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2006).
2006).
These shocks or converging flows are ubiquitously generated, for example, by supernovae, and correspond to energy input on large scales.
These shocks or converging flows are ubiquitously generated, for example, by supernovae, and correspond to energy input on large scales.
In contrast, our analysis of the stability of the transition layer shows another aspect of the dynamics of the multi phase medium that may be created by the thermal instability: the instability of the evaporating front can generate fluctuating motions even without external mechanical forcing.
In contrast, our analysis of the stability of the transition layer shows another aspect of the dynamics of the multi phase medium that may be created by the thermal instability: the instability of the evaporating front can generate fluctuating motions even without external mechanical forcing.
Note, however, that the maximum growth rate of the instability depends sensitively on the velocity of the evaporating flow.
Note, however, that the maximum growth rate of the instability depends sensitively on the velocity of the evaporating flow.
A further detailed study of the nonlinear behavior of the instability will be presented in our next paper.
A further detailed study of the nonlinear behavior of the instability will be presented in our next paper.
We thank Masahiro Nagashima for useful discussions.
We thank Masahiro Nagashima for useful discussions.
This work is supported by the for the 21st Century COE "Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
This work is supported by the Grant-in-Aid for the 21st Century COE "Center for Diversity and Universality in Physics" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
SI is supported by the Grant-in-Aid (No.15740118, 16077202, 18540238) from MEXT of Japan.
SI is supported by the Grant-in-Aid (No.15740118, 16077202, 18540238) from MEXT of Japan.
HK is supported by the 2]st Century COE Program of Origin and Evolution of Planetary Systems in MEXT of Japan.
HK is supported by the 21st Century COE Program of Origin and Evolution of Planetary Systems in MEXT of Japan.
HCOCH2OH(68 19-49-6729 48)) is found peaking at the position of the cavity and showing similar angular sizes lo this, see Table 2.
$_2$ $_{19,49}$ $_{20,48}$ )) is found peaking at the position of the cavity and showing similar angular sizes to this, see Table 2.
This molecule is thus more inumately related with the circumstellar compact dusty disk.
This molecule is thus more intimately related with the circumstellar compact dusty disk.
The observational parameters of all lines are shown in Table 2.
The observational parameters of all lines are shown in Table 2.
We noted from Table 1 and 2, there is à correlation between the deconvolved sizes of the molecular emission and the excitation temperatures in lower energy states of each molecular specie.
We noted from Table 1 and 2, there is a correlation between the deconvolved sizes of the molecular emission and the excitation temperatures in lower energy states of each molecular specie.
The molecular emission [rom transitions characterized by high excitation temperatures show to be very compact.
The molecular emission from transitions characterized by high excitation temperatures show to be very compact.
This correlation could be obviously obtained Lor lines that are optically thick with T5 ~ 77 Τον. where the Tj 1s the brightness temperature, 7? is the filling factor, and T, is the excitation temperature.
This correlation could be obviously obtained for lines that are optically thick with $_b$ $\sim$ $\eta$ $_{ex}$ , where the $_b$ is the brightness temperature, $\eta$ is the filling factor, and $_{ex}$ is the excitation temperature.
In the left panel of Figure 3 the emission of the lines ΗΟΟΟΗ2ΟΗ( 6211-0215) and HC3N(38-37) (v, 20) reveals the presence of a possible warm "companion" located to the northeast of the disk.
In the left panel of Figure 3 the emission of the lines $_2$ $_{13,49}$ $_{12,50}$ ) and $_3$ N(38-37) $\nu_t$ =0) reveals the presence of a possible warm “companion” located to the northeast of the disk.
We have marked the position of this putative companion with a yellow triangle.
We have marked the position of this putative companion with a yellow triangle.
This "companion" is also observed in SO» at the same position, see Figure 3 of ?,, and is associated with a group of water maser spots (?2)..
This “companion” is also observed in $_2$ at the same position, see Figure 3 of \citet[][]{zapataetal2009}, and is associated with a group of water maser spots \citep{Imaietal2002,Eisneretal2002}.
The position of the "companion" was found by fitüng a Guassian to its HCOCH»?OH(62,3 49-62,» 50)) emission.
The position of the “companion” was found by fitting a Guassian to its $_2$ $_{13,49}$ $_{12,50}$ )) emission.
However, it still not clear if the "companion" is real protostar or if this could be the result of the interaction of the outflow with a high density zone of the molecular cloud.
However, it still not clear if the “companion” is real protostar or if this could be the result of the interaction of the outflow with a high density zone of the molecular cloud.
We do not find any clear evidence of outllowing gas activity associated with this possible "companion" (seealso?)..
We do not find any clear evidence of outflowing gas activity associated with this possible “companion” \citep[see also][]{zapataetal2009}.
Although the blueshifted side of the CO(J=3- 2) outflow has not the same position angle as the redshilted one, this seems not be ejected [rom the companion.
Although the blueshifted side of the $J=3-2$ ) outflow has not the same position angle as the redshifted one, this seems not be ejected from the companion.
We show how the blueshilted side appears to be more likely ejected [rom W51 North in Figure 1.
We show how the blueshifted side appears to be more likely ejected from W51 North in Figure 1.
We drew a line that crosses this side of the outflow and points directly to the dusty compact disk.
We drew a line that crosses this side of the outflow and points directly to the dusty compact disk.
The SO» and SiO emission indeed show how the outflow is deviated to where the CO(/=3— 2) is located (?)..
The $_2$ and SiO emission indeed show how the outflow is deviated to where the $J=3-2$ ) is located \citep[][]{zapataetal2009}.
However, more observations in some other molecular outflow tracers are thus necessity to firmly discard the existence of a second outflow in W51 North.
However, more observations in some other molecular outflow tracers are thus necessity to firmly discard the existence of a second outflow in W51 North.
It is interesting to note that the possible companion is not observed in the hotter molecular gas tracer jy49-67 HCOCHOH(6829.48) suggesüng that it may not as warm as the central massive star.
It is interesting to note that the possible companion is not observed in the hotter molecular gas tracer $_2$ $_{19,49}$ $_{20,48}$ ) suggesting that it may not as warm as the central massive star.
Figures 4 and 5 show the position-velocity diagrams (PV-diagrams) of different molecules which trace disünct scales of the disk.
Figures 4 and 5 show the position-velocity diagrams (PV-diagrams) of different molecules which trace distinct scales of the disk.
In Figure 3, we have marked the orientation and position of the PV-cuts.
In Figure 3, we have marked the orientation and position of the PV-cuts.
In the left panel of Figure 3 we show a white line with a 207 that corresponds to the PV-cuts shown in Figure4.
In the left panel of Figure 3 we show a white line with a $^\circ$ that corresponds to the PV-cuts shown in Figure4.
On the other hand, the white line with a P.A.=160° in the right panel corresponds to the the PV-culs shown in Figure 5.
On the other hand, the white line with a $^\circ$ in the right panel corresponds to the the PV-cuts shown in Figure 5.
Figure 4 shows the molecular emission Irom HC3N(38-37) and HCOCH»?OH(68,9.19-67»444) located in the innermost parts of the disk, while the PV-diagrams of molecules as SO(8o-74), and SO»(19,1o- 18444). which trace the oulermost parts, are presented in Figure 5.
Figure 4 shows the molecular emission from $_3$ N(38-37) and $_2$ $_{19,49}$ $_{20,48}$ ) located in the innermost parts of the disk, while the PV-diagrams of molecules as $_{9}$ $_8$ ), and $_2$ $_{1,19}$ $_{0,18}$ ), which trace the outermost parts, are presented in Figure 5.
The PV-diagrams in Figure 4 reveal that the hot molecular gas closer to the massive protostar is Keplerian.
The PV-diagrams in Figure 4 reveal that the hot molecular gas closer to the massive protostar is Keplerian.
In addition, in this figure we have overlaid the PV-diagram of the LTE Keplerian disk modeled in ?.. but without an inner cavity and à smaller size.
In addition, in this figure we have overlaid the PV-diagram of the LTE Keplerian disk modeled in \citet[][]{zapataetal2009}, but without an inner cavity and a smaller size.
Both structures shown a very good correspondence.
Both structures shown a very good correspondence.
The SO(8,-74) and SO»(I9; 10-15) presented in Figure 5 trace much larger structures similar to the ring reported in ?..
The $_{9}$ $_8$ ) and $_2$$_{1,19}$ $_{0,18}$ ) presented in Figure 5 trace much larger structures similar to the ring reported in \citet[][]{zapataetal2009}. .
These molecules in addition show clearly two northwest and southeast high velocity extensions excited by the bipolar outflow mapped
These molecules in addition show clearly two northwest and southeast high velocity extensions excited by the bipolar outflow mapped
stabilised by magnetic diffusivity. η, so that the dispersion relation is|homna (eg. Kitehatinov and Rüddiger 2004).
stabilised by magnetic diffusivity, $\eta ,$ so that the dispersion relation is $\gamma +\eta k^2\approx v_A k$ (e.g. Kitchatinov and Rüddiger 2004).
Therefore. for marginal stability. Vac with A<ff=fr.
Therefore, for marginal stability, v_A with $\lambda < H = f r$.
This criterion translates readily into a minimum magnetic field tr which yields rmG.
This criterion translates readily into a minimum magnetic field = which yields = .
.. using again eqs. (32))
using again eqs. \ref{temp}) )
and (339) above.
and \ref{dens}) ) above.
We require now that the minimum magnetic field for the MRI to work is smaller than the field created in the disk.
We require now that the minimum magnetic field for the MRI to work is smaller than the field created in the disk.
As we have seen. the smallest suitable field amplitude. Sani. is given by the marginal instability criterion applied to scales of order the disk scale.
As we have seen, the smallest suitable field amplitude, $B_{\rm{mri}}$, is given by the marginal instability criterion applied to scales of order the disk scale.
At the same time. magnetic field generation is likely to yield amplitudes of order 2,4.
At the same time, magnetic field generation is likely to yield amplitudes of order $B_{\rm{rad}}$.
This amplitude must be higher than Suni. which happens as soon as
This amplitude must be higher than $B_{\rm{mri}}$, which happens as soon as > 0.46.
The thickness) of the disk has the most dramatic effect on the value of the critical mass. whereas the actual value of the viscous a parameter plays a minor role.
The thickness of the disk has the most dramatic effect on the value of the critical mass, whereas the actual value of the viscous $\alpha$ parameter plays a minor role.
If the disk is rather thin. with f~0.1 for instance. then the mass spans a range of rather large values. going from 128 to 357A7..
If the disk is rather thin, with $f\sim 0.1$ for instance, then the mass spans a range of rather large values, going from $128$ to $357 M_\odot$.
However. in the absence of metals. the properties of the primordial gas are likely to prevent the disk from efficiently cooling. and we expect f to be closer to unity.
However, in the absence of metals, the properties of the primordial gas are likely to prevent the disk from efficiently cooling, and we expect $f$ to be closer to unity.
In that case. taking f.~0.4 as obtained by Mayer and Duschl (2005). the critical mass is much smaller. comprised between roughly 4.3 and 27.8M. for a ranging from 1 to 0.01.
In that case, taking $f\sim 0.4$ as obtained by Mayer and Duschl (2005), the critical mass is much smaller, comprised between roughly $4.3$ and $27.8 M_\odot$ for $\alpha$ ranging from $1$ to $0.01$.
Magnetic fields have probably played a role in Primordial Star Formation. even if starting from a medium free of magnetic fields.
Magnetic fields have probably played a role in Primordial Star Formation, even if starting from a medium free of magnetic fields.
As we have argued. radiation drag or thermal pressure effects are able to generate magnetic seeds in the disk surrounding the central accreting stellar progenitor.
As we have argued, radiation drag or thermal pressure effects are able to generate magnetic seeds in the disk surrounding the central accreting stellar progenitor.
Initially. those seeds are weak too much for being amplitied by small scale turbulence. or even IRI effects on scales comparable to the disk scale.
Initially, those seeds are weak too much for being amplified by small scale turbulence, or even MRI effects on scales comparable to the disk scale.
However. as matter accretion proceeds. the mass of the central object grows. and he gravitational potential it creates deepens. increasing the rotation velocity of the disk.
However, as matter accretion proceeds, the mass of the central object grows, and the gravitational potential it creates deepens, increasing the rotation velocity of the disk.
Eventually. the rotation is fast enough so that ong wavelength MRI modes become unstable. as the minimum magnetic field for MRI becomes smaller than the field generated in he disk.
Eventually, the rotation is fast enough so that long wavelength MRI modes become unstable, as the minimum magnetic field for MRI becomes smaller than the field generated in the disk.
Depending on the properties of the disk. this happens once he mass o"the central object reaches 428A7..
Depending on the properties of the disk, this happens once the mass of the central object reaches $4 - 28 M_\odot$.
Subsequently. MRI dynamo will be at work and will amplify ahe magnetic field.
Subsequently, MRI dynamo will be at work and will amplify the magnetic field.
The field ean then rapidly reach dynamically important values. and magnetically-driven ejection contributes to ower the effective accretion efficiency.
The field can then rapidly reach dynamically important values, and magnetically-driven ejection contributes to lower the effective accretion efficiency.
The actual model of magnetic winds is beyond the scope of this article.
The actual model of magnetic winds is beyond the scope of this article.
but this suggests jat feedback is likely to require magnetically-driven outflows. which could occur during Pop ΠΙ formation already when. as we argued above. the mass of the stellar progenitor is rather small.
but this suggests that feedback is likely to require magnetically-driven outflows, which could occur during Pop III formation already when, as we argued above, the mass of the stellar progenitor is rather small.
Note yat. in their model of protostellar disk in primordial star formation. Tan and Blackman (2004) estimated the power of magnetically driven outflows.
Note that, in their model of protostellar disk in primordial star formation, Tan and Blackman (2004) estimated the power of magnetically driven outflows.
In their study. the outflow feedback effects reduce he star formation efficiency once the protostar reaches roughly OO Δι.
In their study, the outflow feedback effects reduce the star formation efficiency once the protostar reaches roughly 100 $M_\odot$.
Incidentally. concurrent conclusions were reached by Tachida et al. (
Incidentally, concurrent conclusions were reached by Machida et al. (
2006) who simulated the collapse of a magnetized orimordial cloud in rigid rotation. and the subsequent formation of a Pop III star. within the ideal MHD approximation.
2006) who simulated the collapse of a magnetized primordial cloud in rigid rotation, and the subsequent formation of a Pop III star, within the ideal MHD approximation.
Depending on the thigh) initial value (BiniLO°C! for an initial cloud density n.~lO'cm °%) of the magnetic field. their simulations show that magnetically driven jets develop and effectively reduce the accretion rate.
Depending on the (high) initial value $B_{\rm init} \gtrsim 10^{-9} G$ for an initial cloud density $n_{\rm c}\sim 10^3 {\rm cm}^{-3}$ ) of the magnetic field, their simulations show that magnetically driven jets develop and effectively reduce the accretion rate.
Further numerical studies with higher resolution seem necessary to address both the mass ejection rate and the amplitication of magnetic fields in proto-stellar disks or iniially weaker. even vanishing. magnetic seeds.
Further numerical studies with higher resolution seem necessary to address both the mass ejection rate and the amplification of magnetic fields in proto-stellar disks for initially weaker, even vanishing, magnetic seeds.
We iive argued that the interplay between the two modes of star formation. primordial. massive and conventiona. involving all masses. is controlled by £2 and not by Z.
We have argued that the interplay between the two modes of star formation, primordial, massive and conventional, involving all masses, is controlled by $B$ and not by $Z$.
Effective feecback requires that of order 10 percent of the gas accretion rate is channelled into star formation. with an outflow rate that on the average must be of the order of the net star formation rate.
Effective feedback requires that of order 10 percent of the gas accretion rate is channelled into star formation, with an outflow rate that on the average must be of the order of the net star formation rate.
Globally. for the star forming cloud. one expects that Altito~Al,ODN ”- much as is ‘ound in nearby cases of star-forming clouds.
Globally, for the star forming cloud, one expects that $\dot M_{\rm outflow} \sim \dot M_{\ast}\sim 0.1 \dot M_{\rm accretion} $ , much as is found in nearby cases of star-forming clouds.
Moreover. feedback is likely to be responsible for the turbulent support in clouds that lowers the star-formation efficiency and helps to generate the conventional IMF.
Moreover, feedback is likely to be responsible for the turbulent support in clouds that lowers the star-formation efficiency and helps to generate the conventional IMF.
Thus the onset of cloud fragmentation. due eventually but we have argued. not exclusively. to the enhanced role of cooling. would allow field amplification. angular momentum transfer. feedback and low mass star formation.
Thus the onset of cloud fragmentation, due eventually but we have argued, not exclusively, to the enhanced role of cooling, would allow field amplification, angular momentum transfer, feedback and low mass star formation.
Alternatively. suppose we accept the hypothesis that the first stars. were massive objects.
Alternatively, suppose we accept the hypothesis that the first stars were massive objects.
To avoid the empirical objections discussed. above. one would have Oo argue that merging and coagulation of gas clumps resulted in formation of predominantly very massive objects. of characteristic mass i;1000M. whose fate is to form intermediate mass black holes with relatively low nucleosynthetic yields associated with their collapse.
To avoid the empirical objections discussed above, one would have to argue that merging and coagulation of gas clumps resulted in formation of predominantly very massive objects, of characteristic mass $\simgt 1000\rm M_\odot,$ whose fate is to form intermediate mass black holes with relatively low nucleosynthetic yields associated with their collapse.
In this case. the seed tields may come from jets and outflows driven by spin-up of turbulent accretion disk dynamos as a consequence of accretion onto these intermediate mass black holes.
In this case, the seed fields may come from jets and outflows driven by spin-up of turbulent accretion disk dynamos as a consequence of accretion onto these intermediate mass black holes.
One can argue that the first generation of primordial clouds. which cooled predominantly via Lyman alpha emission. preferentially formed IMBHs. since the associated minihalos of mass ~LO‘AL. are promising environments for forming IMBHs in view of the high core accretion rates (Zhao and Silk 2005).
One can argue that the first generation of primordial clouds, which cooled predominantly via Lyman alpha emission, preferentially formed IMBHs, since the associated minihalos of mass $\sim 10^7\rm M_\odot$ are promising environments for forming IMBHs in view of the high core accretion rates (Zhao and Silk 2005).
Accretion disks around the IMBHs provide promising sites for MRI dynamos.
Accretion disks around the IMBHs provide promising sites for MRI dynamos.
cleared polar cavities therefore would the expansion of HII regions and their interaction with the stellar cluster environmenteasier.
cleared polar cavities therefore would the expansion of HII regions and their interaction with the stellar cluster environment.
. The feedback of the energy injection from jets and outflows onto the stellar cluster formation is briefly discussed in ?..
The feedback of the energy injection from jets and outflows onto the stellar cluster formation is briefly discussed in \citet{Bonnell:2006p13823}.
In this study, we investigate different implementation methods for direct stellar irradiation feedback (see Sect. 3))
In this study, we investigate different implementation methods for direct stellar irradiation feedback (see Sect. \ref{sect:Method}) )
and their influence on the stability of radiation pressure dominated cavities.
and their influence on the stability of radiation pressure dominated cavities.
We present in Sect.
We present in Sect.
4 the qualitative outcome and Sects.
\ref{sect:QualitativeResults} the qualitative outcome and Sects.
5 and 6 the quantitative analyses of the simulations.
\ref{sect:QuantitativeResults1} and \ref{sect:QuantitativeResults2} the quantitative analyses of the simulations.
The observed difference in the radiative acceleration of the cavity shell depending on the applied radiation transport method is analytically derived in Sect. 7..
The observed difference in the radiative acceleration of the cavity shell depending on the applied radiation transport method is analytically derived in Sect. \ref{sect:Analytic}.
Finally, a comprehensive comparison section (Sect. 8))
Finally, a comprehensive comparison section (Sect. \ref{sect:Comparisons}) )
to previous simulations, analytic work, and observations as well as a brief summary (Sect. 9))
to previous simulations, analytic work, and observations as well as a brief summary (Sect. \ref{sect:Summary}) )
are provided.
are provided.
We compare the effect of two different radiation transport methods on stellar radiation feedback.
We compare the effect of two different radiation transport methods on stellar radiation feedback.
The goal is to investigate the stability of the shell radiation-pressure-dominated cavity.
The goal is to investigate the stability of the shell radiation-pressure-dominated cavity.
We emphasize that in these simulations we do not aim to provide a complete description of an outflow region around a massive star.
We emphasize that in these simulations we do not aim to provide a complete description of an outflow region around a massive star.
A list of potential limitations and caveats includes: To obtain unambiguous results, the relevant physical length scales have to be resolved by the numerical simulations.
A list of potential limitations and caveats includes: To obtain unambiguous results, the relevant physical length scales have to be resolved by the numerical simulations.
Therefore, we discuss these scales with respect to the resolution of our radiation hydrodynamical simulations.
Therefore, we discuss these scales with respect to the resolution of our radiation hydrodynamical simulations.
We demonstrate that the simulations resolve all relevant length scales sufficiently.
We demonstrate that the simulations resolve all relevant length scales sufficiently.
A study of a radiative Rayleigh-Taylor instability in the shell surrounding the outflow cavity needs to resolve the wavelengths of this instability along the shell discontinuity.
A study of a radiative Rayleigh-Taylor instability in the shell surrounding the outflow cavity needs to resolve the wavelengths of this instability along the shell discontinuity.
At small wavelengths, the radiative Rayleigh-Taylor instability is likely to be suppressed by diffusion, but at large wavelengths — comparable to the physical size of the cavity — the instability occurs.
At small wavelengths, the radiative Rayleigh-Taylor instability is likely to be suppressed by diffusion, but at large wavelengths – comparable to the physical size of the cavity – the instability occurs.
Quantitatively, the shortest unstable wavelength for instance is determined to be 1000 AU for a and 10000 AU for a star (?)..
Quantitatively, the shortest unstable wavelength for instance is determined to be 1000 AU for a and 10000 AU for a star \citep{Jacquet:2011p18452}.
The polar resolution rA0 of the spherical grid in our simulations grows linearly with the radius and is fixed to [email protected].
The polar resolution $r \Delta \theta$ of the spherical grid in our simulations grows linearly with the radius and is fixed to $r \Delta \theta \lesssim 0.1 r$.
The grid size along the shell discontinuity is typically a factor of ten smaller than the shortest unstable wavelength.
The grid size along the shell discontinuity is typically a factor of ten smaller than the shortest unstable wavelength.
Hence, the length scale of the radiative Rayleigh-Taylor instability is fully resolved.
Hence, the length scale of the radiative Rayleigh-Taylor instability is fully resolved.
Two other important length scales are related to the radiative properties of the shell.
Two other important length scales are related to the radiative properties of the shell.