A&A 390, L31-L34 (2002)
DOI: 10.1051/0004-6361:20020939
M. Lemoine
Institut d'Astrophysique de Paris, GReCO, CNRS, 98 bis boulevard Arago, 75014 Paris, France
Received 1 April 2002 / Accepted 19 June 2002
Abstract
It is shown that fusion of neutrons and protons to 4He nuclei occurs in ray burst
outflows in a process similar to big-bang nucleosynthesis in the early Universe. Only the surviving
free neutrons can then decouple kinematically from the charged fluid so that the multi-GeV neutrino signal
predicted from inelastic nuclear n-p collisions is significantly reduced. It is also argued that a
sizeable fraction of ultra-high energy cosmic rays accelerated in
ray bursts should be 4He
nuclei.
Key words: nucleosynthesis - gamma rays: bursts
Multi-wavelength observations of ray bursts in the past
decade have given increasing evidence in favor of the "expanding
fireball'' model (Paczynski 1986, 1990;
Goodman 1986, Shemi & Piran 1990;
Piran 2000; Mészáros 2002 for reviews), in
which a photon - pair plasma loaded with a small admixture of baryons expands
relativistically and converts the initial energy into baryon kinetic
energy. In the internal/external shock scenario (Rees & Mészáros 1992;
Rees & Mészáros 1994; Piran 2000; Mészáros 2002) this kinetic
energy is dissipated in shocks, giving rise to the
ray burst phenomenon. The identity of the
inner engine, the source of energy and the mechanism of collimation remain however
unknown.
One should not expect that only protons are injected in the
accelerating wind. As a matter of fact, all theoretical proposals for
the progenitor involve compact objects, e.g. neutron stars/black
hole mergers, or imploding massive stars, in which the baryon load
must be neutron rich. This has triggered recent interest in the study
of phenomenological consequences of neutron loading in ray
bursts (e.g. Derishev et al. 1999a,b; Bahcall & Mészáros 2000;
Fuller et al. 2000; Mészáros & Rees 2000).
One possible consequence is nucleosynthesis of neutrons and protons to
heavier nuclei. A recent study argued that nucleosynthesis should not
take place as the dynamical timescale (taken to be
s) is too short (Derishev
et al. 1999). However this particular value rather constitutes a strict lower
limit to the dynamical timescale since it implies a source size
3 km. This timescale is nonetheless bounded from above by
the shortest variability timescale observed
10-2 s (Piran 1999).
The purpose of this Letter is to show that nucleosynthesis to
4He can occur and is generically efficient, provided the dynamical
timescale 10-4 s (Sect. 2). Most
available neutrons and protons are then bound into 4He and only the
surviving neutrons can decouple kinematically from the charged fluid
component (Sect. 3). Thus this significantly weakens
the 5-15 GeV neutrino signal expected from
inelastic
collisions (Derishev et al. 1999; Bahcall & Mészáros 2000;
Mészáros & Rees 2000). Furthermore, if
ray bursts
accelerate particles to ultra-high energies
1020 eV
(Levinson & Eichler 1993; Vietri 1995; Waxman 1995; Waxman
2001 for a review), the ultra-high energy cosmic ray spectrum should comprise a
sizeable fraction of 4He nuclei (Sect. 3).
The fireball wind can be modeled as a pair plasma with luminosity
ergs/s injected into a solid angle
,
where
represents the effect
of beaming into two jets of half-opening angle
.
Observations suggest a roughly uniform
among
ray bursts and a varying
with mean value
(Frail et al. 2001; Panaitescu & Kumar 2001;
Piran et al. 2001); for a wind of typical duration
10 s, this yields a total equivalent isotropic output energy
1053 ergs. The bulk Lorentz factor is written
,
and
its saturation value
is the ratio of total
luminosity to baryon outflow (Mészáros et al. 1992). Injection takes place at radius
cm, with initial temperature
MeV
,
using 11/2 for the number of degrees of freedom (photons +
pairs). In the wind frame the ejecta is indeed similar to the early
Universe before big-bang nucleosynthesis (Shemi & Piran 1990). However the physical conditions
are very different: in a
ray burst ejecta, the comoving timescale
10-3 s
and the photon to baryon ratio
,
with
,
to be compared with
s and
for the early
Universe. A large dynamical timescale favors nucleosynthesis but a
high entropy acts against. Big-bang nucleosynthesis is also hampered
by a small neutron to proton ratio due to neutron decay and late
freeze-out of the weak interactions that interconvert protons and
neutrons. Here one expects equal n, p mass fractions
if baryons come from photodissociated nuclei, and neutron
decay is insignificant on a millisecond timescale. If
,
the rate of
weak interactions becomes larger than the fireball expansion rate and
the neutron to proton ratio achieves equilibrium independently of the
initial composition,
.
However this occurs for
,
i.e. for the
most highly beamed or compact
ray bursts, or those with the longest dynamical timescales
(see also Fuller et al. 2000 for a discussion of the impact of a
neutrino flux on the initial chemical composition of the outflow).
Adiabaticity during the acceleration phase implies that the bulk
Lorentz factor
and the temperature T evolve as
,
with r the radial coordinate in the lab
frame. If the flow can be approximated as quasi-one dimensional, the
relativistic Bernoulli equation further implies
(Blandford & Rees 1974). In particular Mészáros et al.
(1993) use
,
which is obviously valid only
in the limit
.
The exact numerical prefactor in
this relation depends on the details of the injection phase at radius
.
In what follows, the early behavior will be
phenomenologically modeled as
,
with
,
which approaches the twin exhaust model of
Blandford & Rees (1974), in which the flow is collimated by outside
pressure forces and accelerates to relativistic velocity through a de
Laval nozzle. A more exact solution requires solving the complex
problem of injection and collimation in the early phase of the
flow. Steps in this direction have been accomplished recently by
Levinson & Eichler (2000) who studied the hydrodynamic collimation of
a
ray burst outflow by a wind emanating from a torus. Their
solution indeed reproduces a nozzle with nearly constant cross-section
at small radii similarly to the Blandford & Rees (1974) model.
The dependence of the bulk Lorentz factor on r is important as it
gives the dynamical expansion timescale
.
Interestingly in this context the scaling
suggests that highly beamed outflows have a longer dynamical
timescale, hence should not show variations on short timescales due to
erasure of inhomogeneities on scales smaller than the sound horizon
.
Nevertheless, in order to
circumvent uncertainties related to the modeling of the early
evolution of the bulk Lorentz factor, all results that follow will be
shown as a function of dynamical timescale
instead of
.
This is justified as
and the entropy are the
two parameters that control the efficiency of
nucleosynthesis. Moreover it was checked that a spherically symmetric
wind with the simple expansion law
gives the same
result than a jet with respect to nucleosynthesis provided the
dynamical timescale and entropy are the same.
In the ultra-relativistic regime
the temperature
decreases exponentially fast with comoving time
.
In the outflow thermal equilibrium is ensured
between all species all along nucleosynthesis, and neutrons remain
coupled to protons through nuclear scattering with velocity averaged
cross-section
mb c (Derishev
et al. 1999). Decoupling occurs when
,
at temperature
,
i.e., well
after nucleosynthesis has taken place (see below).
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Figure 1:
Mass fractions vs. radius for the case
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In order to estimate the final abundances of nuclei synthesized, use
was made of a big-bang nucleosynthesis
numerical code whose time and
entropy evolutions were modified accordingly. This code accounts
successfully for synthesis of elements up to 7Be when compared to
other big-bang nucleosynthesis calculations. An example of the outcome
of nucleosynthesis is shown in Fig. 1 for a ray burst
with fiducial parameters
,
,
L50=1,
initial composition
,
(photodissociated
56Fe nuclei) and dynamical timescale
s. The final mass fractions of elements
produced are
(for 4He),
,
,
and other elements are
produced only in traces.
Although the numerical code used cannot deal with elements beyond mass
9, estimates of the triple
reaction and
Be
C necessary to bridge the mass gaps to
12C and beyond suggest that these three-body reactions do not
have sufficient time to produce heavy elements in abundance (see
however Sect. 3).
![]() |
Figure 2:
Contours of the final 4He mass fraction as a function of
photon to baryon number density
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The bulk of nucleosynthesis occurs at temperature
-0.2 MeV
,
corresponding to
depending on the parameters, and is over by
MeV
.
Nucleosynthesis is favored by a large
and a small
,
and it is thus more
efficient as
decreases (meaning higher baryon density), as
decreases, or as
increases if
(since a large expansion timescale is more critical to
nucleosynthesis than a small entropy). Obviously the efficiency of
nucleosynthesis is maximal for
initially, and less
efficient as this ratio departs from unity. A contour plot of the
final mass fraction of 4He produced is shown in Fig. 2 as a
function of
and
.
In this figure, it
was assumed that initially
.
It shows that
nucleosynthesis is efficient in most of parameter space: most
available neutrons and protons are bound into 4He provided the
dynamical timescale
s if
,
or
s
if
.
One can understand analytically
these results by comparing the expansion rate
to
the deuterium formation rate at temperature T (
):
.
Indeed deuterium production
occurs if
s. The dependence on the
entropy comes through the reverse photodisintegration rate which
inhibits the production of deuterium hence of heavier elements.
One expects nucleosynthesis to be more efficient than above if, for
instance, magnetic energy represents a significant fraction
of
the total energy. The photon number density hence
are then smaller by
while the initial temperature is
only reduced by
.
The present calculation also assumes
a homogeneous steady wind, but inhomogeneities in the wind,
e.g. shells or blobs, would increase the efficiency of
nucleosynthesis, as
where E is
the local energy content.
The feedback of nuclear binding energy release on the fireball
evolution has been neglected in the above calculations. This is
justified since the ratio of entropy density released to total entropy
density (assuming this latter is dominated by the photons) if all
neutrons and protons are instantaneously converted to 4He at
MeV
reads
,
with
and B4=28.3 MeV the 4He binding
energy. Note that in some rather extreme parts of parameter space,
the prerequisite conditions for successful r-process nucleosynthesis
could be satisfied: initial electron fraction
,
s,
and initial temperature
MeV
(e.g. Hoffman et al. 1997; Meyer & Brown 1997). This
presumably requires a highly beamed ejecta
from a
compact source
cm, with a significant fraction of
magnetic energy, and/or a low limiting Lorentz factor
.
This
latter condition can be fulfilled in the outer parts of a jet with
inhomogeneous baryon load as proposed recently (Zhang & Mészáros
2002). Successful r-process nucleosynthesis might thus be able to
operate in
ray bursts outflows, and further studies using
dynamical r-process codes appear mandatory. Other connections
between the site r-process site and
ray bursts have been
proposed by Eichler et al. (1989), Levinson & Eichler (1993)
and Cameron (2001).
An important consequence of successful nucleosynthesis to 4He is to
keep neutrons tied to protons and prevent their kinematical decoupling
when nuclear scattering becomes ineffective. As shown by Bahcall &
Mészáros (2000) and Mészáros & Rees (2000), such decoupling
occurs before the Lorentz factor has saturated to its limiting value
provided
and leads to
relative velocities between neutrons and protons
c; in turn
this leads to pion production in inelastic collisions and thus to a
5-15 GeV neutrino signal. Mészáros & Rees (2000) have further
shown that transverse diffusion of neutrons in inhomogeneously baryon
loaded fireballs can lead to an appreciable multi-GeV neutrino signal
for lower values of
.
However these studies assumed that
nucleosynthesis did not occur. In fact, the inelastic collisions occur
in these scenarii at large radii well after nucleosynthesis, hence
only the surviving free neutrons will be able to decouple.
Consider first an homogeneous fireball. In the absence of
nucleosynthesis, the neutrino signal is proportional to
,
and is thus maximal when
.
The neutron-4He collision cross-section is higher
than the n-p cross-section by the geometrical factor 42/3 but
there are only X4/4 4He nuclei per baryon (X4 denotes as
before the mass fraction). The neutrino signal produced after
nucleosynthesis (
)
is then a factor r of that produced
when X4=0 (no nucleosynthesis) with:
If the jet is inhomogenous, say made of an inner baryon poor jet
surrounded by a baryon rich outer shell (Eichler & Levinson 1999;
Mészáros & Rees 2001), the above discussion still applies if the
outer baryonic wind originates from the central engine with a
temperature
similar to that of the
central jet. In effect nucleosynthesis is then very efficient in the
outer shell since its Lorentz factor and entropy are both lower than
in the jet. If the surrounding shell is "cold'', or if it is neutron
rich and the inner jet proton rich (or vice-versa) one can circumvent
the above argument. However in the case of a jet punching its way
through a collapsar progenitor atmosphere Mészáros & Rees (2000)
have shown that it leads to a very low neutrino signal.
Finally, if ultra-high energy cosmic rays are accelerated in
ray bursts (Levinson & Eichler 1993; Vietri 1995; Waxman
1995; Waxman 2001 and references), one expects in the present context
a significant fraction of these particles to be 4He nuclei. However
nuclei are subject to photodisintegration with cross-section
a few mb for photon energies
20 MeV in the nucleus
rest frame, and during acceleration in internal shocks and production
of the
ray signal a fraction of them will be disrupted.
The calculation of the fraction of nuclei dissociated as a function of
their energy and shock radius is in itself similar to that performed
by Waxman & Bahcall (1997), Guetta et al. (2001) for the
production of a 100 TeV neutrino signal from pion production of
accelerated protons. At the highest energies
eV (observer frame), one can thus write the
photodisintegration rate
,
where
is the
photon energy density,
MeV is the
observed break energy in the
ray spectrum, and
is the inverse energy weighted
photodisintegration cross-section,
mb accounting for 1 and 2 nucleon loss with
respective branching ratios 80%, 20%. The optical depth to
photodisintegration thus reads
,
with
the
radius of emission.
Since features of temporal width
s are emitted at radius
,
this optical depth is unity where features of
width
s are emitted. High energy nuclei
accelerated in shocks at smaller radii are photodisintegrated, while
those accelerated at larger radii are unharmed. One can also show that
low energy nuclei are not photodisintegrated even at small radii. The
overall 4He nuclei energy spectrum is thus subject to
photodisintegration, acceleration and reacceleration in shocks and
adiabatic losses. Acceleration at large radii is in any case crucial
to overcome expansion losses (Waxman 2001). It thus seems reasonable
to expect that a sizeable fraction of 4He nuclei would be present
in the escaping ultra-high energy radiation and the measurement of the
chemical composition might provide further information on the nuclear
processes at work in the
ray burst. Detailed signatures of
these ultra-high energy 4He nuclei will be presented elsewhere.
Note added in proof: While this paper was being refereed, a similar study by Pruet et al. (2002) appeared, with similar conclusions as to the efficiency of nucleosynthesis.
Acknowledgements
It is a pleasure to thank R. Mochkovitch, G. Pelletier and E. Waxman for useful comments and suggestions.