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In section 3, we describe the physical processes that lead to evolution of the neutron star magnetic field. | In section 3, we describe the physical processes that lead to evolution of the neutron star magnetic field. |
In section 4, we describe our neutron star model and give details of calculations of the instability growth rate. | In section 4, we describe our neutron star model and give details of calculations of the instability growth rate. |
Section 5 contains discussion and our conclusions. | Section 5 contains discussion and our conclusions. |
The dipole fields of magnetars are inferred to be in the range ~(0.5—20)x1014 G, based on spindown measurements of SGRs and AXPs ( see ? for a review). | The dipole fields of magnetars are inferred to be in the range $\sim (0.5-20) \times 10^{14} \,$ G, based on spindown measurements of SGRs and AXPs ( see \cite{mereghetti} for a review). |
In order for the magnetic field to be stable in neutron stars, it must contain both toroidal and poloidal components (?).. | In order for the magnetic field to be stable in neutron stars, it must contain both toroidal and poloidal components \citep{tayler}. |
The toroidal component in a stable configuration has a twisted torus shape, and may be an order of magnitude larger than the poloidal component (?).. | The toroidal component in a stable configuration has a twisted torus shape, and may be an order of magnitude larger than the poloidal component \citep{braithwaite09}. |
Large crustal currents associated with the toroidal field would produce significant ohmic heating due to the relatively high resistivity in the outer crust. | Large crustal currents associated with the toroidal field would produce significant ohmic heating due to the relatively high resistivity in the outer crust. |
Ohmic heating can account for the observed trend of surface temperature that increases with surface field observed in neutron stars with B>1015 G (?).. | Ohmic heating can account for the observed trend of surface temperature that increases with surface field observed in neutron stars with $B
> 10^{13} $ G \citep{ponslink}. |
Cooling simulations show that a heating layer in the outer crust, as would arise from current decay, can explain the high surface temperatures of magnetars (?).. | Cooling simulations show that a heating layer in the outer crust, as would arise from current decay, can explain the high surface temperatures of magnetars \citep{kaminker}. |
Current decay is determined primarily by electron-phonon interactions for a temperature T below the melting temperature Tmeit. | Current decay is determined primarily by electron-phonon interactions for a temperature $T$ below the melting temperature $T_{\rm melt}$. |
In this regime, the electrical resistivity scales as T', so that a small increase in temperature leads to increased heat dissipation (Fig. | In this regime, the electrical resistivity scales as $T$, so that a small increase in temperature leads to increased heat dissipation (Fig. |
1). | 1). |
The additional heating raises the temperature further, and a temperature runaway may develop if thermal transport is unable to quench the feedback process. | The additional heating raises the temperature further, and a temperature runaway may develop if thermal transport is unable to quench the feedback process. |
As we show in this paper, this instability can occur in the outer crusts of neutron stars, where the electrical resistivity is relatively high and the thermal conductivity is low. | As we show in this paper, this instability can occur in the outer crusts of neutron stars, where the electrical resistivity is relatively high and the thermal conductivity is low. |
The evolution of the magnetic field in neutron stars after birth is determined primarily by ohmic decay and Hall drift. | The evolution of the magnetic field in neutron stars after birth is determined primarily by ohmic decay and Hall drift. |
The ohmic decay timescale is Tonm—9j17, where L is the typical magnetic field length scale and 7 is the electrical resistivity. | The ohmic decay timescale is $\tau_{\rm ohm} =\eta^{-1} L^2$, where $L$ is the typical magnetic field length scale and $\eta$ is the electrical resistivity. |
A typical value for the outer crust at temperature T=10° K is | A typical value for the outer crust at temperature $T=10^8$ K is |
(1999). | . |
. Varying /inerg corresponds to different definitious of close pairs. | Varying $t_{merg}$ corresponds to different definitions of close pairs. |
The three curves in fig. | The three curves in fig. |
2 t'e exponential laws of the form £,4,4,00)=eqexp(esmni). fitted to the merger fractious for cdillerent linerg. | \ref{mass} are exponential laws of the form $F_{mg}(0)=c_{1}\exp(c_{2}m)$, fitted to the merger fractions for different $t_{merg}$. |
Che parameters used to fit the cata points are c40.058 andο—1.33. e4=0.107 and eo»=—1BL anc c,=0.137 and es=—1.12 for [μοι equal to 0.5 Gyr. 1 Gyr aud 1.5 Gye. respectively. | The parameters used to fit the data points are $c_{1}=0.058$ and$c_{2}=-1.23$ , $c_{1}=0.107$ and $c_{2}=-1.34$, and $c_{1}=0.137$ and $c_{2}=-1.42$ for $t_{merg}$ equal to 0.5 Gyr, 1 Gyr and 1.5 Gyr, respectively. |
In the same euvironment. that is the same final halo mass Mo. £,,4,(0) increases with increasing jerger tiijescale as binaries with larger separatious are iucluded. | In the same environment, that is the same final halo mass $M_{0}$, $F_{mg}(0)$ increases with increasing merger timescale as binaries with larger separations are included. |
The merger iudex i shows ouly weak variation. | The merger index $m$ shows only weak variation. |
For coiputatiolal reasons mergers are ouly resolved: above a ininimunm mass ÀAL,;,. | For computational reasons mergers are only resolved above a minimum mass $M_{min}$. |
Mergers below this lass are neglected. | Mergers below this mass are neglected. |
This correspouds to observations with a imagnitude limited sample ol galaxies. | This corresponds to observations with a magnitude limited sample of galaxies. |
The graphs in fig. | The graphs in fig. |
3aa show the cdeyendence of the merger index mon gap=Mpo/AM,. | \ref{majmer}a a show the dependence of the merger index $m$ on $q_{M}=M_{0}/M_{min}$. |
The filled circles are the results of merger trees with constant. Mj;=LOMAL. aud varying Adj. | The filled circles are the results of merger trees with constant $M_{min}=10^{10} M_{\odot}$ and varying $M_{0}$. |
We compare these resits with siiwtlations for corstall ily;=10HAL. and varying Alpi). represented yw open circles. | We compare these results with simulations for constant $M_{0}=10^{11} M_{\odot}$ and varying $M_{min}$, represented by open circles. |
Τie. value of i1 cepeids οuly on the ralio gay as i=Q.69In(qar)—1.77. | The value of $m$ depends only on the ratio $q_{M}$ as $m=0.69\ln(q_{M})-1.77$. |
Another important questjon is1je Influerce of ie defiuition of major mergers ou tlie mereer ate. | Another important question is the influence of the definition of major mergers on the merger rate. |
The graphs i fig. | The graphs in fig. |
3bb show the dependence of £5,,(0) on different values Of Rinajor. | \ref{majmer}b b show the dependence of $F_{mg}(0)$ on different values of $R_{major}$. |
An event is called iuajor merger if the 1lass Τὰio of the mereiο MODealaxies is |velow Lajor aud larger than 1. | An event is called major merger if the mass ratio of the merging galaxies is below $R_{major}$ and larger than 1. |
As Ronajor increases. Fy)g(O) itcreases. | As $R_{major}$ increases, $F_{mg}(0)$ increases. |
We also iuc tat the mereer iucex a stays roughly coustaut or low Rimajor aud decreases at larger Lingjor- | We also find that the merger index $m$ stays roughly constant for low $R_{major}$ and decreases at larger $R_{major}$. |
A decrease inom with larger Royajor has also beet 'eported by Cottlóbereta.(2001). | A decrease in $m$ with larger $R_{major}$ has also been reported by \citet{got01}. |
. It is a restlt of he adopted iniUlu nass [or merger events. | It is a result of the adopted minimum mass for merger events. |
ve detectable amount of uajor mergers with arge nass ratios dec‘eases [aster with redshift that Or edlal mass mergers. sitice the stnall masses drop faster below te minimi mass. | The detectable amount of major mergers with large mass ratios decreases faster with redshift than for equal mass mergers, since the small masses drop faster below the minimum mass. |
In observe samples of cle»e pairs Roce&Eales(1999) αιd Pattonetal.(2000) found that £5,,(0) increase Νο hey allow for larger liojor. Which agrees with our preclictioS. | In observed samples of close pairs \citet{roch99} and \citet{pat00}
found that $F_{mg}(0)$ increased when they allow for larger $R_{major}$, which agrees with our predictions. |
How clo the theoretica moclels compare to tlie observations? | How do the theoretical models compare to the observations? |
The star in fig. | The star in fig. |
2. is he 1Jeastirec uerger [racticn fo: field galaxies by LeFevreetal.(2000).. who sec Rina>jor{ aud he loca neree: fraction of Pattoetal.(1997). | \ref{mass} is the measured merger fraction for field galaxies by \citet{lef00}, who used $R_{major}=4$ and the local merger fraction of \citet{pat97}. |
. Thes. identified close pai "ons (hose which ine‘ve ona imescale less thet yPg—] Gyr. | They identified close pairs as those which merge on a timescale less then $t_{merg}=1$ Gyr. |
To compare this mereer fraction with otr estimaes οje pneecls ο take into account that he dark halos of iek ealaxies can vary over a ralee of 1lasses axl that he mereer tinescale is stbject to large uncertalutles. | To compare this merger fraction with our estimates one needs to take into account that the dark halos of field galaxies can vary over a range of masses and that the merger timescale is subject to large uncertainties. |
We herfore weightec| the clieren merger ractious of οι Isauple of field galaxies witl lia© lasses Alo between 5x10TAL, and 5xLOM: according to he Press-Sejechiter. preclictions. | We therfore weighted the different merger fractions of our sample of field galaxies with halo masses $M_{0}$ between $5\times10^{11} M_{\odot}$ and $5 \times 10^{12}
M_{\odot}$ according to the Press-Schechter predictions. |
The merger 1xlex in ax the local uerger Tract] Pug(0) for cliferent Ay were caleulatecl using he fitting Lo1nulae as SIOWI by the graphs iu 2 | The merger index $m$ and the local merger fraction $F_{mg}(0)$ for different $M_{0}$ were calculated using the fitting formulae as shown by the graphs in fig. |
and fig. | \ref{mass} and fig. |
Jaa. We varied the the rauge o μαlo masses coitributing to the sauple by ¢lane the lower botud of halo masses from 5x101AZ. to 2.5x10?AL. aud cliaugec linerg Within range of 0.5 - 1.5 Cyr. | \ref{majmer}a a. We varied the the range of halo masses contributing to the sample by changing the lower bound of halo masses from $5\times10^{11}
M_{\odot}$ to $2.5\times10^{12} M_{\odot}$ and changed $t_{merg}$ within the range of 0.5 - 1.5 Gyr. |
The results of this reasonable parameter rauge lie tusice he shaded regiO1 in fig. 2.. | The results of this reasonable parameter range lie inside the shaded region in fig. \ref{mass}. . |
Resultslor larger /4,/:44 correspond to the upper part of the 'eglon and those for larger halo masses lie in the right part of the region. | Resultsfor larger $t_{merg}$ correspond to the upper part of the region and those for larger halo masses lie in the right part of the region. |
A comparison of our results witl the observations shows. that the predicted merger index i and the normalization £5,,4(0) are a facor 2 smaller than observed. | A comparison of our results with the observations shows, that the predicted merger index $m$ and the normalization $F_{mg}(0)$ are a factor 2 smaller than observed. |
acceleration is turned. on ancl off. which is one of the ingredients in overcoming (he known inconsistencies (Drokey1949). | acceleration is turned on and off, which is one of the ingredients in overcoming the known inconsistencies \citep{d49}. |
. The foregoing inconsistency suggests that. unlike the case of svnchrotron radiation. LAE cannot be treated exactly using the Larmor formula. | The foregoing inconsistency suggests that, unlike the case of synchrotron radiation, LAE cannot be treated exactly using the Larmor formula. |
To understand why this is the case. we need {ο consider the conditions under which (4.1)) is valid. | To understand why this is the case, we need to consider the conditions under which \ref{Larmor1}) ) is valid. |
A standard derivation of the Larmor formula involves caleulating the Poynting vector due to the electric ancl magnetic fields of the accelerated charge. and integrating over a lixed. large sphere to find (he power crossing (his sphere. | A standard derivation of the Larmor formula involves calculating the Poynting vector due to the electric and magnetic fields of the accelerated charge, and integrating over a fixed, large sphere to find the power crossing this sphere. |
The power lost bv the particle is al (he retarded time compared with the power escaping. | The power lost by the particle is at the retarded time compared with the power escaping. |
This invalidates any interpretation ol the dependence of the Larmor formula (4.1)) on X in terms of the time dependence of the power radiated. | This invalidates any interpretation of the dependence of the Larmor formula \ref{Larmor1}) ) on $\chi$ in terms of the time dependence of the power radiated. |
However. for the average (24)) over a periodic motion. the average over the actual time and the retarded (ime are equivalent. | However, for the average \ref{Larmor2}) ) over a periodic motion, the average over the actual time and the retarded time are equivalent. |
Ilence. this argument does not invalidate (24)). and so does not explain the inconsistency. | Hence, this argument does not invalidate \ref{Larmor2}) ), and so does not explain the inconsistency. |
The power escaping from the fixed sphere is equated to the power lost bx the particle. | The power escaping from the fixed sphere is equated to the power lost by the particle. |
This assumption is not valid for LAE in general. | This assumption is not valid for LAE in general. |
The motion of the radiation pattern with the particle inside the fixed sphere implies that the total electromagnetic enerev within the sphere is not constant: the radiant energy inside (he fixed sphere is changing systematically as a [function of time. | The motion of the radiation pattern with the particle inside the fixed sphere implies that the total electromagnetic energy within the sphere is not constant: the radiant energy inside the fixed sphere is changing systematically as a function of time. |
Hence the assumption that the power radiated balances the power lost by the particle is not valid over any fixed time interval. | Hence the assumption that the power radiated balances the power lost by the particle is not valid over any fixed time interval. |
ILowever. for periodic motion. provided that one considers only the power averaged over a period. and provided that there is no average drift motion. the Larmor formula should be valid. | However, for periodic motion, provided that one considers only the power averaged over a period, and provided that there is no average drift motion, the Larmor formula should be valid. |
This suggests that the mean power (24)) should be correct for a background particle (but not for a test charge). | This suggests that the mean power \ref{Larmor2}) ) should be correct for a background particle (but not for a test charge). |
therein).. and thus f could be higher. | , and thus $f$ could be higher. |
This was tested in the f=0.18 model C, whose results are shown in the bottom panel of Fig. ΒΙ.. | This was tested in the $f=0.18$ model $_{\rm Lf}$ whose results are shown in the bottom panel of Fig. \ref{fig: Ttest}. |
As expected. the cooling is much stronger for larger f. which leads to a more unstable bow shock — the apex of the bow shock has moved inwards in model C,; and its shape ts far from à smooth curve for both the forward and reverse shock. | As expected, the cooling is much stronger for larger $f$, which leads to a more unstable bow shock – the apex of the bow shock has moved inwards in model $_{\rm Lf}$ and its shape is far from a smooth curve for both the forward and reverse shock. |
The regular appearance and relative stability of the observed bow shock on similar scales (?) suggests that the hydrogen is indeed mostly in atomic form. | The regular appearance and relative stability of the observed bow shock on similar scales \citep{Ueta08} suggests that the hydrogen is indeed mostly in atomic form. |
The emissivities for each of the cooling and heating species included in the simulations are shown in cross-section in Fig. | The emissivities for each of the cooling and heating species included in the simulations are shown in cross-section in Fig. |
B2. for models Ay [top] and D [bottom]. | \ref{fig: Lcross} for models $_{\rm H}$ [top] and D [bottom]. |
These cross-sectional profiles show the location and morphology of the emitting regions for each species more clearly than. the projections shown m Figs. | These cross-sectional profiles show the location and morphology of the emitting regions for each species more clearly than the projections shown in Figs. |
13. and I4.. | \ref{fig: Lproj15} and \ref{fig: Lproj03}. |
Here we also show results for each coolant individually. | Here we also show results for each coolant individually. |
As expected. the processes that involve molecular hydrogen. e.g. Aqsa. tend to emit strongly from the gas in the reverse shock. | As expected, the processes that involve molecular hydrogen, e.g. $\Lambda_{\rm H_2 (r-v)}$, tend to emit strongly from the gas in the reverse shock. |
The ΟΙ fine structure and CO rotational cooling are also more concentrated in the denser. cooler regions of the reverse shock. | The OI fine structure and CO rotational cooling are also more concentrated in the denser, cooler regions of the reverse shock. |
The other species tend to emit more evenly from the entire bow shock. with the atomic cooling greatest in the forward shock. | The other species tend to emit more evenly from the entire bow shock, with the atomic cooling greatest in the forward shock. |
In the slow models. the atomic cooling is largely absent from the cool. dense R-T ‘fingers’. | In the slow models, the atomic cooling is largely absent from the cool, dense R-T `fingers'. |
In the fast model. several species that do emit from both the forward and reverse shocks tend to show a strong ridge of emission just ahead of the contact discontinuity. e.g. the grain. water. atomic. and carbon fine-structure cooling. | In the fast model, several species that do emit from both the forward and reverse shocks tend to show a strong ridge of emission just ahead of the contact discontinuity, e.g. the grain, water, atomic, and carbon fine-structure cooling. |
The lower ISM density in the fast model results in much weaker emission from the bow shock tail than in the slow models. e.g. Ago and Aco.) | The lower ISM density in the fast model results in much weaker emission from the bow shock tail than in the slow models, e.g. $\Lambda_{\rm H_2O (H_2)}$ and $\Lambda_{\rm CO (H_2)}$ . |
We lave given a variation of the pairiug heaps that achieves the same amortized bounds as Fibonacci heaps. except fordecreasc-keg (which still matches Frediman’s lower bound for. what le calls [1].. a generalized pairing heap). | We have given a variation of the pairing heaps that achieves the same amortized bounds as Fibonacci heaps, except for (which still matches Fredman's lower bound for, what he calls \cite{f}, a generalized pairing heap). |
Three important open questions are: | Three important open questions are: |
ο.2/3—2hV3. | $g\chi\rightarrow 4+2\sqrt{3}-2k\sqrt{3}$. |
Let u. fy. and Pg be the values of the functions in question just after our [Iuid element is shocked: then at late times 5250 so {ΕΠ}e>!. | Let $\gamma_0$, $f_0$, and $h_0$ be the values of the functions in question just after our fluid element is shocked; then at late times $\gamma\gg \gamma_0$ so $(f/h)/(f_0/h_0)\sim \gamma^{-1}$. |
Integrating the above differential equations then gives This agrees with the results of Johnson&Melee(1971). for the final Lorentz Factor οἱ the {hued in a strong ulirarelativistic shock propagating into a cold medium with decreasing density. | Integrating the above differential equations then gives This agrees with the results of \cite{johnson71} for the final Lorentz factor of the fluid in a strong ultrarelativistic shock propagating into a cold medium with decreasing density. |
The agreement provides additional support for our claim that the solution outside the star behaves like the solution describing a standard. planar shock up to the initial conditions and the interpretation of the characteristic values A. D. P. IN. | The agreement provides additional support for our claim that the solution outside the star behaves like the solution describing a standard planar shock up to the initial conditions and the interpretation of the characteristic values $R$, $\Gamma$, $P$, $N$. |
Note that the differences between (he initial conditions used in (heir work and in ours are unimportant (o the scaling law relating (he final and initial Lorentz factors of a given fluid element. | Note that the differences between the initial conditions used in their work and in ours are unimportant to the scaling law relating the final and initial Lorentz factors of a given fluid element. |
This result agrees wilh the findings of Tanetal.(2001) concerning the scaling law: partly because ol uncertainty over the different initial conditions. they used numerical simulations to check theyοναμIm result. | This result agrees with the findings of \cite{tan01}
concerning the scaling law: partly because of uncertainty over the different initial conditions, they used numerical simulations to check the $\gamma\sim\gamma_0^{1+\sqrt{3}}$ result. |
Recently. Nakavamia&Shigevama(2005) also investigated the problem of an ultrarelativistic planar shock. | Recently, \cite{nakayama05} also investigated the problem of an ultrarelativistic planar shock. |
While the self-similar solution (μον eive for the Hlow belore breakout is identical to the one in Sari(2005) aud outlined here. they do not give analvtic results for or a physical interpretation of the self-similar solution after breakout. | While the self-similar solution they give for the flow before breakout is identical to the one in \cite{sari05} and outlined here, they do not give analytic results for or a physical interpretation of the self-similar solution after breakout. |
To verily our results numerically. we integrated the time-dependent relativistic hvdrodvnanmic equations using a one-dimensional code. | To verify our results numerically, we integrated the time-dependent relativistic hydrodynamic equations using a one-dimensional code. |
Figure 2 shows curves for 5 as a function of position al a single time before breakout. while Figure 4. shows the time evolution of T. P. and ΔΑ before breakout. | Figure \ref{gprofile_in} shows curves for $\gamma$ as a function of position at a single time before breakout, while Figure \ref{gpnvstime_in} shows the time evolution of $\Gamma$ $P$ , and $N$ before breakout. |
The numerical and analvtie results are in excellent agreement. | The numerical and analytic results are in excellent agreement. |
Figures 3 and 5 respectively show the 5 vs. e profile ancl time evolution of E. P. aud N alter breakout: ihe agreement between numerical and analytic results here confirms the choice of scale (1) alter breakout Chat we discussed in 4.1. | Figures \ref{gprofile_out} and \ref{gpnvstime_out} respectively show the $\gamma$ vs. $x$ profile and time evolution of $\Gamma$, $P$, and $N$ after breakout; the agreement between numerical and analytic results here confirms the choice of scale $R(t)$ after breakout that we discussed in 4.1. |
We have shown that. given an ulirarelativistic shock propagating into a planar polvtropic envelope.the flow upon the shock's emergence Irom the envelope into vacuum follows a | We have shown that, given an ultrarelativistic shock propagating into a planar polytropic envelope,the flow upon the shock's emergence from the envelope into vacuum follows a |
The: gravitational: potential: energy is: GGAT2/FH where Ritter’s:op forma tells us that for] polvtropes a=z—3 (see appendix). | The gravitational potential energy is $-\alpha GM^2/R$ where Ritter's formula tells us that for polytropes $\alpha = {3\over 5-n}$ (see appendix). |
However. for planets aud small bodies the electrons are held in electrically rather than eravitationallv. | However, for planets and small bodies the electrons are held in electrically rather than gravitationally. |
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