Modeling giant extrasolar ring systems in eclipse and the case of J1407b: sculpting by exomoons?

1.8 MB·English

by  Matthew A. Kenworthy

ApJ accepted 2014 Dec 28
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
MODELING GIANT EXTRASOLAR RING SYSTEMS IN ECLIPSE
AND THE CASE OF J1407B: SCULPTING BY EXOMOONS?
M.A. Kenworthy
LeidenObservatory,LeidenUniversity,P.O.Box9513,2300RALeiden
and
E.E. Mamajek
DepartmentofPhysicsandAstronomy,UniversityofRochester,Rochester,NY14627-0171,USA
ApJ accepted 2014 Dec 28
5 ABSTRACT
1
Thelightcurveof1SWASPJ140747.93-394542.6,a∼16MyroldstarintheSco-CenOBassociation,
0
underwent a complex series of deep eclipses that lasted 56 days, centered on April 2007 (Mamajek
2
et al. 2012). This light curve is interpreted as the transit of a giant ring system that is filling up
n a fraction of the Hill sphere of an unseen secondary companion, J1407b (Mamajek et al. 2012; van
a
Werkhoven et al. 2014; Kenworthy et al. 2015). We fit the light curve with a model of an azimuthally
J
symmetric ring system, including spatial scales down to the temporal limit set by the star’s diameter
2 and relative velocity. The best ring model has 37 rings and extends out to a radius of 0.6 AU (90
2
million km), and the rings have an estimated total mass on the order of 100M . The ring system
Moon
has one clearly defined gap at 0.4 AU (61 million km), which we hypothesize is being cleared out by
]
R a <0.8M exosatellite orbiting around J1407b. This eclipse and model implies that we are seeing a
circumpla⊕netary disk undergoing a dynamic transition to an exosatellite-sculpted ring structure and
S
is one of the first seen outside our Solar system.
.
h Subject headings: planets and satellites: rings — techniques: high angular resolution — stars: indi-
p vidual (1SWASP J140747.93-394542.6) — protoplanetary disks — eclipses
-
o
r
t 1. INTRODUCTION Valio 2011), and searches for the transit timing varia-
s
tions caused by attendant exomoons are ongoing (Kip-
a Circumstellar disks of gas and dust are a ubiquitous
[ feature of star formation. Circumstellar gas-rich disks ping et al. 2012, 2013). 1SWASP J140747.93-394542.6
(hereafter J1407) is a pre-main sequence ∼ 16 Myr old
disperse on timescales of <10 Myr, effectively limiting
1 star, 0.9M , V =12.3 mag K5 star at 133 pc associated
therunawaygrowthphaseforgasgiantplanets(Williams
v with the Sc(cid:12)o-Cen OB Association (Mamajek et al. 2012;
& Cieza 2011). The architecture of the resultant plane-
2 tarysystemsisdictatedbythestructureandcomposition van Werkhoven et al. 2014; Kenworthy et al. 2015)1.
5
6 of the disk, its interaction with the young star, and the TheSuperWideAngleSearchforPlanets(SuperWASP)
5 competing formation mechanisms that transfer circum- database (Butters et al. 2010) showed that the star un-
0 stellar material onto accreting protoplanets (e.g. see re- derwent a complex series of eclipses lasting ∼ 56 days
. viewsbyArmitage2011;Kley&Nelson2012). Extended around May 2007 and including a dimming of > 95%.
1 month-toyear-longeclipsesindicatethepresenceoflong Mamajek et al. (2012) and van Werkhoven et al. (2014)
0 liveddarkdisksaroundsecondarycompanions,including proposethattheseeclipsesarecausedbyalargeringsys-
5 (cid:15) Aurigae (Guinan & Dewarf 2002; Kloppenborg et al. tem orbiting an unseen substellar companion, dubbed
1 2010), EE Cep (Mikolajewski & Graczyk 1999; Graczyk J1407b. In the first attempt to model the system us-
:
v et al. 2003; Mikolajewski et al. 2005), a precessing cir- ing nightly averaged photometry, Mamajek et al. (2012)
i cumbinary disk around KH 15D (Hamilton et al. 2005; posited at least four large rings girding J1407b. A more
X
Winn et al. 2006) and three systems recently discov- detailed analysis of the SuperWASP raw data by van
r ered in the OGLE database – OGLE-LMC-ECL-17782 Werkhovenetal.(2014)andremovalofa0.1magampli-
a
(Graczyk et al. 2011), OGLE-LMC-ECL-11893 (Dong tude, 3.2 day periodic variability due to rotational mod-
et al. 2014) and OGLE-BLG182.1.162852 (Rattenbury ulation by star spots shows temporal structure down to
et al. 2014). a limit of 10 minutes (van Werkhoven et al. 2014). Only
Gas planets are thought to form through accretion the 2007 eclipse event is seen in the time series photom-
from circumstellar disks composed of gas and dust. An- etry, and Kenworthy et al. (2015) place constraints on
gularmomentumofthecircumstellardiskmaterialisre- the possible mass and orbital period for this companion.
distributed through the formation of a circumplanetary TheyconcludethatJ1407bisalmostcertainlysubstellar
disk. After the gas is cleared out of the planetary sys- (at >3σ significance), and possibly an exoplanet.
tem, dust in the circumplanetary disk then accretes into The analysis of eclipse light curves to determine the
moons or remains as a ring system within the Roche structure of otherwise unresolved astrophysical objects
limitoftheplanet(Canup&Ward2002;Magni&Cora- is possible for specific cases. Since the first proposal in
dini 2004; Ward & Canup 2010). The transits of giant MacMahon (1908), high speed photometry of lunar oc-
planets with ring systems produces a distinct and de-
tectable light curve (Barnes & Fortney 2004; Tusnski & 1 http://exoplanet.eu/catalog/1swasp_j1407_b/
2 Kenworthy and Mamajek
cultations has been used (White 1987) to determine the theprojectedanglebetweenthenormalofthesecondary
multiplicity of stars close to the Ecliptic, structure of companion’s orbit and the normal of the ring plane is
evolved stars and studies of the Galactic centre. The φ . The obliquity ε of the ring plane is related to
disk
sharp inner ring edge of Saturn’s Encke gap has been these two angles by:
usedtodeducethewavelength-dependentradiusofMira cosε=sini cosφ
disk disk
usinghighspeedphotometryfromCassini(Stewartetal. Theringsareassumedtobeconsiderablythinnerthan
2013). The inverse problem of determining the extended their diameter i.e. a “thin ring” approximation. In Sat-
structure of a foreground object assuming a point-like urn’s ring system, thicknesses from tens of metres down
background source has been used to discern fine struc- to an instrument resolved limit of tens of centimetres
tures in the rings of the gas giant planets, measure the (Tiscareno 2013) have been observed. Pole on, the rings
scaleheightsoftheatmospheresinplanetarybodiessuch formaconcentricsetofcirclescenteredonthesecondary
as Titan and Pluto (e.g. McCarthy et al. 2008), and to companion. Thelightcurveofapoint sourcepassingbe-
determine the shape and orbital properties of Solar sys- hindtheringstructurealonganarbitrarychordisthere-
tem asteroids (e.g. Dunham et al. 1990; Shevchenko & fore symmetric in time about the point of closest ap-
Tedesco 2006). Most recently, two rings were discov- proach of the projected star position to the companion.
eredaroundanasteroidintheSolarsystem(Braga-Ribas What may not be so obvious is that a thin ring system
et al. 2014), where the structure in the rings themselves tiltedatanarbitraryinclinationwillalsoproduceasym-
are unresolved. In this paper, we use knowledge of an metric light curve, regardless of the chord chosen. This
extended background source and multiple ring structure can be seen when one considers that a set of concentric
aroundaforegroundobjecttoderivethegeometricprop- ellipses can be transformed into a set of concentric cir-
ertiesoftheJ1407bringsystem,acandidatecircumplan- cles by a single shear transformation whose shear axis is
etary disk. parallel to the chord.
In Section 2 we present our exoring model and then ThelightcurveI(t)ofasourcewithfiniteangularsize
discuss several features to be expected in a ring system (i.e. the stellar disk of the primary) behind a tilted ring
transit where the ring system is significantly larger than system, however, is not time symmetric (see Figure 1).
the parent star. In Section 3 we present our best fits For each ring boundary, the gradient of the light curve
to the J1407b transit data, and in Section 4 we discuss g(t) is dependent on both the size of the star, and angle
the structure in the most plausible ring models and we betweenthelocaltangentoftheringedgeandthedirec-
posit that they are indirect evidence for exomoons or tion of motion. Ring structures smaller than that of the
exosatellites coplanar with the rings. The large size of stellar diameter are smeared out by the resultant convo-
theringsystempresentsachallengetowhattypeoforbit lution with the stellar disk, resulting in a characteristic
it must have around the primary star, which we address timescale defined by the time taken for the stellar disk
in Section 5. Our conclusions are presented in Section 6. to cross its own diameter t =2R /v.
(cid:63) (cid:63)
The track of the star on the projected ring plane has
2. RINGMODEL
itsclosestapproachattimet withanimpactparameter
b
Our ring model is composed of two parts, which we of b, along with the ring orientation defined by i and
disk
solvesequentiallyforanobservedtransitingringsystem. φ . i is the inclination of the plane of the rings to
disk disk
We assume that the primary star and ring system are theplaneofthesky. φ istherotationoftheringsys-
disk
at a similar distance from the Earth, and that the ring temintheplaneoftheskyinananticlockwisedirection.
system is at least several times larger than the angular These four parameters uniquely define the relationship
sizeoftheprimarystar. Wefirstsolvefortheorientation between epoch of observation t and ring radius r. We
oftheplaneoftheringsystemrelativetoourlineofsight, model an azimuthally symmetric ring of radius r seen in
and we then solve for the transmission of the rings as a projectionbyanellipsewiththesecondarycompanionat
function of radius from the secondary companion, given the origin with a semi-major axis r and semi-minor axis
the geometry of the ring system derived in the previous rcosi. Thesemi-majoraxisoftheellipseisrotatedanti-
step. clockwisefromthex-axisbyanangleφ. Theparametric
equation for a projected ring is then:
2.1. Input parameters
The primary is a star at a distance of d parsecs, with
x(p)=r(cospcosφ −sinpcosi sinφ )
radius R . We approximate the orbit of the secondary disk disk disk
(cid:63)
for the duration of the eclipse as being a straight line,
with constant relative velocity of v. Surrounding the
secondary companion is a ring system in a plane that y(p)=r(cospsinφdisk+sinpcosidiskcosφdisk)
contains both the rings and the equatorial plane of the
where x and y are coordinates on the ring at radius r
secondarycompanion,whichwerefertoastheringplane.
at a given value of the parametric variable p that has a
Theringsarecomposedofindividualparticlesthatorbit
value from 0 to 2π radians.
the secondary companion in Keplerian orbits, assumed
The star moves along a line parallel to the x-axis:
circular. Theseparticlesscatterlightoutofanyincident
beam and in aggregate are approximated by a smooth x =v(t−t )
(cid:63) b
screen with an optical transmission of τ(r) that varies
y =b
(cid:63)
as a function of radial distance r from the centre of the
secondary companion. The rings are assumed to be az- Inadditiontothetimeofclosestprojectedapproachof
imuthallysymmetricandtheinclinationoftheringplane thestartothesecondarycompanion,therearetwoother
asseenfromtheEarthisi (with0obeingface-on)and significant epochs - the epoch when the stellar motion is
disk
Exorings 3
y
Star
(a)
Path of star behind rings Rings
b
✓
✓ ?
k
x
Secondary
I(t)
I
0
(b)
0 t
G(t)
(c)
t
0
t t t
b
? k
Fig. 1.—Thegeometryoftheringmodel. Panel(a)showsaringsysteminclinedatanangleofidisk androtatedfromthelineofrelative
velocity by φdisk. The star passes behind the ring system with impact parameter b at time tb. Panel (b) shows the resultant light curve
I(t) of the star as a function of time, demonstrating how the local ring tangent convolved with the finite sized disk of the star produces
lightcurveswithdifferentlocalslopes. Panel(c)highlightsthethreesignificantepochsintherateofchangeofringradiusr;tb atclosest
projected separation of the star and the secondary, t⊥ where the ring tangent is perpendicular to the direction of stellar motion, and t(cid:107)
wherestellarmotionistangenttothering. t(cid:107) alsomarkswherethestellarpathtouchesthesmallestringradius.
perpendicular to the ring-projected ellipse t , and the
epoch t when the stellar motion is tangen⊥tial to the tanp =(cosi tanφ ) 1
disk disk −
ring-pro(cid:107)jected ellipse. (cid:107)
To find where the tangent of the ring is perpendicular and
to the y-axis, we see where dx/dp = 0. The result is
y(p )
then:
tanθ = (cid:107)
(cid:107) x(p )
tanp =−cosi tanφ (cid:107)
disk disk
⊥ Finally, for distances where the star/secondary com-
Since nested rings only differ in a single scale factor panion distance is much larger than the impact param-
centered on the origin, the loci of all perpendicular tan- eter (i.e where x tends to very large values with y = 0)
gentslieonastraightlinepassingthroughtheorigin,i.e. the gradient dy/dx tends towards an asymptote defined
the geometric centre of the rings. The angle θ can be by:
calculated by substitution: ⊥
y(p ) dy(cid:12)(cid:12) 2(1+cot2φdiskcos2idisk)
tanθ⊥ = x(p⊥) tanθy=0 = dx(cid:12)(cid:12)y=0 = sin2φdisk
⊥
Asimilarderivationgivestheangleforthelinepassing These three regimes are shown in the lower panel of
through the loci of all parallel tangents, θ : Figure 1 as the time of largest gradient, the gradient
(cid:107)
4 Kenworthy and Mamajek
√
touching at zero, and the asymptotic gradient at large 1.22 λ/2d,givinganangularfringeseparationat62±4
positive and negative values along the x-axis. nanoarcsec, about 1000 times smaller than the angular
Asimpleringmodelisuniquelydefinedwithfournum- diameter of the star. Using geometric shadows is there-
bers - the orientation of the ring system (the inclination fore a valid approximation for the model.
and obliquity), the impact parameter b and the epoch of
closest projected approach to the secondary companion
tb. The transmitted intensity of light through a ring at 3.1. Fitting the ring orientation using light curve
radius r with τ(r) is: gradients
I(r)=I e τ(r)
0 − For a given set of ring orientation parameters
Since we do not know if the rings are a single thin
i ,φ ,b,t , we can determine the radial distances
disk disk b
screenofparticlesorareopticallythick,wedonotcorrect
of the rings from the secondary companion (i.e. the ring
τ for the inclination of the ring system.
radius) at any epoch, r(t) = f(i ,φ ,b,t ,t), and
disk disk b
The greatest uncertainty in the model fitting is the di-
also determine dr(t)/dt. The transmission of the disk as
ameter of the star. We therefore express the size of the
a function of r is given by τ(r). Together with a model
rings in units of time, converting back to linear sizes at
of the stellar disk that includes limb darkening (see van
the end of the modeling. With an assumed stellar size
Werkhoven et al. 2014) and the functional form of τ(r),
and relative velocity, the system can be converted back
we can calculate the light curve of a ring system model
into units of length by multiplying by v. A light curve
for any epoch. In practice, we calculate a grid of val-
model for a given star is produced with the diameter
ues of r for the track of the star behind the ring system,
of the star, its relative velocity with respect to the sec-
with a spatial resolution set by the diameter of the star.
ondary companion, the limb darkening parameter, the
The transmission I(r) is calculated for all points in the
orientation and the radial transmission of the ring sys-
gridalongthetrack. Thisgridoffluxvaluesisthencon-
tem.
volved with a model of the stellar disk, and the line of
3. THELIGHTCURVEOFJ1407 pixels along the impact parameter b then represents the
measuredfluxI(t). Weuse25pixelsacrossthediameter
The complex light curve of J1407 was first discussed
ofthestartosamplethelimbdarkeningandstellardisk.
in Mamajek et al. (2012), where intensity fluctuations
ThemeasuredfluxI(t)ofJ1407canbecloselyapprox-
of up to 95% were seen over a 56 day period in April
imated as a sequence of straight lines of different gradi-
and May 2007 towards this young (∼ 16 Myr) K5 pre-
ents (see van Werkhoven et al. 2014, for details). We
main sequence star in photometry taken as part of the
interpret these straight line light curves as a ring edge
SuperWASP Survey (Pollacco et al. 2006; Butters et al.
(betweentworingswithdifferentvaluesoftransmission)
2010). After ruling out other simpler astrophysical ex-
passing across the disk of the star. When the slope of
planations, Mamajek et al. (2012) concluded that the
thelightcurvechanges,thisrepresentsaringedgeeither
light curve was due to the transit of a giant ring system
starting or finishing its transit of the stellar disk. The
orbiting an unseen secondary companion, and that this
measuredgradientsofstraightlinefitstotheJ1407light
ringsystemwasconsiderablylargerthanthediameterof
curve are shown as the black circles in Figure 3. We as-
the central star. The star was simultaneously observed
sumethatalltheringshavewelldefinededgesandhavea
by three of the eight cameras in the SuperWASP South
constant transmission across the width of the ring (with
array, and due to its location in the corner of the field of
some assumed constant τ) - see Figure 2.
view of these three cameras, there is a clear systematic
To understand our ring orientation algorithm, con-
offsetof0.3magnitudesseenbetweenthecamerasinthe
sider a ring system made up of alternately transparent
standarddatareductionphotometry. Adedicatedrepro-
and opaque rings whose radial width would allow com-
cessing of all the raw photometric data successfully re-
pleteobscurationortransmissionofthestellardisk. The
movesboththesystematicoffsetsseeninthelightcurves
transmitted intensity goes from I = I to I = 0 and
and also the much smaller amplitude stellar variability 0
vice versa at a rate determined by the ring velocity v
(van Werkhoven et al. 2014). We use the cleaned pho-
and the local tangent of the ring to the line of stellar
tometric data (van Werkhoven et al. 2014) as the input
motion, defined as parallel to the x-axis in our model. If
for our ring fitting model. The observations over the
the gradient of the light curve is measured close to the
54 day period are not continuous but are interrupted by
midpoint of the transit of a ring edge, and this quantity
the diurnal cycle and cloud cover at the observing site,
is plotted as a function of time, the result is the black
resulting in a completeness of 11.3%.
curve in Figure 3.
Fitting is performed in a two step process - we first
Defining the angle between the tangent of the ring at
constrain the orientation of the ring system using the
point (x ,y ) to the x-axis as θ (see Figure 2):
gradients measured from the light curve (Section 3.1), (cid:63) (cid:63)
and then use these parameters to generate a model of
the ring transmission as a function of radius from the
tanθ =dy/dx
secondary companion (Section 3.2).
The angular diameter of the star is 65.2±9.3µarcsec,
based on a distance d = 133 ± 12 pc and radius of Theareasweptoutbyastraightedgeintimedtacross
0.99±0.11R (van Werkhoven et al. 2014; Kenworthy a stellar disk with no limb darkening is equal to 2R s
(cid:63)
et al. 2015). (cid:12)Treating a transiting ring as a semi-infinite where s = vdtsinθ. Defining the transmission T =
n
knifeedge,andassumingapointsourcebehindtherings, I /I = exp(−τ ), the change in intensity dI is simply
n 0 n
the angular separation between the geometric edge of the change in transmission from T to T over the area
1 2
the ring shadow and the first diffraction maximum is 2R vsinθdt, so the rate of change of intensity is:
(cid:63)
Exorings 5
⌧ ⌧
1 2
✓
v
s
R
?
t
Fig. 2.—Geometryofaringedgecrossingastellardiskofuniformintensity. ThestellardiskisofuniformilluminationwithradiusR(cid:63)
andwedonotshowlimbdarkeninginthisexample. Theringedgemovesatavelocityvacrossthedisk. Theringedgeisatanangleθto
thedirectionofmotionv. Theblackstriphasawidthofs. Thetworingshaveabsorptioncoefficientsofτ1 andτ2.
model point. If we define δ =G(t)−g(t), then our cost
t
(cid:18) func(cid:19)tion ∆ is:
dI(t) 2vsinθ 12−12u+3πu
g(t)= =(T −T )G(t)=(T −T )
dt 1 2 1 2 πR(cid:63) 12−4u n (cid:26)
(cid:88) δ if δ >0
(1) ∆= t t
−50∗δ otherwise
The limb darkening of the star is parameterised by t
t=1
u=0.8(5) for J1407 (Claret & Bloemen 2011).
ThefunctionG(t)representsthemaximumfluxchange The factor of 50 used in the above equation reflects
possible between a fully transparent and fully opaque the error of 2% on the measured light curve gradients,
ring. For rings that have intermediate values of trans- and prevents fitting algorithms from oscillating near lo-
mission, the resultant light gradients will lie underneath calminima. WeuseanAmoebasimplexalgorithm(Press
t
this black curve, and so the curve represents an upper etal.1992)tosolveforthefourfreeparameters. There-
bound on the light gradient for a given ring orientation. sulting behaviour of the cost function is to bring down
Since we do not know τ(r), we can use G(t) as an upper the model curve G(t) so that at least two measured gra-
limit and we search for ring orientations that have all dientpointsfromg(t)lieonG(t),whichmaynotbenec-
measured gradients lie underneath this curve. essarily correct if the data is sparse and the measured
We have n measurements of the light curve gradient slopes do not sample the largest gradient of G(t) at t .
g(t)attimet. ThemodelgradientG(t)iscalculatedfrom InthecaseofJ1407,thelargestobservedgradientisdu⊥r-
Equation 1, where θ is a function of i ,φ ,b,t ,t. ing MJD 54220, and the complete set of parameters for
disk disk b
We calculate a cost function that minimises the differ- the disk geometry is listed in Table 1. It is highly prob-
ence between the model and measured gradients, and able that this is the largest gradient in the ring system
penalizes heavily if the measured point goes above the sincethesecondarycompanionvelocityderivedfromthis
6 Kenworthy and Mamajek
3.0
2.5
y]
a
d
/
L? 2.0
[
nt
e
di
a
gr 1.5
e
v
ur
c
ht
Lig 1.0
0.5
0.0
54150 54200 54250 54300
HJD-2450000[Days]
Fig. 3.—ThemeasuredgradientsinthelightcurveofJ1407plottedasafunctionofMJDofobservation. Thelineshowsthemaximum
allowed gradient of the light curve G(t) for a given set of disk parameters t(cid:107), t⊥, idisk and φdisk. Dotted lines connect the black circle
measuredvaluestothemaximumallowedopencircletheoreticalmaximumvaluesontheblackline.
are estimated by counting the number of slope changes
TABLE 1
identified in the light curve and indirectly implied by
Disk model parameters
the change of the light curve during daylight hours. At
least 24 ring edges are required for the number of gradi-
Model b tb idisk φdisk t(cid:107) v
(d) (d) (deg) (deg) (d) km.s−1 ent changes detected in the J1407 data (van Werkhoven
et al. 2014), but given the sparseness of the photometric
1 3.92 54225.46 70.0 166.1 54220.65 33.0
coverage this number is almost certainly higher.
Using the derived ring geometry parameters, the ab-
gradient presents a challenge to the orbital dynamics for solute value of the time since the closest approach of
rings (see van Werkhoven et al. 2014; Kenworthy et al. the secondary companion, abs(t−t ), is used as the ori-
2015, foradetaineddiscussion). Wethereforeintroduce gin for a graph that displays the o(cid:107)bserved light curve,
anadditionalcostfunctionthatfixesthemidpointofthe the model light curve, and the difference of these two
eclipse light curve t to a user defined value, so that we curves. Displaying the data and model in this way al-
can explore differen⊥t ring geometries that still produce lows a direct visual comparison between the ingress and
reasonable cost functions. the egress of the star about the time of closest approach
The minimum velocity required to cross a limb- of the secondary companion t . In the limiting case of a
darkened star is derived in van Werkhoven et al. (2014), point-likebackgroundsource,(cid:107)theingresslightcurveand
and for the case of J1407, Equation 12 gives a rela- egresslightcurveareidenticalforazimuthallysymmetric
tive velocity of 33km.s−1 for R(cid:63) = 0.99R and L˙max = ring systems. The finite diameter of the star breaks this
3.1L /day. We adopt these values for our(cid:12)model. degeneracy and so the ingress and egress light curve for
∗ a given ring edge radius are not identical. We generate
3.2. Fitting the ring structure an initial estimate of τ(r) using a GUI written in the
The ring radius r(t) can be calculated with values for Python programming language. This is defined as:
i ,φ ,b,t estimatedfromthediskfittingprocedure
disk disk b
of the previous section. We now look for the ring trans-
missionasafunctionofradiusbyusingourmodelofr(t), τ(r)=τ for r <r <r with r =0 and n=1 to N
n n 1 n 0 rings
thestellarradiusandlimbdarkeningprofileofJ1407,and − (2)
an estimate of the transverse velocity v. We adopt the Ring edges and transmission values are interactively
limbdarkeningprofileandparametersofvanWerkhoven added, moved and deleted as appropriate and the ring
et al. (2014) and a stellar radius of 0.99R (Kenworthy model I(t) is generated after each successive operation.
et al. 2015). The number of ring edges in t(cid:12)he light curve Avisualinspectionofthemodellightcurveanditscom-
Exorings 7
with a minimum around MJD 54220. Using photometry
TABLE 2
averaged in 24 hour bins, a ring model with four broad
Table of ring parameters
rings is consistent with data on these timescales (Ma-
majek et al. 2012). On hourly timescales, the large flux
RingOuterEdgeRadius Tau
(106 km) variations are consistent with sharp edged rings cross-
ing over the unresolved stellar disk. We do not find an
30.8 4.65
azimuthally symmetric ring model fit that is consistent
31.3 0.93
32.5 2.18 with all the photometric data at these timescales. Due
33.0 0.52 to the incomplete photometric coverage, there are sev-
34.0 1.47 eral models that fit with similar χ2 values to the data.
35.4 3.61
Inallofthesecases,weseethepresenceofrapidfluctua-
36.2 1.01
37.4 0.24 tions in the ring transmission both as a function of time
38.0 1.07 andasafunctionofradialseparationfromthesecondary
39.1 0.21 companion.
40.6 0.69
Therearecleargapsinalltheringmodelsolutionsex-
42.3 0.40
42.6 1.01 plored. Gaps in the rings of Solar system giant planets
43.5 0.37 are either caused directly by the gravitational clearing
46.6 0.72
of a satellite or indirectly by a Lindblad resonance due
48.0 1.53
49.5 0.38 to a satellite on a larger orbit. The J1407 ring system
50.4 2.66 is larger than its Roche limit for the secondary compan-
51.2 0.03 ion. A search for the secondary companion is detailed in
51.7 1.54
Kenworthy et al. (2015), and the constraints from null
52.9 0.80
53.5 0.00 detections in a variety of methods result in a most prob-
53.9 2.94 able mass and orbital period for the secondary compan-
55.3 0.70 ion. These orbital parameters are summarised in Table
57.2 0.30
3. We take the most probable mass and period for the
59.2 0.59
61.2 0.00 moderaterangeofeccentricitieswithmass23.8MJup and
63.0 0.50 orbital period 13.3 yr, although we note that the period
65.5 0.21 couldbeasshortas10yearsandthemasscanbegreater
66.7 0.11
than 80 M , but mass this is considered highly un-
68.9 0.12 Jup
75.4 0.08 likely with a probability of less than 1.2%. Gaps in the
78.5 0.25 ring system are either seen directly as the photometric
83.0 0.06 fluxfromJ1407returningtofulltransmissionduringthe
90.2 0.52
eclipse,orindirectlyasafitofthemodeltointermediate
transmission photometric gradients. One ring gap with
parison to the data is carried out, and when the mini- photometry is at HJD 54210, seen during the ingress of
mum number of rings are added to the model, the ring J1407behindtheringsystem. Thecorrespondingradius
transmissionvaluesareoptimizedusingaminimumleast for this gap in the disk is seen from 59 million km to 63
squares fit to the data and Amoeba simplex algorithm. million km (indicated in Figure 6), corresponding to an
There are 16489 photometric data points covering orbital period Psat of:
the 2007 observing season of J1407 (MJD 54131.96 to
54306.72). The data of J1407 does not show significant (cid:18) MJ1407b (cid:19)−1/2
changesinfluxphotometryoverthetimescaleof30min- Psat =1.7yr
23.8M
utes, and so to speed up the interactive fitting, the pho- Jup
tometric data is re-binned in 0.02 day (32 minute) inter- If we assume that the gap is equal to the diameter
vals with error estimates calculated from the r.m.s. of of the Hill sphere of a satellite orbiting around the sec-
the photometric data points within each bin. Bins con- ondary companion and clearing out the ring, then an
taininglessthanthreephotometricpointsarediscarded, upper mass for a satellite can be calculated from:
resulting in a data series of 985 points. The photometric
data is sparse (the binned data covers 11.3% of the to-
(cid:18) (cid:19)3 (cid:18) (cid:19)
tal 2007 season), and so no unique solutions for the disk m ≈3M dhill =0.8M MJ1407b
geometry and τ(r) are found.2 sat b 2a ⊕ 23.8MJup
InFigures4and5wepresentonepossibleringsolution
(listed as Model 1 in Table 1) to the J1407 photometric Forthecaseof23.8MJup thiscorrespondstoasatellite
data, where the central eclipse is set at t = 54220.65 mass of 0.8M and an orbital period of 1.7 yr.
MJD. b Theringtra⊕nsmissionatsmallerradiishowsstructures
thatareanalagoustotheKirkwoodgapsintheSolarSys-
4. INTERPRETINGTHEJ1407BMODEL tem, where the smooth radial distribution of asteroids
is interrupted by peturbations due to period resonances
By visual examination there is a clear decrease and
with Jupiter. In the case of the largest ring the Solar
subsequent increase in the transmission of J1407 flux
system around Saturn, the Phoebe ring is not coplanar
2 Whenthelightcurveisfoldedattimeofclosestapproach,the withtheotherrings(Verbisceretal.2009)butliesinthe
coverage approximately doubles to 20%. This implies that there plane of Saturn’s orbit. This is due to the dominance of
areatleast24/0.2≈120ringedgesintheJ1407bsystem. solar perturbations over the gravitational perturbation
8 Kenworthy and Mamajek
Time[days]
54180 54190 54200 54210 54220 54230 54240 54250 54260
1.0
0.8
b= 3.92d
0.6
tb= 54225.46d
0.4 idisk= 70.0o
φ= 166.1o
0.2
t = 54220.65d
k
0.0
-22 -21 -16 -15 -14 -13
height=0.40T
width=0.50d
-10 -9 -8 -7 -6 -5
4 6 10 11 12 25
Fig. 4.— Photometry of J1407. The upper panel shows the light curve of J1407 as the red points with associate error bars. The green
curve is the model fit for Model 1, with t(cid:107) = 54220.65 d (indicated with the vertical dashed line) and v = 33 km.s−1. For the nights
indicatedwiththeinvertedtriangles,thephotometryandmodelfitisenlargedintothepanelsbelow. Eachpanelhasawidthof0.5days
andaheightof0.4intransmission. Thenumberinthetoprighthandcornerrepresentsthenumberofdaysfromt(cid:107).
Exorings 9
0 0
5 5
2 2
4 4
5 5
0 0
4 4
2 2
4 4
5 5
0 0
3 3
2 2 s]
4 4 y
5 5 a
D
[
0
0
0
0
5
4220 4220 D - 24
5 5 HJ
0 0
1 1
2 2
4 4
5 5
0 0
0 0
2 2
4 4
5 5
0 0
9 9
1 1
0 8 6 4 2 014014
1. 0. 0. 0. 0. 0.0.50.0.5
ytisnetnI dezilamroN
rorrE
Fig. 5.—ModelringfittoJ1407data. TheimageoftheringsystemaroundJ1407bisshownasaseriesofnestedredrings. Theintensity
of the colour corresponds to the transmission of the ring. The green line shows the path and diameter of the star J1407 behind the ring
system. Thegreyringsdenotewherenophotometricdataconstrainthemodelfit. Thelowergraphshowsthemodeltransmittedintensity
I(t)asafunctionofHJD.TheredpointsarethebinnedmeasuredfluxfromJ1407normalisedtounityoutsidetheeclipse. Errorbarsin
thephotometryareshownasverticalredbars.
10 Kenworthy and Mamajek
0.2 Orbital rad0iu.4s (AU) 0.6 material. Tests for ring illumination geometry and dust
scattering can then be carried out, to name one possibil-
4 101 es) ity for this. The timescale for J1407 is less than 4 days,
ass withamorespecificnumberdependentondiskgeometry
3 100 ar m and secondary mass of the companion.
Tau2 ass (Lun 5. THEORBITOFJ1407B
1 10−1g m OnlyoneeclipseofJ1407isseeninthepublicallyavail-
n
Ri ablephotometricdata,andsowedonotknowtheorbital
020 30 40 50 60 70 80 90 10−2 periodofJ1407borevenifitisboundtoJ1407inaclosed
Orbital radius (million km) orbit3. We explore two hypotheses for the motion of the
Fig. 6.— The transmission of the ring model as a function of ringsystemrelativetothestarJ1407: (i)theringsystem
radius. Thegreyregionsindicatewherethereinnophotometryto
is unbound to J1407 and is a free-floating planet with a
constrain the model. The blue line indicates the ring gap seen at
61millionkm. Reddotsindicatetheestimatedmassofeachring ring system and (ii) J1407b is in a bound and closed or-
assumingamasssurfacedensityof∼50gcm−2. bit about J1407. Investigations into the latter case are
detailed in van Werkhoven et al. (2014) and Kenworthy
et al. (2015) respectively.
TABLE 3 One estimate of the transverse ring system velocity is
Table of J1407b orbital parameters (from
calculated from the diameter of the star J1407 and the
Kenworthy et al. 2015)
steepestlightcurvegradientseeninthephotometriclight
curve data. Assuming a sharp-edged opaque ring cross-
OrbitalEccentricity 0.7<e<0.9 0.7<e<0.8
ing the disk of the star with the ring edge perpendicular
ProbableperiodP (yr) 27.5 13.3 to the direction of motion, a minimum velocity of 33
ProbablemassM (MJup) 14.0 23.8 km.s−1 is derived (Kenworthy et al. 2015). Combined
with the duration of the eclipses, an estimate of the size
of the ring system can be made. It is this derived trans-
caused by the J2 contribution of Saturn’s gravitational verse velocity and associated ring system size that forms
field. Theplanetβ Pictorisbwasrecentlyshowntohave our central issue with the nature of the system.
arotationperiodof∼8hours(Snellenetal.2014),faster
thanthatoftheothergasgiantsintheSolarsystem. The 5.1. The ring system is an unbound object
β Pictorissystemalsohasanageof22Myr(Mamajek&
Weconsideriftheringsystemisonanunboundtrajec-
Bell2014),similartotheJ1407systemandimplyingthat torywithatangentialvelocityofatleast33km.s 1. The
−
a low mass companion could have a similar fast rotation
May 2007 eclipse is therefore a single event that will not
period, larger oblateness resulting in a larger J contri-
2 berepeatedagainwithJ1407. Ourderivedringmodelis
bution, holding ring structures in the equatorial plane
consistent with the data, yielding a diameter of 1.2 AU
out to larger radii.
fortheringsystem. Weconsidertheunboundhypothesis
as exceptionally unlikely, for two reasons:
4.1. Notable timescales in the giant exoring model
(1) The mean projected separation between stars in
Therearetwoadditionaltimescalesthatareworthnot- the field is ≈103 AU. The probability that a 1 AU scale
ing in this giant exoring model. At the time of t , the object produces an eclipse within the lifetime of Super-
star is close to stationary in r(t) and is moving ta(cid:107)ngen- WASP is exceptionally small.
tiallytotheringatradiusr(t ). Thereisthereforeatime (2)J1407bissubstellarandalmostcertainlyplanetary
interval ∆t where the star (cid:107)is sensitive to variations in mass (Kenworthy et al. 2015). The estimated size of
1
the azimuthal structure around radius r(t ). The length the ring system is two orders of magnitude larger than
of time for this interaction is approximat(cid:107)ely the length the Roche limit for the central substellar mass, and so
of time it takes for the projected disk radius to change theringsoutsidetheRochelimitareexpectedtoaccrete
by the diameter of the star, i.e. where: intosatellitesontimescalesshorterthenGigayears. This
impliesthattheringsystemisconsiderablyyoungerthan
r(t )+(R /cosi )=r(t +∆t ) stars seen in the field.
disk 1
(cid:107) ∗ (cid:107) Direct imaging limits reported in Kenworthy et al.
ThistimescalefortheJ1407exoringsystemisapprox- (2015)constrainsuchafree-floatingobjecttobe8M ,
Jup
imately1dayaroundt . WecannotexaminetheJ1407b assuming an age of 16 Myr and BT-SETTL models (Al-
data for ∆t1 as there i(cid:107)s no recorded photometry within lard et al. 2012). We conclude that the ring system is
3 days of t . bound to J1407 in a closed orbit.
The seco(cid:107)nd timescale relates to ring material orbiting
aroundthesecondarycompanionandthedurationofthe 5.2. J1407b is on a bound orbit
disktransit. Ifthesecondarycompanionislargeandthe
In Kenworthy et al. (2015) a search for J1407b is
impact parameter b small enough, there will be a par-
carried out using photometry, radial velocity measure-
cel of ring material that will transit in front of the star
ments of J1407, direct and interferometric techniques.
atleasttwice. Theorbitalmotionofringmaterialabout
The companion is not detected, but upper limits from
thesecondarycompanion,combinedwiththeorbitalmo-
tion of the secondary companion about the primary star 3 By using the name J1407b for the eclipsing object, we are
will result in two epochs symmetric about t , where the implicitlyassumingthattheringsystemisboundtoJ1407forthe
measured transmission will be of the same p(cid:107)arcel of ring reasonsexpandedonlaterinthissection.