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From (11)) we deduce that P defines an average openiug auele of 77. roughly. | From \ref{cone}) ) we deduce that $P$ defines an average opening angle of $7^o$, roughly. |
We can then interpolate the corresponding LGRD rate or ΙΙΙΟΙ of eveuts in Tables 5 and 6. respectively, which o1 -pred.yyT ⊀∟≺∶↓∖≯∐∿∟≻⋅↱⊐⋝∖↓∩∣∫⇀≼∶↕⊰↕≧↴∖↴⋅∏⋅∶↴∙⊾⋜↧↕⋜⋯⋅↖↽∪↥⋅⊀∖↕∟⋝≺⊔∶↓↸∖⋡∐∿⊔≩∩∣∣ ⇁⋅⇂⋅⋅ LORB events in the past 11 vears. | We can then interpolate the corresponding LGRB rate or number of events in Tables 5 and 6, respectively, which gives $N_{\rm LGRB}^{\rm pred}\!\sim\!(2-5)\times10^{-7}$ LGRBs/yr/galaxy or $N_{\rm LGRB}^{\rm pred}\!\sim\!3-6$ LGRB events in the past 11 years. |
On account of the uncertainties that may affect not only our knowledge of the opening angele of beams or their distiibution. but also the estimated rates of observed LORB eveuts in the local and Unbble Universe. we may consider that the counts lamited to the spectral type interval BO-Bl to O8 predict rates that overlap rather well with the observed ones. | On account of the uncertainties that may affect not only our knowledge of the opening angle of beams or their distribution, but also the estimated rates of observed LGRB events in the local and Hubble Universe, we may consider that the counts limited to the spectral type interval B0-B1 to O8 predict rates that overlap rather well with the observed ones. |
This is the first finie. that au estimate of he LCRDs rates is done on he basis of a well-defined population of stars. 1.6.. Dorn fast rotators with low initial Very recently. 7. has tried to reproduce the observe rates of LCRDs with models of WR stars. | This is the first time that an estimate of the LGRBs rates is done on the basis of a well-defined population of stars, i.e., born fast rotators with low initial Very recently, \citet{georgy2009} has tried to reproduce the observed rates of LGRBs with models of WR stars. |
Although they οἱained ligher values of LOGRBs rates than wha observations ποσα to indicate. these authors couclude tha only a fraction of WC/WO stars amoug those that initially were rapid rotators could produce ganuua ray events. | Although they obtained higher values of LGRBs rates than what observations seem to indicate, these authors conclude that only a fraction of WC/WO stars among those that initially were rapid rotators could produce gamma ray events. |
This fast-rotation areunmeutOo may then focus some mmteres ∪∐↴∖↴↑⋜∐⋅↴∖↴↖↖⇁↕↑∐↑∐↸∖↸⊳∐⋜∐⋅⋜↧↸⊳↑↸∖↥⋅↕↴∖↴↑↕↸∷∖↴≼∐∖↴↸⊳∏↴∖∷∖↴↸∖≼⊔∐↑∐↸∖∐⋅↸∖↴∖↴↸∖∐ The counts of Oc/Be candidates for LORB oxogeuitors as they were done here. however. represents a mere lower limit. | This fast-rotation argument may then focus some interest on stars with the characteristics discussed in the present The counts of Oe/Be candidates for LGRB progenitors, as they were done here, however, represents a mere lower limit. |
Stars with masses correspondius to rest DOU-D1 specval types that rotate at οο,z7 ollowiug the requiremieuts shown in Fie. | Stars with masses corresponding to rest B0-B1 spectral types that rotate at $\Omega/\Omega_{\rm c}\gtrsim0.7$ following the requirements shown in Fig. |
9. then display apparent spectral types cooler by at least one subspectral vpe. | \ref{fig3} then display apparent spectral types cooler by at least one subspectral type. |
A prudent lower limit set at spectral type D2. would revertheless cuhance our predictions by a rough factor 2. as seen from counts displaved iu (6)). or Tables 5 and 6.. ILo | A prudent lower limit set at spectral type B2, would nevertheless enhance our predictions by a rough factor 2, as seen from counts displayed in \ref{nbbe}) ), or Tables \ref{tableLGRBsmeth1} and \ref{tableLGRBlocal}. |
wever. massive fast rotators without ciission lines should also be considered potential LGRBs progcuitors as we see in Sect. 5.2.. | However, massive fast rotators without emission lines should also be considered potential LGRBs progenitors as we see in Sect. \ref{frwel}, |
so that the final predicted rates may widely surpass the observed If the LORBs could also occur in a binary svsteiu (??) under the same conditions as fast rotation and chemical colposition. the rates we estimated above should be multiplied by 1/0.7. | so that the final predicted rates may widely surpass the observed If the LGRBs could also occur in a binary system \citep{cantiello2007, vdh2007} under the same conditions as fast rotation and chemical composition, the rates we estimated above should be multiplied by 1/0.7. |
ILowever. the results from ? sugecst that the binary evolutionary path is unlikely to produceLORBs when they are associated with iietal-poor galaxies MOM | However, the results from \citet{langer2006} suggest that the binary evolutionary path is unlikely to produceLGRBs when they are associated with metal-poor galaxies \citep{kewley2007}. |
The ZAMS rotational velocities of Oe/Be stars in the MW form the of taila true rotational velocity distribution 9(V) that starts at a velocity lut eiven by Vj230.(MAL) (?).. | The ZAMS rotational velocities of Oe/Be stars in the MW form the tail of a true rotational velocity distribution $\Phi(V)$ that starts at a velocity limit given by $V_{\rm L}\!=\!30\times(M/M_{\odot})$ \citep{zorec2007}. |
The tail of the velocity distribution formed by the Oc/Be stars can be represented with a Gaussian function. whose dispersion ds also miass-depencdent aud is roughly eiven by ay—28οAM. (2).. | The tail of the velocity distribution formed by the Oe/Be stars can be represented with a Gaussian function, whose dispersion is also mass-dependent and is roughly given by $\sigma_{\rm V}\!=\!28\times(M/M_{\odot})$ \citep{zorec2007}. . |
This niplies that there are rotators tha can onlv acquire the Be phenomenon at the very end of their MS evolutionary phase. which | This implies that there are rotators that can only acquire the Be phenomenon at the very end of their MS evolutionary phase, which |
calculated. | calculated. |
Here and in all the subsequent analysis, we use simple x? statistics. | Here and in all the subsequent analysis, we use simple $\chi^2$ statistics. |
The problems of using the x? criterion for low photon counting statistics are circumvented following the recipe of Churazovetal.(1996).. | The problems of using the $\chi^2$ criterion for low photon counting statistics are circumvented following the recipe of \cite{1996ApJ...471..673C}. |
Namely, the standard deviation associated with the count rate in a given spectral, spatial or time bin is evaluated using the mean count rate averaged over a large number of “nearby” similar bins. | Namely, the standard deviation associated with the count rate in a given spectral, spatial or time bin is evaluated using the mean count rate averaged over a large number of “nearby” similar bins. |
Typical values of x? per degree of freedom for our models are z1.01. | Typical values of $\chi^2$ per degree of freedom for our models are $\approx 1.01$. |
Given the large number of degrees of freedom (~250000= number of detectors times the number of observations) and the very low signal-to-noise ratio of the annihilation signal in individual observations, this value is not a useful indicator of the “absolute” quality of the model. | Given the large number of degrees of freedom $\sim 250000=$ number of detectors times the number of observations) and the very low signal-to-noise ratio of the annihilation signal in individual observations, this value is not a useful indicator of the “absolute” quality of the model. |
Instead, the change of x? can be used to compare different models or place constraints on the model parameters. | Instead, the change of $\chi^2$ can be used to compare different models or place constraints on the model parameters. |
The resulting contours of χ for both models (as a function of the parameters W; and W;) are plotted in Fig. 4.. | The resulting contours of $\chi^2$ for both models (as a function of the parameters $W_l$ and $W_b$ ) are plotted in Fig. \ref{fig:g2}. |
'The dashed lines correspond to the one-component model (pure 2D Gaussian), while the solid lines correspond to the two-component model Gp(I,b) (see eq. 2)). | The dashed lines correspond to the one-component model (pure 2D Gaussian), while the solid lines correspond to the two-component model $G_D(l,b)$ (see eq. \ref{eq:g2}) ). |
The black dots mark the positions of x? minima. | The black dots mark the positions of $\chi^2$ minima. |
Contours are spaced by Ax?— 3. | Contours are spaced by $\Delta \chi^2=3$ . |
The best-fitting values of the widths are x (if the one-component model is used) and ~8° x (for the two-componentmodel)?. | The best-fitting values of the widths are $\times$ (if the one-component model is used) and $\sim$ $\times$ (for the two-component. |
. Clearly, for the pure 2D Gaussian model the data suggest a significant flattening of the distribution towards the plane. | Clearly, for the pure 2D Gaussian model the data suggest a significant flattening of the distribution towards the plane. |
If an additional component, extended along the plane, is included (the two-component model in eq. 2)), | If an additional component, extended along the plane, is included (the two-component model in eq. \ref{eq:g2}) ), |
then the best-fitting central Gaussian is much more symmetric in | and b. | then the best-fitting central Gaussian is much more symmetric in $l$ and $b$. |
The improvement in x? for the two-component model compared to the one-component model is ~40. | The improvement in $\chi^2$ for the two-component model compared to the one-component model is $\sim 40$. |
The position of the minimum for the two-component model does not depend much on the width of the extended component over b. | The position of the minimum for the two-component model does not depend much on the width of the extended component over $b$. |
We tried for the second component Wp=2°,, Wp=6? and Wp=10° as well as an exponential shape e instead of the Gaussian, and got essentially the same two}best-fitting parameters x )) for the central Gaussian. | We tried for the second component $W_D=2$, $W_D=6$ and $W_D=10$ as well as an exponential shape $\displaystyle e^{-\left\{\frac{|b|}{\rm
W_D}\right\}}$ instead of the Gaussian, and got essentially the same best-fitting parameters $\times$ ) for the central Gaussian. |
We emphasize here that the presence of the second component does not necessarily imply that the Galactic disk is "detected", but rather that a single symmetric Gaussian/exponential component is not a perfect description of the data. | We emphasize here that the presence of the second component does not necessarily imply that the Galactic disk is "detected", but rather that a single symmetric Gaussian/exponential component is not a perfect description of the data. |
An exponential shape eUsel.WeJ with W;~3° and W,~2° can also be used to describe the central component (see Table 1)) instead of a Gaussian. | An exponential shape $\displaystyle e^{-\left\{\frac{|l|}{\rm
W_l}\right\}-\left\{\frac{|b|}{\rm W_b}\right\}}$ with $W_l\sim$ and $W_b\sim$ can also be used to describe the central component (see Table \ref{tab:templates}) ) instead of a Gaussian. |
In fact, thelargest improvement in the x? is achieved when both the central and disk components are described as exponential functions | In fact, thelargest improvement in the $\chi^2$ is achieved when both the central and disk components are described as exponential functions |
1)) aud locally iu the spacetime plots (Figures 6 aud 12)). | \ref{t:sims}) ) and locally in the spacetime plots (Figures \ref{f:st128} and \ref{f:st64r4}) ). |
This hypothesis will be addressed further in future research. comparing in detail the results preseuted here with those from unstratified runs both with aud without mean fields. | This hypothesis will be addressed further in future research, comparing in detail the results presented here with those from unstratified runs both with and without mean fields. |
Due to our choice of periodic vertical boundaries. aud our use of simplified thermodynamics. we have larecly avoided detailed discussion of observational iurplicatious. | Due to our choice of periodic vertical boundaries, and our use of simplified thermodynamics, we have largely avoided detailed discussion of observational implications. |
Such questions are eeucrallvy better addressed by stucdies which include more plivsicallv realistic vertical boundary conditions (0.5.Miller&Stone2000) or more realistic thermodynamics. inchiding the treatment of radiatiou (e.c.Turner2001:Tiroseetal.2006). | Such questions are generally better addressed by studies which include more physically realistic vertical boundary conditions \citep[e.g.][]{ms00} or more realistic thermodynamics, including the treatment of radiation \citep[e.g.][]{tur04,hks06}. |
.. Towever. it is worth briefly notius that our work confirms some iurportant results of earlier studies (seee.g.Draudenu- 2000). | However, it is worth briefly noting that our work confirms some important results of earlier studies \citep[see e.g.][]{bra95,sto96,ms00}. |
. Figures 3. aud 6 show that a sieuificant fraction of the magnetic energv in these sinuulations resides im arge scale mmaeuctic fields that rise buovautlv to the low density regions above the disk midplane. | Figures \ref{f:pscomp} and \ref{f:st128} show that a significant fraction of the magnetic energy in these simulations resides in large scale magnetic fields that rise buoyantly to the low density regions above the disk midplane. |
Blackian&Pessal(2009) arene that the magnetic fold structures hat power accretion disk coronae ust be associated with characteristic leugths that are large compared to he typical turbulent eddies. | \citet{bp09} argue that the magnetic field structures that power accretion disk coronae must be associated with characteristic lengths that are large compared to the typical turbulent eddies. |
If this were not the case. he timescales associated with turbulent diffusion would ο xmnaller than the correspoucding buovant rise tine. παλιο it difficult to transport significant magnetic energv to the coronac. | If this were not the case, the timescales associated with turbulent diffusion would be smaller than the corresponding buoyant rise time, making it difficult to transport significant magnetic energy to the coronae. |
Iu other words. if the corona is a consequence of magnetic field structures that originate within the turbulent disk via the MBI (or other magnetic instabilities). but that dissipate above the disk uicdplane. then these structures nust be of large enough scale to survive the buovaut rise without being shredded by the turbulence within the disk. | In other words, if the corona is a consequence of magnetic field structures that originate within the turbulent disk via the MRI (or other magnetic instabilities), but that dissipate above the disk midplane, then these structures must be of large enough scale to survive the buoyant rise without being shredded by the turbulence within the disk. |
Thus. the results preseuted in this paper provide support to the prevailing paracdieu for N-rav clussion iu accreting svstenis which involves an optically thin. hot corona powered by the dissipation of magnetic fields (c.g.Hardt&Mavascli1993:FieldRogers 1993). | Thus, the results presented in this paper provide support to the prevailing paradigm for X-ray emission in accreting systems which involves an optically thin, hot corona powered by the dissipation of magnetic fields \citep[e.g.][]{hm93a,fr93}. |
. We lave used Athena to examine the effects of stratification on imaguetolbydrodvuamic turbulence driven by the magnuetorotational instability. | We have used Athena to examine the effects of stratification on magnetohydrodynamic turbulence driven by the magnetorotational instability. |
We have shown that stratified simulatious converge as resolution Increases. even dn domains with zero-net-fiux and -10 explicit dissipation. | We have shown that stratified simulations converge as resolution increases, even in domains with zero-net-flux and no explicit dissipation. |
This is coutrary to our own calculations of zero-net-fiux unstratified domains. which do not converge. confirming previous results (Fromane 2009). | This is contrary to our own calculations of zero-net-flux unstratified domains, which do not converge, confirming previous results \citep{fp07,gua09,shb09}. |
. We have also considered calculations with explicit Issipation. and confriued previous results that the uaintenance of sustained turbulence is magnetic Prandtl nuuber dependent. | We have also considered calculations with explicit dissipation, and confirmed previous results that the maintenance of sustained turbulence is magnetic Prandtl number dependent. |
Stratification appears to extend the range for which sustained turbulence develops. aud may allow sustained turbulence at slightly lower Praudtl nuuber for a given Revunolds nmuuber. | Stratification appears to extend the range for which sustained turbulence develops, and may allow sustained turbulence at slightly lower Prandtl number for a given Reynolds number. |
However. the behavior is rather complex with larger variations ancl evolution ou long timescales (greater than LOO orbits). | However, the behavior is rather complex with larger variations and evolution on long timescales (greater than 100 orbits). |
At the highest resolutions considered. (61/77 aud L28/f7) the ratio of total stress to midplane pressure has a inean value of a~0401. but with considerable fluctuation about this mean on long (=50 orbit) timescales. | At the highest resolutions considered $64/H$ and $128/H$ ) the ratio of total stress to midplane pressure has a mean value of $\alpha \sim
0.01$, but with considerable fluctuation about this mean on long $\gtrsim 50$ orbit) timescales. |
Since real astrophysical svstenis are stratified. this somewhat alleviates coucerus that magnetorotational turbulence mught be unable to provide the required augular momentum transport in accretion disks. although values a factor of ten higher have Όσοι inferred in some astroplivsical sources (xineetal.2007). | Since real astrophysical systems are stratified, this somewhat alleviates concerns that magnetorotational turbulence might be unable to provide the required angular momentum transport in accretion disks, although values a factor of ten higher have been inferred in some astrophysical sources \citep{kpl07}. |
. Similarly. it partially alleviates concerus that explicit dissipation may be required in elobal disk simulations at lieh resolution. as stratification and net toroidal fields arise naturally iu such calculations. | Similarly, it partially alleviates concerns that explicit dissipation may be required in global disk simulations at high resolution, as stratification and net toroidal fields arise naturally in such calculations. |
We lave shown that these conclusious do uot depend sensitively ou the vertical or radial dimieusious of the Domains with radial extents of one aud four scale icehts eive the same time averaged values for à and rave nearly identical power spectral deusities for the uaenetic energy. | We have shown that these conclusions do not depend sensitively on the vertical or radial dimensions of the Domains with radial extents of one and four scale heights give the same time averaged values for $\alpha$ and have nearly identical power spectral densities for the magnetic energy. |
Stresses are somewhat more sensitive to variatious in the vertical height of the domain. although lis iiv be related to our assumptions of vertical veriodicity. | Stresses are somewhat more sensitive to variations in the vertical height of the domain, although this may be related to our assumptions of vertical periodicity. |
Increasing the vertical exteut from four to six scale heights results in only a slight increase in the ine aud spatially averaged stresses; as lone as the spatial average is carried out over the same volume (about when using the inner two scale heights). | Increasing the vertical extent from four to six scale heights results in only a slight increase in the time and spatially averaged stresses, as long as the spatial average is carried out over the same volume (about when using the inner two scale heights). |
Qur results ecuerally reproduce the qualitative features ound bv previous authors for stratified systems (c.g.Brandeubureetal.1995:Stone 1996). | Our results generally reproduce the qualitative features found by previous authors for stratified systems \citep[e.g.][]{bra95,sto96}. |
.. This includes oscillations with a periods of =LO orbits in which the horizoutally averaged radial aud toroidal fields alternate sigi. | This includes oscillations with a periods of $\lesssim 10$ orbits in which the horizontally averaged radial and toroidal fields alternate sign. |
Coupled with buovaucy this leads to a characteristic butterfiv diagram in horizoutally averaged space-tinie pots. | Coupled with buoyancy this leads to a characteristic butterfly diagram in horizontally averaged space-time plots. |
A comparison of our results with those of Lesur&Oeilvic(2008) sugeest the mechanisunis for generating the large scale field oscillatious in the stratified iux unstratified domains may be related. | A comparison of our results with those of \cite{lo08} suggest the mechanisms for generating the large scale field oscillations in the stratified and unstratified domains may be related. |
An advantage of deriving numerical algorithms from a variational principle is that conservation laws can be guaranteed. | An advantage of deriving numerical algorithms from a variational principle is that conservation laws can be guaranteed. |
Another advantage is that the algorithms derived from a variational principle are often more stable than other algorithms. | Another advantage is that the algorithms derived from a variational principle are often more stable than other algorithms. |
For example. in the case of smoothed particle hydrodynamics (SPH. for a review see 23). the density may be determined from the continuity equation. and it proves important for stability to combine the SPH continuity equation with the variational principle to deduce equations of motion. | For example, in the case of smoothed particle hydrodynamics (SPH, for a review see \citealt{monaghan92}) ), the density may be determined from the continuity equation, and it proves important for stability to combine the SPH continuity equation with the variational principle to deduce equations of motion. |
We call such a procedure consistent. | We call such a procedure consistent. |
? have derived consistent SPH equations for fluids even when non standard forms of the continuity equation are used. | \citet{bl99} have derived consistent SPH equations for fluids even when non standard forms of the continuity equation are used. |
They include the continuity equation as a constraint on Lagrangian density variations. | They include the continuity equation as a constraint on Lagrangian density variations. |
The resulting equations possess very good stability properties when two fluids with very different densities. for example air and water. are in contact. | The resulting equations possess very good stability properties when two fluids with very different densities, for example air and water, are in contact. |
Other. non consistent. forms of the SPH algorithm. for example with a standard acceleration equation but non standard continuity equation. exhibit instabilities. | Other, non consistent, forms of the SPH algorithm, for example with a standard acceleration equation but non standard continuity equation, exhibit instabilities. |
In the present paper we show how a Lagrangian variational principle can be used to derive the SPMHD (smoothed particle magnetohydrodynamics} equations for ideal MHD. | In the present paper we show how a Lagrangian variational principle can be used to derive the SPMHD (smoothed particle magnetohydrodynamics) equations for ideal MHD. |
Variational equations for continuum MHD have been derived by ? for both the Lagrangian and the Eulerian form of the equations (see also 22. and 2). | Variational equations for continuum MHD have been derived by \citet{newcomb62} for both the Lagrangian and the Eulerian form of the equations (see also \citealt{henyey82,oppeneer84} and \citealt{field86}) ). |
In the Lagrangian form of the equations Newcomb makes use of flux conservation to relate changes in the magnetic field to changes in surface elements. | In the Lagrangian form of the equations Newcomb makes use of flux conservation to relate changes in the magnetic field to changes in surface elements. |
In the present case. where we consider SPH particles. it is not clear how to prescribe such surface elements in a unique way from the particle coordinates. | In the present case, where we consider SPH particles, it is not clear how to prescribe such surface elements in a unique way from the particle coordinates. |
Instead we make use of the induction equation in its Lagrangian form and treat this as a constraint. | Instead we make use of the induction equation in its Lagrangian form and treat this as a constraint. |
An alternative. which we do not explore here. is to begin with plasma physics and prescribe the fields in terms of currents. | An alternative, which we do not explore here, is to begin with plasma physics and prescribe the fields in terms of currents. |
Such an approach would be natural for particle methods (e.g. PIC) which have been so effective for plasma physics where the electrons would be treated as one fluid and the ions as another. | Such an approach would be natural for particle methods (e.g. PIC) which have been so effective for plasma physics where the electrons would be treated as one fluid and the ions as another. |
The plan of this paper is derive the equations of motion from a standard Lagrangian for SPH particles with either. or both. the continuity and induction equations treated as constraints refsec:sphmom)). | The plan of this paper is derive the equations of motion from a standard Lagrangian for SPH particles with either, or both, the continuity and induction equations treated as constraints \\ref{sec:sphmom}) ). |
We then consider the effect of variable smoothing length in the SPH kernels refsec:gradh)) after which we demonstrate by numerical tests that consistent treatment of the variable smoothing length in the SPH equations significantly improves the accuracy of SPMHD shocks and of wave propagation refsec:Dtests)). | We then consider the effect of variable smoothing length in the SPH kernels \\ref{sec:gradh}) ) after which we demonstrate by numerical tests that consistent treatment of the variable smoothing length in the SPH equations significantly improves the accuracy of SPMHD shocks and of wave propagation \\ref{sec:1Dtests}) ). |
Our results complement those obtained in a companion paper (2.. hereafter paper D for non-ideal MHD where artificial dissipative terms were included to handle shocks. | Our results complement those obtained in a companion paper \citealt{pm03a}, hereafter paper I) for non-ideal MHD where artificial dissipative terms were included to handle shocks. |
Variational principles for MHD have been discussed by many authors (e.g. 22223) and the Lagrangian is given by | Variational principles for MHD have been discussed by many authors (e.g. \citealt{newcomb62,henyey82,oppeneer84,field86}) ) and the Lagrangian is given by |
and therefore the fraction of energy in ganimia-ray pulse converted to the creation of a pair-wind is 10"no. | and therefore the fraction of energy in gamma-ray pulse converted to the creation of a pair-wind is $10^{-6} n_0$. |
The overall wind Lorentz factor 54.=LeyfCrit 3. where maeo(Ax£3)Sumy ds the wind mass. | The overall wind Lorentz factor $\gamma_w = E_w/ (m_w c^2) \sim 3$ , where $m_w \sim (4\pi/3)\,r_{end}^3 \next m_p$ is the wind mass. |
There are two factors associated with the wind spherical expansion which have been ignored in the analytical calculations presented in section refdynamies:: the spherical dilution of the incident CRB photon Hux and the interacticons within the pair-wind. | There are two factors associated with the wind spherical expansion which have been ignored in the analytical calculations presented in section \\ref{dynamics}: the spherical dilution of the incident GRB photon flux and the interactions within the pair-wind. |
The former has been considered in refspherical: here we note tha another wav in which it leads to a nonstationary D(r) aux 26r) is that the GRB front radius may increase significantly while a shell of external matter. undergoing acceleraion and pair-enrichment. is within the front. | The former has been considered in \\ref{spherical}; here we note that another way in which it leads to a non-stationary $\npair (x)$ and $\gamma (x)$ is that the GRB front radius may increase significantly while a shell of external matter, undergoing acceleration and pair-enrichment, is within the front. |
Ignoring this elfect is appropriate when the upper Limit to the radius increase curing the front-crossing. 25AAL is less than the front raclius. r. which. after using equation (36)) leads tor2«1077115 em for a=15. FE.=10 erg sí=d. | Ignoring this effect is appropriate when the upper limit to the radius increase during the front-crossing, $2 \gamma_\Delta^2 \Delta$, is less than the front radius, $r$ , which, after using equation \ref{gd}) ), leads to $r > 2\times 10^{15}
T_1^{0.16}$ cm for $\alpha = 1.5$, $E_\gamma = 10^{53}$ ergs, $\varepsilon_p =1 $. |
Collisions within the pair-wind occur because the external medium entering the GRB front at smaller radius is accelerated to a higher Lorentz factor and may overtake a slower shell of external matter entering rw GRB front at a larger radius. | Collisions within the pair-wind occur because the external medium entering the GRB front at smaller radius is accelerated to a higher Lorentz factor and may overtake a slower shell of external matter entering the GRB front at a larger radius. |
A third factor is the esence of the ejecta within the CRB pulse. which sweep-up 10 ρανα! and may drastically. reduce the acceleration wough photon-scatterings if the ejecta Lorentz factor is Lgienifieantly larger than that of the wind. | A third factor is the presence of the ejecta within the GRB pulse, which sweep-up the pair-wind and may drastically reduce the acceleration through photon-scatterings if the ejecta Lorentz factor is significantly larger than that of the wind. |
Thus this factor is of importance only when the lag 9 between cjecta and ye leading edge of the GRB front. r/(2107). is less than the »ulse width. AL at radii r£6LOMPST, cm. and only iEm(4) | Thus this factor is of importance only when the lag $\delta$ between ejecta and the leading edge of the GRB front, $r/(2\Gamma^2)$, is less than the pulse width, $\Delta$, at radii $r \siml 6 \times 10^{15} \Gamma_2^2 T_1$ cm, and only if $\Gamma \gg \gamma (\delta)$. |
The ellects discussed above are taken into account in he results shown in Figureὃν 1. | The effects discussed above are taken into account in the results shown in Figure 1. |
The external medium is ciseretized in shells of thickness much smaller than their radius. which entering the GRB front at αν=0 and are accelerated. compressed. anc pair-Ioaded as described in refevnamies.. | The external medium is discretized in shells of thickness much smaller than their radius, which entering the GRB front at $x=0$ and are accelerated, compressed, and pair-loaded as described in \\ref{dynamics}. |
To include the spherical expansion. the GRB Hus and the shell density are adjusted after cach time-step. | To include the spherical expansion, the GRB flux and the shell density are adjusted after each time-step. |
The progressive merging of shells (interactions within the wind) is accounted for by replacing a pair of colliding shells with one whose Lorentz factor and density are. calculated [rom energy conservation and the jump conditions at shocks. respectively. | The progressive merging of shells (interactions within the wind) is accounted for by replacing a pair of colliding shells with one whose Lorentz factor and density are calculated from energy conservation and the jump conditions at shocks, respectively. |
The location of the ejecta is tracked. allowing for the accumulation of shocked wind shells and compression of the shocked ejecta. | The location of the ejecta is tracked, allowing for the accumulation of shocked wind shells and compression of the shocked ejecta. |
The termination of the wind acceleration. while it is still within the GRB front. due to the collision with the ejecta. is illustrated in. Figure 1 by 1ο rising part of the wind density (y+) and Lorentz factor (<4) atr1013 em. | The termination of the wind acceleration while it is still within the GRB front, due to the collision with the ejecta, is illustrated in Figure 1 by the rising part of the wind density $n\pm$ ) and Lorentz factor $\gamma_\pm$ ) at $r \sim 10^{14}$ cm. |
As the ejecta lag more behind the fronts leading edge. the external medium is more accelerated: ancl pair-Ioaded. | As the ejecta lag more behind the front's leading edge, the external medium is more accelerated and pair-loaded. |
The interactions within the wind and the decrease of the incident lux with radius leac to a Uattening of the wind Lorentz factor at r224101 em. followed by a decrease. | The interactions within the wind and the decrease of the incident flux with radius lead to a flattening of the wind Lorentz factor at $r = 2-4 \times 10^{14}$ cm, followed by a decrease. |
The ejecta drive the forwardshock into a relativistic wind up to regE10 em. thus the dissipation ellicieney of this shock is reduced by the existence of the pairwind up to an observer time F40CLEz)G(nsadf017)DELOL.? s. "Phe pair production becomes negligible"EM at rii;~1077lf em. corresponding. to an observer time fü;c105,4E100L.,? s. As shown in ligure 1. the pair enrichment factor decreases roughly as poxrug 3olor 10715«rc" em. "P | The ejecta drive the forwardshock into a relativistic wind up to $r_{rel} \siml 10^{15}$ cm, thus the dissipation efficiency of this shock is reduced by the existence of the pair-wind up to an observer time $t_{rel} \sim (1+z) (r_{rel}/c\Gamma^2) \siml 10 \Gamma_2^{-2}$ s. The pair production becomes negligible at $r_{pair} \sim 10^{16}$ cm, corresponding to an observer time $t_{pair} \sim 10 t_{rel} \siml 100 \Gamma_2^{-2}$ s. As shown in Figure 1, the pair enrichment factor decreases roughly as $\mu \propto r_{coll}^{-3}$ for $10^{15} < r < 10^{16}$ cm. |
bThis means that most of the radiating leptons at r=rai are the pairs formed abor«orga and not the electrons originally existing in the circumburst medium. | This means that most of the radiating leptons at $r = r_{pair}$ are the pairs formed at $r < r_{pair}$ and not the electrons originally existing in the circumburst medium. |
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