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For completeness. we present here the solution inside the external mecium. | For completeness, we present here the solution inside the external medium. |
By assumption. the external medium has a uniform temperature and density. and a uniform rate of cooling. | By assumption, the external medium has a uniform temperature and density, and a uniform rate of cooling. |
To maintain equilibrium. there has to be some constant source of heat that exactly. compensates for the cooling. | To maintain equilibrium, there has to be some constant source of heat that exactly compensates for the cooling. |
We assume that such a source of heat exists (c.g. cosmic ravs). | We assume that such a source of heat exists (e.g., cosmic rays). |
The rotation @ is non-zero. but it clecays rapidly outward. | The rotation $\Omega$ is non-zero, but it decays rapidly outward. |
“Phe small amount of rotation helps to transport the angular momentum [lux from the star out into the external medium. | The small amount of rotation helps to transport the angular momentum flux from the star out into the external medium. |
Solving the angular momentunir conservation law (3). we obtain the following solution Latin. Omega, ‘Uprttlicted Has QUSS and post=2&8ο | Solving the angular momentum conservation law (3), we obtain the following solution , _3 where _3 = (k , and $p_{\rm ext}=2kT_{\rm
ext}\rho_{\rm ext}/m_p$. |
For i=O. the velocity scales as 2 | For $\dot m\not=0$, the velocity scales as $r^{-2}$. |
The three self-similar solutions written above are special solutions of the basie dilferential equations (2). (3) and (5). which are valid. under specific conditions. | The three self-similar solutions written above are special solutions of the basic differential equations (2), (3) and (5), which are valid under specific conditions. |
To check the validity of these analytical solutions. we have computed numerical solutions of the basic dilferential equations. | To check the validity of these analytical solutions, we have computed numerical solutions of the basic differential equations. |
We use the same code as in ΜΙΝΟΣ with two changes. | We use the same code as in MN01, with two changes. |
First. we switched to the viscosity prescription given in equation (4)). rather than the prescription vo=oclíOg used in AINOL. | First, we switched to the viscosity prescription given in equation \ref{nu}) ), rather than the prescription $\nu = \alpha
c_s^2/\Omega_K$ used in MN01. |
Second. in addition to viscous heating. we included a constant heating rate which we adjusted so as to balance the radiative cooling in the homogeneous external medium (see 822.3). | Second, in addition to viscous heating, we included a constant heating rate which we adjusted so as to balance the radiative cooling in the homogeneous external medium (see 2.3). |
The code uses a relaxation method to solve the onc-dimensional hyelrodvnamic equations. with specified inner and outer boundary conditions. | The code uses a relaxation method to solve the one-dimensional hydrodynamic equations with specified inner and outer boundary conditions. |
Although it. employs the full equations of a two-tempoerature plasma. the results are essentially equivalent to those of a one-tempoerature plasma in the region of interest for this paper. namely the region at large radius where the flow matches onto the external mecdium In the calculations. the How was taken to extend from. an inner radius Ain=3Ry, to Raw=10*Hg. | Although it employs the full equations of a two-temperature plasma, the results are essentially equivalent to those of a one-temperature plasma in the region of interest for this paper, namely the region at large radius where the flow matches onto the external medium In the calculations, the flow was taken to extend from an inner radius $R_{\rm in}=3~R_g$ to $R_{\rm out}=10^7~R_g$. |
Lhe mass accretion rate was taken to be low. ii—2«10.7. in order hat the Low should. correspond to the regime of the hot settling Low solution. | The mass accretion rate was taken to be low, $\dot m=2\times 10^{-5}$, in order that the flow should correspond to the regime of the hot settling flow solution. |
We took the viscosity. parameter to »' à=OL and set the spin of the star to be s=0.3 (i.c... of the Ixeplerian rotation at the stellar surface). | We took the viscosity parameter to be $\alpha=0.1$ and set the spin of the star to be $s=0.3$ (i.e., of the Keplerian rotation at the stellar surface). |
We took the other inner boundary conditions to be the same as in MNOI. | We took the other inner boundary conditions to be the same as in MN01. |
At the outer boundary. we specified the emperature and. density of the external medium. | At the outer boundary, we specified the temperature and density of the external medium. |
Figure 4 shows four solutions. | Figure \ref{f:prof}
shows four solutions. |
Phe external temperature is kept ixecl αἱ T(Rox)=10 lx in all the solutions. but. the external density varies by. a decade and a halt: pCoxi)=202185(,23ο2.510.&1-I0 em7. | The external temperature is kept fixed at $T(R_{\rm ext})=10^8$ K in all the solutions, but the external density varies by a decade and a half: $\rho(R_{\rm ext})= 2.5\times10^9,\
8.1\times10^8,\ 2.5\times10^8,\ 8.1\times10^7$ $^{-3}$. |
We have also clonemE other cuedlations in which we kept post fixed and varied Lis). | We have also done other calculations in which we kept $\rho_{\rm ext}$ fixed and varied $T_{\rm ext}$. |
These give very similar results. | These give very similar results. |
lis. | Fig. |
4 shows that. right next to the star. there is a boundary. laver. where the density rises sharply as one gocs into the star and the temperature drops. suddenly. | \ref{f:prof} shows that, right next to the star, there is a boundary layer, where the density rises sharply as one goes into the star and the temperature drops suddenly. |
We do not analyze this region. | We do not analyze this region. |
Once we are outside the boundary laver. the eas behaves very much according to the analytical solutions discussed in 822. | Once we are outside the boundary layer, the gas behaves very much according to the analytical solutions discussed in 2. |
Starting just outside the boundary laver and extending over a wicle range of radius. the numerical solution exhibits a selsimilar behavior with power-law dependences of the density. emperature and angular velocity. | Starting just outside the boundary layer and extending over a wide range of radius, the numerical solution exhibits a self-similar behavior with power-law dependences of the density, temperature and angular velocity. |
This region corresponds o the self-similar solution of AINOL. | This region corresponds to the self-similar solution of MN01. |
There are. in fact. wo zones. an inner lwo-tempoeratiure Zone. and an outer one-temperature zone (ALNOL). | There are, in fact, two zones, an inner two-temperature zone, and an outer one-temperature zone (MN01). |
The latter corresponds to solution 1 (eq. 6)) | The latter corresponds to solution 1 (eq. \ref{1st-sol}) ) |
discussed in 822.1. | discussed in 2.1. |
The most notable eature of this region is that the density. temperature and angular velocity of the numerical solutions are completeA independent of the outer temperature ancl density. as by the analytical solution. | The most notable feature of this region is that the density, temperature and angular velocity of the numerical solutions are completely independent of the outer temperature and density, as predicted by the analytical solution. |
The slopes of the numerical curves also agree well with the analytical scalings. | The slopes of the numerical curves also agree well with the analytical scalings. |
At a radius sas,~5105.2«107I, 9depending on the outer pressure. see eq. 13)). | At a radius $R_{\rm match}\sim5\times10^4 ... 2\times10^5~R_g$ 9depending on the outer pressure, see eq. \ref{R12}) ), |
solution 1 merges with solution 2 (eq. 11)) | solution 1 merges with solution 2 (eq. \ref{2nd-sol}) ) |
described in 822.2. | described in 2.2. |
Here. the solution does depend on the outer boundary conditions. and it scales roughly according to the slopes derived analytically, | Here, the solution does depend on the outer boundary conditions, and it scales roughly according to the slopes derived analytically. |
At even larger racii /?>Ros~Bo107.2210"A (see eq. 142). | At even larger radii $R > R_{\rm ext}\sim3\times10^5 ... 2\times10^6~R_g$ (see eq. \ref{R23}) ), |
the Dow matches onto the ambient external medium. | the flow matches onto the ambient external medium. |
In this region we have solution 3 (eq. 15)) | In this region we have solution 3 (eq. \ref{3rd-sol}) ) |
described. in 822.3. | described in 2.3. |
As expected. out here only the angular velocity and the racial velocity vary with radius. | As expected, out here only the angular velocity and the radial velocity vary with radius. |
Both have the scalings predicted for solution 3. | Both have the scalings predicted for solution 3. |
In this paper. we have removecl one piece of mystery surrounding the self-similar “hot settling [low or “hot brake” solution discovered. by ALNOL. | In this paper, we have removed one piece of mystery surrounding the self-similar “hot settling flow” or “hot brake” solution discovered by MN01. |
Specifically. we have shown that. the remarkable insensitivity. of the AINOL solution to external boundary. conditions is a consequence of the fact that the solution is insulated. from the outer boundary by the presence ofa second solution. which bridges the gap between the first solution and the external medium. | Specifically, we have shown that the remarkable insensitivity of the MN01 solution to external boundary conditions is a consequence of the fact that the solution is insulated from the outer boundary by the presence of a second solution, which bridges the gap between the first solution and the external medium. |
We derived the form of the second. solution analytically in 822.2 and showed via numerical computations (833. Fig. | We derived the form of the second solution analytically in 2.2 and showed via numerical computations 3, Fig. |
1) that the two solutions together are able to match a wide range of outer boundary. conditions. | 1) that the two solutions together are able to match a wide range of outer boundary conditions. |
This solves one of the nivsteries associated with the hot settling Dow solution. | This solves one of the mysteries associated with the hot settling flow solution. |
There are. however. two other problems that still need o be addressed. | There are, however, two other problems that still need to be addressed. |
First. the solution we have derived. treats he mass accretion rate 5)» as a free parameter. ( | First, the solution we have derived treats the mass accretion rate $\dot m$ as a free parameter. ( |
Indeed. he analvtical solutions were obtained for the limit ) be. for a hot atmosphere.) | Indeed, the analytical solutions were obtained for the limit $\dot m\to0$ , i.e. for a hot atmosphere.) |
What determines ri? | What determines $\dot m$? |
Lt is certainly not the outer boundary. since we have obtained the complete outer solution. | It is certainly not the outer boundary, since we have obtained the complete outer solution. |
The accretion rate must therefore xf determined by an innerboundary condition. | The accretion rate must therefore be determined by an innerboundary condition. |
This is not unexpected. | This is not unexpected. |
In the case of spherical accretion. one recalls hat. while the accretion rate for the transonic solution is determined by the outer boundary conditions. the accretion | In the case of spherical accretion, one recalls that, while the accretion rate for the transonic solution is determined by the outer boundary conditions, the accretion |
The solution is so that. uxiug οαοσα1.020,47. The divergence as C approaches unity is not real. as our ueglect of adiabatic losses and of acctuuulation within the shell are incorrect when the holes close up. | The solution is so that, using $(1-C_f)(1-1.045 C_f)\simeq (1-1.02 C_f)^2$, The divergence as $C_f$ approaches unity is not real, as our neglect of adiabatic losses and of accumulation within the shell are incorrect when the holes close up. |
Likewise values of füapae less than fi, are not realistic. because the wind force is always present: our error in this unit is to assume that the wind energy is thermalizecl when iu fact it remains mostly kinetic if the shell is mostly holes. | Likewise values of $f_{\rm trap,w}$ less than $f_w$ are not realistic, because the wind force is always present; our error in this limit is to assume that the wind energy is thermalized, when in fact it remains mostly kinetic if the shell is mostly holes. |
since f,&0.5. this iuplies that faa&0.22/(1Cy)~1l unless we are cousidering the case of an embedded: region with an extremely non-porous shell Lo€1l. | Since $f_w\simeq 0.5$, this implies that $f_{\rm trap,w}\simeq 0.22/(1-C_f) \sim 1$ unless we are considering the case of an embedded region with an extremely non-porous shell, $1-C_f \ll 1$. |
Such low porosity is imuplausible eiven the turbulent. chuupy nature of the interstellar medi and the fact that pressure-driven shocks tend to run down density eracdieuts and “blow out”. | Such low porosity is implausible given the turbulent, clumpy nature of the interstellar medium and the fact that pressure-driven shocks tend to run down density gradients and “blow out”. |
Even if one started with a perfectly uniforii ISM. the expanding shell is subject to? instability of a pressure-driven slab: moreover if the wind caused the expanding shell to accelerate then the Ravleigh-Tavlor instability would spontaneously create holes iu the shell. reducing C below τητν. | Even if one started with a perfectly uniform ISM, the expanding shell is subject to \citet{Vishniac83a} instability of a pressure-driven slab; moreover if the wind caused the expanding shell to accelerate then the Rayleigh-Taylor instability would spontaneously create holes in the shell, reducing $C_f$ below unity. |
Thus we couclude that fija, is at most a few. and is likely to be small than order uuitv. | Thus we conclude that $f_{\rm trap,w}$ is at most a few, and is likely to be small than order unity. |
Our conchision is cousistent with the numerical simulations of ?.. who also flud that. in a non-uuifori medi. the bulk of the mass around a youug star is not swept up into the thermal wind drveu by ~LO! K clustereas. | Our conclusion is consistent with the numerical simulations of \citet{tenorio-tagle06a}, who also find that, in a non-uniform medium, the bulk of the mass around a young star cluster is not swept up into the thermal wind driven by $\sim 10^7$ K gas. |
Iustead. that gas escapes rapidly through the porous shell. while the bulk of the mass expands more slowly (see their Figure 9). | Instead, that gas escapes rapidly through the porous shell, while the bulk of the mass expands more slowly (see their Figure 9). |
The above calculation depends somewhat ou our estimate of the term Mag. which is uncertain for several reasons: because the phivsies of ablation is uot well understood. because the ablating area could be very different from for instance if the shells structure is amore interestingπμ. than a broken sphere. and because other mechiauisuis like thermal evaporation. photoevaporation. aud cloud disruption can all imject mass. | The above calculation depends somewhat on our estimate of the term $\dot{M}_{\rm abl}$, which is uncertain for several reasons: because the physics of ablation is not well understood, because the ablating area could be very different from $4\pi C_f \rii^2$, for instance if the shell's structure is more interesting than a broken sphere, and because other mechanisms like thermal evaporation, photoevaporation, and cloud disruption can all inject mass. |
Similarly. the deusitv of the ablating gas could be lugher than indicated by our pressurc-balauce areument. since photocvaporation nüght compress the gas (0.8.?).. | Similarly, the density of the ablating gas could be higher than indicated by our pressure-balance argument, since photoevaporation might compress the gas \citep[e.g.][]{bertoldi90}. |
Given these uncertainties. one might consider ey better constrained than Maji for mstance. au upper limit ou the N-rav huuinosity implies a lower limit oun ex. | Given these uncertainties, one might consider $c_X$ better constrained than $\dot{M}_{\rm abl}$ – for instance, an upper limit on the X-ray luminosity implies a lower limit on $c_X$ . |
Takine ον as given aud solving equations (21)) aud (25)) for Py. we find so trapping is muportant when (1CyyeyCpfafd~ OA, | Taking $c_X$ as given and solving equations \ref{HotGasConservation1}) ) and \ref{HotGasConservation2}) ) for $P_X$, we find so trapping is important when $(1-C_f) c_X/v_w < f_w/5 \simeq 0.1$ . |
Protostellar winds represeut a separate. brief. but very intense phase which deserve separate mention. | Protostellar winds represent a separate, brief, but very intense phase which deserve separate mention. |
Bene magnetically launched. they axe strougly collimated aud may far exceed the photon momentum: moreover the entire stellar population generates them. | Being magnetically launched, they are strongly collimated and may far exceed the photon momentum; moreover the entire stellar population generates them. |
Rather than assess them directly we appeal to the treatinent by ?.. who found that protostellar winds are very significant in the formation of the Pleiades and Orion Nebula clusters. but quite insignificant iu the formation of the Arches or more Inassive clusters. | Rather than assess them directly we appeal to the treatment by \citet{matzner07}, who found that protostellar winds are very significant in the formation of the Pleiades and Orion Nebula clusters, but quite insignificant in the formation of the Arches or more massive clusters. |
If the expanding shell traps the iutrared light eiuitted within if. then ifs momentum will exceed. that of the diivius starlighit by a factor where Pug is the pressure of the trapped IR radiation field inside the shell. | If the expanding shell traps the infrared light emitted within it, then its momentum will exceed that of the driving starlight by a factor where $P_{\rm IR}$ is the pressure of the trapped IR radiation field inside the shell. |
We first cousider the highly idealized case of a uniform. non-porous shell C;=1. which provides an upper Tit on fap. before treating leakage through holes. | We first consider the highly idealized case of a uniform, non-porous shell, $C_f = 1$, which provides an upper limit on $\ftrapIR$, before treating leakage through holes. |
When a uniform shell is optically thick to its own thermal enüssiou. it radiates from a photosphere above which the dus-averaged optical depth is 2/3. | When a uniform shell is optically thick to its own thermal emission, it radiates from a photosphere above which the flux-averaged optical depth is 2/3. |
The enmüssion is characterized by the shells effective temperature. which satisfies InroupTil.=L. and the fiux-averaged nass opacity above the photospliere is approximately the Plauck mean #p(Tug). implviug a colunu Syn2ΑΜνι) above the photosphere. | The emission is characterized by the shell's effective temperature, which satisfies $4\pi \rii^2 \sigma_{\rm SB} \Teffsh^4 = L$, and the flux-averaged mass opacity above the photosphere is approximately the Planck mean $\kappa_P(\Teffsh)$, implying a column $\Sigma_{\rm ph}\simeq2/[3\kappa_P(\Teffsh)]$ above the photosphere. |
Shells with “yj,=Mi,< are optically thin to reprocessed light. | Shells with $\Sigma_{\rm sh} = \msh/(4\pi \rii^2)<\Sigma_{\rm ph}$ are optically thin to reprocessed light. |
(Liv?) Thick shells.δα with Xa,>> Man can be treated du the diffusion approximation (Pir) ιο and sg the Rosselaud mean mass opacity). below the photosphere. | Thick shells, with $\Sigma_{\rm sh}\gg \Sigma_{\rm ph}$ , can be treated in the diffusion approximation $d P_{\rm IR}/\kappa_R(T) = - \sigma_{\rm SB} \Teffsh^4 d\Sigma$ with $P_{\rm IR}=a T(\Sigma)^4/3$ and $\kappa_R$ the Rosseland mean mass opacity), below the photosphere. |
The solution for the pressurewithin au optically thick shell is where JP)—fyPirUPigiT). aud Fo+) is. the inverse function of JF. | The solution for the pressurewithin an optically thick shell is where ${\cal F}(P_{\rm IR}) = \int_0^{P_{\rm IR}} dP_{\rm IR}'/\kappa_R(T')$, and ${\cal F}^{(-1)}$ is the inverse function of $\cal F$. |
Evaluating equation (33)) for the To staucdare dust model "A7 for Ry=5.5. we fiud that to good accuracy. where Mayuai is understood to be in gea7. | Evaluating equation \ref{Prad}) ) for the \citet{weingartner01} standard dust model “A” for $R_V=5.5$, we find that to good accuracy, where $\Sigma_{\rm shell}$ is understood to be in $^{-2}$. |
This result shows that radiation trapping can be quite significant. for instance when XaZl aud Ditech>GO TN. as is expected around a Iuuiuous vouug cluster: however this estimate is unrealistically igh wheu radiation can leak away. | This result shows that radiation trapping can be quite significant, for instance when $\Sigma_{\rm shell} \gtsim 1$ and $\Teffsh > 60$ K, as is expected around a luminous young cluster; however this estimate is unrealistically high when radiation can leak away. |
Iu the more realistic case where leakage is important. we can compute Pip dm a manner analogous to our calculation of Py. by balancing the rate at which energy the stars inside the shell add cucrey against the rate at which it leaks out. | In the more realistic case where leakage is important, we can compute $P_{\rm IR}$ in a manner analogous to our calculation of $P_X$, by balancing the rate at which energy the stars inside the shell add energy against the rate at which it leaks out. |
Our treatment here is a simplified version of that given in Appendix D of ?.. | Our treatment here is a simplified version of that given in Appendix D of \citet{mckee08a}. |
We limit our attention to the case where the shell is optically thick (Sopa9 Myn(Toarun)) ou average: in this limit we cau conipute the energy density of the trapped radiation field when the shell is porous. C,« I. siniply by treating the shell as a perfectly opaque sphere with holes iu it. | We limit our attention to the case where the shell is optically thick $\Sigma_{\rm shell}\gg \Sigma_{\rm ph}(\Teffsh)$ ) on average: in this limit we can compute the energy density of the trapped radiation field when the shell is porous, $C_f < 1$ , simply by treating the shell as a perfectly opaque sphere with holes in it. |
Tn this case enerevbalance requires that | In this case energybalance requires that |
moments. | moments. |
For more details the reader is referred to the review article by Stergioulas (2003). | For more details the reader is referred to the review article by Stergioulas (2003). |
The interior and exterior spacetime of a stationary. axisymmetric star is described by a metric in the following form: ολαBre Qut)7 rugduh where v.B.@ and @ are four metric functions to be determined by solving four field equations. | The interior and exterior spacetime of a stationary, axisymmetric star is described by a metric in the following form: dt^2+B^2 dt)^2 ^2), where $\nu,~B, ~\alpha$ and $\omega$ are four metric functions to be determined by solving four field equations. |
In the numerical method of Komatsu (1989. henceforth ΚΕΠ) one defines two auxiliary functions p. Y through the relations v—(yp)/2 and B—οἳ. | In the numerical method of Komatsu (1989, henceforth KEH) one defines two auxiliary functions $\bar \rho$, $\bar \gamma$ through the relations $\nu=(\bar \gamma + \bar \rho)/2$ and $B=e^{\bar \gamma}$. |
Then. three out of the four field equations are written in the following integral forms where rr. ryr. rr. rjr. and S5. Sy and 5, are lengthy source terms. whose expressions can be found in KEH. | Then, three out of the four field equations are written in the following integral forms _0^1 r'^2 ^2(r,r') ) ), _0^1 r'^2 ) _0^1 r'^3 ) ), where r, r'>r, r, r'>r, and $S_{\bar \rho}$, $S_{\bar \gamma}$ and $S_\omega$ are lengthy source terms, whose expressions can be found in KEH. |
In the equations above. 4=cos. while P,(44) denotes the Legendre polynomials and P7'(u) the associated Legendre functions. | In the equations above, $\mu=\cos\theta$, while $P_n(\mu)$ denotes the Legendre polynomials and $P_n^m(\mu)$ the associated Legendre functions. |
The metric function & is determined by an ordinary differential equation. | The metric function $\alpha$ is determined by an ordinary differential equation. |
We compute numerical equilibrium models using the code by Stergioulas and Friedman (1995) (see Nozawa 1998 and Stergioulas 2003 for extensive accuracy tests). | We compute numerical equilibrium models using the code by Stergioulas and Friedman (1995) (see Nozawa 1998 and Stergioulas 2003 for extensive accuracy tests). |
The numerical code uses the CST formulation. in which the KEH equations are written in terms of a compactified coordinate s detined through the relation where rp is the (coordinate) radius of the stellar equator. | The numerical code uses the CST formulation, in which the KEH equations are written in terms of a compactified coordinate $s$ defined through the relation ), where $r_e$ is the (coordinate) radius of the stellar equator. |
This allows the computation of the whole exterior spacetime out to infinity. which is important in detailed comparisons of the numerical metric to the analytic metric. | This allows the computation of the whole exterior spacetime out to infinity, which is important in detailed comparisons of the numerical metric to the analytic metric. |
For a configuration that 1s stationary. axisymmetric. symmetric with respect to reflections in the equatorial plane and asymptotically flat. the spacetime can be characterized by twosets of scalar multipole moments: the even-valued mass moments (Mo.Mo. M4...) and the odd-valued current moments (51.δὲ. Ss... | For a configuration that is stationary, axisymmetric, symmetric with respect to reflections in the equatorial plane and asymptotically flat, the spacetime can be characterized by twosets of scalar multipole moments: the even-valued mass moments $M_0,~M_2,~M_4\dots$ ) and the odd-valued current moments $S_1,~S_3,~S_5\dots$ ). |
) Ryan (1997) presented a method for extracting the multipole moments from the asymptotic formof the metric functions. | Ryan (1997) presented a method for extracting the multipole moments from the asymptotic formof the metric functions. |
The lowest-order appearance of each moment in terms of a power series in |/r is determined by the expansions |. )and | The lowest-order appearance of each moment in terms of a power series in $1/r$ is determined by the expansions ), and . |
LS/LIS transition. | LS/HS transition. |
Since the accretion rate during the LS is very close to the critical value. a slight. increase in the accretion rate by a small factor would produce a stable disc. | Since the accretion rate during the LS is very close to the critical value, a slight increase in the accretion rate by a small factor would produce a stable disc. |
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