###### By Joshua Winn, Princeton University

## Mercury, Venus, Earth, and Jupiter make for an interesting comparative study. A quartet of planets, spanning a wide range in orbital distance, size, mass, temperature, and type of atmosphere. And yet, why are these planets so different? Why is Venus so much hotter than the Earth?

### Reaching Radiative Equilibrium

In order to understand what determines the temperature of a planet, let’s first ask, why is the Earth’s average surface temperature 288 Kelvin, and not much higher or lower? Where does the energy come from, which keeps the Earth from cooling off and approaching absolute zero? The answer is simple. It’s from the Sun, of course. The Earth is absorbing sunlight.

Now, if all the Earth did was keep absorbing sunlight it would keep heating up, and eventually melt, and vaporize. That obviously doesn’t happen because the Earth doesn’t retain all that solar energy. It radiates. As the Earth’s temperature rises, it radiates more power, rising as T to the 4th. At some point the radiated power equals the incoming solar power, at which point the Earth stops heating up. It reaches radiative equilibrium, with no net gain or loss of energy.

### Calculating the Equilibrium Temperature

So how does one calculate the temperature a planet that has achieved radiative equilibrium? To begin with, there’s some flux, F_in, of incoming solar radiation. And the planet is a big target with cross-sectional area π r^{2} that intercepts that flux. So, the incident power is F in times π r^{2}. In equilibrium, this must equal the outgoing power. For simplicity, let’s assume the entire surface of the planet is radiating like a blackbody at a single temperature T, so, according the Stefan-Boltzmann law, the flux is sigma T to the 4th and the total radiated power is sigma T to the 4th times the total surface area, 4 π r^{2}. In radiative equilibrium, we set power in equal to power out, and solve for T. That gives the 4th root of F_in over 4 sigma.

Now, F_in is the solar flux at the orbital distance of the planet, which is equal to Lsun over 4 π a^{2}, the luminosity of the Sun spread out over a giant sphere with a radius equal to the orbital distance, a. We can tidy up a little if we also approximate the Sun as a blackbody, and set Lsun equal to sigma T-sun to the 4th power times 4 π Rsun^{2}. That way, we get some cancellations, leading to our final answer: T-planet equals T-sun times the square root of Rsun over 2a.

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### The Effect of the Atmosphere on Venus and Mercury

If we evaluate the numbers for the case of the Earth, we get 280 Kelvin. That’s not far from the true average of 288 Kelvin, an encouraging sign that our calculation captured the essential physics. We can write the result as a scaling relation: T-planet equals 280 Kelvin divided by the square root of a, expressed in AU. That makes it easy to apply to the other planets in our quartet. We’ll add a column to our chart, called equilibrium temperature, with the result of this calculation.

The equilibrium temperature formula works well for Jupiter, too. But the result for Venus is way off! The calculated temperature is more than 400° too cold! And what about Mercury, where there is no single temperature? Clearly, we’re missing something important in the case of these two planets. What we’re missing are the effects of the atmosphere. Venus has a super-thick atmosphere, and Mercury has none.

### Mercury

First let’s examine this in the case of Mercury. When we derived the equilibrium temperature, we equated the power from the Sun to the power radiated by the entire surface of a spherical planet. So, we assumed that, somehow, the surface maintains a constant temperature. For the Earth and for Jupiter that’s not too bad an assumption, because the global circulation of the atmosphere tends to smooth out any temperature differences. But on Mercury, there’s no way for heat to flow quickly around the surface. So, the day-side gets cooked, and the night-side freezes. The approximation of a constant temperature for Mercury is, thus, inappropriate.

Therefore, let’s recalculate the temperature, using the opposite approximation: zero heat transfer. Every square meter of the surface has its own temperature. For simplicity, let’s take the hottest possible temperature; that’ll be at local noon, when the Sun is directly overhead. The incoming solar flux, F_in, is the same as before, but the outgoing flux is the blackbody radiation at the local surface temperature, sigma T to the 4th, that gives T equals F_in over sigma to the 1/4th power. It’s the same as before, except we’re missing a 4 that used to be in the denominator, which means the result will be Tsun times the square root of R over a, without the 2. So, the temperature we calculate under the assumption of local re-radiation is higher by a factor of the square root of 2.

There’s another thing. For Mercury, the semi-major axis, a, is 0.39 AU. But if we want the absolute hottest temperature we should plug in the distance of closest approach to the Sun, a times 1 -e, where e is the orbital eccentricity. With that correction, we get 708 Kelvin, which agrees pretty well with the data.

### Venus

What about Venus? Here there’s plenty of atmosphere, so a constant surface temperature is a reasonable approximation. So, why didn’t our calculation work? It’s because we ignored the possibility that the atmosphere absorbs and emits radiation, too. Not just the surface.

Venus’s atmosphere is full of carbon dioxide molecules. The Sun’s photons, with energies of a few electron volts, sail through the atmosphere as if it weren’t there, and get absorbed by the surface. The surface temperature rises to a few hundred Kelvin, and starts radiating infrared photons, which can’t escape into space. They get absorbed by the atmosphere.

All of this means that in the case of Venus, we need to modify our calculation of the equilibrium temperature.

### Common Questions about What Determines the Temperature of a Planet

**Q: When does the Earth reach radiative equilibrium?**

At some point the radiated power equals the incoming solar power, at which point the Earth stops heating up. It reaches radiative equilibrium, with no net gain or loss of energy.

**Q: Why is the approximation of a constant temperature for Mercury inappropriate?**

On Mercury, there’s no way for heat to flow quickly around the surface. So, the day-side gets cooked, and the night-side freezes. The approximation of a constant temperature for Mercury is, thus, inappropriate.

**Q: What constitutes Venus’s atmosphere?**

Venus’s atmosphere is full of carbon dioxide molecules. The Sun’s photons, with energies of a few electron volts, sail through the atmosphere as if it weren’t there, and get absorbed by the surface.