2.3. Neutron stars
2.3.1. Magnetohydronamic acceleration of iron nuclei in pulsars
Following earlier ideas [165] (4), Blasi et al. [168] (5) have shown that young magnetized neutron stars in our own Galaxy may be one such astrophysical systems.
Neutron stars - endpoints of stellar evolution - begin their life
rotating rapidly ( ~
3000 rad s-1) and with large surface magnetic fields
(BS
1013
G). The dipole component decreases as the cube of the distance from the
star's surface
B(r) = BS(RS /
r)3, where the radius of the star is
RS
L. At the light cylinder Rlc = c /
~ 107
3k-1 cm,
where
3k
/ 3000 rad
s-1, the dipole field cannot be casually maintained, the
field is mostly azimuthal, with field lines spiraling outwards
[173].
Inside the light cylinder,
the magnetosphere corotates with the star, and the density
(mostly iron peak elements formed during the supernova event that
were stripped off the surface due to strong electric fields) has
the Goldreich-Julian value, nGJ(r) =
B(r)
/ (4
Z e c), where
c is the speed of light
[174]. The
behavior of the plasma outside the light cylinder is still not yet
fully understood
[175,
176,
177,
178],
although some analytical and numerical studies show the development of
kinetically dominated relativistic winds (see e.g.,
[176]).
Blasi et al. [168]
assumed that the magnetic field in the wind zone decreases as
B(r)
BlcRlc / r. For
surface fields of BS
1013 B13 G, the field at
the light cylinder is
Blc = 1010 B13
3k3.
G. The maximum energy of particles that can be contained in the
wind near the light cylinder is
![]() |
(16) |
where Z26
Z / 26.
The typical energy of the accelerated CRs, Ecr,
can be estimated by considering the magnetic energy per ion at
the light cylinder
Ecr
Blc2 /
8
nGJ. At
the light cylinder
nGJ = 1.7 × 1011 B13
3k4 / Zcm-3 which gives
![]() |
(17) |
In this model, as the star spins down, the energy of the cosmic ray particles ejected with the wind decreases. The total fluence of UHECRs between energy E and E + dE is
![]() |
(18) |
where the particle luminosity is
![]() |
(19) |
and < 1 is
the efficiency for accelerating particles at the light cylinder. For
a spin down rate dominated by magnetic dipole radiation, given by
I
=
- BS2 RS6
4 /
6c3 where
I = 1045 g cm2 is the moment of inertia,
the time derivative of the spin frequency is
= 1.7
× 10-5 B132
3k3 s-1, and Eq. (17) gives
![]() |
(20) |
Substituting in Eq. (18), the particle spectrum from each neutron star is
![]() |
(21) |
Taking the confining volume for these particles to be
Vc and the lifetime for confinement to be
tc, the UHECR density is
n(E) =
N(E) tc /
Vc, and
the flux at the surface of the Earth is
F(E) = n(E) c / 4. For a characteristic
confinement dimension of R = 10 R1 kpc,
Vc = 4
R3 / 3
and tc = QR / c, where Q > 1 is a
measure of the how well the UHECR are trapped. With this in mind, the
predicted UHECR flux at the Earth is
[168]
6
![]() |
(22) |
Here, the fact that neutron stars are produced in our Galaxy at a
rate 1 / , where
100
2 yr, but that not
all them (but rather only a fraction
) have the required
magnetic fields, initial spin rates and magnetic field geometry to
allow efficient conversion of magnetic energy into kinetic energy
of the flow, were taken into account. By comparing with
observations, the required efficiency factor,
, can
be estimated, and it only needs to be
4 ×
10-6 Q-1.
The condition that a young neutron star could produce the UHECRs (7) is that Ecr exceeds the needed energy when the envelope becomes transparent (i.e. before the spinning rate of the neutron star decreases to the level where the star is unable to emit particles of the necessary energy), Ecr(ttr) > 1020 E20 eV. This translates into the following condition [168] (8)
![]() |
(23) |
Eq. (23) translates in turn into upper bounds on the
surface magnetic field strength and the star initial spin period
Pi = 2 /
i,
![]() |
(24) |
and
![]() |
(25) |
For M1 = 2 and
E20 = E51 = Z26 =
1, Eq (24) gives
B13 < 15.4, whereas Eq. (25) leads to
Pi
10 ms, not very
restrictive values for a young neutron star, see
for example the Parkes Multibeam Pulsar Survey.
(9)
Magnetars, neutron stars with surface dipole fields on the order of 1015 G [183, 184, 185, 186, 187], were also proposed as plausible sites for the generation of UHECRs [188]. Assuming that they occur occur in all galaxies which form massive stars (then avoiding the large-distance GZK problem), and thus that the UHECR are arriving from outside our own Galaxy, the luminous infrared galaxies are preferred sites to search for a magnetar origin of CRs (see [189] and below the discussion in Section 2.7).
The magnetar model for the acceleration of UHECRs proposed by Arons [188] is a variant of that using neutron stars outlined in the previous Section. The theory predicts an injection of charged particles with maximum energy
![]() |
(26) |
where B* is a magnetar's surface magnetic
field, B15 = B* /
1015 Gauss,
i =
104
4
s-1 is the initial angular velocity of the neutron star,
R* is the
stellar radius, and c is the speed of light. The initial
rotation period is Pi = 0.64 /
4 ms (if
Z = 1 - 2, one requires
Pi < 2 - 3 ms for the model to be viable).
The ions actually gain their energy in the relativistic wind
electromagnetically expelled from the neutron star at distances
r larger than the radii of the star and its magnetosphere. This
avoids catastrophic radiation losses; the electric potential in
the wind is
rE = rB = .
As the star spins down, as in the model by Blasi et al., the
voltage and the maximum particle energies decline. Summing over the
formation and spindown event, one finds a per event injection
spectrum proportional to f (E) = E-1[1 +
(E / Eg)]-1 for
E < Emax. Here Eg measures
the importance
of gravitational wave losses (calculated for a star with static
non-axisymmetric quadrupole asymmetry) in spinning the star down.
When they exert torques larger than the electromagnetic torque,
the star spends less time at the fastest rotation rates (i.e. less
time accelerating the highest energy particles), thus causing a
steepening in the spectral slope at the highest energies. If the
star has an internal magnetic field even stronger than the already
large surface field, equatorial ellipticities
e
in excess of 10-3 can exist, in which case
Eg would be less than
Emax. Three cases of energy loss due to
gravitational radiation (GR) were considered in the model: no GR
loss (
e = 0,
Eg =
); moderate GR loss
(
e = 0.01,
Eg = 3 × 1020 eV); strong GR loss
(
e = 0.1,
Eg = 3 × 1018 eV). If one assumes that
all magnetars have exactly the same
starting voltage (1022.5 V), then the model predicts that the
spectrum E3J(E) should rise with
E above the energy Eg = 2.8 ×
1020 eV, where the GZK loss rate becomes
approximately energy independent (unless the gravitational wave
losses are large)
[188].
This prediction will certainly be testable with PAO.
The total number of particles injected per event is
![]() |
(27) |
for a stellar radius of 10 km and a moment of inertia
I = 1045 cgs. The rate at which galaxies inject UHECR
into the universe in this model then is
cr =
magfast
Ni ngalaxy, where
ngalaxy
0.02 Mpc-3
[190],
Ni is given by Eq. (27), and
magfast
is the birth rate of rapidly rotating magnetars per
galaxy. Multiplication of the source spectrum
q(E)
cr f
(E) by the energy
dependent GZK loss time yields a spectrum received at the Earth in
reasonable accord with the existing observations of UHECR
if
magfast
10-5
yr-1
[188].
That fast magnetar birth rate lies
between 1% and 10% of the total magnetar birth rate inferred for
our galaxy, and about 0.1% of the total core collapse supernova
rate in average star forming galaxy, ~ 10-2 yr-1
[191].
2.3.3. UHECRs from a pulsar in Cygnus OB2?
As discussed in Sec. 2.1, some evidence
may be emerging for a CR accelerator in the Cygnus spiral arm. The HEGRA
experiment has detected an extended TeV
-ray source
in the Cygnus region with no clear counterpart and a spectrum not easily
accommodated with leptonic radiation
[139].
The difficulty to accommodate the spectrum by conventional
electromagnetic mechanisms has been exacerbated by the failure of
Chandra and VLA to detect significant levels of X-rays or
radiowaves signaling acceleration of any electrons
[140].
Especially intriguing is the
possible association of this source with part of Cygnus OB2 itself
[138],
a cluster of several thousands young, hot OB stars with a total mass of
~ 104
M
[192].
At a relatively small distance to Earth,
1.7 kpc, this is the
largest massive Galactic association. It has a diameter of
60 pc and a
core radius of ~ 10 pc. The typical main sequence evolution
lifetime of massive O stars is
(Myr) and a few tens Myr
for massive B stars. Since the O-star population should pass
through the Wolf-Rayet phase and explode as supernovae, very fast
pulsars are expected to be born in explosions of these massive
stars at a rate of about one every ten thousand years.
Apart from the mentioned interpretation of a hadronic production
of the TeV radiation within the winds of outlying OB stars of Cyg
OB2 [138],
it was recently put forward that the TeV
emission reported by HEGRA and the CR anisotropy observed at
about 1018 eV in the direction of the Cygnus region can be
related to a young pulsar and its pulsar wind nebulae (PWN), born
in the Cygnus OB2 association a few ten thousands years ago
[193].
The TeV -ray
emission would originate in the PWN as a result of interactions of high
energy hadrons and/or leptons, whereas there would be a directional CR
signal due to neutrons that are dissolved from heavy nuclei
accelerated by the pulsar.
(10)
Within this model, however,
it is hard to explain the absence of counterparts at lower (EGRET)
energies at the location of the TeV source, as well as the absence
of a stronger X-ray source.
(11) However,
disregarding these difficulties, an interesting consequence for CR
physics can be inferred.
As already discussed
in Sec. 2.3.1, heavy nuclei can attain ultra high
energies through magnetic sling shots inside the neutron star wind zone.
The small part of nuclei, which escaped from the pulsar wind
nebula, are likely to be captured by strong magnetic fields of
dense regions of the OB association. Magnetic field strengths in
dense molecular clouds are expected to be ~ 1 mG
[195].
Thus, nuclei with E / Z ~ 1 EeV
propagate attaining Bohm diffusion
[193].
The resulting time delay of several thousand years
[196]
produces a steepening of the characteristic power law injection spectrum,
E-2, of the pulsar nebula.
(12)
In their random traversal of the OB association, the nuclei
undergo photodisintegration on the far infrared thermal photon
population and liberate neutrons.
![]() |
(28) |
where T is the kinetic temperature of the molecular cloud,
x
E
/ T, and
E
is the target photon energy in degrees K.
The relevant photon energy range is established via standard kinematics
of the photodisintegration process,
![]() |
(29) |
where 5 < E*MeV < 25 is the
energy region for giant dipole resonance contribution in the nucleus
rest frame, and EA / A
EN,
PeV is the lab frame energy/nucleon. For
EN, PeV ~ 103, Eq. (29)
leads to x1 = 50 / T and
x2 = 250 / T, with T in degree K.
Molecular clouds with HII regions have temperatures between 15 and 100
[197],
thus taking an average photodisintegration cross section of 40 mb,
(13) we find
that the nucleus
mean free path lies between 0.80 and 380 pc. This corresponds to a
reaction time between 4 to 1500 yr << time delay, alowing
sufficient neutron production to explain the anisotropy
[54].
The Galactic anisotropy observed by the various collaborations
spans the energy range 0.8 to 2.0 EeV. The lower cutoff specifies
that only neutrons with EeV energies and above have a boosted
c n
sufficiently large to serve as Galactic messengers.
The decay mean free path of a neutron is c
n
n = 10
(En / EeV) kpc, the lifetime being boosted from its
rest-frame value
n = 886
seconds to its lab value via
n =
En / mn. Actually, the broad scale
anisotropy from the direction of the GP reported by Fly's Eye Collaboration
[51]
peaks in the energy bin 0.4 - 1.0 EeV, but
persists with statistical significance to energies as low as 0.2 EeV.
This implies that if neutrons are the carriers of the anisotropy,
there needs to be some contribution from at least one source
closer than 3 - 4 kpc. Interestingly, the full Fly's Eye data include a
directional signal from the Cygnus region which was somewhat lost in
unsuccessful attempts
[198,
199] to
relate it to
-ray
emission from Cygnus X-3. The upper cutoff reflects an important feature
of photodisintegration at the source: heavy nuclei with energies
in the vicinity of the ankle will fragment to neutrons with
energies about an order of magnitude smaller. To account for the
largest neutron energies, it may be necessary to populate the heavier
nucleus spectrum in the region above the ankle.
(14) This is not a
problem - one fully expects the emerging harder extragalactic spectrum to
overtake and hide the steeply falling galactic population. It is
not therefore surprising that in order to fit the spectrum in the
anisotropy region and maintain continuity to the ankle region
without introducing a cutoff, the AGASA Collaboration required a
spectrum
E-3 or steeper
[49].
For every surviving neutron at ~ EeV,
there are many neutrons at lower energy that decay via
n p +
e- +
e.
The proton is bent by the Galactic magnetic
field, the electron quickly loses energy via synchrotron radiation, and the
e travels
along the initial neutron direction, producing a directed TeV energy beam
[54].
The sensitivity of forthcoming neutrino telescopes to this signal is
discussed in Sec. 4.2.
4 Even earlier ideas relating UHECRs with neutron stars can be found in Refs. [166, 167], although these attempts have failed at either reaching the highest energies, or reproducing the spectrum, or reproducing the apparent isotropy of the arrival directions of UHECRs. Back.
5 See [169, 170], also [171], for related research. See [172] for yet another model of UHECR generation in neutron stars, involving planetoid impacts. Back.
6 Note that the predicted
spectrum of Eq. (22) is very flat,
= 1,
which would agree only with the lower end of the plausible range of
observed at ultra-high energies. Propagation effects can produce
an energy dependence of the confinement parameter Q and,
correspondingly, a steepening of the spectrum toward the middle of
the observed range
1
2.
Back.
7 The gyroradius of these UHECRs in a Galactic field of µG strength is considerably less than the typical distance to a young neutron star (~ 8 kpc). Since the iron arrival distribution at 1020 eV probes similar trajectories to protons at a few times 1018 eV, Galactic iron nuclei would show a nearly isotropic distribution with a slight correlation with the Galactic center and disk, at higher energies. Back.
8 A supernova that imparts
ESN = 1051
51 erg to the
stellar envelope of mass Menv = 10 M1
M
is
considered. Also, that the condition for
iron nuclei to traverse the supernova envelope without significant
losses is that
100 g cm-2
[179].
Back.
9 The
Parkes multibeam pulsar survey is a large-scale survey of a narrow
strip of the inner Galactic plane (|b | < 5°, 260°
< l < 50°, see
[180]
and references therein). It
has much greater sensitivity than any previous survey to young and
distant pulsars along the Galactic plane, and it has resulted in
the detection of many previously unknown young pulsars,
potentially counterparts of unidentified
-ray sources,
e.g. [181,
182].
Back.
10 A similar scenario would explain the anisotropy spot towards the Galactic Center [194]. Back.
11 Indeed, location problems may arise as well: the PWN size (given that the TeV source size is 6 pc if indeed located at 1.7 kpc, and that the source is diffuse) would make the PWN substantially larger than both Vela's PWN (~ 0.1 pc at 250 pc) and Crab's (~ 1 pc). Additionally, if one were to associate the nearby EGRET source 3EG 2033+4118 with the putative pulsar itself, it is unclear whether the PWN hypothesis for the TeV source would imply that it is only a one sided PWN (Y. Butt, private comunication). Back.
12 Note that once diffusion has been established, additional Rayleigh steps in the Galactic magnetic field do not change the spectral index (~ -3) significantly. Back.
13 The photoabsorption cross section
roughly obeys the Thomas-Reiche-Kuhn sume rule, i.e.,
d =
60NZ / A mb-MeV
[97].
Back.
14 To produce the highest energy
neutrons (with energy
1018.2
eV) via photodisintegration of medium mass (say, A = 10 - 20)
nuclei, one needs primary particles with energies
1019.2
eV. For a falling spectrum
E-3, the medium mass nucleus population at
1019.2 eV is roughly 3 orders of magnitude smaller than the
diffuse flux at 1018.2 eV. This means that
it is about 1.6 orders of magnitude smaller than the reported CR excess
(which is about 4% of the diffuse flux, see
Sec. 1.3). However, since
each nucleus produces roughly A - Z neutrons, the average
number of liberated neutrons is on the same order of magnitude,
1.6 - log10(A - Z), than the obseved neutron
population at
1018.2 eV.
Back.