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Fig. 7.23 The runaway greenhouse effect. This graph shows the relationship between the solar flux received at the top of the atmosphere (relative to the flux currently received by the Earth: So) and the surface temperature Ts. When Ts exceeds about 550 K, the infrared flux emitted by the planet reaches a threshold. This threshold corresponds to an orbital distance of 0.84 AU from the present-day Sun, and also corresponds to the inner boundary of the habitable zone. Beyond this threshold, any increase, even small, in the irradiation leads to a runaway greenhouse effect, which evaporates the whole reserves of water and takes the surface to temperatures above 1500 K. The flux received by present-day Venus is indicated, as is the flux 4 x 109 years ago (early Venus), the flux that corresponds to rapid escape of H, and the temperature/flux zone where liquid water can exist at the surface. The connection between the flux and distance from the Sun is given on the right-hand vertical axis. Note the break in the temperature axis between 600 and 1400 K (After Kasting, 1988)

be taken into account, for at least two reasons. The first is the escape of this reservoir to space, which may be represented as a relationship between orbital distance and the lifetime of the reservoir of water. The second relates to the fact that water's critical point corresponds to a temperature of 647 K and a pressure of 220 bar. In the case of the Earth, vaporizing the whole of the oceans would produce a pressure of 270 bar, above the critical point Tc: a temperature above Tc is therefore necessary to vaporize the whole ocean. A less massive reservoir could, however, be entirely vaporized at lower temperatures. For example, an ocean 100 metres deep would be vaporized at a temperature of 450 K, and therefore at a distance of about 0.9 AU from the present-day Sun (assuming a cloud-free case).

c. The outer limit of the habitable zone

On Earth, the principal greenhouse gas is water vapour. However, because water vapour is in thermodynamic equilibrium with the reservoir of liquid water at the surface, its abundance in the atmosphere, as well as Ts, is determined by distance from the Sun and by the abundance of the other major greenhouse gas, CO2.

In the case of a planet such as the Earth, with continents that emerge from the ocean and active volcanism, the level of CO2 is also dependent on the distance from the Sun, this being because of the carbonate-silicate cycle and is dependence on Ts. In fact, if the level of CO2 is insufficient to keep Ts > 273 K, water freezes and CO2 can no longer precipitate in the form of carbonate, although it is still being emitted by volcanism. It abundance therefore increases until the frozen surface turns back into liquid. This is how our planet has escaped from phases of worldwide glaciation. If the level of CO2 continues to rise, Ts will increase significantly above 0°C, increasing the quantity of water vapour in the atmosphere. Precipitation will become more and more significant, leading to strong erosion of the rocks and thus increasing the rate at which carbonates form. These two negative-feedback processes tend to stabilize the level of CO2, at least over timescales of several million years, such that Ts remains higher than 273 K. This mechanism has been described in detail by Walker et al. (1981), and probably regulated the Earth's climate throughout the course of its history, and stabilized Ts despite the increase in the Sun's luminosity since it was formed (see also Sect. 4.4.3.3).

So, if we were to move the Earth to an orbit farther from the Sun, the carbonate-silicate cycle would lead to a higher level of CO2. At shown in Fig. 7.24, a temperature of 273 K at 1.2 AU from the present-day Sun, for example, corresponds to a CO2 partial pressure of about 50 mb (about 170 times the current level of CO2).

Orbital distance (AU)

Fig. 7.24 Variation in the surface temperature and the partial pressure of H2O and CO2 with orbital distance for the Earth, within the habitable zone. The concept of a habitable planet and zone arose from the theoretical experiment of modifying the orbital distance a of the Earth, It is assumed here that the carbonate-silicate cycle stabilizes the level of CO2. The partial pressure of CO2 thus increases with orbital distance. At 1.3 AU the formation of clouds of water ice takes place in the atmosphere, and the exact relationship between PCO2 and Ts remains to be determined (Forget et Pierrehumbert, 1997). Towards 0.93 AU, water vapour becomes the principal atmospheric component even in the upper atmosphere, which results in the photodissociation of large quantities of water and the loss of hydrogen to space

Orbital distance (AU)

Fig. 7.24 Variation in the surface temperature and the partial pressure of H2O and CO2 with orbital distance for the Earth, within the habitable zone. The concept of a habitable planet and zone arose from the theoretical experiment of modifying the orbital distance a of the Earth, It is assumed here that the carbonate-silicate cycle stabilizes the level of CO2. The partial pressure of CO2 thus increases with orbital distance. At 1.3 AU the formation of clouds of water ice takes place in the atmosphere, and the exact relationship between PCO2 and Ts remains to be determined (Forget et Pierrehumbert, 1997). Towards 0.93 AU, water vapour becomes the principal atmospheric component even in the upper atmosphere, which results in the photodissociation of large quantities of water and the loss of hydrogen to space

Determining the outer limit of the habitable zone therefore consists of determining the distance above which Ts < 273 K, whatever the level of atmospheric CO2.

Beyond an orbital distance of 1.3 AU (from the present-day Sun), clouds of CO2 ice form in the atmosphere and have a complex influence. The increase in the albedo that they produce tends to decrease Ts, but by scattering the thermal emission emitted by the surface, they produce an additional greenhouse effect, which heats up the surface. According to Forget and Pierrehumbert (1997), the heating dominates and CO2 clouds allow habitability to be maintained beyond 1.3 AU, and perhaps as far out as 2.2 AU. It may be noted that Mars (1.5 AU), although not 'habitable' now, lies inside this limit. If the surface of Mars is not currently habitable, it is mainly for two reasons. On the one hand, the carbonate-silicate cycle does not operate because the internal energy flux is too weak to maintain active volcanism. On the other hand, the weak gravity of Mars, and the absence of a magnetic field have allowed a significant portion of its atmospheric components to escape. The low mass of Mars is thus responsible for its current desert state, and we may assume that a planet with a greater mass (at least half that of the Earth) would have remained habitable at Mars' distance. Numerous geological pieces of evidence appear to show that more than 3800 million years ago, liquid water flowed or lay as open water (or both) on the surface of Mars for long periods, implying a warmer climate (see Sect. 4.4.3.3). According to our definition, Mars was then habitable, or quasi-habitable if the surface temperature was less than 273 K and water was liquid only seasonally or daily. Because of the lower luminosity of the Sun at that time, this situation corresponds to a distance of 1.75 AU from the present-day Sun, which seems to demonstrate that habitability may be maintained beyond 1.3 AU. We may therefore consider the value of 1.75 AU (or 0.32 times the solar energy currently received by Earth) as the outer limit of the habitable zone.

d. The continuously habitable zone

The luminosity of a star increases over the course of its existence, progressively pushing the boundaries of the habitable zone outwards. It is for this reason that Venus probably lay in the habitable zone in the past, but is not included in it at present. So we may also define the Continuously Habitable Zone, over a duration t, as the region that remains habitable for a period of time > t. The choice of t is not obvious, particularly if one wants to compare the continuously habitable zones of different types of star having a wide range of lifetimes. Stars twice as massive as the Sun only live for 109 years, while the lifetime of stars half the mass of the Sun is (theoretically) over 8 x 1010 years. Generally, the boundaries of the continuously habitable zone are calculated for durations of 109 to 5 x 109 years. Figure 7.25 shows the limits of the continuously habitable zone, obtained for t = 109 years for different types of star, determined by using the stellar evolution models by Baraffe etal., 1998.

e. Habitable zones around other stars

Knowing the limits of a habitable zone, calculated for the present-day luminosity of the Sun, can we use this to calculate the boundaries of a habitable zone for any star? This is possible for stars of spectral type (and thus temperature) close to that of the Sun. Here it suffices to use the relationship D* = D0 (L*/L0)1/2, where D* is the distance at which the energy flux from a star of luminosity L* is the same as the solar flux received at a distance D0 from the present-day Sun (of luminosity L0). By replacing D0 by the boundaries of the habitable zone for the present-day Sun, we obtain a good approximation of the boundaries for another star, or for the Sun at a different age.

For stars that have an effective temperature (in other words, colour) that is very different from the present Sun, this approximation is not valid. In fact, the albedo of a planet depends noticeably on the spectral distribution of the energy. The Earth's albedo is about 0.3, but this is not an intrinsic value: the albedo would be different if, with the same atmospheric composition, the Earth were subject to radiation from a hotter (type F) or a colder (type K) star. The hotter the star, the greater the contribution from short wavelengths (UV and visible). Conversely, the cooler the star, the more the maximum emissivity in its spectrum moves towards the near infrared. Because Rayleigh scattering of light varies as X-4, the scattering of incident radiation back into space is more effective, and the albedo is greater, for hot stars. By contrast, the atmospheric components on which habitability depends (H2O and CO2) absorb moderately in the visible, but strongly in the near infrared. So a greater fraction of the incident radiation is absorbed by the planet if the star is cooler. This effect has been studied by Kasting et al. (1993), and is included in calculations of the continuously habitable zone in Fig. 7.25.

F stars

G stars

M stars

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Mercury

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Mercury

F stars

G stars

M stars

0.1 1 Orbital distance (AU)

Fig. 7.25 Boundaries of the Continuously Habitable Zone as a function of the mass of the central star. The oblique line indicates the orbital distance at which a planet with the Earth's mass would be trapped in synchronous rotation, and always turn the same side towards the star, in less than 109 years. (This relationship is valid for a circular orbit only.) The insert shows the relative populations of stars in the Galaxy as a function of their mass. [Diagrams after F. Selsis (limits of the habitable zone) and J.-M. Grießmeier (synchronization time)]

0.1 1 Orbital distance (AU)

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