Planetary geodynamics may have an important influence over planetary habitability and the boundaries of the circumstellar habitable zone (CHZ) in space and time. To investigate this we use a minimal parameterized model of the co-evolution of the geosphere and atmosphere of Earth-like planets around F, G, K and M main sequence stars. We found the CHZ for the present Solar System located between 0.92 and 1.09 au for a 1.0 M$_{\oplus }$ Earth-like planet, extendible to 1.36 au for a 4.0 M$_{\oplus }$ planet. In the literature, the CHZ varies considerably in width and border location, but the outer edges tend to be more spread out than the inner edges, showing a higher difficulty in determining the outer edge. Planetary mass has a considerable effect on planetary geodynamics, with low-mass planets cooling down faster and being less capable of maintaining a rich carbon dioxide atmosphere for several billions of years. Age plays a particularly important role in the width of the CHZ as the CHZ contracts in both directions: from the inner edge (as stellar luminosity increases with time), and from the outer edge (as planetary heat flux and seafloor spreading rate decrease with time). This strongly affects long-lived habitability as the 5 Gyr continuous CHZ may be very narrow or even non-existent for low-mass planets (<0.5 M$_{\oplus }$) and fast-evolving high-mass stars (>1.1 M$_{\odot }$). Because of this, the mean age of habitable terrestrial planets in our Galaxy today may be younger than Earth's age. Our results suggest that the best targets for future surveys of biosphere signatures may be planets between 0.5 and 4.0 M$_{\oplus }$, in systems younger than the Solar System. These planets may present the widest and long-lived CHZ.

Environmental conditions have changed in the past of our planet but were not hostile enough to extinguish life. In the future, an aged Earth and a more luminous Sun may lead to harsh or even uninhabitable conditions for life. In order to estimate the life span of the biosphere we built a minimal model of the co-evolution of the geosphere, atmosphere and biosphere of our planet, taking into account temperature boundaries, CO2 partial pressure lower limits for C3 and C4 plants, and the presence of enough surface water. Our results indicate that the end of the biosphere will happen long before the Sun becomes a red giant, as the biosphere faces increasingly more difficult conditions in the future until its collapse due to high temperatures. The lower limit for CO2 partial pressure for C3 plants will be reached in 170(+ 320, − 110) Myr, followed by the C4 plants limit in 840(+ 270, − 100) Myr. The mean surface temperature will reach 373 K in 1.63(+ 0.14, − 0.05) Gyr, a point that would mark the extinction of the biosphere. Water loss due to internal geophysical processes will not be dramatic, implying almost no variation in the surface ocean mass and ocean depth for the next 1.5 billion years. Our predictions show qualitative convergence and some quantitative agreement with results found in the literature, but there is considerable scattering in the scale of hundreds of millions of years for all the criteria devised. Even considering these uncertainties, the end of the biosphere will hardly happen sooner than 1.5 Gyr.

We show that the periodogram behaves differently for a weakly dependent process with a small monotonic trend and a stationary strongly dependent process. In the former case it has a non-central -distribution with noncentrality parameter tending to 0 uniformly outside intervals . In the latter case it has λα −1 times a central -distribution, 0 < α < 1.

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) = c(m + n)ß, where m is an arbitrary non-negative parameter and c is not 0. For – ½ < ß < 0 the Hurst exponent is shown to be precisely given by 1 + ß. For ß ≦ – ½ and for ß = 0 the Hurst exponent is 0.5, while for ß > 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.