Nereid from space: Rotation, size and shape analysis from Kepler/K2, Herschel and Spitzer observations

4 MB·

by  Cs. Kiss

MNRAS000,1–??(0000) Preprint12January2016 CompiledusingMNRASLATEXstylefilev3.0
Nereid from space: Rotation, size and shape analysis from
Kepler/K2, Herschel and Spitzer observations
Cs. Kiss,1 A. Pa´l,1,2 A. I. Farkas-Taka´cs1,2 Gy. M. Szabo´,1,3,4 R. Szabo´,1
L. L. Kiss,1,4,5 L. Moln´ar,1 K. Sa´rneczky,1,4 Th.G. Mu¨ller,6 M. Mommert,7
and J. Stansberry8
1Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences
6 H-1121 Budapest, Konkoly Thege Miklo´s u´t 15-17, Hungary
1 2E¨otv¨os Lora´nd Tudom´anyegyetem, H-1117 Pa´zma´ny P´eter s´et´any 1/A, Budapest, Hungary
0 3ELTE Gothard Astrophysical Observatory, H-9704 Szombathely, Szent Imre herceg u´t 112, Hungary
2 4Gothard-Lendu¨let Research Group, H-9704 Szombathely, Szent Imre herceg u´t 112, Hungary
5Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, Australia
n
6Max-Planck-Institut fu¨r extraterrestrische Physik, Postfach 1312, Giessenbachstr., 85741 Garching, Germany
a
7Department of Physics and Astronomy, Northern Arizona University, P.O. Box 6010, Flagstaff, AZ 86011, USA
J
8Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, United States
1
1

12January2016
]
P
E
. ABSTRACT
h
In this paper we present an analysis of Kepler K2 mission Campaign 3 observa-
p
tions of the irregular Neptune satellite, Nereid. We determined a rotation period
-
o of P=11.594±0.017h and amplitude of ∆m=0m. 0328±0m. 0018, confirming previous
r shortrotationperiods obtained inground based observations.The similarities oflight
t
s curveamplitudesbetween2001and2015showthatNereidisinalow-amplituderota-
a tion state nowadays and it could have been in a high-amplitude rotation state in the
[
mid 1960’s. Another high-amplitude period is expected in about 30 years. Based on

1 the light curve amplitudes observed in the last 15 years we could constrain the shape
v ofNereidandobtainedamaximuma:caxisratioof1.3:1.Thisexcludesthepreviously
5 suggestedveryelongatedshapeofa:c≈1.9:1andclearlyshowsthatNereid’sspinaxis
9 cannot be in forced precession due to tidal forces. Thermal emission data from the
3 Spitzer Space Telescope and the Herschel Space Observatory indicate that Nereid’s
2 shape is actually close to the a:c axis ratio limit of 1.3:1 we obtained, and it has a
0
very rough, highly cratered surface.
.
1
Key words: methods: observational — techniques: photometric — astrometry —
0
6 planets and satellites: individual: Neptune II Nereid
1
:
v
i
X
1 INTRODUCTION whileinotheryearsnearlyperpendiculartothelineofsight.
r Physicallythiscouldbeexplainedbyforcedprecession,due
a
to Neptune’s tidal torque on a non-spherical Nereid. This
Nereid is a large, ∼350km sized, irregular satellite orbiting
would require high, 1.9:1 or greater axis ratios.
Neptune in a very eccentric and inclined orbit (Dobrovol-
skis 1995; Jacobson 2009). Although its orbital character- Hesselbrock et al. (2013) attempted to model the rota-
istics are well known, there is still a mistery related to its tionofNereidinawaysimilartoSchaeferetal.(2008),but
shape, orientation and rotation rate. There have been sev- alsoconsideringtheeffectofsmallerbodiesinadditiontothe
eralpapersreportingonunusual,insomecaseslargeampli- Sun and Neptune, especially that of Triton. Their ”best es-
tude brightness variations of Nereid on different timescales, timate”solutionisatriaxialellipsoidwithsemiaxisratiosof
from night-to-night variations to changes on annual scales. c/a∼0.5,b/a∼0.6,aninitalobliquityof∼60◦ andaninitial
Theseinvestigationsarenicelysummarisedanddiscussedin rotation period of 144h. Their model predicts ”active”and
Schaefer et al. (2008). In this paper the authors’ preferred ”inactive”periods, depending on the actual direction of the
solution that could explain the photometric variability was mainspinaxis(c-axisaspectanglesof∼90◦and∼0◦,respec-
thatyear-to-yearvariationsarecausedbyNereid’spolepre- tively),matchingthephotometrydatacollectedinSchaefer
cessing,insomeyearsNereid’spolepointingtowardsEarth etal.(2008)qualitatively,however,withsomedifferencesin
(cid:13)c 0000TheAuthors
2 Cs. Kiss et al.
the predicted brightness levels compared with the observed East
ones. They explain this latter discrepancy by an additional 5’
effectofanon-uniformalbedodistribution.Theirmodelpre-
North
dictsa∼15-yeartimebetweenthetwoextremeorientations.
While the shape of Nereid would be essential for the
rotation evolution models, this is not known. Thomas et
al.(1991)derivedasizeof350±50kmfromVoyager-2flyby
observationsinwhichNereidwasresolvedtosomelevel,but
thequalityofthedatawasnotgoodenoughtoconstrainthe
shape (see also Schaefer et al. 2008, for a discussion). Figure1. Thefield-of-viewofKepler,showingthefieldinwhich
BasedonobservationsinAugust2001andAugust2002 NeptuneandNereidwasapparentlymovingduringtheK2Cam-
Grav, Holman & Kavelaars (2003) derived a short rotation paign 3. The stars involved in the determination of the absolute
period of P=11h.52 for Nereid with a small peak-to-peak anddifferentialastrometricsolutionsareindicatedbyredcircles.
lightcurveamplitudeof0m. 029±0.003.Thisrotationperiod ThesmallbluecirclesindicatethepositionofNereidwitha1-day
stepsizethroughouttheobservations.Thetotalsizeofthefieldis
was later questionned by Schaefer et al. (2008) due to the
475×190pixels,i.e.∼32(cid:48)×13(cid:48).ThisimageisshownintheCCD
relatively poor sampling of the rotation curve, and has not
frame, hence the image is flipped with respect to the standard
been included in the analysis by Hesselbrock et al. (2013) view.
either. Terai & Itoh (2013) obtained a very similar light
curve (∆m=0m. 031±0m. 001, P=11h.50±0h.10) based on ob-
servationsin2008,buttheirlightcurvewassparselysampled In the case of Nereid, a nearly parallelogram-shaped
in terms of rotational phase. fieldwasallocatedforKepler/K2observations.Thisfieldin-
As shown above, ground based observations have not cludes the apparent track of both Neptune and Nereid and
placed a strong constraint on the rotation of Nereid. How- has a size of 456×80 pixels. This field is located on Ke-
ever,asithasbeendemonstratedinsomerecentpapers(Pa´l pler module #13 and read out by its output amplifier #2
etal.2015a,b),datafromtheextendedKeplermission(K2, (i.e. channel #42 considering the full mosaic array). In or-
Howell et al. 2014) can be very effectively used to obtain dertoanalyzethefield,weretrievedthe456individualdata
rotational light curves of distant Solar System bodies due frames corresponding to the columns of this parallelogram-
to uninterrupted photometric time series of several tens of shapedCCDarea.Thereafter,webuilttheindividualimage
days in length. Given that Neptune, and therefore Nereid framesonarectangularareahavingasizeof475×190.This
aswell,wasincludedintheCampaign3observationsofthe area safely covers the parallelogram and includes some ad-
K2mission,weattemptedtoobtainthelightcurveofNereid ditionalrowsandcolumnsinordertohavesameextraspace
fromtheseobservations.Inadditiontothelightcurvedata, forframeregistration.Intotal,thecorrespondingdataseries
we collected archival Spitzer Space Telescope and Herschel contains 3336 individual and useable long-cadence frames.
Space Observatory data to detect the thermal emission of The total number of frames for Campaign 3 were 3386, but
Nereidinthemid-andfar-infrared(24–160µm)wavelength 50frameshadtobedroppedduetolackofdataorinappro-
ranges. In this paper, with the synergy of the light curve priate tracking during the exposure.
information and thermal emission data, we put important Due to the lack of a third active reaction wheel, the
constraintsontherotationalstate,shape,aswellasonother positioning jitter of Kepler/K2 is significantly larger than
physical properties of Nereid. what it had been during the main mission. Therefore, the
3336 frames have to be analyzed independently in order to
retrieve the proper astrometric solution as well as to per-
form an image registration needed by the subsequent steps
2 THE KEPLER K2 LIGHT CURVE of differential photometry. To accomplish this astrometry,
we selected nearly a dozen reference stars distributed uni-
2.1 K2 observations
formly in the field. Due to the large size of the field, we did
KeplerobservedNereidduringthethirdcampaignoftheex- not include additional stamps for this purpose (see e.g. the
tended mission, named K2 (Howell et al. 2014). In this K2 case of 2007OR , Pa´l et al. 2015b). These stars were used
10
mission, Kepler targets fields near the Ecliptic, observing to perform both the differential astrometry (i.e. the frame
each field for approximately three months. This quarterly registration) and the absolute astrometry (i.e. finding the
scheduleallowsthecontinuousobservationsofSolarSystem plate solution w.r.t. the J2000 system). The absolute plate
bodiesifsufficientlylargearcsareallocatedintheCCDmo- solution has been derived by cross-matching the pixel co-
saic of the Kepler space telescope. Main-belt asteroids have ordinates of the selected reference stars with the USNO-B
alargeapparentmotion(comparabletoorevenlargerthan catalogue(Monetetal.2003).These11starswerealsoused
the total field-of-view of the space telescope). In addition, to transform Kepler/K2 photometry to USNO-B1 R sys-
main-beltasteroidshaveanon-negligiblenoisesourceonthe tem. The USNO magnitudes of these star cover the range
photometry of stellar targets (Szabo´ et al. 2015). However, of R = 13.2...17.8, nearly homogeneously. Since an unfil-
minor bodies outside the main-belt – such as Centaurs and teredCCDefficiencycurvecanbeconsideredasa“wideR”
trans-Neptunianobjects–canbeobservedwitharelatively band,thecomparisonoftheUSNOmagnitudeswiththein-
low pixel budget. In addition, the pixel budget can even be strumental ones yielded a fit residual of 0.12 magnitudes.
minimized around the stationary point, where masks with Therefore, we can state that the accuracy of this transfor-
a size of ∼ 20×20 pixels can be sufficient to follow trans- mation is in this range.
Neptunian objects up to 10−15 days (Pa´l et al. 2015a). In all other aspects, the data reduction and photome-
MNRAS000,1–??(0000)
Nereid from space 3
try were conducted in a similar manner as it was described
in P´al et al. (2015a) or Pa´l et al. (2015b). Our photometric
pipelineusedforKeplerframereconstruction,sourceextrac-
tion,astrometryandcross-matching,imageregistrationand
differential photometry are based on the various tasks of
theFITSHpackage(Pa´l2012).Thefoldedlightcurve(with
the period of 11.594h, see Sect. 2.2) of Nereid is shown in
Fig. 2. We note here that the formal photometric uncer-
tainities on the individual frames increased from 0.012 up
to 0.022 throughout this ∼67days long campaign. This in-
crement is due to the gradually increasing level of zodiacal
light as the elongation of Nereid decreased from 140◦ down
to74◦.Thisinstrumentaluncertainitycanbecomparedwith
thestatisticalonecomputedfromtherootmeansquarede-
viationofthephotometricdatapointsoneachoftheN =20
binsonthebinnedlightcurve.Thislattermethodgivesusan 150
estimationof0.028−0.034errorforeachframe,i.e.slightly bility
larger than the instrumental estimate. a
b
o
pr
m
ar 100
al
2.2 Light curve analysis e
s
al
Based on previous studies in the literature, the expected of f
m
amplitude of the light curve variations are relatively small. arith 50
Therefore,periodicityinthelightcurvehasbeensearchedby g
o
assuming a sinusoidal function coadded to a linear function ve l
representing the gradual fading of the object (due to phase gati
e
angleanddistancevariations).Thisfunctionwasdefinedas N
0
0 1 2 3 4 5 6
Characteristic frequency (cycle/days)
A+B∆t+Ccos(2πn∆t)+Dsin(2πn∆t), (1)
Figure 2.Upperpanel:FoldedandbinnedlightcurveofNereid
wherenisthesuspectedrotationalfrequencyand∆tisthe asreconstructedfromKepler/K2dataaftersubtractingthelong-
time after 2,457,010JD (chosen in order to minimize the term trend caused by the increasing phase angle and distance.
rounding errors and numerical artifacts). The unknown pa- The binned light curve is shown above with N = 20 bins, the
formal uncertainties of the phase-binned data poins are in the
rameters A, B, C and D can then be derived using lin-
range of 0m. 0027–0m. 0033. The light curve shown in this plot is
ear regression. The parameter space in n is then scanned
normalizedtothebrightnessatT =2,457,010JD,approximately
in the physically relevant domain with a stepsize compa-
atthecenteroftheobservations.Thelightcurveisfoldedwitha
rable to one tenth of the reciprocal timespan of the ob-
periodofP =11.594h,correspondingtothedoubleofthemain
servations (i.e. 0.001d−1). A prominent peak has been de- frequencypeakatn=4.140c/dinthelowerfigurepanel.Lower
tected in the fit residuals at n = 4.140±0.006c/d, as it panel: Frequency spectrum computed from the Kepler/K2 light
can also be well seen in Fig. 2b. The corresponding values curve. The prominent peak at n = 4.140±0.006cycles/day can
for the four coefficients at this peak was found to be A = beeasilyrecognized.
19.3650±0.0006,B =0.00335±0.00003,C =0.0129±0.0008
and D = 0.0101 ± 0.0008. The hypotenuse of C and D,
√
i.e. C2+D2 gives us an estimation for the total ampli- 2.3 Spin axis constraints
tude which is then 0.0164±0.0008. The residual of the fit
The similarity of the 2001, 2008 and 2015 light curves puts
was found to be 0.031magnitudes. Due to the significance
constraints on the actual spin axis orientation. First, due
of this peak, we conclude that the rotation period is either
to the short rotation period with respect to the orbital pe-
P = 5.797±0.008h or its double, P = 11.594±0.017h,
riod(∼360days)theprecessiontimeofthespinaxisislong,
assuming a double-peaked solution. The amplitude of the
at least a few hundred years, even if very elongated body
light curve is 0m. 0164±0m. 0008 (0m. 0328±0m. 0018 peak-to-
and/or high original obliquity are assumed (Dobrovolskis
peak). To correct for phase angle effects, a phase constant
1995). This suggests that the spin axis of Nereid remained
ofk=0.123±0.005magdeg−1 wasalsoderivedfromourK2
in approximately the same direction in the last ∼15years.
data (Nereid was seen at phase angles of 1◦.2≤α≤1◦.9 dur-
However, Neptune (and hence Nereid) moved on its orbit
ing the K2 observations). about 30◦ along the ecliptic in the last 15 years, and there-
The light curve we found is very similar to those ob-
foretheaspectangleϑofthespinaxisshouldhavechanged
tained by Grav, Holman & Kavelaars (2003) and Terai
withrespecttotheobserverevenifitwasotherwisepointing
& Itoh (2013), both in rotation period (11h.52±0h.14
in the same direction in space in this period. The position
and 11h.50±0h.10), and amplitude (0m. 029±0m. 003 and
angleϑdependsontheeclipticcoordinatesofthetarget(λ,
0m. 031±0m. 001). In Sect. 2.3 we analyse in detail the con-
β) as well as on that of the spin axis’ direction (λ , β ):
p p
sequences of the similarities of these light curves obtained
over a ∼15-year period. cosϑ=−sinβ sinβ −cosβ cosβ cos(λ−λ ) (2)
p p p
MNRAS000,1–??(0000)
4 Cs. Kiss et al.
which ∆m corresponds to the observed light curve
amplitues by Grav, Holman & Kavelaars (2003) in
2001-2002 (∆m=0m. 029±0m. 003), Terai & Itoh (2013)
in 2008 (∆m=0m. 031±0m. 001) and by us in 2015
(∆m=0m. 0328±0m. 0018) within the given uncertainties. In
the case of the Terai & Itoh (2013) data we found the orig-
inally quoted uncertainties to be too optimistic. Our anal-
ysis of their photometry gives an amplitude uncertainty of
0m. 0026.Weusedthislattervalueinthespinaxisorientation
analysis.
We originally assumed X in the 0.03 – 0.34 range. The
lowest value of X corresponds to the smallest possible X
value that could produce a light curve compatible with the
observations if ϑ=90◦, while the highest value corresponds
to a ∼2:1 a:c axis ratio, suggested e.g. by Hesselbrock et
al. (2013). The results are plotted in Fig. 3 where the dif-
ferent colours mark the different axis ratio parameter val-
ues. Most of the allowed {λ ,β } pairs belong to a small X
p p
value,i.e.toaslightlyelongatedshapewithanaxisratioof
a:c≈1.06:1. The largest possible axis ratioparameter value
weobtainedisX=0.133.LargerXvaluesarenotallowedby
theamplitudeconstraintsandhencemoreelongatedshapes
(X>0.133) are excluded.
We checked how the light curve amplitude evolves in
Figure 3.Possiblespinaxisgeometryconfigurationsthatrepro- the different cases allowed by the 2001, 2008and 2015 light
ducetheobservedlightcurveamplitudesin2001,2008and2015. curveamplitudeconstraints.InFig.3bwepresentthelight
Upper panel: Allowed ecliptic longitude and latitude of the spin
curveamplitudevariationasafunctionofeclipticlongitude.
axis.ThecolourscorrespondtothedifferentXaxisratioparam-
At the observation epochs between 2001 and 2015 (ecliptic
eters from the lowest (purple) to the highest (red, see the text
coordinates marked by vertical lines in the figure) the light
formoredetails).Lowerpanel:Variationoflightcurveamplitude
curve amplitude does not change notably and remains low,
withtheeclipticlongitudeofNereidassumingastablespinaxis
orientation. The colours correspond to different shapes with X irrespective of the shape of Nereid. While there are many
axis ratio parameters ranging from 0.03 to 0.14, as above. The solutions in which the amplitude remains low (slightly non-
dashedanddash-dottedvertical linesmarktheeclipticlongitude spherical cases, purple curves) along the solar orbit of the
ofthe2001,2008and2015observations,respectively.Thedashed Neptune-Nereid system, there are also cases when Nereid
anddash-dottedhorizontal linesmarktherotationcurveampli- appears notably elongated and the light curve amplitude
tude ranges allowed by the 2001, 2008 and 2015 observations, changes remarkably with the orbital phase around the Sun
respectively.
(casesmarkedwithgreentored).IntheselattercasesNereid
could have had its highest amplitude period in the 1960’s,
andstillcouldhavenotablyhigheramplitudesinthe1980’s
If we assume that the shape of Nereid is a triaxial ellipsoid
thanwhatwecanseetoday.Anotherpeakinthelightcurve
with semi-axes of a>b>c then the object seen at a ϑ spin
amplitude would be expected in ∼30 years. The maximum
axis aspect angle will show a light curve with an amplitude
possible light curve amplitude is ∆m=0m. 13.
of:
(cid:115)
(b/c)2cos2ϑ+sin2ϑ
∆m=2.5log (3)
(b/c)2cos2ϑ+(b/a)2sin2ϑ For the thermal emission models that we discuss in
Sects. 3.3 and 3.4 it is important to see how the spin axis
Here we assumed that the light curve is solely caused by aspect angles and the subsolar latitudes could change with
shape effects and there are no albedo variegations on the time. We plotted the subsolar latitude β as a function of
ss
surface. Due to the small phase angles the phase angle cor- the axis ratio parameter X in Fig. 4a for the allowed con-
rection of the light curve amplitude is negligible. We char- figurations at the epoch of the Spitzer/MIPS observations.
acterise the semi-axis ratios of the triaxial ellipsiod by an Thesubsolarlatitudesshowawelldefinedrelationshipwith
axis ratio parameter X. With this parameter the semi-axes theaxisratioparameter:atlowXvaluesβ isclosetozero
ss
of the ellipsoid are: (equator-onconfigurations)thenitrisesquicklyandreaches
a maximum value of β ≈60◦ for the largest possible val-
b=1, a=(1+X)b, c=(1−X)b (4) ss
ues of X. While β can change notably between the MIPS
ss
and the body rotates around its shortest (c) axis. and PACS epochs (up to ∆β ≈15◦) for low X values, the
ss
With this assumption, and using the equations change is rather small for higher axis ratio parameters (see
above, we calculated those {λ ,β } combinations for Fig. 4b).
p p
MNRAS000,1–??(0000)
Nereid from space 5
very small during the measurements we were not able the
perform any kind of background correction. We clearly de-
tected Nereid in both bands, and the photometry provided
F =22.8±1.7mJy and F =18.6±2.9mJy in-band flux
100 160
densities at 100 and 160µm, respectively.
3.2 Spitzer/MIPS measurements
TheSpitzerSpaceTelescopeobservedNereidinadedicated
observationin2005usingtheMIPScameraat24and70µm
(Rieke et al. 2004). The data were reduced using the same
pipelineaswasusedforthereductionofMIPSdataofCen-
taurs and trans-Neptunian objects by Mueller et al. (2012)
and Stansberry et al. (2008, 2012). Nereid was clearly de-
tectedinbothbands,andweobtainedin-bandfluxdensities
of F =2.56±0.03mJy at 24µm and F =50±11mJy at
24 70
70µm using multi-aperture photometry. The corresponding
images are presented in Fig. 6. At the time of the observa-
tionsNereidwasat∼4(cid:48)separationfromtheverybrightNep-
tune. While the 24µm image (Fig. 6a) does not seem to be
contaminatedbyextendedemissionfeaturesfromtheplanet,
thisisnotthecaseat70µm.Thetwobrightrimsatthetop
and bottom of the 70µm image (Fig. 6b) are caused by the
hexagon structure of the Neptunian point spread function
(PSF), as it is demonstrated in Fig. 6c. There are also less
apparentemissionspikesatthelocationofNereidthatmay
affectthephotometry.ToaccountfortheemissionfromNep-
tuneatthelocationofNereid,wetriedtosubtractNeptune’s
contributionbyscalingatheoretical70µm75KPSF,scaled
totheactualbrightnessofNeptuneinthisbandatthetime
Figure 4. Upper panel: Subsolar latitude (βss) for the allowed oftheobservations.Neptune’sbrightnesswasestimatedus-
geometryandshapeconfigurationsasafunctionoftheaxisratio ingtheNeptunemodelbyR.Moreno(Moreno2012)andwe
parameter X, at the Spitzer/MIPS epoch. Lower panel: Varia- obtained FNeptune=383.2Jy at the effective wavelength of
70
tion of the subsolar latitude between the Spitzer/MIPS and the 71.42µm of this MIPS band. This model has been used for
Herschel/PACSobservationepochsasafunctionoftheaxisratio theHerschel/PACSandSPIREfluxcalibrationandusedto
parameterX. reproducethefluxdensitiesofNeptunewithin2%(Mu¨lleret
al., A&A, submitted). With this correction the main emis-
sion features have disappeared and repeated photometry
3 THE THERMAL EMISSION OF NEREID provided an in-band flux of F(cid:48) =29±16mJy.
70
3.1 Herschel Space Observatory measurements
3.3 NEATM model
We have found Nereid in archival Herschel Space Observa-
tionsdata(proposalID:OT1 ddan01 1).Theoriginaltarget WeusedtheNear-EarthAsteroidThermalModel(NEATM)
in these observations was Neptune, and Nereid was just ac- (Harris 1998) to estimate the thermal emission of Nereid
cidentallyontheimages(seeFig.5).Fourobservationswere using the Spitzer/MIPS 24 and 70µm and the Her-
takenwiththePACS(PhotometerArrayCameraandSpec- schel/PACS 100 and 160µm fluxes, as listed in Table 2.
trometerPoglitschetal.2010)photometercameraonboard TheSpitzer/MIPSandHerschel/PACSobservationsaresep-
the Herschel Space Observatory (Pilbratt et al. 2010), us- aratedbyaboutsixyears,and,aswehaveseenabove,there
ingthe100/160µmfiltercombinationinallfourcases.The wereindicationsthattherotationalaxis,brightnessandthe
datareductionpipelineweusedisthesameastheoneused apparent shape of Nereid may change significantly on this
inthe”TNOsareCool!”HerschelOpenTimeKeyProgram timescale(seeSect.1).However,asdiscussedabove,thesim-
(Mu¨lleretal.2009),describedindetailinKissetal.(2014). ilarity of our newly detected rotation solution to those by
As our aim was to obtain photometry of a point source, Grav, Holman & Kavelaars (2003) and Terai & Itoh (2013)
we used the photProject() task with high pass filtering to indicatesthatNereidhasbeeninalowrotationallightcurve
create maps from the time domain detector data. The im- amplitudeperiodintherecentdecadethatisalsoassociated
ages of the four consecutive measurements were stacked in withrelativelysmallchangesinspinaxisrotationangleand
the co-moving frame of Nereid, and aperture photometry subsolar latitude configurations. Therefore we feel that the
wasperformedonthestacked100and160µmimages.Flux combination of the Spitzer/MIPS and Herschel/PACS data
uncertainties were determined with the implanted source is likely feasible and can better constrain the radiometry
method(Kissetal.2014),butinthiscaseusinga∼80(cid:48)(cid:48)area solutions than analysing them separately.
around Nereid rather than the high coverage regions of the We characterise the radiometry solutions by calculat-
whole map. As the apparent movement of the target was ing the reduced χ2 values of the fits from the observed and
MNRAS000,1–??(0000)
6 Cs. Kiss et al.
Table1.SummaryofNereid’sthermalinfraredobservations.The”identifier”columnreferstoOBSIDinthecaseofPACSandAORKEY
inthecaseofMIPSmeasurements.
Instrument Obs.date identifier duration filtercombination r ∆ α
h
(JD) (s) (µm/µm) (au) (au) (deg)
Herschel/PACS 2456090.655 1342222561 2996 100/160 30.02 29.65 1.85
Herschel/PACS 2456090.691 1342222562 2996 100/160
Herschel/PACS 2456090.726 1342222563 2996 100/160
Herschel/PACS 2456090.679 1342222564 2996 100/160
Spitzer/MIPS 2453539.667 4535808 1012 24 30.07 29.57 1.70
Spitzer/MIPS 2453539.679 4535808 1012 70
Figure 5.NereidontheHerschel/PACS100µm(left)and160µm,co-addedimages,reducedintheco-movingframeofNeptune.The
largeimagesshowthefullareamappedwithNeptuneinthecentre,whilethezoomingsshowthecloseenvironmentofNereid,withthe
moon in the centre of the stamps. At 160µm Nereid was identified by positional matching with other wavelengths from the multiple
sourcesofsimilarbrightness.
model fluxes of a specific effective diameter and beaming magnitudesprovidedbyGrav,Holman&Fraser(2004)and
parameter combination. We consider a solution acceptable Rabinowitz et al. (2007).
ifχ2≤(1+σ)2,whereσ isthestandarddeviationoftheχ2 For the sake of completeness, we derived four kind of
distribution(seee.g.Vileniusetal.2014,forasimilarappli- fits:
cationofthemethod).Albedoandeffectivediameterarenot
independent but are linked by the absolute magnitude. We a) MIPS 24 and 70µm only, original 70µm flux (MIPS
useH =4m. 418±0m. 008,aweightedaverageoftheabsolute epoch). In this case we used input flux derived from the
V
uncorrected 70µm flux, F (70)=55.0±12.7mJy (see Ta-
m
MNRAS000,1–??(0000)
Nereid from space 7
Figure 6.Nereidasobservedat24and70µmwiththeMIPSinstrumentoftheSpitzerSpaceTelescope.Leftpanel:24µmfalsecolour
image. Middle panel: 70µm false colour image; Nereid is the bright compact source in the centre of both images. Right panel: MIPS
70µmintensitycontours(thesameasthemiddlepanel)overlaidonanintensitymap(grayscale)ofamodelimageofNeptunewhichwas
notinthefield-of-view.TheredrectangleshowstheareaoftheMIPS70µmimage.ThepositionofthebrightsourcerelativetoNereid
issetinsuchawaythatitcorrespondstotherelativepositionofNereidandNeptuneatthetimeoftheSpitzer/MIPSobservations.
epoch). In Fig. 7b we plot the regions of the acceptable so-
Detector/ λeff Fi Cλ Fm lution(χ2≤(1+σ)2)inthediameter–beamingparameter
filter (µ)m (mJy) (mJy)
space.ForCasebthisisrepresentedbytheareaenclosedby
MIPS24 23.68 2.56±0.03 0.99±0.01 2.59±0.13 dashed lines. Due to the large error bar of the 70µm flux
MIPS70/u 71.42 50±11 0.91±0.01 55.0±12.7 a wide range of diameters, and an especially wide range of
MIPS70/c 71.42 29±16 0.91±0.01 31.9±17.6
beaming parameters are allowed.
PACS100 100.0 22.8±1.7 0.99±0.01 23.0±2.1
c) PACS 100 and 160µm only (PACS epoch). When only
PACS160 160.0 18.6±2.9 1.03±0.01 18.1±3.0
the PACS measurements are considered the situation is
somewhat different. Here we have a strong correlation of
Table 2.Spitzer/MIPSandHerschel/PACSphotometryresults.
the diameter and η, but the model is not well constrained
The columns of the table are: Detector and filter combina-
due to the lack of short wavelength fluxes (region enclosed
tion; Fi: inband flux, as determined from the corresponding im-
age; λ : effective wavelength of the band; C : Colour correc- by dash-dotted lines in Fig. 7b).
eff λ
tion factor; Fm: final, monochromatic flux, used for radiome- d) PACS and MIPS combined (corrected 70µm flux, both
try modelling. The Fm monochromatic fluxes are obtained as MIPS and PACS epochs). Due to the similarities of the ob-
Fm(λ)=Fm(λ)/Cλ. MIPS 70/u and 70/c refer to the uncor- serving geometries at the dates of the MIPS and PACS ob-
rected and corrected 70µm fluxes, as described in the text. An servations the resulting fluxes of the same NEATM model
instrumental calibration accuracy of 5% was assumed for all de- are almost equivalent for the two epochs, and therefore a
tector/filtercombinationsandaddedquadraticallytothefluxun-
meanobservinggeometrycanbesafelyappliedonthecom-
certainties.
bined dataset (see Table 1). When all four data points are
considered, the NEATM model becomes well constrained.
The common solution (gray contour ellipses in Fig. 7b)
ble 2). The NEATM fits resulted in a best fit diameter of is compatible with both the MIPS-only and the PACS-
D=605±95km, η=1.36±0.16, and pV =0.081±0.027. (see only solutions (Cases b and c). In this case we obtain
thecorrespondingSED,plottedwithdottedline,inFig.7a). D=345±15km,η=0.76±0.02,andp =0.25±0.02.Thisef-
V
This diameter is by far larger than any conceivable size for fectivediameterisverysimilartotheVoyager-2flybyvalue
Nereid, and clearly indicates that the MIPS 70µm flux is of 350±50km (Thomas et al. 1991). The low χ2 value ob-
strongly contaminated, most likely by the diffraction spikes tainedforthiscombinedsolutionconfirmsthattheη values
of Neptune, as discussed in Sect. 3.2 earlier. Therefore this related to the two epochs are the same to a level that they
70µm input flux and the related solution is not considered cannot be distinguishedwith the current flux uncertainties.
any further in our analysis.
b) MIPS 24 and 70µm only, corrected 70µm flux (MIPS We consider the combined dataset and the solution of
MNRAS000,1–??(0000)
8 Cs. Kiss et al.
roughness level of ρ=0.6 (Lagerros 1998). Such a surface
can also be constructed by considering 90◦ hemispheral
craters with 50 per cent surface coverage. To obtain this
lowη,itis,however,requiredinadditiontothemoderately
high surface roughness that either the thermal inertia is
extremely low (Γ<0.1Jm−2s−1/2K−1) or the subsolar
latitude is β ≈90◦ (we see the spin-axis near to pole-on).
ss
For a near to pole-on configuration the thermal intertia
should still be low, but it can be a value somewhat higher
than in general case, typically Γ≈2Jm−2s−1/2K−1
may be possible. The other possibility, β ≈90◦ is not
ss
supported by the light curve constraints, as a maximum
value of β ≈60◦ is obtained in the spin axis analysis (see
ss
Sect. 2.3 and Fig. 4).
If we allow for a high surface roughness (ρ=0.9
that can be achieved with 90◦ craters at 100% cover-
age), different sub-solar latitudes become possible. E.g. for
β =0◦ η=0.76 results in a very low thermal inertia of
ss
Γ=0.5Jm−2s−1/2K−1.Lellouchetal.(2013)foundanav-
eragethermalinertiaofΓ=2.5±0.5Jm−2s−1/2K−1among
Centaurs and trans-Neptunian objects in the heliocentric
distance range of 25au<r <41au, with a strong sugges-
h
tion of decreasing Γ with increasing heliocentric distance.
EvenhighervaluesofΓareexpectedamongtheicymoonsof
theoutergiantplanets.Thismayindicatethattheobserved
values of the beaming parameter may rather be explained
byahighersubsolarlatitude(β ≈60◦)thanbyaverylow
ss
thermal inertia value of the surface. This also suggests that
theshapeofNereidmaybeelongatedtosomelevel,because
highβ valuesarefoundforlargeraxisratioparametersin
ss
Sect. 2.3.
Figure 7. Upper panel: Best-fit NEATM model of the ther-
mal emission of Nereid with the observed and colour corrected
fluxesoverplotted(Cased,diamondsymbolsandsolidline).The
3.4 Thermophysical modelling
χ2=(1+σ)2NEATMfitsaredrawnbygraycurves.TheNEATM
fitusingtheuncorrectedMIPS70µmflux,correspondingtoCase We also used a thermophysical model (TPM, see Lagerros
a,isalsoplottedwithadottedcurve.Theuncorrected70µmplot
1996, 1997, 1998; Mu¨ller & Lagerros 1998, 2002). to char-
ispresentedbyatriangleandslightlyshiftedinwavelengthfrom
acterise the thermal emission of Nereid. The TPM calcu-
the uncorrected point for clarity. Lower panel: Reduced χ2 con-
lates the temperature distribution on the surface of a body
tourmapoftheNEATMmodelfitsasafunctionoftheeffective
for a specific shape, illumination, observing geometry, spin
diameter and the beaming parameter η. The outermost contour
of 1.7 correspond to the reduced χ2 limit of acceptable models axisdirectionandrotationperiod.Themodelconsidersthe
with four data points and two model parameters (diameter and thermalpropertiesofthesurfacedirectly,includingthermal
η) inertia and surface roughness.
We looked for solutions that matches the observed
Spitzer/MIPS and Herschel/PACS far infrared fluxes the
Casedasthemostacceptablesize,albedoandbeamingpa- best,usingtheinputfluxeslistedinTable2.Asinthecase
rameterforNereid.Itissupportedbythelightcurveanalysis oftheNEATMmodel,wecharacterisethegoodnessoffitby
/ spin axis constraints and none of the other NEATM solu- calculatingthereducedχ2valuesandwerequirethataccept-
tions(Casesbandc)contradictwithsolutiondconsidering ablesolutionsshouldfulfiltherequirementofχ2≤(1+σ)2.
r
all the errors in the NEATM model parameter determina- Low (ρ=0.1) to intermediate (ρ=0.5) surface rough-
tion. ness cannot explain the observed fluxes as these values re-
Using our favoured radiometry solution of sult in very high (χ2(cid:29)3) reduced χ2. All acceptable solu-
r
D=345±15km, η=0.76±0.02, and following the rela- tions(χ2<1.7)arerelatedto”hotmodelsettings”,i.e.,very
r
tionsofthermalparameter,beamingparameterandsurface highroughness(ρ≥0.7)combinedwithextremelylowther-
roughness described in Spencer et al. (1989) and Spencer mal inertia (Γ(cid:28)1Jm−2s−1/2K−1), except for configura-
(1990), we can constrain the thermal properties (thermal tions near to pole-on where hot temperatures are reached
inertia and surface roughness) of Nereid using the NEATM for a wider range of thermal inertias, as in these cases
solutions to some level. In the framework of the above- heat is not transported to the night side. All acceptable
mentioned models, a beaming parameter below unity can settings (χ2<1.7) produce size and albedo solutions with
r
be explained by surface roughness effects. The minimum D =353–362km and p =0.23–0.25.
eff V
level of surface roughness beaming contribution required Assuming that Nereid has a thermal inertia in
to obtain η=0.76 (our best-fit value) is δη=0.24 (see eq. the range of 1–5Jm−2s−1/2K−1, comparable to typical
7 in Spencer 1990) that corresponds to an r.m.s. surface TNOs/Centaurs at similar distances (see Lellouch et al.
MNRAS000,1–??(0000)
Nereid from space 9
even when the ”nominal”Γ=5Jm−2s−1/2K−1thermal in-
ertia values are used. However, in these cases we obtain a
somewhatsmallersizeofD=326kmandahigheralbedoof
p =0.29.
V
4 DISCUSSION
The most important question about Nereid, as discussed in
the literature, was its shape and the precession behaviour
of its spin axis. This was modelled in detail by Schaefer et
al. (2008) and Hesselbrock et al. (2013), with the main aim
toexplainthelargeamplitudebrightnessvariationsseenon
differenttimescales.Thesemodelsassumedalongrotation
period (72–144h) which was needed to achieve precession
timescales of around a decade.
Oneimportantfindingofourpaperisthatweconfirmed
theshort(11h.594)rotationperiodobservedearlierbyGrav,
Holman & Kavelaars (2003) and Terai & Itoh (2013). As it
has been mentioned previously, a short rotation period im-
pliesalongspinaxisprecessiontime,asP ∝P2 /P ,
prec orb spin
where P is the precession period, P is the orbital pe-
prec orb
riod around Neptune, and P is the rotation period (see
spin
e.g. Hesselbrock et al. 2013). P values calculated from
prec
the 11h.594 rotation period are at least an order of magni-
tude longer than those assumed by Schaefer et al. (2008)
and Hesselbrock et al. (2013), even if an extremely elon-
gated body is assumed. Therefore the spin axis orientation
couldnotchangesignificantlyinthelast∼15years,andthe
precession of the spin axis cannot be the reason behind the
large flux variations observed.
InSect.2.3weusedthisstabilityofthespinaxisorien-
tation andobtained amaximum axis ratio of a:c=1.3 from
thelightcurveamplitudeanalysis.AccordingtoHesselbrock
Figure8.Upperpanel:BestfitTPMmodelofthethermalemis-
et al. (2013), forced precession of Nereid could be feasible
sion of Nereid for an axis ratio parameter of X=0.0133 and a
spin axis orientation of λp=320◦ and βp=32◦. The solid and only if the moon is considerably elongated with an a:c axis
dashed lines correspond to the observing geometries at the Her- ratio of ∼1.9:1. Such a high axis ratio is excluded by our
schel/PACS and Spitzer/MIPS observation epochs, respectively. results, consistently with the long precession timescales ob-
Lowerpanel:Temperaturedistributionofthemodelabove. tained.
The shape solution favoured by the thermal analysis
(axis ratio parameter of X=0.133) is also the one that
2013),onlymodelsettingswithaspinaxisdirectioncloseto provides the maximum possible light curve amplitude of
”pole-on”,{λ ,β }={320±30◦,0±30◦},produceacceptable ∆m≈0m. 13, with a peak in the 1960’s (see Sect. 2.3). Even
p p
flux predictions. The best-fit case of these models provided thissolutionisunabletoexplaintheextremelylargebright-
an effective size of D=357±13km and a geometric albedo ness variations observed some decades ago (up to ∆m of
of p =0.24±0.02. 0m. 5, see fig. 4 in Schaefer et al. 2008, for a summary of
V
OurTPManalysisabovesuggestedahighsubsolarlat- thesedata).Whatevercausedthesevariations,itcannotbe
itude and a correspondingly low spin axis aspect angle. Ac- theforcedprecessionofaveryelongatedNereid,asthissce-
cording to the spin-axis constraints discussed in Sect. 2.3 nario is inconsistent with our present data.
high β may occur for the highest possible axis ratio pa- The long rotation period assumed by Schaefer et al.
ss
rametersonly.ThisfavoursshapesolutionsofX≈0.13,and (2008) and Hesselbrock et al. (2013) was partly based on
a corresponding subsolar latitude of β ≈60◦ (see Fig. 4). the expectation that if Nereid was formed around Neptune
ss
We have chosen the highest possible shape parameter value orcapturedatearlytimes,thenitsoriginalrotation–which
ofX=0.133(a:c=1.3:1)andthecorrespondingsubsolarlat- couldhavehadaperiodofafewhours–shouldhaveslowed
itude of β =58◦, and tested the feasibility of this config- down considerably, even if Nereid is just slightly elongated.
ss
uration in a TPM model. When high roughness (ρ=0.9) In this sense, Nereid rotates fast today – this may favour a
is assumed we obtained an acceptable χ2 value of 1.2, late capture (i.e. Nereid did not have enough time to slow
r
but this is associated with a very low thermal inertia of down considerably) or indicate some other influence, like a
Γ=0.5Jm−2s−1/2K−1.Forthissolutionthebestfiteffec- collision, that may overwrite the rotational state. We note
tive diameter is D=335km, with a correspondig geomet- that Grav, Holman & Fraser (2004) suggested that Hal-
ric albedo of p =0.275. Application of an extreme rough- imede, a small moon of Neptune, may be a fragment of
V
ness of ρ=1.0 provides very low reducedχ2 values of ∼0.5, Nereid broken off during a collision. This scenario for the
MNRAS000,1–??(0000)
10 Cs. Kiss et al.
originofHalimedeissupportedbytheirsimilarcoloursand
the relatively high probability of a collision between Nereid
andHalimedeinthetimespanbetweentheformationofthe
Solar System and today (Holman et al. 2004).
As discussed in detail in Schaefer et al. (2008), the
Voyager-2 images analysed by Thomas et al. (1991) could
hardly be used to put constraints on the actual shape
of Nereid, i.e. we did not have enough information to
tell whether Nereid was spherical or notably non-spherical.
The combination of the spin axis constrains and both the
NEATM and TPM radiometry results favors a moderately
elongated shape with a axis ratio parameter X≈0.13, and
correspondingly a present subsolar latitude of β ≈60◦(a
ss
possible shape solution using X=0.133 is presented in
Fig. 8b). The moderately elongated shape putforwardhere
for Nereid is feasible regarding observational constraints.
However,foramoonof∼350kmindiameterashapecloser
tosphericalmaybemorelikely(seee.g.Schaeferetal.2008)
basedontheshapeinformationofgiantplanetmoonsofsim-
ilarsize.Mostoftheselarge(>100kminradius)moonsare Figure 9. Albedo-colour diagram of Centaurs and trans-
round to a few per cent (better than 10%) accuracy. Our Neptunian objects, also presenting the colours and albedos of
Nereid (brown triangle, data from this paper and Grav, Hol-
preferred solution for Nereid is further away from a perfect
man & Fraser 2004) and Hyperion (purple triangle, Hicks et al.
sphere. One exception among these satellites may be the
2008; Thomas 2010). The colours and albedos of the other ob-
Saturn moon, Hyperion, which is highly elongated despite
jectsaretakenfromtheoriginalresourcescitedinLacerdaetal.
that it is similar in size to Nereid (about 205×130×110km
(2014).ThecoloursofthesymbolsarethesameasinLacerdaet
semi-axes).InthecaseofHyperiontheelongatedshapeisex- al. (2014): red dots – bright-red group, blue dots – dark-neutral
plainedbythehighporosityoftheinterior(∼40%)thatalso group, orange dots – Haumea-type objects, green-dots – largest
leadstoa’sponge-like’,highroughnesssurfacewherecraters TNOs, black-dots – objects with large uncertainties and
remain nearly unchanged over billions of years (Thomas et ambiguous surface type.
al. 2007). A porous internal structure and high roughness
surface similar to that of Hyperion may explain well both
theelongatedshapeandtheradiometryanalysisresultsob- 5 SUMMARY
tained for Nereid in the present paper.
InthispaperwepresentedspacebornobservationofNereid,
performed by the Kepler Space Telescope in the frame-
In a recent paper Lacerda et al. (2014) presented an workoftheextendedK2mission,andresultsobtainedfrom
analysis of colours and albedos of Centaurs and trans- archival infrared data of the Spitzer Space Telescope and
Neptunian objects, and identified two main groups whose the Herschel Space Observatory. From the Kepler K2 data
existence can be considered as an evidence for a composi- we obtained a light curve that was the same in period and
tionaldiscontinuityintheyoungSolarSystem.Forcompar- similar in amplitude to those obtained from ground based
ison,weplottedthecolourandalbedoofNereidinadiagram observations in 2001 and 2008. These observations together
(Fig. 9) similar to fig. 2 in Lacerda et al. (2014), presenting contrainthepossiblerotationstatesofNereidverywell.Ac-
these dark-neutral and bright-red objects. Colours are rep- cording to these results, Nereid is in a low amplitude ap-
resentedbyspectralslopes,calculatedinthesamewayasin parent light curve state nowadays, but may have been in
Lacerda et al. (2014). If Nereid was either formed around a much larger amplitude state some decades ago. We man-
Neptune or captured during Neptun’s outward migration agedtoexcludeveryelongatedshapeswithaxisratiosabove
from the 20-30au distance range in the early Solar System, 1.3:1;thisalsomeansthatNereidcannotbeinaforcedpre-
it is expected to exhibit a surface similar to the objects in cessionstateduetotidalforcesasitisnotelongatedenough
the dark-neutral group. Indeed, Nereid is close in colour to for this process. This is a robust result as we assumed that
thetypicalcoloursofobjectsinthisgroup,butatthesame thelightcurveofNereidissolelycausedbyshapeeffectsand
time,itsalbedoissignificantlylargerthanthetypicalvalue albedo variegations existing on the surfce would just make
inthisgroup,evenlargerthanthealbedosofmostobjectsin the moon more spherical in this respect.
the bright-red group. If Nereid was a member to the dark- WeconfirmedthesizeofNereidobtainedfromVoyager-
neutral group, it would have the brightest surface among 2flybydatabyanindependentmethod–radiometrybased
these objects. While Nereid is at the edge of the distribu- on infrared data. Both the NEATM and TPM thermal
tions of objects of both major groups in the albedo-colour emission models resulted in similar effective size (D=335–
plain, it is rather close in these characteristics to the Sat- 345km) and albedo values (p =0.25–0.27). Both methods
V
urnian moon Hyperion (purple triangle in Fig. 9). As men- indicate very high roughness (likely ρ≈0.9) independently
tioned earlier, the surface roughness and internal structure of shape, i.e. a surface with deep craters and very high sur-
of Nereid may as well be similar to those characteristic of facecoverage.Usingthelightcurveandthermalemissionre-
this irregular satellite. sultstogetherweobtainalikelymoderatelyelongatedshape
MNRAS000,1–??(0000)