89. The Greenhouse Effect on Mars

In my previous three posts I have explained how the Greenhouse Effect (GHE) works on Earth, and how it is affected by changes to the carbon dioxide (CO₂) concentration in the atmosphere. The problem with studying the GHE on Earth, though, is that its operation is complicated by the presence of large amounts of water vapour in the atmosphere. As water vapour also has a broader absorption band and higher atmospheric concentration than CO₂, this means that changes in the CO₂ concentration are less important than they would be otherwise. If we want to understand and measure the GHE just due to carbon dioxide, then we need an environment with high levels of CO₂ but low levels of other greenhouse gases. In this respect one of the best places to study is Mars. 

The Atmosphere of Mars

The Martian atmosphere has some similarities with Earth but many differences. It contains many of the same gases (nitrogen, oxygen, water vapour, argon, CO₂), but the proportions are vastly different. The atmosphere of Mars is 96% CO₂ with about 2% nitrogen and 2% Argon (although different pages on Wikipedia give slightly different values such as 95% CO₂ and 3% nitrogen). There are also trace levels (< 0.1%) of other gases such as water vapour (210 ppm) and oxygen (0.15%). 

The other main difference in terms of the atmosphere is the pressure at the surface. At 610 Pa this is only 0.60% of the surface pressure on Earth, and as about 96% of this is CO₂, this means that there is a surface density of 3610 mol/m² of CO₂ on Mars compared to only 150 mol/m² on Earth. So any outgoing radiation from the surface of Mars has 24.06 times as much carbon dioxide gas to penetrate, in order to escape into outer space, compared to on Earth. What this means in practice is that the Greenhouse Effect in the Martian atmosphere should be easier to analyse because it can only have one source - CO₂.

 

The Energy Balance for Mars

Mars is also approximately 52% further from the Sun than is Earth, so one might expect that to mean that its surface is colder. This is true, but not as much as it should be based on distance alone. 

The solar radiation flux entering the Martian atmosphere is only 586 W/m² compared to the 1360 W/m² that irradiates the Earth. As this radiation is spread over a surface area (4πr²) that is four times greater than the cross-sectional area of the planet (πr²) in each case, this means that the average radiation flux at the Martian surface is 143.5 W/m². Yet this is only 22% less than the 184 W/m² that reaches the Earth's surface. This is because nearly 50% of incident radiation on Earth is either reflected by clouds or is absorbed by ozone and water vapour in the upper atmosphere. But there are no clouds, ozone or significant water vapour on Mars.

Then there is the issue of surface albedo or Bond albedo. This is the proportion of incident radiation that is reflected by the planet back out into space without being absorbed, either from clouds or the surface. For Earth this is about 31%; for Mars it is only 25%. This means that the average surface absorption on Mars is 108 W/m² compared to 161 W/m² on Earth. So while Mars only receives 43% of the solar radiation that Earth does, after absorption and reflection the Martian surface receives 67% of the radiation that the Earth's surface does. That is a relative increase of more than 50% for Mars which partially compensates for its greater distance from the Sun.

Calculating the Surface Temperature of Mars

If we now invoke the Stefan-Boltzmann law (see Eq. 13.1 in Post 13),

I = σT⁴

(89.1)

where I is the surface radiation flux, σ is the Stefan-Boltzmann constant, and T is the surface temperature in kelvins, we see that a surface radiation flux of 108 W/m² equates to a mean surface temperature on Mars of 209 K (or -64°C). Yet the true mean temperature is thought to be about 215 K (or -58°C). The difference is due to the Greenhouse Effect. 

For comparison, on Earth a solar flux at the surface of 161 W/m² would equate to a mean surface temperature of 231 K (or -42°C), yet the true mean temperature is about 289 K (or +16°C). So the GHE on Earth adds 58°C of warming while on Mars it only adds 6°C. Yet there is almost twenty-five times more carbon dioxide on Mars (3690 mol/m²) than on Earth (150 mol/m²), so you would expect the greenhouse effect due to CO₂ to be stronger. But how much stronger? The answer is: not very. In fact it is significantly less than the actual measured value on Earth.

Calculating the Strength of the GHE on Mars

In Post 87 I showed how the width of the CO₂ absorption band at 15 µm can be determined using the known concentration of CO₂ (N_o), its scattering or absorption cross-section (σ_s), the quantized frequency of rotation of the CO₂ molecules (B), and the temperature (T). From this it is possible to estimate the critical CO₂ concentration needed to absorb most of the infra-red radiation (N_th). The term N_th can be estimated to be 0.5 mol/m² based on the value of σ_s, while N_o will be about half the total CO₂ concentration, so N_o = 1845 mol/m². From this we can estimate the maximum number of excited rotational states in the absorption band, J_th, using (see Post 87)

(89.2)

where k is Boltzmann's constant, h is Planck's constant and Z is a normalization constant (see Eq. 87.2 in Post 87) equal to 185.5 in this case. The ratio term kT/hB has the value 185.3 (as hB = 0.1 meV and T = 215 K) and is always approximately equal to the Z value. 

The result we obtain is that J_th = 36.3, which, given that the spacing of the bands is 0.2 meV, means that the absorption band has a width of 14.52 meV and extends from a wavelength of 13.78 µm to 16.44 µm. This compares to a calculated range of 14.00 µm to 16.14 µm for the same band on Earth, although the measured width on Earth is actually found to be from about 13.35 µm to 17.35 µm. So even though the calculated width of the 15 µm absorption band for Mars is slightly larger than the equivalent for Earth, it is not significantly greater. But it is significantly less than the measured band width on Earth.

 

Fig. 89.1: The electromagnetic emission spectrum for the surface of Mars at a mean temperature of 215K (blue curve) together with the absorption profile due to CO₂ between 13.78 µm and 16.44 µm (red curve).

The impact of the high Martian atmospheric CO₂ concentration on the radiation feedback is demonstrated in Fig. 89.1 above. The red curve indicates the proportion of the outgoing infra-red radiation (blue curve) that is reflected by the CO₂ molecules and it amounts to f = 12.5%. This is slightly more than the 10% seen for the GHE due to backscattering from CO₂ on Earth (see Post 87) despite the absorption band being further from the peak in the emission spectrum due to the lower surface temperature on Mars.

Calculating the Temperature Rise

The radiation feedback of  f = 12.5% shown in Fig. 89.1 equates to a 14.3% increase in the surface radiation and also of T⁴ (because of Eq. 89.1), which then equates to a 3.4% increase in T. So if the initial surface temperature on Mars without the GHE was 209 K, the temperature with the GHE included will be 3.4% greater, or 216.1 K. This means that the temperature rise at the surface due to the Greenhouse Effect is expected to be 7.1 K, which is pretty close to the observed value of around 6 K (or 6°C). The reason for the small difference could be the limited availability of accurate Mars temperature data, or the uncertainty in the value of the CO₂ absorption cross-section, σ_s.

We know how much radiation from the Sun is arriving at Mars to high accuracy, but knowing how much is being absorbed by the planet surface is more difficult as this depends on an accurate measurement of the Bond albedo. However, conventional astronomical telescopes should be able to measure the reflected radiation to pretty good accuracy as well. That allows us to estimate the expected mean surface temperature without the GHE to fairly high precision. The problem is knowing what the actual surface temperature is with the GHE in operation. This is a difficult enough calculation to do on Earth where we have over 16,000 weather stations measuring the surface temperature on a daily basis, and numerous satellites in orbit. Sadly, none, or very little, of this exists for Mars.

So far in this blog post I have assumed a value of 215 K for the mean surface temperature of Mars, but some reports have put it as high as 225 K (or as low as 210 K). In which case Z = 194.15 and J_th = 37.0. This leads to a 15 µm band stretching from 13.76 µm to 16.47 µm, and a feedback factor of f = 13.0%. Under these circumstances the warming from the Greenhouse Effect increases slightly, but only to 7.7°C. 

 

A Comparison with Earth

For comparison, it is instructive to hypothecate the extent of warming on Earth if its atmosphere also contained 3690 mol/m² of CO₂. In that case Z = 249.3 and J_th = 41.6, which leads to a 15 µm band stretching from 13.62 µm to 16.67 µm, and a feedback factor of f = 14.1%. The resulting predicted temperature rise due to CO₂ would be 10.74 K, which is 3.26°C less than the 7.48°C rise currently predicted for Earth as was shown in Post 87). So a twenty-five fold increase in the CO₂ concentration would only result in a 3.26°C temperature increase, although as I showed in Post 87, masking by water vapour would probably reduce this by 75% to only 0.8°C. 

It is a point of note that the density of CO₂ molecules on Mars (3690 mol/m²) is more than double the combined density of all the greenhouse gases on Earth (1560 mol/m²), yet it results in a temperature rise of 5°C - 7°C that is almost ten times less than the 58°C observed for Earth. This is mainly because most of the GHE on Earth is due to water vapour as the width of the CO₂ absorption band is so much less than that for water vapour. Even increasing the concentration of CO₂ on Mars by a factor of twenty-five cannot appreciably change this.

Summary and Conclusions

What I hope I have shown in this post is that Mars is a good test bed for studying the Greenhouse Effect (GHE). Knowing only its albedo, the atmospheric concentration of CO₂, and the intensity of radiation arriving from the Sun, it is possible to accurately predict the temperature rise due to the Greenhouse Effect. This I have predicted to be about 7°C, in close agreement with the current estimate based on observational data (6°C). And this is despite the significant uncertainty over the true measured value of the mean surface temperature on Mars.

The reduced GHE on Mars relative to the Earth occurs despite its much higher (i.e. 24 times greater) atmospheric CO₂ concentration. This in turn suggests that the increasing levels of atmospheric CO₂ we are currently seeing on Earth will produce only slight temperature increases in the future. 

Yes, Mars has its own complicating factors. Heat retention on Mars is limited by the thin atmospheric blanket compared to Earth. This means that the planet does not retain heat very well, but conversely it means that the atmosphere will warm quickly when heated by the Sun. For this reason it may be better to consider Mars under direct solar illumination in daytime. Under these conditions the peak solar flux at the surface of Mars will be four times greater than stated above, or 432 W/m². This will equate to a peak surface temperature of 295 K (or +22°C). Yet the actual maximum temperature is reported to be around 303 K to 308 K (or +30°C to +35°C). So on this measure the warming from the Greenhouse Effect in daytime near the equator appears to be in the range 8°C to 13°C. Yet the predicted value based on a calculation of the width of the 15 µm absorption band is found to be 10.9°C, in other words in the mid-range of the observed values. Once again this is still much less than the total warming seen on Earth and comparable to the contribution to Earth's GHE seen just from CO₂.

85. The Greenhouse Effect

In the overall debate over global warming probably the most contentious area for many is the scientific validity of The Greenhouse Effect. Whether on Twitter or on climate sceptic sites like WUWT, there are many commenters who simply refuse to accept it, or fail to understand it. In fact many climate scientists (particularly non-physicists) also do not fully understand it, or misrepresent it. In this post I will outline some of the common misconceptions about it, and then explain how the greenhouse effect actually arises. 

 

Myth 1: Carbon dioxide causes an increased heating of the atmosphere

As the atmosphere can absorb heat that is radiated from the surface of the planet, the claim here is that the presence of greenhouse gases like carbon dioxide (CO₂) increases the amount of heat that the atmosphere can store. This is true(ish), but the amount of additional heat or thermal energy stored by CO₂ is so small relative to the total that it is irrelevant.

The key property here is the molar heat capacity of the gas. Every gas in the atmosphere has a different heat capacity, this being the additional amount of thermal energy stored in that gas per unit increase in temperature. However, they are generally very similar in magnitude (see here), although the molar heat capacity of CO₂ is about 25% greater those of nitrogen (N₂) and oxygen (O₂) due to its additional degrees of freedom in accordance with the equipartition theorem. But carbon dioxide only comprises about 0.042% of all the molecules in the atmosphere, so it can only increase the total heat capacity of the atmosphere by an insignificant 0.01%. But more importantly, even this small increase is irrelevant because it is the temperature of the atmosphere that determines its rate of thermal emission, not the quantity of heat that it stores. 

The amount of heat stored merely determines the rate at which the atmosphere will cool at night. This is why planets with thick atmospheres, like the Earth and Venus, cool less at night than planets with thin atmospheres like Mars. Their thick atmospheres mean that they store a lot more energy at a given temperature, but it is the temperature that determines the rate of energy loss. So two planets at the same temperature will cool at different rates if they have different densities of atmosphere, even though the initial rate of energy loss (as set by the temperature and the Stefan-Boltzmann law) will be the same for each. This is because the planet with the thinner atmosphere will run out of stored energy first.

Myth 2: The greenhouse effect is the result of a hot atmosphere heating the Earth's surface

This myth is related to, and dependent on, Myth 1. If the atmosphere is getting hotter because the CO₂ is trapping and storing heat emitted by the Earth's surface, then the temperature of the atmosphere will increase. Eventually the atmosphere will become hotter than the surface and so it will begin reheating the surface. So the surface temperature will also increase. Except this is not how greenhouse gases work. 

They don't trap heat for long periods, but instead just reflect it back to the surface. In essence they behave like a thermal mirror. The result is that the surface gets reheated by the atmosphere, but not because the atmosphere is hotter than the surface. The lower atmosphere, or troposphere, is never hotter than the surface. It is just that the photons of infra-red radiation emitted by the surface bounce off the CO₂ molecules, and some then get reflected back to the surface and reheat it.

Myth 3: Thermal radiation cannot move from a cold object to a hotter one

One reason many climate sceptics appear to reject the concept of the greenhouse effect is that they feel it violates basic principles of physics, not least the second law of thermodynamics. One of the many versions of this law states that net heat flow is always from a hot object to a cooler one, and not in the reverse direction. Many thus misinterpret this law because they fail to appreciate the importance of the term "net". It is not that heat or thermal energy cannot flow from a cold object to a hotter one: it does. In fact all objects emit (and absorb) thermal radiation irrespective of their temperature; the Stefan-Boltzmann law tells us that (see Post 12). The key point is that hotter objects emit more. In fact the Stefan-Boltzmann law dictates that the amount of radiation emitted is proportional to the fourth power of the thermodynamic temperature T measured in kelvins (see Eq. 12.6 in Post 12).

What the greenhouse effect does is increase the amount of energy that moves in the opposite direction by enabling the atmosphere to reflect back energy emitted by the surface. But the amount reflected back is always less than 100% of that emitted by the surface, so this still means that more energy is moving from the surface up into the atmosphere than is moving in the opposite direction. Thus, the net heat flow is still upwards into the atmosphere, moving from hot to cold. Consequently the second law of thermodynamics still holds.

Myth 4: There is too little carbon dioxide in the atmosphere to make a difference

Currently the concentration of carbon dioxide in the atmosphere is about 420 ppm, or 0.042% of all the gas molecules. This looks like a small number, but there are a lot of molecules in the atmosphere. In fact there are are 0.357 million moles of gas for every square metre of the Earth's surface (1 mole = 6.02 x 10²³ atoms or molecules). So 0.042% of that equates to 150 moles of carbon dioxide per square metre, or 9.04 x 10²⁵ molecules of carbon dioxide per square metre. 

As these molecules are typically 0.33 nm in diameter, this still means that every infra-red photon emitted from the surface of the Earth will expect to collide with at least ten million CO₂ molecules before it can escape into outer space. Sooner or later one of these molecules with absorb it and then re-emit it, and half of these re-emissions will be in a reverse direction towards the Earth's surface. That is why so few infra-red photons can escape. 

The exact number of collisions depends on the scattering cross-section of the molecule at the relevant wavelength of radiation. This cross-section represents the effective area of the molecule that the photon of radiation sees, or alternatively the actual area of the molecule multiplied by the probability of being absorbed at each potential collision. So while the actual cross-sectional area of the carbon dioxide molecule is about 10^-19 m², the effective area as measured by spectroscopy is much less, around 4x10^-24 m². This means that the number of collisions each photon can expect to make with a CO₂ molecule before it can escape into outer space is only about 360 (i.e. 9.04 x 10²⁵ x 4 x 10^-24). 

This, though, is still more than enough to block the emission path of virtually every photon with the necessary wavelength. In fact most will be blocked within 30 m of the surface (the result of dividing the effective thickness of the atmosphere of 10 km by 360). But as Fig. 85.1 below shows, only photons with wavelengths close to the CO₂ absoption bands at 2 µm, 2.7 µm, 4 µm and 15 µm can be absorbed by the carbon dioxide. For the rest the CO₂ molecules will be completely transparent.

Fig. 85.1: The absorption spectrum of different greenhouse gases in the visible and infra-red.

How the greenhouse effect really works

The starting point is electromagnetic radiation from the Sun which heats up the surface of the planet. As the surface warms it also gives off radiation, but because the temperature of the Earth's surface (288 K) is much less than that of the Sun (5778 K), the energy, or frequency, of the radiation emitted is much less than that of the incoming radiation. That means that its wavelength is longer - typically about twenty times greater. So while the incoming radiation from the Sun is mainly in the visible part of the electromagnetic spectrum (see the red curve in Fig. 85.1), the radiation emitted by the Earth's surface is generally in the infra-red (see the blue curve in Fig. 85.1).

The effect of greenhouse gases like carbon dioxide is to reflect back some of the outgoing infra-red radiation. This then gets re-absorbed by the surface and heats it further. This is the greenhouse effect. The key point is that the outgoing radiation is merely being reflected back by collisions with carbon dioxide (or water) molecules in the atmosphere. So these greenhouse gas molecules act like a mirror. In the next post I will outline the mechanism and mathematics of this in more detail. Any additional heating of the atmosphere only comes later as a result of the additional heating of the surface.

The result of this reflection of outgoing radiation is that the amount of radiation hitting the Earth's surface and being absorbed increases. For example, suppose the amount of radiation from the Sun that is being absorbed by the Earth's surface is I_o. As I showed in Post 13 (The Earth's energy budget) this equates to about 161 W/m². With no greenhouse effect in place the outgoing radiation will balance the incoming radiation, and so according to the Stefan-Boltzmann law (see Eq. 12.6 in Post 12) the surface temperature will be 231 K (or -42°C). 

However, if a fraction p of the initial outgoing radiation is reflected back (i.e. pI_o), then the total incoming radiation will be (1 + p)I_o . This will heat up the surface even more and result in a higher emission temperature, which in turn will increase the total intensity of the outgoing radiation I_T. If we then assume that the greenhouse gases reflect back a fraction f of the outgoing radiation, then the following equation must hold. 

(85.1)

This basically states that the total outgoing radiation must balance the incoming radiation plus the fraction of the outgoing radiation that is reflected back I_r = f I_T. Rearranging Eq. 85.1 gives the result

(85.2)

We know that the current mean surface temperature of the Earth is approximately 289 K (or 16°C), so this allows us to calculate I_T using the Stefan-Boltzmann law

I = σT⁴

(85.3)

where T is the temperature in kelvins and σ is the Stefan-Boltzmann constant. The result we obtain is that I_T = 396 W/m², and as I_o = 161W/m², this then implies that f = 0.59. In other words, the greenhouse gases reflect back about 59% of all outgoing infra-red radiation. 

Knowing the values of I_T and I_o also allows us to use Eq. 85.3 to determine the temperature of the Earth's surface both with and without the reflected radiation. These values will be 289 K and 231 K respectively. So the reflected radiation due to the Greenhouse Effect has increased the Earth's temperature by 58 K.