Gravitational Bending of Light Near Compact Objects

4 Pages·0.18 MB·English

by  Andrei M. Beloborodov

DRAFTVERSIONFEBRUARY5,2008
PreprinttypesetusingLATEXstyleemulateapjv.04/03/99
GRAVITATIONALBENDINGOFLIGHTNEARCOMPACTOBJECTS
ANDREIM.BELOBORODOV1
CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto,60St.GeorgeStreet,Toronto,ONM5S3H8,Canada
DraftversionFebruary5,2008
ABSTRACT
A photonemitted near a compactobjectat an angle α with respect to the radial direction escapes to infinity
at a differentangle ψ > α. This bending of light is caused by a strong gravitationalfield. We show that, in a
Schwarzschildmetric,theeffectisdescribedby1 cosα=(1 cosψ)(1 r /R)whereR/r istheemission
2 − − − g g
radiusinSchwarzschildunits. TheformulaisapproximateanditappliesatR 2r only,howeverattheseradii
0 ≥ g
ithasamazingaccuracy,fullysufficientinmanyapplications. Asoneapplicationwedevelopanewformulation
0
forthelightbendingeffectsinpulsars. Itrevealsthesimplecharacteroftheseeffectsandgivestheirquantitative
2
description with practically no losses of accuracy (for the typicalradius of a neutronstar R = 3rg the error is
n 1%). ThevisiblefractionofastarsurfaceisshowntobeS /4πR2 =[2(1 r /R)]−1whichis3/4forR=3r .
a v − g g
Theinstantaneousfluxofapulsarcomesfromoneortwoantipodalpolarcapsthatrotateinthevisiblezone.The
J
pulseproducedbyoneblackbodycapisfoundtobesinusoidal(lightbendingimpactsthepulseamplitudebutnot
8
its shape). When bothcapsare visible, the pulse showsa plateau: the variablepartsof the antipodalemissions

precisely cancel each other. The pulsed fraction of blackbodyemission with antipodalsymmetry has an upper
1
v limitAmax =(R 2rg)/(R+2rg). PulsarswithA>Amax mustbeasymmetric.
−
7 Subjectheadings: gravitation—radiationmechanisms:general—relativity—stars: neutron—(stars:)
1 pulsars:general—X-rays:binaries
1
1
0
2
0
1. INTRODUCTION 2. THECOSINERELATION
/
h
Lightbendingbygravityisaclassicaleffectofgeneralrela- Let an emitter be located at point E at a radius R (Fig. 1).
p
tivity(e.g. Misner,Thorne,&Wheeler1973). Inaspherically A photon emitted along the radial direction escapes radially.
-
o symmetricgravitationalfield, theexactbendingangleisgiven A photon emitted at some angle α with respect to radius (α
r byanellipticintegral(seeAppendix). Theeffectplaysimpor- is measured by the local observer at E) escapes along a bent
t
s tant role in astrophysics; e.g. weak bending gives rise to the trajectory. We willdescribethetrajectoryinpolarcoordinates
a gravitationallensingphenomenon.Strongbendingoccursnear (r,ψ)whereψ =0fortheescapedirection. Thestandardway
:
v compact objects (black holes and neutron stars) and crucially ofcomputingα(ψ)isgiveninAppendix. Insteadonecanuse
i affectstheiremission. theapproximaterelation
X
In this Letter we give a simple formula for the bendingan-
r glethatreplacestheellipticintegralwithhighaccuracyatradii
a
Rthe≥gr2arvgit.atHinegresprhge=rica2lGoMbje/cct2oifsmthaessScMhw.aWrzescahpipldlyrathdiisusfoor-f 1 cosα=(1 cosψ) 1 rg . (1)
− − − R
mulatoneutronstarsanddevelopanewsimpleformalismfor (cid:16) (cid:17)
the lightbendingeffectsin pulsars. We illustrate with pulsars
thathavetwopoint-likeantipodalspotsemittingthermalradia-
The accuracy of relation (1) is remarkably high. It gives the
tion,andcomparetheresultswiththeexacttheory(Pechenick,
bendingangleβ =ψ αwithasmallfractionalerrore=δβ/β
Ftaclas, & Cohen 1983). The advantage of our formalism is −
showninFigure2. ForexampleforR=3r themaximumer-
that it gives a clear understanding of the bending effects and g
describes them analytically with almost no losses of accuracy rorisemax =3%(atα=90o)ande<1%forα<75o. High
accuracyismaintainedforR > 2r . ForR < 2r theapprox-
aslongasstarswithR>2r areconsidered.Theexacttheory g g
g imationisnotapplicable: thencosα = 0wouldcorrespondto
welldescribesstarswithanyR,includingR < 2r wherethe
bendingangleβ canexceed90o,howeveritrequiregsacompli- ceoscsaψpin<g−fro1m. WRe>the2rrefowrehelirmeβito<ur90coo.nsiderationtoradiation
cated numerical treatment. Standard models of neutron stars g
Thehighaccuracyofequation(1)isastrikingpropertyofthe
predict R > 2r with a typical R 3r (e.g. Shapiro &
g ≈ g Schwarzschildspacetime. Thisfunctionalshapecannotbeun-
Teukolsky 1983). Our formalism then works with 1% ac-
curacyandshouldbesufficientinmanypracticalpro∼blems. It derstoodasalinearexpansioninrg/R(thatexpansionisgiven
inAppendixanditsaccuracyismuchworse). Equation(1)can
shouldalso be goodforaccretiondisksaroundSchwarzschild
be understood as follows. Let us consider x = 1 cosα as
blackholes:thediskinneredgeisat3rgandthebulkofemitted a small parameter and expandy = 1 cosψ in te−rms of xk.
radiationdoesnotentertheregionR<2rg. Aftersomealgebrawefind −
1AlsoatStockholmObservatory,SE-10691Stockholm,Sweden
andAstro-SpaceCenterofLPI,Profsojuznaja84/32,Moscow117810,Russia
1
2 BENDINGOFLIGHTNEARCOMPACTOBJECTS
3
x 1 x
2
y = u
1−u − "112(cid:18)1−u(cid:19)
4
1 5 x
5
+ u +O(x ) , (2)
224(cid:18)3 − (cid:19)(cid:18)1−u(cid:19) #
where u = r /R. The linear term is just equation (1); it de-
g
scribesthestandardsolidangletransformationdΩ0 =2πdx
→
dΩ = 2πdy along the radial ray. The interesting property of
equation (2) is that the x2-term vanishesand the higher order
termshavesmallcoefficients.Itexplainstheextremelyhighac-
curacyofequation(1)atsmalluand/orx(Fig.1). Onlywhen FIG.1.—Examplephotontrajectory.TheemissionpointEisatR=2.5rg
and the emission angle is cosα = 0.1; the corresponding impact param-
both x 1 and u 0.5 there is a sizable correction to the
linearte→rmandhere→epeakssharply.Thelinearexpansioninu βete=r bψ−=α3≈.213r5go..TIhnetahpisprcoaxsiemactoestψraje=ctor−y0(.e4q8.23)ainsdalsthoeplboetnteddinhgeraenagnlde
(alsoderivedinAppendix)readsy =x(1+u)+O(u2). itisnotdistinguishablefromtheexacttrajectory.
Thoughthecosinerelation(1)issufficientforoursubsequent
discussion, note that it also gives a simple description for the
shape of a photon trajectory with a given impact parameter b
(Fig. 1). Equation(1)canbewrittenatanypoint(r,ψ)along
the trajectory if α measures the angle betweenthe photonve-
locity and the local radial direction. Combining (1) with (10)
wefindr2 =b2[sin2ψ+(1 cosψ)2r /r]−1and
g
−
r2(1 cosψ)2 b2 1/2 r (1 cosψ)
r(ψ)= g − + g − .(3)
" 4(1+cosψ)2 sin2ψ# − 2(1+cosψ)
Trajectories(3)areveryclosetotheexactones. Thefractional
errorofratagivenψisδr/r <4 10−3aslongasr >2r .
g
×
3. THEVISIBLESURFACEOFASTARANDTHEOBSERVEDFLUX
Considera sphericalcompactstar viewedby anobserverat
distanceD R. Choosesphericalcoordinates(R,ψ,φ)with
≫
thepolaraxisψ = 0directedtowardtheobserver. Whatisthe
observedfluxdF fromasurfaceelementdS ofthestar? First
ofalloneshoulddecidewhetherdS isvisible. Inaflatspace-
time the visibility conditionwould be simply µ = cosψ > 0
whereψ isthepolarangleofdS. Thestargravitycreatescur-
vature that allows the observer to see regions with negative µ
downtoacriticalµ thatdefinesthedarksideofthestar. The
v
observedlightfromthecircleµ=µ isemittedtangentiallyto
v
thestar (α = 90o). Exactcalculationsofα(ψ) viatheelliptic
FIG.2.—Accuracyofthecosinerelation(1).Thisrelationgivesthebendig
integralyieldtabulatedµv,e.g.µv = 0.886, 0.633, 0.484 angleβ = ψ−αwithanerrore = δβ/β whichdependsontheemission
− − −
forR/rg = 2,2.5,3. Fromequation(1)wefindimmediately radiusR/rgandtheemissionangleα.Herethecontoursofe=const(with
the approximateµ = r /(R r ). Thevisiblefractionof logarithmicstep0.5)areplottedontherg/R−αplane.
v g g
thesurfaceisS /4πR2−=(1 µ−)/2=[2(1 r /R)]−1,e.g.
v v g
− −
weget0.75forR=3r whiletheexactvalueis0.742.
g
Appendix gives the standard formula for the observed flux
When the local intensity of radiation I0(α) is known for
froma visibledS (eq.11). Itrequiresthe knowledgeofα(ψ)
all dS, equation (4) can be integrated over the visible surface
andtheexacttheoryleadstoacomplicatednumericaltreatment
µ > µ togetthetotalobservedflux. Forexample,supposea
v
oftheproblem(Pechenicketal. 1983). Insteadweusetheap-
starhasahomogeneoushotspotofarbitrarysizeandshapethat
proximateequation(1);thenequation(11)reads
emitsthermalradiation[I0(α) = const]. Denotethespotarea
r 2 dS by s = dS and the corresponding projected area by s⊥ =
dF = 1− Rg I0(α)cosαD2, (4) theµdinSte(girRfaolsnalyreatapkaertnoofvtehrethsipsoptairst)i.nTthheenvitshiebloebzseornveedµfl>uxµivs
winhgelyrescimospαle.=Aµp(a1rt−(cid:16)frrogm/Rth)e(cid:17)+rerdgs/hRift. fEaqctuoarti(o1n(4)ris/Rsu)r2prtihse- FsR⊥=<(0I0w/Dhe2n)(th1e−sprogt/iRs)in2[tsh⊥e(r1eg−iorng/µRv)<+µsr<g/0R. ]. Notethat
g
−
elementdS emitsasiftheproblemwereNewtonianwithψre-
placedbyα.
BELOBORODOV 3
4. PULSARS
Consideraneutronstarwithtwoantipodalhotspotsassoci-
ated with polar caps of the star magnetic field. Assume here
that the spot size is small comparedto the star radiusR. The
starisspinninganditsmagneticaxisisinclinedtothespinaxis
byanangleθ 90o. Thereforethespotsperiodicallychange
≤
theirpositionandinclinationwithrespecttoadistantobserver
andtheobservedradiationpulsates.
Thespotwhosecircleofrotationisclosertotheobserverwill
becalled“primary”andtheothersymmetricspot“antipodal”.
Denote the unit vector pointing toward the observer by d and
theanglebetweendandthespinaxisbyi. Tocomputethera-
diationfluxesfromthe primaryandantipodalspotsoneneeds
toknowtheirinclinations: µ = n dandµ¯ = n¯ d,wheren
andn¯ = narethe spotnormals.· Whenthestar·executesits
−
rotation,µvariesperiodicallybetweenµmin = cos(i+θ)and
µmax =cos(i θ),
−
µ(t)=sinθsinicosΩt+cosθcosi. (5)
FIG. 3.— Location of classes I–IV on the i−θ plane. The angle ξ is
definedsothatcosξ = rg/(R−rg), e.g. ξ = 60o forR = 3rg. The
Here Ω is the angular velocity of the pulsar; t = 0 is chosen classes are determined by the relative positions of µmin = cos(i + θ),
whenµ=µmax. µmax = cos(i−θ), κ ≡ rg/(R−rg), and −κ. ClassI:µmin > κ.
Theantipodalspotisneverseeninsuchapulsar(andtheprimaryspotisvisi-
Considertheblackbodycomponentofthepulsaremission;it
bleallthetime).Theobservedblackbodypulsehasaperfectsinusoidalshape.
has isotropic intensity I0(α) = const. Using (4) we find the ClassII:−κ<µmin<κ<µmax.Theprimaryspotisseenallthetimeand
observedfluxesF andF¯ fromtheprimaryandantipodalspots theantipodalspotalsoappearsforsometime.Thepulsehasasinusoidalpro-
(whentheyarevisible) fileF(t)interruptedbytheplateau(7).ClassIII:µmin <−κ.Theprimary
spotisnotvisibleforafractionofperiod(andthenonlytheantipodalspotis
seen). SuchapulsarshowsaprimarysinusoidalprofileF(t)interruptedby
F =µ 1 rg + rg, F¯ = µ 1 rg + rg. (6) tF¯he(tp)lafrteoamut(h7e)aanndtipthoedapllastpeoatu.iTshinetesurrbu-pptueldsebyocacuwresawkehresninounsloyidthaelsaunbti-ppoudlsael
F1 − R R F1 − − R R spotisinthevisiblezone.ClassIV:−κ<µmin,µmax<κ.Bothspotsare
(cid:16) (cid:17) (cid:16) (cid:17) seenatanytime. Thentheobservedblackbodyfluxisconstant(eq.7). Only
anisotropicemissionwithI0(α)6=constcancontributetopulsations.
HereF1 = (1 rg/R)2I0s/D2,sisthespotarea,andpoint-
−
like-spotapproximationis adopted. For the antipodalspotwe
substitutedµ¯ = µ. Theobserverseestheprimaryspotwhen
−
µ > µ and the antipodal spot when µ¯ > µ ( 3). De-
v v
§
note κ = r /(R r ) = µ . Both spots are seen when
g g v
− | |
κ<µ<κandthentheobservedfluxis
−
r
Fobs =F +F¯ =2 gF1 =const ! (7)
R
Hence the blackbody pulse must display a plateau whenever
bothspotsareinthevisiblezone.Weemphasizethehighaccu-
racyofequations(6)and(7),thoughtheexacttheorydoesnot
givetheidealplateau.ThefractionalerrorδinFobsismaximal
whenµ=µv,e.g.δmax 1.3%forR=3rg.ForR=2rgour
≈
formalismpredictsthatthewholesurfaceisvisibleandFobs is
givenbyequation(7)atanyθ,i,µ;thenδmax 6%.
AtanytimeFobstakesoneofthreevalues:F≈,F¯,orF +F¯.
A pulsar shows the plateau (7) if cos(i + θ) < κ. A pulsar
witharbitraryiandθ belongstooneoffourclassesdescribed
in Figures 3 and 4. It is easy to compute the pulsed fraction
A=(Fmax Fmin)/(Fmax+Fmin)foreachclass,
−
(µmax−µmin)/(µmax+µmin+2κ)classI FIG. 4.— Blackbody pulse profiles for classes I–IV defined in Fig. 3.
A= (µmax κ)/(µmax+3κ) classII,III (8) R = 3rg ischosenandthefluxisnormalized toF1 (themaximumpossi-
(0 − classIV blefluxthatisobservedwhentheprimaryspotisviewedfaceon). ClassIis
exemplifiedwithθ =20oandi=30o,classIIwithθ =30oandi=60o,
classIIIwithθ=60oandi=80o,andclassIVwithθ=20oandi=80o.
AreachesitsmaximumAmax ifthepulsarhasµmax = 1and Theapproximatemodeldevelopedinthispaperisshownbysolidcurvesand
theexactmodelisshownbydottedcurves. Theplateauhasthesamelevelin
µmin <κ. ThismaximumisAmax =(R−2rg)/(R+2rg). allcases:Fp/F1=2rg/R=2/3.
4 BENDINGOFLIGHTNEARCOMPACTOBJECTS
5. CONCLUSIONS toanisotropicemissivitywithapreferentialdirectionalongthe
A new formulation for the light bending effect is given in field. Inanycase,calculationsofI0(α)separatefromthelight
bending effects. Radiation that propagates freely through the
thisLetter. Itmakesuseofanapproximatesimpleformulathat
pulsarmagnetosphereisdescribedbyequations(1)and(4)for
describesphotontrajectorieswithremarkableaccuracy(eq. 1).
This formulation is applied to pulsars where light bending anyI0(α).
is knownto be a crucialeffect. The developedanalyticalthe-
APPENDIX:LIGHTPROPAGATIONINA
ory replaces the previously used complicated numerical treat-
SCHWARZSCHILDMETRIC
mentwithhighaccuracy,sufficientforpracticalpurposes. For
example, it immediatelygivesthe observedpulseprofile fora
ForagivenphotontrajectoryletuschooseSchwarzschildco-
rotating neutron star with two antipodal hot spots (eqs. 6 and
ordinatesxk = (t,r,θ,ψ)sothatthetrajectoryisintheplane
7). These analytical expressions highlight the effects of light
θ = 90o. Let uk = dxk/dλ be the 4-velocity of the photon
bending in pulsars: (i) If the antipodalspot is never seen, the
(λ is an affine parameter, λ = r at r ). The trajectory
blackbody pulse is perfectly sinusoidal. The bending affects has integrals u = 1 and u = b. Fro→m u∞iu = 0 one finds
thepulsedfractionAonly(eq.8).(ii)Thevisiblesurfaceofthe t ψ i
star is significantly enhanced, Sv/4πR2 = [2(1−rg/R)]−1, e(uscra)p2e=dir1e−ctio(bn2./Trh2e)(n1a−trardgi/urs)R. Ltheteupshcohtoonosheasψ = 0forthe
and both spots may be seen for a fraction of period or all
tFhpe =tim(e2.rgT/hRen)Ft1hewbhlearcekbFo1diyspthuelsfleusxhofwrosmaopnleatsepaoutFvoiebws e=d ψ = ∞ −uψ dr = ∞ dr 1 1 1 rg −1/2.
fnaocteboens.m(iailil)erThtheaonbFser.veTdhfleulxatctearnnisotaepxucreeeldyFre1laatnivdisitticcaenf-- ZR ur ZR r2 (cid:20)b2 − r2 (cid:16) − r (cid:17)(cid:21) (9)
p
fect: F = 0 in the Newtonian limit r /R 0. The max-
p g
→
imum pulsed fraction of a pulsar with antipodal symmetry is Theemissionangleα(betweentheemittedphotonandthelocal
Amax =(F1−Fp)/(F1+Fp)=(R−2rg)/(R+2rg). This radialdirection)istanα= uψuψ/√urur whichyields
isasevereconstraint,e.g. Amax =1/5forR=3rg insteadof
NewtonianAmax =1. pb r
Withinafewpercenttheresultsagreewiththepreviousnu- sinα= 1 g. (10)
R − R
mericalworks. Thenewformalism,however,ismuchsimpler r
inuse.Itgivesacleardescriptionofthepulseprofilesforgiven
Combining equations (9) and (10) one can compute numeri-
i, θ, andR/r andmakesiteasierto comparethetheorywith
g callytherelationbetweenψandαforagivenR. Atsmallu=
observations and constrain pulsar parameters. We focused in
r /Ronecanexpandequation(9)inuandkeepthelinerterm
tg(hleis3s)L.δθeNtt<oerte1oh0neorp.euTtlhshaaertsethxweteitpnhlsaistomenaautlolishbopigtressdppiooctttessdiwswisthhtreanoigpbheonttfihonrgswpaaonrtds- o2gn(1ly. Ucosisnαg)thoeneeqguetasliψty=0sαin+α(usi(n1α−cozs3α)()1/−sinzα2)+−3O/2(duz2)=.
a§refullyvisibleandtheupperlimitAmax willremainvalidfor C−onsider a star of radiRus R with−a surface radiation inten-
blackbodyemissionwithantipodalsymmetry. sity I0(α) which can vary over the surface. The observer lo-
Caps of real pulsars (both radio and X-ray) do not emit as cation is given by a radius-vector D (D R). A (visible)
a blackbody and their I0(α) need not be isotropic. In partic- surfaceelementdS = R2dµdφisobserved≫atimpactparame-
ular: (1) X-ray pulsars show Comptonized power-law spec- ters(b,b+db)andsubtendsasolidangledΩ=bdbdφ/D2on
tra, sometimesalongwithaclearblackbodycomponent(asin theobserversky. Hereµ = cosψ andφisanazimuthalangle
SAXJ1808.4-3658).Lightbendingaffectsthetwocomponents correspondingto rotationaroundD; b dependson µonly,not
differently: the blackbody pulse from one spot remains sinu- onφ. TheobservedradiationfluxfromdS isdF =IdΩwhere
soidalwhiletheComptonizedpulseisdistorted.Thisdistortion I =(1 rg/R)2I0[IandI0arerelatedthroughI/ν4 =I0/ν04
is describedby equation(4) andit dependson the anisotropic andthe−frequencyredshiftν/ν0 = (1 rg/R)1/2,seeMisner
−
intensity I0(α) of the Comptonized emission. (2) For pul- etal. 1973].Usingequation(10)onegets
sarsofhighspin(withmsperiods)theDopplereffectchanges
I0(α). It is easily added to the model by the corresponding Ib db dS rg dcosαdS
transformationofI0 fromthespotrestframetothelocalnon- dF = R2|dµ|D2 = 1− R I0(α)cosα dµ D2.
rotatingframe. (3)AstrongmagneticfieldB 1012 Gleads (cid:16) (cid:17) (11)
∼
REFERENCES
Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation, (San Shapiro, S. L., & Teukolsky, S. L. 1983, Black Holes, White Dwarfs, and
Francisco:Freeman) NeutronStars,(NewYork:Wiley-Interscience)
Pechenick,K.R.,Ftaclas,C.,&Cohen,J.M.1983,ApJ,274,846