ReadingTimeMachine/rtm-sgt-ocr-v1 · Datasets at Hugging Face

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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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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:

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.