C).
• Now if cracked the mystery:

\\ [T_ \\ Oplus = \\ sqrt [4] {\\ frac {3.85 \\ times 10 ^ {26} \\ mathrm {W}} {(16 \\ pi) (5.67 \\ times 10 ^ {-8} \\ mathrm {W} \\ mathrm {m} ^ {-2} \\ mathrm {K} ^ {-4}) (1.5 \\ times 10 ^ {11} \\ mathrm {m}) ^ 2}} \\]

\\ [T_ \\ Oplus = 278 \\ mathrm {K} = 5 ^ \\ mathrm {o} \\ mathrm {C} \\]

more impressive results, is not it?