Astron. Astrophys. 343, 953–965 (1999) ASTRONOMYAND

ASTROPHYSICS

Numerical hydrodynamic simulationsof molecular outflows driven by Hammer jets

Roland Volker1, Michael D. Smith2,1, Gerhard Suttner1, and Harold W. Yorke3,1

1 Astronomisches Institut der Universitat Wurzburg, Am Hubland, D-97074 Wurzburg, Germany2 Armagh Observatory, College Hill, Armagh BT61 9DG, Ireland3 Jet Propulsion Laboratory, MS 169–506, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

(smith@star.arm.ac.uk; suttner@ voelker@ astro.uni-wuerzburg.de; yorke@lear.jpl.nasa.gov)

Received 13 August 1998 / Accepted 15 December 1998

Abstract. Very young protostars eject collimated jets of molec-ular gas. Although the protostars themselves are hidden, some oftheir properties are revealed through the jet dynamics. We heremodel velocity shear, precession, pulsation and spray withindense jets injected into less-dense molecular clouds. We inves-tigate the Hammer Jet, for which extreme velocity variationsas well as strong ripping and spray actions are introduced. Athree dimensional ZEUS-type hydrodynamics code, extendedwith molecular physics, is employed.

Jet knots, previously shown to be compact in simulations ofsmoother jets, now appear as prominent bow shocks in H2 and asbullets in CO emission lines. High proper motions are predictedin the jet. In the lobes we uncover wide tubular low-velocity COstructures with concave bases near the nozzle. Proper motionvectors in the lobes delineate a strong accelerated flow awayfrom the head with some superimposed turbulent-like motions.The leading bow is gradually distorted by the hammer blowsand breaks up into mini-bow segments. The H2 emission lineprofiles are wide and twin-peaked over much of the leading bow.

On comparison with the simulations, we identify observedoutflows driven by various dynamical types of jet. Shear is es-sential to produce the jet bows, spray or precession to widen theoutflows and hammer blows to generate knotty jets. We iden-tify the proper motions of maser spots with the pattern speed ofdensity peaks in the inner jet and shell.

Key words: hydrodynamics – shock waves – ISM: clouds –ISM: jets and outflows – ISM: molecules – radio lines: ISM

1. Introduction

Molecular jets and outflows appear during a critical phase inthe formation of a star in which the mass of an envelope ac-cretes onto a growing protostellar core (e.g. Andre et al. 1993,Bachiller 1996, Padman et al. 1997). The jets may well be theagents which channel away excess angular momentum to allowthe collapse to proceed (Shu et al. 1994, Bontemps et al. 1996).

Send offprint requests to: M.D. Smith

They may also promote the disruption and dispersal of the em-bedding cloud and so limit the eventual stellar mass (Velusamy& Langer 1998). Furthermore, the properties of the extendedjets take on extra significance because the stars themselves arehighly obscured even in the infrared (Andre et al. 1993). Thussurveys for outflows in the infrared can be used to locate theyoungest protostars (Stanke et al. 1998). Ideally, we would liketo detect or place useful constraints on the angular momentumand magnetic field in the outflow, since these quantities playcrucial roles in the interaction between the accretion disk, mag-netosphere and the wind (e.g. Camenzind 1997, Shu & Shang1997, Ouyed et al. 1997). This is, however, not yet observa-tionally feasible. Here, we ask: what other constraints on howan outflow must have formed, how it has developed and how itprovides feedback on the environment in which it is created, canat present be derived? With this aim we develop the Wurzburgmolecular outflow model (Suttner et al. 1997, Smith et al. 1997a)to examine highly variable and non-uniform jets.

Many qualitative properties of the earliest outflows fromprotostars have now been ascertained (Tables 1 and 2). Theseoutflows are often driven by highly supersonic pulsed jets. Theyare well collimated and extremely long. Despite being deeplyenshrouded by molecular gas, the warm shocked regions canbe observed in the infrared H2 lines, thus revealing the presentimpact regions. The cool accumulated material can be observedin the submillimetre CO lines, delineating the total swept-upand swept-out material. Some of the constraints imposed arequantified in Sect. 2.

The theory of molecular jets is still being established. Ourmodel involves three dimensional hydrodynamic calculationsof supersonic molecular jets into molecular environments (Sut-tner et al. 1997). The ZEUS-type code encompasses the basicmolecular characteristics: strong cooling, dissociation in fastshocks and reformation in dense dusty environments. It has sofar proven to be remarkably successful, as signified by the num-ber of ticks in the two tables. Nevertheless, the results sug-gest some extra dynamical input is required: we need to simu-late more active jets to determine if bow-shaped jet knots andwider outflows are feasible. Specifically, we here investigate(1) hammer-like pulsations, (2) velocity shear, (3) directional

954 R. Volker et al.: Simulations of molecular outflows driven by Hammer jets

Table 1.Molecular Hydrogen signatures, typical examples and the successes (ticks) and failures (crosses) of the Suttner et al. 1997 jet model

Multiple Bows Complex Filamentation Wide Turbulent JetL 1634: Davis et al. 1997a HH 90/91: Davis et al. 1994b HH 110: Davis et al. 1994bCepheus A: Hartigan et al. 1996 HH 2: Davis et al. 1994a

Twin/single-peaked line Profiles Extremely Broad Profiles Constant ExcitationL 1448: Davis & Smith 1996 DR 21: Garden et al. 1991 Cepheus E: Eisloffel et al. 1996Cepheus E: Eisloffel 1997 HH 7-11: Carr 1993 L 1448: Davis & Smith 1995

Knots/mini-bows in jets Asymmetric Lobes High Proper MotionsHH 111: Gredel & Reipurth 1993 L 1660: Davis et al. 1997a HH 46/47: Micono et al. 1998HH 212: Zinnecker et al. 1998 L 1551: Davis et al. 1995 HH 111: Coppin et al. 1998

/ /

Table 2. Signatures of well-collimated CO bipolar outflows and the successes (ticks) and failures (crosses) of the Smith et al. jet model.NGC 2264G (Lada & Fich 1996) displays most of these signatures. Note that these are examples of a class of outflows. It is not clear howcommon counter-examples may prove.

Collimation increases Red/blue lobe symmetry Power-law line profileswith Velocity (high-v) (γ = −1.3 to−2.0)

Hubble Law Strong Forward Motion High-speed CO Bullets(velocity∝ distance) IRAS 03282+3035: Bachiller et al. 1991

High-speed CO Jets Tubular low-speed structure Helical appearanceHH 211: Guilloteau et al. 1997 HH 111: Nagar et al. 1997 L 1157: Gueth et al. 1996

P

P

θ0v (0,t)

0v (R,t)

ψ

precession

pulsationspray nozzle

velocity shear

Fig. 1. Nozzle geometry for the 3D simulations

changes and (4) wide opening angles, and present their signa-tures, including line images, proper motions and spectroscopicproperties.

A comparison to other protostellar jet models was presentedby Smith (1998). Frank et al. (1998) have also reviewed as-trophysical jet simulations. In addition, three-dimensional hy-dromagnetic jets with atomic radiative cooling have now beenpresented by Cerqueira, de Gouveia Dal Pino & Herant (1997).Molecular jets are, however, particularly difficult to simulate nu-

merically because of the high compression behind the strongly-radiative shocks. Nevertheless, Raga et al. (1995) managed tosimulate an atomic jet drilling into a molecular cloud in 2D byemploying an adaptive grid technique. Synthetic images frommolecular emission lines have also been generated on assum-ing the shock physics across paraboloidal bow shocks (Smith1991).

We begin by placing the physical and dynamical propertiesstudied here into their context. Radiative bow shocks drivenby jets were analyzed by Raga (1988) through simulations intwo dimensions. Complete outflows were simulated by Blondin,Fryxzell & Konigl (1990). They demonstrated the formation ofextended nose cones and disrupted, deformed bows. GouveiaDal Pino & Benz (1993) and Stone & Norman (1994) steppedup the simulations to three-dimensions (3D) and found that theshell disrupted into knots and filaments but nose cones wereabsent. These findings were confirmed for strongly-radiativemolecular fluids in full 3D in the work of Suttner et al. (1997).

Jet structure could be generated by internal fluid instabilities,external triggers or source variations. Blondin et al. found thatthe reflecting pinch modes of the Kelvin-Helmholtz instabilitywere suppressed in strongly-cooling jets. Similarly, Chernin etal. (1994) found that a low-density cocoon of gas shields highMach number jets. The pulsation model has received most at-tention (e.g. Raga et al. 1990; Kim & Raga 1991; Hartigan &

R. Volker et al.: Simulations of molecular outflows driven by Hammer jets 955

Fig. 2. The boundary condition for the jet velocity of model 3D-5: thecentral input velocity as a function of time (in years).

Raymond 1993; Stone & Norman 1993; Biro & Raga 1994;Smith et al. 1997b). Note that non-periodic variations, whichwill be included here, have been considered by Raga (1992),who obtained some self-similar solutions for the separation ofknots.

Precession and directional variability of strongly coolingand ballistic jets were investigated by Raga et al. (1993), Biro,Raga & Canto (1995) and Cliffe et al. (1996), amongst others.Biro et al. demonstrated the proper motion vectors for densitypeaks in an adiabatic jet. Raga (1988) presented the proper mo-tions from simulations of bow shocks driven by solid obstacles.Here we shall model the proper motions in strongly coolingjet-driven flows.

Other topical questions concern molecule formation on thesmall-scale (e.g. Ruden, Glassgold & Shu 1990; Glassgold, Ma-mon & Huggins 1991) and the continued growth of the gigan-tic outflows (Bally & Devine 1994; Eisloffel & Mundt 1997;Reipurth et al. 1997). We intend to approach these themes withnumerical simulations after the basic dynamical properties areestablished.

2. The Hammer jet model

2.1. Limitations

Only the jet-driven outflow model is simulated here. Thus weassume that a collimated jet of gas is switched on and main-tained. (The alternative model for collimated flows involveswinds focussed on quite large scales (Barral & Canto (1981)).This model for molecular jets has been developed and appliedby Suttner et al. (1997) and Smith et al. (1997a). To summa-rize: (1) three-dimensional simulations are performed, (2) a hy-drodynamic code is employed, (3) the jet and environment areinitially fully molecular, (4) the jet is over-dense by a factor of10, (5) the jet power is variable but non-evolving (i.e. no sys-tematic long-term variation), (6) the environment is uniform inall parameters. Given these limitations, the ability to reproduceseveral observed features is encouraging.

The conception followed here, presented in more detail bySmith (1998), is that the Class 0 phase is accompanied bydense molecular jets. The jets may also contain a high frac-tion of atomic hydrogen simply because molecule formation isnot complete. During this phase large eruptions occur on timescales of 50–1000 years but the mass outflow over each 1000years is held constant (some fraction of the accreting mass isejected). This phase lasts a total of at least 10,000 years. In the

following Class 1 phase the jet density is considerably lowerand molecules cannot form within the jet; the environment hasbeen swept away. In the Class 0 phase the jet possesses ballisticproperties whereas in the Class 1 phase it interacts strongly withthe cavity (e.g. through the growth of non-linear waves formedby the Kelvin-Helmholtz instability).

We shall here assume the inflowing hydrogen gas to be100% molecular. The shocks which form in the pulsed jets arestrong enough near the nozzle to dissociate at least some of themolecules, leading to jet knots with a mixture of atomic andmolecular gas. In sheared jets, the knots appear as bow shockswith hot atomic gas at the apices and warm molecular gas alongthe flanks.

The calculations were carried out on a Cartesian grid thatcontains500 × 80 × 80 zones. The whole grid covers an inte-gration domain of(12.5 × 2.0 × 2.0) 1016 cm3. A nozzle withradiusR = 1.625 · 1015 cm was placed at the center of one sideof the grid.

We are limited by the grid resolution and the molecularcooling time scale to simulate on rather short time scales andlength scales. Examples of small jet outflows include HH 211(2 1017 cm), HH 24 (6 1016 cm and L 483 (2 1017 cm) (see Ta-ble 5 of Davis et al. 1997a). We compensate by consideringquite rapid variations. Thus we find that the end of the gridis reached in typically 600 years. However, (1) the dynamicsare controlled only by the fact that strong radiative cooling ispresent, and (2) the infrared structure is dominated by narrowshocked regions. Hence,provided that the chemistry does notintroduce a further time scale, we believe that the flow pat-terns and resulting images should be scalable. Specifically, wefind that molecule reformation is negligible during the simula-tion: the reformation time scale, assuming H2 reforms on cooldust grains, is roughly1017/n s, where n is the atomic density.Hence, clumps of atomic gas of density106 cm−3 would beconverted to molecular clumps on a time scale of 3000 years.However, we actually find, because the atomic gas is formedin fast shocks through collisional dissociation of the molecules,instead of cooling and collapsing into clumps, the atomic gassimply fills a large low-density cavity. This gas then does notreassociate.

Speeds in observed molecular jets range from 100 to500km s−1. This range is derived from proper motions of H21-0 S(1) knots (Micono et al. 1998, Coppin et al. 1998). Herewe take the lower limit, consistent with our first goal of simu-lating the first stages in the outflow formation. Higher speeds ofmolecular jets were considered in two dimensional simulationsby Suttner (1997). No noteworthy new structures were detected.

2.2. Precession

Five three dimensional simulations have been performed (Ta-ble 3). These include two simulations previously analyzed bySuttner et al. (1997) and three new simulations. The first twosimulations (3D-1 & 3D-2) were of uniform and low-amplitudepulsation jets with small precession angles. They are rediscussedhere to contrast the new simulations. Furthermore, we have im-

956 R. Volker et al.: Simulations of molecular outflows driven by Hammer jets

Fig. 3. The density of H2 molecules from models 3D-1 (upper) to 3D-5 (lower). Note the black edges where molecules are destroyed in fastshocks. The XY plane which includes the jet axis are displayed.

Table 3.Parameter for the three dimensional calculations. Here nm is the ambient nucleon density.

shear pulsation precession spraymodel nm/104 cm−3 nj/nm vj/km s−1 v0(R)/v0(0) ∆v/vj P/yr θ P/yr ψ

3D-1 1.4 10 100 0.0 0.00 – 1o 100 0o

3D-2 1.4 10 100 0.0 0.30 50 1o 100 0o

3D-3 1.4 10 100 0.5 0.90 50 1o 46 0o

3D-4 1.4 10 100 0.5 0.90 40 2o 36 2o

3D-5∗ 1.4 10 55–160∗ 0.5 0.82–0.94∗ 12 1.75o 26 2o

∗ See Fig 2 for the input velocity.

proved the technique to analyze the data (when making imageswe take advantage of all the information in the second-ordercalculations).

Precession, or more appropriately, directional wandering, issuspected from the spatial structure of several molecular out-flows including Cepheus E (Eisloffel et al. 1996), RNO 43(Bence, Richer & Padman 1996), RNO 15-FIR (Davis et al.1997b). Cliffe, Frank & Jones (1996) have studied the preces-sion of heavy jets with numerical simulations, but with adiabaticequations of state. Hence, it is not clear that their conclusion,

that precessing jets match better the morphology and momen-tum distribution of molecular outflows, is valid.

2.3. High-AMplitude Multiple ERuptions (HAMMER)

We superimpose a strong sinusoidal perturbation onto the uni-form jet velocityvj at the nozzle:

v0(0, t) = vj + ∆v sin(

2π

Pt

), (1)

R. Volker et al.: Simulations of molecular outflows driven by Hammer jets 957

Fig. 4.The 1-0 S(1) H2-emission images of models 3D-1 (top) to 3D-5 (bottom) taken from near the end of the runs. The fluxes were calculatedby the method of Suttner et al. (1997). The XY projection is shown here.

wherev0(0, t) is the velocity in the center of the nozzle (r =0). Hammer-type pulsations in the injection velocity are thusintroduced in the three new models (Table 3. The variations aretypically 90%. This causes the material to rapidly pile up intoknots in the jet and produces bullet-type outflows (see Sect. 3).

2.4. Shear and spray

Jet shear (see Fig. 1) is quite plausible on theoretical grounds.Winds focussed by shocks may well have higher central veloci-ties since the strength of the deflecting shock is angle-dependent.In extended disk-wind models the wind speed reflects the origi-nal Keplerian speed. In fast stellar winds asymmetries are relatedto the rotation of the star. Shear is difficult to detect but has beenobserved in the HH 47 jet (Hartigan et al. 1993).

We apply a velocity variation over the cross section of thejet inflow:

v0(r, t) =(

1 − r2

2R2

)v0(0, t) . (2)

Finally, the jet beam was forced to precess at the nozzle with aspecified opening angle. The parameters used for the individualmodels are listed in Table 3.

3. Results

Cross sections of the molecular density of all the models aredisplayed in Fig. 3. Note:

• The black tips are the regions where molecular destructionis complete. This occurs mainly at the locations of the strongshocks: at the leading edge and at the mini-bows along the shell.• Low molecular densities also occur within the cavity, i.e. be-tween the dense jet and the dense shell.• High molecular densities are restricted to thin sheets whereoblique shocks have swept up the gas.• Knots of atomic gas are found along the jet axis, as moleculesare destroyed as they enter the strong shock at the beginning ofthe jet. Downstream, the jet shocks weaken and the moleculessurvive (e.g. see Smith et al. 1997b).• The introduction of a spray angle significantly widens the out-flow shell and cavity as well as broadening the jet bow shocks.• The cause of the rapid growth of mini-bows along the shellwalls are the hammer blows from the jet. The blows acceleratethe knots, which then grow via the Rayleigh-Taylor instability.This appears to intensify wave structures originally produced inthe bow via the “thin-shell” instability (see Mac Low 1998).• The jets propagate within low-density cavities, unaffected bythe ambient medium.

958 R. Volker et al.: Simulations of molecular outflows driven by Hammer jets

Fig. 5. H2 1-0 S(1) line profiles from se-lected regions in the bow shock of model3D-4. The projection angle is 60◦ to theline of sight. The abscissa runs from−20to 80km s−1, with the vertical dashed lineat the rest velocity. The relative peak inten-sities are indicated.

Fig. 6. H2 1-0 S(1) line profiles from se-lected regions in the bow shock of model3D-4. The projection angle is 30◦ to theline of sight. The abscissa runs from−10to 140km s−1, with the vertical dashed lineat the rest velocity.

The infrared images (Fig. 4) bear little resemblance to themolecular distributions. The distributions of warm shockedmolecules possess the following properties.• A sequence of increasing jet brightness is produced, as ex-pected. Without pulses (top box) the jet is invisible. With highvariability (bottom box) an almost continuous infrared jet isgenerated.• Jet shear is necessary to generate bow shocks in the jet in the1-0 S(1) H2 line. Pulsation alone (model 3D-2) is not sufficient.• A complex emission structure is produced in the head regionthrough high shear and hammer blows. Most of this emissionarises from sheet structures confined to a disturbed bow-shellstructure.

• Precession distorts the arrangement of bow shocks, which thenappear slightly off-axis.• Wave patterns are generated in and behind the main head.Close analysis reveals that they lie near the inner shell surface,generated by the Kelvin-Helmholtz instability due to relativefluid motions.

High-resolution spectroscopy of the H2 line provides a testfor bow shock models. We present the line profiles for model3D-4 in Figs. 5 and 6. These can be directly compared to thoseof model 3D-1 published by Suttner et al. (1997, Fig. 18). Thehammer model leads to twin-peaked line profiles over an ex-tended region (e.g. over the edge C, D, E & J). This is due tothe shocked jet layers which reach the termination region and

R. Volker et al.: Simulations of molecular outflows driven by Hammer jets 959

Fig. 7. The CO–emission from theJ = 1 - 0rotational transition (i.e. 0-0 R(1)) of model 3D-4. Three radial velocity intervals are displayed. Theangle between the jet axis and the line of sight is 60◦.

Table 4. Total luminosities and excitation temperatures near the end ofeach simulation. The percentage of the total jet mechanical luminosityLm is given in parentheses; the model time of the simulation is in thefinal column.

modelH2 1−0 S(1)

L/L�H2 2−1 S(1)

L/L�CO 0−0 R(1)

L/L� Lm/L� Tex/K t/yr

3D-1 9.0 · 10−4

(0.5%)1.1 · 10−4

(0.06%)1.5 · 10−6

(0.0008%)0.20 2300 600

3D-2 2.7 · 10−3

(1.2%)2.8 · 10−4

(0.13%)1.3 · 10−6

(0.0006%)0.22 2200 604

3D-3 3.0 · 10−3

(0.8%)6.4 · 10−4

(0.18%)8.2 · 10−7

(0.0002%)0.36 3000 457

3D-4 5.8 · 10−3

(1.6%)8.3 · 10−4

(0.23%)1.2 · 10−6

(0.0003%)0.36 2500 553

3D-5 5.0 · 10−4

(—)1.0 · 10−4

(—)9.0 · 10−8

(—)— 3600 462

are reshocked as they decelerate. In model 3D-1, the uniformjet, twin peaks were only generated very close to the bow apex.Note that the mini-bow structure C possesses the most com-plex line profiles, whereas Knots G & H arerelatively narrow.Hence, Hammer Jets produce rather turbulent bow heads as willbecome evident when we study the kinematics below.

The CO emission is dominated by a smooth tubular structureinto which the ambient gas has been swept up over a consid-erable time. The CO submillimetre images for model 3D-4 isshown in Fig. 7. The jet bullets are present in CO but weak. Ajet component and a bow component, however, can dominate inthe high radial velocity channels.

Outflows generated by Hammer Jets (3D-3, 4 & 5) growfaster than the more uniform jet-driven outflows (see Column 7of Table 4). The bow of model 3D-3 advances approximately

31% quicker than the 3D-1 bow. To explain this, the factors toconsider are the extra momentum flow rate due to the pulsations,the lower average momentum flow due to the shear and theballistic propagation of the jet core. Time and space integrationsyield the result that the specific momentum flow rate in model3D-3 is 12% higher than in model 3D-1, therefore not sufficientto explain the fast growth of model 3D-3. Hence, it is the ballisticmotion of the jet core which is responsible. This is confirmed byinspecting Fig. 4, which shows a highly streamlined bow shockdue to the fast jet core propagation.

The spray jet model advances, of course, slower. The dif-ference in advance speed between models 3D-3 and 3D-4 re-mains quite small, however, since the jets are still ‘over-dense’,and thus the advance speed U is only a weak function of thejet speed (note that the classical momentum balance yieldsU = vj/(1 +

√η) whereη = nm/nj). Note also that the

advance speed and the excitation temperature are correlated.One may naively expect the presence of a greater quantity ofhotter gas to lead to higher H2 excitation (as measured by the 2-1S(1)/1-0 S(1) ratio). This reasoning is not absolutely straightfor-ward, however, since the excitation should depend on theratio ofwarm and cool molecular gas which in turn depends on the bowshock shape or the spectrum of shocks produced in the turbulentwake, not on the absolute bow speed (since higher bow speedsonly lead to more dissociation). We have previously found thatthe highest excitation is produced in the turbulent flanks ratherthan near the bow’s leading edge (Suttner et al. 1997). We awaita more detailed study of supersonic turbulence dissipation toshed light on this issue.

The 1-0 S(1) luminosity is typically 1% of the available me-chanical jet power (Column 2 of Table 4). Total H2 luminosities

960 R. Volker et al.: Simulations of molecular outflows driven by Hammer jets

Fig. 8.A CO position-velocity diagram for model 3D-4. The line emis-sion from the submillimetre rotational transition J=1-0 is displayed and,to ease comparison, the full bipolar outflow is simulated. The angleαis the angle between the jet direction and the sky plane.

are typically 20 times higher (Smith 1995). Hence, about 20%of the power is radiated from hot molecular hydrogen gas. (Notethat H2O cooling has not been included in these calculations andmay well be a strong coolant of the warm gas.) There is also noclear trend of decreasing CO J = 1-0 flux with increasing activ-ity in the models, although the CO flux is extremely low in thehighly-variable 3D-5 simulation.

Finally, we demonstrate the appearance of CO bullets ona position-velocity diagram. Fig 8 shows that a series of ra-dial spikes are produced, which continue out to high veloci-ties. This is superimposed on the ‘Hubble-type’ low-velocitybehaviour (i.e. the triangular contours) which dominated theposition-velocity diagrams of the more uniform outflows (Smithet al. 1997a).

4. Kinematics

Protostellar jets often contain numbers of well aligned knots.The origin of this phenomenon is still not fully understood.Individual knots possess high proper motions away from the jetsource (Neckel & Staude 1987). This rules out jet models whichassume that the knots are stationary shocks (Konigl 1982). Twopossible mechanisms are left to discuss. In the first scenariothe knots are interpreted as crossing shock waves excited byKelvin-Helmholtz instabilities that occur at the edge of the jetchannel (Norman et al. 1982). The alternative model considersthe knots to be the result of a time variable jet source (Kofman& Raga 1992).

Efforts have been made to decide between these two mod-els by examining the kinematic properties. In detailed studiesEisloffel and Mundt (1992, 1994, 1995) measured the propermotions of knots in several well known jet systems. From theseoptical observations they calculated the absolute values of thevelocity of the pattern movement. This pattern speedvp is com-pared to the fluid speedvf derived from radial velocity mea-surements. The pattern speedvp is mostly observed to be lower.

Fig. 9. A high-resolution one-dimensional study of proper motionsfrom model 1D-3, as described in Table 5. At the time of 55.7 yr thejet has propagated out to 1.83 1016 cm.

Fig. 10.The kinematics of the density peaks of model 3D-3. Note thatthe top panel displays thelogarithm of the ratio of pattern to fluidspeeds, shown individually below.

The measured ratiosζ = vp/vf usually range from0.4 to1.0. Insome of the jet systems theζ-values increase with propagationdistance. These results seem to be in agreement with modelswhich describe the knots as running shock waves.

However, the question remains as to the origin of the shocks.For this reason we investigated here the kinematics of our nu-merical jet models. During a calculation the positions of ex-trema of different variables were detected in several successivetime steps. For numerical reasons these peaks cannot move fur-ther than a single grid cell spacing in one step. It is therefore astraightforward task to follow individual peaks and to determinethe direction and the velocities of their motion.

R. Volker et al.: Simulations of molecular outflows driven by Hammer jets 961

Table 5. Parameters and resulting knot velocities of the one dimensional models.

model nm/104 cm−3 nj/nm vj/km s−1 ∆v/vj P/yr shapenumericvk/vj

analyticvk/vj deviation

1D-1 1.4 10 100 0.3 5 sinusoidal 1.06 1.045 1.4%1D-2 1.4 10 100 0.6 5 sinusoidal 1.27 1.180 7.6%1D-3 1.4 10 100 0.9 5 sinusoidal 1.50 1.405 6.8%1D-4 1.4 10 100 0.3 5 sawtooth 1.02 1.030 1.0%1D-5 1.4 10 100 0.6 5 sawtooth 1.10 1.120 1.8%1D-6 1.4 10 100 0.9 5 sawtooth 1.21 1.270 4.7%

Fig. 11.The kinematics of the density peaks of model 3D-4. Note thatthe top panel displays thelogarithm of the ratio of pattern to fluidspeeds, shown individually below.

Fig. 12.The relationship between the four speeds: pattern (p), fluid (f),radial (r) and tangential (t).

4.1. One-dimensional proper motions

We present here the first numerical simulations of proper mo-tions in radiative jets. The proper motions of knots along theaxis of a jet are the ‘pattern speeds’. Biro (1996) has studiedthe pattern speed of an isolated internal working surface. Anumerical simulation demonstrated the deceleration due to theambient medium. Full jet simulations have not reproduced thisbehaviour, probably because cocoon densities are extremely low(Stone & Norman 1993; Suttner et al. 1997).

We begin with a one dimensional calculation, as shown inFig. 9. The variation of the jet velocity produces periodic dense

knots along the jet axis (upper panel). These material fragmentsexpand with increasing distance from the source and form acomplex structure of shock waves (Smith et al. 1997). The sec-ond panel displays the propagation velocities of density max-ima and of extreme density gradients in comparison to the fluidmotion. The one dimensional models (Table 5) show that theaverage pattern speed is always higher than the predicted knotvelocity

vk =

∫ P

0 ρ0(t)v20(t) dt∫ P

0 ρ0(t)v0(t) dt, (3)

wherev0(t) andρ0(t) are the time dependent velocity and den-sity of the material ejected by the jet source. This equation isbased on the assumption that at great distances from the originthe knots contain most of the mass and the momentum producedby the source during a periodP . The systematic deviation be-tween the numerical and the analytical results suggests that themethod which is used to release the momentum onto the inte-gration area contains numerical discrepancies.

The ratios between pattern and fluid speed are all close toone, as one can expect from analytic studies of vertical shockwaves (see Appendix). At the jet shocks where the gas overtakesthe shock wave we find values less than one and, vice versa, wefind values greater than one at the bows.

4.2. Three-dimensional density peaks

In the 3D-models the situation is different. Here we find that themeasured points split up into a fast and a slow component. Thisseparation can clearly be seen in Fig. 10. The individual plotsdisplay the absolute values of the calculated fluid and patternvelocities of the density peaks in model 3D-3 and the corre-sponding ratios. The low velocity component is formed fromdensity maxima in the bow shock that surrounds the whole jet.The velocities range from only a fewkm s−1 in the region closeto the origin up to about100 km s−1 at the tip. Because the bowshock is oblique, only the normal component of the fluid speedis effected. Thus, one expects velocity ratios greater than one.In agreement with this idea, we find increasing ratios away fromthe apex. Whereas in the head region most of the values rangefrom 1 to 5, in the wings, where the shock is quite oblique, ratiosup to 40 are reached (as clarified in the Appendix).

The high velocity component forms several groups whichcan be assigned to the pulses in the jet flow. The velocity in-

962 R. Volker et al.: Simulations of molecular outflows driven by Hammer jets

creases towards the tips of the pulses. Thus they behave likesmall bow shocks. The values range from 70 to 150km s−1.The corresponding velocity ratios are close to 1, but exhibit asmall variation (0.8 < ζ < 1.4). However, we find greater vari-ations in other models. The density peaks that belong to the jetknots in model 3D-5 (Fig. 11) possess ratios up toζ = 2. Thekinematics of all models show in principle similar characteris-tics. The different velocities and ratios depend on the boundarycondition for the jet inflow.

4.3. Infrared proper motions

Using the method described above we studied the developmentof the emission structure of our models. Fig. 14 presents the mo-tion of intensity peaks of model 3D-4, calculated from consec-utive simulated H2 1-0 S(1) images. In the first plot an emissionmap is displayed that shows the jet at an orientation angle ofα = 30o. The arrows illustrate the direction and the speed ofthe pattern motionsvp calculated by reprojecting the measuredtangential velocitiesvt onto the jet axis (Fig. 12). The lowercaptions reveal the corresponding absolute values of the patternand fluid speeds and the resulting ratios. The required tangentialvt velocities were taken from the model data. The “measured”kinematics are comparable to the motion of the density maxima.Within the knots high velocities (120–160 km s−1) along the jetaxis appear. As in the example above the ratios of pattern to fluidspeed are close to one here but posses a certain variation. In thehead region, however, one can find in addition lower velocitieswith strongly differing ratios. The values forζ range from nearlyzero up to 4. Owing to lack of emission in the wings of the bowshock far away from the jet apex, a low velocity component isnot observed in the H2 1-0 S(1) line.

To demonstrate the motion within the bow shocks of our nu-merical models, we displayed this region in a reference framemoving with the jet. Fig. 13 reveals the relative pattern veloc-ities near the apex of model 3D-4. The average pattern speedof the jet has been subtracted. The knots posses the tendencyto accelerate progressively away from the leading edge backtowards the source. However, the flow pattern is more intricate,e.g. in a few cases motions towards the apex occur. These in-frared models closely resemble the properties that are observedin the optical (Eisloffel et al. 1994).

5. Conclusions

We have studied outflows driven by highly-variable HammerJets and compared them to relatively uniform jet-driven out-flows. We have presented a range of observable infrared andsubmillimetre properties. Many results for uniform jets remainvalid for Hammer Jets (e.g. CO line profile structures and the‘Hubble-type’ expansion law). Major new predictions, however,are as follows.

Jet results. Long strings of knots, visible as CO bullets and H2knots, are generated in highly-variable jets. Continuous H2 jetsare not produced. H2 bow shocks are produced when a velocity

shear is present. The CO bullets can be identified spectroscopi-cally on position-velocity diagrams.

Bipolar Lobes. Hammer jets generate more complex filamen-tary infrared lobe structures. CO images are dominated by wider,tubular structures at low velocities. H2 line profiles are oftentwin-peaked in the lobe, whereas for uniform jets twin-peakedprofiles are located only immediately behind the bow apex.

Proper Motions. We have presented detailed simulations ofproper motions. Although done specifically for the H2 infraredemission knots, the flow directions in the bows resemble the pat-terns produced in the optical studies (Eisloffel & Mundt 1992):some acceleration away from the head plus a turbulent compo-nent. The proper motions are very high in the jet, with the ratioof pattern to fluid speed close to unity.

This model unites the bullet and jet driven scenarios. Smoothjets may well drive the HH 211 outflow, where no H2 jet hasbeen detected. In contrast, HH 111, with a long string of jetbows, would be hammer type. This does not imply that HH 111possesses high-amplitude velocity variations in the visible jet –jet shocks weaken rapidly with distance. The weakening is am-ply demonstrated in Fig. 9. Tubular CO structures, remarkablylike those presented in Fig. 7, have been recently discovered inHH 111 (Nagar et al. 1997) and HH 211 (Guilloteau et al. 1997).The CO bullets in HH 111, are, however, found on a larger scalethan the H2 jet bows. This could imply that the most massiveknots are formed downstream, perhaps involving the gradualaccumulation of smaller bullets. Prominent gaps between theinner knots and the driving source are found on the infrared H2images of HH 111 and HH 212. In the present model 3D-5, wehave uncovered striking gaps. Fig. 15 displays the outflow at atime of 414 years, towards the end of a low period in the jetpower. This gap has disappeared 50 years later, as displayedin Fig. 4. This is an alternative interpretation for the gaps; highextinction of the inner infrared jets due to a dense core aroundthe protostar is expected.

Proper motion studies in the near infrared are now just pos-sible (Noriega-Crespo et al. 1997, see also Table 1). High speedsdirected roughly along the jet axis are indeed being found in thejets. Proper motions traced by atomic line emission, however,lead to pattern/fluid speed ratios which are often well underunity and variable along the jet axis (Eisloffel & Mundt 1992,1995). We predict thatdense molecular/atomic jetswill havepattern/fluid speed ratios near unity. Light pressure-confinedjets, in contrast, are susceptible to the Kelvin-Helmholtz in-stability and modes can grow locally rather than being rapidlyadvected.

High resolution infrared spectroscopy along jets and withinbow shocks should begin to reveal new features. The high-speedjet should be detected just where it impacts the bow regionthrough the high-velocity bump in the 1-0 S(1) line profile (seeFigs. 5 and 6).

Masers in outflows probably correspond to shock-compressed layers, edge-on to the observer. Maser observations

R. Volker et al.: Simulations of molecular outflows driven by Hammer jets 963

Fig. 10. 3D 4 Bow

Fig. 13. Proper motions and fluidspeeds in the bow shock of the 3D-4simulation of the emission from the1-0 S(1) H2 line as seen in the XY-plane.

Fig. 14. Proper motions and fluidspeeds in the 3D-4 simulation of theemission from the 1-0 S(1) H2 line,as seen in the XZ-plane.

Fig. 15.The hydrocode simulation of a pulsating and outbursting jet, model 3D-5, demonstrates (at 414 years) a gap followed by complex knotstructures. The infrared H2 1-0 S(1) emission is displayed.

provide constraints on both proper motions and radial veloci-ties (Gwinn, Moran & Reid 1992). Mac Low et al. (1994) testedthe proposal that masers could be identified with density peakswithin a jet-driven outflow. The simulated pattern speeds of thedensity peaks shown here in Fig. 11 are indeed strikingly similarto those observed for maser proper motions in the W49N out-

flow (Gwinn et al. 1992). This is the first direct comparison forthe ‘bullet model’ (since Mac Low et al. (1994) investigated theinstantaneous velocities i.e. the fluid speeds, not the proper mo-tions). On small scales, hypersonic turbulence is inferred fromthe observed maser properties (Gwinn 1994). Confirmation will

964 R. Volker et al.: Simulations of molecular outflows driven by Hammer jets

require higher resolution simulations to resolve hypersonic jet-driven turbulence within the dense layers.

Several problems listed in Tables 1 and 2 remain to be solved.We have made no detailed attempt to model the excitation ofthe hydrogen molecules or other molecular species. A physicaldescription of supersonic turbulence in jets is still lacking al-though one means to generate it by fluid instabilities is there (e.g.Rossi et al. 1997; Stone et al. 1997; Downes & Ray 1998). Themagnetic field has also been ignored. Ambipolar diffusion maybe important in molecular jets, heating the gas even in uniformjets. Evolution in the spray angle and the jet velocity should bemodeled since outflow ages and protostellar formation ages arecomparable.

Finally, we return to the initial goal: what can we now sayabout the protostar itself? First, the ejected material is consis-tent with a simple hydrodynamic flow: laminar and center-filledrather than shell-type. There is some evidence for moderateshear and spray being introduced, either at the source or dur-ing the propagation through the first1015 cm. Small variationsin the ejection direction are also consistent. The outburst timescale is given by the inter-knot spacing divided by the knots’velocity. Outbursts must generate jet powers which vary by highfactors. Jets with multiple time scales for the power variations(see model 3D-5) generate a fuller jet structure. These varia-tions could arise from the thermal instabilities in accretion disksthought to be responsible for FU Ori objects (Bell & Lin 1994).Note, however, that we have here specifically modeled velocityvariations, holding the density fixed. An accretion disk instabil-ity would probably lead to variations in both parameters. Morework is necessary to specifically identify FUor outburst signa-tures with jet structures and thus to establish the inflow-outflowconnection in the earliest protostars.

Acknowledgements.We sincerely thank C. Davis, J. Eisloffel, M.-M.Mac Low and H. Zinnecker for helpful discussions. We also thankthe DFG for financial support. The calculations were performed onCRAY computers at the LRZ Munchen, at the HLRZ Julich, and at theRechenzentrum der Universitat Wurzburg.

Appendix

We here develop the basic relation between the propagationspeedvs of the shock wave and the velocityvg of the deflectedgas. For this purpose we assume an oblique shock that runs intoa steady medium. Fig. A1 shows the shock propagating alongthex-axis in a frame which is moving with the shock wave. Theunshocked gas moves towards the shock front and is deflectedaway from thex-axis when it penetrates the discontinuity.

On the surface of the shock front the jump conditions forthe hydrodynamical variables have to be satisfied. In this casethe mass fluxρvn through the shock front is continuous and thetangential component of the gas velocityvt is preserved.

ρ1 v1 · en = ρ2 v2 · en , (A.1)

v1 · et = v2 · et . (A.2)

ρ2

ρ1

e t

e n

φ

y

x2 sv = v - v

sv = - v

g

1

shoc

k w

ave

Fig. A1. Oblique shock running into a steady medium

Solving for the velocity component of the compressed gas, weobtain

vx2 = vx1 [ 1 − ( 1 − β2 ) sin2 φ ] , (A.3)

vy2 = vx1 ( 1 − β−2 ) sin φ cos φ , (A.4)

where

β =√

ρ2

ρ1. (A.5)

In a stationary frame we have to take into account the motionof the shock wave.

vx1 = −vs , (A.6)

vx2 = vxg − vs , (A.7)

vy2 = vyg . (A.8)

This transformation allows us to derive an equation for the ratioζ of the velocity of the shock wavevs and the velocity of thedeflected gasvg.

ζ =vs

vg=

1( 1 − β−2 ) sin φ

. (A.9)

Note thatβ is a function of the normal component of theMach number of the inflowing gas. In strong shocks the de-flected gas is so strongly compressed thatβ far exceeds unity.Thus,vs ≈ vg for vertical shock waves (φ = 90o). For obliqueshocks (φ < 90o) the ratioζ increases with decreasingφ.

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