## For Further Reading

Finding appropriate spectroscopic data appropriate to a novel planetary atmosphere can be a real challenge. A wealth of specialized spectroscopic data can be found in the Journal of Quantitatitive Spectroscopy and Radiative Transfer. The reader is directed especially to their 2008 special issue on planetary atmospheres (see (Rothman L 2008, JQSRT doi 10.1016 j.jqsrt.2008.02.002). In addition, laboratory and theoretical spectroscopy related to planetary problems can often be found in the journal...

## Simple models of sea ice in equilibrium

There are many circumstances in which one would like to know the thickness attained by a layer of ice floating on an ocean once it comes into a state of thermal equilibrium. This problem is relevant to the state of the sea ice cover which forms in the polar regions on Earth today and in other icy climates. It is also relevant to the thick-ice regimes prevailing on a globally glaciated Snowball Earth. Another application is the determination of the ice-crust thickness for icy moons such as...

## Po

Where iH2 is the mass path of Hydrogen in the box, in kg m2, and p0 is a standard pressure. The coefficient kH2 has dimension m2 kg and can be used in precisely the same way as the absorption 0 100 200 300 400 500 600 700 800 Wavenumber (cm ') 0 100 200 300 400 500 600 700 800 Wavenumber (cm ') Figure 4.23 The N2 N2 and N2 H2 collision-induced continuum absorption coefficients as a function of temperature (indicated on curves). The coefficients are given at an N2 partial pressure of 100mb....

## N

Sin(5) cos( 5) cos(k) sin(Y) (7.10) In the limit of small obliquity, this equation reduces to 5 y cos K(t). For a circular orbit, K(t) Qt, where is the orbital angular velocity (2n divided by the orbital period). In this special case, the subsolar latitude varies cosinusoidally over the year, with amplitude given by the obliquity. This is actually not a bad approximation even for the roughly 23o current obliquity of Earth and Mars, agreeing with the true value to two decimal places. At the...

## Pleistocene Glacial Interglacial cycles

Now we'll take a closer look at what's been going on in the past five million years. The earlier portion of this time is known as the Pliocene epoch, and the latter portion, beginning around 1.8 million years ago, is the Pleistocene. The choice of the Pliocene-Pleistocene boundary is based on Figure 1.11 Evolution of geography over the Phanerozoic 1.8. The numbers give the time in millions of years ago. Figure 1.11 Evolution of geography over the Phanerozoic 1.8. The numbers give the time in...

## Thermal inertia for a mixedlayer ocean

The concept of thermal inertia is well illustrated by consideration of heat storage in the mixed layer of an ocean. Consider a layer of incompressible fluid with density p and specific heat cp, which is well-mixed by turbulence to a depth H. The assumption of well-mixedness implies that any heating or cooling applied to the surface is distributed instantaneously throughout the depth of the mixed layer, whose temperature thus remains uniform. Let S(t) be the solar flux heating the mixed layer,...

## OLR and backradiation for an optically thin grey atmosphere

The OLR and surface back-radiation for an optically thin grey atmosphere are obtained by integrating Eq. 4.27 over all frequencies. The result is I+ (1 - tto)'+(0) + T aT4 I-,s (1 - tx)I-,x + T aT4 where the mean atmospheric emission temperature is given by The first term in the expression for is the proportion of upward radiation from the ground which escapes without absorption by the intervening atmosphere, while the second is the emission to space added by the atmosphere. In the expression...

## Optical thickness and the Schwarzschild equations

Although the radiation field varies in space only as a function of pressure, p, its intensity depends also on direction. Let I(p, n, v) be the flux density of electromagnetic radiation propagating in direction n, measured at point p. This density is just like the Planck function B(v, T), except that we allow it to depend on direction and position. The technical term for this flux density is spectral irradiance. Now we suppose that the radiation propagates through a thin layer of atmosphere of...

## Digression About climate proxies 171 Overview of proxy data

Instrumental records of climate - that is, records of measurements of temperature and other quantities by scientific instruments - date back at most a few hundred years. The first accurate thermometer was invented in 1654 by Ferdinando II de'Medici, and two hundred years passed before anything like a global network of reliable temperature measuring stations began to become available. Written historical records of such events as frost dates, encounters with sea ice and depictions of mountain...

## OLR AaT p14 AaLR4494

ypV ln (p* J f20gsJKo) 4 v where p* is defined as before and A' is an order unity constant which depends on L R An examination of the g dependence of the calculated Kombayashi-Ingersoll limit in Figure 4.37 shows that over the range 1m s2 < g < 100m s2, the numerically computed dependence can be fit almost exactly with this formula if we take A' .7344 and ko .055 (assuming po 104Pa). Though the pressure dependence of absorption causes the limiting OLR to vary more slowly with g than was...

## Similarity theory for the surface layer

The surface layer theory based on dimensional analysis tells us most of what we need to know, but it doesn't tell us how the drag coefficient depends on the height at which the top-of-layer conditions are applied, nor does it say precisely how the coefficient depends on stratification or the surface roughness. We will now re-do the surface layer theory using a more precise form of the similarity assumption. The most important thing we will get out of this is a quantification of the suppression...

## Spectroscopic properties of selected greenhouse gases

Now we will provide a survey of the infrared absorption properties of a few common greenhouse gases, with particular emphasis on the big 3 that determine much of climate evolution of Earth, Mars and Venus from the distant past through the distant future CO2, H2O and CH4. In each case, the spectral results shown are for the dominant isotopic form of the gas, e.g. 12C16O2 for the case of carbon dioxide. Other isotopic forms (isotopologues) can have significantly different spectra, particularly...

## Goldilocks in space Earth Mars and Venus

Until well into the 1960's, science fiction stories about Venus generally portrayed it as a steamy jungle planet, but one where intrepid explorers could perhaps survive unprotected on the surface. The idea of a jungle and breathable air was of course unfounded speculation, but the general picture of the climate was not wholly without merit. After all, the dense reflective cloud deck of Venus was readily observable - it is what makes Venus so bright as the evening star - and the reflection of...

## [w cpT 22gsrs 828

Note that 2gsrs is the square of the escape velocity from the surface, and when multiplied by rs r it becomes the square of the escape velocity from radius r. Let us defer for the moment the business of making sense of the term SQ dr - which is a bit of a mathematical monstrosity - and explore the adiabatic case, for which SQ 0. In this case, the expression E -w2 + cpT - -2g.srs r (8.29) must be independent of r. E is the energy per unit mass of the fluid, the three terms representing kinetic...

## T Earth m

Given the weak variation of the factors multiplying nh (Earth) in typical cases, a good rule of thumb for use in making crude estimates of escape fluxes is to simply assume the homopause density to be the same as Earth's, though of course where observations are available it is better to use the observed value. For most escape calculations, it is not necessary to know the homopause altitude, though it can be estimated from the lower atmosphere scale height if it is desired. The homopause...

## Pure radiative equilibrium for a grey gas atmosphere

For the temperature profiles discussed in Sections 4.3.2 and 4.3.3, the net infrared radiative heating computed from Eq. 4.14 is nonzero at virtually all altitudes generally the imbalance acts to cool the lower atmosphere and warm the upper atmosphere. In using such solutions to compute OLR and back-radiation, we are presuming that convective heat fluxes will balance the cooling and keep the troposphere in a steady state. The upper atmosphere will continue to heat, and ultimately reach...

## Milankovic cycles on Earth

Earth's precessional cycle is shown in Figure 7.12. The precession angle increases at a nearly constant rate, completing a cycle every 22,000 years. Though the variation in rate is not evident over any one cycle, the rate is not exactly constant, and therefore the phase drifts over the course of hundreds of thousands of years. The precessional cycle is very rapid, and the precession angle has changed markedly even over historical times. Eight thousand years ago, when the first Sumerians poured...

## A homebrew radiation model

We have now laid out all the ingredients that go into a real gas radiation model, and are ready to begin assembling them. The ingredients are A means of computing the band-averaged transmission over a specified wavenumber range The band-averaged integral (Eq. 4.11,4.12, or 4.13) giving the band-averaged solution to the Schwartzschild equation in terms of the preceding transmission functions Divide the spectrum into bands of a suitable width Prepare in advance Malkmus coefficients or exponential...

## Forams to the rescue

As it happens, Nature has provided a handy way of determining the isotopic composition of past ocean waters, via the good works of single-celled shelly amoeba-like organisms known as foramanifera, nicknamed forams (see Fig. 1.6). These creatures build distinctive calcium carbonate (CaCO3) shells which record the state of the water in which they grew. Because the shells have such diverse and unmistakable shapes, it is easy to recognize and select out the species which live at the the depth level...

## Behavior of the bandaveraged transmission function

Although the absorption spectrum has very complex behavior, the band-averaged transmission function averages out most of the complexity. The definition of the transmission guarantees that it decays monotonically as p1 p21 increases and the path increases, but in addition the decay is invariably found to be smooth, proceeding without erratic jumps, kinks or other complex behavior. This smoothness is what makes computationally economical radiative transfer solutions possible, and the various...

## Effects of atmospheric solar absorption

On the present Earth the idealized picture of climate in which all solar absorption occurs at the ground is useful, but even for the present Earth about 20 of solar radiation is absorbed within the atmosphere. For other atmospheres, the proportion absorbed in the atmosphere could be much greater. The effect of this absorption on climate depends very much on the vertical distribution of the absorption, and that is what we will explore here for selected real gases. The two key questions we have...

## Global warming and the surface budget fallacy

A common fallacy in thinking about the effect of doubled CO2 on climate is to assume that the additional greenhouse gas warms the surface by leaving the atmospheric temperature unchanged, but increasing the downward radiation into the surface by making the atmosphere a better infrared emitter. A corrollary of this fallacy would be that increasing CO2 would not increase temperature if the lower atmosphere is already essentially opaque in the infrared, as is nearly the case in the Tropics today,...

## About Extreme Ultraviolet

Extreme ultraviolet (EUV) consists of electromagnetic radiation with wavelengths between 0.12 pm and 0.01 pm. EUV makes up only a tiny part of the spectrum of stars with photospheric temperatures under 10000 K - main sequence stars of spectral class B,A,F,G,K and M. (Recall that our Sun is a class G star). It nonetheless fuels the chemistry and physics of the outer atmosphere. EUV photons have sufficient energy to break up otherwise stable atmospheric compounds, allowing their components to...

## The absorption spectrum of real gases

We will now take a close look at the absorption properties of CO2, in order to introduce some general ideas about the nature of the absorption of infrared radiation by molecules in a gas. Continuing to use CO2 as an example, these ideas will be developed in Sections 4.4.3,4.4.4 and 4.4.6 into a computationally efficient means of calculating infrared fluxes in a real-gas atmosphere. A survey of the spectral characteristics of selected other greenhouse gases will be given in Sections Figure 4.7...

## DTs GTs

By definition, G 0 at an equilibrium point Teq. Suppose that the slope of G is well-defined near Teq - in formal mathematical language, we say that G is continuously differentiable at Teq, meaning that the derivative of G exists and is a continous function for Ts in some neighborhood of Teq. Then, if dG dTs < 0 at Ts, it will also be negative for some finite distance to the right and left of Ts. This is the case for points a and c in the net flux curve sketched in Fig. 3.9. If the temperature...

## Tropopause height for real gas atmospheres

The radiative equilibrium solutions discussed in the preceding section are all unstable near the ground. As convection sets in, it will mix away the unstable layer and replace it by an adiabat the well-mixed region is the troposphere. The change in lower level temperature profile, however, will alter the upward radiation which heats the stratosphere, and therefore cause temperature changes even above the layers reached directly by convection. When all this sorts itself out, how deep is the...

## Pure CO2 atmospheresPresent and Early Mars and Venus

Figure 4.30 shows the OLR as a function of surface pressure for a pure CO2 atmosphere subject to Martian gravity. The results span the range of surface pressure from those similar to the thin atmosphere of present Mars up to the thick atmospheres commonly hypothesized for Early Mars 6. The calculations were carried out for a fixed surface temperature of 270K, since we are 6 There is no strong reason to exclude the possibility of a substantial amount of N2 in the Early Martian atmosphere....

## Erosion by solar wind

Solar wind erosion is a form of nonthermal escape energized by solar wind particles instead of EUV photons. The corona is basically the exosphere of the Sun, and the solar wind is nothing more nor less than hydrodynamic escape of the Solar atmosphere, which is primarily hydrogen ionized to protons. The mechanism is general and applies to virtually all stars, though we will not attempt to discuss here how the stellar wind characteristics vary from star to star, nor the way the stellar wind...

## Nonthermal escape

Using Planck's constant, the energy of an EUV photon with wavelength 0.05 pm is 4 10-18J. This is sufficient to dissociate the components of many molecules, and to knock off electrons from just about anything - a process called ionization which produces charged particles in the outer atmosphere. In fact, ionization is the principle means by which absorbed EUV heats the outer atmosphere, since ejection of an electron increases the kinetic energy of the ion left behind, as well as imparting...

## Diffusion limited escape

The efficiency of escape of material that reaches the exobase is not necessarily the controlling factor determining atmospheric mass loss. For mass to escape from the exobase, it must first be delivered to the exobase, and in many circumstances the rate of transport of mass to the exobase is the limiting factor. When a minor consituent of an atmosphere is escaping, it must first diffuse through the dominant component on its way to the exobase, and even if the escape from the exobase is very...

## H2oso2h2s

Figure 4.11 Some polar triatomic molecules. Two different modes of rotation are indicated for the H2O molecule. There is a third mode of rotation about an axis perpendicular to the page. Figure 4.11 Some polar triatomic molecules. Two different modes of rotation are indicated for the H2O molecule. There is a third mode of rotation about an axis perpendicular to the page. think of the interaction in semiclassical terms. The reason is that the wavelength of infrared is on the order of 10 pm,...

## Walk the line

An individual spectral line is described by a line position (i.e. the wavenumber at the center), a line shape, a line strength (or intensity), and a line width. The line shape is described by a nondimensional function of nondimensional argument, f (x), normalized so that the total area under the curve is unity. The contribution of a single spectral line to the absorption coefficient for substance G can then be written KG(v,p,T ) Sf (v vc) (4.60) where vc is the frequency of the center of the...

## Greenhouse effect of CO2 vs CH4

There is considerable interest in the idea that on the Early Earth methane may have taken over much of the role of CO2 in offsetting the Faint Young Sun. In part this interest is due to rather sketchy geochemical evidence that at some times in the Archaean CO2 concentrations may not have been high enough to do the trick, but regardless of whether the evidence actually demands a relatively low-CO2 atmosphere, possibilities abound that in an anoxic atmosphere methane could build up to high...

## Effect of atmospheric solar absorption on pure radiative equilibrium

Now we will examine how the absorption of solar radiation within an atmosphere affects the temperature structure of the atmosphere in radiative equilibrium. The prime application of this calculation is to understand the thermal structure of stratospheres. Under what circumstances does the temperature of a stratosphere increase with height The effect of solar absorption on gas giant planets like Jupiter is even more crucial. There being no distinct surface to absorb sunlight, all solar driving...

## Basic concepts

The problem of permanently removing a molecule from a planet's atmosphere is much the same as the problem of sending a rocket from Earth to Mars one must impart enough velocity to the object, and in the right direction, to allow the object to overcome the potential energy at the bottom of the gravitational well, and still have enough kinetic energy left over to allow the object to continue moving away. This leads to the central concept of escape velocity, which is the minimum velocity an object...

## Mach Number

Movement of the energy curve through the sonic point means that the gradient of M(r) and w(r) have square-root singularities there - a consequence of the violation of the transonic rule. From a physical standpoint, what is important is not so much the singularity as the fact that solutions cease to exist altogether once r is moved past the sonic point. An examination of Eq. 8.32 shows that the temperature decreases as the sonic point is approached this increases the Mach number by decreasing...

## Water and ice clouds

Now we'll take a closer look at the way Earth's water and water-ice clouds effect the radiation budget, taking account of the balance between the shortwave albedo effect of clouds which act to cool the planet and the longwave cloud greenhouse effects which act to warm the planet. Much of the general behavior in evidence on Earth applies equally well to water clouds on other planets, or for that matter to any cloud-forming substance which is strongly absorbing in the infrared but fairly...

## The past 70 million years

Figure 1.8 shows the paleogeography at the end of the Cretaceous, 65 million years ago. The continent of Antarctica has approached the South Pole, and will continue to drift over the next 40 million years or so until it is more nearly centered on the pole. There is open water at the North Pole in the late Cretaceous, and the open Arctic Ocean continues throughout the subsequent time through the present. The modern continents of North and South America, Eurasia, and Africa are still early in...

## Ppi Rpicppii pp2 Rp2cpp2i215

P(pi)( pi )R(pi) cp(pi)-1( po )R(p2) cP(p2)-1 < p(p2 ) (2.16) po p2 This yields the same criterion as the correct criterion given in Eq. 2.14 only if I cp is constant. The lack of a globally valid potential density complicates the precise analysis of the static stability of inhomogeneous atmospheres. Strictly speaking, one needs to examine potential density profiles for a range of different po covering the atmosphere. In practice, the potential density based on a single po can often provide a...

## Hydrodynamic escape

Hydrodynamic escape is basically a more efficient means of deploying the energy available to the atmosphere in order to assist escape. The energy involved still comes from EUV absorption or the general thermal energy of the atmosphere, but instead of this accumulating in a more or less random set of motions, in some circumstances the energy can sustain a mean outward escaping flow which carries fluid to space without wasting energy on motions directed toward the planet or on a population of...

## Isotopic proxies

The chemical properties of an element are primarily determined by the number of protons in the nucleus (the atomic number, which also determines the configuration of the electron cloud. Nucleii also contain neutrons, and atoms having the same atomic number can appear in forms with different numbers of neutrons. These differing forms are known as isotopes of the element. Isotopic proxies have proved to be a versatile source of information about past climates. Some isotopes are unstable, and...

## The twostream equations with scattering

The two stream approximations to the full scattering equation are derived from Eq. 5.14 and Eq. 5.15 by constraining the angular distribution of the radiation in such a way as to allow all integrals appearing in these equations to be written in terms of either I+ + I_ or I+ I_. In the resulting equations, flux in the upward stream is absorbed, or scattered into the downward stream, at a rate proportional to the upward stream intensity, and similarly for the downward stream. The two-stream...

## Impact erosion

The two main cases in the solar system where theories of atmospheric evolution call for massive atmospheric loss are the problem of water loss on Venus and the loss of a hypothetical dense CO2 atmosphere on Mars. For Titan the question is the converse - accounting for the lack of atmospheric N2 loss, and that is plausibly accounted for by the solar wind shielding provided by Saturn's magnetic field and the low EUV flux at such large distances from the Sun. There may have been blowoff early in...

## Pure radiative equilibrium for real gas atmospheres

Pure radiative equilibrium amounts to an all-stratosphere model of an atmosphere, and is a counterpoint to the all-troposphere models we have been discussing. Real atmospheres sit between the two extremes, sometimes quite near one of the idealizations. In this section we will focus on pure infrared radiative equilibrium. The effects of solar absorption in real gases will be taken up in From simple analytic solutions, we know essentially all there is to know about pure radiative equilibrium for...

## Greenland GISP Ice Core O

Figure 1.14 Oxygen isotope data from the GISP-2 Greenland ice core. Larger (less negative) values correspond to warmer temperatures. Figure 1.14 Oxygen isotope data from the GISP-2 Greenland ice core. Larger (less negative) values correspond to warmer temperatures. of a glacial lake into the ocean, but generally speaking the mechanisms of both the Younger Dryas and of millennial variability remain as Big Questions that are yet to be resolved. This is an especially important question because the...

## Real gas OLR for alltroposphere atmospheres

Calculation of OLR is one of the most fundamental steps in determining a planet's climate. Now that we are equipped with an ability to compute the OLR for real gases, we can revisit some of our old favorite problems - Snowball Earth, the Faint Young Sun, Early Mars, and so forth -but this time relate the results to the actual atmospheric composition. In this section we present results for the all-troposphere model introduced in Section 4.3.2, occasionally limiting the upper air temperature drop...

## Hydrogen and Oxygen isotopes in sea water and marine sediments

We will turn our attention now to the isotopes of hydrogen and oxygen contained in water and in sediments precipitated from the water column. We'll learn what the concentration of these isotopes tells us about the volume of glacier ice and the temperature of various parts of the ocean. Normal water is H2l6O, but other isotopes of hydrogen or oxygen can substitute for the most prevalent isotopes, leading to various forms of heavy water, notably HD16O and H28O. Figure 1.5 Sketch showing how the...

## The Grey Gas Model

We will see in Section 4.4 that for most atmospheric gases k, and hence the optical thickness, has an intricate dependence on wavenumber. This considerably complicates the solution of the radiative transfer equations, since the fluxes must be solved for individually on a very dense grid of wavenumbers, and then the results integrated to yield the net atmospheric heating, which is the quantity of primary interest. The development of shortcuts that can improve on a brute-force integration is an...

## Thermodynamics in a Nutshell

The atmospheres which are our principal objects of study are made of compressible gases. The compressibility has a profound effect on the vertical profile of temperature in these atmospheres. As things progress it will become clear that the vertical temperature variation in turn strongly influences the planet's climate. To deal with these effects it will be necessary to know some thermodynamics - though just a little. This chapter does not purport to be a complete course in thermodynamics. It...

## Distribution of incident solar radiation

The geographical variations of temperature are driven by variations in the amount of sunlight falling on each square meter of surface, and also by variations in albedo. Seasonal variations are driven by changes in the geographical distribution of absorbed sunlight as the planet proceeds through its orbit. Therefore, the starting point for any treatment of seasonal and geographical variation must be the study of how the light of a planet's sun is distributed over the spherical surface of the...

## Upward Flux Wm

Figure 3.7 The Earth's observed zonal-mean OLR for January, 1986. The observations were taken by satellite instruments during the Earth Radiation Budget Experiment (ERBE), and are averaged along latitude circles. The figure also shows the radiation that would be emitted to space by the surface (aTf) if the atmosphere were transparent to infrared radiation. condensed substances absorb infrared as well as water does. Liquid methane (imporant on Titan) and CO2 ice (important on present and early...

## Scattering by particles

Rayleigh theory tells us everything we need to know about scattering from the gas making up an atmosphere, but to deal with cloud and aerosol particles, we need to know about scattering from objects that are not small compared to a wavelength, and indeed could be considerably larger than a wavelength, as is the case for visible light scattering from water or ice clouds on Earth. The answer is provided by Mie theory 2, which is a general solution for scattering of an electromagnetic wave from a...

## A first look at the runaway greenhouse

We have seen in Chapter 2 that the mass of an atmosphere in equilibrium with a reservoir of condensed substance (e.g. a water ocean) is not fixed. It increases with temperature in accordance with the dictates of the Clausius-Clapeyron relation. If the condensible substance is a greenhouse gas, then the optical thickness increases with temperature. This tends to reduce the OLR, offsetting or even reversing the tendency of rising temperature to increase the OLR. What are the implications of this...

## Some elementary orbital mechanics

Sir Isaac Newton showed that the orbit of a single planet revolving about its star takes the form of an ellipse, with a focus of the ellipse at the center of mass of the system. Since stars are typically much more massive than their planets, the center of mass for most purposes is identical to the center of the star. The elliptical nature of orbits has an important effect on the seasonal cycle, since the planet is farther from its sun at some parts of the year than it is at others. This makes...

## Elementary models of radiation balance

Our objective is to understand the factors governing the climate of a planet. In this chapter we will be concerned with energy balance and planetary temperature. Certainly, there is more to climate than temperature, but equally certainly temperature is a major part of what is meant by climate, and greatly affects most of the other processes which come under that heading. From the preceding chapter, we know that the temperature of a chunk of matter provides a measure of its energy content....

## Milankovic cycles on Mars

As expected from general mechanical considerations, Mars has Milankovic cycles analogous to those of Earth. Mars' cycles differ in some key respects, because of the lack of a massive moon, and because of the proximity of Jupiter. As for Earth, the precession angle of Mars increases at a nearly constant rate. However, because Mars does not have a moon as massive as Earth's, the precession is dominated by Solar gravity, and is slower. The Mars precessional cycle has a period of approximately...

## Scattering by molecules Rayleigh scattering

Rayleigh scattering theory is a classical (i.e. non-quantum) electromagnetic scattering theory which began life as a theory for scattering of an electromagnetic plane wave from a small sphere with real index of refraction n. Small in this context means small compared to the wavelength of the light being scattered. The scattering calculation is quite simple in the Rayleigh limit because the incident electric field is nearly constant over the particle, which makes it simple to compute the induced...

## Dr P pgs r812

As in the treatment of the hydrostatic relation in Chapter 2, the equation is closed by using the ideal gas law, p pRT, where R is the gas constant for the mixture making up the uppermost part of the atmosphere. In escape problems it is often more convenient to deal with particle number density rather than mass density. The particle density n(r) is obtained by dividing p by the mass of a molecule, m. If the upper atmosphere is isothermal with temperature To, then the solution expressed as...

## Hothouse and icehouse climates over the Phanerozoic

The Cretaceous hothouse climate and the Pleistocene icehouse climate represent opposite extremes of the Earth's typical climate state of the past half billion years. Going back further in time, the Snowball Earth represents an ultra-extreme on the cold end going further afield in space, the runaway greenhouse represents an ultra-extreme on the hot end, though one that evidently never occurred on Earth. The Earth has experienced many individual hothouse and icehouse episodes in the past half...