Ultraviolet image of Venus’ clouds as seen by the Pioneer

Venus Orbiter, Feb. 26, 1979 (NASA).

Scientists have known that the surface of Venus is extremely hot since the first probes flew by the planet in the 1960s. Venus’ hot surface is presently understood to be a direct result of the composition of the atmosphere – Venus’ atmosphere is nearly 97% carbon dioxide (CO_{2}). CO_{2} is known to be a greenhouse gas, and the same optical properties that make it a greenhouse gas are what’s responsible for Venus’ high surface temperature.

But there are people who reject the idea that CO_{2} could be the cause of greenhouse warming on the Earth. They have come up with a number of interesting hypotheses for how Venus’ surface could be so hot without CO_{2}-induced greenhouse warming.

Over the next few days, I will examine the most common claims about Venus’ surface temperature made by climate disruption deniers and look at whether or not their claims stand up to some basic physical tests. In the process, I’ll show that every one of the alternate hypotheses for Venus’ surface temperature is incorrect. Today I’ll first introduce some scientific basics and key physical concepts and then I’ll follow up with how scientists know Venus’ surface temperature is hotter than it should be if it lacked an atmosphere.

It will be impossible to avoid mathematics entirely, so I’m not even going to try. Instead, I’ll try to explain where the equations come from and why we’re using them as I run through the calculations. Please don’t hesitate to ask questions in the comments – I’ll run updates to the posts in order to clarify things if I need to.

Surface photographs from the Soviet Venera 14 spacecraft. The Venera 14 lander became the second Venus surface

probe to transmit color images after setting down on 5 May 1982. (NASA)

Before we get started, I need to define a whole bunch of concepts and describe how the variables in the equations are written. I’m going to define everything I think is critical up front so we don’t have to do it repeatedly later. Be aware that I’m not an astrophysicist, so I’m probably not using the exact variable names (*J* where it should be *I*, *m* where it should be μ). That’s part of the reason I’m defining everything up front.

**Scientific notation**: Numbers can be represented several different ways, but scientific notation is one of the better ways, especially when you’re dealing with really big or really small numbers. For example, it’s easier to write 3 x 10^{8} than it is to write 300,000,000. Similarly, it’s easier to write 6.669 x 10^{-8} than it is to write 0.00000006669. Only how the number is written is different.

In addition, scientific notation lets us more easily keep track of how accurate the number is, something that’s important in science. 6.669 x 10^{-8} implicitly means that the number is accurate to at least 1 part in 1000. Writing 6.669 x 10^{-8} as 6.67 instead implies that the number is accurate to 1 part in 100. Similarly, 6.7 is considered accurate to 1 part in 10. Scientific notation gives the scientist a quick way to estimate how accurate the calculations are. Note that, to one unit of precision, 6.669 x 10^{-8} is equal to 7 x 10^{-8}, a very inaccurate number indeed.

Finally, most scientists (and this series of posts) tend to use scientific notation only for really large or really small numbers, using normal notation (10, not 10^{1}) for numbers that are easy to write (usually between .001 and 1000, but with some variation from one scientist to another).

**Superscripts and subscripts**: Superscripts are used to represent mathematical operations and reciprocals.

- Example 1: A x A = A
^{2} - Example 2: 1/C = C
^{-1} - Example 3: 10
^{-8}= 1/100,000,000

Subscripts are used to differentiate different variables from each other in order to keep track of related, but different, numbers.

- Example 1: Radius of the sun = r
_{Sun} - Example 2: Radius of Venus = r
_{Venus}

Note that in the text, I’ll italicize variables that I’m describing and defining, like this: “*r _{S}* is the radius of the sun,

*r*is the radius of Venus,” and so on. This is to make it clearer what is a variable and what is just text.

_{V}**Common Constants and Variables**: These posts use many different variables and physical constants repeatedly throughout. Here are some of the more common ones:

- Speed of light in a vacuum:
*c*= 299792458 m·s^{-1}(exact) - Plank’s Constant:
*h*= 6.62606896 × 10^{−34}J·s - Boltzman’s Constant:
*k*= 1.3806504 × 10^{−23}J·K^{-1} - Stefan-Boltzman constant:
*σ*= 6.669 x 10^{-8}W·m^{-2}·K^{-4}) - Universal Gravitation constant:
*G*= 6.674 x 10^{-11}N·m^{2}·kg^{-2} - Avagadro’s number:
*N*= 6.023 x 10_{A}^{23}items·mol^{-1} - Gas constant:
*R*= 8.314 J·mol^{-1}·K^{-1} - Mean radius of the Sun:
*r*= 6.955 x 10_{Sun}^{8}m - Mean radius of Venus:
*r*= 6.052 x 10_{Venus}^{6}m - Mean orbital distance from Venus to the Sun:
*r*= 1.0821 x 10_{Vorbit}^{11}m - Mass of Venus:
*m*= 4.87 x 10_{Venus}^{24}kg - Mass of the Sun:
*m*= 1.99 x 10_{Sun}^{30}kg - Approximate thickness of Venus’ crust:
*l*= 50 km_{crust} - Average surface temperature of the Sun:
*T*= 5778 K_{Sun} - Average surface temperature of Venus:
*T*= 735 K_{Venus} - Estimated temperature of Venus’ core:
*T*= 7000 K_{Vcore} - Specific heat capacity of silica (SiO
_{2}):*κ*= 703 J·kg_{silica}^{-1}·K^{-1} - Thermal conductivity of silica:
*k*= 1.38 W·m_{silica}^{-1}·K^{-1} - Approximate density of silica:
*D*= 2203 kg·m_{silica}^{-3} - Molar mass of silica:
*M*= 2.81 x 10_{silica}^{-2}kg·mol^{-1} - Heat of fusion of silicon:
*H*= 5.021 x 10_{fuse-Si}^{4}J·mol^{-1}

This table will be repeated at the bottom of each post so that you don’t have to come back to this post if you’re looking up what some variable in an equations means.

**Black body**: A black body is an ideal object that absorbs energy perfectly and that radiates energy perfectly into a volume of space according according to Planck’s Law, given by

(Eqn. 1.1)

where *I* = power spectral density of emitted radiation (photons), λ = the wavelength of the radiated photon, *T* = temperature of the radiating object in Kelvin, and the other variables/constants are defined above.

The graph at right shows two curves of interest, namely the curve for the black body emission spectra of the surface of the Sun (~5778 K) and the surface of Venus (~735 K). Note that the two vertical scales are NOT the same (click for a larger version).

When Planck’s Law is integrated over the entire spectrum, the result is known as the Boltzman black body radiation equation:

(Eqn. 1.2)

where *J* = irradiance from a surface in W·m^{-2}, *T* = temperature of the surface, and σ is defined above This equation shows that the total energy radiated from the surface of an object is a function of the fourth power of the temperature of the object, so hot objects radiate a **lot** more than cold objects do. The implications of this will become clearer as we work through the equations below and over the following days.

A black body represents an ideal object and is not physically possible – all objects from a person to a star radiate energy according to their temperature as well as their composition, gravity, pressure, and so on. However, the black body approximation of real objects will be good enough for most of what we need, and it’s always a good starting point. After all, if something doesn’t work in an ideal situation, it’s certainly not going to work in a non-ideal situation.

**Kelvin temperature scale**: In daily life, most people use either the Fahrenheit or Celsius temperature scales. Scientists and engineers tend to use a different scale, known as the Kelvin temperature scale. In Kelvin, all temperatures are above 0 – there is no “-10 °C” in the Kelvin scale, because the coldest temperature possible is 0 K, also known as “absolute zero.”

To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. Thus normal body temperature (98.6 °F, 37 °C) is 310.15 K. Note that there is no such thing as “degrees” Kelvin – Kelvin is the unit itself.

Now that we’re done with all the preliminary stuff, how about a musical interlude to refresh our brains before we start looking at the real science?

The first step of calculating how hot Venus *should* be if it were a black body is to calculate how much solar energy is reaching the orbit of Venus, and to do that we need to know how much total power the Sun emits from its surface. Equation 1.2 tells us how much power is emitted per square meter of area, and given the Sun is a sphere, we can multiply Equation 1.2 times the area of a sphere to get our total power output:

(Eqn 2.1)

where *P _{Sun}* is the power radiated by the Sun and the other variables are described above.

Now we know how much power is radiated by the surface of the Sun, we can calculate what the irradiance of that power is once it reaches the orbit of Venus. The orbit of Venus is essentially a circle, so to get the irradiance (power per square meter, W·m^{-2}) again we need to divide the total power by the area of a circle that with a radius equal to the semi-major axis of Venus’ orbit.

(Eqn 2.2)

Substituting the actual values for the variables and constants in Equation 2.2, we get the irradiance *J _{Vorbit}* = 3071 W·m

^{-2}.

Diagram of key variables of the Sun/Venus system

The third step to calculate how hot Venus should be is to calculate how much of that power incident upon Venus’ surface is actually absorbed. If Venus is a black body, then every photon of light from the Sun is absorbed by the surface. If you picture a perfectly black ball, then you can see that we can make a simplifying assumption, namely that the ball absorbs as if it were a flat circle instead of a sphere. This makes the math much simpler, but at the cost of a little realism. We’ll make this assumption now and come back later to see if it was reasonable.

Treating Venus as a black body that absorbs as if it were a circular area with a radius equal to the radius of the planet, we determine that the total power absorbed by Venus is:

(Eqn 2.3)

where *P _{Vabsorb}* is the power absorbed by the planet Venus and the other values are defined above. The total power absorbed by Venus would thus be 3.533 x 10

^{17}W.

Kirchoff’s Law requires that energy absorbed must equal energy emitted, so if Venus were a black body, then we know how much power Venus would have to radiate; 3.538 x 10^{17} W. However, while Venus absorbs energy as a disk, it re-radiates energy as a sphere just like the Sun does, and according to the same basic equation – power divided by the surface area of the planet:

(Eqn 2.4)

This tells us how much power per square meter is radiated by the planet Venus assuming it is a black body. But we’re interested in knowing what Venus’ temperature would be in the black body assumption, so we want to get the power radiated by Venus in terms of Venus’ own temperature. To do that, we substitute Equation 1.2 into Equation 2.4

(Eqn 2.5)

and from Equation 2.5, we can solve for the black-body temperature of Venus *T _{V}*

(Eqn 2.6)

Note that this equation is slightly different from the normally derived equation which has an albedo term included, but it’s a reasonable starting approximation.

The temperature of a black body Venus should be 327.5 K, or 54.5 °C.

In reality, the surface temperature of Venus is actually about 735 K. Using Eqn. 1.2 again to estimate how much Venus’ surface would be radiating as a black body, we get approximately 1.946 x 10^{4} W·m^{-2}. Dividing this by the calculated black body radiation for Venus from Equation 2.4, we find that Venus appears to be radiating over 25x more power than it’s receiving from the Sun.

Clearly, something isn’t right – Kirchoff’s Law says that a body cannot re-emit more energy than it receives, so something else must be going on. There are only three possibilities.

- The equations I’m using aren’t sufficiently detailed or accurate to draw the kind of conclusions I’m drawing.
- Venus is hot due to internal heating of some sort.
- Venus is hot due to the structure and/or composition of its atmosphere.

The first possibility is easy to deal with. First off, we know that Venus reflects a massive amount of the solar energy that’s incident upon it – you can tell that just by looking at the planet with a telescope and noticing that it’s pretty bright and white. Scientists have measured how much energy Venus reflects and found that about 75% of all solar energy is reflected by Venus’ cloudy atmosphere before that energy could heat up the surface in a black body fashion. If we reduced the amount of energy that actually reaches Venus’ surface by about 75%, then our 25x energy problem in the black body case becomes a 100x energy problem for a real Venus.

Similarly, when you treat Venus as a sphere instead of a disk, you get more reflection off the high latitudes (poles) than off the low latitudes (equator). Basically, the poles essentially behave more like mirrors than the equatorial areas do. However, this also reduces the amount of energy that reaches Venus’ surface, increasing the amount of “missing energy” that we have to account for. Again, if we can’t make the equations work for the perfect black body case, then there’s no way that the equations will work for more accurate models that include albedo and/or treat Venus as a sphere.

This leaves two possibilities – either Venus is hot because something in its core is keeping it that way (radioactive decay, heat of formation/young Venus, or collisional heat) or because something about its atmosphere is preventing the surface from radiating all that heat away.

Tomorrow we look at the internal heating possibilities.

Topographic Map of Venus from Pioneer Venus, Mercator projection (NASA).”

*Environment & Nature, Science*

Randy McCarthy

May 2, 2011

The temperature of the sun’s corona is 100 times hotter than the data presented, don’t really belive in the greenhouse theory, not for venus nor for this planet.

Brian Angliss

May 3, 2011

The sun’s corona is not radiative as compared to the surface of the sun. You can see this obviously every time you observe the sun – unless you’re looking at an eclipse or mask out the solar disk on a telescope, you can’t even see the corona.

Ross McLeod

September 4, 2011

Isn’t the value you quoted for the Stefan-Boltzman constant in error ? I’ve usually seen 5.67 x 10 -8.

Applying this to your calculations reduces 3071 down to 2 640 W/m2 – a figure I’ve seen quoted fairly regularly.

It only serves to make the energy imbalance worse.

And just using the incident energy of say 2640 W/m2 and stefan-boltzman equation couldn’t Venus Temperature be 464 K for 5.67 x 10 -8. Or 482 K if we use your 3071 or 463 if we use 3071 and your quoted constant ?

Brian Angliss

September 6, 2011

Yes, it appears that I used the wrong value and as a result I need to go back through and correct the calculations. The overall conclusion doesn’t change because, as you noted, the imbalance gets worse. But that doesn’t forgive an embarrassing math error.