Table Of ContentMon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
Polarization evolution accompanying the very early sharp
decline of GRB X-ray afterglows
Yi-Zhong Fan1,2,3⋆, Dong Xu4 and Da-Ming Wei1,2,5
8 1PurpleMountainObservatory,ChineseAcademyofSciences,Nanjing210008, China
0 2NationalAstronomicalObservatories,ChineseAcademyofSciences, Beijing100012, China
0 3NielsBohrInstitute, NeilsBohrInternational Academy,Blegdamsvej17,DK-2100Copenhagen, Denmark
2 4DarkCosmologyCentre,NielsBohrInstitute, UniversityofCopenhagen, JulianeMariesVej30,2100Copenhagen, Denmark
n 5JointCenter forParticleNuclearPhysicsandCosmologyofPurpleMountainObservatory-NanjingUniversity,Nanjing210008, China.
a
J
1
2 Accepted2007December19.Received2007December04;inoriginalform2007October 02
]
h
ABSTRACT
p
- In the synchrotron radiation model, the polarization property depends on both the
o configuration of the magnetic field and the geometry of the visible emitting region.
r
t Some peculiar behaviors in the X-ray afterglows of Swift gamma-ray bursts (GRBs),
s
such as energetic flares and the plateau followed by a sharp drop, might by highly
a
[ linearly-polarized because the outflows powering these behaviors may be Poynting-
fluxdominated.Furthermore,thebroken-downofthesymmetryofthevisibleemitting
2
v regionmaybehidingincurrentX-raydataandwillgiverisetointerestingpolarization
7 signatures. In this work we focus on the polarization accompanying the very early
1 sharpdecline ofGRB X-rayafterglows.We show thatstrongpolarizationevolutionis
5
possible in both the high latitude emission model and the dying centralengine model
1
which are used to interpret this sharp X-ray decline. It is thus not easy to efficiently
.
1 probethephysicaloriginoftheveryearlyX-raysharpdeclinewithfuturepolarimetry.
0
Strong polarization evolution is also possible in the decline phase of X-ray flares and
8
in the shallow decline phase of X-ray light curves characterized by chromatic X-ray
0
: VS. Optical breaks. An XRT-like detector but with polarization capability on board
v
a Swift-like satellite would be suitable to test our predictions.
i
X
Key words: Gamma Rays: bursts − polarization − GRBs: jets and outflows −
r
a radiation mechanisms: nonthermal
1 INTRODUCTION
ThesuccessfullaunchoftheSwiftsatelliteinNov2004openedanewwindowtorevealwhathasbeenhappeningintheearly
afterglow phaseofgamma-ray bursts(GRBs).Assummarized inZhangetal.(2006) andNouseketal.(2006), inacanonical
SwiftGRBX-rayafterglowlightcurvesomeinterestingfeaturesareemerging(detectedinagoodfractionofbutnotallbursts),
including a very early sharp decline (phase-I), a shallow decline or even plateau, (phase-II), preceding the conventional or
so-called “normal” decay phase (phase-III & -IV), and the energetic X-ray flares (phase-V). Various modifications of the
standard GRB afterglow model have been put forward to explain the observations (see M´esz´aros 2006; Piran & Fan 2007;
Zhang 2007 for recent reviews). Although the underlying physical processes are not very clear, it is widely suggested that
the X-ray Phases I, II and V may be relevant to prolonged activities of the central engine while the simultaneous optical
afterglowisdominatedbyGRBforwardshockemission(Fan&Wei2005;Zhangetal.2006;Ghisellinietal.2007;Panaitescu
2007). If so, the early time UV/optical and X-ray polarization behaviors should be independent and may be very different
from each other. In some models, the outflows powering energetic flares and X-ray plateaus followed by a sharp drop are
Poynting-flux dominated (e.g., Gao & Fan 2006; Giannios 2006; Troja et al. 2007), thus high-linear polarization of these
synchrotron radiation photonsis expected (Fan, Zhang & Proga 2005).
⋆ E-mail:[email protected]
2
Excepttheconfigurationofthemagneticfield,thegeometry oftheemittingareacaninfluencethepolarization property
of the synchrotron radiation as well. The broken-down of the symmetry of the visible emitting region may be hiding in the
peculiar X-ray data and can give rise to interesting polarization signatures. Motivated bythis idea, in section 2 we calculate
the polarization property of Phase-I and show that strong evolution is likely. In section 3, we argue that similar phenomena
are also expected in the decline phase of the X-ray flares and possibly in the late time part of Phase-II, especially those
associated with a chromatic break in optical band. We also discuss the detection prospect there. In section 4, we summarize
ourwork with some discussions.
2 THE POLARIZATION EVOLUTION ACCOMPANYING PHASE-I
Forphase-I,twoleadinginterpretationsare:(i)theseX-rayphotonsareactuallythehighlatitudeemissionofamaininternal
shock pulse (Fenimore et al. 1996; Kumar & Panaitescu 2000; Zhang et al. 2006) while the central engine has died; and (ii)
theseX-rayphotonsarepowered bytheweaker andweaker activityof thecentralengine, i.e., thesharp X-raydeclinetraces
theactivityofthedyingcentralengine(Fan&Wei2005;Piran&Fan2007)1.Theinterpretation(i)hasbeenwidelyaccepted
becausesomeX-raydeclinescanbeadjustedas∝(t−t0)−(2+βX) predictedbythehighlatitudeemission model(Liangetal.
2006; Yamazaki et al. 2006; Zhang, Liang & Zhang 2007a), where β is the X-ray spectral index, and t is a free parameter
X 0
parameterizingthebeginningofacertaininternalshockpulse2.Suchatest,however,isnotconclusive.AsshowninSakamoto
et al. (2007), for the very sharp X-ray decline, the data itself can not well constrain the function form (t−t )−k. In many
0
events,kcanbetakenasaconstant(forexample,k∼2)andthusisnotrelevanttoβ .Additionally,itisnoteasytoseewhy
X
theGRBcentralengineturnsoffabruptlywhilethenumericalsimulationactuallyshowsthecontrast(MacFadyenetal.2001;
Zhang, Woosley & Heger 2007b). Interpretation (i) also fails to interpret some slowly decaying phase-I as in GRB 060614
(Butler & Kocevski 2007;Zhanget al. 2007a;Xu et al. 2008).Asforinterpretation(ii),therehasbeennostrictobservational
test so far though it seems possible for any energetic burst phenomenon.
Wetakeintoaccountbothmodelsinourcalculationandexaminethepossibilitytodistinguishbetweenthesetwomodels
with futureX-ray polarimetry.
2.1 High latitude emission model
Ifinterpretation(i)iscorrect,i.e.,thesharpX-raydeclineisduetothehighlatitudeemission ofamaininternalshockpulse,
there would be very interesting polarization signals. As is known, GRB outflow is likely to be jetted. In this work we only
consider theuniform jet model. Following Ghisellini & Lazzati (1999), weassume that thehalf-opening angle of theejecta is
θ,andthatthelineofsight(L.o.S.)makesanangleθ withrespecttothejet’scentralaxis(C.A.,seeFig.1aforillustration).
j v
The probability of observing the ejecta along its C.A. is vanishingly small since it corresponds a very small solid angle. For
typical bright GRBs, the L.o.S. is likely to be within the cone, i.e., θ < θ. At early time, the high latitude emission is
v j
from zones around the L.o.S. satisfying θ 6 θ −θ , so the net polarization of the detected photons vanishes because of the
j v
symmetry; but at later time, the high latitude emission is from θ > θ −θ , the symmetry is broken and the observed net
j v
polarization is not zero any longer.
In this model, thephotonsarrived at t were emitted from
θ≈(2ct/R )1/2, (1)
0
where R ∼R is the radius of the main internal shock pulse and c is the speed of light. Below we adopt the simplified
0 prompt
calculation of Ghisellini & Lazzati (1999). The notations used in Fig.1b are related to R , θ and θ as R ≈ R sinθ and
0 j v 0 j
R ≈R sin(θ −θ ). When ρ>R , thering is not a complete circle. The missing parts are in (0, ψ ) and (2π−ψ , 2π) as
1 0 j v 1 1 1
shown in Fig.1. Approximately,we have(Ghisellini & Lazzati 1999)
ψ1 ≈ π2 −arcsin[2R21ρR(R−−RR21−1)ρ2], for ρ>R1; (2)
( 0, for ρ<R1,
1 Thesetwointerpretations canbealsoappliedtothesharpdeclineoftheX-rayflares.
2 In this interpretation, the duration of the main prompt emission pulse should be in order of the timescale of whole burst otherwise
the total high latitude emission of those early pulses decay too fast to account for the observation (see Fig.2 of Fan & Wei 2005 for
illustration). This is somewhat unusual in the standard internal shock model, inwhich the typical variability timescale isjust inorder
of milliseconds. A possible solution of this puzzle is to assume that the prompt γ−ray emissionare powered by some magnetic energy
dissipationprocessesatadistance Rprompt∼1015−1016 cmtothecentralengine,sotheemissiondurationisgovernedbytheangular
timescale∼Rprompt/(2Γ2c)∼6(Rprompt/1015 cm)(Γ/50)−2 sec,whereΓisthebulkLorentzfactoroftheoutflow.
3
V
j
C.A.
(a) L.o.S.
R
C.A.
R1
L.o.S.
1
(b)
Figure 1.(a)ThehighlatitudeemissionmodelfortheX-raydecline.Thelargertheanglewithrespecttothelineofsight(L.o.S),the
latter the emission arrives at us because of the geometry effect. (b) Sketch of the geometrical set-up used to compute the polarization
signal(seealsoGhisellini&Lazzati 1999).
where ρ≈R sinθ.
0
Inprinciple,theGRBoutflowcouldbeeitherbaryon-richorPoynting-fluxdominated.Fortheformerthemagneticfield
is generated in the shock front and is likely to be random. For the latter the magnetic field is coherent in a large scale. The
resulting polarization light curvesare expected to be different in thesetwo cases, as shown below.
Random magnetic field. The net polarization of the emission from ρ−ρ+dρ can be approximated by (See the
Appendixfor the derivation)
(2ζ2−1)sin2α |sin(2ψ )|
Π(ρ) ≈ Π 1 , (3)
02ζ2sin2α+(1+cos2α) 2π−2ψ1
where Π is the maximum polarization at one point, α is the viewing angle with respect to the fluid velocity in the outflow
0
comoving frame (see Figure A1 for illustration), which is governed by cosα = (cosθ−β)/(1−βcosθ), and the parameter
ζ2 =<B2 >/<B2 >denotesthelevelofanisotropyofthemagneticfielddistribution,whereB andB aremagneticfield
k ⊥ ⊥ k
perpendicularandparallel tothenormaloftheshockplane.β=(1−1/Γ2)1/2 isthevelocityoftheGRBejectainunitsofc.
Wenotethatineq.(3)Π(ρ)∝sin2α/(1+cos2α)ifζ2 ∼0.ForΓ(θ −θ )≫1,wehaveΠ(ρ)∝2Γ2(θ −θ )2/[1+Γ4(θ −
j v j v j
θ )4]≪1. Theresulting netpolarization can thusbeneglected. But for2ζ2sin2α≫1+cos2α,thesecond term in theright
v
side of eq.(3) is →1. Significant net polarization is possible.
Asθ>(θ −θ ),theobserversjustdetected∼(π−ψ )/π partofthecircle,theX-rayfluxwilldeclinemoresteeplythan
j v 1
t−(2+βX) and is governed by
π−ψ (t)
F (t)∝ 1 t−(2+βX). (4)
X π
We plot in Fig.2 the sharp X-ray decline and the corresponding polarization light curves. When the emission is from a
circle ring,thenet polarization is zerobecauseof thesymmetry.Asθ isgetting larger andjust part oftheringis visible,the
observedpolarization degreeisnotzeroanylongeruntilρ2→2R R−R2 (i.e.,ψ →π/2).Atthepoint ofψ =π/2,thenet
1 1 1 1
polarization becomeszeroagain according toequation(3),becausethepolarization direction abruptlychangesbyπ/2.After
thispoint, the polarization increases again. But theflux may be too weak to havea significant polarization measurement.
Homogenous magnetic field. In this case, the instantaneous stokes parameters (note that V =0 as the polarization
is linear) and thepolarization degree are given by(Granot & K¨onigl 2003):
Q 2π−ψ1[(1−y)2cos2φ+sin2φ]ǫ/2 cos(2θp) dφ
U ψ1 1+y sin(2θp)
(cid:26) (cid:27) =Π (cid:26) (cid:27) , (5)
I 0R 2π−ψ1[(1−y)2cos2φ+sin2φ]ǫ/2dφ
ψ1 1+y
and R
U2+Q2
Π= , (6)
I
p
where θ =φ+arctan(1−ycotφ), y =Γ2θ2 and ǫ=(1+β )/2 (Granot & K¨onigl 2003). Again the polarization light curve
p 1+y X
4
3
10
s) 102
nit
u
y 1
ar 10
bit
ar
x ( 100
u
Fl
-1
10
0.5
ee 0.4
gr Homogenrous
de magnetic field
n 0.3
o
ati
z
ari 0.2 Random( mag)netic
Pol field =2
0.1
0.0
10 100
t (sec)
Figure 2. The sharp X-ray decline (upper panel) and the corresponding polarization light curves (lower panel) after a time-shift
correction,inthecaseofthehighlatitudeemissionmodel.ForthisfigureweassumedΠ0=60%,Γ=50,θj=0.1,θv=0.07,R0=1015
cmandβX =1.15.
hasbeenplottedinFig.2,whichissimilartothatoftherandommagneticfieldcaseexceptaninitialhighpolarizationdegree.
Thisstronglysuggests thatthelatetimenon-zeropolarization ismainly contributedbythegeometry effect,asinthecaseof
arandommagneticfield.Ifthesymmetryhasnotbeenbrokendown,thelatetimenetpolarizationforanorderedmagnetized
outflow disappears, as already found in Nakar, Piran & Waxman (2003).
2.2 Dying central engine model
Generally, in the dying central engine model there are two possibilities. (1) If the outflow powering the fast decaying X-ray
tail emission always has a Lorentz factor Γ > 1/(θ −θ ), no strong polarization evolution is expected unless the physical
j v
composition oftheoutflowhaschangedalot (e.g.,from byrondominated toPoynting-fluxdominated).Wethusmaybeable
todistinguishbetweenthehighlatitudeemissionmodelandthedyingcentralenginemodelwiththefutureX-raypolarimetry.
(2)ThebulkLorentzfactoroftheoutflowdropssoquicklyastosatisfyΓ<1/(θ −θ )inashortperiod.WhenΓ<1/(θ −θ ),
j v j v
thesymmetryof thevisibleemittingarea isbrokenandinteresting polarization signature beginstoexist.Sincethere islittle
information about the evolution of Γ with t, we here simply assume Γ ≈ 150(t/10)−1 for 10 < t < 200 sec and θ and θ
j v
areconstant. Again,we calculate thepolarization evolution in thecases ofrandom magnetic fieldand Homogenous magnetic
field. But now the calculation should be due to the emission area within θ 6 1/Γ, rather than from a ring at θ in the high
latitudeemissionmodel.Therefore,wecalculatethequantitiesof(Q, U, I)alongtheradiusρandintegratethemoverρ.Such
calculations have been carried out by Ghisellini & Lazzati (1999) and Granot & K¨onigl (2003), respectively. Our numerical
results are presented in Figure 3. One can see that polarization evolution is evident and shares some similarity with that in
Fig.2, particularly in the case of random magnetic filed, though the polarization degree of high latitude emission model is
larger than that of the dying central engine model. This is reasonable since in the former case the emission is from the ring
whileinthelattercasetheemission isfrom θ61/Γ.Inthecaseofhomogenous magnetic filed,theevolutionissmallbecause
of thesignificant polarization background(thedegreeis ∼0.5) caused bytheordered magnetic field from thecentralsource.
3 PROSPECT OF DETECTING THE X-RAY POLARIZATION
OurcalculationsinSection2areactuallyapplicabletomorephasesintheGRBX-rayafterglowlightcurve.Oneisthedecline
phaseofX-rayflares(inPhase-V),whichisalsoattributedtothehighlatitudeemissionmodelorthedying(re-)activityofthe
central engine. The other is the late phase of the X-ray flattening (in Phase-II). Recently Ghisellini et al. (2007) interpreted
Phase-II,particularlythoseshowchromaticbreaksinX-rayandopticalbands,astheprolongedemissionofthecentralengine.
The X-ray flattening ceases to exist when the bulk Lorentz factor of the outflow is so small that one sees the edge of the
5
0.04
Random (magn)etic
field =2
e0.03
e
gr
e
d
n
o
ati0.02
z
ari
ol
P
0.01 Homogenrous
magnetic field
-0.5
0.00
10 100
t (sec)
Figure3.Thepolarizationlightcurvesinthecaseofdyingcentralenginemodel.ThephysicalparametersarethesameasthatofFig.2
except we assume Γ= 150(t/10)−1 for 10 < t< 200 sec. Please note that the dashed line is not the real polarization degree unless a
factor0.5hasbeenadded.
outflow.Iftheseinterpretationsarecorrect, strongpolarization evolutionshouldbepresentattheendofPhase-IIandinthe
decline of Phase-V.
Measuringpolarization isofgrowinginterestinhighenergyastronomy.Newtechnologiesarebeinginvented,andseveral
polarimeter projects are under construction. In the X-ray band, the ongoing projects include XPE (Elsner et al. 1997),
PLEXAS(Marshelletal.1998), POLAR(Produitetal.2005),XPOL(Costaetal.2006),POET(Polarimeters forEnergetic
Transients,seeMcConneletal.2007),andsoon.Forexample,theplannedPOETincludestwoinstruments:GRAPE(Gamma-
RayPolarimeterExperiment)andLEP(LowEnergyPolarimeter,workinginsoftX-rayband).Soitmaybesuitabletodetect
thepolarizationsignaturespredictedintheearlyX-rayafterglowofGRBs.Animportantissueiswhetheranyofthesedetectors
couldperformapromptslewtotheGRBslocalizedbytheγ−raymonitor.AnidealinstrumentwouldbeanXRT-likedetector
with polarization capability on board a Swift-likeGRB mission.
4 SUMMARY & DISCUSSION
ManysurprisesinGRBafterglows, mainlyin theX-rayband,havebeenbroughtsincethesuccessfullaunchofSwiftin2004.
Though not clearly understood yet, the (re-)activity of GRB central engine is likely to play an important role on producing
afterglows, as speculated by Katz, Piran & Sari (1998). The different temporal behaviors in the UV/optical and X-ray
afterglowfavorthatthepeculiarX-rayemission tracestheprolongedactivityofthecentralenginewhiletheopticalafterglow
is dominated by the forward shock emission (Fan & Wei 2005; Ghisellini et al. 2007). If true, the polarization behaviors of
the early X-ray and optical afterglow may be independent and different. At such an early time, the linear polarization of
theoptical forward shock emission is expected to be very small (Ghisellini & Lazzati 1999) unless theinterstellar medium is
magnetized (Granot & Konigel 2003), because the Lorentz factor of the outflow is so large that the visible emitting region
is well within the cone of the ejecta and is thus symmetric. But in the X-ray band, interesting polarization signatures may
bepresent.For example,Fan et al. (2005) argued that some X-ray flaresmight haveahigh linear polarization because these
newoutflows are likely tobe Poynting-fluxdominated. Such an argument is also applicable totheX-ray plateau followed by
a sharp drop as detected in GRB 070110.
In this work, we focus on the polarization properties of the very early sharp decline of X-ray afterglows (Phase-I).
In the high latitude emission model, the later the emission, the larger the angular θ (see Figure 1 for details). So at late
time, the symmetry of the emitting ring is broken when part of the emitting ring is out of the cone of the ejecta. Strong
polarization evolution thus emerges (see Section 2.1). For comparison, results from the dying central engine model are a bit
more complicated. If theoutflow powering thefast decaying X-ray tail emission always has a Lorentz factor Γ>1/(θ −θ ),
j v
no strong polarization evolution is expected unless the physical composition of the outflow changed significantly. However,
if the Lorentz factor drops rapidly to satisfy Γ < 1/(θ −θ ), then polarization evolution is evident. In this scenario, if the
j v
magnetic field involved is shock-generated and is thus random, then strong polarization evolution is expected; while if the
magneticfieldinvolvedisbroughtfrom thecentralengineandisthuslarge-scale ordered,thenweak polarization evolutionis
expected due to the significant polarization background (the degree is ∼0.5). One of our goals to calculate the polarization
6
of Phase-I isto seewhetherit’s possible, via X-raypolarimetry, todistinguish between thehigh latitudeemission modeland
thedyingcentralenginemodel.Thisgoalisnoteasytofulfillinviewoftheabovecomparison.However,ourcalculationdoes
support that strong polarization evolution is likely to accompany Phase-I, the decline phase of X-ray flares (Phase-V) and
possiblythelatepartofPhase-II.Fruitfulnovelfeatures areexpectedtoappearinGRBX-rayafterglowpolarimetry,though
detectionofthemisbeyondthecapabilityofcurrentmonitors.WehavewitnessedthebreakthroughmadebySwiftXRTand
wearelikelytoexperienceit oncemoreinthecomingX-raypolarimetry era.OnebyproductofthefutureX-raypolarimetry
is to probe the cosmological birefringence effect that arises in some quantum gravity models (Fan, Wei & Xu 2007 and the
references therein).
Finallywe’dliketopointoutthattheopticalflaresinafewbursts(e.g.,B¨oeretal.2006) mayalsohaveacentralengine
origin (Wei,Yan & Fan 2006). Interesting polarization signatures are also expected.
ACKNOWLEDGMENTS
Wethanktherefereeforher/hisconstructivecomments.We’dliketothankJ.Gorosabel,J.Hjorth,J.FynboandJ.Sollerman
forvaluablecomments,andB.ZhangandX.F.Wuforcommunication.ThisworkissupportedbytheNationalScienceFoun-
dation(grant10673034,10621303)ofChina,theNationalBasicResearchProgramofChina(973program2007CB815404),and
aspecial grant ofChineseAcademyofSciences(Y.Z.FandD.M.W) andbya(postdoctoral) grant from theDanishNational
Science Foundation (Y.Z.F). DX is at theDark Cosmology Centre funded bythe Danish National Research Foundation.
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7
z (velocity)
0
obs
B y
x
Figure A1.Thecoordinatesystemtooptimizethederivation(seealsoSari1999).
APPENDIX A: DERIVATION OF EQUATION (3) IN THIS WORK
Obviously,firstweneedtoestablish asuitablecoordinate systemtooptimize thederivation.Asstated in Sari(1999), at any
point in the shock front thereis a preferred direction, theradial direction, in which thefluidmoves. Wecall thistheparallel
directionandchoosethez-directionofthefluidlocal framecoordinatesystemtobeinthatdirection.Thetwoperpendicular
directions(x,y)areassumedequivalent,i.e.,thesystemisisotropicintheplaneperpendiculartothedirectionofmotion.We
chose the x-direction to be in the plane that contains the z-direction and the direction towards the observer n, as shown in
FigureA1.Supposenowthat themagneticfield hassphericalcoordinates (θ,ϕ) in thatframe. Aquitegeneral description of
the distribution of the magnetic field in such anisotropic system would be to allow different values of the mabgnetic field as
function of the inclination from the preferred direction B = B(θ) as well as a probability function for the magnetic field to
bein each given inclination f(θ).
The relevant component of the magnetic field is that perpendicular to the observer i.e. Bsin(δ), where δ is the angle
betweenthedirectionofthemagneticfieldandtheobserver.Thiswillproducepolarization Π inthedirectionperpendicular
0
both to theobserverand tothe magnetic field i.e. in thedirection nˆ×Bˆ. However, thispolarization should be averaged due
to contributions from magnetic fields oriented differently. On the other hand, because light is a kind of transverse wave, its
polarization should be perpendicular to its propagation. Therefore, we project the polarization to yˆ(i.e., the component Q)
and nˆ×yˆ(i.e., the component U), respectively.
Assuming that theemission is proportional to a certain power of the magnetic field, ∝Bǫ, the total polarization from a
point-likeregion is then3
Q=Π cos(2η)[B(θ)sinδ]ǫf(θ)sinθdφdθ, (A1)
0
Z
and
U =Π sin(2η)[B(θ)sinδ]ǫf(θ)sinθdφdθ, (A2)
0
Z
and
I =Π [B(θ)sinδ]ǫf(θ)sinθdφdθ. (A3)
0
Z
Since nˆ=(sinα, 0, cosα) and Bˆ =(sinθcosφ, sinθsinφ, cosθ), we have
Πˆ =nˆ×Bˆ =(−cosαsinθsinφ, cosαsinθcosφ−sinαcosθ, sinαsinθsinφ).
With yˆ=(0, 1, 0), we obtain
Πˆ ·yˆ cosαsinθcosφ−sinαcosθ
cosη= = ,
|Πˆ| [sin2θsin2φ+(cosαsinθcosφ−sinαcosθ)2]1/2
3 wherewehavetakenintoaccount thefactthat yˆ,nˆ×yˆandΠˆ =nˆ×Bˆ areallonthesameplanewhichisperpendicularto nˆ.
8
and
sinθsinφ
sinη= .
[sin2θsin2φ+(cosαsinθcosφ−sinαcosθ)2]1/2
From theforms of cosη and sinη , we have
(cosαsinθcosφ−sinαcosθ)2−sin2θsin2φ
cos(2η)=2cos2η−1= , (A4)
sin2θsin2φ+(cosαsinθcosφ−sinαcosθ)2
and
2sinθsinφ(cosαsinθcosφ−sinαcosθ)
sin(2η)=2sinηcosη= . (A5)
sin2θsin2φ+(cosαsinθcosφ−sinαcosθ)2
Now substituteequation (A5) into equation (A2),we get
U = Π sin(2η)[B(θ)sinδ]ǫf(θ)sinθdφdθ
0
Z
∝ sin2θsinφ(cosαsinθcosφ−sinαcosθ)|Πˆ|ǫ−2f(θ)dθdφ
Z
π 2π
∝ sin2θf(θ)dθ F(φ,α,θ)dφ. (A6)
Z0 Z0
where F(φ,α,θ)≡ sinφ(cosαsinθcosφ−sinαcosθ) , and we havetaken sinδ=|nˆ×Bˆ|=|Πˆ|.
[sin2θsin2φ+(cosαsinθcosφ−sinαcosθ)2](2−ǫ)/2
To calculate the integration for F(φ,α,θ), we consider
2π π/2 π 3π/2 2π
F(φ,α,θ)dφ=[ + + + ]F(φ,α,θ)dφ. (A7)
Z0 Z0 Zπ/2 Zπ Z3π/2
In regions of φ ∈ (0, π/2) and φ ∈ (3π/2, 2π), sinφ changes the sign but cosφ does not. So in the above integral the
contributionfrom theangleφhasbeencancelled outbythatfrom 2π−φ.Similarresultsholdforthecontributionofregions
of φ∈(π/2, π) and φ∈(π, 3π/2). It is then straightforward toget
2π π/2 2π π 3π/2
U ∝ F(φ,α,θ)dφ=[( + )+( + )]F(φ,α,θ)dφ=0. (A8)
Z0 Z0 Z3π/2 Zπ/2 Zπ
We thus have proved that because of the isotropy of the magnetic field inthe (x,y) direction, the polarization of radiation
emitted from a point-like region (after averaging on magnetic field orientation) must be in the direction perpendicular to the
z-axis and to the observer, i.e., in the yˆdirection.
In the case of ǫ= 2, both Q and I can be analytically integrated. With the relations 2πsin2φdφ= 2πcos2φdφ= π,
0 0
we have
R R
π 2π
Q ∝ f(θ)sinθdθ B2[(cosαsinθcosφ−sinαcosθ)2−sin2θsin2φ]dφ
Z0 Z0
π
= πsin2α B2f(θ)sinθ[2cos2θ−sin2θ]dθ, (A9)
Z0
Notethat <B2 >= πB2f(θ)sinθcos2θdθ and <B2 >= πB2f(θ)sin3θdθ, we have
k 0 ⊥ 0
R <B2 > R
Q∝2πsin2α(<B2 >− ⊥ ).
k 2
Similarly,
π
I = f(θ)sin2θB2|Πˆ|2dθ=2π(sin2α<B2 >+(1+cos2α)<B2 >/2).
k ⊥
Z0
Therefore, thenet polarization is given by(see also Gruzinov 1999 and Sari 1999)
P = Q =Π sin2α(<Bk2>−<B⊥2 >/2) . (A10)
point I 0sin2α<B2 >+(1+cos2α)<B2 >/2
k ⊥
In thehigh latitudemodel for theX-raysharp decline, theemission is from an angle α (or,equivalently ρ).Asshown in
Ghisellini&Lazzati(1999),itisconvenienttowritethepolarizationvectorasacomplexnumberP =P e2iθp andintegrate
point
it between ψ and 2π−ψ .Here θ is theposition angle of thelinear polarization in theobserver frame. The polarization of
1 1 p
a generic ring is thus
9
Π(ρ)= ψ21π−ψ1PpointIdθp =Π (2ζ2−1)sin2α |sin(2ψ1)|, (A11)
R ψ21π−ψ1Idθp 02ζ2sin2α+(1+cos2α) 2π−2ψ1
where ζ2 =R<B2>/<B2 >.
k ⊥
Till here, we thushaveproved equation (3) in this work.
