ETAPA DE FINALIZACIÓN

Observational results suggest a cut-off in the energy spectrum at around1 10GeV. The cut- off is explained differently whether emission comes from the inner magnetosphere or the outer magnetosphere.

The Outer Magnetosphere Case: Curvature Radiation Cooling

Even if the potential drop can be10 15

V (see Sect. 2.6.3), electrons are not necessarily accelerated up to 10

15

eV because they lose energy mainly through curvature radiation. The maximum energy of the electron determines the cut-off energy of the gamma-ray energy spectrum in the outer magnetosphere case.

2.7 Non-thermal Radiations in the Pulsar Magnetosphere 51

Figure 2.17: Basic energy spectra of a young pulsar. The emission region is assumed to be in the outer magnetosphere. Thermal (kT), synchrotron (Sy), and curvature (CR) radiation explain the entire energy spectrum. The dashed line (CS) is inverse Compton emission which may play a role in specific conditions. A power law spectrum was assumed for electrons, which may not be the standard case. The dotted line is the spectrum of curvature radiation from monoenergetic electrons. Figure adopted from [155].

The energy loss in unit time would ben E

, where

n is the number of emitted photons per unit of time and

E

is their mean energy. From Eq. 1.62 and Eq. 1.64, n E e 4 B 2 urv 2 m 2 e 3 = e 2 4 R 2 urv (2.28) 410 3 1 R urv =m ! 2 4 eV =s (2.29)

The maximum would be derived as follows: n E = eE k (2.30) max = 50 eE k (eV =m) ! 1=4 q R urv =m (2.31) SubstitutingE k =310 6 V/cm, andR urv =10 8

cm, which are reasonable values for the Crab pulsar, one can obtain max

' 210 7

(10 TeV). It is consistent with many other calculations (see e.g. [155], [104] and [93]). From Eq. 1.62, the energy of the curvature radiation photon from the maximum energy electron would be6:5GeV.

It should be noted that, with E k

= 310 6

V/cm, the electron energy of 10 TeV can be achieved with 33 km of acceleration. Considering the co-rotation radius of the Crab pulsar (R

L

=1500km), accelerated electrons should be nearly monoenergetic at around the maximum energy.

Inner Magnetosphere Case: Pair Creation by Strong Magnetic Fields

In the inner magnetosphere, the maximum Lorentz factor of the accelerated electrons is similar to the one in the outer magnetosphere (see e.g. [91]). On the other hand, the curvature of the

52 2. Pulsars

magnetic field is smaller than that in the outer magnetosphere. Therefore, one might expect even higher energy of photons via curvature radiation from the inner magnetosphere than from the outer magnetosphere. However, the strong magnetic field can absorb the high energy photons via the magnetic pair creation process, which actually determines the cut-off in the observed gamma-ray spectrum.

One photon in a field-free vacuum cannot produce an electron-positron pair even if the photon energy is higher than 2m

e

2

, because both energy and momentum cannot be conserved at the same time. However, if the magnetic field participates in momentum and energy transfer, it becomes possible to create an electron-positron pair from a single photon. The energyE

Bwhich has to be transferred from the magnetic field, would be estimated as follows:

E B 2E e E 2m 2 e 4 E (2.32) assuming that the high energy photon with the energy E

and momentum P

produce an elec- tron and a positron with the energy of E

e and the momentum 1=2P

, parallel to the photon momentum. On the other hand, the cyclotron energy states of an electron under magnetic field are discrete

h!

g

(N+1=2) (2.33)

Thus, the electron and the positron gain energy from the magnetic field by h!

g, where ! g = eB=m e is the gyro-frequency. h ! g is equal to m e 2

if the magnetic field strength is the critical magnetic field strengthB

r =m 2 e 3 =eh=4:410 13

G. From this argument and Eq. 2.32 2m 2 e 4 E <m e 2 B ? B r (2.34) would roughly be the condition for the magnetic pair creation, where B

? is the magnetic field strength perpendicular to the electron motion. Therefore, one can define a useful dimensionless parameter = E 2m e 2 B ? B r (2.35) If<< 1then pair creation should not happen while it should happen if1or>1.

The detailed calculations were first done independently by Toll [182] and Klepikov [113] and later confirmed by many authors such as [72] and [185]. Among them, T. Erber [72] provided a useful calculation for the attenuation coefficient as a function of. It shows that gamma-rays can escape the pulsar magnetosphere when < 0:1 is fulfilled throughout the propagation. In other words, from Eq 2.35, the following condition:

E m e 2 < 0:2B r B ? = 0:2B r Bsin (2.36) must be fulfilled throughout the trajectory in the pulsar magnetosphere, where is the angle between the photon propagation direction and the magnetic field. In a pulsar magnetosphere,

2.7 Non-thermal Radiations in the Pulsar Magnetosphere 53

gamma-ray photons are emitted tangentially to the magnetic field ( =0). As long as a photon travels parallel to the magnetic fields, pair creation will never happen. However, since the dipole magnetic fields are curved, increases with distance. On the other hand, the magnetic field strength rapidly decreases with the distance from the stellar surface.

The maximum energyE max

(r)of gamma-rays that were emitted at distancerfrom the pulsar and reach the Earth can be roughly estimated as follows: one can assume that the magnetic fields are uniformly curved with the same strength in a scale of the pulsar radius R

0 = 10

6

cm and that, due to the rapid decrease of the field strength, a photon which could travel up to the distance R

0 will never cause the magnetic pair creation afterwards. The radius of field curvature near the last closed line can be approximated as

(r)= p R L r = q Pr=(2)(see [34]). If a photon is emitted tangentially to the magnetic field, after traveling byR

0, then sin ' R 0 = . Therefore, from Eq. 2.36 E max (r) = 0:2B r m e 2 Bsin ' 0:2B r m e 2 B 0 (R 3 0 =r 3 )(R 0 = ) (2.37) ' 7 p P r R 0 7=2 B r B 0 MeV (2.38)

Accurate calculations, taking into account the effect of general relativity can be summarized in a similar formula (see [90], [33] and [34]):

E max (r) ' 40 p P r R 0 7=2 B r B 0 MeV (2.39)

In the case of the Crab pulsar, substitutingB 0

=410 12

G andP =0:034sec, thenE max

(r = R

0

)will be 0.3 GeV. This value can change depending on the emission distancer.

The photon splitting process, which is forbidden in a field-free vacuum by the Furry theorem (see [81])

!+ (2.40)

is allowed in the presence of a strong magnetic field. This process, therefore, limits the attainable energy of photons. Photon splitting can dominate magnetic pair creation in a certain energy range only whenB

0

> 0:3B

r (see [90]), which is rarely the case for pulsars except for some special ones such as PSR B 1509-58 (B

0

'310 13

G).