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Solar evolution stronomy textbooks tell us that one day the Sun will expand to become a red supergiant of enormous size, finally swallowing its inner planets, including Earth. However, recent solar evolution models, which account for a realistic amount of mass loss, suggest a (slightly) less catastrophic future for our planet. A I Solar evolution and the distant future of Earth Peter Schröder, Robert Smith and Kevin Apps take a speculative look at what the future may hold as the Sun becomes a supergiant. 6.26 t has long been known, and all recent stellar evolution computer models agree with this, that solar-type stars at the end of their “lives” expand to red supergiants of several hundred times their initial diameter. This is a reaction of the outer layers of the star to the significant changes deep down in the stellar core, which contracts significantly after it has consumed its reservoir of hydrogen fuel. In fact, any solar-type star goes through two such phases of expansion. The first one ends with the ignition of He-burning in the core, after which the stellar core immediately goes into a less condensed state and the outer layers become less expanded. Stars in this first expansion phase are identified as RGB (Red Giant Branch) stars in the Hertzsprung–Russell (HR) diagram, the common astrophysicist’s plot of stellar luminosity against temperature. After the stellar core has consumed its helium, the core must contract yet again, which leads to the second expansion of the outer layers. Supergiants in this phase are slightly less cool and red, and they have been identified as AGB (Asymptotic Giant Branch) stars in the HR diagram. RGB and AGB supergiants both become not only much larger but also much more luminous than the original star used to be. These supergiants obtain their energy mostly from a hydrogen-burning shell around their dense core of nuclear ashes. As the core becomes denser and more massive with time, rising temperature and pressure in the hydrogen-burning shell increase the energy output. Astronomy textbooks often have the following scenario for the end of the Earth (e.g. Prialnik 2000 p10, Hellier 2000 p45). In about 7.5 billion years’ time, our planet is expected to be swallowed by the solar red supergiant when it has expanded beyond the orbit of Earth. Figure 1 illustrates the classical evolution model of the solar radius by comparison to the orbital radii of the inner planets, according to our computations with a well-tested evolution code (see Pols et al. 1998 for a description of the code and further citations). To make things worse, already 2 billion years earlier, in about 5.5 billion years from today, the increasing December 2001 Vol 42 Solar evolution radius (logarithmic) 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 12100 12125 12150 12175 12 200 12 225 12 250 12 275 12 300 age (106 years) solar luminosity will have doubled and turned our planet into a no-go zone for life. Eventually, on the tip AGB, solar luminosity will peak at several thousand times its present value. The AGB expansion comes to an end only when too little envelope mass is left to feed the hydrogen shell-burning zone. At that point, the star disperses the remains of its thin outer layers, ends its active energy production, and only leaves behind its hot core, which would be observed eventually as a “white dwarf”. For a long time, computer models of solar and stellar evolution have differed only in marginal (but labour-intensive) details. One of these important details is the exact combined opacity of metal and molecular lines, resulting in slightly different radii for the extreme RGB and AGB phases of solar evolution models. But the general picture is mostly still agreed upon, at least in most textbooks: an abrupt end and complete disintegration of planet Earth by merging with the solar supergiant seemed inevitable. (For a somewhat more massive star with a planetary system, the star might develop a thick circumstellar envelope that would affect a much larger fraction of the system: there is tentative evidence from the presence of water vapour [Melnick et al. 2001] that the carbon star IRC+10216 is evaporating its equivalent of the Kuiper belt, the source in the solar system of the short-period comets.) There has been one remarkable exception. Sackmann et al. (1993) considered an important extra detail – mass loss – and their model of the future Sun held good news: Earth may survive after all, at least as a planet! But their predicted escape is a narrow one and depends on precisely the right amount of mass loss and other details. So, before drawing any conclusions for the long-term prospects of planetary real estate, we thought it was time to revisit this point with our code, in a fresh and independent approach, and to elaborate the details of the emerging picture of our distant future. At the mercy of solar mass-loss The main point, which indeed makes all the difference, is stellar mass-loss. This is an onDecember 2001 Vol 42 1: The evolution of the solar radius without mass loss, compared to the orbits of the inner planets, for the critical time during the RGB and AGB supergiant stages of solar evolution. The yellow line shows the radius of the Sun in units of its present radius, while the horizontal lines are the radii of the orbits of the four inner planets in the same units. Mercury (red) and Venus (orange) are swallowed by the Sun during the RGB phase, while the Earth (green) and Mars (magenta) succumb at the later AGB phase about 120 million years later, when the Sun becomes considerably larger. Note that the vertical scale gives the logarithms of the various radii. Thus the radius of the Sun is already 10 times its present radius at the start of this phase of evolution and increases to nearly 1000 times at the AGB tip. going process while a star is a red supergiant. The low gravity in combination with energy deposition into the supergiant atmosphere by complex hydrodynamic processes and by absorption of stellar radiation leads to a “cool wind” that steadily carries away mass from the outer layers of the supergiant. The mass-loss rate increases with the expansion of the supergiant as a result of the decreasing surface gravity and the final stages of the supergiant, which come with significant mass loss, are shortlived. However, for a star with a mass like the Sun, we find that the moderate loss during the earlier giant stages on the RGB already removes a considerable amount of mass, mainly during the final approach to the RGB tip. This has long been known from modelling Heburning stars in the old and star-rich globular clusters. Depending on the specific model assumptions and estimated age, evolutionary models of those He-burning stars in globular clusters give them an initial mass of about 0.9 M (M is the symbol for the present solar mass; we shall use M or M(t) for the mass of the Sun at a general time) and a present mass of just under 0.6 M. Globular cluster stars have already been used by Sackmann et al. (1993) to estimate the expected mass loss M· of the late stages of the Sun. They adopted the Reimers (1975) parameterization, assuming that mass-loss depends only on the stellar luminosity L, mass M and radius R (all taken in solar units), namely: LR M· = –η(4×10–13 M /yr) (1) M with a preferred η of 0.6, but they also tried η = 0.4 and 0.8. In our attempt to model stellar mass-loss in a uniform, canonical way, we took an independent approach to several details which are relevant to the maximum radius of the supergiant Sun. As described in more detail in Schröder et al. (1999), apart from modern opacities and neutrino loss rates, we use mixing parameters that are carefully calibrated with precise properties of real stars. In particular, we made a gravity-dependent adjustment of the mixing length to match modern observations of effective temperatures of tip-AGB stars. By comparison, the Sackmann et al. models appear to be too warm by about 10%, which on its own would result in 20% smaller radii. We then revisited mass-loss studies from well-observed K supergiants. In fact, modern work (Baade 1998) finds values about 5 times lower (corresponding to η ≈ 0.2) than the original work of Reimers, three decades ago, would suggest. With η = 0.2, our models of stars like the Sun, including stars a little less massive, reach He-burning after a mass lost on the RGB of around 0.3 M. This is consistent with globular cluster observations, but remains a rather conservative estimate, since giants with solar metallicity will be cooler and may lose even more mass than the metal-poor stars in globular clusters did. Our models also consider dust-driven mass loss which, however, does not become important in the case of the Sun. Specifically for the peak of the Sun’s expansion (figure 2), the solar supergiant will have already lost almost 0.2 M , similar to the predictions made by Sackmann et al. (1993). This still conservative amount of mass-loss works in favour of the Earth in two ways: ● Assuming that the Earth conserves its orbital angular momentum, the orbit of the Earth will expand proportional to 1/M(t), to 185 million km. This is comfortably enough to escape the predicted maximum solar RGB radius of 168 million km, although the logarithmic difference of only 0.04 is barely detectable in figure 2. Interestingly, the Sun’s increase in radius is actually larger in the RGB phase than in the case without mass loss, because of the decreased surface gravity after mass loss. It is only the expansion of the Earth’s orbit that saves it from being engulfed. ● There will be much less mass left in the outer solar layers to feed the hydrogen shell-burning zone during the second expansion of the Sun. Our model computations for the solar AGB supergiant suggest a maximum AGB solar radius of 172 million km, much less than in the case with no mass loss and barely larger than in the RGB phase. At the same time, the Sun 6.27 Solar evolution 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 12100 12120 12140 12160 12180 12220 12220 12240 12260 12280 12300 age (106 years) 2: The evolution of the solar radius, taking account of mass loss. The yellow line shows the solar radius in units of its present value, while the green line shows the Earth’s orbital radius in the same units; both of these curves are plotted on a logarithmic scale, and can be directly compared to figure 1. The blue line shows (on a linear scale) the mass of the Sun in units of its present mass. The changes in the Sun’s mass can be seen to occur principally in two brief phases near the RGB and AGB tips, and these rapid changes are reflected in a rapid expansion of the Earth’s orbit at these phases, which just keeps ahead of the expanding photosphere of the Sun. Table 1: No place like home The range of temperatures and times for which human life might survive on various planetary bodies. will have lost more mass and weigh only 0.68 M which enlarges the orbit of the Earth to 220 million km. Hence, we find that the inclusion of solar mass-loss keeps planet Earth clear of the solar photosphere by a small but crucial margin at all phases of the Sun’s evolution. While the solar luminosity will have increased to about 2800 times (tip RGB) and 4200 times (tip AGB) its present value, the solar photospheric temperature will have dropped from 5800 K today to only about 2700 K for the supergiant. Compared to the earlier predictions of Sackmann et al. (1993), our results are quite similar, but there are differences in the details of the picture. First of all, our models are a more realsitic match to modern observations – in particular, they predict a slightly cooler solar supergiant. As a result of all the differences, we find that the future mass-loss is more concentrated in the very tips of the RGB and the AGB than is predicted by Sackmann et al. and so is the expansion of the orbits. For Venus, this has fatal consequences: while Sackmann et al. predicted that it would escape along with the Earth, our models suggest otherwise. The losers: Mercury and Venus The inner planets Mercury and Venus will not benefit from the moderation of our distant future by solar mass-loss. Mercury will be consumed rapidly by the swelling Sun, long before the Sun gets anywhere near the tip of the RGB, and indeed before the orbit has a chance to expand significantly. Venus has pretty much the same size and mass as Earth, and it is an intriguing example of what was thought to happen to us in the future. The current orbital distance of Venus, 108 million km, will expand due to the predicted solar mass loss to 134 million km by the time the Sun approaches the tip of the RGB. That is not enough to escape the predicted maximum solar RGB radius of 168 million km and Venus will find itself some 30 million km below the Sun’s 6.28 Planet (satellite) Range of Range o photosphere. ages Once inside the stellar photoTplanet /Tplanet(0) in 109 years sphere, the orbit of Venus will decay rapidly because of gas Mars 1.29–1.43 11.6–11.7 drag. From our evolution model Jupiter (Europa) 2.25–2.50 12.07–12.10 of the future RGB Sun, we find a Saturn (Titan) 3.18–3.53 12.139–12.147 gas density in the outer layers of Uranus (Oberon) 5.09–5.66 12.162–12.164 about 2.4×10–6 kg m–3. The drag Neptune (Triton) 8.44–9.38 never force on a spherical body moving through a gaseous medium can be expressed as (Weidenschilling 1977): our moving out into interstellar space, to find F = 0.5πCD ρv2 r 2 another star system? To answer this, we can (2) make a simple calculation that will enable us to where CD is a dimensionless coefficient predict the temperature at the surface of any approximately equal to 0.44, ρ is the density of planet or satellite as the Sun expands. the gas, v is the velocity of the body relative to Suppose that the luminosity of the Sun is L the gas and r is its radius. If we assume v to be and that the (black-body, or effective) temperequal to the orbital velocity of Venus, neglectature of a planet or satellite of radius R at a ing any rotational velocity of the giant star distance D from the centre of the Sun is T; L which is likely to be much lower, and consider and T are in general time-dependent as the Sun Venus’s current mass and radius, we obtain a evolves. If the planet has an albedo A, then it deceleration of about 1.5×10–8 m s–2. absorbs a fraction (1–A) of the incident radiaThis indicates that Venus’s orbit will decay tion and in equilibrium between absorption with a timescale of less than 105 years. In fact, and reradiation we have: the orbit will shrink by 0.01 AU in just about L 1000 years, and the gas drag will then increase 2 (1–A)πR = 4πR2 σT 4 (3) 4π D2 due to both the rising gas density and the rising orbital velocity. Hence, Venus clearly will where σ is the Stefan–Boltzmann constant. not survive the 5×105 years duration of the This is more usefully written in terms of the radius and effective temperature of the Sun, remaining RGB phase. However, the orbital RSun and TSun , as: decay time of a planet like Venus or Earth is longer than brief radius changes due to such R 1/2 TSun (4) T = (1–A)1/4 Sun short-lived events as thermal pulses.These can 2D therefore be ignored in this respect: even if the where the subscript “Sun” is used to avoid tenuous outer envelope of the supergiant confusion with the planetary values. Here we briefly swept over the Earth, during a pulse, have used the definition of the effective temthe encounter would be too brief to cause any perature of the Sun, L = 4πRSun2 σTSun4 . significant orbital change. In fact, we do not For the present Sun, this gives a mean temresolve thermal pulses for reasons of simplicity perature for the Earth (assuming an albedo of and computing speed. 0.3, although the result is not very sensitive to the exact value) of 255 K. This temperature When it is time to move needs some explanation. First of all, it is too low by about 10%, because it takes no account Is there anywhere in the solar system that of the Earth’s greenhouse effect (or of its interwould be safe at the height of the Sun’s luminal radioactive heat sources), but we shall use nosity, or does our survival as a race depend on December 2001 Vol 42 Solar evolution 2.0 7 (a) Tplanet /Tplanet(0) Tplanet / Tplanet (0) 1.6 1.4 1.2 5 4 3 2 1.0 0.8 (b) 6 1.8 0 2000 4000 6000 age (106 years) 8000 10000 12 000 1 12 000 12 050 12 200 12 100 12 150 age (106 years) 12 250 12300 3: The surface temperature of any planet (or satellite) relative to its present temperature. (a) The predicted variation for the first 12 billion years of the Sun’s life, up to the point where it starts to climb the RGB. The Sun is at about 4.5 billion years. (b) An expanded look at what happens on the RGB and AGB. The maximum ratio is about 6.6. There is a period of about 100 million years between the RGB and the AGB when it would be possible (maybe even necessary) to survive if we moved back closer to the Sun, but none of the planets in that region would by that time be worth inhabiting. Not only would there have been a greatly enhanced solar wind during the ascent of the RGB but the atmospheres of the planets would have been evaporated long before that by the strong temperature rise (Jones 1999, p360). it as a rough guide. The actual mean temperature of the Earth is about 288 K, or 15 °C, and in what follows we shall assume that the changes in temperature calculated from the above equation are about right but that the present Earth’s temperature is 288 K. Secondly, whether we take it as 255 K or 288 K, the temperature is an average over the whole Earth, from the very hot equatorial regions to the freezing polar ones, a range of more than 40 °C. A 30° rise in the mean temperature would make the equator uninhabitable, but would leave habitable zones near the poles. This immediately raises the question of how soon the Earth will be uncomfortably hot everywhere, and where we might go then. Is it possible to hop outwards from one planet or satellite to the next, always keeping ahead of the Sun? To find out, we need to know how the temperature of any planet varies with time as the Sun evolves. Taking proper account of the increasing orbital radius which results from the mass loss from the Sun, it can be shown that: L 1/4 M 1/2 Tplanet = Tplanet(0) (5) L M where Tplanet(0) is the present temperature of the planet and T, L and M are time-dependent. This function is illustrated in figure 3. As a first application, let us ask how long it will take for the temperature of the Earth to rise by 5° (the rise anticipated in the next century or so if the current human-induced greenhouse effect continues unchecked). The equation predicts that it will take the evolving Sun about 0.8 billion years to produce this rise – so human activity may be accelerating astronomical effects by a factor of about 10 million. The observant reader will note from figure 3(a) that, on this model, the mean temperature of the Earth when the solar system was formed was substantially less than it is now. A detailed calculation shows that it was only 233 K, cold enough for the oceans to have been frozen solid – how, then, did life form as early as 650 million years after the Earth itself? It is December 2001 Vol 42 now generally believed that the Earth’s early atmosphere contained much more carbon dioxide than it does today and that a greatly increased greenhouse effect kept the Earth’s temperature high enough for liquid water to be present. The carbon dioxide content declined slowly, at a rate that kept pace with the slowly increasing input of energy from the Sun, so that the temperature of the Earth seems to have stayed fairly constant over its entire life to date (Jones 1999); this remarkable result led to the hypothesis by James Lovelock that the whole Earth and its biosphere is a self-sustaining system: the Gaia hypothesis. If correct, this could lead to the Earth resisting, for longer than one might naively expect, the gradual rise in heating by the Sun in the future. However, it seems inconceivable that even the Gaia effect could continue to be effective for long after the Earth is hot enough for the oceans to boil. Let us suppose that all forms of life will be extinguished by the time the Earth’s mean temperature has reached 380 K (a rise of 125°). Figure 3 predicts that this will happen at an age of 11.8 billion years, as the Sun starts its ascent of the red giant branch. Of course, this means that humans will need to leave much earlier in order to continue living in something like our present conditions – probably before the model temperature reaches 300 K (a rise of 45° in the mean temperature to about 60 °C for the real Earth, as opposed to the too-cool model); this occurs at 10.2 billion years. Encouragingly, this gives us 5.7 billion years to figure out how to move house. Let us suppose that we want to find a planet or satellite with a mean temperature of 285 K (12 °C) (here we neglect any greenhouse effect, because Mars and the satellites of the outer planets probably don’t have the right conditions), and that human life can live there for mean temperatures between 270 K and 300 K. Then figure 3 tells us the range of times for which the conditions are right on various bodies; the results are given in table 1, assuming that the current albedo of the body does not change. (In practice, the albedo is very likely to change, but we do not attempt to model that: table 1 is thus only illustrative.) Table 1 shows our dilemma in a stark way – there are periods when we could in principle survive on one of the outer bodies, but there are also long periods when none of these bodies is at a suitable temperature. In particular, we shall have to leave the Earth before even Mars is warm enough. We had better get used to the idea that we shall need to build our own survival capsules – the planets are simply too far apart for planethopping to be a viable solution. Perhaps this is the ultimate justification for developing the International Space Station. ● Klaus-Peter Schröder and Robert Connon Smith are in the Astronomy Centre of the University of Sussex, Brighton; Kevin Apps has recently graduated from there with an MPhys in physics with astrophysics. Acknowledgement. We owe the idea for this look at the distant future of Earth to a supposedly innocent question posed by our colleague Andrew Liddle, when preparing for a public lecture. We thank a referee for drawing our attention to the article by Sackmann, Boothroyd and Kraemer. References Baade R 1998 Ultraviolet Astrophysics: beyond the IUE Final Archive ed. Harris B, ESA SP-413 325. Hellier C 2001 Cataclysmic Variable Stars: how and why they vary Springer and Praxis, Chichester. Iben I, Livio M 1993 PASP 305 1373. Jones B W 1999 Discovering the Solar System John Wiley & Sons 332. Melnick G J et al. 2001 Nature 412 160. Reimers D 1975 Problems in Stellar Astrophysics and Envelopes ed. Baschek B et al. Springer. Sackmann I-J et al. 1993 Ap J. 418 457. Schröder K-P, Winters J M, Sedlmayr E 1999 A&A 349 898. Pols O R et al. 1998 MNRAS 298 525. Prialnik D 2000 An Introduction to the Theory of Stellar Structure and Evolution Cambridge University Press. Weidenschilling S J 1977 MNRAS 180 57. 6.29