Likewise a similar empirical limit can be defined for the outer edge This early

Likewise a similar empirical limit can be defined for

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Likewise, a similar empirical limit can be defined for the outer edge. This “early Mars limit” is based on geological evidence that Mars may have been a habitable planet ~3.8 Ga, which occurred when the Sun was ~75% as bright as today. The flux received by present Mars is ~43% of that of Earth’s, according to the inverse square law (and Equation 3). Thus, S EFF in this case is ~0.32 (0.75 × 0.43), yielding an early Mars limit distance ( L/L sun = 1) of ~1.77 AU according to Equation (3). These Figure 2. Plot of the effective stellar flux ( S EFF ) versus CO 2 partial pressure for F0 (7200 K, blue), G2 (5800 K, black), and M8 (2600 K, red) stellar types. S EFF decreases from the right to the curve minima because stellar radiation decreases farther away from the Sun, requiring a stronger greenhouse effect (higher CO 2 pressures via the carbonate–silicate cycle) to maintain habitable surface temperatures. S EFF increases beyond the curve minimum because the planet cools at the larger distances associated with increased Rayleigh scattering and CO 2 condensation. Thus, the maximum greenhouse effect of CO 2 occurs at the curve minima (based on calculations from Kasting et al. [ 1 ] and Kopparapu et al. [ 26 ]). Such a carbonate–silicate cycle may help explain the faint young Sun paradox, which is the perceived contradiction between abundant evidence for warm conditions on the early Earth (~2–3.9 Ga) in spite of a fainter young Sun [ 41 ]. It was originally thought that CO 2 levels were not high enough (<25 times that of today’s) to warm the planet at ~2.5 Ga, according to earlier reconstructions of paleosoil data [ 42 ]. At these later times, additional explanations (e.g., lower cloud cover, fractal hazes, higher CH 4 , and/or H 2 concentrations) may have been necessary to completely resolve the paradox (e.g., [ 43 46 ]). However, updated estimates indicate that these CO 2 levels could have been underestimated by about an order of magnitude, suggesting that higher pCO 2 may have been enough to resolve the paradox without invoking additional mechanisms [ 47 ] Plus, very high atmospheric CO 2 pressures could have resolved the paradox at the earliest times (~3.9 Ga) as well [ 45 ]. 2.5. The Classical Habitable Zone with Empirical Limits Alternatively, complementary HZ limits can be derived that are not based on models, but rather on empirical observations of our solar system. For example, the inner edge of this empirical HZ is defined by the stellar flux received by Venus when we can exclude the possibility that it had standing bodies of water on its surface (~1 Ga; [ 1 ]). That is, if Venus had surface fluvial features suggestive of a once habitable planet, they have been absent for at least 1 billion years given that the present surface is free of such features. The flux received at Venus’s orbit today is ~1.92 times that received by the Earth. According to solar evolution models [ 48 ], the Sun was only 92% as bright ~1 Ga, while Venus then had received (0.92 × 1.92) ~1.77 times the energy received by the Earth today. Thus, the corresponding recent Venus limit today is at ~0.75 AU, according to Equation (3).
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