Douce A.P. Thermodynamics of the Earth and Planets

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618 Topics in atmospheric thermodynamics

tail represents molecules with enough velocity to exceed the escape velocity. The giant

planets can therefore be expected to be rich in the lightest volatiles, hydrogen and helium,

which is indeed the case. By comparison, Titan has a much higher ^ value, so that even

though its temperature, given by its solar distance, is similar to that of Saturn, its atmosphere

is dominated by the much heavier species N

. The value of the parameter for the Galilean

satellites of Jupiter is higher still, consistent with the fact that not even N

molecules have

remained gravitationally bound.

Among the three large terrestrial planets Earth has a significantly lower ^ value than

Venus and Mars. This is part of the explanation for why Earth is so much richer in water

than the other two planets. However, the fact that Europa, Ganymede, Callisto and even the

small icy moons of Saturn are richer in water than Earth suggests that the temperature at

which a volatile species condenses and/or freezes is also an important factor in determining

whether H

O can remain gravitationally bound to the planet.

The apparently simple physical picture suggested by Fig. 13.1 becomes complicated

when we consider it in detail. For instance, the characteristic temperature that determines

whether molecules have enough kinetic energy to escape is not the temperature at the

planet’s surface (which I used in the figure), but rather the temperature at an elevation at

which atmospheric density is low enough that molecular collisions are unlikely, so that a

molecule that is moving faster than the escape velocity will indeed escape before it has a

chance of colliding with another molecule. The elevation where this becomes true, and the

temperature at that elevation, are not as simply defined as the surface temperature (or the

equilibrium temperature, Worked Example 13.1). Also, the escape mechanism on which

equation (13.1) is based, which is known as thermal escape, is not the only way in which a

planet can lose its atmosphere, and perhaps not even the dominant way. Further discussion of

these topics is beyond the scope of this book but clear mathematical descriptions, including

rigorous treatments of thermal and other escape mechanisms, can be found, for example,

in Hunten (1973), Chamberlain and Hunten (1987), Zahnle and Kasting (1986), Zahnle et

al. (1990) and Bohren and Albrecht (1998).

Should we expect the mass of a planet’s atmosphere to be related in some simple fashion

to the parameter ^? Atmospheric pressure at a planet’s surface is given by the weight of the

gas column. Calling the atmospheric mass density per unit of area m

, the mean atmospheric

pressure at the planet’s surface, P

, is given by:

g, (13.2)

where g is the gravitational acceleration at the planet’s surface, assumed to remain constant

throughout the atmospheric thickness. Substituting numerical values of P

and g for the

five solid planetary bodies with substantial atmospheres we find that atmospheric mass

densities vary by six orders of magnitude, as follows:

Venus ∼1.1 ×10

kg m

−2

Titan ∼1.1 ×10

kg m

−2

Earth ∼1.0 ×10

kg m

−2

Mars ∼1.7 ×10

kg m

−2

Triton ∼2.1 ×10

kg m

−2

619 13.1 Gravitational binding of planetary atmospheres

Comparing these values with Fig. 13.1 reveals no correlation between atmospheric mass

and ^. What determines atmospheric mass, then? We begin by considering planetary accre-

tion, the bulk of which must have taken place while the growing solid bodies were immersed

in the solar nebula. Rocky planets are likely to have had primordial atmospheres of solar

composition, i.e. dominated by hydrogen and helium. These primordial atmospheres persist

in the giant planets (which have a rocky core – the image that comes to mind is that of a

dandelion), but they have been lost from the solid planets (think of dandelions after you

blow on them). Loss of primordial atmospheres probably took place by a combination of

gradual processes, such as thermal and non-thermal escape (see Hunten, 1973; Chamber-

lain & Hunten, 1987), and catastrophic processes such as atmospheric blowoff caused by

large impacts (Pepin, 1997). This physical picture is supported by the observed abundances

and isotopic compositions of noble gases in planetary atmospheres (Pepin, 2006). It is

unlikely that whatever atmosphere the proto-Earth may have had would have been capable

of surviving the Moon-forming event.

Present day atmospheres are therefore “secondary”, in the sense that they were acquired

after, and perhaps long after, planetary accretion was substantially completed, and the pri-

mordial atmospheres had been lost. The present day atmospheric masses and compositions

are therefore determined by the relative rates of addition, removal and modification of

individual volatile species since loss of the nebular atmosphere. Buildup of the secondary

atmospheres must have occurred via a combination of late accretion of volatile-rich mate-

rials (e.g. comets) and volcanic outgassing. The rates of these two processes during the

formative stages of the present-day atmospheres, almost certainly more than 4 billion years

ago, are very difficult to pin down with any degree of certainty. Yet those rates are largely

responsible for determining the initial masses and chemical compositions of the secondary

atmospheres. Crucially, the compositions of volcanic gases and cometary volatiles are likely

to have been different, and are poorly constrained.

Removal of volatile species is perhaps easier to constrain. It takes place by three distinct

pathways: escape to space, condensation, and chemical reaction with the planet’s surface

materials. As we saw above, the effectiveness of the first of these pathways depends at least

in part on the planet’s mass (i.e. its gravitational attraction) and distance from the Sun (i.e.

temperature), and may selectively remove light volatile species. For instance, we can expect

that molecular and atomic hydrogen escape planetary atmospheres much more effectively

than, say, molecular nitrogen or carbon dioxide, and that hydrogen loss will be more severe

from Venus and Mars than from Earth (Fig. 13.1). Hydrogen-bearing molecules such as

O and CH

are photodissociated in the upper atmosphere (Section 12.4.2). Escape of the

resulting hydrogen atoms is equivalent to an irreversible loss of water or methane, and an

increase in the oxidation state of the planet’s surface. Replenishment of these species in the

atmosphere by evaporation of liquid or solid reservoirs provides a continuous pathway for

planetary desiccation and oxidation (Chapter 14), which may have gone to near completion

in Venus and perhaps somewhat less in Mars.

Condensation of volatile species removes them from the atmosphere and tends to protect

them from escape. The process is particularly efficient if volatile species freeze, as in

Europa, Ganymede, Callisto and Triton. The surfaces of these bodies can be thought of

as collapsed atmospheres, that have been protected from escape by virtue of the very low

vapor pressures in equilibrium with solid phases. Since the rate of photodissociation is

proportional to concentration (e.g. equation (12.83)), the low vapor pressure of H

Oin

equilibrium with a planetary surface composed of ice hinders hydrogen loss.

620 Topics in atmospheric thermodynamics

Volatile species can also be removed from the atmosphere by reaction with surface mate-

rials. One example is precipitation of carbonates by reaction of atmospheric CO

with

aqueous cations leached from silicate minerals. This process is thought to have scavenged

10–60 bars’ worth of CO

from the terrestrial atmosphere (see for example, Walker, 1985;

Kasting & Ackerman, 1986; Kasting, 1987; Tajika & Matsui, 1993), which would other-

wise have present-day composition, mass, surface pressure and temperature not all that

different from those of Venus (and Gustav Mahler would not have existed, which would

have made the universe a much poorer place). Another example is the effect of Fe

as a

sink for atmospheric O

. Oxidation of Fe

has sequestered oxygen liberated by photodis-

sociation of H

O in the Martian atmosphere over billions of years, generating the strongly

oxidized Martian surface. At least in the case of Earth, present day atmospheric composi-

tion is also the result of modification by biological activity, including photosynthesis and

also unnecessary and irresponsible use of pick-up trucks, minivans, jet-skis, snowmobiles,

street lighting, insanely cold air conditioning, suffocating heating, and other sources of

anthropogenic greenhouse gases.

13.2 Equilibrium thermodynamics in a gravitational ﬁeld

The equations of equilibrium thermodynamics that we have discussed to this point ignore

the existence of a gravitational field. This is generally acceptable locally, for example,

in a laboratory setting or at the scale of heterogeneous phase equilibrium in a planetary

body. However, if we are interested in the equilibrium distribution of species in a planetary

atmosphere, then the effect of the gravitational force must be taken into account. Each of

the terms in the fundamental equation for Gibbs free energy:

dG =−SdT +V dP +



(13.3)

is the product of a pair of conjugate variables, one intensive and the other extensive. In

every case the intensive variable can be thought of as a field, such that gradients in the

field drive displacement of the corresponding extensive quantity. Thus, a temperature gra-

dient generates heat flow, or, equivalently, entropy flow; a pressure gradient causes volume

change, and a gradient in chemical potential drives mass transfer. Another way of stating

these conditions is that dT = 0 implies thermal equilibrium, dP = 0 implies mechanical

equilibrium relative to expansion work, and dµ = 0 implies chemical equilibrium.

If the system is immersed in a non-uniform gravitational field then there is an additional

contribution to its energy, which arises from the work associated with displacement of matter

in the gravitational field. This is mechanical work, but it is distinct from the expansion work

that is encapsulated in the V dP term in equation (13.3). If Φ is the gravitational potential (=

gravitational potential energy per unit mass, equation (1.8)) at a point, and m

the molecular

weight of component i, then the product m

Φdn

is the work associated with an infinitesimal

change in the amount of component i at the point. This contribution must be included in

the Gibbs free energy of a system embedded in a gravitational field, which now becomes:

dG =−SdT +V dP +







. (13.4)

621 13.2 Equilibrium thermodynamics in a gravitational ﬁeld

By analogy with equation (5.24), a system in a gravitational field is in equilibrium relative

to transfer of component i if:





=0 (13.5)

or, as molecular weight is a constant:

dµ

dΦ =0. (13.6)

In the absence of a gravitational field, or if the field can be considered to be uniform,

dΦ = 0 and we recover (5.24). However, if dΦ = 0, which is the general case, and is

in particular true in the neighborhood of planetary bodies, then equilibrium with respect

to mass transfer requires that dµ = 0. Thus, equilibrium distribution of matter in a non-

vanishing gravitational field implies a gradient in chemical potentials. Moreover, because

chemical potential is in general a function of temperature, pressure and composition, it

follows that gradients in at least some of these variables must exist in matter at equilibrium

in a gravitational field.

We can write the total change in chemical potential of component i as follows:

dµ



∂µ

∂T



P ,X

dT +



∂µ

∂P



T ,X

dP +



j=i



∂µ

∂X



P ,T

(13.7)

or, by using (5.37) and (5.38), in terms of partial molar entropy and partial molar volume:

dµ

=−s

dT +v

dP +



j=i



∂µ

∂X



P ,T

. (13.8)

Substituting in (13.6):

−s

dT +v

dP +



j=i



∂µ

∂X



P ,T

dΦ =0, (13.9)

which is the equilibrium condition for each component i of a multicomponent phase

immersed in a gravitational field. Summing over all components we get the equilibrium

condition for the phase:

−SdT +VdP +









j=i



∂µ

∂X



P ,T





+MdΦ = 0, (13.10)

where S and V are the molar entropy and volume of the phase, and M is the molecular weight

of the phase (=weighted average of the molecular weights of the phase components). From

the Gibbs–Duhem equation (6.7) we have:





dµ



P ,T









j=i



∂µ

∂X



P ,T





=0. (13.11)

So (13.10) simplifies to:

−SdT +VdP +MdΦ = 0. (13.12)

622 Topics in atmospheric thermodynamics

These differential equations are general, but they are not necessarily simple to solve. In

particular, the entropy term always introduces significant conceptual and computational

difficulties. Here we restrict ourselves to isothermal processes. At constant temperature,

and noting that density, ρ = M/V, we re-write (13.12) as follows:

dΦ

=−ρ, (13.13)

which is the condition of hydrostatic equilibrium. In order to see this, use equation (1.8)to

calculate dΦ/dr, apply the chain rule to calculate dP/dr, and compare to equation (2.34).

13.2.1 Pressure in a one-component isothermal atmosphere.

Atmospheric scale height

Let us assume that a planetary atmosphere is composed of a single ideal gas species, so

that M is constant with elevation, and that the atmosphere is isothermal (see also Exercise

13.1). From the ideal gas EOS we have:

ρ =

. (13.14)

Substituting in (13.13):

=−

dΦ (13.15)

and integrating:

=exp



−

(

Φ −Φ

)



, (13.16)

where P

, Φ

could be the pressure and gravitational potential at the planet’s surface, for

instance. With this convention, as we move up in the planet’s atmosphere it is Φ>Φ

(recall equation (1.8)), so that P < P

, as expected. Using the approximate expression for

gravitational potential energy that we discussed in Worked Example 1.1, and using h > 0

for elevation above the planet’s surface we get:

P = P

exp



−

Mgh



. (13.17)

Provided that atmospheric temperature does not vary greatly, so that it makes sense to

talk of a “characteristic” temperature for the atmosphere, this equation provides a first

approximation to the variation in atmospheric pressure with elevation. In particular, it leads

to the definition of the scale height of the atmosphere, H :

H ≡

(13.18)

so that:

P = P

exp



−



(13.19)

623 13.2 Equilibrium thermodynamics in a gravitational ﬁeld

Thus, when elevation changes by H atmospheric pressure varies by a factor e. This gives

an indication of how quickly an atmosphere “fades” with elevation: the greater the scale

height, the more the atmosphere extends into space. Substituting appropriate values for the

five planetary bodies with substantial atmospheres in equation (13.18) we calculate scale

heights of: 7.9 km for Earth, 10.7 km for Mars, 13.3 km for Triton, 14.9 km for Venus and

19.8 km for Titan. Of all the bodies with air in the Solar System the Earth is the one that

“holds” its atmosphere closer to the solid surface. At the other end are Venus and Titan,

that have the most “stretched out” atmospheres, although for different reasons (see equation

(13.18)): Venus because of its high temperature, and Titan because of its low gravitational

acceleration. In the case of Triton the low gravitational acceleration is offset by the very

low temperature. Note that the definition of scale height, equation (13.18), is similar to

that of the parameter ^, equation (13.1), except that H is a dimensional quantity, and the

composition of the atmosphere is accounted for by means of the mean molecular weight,

M . The scale height is, however, undefined for an airless body, but ^ is independent of the

existence of an atmosphere.

13.2.2 Compositional stratification in a fluid immersed in a gravitational field

Consider a fluid composed of an arbitrary number of chemical species. The chemical

potential of species i is given by:

=µ

0,i

+RT ln f

(13.20)

from which, at constant temperature:

dµ

=RT d ln f

. (13.21)

Substituting in (13.6):

RT d ln f

dF =0 (13.22)

and integrating:

i,0

exp



−

(

Φ −Φ

)



(13.23)

where f

i,0

is the fugacity at the reference level, not the standard state fugacity. This rela-

tionship is general, and must be satisfied by every species in a multicomponent gas phase

immersed in a gravitational field. Suppose that the atmosphere behaves as an ideal gas. We

can then substitute partial pressure for fugacity, which simplifies things considerably. By

using, as in the previous example, h for elevation relative to the planet’s surface we find:

i,0

exp



−



, (13.24)

where p

is the partial pressure of species i.

If we now consider two species in the gas phase, call them 1 and 2, with differentmolecular

weights we see that the ratio between the partial pressures of the two species varies as a

function of elevation as:

1,0

2,0

exp



(

−m

)



, (13.25)

624 Topics in atmospheric thermodynamics

0 0.02 0.04 0.06 0.08 0.1 0.1

X (CH

)

Elevation over Titan’s surface (km)

0 0.5 1 1.5

Atmospheric pressure (bar)

X(CH

)

Fig. 13.2

Variation with height of atmospheric pressure and methane mol fraction in Titan’s atmosphere, assuming surface

pressure =1.5 bar and constant temperature =90 K. H is the atmospheric scale height (equation (13.18)).

which shows that the composition of the gas phase will be stratified according to the

molecular weights (or densities) of the component species. For instance, if m

> m

, then

the ratio p

increases with elevation. The gas becomes enriched in the lighter component

towards the top – which of course we already knew, but we can now quantify the effect.

Equation (13.25) shows that the extent to which a fluid phase fractionates in a gravitational

field varies directly with the difference in molecular weights of the component species and

with the gravitational acceleration, and inversely with temperature.

Figure 13.2 shows an application to Titan’s atmosphere. We assume that the atmosphere

consists of a binary mixture of CH

and N

, with X(CH

) at the surface equal to 0.05,

a surface pressure of 1.5 bar and a characteristic atmospheric temperature of 90 K. The

partial pressure of each gas as a function of elevation is calculated with (13.24), the

mol fraction of methane is X(CH

) = p(CH

)/(p(CH

) + p(N

)), and the total atmo-

spheric pressure is p(CH

) + p(N

). The latter value is virtually indistinguishable from

the value calculated with (13.19) because, even if the methane/nitrogen ratio changes sub-

stantially with elevation, nitrogen is always the dominant species. The mol fraction of

methane more than doubles between the surface and the upper atmosphere, which is impor-

tant given that methane is destroyed by photochemical reactions in the upper atmosphere

(Section 12.4.2), leading to irreversible hydrogen loss. Since the photochemical reaction

rate is proportional to methane concentration, atmospheric stratification enhances the rate of

hydrogen loss.

625 13.3 Radiative energy transfer

13.3 Radiative energy transfer

Although advection is an important heat transfer mechanism in some atmospheric layers,

solar heating of atmospheres and planetary surfaces, and heat loss of planets to space, take

place by radiation. We therefore devote this section to a discussion of radiative heat transfer,

and apply it to construct a simple quantitative model of greenhouse warming.

13.3.1 Fundamental concepts and equations of thermal radiation

All materials, at any temperature, emit and absorb electromagnetic radiation. The rates

at which a body emits and absorbs electromagnetic energy are not necessarily the same,

however. If they are then there is no net conversion between internal and electromagnetic

energies, but if, say, the rate of emission is higher than the rate of absorption then internal

energy is converted to electromagnetic energy, and conversely if absorption outpaces emis-

sion. At the microscopic level, conversion between internal energy and electromagnetic

radiation corresponds to exchanges between the vibrational, rotational and/or translational

kinetic energy modes of atoms and molecules on one side and energy of photons on the

other.

Here we will assume that all radiant surfaces behave as diffuse emitters, which means

that they emit radiation with the same intensity in all directions. This is not necessarily

true in nature, but this simplification makes it possible to avoid a significant amount of

terminology, algebra and solid geometry, and concentrate on the fundamental physics of

radiative heat transfer. We define the irradiance, F, as the total flux (energy per unit area

per unit time) of electromagnetic radiation over the entire spectrum and traveling in all

directions. The monochromatic irradiance, F

, is the energy flux for radiation of a single

wavelength λ. The two variables are obviously related by:

F =



∞

dλ. (13.26)

It is often necessary to distinguish between emitted and incident energy flux, which we will

do explicitly (special terms exist for these different quantities, but we will not introduce

them here).

Electromagnetic radiation travels unimpeded in vacuum, but it interacts with matter.

Interactions between radiant energy and matter can be described in terms of the absorptivity,

, the transmissivity, Θ

, and the reflectivity, R

, of the medium, all of which are functions

of wavelength. These macroscopic parameters arise from microscopic interactions between

photons and particles of matter (molecules for the range of wavelengths of interest in

planetary processes, see Section 13.3.3). Each of these parameters varies between 0 and 1

and represents the fraction of the irradiance of wavelength λ that is absorbed, transmitted

and reflected, respectively. Their sum is always equal to 1, i.e.:

+Θ

=1 (13.27)

and also:

A +Θ +R =1, (13.28)

626 Topics in atmospheric thermodynamics

where A, Θ and R are the total absorptivity, transmissivity and reflectivity, integrated over

all wavelengths. These equations state energy conservation: the total incident flux of radiant

energy must be accounted for in terms of a fraction that is reflected, a fraction that is absorbed

and a fraction that is transmitted.

The spectrum and intensity of the electromagnetic radiation emitted by a body depends

on its temperature and on another macroscopic parameter called the emissivity, / (/ ≤ 1),

which is a function of temperature and of wavelength. Ablack body is defined as a substance

that emits and absorbs the maximum possible intensity of radiation at all wavelengths and

in all directions. Thus, for a black body A

= /

=1 and Θ

= R

= 0, for all wavelengths

and at all temperatures. The black body monochromatic emission flux, F

∗

, also called the

spectral emissive power, is given by Planck’s radiation law:

∗

2πhc



λk

−1



, (13.29)

where h is Planck’s constant, k

is Boltzmann’s constant and c is the speed of light. This

equation was first proposed semi-empirically by Planck in 1901 and became one of the

foundational pillars of quantum mechanics. Its derivation is beyond the scope of this book

but can be found, for example, in Incropera and DeWitt (1996) or Jones (2000). Equation

(13.29) is a function of two variables, temperature and wavelength. It yields the spectrum

of the electromagnetic radiation emitted by a black body at a constant temperature T, i.e.

the distribution of emitted energy flux as a function of wavelength (Fig. 13.3). The curve is

a “skewed bell”. From the peak, radiation flux falls off steeply towards shorter wavelengths

and more gently towards longer wavelengths. The wavelength at which the maximum flux

occurs can be found by taking the derivative dF

∗

/dλ and equating it to zero (see Exercise

13.3). The result, known as Wien’s displacement law, is:

peak

≈

2898

(13.30)

with T in Kelvin and λ in µm. The peak shifts towards shorter wavelengths with increasing

temperature.

Figure 13.3 shows the regions into which the electromagnetic spectrum is conventionally

subdivided within the interval 10

−2

–10

µm. The Sun radiates (approximately) as a black

body at a temperature of ∼6000 K. Because radiation emitted at this temperature peaks

at ∼0.5 µm it was evolutionarily advantageous for life on Earth to develop acute sensory

organs that respond to wavelengths in this region. For this reason we call the 0.4–07 µm

range visible radiation. It also happens that animal perception senses radiation extending

from the near ultraviolet to the near infrared (roughly, 0.1–10 µm) as heat. Conventionally,

we extend this range to the far infrared (∼100 µm) and call the range 0.1–100 µm thermal

radiation. However, any electromagnetic radiation that has a spectrum given by Planck’s

equation (13.29) is strictly speaking thermal radiation.

The integral of F

∗

over all wavelengths (i.e. the area under the curves in Fig. 13.3)

yields the total black body emission flux, F

∗

, at temperature T and across the full spectrum.

Substituting (13.29)in(13.26) and solving the integral (see Exercise 13.4) we get:

∗



∞

∗

dλ =σT

. (13.31)

627 13.3 Radiative energy transfer

0.01 0.1 1 10 100 1000 1000

–17

–12

–7

(MW m

–2

µm

–1

)

–2

Wavelength (µm)

ultraviolet

infrared microwavev.

thermal radiation

6000 K

278 K

35 K

2.7 K

Fig. 13.3

Blackbody emission power spectrum calculated with Planck’s radiation law (equation (13.29)) for the temperature of

the solar photosphere (∼6000 K), the terrestrial equilibrium temperature (278 K), the temperature at which the

Earth’s internal energy ﬂux would radiate to space if the Sun disappeared (35 K) and the cosmic microwave

background radiation (2.7 K). The narrow wavelength interval labeled “v.” is what we call visible radiation, because

we evolved around a star whose spectrum peaks in this region, making sensory organs that respond to these

wavelengths evolutionarily advantageous.

This is Stefan–Boltzmann’s law (equation (2.1), with / = 1). From the integral it also

follows that the Stefan–Boltzmann constant, σ , is given by (Exercise 13.4):

σ =

−3

−2

. (13.32)

13.3.2 Radiant energy exchange

The geometry sketched in Fig. 13.4 yields some results that are useful in solving problems

of thermal radiation that arise in planetary sciences. We consider a spherical body of radius

and surface area a

, at a uniform temperature T

, concentric with a spherical cavity of

radius r

and surface area a

, at temperature T

. We assume that the intervening space

has transmissivity Θ =1 and we will initially also assume that both the sphere and the cavity

are black bodies, i.e. A

= /

= 1 for all λ. From equation (13.31) and the surface areas