Distant future of the Sun and Earth revisited

0.25 MB·English

by  Klaus-Peter Schroder

Mon.Not.R.Astron.Soc.000,1–10(2008) Printed3February2008 (MNLATEXstylefilev2.2)
Distant future of the Sun and Earth revisited
K.-P. Schr¨oder1⋆ and Robert Connon Smith2†
1Departamento de Astronom´ıa, Universidad de Guanajuato, A.P. 144, Guanajuato, C.P. 36000, GTO, M´exico
2Astronomy Centre, Department of Physics and Astronomy, Universityof Sussex, Falmer, Brighton BN19QH, UK
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Accepted 2008....;Received200 ....;inoriginalform2007 September 25
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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 Schr¨oder & 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
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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, Schr¨oder, 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 rˆole of clouds, and their interaction with galactic
† E-mail:[email protected](RCS) cosmicrays(CR),mayalsobeimportant:thereisnowsome
2 K.-P. Schr¨oder 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 (Schr¨oder & 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, seeSchr¨oderetal. 1997, Polset al.
hanced by the fact (SSA) that when mass loss is included 1997, Schr¨oder 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
bySchr¨oder,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 su-
pergiant 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(seeSchr¨oder&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 atmo-
spheric 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)ofSchr¨oder&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 Schr¨oder &
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, Schr¨oder & 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. Schr¨oder 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 evolu-
tion 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 Schr¨oder 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. Schr¨oder 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 madebydS´anchez-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 Schr¨oder,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 Schr¨oder 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 de-
scription ofthedensitystructureofthefuturecool solargi-
ant.Fortunately,asitturnsout(seebelow),dynamicaldrag
will playonly aminorrˆole, verynearthesolar giant photo-
sphere, 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 signif-
icant 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 Schr¨oder 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. Schr¨oder 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 Schr¨oder & 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 Schr¨oder & 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 Schr¨oder 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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