Obliquity Variability of a Potentially Habitable Early Venus

3.4 MB·

by  Jason W. Barnes

DRAFTVERSIONFEBRUARY2,2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
OBLIQUITYVARIABILITYOFAPOTENTIALLYHABITABLEEARLYVENUS
JASONW.BARNES1,BILLYQUARLES2,3,JACKJ.LISSAUER2,JOHNCHAMBERS4,MATTHEWM.HEDMAN1
DraftversionFebruary2,2016
ABSTRACT
Venus currently rotates slowly, with its spin controlled by solid-body and atmospheric thermal tides. However,
conditions may have been far different 4 billion years ago, when the Sun was fainter and most of the carbon within
Venuscouldhavebeeninsolidform,implyingalow-massatmosphere. Weinvestigatehowtheobliquitywouldhave
6
1 variedforahypotheticalrapidlyrotatingEarlyVenus.Theobliquityvariationstructureofanensembleofhypothetical
0 EarlyVenusesissimplerthanthatEarthwouldhaveifitlackeditslargeMoon(Lissaueretal.2012),havingjustone
2 primary chaotic regime at high prograde obliquities. We note an unexpected long-term variability of up to ±7◦ for
retrograde Venuses. Low-obliquity Venuses show very low total obliquity variability over billion-year timescales –
n
comparabletothatoftherealMoon-influencedEarth.
a
J
Subjectheadings:planetsandsatellites: Venus
0
3
1. INTRODUCTION high obliquity can act to stave off snowball states (Spiegl et al.
] 2015) and extreme obliquity variations may act to expand the
P
TheobliquityΨ—definedastheanglebetweenaplanet’sro- outeredgeofthehabitablezone(Armstrongetal.2014)bypre-
E
tationalangularmomentumanditsorbitalangularmomentum— ventingpermanentsnowballstates.
.
hisafundamentaldynamicalpropertyofaplanet.Aplanet’sobliq-
Thusknowledgeofaplanet’sobliquityvariationsmaybecrit-
puityinfluencesitsclimateandpotentialhabitability. Varyingor-
ical to the evaluation of whether or not that planet provides a
-bitalinclinationsandprecessionoftheorbit’sascendingnodecan
o long-term habitable environment. A planet’s siblings affect its
alter obliquity, as can torques exerted upon a planet’s equatorial
r obliquityevolutionprimarilyvianodalprecessionoftheplanet’s
tbulgebyotherplanets. TheEarthexhibitsarelativelystableand
sbenign long-term climate because our planet’s obliquity varies orbit. Obliquity variations become chaotic when the precession
aonly of order ∼3◦. As a point of comparison, the obliquity of periodoftheplanet’srotationalaxis(26,000yearsforEarth)be-
[Marsvariesoveraverylargerange—∼0◦–60◦(Touma&Wis- comes commensurate with the nodal precession period of the
planet’s orbit (∼100,000 years for Earth). Secular resonances,
1dom1993;Laskaretal.1993,2004).
v those that only involve orbit-averaged parameters as opposed to
3 Changes in obliquity drive changes in planetary climate. In mean-motion resonances for which the orbital periods are near-
5thecasewherethoseobliquitychangesarerapidand/orlarge,the commensurate,typicallyclustertogetherintheSolarSystemsuch
0resultingclimateshiftscanbecommensuratelysevere(see, e.g., thatifyouarenearonesecularperiod,youarelikelynearothers
0Armstrong et al. 2004). The Earth’s present climate resides at as well. And those clusters of secular resonances act to drive
0a tipping point between glaciated and non-glaciated states, and chaosthatincreasestherangeofaplanet’sobliquityvariations.
.thesmall∼3◦ changesinourobliquityfromMilankovicCycles
2 ThegravitationalinfluenceofEarth’sMoonspeedsthepreces-
drives glaciation and deglaciation of northern Europe, Siberia,
0 sion of our rotation axis, and stabilizes our obliquity. Without
andNorthAmerica(Milankovicˇ1998). Theseglacial/interglacial
6
this influence Earth’s rotation axis precession would have a pe-
1cyclesreducebiodiversityinperiodicallyglaciatedArcticregions
riodof∼100,000years,closeenoughtocommensurabilityasto
:(e.g., Araújo et al. 2008; Hawkins & Porter 2003; Hortal et al.
v2011).Theresultinginsolationshiftsjoltclimaticpatternsworld- drivelargeandchaoticobliquityvariability(Laskaretal.1993).
i Thoughourpreviouswork(Lissaueretal.2012)showedthatsuch
Xwide,causingspeciesinaffectedregionstomigrate,adapt,orbe
variationswouldnotbeaslargeasthoseofMars,thedifference
renderedextinct.
r betweencommensurateprecessionsandnon-commensuratepre-
a
Perhapsparadoxically,large-amplitudeobliquityvariationscan cessionsisstark.
also act to favor a planet’s overall habitability. Low values of
Atobe&Ida(2007)investigatedtheobliquityevolutionofpo-
obliquitycaninitiatepolarglaciationsthatcan,intherightcondi-
tentiallyhabitableextrasolarplanetswithlargemoons,following
tions,expandequatorwardtoenvelopanentireplanetliketheill-
onworkby(Atobeetal.2004)showingthegeneralizedinfluence
fated ice-planet Hoth in The Empire Strikes Back (Lucas 1980).
of nearby giant planets on terrestrial planet obliquity in general.
Indeed,ourownplanethasexperiencedso-calledSnowballEarth
Brasser et al. (2014) studied the obliquity variations in for the
states multiple times in its history (Hoffman et al. 1998). Al-
specificsuper-EarthHD40307g.
though high obliquity drives severe seasonal variations, the an-
nualaveragefluxateachsurfacepointismoreuniformonahigh- To expand the general understanding of potentially habitable
obliquity world than the equivalent low-obliquity one. Hence worlds’ obliquity variations, we use the only planetary system
that we know well enough to render our calculations accurate:
1Department of Physics; University of Idaho; Moscow, ID 83844-0903,
USA;ResearcherID:B-1284-2009;email:[email protected] our own. In this paper, we analyze the obliquity variations of a
2Space Science and Astrobiology Division MS 245-3; NASA Ames Re- hypothetical Early Venus as an analog for potentially habitable
searchCenter;MoffettField,CA94035,USA exoplanets.
2DepartmentofPhysicsandPhysicalScience,TheUniversityofNebraska
atKearney,Kearney,NE68849,USA VenuswaslikelyintheSun’shabitablezone4.5Gyrago,when
4DepartmentofTerrestrialMagnetism;CarnegieInstitutionofWashington;
the Sun was only 70% its present luminosity (Sackmann et al.
Washington,DC20015-1305,USA
2 Barnes,Quarles,Lissauer,Chambers,&Hedman
1993). Such an Early Venus could well have had a low-mass We select orbital initial conditions with respect to the J2000
atmosphere (with most of the planet’s carbon residing within epochwheretheEarth-Moonbarycenterresidescoplanarwiththe
rocks),andtideswouldnotyethavesubstantiallydampeditsspin ecliptic.Li&Batygin(2014b)andBrasser&Walsh(2011)inves-
rate. In fact, Abe et al. (2011) suggest that the real Venus may tigatedhowobliquityvariationsareaffectedbyalternateearlySo-
havebeenhabitableasrecentlyas1Gyrago,providedthatitsini- larSystemarchitectures,specificallytheNicemodel(Morbidelli
tialwatercontentwassmall(asmightresultfromimpact-driven et al. 2007). As an investigation of the long-term characteris-
dessication,asperKurosawa(2015),orbecausetheplanetislo- ticobliquitybehavior, however, weinsteadelecttointegratethe
catedwellinteriortotheiceline). presentSolarSystemorbits,whichareknowntomuchhigherac-
curacy.
In this work we numerically explore the obliquity variations
of Early Venus with a parameter grid study that incorporates a Lissauer et al. (2012) showed that chaotic variations in obliq-
widevarietyofrotationratesandobliquities. Notethatthiswork uity for a Moonless Earth can manifest from slightly different
is not intended to study Venus’ actual historical obliquity state, initialorbits. Wethusremovethiseffectbyusingacommonor-
information about which has been destroyed by its present tidal bital solution for all of our simulations. We assume the density
equilibrium (Correia et al. 2003; Correia & Laskar 2003). In- ofourhypotheticalVenustobethesameastherealVenus,5.204
stead,weuseVenuswithawiderangeofassignedrotationrates g/cm3. However, we assume a moment of inertia coefficient to
and initial obliquities as an analog for habitable exoplanets and bethesameasthatfortherealEarth(0.3296108;Ahrens1995)
to explore what types of obliquity behavior were possible for a given that a truly habitable Venus would likely have a different
potentiallyhabitableEarlyVenus. Ourmethodsbuildonthoseof internalstructurethantherealone. Wedonotvarythemoment
Lissaueretal.(2012)andaredescribedinSection2. Weprovide ofinertiawithrotationperiod.
qualitativeandquantitativedescriptionsofthedrivesofobliquity
We consider a range of rotation periods of between 4 and 36
variationsandchaosinSection3. Resultsofoursimulationsare
hours. The short end is set by the rotation speed at which the
presented in Section 4, and we conclude in Section 5. Readers
planetwouldbenearbreakup,whereourDarwin-Radauandax-
interestedprimarilyintheresultsmightconsiderjumpingtoSec-
isymmetricassumptionsbreakdown. Thelongerlimitrepresents
tion4,whilethosealsointerestedinthephysicsofwhyobliquity
a value 50% longer than Earth’s rotation, which itself has been
variescanaddSection3.
tidally slowed over the past 4.5 Gyr. In the epoch of Solar Sys-
tem history that we consider, Earth’s own day was significantly
shorterthanitistoday.
2. METHODOLOGY
Alongwithvariousrotationrates,wealsoconsiderinitialobliq-
2.1. Approach uityvaluesΨthatrangefrom0◦ to180◦. Planetswithobliquity
between90◦ and180◦ rotateretrogradetotheirorbitalmotions.
WetracktheevolutionofobliquityforthehypotheticalVenus Obliquityalonedoesnotcompletelydeterminetheorientationof
computationally, usinga modifiedversionof themixed-variable aplanet’sspinaxisinspace(unlessΨ=0◦ orΨ=180◦). There-
sympletic (MVS) integration algorithm within the mercury fore, for each obliquity we also consider various initial axis az-
package developed by Chambers (1999). The modified algo- imuths, ϕ, which correspond to the direction that the spin pole
rithmsmercury(forspin-trackingmercury)explicitlycalcu- points.
lates both orbital forcing for the 8-planet Solar System and spin
torques on one particular planet in the system from the Sun and Inordertogeneratetheproperinitialspinstates,wedefinethe
siblingplanetsfollowingTouma&Wisdom(1994). Ourexplicit angles of obliquity and azimuth. Lissauer et al. (2012) used a
numerical integrations represent an approach distinct from the similarapproach;however,thatpreviousstudywasfortheEarth-
frequency-mapping treatment employed by Laskar et al. (1993). Moonbarycenterwithzeroinitialinclinationrelativetotheeclip-
See Lissauer et al. (2012) for a complete mathematical descrip- ticplane.Incontrast,thedefinitionofspindirectionforanyother
tionofourcomputationaltechniques. planetrequirestwoadditionalrotationsthatincludethatplanet’s
inclination,i,anditsascendingnode,Ω,bothrelativetotheJ2000
The smercury algorithm treats the putative Venus as an ax- ecliptic.ThusthegeneralrotationmatricesR andR canbeused
1 2
isymmetricbody. Insodoing,weneglectbothgravitationaland todefinethedesiredobliquity,Ψ,andazimuth,ϕ:
atmospheric tides. Tidal influence critically drives the present-
day rotation state of real Venus (Correia & Laskar 2001). We 1 0 0  cosβ sinβ 0
areinterestedinanearlystageofdynamicalevolution,however,
R1(γ)=0 cosγ −sinγand R2(β)=−sinβ cosβ 0 .
where the tidal effects do not dominate. Therefore we consider
0 sinγ cosγ 0 0 1
onlysolarandinterplantarytorquesontherotationalbulge. The
(1)
simultaneousconsiderationoftidalanddynamicaleffectsisout-
The azimuthal angles, ϕ and Ω, undergo rotations via R with
sidethescopeofthepresentwork. 2
β=ϕorβ=Ω,wherethealtitudinalanglesarerotatedusingR
1
In the case of differing rotation periods, we incorporate the withγ=Ψandγ=i.
planet’sdynamicaloblatenessanditseffectsontheplanet’sgrav-
itationalfield. Theseeffectsmanifestastheplanet’sgravitational
coefficient J , values for which we determine from the Darwin- 3. OBLIQUITYEVOLUTION
2
Radau relation, following Appendix A of Lissauer et al. (2012).
Additionally,asinLissaueretal.(2012)weemploy“ghostplan- Aplanet’sobliquity,Ψ,isdefinedasthemagnitudeofthean-
ets” to increase the efficiency of our calculations — essentially gular distance between the direction of the angular momentum
wecalculateplanetaryorbitsjustonetime,whileassumingava- vector for a planet’s spin and that for its orbit. Therefore obliq-
rietyofdifferenthypotheticalVenusesforwhichwecalculatejust uity can change if either of those two vectors change direction:
the obliquity variations. We neglect (the very small effects of) (1)therotationalangularmomentumvectoror(2)theorbitalan-
generalrelativityandstellarJ . gularmomentumvector. Let’sconsidereachinturn.
2
2.2. InitialConditions 3.1. RotationalAngularMomentum
EarlyVenusObliquityVariations 3
Rotation LR93a CLS03 Lissaueretal.(2012)
Period(hr) J2 (''/yr) J2 (''/yr) J2 (''/yr)
4 1.405422e-02 99.94621 4.694002e-02 334.75028 4.692702e-02 334.63436
8 3.504570e-03 49.84532 1.036030e-02 147.76787 1.034730e-02 147.57222
12 1.555255e-03 33.18048 4.526853e-03 96.84904 4.513853e-03 96.56422
16 8.739020e-04 24.85893 2.536138e-03 72.34533 2.523138e-03 71.96950
20 5.588364e-04 19.87076 1.623182e-03 57.87820 1.610182e-03 57.41067
24 3.878196e-04 16.54781 1.129453e-03 48.32779 1.116453e-03 47.76823
30 2.480000e-04 13.22734 7.266291e-04 38.86438 7.136291e-04 38.16642
36 1.721063e-04 11.01537 5.082374e-04 32.62021 4.952374e-04 31.78363
Table1
Valuesforthezonalharmonic(J2)and‘precession’constant(α)determinedfromtherotationperiodforthemodelspresentedinLaskar&Robutel(1993),Correiaetal.
(2003),andLissaueretal.(2012).
7000
6000
) Mercury
m
5000 Venus
k
(q4000 Moonless Earth
e
R Mars
3000
2000
-1
2
J -2

0
1
g-3
o
l
-4
-5
80
300 r)60
y
yr)240 ''/40
''/180 (α20
( 0
α120
16 18 20 22 24 26
60 Rotation period (hrs)
0
10 20 30 40
Rotation period (hrs)
Figure1. Illustrationofhowthederivedstartingvaluesfortheequatorialradiusduetorotation(Req),zonalharmonic(J2),and‘precession’constant(α)varyinresponse
totheinitialrotationperiodfrom4to48hrsusingtheformalismgivenbyLissaueretal.(2012).Curvesareprovidedconsideringeachoftheterrestrialplanets(Mercury,
Venus,MoonlessEarth(asingleplanetwiththemassoftheMoonaddedtothatoftheEarth),andMars).
Torques on a planet’s rotational bulge from the Sun primar- InTable1weshowthevaluesofVenus’zonalharmonic(J )and
2
ilydrivechangesinthedirectionofthatplanet’srotationalaxis. precession constant (α) for both the present study and previous
Because the star must always be located within the plane of the workforarangeofrotationperiods. Ourvaluesstronglyresem-
planet’s orbit, however, these changes cannot directly alter the ble those of Correia et al. (2003) but differ substantially from
planet’sobliquityΨ. Instead,thestellartorqueinducestheplan- thoseusedbyLaskar&Robutel(1993)5. Figure1grapicallyrep-
etaryrotationaxistoprecessaroundtheorbitnormal,constantly resentstheequatorialradius,J ,andprecessionconstantαforour
2
changing the axis azimuth ϕ, but leaving the obliquity Ψ un- hypotheticalEarlyVenusesasafunctionoftheirrotationperiod.
changed. This effect is called the precession of the equinoxes.
Ingeneral,axialprecessionforVenusoccursabouttwiceasfast
Itiswhythedateoftheequinoxslowlycreepsforwardovertime,
asaxialprecessionforanequivalentplanetat1AU.Becausethe
andwhyPolarishasnotalwaysbeenneartheEarth’snorthpole
(see,e.g.,Karttunen2007). Sun’s gravity drives√axial precession, the fact that Venus’ semi-
major axis is nearly 2 AU explains the factor of 2 faster axial
The rate of axial precession depends on the planet’s dynami- precession. Functionally,forobliquityvariations,theSunspeeds
caloblatenessgravitationalcoefficientJ , themassanddistance Venus’axialprecessioninasimilarmannerthattheMoonspeeds
2
from the Sun, and the planet’s moment of inertia. The rate also
dependsweaklyontheobliquityitself;thereforeprecessionrates 5Laskar&Robutel(1993)providesaformalismtoderivethevalueofα,but
doesnotindicateaprecisedeterminationoftheequatorialflattening(C−A). In
aretypicallygivenintermsoftheprecessionconstantα,where C
ordertodeterminetheappropriatestartingvalues, weproduceapowerlawfit
usingtheirFigure5b.Fromthispowerlaw,theinitialvaluesofα(andhenceJ2)
ϕ˙ =αcos(Ψ) . (2) arereducedbyafactorof∼3.
4 Barnes,Quarles,Lissauer,Chambers,&Hedman
into near-commensurability with precession rates of the orbital
2
ascendingnode,leadingtochaoticobliquityevolution.
Ve
n
us
3.2. OrbitalAngularMomentum
1 Inasingle-planetsystem, neglectingtidaleffectsandthoseof
stellaroblateness,aplanet’srotationalaxiswouldmerrilyprecess
h aroundinazimuthϕataconstantrate,butitsobliquityΨwould
rt
a never change because the orbit plane would remain fixed. Thus
E
animportantmechanismforalteringplanetaryobliquityinvolves
p
0 theevolutionoftheorbit. Becauseobliquityistherelativeangle
between the rotation axis and the orbit normal, either changes
in the direction that the axis points in space or changes in the
direction of the orbit normal can each alter obliquity (see, for
instance,Armstrongetal.2014,Figure1).
−1
WeillustratetheevolutionoftheorbitplanesofbothVenusand
Earthfromananalytical,secularcalculationinFigure2.Figure2
showsthevariationsindirectionoftheorbitalangularmomentum
vectors over 500,000 years. The motions are of similar magni-
−2
tude. BecauseEarthandVenushavesimilarmassesandbecause
they each provide the primary influence on the orbital evolution
−2 −1 0 1 2 oftheother(e.g.,Murray&Dermott2000),theirorbitalpreces-
q
sions are qualitatively similar. Interestingly, Mercury drives the
second most important influence on both Venus and Earth ow-
Figure2. This plot shows the projected direction in which Venus’ (yellow)
andEarth’s(blue)orbitalangularmomentumpointsasitvariesoverthecourse ingtoitshighorbitalinclinationrelativetoboththeecliptic(the
of 500,000 years from the present day. The projection is in p-q space, with plane of Earth’s orbit) and the invariable plane (the plane of the
p≡Isin(Ω)andq≡Icos(Ω)whereΩisthelongitudeoftheascendingnode netangularmomentumoftheentireSolarSystem).
oftheorbitandIistheorbitalinclination.Theindicatedmotionrepresentsnodal
precession,whereaplanet’sorbitreorientsinspacelikeacoinspinningdownon TheorbitalvariationsofVenusandEarthinvolvesomechanges
adesktop.
in the orbital inclination of the two planets, represented by the
distanceofthelinesinFigure2fromtheorigin. Theprimaryef-
d
o 100,000 fect, though, iscounterclockwisenear-circularchangesthatcor-
eri Earth respondtotheprecessionoftheorbitthroughspace. Wecallthat
P effect nodal precession, as it drives monotonic increases in the

n elementknownastheorbit’sascendingnode,theangleatwhich
o 80,000
si theplanetcomesupthroughthereferenceplanefrombelow.
s
ces) We show the effective period of the nodal precession of the
rear60,000 orbits of Venus and Earth in Figure 3. Although the precession
Pe ratechangesastheorbitinclinationsofeachplanetvary,thelong-
al y termaverageprecessionrateforbothplanetsisinthevicinityof
odrth Venus ∼70,000years.
Na40,000
E

s(
u 3.3. SpinChaos
o
e
n 20,000 Through the integration of a secular solution and frequency
a
t analysis, Laskar & Robutel (1993) and Laskar (1996) showed
n
a thatchaoscanbeinducedwhentheaxial(spin)precessionalfre-
t
s quencies are commensurate with the secular eigenmodes of the
n 0
I SolarSystem(thedriversofnodalprecession). Specificallywhen
0 0.2 0.4 0.6 0.8 1.0
thespinprecessionfrequencycrossestheeigenmodesassociated
Time (Myr)
with secular frequencies s1 – s8 (0−26''/ yr) that are associated
withorbitalvariationsoftheplanets(includingnodalprecession),
Figure3. WhileFigure2showsthedirectionthattheorbitpolesofVenusand
Earth point to, it lacks a timescale. This plot provides such a timescale, as it chaoticobliquityevolutioncanresult. Asaresultofthisinterac-
depictstheinstantaneousprecessionperiod(i.e.,2π/dϕ forbothVenus(yellow) tion, the obliquity of our hypothetical Venus can vary substan-
dt
andEarth(blue)overamillionyears. BecauseVenusandEartheachrepresent tially. Weaker secular frequency eigenmodes can produce addi-
theprimaryinfluenceonthenodalprecessionoftheotherandbecausetheyare
tional chaotic zones, albeit smaller in amplitude and potentially
ofcomparablemass,thelong-termaverageprecessionratesforthetwoareabout
thesameat∼70,000years. withalongertimescaletodevelop(e.g., Li&Batygin2014a).
Figure 4a illustrates the chaotic zones as a function of initial
Earth’s. obliquity Ψ for a hypothetical Venus with a 20-hr rotation pe-
0
riod. They-axisrepresentstheactualaverageaxial(spin)preces-
An expectation might be that Early Venus’ obliquity varia-
sionrateϕ˙ inarcsecondsperyear. Positiveratesherecorrespond
tionsshouldmorecloselyresemblethatofreal-lifeEarthwiththe
to clockwise precession as viewed from above the orbit normal;
MoonthanthatofthemoonlessEarthfromLissaueretal.(2012).
negativeratescorrespondtocounterclockwiseprecession,asoc-
CircumstancesthatacttoslowVenus’axialprecessionfromthat cursforobliquitiesΨ>90◦(retrograderotation).
oftheprecessionconstant—suchasasmallerrotationalbulge,or
ahigherobliquity—couldacttobringtheaxialprecessionrate In general the curve of precession rates in Figure 4a varies as
EarlyVenusObliquityVariations 5
60 60
VENUS (a) VENUS (b)
40 40
s6
20 20
s3
s4
)
r s2
y
/ s1
yr) y ('' s8s7
/ 0 c 0 s5
(''˙ϕ uen
q
e
r
F
r1
20 20
r2
40 40
r3
r4
60 60
0 15 30 45 60 75 90 105 120 135 150 165 180 10 8 6 4 2 0
ψ Power
0
Figure4. Precessionfrequencies(a)forahypotheticalVenuswitharotationperiodof20hours. Panel(a)showstheaverageprecessionrateϕ˙ asafunctionofinitial
obliquityΨ0.Chaoticzonesappearforobliquitiesrangingfrom60◦−90◦thatcorrelatestotheprecessionfrequenciesrangingfrom0–26''/yrshowingcorrespondence
withthemainsecularorbitalfrequencies(b)oftheSolarSystem(Laskar&Robutel1993;Laskar1996).Thepowershowninthex-axisofpanel(b)islogarithmic.
a smooth cosine from +α to −α, as expected from Equation 2. bracket the 0''/yr-26''/yr chaos region for the 20-hour-rotation
However between ∼ 0''/yr and 26''/yr the obliquity becomes Venus,correlatingwiththechaoticzonesinFigure4a. Theareas
chaotic, ranging freely over this span as a function of time re- labeledr1-r4areclustersoflower-graderetrogradepeaksinthe
gardlessofwhereinthatregiontheinitialobliquitywouldplace frequencypowerspectrumthatwewilldiscussfurtherinSection
it.Forthisrotationrate,theprimarychaoticobliquityzoneranges 4.2.3.
fromΨ∼60◦toΨ=90◦.
Weshowasimilarplotfora24-hour-rotationMoonlessEarth
Thepowerspectrumoftheorbitangularmomentumdirection inFigure5forcomparison. TheMoonlessEarthplotshowsthe
vector (like that shown in Figure 2) is shown in Figure 4b. The previously known chaotic regions in ϕ˙ space, though their cor-
peaksinthispowerspectrumlabeleds1-s8correspondtoknown relation with the secular eigenfrequences is poorer than the hy-
Solar System secular eigenfrequencies that result from the 8 in- pothetical 20-hour-rotation Venus case. Li & Batygin (2014a)
teracting Solar System planets. The secular eigenfrequencies showedthatwhilethechaoticrangeofobliquitiesforaMoonless
6 Barnes,Quarles,Lissauer,Chambers,&Hedman
60 60
MOONLESS EARTH (a) MOONLESS EARTH (b)
40 40
s6
20 20
s3
s4
)
r s2
y
/ s1
(''/yr) 0 ency ('' 0 s8s7 s5
p u
q
e
r
F
r1
20 20
r2
40 40
r3
r4
60 60
0 15 30 45 60 75 90 105 120 135 150 165 180 10 8 6 4 2 0
ψ Power
0
Figure5. SimilartoFigure4,hereweshowtheprecessionfrequencies(a)andthepowerspectrumoftheorbitalangularmomentumvector(b)fora24-hour-rotation
MoonlessEarthforcomparisonwithourhypotheticalVenuses.
EarthdoesindeedextendfromΨ=0◦uptoΨ∼85◦,thechaotic 4.1. CoarseGrid
behaviorisnotuniformthroughoutthatrange.
WeinitiallyexploretheobliquityofhypotheticalEarlyVenuses
In fact, Li & Batygin (2014a) find two separate and indepen-
by numerically integrating the obliquity variations forward to
dentmajorchaoticzones: onefromΨ=0◦ toΨ=45◦, andone
+1 Gyr and backward to -1 Gyr over a coarse grid of rotation
from Ψ=65◦ to Ψ=85◦. While the region between these two
ratesandinitialobliquities. Weshowasummaryoftheresulting
majorzonesisalsochaotic,itisonlyweaklychaotic. Thatcon-
obliquityvariationsasafunctionofinitialobliquityandrotation
nectingregionservesasanarrow‘bridge’acrosswhichitispossi-
rateinFigures6and7. Thedifferencebetweenthetwofiguresis
bleforplanetstotraverse,thoughonlywithsubstantiallyreduced
theazimuthaldirectioninwhichtherotationaxisinitiallypoints,
probability(Li&Batygin2014a).
whicheffectivelycorrespondstowheretheplanetisinitsrotation
axisprecession(i.e.,theprecessionoftheequinoxes).
Weconsidertheresultsinthecontextofthevaluesforthepre-
4. NUMERICALRESULTS
EarlyVenusObliquityVariations 7
Initial Obliquity (deg.) -- ϕ = 0
o ◦
e
d 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
a 90
P = 4 hrs
r
g 60
o
r 30 +/- 1 Myr +/- 100 Myr +/- 1 Gyr
t
e 0
r P = 8 hrs

f 60
i
Ψ 30

- 0
0 P = 12 hrs
8 60
1
30
;
e
d 0
a P = 16 hrs
r 60
g
o 30
r
p 0
P = 20 hrs
f
i 60
Ψ
30

-
- 0

) P = 24 hrs
.
g 60
e
d 30
( 0
e P = 30 hrs
g 60
n
a 30
R
0
y P = 36 hrs
uit 60
q 30
bli 0
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
O
Initial Obliquity (deg.)
Figure6. ObliquityvariationofahypotheticalyoungVenusconsideringforvariousinitialobliquities(Ψ)androtationperiods(P).Wecalculatethevariationsfortwo
differentinitialazimuths: ϕ0=0◦isshownhere,andϕ0=180◦isshowninFigure7. Thecoloredbarsindicatetherangeofobliquityvariationover±1Myr(green),
±100Myr(red),and±1Gyr(blue).Obliquitiesgreaterthan90◦areconsideredtospininretrogradeandthoselessthan90◦areprograderelativetotheorbitalmotion.
Thelargestvariationsoccurforhighprogradeinitialobliquity,withthelargervariationsextendingtolowerinitialobliquityforslowerrotation. Someinitiallyhigh
progradeobliquitiesobtainretrograderotation,buttheydosotemporarily,withamaximumobliquityof∼95◦.
cessionconstantαshowninFigure1. ArapidVenusspinperiod chaoticregionsatlowerobliquity,butratherslightlyreducesthe
of 4 hours drives a considerably large equatorial bulge (oblate- maximum obliquity at the top of the chaotic zone and substan-
ness and J ), which in turn leads to a high precession constant tially reduces the minimum obliquity at the bottom of the zone,
2
of 335''/yr. As this value greatly exceeds any of the frequencies leading to a wider zone overall. Hypothetical Venuses that start
of significant power in the orbital angular momentum direction anywhere within the chaotic region have their obliquities vary
powerspectrum(Figure4b), nearlyalloftheresultingobliquity across the entire range from the lower limit to near 90◦ over a
variationsremainwithintightranges(±∼2◦,similartopresent- Gyr.
dayEarthobliquityvariabilitywiththeMoon)andnon-chaotic.
ThesesametrendscontinueasweproceeddownFigures6and
At very high obliquity, however, near-resonant conditions can 7tolongerrotationalperiods. From20-hrupthrough36-hrrota-
occurduetothecosΨdependenceinEquation2forrotationaxis tionperiods,theoverallextentofthechaoticzoneathighobliq-
precession. HenceforinitialconditionswithΨ =85◦ andΨ = uities grows. The upper limit of the primary chaotic region is
0 0
90◦ weseemoderatelyvariableandchaoticobliquityvariations, always above Ψ∼85◦, but the lower boundary extends all the
evenforthisfast4-hourrotationperiod. waydowntobelowΨ=45◦fora36-hoursiderialrotation.
Retrograde rotations for the 4-hour rotation period (Ψ>90◦) Interestingly, a new, more weakly chaotic region also appears
show very low variability. Similarly small variations were seen atslowerrotationrates.At20,24,30,and36hours,somesmaller
forretrogradeMoonlessEarths(Lissaueretal.2012). initialobliquitiesΨ belowtheedgeoftheprimarychaoticzone
0
showmoderatelyvariableobliquities. Whentheinitialobliquity
Proceedingtoslowerrotationratesof8,12,and16hours,the isΦ =50◦ inthe24-hourrotationcaseatϕ =180◦ (Figure7),
low-obliquityendofthechaoticregiondropstoΨ=75◦,Ψ=70◦, fori0nstance,theobliquityvariesintherange045◦≤Φ≤60◦over
and Ψ=65◦ respectively for initial axis azimuth of ϕ =180◦
0 ±1Gyr.
in Figure 7 (with similar results at ϕ =0◦ in Figure 6). This
0
downward expansion of the chaotic zone is consistent with the In the 30-hour and 36-hour period cases, this lower-obliquity
effectsoflowerobliquityonprecessionratefromEquation2. As weaklychaoticzonegrows.At36-hours,itincludesalloftheini-
the slower rotation reduces the planet’s J , it also diminishes its tialobliquitiessmallerthantheprimaryzone,from0◦≤Φ≤35◦.
2
precessionconstantα.HencealowerobliquityvalueΨcanresult Hypothetical Venuses with initial obliquities inside this weaker
in similar rotation axis precession rates as the high-obliquity 4- zone show increased obliquity variability at the ±∼15◦ level.
hourrotationcase. ButwiththeexceptionoftheΨ =10◦,ϕ =180◦ casethetotal
0 0
±1 Gyr variability does not encompass the entire extent of the
Importantly the slower rotation rate does not introduce new
weakerchaoszone. Thesecasesonlyshowmoderatelyincreased
8 Barnes,Quarles,Lissauer,Chambers,&Hedman
Initial Obliquity (deg.) -- ϕ = 180
o ◦
e
d 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
a 90
P = 4 hrs
r
g 60
o
r 30 +/- 1 Myr +/- 100 Myr +/- 1 Gyr
t
e 0
r P = 8 hrs

f 60
i
Ψ 30

- 0
0 P = 12 hrs
8 60
1
30
;
e
d 0
a P = 16 hrs
r 60
g
o 30
r
p 0
P = 20 hrs
f
i 60
Ψ
30

-
- 0

) P = 24 hrs
.
g 60
e
d 30
( 0
e P = 30 hrs
g 60
n
a 30
R
0
y P = 36 hrs
uit 60
q 30
bli 0
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
O
Initial Obliquity (deg.)
Figure7. SameasFigure6,butforϕ=180◦.
obliquity variability, similar to that of Moonless Earths, which past).
havebroadlycomparablenodalandaxialprecessionrates.
TheretrogradeΨ =120◦,Ψ =140◦,andΨ =160◦casesdis-
0 0 0
Although rapidly-rotating retrograde Early Venuses lack the playbehaviorqualitativelydifferentfromanyseenintheMoon-
large-scalevariationsfoundforhighprogradeobliquities,insome lessEarthcase,ontheotherhand. Inthesecases,ourhypotheti-
simulationstheirobliquitiesvarymuchmorethanthoseofEarth calVenus’obliquityvarieswithinarelativelytight±2◦ bandon
would have if it lacked a large moon and rotated in the retro- both short- and medium-term timescales. On longer timescales
gradesense. Inthe20-hour-rotation,ϕ =180◦case,forinstance approaching 10-100 Myr, however, the center of that tight band
0
(Figure 7), obliquity variations of similar magnitudes to those wandersaroundinobliquityΨspaceupto±10◦(intheΨ =140◦
0
in the weakly chaotic low-obliquity regime appear at Ψ =155, andΨ =160◦cases;theΨ =120◦caseislessadventurous).
0 0 0
Ψ =160, and Ψ =165 for instance. We did not expect to find
0 0 These odd retrograde cases and the chaotic Ψ = 60◦ and
chaoticobliquitybehaviorforretrograderotationsgiventhehigh 0
Ψ =80◦situationaredistinct. Intheprimarychaoticzoneobliq-
degreeofstabilityfoundinretrogradeMoonlessEarths(Lissauer 0
uity varies within a single chaotic subregion while periodically
etal.2012).
andvaryrapidlytraversingwidechaotic‘bridges’(Li&Batygin
2014a) to neighboring chaotic subregions. These transitions be-
4.2. CloserLookatVenuswitha20-HourRotationPeriod tweensubregionslastforonlyoforderasingleprecessionperiod,
or∼70,000yearsforhypotheticalEarlyVenus. Incontrast,the
4.2.1. Time-Series retrograderotatorswithΨ0=140◦andΨ0=160◦continuerapid,
short-term variations on 105-year timescales. But they slowly
Focusing on the results for Venus with a 20 hr rotation pe- varyinobliquityon107-yeartimescalesinsteadofnearlyinstan-
riod, which show unexpected chaos for some retrograde obliq- taneous alterationof their variations intoa new regime asin the
uities, Figure 8 shows the full ±1 Gyr time histories for the primarychaoticzone.
obliquity Ψ of hypothetical Venuses with initial obliquities Ψ
0 A glance back at Figure 4 shows that the unexpected retro-
spaced out every 20◦. The low initial obliquity cases Ψ =0◦,
0 gradelong-termvariabilitycorrespondstothelocationsofweak
20◦,and40◦,haveobliquitiesthatvarywithinnarrowrangesand
frequencies in Venus’ orbit variations. However, the power as-
show no chaotic long-term behavior. Similarly, the Ψ =100◦
0 sociated with those peaks is a factor of 106 lower than the pri-
andΨ =180◦ casesvaryuniformlywithinatightbandwithno
0 mary Solar System s eigenfrequencies. Furthermore, in the 24-
chaoticbehavioroneitherthemedium-orlong-term.
hour Moonless Earth case shown in Figure 5 for comparison,
The Ψ = 60◦ and 80◦ cases are within the primary chaotic similarpeaksinEarth’sorbitalangularmomentumdirectionfre-
0
region. These two cases bounce around between three smaller quencydonotyieldcorrespondingvariabilityforretrograderota-
chaoticsubregions(exceptfortheΨ =80◦ casewhichmanages tors. Similarly, notallhypotheticalVenusesshowthisbehavior;
0
to find its way out of the chaotic region beyond 550 Myr in the retrograderotatorswith4and8hourperiodsshowlittlelong-term
EarlyVenusObliquityVariations 9
Figure8. VariationsofhypotheticalVenus’sobliquity1Gyrintothefutureandpastforarangeofinitialobliquitieswithaninitial20hourrotationperiodandinitial
azimuthof180◦.Theinitialobliquitiesarefrom0◦−180◦forincrementsof20◦andhavebeencolor-codedtodistinguishbetweenoverlappingregions.
obliquityvariation. prograde initial conditions, the opposite is true once the planets
flip over into regrograde rotation at Ψ > 90◦. At retrograde
0
obliquities the Moonless Earth shows minimal obliquity varia-
4.2.2. FineGridinObliquity
tionsevenover100Myrtimescales.
Tofurtherinvestigatethisunexpectedchaoticobliquityevolu- Retrograde 20-hour Venus, on the other hand, shows broadly
tion in retrograde rotators, we ran additional obliquity variation stable obliquities but with four modestly more variable regions
integrations at very high resolution in initial obliquity Ψ0. In centeredaroundΨ0=100◦,Ψ0=120◦,Ψ0=135◦,andΨ0=158◦.
this section we analyze the 20-hour-rotation-period Early Venus Theseregionsseemtocoincidewiththelow-powerpeaksinor-
specificallybecauseitshowsthestrongestanomalousretrograde bital frequency space shown in Figure 4. However, we do not
variability. Figure9showsourresultsusingagridof361differ- at present understand why these peaks are important for Venus
entΨ0 valuesspacedoutevery0.5◦. Inthisportionoftheinves- butnotforMoonlessEarths,whichshowsimilarpeaksinorbital
tigation we elect to integrate out only to ±100 Myr to allow for frequencyspace.
improvedresolutioninΨ withinouravailablecomputingpower.
0
We show the results of this fine Ψ integration in Figure 9.
0 4.2.3. FineGridinRotationPeriod
We also show the analogous plot for Moonless Earths in Figure
10,seeingasLissaueretal.(2012)didnotperformsuchahigh-
For one last exploration of this unexpected retrograde behav-
resolutiongridofsimulations.
ior we do another set of integrations at high resolution, but this
Similartothecoarser-griddedLissaueretal.(2012)result,the time in rotation-period space. Starting with Ψ = 23.45◦ and
0
MoonlessEarthshowsmoderatelywide±∼10◦obliquityvaria- Ψ = 180◦−23.45◦, we show obliquity variations as a func-
0
tionsfromΨ =0◦ throughΨ =55◦ over±100Myr. Moredis- tion of rotation rate in high resolution in Figure 11. This plot
0 0
tinctchaoticsubregionsthenextenduptoΨ =85◦. Evenwhen showstheobliquityvariationsforhypotheticalVenusesoverthree
0
viewedatthishighresolution,though,theMoonlessEarthshows timescales:±1Myr(green),±100Myr(red),and±1Gyr(blue).
nosignsofanomalousbehaviorforretrograderotations(although
ForaretrogradeEarth-likeobliquityofΨ =156.55◦=180◦−
inretrospectFigure10fromLissaueretal.(2012)mayshowthe 0
23.45◦,threeindependentwide-variabilityregionsoccuratrota-
incipientonsetofsuchvariationsat1Gyrtimescales).
tion periods P of 18.8-21 hours, 23.5-28 hours, and 31.5-36+
rot
The 20-hour hypothetical Venus in Figure 9 shows very tight hours. Thesecorrespondrespectivelytother4,r3,andr2retro-
obliquityrangesfromΨ =0◦ throughΨ ∼55◦ orso,followed gradeprecessionfrequenciesfromFigure9b.Presumablyanother
0 0
bythesinglelargeprimarychaoticregionfromΨ =60◦toΨ = similarregionexistsforevenlongerrotationperiodscorrespond-
0 0
90◦ as discussed above. The smaller chaotic subregions reveal ingtor1.
themselveswhenlookingatshorter1Myrtimescales(green).
All told, while unexpected, these chaotic zones at retrograde
Althoughhypothetical20-hourVenusshowsasomewhatsim- rotationsonlyshowmodesttotalvariability—±7◦ intheworst
plerchaoticobliquityvariationstructurethanMoonlessEarthfor case over 1 Gyr. Understanding their origin is important for
10 Barnes,Quarles,Lissauer,Chambers,&Hedman
60
(a) VENUS max
40 mean
min
20
)
r
y
/ 0
(''
p
20
40
60
(b) 1 Myr
160
100 Myr
140
120
100
ψ 170
80
160
60
150
40
140
20
130
130 140 150 160 170
0
0 20 40 60 80 100 120 140 160 180
ψ
0
Figure9. Maximum,mean,andminimumvariationsdesignatedbytheup-triangle,filledcircle,anddowntriangle,respectively,in(a)theprecessionconstantand(b)the
obliquityforahypotheticalVenuswitha20hrrotationperiod. Thepointsarecolorcodedtosignifyhowtheparameterschangeover1Myr(green)and100Myr(red).
Theinsetin(b)showspotentialchaoticzonesthatappeartobepresentinsmallrangesofretrogradeobliquity.
evaluating the suggestion of Lissauer et al. (2012) that “if ini- habitable worlds. We show that even in the Solar System the
tial planetary rotational axis orientations are isotropic, then half increasedrotationalaxisprecessionratedrivenbyVenus’closer
of all moonless extrasolar planets would be retrograde rotators, proximity to the Sun is sufficient to push Venus into a benign
and these planets should experience obliquity stability similar obliquity variability regime. Indeed, Figure 4 for example indi-
to that of our own Earth, as stabilized by the presence of the catesthatforpresent-Earth-likeinitialobliquities(Ψ =23.45◦),
0
Moon."Whileourresultsshowthatthemostvariableretrograde the overall obliquity variability over 100 Myr for Venus with a
hypotheticalVenusesaremorestablethanthestandardMoonless 20-hour rotation period is similar to that for the real Earth with
Earths, the same may not be true for retrograde-rotating planets theMoon.
inallcases.
More rapid rotational axis precession will naturally result on
planets in the habitable zones of lower-mass stars. While these
5. CONCLUSIONS
stars’ gravity is proportionally lower, their disproportionately
fainter luminosities drive the habitable zone inward from that
We investigate the variations in obliquity that would be ex-
aroundthepresent-daySun. Thus,forsimilarorbitaldrivingfre-
pectedforhypotheticalrapidlyrotatingVenusfromearlyinSolar
quencies–i.e.,acloneoftheSolarSystemaroundalower-mass
System history. These hypothetical Early Venuses allow us to
star–stellargravityalonewouldbesufficienttopushahabitable
investigatetheconditionsunderwhichVenus’climatemayhave
planet’sobliquityvariationsintoabenignregime.
been sufficiently stable as to allow for habitability under a faint
youngSun. Of course tides provide a drawback to using stellar proximity
tospeedrotationalprecession. Inanyrealsystem,inadditionto
Additionally, they also serve as a comparitor for potentially
theobliquityvariationsthatwedescribehere,tideswillsimulta-
habitable terrestrial planets in extrasolar systems. While previ-
neouslyacttouprightaplanet’srotationaxisandslowitsrotation
ousworkonamoonlessEartheffectivelymodeledasinglepoint
rateTidaleffectswillbeevenmoreimportantonhabitablezone
of comparison, the present work provides a second comparator
planetsaroundlowermassstarsthantheyareforEarthandVenus
fromwhichwecanstarttoimagineamoregeneralresult. These
aroundtheSun.
intensive,single-planetstudiescomplementthoseofgeneralized
systems(Atobe&Ida2007). Henceapotentialavenueforfutureworkwillbetocouplethe
adiabaticobliquityvariationsthatwedescribeheretotidaldissi-
We show that while retrograde-rotating hypothetical Venuses
pationovertime. Giventhatthenaturalvariabilitywithinchaotic
showshort-andmedium-termobliquitystability,anunusualand
zones is much more rapid than tidal timescales, we suspect that
as-yet-understoodlong-terminteractiondrivesvariabilityofupto
theprimaryeffectoftideswillbethroughrotationalbraking. A
±7◦overGyrtimescales.
planetwithslowingrotationcouldtraversethroughvariousobliq-
Theverylowvariabilityoflow-obliquityhypotheticalVenuses uitybehaviorregimesoveritslifetime. Suchaplanetmightthen
over a range of rotation rates provides additional evidence that potentially have multiple possible interesting and chaotic path-
massivemoonsarenotnecessarytomuteobliquityvariabilityon waystowardtidallocking,asopposedtothesimplerslowobliq-