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A&A 377, 589608 (2001)DOI: 10.1051/0004-6361:20011145c ESO 2001

Astronomy&

Astrophysics

Atomic T Tauri disk winds heated by ambipolar diffusion

I. Thermal structure

P. J. V. Garcia1,2, , J. Ferreira3, S. Cabrit4, and L. Binette5

1 Centro de Astrofsica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal2 CRAL/Observatoire de Lyon, CNRS UMR 5574, 9 avenue Charles Andre, 69561 St. Genis-Laval Cedex, France3 Laboratoire dAstrophysique de lObservatoire de Grenoble, BP 53, 38041 Grenoble Cedex, France4 Observatoire de Paris, DEMIRM, UMR 8540 du CNRS, 61 avenue de lObservatoire, 75014 Paris, France5

Instituto de Astronoma, UNAM, Ap. 70-264, 04510 D. F., Mexico

Received 21 March 2001 / Accepted 3 August 2001

Abstract. Motivated by recent subarcsecond resolution observations of jets from T Tauri stars, we extend thework of Safier (1993a,b) by computing the thermal and ionization structure of self-similar, magnetically-driven,atomic disk winds heated by ambipolar diffusion. Improvements over his work include: (1) new magnetized cold jetsolutions consistent with the underlying accretion disk (Ferreira 1997); (2) a more accurate treatment of ionizationand ion-neutral momentum exchange rates; and (3) predictions for spatially resolved forbidden line emission (maps,long-slit spectra, and line ratios), presented in a companion paper, Garcia et al. (2001). As in Safier (1993a), weobtain jets with a temperature plateau around 104 K, but ionization fractions are revised downward by a factorof 10100. This is due to previous omission of thermal speeds in ion-neutral momentum-exchange rates and todifferent jet solutions. The physical origin of the hot temperature plateau is outlined. In particular we presentthree analytical criteria for the presence of a hot plateau, applicable to any given MHD wind solution where

ambipolar diffusion and adiabatic expansion are the dominant heating and cooling terms. We finally show that,for solutions favored by observations, the jet thermal structure remains consistent with the usual approximationsused for MHD jet calculations (thermalized, perfectly conducting, single hydromagnetic cold fluid calculations).

Key words. ISM: jets and outflows stars: pre-main sequence MHD line: profiles accretion disks

1. Introduction

Progresses in long slit differential astrometry techniquesand high angular resolution imaging from Adaptive Opticsand the Hubble Space Telescope have shown that the highvelocity forbidden emission observed in Classical T TauriStars (CTTSs) is related to collimated (micro-)jets (e.g.Solf 1989; Solf & Bohm 1993; Ray et al. 1996; Hirth et al.1997; Lavalley-Fouquet et al. 2000; Dougados et al. 2000;Bacciotti et al. 2000). Although outflow activity is knownto decrease with age (Bontemps et al. 1996), CTTSs stillharbor considerable activity (e.g. Mundt & Eisloffel 1998)and present the advantage of not being embedded. It isnow commonly believed that such jets are magneticallyself-confined, by a hoop stress due to a non-vanishingpoloidal current (Chan & Henriksen 1980; Heyvaerts &Norman 1989). The main reason lies in the need to pro-

Send offprint requests to : P. J. V. Garcia,e-mail: [email protected] CAUP Support Astronomer, during year 2000, at the Isaac

Newton Group of Telescopes, Sta. Cruz de La Palma, Spain.

duce highly supersonic unidirectional flows. Indeed, thisrequires an acceleration process that is closely related tothe confining mechanism. The most promising models ofjet production rely therefore on the presence of large scalemagnetic fields, extracting energy and mass from a ro-

tating object. However, we still do not know preciselywhat the jet driving sources are. Moreover, observed jetsharbor time-dependent features, with time-scales rang-ing from tens to thousands of years. Such time-scales aremuch longer than those involving the protostar or the in-ner accretion disk. Therefore, although the possibility re-mains that jets have a non-stationary origin (e.g. Ouyed& Pudritz 1997; Goodson et al. 1999), only steady-statemodels will be addressed here.

Stationary stellar wind models have been developed(e.g. Sauty & Tsinganos 1994), however observed correla-tions between signatures of accretion and ejection clearly

show that the disk is an essential ingredient in jet for-mation (Cohen et al. 1989; Cabrit et al. 1990; Hartiganet al. 1995). Therefore we expect accretion and ejection tobe interdependent, through the action of magnetic fields.

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590 P. J. V. Garcia et al.: Atomic T Tauri disk winds heated by ambipolar diffusion. I.

There are mainly two classes of stationary magnetized diskwind models, depending on the radial extent of the wind-producing region in the disk. In the first class (usuallyreferred to as disk winds), a large scale magnetic fieldthreads the disk on a large region (Blandford & Payne1982; Wardle & Koenigl 1993; Ferreira & Pelletier 1993,1995; Li 1995, 1996; Ferreira 1997; Krasnopolsky et al.1999; Casse & Ferreira 2000a,b; Vlahakis et al. 2000). Sucha field is assumed to arise from both advection of interstel-lar magnetic field and local dynamo generation (Rekowskiet al. 2000). In the second class of models (referred toas X-winds), only a tiny region around the disk inneredge produces a jet (Camenzind 1990; Shu et al. 1994,1995, 1996; Lovelace et al. 1999). The magnetic field isassumed to originally come from the protostar itself, af-ter some eruptive phase that linked the disk inner edge tothe protostellar magnetosphere. Note that in both mod-

els, jets extract angular momentum and mass from theunderlying portion of the disk. However, by construction,disk-winds are produced from a large spread in radii,while X-winds arise from a single annulus. Apart fromdistinct disk physics, the difference in size and geometry ofthe ejection regions should also introduce some observablejet features. Another scenario has been proposed, wherethe protostar produces a fast collimated jet surroundedby a slow uncollimated disk wind or disk corona (Kwan& Tademaru 1988, 1995; Kwan 1997), but such a scenariostill lacks detailed calculations.

So far, all disc-driven jet calculations used a coldapproximation, i.e. negligible thermal pressure gradi-ents. Therefore, each magnetic surface is assumed eitherisothermal or adiabatic. But to test which class of modelsis at work in T Tauri stars, reliable observational predic-tions must be made and the thermal equilibrium needsthen to be solved along the flow. Such a difficult task isstill not addressed in a fully self-consistent way, namely bysolving together the coupled dynamics and energy equa-tions. Thus, no model has been able yet to predict the gasexcitation needed to generate observational predictions.

One first possibility is to use a posteriori a simple pa-rameterization for the temperature and ionization frac-tion evolution along the flow. This was done by Shang

et al. (1998) and Cabrit et al. (1999) for X-winds and diskwinds respectively. These approaches are able to predictthe rough jet morphology, but do not provide reliable fluxand line profile predictions, since the thermal structurelacks full physical consistency.

The second possibility is to solve the thermal evolutiona posteriori, with the difficulty of identifying the heatingsources (subject to the constraint of consistency with theunderlying dynamical solution). Several heating sourcesare indeed possible: (1) planar shocks (e.g. Hartigan et al.1987, 1994); (2) oblique magnetic shocks in recollimatingwinds (Ouyed & Pudritz 1993, 1994); (3) turbulent mixing

layers (e.g. Binette et al. 1999); and (4) current dissipationby ion-neutral collisions, referred to as ambipolar diffusionheating (Safier 1993a,b). A further heating scenario (notyet explored in the context of MHD jets and only valid

in some environments) is photoionization from OB stars(Reipurth et al. 1998; Raga et al. 2000; Bally & Reipurth2001). Of all these previous mechanisms only ambipolardiffusion heating allows minimal thermal solutions, inthe sense that the same physical process non-vanishingcurrents is responsible for jet dynamics and heating. Asa consequence no additional tunable parameter is invokedfor the thermal description. Furthermore, Safier (1993b)was able to obtain fluxes and profiles in reasonable agree-ment with observations. In this paper, we extend the workof Safier (1993a,b) by (1) using magnetically-driven jetsolutions self-consistently computed with the underlyingaccretion disk, and (2) a more accurate treatment of ion-ization using the Mappings Ic code and ion-neutral mo-mentum exchange rates which include the thermal con-tribution. In a companion paper (Garcia et al. 2001,hereafter Paper II), we generate predictions for spatially

resolved orbidden line emission maps, long-slit spectra,and line ratios.

This article is structured as follows: in Sect. 2 we in-troduce the dynamical structure of the disk wind understudy, and present physical values of the density, veloc-ity, magnetic field, and Lorentz force along streamlines;in Sect. 3 we describe the physical processes taken intoaccount in the thermal evolution computations, whose re-sults are presented and discussed in Sect. 4. Conclusionsare presented in Sect. 5. Some important derivations, dustdescription and consistency checks of our calculations arepresented in the appendices.

2. Dynamical structure

2.1. General properties

A precise disk wind theory must explain how much matteris deviated from radial to vertical motion, as well as theamount of energy and angular momentum carried away.This implies a thorough treatment of both the disk interiorand its matching with the jets, namely to consider magne-tized accretion-ejection structures (hereafter MAES). Theonly way to solve such an entangled problem is to take intoaccount all dynamical terms, a task that can be properly

done within a self-similar framework.In this paper, we use the models of Ferreira (1997)

describing steady-state, axisymmetric MAES under thefollowing assumptions: (i) a large scale magnetic field ofbipolar topology is threading a geometrically thin disk;(ii) its ionization is such that MHD applies (neutrals arewell-coupled to the magnetic field); (iii) some active turbu-lence inside the disk produces anomalous diffusion allow-ing matter to cross the field lines. Two extra simplifyingassumptions were used: (iv) jets are assumed to be cold,i.e. powered by the magnetic Lorentz force only (the cen-trifugal force is due to the Lorentz azimuthal torque), with

isothermal magnetic surfaces (the midplane temperaturevarying as T0 r1) and (v) jets carry away all disk an-gular momentum. This last assumption has been removedonly recently by Casse & Ferreira (2000a).

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P. J. V. Garcia et al.: Atomic T Tauri disk winds heated by ambipolar diffusion. I. 591

All solutions obtained so far display the same asymp-totic behavior. After an opening of the jet radius lead-ing to a very efficient acceleration of the plasma, the jetundergoes a refocusing towards the axis (recollimation).All self-similar solutions are then terminated, most prob-ably producing a shock (Gomez de Castro & Pudritz 1993;Ouyed & Pudritz 1993). This systematic behavior couldwell be imposed by the self-similar geometry itself andnot be a general result (Ferreira 1997). Nevertheless, sucha shock would occur in the asymptotic region, far awayfrom the disk. Thus, we can confidently use these solu-tions in the acceleration zone, where forbidden emissionlines are believed to be produced (Hartigan et al. 1995).

2.2. Model parameters

The isothermal self-similar MAES considered here are de-

scribed with three free dimensionless local parameters (seeFerreira 1997, for more details) and four global quantities:(1) the disk aspect ratio

=h()

(1)

where h() is the vertical scale height at the cylindricalradius ;(2) the MHD turbulence parameter

m =m

VAh(2)

where m is the required turbulent magnetic diffusivityand VA the Alfven speed at the disk midplane; this dif-fusivity allows matter to cross field lines and therefore toaccrete towards the central star. It also controls the am-plitude of the toroidal field at the disk surface.(3) the ejection index

=d ln Macc()

d ln (3)

which measures locally the ejection efficiency ( = 0 in astandard accretion disk), but also affects the jet opening(a higher translates in a lower opening);(4) M the mass of the central protostar;

(5) i the inner edge of the MAES;(6) e the outer edge of the MAES, a standard accretiondisk lying at greater radii. This outer radius is formallyimposed by the amount of open, large scale magnetic fluxthreading the disk and producing jets;(7) Macc, the disk accretion rate fueling the MAES andmeasured at e.

For our present study, we keep only and Macc asfree parameters and fix the values of the other five asfollows: The disk aspect ratio was measured by Burrowset al. (1996) for HH 30 as 0.1 so we fix = 0.1. TheMHD turbulence parameter is taken m = 1 in order to

have powerful jets (Ferreira 1997). The stellar mass is fixedat M = 0.5 M, typical for T Tauri stars, and the innerradius of the MAES is set to i = 0.07 AU (typical diskcorotation radius for a 10 days rotation period): inside

this region the magnetic field topology could be signif-icantly affected by the stellar magnetosphere-disk inter-action. The outer radius is kept at e = 1AU for consis-tency with the one fluid approximation (Appendix C) andthe atomic gas description. Regarding atomic consistency,Safier (1993a) solved the flow evolution assuming iniciallyall H bound in H2. He found H2 to completely dissociate atthe wind base, for small 0. However, after a critical flowline footpoint H2 would not completely dissociate, there-fore affecting the thermal solution. This critical footpointwas at 3 AU for his MHD solution nearer our parameterspace.

We note that our two free parameters are still boundedby observational constraints: mass conservation relates theejection index to the accretion/ejection rates ratios,

2MJ Macc ln e

i

(4)

The observational estimates for the ratio of mass outflowrate by mass accretion rate are MJ/Macc 0.01 (Hartiganet al. 1995). The uncertainties affecting these estimatescan be up to a factor of 10 (Gullbring et al. 1998; Lavalley-Fouquet et al. 2000). The range of ejection indexes consid-ered here (0.0050.01) is kept compatible with Hartiganscanonical value. The accretion rates Macc are also keptfree but inside the observed range of 108 M yr

1 to105 M yr

1 in T Tauri stars (Hartigan et al. 1995).Table 1 provides a list of some disk and jet parameters.

These local parameters were constrained by steady-state

requirements, namely the smooth crossing of MHD criticalpoints. Disk parameters are useful to give us a view ofthe physical conditions inside the disk. Thus, the requiredmagnetic field B0 at the disk midplane and at a radialdistance 0 is

B0 = 0.3

MM

14

Macc

107 M yr1

12 0

1 AU

2 54

G.

(5)

The global energy conservation of a cold MAES writes

Pacc = 2Pjet + 2Prad (6)

where Pacc is the mechanical power liberated by the accre-tion flow, Pjet the total (kinetic, thermal and magnetic)power carried away by one jet and Prad the luminosity ra-diated at one surface of the disk. For the solutions used,the accretion power is given by

Pacc = GMMacc

2i= 0.1

MM

Macc

107 M yr1

i

0.07AU

1L (7)

where L is the solar luminosity and the efficiency fac-tor = (i/e) (i/e) depends on both the local

ejection efficiency and the MAES radial extent. Typicalvalues for our solutions are 0.9. The ratio Pjet/Prad

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Table 1. Isothermal MAES parameters. With = 0.1 andm = 1, the only parameter remaining free is . Here, the mag-netic lever arm , mass load and initial jet opening angle 0are presented to ease comparison with Safiers models. Howeverthese parameters do not uniquely determine the MHD solution.

Solution Pjet/Prad 0()

A 0.010 0.729 1.46 0.014 41.6 50.6B 0.007 0.690 1.46 0.011 59.4 52.4C 0.005 0.627 1.52 0.009 84.2 55.4

Fig. 1. Several wind quantities along a streamline for modelA (long-dashed line), B (solid) and C (dashed): jet nuclei den-sity n, velocity, magnetic field, and Lorentz force. For the lat-ter three vectors, poloidal components (vp, Bp, ( J B )p) areplotted in black and toroidal components (v B, (J B ))in red. The field line is anchored at 0 = 0.1 AU, around a1 M protostar, with an accretion rate Macc = 10

6 M yr1.

is given in Table 1. The jet parameters, mass load and magnetic lever arm , have the same definition as in

Blandford & Payne (1982). They are given here to allowa comparison with the solutions used in Safiers work.

2.3. Physical quantities along streamlines

In order to obtain a solution for the MAES, a variable sep-aration method has been used allowing to transform theset of coupled partially differential equations into a setof coupled ordinary differential equations (ODEs). Hence,the solution in the (, z) space is obtained by solving fora flow line and then scaling this solution to all space.Once a solution is found (for a given set of parameters

in Sect. 2.2), the evolution of all wind quantities Q alongany flow line is given by:

Q(, z) = Q0(0)Q() (8)

where the self-similar variable = z/0 measures theposition along a streamline flowing along a magnetic sur-face anchored in the disk at 0. In particular, the flowline shape equation is given by (z) = 0(), wherethe function () is provided by solving the full dynami-cal problem. In Fig. 1 we plot the values of the jet nucleidensity n and the poloidal and toroidal components of jetvelocity, magnetic field and Lorentz force. This is donefor our 3 models, along a streamline with 0 = 0.1 AU,M = 1 M and Macc = 10

6 M yr1. Values for other

0 and Macc can easily be deduced from these plots usingthe following scalings

n MaccM12

3

2

0

v M12 12

0

B

M

12

accM14

254

0 (9)

J M12accM14 294

0 .

The terminal poloidal velocity is vp,

GM/0so that solutions with smaller opening angles also reachsmaller terminal velocities, with higher terminal densities.The point where v reaches a minimum is also the pointwhere the jet reaches its maximum width (we call it rec-ollimation point), before the jet starts to bend towardsthe axis. The numerical solution becomes unreliable as wemove away from this point. The MHD solution is stoppedat the super-Alfvenic point, which is reached nearer forhigher . An illustration of the resulting (, z) distribu-

tion of density n and total velocity modulus for model Acan be found in Fig. 1 of Cabrit et al. (1999).

3. Flow thermal and ionization processes

3.1. Main equations

Under stationarity, the thermal structure of an atomic(perfect) gas with density n and temperature T is givenby the first law of thermodynamics:

P v + Uv = , (10)

where P = nkT is the gas pressure, U = 32 nkT its internalenergy density, v the total gas velocity and and arerespectively the heating and cooling rates per unit volume.Since during most of the flow the ejected gas expands, wecall the term adia = P v the adiabatic cooling.

The gas considered here is composed of electrons,ions and neutrals of several atomic species, namely n =ne + ni + nn where the overline stands for a sum overall present chemical elements. We then define the densityof nuclei n = ni + nn and the electron density ne = fen.Correspondingly, the total velocity v appearing in Eq. (10)must be understood as the barycentric velocity. As usual

in one-fluid approximation, we suppose and verify it inSect. C.1 all species well coupled (through collisions),so that they share the same temperature T. We also as-sume that no molecule formation occurs, so that mass

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conservation requires

nv = 0. (11)Under stationarity, the gas species ionization state evolves

according to the rate equations,DfAi

Dt= v fAi =

RAi

n, (12)

DfeDt

= v fe =A

iNAi=1

iRAi

n, (13)

subjected to the elemental conservation constraint,

iNAi=0

RAi = 0. (14)

In the above equations fAi = nAi/n is the population

fraction of the chemical element A, i times ionized, RAithe rate of change of that element and NA is its maxi-mum ionization state. Note that RAi is a function of allthe species densities nAi , the temperature and the radi-ation field. In order to obtain the gas temperature andionization state, we must solve energy equation coupledto the ionization evolution. This task requires to specifythe ionization mechanisms as well as the gas heating ()and cooling () processes (other than adia).

3.2. Ionization evolution

3.2.1. The Mappings Ic codeWe solve the gas ionization state (Eqs. (12) to (14)) usingthe Mappings Ic code Binette et al. (1985); Binette &Robinson (1987); Ferruit et al. (1997). This code considersatomic gas composed by the chemical elements H, He, C,N, O, Ne, Fe, Mg, Si, S, Ca, Ar. We also added Na (whoseionization evolution is not solved by Mappings Ic), as-suming it to be completely ionized in Na ii. Hydrogen andHelium are treated as five level atoms.

The rate equations solved by Mappings Ic includephotoionization, collisional ionization, secondary ioniza-tion due to energetic photoelectrons, charge exchange, re-

combination and dielectronic recombination. This is incontrast with Safier, who assumed a fixed ionization frac-tion for the heavy elements and solved only for the ioniza-tion evolution of H and He, considering two levels for Hand only the ground level for He.

The adopted abundances are presented in Table 2.In contrast with Safier, we take into account heavy el-ement depletion onto dust grains (see Sect. 3.2.3 andAppendix B) in the dusty region of the wind.

3.2.2. Photoionizing radiation field

For simplicity, the central source radiation field is de-scribed in exactly the same way as in Safier and we referthe reader to the expressions (C1C10) presented in hisAppendix C. This radiation field is diluted with distance

Table 2. Abundances by number of various elements with re-spect to hydrogen. The notation 3.55(4) means 3.55 104.Column Z gives solar abundances, from Savage & Sembach(1996). Column ZDL84 elemental abundances locked in grainsfor a MRN-type dust distribution, from Draine & Lee (1984).

Column Zd gives the abundances locked in grains in the diffuseclouds toward Ophiuchi (Savage & Sembach 1996)), com-puted using the solar gas phase abundances from the same au-thors. The depleted abundances adopted here are Z = ZZd.

Element Z ZDL84 Zd

H 1.0

He 1.0(1)

C 3.55(4) 3.0(4) 2.17(4)

N 9.33(5) 1.39(5)

O 7.41(4) 1.36(4) 3.39(4)

Ne 1.23(4)

Fe 3.24(5) 3.01(5) 3.22(5)

Mg 3.80(5) 3.71(5) 3.69(5)

Si 3.55(5) 3.33(5) 3.37(5)

S 1.86(5)

Ca 2.19(6) 2.19(6)

Ar 3.63(6) 2.43(6)

Na 2.04(6) 1.81(6)

but is also absorbed by intervening wind material ejectedat smaller radii.

We treat the radiative transfer as a simple absorptionof the diluted central source, namely

4J(r, ) =L

()

4r2 e(r,), (15)

where J(r, ) is the local mean monochromatic intensityat a spherical radius r and angle with the disk vertical ,L() is the emitted luminosity of both star and boundarylayer and (r, ) the optical depth towards the centralobject.

We now address the question of optical depth. Inour model, the flow is hollow, starting from a ring lo-cated at the inner disk radius i and extending to theouter radius e. The jet inner boundary is therefore ex-posed to the central ionizing radiation, which produces

then a small layer where hydrogen is completely pho-toionized. The width r of this layer can be computedby equating the number of emitted H ionizing photons,Q(H0) =

12

0

Ld/h, to the number of recombina-

tions in this layer, n2HB(T)2r2r for our geometry. We

found that r r, and thus assume that all photons ca-pable of ionizing hydrogen are exhausted within this thinshell. Furthermore, there is presumably matter in the in-ner hollow region, so the previous considerations areupper limits.

For the heavy elements, photoionization optical depthsare negligible, due to the much smaller abundances, and

are thus ignored. The opacity is assumed to be dom-inated by dust absorption (see Appendix B for details).Dust will influence the ionization structure at the base ofthe flow, where ionization is dominated by heavy elements.

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To summarize, the adopted radiation field is a centralsource absorbed by dust, with a cutoff at and above theHydrogen ionization frequency.

3.2.3. Dust properties and gas depletionSafier showed that if dust exists inside the disk, thenthe wind drag will lift the dust thereby creating a dustywind. Our wind shares the same property. We model thedust (Appendix B) as a mix of graphite and astronomi-cal silicate, with a MRN size distribution and use for thedust optical properties the tabulated values of Draine &Lee (1984), Draine & Malhotra (1993), Laor & Draine(1993). For simplicity we assumed the dust to be station-ary, in thermodynamic equilibrium with the central radia-tion field and averaged all dust quantities by the MRN sizedistribution.

In addition, we take into account depletion of heavy el-ements into the dust phase. This effect was not consideredby Safier. In Table 2 we present the dust phase abundancesneeded to maintain the MRN distribution (Draine & Lee),and our adopted depleted abundances, taken from obser-vations of diffuse clouds toward Ori (Savage & Sembach1996). These are more realistic, although presenting lessdepletion of carbon than required by MRN. Depletion hasonly a small effect on the calculated wind thermal struc-ture, but can be significant when comparing to observedline ratios based on depleted elements.

3.3. Heating and cooling mechanisms

3.3.1. MHD heating

The dissipation of electric currents J provides a local heat-ing term per unit volume MHD = J (E+ vc B), whereE and B are the electric and magnetic fields, v the fluidvelocity and c the speed of light. In a multi-componentgas, with electrons and several ion and neutral species,the generalized Ohms law writes

E+v

cB = J

n

2 1c2

(JB) Bminniin

Peene+J

B

enec, (16)

where is the fluid electrical resistivity, and n arethe total and neutral mass density, min the reduced ion-neutral mass, ni the ionic density and in the ion-neutralcollision frequency. The overline stands for a sum over allchemical elements relevant to a given quantity. These lastquantities depend on the gas ionization state, the temper-ature and the momentum exchange rate coefficients. Thereader is referred to Appendix A for the expressions ofthese coefficients and the uncertainties affecting them.

The first term appearing in the right hand side of the

generalized Ohms law is the usual Ohms term, while thesecond describes the ambipolar diffusion, the third is theelectric field due to the electron pressure and the last isthe Hall term. This last effect provides no net dissipation

in contrary to the other three. It turns out that the dis-sipation due to the electronic pressure is quite negligibleand has been therefore omitted (Appendix C). Thus, theMHD dissipation writes

MHD = J2 +

n

2 1c2

JB2

minniin Ohm + drag. (17)

The first term, Ohmic heating Ohm, arises mainly fromion-electron drag. The second term is the ambipolar diffu-sion heating drag and is mainly due to ion-neutral drag.This last term is the dominant heating mechanism inSafiers disk wind models, as well as in ours.

An important difference with Safier is that we takeinto account thermal speeds in ion-neutral momentum ex-change rate coefficients. This increases in, and results insignificantly smaller ionization fractions (Sect. 4.8).

3.3.2. Ionization/recombination cooling

Both collisional ionization cooling col and radiative re-combination cooling rec effects are taken into account byMappings Ic. These terms are given by,

col =A

i

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will allow us to compute emission maps and line profilesfor comparison with observations (see Paper II). We in-clude cooling by hydrogen lines, rad(H), in particular H,which could not be computed by Safier (two-level atomdescription).

3.3.4. Other minor heating/cooling mechanisms

Several processes, also computed by Mappings Ic, ap-peared to be very small and not affecting the jet ther-mal structure. We just cite them here for completeness:free-free cooling and heating, two-photon continuum andCompton scattering.

We ignored thermal conduction, which could be rele-vant along flow lines, the magnetic field reducing the gasthermal conductibility in any other direction. Also ignoredwas gas cooling by dust grains and heating by cosmic rays.

We checked a posteriori that these three terms indeed havea negligible contribution (see Appendix C.2).

3.4. Numerical resolution

In our study, flow thermodynamics are decoupled from thedynamics cold jet approximation. The previous Eqs. (10)to (14) can then be integrated for a given flow pattern.The dynamical quantities (density, velocity and magneticfields) are given by the cold MHD solutions presented inSect. 2. For the steady-state, axisymmetric, self-similarMHD winds under study, any total derivative writes

DDt

= (v ) = vz0

dd

(21)

where vz is the vertical velocity and = z/0 is the self-similar variable that measures the position along a flowline anchored at 0. With this in hand, and the massconservation condition for an atomic wind (Eq. (11)), theenergy Eq. (10) becomes an ODE along the flow line:

d ln T

d=

2

3

d ln n

d d

dln(1 + fe) +

32 nkT

vz0

23

d ln n

d 1 MHD + ( )Map

adia (22)The term d ln(1+ fe)/d (due to variations in internal en-ergy density) is in fact negligible and has not been imple-mented for computational simplicity (see Appendix C.2).The term ( )Map = P rad col rec is providedby Mappings Ic. The wind thermal structure is computedby integrating along a flow line the energy Eq. (22) cou-pled to the set of electronic population equation (Eqs. (12)to (14)).

3.4.1. Initial conditions

The integration of the set Eqs. (12) to (14) and (22) alongthe flow is an initial value problem. Thus, some way toestimate the initial temperature and populations must bedevised. All calculations start at the slow-magnetosonic

(SM) point, which is roughly at two scale heights abovethe disk midplane (for the solutions used here).

To estimate the initial temperature, Safier equated thepoloidal flow speed at the SM point to the sound speed.Although this estimation agrees with cold flow theory, itis inconsistent with the energy equation which is used fur-ther up in the jet. Our approach was then to compute theinitial temperature and ionic populations assuming that

DT

Dt

=s

= 0 andDnAi

Dt

=s

= 0 (23)

is fulfilled at = s. We thus assume for convenience that,at the base of the jet, there is no strong variations neitherin temperature nor in ionization fractions. Physically thismeans that the gas is in ionization equilibrium, at theobtained temperature, with the incoming radiation field.The temperature thus obtained is always smaller than that

provided by the SM speed. This is due to the large openingof the magnetic surfaces, providing a dominant adiabaticcooling over all heating processes. For numerical reasonsthe minimum possible initial temperature was set to 50 K.It is noteworthy that the exact value of the initial tem-perature only affects the base of the flow, below a fewthousand degrees (see Appendix C.3). These regions aretoo cold to contribute significantly to optical line emission,leaving observational predictions unaffected.

The initial populations are computed by Mappings Icassuming ionization equilibrium with the incoming radi-ation field. However for high accretion rates and for theouter zones of the wind, dust opacity and inclination ef-fects shield completely the ionizing radiation field. Thetemperature is too low for collisional ionization to be ef-fective. The ionization fraction thus reaches our prescribedminimum all Na is in the form Na ii (Table 2) and allthe other elements (computed by Mappings Ic) neutral.However, soon the gas flow gains height and the ioniza-tion field is strong enough such that the ionization is self-consistently computed by Mappings Ic.

3.4.2. Integration procedure

After obtaining an initial temperature and ionization state

for the gas we proceed by integrating the system of equa-tions. In practice the ionization evolution is computed byMappings Ic and separately we integrate Eq. (22) witha Runge-Kutta type algorithm (Press et al. 1988). Wemaintain both the populations and Mappings Ic cool-ing/heating rates per n2 fixed during each temperature in-tegrating spatial step. After we call Mappings Ic to evolvethe populations and rates, at the new temperature, duringthe time taken by the fluid to move the spatial step. Thisstep is such, that the RK integration has a numerical ac-curacy of 106 and, that the newly computed temperaturevaries by less than a factor of 104. Such a small varia-

tion in temperature allows us to assume constant ratesand populations while solving the energy equation. Wechecked a few integrations by redoing them at half thestep used and found that the error in the ionic fraction is

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Fig. 3. Ion abundances with respect to hydrogen (fAi = nAi/nand thus fAi depends also on the abundances) along the flowline versus z/0 for model B in out of ionization equilib-rium, with Macc = 10

6 M yr1. The jump at 0.5 is due

to depletion as the gas enters the sublimation surface.

and expands. Similarly, the rise in ne in the recollimationzone is due to gas compression. A remarkable result isthat, as long as ionization is dominated by hydrogen (i.e.fp 104), ne is not highly dependent ofMacc, increasingby a factor of 10 only over three orders in magnitude inaccretion rate. This indicates a roughly inverse scaling offp with Macc (bottom panels of Fig. 2), a property alreadyfound by Safier which we will discuss in more detail later.

In regions at the wind base where fp < 104, variationsofne are linked to the detailed photoionization of heavy el-ements which are then the dominant electron donors. The

respective contributions of various ionized heavy atomsto the electronic fraction fe is illustrated in Fig. 3 forMacc = 106 M yr

1. While O ii and N ii are stronglycoupled to hydrogen collisional ionization through chargeexchange reactions, the other elements are dominated byphotoionization. The sharp discontinuity in C ii and Na iiat the wind base for 0 = 0.1 AU is caused by the cross-ing of the dust sublimation surface by the streamline (seeAppendix B). Inside the surface we are in the dust sub-limation zone where heavy atoms are consequently notdepleted onto grains and hence have a higher abundance.In contrast, for 0 = 1 AU, the flow starts already outside

the sublimation radius, in a region well-shielded from theUV flux of the boundary-layer, where only Na is ionized.Extinction progressively decreases as material is liftedabove the disk plane and sulfur, then carbon, also becomecompletely photoionized.

4.3. Heating and cooling processes

The heating and cooling terms along the streamlines forour out of equilibrium calculations are plotted in Fig. 4 for0 = 0.1 and 1 AU, and for two values of Macc = 10

6

and 107 M yr1.

Before the recollimation point, the main cooling pro-cess throughout the flow is adiabatic cooling adia, al-though Hydrogen line cooling rad(H) is definitely notnegligible. The main heating process is ambipolar diffusion

drag. The only exception occurs at the wind base for small

0 0.1 AU and large Macc 106 M yr1, where pho-toionization heating P initially dominates. Under suchconditions, ambipolar diffusion heating is low due to thehigh ion density, which couples them to neutrals and re-duces the drift responsible for drag heating. However, Pdecays very fast due to the combined effects of radiationdilution, dust opacity, depletion of heavy atoms in thedust phase, and the decrease in gas density. At the sametime, the latter two effects make drag rise and becomequickly the dominant heating term. In the recollimationzone, the main cooling process is hydrogen line coolingrad(H), and the main heating term is compression heat-ing (adia is negative).

A striking result in Fig. 4, also found by Safier, is thata close match is quickly established along each stream-line between adia and drag, and is maintained until the

recollimation region. The value of where this balance isestablished is also where the temperature plateau starts.We will demonstrate below why this is so for the class ofMHD wind solutions considered here.

4.4. Physical origin of the temperature plateau

The existence of a hot temperature plateau where adiaexactly balances drag is the most remarkable and robustproperty of magnetically-driven disk winds heated by am-bipolar diffusion. Furthermore, it occurs throughout sev-eral decades along the flow including the zone of the jet

that current observations are able to spatially resolve.In this section, we explore in detail which generic prop-

erties of our MHD solution allow a temperature plateauat 104 K to be reached, and why this equilibrium maynot be reached for other MHD wind solutions.

4.4.1. Context

First, we note that the energy equation (Eq. (22)) in theregion where drag heating and adiabatic cooling are thedominant terms (which includes the plateau region) canbe cast in the simplified form:

d ln T

d ln = 2

3

d ln n

d ln drag

adia 1

= 1 G()

F(T) 1

, (24)

where:

()1

23

d ln n

d ln

, (25)

remains positive before recollimation and depends only onthe MHD solution,

G() = 1c2

JB2n2(v )n

M2Macc0

(26)

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Fig. 4. Heating and cooling processes (in erg s1 cm3) along the flow line versus z/0 for model B. Top figures for

Macc = 10

6

M yr

1

and bottom ones for

Macc = 10

7

M yr

1

. Ambipolar heating and adiabatic cooling appear to be thedominant terms, although Hydrogen line cooling cannot be neglected for the inner streamlines.

is a positive function, before recollimation, that dependsonly on the MHD wind solution, and

F(T) = kT(1 + fe)minfifn inv

n

2(27)

also positive, depends only on the local temperature andionization state of the gas. The functions G and F separatethe contributions of the MHD dynamics and ionization

processes in the final thermal solution.The function is roughly constant and around unity

before recollimation, it diverges at the recollimationpoint and becomes negative after it. Throughout theplateau 1.

The wind function G is plotted in the center panel ofFig. 5 for our 3 solutions. It rises by 5 orders of magnitudeat the wind base and then stabilizes in the jet region (untilit diverges to infinity near the recollimation point). Thephysical reason for its behavior is better seen if we notethat the main force driving the flow is the Lorentz force:

G Dv

Dt2

D

Dt1

. (28)

For an expanding and accelerating flow (D/Dt)1 is anincreasing function. At the wind base the Lorentz force

accelerates the gas thus causing a fast increase in G. Oncethe Alfven point is reached, the acceleration is smaller andDv/Dt decreases. However this decrease is not so abruptas in the case of a spherical wind, because the Lorentzforce is still at work, both accelerating and collimating theflow. This collimation in turn reduces the rate of increaseof (D/Dt)1. The stabilization of G observed after theAlfven point is thus closely linked to the jet dynamics.

The ionization function F is in general a risingfunction of T and is plotted in the left panel of Fig. 5under the approximation of local ionization equilibrium.Two regimes are present: in the low temperature regime,fi fp is dominated by the abundance of photoionizedheavy elements and F(T) T fi increases linearly with T,for fixed fi. The effect of the UV flux in this region is toshift vertically F(T): for a low UV flux regime only Na isionized and fi f(Na ii); for a high UV flux regime wereCarbon is fully ionized, fi f(C ii). In the high tem-perature regime (T 8000 K) where hydrogen collisionalionization dominates, fi fp, and F(T) T fp becomesa steeply rising function of temperature, until hydrogen isfully ionized around T 2 104 K. The following secondrise in F(T) is due to Helium collisional ionization. As wego out of perfect local ionization equilibrium the effect is

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Fig. 5. Left: function F(T) in erg g cm3 s1 versus temperature assuming local ionization equilibriumand an ionization flux thationizes only all Na and all C. Center: function G() in erg g cm3 s1 for models A, B, C (bottom to top), Macc = 10

6 M yr1

and 0 = 0.1 AU. Right: temperature for model B from the complete calculations in ionization equilibrium (dashed), andassuming T = T as given by Eq. (30) ( dash-dotted). 0 = 0.1 AU and accretion rates Macc are 10

8 to 105 M yr1, from

top to bottom.

to decrease the slope of F(T) in the region where H ion-ization dominates. In the extreme situation of ionizationfreezing, F(T) becomes linear again as in the photoionizedregion: F(T) T fp,freezed.

4.4.2. Conditions needed for a hot plateau

The plateau is simply a region where the temperature doesnot vary much,

d ln T

d ln = with || 1. (29)Naively, temperatures T() defined by,

F(T) = G() drag = adia, (30)will zero the right hand side of Eq. (24) and thus sat-isfy the plateau condition. This equality is the first con-straint on the wind function G, because there must exista temperature T such that the equality holds. Howeverthis condition is not sufficient. Indeed the above equalitydescribes a curve1 T() which must be flat in order tosatisfy the plateau condition (Eq. (29)). Therefore the re-

quirement of a flat T translates in a second constraint onthe variation of G with respect to F. This constraint isobtained by differentiating Eq. (30). We obtain

d ln G

d ln =

d ln F

d ln T

T=T

(31)

and after using (Eq. (29)):d ln Gd ln

d ln Fd ln T

T=T

(). (32)

Thus only winds where the wind function G varies muchslower than the ionization function F will produce aplateau. This is fulfilled for our models: below the Alfven

1 This is only true if F is a monotonous function of T. Ascan be seen from Fig. 5 this is true for almost all its domain.

surface, G varies a lot, but collisional H ionization is suf-ficiently close to ionization equilibrium that F(T) stillrises steeply around 104 K (Fig. 5). For our numericalvalues of G, within our range of Macc and 0, we haveT 104 K and thus |d ln G/d ln | |d ln F/d ln T|.Further out, where ionization is frozen out, we haved ln F/d ln T = 1 (because F T fp,freezed) but it turnsout that in this region G is a slowly varying function of,and thus we still have |d ln G/d ln | |d ln F/d ln T|.

Finally, a third constraint is that the flow must quickly

reach the plateau solution T() = T() and tend tomaintain this equilibrium. Let us assume that T = Tis fulfilled at = 0, what will be the temperature at = 0(1 + x)? Letting T = T(1 + ) and assuming 1, Eq. (24) gives us = exp(x), which providesan exponential convergence towards T = T as long as

=1

d ln F

d ln T

T=T

> 0 . (33)

Note that depends mainly on the MHD solution (inde-pendent ofMacc/0) and that the steeper the function F,the faster the convergence. The above criterion is always

fulfilled in the expanding region of our atomic wind so-lutions, where > 0 and F increases with temperature.The physical reason for the convergence can be easily un-derstood in the following way: it can be readily seen thatif at a given point T > T(), then G()/F(T) < 1 andthe gas will cool (cf. Eq. (24) with > 0). Conversely, ifT < T(), the gas will heat up. Thus, for > 0, the factthat F(T) is a rising function introduces a feedback thatbrings and maintains the temperature close to its localequilibrium value T(), and adia close to drag.

We conclude that three analytical criteria must be metby any MHD wind dominated by ambipolar diffusion heat-

ing and adiabatic cooling, in order to converge to a hottemperature plateau:(1) Equilibrium: the wind function G must be such thatF(T) = G() is possible around T 104 K;

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(2) Small temperature variation: the wind function G()must vary slower than the ionization function F(T) suchthat |d ln G/d ln | |d ln F/d ln T|: (i) the wind must bein ionization equilibrium, or near it, in regions where G isa fast function of ; (ii) once we have ionization freezing,G must vary slowly, with |d ln G/d ln | 1;(3) Convergence: > 0, i.e. 2

3(d ln n/d ln )

(dln F/d ln T)|T=T < 0, which is always verified for anatomic and expanding wind.

4.4.3. Comments on other MHD winds

Not all types of MHD wind solutions will verify our firstcriterion. Physically, the large values of G() observed inour solutions indicates that there is still a non-negligibleLorentz force after the Alfven surface. In this region(which we call the jet) the Lorentz force is dominated

by its poloidal component which both collimates and ac-celerates the gas (Fig. 1). The gas acceleration translatesin a further decrease in density contributing to a furtherincrease in G. Models that provide most of the flow accel-eration before the Alfven surface might turn out to havea lower wind function G(), not numerically compatiblewith the steep portion of the ionization function F(T).These models would not establish a temperature plateauaround 104 K by ambipolar diffusion heating. They wouldeither stabilize on a lower temperature plateau (on the lin-ear part ofF(T)) if our second criterion is verified, or con-tinue to cool ifG varies too fast for the second criterion tohold. This is the case in particular for the analytical windmodels considered by Ruden et al. (1990), where the dragforce was computed a posteriori from a prescribed velocityfield. The G function for their parameter space (Table 3of Ruden et al.) peaks at 1048 erg g cm3 s1 at 3 Rand then rapidly decreases as G r1 for higher radii.This translates into a cooling wind without a plateau.

4.5. Scalings of plateau parameters with Macc and 0

The balance between drag heating and adiabatic cooling(Eq. (30)) can further be used to understand the scal-ings of the plateau temperature T and proton fraction

fp, with the accretion rate Macc and flow line footpointradius (0). In the plateau region, ionization is intermedi-ate, i.e., sufficiently high to be dominated by protons butwith most of the Hydrogen neutral. Under these conditionswe have F(T) Tfp,. On the other hand, self-similardisk wind models display G() (Macc0)1 (Eq. (26)).Therefore, we expect

Tfp 1Macc 0

(34)

This behavior is verified in the left panel of Fig. 6 for ourout-of-equilibrium results for the 3 wind solutions.

To predict how much of this scaling will be absorbed byT and how much by fp,, Safier considered the ionizationequilibrium approximation: for the temperature range ofthe plateau (T 104 K) fp fi is a very fast varying

Fig. 6. Verification of the plateau scalings (Eq. (34)). Left: weplot the measured Tfp, versus (Macc0)

1 for all models inout of ionization equilibrium. The evolution is linear exceptfor the very edges of the (Macc0)

1 domain. This is due tothe failure of our assumptions: in the lower edge fe < 10

4

and thus it isnt dominated by H; in the upper edge fe > 0.1and thus the small ionization fraction approximation fails. Thecrosses are from Safier; because of a smaller momentum trans-

fer rate coefficient they are quite above model C. Right: weplot the measured fp, versus (Macc0)

1 for Model B inout of ionization equilibrium (solid) and ionization equilibrium(dashed). Note that all the variation is absorbed by fp, andthus fp, (Macc 0)

1. A straight line is also plot for com-parison.

function of T that can be approximated as fp Ta witha 1. One predicts that T (Macc 0)1/(a+1) whilefp, (Macc 0)a/(a+1) (Macc 0)1. Hence, the in-verse scaling with (Macc0) should be mostly absorbed

by fp,, while the plateau temperature is only weakly de-pendent on these parameters. This is verified in the rightpanel of Fig. 6, where fp, in ionization equilibrium is

plotted as a function of (Macc 0)1. The predicted scal-ing is indeed closely followed.

Let us now turn back to the actual out-of-equilibriumcalculations. At the base of the flow we find that the windevolves roughly in ionization equilibrium (see Fig. 2), how-ever at a certain point the ionization fraction freezes atvalues that are near those of the ionization equilibriumzone. This effect implies that the ionization fraction shouldroughly scale as the ionization equilibrium values at the

upper wind base. This in indeed observed in Fig. 6. Thismemory of the ionization equilibrium values by fp (as ob-served in the solar wind by Owocki et al. 1983) is the rea-son why the scalings offp, with (Macc0) remain correct.We computed for our solutions the scalings and found formodel B: fp, M0.76acc , fp, 0.830 , T M0.13accand no dependence of T on 0, confirming the memoryeffect on the ionization fraction only.

Finally, we note that for accretion rates in excess of afew times 105 M yr

1, the hot plateau should not bepresent anymore: Because of its inverse scaling with Macc,the wind function G remains below 1047 erg g cm3 s1,

and F = G occurs below 104

K, on the linear low-temperature part of F(T) where fi f(C ii) (see Fig. 5).These colder jets will presumably be partly molecular.Interestingly, molecular jets have only been observed so

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Fig. 7. Out of ionization equilibrium evolution of wind quan-tities in the plateau. Points are for all flow lines (0 =0.071.3i AU and i = 0, 1, ..., 10) and accretion rates (Macc =10i M yr

1 with i = 5, 6, 7, 8). We plot, fp, versus T forall models C (red), B (black), A (blue) and in red the valuesexpected from ionization equilibrium. The dashed/dotted linesare for models without depletion, the solid lines are for modelswith depletion. The accretion rates increase in the directiontop right to bottom left. The thick solid curve traces fp inionization equilibrium, while the two dotted lines embrace thelocus of our MHD solutions.

far in embedded protostars with high accretion rates (e.g.Gueth & Guilloteau 1999).

4.6. Effect of the ejection index

The importance of the underlying MHD solution is illus-trated in Fig. 5. The ejection index is directly linked tothe mass loaded in the jet (, see Table 1 and Ferreira1997). Thus a higher translates in an stronger adiabaticcooling because more mass is being ejected. The ambipolar

diffusion heating is less sensitive to the ejection index, be-cause the density increase is balanced by a stronger mag-netic field. Hence, the wind function G() decreases withincreasing . As a result, the plateau temperature and ion-ization fraction also decrease (see Eq. (34)).

In Fig. 7 we summarize our results for the three modelsby plotting the plateau fp, versus T, for several Macc(values of 0 are connected together). In this plane, ourMHD solutions lie in a well-defined strip located belowthe ionization equilibrium curve, between the two dottedcurves. For a given model, as Macc increases, the plateauionization fraction and the temperature both decrease, as

expected from the scalings discussed above, moving themodel to the lower-left of the strip. Increasing the ejectionindex decreases G(), and it can be seen that this has asimilar effect as increasing Macc (Eq. (34)).

Fig. 8. H H+v in cm3 s1 using Draine expression (solid),

using Geiss & Buergi expression (see Appendix A.2) (dash)and Draine expression but ignoring the thermal velocity (dot).

4.7. Depletion effects on the thermal structure

In our calculations we take into account depletion of heavyspecies into the dust phase. We ran our model with andwithout depletion and found these effects to be minor.Changes are only found when fp 104. The temperaturewithout depletion is slightly reduced (the higher ioniza-tion fraction reduces drag) and as a consequence fp is alsosmaller. Normally these changes affect only the wind base,as the temperature increases fp dominates the ioniza-tion and we obtain the same results for the plateau zone.

However for high accretion rates (Macc = 105 M yr1)in the outer wind zone (large 0) we still have fp 104

for the plateau and thus the temperature without deple-tion is reduced there.

4.8. Differences with the results of Safier

The most striking difference between our results andthose of Safier is an ionization fraction 10 to 100 timessmaller. This difference is mainly due to both differentH H+v momentum transfer rate coefficients and dy-namical MHD wind models.

The critical importance of the momentum transfer ratecoefficient ( H H+v ) for the plateau ionization fractioncan be seen by repeating the reasoning in the previous sec-tion but including the momentum transfer rate coefficientin the scalings. We thus obtain

T fp, 1H H+v Macc0 (35)

This shows that, because the freezing of the ionizationfraction is correlated to the ionization fraction at the baseof the wind (which is in ionization equilibrium), fp willscale with the momentum transfer rate coefficient value.

This means that if the momentum transfer rate coefficientis larger, there is a better coupling between ions and neu-trals and hence a smaller drag heating. For the calculationof the H H+v , Safier ignored the contribution of the

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thermal velocity in the collisional relative velocity. Thisconsiderably reduces H H+v and thus, increases fp. InFig. 8 we plot the corresponding momentum transfer ratecoefficient values. It can be seen that ignoring the thermalcontribution to the momentum transfer rate coefficient de-creases it typically by a factor of6. We also plot in thisfigure the value obtained by Geiss & Buergi (1986) illus-trating the uncertainties in the momentum transfer ratecoefficient (more on this in Appendix A.2).

5. Concluding remarks

We performed detailed non-equilibrium calculations ofthe thermal and ionization structure of atomic, self-similar magnetically-driven jets from keplerian accretiondisks. Current dissipation in ion-neutral collisions am-bipolar diffusion , was assumed as the major heating

source. Improvements over the original work of (Safier1993a,b) include: a) detailed dynamical models by Ferreira(1997) where the disk is self-consistently taken into ac-count but each magnetic surface assumed isothermal; b)ionization evolution for all relevant heavy atoms us-ing Mappings Ic code; c) radiation cooling by hydrogenlines, recombination and photoionization heating usingMappings Ic code; d) H-H+ momentum exchange ratesincluding thermal contributions; and e) more detailed dustdescription.

We obtain, as Safier, warm jets with a hot tempera-ture plateau at T 104 K. Such a plateau is a robustproperty of the atomic disk winds considered here for ac-cretion rates less than a few times 105 M yr

1. Itis a direct consequence of the characteristic behavior ofthe wind function G() defined in Eq. (26): (i) G() in-creases first and becomes larger than a certain value fixedby the minimum ionization fraction (see Fig. 5); (ii) G()is flat whenever ionization freezing occurs (collimated jetregion). More generally, we formulate three analytical cri-teria that must be met by any MHD wind dominated byambipolar diffusion heating and adiabatic cooling in orderto converge to a hot temperature plateau.

The scalings of ionization fractions and temperaturesin the plateau with Macc and 0 found by Safier are recov-

ered. However the ionizations fractions are 10 to 100 timessmaller, due to larger H-H+ momentum exchange rates(which include the dominant thermal velocity contributionignored by Safier) and to different MHD wind dynamics.

We performed detailed consistency checks for our solu-tions and found that local charge neutrality, gas thermal-ization, single fluid description and ideal MHD approxi-mation are always verified by our solutions. However atlow accretion rates, for the base of outer wind regions(0 1 AU) and increasingly for higher , single fluidcalculations become questionable. For the kind of jets un-der study, a multi-component description is necessary for

field lines anchored after 0 > 1 AU. So far, all jet calcu-lations assumed either isothermal or adiabatic magneticsurfaces. But our thermal computations showed such anincrease in jet temperature that thermal pressure gradi-

ents might become relevant in jet dynamics. We thereforechecked the cold fluid approximation by computing theratio of the thermal pressure gradient to the Lorentz force,along () and perpendicular () to a magnetic surface.Both ratios increase for lower accretion rates and outerwind regions. We found that for some solutions, thermalpressure gradients play indeed a role, however only at thewind base (possible acceleration) and in the recollimationzone (possible support against recollimation). Fortunately,(as will be seen in a companion paper, Paper II), the dy-namical solutions which are found inconsistent are alsothose rejected on an observational ground. Therefore, itturns out that the models that best fit observations areindeed consistent.

Acknowledgements. P. J. V. G. acknowledges financial supportfrom Fundacao para a Ciencia e Tecnologia by the PRAXISXXI/BD/5780/95, PRAXIS XXI/BPD/20179/99 grants. Thework of LB was supported by the CONACyT grant 32139-E.We thank the referee, Pedro N. Safier, for his helpful comments.We acknowledge fruitful discussions with Alex Raga, ElianaPinho, Pierre Ferruit and Eric Thiebaut. We are grateful toBruce Draine for communicating his grain opacity data, and toPierre Ferruit for providing his C interface to the Mappings Icroutine. P. J. V. G. warmly thanks the Airi team and his ad-viser, Renaud Foy, for their constant support.

Appendix A: Multicomponent MHD equations

A.1. Single fluid description

Let us consider a fluid composed of species of numeri-cal density n, mass m, charge q and velocity v. Allspecies are assumed to be coupled enough so that theyhave the same temperature T. To get a single fluid de-scription, we then define

= nm

v = mnv

P = nkBT (A.1)

J = nqv

as being the flow density , velocity v, pressure P and cur-rent density J. We consider now a fluid composed of threespecies, namely electrons (e), ions (i) and neutrals (n). Theequations of motion for each species are

nDvnDt

= Pn nG + Fen + Fin (A.2)

iDviDt

= Pi iG + Fei + Fni+ qini

E+

vi

cB

(A.3)

eDveDt

= Pe eG + Fie + Fnee ne E+

ve

cB (A.4)

where G is the gravitational potential and the colli-sional force of particles on particles is given byF = mn(v v), m = mm/(m + m)

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being the reduced mass, = n v the collisionalfrequency and v the averaged momentum transferrate coefficient.

A single fluid dynamical description of several speciesis relevant whenever they are efficiently collisionally cou-pled, namely if they fulfill v=e,n,iv v. Under thisassumption and using Newtons principle (,F = 0),we get the usual MHD momentum conservation equationfor one fluid

Dv

Dt= P G + 1

cJB (A.5)

by adding all equations for each specie. The Lorentz forceacting on the mean flow is

1

cJB = (1 + X)(Fin + Fen) , (A.6)

where X = i/n. Even if the bulk of the flow is neutral,collisions with charged particles give rise to magnetic ef-fects. In turn, the magnetic field is coupled to the flow bythe currents generated there. This feedback is provided bythe induction equation, which requires the knowledge ofthe local electric field E. Its expression is obtained fromthe electrons momentum equation

E+ve

cB = 1

ene

mieniievie + mnennnevne

Pe

ene(A.7)

where v vv is the drift velocity between the twospecies. Due to their negligible contribution to the massof the bulk flow, all terms involving the electrons iner-tia have been neglected (electrons quite instantly adjustthemselves to the other forces).

All drift velocities can be easily obtained. The electron-ion drift velocity is directly provided by vie = J/ene.Using Eq. (A.6) and noting that een iin we getthe ion-neutral drift velocity

vin =1cJB

(1 + X)minniin+

menneenminniin

J

ene (A.8)

On the same line of thought, the electrons velocity is ve =v (v ve) where

v ve = vn ve1 + X

+X

1 + X(vi ve) (A.9)

Jene

1(1 + X)2

1cJB

minniin (A.10)

Gathering these expressions for all drift velocities, we ob-tain the generalized Ohms law

E+v

c B = J+1

c

J

B

ene Peene

1(1 + X)2

1c2 (JB) B

minniin(A.11)

where = (mnennne + mieniie)/(ene)2 is the electrical

resistivity due to collisions. The corresponding MHD heat-ing rate writes

MHD = J

E

= J

E+

v

cB (A.12)

where E is the electrical field in the comoving frame. Thisexpression leads to Eq. (17).

The generalization of this derivation for a mixtureof several chemical elements has been done in a quitestraightforward way. The bulk flow density becomes =i + n + e, where the overline stands for a sum overall elements (ions and neutrals), with X = i/n. Theneutrals and ions velocities are means over all elements,vn,i

n,i n,ivn,i/n,i. The conductivity and colli-

sion terms are also sums over all elements, namely =(mnennne + mieniie)/(ene)

2 and minniin, and are com-puted using the expressions for the collision frequencies.

A.2. Momentum transfer rate coefficient

For ion-electron collisions we use the canonical fromSchunk (1975), summed over all species:

mieniie = mene2e4

3(kBTe)2

8kBTeme

i

niZ2i ln i (A.13)

with the Coulomb factor i = (3/2Zie3)

(kBTe)3/ne.For the collisions between electrons and neutrals we

use the expression of Osterbrock (1961) for the colli-sional momentum transfer rate coefficient between a neu-

tral and a charged particle, which corrects the classical one(e.g., Schunk 1975) for strong repulsive forces at close dis-tances. Its expression is v n,ie = 2.41e

n/mn,ie,

where the polarizabilities n used are also taken fromOsterbrock. We thus obtain

mennnne = 2.41 e ne

n=H,He

nn

men . (A.14)

Finally, it is mainly the ion-neutral collision momentumtransfer rate coefficient determines the ambipolar diffusionheating. It can be computed with the previous momentumtransfer rate coefficient expression. However as noted byDraine (1980) the previous expression underestimates at

high velocities. Thus, as Draine, we take the hard spherevalue for the cross-section (S 1015 cm2) whenever itis superior to the polarizability one. For intermediate tohight ionizations (fH+ 10

4) the dominant ion-neutralcollisions are those between H-H+. Charge exchange ef-fects between these two species will amplify H H+vabove the values expected by polarizability alone and thusit is computed separately (Eqs. (A.16) and (A.17)). Wethus obtain for ion-neutral collisions

minniin =1

2mHnH+nH H H+v

+ i>H+

nimiH max(S v;

v

H,i)

+i

nimiHe max(S v; v He,i) (A.15)

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-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

Tsub=1500K - MRN mix

Tsub=1400K - silicate

Tsub=1750K - graphite

Flow lines

Disk surface

AU

AU

Fig. B.1. Dust sublimation surfaces geometry for the adoptedradiation field.

where v =

8kBT/min + v2in. For v < 1000 km s1.

The value of H H+v which we used is given byDraine (1980),

H H+v1 cm3 s1

3.26 109 v < 2 km s12.0 109 v0.73 v 2 km s1 (A.16)

Safier (1993a) used the expression v = vin which, as dis-cussed in Sect. 4.8, results in a smaller momentum transferrate coefficient. Geiss & Buergi (1986) computed anotherexpression of the H-H+ momentum transfer rate coeffi-cient, which provides

H H+v = 1.12 108 T12

4 (1 0.12 log T4)2+

2.4 0.34(1 + 2 log T4)2 109 cm3 s1. (A.17)

In Fig. 8 we compared both momentum transfer rate coef-ficients, they typically differ in 40%, which can be used asan estimate of their accuracy. It is thus the uncertainty inthe H-H+ momentum transfer rate coefficient that domi-nates the final intrinsic uncertainty of our calculations.

Appendix B: Dust implementation

As shown by Safier if there is dust in the disk, the windis powerful enough to drag it along. Thus disk winds aredusty winds. Dust is important for the wind thermal struc-ture mainly as an opacity source affecting the photoion-isation heating at the wind base. To compute the dustopacity we need a description of its size distribution, itswavelength dependent absorption cross-section and the in-ner dust sublimation surface. In the inner flow zones andfor high accretion rates the strong stellar and boundarylayer flux will sublimate the dust, creating a dust free in-

ner cavity (see Fig. B.1). Results on the evolution of dustin accretion disks by Schmitt et al. (1997) show that atthe disk surface the initial dust distribution isnt muchaffected by coagulation and sedimentation effects. Thus

we assume a MRN dust distribution (Mathis et al. 1977;Draine & Lee 1984):

dni = nH Ai a3.5 da (B.1)

where dni is the number of particles of type i (astronom-ical silicate Sil or graphite C) with sizes in [a, a + da],and 0.005 m a 0.25 m, ASil = 1025.11 cm2.5 H1and AC = 1025.16 cm2.5 H1. We then proceed by av-eraging all relevant grain quantities function of size andspecies (Fi(a)) by the size/species distribution,

Fi(a) a =amaxamin

i=Sil,C

Fi(a)dniNT

(B.2)

In order to compute the sublimation radius, some descrip-tion of the dust temperature must be made. For simplicity,we assume the dust to be in thermodynamic equilibrium

with the radiation field, the dominant dust heating mech-anism. In our case, the central source radiation field willdominate throughout the jet, except probably in the rec-ollimation zone, where the strong gas emission overcomesthe central diluted field. However in this region dust is nolonger relevant for the gas thermodynamics and we willtherefore only consider dust heating by the central source.The dust temperature Tgr for a grain of size a is obtainedby equating the absorbed to the emitted radiation (e.g.,Tielens & Hollenbach 1985),

4a2 Qa (Tgr)T4gr = a2

0

Qabsa ()4Jd (B.3)

where a is the grain size Qa (Tgr) is the Planck-averagedemissivity (Draine & Lee 1984; Laor & Draine 1993;Draine & Malhotra 1993), is the Stefan-Boltzmann con-stant, Qabsa () is the dust absorption efficiency

2 and 4Jis the central source radiation flux at the grain positiongiven by Eq. (15). Averaging out the previous equation bythe size/species distribution (Eq. (B.2)) we obtain,

4 Qema (Tgr) T4gr =

0

Qabsa ()F()e(r,)d (B.4)

where we describe the central source flux by F() which

is attenuated only by the dust opacity . For simplicityF() is taken as exactly the same as in Safier, i.e. a classi-cal boundary layer (Bertout et al. 1988). The sublimationradius is obtained from the previous expression by notingthat at its position = 0,

rsub()R

=g() Qabsa (T) T4 + gbl() Qabsa (Tbl) T4bl

4 Qema (Tsub) T4sub(B.5)

where Tbl and T are the boundary layer and star tem-peratures, R the stellar radius and gbl()/g() are

the dependent terms of the radiation field (given in2 The dust absorption efficiency is related to the dust ab-

sorption cross-section by absa, = a2Qabsa ().

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Bertout et al. and Safier). We assume a dust sublimationtemperature Tsub of 1500 K.

With the dust sublimation radius in hand we can nowproceed to compute the dust optical depth defined as,

(r, ) =rrin()

abs (r)dr =

rrin()

nH(r, )()adr

(B.6)

where rin() is the radius inside which there is no dust.This radius is given by the inner flow line ri() andby the sublimation radius rsub() (see Fig. B.1) suchthat rin() = max( rsub ; ri). The dust absorption cross-section (()a) is,

()a =

amaxamin

a2

QabsSil (a, )ASil

+ QabsC (a, )ACa3.5da. (B.7)Using the self-similarity of nH(r, ) we can integrateEq. (B.6) to obtain,

(r, ) = nH(r, ) () 2r r

rin() 1

(B.8)

which was used in Eq. (15). We note that at large dis-tances from the source, the optical depth converges to afinite value, proportional to Macc and whose variation isfunction of the self similar wind solution and central sourceradiation field. Thus for high accretion rates, although thecentral source radiation hardens, the outer zones of the

wind base are less photoionized than for smaller accretionrates.

Appendix C: Consistency checks

C.1. Dynamical assumptions

First, local charge neutrality is always achieved. For exam-ple, we achieve a maximum Debye length of rD 105cmat the outer radius of the recollimation zone (model C,lowest Macc).

Second, single fluid approximation requires that rela-tive velocity drifts of all species ( = ions, electrons, neu-

trals) vv/v are smaller than unity. These drifts arehigher for lower accretion rates and at the outer wind base(due to the decrease in density and velocity, see Eq. (9)).In Fig. C.1 we present the worst case for the drift veloci-ties, showing that our jets can be indeed approximated bysingle fluid calculations.

We assumed gas thermalization, which is achieved onlyif collisional time-scales between species = 1/are much smaller than the dynamical time-scale dyn =0(dvz/d)1. In the collision network considered here,the longer time-scales involve collisions with neutrals.However, even in the worst situation (see Fig. C.1), after

the wind base they remain comfortably below the abovedynamical time-scale.Our dynamical jet solutions were derived within the

ideal MHD framework. This assumption requires that all

Fig. C.1. Top left: we plot the relevant drift speeds nor-malized to the fluid poloidal velocity. The worst case of onefluid approximation violation is obtained for model C, Macc =108 M yr

1 and 0 = 1 AU. Top right: we plot the versus dynamical dyn time-scales (s) versus , normalized tothe Keplerian period at the line footpoint (for a 1 M star).We plot only the longer time-scales (in = ni/fi and en =ne/fe). The worst case is obtained again for model C, Macc =108 M yr

1 and 0 = 1 AU. Bottom left: ideal MHD testsfor the worst situation (model C, Macc = 10

8 M yr1 and

0 = 1 AU). As expected from our heated winds, the am-bipolar diffusion term is the dominant one. Bottom right:ratios of the thermal pressure gradient to the Lorentz forceversus , along (, solid) and across (, dash) a magneticsurface anchored at 0. The worst case for is obtained for

model A, 0 = 1 AU, Macc = 108 M yr

1, the best formodel A, 0 = 0.1 AU and Macc = 10

5 M yr1. With

respect to , the worst case is for model C, 0 = 1 A Uand Macc = 10

8 M yr1, while the best is for model A,

0 = 0.1 AU and Macc = 105 M yr

1. Although definitelynot negligible in some models, those compatible with obser-vations do not show important deviations from the cold jetapproximation.

terms in the right hand side of the generalized Ohms law(Eq. (A.11)) are negligible when compared to the electro-motive field vB/c. We consider Ohms term J, Hallseffect J B/cene and the ambipolar diffusion term( n )

2(JB) B/c2minniin (effects due to the elec-tronic pressure gradient are small compared to the Lorentzforce Halls term ). In Fig. C.1 we present the worstcasefor our ideal MHD checks. We find that deviations fromideal MHD remain negligible, despite the presence of am-bipolar diffusion. As expected, this is the dominant diffu-

sion process in our (non turbulent) MHD jets. Ambipolardiffusion is larger for low accretion rates and at the outerwind base (because the ratio of the ambipolar to the elec-tromotive term scales as (Macc fi)1).

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The worst case for the previous three tests is,as expected, for the model that attains the low-est density: Model C, with the lowest accretion rate(Macc = 10

8 M yr1) and at the outer edge footpoint

(0 = 1 AU).The dynamical jet evolution was calculated under the

additional assumption of negligible thermal pressure gra-dient (cold jets). Since it is the gradient that provides aforce, one should not just measure (along one field line)the relative importance of the gas pressure to the mag-netic pressure (usual = P/(B2/8) parameter). Instead,we compare the thermal pressure gradient to the Lorentzforce, along () and perpendicular () to the flow,namely

=P

F= c

vp Pvp (JB) (C.1)

= PF

= c a Pa (JB) (C.2)

Here a(, z) is the poloidal magnetic flux function,hence a is perpendicular to a magnetic surface. High val-ues of imply that the thermal pressure gradient playsa role in gas acceleration, whereas high values of showthat it affects the gas collimation.

In Fig. C.1 we plot the worst case of cold fluid viola-tion and best case of cold fluid validity. Again the worstcase appears at lower accretion rates and in outer windzones. It can be seen that high values of and canbe attained, hinting at the importance of gas heating onjet dynamics (providing both enthalpy at the base of thejet and/or pressure support against recollimation furtherout). We underline that models inconsistent with the coldfluid approximation are those found to have the largest dif-ficulty in meeting the observations (Paper II). Conversely,models that better reproduce observations also fulfill thecold fluid approximation. For those models, the thermalpressure gradient appears to be fairly negligible with re-spect to the Lorentz force.

C.2. Thermal assumptions

Finally, we check that all ignored heating/cooling pro-cesses are not relevant when compared to adiabatic coolingand ambipolar diffusion heating.

The first ignored process is the term 32 knTDfe/Dt.This term decreases for increasing accretion rate and0 due to the lower ionizations found in these regions.It is plotted in Fig. C.2 for the worst case (model C,Macc = 108 M yr

1 and 0 = 0.1 AU). There, itreaches at most 13% of adia. Typical values for higheraccretion rates are only 0.1% of adia.

Next we consider heating/cooling of the gas by col-lision with dust grains, given by Hollenbach & McKee

(1979):

gr = n ngrgr

8kBT

mHf(2kBTgr 2kBT) (C.3)

Fig. C.2. Ignored heating/cooling terms (in erg s1 cm3).Left: we compare the cooling term 3

2knTDfe/Dt (dashed)

with adiabatic cooling adia (solid) for the worst case (modelC, Macc = 10

8 M yr1 and 0 = 0.1 AU). Right: grain

heating/cooling rate |gr| (dashed) compared with ambipolardiffusion heating drag (solid) for the worst case (model A,Macc = 10

5 M yr1 and 0 = 1 AU). gr is only significant

at the very base of the wind, and will not affect the thermalstate further out in the jet.

where ngrgr is computed from the adopted MRN dis-tribution, and f = 0.16 is the sticking parameter thattakes into account charge and accommodation effects fora warm gas (Hollenbach & McKee 1979). With these val-ues the grain heating/cooling becomes,

gr = 4.78 1034n2(Tgr T) erg s1 cm3. (C.4)This term increases with increasing 0 and accretion rate.In Fig. C.2 we compare it (in absolute value) with dragin the case where its contribution is the most important(model A, Macc = 10

5 M yr1 and 0 = 1 AU). gr is

initially positive (dust hotter than the gas), but changessign at 0.4, where the gas becomes hotter than thedust, becoming an effective cooling term. It is only sig-nificant at the very base of the wind, where it exceedsthe ambipolar diffusion heating term by a factor 3.5.However, this effect will not have important consequencesin terms of observational predictions: we will show in nextsection that the thermal state in the hot plateau (whereforbidden line emission is excited) is not sensitive to theinitial temperature. Furthermore, the outer streamlines at

0 1 AU contribute much less to the line emission thanthe inner ones. At lower accretion rates 106 M yr1,gr is always 10% of drag.

Heating due to cosmic rays, which could be impor-tant in the outer tenuous zones of the wind is (Spitzer &Tomasko 1968),

cr = nHE = 1.9 1028 nH erg s1 cm3 (C.5)where is the ionization rate which we took as = 3.5 1017 s1 (Webber 1998) and E = 3.4 eV is the averagethermal energy transmitted to the gas by each ionization(Spitzer & Tomasko 1968). This effect is at most 3.6 106

times drag for model A,

Macc = 105

M yr1

and0 = 0.1 AU.Finally the thermal conductivity along magnetic field

lines was computed with the Spitzer conductivity for a

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Fig. C.3. Effect of initial temperature on the thermal evolu-tion for model B, Macc = 10

6 M yr1. The several initial

temperatures are in dashed 50 K, 100 K, 500 K, 1000 K, 2000 Kand 3000 K. In solid we plot the solution obtained by our stan-dard initial conditions. Note that the almost vertical evolutionof the temperature for very low initial temperatures is not anartifact. The field line anchored at 0 = 0.1 AU crosses thesublimation surface at 0.5.

fully ionized gas (Lang 1999) and is irrelevant (at most106 of the adiabatic cooling term) the maximum be-ing achieved at the recollimation zone where the physicalvalidity of our MHD solutions ends. It should be pointedout that Nowak & Ulmschneider (1977) compute the ther-mal conductivity for a partially ionized mixture in ion-ization equilibrium and found that for low temperaturesT 1034K the Spitzer expression underestimates theconductivity by a factor 102. However this is still too smallto be important.

C.3. Dependence on initial conditions

Formally, our temperature integration is an initial valueproblem. In the absence of a self-consistent descriptionof the disc thermodynamics, there is some freedom in theinitial temperature determination. It is therefore crucial tocheck that the subsequent thermal evolution of the winddoes not depend critically on the adopted initial value.

Safier obtained the initial temperature by assumingthe poloidal velocity at the slow magnetosonic point(vp,s) to be the sound speed for adiabatic perturbations

Ts = mHv2

p,s/kB. Here, we have chosen to compute theinitial temperature assuming local thermal equilibriumDT /Dt = 0. Our method produces lower initial temper-atures than Safier due to adiabatic cooling.

For high accretion rates Macc 106 M yr1 ourinitial temperature versus 0 has a minimum at the be-ginning of the dusty zone: inside the sublimation cavity,the thermal equilibrium is between photoionization heat-ing and adiabatic cooling. Just beyond the dust sublima-tion radius, photoionization heating is strongly reduced,but the ionization fraction is still too high for efficient dragheating, resulting in a low initial equilibrium temperature.

The initial ionization fraction is similarly determinedby assuming local ionization equilibrium DfiA/Dt = 0for all elements. It decreases with 0. For Macc =105 M yr

1 and 0 0.8 AU, the initial ionizationfraction is set to a minimum value by assuming that Nais fully ionized, which is somewhat arbitrary. However, asgas is lifted up above the disk plane, the dust opacity de-creases and the gas heats up, so that ionization becomesdominated by other photoionized heavy species and by

protons, all computed self consistently.In order to check that our results do not depend on

the initial temperature, we have run model B for a broadrange of initial temperatures. As shown in Fig. C.3), wefind that the thermal and ionization evolution quickly be-comes insensitive to the initial temperature. If we startwith a temperature lower than the local isothermal condi-tion, the dominant adiabatic cooling is strongly reduced,and the gas strongly heats up, quickly converging to ournominal curve. If we start with a higher initial tempera-ture, adiabatic cooling is stronger, and we have the charac-teristic dip in the temperature found by Safier. Our choice

of initial temperature has the advantage of reducing thisdip, which is somewhat artificial (see Fig. C.3). In eithercase, we conclude that our results are robust with respectto the choice of initial temperature. In particular, the dis-tance at which the hot plateau is reached, which has acrucial effect on line profile predictions, is unaffected.

References

Bacciotti, F., Mundt, R., Ray, T. P., et al. 2000, ApJL, 537,L49

Bally, J., & Reipurth, B. 2001, ApJ, 546, 299Bertout, C., Basri, G., & Bouvier, J. 1988, ApJ, 330, 350Binette, L., Cabrit, S., Raga, A., & Canto, J. 1999, A&A, 346,

260Binette, L., Dopita, M. A., & Tuohy, I. R. 1985, ApJ, 297, 476Binette, L., & Robinson, A. 1987, A&A, 177, 11Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883Bontemps, S., Andre, P., Terebey, S., & Cabrit, S. 1996, A&A,

311, 858Burrows, C. J., Stapelfeldt, K. R., Watson, A. M., et al. 1996,

ApJ, 473, 437Cabrit, S., Edwards, S., Strom, S. E., & Strom, K. M. 1990,

ApJ, 354, 687Cabrit, S., Ferreira, J., & Raga, A. C. 1999, A&A, 343, L61Camenzind, M. 1990, Rev. Mod. Astrophys., 3, ed. G. Klare

(Springer-Verlag, Berlin)Casse, F., & Ferreira, J. 2000a, A&A, 353, 1115Casse, F., & Ferreira, J. 2000b, A&A, 361, 1178Chan, K. L., & Henriksen, R. N. 1980, ApJ, 241, 534

8/7/2019 aa1260

20/20


Cohen, M., Emerson, J. P., & Beichman, C. A. 1989, ApJ, 339,455

Dougados, C., Cabrit, S., Lavalley, C., & Menard, F. 2000,A&A, 357, L61

Draine, B. T. 1980, ApJ, 241, 1021

Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89Draine, B. T., & Malhotra, S. 1993, ApJ, 414, 632Ferreira, J. 1997, A&A, 319, 340Ferreira, J., & Pelletier, G. 1993, A&A, 276, 625Ferreira, J., & Pelletier, G. 1995, A&A, 295, 807Ferruit, P., Binette, L., Sutherland, R. S., & Pecontal, E. 1997,

A&A, 322, 73Garcia, P. J. V., Cabrit, S., Ferreira, J., & Binette, L. 2001,

A&A, 377, 609, Paper IIGeiss, J., & Buergi, A. 1986, A&A, 159, 1Gomez de Castro, A. I., & Pudritz, R. E. 1993, ApJ, 409, 748Goodson, A. P., Bohm, K., & Winglee, R. M. 1999, ApJ, 524,

142Gullbring, E., Hartmann, L., Briceno, C., & Calvet, N. 1998,

ApJ, 492, 323Hartigan, P., Edwards, S., & Ghandour, L. 1995, ApJ, 452, 736Hartigan, P., Morse, J. A., & Raymond, J. 1994, ApJ, 436, 125Hartigan, P., Raymond, J., & Hartmann, L. 1987, ApJ, 316,

323Heyvaerts, J., & Norman, C. 1989, ApJ, 347, 1055Hirth, G. A., Mundt, R., & Solf, J. 1997, A&AS, 126, 437Hollenbach, D., & McKee, C. F. 1979, ApJS, 41, 555Krasnopolsky, R., Li, Z., & Blandford, R. 1999, ApJ, 526, 631Kwan, J. 1997, ApJ, 489, 284Kwan, J., & Tademaru, E. 1988, ApJL, 332, L41Kwan, J., & Tademaru, E. 1995, ApJ, 454, 382Lang, K. R. 1999, Astrophysical Formulae, vol. I (Springer

Verlag)Laor, A., & Draine, B. T. 1993, ApJ, 402, 441Lavalley-Fouquet, C., Cabrit, S., & Dougados, C. 2000, A&A,

356, L41Li, Z. 1995, ApJ, 444, 848Li, Z. 1996, ApJ, 465, 855Lovelace, R. V. E., Romanova, M. M., & Bisnovatyi-Kogan,

G. S. 1999, ApJ, 514, 368Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217,

425

Mundt, R., & Eisloffel, J. 1998, AJ, 116, 860Nowak, T., & Ulmschneider, P. 1977, A&A, 60, 413Osterbrock, D. E. 1961, ApJL, 134, 270Ouyed, R., & Pudritz, R. E. 1993, ApJ, 419, 255Ouyed, R., & Pudritz, R. E. 1994, ApJ, 423, 753

Ouyed, R., & Pudritz, R. E. 1997, ApJ, 484, 794Owocki, S. P., Holzer, T. E., & Hundhausen, A. J. 1983, ApJ,

275, 354Raga, A., Lopez-Martn, L., Binette, L., et al. 2000, MNRAS,

314, 681Ray, T. P., Mundt, R., Dyson, J. E., Falle, S. A. E. G., & Raga,

A. C. 1996, ApJL, 468, L103Reipurth, B., Bally, J., Fesen, R. A., & Devine, D. 1998,

Nature, 396, 343Rekowski, M. V., Rudiger, G., & Elstner, D. 2000, A&A, 353,

813Ruden, S. P., Glassgold, A. E., & Shu, F. H. 1990, ApJ, 361,

546Safier, P. N. 1993a, ApJ, 408, 115Safier, P. N. 1993b, ApJ, 408, 148Sauty, C., & Tsinganos, K. 1994, A&A, 287, 893Savage, B. D., & Sembach, K. R. 1996, ARA&A, 34, 279Schmitt, W., Henning, T., & Mucha, R. 1997, A&A, 325, 569Schunk, R. W. 1975, Planet. Space Sci., 23, 437Shang, H., Shu, F. H., & Glassgold, A. E. 1998, ApJL, 493,

L91Shu, F., Najita, J., Ostriker, E., et al. 1994, ApJ, 429, 781Shu, F. H., Najita, J., Ostriker, E. C., & Shang, H. 1995, ApJL,

455, L155Shu, F. H., Najita, J., Ostriker, E. C., & Shang, H. 1996, ApJL,

459, L43Solf, J. 1989, in Low Mass Star Formation and Pre-main

Sequence Objects, 399Solf, J., & Bohm, K. H. 1993, ApJL

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