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Mon.Not.R.Astron.Soc.000,1–10(2008) Printed3February2008 (MNLATEXstylefilev2.2) Distant future of the Sun and Earth revisited K.-P. Schröder1⋆ and Robert Connon Smith2† 1Departamento de Astronom´ıa, Universidad de Guanajuato, A.P. 144, Guanajuato, C.P. 36000, GTO, México 2Astronomy Centre, Department of Physics and Astronomy, Universityof Sussex, Falmer, Brighton BN19QH, UK 8 0 0 Accepted 2008....;Received200 ....;inoriginalform2007 September 25 2 n a ABSTRACT J We revisitthe distantfuture ofthe Sun and the solarsystem,basedon stellarmodels 5 computed with a thoroughly tested evolution code. For the solar giant stages, mass- 2 loss by the cool(but not dust-driven)wind is consideredin detail. Using the new and well-calibrated mass-loss formula of Schröder & Cuntz (2005, 2007), we find that the ] h masslostbytheSunasanRGBgiant(0.332M⊙,7.59Gyfromnow)potentiallygives p planet Earth a significant orbital expansion, inversely proportional to the remaining - solar mass. o According to these solar evolution models, the closest encounter of planet Earth r t with the solar cool giant photosphere will occur during the tip-RGB phase. During s a this critical episode, for each time-step of the evolution model, we consider the loss [ of orbital angular momentum suffered by planet Earth from tidal interaction with the giant Sun, as well as dynamical drag in the lower chromosphere. As a result of 1 this, we find that planet Earth will not be able to escape engulfment, despite the v positive effect of solar mass-loss. In order to survive the solar tip-RGB phase, any 1 hypotheticalplanetwouldrequireapresent-dayminimumorbitalradiusofabout1.15 3 0 AU. The latter result may help to estimate the chances of finding planets around 4 White Dwarfs. . Furthermore, our solar evolution models with detailed mass-loss description pre- 1 dict that the resulting tip-AGB giant will not reach its tip-RGB size. Compared to 0 other solar evolution models, the main reason is the more significant amount of mass 8 0 lost already in the RGB phase of the Sun. Hence, the tip-AGB luminosity will come : shortofdrivingafinal,dust-drivensuperwind,andtherewillbenoregularsolarplan- v etarynebula (PN).Thetip-AGB ismarkedbyalastthermalpulse andthe finalmass i X loss of the giant may produce a circumstellar (CS) shell similar to, but rather smaller than, that of the peculiar PN IC 2149 with an estimated totalCS shell mass of just a r a few hundredths of a solar mass. Key words: Sun: evolution – Sun: solar-terrestrial relations – stars: supergiants – stars: mass loss – stars: evolution – stars: white dwarfs 1 INTRODUCTION Apps 2001 (hereafter SSA)), and has been discussed very recently by Laughlin (2007). Climatechangeandglobalwarmingmayhavedrasticeffects on the human race in the near future, over human time- Theoretical models of solar evolution tell us that the scalesofdecadesorcenturies.However,itisalsoofinterest, Sun started on the zero-age main sequence (ZAMS) with and of relevance to the far future of all living species, to a luminosity only about 70% of its current value, and it considerthemuchlonger-termeffectsofthegradualheating has been a long-standing puzzle that theEarth seems none of the Earth by a more luminous Sun as it evolves towards thelesstohavemaintainedaroughlyconstanttemperature its final stage as a white dwarf star. This topic has been over its life-time, in contrast to what an atmosphere-free exploredonseveraloccasions (e.g.Sackmann,Boothroyd & model of irradiation would predict. Part of the explanation Kraemer 1993, Rybicki & Denis 2001, Schröder, Smith & may be that the early atmosphere, rich in CO2 that was subsequentlylockedupincarbonates,keptthetemperature up by a greenhouse effect which decreased in effectiveness at just theright ratetocompensate for theincreasing solar ⋆ E-mail:[email protected](KPS) flux. The rôle of clouds, and their interaction with galactic † E-mail:[email protected](RCS) cosmicrays(CR),mayalsobeimportant:thereisnowsome 2 K.-P. Schröder and R.C. Smith evidence(Svensmark2007; butseeHarrison et al.2007 and hassincebeenfurtherimprovedand calibrated rathercare- Priest et al. 2007) that cosmic rays encourage cloud cover fully against observation, so that we believe that it is cur- at low altitudes, so that a higher CR flux would lead to a rently the best available representation of mass loss from higher albedo and lower surface temperature. The stronger stars with non-dusty winds (Schröder & Cuntz 2005, 2007 solarwindfromtheyoungSunwouldhaveexcludedgalactic – see Section 2, where we explore the consequences of this cosmicrays,socloudcoverontheearlyEarthmayhavebeen improved mass-loss formulation). less thannow, allowing thefull effect ofthesolar fluxto be However, although we have considerably reduced the felt. uncertainties in the mass-loss rate, there is another factor What of the future? Although the Earth’s atmosphere that works against the favourable effects of mass loss: tidal may not be able to respond adequately on a short time- interactions. Expansion of the Sun will cause it to slow its scale to the increased greenhouse effect of carbon dioxide rotation, and even simple conservation of angular momen- andmethanereleased intotheatmospherebyhumanactiv- tum predicts that by the time the radius has reached some ity,thereis still thepossibility, represented by James Love- 250 times itspresent value(cf. Table 1) therotation period lock’sGaiahypothesis(Lovelock1979,1988,2006),thatthe of theSunwill haveincreased to several thousand years in- biosphere may on a longer time-scale be able to adjust it- stead of its present value of under a month; effects of mag- selftomaintainlife.Somedoubthasbeencastonthatview neticbrakingwilllengthenthisperiodevenmore.Thisisso byrecentcalculations(Scaife,privatecommunication,2007; much longer than the orbital period of the Earth, even in fordetails, seee.g. Cox etal. 2004, Bettset al.2004) which its expanded orbit, that the tidal bulge raised on the Sun’s suggest that,on thecentury timescale, theinclusion of bio- surface by the Earth will pull the Earth back in its orbit, spheric processes in climate models actually leads to an in- causing it to spiral inwards. crease in carbon dioxide emissions, partly through a feed- This effect was considered by Rybicki & Denis (2001), backthatstartstodominateasvegetationdiesback.Inany whoarguedthatVenuswasprobablyengulfed,butthatthe case, it isclear thatthetimewill comewhen theincreasing Earthmightsurvive.AnearlierpaperbyRasioetal.(1996) solar fluxwill raisethemean temperatureoftheEarthtoa also considered tidal effects and concluded on the contrary levelthatnotevenbiological orotherfeedback mechanisms that the Earth would probably be engulfed. However, the can prevent.There will certainly be a point at which life is Rybicki & Denis calculations were based on combining an- no longer sustainable, and we shall discuss this further in alytic representations of evolution models (of Hurley, Pols Section 3. & Tout 2000) with the original Reimers’ mass-loss formula After that, the fate of the Earth is of interest mainly rather than on full solar evolution calculations with a well- insofarasittellsuswhatwemightexpecttoseeinsystems calibratedmass-lossformulation.TheRasioetal.paperalso thatweobservenowatamoreadvancedstageofevolution. employedtheoriginalReimers’formula,andbothpapersuse WeexpecttheSuntoendupasawhitedwarf–doweexpect somewhatdifferenttreatmentsoftidaldrag.Wehavethere- there to be any planets around it, and in particular do we forere-consideredthisproblemindetail,withourownevolu- expect any small rocky planets like theEarth? tionary calculations and an improved mass-loss description Thequestion ofwhethertheEarth surviveshasproved as thebasis; full details are given in Sections 2 and 4. somewhat tricky to determine, with some authors arguing that the Earth survives (e.g. SSA) and others (e.g. Sack- mann et al. 1993) claiming that even Venus survives, while 2 SOLAR EVOLUTION MODEL WITH MASS generaltextbooks(e.g.Prialnik2000,p.10)tendtosaythat LOSS the Earth is engulfed. A simple model (e.g. SSA), ignoring mass loss from the Sun, shows clearly that all the planets Inordertodescribethelong-termsolarevolution,weusethe out to and including Mars are engulfed, either at the red Eggleton evolution code (Eggleton 1971, 1972, 1973) in the giantbranch(RGB)phase–MercuryandVenus–oratthe version described by Pols et al. (1995, 1998), which hasup- laterasymptoticgiantbranch(AGB)phase–theEarthand dated opacities and an improved equation of state. Among Mars. However, the Sun loses a significant amount of mass otherdesirablecharacteristics, hiscode usesaself-adapting during its giant branch evolution, and that has the effect mesh and a ∇-based prescription of “overshooting”, which that the planetary orbits expand, and some of them keep hasbeenwell-testedandcalibratedwithgiantstarsineclips- ahead of the advancing solar photosphere. The effect is en- ingbinaries(for details, seeSchröderetal. 1997, Polset al. hanced by the fact (SSA) that when mass loss is included 1997, Schröder 1998). Because of the low mass and a non- thesolarradiusatthetipoftheAGBiscomparabletothat convective core, solar evolution models are, however, not at the tip of the RGB, instead of being much larger; Mars subject to any MS (main sequence) core-overshooting. In certainlysurvives,anditappears(SSA)thattheEarthdoes use, the code is very fast, and mass-loss is accepted simply also. as an outerboundary condition. The crucial question here is: what is the rate of mass As already pointed out by VandenBerg (1991), evolu- loss in real stars? Ultimately this must be determined from tion codes have the tendency to produce, with their most observations, but in practice these must be represented evolved models, effective temperatures that are slightly by some empirical formula. Most people use the classical higher than the empirically determined values. The reason Reimers’formula(Reimers1975, 1977), butthereisconsid- lies, probably, in an inadequacy of both low-temperature erableuncertaintyinthevaluetobeusedforhisparameter opacitiesandmixing-lengththeoryatlowgravity.Withthe η,anddifferentvaluesareneededtoreproducetheobserva- latter, we should expect a reduced efficiency of the con- tionsindifferentparameterregimes.Inourowncalculations vective energy transport for very low gravity, because the (SSA)weusedamodificationoftheReimers’formula,which largesteddiesarecutoutoncetheratioofeddy-sizetostellar Distant future of Sun and Earth 3 radiushasincreasedtoomuchwithg−1.Hence,asdescribed Table 1.Mainphysicalpropertiesofcharacteristicsolarmodels bySchröder,Winters&Sedlmayr(1999),ourmixing-length parameter, normally α = 2.0 for logg < 1.94, receives a Phase Age/Gy L/L⊙ Teff/K R/R⊙ MSun/M⊙ small adjustment in theform of a gradual reduction for supergiant models, reaching α = 1.67 at logg = 0.0. With ZAMS 0.00 0.70 5596 0.89 1.000 present 4.58 1.00 5774 1.00 1.000 this economical adjustment, our evolution models now give MS:hottest 7.13 1.26 5820 1.11 1.000 abettermatchtoempiricallydeterminedeffectivetempera- MS:final 10.00 1.84 5751 1.37 1.000 turesof very evolved late-typegiants and supergiants, such RGB:tip 12.17 2730. 2602 256. 0.668 asα1 Her(seeSchröder&Cuntz2007,Fig.4inparticular), ZA-He 12.17 53.7 4667 11.2 0.668 and even later stages of stellar evolution (Dyck et al. 1996, AGB:tip 12.30 2090. 3200 149. 0.546 and van Belle et al. 1996, 1997). AGB:tip-TP 12.30 4170. 3467 179. 0.544 The evolution model of theSunpresented hereusesan opacity grid that matches the empirical solar metallicity of (note: 1.00AU=215R⊙) Anders&Grevesse (1989), Z =0.0188, derivedfrom atmospheric models with simple 1D radiative transfer – an ap- giant branch” in the HRD) – at first very gradually, but proachconsistent withourevolutionmodels.Togetherwith thenaccelerating. Atanageof12.167Gy,theSunwill have X = 0.700 and Y = 0.2812, there is a good match with reachedthetipof theRGB,with amaximumluminosity of present-day solar properties derived in the same way (see 2730L⊙. Polsetal.1995).Wenotethattheuseof3D-hydrodynamic In order to quantify the mass-loss rate of the evolved, modellingofstellaratmospheresandtheirradiativetransfer cool solar giant ateach time-step,weusethenewandwell- mayleadtoasignificantlylowersolarabundancescale(e.g., calibrated mass-loss formula for ordinary cool winds (i.e., Asplund, Grevesse & Sauval 2005, who quote Z = 0.0122), notdrivenbydust)ofSchröder&Cuntz(2005, 2007).This but these lower values are still being debated, and create relationis,essentially,animprovedReimers’law,physically some problems with helioseismology. Of course, usinglower motivatedbyaconsideration of global chromosphericprop- metallicities with an evolution code always results in more erties and wind energy requirements: cZomoupracctodanedwhooutltderpslatienlllayrfmaioldteolsr.eHpreondcue,ceiftwheeupsreedseanlto-wdaeyr M˙ = ηL∗R∗ Teff 3.5 1+ g⊙ (1) Sun, and the reliability of more evolved models with lower M∗ (cid:16)4000K(cid:17) (cid:16) 4300g∗(cid:17) Z must therefore also beseriously doubted. withη=8×10−14M⊙ y−1, g⊙=solarsurfacegravitational The resulting solar evolution model suggests an age of acceleration, and L∗, R∗, and M∗ in solar units. thepresent-dayMSSunof4.58Gy(±0.05Gy),countedfrom This relation was initially calibrated by Schröder & its zero-age MS start model, which is well within the range Cuntz (2005) with the total mass loss on the RGB, using ofcommonlyacceptedvaluesfortherealageoftheSunand theblue-end(i.e.,theleastmassive)horizontal-branch(HB) thesolarsystem(e.g.Sackmannetal.1993).Ourmodelalso stars of globular clusters with different metallicities. This confirms some well-established facts: (1) The MS-Sun has method avoids the interfering problem of temporal mass- alreadyundergonesignificantchanges,i.e.,thepresentsolar loss variations found with individual giant stars and leaves luminosityLexceedsthezero-agevalueby0.30L⊙,andthe an uncertainty of the new η-value of only 15%, just under zero-age solar radius R was 11% smaller than the present theindividual spread of RGB mass-loss required to explain value.(2)TherewasanincreaseofeffectivetemperatureTeff thewidth of HBs. from, according toourmodel, 5596Kto5774K(±5K).(3) Later, Schröder & Cuntz (2007) tested their improved ThepresentSunisincreasingitsaverageluminosityatarate mass-loss relation with six nearby galactic giants and su- of1%inevery110millionyears,or10%overthenextbillion pergiants, incomparison with four other,frequentlyquoted years.Allthisiscompletelyconsistentwithestablishedsolar mass-lossrelations.AllbutoneofthetestedgiantsareAGB models likethe oneof Gough (1981). stars, which have (very different) well-established physical Certainly,thesolarMS-changesandtheirconsequences properties and empirical mass-loss rates, all by cool winds for Earth are extremely slow, compared to the current cli- notdrivenbyradiation-pressure ondust.Despitetheafore- mate change driven by human factors. Nevertheless, solar mentionedproblemwiththeinherenttime-variabilityofthis evolution will force global warming upon Earth already in individual-star-approach, the new relation (equation (1)) the“near”MSfutureoftheSun,longbeforetheSunstarts was confirmed to give the best representation of the cool, its evolution as a giant star (see our discussion of the hab- but not “dust-driven”stellar mass-loss: it was the only one itable zone in Section 3). that agreed within theuncertainties (i.e., within a factor of Atanageof7.13Gy,theSunwillhavereacheditshigh- 1.5 to 2) with the empirical mass-loss rates of all giants. estTeff of5820K,ataluminosityof1.26L⊙.Fromthenon, Hence, since the future Sun will not reach the critical lu- the evolving MS Sun will gradually become cooler, but its minosity required by a “dust-driven” wind (see Section 5), luminosity will continue to increase. At an age of 10.0Gy, wehereapplyequation(1)todescribeitsAGBmass-lossas thesolareffectivetemperaturewillbebackatTeff =5751K, well as its RGBmass-loss. while L = 1.84L⊙, and the solar radius then will be 37% The exact mass-loss suffered by the future giant Sun larger than today. Around that age, the evolution of the has, of course, a general impact on the radius of the solar Sunwillspeedup,sincethesolarcorewillchangefromcen- giant, since the reduced gravity allows for an even larger tral hydrogen-burning to hydrogen shell-burning and start (and cooler) supergiant. The luminosity, however, is hardly to contract. In response, the outer layers will expand, and affectedbecauseitismostlysetbytheconditionsinthecon- theSunwill start climbingup theRGB(the“red” or“first tracting core and the hydrogen-burning shell. In total, our 4 K.-P. Schröder and R.C. Smith solar evolution model yields a loss of 0.332M⊙ by the time the tip-RGB is reached (for η = 8×10−14M⊙y−1). This is a little more than the 0.275M⊙ obtained by Sackmann et al. (1993), who used a mass-loss prescription based on the original, simple Reimers’ relation. Furthermore, our evolution model predicts that at the very tip of the RGB, the Sun should reach R = 256R⊙ = 1.2AU (see Fig. 1), with L = 2730L⊙ and Teff = 2602K. More details are given in Table 1. By comparison, a prescription of the (average) RGB mass-loss rate with η = 7×10−14M⊙y−1, near the lower errorlimitofthemass-losscalibration withHBstars,yields a solar model at the very tip of theRGB with R=249R⊙, L = 2742L⊙, Teff = 2650K, and a total mass lost on the RGBof 0.268M⊙.Withη=9×10−14M⊙y−1,ontheother hand, the Sun would reach the very tip of the RGB with R=256R⊙, L=2714L⊙, Teff =2605K, and will have lost Figure 1. Solar radius evolution during the RGB and AGB phases. Included for comparison (dashed curve) is the potential a total of 0.388M⊙. While these slightly different possible orbital radius of planet Earth, taking account of solar mass loss outcomes of solar tip-RGB evolution – within the uncer- butneglectinganylossoforbitalangularmomentum.Thelabels taintyofthemass-loss prescription–requirefurtherdiscus- on the curve for the solar radius show the mass of the Sun in sion, which we give in Section 4.3, the differences are too unitsofitspresent-daymass. small to beobvious on thescale of Fig. 1. With the reduced solar mass and, consequently, lower gravitational attraction, all planetary orbits – that of the priormasslossfromthegiantSunisessentialformodelling Earth included – are bound to expand. This is simply thisphase reliably. a consequence of the conservation of angular momentum Λ = M v r , while the orbital radius (i.e. r ) adjusts E E E E E toa newbalance between centrifugal force and thereduced 3 EVOLUTION OF THE HABITABLE ZONE gravitational force of the Sun, caused by the reduced so- The Earth currently sits in the ‘habitable zone’ in the so- yttlahhireaeltmdssmraErsasElliMes∝r1SΛ(.u57n2E0(×/tA)M.1U0SS−uufno1b(r4st)t)ti.hatuFnetodcirnalgtashervgisEηerc==o(n9p8s×e×rGv1Ma10t0−iS−v1ue41n4)(cMtav)sa/⊙elruy,Ee−ws1ien.ofifFΛnoηEdr, lEaararertsfhyavs–toeumirna,bptlaheratftoicruisll,iafreth.teThehreeagrvieoenraargeinevpawrlhaioincuehstapcroryencdtieistmeiopdneesrfianotinutriotehn–es of ‘habitability’ in the literature, and a useful overview of we find, respectively, r = 1.37AU and r = 1.63AU, so E E habitablezonesinthewidercontextofextrasolarplanetary in all cases the orbital radius is comfortably more than the systemsisgivenbyFrancketal.(2002).Forthecurrentpa- solar radius, when angular momentum is conserved. per, a convenient definition is that a planet is habitable if Section 4.1 provides a treatment of the more realistic theconditionsonitallowthepresenceofliquidwateronits case,inwhichangularmomentumisnotconserved.Wehave surface.Thismayallowextremesoftemperaturethatwould takengreat careindeterminingthemass-loss andotherpa- make life uncomfortable if not impossible for humans, but rametersforourmodels,becausethebestpossiblemodelsof the argument is that life of any kind (at least any kind we theevolutionofsolarmassandradiusthroughthetip-RGB know about at present) requires water at some stage in its phase are required to providereliable results. life cycle. We shall adopt that definition in this paper, but Thesignificantsolar RGBmassloss willalso shapethe note that even with that apparently simple definition it is later solar AGB evolution. Compared with models without not straightforward to calculate the width of the habitable mass loss, the AGB Sun will not become as large and lu- zone. minous, and will be shorter-lived, because it lacks envelope Itmaybeinstructivetobeginwith acalculation of the massforthecoreanditsburningshellsto“eat”into.Infact, mean planetary temperature in terms of a spherical black the solar tip-AGB radius (149R⊙) will never reach that of body,by assuming that the planetary body absorbs the so- the tip-RGB (see Fig. 1), and AGB thermal pulses are no larfluxinterceptedbyits(circular)cross-sectionalareaand threat to any planet which would have survived the tip- re-emits it spherically symmetrically at a black body tem- RGB. Our evolution code resolved only the two final and peratureT. Then (cf. SSA)T is given by most dramatic thermal pulses (cf. Section 5). Theregulartip-AGBluminosityof2090L⊙ willnotex- T = (1−A)1/4 R 1/2Teff ceed the tip-RGB value, either. Hence, as will be discussed 2D (cid:16) (cid:17) idnusSte-cdtriiovnen5,suthpeertwipin-AdGbButSwuinllwstiallynsohtodrtevoeflothpeacrsiutsictaailnleud- = 0.0682(1−A)1/4 R 1/2 1AU 1/2Teff (2) minosity required by dust-driven winds (see Schröder et al. (cid:18)R⊙(cid:19) (cid:16) 2D (cid:17) 1999). The very tip of the AGB coincides with a thermal where D is the distance of the body from the centre of the pulse (TP), after which the giant briefly reaches a peak lu- Sun,RistheradiusoftheSun,AistheBondalbedoofthe minosity of 4170L⊙, but at a higher Teff =3467K than on Earth and Teff is the effective temperature of the Sun. On the RGB (see Table 1 and Section 5), keeping the radius thatbasis,takingTeff =5774KandR=R⊙ (Table1),and down to 179R⊙. Again, the best possible treatment of all A=0.3(Kandel&Viollier2005),wefindT(1AU)=255K. Distant future of Sun and Earth 5 ButtheactualmeantemperatureoftheEarthatpresentis momenttheshort-timescale(decadestocenturies)problems 33Kwarmer,atT =288K.Thisdemonstratesthewarming currently being introduced by climate change, we may ex- effect of our atmosphere, which becomes significantly more pecttohaveaboutonebillion yearsoftimebeforethesolar important with higher temperature(see below). fluxhasincreased bythecritical 10% mentioned earlier. At In fact, there are various complex, partly antagonistic thatpoint,neglectingtheeffectsofsolarirradiance changes atmospheric feedback mechanisms (for example, the green- on the cloud cover, the water vapour content of the atmo- houseeffect,thevariationofplanetaryalbedowiththepres- sphere will increase substantially and the oceans will start enceofclouds,snowandice,andthecarbonate-silicatecycle to evaporate (Kasting 1988). An initially moist greenhouse which determines the amount of carbon dioxide in the at- effect (Laughlin 2007) will cause runaway evaporation until mosphere) that act tochangethesurface temperaturefrom the oceans have boiled dry. With so much water vapour in what it would be in the absence of an atmosphere. These theatmosphere,someofitwillmakeitswayintothestrato- mechanismshavebeencarefullydiscussedbyKasting,Whit- sphere. There, solar UV will dissociate the water molecules mire & Reynolds (1993), who conclude that a conservative intoOH and free atomic hydrogen,which will gradually es- estimate of the current habitable zone (HZ) stretches from cape, until most of the atmospheric water vapour has been 0.95AUto1.37AU.Weshall adopt theirresult for thelim- lost.Thesubsequentdrygreenhousephasewillraisethesur- ited purposes of this paper. It can be adjusted in a simple- facetemperaturesignificantlyfasterthanwouldbeexpected mindedway toallow fortheevolutionoftheSunbyscaling from our very simple black-body assumption, and the ulti- theinner and outer HZ radii rHZ,i, rHZ,o with thechanging mate fate of the Earth, if it survived at all as a separate solar luminosity LSun(t): rHZ ∝ LSun(t). In this way, the body(cf.Section4),wouldbetobecomeamoltenremnant. respectivecriticalvaluesofsolarpirradiancederivedbyKast- ing et al. (1993) for the innerand outer edge of theHZ are maintained. 4 THE INNER PLANETARY SYSTEM Certainly,withthe10%increaseofsolarluminosityover DURING TIP-RGB EVOLUTION the next 1Gy (see previous section), it is clear that Earth will come to leave the HZ already in about a billion years After 12 Gy of slow solar evolution, the final ascent of the time, since the inner (hot side) boundary will then cross RGB will be relatively fast. The solar radius will sweep 1AU. By the time the Sun comes to leave the main se- through the inner planetary system within only 5 million quence,aroundanageof10Gy(Table1),oursimplemodel years, by which time the evolved solar giant will have predicts that the HZ will havemoved out to the range 1.29 reached the tip of the RGB and then entered its brief (130 to 1.86AU. The Sun will have lost very little mass by that million year)He-burningphase.Thegiantwillfirstcometo time,sotheEarth’sorbitalradiuswillstillbeabout1AU– exceedtheorbital sizeofMercury,thenVenus.By thetime left far behind by the HZ, which will instead be enveloping itapproachesEarth,thesolarmass-lossratewillreachupto theorbit of Mars. 2.5×10−7M⊙y−1 and lead to some orbital expansion (see By the time the Sun reaches the tip of the RGB, at Section2).Buttheextremeproximityoftheorbitingplanet 12.17Gy,theEarth’sorbitalradiuswillonlyhaveexpanded to the solar photosphere requires the consideration of two toat most 1.5AU,butthehabitablezonewill havearange effects, which both lead to angular momentum loss and a of 49.4 to 71.4AU, reaching well into the Kuiper Belt! The fatal decrease of theorbital radius of planet Earth. positions of the HZ boundaries are not as well determined as these numbers suggest, because in reality the scaling for 4.1 Tidal interaction the boundaries of the HZ almost certainly depends also on how clouds are affected by changes in the solar irradiance. For the highly evolved giant Sun, we may safely assume Theseeffectsarecomplex and uncertain(cf.Kasting 1988), (cf. Section 1) that it has essentially ceased to rotate, after and may increase or decrease the speed at which the HZ nearly2billionyearsofpost-MSmagneticbrakingactingon drifts outwards. But none the less it seems clear that the the hugely expanded, cool RGB giant. Consequently, any HZ will move out past the Earth long before the Sun has tidal interaction with an orbiting object will result in its expanded very much, even if the figure of one billion years suffering a continuous drag by the slightly retarded tidal is a rather rough estimate of how long we have before the bulges of the giant solar photosphere. Earth is uninhabitable. AsshowninSection2,theorbitalradiusofplanetEarth Inotherplanetarysystemsaroundsolar-typestars,con- r dependsontheangularmomentumsquared,bytheequa- E ditionsmaybedifferent,anditmayevenbepossibleforlife tion to start during a star’s post-main-sequence evolution, if a Λ2(t) planet exists at a suitable distance from thestar. This pos- r = E . (3) sibility is discussed by Lopez, Schneider & Danchi (2005), E ME2 GMSun(t) whoalso giveageneraldiscussion of theevolution ofhabit- Hence, the terrestrial orbit reacts quite sensitively to any ablezoneswithtime.However,theyusetheevolutionmod- loss of angular momentum,by shrinking. els of Maeder & Meynet (1988), which do not agree as well The retardation of the tidal bulges of the solar photo- as ours with the colours and observed Teff’s of the red gi- sphere will be caused by tidal friction in the outer convec- antsinstarclusters(see,e.g., illustrations givenbyMeynet tive envelope of the RGB Sun. This physical process was etal.1993),andwhichpredictaverydifferentbehaviourfor analyzed,solvedandappliedbyJ.-P.Zahn(1977,1989,and thesolarradius;sotheirresultsarenotdirectlycomparable otherworkreferredtotherein),andsuccessfullytestedwith with ours. thesynchronizationandcircularization ofbinarystarorbits What will happen on theEarth itself? Ignoring for the byVerbunt&Phinney(1995).Inaconvectiveenvelope,the 6 K.-P. Schröder and R.C. Smith main contribution to tidal friction comes from the retarda- In the case of supersonic motion (with a Mach num- tion of the equilibrium tide by interaction with convective ber1 oftheorderof2to3)inagaseousmedium,dynamical motions.Foracircularorbit,theresultingtorqueΓexerted friction consists in about equal shares of the collisionless, on planet Earth by the retarded solar tidal bulges is given gravitationalinteractionwithitswakeandofthefrictionit- by (Zahn 1977; Zahn 1989, Eq.11): self.Inherstudy,Ostriker(1999,Fig.3)findsthatthedrag force exerted on the object in motion is λ2 2 2 RSun 6 Γ=6 q MSunRSun (Ω−ω). (4) tf (cid:16) rE (cid:17) F =λ 4πρ(GM /c )2 (5) d d E s Here, the angular velocity of the solar rotation is sup- posed to be Ω = 0, while that of the orbiting Earth, whereλ isoftheorderof1to3.Thenumericalsimulations ω(t) = 2π/PE(t) = Λ−3(t)ME3(GMSun(t))2, will vary both madebydSánchez-Salcedo&Brandenburg(2001)areingen- with the decreasing angular momentum Λ(t) (= 2.67 × eral agreement with the results of Ostriker (1999). Here c s 1040kgm2s−1 at present) and with the solar mass in the is the speed of sound, which in a stellar chromosphere is final solar RGB stages. The exerted torque scales with about8kms−1,andρisthegasdensity(SIunits).Thelat- the square of the (slowly increasing) mass ratio q(t) = ter quantity is the largest source of uncertainty, as we can ME/MSun(t) (= 3.005 × 10−6 at present), because q only make guesses (see below) as to what the gas density determines the magnitude of the tidal bulges. tf(t) = in the lower giant solar chromosphere will be. The angular (MSun(t)RS2un(t)/LSun(t))1/3 ≈O(1y)istheconvectivefric- momentum loss resulting from thisdrag is simply tiontime(Zahn1989, Eq.7),andthecoefficient λ2 depends onthepropertiesoftheconvectiveenvelope.Forafullycon- dΛ/dt=−F r , (6) d E vective envelope (Zahn 1989, Eq.15), with a tidal period ≈ O(1y), comparable to 2tf, we may use λ2 ≈ 0.019α4/3 ≈ and the corresponding life-time of the orbital angular mo- 0.038 (with a convection parameter of our tip-RGB solar mentum is τ =Λ/|dΛ/dt|, as above. model of α ≈ 1.7). This coefficient appears to be the main Forthelower chromosphereof theKsupergiant ζ Aur, source of uncertainty(seeSection 4.3),because it is related employing an analysis of the additional line absorption in to thesimplifications of themixing length theory (MLT). the spectrum of a hot companion in chromospheric eclipse, Withthepropertiesofthetip-RGBSun,atypicalvalue Schröder,Griffin&Griffin(1990)foundanaveragehydrogen of the tidal drag acting on planet Earth is Γ = dΛ/dt = particle density of 7×1011cm−3 at a height of 2×106km. −3.3×1026kgm2s−2, which gives a typical orbital angular Alternatively,wemaysimplyassumethatthedensityofthe momentum decay time of τ =| Λ/Γ |= 2.6×106y. This is lowersolarchromospherescaleswithgravityg,whichwillbe comparabletothetimespentbytheSunnearthetip-RGB; lowerby4.7ordersofmagnitudeonthetip-RGB,whilethe since a loss of only ≈ 10% of the angular momentum will densityscale-height scales withg−1 (asobservations ofcool be sufficient to reduce the orbital radius (by 20%) to lower giant chromospheres seem to indicate, see Schröder 1990). it into the solar giant photosphere, this order-of-magnitude ThechromosphericmodelsofbothLemaireetal.(1981)and calculationillustratesclearlythattidalinteractioniscrucial. Maltbyetal.(1986)suggestparticledensitiesoftheorderof Its full consideration requires a timestep-by-timestep com- 1017cm−3 at a height of 100km, and a scale height of that putation of the loss of orbital angular momentum; at each orderforthepresent,lowsolarchromosphere.Scaledtotip- time-step of the solar evolution calculation, we use equa- RGB gravity, that would correspond to a particle density tion (4), together with the radii and masses of our solar of 2×1012cm−3, or ρ ≈ 4×10−9kgm−3, at a height of evolutionmodel,tocomputethechangeinangularmomen- 5×106km (0.03 AU), and a density scale height of that tum, and then use equation (3) to compute the change in same value. the orbital radius, and hence the new orbital period of the For the computation of the orbital angular momentum Earth.Section 4.3 presentstheresult, which also takesinto lossoftheEarth,presentedbelow(seeFigures2and3),we account the relatively small additional angular momentum apply the latter, rather higher values of the future chromo- losses by dynamical drag, as discussed in the nextsection. sphericgasdensity,togetherwiththe(alsomorepessimistic) assumptionofλ =3(usingc =8kms−1).Thetypicalan- d s gular momentum decay-time by dynamical friction in the 4.2 Dynamical friction in the lower chromosphere low(h≈0.03AU)chromosphereofthetip-RGBsolar giant is 14 million years – significantly longer than that for tidal A further source of angular momentum loss by drag is dy- interaction.Hence,thisillustratesthatdynamicalfrictionis namical friction, from which any object suffers in a fairly of interest only in the lowest chromospheric layers, adding close orbit, by its supersonic motion through the gas of there just a little to the drag exerted by tidal interaction. thethenveryextended,cool solar giant chromosphere.Ina None the less, we include it, using equations (5) and (6) to different context, dynamical drag exerted by a giant atmo- calculate the additional angular momentum change to be spherehasalreadybeenconsideredbyLivio&Soker(1984). included in equation (3). But the specific problem here is to find an adequate description ofthedensitystructureofthefuturecool solargi- ant.Fortunately,asitturnsout(seebelow),dynamicaldrag will playonly aminorrôle, verynearthesolar giant photosphere, and the total angular momentum loss is dominated by the tidal interaction described above. An approximate 1 NotethatvE∝MSun(t),andsotheMachnumberissomewhat treatment ofthedragistherefore adequate,and weusethe lowerthanwouldbeexpectedfromthepresentorbitalvelocityof recent study by Ostriker(1999). theEarthofabout30kms−1. Distant future of Sun and Earth 7 ignitionandnotreachthehorizontalbranchatall.Andthe full width of the HB towards lower Teff is achieved already with an η of 7×10−14M⊙y−1. Furthermore, the benefit of larger orbits with a reduced solar mass is to some extent compensated for by a larger solar giant. Dynamical drag does not become important until the planet is already very near the photosphere, i.e., after tidal drag has already lowered the orbit. Hence, the most significant uncertainty here comes from the scaling of the tidal friction coefficient λ2 (of Zahn, 1989). For this reason, we computed several alternative cases, and from thesewe find: (1) With themass-loss rate unchanged,thevalueof λ2 would have to be significantly smaller for an escape from the “doomsday” scenario, i.e., less than 0.013, instead of our adopted value of 0.038. But Zahn’s scaling of λ2 has been empirically confirmed within a factor of 2, if not bet- Figure 2.Thefinal 4millionyearsofsolarevolutionbeforethe ter (see Verbunt & Phinney, 1995). Very recently, realistic tip-RGB,showingtheradiioftheSunandoftheorbitofplanet 3D simulations of the solar convection havealso resulted in Earth (dashed curve) – taking account of angular momentum aneffectiveviscositywhichmatchesthatofZahn’sprescrip- lossesbytidalinteractionandsupersonicdrag.Thelabelsonthe tion surprisingly well (Penev et al. 2007). And Rybicki & solarradiustrackgivevaluesofMSun(t)/M⊙,asinFigure1. Denis (2001), by comparison, used a value (K2 = 0.05 in the notation of their very similar calculation of tidal angu- larmomentumloss)whichisentirelyconsistentwithZahn’s 4.3 “Doomsday” confirmed scaling of λ2 for a convection parameter of α=2. Asexplained intheprevioustwosections, weuseequations (2) We then considered solar evolution models with a (3)to(6)tocompute,ateachtime-stepofourevolutionary reasonably larger mass-loss rate (η = 9×10−14M⊙y−1) in calculation,adetaileddescriptionoftheorbitalevolutionfor combination with tidal friction coefficients of 1/1, 2/3 and planetEarthinthecriticaltip-RGBphaseoftheSununder 1/2 oftheonegivenbyZahn.Ineach of thesecases, planet the influence of tidal interaction and dynamical drag. The Earth would not be able to escape doomsday but would resulting evolution both of the orbital radius of the Earth face adelayed engulfment bythesupergiant Sun–470,000, and of the radius of thesolar giant is shown in Fig. 2. This 230,000 and 80,000 years before the tip-RGB is reached, shows that,despite the reduced gravity from a less massive respectively. tip-RGBSun,theorbitoftheEarthwillhardlyevercometo (3) Finally, we checked the outcome for a reasonably exceed 1AU by a significant amount. The potential orbital lowermass-loss rate(η=7×10−14M⊙y−1)incombination growth given by the reduced solar mass is mostly balanced with the same tidal friction coefficients as above. The en- and,eventually,overcomebytheeffectsoftidalinteraction. gulfment would then happen rather earlier than with more Neartheveryend,supersonicdragalsobecomesasignificant mass-loss – 540,000, 380,000 and 270,000 years before the source of angular momentum loss. tip-AGBis reached. AsshownbyFig.2,engulfmentandlossofplanetEarth These computations confirm that reducing the solar will take place just before the Sun reaches the tip of the mass enlarges the planetary orbit more than the tip-RGB RGB,7.59Gy(±0.05Gy)fromnow.Accordingtoourcalcu- solar radius, so that the best way to avoid the doomsday lation,itoccurswhentheRGBSunhasstillanother0.25AU scenario would betohaveas high amass-loss rateas possi- togrow,about500,000yearsbeforethetip-RGB.Ofcourse, ble.However,webelievethatthevalueofηincase(2)above MercuryandVenuswillalreadyhavesufferedthesamefate already is as high as it can be without violating agreement asEarthsometimebefore–respectively,3.8and1.0million of evolved models with observations, and that the smallest years earlier. value used there for the tidal friction coefficient is also at Asmentioned intheintroduction,asimilar calculation the limits of what is allowed by observational constraints. was already carried out in the context of extra-solar plan- Theonly possible escapewould beif oursolar giant models ets by Rasio et al. (1996), who basically came to the same weretoocool(byover100Kincase2),andthereforelarger conclusions;theirfig.2isreminiscentofours.Theyalsoem- thantherealSunwillbe.Hence,toavoidengulfmentbythe ployedtheorbitaldecayratepredictedbyZahn’stheory,but tip-RGBSunwouldrequirethatallthreeparameters(η,λ2 their solar evolution model used the old Reimers mass-loss andTeff)wereatoneedgeoftheiruncertaintyrange,which relation, and they did not make any adjustments to match seems improbable. Rather, our computations confirm, with theeffectivetemperaturesfoundempiricallyatthetipofthe reasonable certainty,theclassical “doomsday” scenario. giant branches (see Section 2). Dotheremaininguncertaintiesallow thepossibility for 4.4 “Doomsday” avoidable? Earth to escape the “doomsday” scenario? As far as the mass-loss alone is concerned, this seems unlikely:according Even though this is an academic question, given the hos- to the study of HB stars in globular clusters by Schröder tile conditions on the surface of a planet just missing this & Cuntz (2005), η is remarkably well constrained and can- “doomsday”scenario,wemayask:whatistheminimumini- not exceed 9×10−14M⊙y−1, or the total RGB mass-loss tial orbital radius of a planet in order for it to “survive”? wouldbecomesolargethatthetip-RGBstarwouldmissHe- Fig.3shows,bythesamecomputationascarriedoutforFig. 8 K.-P. Schröder and R.C. Smith Figure3.AsFig.2,butforaplanetwithapresentorbitalradius Figure4.Solarmasslossduringthefinal1millionyearsonthe of1.15AU. AGBwillremainmainlyoftheorderof2×10−7M⊙y−1andnot provide sufficient CS shell mass to form a regular PN. Only the lasttwoTP’s(tip-AGBandpost-AGB,seetext) areresolved. 2, that an initial orbital radius of 1.15AU is sufficient for anyplanettopassthetip-RGBofastarwithMi =1.0M⊙. Since,asshowninSection5,thetip-AGBSunwillnotreach thatcouldmoveslowlyoutwardstomaintainhabitablecon- any similarly large extent again, such a planet will eventu- ditions would, on cost and energy grounds, necessarily be ally be orbiting a WhiteDwarf. confinedtoasmallfractionofthehumanpopulation–with A more general discussion of planetary survival during all the political problems that that would produce – plus post-main-sequence evolution has been given by Villaver & perhaps a tiny proportion of other species. None the less, Livio(2007),whosuggest thataninitialdistanceofatleast theasteroidal fly-byschemehasitsownproblems,notleast 3AU is needed for the survival of a terrestrial-size planet thedanger of a benign close approach turninginto a catas- when one also takes into account the possible evaporation trophic accidental collision, and possibly also triggering or- of the planet by stellar heating. However, they use stellar bital instability – cf. also Debes& Sigurdsson (2002). models and mass-loss rates that have the maximum radius and mass loss occurring on the AGB. That has been the expected result for many years, but is quite different from 5 TIP-AGB SOLAR EVOLUTION what we find (Section 5 and Table 1) with the improved mass-loss formulation of Schröder & Cuntz (2005, 2007). Thelossof1/3ofthesolarmassduringtherisetothetipof Hence,Villaver&Livio’sresultsmaybeundulypessimistic. theRGBwillmakeasignificantimpactonthefurtherevolu- Inanycase,itisclearthatterrestrialplanetscansurvive tionasanAGBstar.Thereisverylittleshellmassleft,into if sufficiently far from their parent star. If it were possible which the two burning shells (H, followed by He) can ad- to increase the orbital radius from its initial value, then an vance(onaradial mass scale). Hence,theC/O corecannot increase of only 8% of angular momentum should yield the grow as much as with a conservative model without mass pre-RGB orbital size required by planet Earth to escape loss, and the whole core region will not contract as much, engulfment. Is that conceivable? either. Consequently, the AGB luminosity, determined by An ingenious scheme for doing so which, in the first thedensityandtemperatureintheH-burningshell,willnot place, could increase the time-scale for habitation by in- reachashighlevelsasinaconservativeAGBmodel,andnei- telligent life for the whole of the Sun’s MS life-time, was ther will the AGB radius of the late future Sun (see Table proposed by Korycansky, Laughlin & Adams (2001). They 1). pointed out that a suitable encounter of the Earth every Accordingtoourevolutionmodel,theregulartip-AGB 6000 years or so with a body of large asteroidal mass could evolution will be ended with a luminosity of only 2090L⊙, bearrangedtomovetheorbitoftheEarthoutwards;Kuiper Teff = 3200K, and R = 149R⊙. The AGB mass-loss rate, Beltobjectsmightbethemostsuitable.Theenergyrequire- according to the relation of Schröder & Cuntz (2005), will mentscouldbereducedbyincorporatingadditionalencoun- reach only 2.0×10−7M⊙y−1 (see Fig. 4), since the lumi- ters with Jupiter and/or Saturn. Although still very large nositywillnotbesufficient todriveadust-drivenwind(see bytoday’sstandards,theenergyrequirementsremainsmall Schröder et al. 1999). Also, even if it did: only a little shell compared to those for interstellar travel. mass will have been left to lose after the RGB phase, only On the face of it, this scheme seems far-fetched, but 0.116M⊙. Korycansky et al. (2001) show that it is in principle pos- Hence, for this non-dust-driven AGB solar mass-loss, sible, both technically and energetically, although currently we have adopted the same mass-loss description as above somewhat beyondourtechnical capabilities; however, there (equation(1)).Thismass-loss,incombinationwithoursolar isnoimmediatehurrytoimplementthescheme,whichcould evolution model, yields thefollowing prediction: during the await the development of the relevant technology. It would final30,000yontheverytip-AGB,whicharecrucialforany have the advantage of improving conditions for the whole build-up of sufficient CS (circumstellar) material to form biosphere,whereas anyschemeforinterplanetary ‘life rafts’ a PN, the solar giant will lose only 0.006M⊙. A further Distant future of Sun and Earth 9 0.0015M⊙ will be lost in just 1300 years right after a final ACKNOWLEDGMENTS thermal pulse (TP) on the tip-AGB. That marks the very KPS is grateful for travel support received from the As- end of AGB evolution, and it allows the solar supergiant tronomy Centre at Sussex through a PPARC grant, which briefly to reach a luminosity of 4170L⊙ and R = 179R⊙, with amass-loss rateof 10−6M⊙y−1,butwith Teff already enabled the authors to initiate this research project in the summer of 2006. We further wish to thank Jean-Paul Zahn increased to 3467K. Again, there will be no involvement for very helpful comments on his treatment of tidal friction of a dust-driven wind. Since common PNe and their dusty andAdamScaifeoftheMetOffice’sHadleyCentreforsug- CSenvelopesrevealadust-drivenmass-loss historyofmore like 10−5 to 10−4M⊙y−1 during the final 30,000 years of gesting changes toSections 1 and 3. tip-AGBevolution, we must conclude that theSun will not form such a PN. 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