A Simple Analytic Treatment of the Intergalactic Absorption

We derive a new and user friendly simple analytic approximation for determining the effect of intergalactic absorption in the energy range 0.2-2 TeV and the redshift range 0.05-0.4. In these ranges, the form of the absorption coeeficient is approximately l

ASimpleAnalyticTreatmentoftheIntergalacticAbsorption

6

2

v

o

N

3

1

3

v

01

1

8

6

/

h

p

-

o

rt

sa

:v

i

X

r

aE ectinBlazarGamma-RaySpectraF.W.SteckerNASA/GoddardSpaceFlightCenterstecker@milkyway.gsfc.nasa.govS.T.ScullyDepartmentofPhysics,JamesMadisonUniversityscullyst@jmu.eduReceivedAstrophysicalJournalLetters652,L9(2006)

We derive a new and user friendly simple analytic approximation for determining the effect of intergalactic absorption in the energy range 0.2-2 TeV and the redshift range 0.05-0.4. In these ranges, the form of the absorption coeeficient is approximately l

ABSTRACT

Wederiveanewanduserfriendlysimpleanalyticapproximationfordeter-

miningthee ectofintergalacticabsorptionontheγ-rayspectraofTeVblazarstheenergyrange0.2TeV<Eγ<2TeVandtheredshiftrange0.05<z<0.4.Intheseranges,theformoftheabsorptioncoe cientτ(Eγ)isapproximatelylogarithmic.Thee ectofthisenergydependenceistosteepenintrinsicsourcespectrasuchthatasourcewithanapproximatepower-lawspectralindexΓsisconvertedtoonewithanobservedspectralindexΓo Γs+ Γ(z)where Γ(z)isalinearfunctionofzintheredshiftrange0.05–0.4.Weapplythisapproxi-mationtothespectraof7TeVblazars.

Subjectheadings:Gamma-rays:general–blazars:individual(Mkn180,

PKS2005-489,PKS2155-304,H2356-309,1ES1218+30,1ES1101-232,PG

1553+113)

1.Introduction

Stecker,Malkan&Scully(2006)(SMS)haveusedrecentSpitzerobservations(LeFloc’hetal.2005,Perez-Gonzalezetal.2005)alongwithotherdataongalaxyluminosityfunctionsandredshiftevolutioninordertomakeadetailedevaluationoftheintergalacticphotondensityasafunctionofbothenergyandredshiftfor0<z<6forphotonenergiesfrom.003eVtotheLymanlimitcuto at13.6eVinaΛCDMuniversewith Λ=0.7and m=0.3.Theythenusedtheircalculatedintergalacticphotondensitiestocalculatetheopticaldepthoftheuniverse,τ,forγ-rayshavingenergiesfrom4GeVto100TeVemittedbysourcesatredshiftsfrom0to5.Theyalsogaveaparametric twithnumericalcoe cientsforτ(Eγ,z).

We derive a new and user friendly simple analytic approximation for determining the effect of intergalactic absorption in the energy range 0.2-2 TeV and the redshift range 0.05-0.4. In these ranges, the form of the absorption coeeficient is approximately l

Asanexampleoftheapplicationofthedetailednumericaldeterminationoftheopticaldepth,theycalculatedtheabsorbedspectrumoftheblazarPKS2155-304atz=0.117andcompareditwiththespectrumobservedbytheH.E.S.S.airCherenkovγ-raytelescopearray.ItwasnotedthatasteepeninginanE 2power-lawdi erentialphotonsourcespectrumforPKS2155-304wouldbesteepenedbyapproximatelyonepowertoE 3aswouldbeproducedbyaγ-rayopacitywithalogarithmicenergydependence.

Thepurposeofthisletteristogeneralizethisresultbydetermining tsfor

approximatingτ(Eγ,z)bylogarithmicfunctionsinordertopredictpower-lawsteepeningsinassumedblazarsourcespectraoftheformE ΓstoobservedspectraoftheformE ΓowhereΓo Γs+ Γand Γisdeterminedtobealinearfunctionofz.

2.Calculation

Inordertoderiveourresults,we ttheresultsfromSMSforτ(Eγ,z)toaformwhichisassumedtobelogarithmicinEγintheenergyrange0.2TeV<Eγ<2TeVandwhichhasalineardependenceonzovertherange0.05<z<0.4.Itisimportanttonotethatourlinear ttothezdependenceisbothqualitativelyandquantitativelydi erentfromthelineardependenceonredshiftwhichwouldbeobtainedforsmallredshiftsz<<1andwhichsimplycomesfromthefactthatforsmallzwhereluminosityevolutionisunimportantandwhereτ∝d,withthedistanced cz/H0∝z,ourquantitaitve tforthehigherredshiftrange0.05<z<0.4comesfromthemorecomplexcalculationsbasedonthemodelsofSMS.Forthisreason,ourlinear tsarenotsimplyproportionaltoredshift.

Wechoosetheformofour ttingfunctioninnbothredshiftandenergytobegivenby

τ(Eγ,z)=(A+Bz)+(C+Dz)ln[Eγ/(1TeV)],

We derive a new and user friendly simple analytic approximation for determining the effect of intergalactic absorption in the energy range 0.2-2 TeV and the redshift range 0.05-0.4. In these ranges, the form of the absorption coeeficient is approximately l

whereA,B,CandDareconstants.

Ifwethenpostulateanintrinsicsourcespectrumwhichcanbeapproximatedbyapowerlawoverthislimitedenergyrangeofonedecade,

ΓsΦs(Eγ) KEγ,

thespectrumwhichwillbeobservedattheEarthfollowingintergalacticabsorptionwillbeofthepower-lawform

(Γs+C+Dz)Φo(Eγ)=Ke (A+Bz)Eγ.

Thiscanbecomparedwiththeempiricallyobservedspectrumwhichisusuallypresentedintheliteraturetobeofthepower-lawform.Theobservedspectralindex,Γo,willthenbegivenby

Γo=Γs+ Γ(z)

wheretheintrinsicspectralindexofthesourceissteepenedby Γ(z)=C+Dz.

Table1.OpticalDepthParameters

EvolutionModelABCD

We derive a new and user friendly simple analytic approximation for determining the effect of intergalactic absorption in the energy range 0.2-2 TeV and the redshift range 0.05-0.4. In these ranges, the form of the absorption coeeficient is approximately l

TheparametersA,B,C,andDobtainedby ttingtheopticaldepthsderivedforthefastevolution(FE)andbaseline(B)modelsofSMSaregiveninTable1.Figures1and2showtheexcellent tsoftheseparameterstoalineardependenceinbothlnEγandredshift.

12

10

8

6

4

2

0.20.30.50.7

Energy TeV 11.52Τ

Fig.1.—Linear tsinlnEγtotheopticaldepthscalculatedinSMS(pointsshown)fortheirfastevolution(FE)caseintheenergyrange0.2TeVto2.0TeVforredshiftsof(bottomtotop)0.05,0.10,0.15,0.20,0.25,0.30,0.35,and0.40.Asimilarresultisobtainedforthe …… 此处隐藏：8174字，全部文档内容请下载后查看。喜欢就下载吧 ……