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Shell-tvpe SNR RX J1713.7-3946 is one of the most widelv studied SNRs with perhaps the best multi- observations. | Shell-type SNR RX J1713.7-3946 is one of the most widely studied SNRs with perhaps the best multi-wavelength observations. |
The observational data span from radio (Lazendicetal.2004).. inliaved (Benjaminοἱal.2003:Aceroet2009).. X-ray. Cxovamaetal.1997:Uchivamaetal.2003:Cassam-Chenai 2004).. GeV s-ray (Abdoοἱal.2011).. to TeV s-rax band (Muraishietal.2000:Enomoto2002:Aharonian 2006).. | The observational data span from radio \citep{2004ApJ...602..271L}, infrared \citep{2003PASP..115..953B, 2009A&A...505..157A}, X-ray \citep{1997PASJ...49L...7K,2003A&A...400..567U,2004A&A...427..199C}, GeV $\gamma$ -ray \citep{fermi:rxj1713}, to TeV $\gamma$ -ray band \citep{2000A&A...354L..57M,2002Natur.416..823E,2006A&A...449..223A}. . |
Recent observations. especially (he A-ray emission delected by Suzaku (Tanakaetal.2008) and TeV οταν emission measured bv HESS (Aharonianetal.2007).. eive the energy spectra and images of this SNB. with very high equality. which makes detailed modelings of the emission mechanism plausible (Morlinoetal.2009a:Fang2009:Fan2010a:Zirakashvili&Aharonian2010:Ellisonetal.2010;Fan 2010b).. | Recent observations, especially the X-ray emission detected by $Suzaku$ \citep{2008ApJ...685..988T} and TeV $\gamma$ -ray emission measured by $HESS$ \citep{2007A&A...464..235A}, give the energy spectra and images of this SNR with very high quality, which makes detailed modelings of the emission mechanism plausible \citep{2009MNRAS.392..240M,2009MNRAS.392..925F,2010MNRAS.406.1337F,
2010ApJ...708..965Z,2010ApJ...712..287E,2010A&A...517L...4F}. |
The newly reported data from Fermi elal.2011). also set strict constraints on the nature of the radiation from this SNR. | The newly reported data from $Fermi$ \citep{fermi:rxj1713} also set strict constraints on the nature of the radiation from this SNR. |
Basie results of recent studies of this SNR may be summarized briefly as the following. | Basic results of recent studies of this SNR may be summarized briefly as the following. |
5 The wide range TeV οταν spectrum [avors a hadronic origin of the high energy emission (Aharonianetal.2006:Drurv2009:Morlinoοἱ 2010).. | The wide range TeV $\gamma$ -ray spectrum favors a hadronic origin of the high energy emission \citep{2006A&A...449..223A,
2009A&A...496....1D,2009MNRAS.392..240M,2009MNRAS.392..925F,
2010A&A...511A..34B}. |
This scenario is also in line with the long standing view that SNRs are the most important CR. accelerators (Axford1931).. | This scenario is also in line with the long standing view that SNRs are the most important CR accelerators \citep{1981ICRC...12..155A}. |
However. there is a strong correlation between the X-ray image and TeV 5-rav image. favoring a leptonic origin of the multi-waveleneth emission (Aharonianetal.2006:Acero2009).. | However, there is a strong correlation between the X-ray image and TeV $\gamma$ -ray image, favoring a leptonic origin of the multi-wavelength emission \citep{2006A&A...449..223A,
2009A&A...505..157A}. |
Plaga(2008) also claims that the lack of spatial correlation between 5-ravs and (the molecular cloud in the vicinity ofSNR RX J1713.7-3946 argues against the hadronie scenario. | \cite{2008NewA...13...73P} also claims that the lack of spatial correlation between $\gamma$ -rays and the molecular cloud in the vicinity ofSNR RX J1713.7-3946 argues against the hadronic scenario. |
Furthermore the lack of thermal | Furthermore the lack of thermal |
that after tidal mass loss). | that after tidal mass loss). |
The characteristic timescale for mass loss is then where we have introduced a parameter f- to allow this timescale to be adjusted to match numerical results. | The characteristic timescale for mass loss is then where we have introduced a parameter $f_\tau$ to allow this timescale to be adjusted to match numerical results. |
We expect f-~I. | We expect $f_\tau\sim 1$. |
Using the model described above. we can compute the mass ost bv the satellite due to tidal forces. and 1e resulting density profile of the remaining bound material. | Using the model described above, we can compute the mass lost by the satellite due to tidal forces, and the resulting density profile of the remaining bound material. |
This new density profile will give rise to a new potential (7). weaker han that of the original satellite. | This new density profile will give rise to a new potential $\Phi(r)$, weaker than that of the original satellite. |
As such. if we apply our same calculation of mass loss to the remaining bound xwiicles. using this new potential. we expect that further mass loss will occur. | As such, if we apply our same calculation of mass loss to the remaining bound particles, using this new potential, we expect that further mass loss will occur. |
This process can. in principle. be repeated. | This process can, in principle, be repeated. |
inféniliim.. Assuming that the potential changes on à timescale of approximately £444) we can use our mocel to compute mass loss as a function of time. | Assuming that the potential changes on a timescale of approximately $\langle t_{\rm dyn}\rangle$ we can use our model to compute mass loss as a function of time. |
Essentially. we are breaking up the continuous process of mass loss into discrete intervals (each of order the dynamical time). | Essentially, we are breaking up the continuous process of mass loss into discrete intervals (each of order the dynamical time). |
We compute the total mass loss over cach interval. before upelating the satellite density. profile and potential prior to computing mass loss for the next interval. | We compute the total mass loss over each interval, before updating the satellite density profile and potential prior to computing mass loss for the next interval. |
We will refer to these intervals as “mass loss iterations". | We will refer to these intervals as “mass loss iterations”. |
Our model. while a vast. improvement over the "classic? calculation of tidal mass loss. has its own limitations: | Our model, while a vast improvement over the “classic” calculation of tidal mass loss, has its own limitations: |
over cluster temperature as obtained from e.g. X-ray observations for constant CRp scaling (model 1). | over cluster temperature as obtained from e.g. X-ray observations for constant CRp scaling (model 1). |
We include the evolution of the emission with time as lines in black (gray) for z<0.48 (z> 0.48). | We include the evolution of the emission with time as lines in black (gray) for $z<0.48$ $z>0.48$ ). |
Further we plot a number of recent observations. | Further we plot a number of recent observations. |
The sample follows the correlation closely at redshift zero, and only two clusters show significant deviation from the correlation at temperature larger than 5 keV, and only at high redshift. | The sample follows the correlation closely at redshift zero, and only two clusters show significant deviation from the correlation at temperature larger than 5 keV, and only at high redshift. |
In simulations, temperature is a less merger sensitive mass estimator compared to the X-ray luminosity, in terms of cluster mergers. | In simulations, temperature is a less merger sensitive mass estimator compared to the X-ray luminosity, in terms of cluster mergers. |
We therefore conclude, that the bimodality observed in large clusters is not a result of biased mass estimation. | We therefore conclude, that the bimodality observed in large clusters is not a result of biased mass estimation. |
the evolution of the global energy-density. which is that corresponding to the IDM ansanz: Prior to the present epoch we have that pla)xa7. while at late enough times (a>> I) the above integral converges. which implies that the corresponding global density tends to evolve again as the usual dark matter (see Weinberg 2008). with where fy is the present age of the Universe. | the evolution of the global energy-density, which is that corresponding to the IDM ansanz: Prior to the present epoch we have that $\rho(\alpha) \propto
\alpha^{-3}$, while at late enough times $\alpha \gg 1$ ) the above integral converges, which implies that the corresponding global density tends to evolve again as the usual dark matter (see Weinberg 2008), with where $t_{0}$ is the present age of the Universe. |
The latter analysis. relevant to the usual weakly interacting massive particle case - Weinberg (2008). leads to the conclusion that the annihilation term has no effect resembling that of dark energy. but does affect the evolution of the self interacting DM component. with the integral in the denominator rapidly converging to a constant (which does depend on the annihilation cross-section).V)/3H: | The latter analysis, relevant to the usual weakly interacting massive particle case - Weinberg (2008), leads to the conclusion that the annihilation term has no effect resembling that of dark energy, but does affect the evolution of the self interacting DM component, with the integral in the denominator rapidly converging to a constant (which does depend on the annihilation cross-section).: |
: For the case of a non-perfect DM fluid (1e.. | For the case of a non-perfect DM fluid (ie., |
having up to the present time a disequilibrium between the arathilation and particle pair creation processes) we can either have a positive or a negative effective pressure term. | having up to the present time a disequilibrium between the annihilation and particle pair creation processes) we can either have a positive or a negative effective pressure term. |
Although the latter situation may or may not appear plausible. even the remote such possibility. 1e.. | Although the latter situation may or may not appear plausible, even the remote such possibility, ie., |
the case for which the DM particle creation term is larger than the annihilation term («p—V.« 0). is of particular interest for its repercussios on the global dynamics of the Universe (see for example Zimdahl et al. | the case for which the DM particle creation term is larger than the annihilation term $\kappa \rho^{2}-\Psi<0$ ), is of particular interest for its repercussions on the global dynamics of the Universe (see for example Zimdahl et al. |
2001: Balakin et al. | 2001; Balakin et al. |
2003). | 2003). |
4 It is interesting to note that this case can be viewed as a generalization of the gravitational matter creation model of Prigogine et al. ( | It is interesting to note that this case can be viewed as a generalization of the gravitational matter creation model of Prigogine et al. ( |
1989) [see also Lima et al. | 1989) [see also Lima et al. |
2008 and references therein] in which annihilation processes are also included. although the matter creation component dominates over annihilations. | 2008 and references therein] in which annihilation processes are also included, although the matter creation component dominates over annihilations. |
In such a scenario. as well as in any interacting dark-matter model with a left-over residual radiation. a possible contribution from the radiation products to the global dynamics is negligible. as we show in appendix A. In general. for «# Oand V.¥0 it is not an easy task to solve analytically eq. (7)). | In such a scenario, as well as in any interacting dark-matter model with a left-over residual radiation, a possible contribution from the radiation products to the global dynamics is negligible, as we show in appendix A. In general, for $\kappa \ne 0$ and $\Psi \ne 0$ it is not an easy task to solve analytically eq. \ref{bol2}) ), |
which ts a Riceati equation. due to the fact that it is a non-linear differential equation. | which is a Riccati equation, due to the fact that it is a non-linear differential equation. |
However. eq.(7)) could be fully solvable if (and only if) a particular solution is known. | However, \ref{bol2}) ) could be fully solvable if (and only if) a particular solution is known. |
Indeed. we find that for some special cases regarding the functional form of the interactive term. such as Y=ία.Η). we can derive analytical solutions. | Indeed, we find that for some special cases regarding the functional form of the interactive term, such as $\Psi=\Psi(\alpha,H)$, we can derive analytical solutions. |
We have identified two functional forms for which we can solve the previous differential equation analytically. only one of which is of interest since it provides ao«7? dependence of the scale factor (see appendix B). | We have identified two functional forms for which we can solve the previous differential equation analytically, only one of which is of interest since it provides a $\propto a^{-3}$ dependence of the scale factor (see appendix B). |
This is: Although. the above functional form was not motivated by some physical theory. but rather phenomenologically by the fact that it provides analytical solutions to the Boltzmann equation. its exact form can be justified within the framework of IDM (see appendix C). | This is: Although, the above functional form was not motivated by some physical theory, but rather phenomenologically by the fact that it provides analytical solutions to the Boltzmann equation, its exact form can be justified within the framework of IDM (see appendix C). |
The general solution of equation (7)) for the total using eq.(11)). 1s: where the kernel function F(a) has the form: Note that «C, has units of Gyr!. while n. Cj and C» are the corresponding constants of the problem. | The general solution of equation \ref{bol2}) ) for the total energy-density, using \ref{int1}) ), is: where the kernel function $F(\alpha)$ has the form: Note that $\kappa {\cal C}_1$ has units of $^{-1}$, while $m$, ${\cal C}_{1}$ and ${\cal C}_2$ are the corresponding constants of the problem. |
Obviously. eq.(12)) can be seen as where p.=Cia" is the density corresponding. to the residual "matter creation”. resulting from a_ possible disequilibrium between the particle creation and annihilation processes. while p. can be viewed as the energy density of the self-interacting dark matter particles which are dominated by the annihilation processes. | Obviously, \ref{sol1}) ) can be seen as where $\rho_{c}=C_{1}\alpha^{m}$ is the density corresponding to the residual ”matter creation”, resulting from a possible disequilibrium between the particle creation and annihilation processes, while $\rho^{'}$ can be viewed as the energy density of the self-interacting dark matter particles which are dominated by the annihilation processes. |
This can be easily understood if we set the constant C, strictly equal to zero. implying that the creation term is negligible. which reduces the current solution 14)) to that of eq.(9)). | This can be easily understood if we set the constant ${\cal C}_{1}$ strictly equal to zero, implying that the creation term is negligible, which reduces the current solution \ref{gg33}) ) to that of \ref{bern}) ). |
Note that near the present epoch as well as at late enough times (α>>1). as also in2. the p evolves as the usual dark matter (see also Weinberg 2008). | Note that near the present epoch as well as at late enough times $\alpha \gg 1$ ), as also in, the $\rho^{'}$ evolves as the usual dark matter (see also Weinberg 2008). |
Finally. if both κ and V tend to zero. the above cosmological model reduces to the usual Einstein-deSitter model 7). | Finally, if both $\kappa$ and $\Psi$ tend to zero, the above cosmological model reduces to the usual Einstein-deSitter model ). |
Note that. due to p.>0. the constant C» obeys the following restriction: Evaluating now eq.(12)) at the present time (@=1. F(a)= 1). we obtain the present-time total cosmic density. which ts: po2Ci+1/Cs. with Cj2O and C»>0.—V/3H: | Note that, due to $\rho^{'}>0$, the constant ${\cal C}_{2}$ obeys the following restriction: Evaluating now \ref{sol1}) ) at the present time $\alpha=1$, $F(\alpha)=1$ ), we obtain the present-time total cosmic density, which is: $\rho_0={\cal C}_{1}+1/{\cal C}_2\;$, with ${\cal C}_{1} \ge 0$ and ${\cal C}_{2}>0$.: |
: In this scenario we assume that the annihilation term is negligible [s=O and f(a)=0] and the particle creation term dominates. | In this scenario we assume that the annihilation term is negligible $\kappa=0$ and $f(\alpha)$ =0] and the particle creation term dominates. |
Such a situation is mathematically equivalent to the gravitational DM particle creation process within the context of non-equilibrium thermodynamics Prigogine et al. ( | Such a situation is mathematically equivalent to the gravitational DM particle creation process within the context of non-equilibrium thermodynamics Prigogine et al. ( |
1989). the important cosmological repercussions of which have been studied in Lima et al. ( | 1989), the important cosmological repercussions of which have been studied in Lima et al. ( |
2008 and references therein). | 2008 and references therein). |
Using our nomenclature and «=0. eq.(7)) becomes a first order linear differential equation. a general solution of which ts: The negative pressure can yield a late accelerated phase of the cosmic expansion (as in Lima et al. | Using our nomenclature and $\kappa=0$, \ref{bol2}) ) becomes a first order linear differential equation, a general solution of which is: The negative pressure can yield a late accelerated phase of the cosmic expansion (as in Lima et al. |
2008). without the need of the required. in the “classical” cosmological models. dark energy. | 2008), without the need of the required, in the “classical” cosmological models, dark energy. |
Below. we investigate the conditions under which eqs. (12) | Below, we investigate the conditions under which eqs. \ref{sol1}) ) |
and (16)) could provide accelerating solutions. similar to the usual dark energy case. | and \ref{sol7}) ) could provide accelerating solutions, similar to the usual dark energy case. |
of the event is also luted by the possibility that shell b dissipates its entire enerey before the forward shock passes through shell ο. | of the event is also limited by the possibility that shell $b$ dissipates its entire energy before the forward shock passes through shell $a$. |
Using similar reasoning for the reverse shock. we obtain the comoving timescales for the interactions at the forward aud reverse shocks: Following the passage of the reverse aud forward shocks through the shells. adiabatic expansion quickly cuds subsequent acceleration and emission (απο2008). | Using similar reasoning for the reverse shock, we obtain the comoving timescales for the interactions at the forward and reverse shocks: Following the passage of the reverse and forward shocks through the shells, adiabatic expansion quickly ends subsequent acceleration and emission \citep{der08}. |
. The maxinuun euergev of particles accelerated at the forward aud reverse shocks is given bw νε=NZOUBFpAMES gu | The maximum energy of particles accelerated at the forward and reverse shocks is given by $E_{max,f(r)} \cong Ze\Gamma B^\prime_{f(r)}c\bar\beta_{f(r)}\Delta t^\prime_{FS(RS)}$ . |
The general expression can be written as The general expression for the maxinuuu radiative cficiency eivine the internal euergyv clissipated in the forward and reverse shocks can be written as For the differcut cases. we obtain the following results: The masximaun particle energy for the various cases ds always proportional to the mubrella function. ((1)). derived. from clementary principles. but inultiplied. bv a cocficicut =O(1). | The general expression can be written as The general expression for the maximum radiative efficiency giving the internal energy dissipated in the forward and reverse shocks can be written as For the different cases, we obtain the following results: The maximum particle energy for the various cases is always proportional to the umbrella function, \ref{Emax}) ), derived from elementary principles, but multiplied by a coefficient $\lesssim {\cal O}(1)$. |
The ability of a shell collision to accelerate particles to the highest energies is conditioned on very specific behaviors of the shells. uamely that the second shell is much faster than the first (ppo>1). and that the tine t. between shell ejectious is a small factor larger than the duration Δον of the event forming shell @ (as expressed by the term (fayAte.)Δι in the denominators of ((10)) (16))). | The ability of a shell collision to accelerate particles to the highest energies is conditioned on very specific behaviors of the shells, namely that the second shell is much faster than the first $\rho_\Gamma \gg 1$ ), and that the time $t_{*d}$ between shell ejections is a small factor larger than the duration $\Delta t_{*a}$ of the event forming shell $a$ (as expressed by the term $(t_{*d} - \Delta t_{*a})/\Delta t_{*a}$ in the denominators of \ref{Emax1f}) ) – \ref{Emax3r}) )). |
The most favorable regime for particle acceleration to the highest enereles occurs for the case of à RRS aud RES wheu the energies aud Iunuinosities of the two shells are about equal. | The most favorable regime for particle acceleration to the highest energies occurs for the case of a RRS and RFS when the energies and luminosities of the two shells are about equal. |
This also gives the highest radiative efficiencies. | This also gives the highest radiative efficiencies. |
The main requireiieut is a large contrast between the Lorentz factors of the two shells (Beloborodov2000:I&u-mar&Piran 2000). | The main requirement is a large contrast between the Lorentz factors of the two shells \citep{bel00,kp00}. |
. The highest radiative. Ποιος coincides with approximately equal energies and luninositics for the cases ofa RRS aud RES. aud à RRS aud NFS. | The highest radiative efficiency coincides with approximately equal energies and luminosities for the cases of a RRS and RFS, and a RRS and NFS. |
In the case ofa NRS aud RFS. where Lj29L, is equired for validity of this asvinptote. a much larger energy in shell 5 thau shell «à is required for πλακα radiative efficiency at the reverse shock. as shown by (153). | In the case of a NRS and RFS, where $L_b \gg L_a$ is required for validity of this asymptote, a much larger energy in shell $b$ than shell $a$ is required for maximum radiative efficiency at the reverse shock, as shown by \ref{Emax2r}) ). |
Energy dissipation in this case would. however. more likely be dominated by the forward shock. | Energy dissipation in this case would, however, more likely be dominated by the forward shock. |
Iàiucinatic linitatious ensure that the radiative efficiency is poor for dissipation at either the forward or reverse shocks for the case of à NRS aud NFS. ((13)). depeudiug ou the precise energies in each of the shells. | Kinematic limitations ensure that the radiative efficiency is poor for dissipation at either the forward or reverse shocks for the case of a NRS and NFS, \ref{Emax4f}) ), depending on the precise energies in each of the shells. |
Supposing that the engines of GRBs and blazars or. for that matter. nücroquasars. eject shells with such properties (whichisnecessaryiuthecaseofGRBstonalshockseenario:ef.Tolkaetal. 2006).. then we can construct a diagram illustrating the viability of various sources to accelerate UITECTs. | Supposing that the engines of GRBs and blazars or, for that matter, microquasars, eject shells with such properties \citep[which is necessary in the case of GRBs to explain their high $\gamma$-ray radiative
efficiency in an internal shock scenario; cf.][]{iok06}, then we can construct a diagram illustrating the viability of various sources to accelerate UHECRs. |
Iu Fie. | In Fig. |
2 we plot apparent huuinositv as a function of Loreutz factor for the acceleration of 1029 eV protons (leavy solid curve) aud | 2 we plot apparent luminosity as a function of Lorentz factor for the acceleration of $10^{20}$ eV protons (heavy solid curve) and |
As they contract. their internal temperature increases. | As they contract, their internal temperature increases. |
At early times the rate of this contraction is controlled by the radiative cooling rate of the embryo — which is by the rate at which the embryo can get rid of the excess energy. | At early times the rate of this contraction is controlled by the radiative cooling rate of the embryo – which is by the rate at which the embryo can get rid of the excess energy. |
However. when temperature 754422000 IX is reached. molecular hyelrogen disassociates. | However, when temperature $T_{\rm 2nd} \approx 2000$ K is reached, molecular hydrogen disassociates. |
This process is an cllicient energy sink. which allows the embryo to contract rapidlv — in fact collapse hydrodvnamically without the need to racdiate the energy away. | This process is an efficient energy sink, which allows the embryo to contract rapidly – in fact collapse hydrodynamically – without the need to radiate the energy away. |
Ehe embryo collapse stops ον at much higher densities. and temperatures as high as 107 Ix. at which point hydrogen is ionised. | The embryo collapse stops only at much higher densities, and temperatures as high as $10^4$ K, at which point hydrogen is ionised. |
The embryo must then continue a slower contraction. again regulated by the rate at which its energy is radiated away. | The embryo must then continue a slower contraction, again regulated by the rate at which its energy is radiated away. |
The collapse is known as the "second collapse” in the star formation literature. (Larson 1969).. when the “first cores? of masses 5037; collapse (Masunaga&Inutsuka2000). to become “second cores". which are the proper proto-stars. | The collapse is known as the “second collapse” in the star formation literature \citep{Larson69}, when the “first cores” of masses $\sim 50 M_J$ collapse \citep{Masunaga00} to become “second cores”, which are the proper proto-stars. |
In the TD hypothesis for planet formation. the second collapse may be the last step to making a eas giant planet. | In the TD hypothesis for planet formation, the second collapse may be the last step to making a gas giant planet. |
Llowever. as we show below. this final step is not automatically successful — planets continuing to migrate rapidly towards their parent stars may still be disrupted at R~OL AU. | However, as we show below, this final step is not automatically successful – planets continuing to migrate rapidly towards their parent stars may still be disrupted at $R \sim 0.1$ AU. |
We sugeest this process as a wav of forming the hot Super Earths observed by theAcplíer mission (Boruckietal. 2011). | We suggest this process as a way of forming the hot Super Earths observed by the mission \citep{BoruckiEtal11}. |
. 1n analogy to the star formation literature. we refer to the CEs that are mainly molecular. embryos temperature 7). Zoya. as the “first CEs”: those where Ho is clisassociated are termed. "second Giles” instead. | In analogy to the star formation literature, we refer to the GEs that are mainly molecular, embryo's temperature $T_e < T_{\rm 2nd}$ , as the “first GEs”; those where $_2$ is disassociated are termed “second GEs” instead. |
‘To illustrate the main point of this paper. we caleulate the contraction of a giant embryo with “typical” parameter values (o...thosethatappearquitereasonabletousfora 2010a). | To illustrate the main point of this paper, we calculate the contraction of a giant embryo with “typical” parameter values \citep[e.g., those that appear
quite reasonable to us for a solar metalicity disc around a $\sim$ solar
mass star; see][]{Nayakshin10c}. |
. In particular. the embryo mass is AZ,=10M;. the normalised dust opacity is Αν=0.5. and the grain mass fraction f,=0.01. | In particular, the embryo mass is $M_e = 10
M_J$, the normalised dust opacity is $k_* = 0.5$, and the grain mass fraction $f_g = 0.01$. |
Ehe embryo is initialised as a first core of same mass (seeNavakshin2010c). | The embryo is initialised as a first core of same mass \citep[see][]{Nayakshin10a}. |
. Figure 1. shows the time evolution of the embrvo's central temperature (solid. in units of 10 lk). the gas density (dotted. in units of 10.7 g 7). and the outer radius of the embryo. 7. (dashed. in units of 1 AW). | Figure \ref{fig:fig1} shows the time evolution of the embryo's central temperature (solid, in units of $10^3$ K), the gas density (dotted, in units of $10^{-8}$ g $^{-3}$ ), and the outer radius of the embryo, $r_e$, (dashed, in units of 1 AU). |
The calculation is carried out with an updated version of the 1D eas-clust erains radiative hyvedrodynamics code of Navakshin (2010c.h).. | The calculation is carried out with an updated version of the 1D gas-dust grains radiative hydrodynamics code of \cite{Nayakshin10a,Nayakshin10b}. |
Instead. of using an ideal gas equation of state with +=5/3. the code now uses the equation. of state appropriate for molecular hydrogen. including disassociation and rotational anc vibrational degrees of freedom for I». with the orto-hyvdrogen to para-hyerogen ratio fixed at 3:1 (cf.Boleyetal.2006). | Instead of using an ideal gas equation of state with $\gamma = 5/3$, the code now uses the equation of state appropriate for molecular hydrogen, including disassociation and rotational and vibrational degrees of freedom for $_2$, with the orto-hydrogen to para-hydrogen ratio fixed at 3:1 \citep[cf.][]{BoleyEtal06}. |
. Despite the updated equation of state. the evolution of the first embryo is quite similar to that of the cases studied in Navakshin(2010b). | Despite the updated equation of state, the evolution of the first embryo is quite similar to that of the cases studied in \cite{Nayakshin10b}. |
. This may not be entirely. surprising eiven the similar insensitivitv of the first (gas) cores to the equation of state as found by Masunagaetal.(1998):Ma-sunaga&Inutsuka (2000). | This may not be entirely surprising given the similar insensitivity of the first (gas) cores to the equation of state as found by \cite{Masunaga98,Masunaga00}. |
. Phe embryo contracts and heats up. whereas cust grains grow. | The embryo contracts and heats up, whereas dust grains grow. |
By time /~1000. vers. the erains increase in size to about 20 em. | By time $t\sim 1000$ yrs, the grains increase in size to about 20 cm. |
Their density exceeds hat of the gas in the centre of the embryo: they. become sell-eravitating and form a solid core of mass AL.~SAL). | Their density exceeds that of the gas in the centre of the embryo; they become self-gravitating and form a solid core of mass $M_{\rm c} \sim 5\mearth$. |
In Figure 1.. the solid core formation is notable by the imp in the central temperature. | In Figure \ref{fig:fig1}, the solid core formation is notable by the bump in the central temperature. |
After the core formation. he central region becomes hotter than grain vaporisation emperature of zz1400 Ix. evaporating the grains. and thus erminating further core growth (seeNavakshin2010b.fordetailsonthisnegativefeedback loop).. | After the core formation, the central region becomes hotter than grain vaporisation temperature of $\approx 1400$ K, evaporating the grains, and thus terminating further core growth \citep[see][for details on this negative feedback
loop]{Nayakshin10b}. |
The central region also expancs slightly. | The central region also expands slightly. |
Most of the HZ is however unallected » the solid core in this case. and the curves resume their otherwise monotonic behaviour a few hundred. vears later. | Most of the GE is however unaffected by the solid core in this case, and the curves resume their otherwise monotonic behaviour a few hundred years later. |
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