1. Slot Gap Emission from Pulsar Magnetospheres
Kouichi HIROTANI
MPI für Kernphysik, Heidelberg
March 17, 2005, TIARA
It’s my great pleasure to give a talk to you.
My topic will deal with the particle acceleration mechanisms in rotating neutron star magnetospheres.

2. Pulsar Activities §1 Introduction Pulsar Wind
Pulsed (beamed) Radio Emission
Pulsed (beamed) High Energy Emission
Chandra
HST
It is commonly accepted that isolated pulsars can be activated by the electro-motive force exerted on the spinning neutron star surface.
Most of the energy is carried by the wind and dissipated in the nebula.
And a very small portion of it is emitted as radio pulsation.
The rest of the energy is mainly converted into pulsed high-energy emission within the magnetosphere.
Komissarov & Lyubarksy

3. The seven highest-confidence g-ray pulsars
So far, OSSE and EGRET experiments on board Compton Gamma-Ray Observatory have detected pulsed signals from at least seven rotation-powered pulsars.

4. §1 Introduction: CGRO observations
g-ray pulsars emit radiation in a wide frequency range:
double peaks
It has bee revealed that these gamma-ray pulsars emit radiation in a wide frequency range from radio, optical, X-ray, and gamma-rays. Each column represents the pulse profiles of individual pulsars.
Fro the left, they are the Crab pulsar, B , the Vela pulsar, B ,…
It should be noted that most of the gamma-ray light curves exhibit double peaks with intense emission between them.
bridge
Thompson 2003, astro-ph/

5. Broad-band spectra Power peaked in g-rays
No pulsed emission above 20 GeV
High-energy turnover
Increase in hardness with age
Thermal component appears in older pulsars
Their broad-band spectra show the following properties.

6. §1 Introduction: CGRO observations
3 candidate g-ray pulsars (statistical probability ~10-4)
B *
Ramnamurthy et al.
1996, ApJ 458, 755 B
Kaspi et al. 2000,
ApJ 528, 445 J
Kuiper et al. 2000,
A&A 378, 918
* not in the EGRET
catalog as a source
by spatial analysis
In addition to the seven highest-confidence gamma-ray pulsars, there exist three other candidate pulsars having statistical probability occurring by chance is about …
They are B reported by …

7. §1 Introduction (cont’d)
Such high-energy photons are probably emitted by ultra-relativistic particles via curvature radiation and/or inverse Compton scatterings.
What energizes the particles ?

8. §2 Pulsar as a unipolar inductor (cont’d)
In a rotating NS magnetosphere, the Goldreich-Julian charge density is induced for a static observer. Decoupling E into E^ and E||, we obtain the following Maxwell eq:

9. §2 Pulsar as a unipolar inductor (cont’d)
Since E^ is perpendicular to B, the term
is essential for particle acceleration. We thus obtain
rGJ changes sign
at the null surface.
Fig. Goldreich-Julian charge distribution in NS magnetosphere

10. §2 Pulsar as a unipolar inductor (cont’d)
If re deviates from rGJ in some region, an acceleration electric field E|| is exerted along the B field lines.
If the potential drop in such a region attains a certain value ( >1013 V ), particles will be accelerated ( G>107 ) to emit high-energy photons.
Next question:
Where is the possible place of particle acceleration?

11. §3 Accelerator Models A long-standing issue: Polar-cap model vs.
Outer-gap model
(Back to solution)
Fig. Two magnetospheric accelerators.

12. §3 Accelerator positions (cont’d)
The modulation of the GeV light curves

13. §3 Accelerator models (cont’d)
The modulation of the GeV light curves testifies to the γ-ray production either at the
(1) polar cap (Harding et al. 1978, ApJ 225, 226;
Daugherty & Harding 1982 ApJ 252, 337;
1996 ApJ 458, 278;
Sterner et al. 1995, ApJ 445, 736),
or in the
(2) outer magnetosphere
(Cheng, Ho, Ruderman 1986, ApJ 300, 500;
300, 522
other refs…later.)

14. §3 Accelerator models: Outer-gap model
Various properties of high-energy emissions (e.g., double-peak light curves with strong bridges, phase-resolved spectra of
the Crab pulsar,
Cheng, Ruderman,
Zhang 2000,
ApJ 537, 964)
have been explained
with outer-gap models.

15. §3 Outer-gap Models: Previous works
Following Cheng, Ho, Ruderman (1986, ApJ 300, 500; 300, 522), Chiang & Romani (1992, ApJ 400, 629; 1994, ApJ 436, 754) demonstrated a broad, irregularly shaped emission beam, by assuming an outwardly unidirectional g-ray flux from a 3-D magnetosphere.
Romani (1996, ApJ 470, 469) estimated the evolution of GeV -photon-production efficiency, and considered the phase-resolved specra for the Vela pulsar, assuming Vgap∝θgap/ θopen.
Zhang & Cheng (1997, ApJ 487, 370) considered the high-energy emission from Geminga, B , by estimating the trans-field gap size by imposing that the surface, thermal X-rays and the curvature-radiated g-rays marginally satisfy the pair-production criteria.

16. §3 Outer-gap Models: Problems
However… the hypothesized gap geometry does not satisfy the Maxwell equation.
Created pairs are immediately discharged by the strong E|| .
Because of this discharge, the original E|| is screened at least partially.

17. §3 Outer-gap Models: Problems (cont’d)
This screening effect is particularly important at the null surface, where rGJ vanishes. + -
Only e-’s exist at the inner BD when E|| >0. Thus, a copious pair creation (~ GJ rate) leads to
re= e(n+-n-)
= -en-~ -WB/(2pce)
In the Poisson equation
a positive -rGJ must cancel the negative re , so that E|| increases monotonically near
the inner boundary.

18. §3 Outer-gap models: Problems (cont’d)
Screening effect is important at the null surface.
Charge density
distance along B field
If the inner boundary is located near the null surface,
| re | > | rGJ | holds.
Thus, the Maxwell equation
shows that dE||/ds is always negative at the inner boundary.
determined by geometry

19. §3 Outer-gap models: Problems (cont’d)
On the other hand, if the inner boundary is located near the stellar surface, -rGJ > | re | leads to dE||/ds >0 at the inner boundary.
Therefore, E|| can have a single-sign value in the gap.
If inner BD is located near the NS.
If inner BD is located near the null surface.

20. §3 Outer-gap models: Problems (cont’d)z
In short, the hypothesized outer-gap geometry is inconsistent with the Poisson equation.

21. §4 New accelerator model
On these grounds, to study accelerator models more quantitatively, I solve the set of Maxwell & Boltzmann equations in pulsar magnetospheres on 2-D poloidal plane.
On these grounds, to study …

22. §4 New accelerator model
Let us first describe the physical processes that take part in a stationary pair-production in an outer gap.
e+ and e- are accelerated by E||
Relativistic e+/e- emit g-rays via
synchro-curvature, and IC processes
g-rays collide with soft photons/B field to materialize as pairs in the accelerator

23. §4 New gap model: Maxwell equation
The Poisson equation for the electrostatic potential ψ is given by
N+/N-: distrib. func. of e+/e-
G : Lorentz factor of e+/e-

24. §4 New gap model: Particle Boltzmann eqs.
Distribution functions of e±’s having momentum p, obeys
N±: distrib. func. of e+/e- with Lorentz factor G= p/mec

25. §4 New gap model: Particle Boltzmann eqs.
N±: distrib. func. of e+/e- with Lorentz factor G= p/mec
G: distr. func. of g-rays with energy Eg and momentum k.
ηp: pair-production rate (particles/s/γ-ray)
ηIC: ICS transfer rate between (γ-rays/s/particle)

26. §4 New gap model: g-ray Boltzmann eqs.
Distribution functions of g-rays with momentum k, obeys
G: distribution func. of g-rays with 4-momentum (Eg,k)
ηIC: curvature radiation rate (g-rays/s/particle)
ηSC: synchro-curvature radiation rate (particles/s/g-ray)
N±: distrib. func. of e+/e- with Lorentz factor G= p/mec

27. §4 New gap model: Boundary conditions
To solve the set of Maxwell & Boltzmann equations, we must impose appropriate BCs.
lower boundary
= last-open field line

28. §4 New gap model: Boundary conditions
To solve the Poisson eq. for electrostatic potential Y,
we impose

29. §4 New gap model: Boundary conditions
To solve particle/g-ray
Boltzmann eqs., we impose
at the inner boundary
That is, no outwardly migrating particles (i.e., positrons) and no outgoing g-rays are injected across the BD.

30. §4 New gap model: Boundary conditions
At the
outer boundary,
we impose
That is, no particle/g-ray injection across the BD.

31. §4 New gap model: Boundary conditions
To determine both inner and outer BDs, we additionally impose the following two conditions:

32. §4 New accelerator model
Three free parameters:
magnetic inclination (assumed to be 45o),
azimuthal gap width [rad], Df
fraction of soft-photon field illuminating gap, fr
g-ray flux Df
g-ray spectral shape fr
In this scheme, there are three adjustable parameters.
Other quantities such as gap geometry, acceleration electric field, particle density and energy distribution, g-ray flux and spectrum, created pairs outside of the gap, are determined if we specify these 3 parameters.

33. §5 Application to the Crab Pulsar
I applied the theory to the Crab pulsar.
It is found that the gap extends from the vicinity of the N.S. to the outer magnetosphere.
Cf. conventional models:

34. §5 Crab Pulsar: acceleration field
For comparison, let us briefly consider the rectilinear approximation.

35. §5 Crab Pulsar: acceleration field
In the rectilinear approximation, E|| is roughly constant throughout the gap.
50%
2*107
17%, 83%
Cheng, Ho, Ruderman’s prediction for a vacuum gap appears around 1.5 * 10^8 V/m.
Thus, the acceleration electric field is partially screened by the discharge of produced pairs in the gap.
17%, 83%
107

36. §5 Crab Pulsar: acceleration field
Deviation from rectilinear approx. acts to partially screen E||.
Then, what occurs is the rectilinear approximation breaks down due to field line curvature?

37. §5 Crab Pulsar: acceleration field
Thus, the screening effect is particularly important in the inner region where the curvature radius is small.
Because,… if curvature radius is infinity,
rectilinear approx. is OK…

38. §5 Crab Pulsar: acceleration field
Thus, the screening effect is particularly important in the inner region where the curvature radius is small.
50%
25%, 75%

39. §5 Crab Pulsar: acceleration field
As a result, E|| increases outwards.
50%
25%, 75%

40. §5 Crab Pulsar: Predicted Spectrum
EGRET observations are reproduced (below GeV)
if fr < That is, the gap is not illuminated by the magnetospheric soft photons.
Thus, pulsation not detectable above 30 GeV.
Flat spectrum below GeV by secondary and higher-generation synch. rad.
fr=10-3
fr=10-4
fr=0

41. §5 Crab Pulsar: Magnetization Parameter
Creation rate, dN/dt = particles s-1.
Consistent with the requirement of the “s problem”.
Finally let us examine the pair production cascade outside of the gap. Copious pair production takes place close to the star when ingoing gamma-rays collide with the strong B field. The production rate attains… The produced pairs have ingoing momenta initially; however, they will lose energy very quickly by synchrotron radiation in the strong B field to become non-relativistic. Such non-relativistic particles are resonantly scattered by the surface X-rays to migrate outwards. As a result, a substantial particle outward flux is realized above the gap. The pulsar wind will have a particle injection rate at nearly 10^40 per second. It is interesting to compare this result with what is required from the so-called sigma-problem.
So that the fluctuating component of B field may be dissipated by reconnection to accelerate the flow,
dN/dt > 1040 is required (Kirk & Skjaeraasen 2003).

42. §5 Crab Pulsar: Magnetization Parameter
Creation rate, dN/dt = particles s-1.
s~105 at light cylinder
meridional distribution of
light cylinder
gap
2D MHD
wind model
without
reconnection
Computing the sigma parameter distribution on the meridional plane, we find that its typical value to be about or slightly less than
equator
mag. pole

43. Summary
A stationary pair-production cascade in pulsar magnetospheres is self-consistently solved from the set of Maxwell & Boltzmann eqs. on 2-D poloidal plane.
The gap extends from the vicinity of the star to outer magnetosphere above the last-open field line.
g-ray spectrum is reproduced for the Crab pulsar.
Above 30 GeV, pulsed flux lies well below the current and future ground-based telescopes.
Pair production attains nearly 1040 particles s-1.
s-problem may have been solved.
3-D extension is necessary to compute pulse profiles and phase-resolved spectra.

44. 2D MHD wind model Formation of torus in equatorial flow
Komissarov & Lyubarksy 2004
Formation of torus in equatorial flow
initial s0~ is acceptable
widely accepted value s0~104 gives too thick equatorial totus (100 times larger)
                                                      
observation

45. Compton Gamma Ray Observatory

46. EGRET on board CGRO Energy range: 30 MeV to 30 GeV
10 to 20 times more sensitive than previous detectors operating at same energies
High-energy gamma rays enter the chambers and produce electron-positron pairs which cause sparks. The path of the particles is recorded allowing the determination of the direction of the original gamma ray. The particle energies are recorded by a NaI crystal beneath the spark chambers providing a measure of the original gamma-ray energy.
Energy range: 30 MeV to 30 GeV
Technique: high-voltage gas-filled spark chambers
Targets: diffuse g-ray emission, g-ray bursts, cosmic rays,
pulsars, and blazars.

47. Multi-wavelength spectra of 7 g-ray pulsars (pulsed emission)
Maximum of emission in hard X-ray and g-ray energies.
High-energy spectral cutoff.

48. Cutoff energy vs. surface B field

49. P-Pdot diagram

50. Lg vs. Lspin

51. Light curves: total vs. hard g-rays

52. g-ray pulsar observability

53. Seven highest-confidence g-ray pulsars

54. Multi-wavelength detections of high-energy pulsars

55. High-energy Lightcurves