Table Of ContentAstronomy&Astrophysicsmanuscriptno.13065 (cid:13)c ESO2010
January4,2010
LettertotheEditor
The lunar phases of dust grains orbiting Fomalhaut
M.Min1,M.Kama2,C.Dominik2,3,andL.B.F.M.Waters2,4
1 AstronomicalInstituteUtrecht,UniversityofUtrecht,P.O.Box80000,NL-3508TAUtrecht,TheNetherlands
2 AstronomicalInstituteAntonPannekoek,UniversityofAmsterdam,Kruislaan403,1098SJAmsterdam,TheNetherlands
3 AfdelingSterrenkunde,RadboudUniversiteitNijmegen,Postbus9010,6500GLNijmegen,TheNetherlands
4 InsituutvoorSterrenkunde,K.U.Leuven,Celestijnenlaan200D,B-3001Leuven,Belgium
ReceivedAugust5,2009;acceptedDecember18,2009
0 Abstract
1
0 OpticalimagesofthenearbystarFomalhautshowaringofdustorbitingthecentralstar.Thisdustisinmanyrespectsexpectedtobe
2 similartothezodiacaldustinthesolarsystem.Theringdisplaysaclearbrightnessasymmetry,attributedtoasymmetricscatteringof
n thecentralstarlightbythecircumstellardustgrains.RecentmeasurementsshowthatthebrightsideoftheFomalhautringisoriented
a awayfromus.Thisimpliesthatthegrainsinthissystemscattermostofthelightinthebackwarddirection,insharpcontrasttothe
J forward-scatteringnatureofthegrainsinthesolarsystem.Inthisletter,weshowthatgrainsconsiderablylargerthanthosedominating
thesolarsystemzodiacaldustcloudprovideanaturalexplanationfortheapparentbackwardscatteringbehavior.Infact,weseethe
4
phasesofthedustgrainsinthesamewayaswecanobservethephasesoftheMoonandotherlargesolarsystembodies.Weoutline
howthetheoryofthescatteringbehaviorofplanetesimalscanbeusedtoexplaintheFomalhautdustproperties.Thisindicatesthat
]
P theFomalhautdustringisdominatedbyverylargegrains.ThematerialorbitingFomalhaut,whichisatthetransitionbetweendust
E andplanetesimals,can,withrespecttotheiropticalbehavior,bestbedescribedasmicro-asteroids.
. Keywords.Keywordsshouldbegiven
h
p
-
o
1. Introduction the conclusion that the bright side of the ring should be the
r
t onetiltedtowardstheobserver.UsingtheVeryLargeTelescope
s Optical images of some nearby stars show the presence of
Interferometer (VLTI), the rotation axis of the central star and
a
orbiting solid material through the starlight it scatters (e.g
[ subsequently the inclination axis of the system was inferred
Kalasetal., 2005). This has been interpreted as evidence of an
(LeBouquinetal., 2009). This inclination is opposite to pre-
1 evolvingplanetarysystem.Collisionsbetweenlargebodiessuch
vious assumption in that the direction of scattering is reversed
v as asteroids are believed to be responsible for the production
fromforward-dominatedtobackward-dominated.Thiscontrasts
6 of these small solid particles. This is similar to our own so-
withthecurrentconsensusinthefieldoflightscattering.
1 lar system, in which interplanetary dust grains are found that
5 Arelativelysimpleexplanationforthiscanbefoundbycon-
are thought to originate from asteroids and comets. The size
0 sidering bodies much larger than interplanetary dust grains. As
andchemicalcompositionofthesegrainsdeterminethewayin
. an example let us take the Moon, which for all practical pur-
1 whichtheyreflectstarlight.Laboratorymeasurementsandlight
posescanbeconsideredagiganticdustgrain.However,itisev-
0 scatteringtheoryshowthatsmallsolidparticlesscattermostin-
ident that the Moon is bright when viewed in a backward scat-
0 cident light in the forward direction, in agreement with astro-
1 tering situation (full Moon) and it is dark in the forward scat-
nomicalobservations.However,recentobservationsofthedust
: tering case (new Moon). The question then arises: is the Moon
v orbitingthenearbystarFomalhauthaveshownforthefirsttime
trulybackscattering?Theansweris“no”.Theapparentsurprise
i that the grains in this system scatter most of the light in the
X that the Moon is forward scattering is caused by the sharp and
backward direction (LeBouquinetal., 2009). This is in sharp inmostcasesinvisiblediffractionpeakitcauses.FortheMoon
ar contrast with the current understanding of light scattering and this diffraction spike, caused by interference effects due to the
challengestheanalogybetweeninterplanetarydustandthedust
shadowoftheMoon,issonarrowlyforwardpeaked thatitcan
observedinFomalhaut.Inthispaper,weshowthatanexcellent
onlybeseenintheforward-mostfractionofadegree1,i.e.when
matchtotheobservedbackscatteringbehaviorofthematerialor-
theSunisrightbehindtheMoon.Forallpracticalpurposes,the
bitingFomalhautcanbeobtainedbyusingthetheoryofregolith Moonappearstobebackscattering,anddiffractioncanbeomit-
coveredsurfaces.
ted.
Forsmallergrains,thediffractionspikeoftendominatesthe
overallscatteringbehavioroverabroaderrangeofscatteringan-
2. Apparentbackwardscattering
gles. It appears that the disk around Fomalhaut contains dust
ThecircumstellarringofFomalhautasobservedbytheHubble grains that behave backward-reflecting like asteroidal bodies.
space telescope clearly shows a dark side and a bright side This must be because the grains are very large, making the
(Kalasetal., 2005). Regarding the orientation of the ring, the diffraction spike so narrowly forward-peaked that it is outside
general forward scattering behavior of dust grains has led to
1 For objects the size of the Moon, the diffracted energy at visible
Sendoffprintrequeststo:M.Min,e-mail:[email protected] wavelengthsisconfinedtowithintheforward-mostµ-arcsecond.
2 M.Minetal.:ThelunarphasesofdustgrainsorbitingFomalhaut
ometricalopticscanbeapplied,andinadditionweonlyconsider
surfacereflections,i.e.transmittedraysareignored.Werealize
that using geometrical optics is a rough approximation for the
grainsizesweareconsidering.Nevertheless,sinceweareinter-
estedonlyintheoverallshapeoftheangularscatteringfunction
mostly at intermediate scattering angles, the approximation is
not so bad for grain sizes a few times the wavelength of radia-
tion,especiallyforparticleswithaconsiderableimaginarypart
of the refractive index (Wielaardetal., 1997). In this approxi-
mationthescatteringpropertiesoftheparticlesareindependent
oftheirsizeandshapeaslongastheyhaveaconvexsurface.To
Figure1.Angularscatteringfunctionsfortheregolithparticles, compute the reflections we need to specify the refractive index
showing the relative intensity scattered at a scattering angle θ. of the regolith particles which we take to be m = 1.6 + 0.1i,
The pure reflectance function (solid, black curve) is given, and thought to be typical for cosmic materials. However, we find
diffractioneffectsareaddedforparticleswithanarrowsizedis- that the final result is hardly influenced by modest variations
tribution around 3µm grains (dashed curve). The exact scatter- in the refractive index (with the real part ∼ 1.5 − 1.8 and the
ing function for this narrow size distribution of spheres as ob- imaginary part . 0.3). The angular scattering function for the
tainedfromtheMietheoryisalsoshown(solid,greycurve). regolithparticleswhichresultsfromthisisdominantlyforward-
scattering,whiletheirsinglescatteringalbedois11%(omitting
diffraction).InFig.1weshowthreedifferentcurves:theangu-
theobservablerangeofangles.FortheFomalhautsystem,which lar scattering function omitting diffraction (solid, black curve),
hasaninclinationof25◦,therangeofobservablescatteringan-
the angular scattering function with added contribution from
gles is 25◦ < θ < 155◦. Kalasetal. (2005) already noted that diffraction by a narrow size distribution of spheres with diam-
theapparentsinglescatteringalbedoofthedustgrainsisinthe
etersaround3µm(dashedcurve),andthefullangularscattering
rangeof5−10%,whichislowforanykindofdustgrains.Such
functioncomputedusingtheMietheoryforthisnarrowsizedis-
values are more typical for the geometric albedo of asteroids.
tribution(solid,greycurve).Allcurvesareplottedasafunction
The geometric albedo of asteroids does not include the contri- ofthescatteringangle,i.e.0◦isforward-scatteringwhile180◦is
bution fromdiffraction.For largegrains,itiseasilyshown that
backward-scattering. Note that the curve obtained using reflec-
when diffraction is included, the single scattering albedo is al- tionplusdiffractionisquitesimilartothatobtainedusingthefull
ways larger than 50% (vandeHulst, 1957, Chapter 12). Thus,
Mietheory.Itcanbearguedthat3µmisratherlargeforthere-
thetruesinglescatteringalbedoofthegrains,averagedoverall
golithparticlescoveringthegrainswediscussbelow.However,
angles and including diffraction, must be much higher than the
itdoesallowthegrainstobecoveredbyorcomposedofthesere-
observedvalue,indicatingthatmostofthescatteredlightisnot
golithparticles.Theparametersderivedaboveareusedasinput
detected.
for the optical properties of the regolith particles in the Hapke
theory.
3. Modelingthescatteringfunctionofasteroidal
bodies 3.2.Thegrainsasawhole
Below we treat the dust grains in the Fomalhaut system as Theresultingsinglescatteringpropertiesoftheregolithparticles
micro-asteroids, meaning that we use the theory of reflectance obtained in the previous paragraph are inserted into the Hapke
byregolith-coveredasteroidalsurfacestocomputetheirscatter- theory.Notethatweusethetheoryinitsmostsimpleform,i.e.
ingbehavior.Forthereflectanceofasteroidalsurfaces,weusean we ignore effects of macroscopic roughness and the opposition
analyticalmodelforthebidirectionalreflectance(Hapke,1981). effect.Inthatcase,theonlyremainingparametersarethesingle
Forsimplicityweusethetheoryomittingeffectsofmacroscopic scattering albedo and the angular scattering function of the re-
roughness as discussed in Hapke (1984). This basically means golithparticlesobtainedabove.Aswewillshowbelow,wecan
that we set the macroscopic roughness parameter as discussed findanalmostperfectfitwithoutconsideringadditionalparame-
in that paper to zero. In addition to this, we ignore the oppo- tersonmacroscopicroughnessandoppositioneffect.Thismeans
sitioneffect. This mainly plays a role near backward-scattering that using only the measurements available so far, we cannot
andthuslargelyfallsoutsidetherangeofscatteringanglesthat constrainthemanddecidedtousethemostsimpleformthatcan
wehaveaccesstointheFomalhautsystem.Tointegrateoverthe still reproduce the observations. Much more accurate observa-
surfaceofthebody,weuseMonteCarloraytracing. tionsoftheangularscatteringfunctionoftheFomalhautgrains
areneededtocompareallHapkeparameterstothosederivedfor,
forexample,theMoonorsolarsystemasteroids.
3.1.Theregolithparticles
In Fig. 2 we show the resulting angular scattering function
Asinputinthesecomputations,weneedtoknowthealbedoand of the model Fomalhaut dust grains. The curves show the rel-
angular scattering function for the regolith particles that cover ative intensity scattered at an angle θ. The grey area indicates
thesurfaceofthegrain.Notethatsincetheregolithparticlesare theobservablerangeofscatteringanglesfortheFomalhautsys-
closelypacked,weneedtoknowtheseparametersomittingthe tem.NotethatthegrainswesimulateusingtheHapketheoryap-
contributionfromdiffraction(Hapke,1981).Thecontributionof peartobepredominantlybackward-scattering(weseethecres-
diffractioncanbeseparatedwhenwecomputetheangularscat- cents of the grains), while the small regolith particles covering
tering function and single scattering albedo using geometrical the surface of these larger grains are predominantly forward-
optics. More exact computational methods like the Mie theory scattering, as discussed above. The angular scattering function
donotallowtheseparationofthecontributionfromdiffraction. resultingfromonlyusingtheHapkereflectancetheory,i.e.with-
Wethusassumetheregolithparticlesarelargeenoughsothatge- out including diffraction, is shown by the long dashed line. To
M.Minetal.:ThelunarphasesofdustgrainsorbitingFomalhaut 3
Figure2.Angularscatteringfunctionsforlargedustgrains,showingtherelativeintensityscatteredatascatteringangleθ.Thepure
reflectance function (long-dashed curve) is given, and diffraction effects are added for 10, 30 and 100µm grains (dotted, dashed
and solid curves). The empirical scattering function of the Fomalhaut disk grains from Kalasetal. (2005) is also shown (thick
grey curve, albedo chosen to match the computations). The grey area indicates the observable range of scattering angles for the
Fomalhautsystem.
simulate the finite size of the dust grains, we added the con- afewmicronareheavilydepletedinthesystem.Also,Dentetal.
tribution from diffraction using a diameter of the grains of 10, (2000)findthatlargegrainsdominatethethermalemission.Itis
30 and 100µm in dotted, dashed and solid lines, respectively. remarkable that such large grains are observed directly in opti-
For the diffraction, we took simple Fraunhofer diffraction by cal light in an astronomical object. The reason for this is that
a spherical aperture. To get rid of the resonance structures as- while the dust mass in many systems is dominated by large
sociated with Fraunhofer diffraction, averaging was performed grains (see e.g. Wilneretal., 2005; Testietal., 2003), they are
over a narrow size distribution (a flat distribution from 0.8 to normallyover-shonebyacomponentofsmallgrains.Thiscom-
1.2 times the average size). Grains of 100µm diameter clearly ponent,thoughlessmassive,dominatestheopticalcrosssection,
fit the best to the empirical angular scattering function (shown and therefore appearance, of the disk. The fact that very large
bythethickgrayline)inagreementwithDentetal.(2000).For grains are so visible on a global scale in the Formalhaut disk
theobservedscatteringfunctionofthegrainsorbitingFomalhaut meansthatsmallgrainshavebeencleanedoutfromthisdiskex-
we used the Henyey-Greenstein parameterization obtained by tremely efficiently, and that all there is left are these very large
Kalasetal. (2005), where we simply switched the forward and grains,directlyprobedbyreflectedstellarlight.
backwardscatteringdirections,i.e.switchingtheasymmetrypa- Weconcludethatweareobservingthetransitionpartofpa-
rametergfrom+0.2to−0.2.Theangularscatteringfunctionob- rameterspace,goingfromdustscatteringtoplanetesimalreflec-
tainedinthiswaywasmultiplicativelyscaledtomatchthecom- tion. In fact, one might conclude that, almost 400 years after
putations. This scaling factor gives the single scattering albedo GalileoobservedthecrescentofVenus,weareseeingthecres-
ofthegrains.Wefoundthatwhenweintegratedtheangularscat- centsofthelargedustgrainsinthediskaroundFomalhaut.
teringfunctionresultingfromtheHapketheoryovertheangles
accessibleintheFomalhautsystem(theareaindicatedingray), Acknowledgements. Weareindebtedtothereferee,LudmillaKolokolova,for
constructivecommentsleadingtoasignificantimprovementofthepaper.
the albedo of the grains averaged over the observable range of
angles is 5%. This is consistent with the low albedo found by
Kalasetal.(2005). References
Ifweconsidertheassumptionsontheregolithparticlesand
the Hapke theory it is clear that the model we present is quite Dent,W.R.F.,Walker,H.J.,Holland,W.S.,&Greaves,J.S.2000,MNRAS,
314,702
rough.However,afirmoutcomeofthemodelisthatfortherange
Hapke,B.1981,J.Geophys.Res.,86,3039
ofscatteringanglesweobservethescatteringisdominatedbyre- Hapke,B.1984,Icarus,59,41
flectionratherthanbydiffraction.Thisimplies,asshownabove, Kalas,P.,Graham,J.R.,&Clampin,M.2005,Nature,435,1067
thatthegrainsarelargerthan100µmindiameter.Thisoutcome LeBouquin,J.-B.,Absil,O.,Benisty,M.,etal.2009,A&A,498,L41
doesnotdependontheassumptionsinthemodelandistherefore Stapelfeldt,K.R.,Holmes,E.K.,Chen,C.,etal.2004,ApJS,154,458
Testi,L.,Natta,A.,Shepherd,D.S.,&Wilner,D.J.2003,A&A,403,323
asolidconclusion.
vandeHulst,H.C.1957,LightScatteringbySmallParticles(NewYork:Wiley)
Wielaard, D. J., Mishchenko, M. I., Macke, A., & Carlson, B. E. 1997,
Appl.Opt.,36,4305
4. Conclusion Wilner,D.J.,D’Alessio,P.,Calvet,N.,Claussen,M.J.,&Hartmann,L.2005,
ApJ,626,L109
We conclude that the scattering surface in the disk around
Formalhautisdominatedbygrainsofatleast100µminsize.Our
findingsagreewiththefactthattheinfraredspectrumisfeature-
less(Stapelfeldtetal.,2004),indicatingthatgrainssmallerthan
