A reappraisal of parameters for the putative planet PTFO 8-8695b and its potentially precessing parent star

0.87 MB·

by  Ian D. Howarth

MNRAS000,000–000(0000) Preprint28January2016 CompiledusingMNRASLATEXstylefilev3.0
A reappraisal of parameters for the putative planet
PTFO 8-8695b and its potentially precessing parent star.
Ian D. Howarth(cid:63)
Dept. of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
Accepted2016January27.Received2016January23;inoriginalform2015November16
6
1
ABSTRACT
0
Published photometry of fading events in the PTFO 8-8695 system is modelled using
2
improved treatments of stellar geometry, surface intensities, and, particularly, gravity
n
darkening, with a view to testing the planetary-transit hypothesis. Variability in the
a
morphologyoffadingeventscanbereproducedbyadoptingconvective-envelopegrav-
J
itydarkening,butnear-criticalstellarrotationisrequired.Thisleadstoinconsistencies
7
withspectroscopicobservations;themodelalsopredictssubstantialphotometricvari-
2
ability associated with stellar precession, contrary to observations. Furthermore, the

empirical ratio of orbital to rotational angular momenta is at odds with physically
]
R plausible values. An exoplanet transiting a precessing, gravity-darkened star may not
S be the correct explanation of periodic fading events in this system.
.
h Key words: stars: individual: PTFO 8-8695 – stars: planetary systems
p
-
o
r
t
s 1 INTRODUCTION mined, multiple plausible solutions of the intervening pre-
a
cessional motion exist (as had been anticipated by Barnes
[ Although the number of confirmed exoplanets is now in
etal.).BoththeBarnesetal.andtheKamiakaetal.models
the thousands, the discovery by van Eyken et al. (2012)
1 predictthat,asobserved,transitsshouldnotoccuratsome
of a possible hot Jupiter transiting the M-dwarf T-Tauri
v epochs, as a consequence of orbital precession.
star PTFO 8-8695 is of particular interest. Not only is the
0 Ciardi et al. (2015) have recently published follow-up
4 system exceptionally young (∼3 Myr; Bricen˜o et al. 2005), observationswhichdemonstratefurthertransit-likefeatures
5 whichisofsignificanceinthecontextoftimescalesforplan-
in the light-curve with the correct orbital phasing, albeit
7 etary formation and orbital evolution, but also it exhibits
at epochs not consistent with the specific precession model
0 variability and asymmetry in the transit light-curves ob-
advanced by Barnes et al. (2013); but while this paper was
. served in the two seasons of the Palomar Transient Fac-
1 being prepared, Yu et al. (2015) reported additional obser-
0 tory Orion project (PTFO; van Eyken et al. 2011). While vations which challenge the Barnes et al. framework. Thus
6 part of the variability may arise through intrinsic stellar ef-
PTFO 8-8695b remains, at best, only a candidate planet.
1 fects (such as starspots), Barnes et al. (2013) offered an in-
The purpose of the present note is to examine this issue
: sightful and credible interpretation that requires precession
v through more-detailed modelling of the stellar emission, to
of the orbital and rotational angular-momentum vectors on
i test, in particular, the gravity-darkening hypothesis.
X short timescales (∼102 d, to account for the variable tran-
r sitdepth)coupledwithasignificantlygravity-darkenedpri-
a mary (to generate the light-curve asymmetry).
2 MODEL
Barnesetal.(2013)constructedadetailednumericalre-
alisationofthismodel,includingperiodicprecession,which 2.1 Motivation
reproducedthevariablelight-curveextremelywell.Because
BothBarnesetal.(2013)andKamiakaetal.(2015)adopted
oftheirinterestinphysicallymodellingtheprecession,they
classical von Zeipel (1924) gravity darkening, in which the
constrained their model fits by adopting specific values for
emergent flux is locally proportional to gravity; that is,
the stellar mass; and in order to reduce the number of free
parameters they assumed (with some observational justifi- T ∝gβ
eff((cid:96))
cation)synchronousrotation.Kamiakaetal.(2015)relaxed
this assumption, and showed that, while the system geom- withβ =0.25 (where Teff((cid:96)) is to be understood asthe local
etry at the two observed epochs is reasonably well deter- effective temperature).
However, von Zeipel’s derivation was based on consid-
erationofabarotropicenvelopeinwhichenergytransportis
(cid:63) E-mail:[email protected] diffusive – i.e., radiative. Lucy (1967) argued that for stars
(cid:13)c 0000TheAuthors
2 Ian D. Howarth
withconvectiveenvelopesthegravity-darkeningexponentβ
is expected to be considerably smaller; this argument cer-
tainly applies in the case of PTFO 8-8695 (spectral type
∼M3; Bricen˜o et al. 2005).
Recent work suggests that von Zeipel’s ‘law’ may over-
estimate gravity darkening even in radiative envelopes (Es-
pinosa Lara & Rieutord 2011); and, while it may be ar-
gued that, in respect of convective envelopes, “nothing is
clear”(Rieutord2015),itissurelythecasethatthegravity-
darkeningexponentwillbelessthaninradiativeenvelopes.
The limited empirical evidence is broadly consistent with
Lucy’s estimate of β (cid:39) 0.08 (e.g., Rafert & Twigg 1980;
Pantazis&Niarchos1998;Djuraˇsevi´cetal.2003,2006),and
it is this value that will be adopted here.
Figure 1.Spin-orbitgeometry;they axiscoincideswithL,the
Thebest-fitparametersderivedbyBarnesetal.(2013;
sum of the orbital and stellar-rotation angular momenta (unit
their Table 2)1 imply a ratio of rotational angular veloc- vectorsoˆ,ˆs),whichlieinthexy plane.Thelineofsight,‘los’,is
ity to the critical, or break-up, value of ω/ωc (cid:39) 0.70 ± shownforprecessionalphaseψ;itisoffsetfromthetotalangular-
0.04, and thence an equatorial:polar temperature ratio of momentumvectorLbytheangleiL.Theobserver’s-frameincli-
∼ 0.90±0.02. To achieve the same temperature contrast nationanglesio(betweenthelineofsightandoˆ)andis(between
with β = 0.08 (and hence to achieve roughly the same de- thelineofsightandˆs)arenotshownexplicitly.
gree of light-curve asymmetries) requires significantly more
rapid rotation: ω/ω (cid:39) 0.95±0.02. Of course, any change
c
in ω/ω leads to changes in the shape of the star (and has
c
implicationsfortherotationperiodandtheprojectedequa-
Orbital motion is retrograde with respect to the stellar ro-
torial rotation velocity, v sini ), so it is not necessarily ob-
e s tation for ϕ +ϕ >π/2, prograde otherwise (where the ϕ
vious that a consistent solution to the light-curves can be s o
angles are defined in Fig. 1).
achievedwithamoreplausiblegravity-darkeningexponent.
Simple precession amounts to a rotation of o and s
about L; observationally, this is equivalent to counter-
rotation of the line of sight. Rather than impose a physi-
2.2 Implementation cal model of precession (which requires assumptions about
quantities such as the stellar moment of inertia), in the
To examine this issue, a modified version of the code
present work the precessional angle ψ is left as a free pa-
for spectrum synthesis of rapidly rotating stars described
rameter for each epoch of observation.
by Howarth & Smith (2001) has been used. The code,
exoBush,2 simply divides the rotationally distorted stellar
surface into a large number of facets; evaluates the local
temperatureandgravityateachpoint;andsumstheinten-
sities I(λ,µ,T,g)3 to produce a predicted flux, taking into 2.2.2 Stellar properties
account occultation by an opaque, nonluminous transiting
Barnesetal.(2013)approximatedthegeometryoftherota-
body of assumed circular cross-section (e.g., an exoplanet).
tionallydistortedstarbyanoblatespheroid;here,thestellar
surface is computed as a time-independent Roche equipo-
tential (cf., e.g., Howarth & Smith 2001), neglecting any
2.2.1 Spin-orbit geometry
gravitational effects of the companion. The global effective
The spin-orbit geometry is conveniently considered in a temperature is defined by
right-handed co-ordinate system defined by the angular-
(cid:90) (cid:30)(cid:90)
momentum vectors, as illustrated in Fig. 1. The convention T4 = T4 dA dA
eff eff((cid:96))
adoptedhereisthattheplanetary-orbitandstellar-rotation
angular-momentum vectors o, s lie in the xy plane, with
where the integrations are over the distorted surface area,
theirvectorsumLdefiningtheyaxis.Stellarrotationisas-
taking into account gravity darkening. For very rapid rota-
sumedtobeprograde(achoicethatisnecessarilyarbitrary,
tors this may not correspond to any particular ‘observed’
leading to ambiguities in several model parameters; cf. the
temperature (since the integrated line and continuum spec-
captiontoFig.3),andtheinclinationoftherotationaxisto
tra will not precisely match any single-star standards), but
thelineofsightisrequiredtobeintherange0≤i ≤π/2.
s
it is at least a well-defined quantity.
Values of T =3470 K and polar gravity log(g)=4.0
eff
(cgs) are adopted here, following Bricen˜o et al. (2005) and
1 ‘The’radiustabulatedthereinistheequatorialvalue(Barnes,
Barnes et al. (2013). These values enter the analysis only
personalcommunication).
through the calculation of the surface intensities, discussed
2 The etymology may be elucidated by an internet search for ‘a
below;otherwise,noassumptionsaremade,orarerequired,
thousandpointsoflight’,whichis,conceptually,howthetilingof
inrespectofspecificvaluesofthemassorpolarradius,and
thestellarsurfaceisperformed.
3 Hereµ=cosθ,whereθistheanglebetweenthesurfacenormal otherreasonablechoicesforlog(g)wouldhavenoimportant
andthelineofsight. effect on the results.
MNRAS000,000–000(0000)
PTFO 8-8695 3
Table 1. Summary of Markov-chain Monte-Carlo results, for
gravitydarkeningfixedatβ=0.08.
Parameter Distribution [Best]
Rs/a 0.518 +0.023/−0.031 0.515
Rp/a 0.1050 +0.0087/−0.0094 0.1045
ω/ωc 0.954 +0.012/−0.015 0.953
iL(◦) 111.9 +3.6/−4.8 111.6
ϕo(◦) 0.8 +2.0/−0.7 0.2
ϕs(◦) 108.2 +7.7/−4.9 110.2
ψ(◦,2009) 272.1 +2.7/−2.4 272.1
ψ(◦,2010) 298.1 +5.1/−4.3 298.8
ϕo+ϕs(◦) 109.1 +7.6/−4.5 110.4
is(◦,2009) 81.4 +2.8/−3.2 80.8
is(◦,2010) 57.9 +4.4/−4.9 56.8
io(◦,2009) 111.9 +3.6/−4.8 111.6
Figure 2.TheR-bandsurface-normalintensityasafunctionof io(◦,2010) 112.3 +3.8/−4.8 111.7
effectivetemperature.Filledandopencirclesaresolar-abundance
model-atmosphere results for log(g) = 4.5 and 2.5 dexcms−2, ‘Distribution’resultsarethemedianand95%confidenceintervals
respectively;thecontinuouscurveisthe(monochromatic)black- (from106MCMCreplications),whilethefinalcolumnlistsvalues
body result. Note that the y scale is logarithmic; the model- fortheindividualtrialmodelyieldingthesmallestrmsresiduals.
atmosphereintensitiescanshowlargedeparturesfromblack-body Rs/a and Rp/a are the stellar polar radius and the planetary
behaviour. radiusinunitsoftheorbitalsemi-majoraxis;ω/ωcistheratioof
stellar angular rotation to the critical value. Angles are defined
in Fig. 1; the sum ϕo +ϕs, and the stellar-rotation & orbital
inclinations, is & io, are derived quantities, not free parameters
inthemodel.
2.2.3 Intensities 3 ANALYSIS
Since the temperature and surface gravity must vary sig- 3.1 Observations
nificantly over the stellar surface (in order to generate
vanEykenetal.(2012)reportedR-bandobservationsof 11
the observed transit-curve asymmetries), the dependence
separatetransitsinthe2009/10observingseason,andafur-
of the emergent intensity on these quantities is of inter-
ther six in December 2010. Barnes et al. (2013) detrended
est. In this work, R-band surface intensities were evalu-
andaveragedtheseresultstoproducemean‘2009’and‘2010’
atedasI(µ,T ,g)byinterpolationinthe‘quasi-spherical’
eff((cid:96)) light-curves.Inordertoapproachascloseasreasonablypos-
limb-darkening coefficients (LDCs) published by Claret,
siblealike-for-likecomparisonwiththeirresults,theBarnes
Hauschildt & Witte (2012), supplemented by surface-
et al. composite light-curves were digitized and form the
normal intensities kindly provided by Antonio Claret. His
basis of the present analysis; the Barnes et al. ephemeris is
4-coefficient LDC parametrization (Claret 2000), which re-
consequently also adopted, as a fixed quantity. Because the
produces the actual I(µ) distributions extremely well, was
dispersion in the data appears not to be purely stochastic,
used.
all points were equally weighted.
The intensities derive from solar-abundance, line-
blanketed,non-LTEPhoenixmodelatmospheres(cf.Claret
et al. 2012). Fig. 2 shows that the model-atmosphere emis-
sivitiescandepartsubstantiallyfromblack-bodyresults,by
up to a factor ∼10 at 2.5 kK. Thus although the principal
3.2 Methodology
intention of the present analysis is to consider the conse-
quences of an appropriate treatment of gravity darkening, In the model, the stellar geometry is determined by ω/ω
c
the use of model-atmosphere intensities also represents a and by R /a, the polar radius expressed in units of the or-
s
noteworthy if minor technical improvement over previous bital semi-major axis; the exoplanet is characterized by its
work, which adopted black-body fluxes coupled to a single, normalized radius, R /a. Orbital/viewing geometry is de-
p
global, two-parameter limb-darkening law. fined by the angles ϕ , ϕ , i , and ψ (Fig. 1). The analysis
o s L
FortheBarnesetal.(2013)best-fitmodels,theimplied requires all free parameters to be the same at both epochs
equator–poletemperaturerangeis∼3650–3350K(forT = of observation, excepting the viewing angles ψ.
eff
3470K);overthistemperaturerange,themodel-atmosphere Preliminarycomparisonsbetweenthemodelandobser-
R-band surface-normal intensity ratio is ∼0.47, while the vationswerecarriedoutbyusingasimplegridsearch,guided
black-bodyratiois∼0.57.Relaxingtheassumptionofblack- by the Barnes et al. (2013) results. This pilot survey of pa-
body emission is therefore liable to counteract the drive to rameterspacewasfollowedbyaMarkov-chainMonte-Carlo
larger values of ω/ω required by adopting a smaller value (MCMC) analysis using a standard Metropolis-Hastings al-
c
for the β exponent. gorithm(defaultingto106 replicationsanduniformpriors).
MNRAS000,000–000(0000)
4 Ian D. Howarth
Figure 3. Upper panels: system geometry at the two epochs,
for the ‘best’ solution summarized Table 1; the unit of length Figure4.Upperpanel:Equatorialrotationvelocityasafunction
is the polar radius. The colour coding of blue- and red-shifted of mass, for ω/ωc = 0.954 and three possible rotation periods.
stellar hemispheres correspondsto prograderotation; retrograde Thevesinis measurementreportedbyvanEykenetal.(2012)is
rotationwouldgiverisetoidenticallight-curves,aswouldmirror shown(withits1-σ error)atthemassrangeadoptedbyBarnes
images of these panels. The locations of the transiting body at etal.(2013).
orbital phases 0.0 and 0.25 are shown, to indicate the direction Lowerpanel: correspondingpolarradii,togetherwiththeZAMS
oforbitalmotion.Theprojectionofthetotalangular-momentum mass–radiusrelationship(followingEkeretal.2015;Bertellietal.
vectorontotheplaneoftheskyisalignedwiththe−xaxisineach 2008).Thephotometricpolarradiusof∼1.0±0.2R(cid:12)isindicated
panel(andalmostcoincideswiththeorbitalangular-momentum (§4).
vector).
Centre panels: corresponding normalized model R-band light-
curvesandobservations.
Bottom panels: the poorer ‘best-fit’ models obtained with might expect, given the number of free parameters), but it
angular-momentum ratios Ls/Lo constrained to values of 1 and is also suggestive of possible limitations of the model.
2.5(q.v.§4.2).
4 DISCUSSION
3.3 Results
Phenomenologically, the solution obtained here provides a
A test run with β = 0.25 initially recovered essentially the reasonably satisfactory match between observed and pre-
same geometry as found by Barnes et al. (2013), although dicted normalized light-curves; however, it has physical im-
after ∼ 3×105 MCMC replications the fit migrated to an plicationswhichcastdoubtonthecompleteness,orcorrect-
unphysical(thoughstatisticallymarginallybetter)solution, ness, of the underpinning model.
involving a grazing transit of a planet ∼5× larger than its
parent star.
Theβ =0.08rundidnotsufferthisproblem,returning 4.1 Angular-momentum expectations
the parameter set summarized in Table 1. Fig. 3 illustrates
the implied geometry, and confronts the predicted light- The magnitude of the stellar-rotation angular momentum
curvewiththedata.Disappointingly,thetechnicalimprove- for a star of mass Ms and polar radius Rs is
ments to the basic Barnes et al. (2013) model implemented L =Iω(cid:39)β2M R2ω
here lead to a somewhat poorer overall match than they s g s s
achieved. In part, this is a consequence of requiring a con- where I is the moment of inertia, ω is the rotational fre-
sistent parameter set at both epochs (the two datasets can quency, and β is the fractional radius of gyration. A non-
g
be matched extremely well if modelled separately, as one rotating0.4M starapproachingthezero-agemainsequence
(cid:12)
MNRAS000,000–000(0000)
PTFO 8-8695 5
has β2 (cid:39)0.19 (Claret 2012), giving thesmallestallowedvalues.Figure3illustratestheoutcomes
g
oftwosuchexperiments,oneinwhichL /L wasfixedatthe
Ls =1.58×1043(cid:20) Ms (cid:21)3/2(cid:20)Rs (cid:21)1/2(cid:20)ω (cid:21)(cid:20) βg2 (cid:21) Barnesetal.valueof2.5,andoneinwhischiotwasrequiredto
kg m2 s−1 0.4M(cid:12) R(cid:12) ωc 0.19 be ≥1 (with the outcome that the chain settled on a value
very close to 1). Neither of these models, nor any others
where each bracketed term is intended to be of order unity
examined, can be considered as giving satisfactory fits.
(using values for mass and radius based on discussions in
van Eyken et al. 2012 and Barnes et al. 2013).
The magnitude of the planetary-orbit angular momen-
4.3 Consequences of stellar precession
tum for a planet of mass M with semi-major axis a and
p
orbital frequency ω (=2π/P ) is In the basic Barnes et al. (2013) model explored here, a
orb orb
largepartofthelight-curvevariabilitybetweenepochsarises
L =M a2ω ,
o p orb through precessional ‘nodding’ of the star (almost inde-
(cid:20) G (cid:21)2/3 pendently of the orbital angular-momentum issue discussed
(cid:39)M M2/3P1/3 √ (for M (cid:28)M );
p s orb 2π p s above). This nodding gives rise to two potentially observ-
able diagnostics. First, because of changes in sini , vari-
s
numerically, for P =0.4484d (van Eyken et al. 2012),
orb ability is expected in the projected equatorial rotation ve-
L (cid:20) M (cid:21)(cid:20) M (cid:21)2/3 locity, vesinis (by a factor ∼ 1.2 between the 2009 and
p =1.47×1042 p s 2010epochs).Thismaybeeasiertostudyspectroscopically
kg m2 s−1 3M 0.4M(cid:12) thantheRossiter–McLaughlineffect,becausethevariability
whence the rotational:orbital a(cid:88)ngular-momentum ratio is timescaleisverymuchlonger(allowingacquisitionofbetter
data).
L (cid:20) β2 (cid:21)(cid:20)ω (cid:21)(cid:20) M (cid:21)5/6(cid:20)R (cid:21)1/2(cid:20)3M (cid:21)
s (cid:39)10.7 g s s . Secondly, because the hotter polar regions of the star
L 0.19 ω 0.4M R M
o c (cid:12) (cid:12) p(cid:88) are presented towards the observer in 2010, the system is
Themajorsourceofuncertaintyinthisratioistheplanetary predicted to be brighter, by as much as ∆R=0m. 30 for the
mass, which is constrained only by the van Eyken et al. ‘best’solutionofTable1.ThePTFOphotometryisinclear
upper limit (M ≤[5.5±1.4]M ), but the bracketed terms contradiction with this prediction. Although there is signif-
p
are,cumulatively,unlikelytodifferfromunitybymorethan icant non-orbital variability, the observations shown by van
(cid:88)
perhaps a factor ∼3 or so.4 Eyken et al. (2012) fall in the range R (cid:39) 15.20±0.05 at
both2009and2010epochs(theirFigs.2and3;theextreme
peak-to-peakrangeisonly0m. 17).Thisisastrongargument
4.2 Angular-momentum results againstthebasicfoundationoftheBarnesetal.model:any
significantchangesinthetransitmorphologyresultingfrom
TheempiricalresultssummarizedinTable1,obtainedinthe
precessionofagravity-darkenedstararenecessarilyaccom-
absence of any constraint on the angular-momentum ratio,
panied by changes in the overall brightness5 – which is not
yield
observed.
(cid:20) (cid:21)
L sin(ϕ )
s ≡ p =0.014+0.036
L sin(ϕ ) −0.013
o s
4.4 Stellar rotation
(median, 95% confidence intervals). This is discrepant, by
almost three orders of magnitude, with the prediction of As anticipated, the solution with β = 0.08 requires a large
§4.1; furthermore, the negligible orbital precession implied (andreasonablywell-defined)valueforω/ωc.Theassociated
by the small value of ϕ is inconsistent with the absence of valuesofstellarmass,radius,andequatorialrotationarenot
o
transits at some epochs (e.g., Kamiaka et al. 2015). independent,butitisstraightforwardtocomputeconsistent
Reasonably extensive sampling of parameter space, in- sets of values for given ω/ωc and rotation period Prot. van
cludingseveraltensofmillionsofMCMCreplicationsstart- Eyken et al. (2012) found a signal with P =0.448 d in out-
ingfrommultipleinitialparametersets,encouragestheview of-transit photometry, suggesting the possibility of approx-
that the solution summarized in Table 1 locates the global imate rotational/orbital synchronization; Fig. 4 illustrates
minimuminχ2hyperspace.However(andparticularlygiven the stellar equatorial rotation velocity and polar radius as
that the model is constrained by observations only two functions of mass for this Prot, and for values that are a
epochs),thequestionarisesastowhetheraphysicallybetter factor two different in each direction.
model may exist with lower, but still acceptable, statistical For V0 (cid:39) 16.1, Teff (cid:39) 3.5 kK, and d (cid:39) 330 pc
probability – that is, does a preferable solution occur at a (Bricen˜o et al. 2005), the effective stellar radius must be
local χ2 minimum? ∼ 1.1±0.2 R(cid:12) (polar radius ∼ 1.0±0.2 R(cid:12)), as judged
Toinvestigate this issue,further solutions were sought, from marcs and atlas model-atmosphere fluxes (Gustafs-
againthroughtheMCMCprocessbutimposingavarietyof son et al. 2008; Howarth 2011; see also Barnes et al. 2013).
constraintsontheLs/Lo ratio.Inalltheseexperiments,the Supposing the stellar mass to be ∼0.4±0.05 M(cid:12) (Bricen˜o
angular-momentumratiowasfoundalwaystodrivetowards etal.2005;Barnesetal.2013),rotationmustindeedbeclose
to, or somewhat faster than, synchronous to match this ra-
dius (Fig. 4), which in turn implies an equatorial rotation
4 The Barnes et al. (2013) ‘joint solution’ (their Table 3) has velocity v (cid:38)160 kms−1.
e
Ls/Lo(cid:39)2.5,consistentwiththeiradoptionoftheSolarvaluefor
squared normalised radius of gyration (the ‘moment of inertial
coefficient’intheirterminology),β2=0.059. 5 TheBarnesetal.β=0.25solutionimplies∆R(cid:39)0m. 2.
g
MNRAS000,000–000(0000)
6 Ian D. Howarth
van Eyken et al. (2012) report vesinis = 80.6 ± ClaretA.,2000,A&A,363,1081
8.1 km s−1 from observations obtained in 2011 February. ClaretA.,2012,A&A,541,A113
If we suppose the inclination at that epoch to be close to ClaretA.,HauschildtP.H.,WitteS.,2012,A&A,546,A14
the 2010 December value, then v (cid:39) 95 km s−1. Rapid ro- Djuraˇsevi´cG.,Rovithis-LivaniouH.,RovithisP.,GeorgiadesN.,
e
Erkapi´cS.,Pavlovi´cR.,2003,A&A,402,667
tation may lead to underestimation of v sini (because a
e s
Djuraˇsevi´cG.,Rovithis-LivaniouH.,RovithisP.,GeorgiadesN.,
consequence of gravity darkening is relatively low visibility
Erkapi´cS.,Pavlovi´cR.,2006,A&A,445,291
ofequatorialregions;cf.,e.g.,Townsend,Owocki&Howarth
EkerZ.,etal.,2015,AJ,149,131
2004), but the discrepancy between observed and expected
EspinosaLaraF.,RieutordM.,2011,A&A,533,A43
equatorial velocities is too large to be explained by this ef- Gustafsson B., Edvardsson B., Eriksson K., Jørgensen U. G.,
fect. Though less secure than the photometric constraint, Nordlund˚A.,PlezB.,2008,A&A,486,951
this is therefore a further source of conflict between the HowarthI.D.,2011,MNRAS,413,1515
model and observations. HowarthI.D.,SmithK.C.,2001,MNRAS,327,353
KamiakaS.,etal.,2015,PASJ,67,94
LucyL.B.,1967,Z.Astrophys.,65,89
PantazisG.,NiarchosP.G.,1998,A&A,335,199
5 CONCLUSION RafertJ.B.,TwiggL.W.,1980,MNRAS,193,79
RieutordM.,2015,preprint, (arXiv:1505.03997)
The ingenious ‘precession + gravity darkening’ model pro-
TownsendR.H.D.,OwockiS.P.,HowarthI.D.,2004,MNRAS,
posedbyBarnesetal.(2013)tointerprettransitphotometry
350,189
of PTFO 8-8695 has been tested using a more appropriate
YuL.,etal.,2015,ApJ,812,48
characterizationofgravitydarkening,alongwithwithmore vanEykenJ.C.,etal.,2011,AJ,142,60
sophisticatedtreatmentsofsurfaceintensitiesandstellarge- vanEykenJ.C.,etal.,2012,ApJ,755,42
ometry. vonZeipelH.,1924,MNRAS,84,665
Although the normalized transit light-curves can still
beadequatelyreproducedbythemodel,thesolutionoffered
here has an implausibly small ratio of rotational to orbital
angular momenta. While other, physically more acceptable
solutionsarenotcompletelyruledout,reasonablyextensive
explorationofparameterspacehasfailedtolocateanysuch
solution.
Independently of this issue, the adoption of a smaller
gravity-darkening exponent than previously assumed leads
inexorably to the requirement of near-critical stellar rota-
tion. Such rapid rotation raises two further, and more gen-
eral, difficulties for the model. First, given the ‘known’ ra-
dius,theprojectedrotationalvelocityispredictedtobeap-
proaching a factor two greater than observed. Secondly, a
substantially gravity-darkened star must exhibit significant
photometricvariabilityassociatedwithprecessionofthero-
tation axis; no such variability is observed.
Collectively, these results suggest that either the basic
modelomitsimportantphysics,orthataconventionaltran-
sitingexoplanetisnotthecorrectexplanationforthefading
events in the PTFO 8-8695 system.
ACKNOWLEDGMENTS
IthankAntonioClaretforkindlyprovidingabsoluteintensi-
tiestosupplementhispublishedlimb-darkeningcoefficients;
Ingo Waldmann & Michel Rieutord for enlightening discus-
sions; and Jason Barnes & Julian van Eyken for helpful ex-
changes (the former both as pre-submission correspondent
and constructive referee).
REFERENCES
Barnes J. W., van Eyken J. C., Jackson B. K., Ciardi D. R.,
FortneyJ.J.,2013,ApJ,774,53
BertelliG.,GirardiL.,MarigoP.,NasiE.,2008,A&A,484,815
Bricen˜oC.,CalvetN.,Hern´andezJ.,VivasA.K.,HartmannL.,
DownesJ.J.,BerlindP.,2005,AJ,129,907
CiardiD.R.,etal.,2015,ApJ,809,42
MNRAS000,000–000(0000)