Why Hart Found Narrow Ecospheres—A Minor Science Mystery Solved

Abstract

Aims: To explain why two NASA computer simulation studies in the 1970s (Hart, 1978, 1979) briefly rocked the subfield of astrobiology and SETI studies by showing very narrow habitable zones (HZs) for solar-type stars. Although other studies later supported wider HZs, it was never clear why the Hart simulations found the narrow limits they did. Results: Investigation of the state of climate studies and radiative transfer models in the period 1960–1970 provides a likely explanation. Conclusion: Hart's findings were in line with earlier results, preventing him from noticing that his radiation model was inadequate. Key Words: Atmosphere—Carbon dioxide—Habitable zone—Radiative transfer. Astrobiology 15, 327–330.

Introduction

Strughold (1953) coined the word “ecosphere” (from the Greek oικoς, house) for the distance range near a star that can hold habitable planets. Too close to the star, the planet will be too hot. Too far, and the planet will be too cold. Rampino and Caldeira (1994) called this the “Goldilocks problem.”

Early estimates of the size of the Sun's ecosphere (Strughold, 1953, 1955; Huang, 1959) were vague. It might extend from 0.7 to 1.5 AU. Venus and Mars might both be habitable, at least for some forms of life.

By the early 1960s, it was clear that Venus was too hot and Mars too cold and airless. Dole (1964) found that mean global annual temperatures of 273–303 K were limits for human habitability, then used a static climate model to estimate a 0.86–1.24 AU solar ecosphere (or under special circumstances, 0.725–1.24).

Hart (1978, 1979) observed that, since stars increase in luminosity across the main sequence, the habitable zone (HZ) moves outward with time. The region where things are clement over long periods is the narrower continuously habitable zone (CHZ). A planet too close undergoes a runaway greenhouse effect to wind up a carbon dioxide desert like Venus, even if present Earth, moved instantaneously to that location, would have mild temperatures. A planet too far from Sol undergoes runaway glaciation. Hart came up with the small range of 0.95–1.01 AU for the solar CHZ, later (1979) revised to 0.958–1.004 AU.

Hart's findings alarmed searchers for extraterrestrial life and would-be colonizers of the Galaxy. Note this excerpt from a semipopular work by Asimov (1980):

If Hart's computer simulation of Earth's past history is accurate, then it is very likely that no planet at all will form within the ecosphere…all the planets near the star will be Venuslike or Marslike…The probability of a planet within the ecosphere would then be close to 0.0.

This represents a failure to think quantitatively. It was usually assumed in the 1960s and 1970s that planetary orbits were spaced roughly evenly on a logarithmic scale, as in the Solar System (Dole, 1960, 1970; Isaacman and Sagan, 1977). This assumption is still in common use (cf. Kasting et al., 1993, Kasting, 2010).

With logarithmic spacing between orbits, the difference in spacing between Dole's HZ ecosphere (1.24/0.86) and Hart's CHZ (1.004/0.958), compared to a mean spacing ratio of 1.73 in the Solar System, indicates a decrease in the probability of one planet in the ecosphere from 67% to 8.6%, a factor of 7.8. Compared to Dole's estimate of 600 million habitable planets in the Milky Way galaxy, this would leave 77 million such planets, still a substantial number.

But the effect of Hart's articles was disproportionate. Climate scientists Schneider and Thompson (1981) felt compelled to respond that the scientific knowledge available was too uncertain for such estimates. Schneider (1983, personal communication) felt that Hart's conclusions were “completely unjustified.”

In 1981, Walker et al. noted a stabilizing feedback that prevented runaway glaciation at the outer boundary. A more ice-covered Earth has less weathering of rock, less carbonate gets washed to the sea, and with volcanic activity, CO₂ accumulates in the atmosphere, eventually reversing the glaciation. In a series of papers (Kasting, 1989, 1993; Kasting and Toon, 1989, Kasting et al., 1993), Kasting and others used this information to derive a wider solar CHZ, 0.95–1.15 AU. Kasting defined this as a “4.5 GY” CHZ, referring to Earth's age; a planet in that distance range would have a stable climate for at least that long. His HZ was wider still, with an inner limit as close as 0.85 AU and an outer limit from 1.37 to 1.67 AU depending on various assumptions. A recent revision of this model (Kopparapu et al., 2013) makes the limits 0.99–1.70 AU.

In 1997, Forget and Pierrehumbert showed that high-altitude CO₂ ice clouds could extend the outer HZ limit as far as 1.7 AU, and Mischna et al. (2000) made it 2.4 AU. This, of course, puts glaciated Mars well inside the HZ. But because of the small size of Mars, it cooled off early, plate tectonics never got started, and there are no active volcanoes to add CO₂ to the atmosphere. Thus, Walker's feedback does not exist for it. If Mars were Earth-sized, it might be habitable.

Many more estimates of CHZ (and HZ) size have been made since (see, e.g., Abe et al., 2011; Kasting et al., 2014). The inner boundary was not in much dispute; most modern estimates are still close to Hart's. [One exception is a recent paper by Zsom et al. (2013), which puts the inner limit as far in as 0.38 AU for a planet with low water content, very low relative humidity, and very high surface albedo.] But estimates for the outer boundary have changed greatly, and Hart's were noticeably out of line with the others. It was an advance, as Hart had done, to take account of thermal runaways in a planet's history to find CHZ boundaries. It was another advance to take into account the stabilizing feedback of Walker et al. Nonetheless, a certain amount of mystery continued to surround Hart's findings. As Kasting (2010) put it: “Exactly why [Hart's] model failed to recover from runaway glaciation is not clear. It was a highly simplified model, though, and its treatment of both radiation and convection left much to be desired….”

A look at some details of Hart's model, plus knowledge of the state of climate science and radiative transfer theory in the period 1960–1970, offers important clues.

Hart's Temperature Model

One must keep in mind that available computer power was radically less in 1978 than today. Hart's computer simulation was likely written in the procedural language Fortran IV and either punched into cards to be read in batch mode by a mainframe computer such as an IBM 360/70 or entered via a teletype or CRT-screen text editor for submission to a minicomputer such as the DEC PDP-11. Although radiative-convective models (RCMs) of Earth's atmosphere had existed since 1964, a geohistorical model such as Hart's would have needed far too much computer time to repeatedly run an RCM. His Earth history simulation used a time step of 2.5 million years and an Earth age of 4.5 billion years, requiring 1800 iterations of the main processing loop. A detailed temperature model would have been prohibitively costly.

He therefore substituted an iterative semigray model. The equation used for planetary surface temperature was based on the Milne-Eddington approximation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}T_{ \rm s} = T_{ \rm eff} ( 1 + 0.75 \tau ) ^{0.25} \tag{1}\end{align*} \end{document}

where T _s is surface temperature, T _eff radiative equilibrium temperature (from sunlight and albedo alone), and τ infrared optical thickness. Hart estimated τ for present-day Earth at 2.49, which for T _eff=255 K results in a surface temperature of 332 K. This would be true for a purely radiative situation, but of course conduction, convection, and evapotranspiration act to cool Earth's surface at the expense of the atmosphere. Hart therefore modified the equation to include a “convection factor” F _conv=0.43: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}( \Delta T ) _{ \rm green} = [ ( 1 + 0.75 \tau ) ^{0.25} - 1 ] T_{ \rm eff} F_{ \rm conv} \tag{2}\end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}T_{ \rm s} = T_{ \rm eff} + ( \Delta T ) _{ \rm green} \tag{3}\end{align*} \end{document}

Hart treated water vapor and carbon dioxide as the only greenhouse gases for present-day Earth. He added ammonia and methane for a primitive Earth assumed, in line with theory at the time, to have a reducing atmosphere. For present-day Earth, he found τ_H2O=2.34 and τ_CO2=0.15, for relative contributions of 94% and 6% to Earth's greenhouse effect, respectively.

In fact, RCM studies show the clear-sky greenhouse effect to be about 26% from carbon dioxide (Ramanathan and Coakley, 1978, Kiehl and Trenberth, 1997).

For a Mars-like planet where almost all the greenhouse is due to CO₂ and very little to H₂O, a temperature model that undervalues the CO₂ contribution will clearly be more easily prone to runaway glaciation as sunlight is reduced. The questions therefore arise as to whether Hart knew he was treating CO₂ as too weak a greenhouse gas and if not, why not. The answers may be (1) he did not realize it and (2) he failed to do so because his results were in line with climatological thinking of the time.

Hart's Predecessors

It is important to realize that narrow CHZ boundaries were actually found by researchers prior to Hart but that no one considered both limits together until Hart did. In 1969, Ingersoll used a gray model to explain how the runaway greenhouse effect might have taken place on Venus. A year later, Rasool and De Bergh (1970) estimated the inner boundary of the solar CHZ at 0.93–0.96 AU based on such considerations.

It was also in 1969 that Budyko found an outer CHZ boundary of 1.008 AU, based on a one-dimensional (in latitude) energy-balance climate model. The same year, Sellers, apparently without knowledge of Budyko's work, estimated the outer boundary at 1.01–1.025 AU, also based on an energy-balance model. These models were considered state-of-the-art at the time.

Thus, years before Hart's publications, careful review of the climatology and planetary astronomy literature would already have implied a HZ stretching from 0.93 to 0.96 AU at the inner boundary to 1.008–1.025 at the other, for a spacing ratio of 1.05–1.10. Hart's 1978 paper cites both Budyko's paper and Sellers'. His simulation only reinforced a conclusion already in the professional literature. In short, he had no reason to question his results. The “ice catastrophe” at the outer CHZ boundary had been “in the air” in professional circles for many years; he got the answer he almost certainly expected.

The only remaining question is why his figures for the optical effect of CO₂ were off. This, too, may have a traceable answer. Hart cites the seminal work of Elsasser and Culbertson (1960) in his references, though he inexplicably leaves off Culbertson's name.

Hart computed partial optical thickness τ_i due to each absorber i as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\tau_i = q_i^{ \ast} \ m_i^{1 / 2} \ ( P / P_0 ) ^{1 / 2} ( T_0 / T ) ^{1 / 4} \tag{4}\end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q_i^{ \ast}$$ \end{document} is a constant, different for each absorber, m_i is the airborne mass of that absorber, and P and T are pressure and temperature relative to reference values P ₀ and T ₀.

The paper does not explicitly list any of the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q_i^{ \ast}$$ \end{document} values, which must have represented weighted means of the absorption coefficients. However, it does list tables of Earth's status at different times in geological history according to the simulation, giving the partial pressure of each absorber, P, and T. From these, it was possible to work backward, with simultaneous equation and linear regression methods, to recreate the absorption coefficients used (Table 1).

Table 1.

Reconstructed Estimates of Hart's Semigray Absorption Coefficients

Absorber	Name	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\rm q_i^{ \ast}$$ \end{document} (Dimensionless)
CH₄	methane	3.41×10⁻⁹
CO₂	carbon dioxide	3.06×10⁻⁹
H₂O	water vapor	2.08×10⁻⁸
NH₃	ammonia	1.02×10⁻⁷

The coefficient for water vapor is 6.8 times higher than that for carbon dioxide. Coupled with the greater mass of water vapor in Earth's atmosphere, the smaller contribution from CO₂ follows inevitably.

It is interesting to compare estimates of semigray absorption coefficients in the CO₂ band most relevant to terrestrial planet temperatures, the 15-micron band. I have here translated units from the originals (Table 2).

Table 2.

Estimates of CO₂ 15 μ Band Absorption Coefficients

Source	Wavelength(s) (μ)	k_a (m² kg⁻¹)
Elsasser and Culbertson, 1960	14.1	0.0970
	14.3	0.233
	14.5	0.509
	14.7	1.06
	14.9	1.37
	15.2	1.22
	15.4	0.736
	15.6	0.314
	15.9	0.125
Hanel et al., 1963	14.0–14.5	0.04
	14.5–15.5	0.2
	15.5–16.0	0.04
Gonima, 1992	14.0–16.0	20
Evans, 2001	14.278	16.3

The geometric mean of Elsasser and Culbertson's figures in the 14–16 μ range is 0.438 m² kg⁻¹. The mean figure for Hanel et al. (1963) is 0.07 m² kg⁻¹, while Gonima (1992) finds 20 m² kg⁻¹, and Evans (2001) reports a point measurement of 16.3 m² kg⁻¹.

Clearly, there is a great discrepancy here. Figures from the 1960s are very low compared to later ones.

For a researcher to estimate semigray band absorptance in a terrestrial planet atmosphere, the usual procedure now is to apply sophisticated line-by-line software to the HITRAN or HITEMP line data compilations (Rothman et al., 2013). From individual line data, one can estimate a mean absorption coefficient in a given wavelength (or frequency/wavenumber) domain.

However, HITRAN was not widely available before 1973. The estimates of Elsasser and Culbertson were based on a mathematical treatment of laboratory test data on carbon dioxide in glass tubes, with nitrogen as a line-broadening agent. Either because of deficiencies in the mathematical model or problems with the laboratory procedure, they concluded that carbon dioxide was a much weaker greenhouse gas than it is now considered to be. Similar considerations undoubtedly arose with the work of Hanel et al. (1963). By the time of Gonima's paper (1992), both better models and better data were available.

Conclusion

Because of (1) earlier work that showed Earth orbiting very near the outer edge of a narrow CHZ, in danger of runaway glaciation if solar illumination fell by even a few percent, and (2) inadequate treatment of radiation transfer in the source material available to him, Hart's planetary temperature model did indeed suffer from serious deficiencies, as proposed by Kasting (2010). A more sophisticated semigray model might have provided much different results, even without considering the stabilizing carbonate-silicate feedback found by Walker et al. (1981).

Footnotes

Acknowledgments

The late Michael H. Hart was generous with his time and conversation about his 1978 study when the author of this work was simply an interested high school student. The late Stephen H. Schneider was also helpful in discussing that paper. Dr. James F. Kasting gave extremely useful criticism both before submission of this paper and as a reviewer.

Author Disclosure Statement

No competing financial interests exist.

Abbreviations Used

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