High-energy Emission from Pulsar Outer Magnetospheres: Two-dimensional Electrodynamics and Phase-averaged Spectra

0.19 MB·English

by  Kouichi Hirotani

High-energy Emission from Pulsar Outer Magnetospheres
Kouichi Hirotani
ASIAA/National Tsing Hua University - TIARA,
PO Box 23-141, Taipei, Taiwan1
[email protected]
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n ABSTRACT
a
J We investigate particle accelerators in rotating neutron-star magnetospheres, by simultane-
4 ously solving the Poisson equation for the electrostatic potential together with the Boltzmann
2 equationsforelectrons,positronsandphotonsonthepoloidalplane. Applyingtheschemetothe
threepulsars,Crab,Vela andPSRB1951+32,we demonstratethatthe observedphase-averaged

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spectra are basically reproduced from infrared to very high energies. It is found that the Vela’s
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spectrumin10-50GeVissensitivetothethree-dimensionalmagneticfieldconfigurationnearthe
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7 light cylinder; thus, a careful argument is required to discriminate the inner-gap and outer-gap
6 emissions using a gamma-ray telescope like GLAST. It is also found that PSR B1951+32 has a
1 largeinverse-ComptonfluxinTeVenergies,whichistobedetectedbyground-basedairCerenkov
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telescopes as a pulsed emission.
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0 Subject headings: gamma-rays: theory – magnetic fields – methods: numerical – pulsars: individ-
/
h ual(B1951+32,Crab,Vela)
p
- 1. Introduction zone. Ininner-gap(IG)models,whichadoptspar-
o
ticle accelerationwithin severalneutron-starradii
r Pulsarsformthesecondmostnumerousclassof
t above the polar-cap surface, the energetics and
s objects detected in high-energy γ-rays. To date,
a pair cascade spectrum have had success in repro-
: six have been detected by the Energetic Gamma ducing the observations (e.g., Daugherty & Hard-
v
Ray Experiment Telescope (EGRET) aboard the
i ing 1982, 1996). However, the predicted beam
X Compton Gamma Ray Observatory. The γ-ray
size is too small to produce the wide pulse pro-
r pulsations observed from these objects are par- files that are observed from high-energy pulsars.
a
ticularly important as a direct signature of non-
Seekingthepossibilityofawidehollowconeemis-
thermalprocessesinrotatingneutron-starmagne-
sionduetoflaringfieldlines,MuslimovandHard-
tospheres, and potentially should help to discrim-
ing (2004) extended the idea of Arons (1983) and
inate among different emission models.
proposed a slot-gap (SG) model, in which emis-
The pulsar magnetosphere can be divided into sion takes place very close to the last-open field
two zones: The closed zone filled with a dense line fromthe stellarsurfacetothe outermagneto-
plasmacorotatingwiththestar,andtheopenzone sphere. Since the SG model is an extension of the
inwhichplasmaflowsalongtheopenfieldlinesto IG model into the outer magnetosphere, a nega-
escape through the light cylinder. The last-open tivemagnetic-field-alignedelectricfield,E ,arises
k
field lines form the border of the open magnetic if the magnetic moment vectorpoints in the same
field line bundle. In all the pulsar emission mod- hemisphere as the rotation vector. However, the
els, particle acceleration takes place in this open electric current induced by the negative E does
k
nothaveaself-consistentclosurewithinthemodel
1Postal address: TIARA, Department of Physics, Na- (Hirotani 2006, hereafter H06).
tional Tsing Hua University, 101, Sec. 2, Kuang Fu
To contrive an accelerator model that predicts
Rd.,Hsinchu,Taiwan300
1
an outward current in the lower latitudes (within tions, we obtain
the open zone), it is straightforward to extend
outer-gap (OG) models (Cheng et al. 1986; Ro- ckr ∂g +ckθ ∂g =S (r,θ,c|k|,kr,kθ), (2)
mani & Yadigaroglu1995,hereafter RY95; Cheng |k|∂r |k|∂θ γ
et al. 2000) into the inner magnetosphere. Ex-
tending several OG models (Hirotani et al. 2003; where the wave numbers kr and kθ are given by
Takata et al. 2004), H06 demonstrated that the the ray path, (r,θ) are the polar coordinates, and
gap extends from the stellar surface to the outer thedimensionlessphotondistributionfunctiongis
magnetosphere, that the positive E extracts normalized by the GJ number density at the stel-
k
ions from the star as a space-charge-limited flow lar surface. ICS, synchro-curvature emission, and
(SCLF), and that most photon emission takes the absorption are contained in Sγ. In H06, the
place in the outer magnetosphere because E is rate of synchrotron emission by secondary pairs
k
highlyscreenedinsidethenullsurfaceowingtothe created outside the gap, was calculated assuming
discharge of the created pairs. In the present let- aconstantpitchangle. However,itturnsoutthat
ter, we formulate the scheme (Beskin et al. 1992) only 17 % of the initial particle energy is con-
in § 2, apply it to three rotation-powered pulsars verted into radiation. In this letter, we corrected
in § 3, and give a brief discussion in § 4. this problem by computing the pitch angle evo-
lution of radiating particles, which increases the
2. Gap Electrodynamics synchrotron cooling time and hence recovers the
time-integrated, radiated energy to 100 % of the
Wefollowtheschemedescribedin§2ofH06to initial particle energy. Note that the gap electro-
solvethesetofMaxwellandBoltzmannequations. dynamicsinvestigatedinH06remainscorrect,de-
The firstequationwehavetoconsideris the Pois- spite the insufficient secondarysynchrotronfluxes
sonequationfortheelectro-staticpotential,Ψ. As in H06.
space is limited, we presentits Newtonian expres- TosolvethePoissonequation,weimposeΨ=0
sion,−∇2Ψ=4π(ρ−ρGJ). Iftherealchargeden- atthe lower,upper,andinner (s=0)boundaries,
sity, ρ, deviates from the Goldreich-Julian (GJ) and E = −∂Ψ/∂s = 0 at the outer boundary.
k
charge density, ρGJ, in some region, an acceler- Ion extraction rate is regulated by the condition
ation electric field Ek ≡ −∂Ψ/∂s arises, where s Ek = 0 at s = 0. We parameterize the trans-field
designatesthedistancealongamagneticfieldline. thickness of the gap with hm. If hm = 1 the gap
The second equationwe have to consider is the existsalongalltheopenfieldlines,whileifhm ≪1
Boltzmannequationsfore±’s. Assuming asteady the gap becomes transversely thin.
state inthe frame ofreferencecorotatingwith the
magnetosphere, we obtain 3. Application to Individual Pulsars
∂n± dp∂n± dχ∂n± We apply the theory to three γ-ray pulsars,
ccosχ + + =S , (1)
∂s dt ∂p dt ∂χ ± Crab,Vela,andB1951+32,focusingontheir pho-
tonspectra. Evennearandoutsidethelightcylin-
wherecdenotesthe speedoflight,n+ (orn−)the der, photon emission and absorption occur effec-
dimensionless positronic (or electronic) distribu-
tively; thus, equations (1) and (2) are solved in
tion function normalized by the local GJ number 0 < s < 16̟LC, where Ek = 0 in ̟ > 0.9̟LC;
density. The temporal derivatives of momentum
̟LC ≡ c/Ω denotes the light cylinder radius, Ω
and pitch angle, dp/dt and dχ/dt, and the colli-
the stellar rotation frequency, and ̟ the distance
siontermS± areexplicitlygiveninH06. Synchro- from the rotation axis. The field line geometry in
curvatureradiation-reactionforceisincludedasan
0.9̟LC < ̟ < 2̟LC mimics the aligned dipole
externalforceindp/dt,whiletheeffectsofinverse- in the force-free magnetosphere (Contopoulos et
Compton scatterings (ICS) and (one-photon and
al. 1999; Gruzinov 2005). In ̟ >2̟LC, the field
two-photon) pair creation processes are in S .
± lines are assumed to be straight so that they con-
The third equation we have to consider is the nect smoothly at ̟ =2̟LC.
Boltzmann equations for photons. Assuming a
steady state, and neglecting azimuthal propaga-
2
3.1. Crab pulsar 3.2. Vela pulsar
WepresenttheresultsfortheCrabpulsarinfig- Next, we present the results for the Vela pul-
ure1, adoptingamagnetic inclinationofαi =75◦ sar in figure 2 (left). Taking an angle average
andadipolemomentofµ=4×1030Gcm3,which over 75◦ < θ < 103◦ (solid line), we can repro-
is close to the value (3.8× 1030Gcm3) deduced ducetheobservedpulsedspectrum,exceptforthe
from the dipole radiation formula. It follows that RXTEresults. Theprimarycurvaturecomponent
the observed pulsed spectrum from IR to VHE appears between 100 MeV and 10 GeV, while the
can be reproduced, provided that we observe the secondary and tertiary synchrotron components
photons emitted into 75◦ <θ <103◦, where θ de- appear below 100 MeV. ICS is negligible for the
notes the photon propagation direction measured Vela pulsar because of its weak magnetospheric
from the rotational axis. Because of the aberra- emission. Similar spectral shapes are obtained for
tion of light, it is reasonable to suppose that pho- super-GJsolutions,eventhoughwehavetoadjust
tons emitted in a certain range of θ comes into hm to obtain an appropriate flux.
our line of sight in an obliquely rotating three- In the right panel, we compare the present re-
dimensionalmagnetosphere(e.g.,RY95). Theflux sults with IG (dotted) and OG (dashed) models,
rapidly decreases with decreasing θ for 75◦ <θ < where the dash-dotted line denotes the averaged
93◦, because Ek is highly screened in the inner flux for 75◦ < θ < 107◦. It follows that the spec-
part of the gap. Nevertheless, an average over trum between 10 and 100 GeV crucially depends
75◦ < θ < 103◦, which includes negligible flux on the angles in which the photons that we ob-
between 75◦ < θ < 89◦, achieves the current ob- serve are emitted. Thus, to quantitatively predict
jective, because the spectral normalizationcan be the γ-rayemissionfromthe outer magnetosphere,
fitted (within a factor of a few) without changing it is essential to examine the three-dimensional
the spectral shape, by adjusting hm. magneticfieldstructurenearandoutsidethelight
ICS takes place efficiently near and outside the cylinder.
light cylinder (Aharonian & Bogovalov 2003), be-
cause the magnetospheric IR photons, which are 3.3. PSR B1951+32
emitted along convex field lines, collide with the
Thirdly and finally, we present the spectrum
gap-accelerated positrons near the light cylinder,
of B1951+32 in figure 3 (left). It follows that
where the field lines are concave. Most of such
theROSATandEGRETfluxesarereproducedby
upscattered photons, as well as some of the high-
taking the flux average in 75◦ < θ < 103◦ (solid
energy tail of the curvature component, are ab-
line), except for 17 GeV flux, which was derived
sorbed by the γ-γ collisions. As a result, there is
from only two photons. Because of its weak mag-
a gradual turnover around 10 GeV, which forms
netic field, less energetic synchrotron photons re-
a striking contrast with the steep turnover pre-
duce the absorption of the ICS component. The
dictedinIGmodelsduetomagneticpaircreation.
reduced absorption results in a small synchrotron
The primary curvature component appears be-
flux, whichfurther reducesthe absorptionoutside
tween 100 MeV and 10 GeV, while the secondary
the light cylinder, leading to a large, unabsorbed
synchrotroncomponentappearsbelowafewMeV.
TeVfluxes. Italsofollowsthatthespectrumturns
BetweenafewMeVand100MeV,theICScompo-
overatlowerenergythantheIGmodelprediction
nent dominates, because the secondary pairs that
(dotted curve in the right panel; Harding 2001).
havebeen cooleddownbelow a few hundredMeV
It should be noted that the predicted ICS flux
efficiently up-scatter magnetospheric UV and X-
(above 100 GeV) represents a kind of upper lim-
rays to lose energy. Similar spectral shapes are
its,becauseitisobtainedbyassumingthatallthe
obtainedfordifferentvaluesofαi,µ,hm,provided
magnetospheric synchrotron photons illuminate
that the created current is super GJ, by virtue of
the equatorial region in which gap-accelerated
the negative feedback effects (H06).
positrons are migrating. Some of such VHE
photons materialize as energetic secondary pairs,
emitting the synchrotron component between a
few keV and 100 MeV. Between 100 MeV and
3
30 GeV, the primary curvature component domi- lines aremore or less straightnear the lightcylin-
nates, which represents the absolute flux (instead der, as demonstrated by Spitkovsky (2006, fig. 2)
of upper limits). Some of such curvature photons for an oblique rotator, the equatorial region may
haveenergiesabove10GeVandareefficientlyab- not be efficiently illuminated. In this case, the
sorbedtomaterializeaslessenergeticpairs,which VHE flux will decrease from the current predic-
emitsynchrotronradiationbelowafewkeV.Thus, tion. There is room for further investigation how
the spectrum below a few keV also represents the to extend the present analysis into three dimen-
absolute flux. sions, and to combine it with time-dependence
three-dimensional force-free electrodynamics.
4. Summary and Discussion The author is grateful to Drs. N. Otte,
R. Taam, J. Takata for helpful suggestions. This
Tosumup,theself-consistentgapsolutionsba-
workis supported by the TheoreticalInstitute for
sically reproduce the observed power-law spectra
Advanced Researchin Astrophysics (TIARA) op-
below a few GeV for the three pulsars examined.
erated under Academia Sinica and the National
The cut-offspectrabetween10GeVand100GeV
Science Council Excellence Projects program in
strongly reflect the three-dimensional magnetic
Taiwan administered through grant number NSC
field configuration near the light cylinder; thus, a
95-2752-M-007-006-PAE.
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5
Crab
bestfit (q <103o)
Fig. 1.— Phase-averaged spectrum of the mag-
netospheric emissions from the Crab pulsar for
αi =75◦, µ=4×1030Gcm3 and hm =0.12. The
thickdashed,dash-dotted,dottedanddash–triple-
dotted lines represent fluxes into 93◦ < θ < 97◦,
97◦ < θ < 101◦, 101◦ < θ < 105◦,105◦ < θ <
109◦; the thin dashed, dash-dotted, dotted ones
109◦ < θ < 113◦,113◦ < θ < 117◦, 117◦ < θ <
121◦. The thick solid line represents the averaged
flux in 75◦ < θ < 103◦. See Eikenberry et al.
(1997) for IR-UV data; Knight (1982), Weisskopf
et al. (2004), and Mineo et al. (2006) for X-rays;
Nolan et al. (1993), Ulmer et al. (1995), Much et
al.(1995),Fierroetal.(1998),Kuiperetal.(2001)
for 10 MeV–20GeV; Borione et al. (1997), Tan-
imori et al. (1998), Hillas et al. (1998), Lessard
et al. (2000), and de Naurois et al. (2002) for the
upper limits above 50 GeV.
6
Vela
bestfit (q < 103o)
Vela
this work (q < 103o)
this work (q < 107o)
OG model
IG model
Fig. 2.— Phase-averagedVela spectrum for µ =
4 × 1030Gcm3, αi = 75◦ and hm = 0.21. See
Manchester et al. (1980) for optical data (open
squares), Harding et al. (2002) for X-rays (filled
triangles),Kanbachet al. (1994),Ramanamurthy
et al. (1995b), Fierro et al. (1998), Thompson et
al. (1999) for γ-rays (open circles). Left: Same
figureasfigure1. Right: Closeupabove100MeV
to compare with IG (Harding & Daugherty 1993)
and traditional OG (Romani 1996) models. The
solid line is same as the left panel.
7
bestfit (q < 103o)
EGRET
+ 75GeV cutoff
this work
(q < 103o)
this work
(q < 107o)
IG model
Fig. 3.— Phase-averaged spectrum of
PSR B1951+32 for µ = 2×1029Gcm3, αi = 75◦
and hm = 0.39. Lines represent same quanti-
ties as figure 2 unless notified. See Clifton et
al. (1988) for IR upper limits (open squares),
Becker& Tru¨mper (1996),Safi-Harbet al. (1995)
and Chang & Ho (1997) for X-ray data (open
triangles), Ramanamurthy et al. (1995a), for
100 MeV-15 GeV data (open circles), Srinivasan
et al. (1997) for VHE upper limits.
8