Luminosity Evolution of Gamma-ray Pulsars

0.34 MB·English

by  Kouichi Hirotani

Received 2012 November 19; accepted 2013 January 22
PreprinttypesetusingLATEXstyleemulateapjv.08/22/09
[email protected]
LUMINOSITY EVOLUTION OF GAMMA-RAY PULSARS
Kouichi Hirotani1
TheoreticalInstitute forAdvanced ResearchinAstrophysics(TIARA),AcademiaSinica,InstituteofAstronomyandAstrophysics
(ASIAA),POBox23-141,Taipei,Taiwan
Received 2012 November19; accepted 2013 January 22
ABSTRACT
3 We investigate the electrodynamic structure of a pulsar outer-magnetospheric particle accelerator
1 and the resultant gamma-ray emission. By considering the condition for the accelerator to be self-
0 sustained, we derive how the trans-magnetic-fieldthickness of the accelerator evolves with the pulsar
2
age. It is found that the thickness is small but increases steadily if the neutron-star envelope is

n contaminatedbysufficientlightelements. Forsuchalightelementenvelope,thegamma-rayluminosity
a oftheacceleratoriskeptapproximatelyconstantasafunctionofageintheinitialtenthousandyears,
J formingthelowerboundoftheobserveddistributionofthegamma-rayluminosityofrotation-powered

4 pulsars. If the envelope consists of only heavy elements, on the other hand, the thickness is greater
2 butincreasesless rapidlythanwhata lightelementenvelopehas. Forsuchaheavyelementenvelope,
the gamma-ray luminosity decreases relatively rapidly, forming the upper bound of the observed

] distribution. The gamma-rayluminosity of a generalpulsar resides between these two extreme cases,
E
reflecting the envelope composition and the magnetic inclination angle with respect to the rotation
H axis. The cutoff energy of the primary curvature emission is regulated below several GeV even for
. young pulsars, because the gap thickness, and hence the acceleration electric field is suppressed by
h
the polarization of the produced pairs.
p
Subject headings: gammarays: stars—magneticfields—methods: analytical—methods: numerical
-
o — stars: neutron
r
t
s
a 1. INTRODUCTION observed light curves (Abdo et al. 2010) favor fan-like
[ The Large Area Telescope aboard Fermi Gamma-ray emission geometry, which scan over a large fraction of
SpaceTelescope(Atwood et al.2009)hasprovedremark- the celestial sphere, and because the Crab pulsar shows
1 pulsed photons near and above 100 GeV (Aliu et al.
ably successful at discovering rotation-powered pulsars
v
2011;Aleksi´c et al.2011a,b),whichrulesoutanemission
7 emitting photons above 0.1 GeV. Thanks to its superb
from the lower altitudes, where strong magnetic absorp-
1 sensitivity, the number of gamma-ray pulsars has in-
tion takes place for γ-rays above 10 GeV. Consequently,
7 creased from six in Compton Gamma Ray Observa-
higher-altitude emission models such as the outer-gap
5 tory era (Thompson 2004) to more than one hundred
model (Cheng et al. 1986a,b), the high-altitude slot-gap
. (Nolan2012). Plottingtheirbestestimateofthegamma-
1 model (Muslimov & Harding 2004), or the pair-starved
ray luminosity, L , against the spin-down luminosity,
0 γ polar-capmodel(Venter et al.2009),gatheredattention.
3 Lspin = 4π2IP˙P−3, Abdo et al. (2010) found the im- Itisnoteworthythattheouter-gapmodelispresentlythe
1 portant relation that Lγ is approximately proportional onlyhigher-altitudeemissionmodelthatissolvablefrom
v: to Lspin0.5 (with a large scatter), where I refers to the thebasicequationsself-consistently(Hirotani2011a). In
i neutron-star (NS) moment of inertia, P the NS rota- the present paper, therefore, we focus on the outer-gap
X tional period, and P˙ its temporal derivative. How- modelandderivetheobservedrelationshipL L 0.5
γ spin
∝
r ever, it is unclear why this relationship arises, in spite both analytically and numerically.
a of its potential importance to discriminate pulsar emis- We schematically depict the pulsar outer-
sion models such as the polar-cap model (Harding et al. magnetospheric accelerator (i.e., the outer gap) in
1978; Daugherty & Harding 1982; Dermer 1994), the figure 1. As the NS rotates, there appears the light
outer-gapmodel(Chiang & Romani1992;Romani1996; cylinder, within which plasmas can co-rotate with the
Zhang & Cheng 1997; Takata et al.2004; Hirotani2008; magnetosphere. The magnetic field lines that become
Wang et al. 2011), the pair-starved polar-cap model tangential to the light cylinder at the light cylinder
(Venter et al. 2009) (see also Yuki & Shibata (2012) radius, ̟ = cP/2π, are called the last-open magnetic
LC
for the possible co-existence of such models), and field lines, where c refers to the speed of light. Pairs
the emission model from the wind zone (Petri 2011; are produced via photon-photon pair production mostly
Bai & Spitkovski 2010a,b; Aharonian et al. 2012). near the null-charge surface and quickly polarized by
Recent gamma-ray observations suggest that the the magnetic-field-aligned electric field, E , in the
k
pulsed gamma-rays are emitted from the higher alti- gap. In this paper, we assume that the rotation and
tudes of a pulsar magnetosphere. This is because the magnetic axes reside in the same hemisphere to obtain
E > 0, which accelerates positrons (e+’s) outwards
k
1Postaladdress: TIARA,Department ofPhysics,National Ts- while electrons (e−’s) inwards. These ultra-relativistic
ing Hua University, 101, Sec. 2, Kuang Fu Rd.,Hsinchu, Taiwan
300 particles have Lorentz factors, γ 107.5, to emit
∼
2 Hirotani
a Upper boundary where h denotes the Planck constant, ~ h/2π. Once
m B field line, h is obtained, we can readily compute≡the γ-ray lu-
q =(1-h )q max m
* m * minosity of curvature radiation from an outer gap by
q
1 l l (Hirotani 2008)
(r1,q1) 1 2 (r2,q2)
Neutron µ2Ω4
star Outer gap L 2.36(νF ) 4πd2f 1.23f h3 , (4)
qLC (r,q ) γ ≈ ν peak× Ω ≈ Ω m c3
0 0
where f , which has been conventionally assumed to be
Last-open Ω
Null-charge B field line, er approximately unity, refers to the flux correction factor
surface q =q max nd (Romani, R. & Watters 2010), and Ω = 2π/P the rota-
* * yli
c tion angular frequency of the NS. Here, it is assumed
Middle-latitude B field line, ht
q*=(1-hm/2)q*max Lig that the current density flowing in the gap is compa-
rable to the Goldreich-Julainvalue (Goldreich, & Julian
Light cylinder raiuds, v =c/W 1969). The last factor, µ2Ω4/c3 is proportional to the
LC spin-down luminosity, L . Therefore, the evolution
spin
Fig. 1.— Side view of an outer gap. The neutron star (filled law,L L0.5 ,iscruciallygovernedbytheevolutionof
circle on the left) obliquely rotates around the vertical axis with γ ∝ spin
magnetic inclination angle α. The thin solid curves denote the hm as a function of the NS age, t.
magneticfieldlines,whilethedashed straightlinethe null-charge The evolution of h is essentially controlled by the
m
surface, on which the magnetic field lines become perpendicular photon-photon pair production in the pulsar magneto-
to the rotation axis. Outside the light cylinder (the vertical long
sphere. To analytically examine the pair production, we
dashedline),plasmasthatarefrozentothemagneticfieldlinescan
onlymigrateoutwards(asapulsarwind)becauseofthecausality assume the static dipole magnetic field configurationfor
requirement in special relativity. The light-cylinder radius, ̟LC, simplicity, and consider the plane on which both the ro-
becomes typically a few or several hundred neutron-star radii for tational and magnetic axes reside (fig. 1). On this two-
young pulsars. In modernouter-gap models (Hirotanietal.2003;
dimensional latitudinal plane, the last-open field line in-
Takataetal.2004;Hirotani2006a;Takataetal.2004),itisproved
thattheoutergapextends betweenthestellarsurface(becauseof tersects the NS surface at magnetic co-latitudinal angle
anegativechargedensityintheloweraltitudes)andthevicinityof θmax (measuredfromthe magnetic dipole axis)that sat-
thelightcylinder(becauseofapositivechargedensityinthehigher ∗
isfies
altitudes). Thus, typical inward (or outward) photons propagate
distancel1 (orl2)beforeescapingfromthegap(shadedregion). sin2θ∗max = sin2(θLC−α), (5)
r ̟ /sinθ
∗ LC LC
photons efficiently by the curvature process. wherer denotestheNSradius,θ theangle(measured
∗ LC
from the rotation axis) of the point where the last-open
2. ANALYTICALEXAMINATIONOFOUTER-GAP field line becomes tangential to the light cylinder, and
LUMINOSITY
α the inclination angle of the dipole magnetic axis with
Inthis section,weanalyticallyderivehowthe gamma- respecttotherotationaxis. Amagneticfieldline canbe
ray luminosity of an outer gap evolves with time. In the specifiedbythemagneticco-latitude(measuredfromthe
outermagnetosphere,onlythedipolecomponentremains dipole axis), θ , whereit intersectsthe stellarsurface. A
∗
in the magnetic field; thus, the inhomogeneous part of magneticfieldlinedoesnotclosewithinthelightcylinder
the Maxwell equation (i.e., the Poisson equation for the (i.e.,opentolargedistances)if0<θ <θmax. Thus,the
∗ ∗
electro-static potential) gives the magnetic-field-aligned last-openfieldlines, θ =θmax, correspondsto the lower
∗ ∗
electric field (Hirotani 2008), boundary, which forms a surface in a three-dimensional
µ magnetosphere, of the outer gap.
Ek ≈ 2̟3 h2m, (1) Let us assume that the gap upper boundary coin-
LC cides with the magnetic field lines that are specified by
whereµdenotestheNSmagneticdipolemoment,andhm θ∗ = (1−hm)θ∗max. Numerical examinations show that
the trans-magnetic-field thickness of the gap. Since the hm, indeed, changes as a function of the distance along
Poisson equation is a second-order differential equation, the field line and the magnetic azimuthal angle (mea-
E isproportionaltoh 2. Electrons(e−’s)andpositrons sured counter-clockwise around the dipole axis). Nev-
k m
(e+’s)arecreatedviaphoton-photon(andsometimesvia ertheless, except for young pulsars like the Crab pulsar,
an assumption of a spatially constant h gives a rela-
magnetic)pairproduction,being subsequentlypolarized m
tively good estimate. Thus, for an analytical purpose,
by E and accelerated in the opposite directions, to fi-
k we adopt a constant h in this analytical examination.
nally attain the terminal Lorentz factor m
In this case, we can specify the middle-latitude field line
γ = 3ρ2cE 1/4, (2) binygthofeEmagdnueetitcocoth-leatpitouladreizθa∗ti=on(1o−f hthme/p2)rθo∗mduaxc.edScpraeierns-,
2e k k
(cid:18) (cid:19) takesplacemostlyintheloweraltitudes. Itis,therefore,
appropriate to evaluate the screening of E around the
where ρ refers to the radius of curvature of particle’s k
c
motion in the three-dimensional magnetosphere, e the point(r0,θ0) wherethe null-chargesurfaceintersects the
charge on the positron. Photons are radiated by such middle-latitude field line (fig. 1).
ultra-relativistic e±’s via curvature process with charac- An inwardly migrating electron (or an outwardly mi-
teristic energy, grating positron) emits photons inwards (or outwards),
3 γ3 whichpropagatethe typicaldistance l1 (or l2)before es-
hν = ~c , (3) caping from the gap. Denoting the cross section of the
c
2 ρ
c
Luminosity Evolution of Gamma-ray Pulsars 3
inward(oroutward)horizontallinefromthepoint(r ,θ ) or
0 0
and the upper boundary as (r1,θ1) (or as (r2,θ2)), and τ2 =l2F2σ2/c, (15)
noting r cosθ =r cosθ =r cosθ , we obtain
0 0 1 1 2 2
whereF andF denotetheX-rayfluxinsideandoutside
1 2
l =r cosθ (tanθ tanθ ), (6) of(r ,θ ),respectively;σ andσ arethepair-production
1 0 0 0− 1 0 0 1 1
cross section for inward and outward photons, respec-
l2 =r0cosθ0(tanθ2 tanθ0). (7) tively. Thus, a single e− or e+ cascades into
−
Along the upper-boundary field line, we obtain eE F
(N ) τ = k 1l 2σ (16)
γ 1 1 1 1
sin2(θ α) sin2(θ α) sin2[(1 h )θmax] hνc c
1− = 2− = − m ∗ ,
r r r pairs or into
1 2 ∗
(8)
eE F
whereas along the middle-latitude field line, we obtain (N ) τ = k 2l 2σ (17)
γ 2 2 2 2
hν c
c
sin2(θ α) sin2[(1 h /2)θmax]
0− = − m ∗ . (9) pairs within the gap. That is, a single inward-migrating
r0 r∗ e− cascades into pairs with multiplicity (N ) τ . Such
γ 1 1
wCeomfinbdinitnhgatthθes(e<twθo)eaqnudaθtio(n>s,θan)dcannotbineggiθv∗menaxby≪th1e, pourotwduacrded-mpigarirastinagree+poclaasrcizaeddesbinytoEpk.airsEwacithhrmetuultrinpilnicg-,
1 0 2 0
solution θ that satisfies ity(N ) τ inoutermagnetosphere. Asaresult,asingle
γ 2 2
cosθsin2(θ α)= 1−hm 2cosθ sin2(θ α), winawradrde−e’−s,cwahscicahdesshoevueldntbueaclloymientuon(iNtyγf)o1rτ1th·e(Ngγa)p2τt2oibne-
0 0
− (cid:18)1−hm/2(cid:19) − self-sustained. Therefore, in a stationary gap, the gap
(10) thickness h is automatically regulated so that the gap
m
where θ0 is given by closure condition,
tanθ = 1 3tanα+ 9tan2α+8 . (11) (Nγ)1τ1 (Nγ)2τ2 =1, (18)
0 ·
2
(cid:16) p (cid:17) may be satisfied.
Thus, if we specify α, we can solve θ =θ1 and θ =θ2 as Approximatelyspeaking,asingle,inward-migratinge−
a function of hm by equation (10). Substituting these θ1 emits(Nγ)1 104 curvaturephotons,aportionofwhich
and θ2 into equations (6) and (7), we obtain l1 and l2, head-on colli∼de the surface X-ray photons to material-
where r0 depends on P. ize as pairs with probability τ1 10−3 within the gap.
tioInf h(m10)≪ar1o,uwnedcθan=exθp0a,nwdhtehree lθef=t-hθa1ndfosridinewoafredqu(oar- tThheusg,apa.sinEgalcehe−procadsuccaeddese+introe∼t(uNrnγ)1oτu1tw∼ar1d0s ptoairesmiint
θ = θ2 for outward) γ-rays to find θ2 θ0 = θ0 θ1 (N ) 105 photons, which materialize as pairs with
− − ∝ γ 2
√hm. Thatis,the leadingtermsintheexpansionvanish probab∼ility τ 10−6 by tail-on colliding with the sur-
2
and we obtain l = l from the next-order terms, which ∼
1 2 faceX-rays. Inanotherword,(N ) τ 10holdsregard-
are quadratic to θ θ . Assuming L t−β, where γ 1 1 ∼
0 X less of the nature of the pair production process (e.g.,
β 0.48 is approp−riate for t < 104 yea∝rs for a light-
either photon-photon or magnetic process (Takata et al.
ele≈ment-envelope NS and for t < 105 years for a heavy-
2010)) in the lower altitudes, because it is determined
element NS, we find hm P5/6µ−1/6tβ/2, and hence by the pair-production efficiency in the outer magneto-
∝
Lγ ∝P−3/2µ3/2t3β/2. Sincethedipoleradiationformula sphere (Nγ)2τ2 ∼ 0.1, which is always due to photon-
gives P µt1/2, we obtain L µ0t3(β−1/2)/2 t−0.03. photon pair production.
γ
Thus, w∝hen the gap is very thi∝n, which is expe∝cted for In general,(Nγ)1, τ1, (Nγ)2, τ2 are expressedin terms
a light-element-envelope NS, Lγ little evolves with the of hm, P, µ, T, and α, where T denotes the NS surface
pulsar age, t. temperature. Note that we can solve P = 2π/Ω as a
However,if h >0.2,say,the rapidly expanding mag- functionoftheNSage,t,fromthespin-downlaw. Thus,
m
netic flux tube gives asymmetric solution, l > l . That specifying α and the cooling curve, T = T(t), we can
2 1
is,thethirdandhigherordertermsintheexpansioncon- solvehmasafunctionoftfromthegapclosurecondition,
tribute significantly compared to the quadratic terms. (Nγ)1τ1(Nγ)2τ2 = 1. Note also that the spin-down law
Thus, we must solve equation (10) for θ (= θ1 or θ2) readily gives the spin-down luminosity, Lspin P˙P−3,
without assuming θ θ0 1, in general. as a function of t, once P = P(t,α) is solved. ∝On these
| − |≪
Let us now consider the condition for a gap to be self- grounds,L canberelatedtoL withanintermediate
γ spin
sustained. A single ingoing e− or an outgoing e+ emits parametert,ifwespecifythecoolingcurveandthespin-
down law.
(N ) =eE l /(hν ) (12)
γ 1 k 1 c Substituting equations (16) and (17) into (18), we ob-
or tain
eE √F σ F σ
(Nγ)2 =eEkl2/(hνc) (13) k 1 1 2 2l1l2 =1, (19)
hν c
c
photons while running the typical distance l or l , re-
1 2
spectively. Such photons materialize as pairs with prob- where
ability r 2 ∞ B (T)
∗ ν
τ =l F σ /c (14) F σ =π(1 µ ) σ (ν,ν ,µ ) (20)
1 1 1 1 i i − i r hν P γ i
(cid:18) i(cid:19) Zνth,i
4 Hirotani
withi=1,2; ν denotesthe γ-rayfrequency,andB (T) To describe the evolution of P = P(t) = 2π/Ω(t), we
γ ν
the Planck function. We have to integrate over the soft adopt in this paper
photon frequency ν above the threshold energy
µ2Ω4
2(m c2)2 IΩΩ˙ =C (28)
hν = e , (21) − c3
th,i
(1 µ )hν
− i γ where C = (2/3)sin2α for a magnetic dipole braking,
wheremec2referstotherest-massenergyoftheelectron. while C =1+sin2α for a force-free braking (Spitkovski
The cosine of the collision angle µi becomes 1 µ1 = 2006). Assuming a magnetic dipole braking, we obtain
−
1 sinθ for outward (or 1 µ =1+sinθ for inward)
0 2 0
γ-−rays. Thatis,collisionsta−keplace head-on(or tail-on) P =39.2msµ I−1/2(t/103years)1/2, (29)
30 45
for inward (or outward) γ-rays. The total cross section
is given by where µ µ/(1030G cm3) and I I/(1045g cm2).
30 45
≡ ≡
Thus, if we specify a cooling scenario, T = T(t), equa-
3 1+v
σP = σT(1 v2) (3 v4)ln 2v(2 v2) , tion (27) gives hm as a function of t. Note that the α
16 − − 1 v − − dependence of the spin-down law is not essential for the
(cid:20) − (cid:21)
(22) present purpose; thus, C =2/3 is simply adopted. Once
where σT denotes the Thomson cross section and hm = hm(t) is obtained, equation (4) readily gives Lγ
as a function of t, and hence of L . It is worth not-
spin
2 (mec2)2 ing thatthe heatedpolar-capemissionis relativelyweak
v 1 . (23)
≡s − 1 µi hνhνγ comparedtothecoolingNSemission,exceptformillisec-
− ond or middle-aged pulsars.
Pair production takes place when the γ-rays collide Let us now consider the cooling scenario. Since
with the surface X-rays in the Wien regime, that is, at the mass of PSR J1614-2230 is precisely measured to
hν ≫kT. Anaccurateevaluationofσ2requiresacareful be 1.97M⊙ (i.e., 1.97 solar masses), and since other
treatment of the collision geometry, because the thresh- three NSs (4U1700-377, B1957+20, and J1748-2021B)
old energy, hνth,2, strongly depends on the tiny collision (Lattimer & Prakash 2001) are supposed to be heavier
angles. Inthenumericalmethod(nextsection),thepair- than 2.0M , we exclude the equation of state (EOS)
⊙
production absorption coefficient is explicitly computed that gives smaller maximum mass than 1.95M . That
⊙
at each point in the three-dimensional pulsar magneto- is, we do not consider the fast cooling scenario due to
sphere by equation (50). However, in this section, for directUrcaprocessinahyperon-mixed,pion-condensed,
analytical purpose, we simply adopt the empirical rela- kaon-condensed, or quark-deconfined core, which gives
tion, softer EOS and hence a smaller maximum mass. Even
F σ F σ =ǫ 1 µ σ F , (24) withoutexoticmatters,thedirectUrcaprocessmayalso
1 1 2 2 1 T X
− becomeimportantinthecoreofaNSwiththemassthat
where ǫ = 0.p004, 0.01, and 0p.038 for α = 45◦, 60◦, and isslightly lessthanthe maximummass. However,in the
75◦,respectively;1 µ 2. TheX-rayfluxisevaluated
1 present paper, we exclude such relatively rare cases and
− ≈
at (r ,θ ) such that
0 0 adopt the canonical value, 1.4M , as the NS mass.
⊙
L 1 On these ground, we adopt the minimal cooling sce-
X
FX = 2.82kT 4πr 2, (25) nario (Page et al. 2004), which has no enhanced cooling
0 that could result from any of the direct Urca processes
where L refers to the luminosity of photon radiation andemploysthestandardEOS,APREOS(Akmal et al.
X
from the the cooling NS surface. For a smaller α, the 1998). Within the minimal cooling scenario, the cool-
point(r ,θ )islocatedinthehigheraltitudes,wherethe ing history of a NS substantially depends on the com-
2 2
magnetic field lines begin to collimate along the rota- position of the envelope, which is defined as the upper-
tion axis, deviating from the static dipole configuration. most layer extending from the photosphere down to a
Thus, the collision angles near the light cylinder, and boundary where the luminosity in the envelope equals
henceσ decreaseswithdecreasingα. Theexplicitvalue the total surface luminosity of the star. An envelope
2
of ǫ can be computed only numerically, solving the pho- contains a strong temperature gradient and has a thick-
ton specific intensity from infrared to γ-ray energies in ness around 100 meters. We adopt the cooling curves
the three-dimensional pulsar magnetosphere. giveninPage et al.(2004)andconsiderthe twoextreme
The last factor, l l , in the left-hand side of equa- cases: light element and heavy element envelopes. Be-
1 2
tion (19) is given by cause of the uncertainty in the modeling of neutron 1S
0
(i.e., spin-singlet state) Cooper paring temperature (as
l1l2 =r02cos2θ0(tanθ0 tanθ1)(tanθ2 tanθ0) (26) afunction ofneutronFermimomentum) andproton1S
− − 0
Thus, equation (19) gives paring temperature (as a function of proton Fermi mo-
mentum), the predicted L distributes in a ‘band’ for
X
eEk LX/c each chemical composition of the envelope. When a NS
ǫ 1 µ σ
hν 2.82kT − 1 T is younger than 104.4yr, a light element envelope (con-
c
cos2θ (tanpθ tanθ )(tanθ tanθ )=1, (27) taminatedbye.g.,H,He, C,orOwithmassesexceeding
× 0 0− 1 2− 0 10−6M⊙) has a higher surface temperature, and hence
where the r0 dependence vanishes. Substituting equa- a greater luminosity, LX, compared to a heavy element
tions(1),(2),(3)into(27),wecansolveh asafunction envelope (contaminated by light elements with masses
m
of L /kT, P, and µ. lessthan10−16M ),becausetheheattransportbecomes
X ⊙
Luminosity Evolution of Gamma-ray Pulsars 5
Analytical solution of outer-gap thickness Analytical outer-gap solutions
000...888
ess, hm 000...666 a = 60o -1)g s 104 years a =45o
p transverse thickn 000000......222444 HHHeeeaaavvvyyy eeellleeemmmeeennnttt eeennnvvveeellloooppp LLLiiiggghhhttt eeellleeemmmeeennnttt eeennnvvveeellloooppp (eruminosity, Lg 1036 LLgg==LLssppiinn aaaaa =====6644400555ooooo 110033 yyeeaarrss
Outer-ga 0102 103 104 105 ma-ray l aaa ===666000ooo
Neutron star age (years) m1035
a
Fig. 2.— Analytically solved gap thickness. The (blue) dot- d g
attoenddnuaacnldehoen(arveCydo)oelpdeemarsehpneatdriencnguvremvleoospdeersle,sp,rraeersseepnertcetpihrvemesleyn=.teUdhmnbc(yetr)tthafeoinrbtliaueseliadgnuhdet Pulse 10P3u5lsar spin-1d0o3w6n lumino1s0it3y7, L (e1r0g3 s8-1)
spin
red shaded ‘bands’. Magnetic inclination angle is assumed to be
α=60◦. Fig. 3.— Analyticallysolvedgapluminositywithuncertainties
more efficient in the former. As the NS ages,a star with due to nucleon Cooper paringmodels. The thick and thin dotted
(or dashed) curves represent the evolution of the gap luminosity
a light element envelope quickly loses its internal energy
for a light (or a heavy) element envelope. Two cases of magnetic
viaphotonemission;asaresult,after104.6yr,itbecome inclination angle, α = 45◦ and 60◦, are depicted as labeled. The
less luminous than that with a heavy element envelope. greenfilledcirclesdesignatesthenormalgamma-raypulsars,while
A realistic cooling curve will be located between these the blue filled squares do those detected by the gamma-ray blind
searchtechnique.
two extreme cases,depending on the actual composition
of the NS envelope.
method. To this end, we adopt the modern outer-gap
We present the solved h for a light and a heavy el-
m model(Hirotani2011a)andsolvethesetofMaxwelland
ement envelope in figure 2, adopting µ = 3.2, which
30 Boltzmann equations self-consistently and compute E ,
gives the magnetic field strength of 4.1 1012 G at the k
magnetic pole, where r = 11.6 km is ×used. It follows distribution functions of e±’s, and the photon specific
∗
that the gap becomes thinner for a light element case intensity at each point in the three-dimensional pulsar
(between the thin and thick dotted curves, blue shaded magnetosphere. We considernotonly the whole-surface,
region) than for the heavy element cases (between the cooling NS emission but also the heated polar-capemis-
thin and thick dashed curves, red shaded region). This sion as the photon source of photon-photon pair pro-
is because the more luminous photon field of a light ele- duction in the numerical analysis. The former emission
ment envelope leads to a copious pair production, which component is given as a function of the pulsar age from
prevents the gap to expand in the trans-field direction. the minimum cooling scenario, in the same manner as
As a result, the predicted L becomes less luminous for in the analytical examination, while the latter emission
γ
a light element envelope than a heavy one. component is solved consistently with the energy flux of
Infigure 3, we presentthe analyticalresultsofL ver- the e−’s falling on to the pulsar polar-cap surface. The
γ
susL asthe dotted(ordashed)curveforalight(ora method described below was also used in recent theo-
spin
heavy) element envelope. As the pulsar spins down, L reticalcomputation of the Crabpulsar’s γ-rayemissions
γ
evolvesleftwards. In this section, for analyticalpurpose, (Aleksi´c et al. 2011a,b).
wearenotinterestedinthedependenceontheobserver’s Let us present the basic equations that describe a
viewing angle, ζ. Thus, in equation (4), we simply put pulsar outer-magnetospheric accelerator, extending the
f = 1. It is interesting to note that L little evolves method first proposed for black-hole magnetospheres
Ω γ
for a light element envelope, because of h 1, as dis- (Beskin1992;Hirotani & Okamoto 1998). Aroundaro-
m
cussedafterequation(11). Preciselyspeakin≪g,weobtain tating NS, the backgroundgeometry is described by the
h t0.62 and hence L L 0.07 (or h t0.50 and space-time metric (Lense & Thirring 1918)
m γ spin m
∝ ∝ ∝
hence Lγ Lspin0.25) for a light (or a heavy) element ds2 =g dt2+2g dtdϕ+g dr2+g dθ2+g dϕ2, (30)
∝ tt tϕ rr θθ ϕϕ
envelope. That is, although h is smaller, h increases
m m
more rapidly in a light element case than in a heavy where
eemlemenetntencvaesleo,pwe.hiAchtelantaebrlesstaagec,onts>tan1t04Lyγefaorrs,abloigthhtheml- gtt ≡ 1− rrg c2, gtϕ ≡acrrg sin2θ, (31)
andLγ increasewithdecreasingLspin foralightelement (cid:16) (cid:17)
r −1
envelope, because LX/(kT), and hence the pair produc- g 1 g , g r2, g r2sin2θ;
rr θθ ϕϕ
tion rate, rapidly decreases with increasing t. On the ≡− − r ≡− ≡−
other hand, for a heavy element envelope, their greater (cid:16) (cid:17) (32)
hm results in a monotonically decreasing Lγ with de- M denotestheNSmass,rg ≡2GM/c2theSchwarzschild
creasing L . radius,anda IΩ/(Mc)thestellarangularmomentum.
spin ≡
At radial coordinate r, the inertial frame is dragged at
3. NUMERICALEXAMINATIONOFOUTER-GAP angularfrequency ω gtϕ/gϕϕ =0.15ΩI45r6−3,where
ELECTRODYNAMICS I I/1045 erg cm2≡, a−nd r r /10km.
45 6 ∗
≡ ≡
Let us develop the analytical examination and look First,letusderivethePoissonequationfortheelectro-
deeper into a self-consistent solution by a numerical staticpotentialunderthespace-timegeometrydescribed
6 Hirotani
just above. Let us consider the Gauss’s law, both the rotation and the magnetic axes reside, we ob-
tain the Poisson equation (Hirotani 2006a,b),
1 √ g 4π
Ftµ = ∂ − gµν( g F +g F ) = ρ,
∇µ √−g µ(cid:20) ρ2w − ϕϕ tν tϕ ϕν (cid:21) (3c32) −c2ρg2ϕϕ gss∂s2+gθ∗θ∗∂θ2∗ +gϕ∗ϕ∗∂ϕ2∗
w
wdihceesreru∇ndoevneortet,srt,heθ,cϕov;a√riangt=derivgativge,ρt2he=Gcrre2eksininθ- +(cid:0)2gsθ∗∂s∂θ∗ +2gθ∗ϕ∗∂θ∗∂ϕ∗ +2gϕ∗s∂ϕ∗∂s Ψ
and ρ2w ≡ gt2ϕ −gttgϕϕ, ρ t−he reapl chrarrgθeθ dwensity. The − As∂s+Aθ∗∂θ∗ +Aϕ∗∂ϕ∗ Ψ=4π(ρ−ρGJ), (cid:1)(39)
electromagnetic fields observed by a distant static ob-
whe(cid:0)re (cid:1)
serveraregivenby(Camenzind1986a,b)E =F , E =
Fθt, Eϕ = Fϕt, Br = (gtt + gtϕΩ)Fθϕr/√−gr,tBθθ = gi′j′ =gµν∂xi′ ∂xj′ =grr∂xi′ ∂xj′ +gθθ∂xi′ ∂xj′
(g +g Ω)F /√ g, B = ρ2F /√ g,whereF ∂xµ ∂xν ∂r ∂r ∂θ ∂θ
∂µtWAt νe−atϕs∂sνuAmµϕeratnhda−tAµthdeeϕneoletce−tsrotwhmearvgθencettoi−cr pfioetlednstiaarl.eµνun≡- k0 ∂xi′ ∂xj′, (40)
−ρ2 ∂ϕ ∂ϕ
changedintheco-rotatingframe. Inthiscase,itisconve- w
nient to introduce the non-corotationalpotential Ψ that
satisfies Ai′ c2 ∂ gϕϕ√ ggrr∂xi′
Fµt+ΩFµϕ =−∂µΨ(r,θ,ϕ−Ωt), (34) ≡√−g ( r" ρ2w − ∂r #
afwrnehdqeurθee,nµctyh=eΩtm,.ra,Iθgm,nϕept.oicsIifnfiFgeAldtth+aritgΩitdFhlAyeϕNro=Sta0stuehsrofalwdcisethfaorapnAegruf=elacrtr +∂θ"gρϕ2wϕ√−ggθθ∂∂xθi′#)− c2ρg2wϕϕρk2w0 ∂∂2ϕx2i′;(41)
conductor, F +ΩF = 0, we find that the NS sur-
θt θϕ the coordinate variables are x1 = r, x2 = θ, x3 = ϕ,
face becomes equi-potential, ∂ Ψ = ∂ Ψ+Ω∂ Ψ = 0.
However, in a particle acceleratθion regiton, the mϕagnetic x1′ =s,x2′ =θ∗,andx3′ =ϕ∗. Notethatthisformalism
field does not rigidly rotate, because F +ΩF devi- is applicable to arbitrary magnetic field configurations,
At Aϕ
ates from 0. The deviation is expressed in terms of Ψ, and also that equation (35) gives Ek = −(∂Ψ/∂s)θ∗,ϕ∗.
whichgivesthestrengthoftheaccelerationelectricfield, Causality requires that a plasmas can co-rotate with
the magnetic field only in the region that satisfies k
0
B Bi B g +2g Ω+g Ω2 > 0 (Znajek 1977; Takahasi et a≡l.
E E = (F +ΩF )= ( Ψ), (35) tt tϕ ϕϕ
k ≡ B · B it iϕ B · −∇ 1990). That is, the concept of the light cylinder is gen-
eralizedinto the light surface on whichk vanishes. The
0
whichismeasuredbyadistantstaticobserver,wherethe
effects of magnetic field expansion (Scharlemann et al.
Latin index i runs over spatial coordinates r, θ, ϕ.
1978; Muslimov & Tsygan 1992) are contained in the
Substituting equation (34) into (33), we obtain the
coefficients of the trans-field derivatives, gθ∗θ∗, gθ∗ϕ∗,
Poissonequation for Ψ,
gϕ∗ϕ∗. In what follows, we adopt the vacuum, rotat-
c2 √ g ing dipole solution (Cheng et al. 2000) to describe the
− √ g∂µ ρ−2 gµνgϕϕ∂νΨ =4π(ρ−ρGJ), (36) magnetic field configuration.
− (cid:18) w (cid:19) Second, let us consider the particle Boltzmann equa-
where tions. At time t, position r, and momentum p, they
become,
c2 √ g
ρGJ ≡ 4π√−g∂µ(cid:20) ρ−2w gµνgϕϕ(Ω−ω)Fϕν(cid:21) (37) ∂∂Nt± +v·∇N±+ qE+ vc ×B · ∂∂Np± =S±(t,r,p),
denotes the general relativistic Goldreich-Julian charge (cid:16) (cid:17) (42)
density. If ρ = ρGJ holds everywhere, Ek vanishes in where N+ (or N−) denotes the positronic (or electronic)
the entire region,providedthat the boundaries are equi- distribution function; v p/(m γ), m refers to the rest
e e
potential. However,ifρ deviatesfromρGJ inanyregion, mass of the electron, an≡d q the charge on the particle.
E appears around the region with differentially rotat-
k TheLorentzfactorisgivenbyγ 1/ 1 (v /c)2. The
einqguamtiaognne(t3i7c)fireelddulcinesest.oInthtehespheicgihael-rreallatittiuvidsetsic, rex≫prregs-, creoslleinsitotnhteearmppSea+ri(nograSn−d)dciosnapsipste≡saroifnptghrea−tteersm|ofs|pthoasittrroenps-
sion (Goldreich, & Julian 1969; Mestel 1971),
(or electrons) at r and p per unit time per unit phase-
Ω B (Ω r) ( B) space volume. Since we are dealing with high-energy
ρ · + × · ∇× , (38) phenomena, we consider wavefrequencies that are much
GJ
≡− 2πc 4πc
greater than the plasma frequency and neglect the col-
which is commonly used. lective effects.
Instead of (r,θ,ϕ), we adopt the magnetic coordinates It is noteworthy that the particle flux per magnetic
(s,θ ,ϕ ), where s denotes the distance along a mag- flux tube is conserved along the flow line if there is
∗ ∗
netic field line, θ and ϕ represents the magnetic co- no particle creation or annihilation. Thus, it is con-
∗ ∗
latitude and the magnetic azimuthal angle, respectively, venient to normalize the particle distribution functions
of the point where the field line intersects the NS sur- by the Goldreich-Julian number density such that n =
±
face. Defining that θ = 0 corresponds to the magnetic N /(ΩB/2πce), where denotes that the quantity is
∗ ±
h i hi
axis and that ϕ = 0 to the latitudinal plane on which averagedin a gyration. Imposing a stationary condition
∗
Luminosity Evolution of Gamma-ray Pulsars 7
∂/∂t+Ω∂/∂φ=0intheco-rotatingframe,wecanreduce coefficientsformagneticandphoton-photonpairproduc-
the particle Boltzmann equations into (Hirotani et al. tion processes, are given by
2003)
e2/~cm c 8 1
α = e B′ exp
∂n± ∂n± ∂n± γB 4.4 ~ ⊥ −3ǫ B′
ccosχ +p˙ +χ˙ =SIC+Sp, (43) (cid:18) γ ⊥(cid:19)
∂s ∂p ∂χ
δ(γ γ )δ(χ χ ), (49)
0 0
× − −
where the upper and lower signs correspond to the 1 cosθ ∞ dF ∂2σ
c s γγ
positrons (with charge q = +e) and electrons (q = e), αγγ= − dǫs , (50)
respectively, χ denotes the pitch angle of gyrating p−arti- 2 Zǫth dǫs dγdχ
cles, and p p =m c γ2 1. Since pair annihilation where B′ Bsinθ /B ; explicit expression of
e ⊥ c cr
IisCnsecgalitgtiebrlien≡gin|tae|rpmu,lsSarICmp,aagnnde−ttohsepphaeirre,cSre±atcioonnsitsetrsmo,fSthpe, HηIγiCroatanndi e∂t2σalγ.γ≡/(d2γ00d3χ))a.re Igniveenquiantiothne(l4it9e)r,atBur′⊥e (ceo.gn.-,
in the right-hand side. When a particle emits a photon tains the collision angle, θ , between the photon and
c
via synchro-curvature process, the energy loss ( GeV) the magnetic field. In equation (50), θ does that be-
∼ c
is small compared to the particle energy ( 10 TeV); tween the two photons. The IC redistribution function
thus, it is convenient to include the back rea∼ction of the ηγ (ǫ ,γ,µ ) represents the probability that a particle
IC γ c
synchro-curvatureemissionas a friction termin the left- with Lorentz factor γ up-scatters photons into energies
hand side (Hirotani 2006a). In this case, the character- between m c2ǫ and m c2(ǫ +dǫ ) per unit time when
e γ e γ γ
istics of equation (43) in the phase space are given by the collision angle is cos−1µ , where ǫ = hν/(m c2)
c γ e
refers to the dimensionless photon energy. On the other
P
p˙ qE cosχ SC, (44) hand, ηe (γ ,γ ,µ )representsthe probabilityforapar-
≡ k − c IC i f c
ticle to change its Lorentz factor from γ to γ in a sin-
i f
qE sinχ ∂(lnB1/2) gle scattering. We thus obtain ηe (γ ,γ ,µ )=ηγ (γ
χ˙ k +c sinχ, (45) IC i f c IC i−
γ ,γ ,µ ) by energy conservation.
≡− p ∂s f i c
Third, let us consider the radiative transfer equation.
where the synchro-curvature radiation force, PSC/c The variation of specific intensity, Iν, along a ray is de-
(Cheng & Zhang 1996; Zhang & Cheng 1997), is in- scribed by the radiative transfer equation,
cluded as the friction; the particle position s is related
dI
with time t by s˙ = ds/dt = ccosχ. For outward- (or ν = α I +j , (51)
ν ν ν
inward-) migrating particles, cosχ>0 (or cosχ<0). If dl −
Ek = 0, particles will be reflected by magnetic mirrors, wherelreferstothedistancealongtheray,αν andjν the
whichareexpressedbythesecondtermintheright-hand absorption and emission coefficients, respectively. Both
sideofequation(45. Ifweintegraten± overpandχ,and αν and jν are a function of l, photon energy Eγ, and
multiply the local GJ number density, ΩB/2πce, we ob- propagationdirection(kθ,kϕ),wherekµdenotesthepho-
tain the spatial number density of particles. Therefore, ton momentum four vector (µ=t,r,θ,ϕ).
we can express the real charge density ρ as Equation (51) can be solved if we specify the photon
ΩB propagation in the curved space time. The evolution of
ρ= (n+ n−)dγdχ+ρion, (46) momentum and position of a photon is described by the
2πc −
ZZ Hamilton-Jacobi equations,
where n are a function of s, θ , ϕ , γ, χ; ρ refers to
± ∗ ∗ ion dk ∂k dk ∂k
the charge density of ions, which can be drawnfrom the c r = t, c θ = t, (52)
NS surface as a space-charge-limited flow (SCLF) by a dl − ∂r dl − ∂θ
positive E (Hirotani 2006a). dr ∂k dθ ∂k
k c = t, c = t. (53)
In equation (43), the collision terms are expressed as dl ∂k dl ∂k
r θ
ǫn−1<γ Since the metric (eqs. [30]–[32]) is stationary and ax-
S ηγ ( ǫ ,γ,µ ) n dγdχ, isymmetric, the photon energy at infinity k and the az-
IC≡− IC h ni c ± imuthal wave number k are conserved atlong the ray.
n ZZ ϕ
X −
When a particle is rotating with angular velocity ϕ˙ and
+ ηIeC(γi,γ,µc) n±dγdχ, (47) emit a photon with energy Elocal, kt and kϕ are re-
−
Xi ZZ latedtothesequantitiesbytheredshiftrelation,Elocal =
(dt/dτ)(k +k ϕ˙),wheredt/dτ issolvedfromthe defini-
and t ϕ
2πce I tionofthe propertime, (dt/dτ)2(g +2g ϕ˙+g ϕ˙2)=
ν tt tϕ ϕϕ
Sp (αγB +αγγ) dω˜dν, (48) 1. The dispersion relation kµk = 0, which is quadratic
≡ ΩB hν µ
Z Z to k ’s (µ = t,r,θ,ϕ), gives Hamiltonian k in terms of
µ t
where hǫni = (ǫn−1 +ǫn)/2 represents the typical pho- r, θ, kr, kθ, and kϕ. Thus, we have to solve the set of
tonenergywithintheinterval[m c2ǫ ,m c2ǫ ],µ the four ordinary differential equations (52) and (53) for k ,
e n−1 e n c r
cosine of the collision angle between the particles and k , r, andθ. Whenphotons are emitted, they arehighly
θ
the soft photons, I the specific intensity of the radia- beamedalongtheparticle’smotion;thus,theinitialcon-
ν
tion field, ω˜ the solid angle into which the photons are ditions of (kr,kθ,kϕ) are givenby the instantaneous par-
propagating. To compute S at each point, we have to ticle’s velocity measured by a distant static observer. It
p
integrate I /(hν) in alldirections to calculate the differ- may be helpful to give the initial conditions of ray trac-
ν
entialphotonnumberflux, I /(hν)dω˜. Theabsorption ing in the limit r r and γ 1, because photons are
ν g
≫ ≫
R
8 Hirotani
emittedmostlyintheoutermagnetosphere. Inthislimit, the open field lines cross the PC surface at magnetic co-
theorthonormalcomponentsofthee+’sande−’sinstan- latitudes θmin = (1 h )θmax < θ = (1 h)θ max <
taneous velocity are given by (Mestel 1985; Camenzind θmax (i.e., ∗h > h >−0)m, an∗d that ∗θ = 0−(i.e., ∗h = 1)
∗ m ∗
1986a,b) corresponds to the magnetic axis.
We also have to consider the boundary conditions for
vr Br vθˆ Bθˆ vφˆ Bϕˆ ̟ the hyperbolic type equations (43) and (51). (Eq. [51]
=f , =f , =f + (54)
c v c c v c c v c ̟ itself is an ordinary differential equation; however, it is
LC
equivalent to solving the Boltzmann equation of photon
in the polar coordinates, where distributionfunction.) Weassumethate±’sandphotons
are not injected into the gap across either the inner or
̟ Bϕˆ ̟ 2 Bp 2 the outer boundaries. However, if the created current
f 1 ; (55)
v ≡−̟LC B ±s −(cid:18)̟LC(cid:19) (cid:18) B (cid:19) batectohmeeNs gSresautrefractehatno tdhreawGJiovnaslufer,oma ptohseitisvuerfEackearaisseas
SCLF until E almost vanishes at the surface.
B (Br)2+(Bθˆ)2. The upper (or the lower)sign of k
p ≡ Tosumup,wesolvethesetofpartialandordinarydif-
fv (eqq. [55]) corresponds to the outward (or the inward) ferentialequations(36),(43), and(51)underthe bound-
particle velocity. Note that Bϕˆ < 0 holds in ordinary ary conditions described in the three foregoing para-
situation. graphs. By this method, we can solve the acceleration
Fourth and finally, let us impose appropriate bound- electric field E , particle distribution functions n , and
k ±
ary conditions to solve the set of Maxwell (i.e., Poisson) the photon specific intensity I (from hν = 0.005 eV to
ν
andBoltzmannequations. Westartwithconsideringthe 50 TeV), at each position in the three-dimensional mag-
boundary conditions for the elliptic type equation (36). netosphere of arbitrary rotation-powered pulsars, if we
Wedefinethattheinnerboundary,s=0,coincideswith specify P, µ, α, and kT, where the surface tempera-
the NS surface, on which we put Ψ=0 for convenience. ture kT is necessaryto compute the photon-photon pair
The outer boundary, s = sout(θ∗,ϕ∗), is defined as the productionthroughthe differentialflux dFs/dǫs inequa-
place where Ek changes sign near the light cylinder. tion(50). Weadopttheminimumcoolingscenariointhe
Its locationis solvedself-consistently as a free-boundary same manner as in 2.
problemandappearsnearthe placewhere ∂(ρGJ/B)/∂s In figure 4, we pl§ot the result of Lγ as a function of
vanishes due to the flaring up of the field lines towards L as the dash-dotted (or solid) curve for a light (or a
spin
the rotation axis (eq. [68] of Hirotani (2006a)). At each heavy) element envelope, where µ = 3.2 is adopted in
30
ϕ∗, the lower boundary θ∗ =θ∗max(ϕ∗) is assumed to co- thesamemannerasintheanalyticalexamination. Itfol-
incide with the last open field line, which is defined by lows that these numerical solutions are consistent with
the condition that √−grrBrsinθ +√−gθθBθcosθ = 0 the analytical ones, and that Lγ decreases slowly until
is satisfied at the light cylinder. We can compute the 104.5 years. The physical reason why L increases with
γ
potentialdropalongeachfieldline, ∆Ψ(θ ,ϕ ) Ψ(s=
0) Ψ(s = s ) = Ψ(s = s ), by ∗inte∗gra≡ting E decreasingLspin att>104yearsforalightelementenve-
− out − out k lope,isthesameasdescribedattheendof 2. Arealistic
along the field line. The maximum value of ∆Ψ(θ ,ϕ ) §
∗ ∗ NS will have an envelope composition between the two
is referred to as ∆Ψ .
max extreme cases, light and heavy elements. Thus, the ac-
Let us consider the upper boundary, θmin =
∗ tualLγ’swilldistributebetweentheredsolid(ordashed)
θ∗min(s,ϕ∗). At each (s,ϕ∗), Ek vanishes on the last- and the blue dash-dotted (or dotted) curves. However,
open field line, θ = θmax (i. e., at the lower bound- after L approaches L (thin dashed straight line; see
∗ ∗ γ spin
ary), and increases with decreasing θ in the latitudi- Wang & Hirotani (2011) for the death line argument),
∗
nal direction (i. e., towards the magnetic axis). The the outer gap survives only along the limited magnetic
acceleration field E peaks around the middle-latitudes, field lines in the trailing side of the rotating magneto-
k
θ [θmax+θmin]/2=(1 h /2)θmax, andturns to de- sphere because of a less efficient pair production; as a
cr∗e≈aset∗owards∗theupperb−ounmdary.∗Atsomeco-latitude, result, Lγ rapidly decreaseswith decreasingLspin. For a
Ek eventually decreases below 0.01∆Ψmax/̟LC; we de- smaller α, even for a light element envelope, Lγ mono-
fine this co-latitude as the gap upper boundary, θmin = tonically decreases as the dash-dot-dot-dot curve shows,
∗ because the gap is located in the higher altitudes, and
θmin(s,ϕ ). To specify the magnetic field line at each
∗ ∗ because the less efficient pair production there prevents
ϕ , it is convenient to introduce the dimensionless co-
∗ theproducedelectriccurrenttoincreasewithdecreasing
latitude,
age around t 104.5 years.
h≡1−θ∗/θ∗max(ϕ∗). (56) ∼
For example, h=0 specifies the last-open field line, and 4. EXPONENTIALCUTOFFENERGY
h = 0.1 does the field line having the foot point on the
Itisalsoworthexaminingthe cutoffenergy,E ,of
polar-cap(PC)surfaceatθ =0.9θmax ateachmagnetic cutoff
∗ ∗ the gamma-ray spectrum. We plot E as a function
azimuthal angle ϕ . At the upper boundary, we obtain cutoff
∗ ofthe magnetic-fieldstrengthatthe lightcylinder, B ,
LC
h=h (s,ϕ ) 1 θmin(s,ϕ )/θmax(ϕ ). (57) in figure 5. In the analyticalcomputation(dash, dotted,
m ∗ ≡ − ∗ ∗ ∗ ∗ and dash-dot-dot-dot curves), only the curvature pro-
To solve the Poissonequation, we put Ψ=0 at h=h , cess (Hirotani 2011a) from the particles produced and
m
or equivalently, at θ = θmin(s,ϕ ). If h 1, the gap accelerated in the outer gap, is considered as the pho-
∗ ∗ ∗ m ≪
becomes thin, whereas h 1 indicates that the gap ton emission process, and no photon absorption is con-
m
∼
is threaded by most of the open-field lines. Note that sidered. In the numerical computation (solid and dash-
Luminosity Evolution of Gamma-ray Pulsars 9
8
4
10
-1g s) 1036 105 years V) 6 105 years years
minosity, L(erg 11003345 104 years 103 years off energy (Ge 42 103 years
u ut
y l a Envelop Method C
mma-ra 1033 666000oooHLHieegaahvvtyy NANunumamlyeetrriiicccaaalll 0 103 104 105 106
ed ga 1032 6405ooLLiigghhtt NAunmaleyrtiiccaall Magnetic field at light cylinder (gauss)
uls
P 1034 1035 1036 1037 1038 Fig.5.— Cutoff energy of γ-ray spectrum. The dashed and
dotted curves denote the cutoff energies obtained analytically for
Pulsar spin-down luminosity, L (erg s-1) heavy andlightelement envelopes, respectively, whereas the solid
spin
and dash-dotted curves do those obtained numerically for heavy
Fig.4.— Same figure as figure 3, but numerical results are and light element envelopes, respectively. For these four cases,
plotted as dash-dotted and solid curves for a light and a heavy α = 60◦ is assumed. For comparison, an analytical solution for
element envelope, respectively. For comparison, anumerical solu- α=45◦ andalightelement envelope, isalsoplotted asthe dash-
tion for α = 45◦ and a light element envelope, is also plotted as dot-dot-dotcurve.
thedash-dot-dot-dotcurve. Toavoidcomplications, weadoptthe
geometrical meanof the upper-bound LX and lower-boundLX of
cao‘oblianngd’c,urwvheischforreeflaecchtschtehmeiucanlcecrotmaipnotiseitsioinn,tihnestenaudcloeofndeCpoicotpinegr or) 103 Heavy element, 105 yrs
pairingmodels. act 102 Heavy element, 104 yrs
dottedcurves),ontheotherhand,synchro-curvatureand ction f 10 Light element, 105 yrs
inverse-Compton processes from the particles not only orre Light
produced and accelerated in the gap but also cascaded x c 1 element,
oauntdsidbeotthhetghaep,pahroetcoonn-psihdoetroendaasntdhememaginsseitoicnppraoicrespsreos-, (flu 0.1 104 yrs
fW 0 0.2 0.4 0.6 0.8 1
duction processes are taken into account when comput-
ing the photon propagation. Therefore, for very young cosz
pulsars, a strong photon-photon absorption (and the Fig.6.— Flux correction factor, fΩ, as a function of the ob-
subsequent production of lower-energy photons via syn- server’s viewing angle, ζ, with respect to the rotation axis. The
chrotronand synchrotron-self-Comptonprocesses)takes (red)thicksolidanddashedcurvesdenotefΩatpulsarage104and
place even in the higher altitudes. As a result, spectrum 105 years, respectively, foraheavy element envelope, whereas the
(blue)thinsolidanddashedcurvesdothoseat104 and105 years,
can no longer be fitted by power-law with exponential-
respectively,foralightelementenvelope. Theabscissadenotesthe
cutoff functional form. Thus, for t < 104 years, the fit- cosine of ζ; thus, ζ distributes randomly along the abscissa with
tedcutoffenergiesarenotplottedfornumericalsolutions uniformprobability. Forallthecases,α=60◦ isassumed.
(i.e., solid and dash-dotted curves). the middle and lower altitudes, resulting in a stronger
Itfollowsthatthepresentoutergapmodelexplainsthe flux near the rotational equator, 66◦ <ζ <73◦.
observed tendency that E increases with increasing
cutoff
B ,becausetheGoldreich-Julianchargedensity(inthe 6. DISCUSSION
LC
gap) increases with increasing BLC. It also follows that To sum up, a light element envelope approximately
Ecutoff is regulated below several GeV, because copious corresponds to the lower bound of the (observationally
pair production leads to hm 1 for young pulsars (i.e., inferred)gamma-rayluminosityofrotation-poweredpul-
≪
for strong BLC). sars, whereas a heavy element one to the upper bound.
The scatter of the intrinsic gamma-ray luminosity is
5. FLUXCORRECTIONFACTOR
physically determined by the magnetic inclination an-
Finally,letusinvestigatethefluxcorrectionfactor,fΩ. gle, α, and the envelope composition. The cutoff energy
To infer the γ-ray luminosity, Lγ =4πfΩFγd2, from the of the primary curvature emission is kept below several
observed flux, Fγ, one has conventionally assumed that GeV even for young pulsars, because the gap trans-field
thefluxconversionfactorisapproximatelyunity,fΩ 1, thickness,andhencetheaccelerationelectricfield,issup-
≈
where d denotes the distance to the pulsar. However, pressed by the polarization of the produced pairs in the
now we can compute fΩ explicitly as a function of the lower altitudes.
observer’s viewing angle, ζ, using the three-dimensional Toconverttheobservedγ-rayfluxintoluminosity,L ,
γ
numerical solution. In figure 6, fΩ(ζ) is depicted for one has conventionally assumed fΩ = 1. For example,
heavy and light element envelopes at 104 and 105 years. theerrorbarsoftheobservationaldatapointsinfigure4,
It follows that the error of f = 1 is kept within a fac- do not contain any uncertainties incurred by f . Never-
Ω Ω
tor of 3 with a probability greater than 50%. When the theless, if α and ζ can be constrained, we can estimate
pulsar is young, E is highly screened in the middle and L more accurately, by applying the present quantita-
k γ
loweraltitudes;asaresult,mostγ-raysareemittedfrom tive outer-gap calculations. It is noteworthy that L ’s
γ
thehigheraltitudestoappearwithintheobserver’sview- giveninfigures3and4littledependontheNSmagnetic
ing angle 46◦ < ζ < 53◦ (i.e., 0.6 < cosζ < 0.7). How- moment, µ. This is particularly true for a light element
ever, as the pulsar ages, more γ-rays are emitted from case, which has h 1, by the reason described after
m
≪
10 Hirotani
equation(11). Whatismore,withanadditionaldetermi- The author is indebted to Dr. A. K. Harding for valu-
nation of d (e.g., by parallax observations), we can infer able discussion on the results. He also thanks ASPEN
thecompositionofindividualNSenvelopes,byusingthe Center for Physicsfor providingprecious opportunity to
constrained flux correction factor, f (fig. 6). We hope debate the main topic of this letter. This work is partly
Ω
toaddresssuchaquestionasthedeterminationofαand supported by the Formosa Program between National
ζ,andhencef ,forindividualpulsars,bymakingan‘at- Science Council in Taiwan and Consejo Superior de In-
Ω
las’ of the pulse profiles and phase-resolvedspectra that vestigaciones Cientificas in Spain administered through
are solved from the basic equations in a wide parameter grant number NSC100-2923-M-007-001-MY3.
space of P, µ, T, α, and ζ, and by comparing the atlas
with the observations.
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