Generation of radiative knots in a randomly pulsed protostellar jet

Generation of radiative knots in a randomly pulsed protostellar jet

II. X-ray emission
Key Words.:
Hydrodynamics; ISM: Herbig-Haro objects; ISM: jets and outflows; X-rays: ISM
1

Abstract

Context:Protostellar jets are known to emit in a wide range of bands, from radio to IR to optical bands, and to date also about ten X-ray emitting jets have been detected, with a rate of discovery of about one per year.

Aims:We aim at investigating the mechanism leading to the X-ray emission detected in protostellar jets and, in particular, at constraining the physical parameters that describe the jet/ambient interaction by comparing our model predictions with observations available in the literature.

Methods:We perform 2D axisymmetric hydrodynamic simulations of the interaction between a supersonic jet and the ambient medium. The jet is described as a train of plasma blobs randomly ejected by the stellar source along the jet axis. We explore the parameter space by varying the ejection rate, the initial Mach number of the jet, and the initial density contrast between the ambient medium and the jet. We synthesized from the model the X-ray emission as it would be observed with the current X-ray telescopes.

Results:The mutual interactions among the ejected blobs and of the blobs with the ambient medium lead to complex X-ray emitting structures within the jet. The X-ray sources consist of several components: irregular chains of knots; isolated knots with measurable proper motion; apparently stationary knots; reverse shocks. The predicted X-ray luminosity strongly depends on the ejection rate and on the initial density contrast between the ambient medium and the jet, with a weaker dependence on the jet Mach number.

Conclusions:Our model represents the first attempt to describe the X-ray properties of all the X-ray emitting protostellar jets discovered so far. The comparison between our model predictions and the observations can provide a useful diagnostic tool necessary for a proper interpretation of the observations. In particular, we suggest that the observable quantities derived from the spectral analysis of X-ray observations can be used to constrain the ejection rate, a parameter explored in our model that is not measurable by current observations in all wavelength bands.

1 Introduction

In the past decade, high-energy emission from protostellar jets (originally suggested by Pravdo & Marshall 1981) has been discovered, taking advantage of both the high spatial resolution of Chandra and the high effective area of XMM-Newton. To date, about ten X-ray emitting Herbig-Haro (HH) objects are known and the main properties of their emission are reviewed in Table 1 (see also Bonito et al. 2007).

object Reference
[ks] [ erg s] [ K] [arcsec] [AU] [pc]
HH 2 LM Pravdo et al. (2001)
HH 154 LM Favata et al. (2006)
HH 80/81 HM Pravdo et al. (2004)
HH 168 HM Pravdo et al. (2009)
HH 210 HM Grosso et al. (2006)
HH 540 LM Kastner et al. (2005)
HH 216 - Linsky et al. (2007)
DG Tau LM Güdel et al. (2008)
Z CMa HM Stelzer et al. (2009)
TKH 8 LM Tsujimoto et al. (2004)

is the exposure time of the observations (or the total exposure time adding together several observations, as in the case of DG Tau); are the collected photons; is the X-ray luminosity in the keV band; is the best fit temperature; is the linear size of the X-ray source; is the distance of the X-ray source from the central protostar; is the distance of the HH object; indicates the low-mass or high-mass young stellar object from which the jet originates.

Improved estimates of and from Joel Kastner and Ettore Flaccomio, private communication The first value is associated with the South-Western jet and the second with the North-Eastern jet in DG Tau Values derived from SIMBAD See also Bally et al. (2003) See also Tsuboi et al. (2001)

Table 1: Physical properties observed in X-ray emitting HH jets.

The X-ray sources detected in HH jets are characterized by different morphologies, luminosities, and locations within the jet. In some cases, the X-ray emitting region is located at the base of the (optical) jet, near the protostar from which the jet originates. This is the case, for instance, for the low-mass (LM) HH 154 and DG Tau jets, both in Taurus (see Bally et al. 2003; Güdel et al. 2005). In other cases, the X-ray source is located further away from the protostar as, for example, for the high-mass (HM) HH 80/81 jet. As for the morphology, some of the X-ray sources cannot be resolved by current instruments (and appear point-like), some others appear to be elongated. In one case (HH 154) a knotty X-ray source is found (resembling the knotty morphology commonly observed in the optical emission from HH jets) which consists of an elongated tail with a measurable proper motion away from the stellar source and an apparently stationary point-like source (Favata et al. 2006).

Current models, aimed at explaining the origin of X-ray emission from protostellar jets, considered a continuous supersonic jet propagating through a homogeneous interstellar medium (Bonito et al. 2004, 2007). These models succeeded in predicting X-ray sources associated with jets with luminosity and proper motion in good agreement with the observations. They failed, however, in reproducing the complex knotty morphology of the X-ray source detected in HH 154 (Favata et al. 2006).

In a previous paper (Bonito et al. 2010; hereafter Paper I), we proposed an improvement to these models by considering the scenario based on a pulsed jet, i.e. a jet characterized by an ejection velocity varying randomly in time, and interacting with an initially homogeneous ambient medium. The aim was to investigate the origin of the irregular knotty structure observed in protostellar jets in different wavelength bands and the complex interactions occurring among blobs of plasma ejected from the stellar source. Our analysis showed that the mutual interactions of blobs ejected at different epochs and with different speed naturally reproduce the irregular pattern of knots observed along the jet axis in many HH objects and lead to a variety of plasma components which cannot be described by models of a jet ejected with a sinusoidal variable velocity (e.g. Raga et al. 2007).

Here we further explore the model of a randomly ejected pulsed jet presented in Paper I with the aim to investigate the properties of its X-ray emission and how these properties may depend on the physical conditions in which the jet evolves. In fact, X-ray emission from jets is observed in HH objects at different evolutionary stages and with different masses: for instance, HH 154 originates from a LM binary system of class 0 and class I sources; DG Tau is a more evolved LM classical T Tauri star; HH 80/81 originates from a HM protostar. Given the wide range of conditions in which X-ray emission from jets originates, it is important to address the following issues: where and when is the X-ray emission expected to arise from HH jets? How common is the high-energy emission from HH jets?

The paper is organized as follows: in Sect. 2 we present our model; in Sect. 3 we show our results concerning the morphology and luminosity in the X-ray band; in Sect. 4 we discuss our results and in Sect. 5 we draw our conclusions.

2 The model

We model the evolution of a protostellar jet ejected with a supersonic random speed and ramming into an initially homogeneous ambient medium. A detailed description of our model can be found in Paper I to which the reader is referred for more details. Briefly, we performed 2D hydrodynamic simulations of the interaction between the jet and the ambient medium in a cylindrical coordinate system, , by solving the equations of conservations of mass, momentum, and energy, using the FLASH code (Fryxell et al. 2000). The model takes into account the radiative losses from optically thin plasma and the thermal conduction in both the Spitzer (Spitzer 1962) and saturated (Cowie & McKee 1977) regimes.

Axisymmetric boundary conditions are imposed along the jet axis (consistent with the adopted symmetry), inflow boundary conditions at the base for , where is the radial distance from the jet axis in cylindrical coordinates and AU is the jet radius, and outflow boundary conditions elsewhere (see also Paper I). The spatial resolution achieved in our simulations is about AU, i.e. about times the resolution of Chandra/ACIS-I observations at a distance of pc (i.e. roughly the distance of the closest star-forming region; SFR).

The time covered by our simulations ranges between yr and yr, depending on the initial setup of the model. The jet is described as a train of blobs, each lasting for yr, with an ejection rate corresponding to a time interval between the ejection of two consecutive blobs yr. The initial setup is derived from the analysis of the optical and X-ray data and from the results of previous models (Bonito et al. 2007). Each blob is ejected with a random velocity directed along the jet axis (the -axis), sampled from an exponential distribution, and with a maximum velocity , corresponding to the initial Mach number , where is the isothermal sound speed (see Paper I for more details).

We explore both the initial light jet scenario (a jet initially less dense than the unperturbed ambient medium; hereafter LJ runs), and the initial heavy jet scenario (a jet initially denser than the ambient medium; hereafter HJ runs). Note that the concept of light and heavy jet is referred to the initial conditions of our simulations. In fact, as already discussed in Paper I, the density contrast between the ambient medium and the blob can vary during the jet/ambient evolution, the density of the medium varying by several orders of magnitude after the first high-speed blob has perturbed the whole computational domain (see Fig. 3 of Paper I). In other words, the ejected blobs can be denser or less dense than the medium in which they propagate, in each of the simulations considered here. Table 2 summarizes the physical parameters characterizing the simulations. In all the models we assume: AU as initial jet radius and we derive K as initial ambient temperature, assuming initial pressure balance between the jet and the ambient medium.

Model
[yr] [km s] [K] [cm]
LJ0.5-M1000 10 - 4680
LJ2-M1000 10 - 4680
LJ8-M1000 10 - 4680
LJ2-M100 10 - 470
LJ2-M300 10 - 1400
LJ2-M500 10 - 2340
HJ2-M300 10 - 1400
HJ2-M500 10 - 2340
HJ2-M1000 10 - 4680

is the time interval between two consecutive blobs, is the ambient-to-jet density contrast, is the Mach number of the first blob, is the velocity of each ejected blob (here we indicate the range of values randomly generated by our model), is the initial jet temperature, and is the ambient density.

Table 2: Summary of the initial physical parameters characterizing the simulations (see text for details).

3 Results

The dynamics and energetics of the randomly ejected pulsed jet have been extensively described in Paper I. Here we focus on the analysis of the X-ray emission predicted to arise from the pulsed jet. The X-ray emission is synthesized from the model results by adopting the methodology described in Bonito et al. (2007) that allows us to derive images and spectra of the X-ray sources associated with the jets. Briefly, we first recover the 3D spatial distributions of density and temperature by rotating the corresponding 2D distributions around the symmetry axis (). Then we integrate along the line of sight2, under the assumption of an optically thin plasma, and derive the emission measure distribution as a function of the temperature, EM3. Using available spectral emission codes and taking into account both the instrumental response and the interstellar absorption, we derive the X-ray spectra, the morphology, and the evolution of the X-ray luminosity, , to be compared directly to the parameters derived from the observations.

3.1 Morphology of X-ray jets

Figure 1: X-ray image of the jet synthesized from the pulsed jet model in the [] keV band (left panel) and corresponding mass density distribution (right panel) yr after the beginning of the simulation in run LJ0.5-M1000. The blue arrows superimposed on each panel mark the same positions in the X-ray map and mass density distribution. The size of the jet is about AU, corresponding to at pc.

Figure 2: Evolution of the X-ray emission as it would be observed with Chandra/ACIS-I in run LJ0.5-M1000. The head of the jet is outside the spatial domain. Two consecutive panels are separated by a time interval of yr, the first panel corresponding to yr since the beginning of the jet/ambient interaction. The spatial scales are shown in the first frame as well as the three main sources highlighted by the arrows: knot A, B, and C.

We investigate the morphology of the X-ray sources associated with jets, by simulating observations with Chandra/ACIS-I, namely the current X-ray instrument with the highest spatial resolution. As an example, Fig. 1 shows the X-ray emission in the [] keV band (left panel) and the corresponding mass density distribution (right panel) for run LJ0.5-M1000, assuming a distance of pc and an absorption column density cm, as derived for the HH 154 jet. In general the morphology of the source is quite complex showing several X-ray emitting plasma components: knots resulting from the interaction among different blobs4 of plasma ejected from the stellar source (for instance see left panel in Fig. 1), reverse shocks interacting with outgoing knots, stationary knots, oblique structures.

The most striking feature in Fig. 1 (left panel) is the irregular pattern of knots aligned along the jet axis and possibly interacting with each other, analogous to the well-known optical knotty structure observed within HH jets. Such a complex chain of X-ray knots originates from the interaction of different ejected blobs with each other and with the inhomogeneous medium in which they propagate. In fact, as explained in Paper I, the initially homogeneous ambient medium becomes quickly inhomogeneous because the blobs ejected with different speeds and at different epochs perturb the medium and strong variations of the pre-shock conditions occur (see also Fig. 3 in Paper I). Chains of knots are particularly evident in cases of high ejection rates; for instance, we found up to six knots for the case with yr (see left panel in Fig. 1). The X-ray sources along the jet axis are associated with strong shocks visible in the density distribution (see right panel of Fig. 1). In general, the brightest X-ray knots occur at the base of the jet and originate from the ejection of single blobs of plasma with high speed into the ambient medium. X-ray emission from isolated knots created by a single high-speed ejected blob is common to all the runs discussed here.

In principle, it is possible to derive the proper motion of modeled X-ray knots and compare it to the measured or expected proper motion from observed X-ray sources in protostellar jets. Our simulations have shown that the knot speed ranges between and km s (i.e. between ”/yr and ”/yr at pc). As an example, Fig. 2 shows the evolution of the X-ray emission for run LJ0.5-M1000; the proper motion of X-ray emitting knots A, B, and C is measurable in a few years. The knots speed deduced from these measurements ranges between about km s for the C knot at the base of the jet, and km s (i.e. ”/yr at pc) for the bright A knot. It is worth noting that the faint and slow C knot at the base of the jet is persistent throughout the evolution of the jet shown in Fig. 2, due to continuous fueling by subsequent ejected blobs; on the other hand the brightest source (A knot), which shows the highest proper motion, cools down and its X-ray luminosity drops down by about an order of magnitude after about two years. Unfortunately, up to now, the only X-ray emitting jet for which it was possible to measure the proper motion of the X-ray source ( km s) is HH 154 (Favata et al. 2006). For this case, our model predictions are in good agreement with the observations being the characteristic values of the average knot speeds derived from our model consistent with a few hundreds km/s (see Fig. 10 of Paper I). Nonetheless, our analysis suggests that, in general, a measurable proper motion of X-ray knots is expected and our predictions can be challenged in future observations (see, however, possible misinterpretations of observations discussed in Sect. 4.2).

Additional contributions to X-ray emission may come from a variety of complex plasma structures formed within the jet due to interactions among supersonic blobs, shocks, and the cocoon enveloping the jet. In particular, oblique structures can form at the cocoon (see also Paper I), as well as reverse shocks traveling in the opposite direction of ejected blobs (i.e. toward the stellar source) and possibly interacting with them. Under certain circumstances, shocks formed within the jet may show no proper motion on time-scales of a few years. As an example, Fig. 3 shows in run LJ2-M1000 an almost stationary X-ray source over a time-scale of about yr due to the interaction of a reverse shock with outgoing plasma blobs.

The position of the X-ray sources can be compared to that of expected optical knots, by deriving density maps of plasma with temperature ranging between K, which is a proxy of optical emission (see Paper I). As an example, Fig. 4 shows the evolution of both expected optical knots and X-ray sources in run HJ2-M300. In general, the X-ray sources are not co-spatial with optical knots. In fact, the figure shows a weak X-ray source located at the base of the chain of optical knots, showing a stratification of the X-ray and optical emission.

Figure 3: Zoom of the X-ray emission at the base of the jet for run LJ2-M1000 over a time baseline of about 5 yr. The spatial scale is shown in the first frame ( AU correspond to at pc).

Figure 4: Evolution of both expected optical knots (odd panels) and X-ray emission (even panels) in run HJ2-M300. The contours of the brightest regions of the X-ray source in the last panel are superimposed on the corresponding optical image. The spatial scale is shown in the first panel.

3.2 X-ray light curves

We derived the X-ray luminosity, , and its evolution for all the runs listed in Table  2. Our model predicts significant X-ray emission in the light jet scenario (LJ runs in Table 2) and very faint or no X-ray emission in the heavy jet scenario (HJ runs in Table 2; compare the different scales in Fig. 1 and Fig. 4). In this section, we therefore focus on LJ runs that predict X-ray emission detectable with current X-ray observatories.

In our simulations, the maximum value of is reached when the first blob (characterized by the highest allowed velocity) is ejected in the unperturbed ambient medium. Since such a high-speed blob and its interactions with subsequent ones cause the formation of very bright sources that are clearly due to the initial transient configuration, we removed the contribution to due to the first three blobs ejected5. Figure 5 shows as a function of time for runs LJ0.5-M1000 (solid line in black), LJ2-M1000 (dashed line in red), and LJ8-M1000 (dashed-dotted line in green), after the transient phase has been removed. Typical values of range between erg and a few erg , in good agreement with almost all the observations of X-ray emitting HH jets.

Figure 5: Evolution of the X-ray luminosity derived from runs LJ0.5-M1000 (solid line in black), LJ2-M1000 (dashed line in red), and LJ8-M1000 (dashed-dotted line in green). Time corresponds to yr, yr, and yr from the beginning of the simulation for run LJ0.5-M1000, LJ2-M1000, and LJ8-M1000, respectively.

Figure 5 shows that depends on the velocity of the ejected blobs and on the ejection rate . In particular, the X-ray luminosity decreases by three orders of magnitude from yr to yr, whereas it can vary by a factor in each simulation due to the velocity variations of the ejected blobs. This small variation of the X-ray luminosity with the velocity is also found by comparing the values of derived from simulations with the same ejection rate () and different Mach numbers, (runs LJ2-M100, LJ2-M300, and LJ2-M500 in Table 2). We conclude, therefore, that the critical parameter in determining the X-ray luminosity of a protostellar jet is the ejection rate of plasma blobs rather than their velocities: the higher the ejection rate, the brighter the X-ray source associated with the jet. In fact, the X-ray emission originates mainly from multiple interactions among knots and blobs which increase for higher ejection rates.

4 Discussion

4.1 Visibility of X-ray emission from HH jets

Although hundreds of HH jets have been revealed in several bands6 (from radio to IR, to optical wavelength bands) up to now, only ten HH objects are known to emit also in X-rays (Table 1; see also Bonito et al. 2007). The small fraction of jets visible in the X-ray band poses a problem because, in principle, all the high speed jets should emit X-rays. Of course, it is possible that X-ray emission from HH jets can be observed only under favorite conditions. In fact, X-ray emitting jets have to be sufficiently luminous in order to be detected in far away SFRs. Chandra observations of HH 154, the most luminous X-ray emitting HH jet detected in the nearest SFR, collected about counts in ks (Favata et al. 2006); an even worse statistics have been obtained for the less luminous X-ray jet revealed in the same SFR and associated with DG Tau (Güdel et al. 2008). These sources, therefore, are intrinsically faint and analogous ones located at larger distances cannot be detected with current instruments. It is therefore not surprising that X-ray jets located further away are very bright, as in the case of the most luminous X-ray emitting HH object revealed up to now, HH 80/81 (Pravdo et al. 2004).

Our simulations have shown that the necessary conditions to have detectable X-ray emission from a randomly pulsed jet are both a high ejection rate and a high density of the medium in which subsequent plasma blobs are ejected. As an example, in a light jet (i.e. the scenario leading to detectable X-ray emission) the X-ray emission produced by a single blob interacting with the surrounding medium can be rather faint, or not present at all, if the ambient-to-blob density contrast is low, unless the blob propagates with very high speeds ( km s).

As for the optical emission, the train of blobs forming the jet builds up a cocoon enveloping the jet and the subsequently ejected blobs. In the light jet scenario, such a cocoon is characterized by high mass density and is the dominant component in the optical emission, thus making negligible in this band any contribution from internal knots. Conversely, in the heavy jet scenario, the density of the cocoon is much lower than that of the ejected blobs and the main contribution to optical emission comes from the dense knots formed within the jet (see Fig. 4). Our model therefore predicts that light jets can lead to X-ray emission to observed levels, but do not produce observable optical knots, whereas heavy jets reproduce the chains of optical knots commonly observed in almost all the known protostellar jets, but do not lead to detectable X-ray emission.

The different characteristics of light and heavy jets, described above, originates from the fact that in the former ones the traveling knots are decelerated by a factor of about three by the surrounding dense medium, whereas in the latter ones the knots are only slightly decelerated by the medium in which they propagate (Bonito et al. 2007). New ejected blobs therefore travel into a medium constituted by strongly decelerated blobs in light jets and by several high-speed blobs in heavy jets. The average relative velocity between two consecutive blobs therefore is expected to be higher in light jets than in heavy jets, leading in general to X-ray emission in the former and to optical emission in the latter case as a consequence of blob collisions.

At variance with our model predictions, however, all X-ray emitting jets show also a knotty morphology in the optical band. A possible way to reconcile model predictions and observations (i.e. reproducing both X-ray and optical knots in a single run) is to consider the generation of knots into a medium partly constituted by high-speed blobs and partly by strongly decelerated blobs. In such a case, the relative velocity between two consecutive blobs is expected to be small on average in the part with high-speed blobs, leading to optical emission, and large in the part with decelerated blobs, leading to X-ray emission. This scenario may be reproduced by considering an ejection direction varying in time. If the blob is ejected into a co-moving medium, filled of previously ejected high-speed blobs, the average relative velocities between two blobs is expected to be small, leading to optical knots. Otherwise, if the blob is ejected into an almost stationary medium (because of the strong deceleration of previously ejected blobs), the average relative velocities between blobs is high and the knots resulting from the blob interactions may be observed in the X-rays. The above scenario is supported by the evidence that the location of the X-ray source in HH 154 is not completely aligned with the optical jet, as shown in Fig. 13 of Bonito et al. (2008).

4.2 Interpretation of X-ray observations of HH jets

As already discussed above, X-ray emitting jets are faint sources and only limited spectral and morphological analysis can be performed with the instruments in operation at the present time. The modeling of the X-ray emission from protostellar jets can therefore be an important tool in the interpretation of observations and may provide crucial information to unveil the nature of the X-ray emission.

Our model has shown that a specific feature of a randomly pulsed jet is the mutual interaction between plasma blobs that leads to irregular patterns of knots aligned along the jet axis, possibly interacting with each other. The knots emit preferentially in the X-ray band in light jets and in the optical band in heavy jets. However, although chains of optical knots are commonly observed in almost all HH jets, current X-ray observations do not allow to reveal the predicted knotty structure of X-ray jets due to limited statistics. The only jet for which it was possible to analyze the morphology of the X-ray source is HH 154, thanks to its proximity and brightness (Favata et al. 2006). In this case, a knotty structure in the X-ray band has been revealed in agreement with our model predictions and somewhat correlated to the observed knotty structure in the optical band (Bonito et al. 2008). Such a feature is probably common to all X-ray emitting jets but is difficult to observe because of limited statistics.

For instance, a knotty structure of the X-ray jet may be hidden in the current observations of DG Tau. Güdel et al. (2008) analyzed the X-ray source associated with this jet and needed to co-add several different observations collected over a time-scale of 3 years to improve the statistics. Consequently, any information on the spatial structure of the X-ray source has been averaged over 3 years, making it impossible to reveal its true morphology. In the co-added image, these authors noted an elongated X-ray source with a spatial scale of about and originating from the central protostar. However, the observations of DG Tau cannot exclude that the X-ray source is constituted by several outgoing knots. Figure 6 shows the evolution of the X-ray emission for run LJ0.5-M1000 at the same resolution as the Chandra/ACIS-I instrument (panels 1-8): once several different frames separated in time by a few years have been added together (see last two panels in the figure), the knotty morphology clearly recognized in the single frames cannot be recovered in the summed image that shows an almost continuous X-ray emitting structure along the jet axis, with spatial scales of about , similar to that derived by Güdel et al. (2008). In fact, even if the proper motion of the single X-ray knot may be negligible in a few years (as explained by Güdel et al. 2008), it is important to take into account the fast variability of the structures within the jet and the radiative time-scales, which determine the lifetime of the knots. A knot can disappear over this time-scale and new ejected blobs can form new observable knots.

Figure 6: Evolution of the X-ray emission in the LJ0.5-M1000 run. First to eighth panels: two consecutive panels are separated by a time interval of yr, the first panel corresponding to yr since the beginning of the jet/ambient interaction and the last to yr. The ninth panel shows the yr frame, yr frame, yr frame, and yr frame added together (the spatial scale is also shown: AU correspond to at pc). The tenth panel shows the first to eighth frames added together.

Another important contribution to the X-ray emission from HH jets comes from reverse shocks traveling in the opposite direction of ejected blobs and possibly interacting with them (see Sect. 3.1). These features are in general less frequent and short lived (lasting for less than about yr) than the X-ray knots previously discussed, but are not negligible. They predict X-ray sources within the jet with no appreciable proper motion and may explain, for instance, the apparently stationary X-ray source detected at the base of the optical jet HH 154 over a time baseline of yr (Favata et al. 2006; Bonito et al. 2008). An example of this feature is in Fig. 3 that shows an X-ray emitting source roughly at the same position due to a reverse shock powered by subsequent interactions with outgoing plasma blobs. A series of observations of this source on a baseline of yr may erroneously interpret the multiple interaction of the shock with subsequent blobs as a stationary X-ray source located at the base of the jet. This example shows how the comparison between model predictions and observations may be a useful tool in the data interpretation.

4.3 Comparison between model predictions and observations of X-ray emitting jets

X-ray observations of HH jets allow us to derive the luminosity and best-fit temperature of the X-ray sources associated with the jets (see Table 1). The comparison between these observed values and those derived from our model may contribute with useful information on the properties of X-ray emission from protostellar jets. Figure 7 shows the X-ray luminosity, , versus the best-fit temperature, , as a function of the ejection rate, derived from the analysis of spectra synthesized from runs LJ0.5-M1000, LJ2-M1000, and LJ8-M1000 (assuming high statistics, total counts; see details in Bonito et al. 2007). The symbols and the bars in the figure represent the median values of and and the th and th percentiles ranges, respectively. Each simulation predicts a wide range of variation for both the and the possibly related to the complex knotty morphology and fast variability (a few years) of X-ray jets. As expected, the most energetic case considered ( yr) leads to the highest luminosity and temperature: the higher , the lower and .

Figure 7: X-ray luminosity as a function of best-fit temperature for the three cases of ejection rate considered: (diamond), (triangle), and (square) yr. The dots superimposed to the figure mark the values derived from the observations of all known X-ray emitting HH jets. The error bars of observed values are given if reported in the literature. The X-ray luminosity of Z CMa is not well constrained and only a lower limit is reported (pink dotted vertical line).

Figure 7 also shows the X-ray luminosity and temperature derived from X-ray observations for all the cases in which some indication of and has been reported in the literature (see Table 1). Note that error bars are given in Fig. 7 if reported in the literature. The dashed line superimposed to the figure is arbitrary and separates HH objects characterized by X-ray emission close to the base of the jet (within AU from the stellar source; to the right) from HH objects showing X-ray emission at large distances (several thousands of AU) from the young stellar object (YSO) from which the jet originates (to the left; see Table 1). Figure 7 shows that HH jets with X-ray emission at the base of the jet have luminosities and temperatures in nice agreement with our model predictions. In particular, the HH 154 jet is well described by a pulsed jet with an ejection rate of two years. Remarkably, this result is in good agreement with the independent estimate of morphological evolution time-scale of the optical knots discussed by Bonito et al. (2008). DG Tau is the weakest X-ray emitting jet (see Table 1) and is consistent with the case of a jet with low ejection rate (i.e. the case that shows the lowest luminosities). Conversely, HH jets with X-ray emission at large distances from the YSO (namely HH 80/81, HH 210, and HH 2) are characterized by very low values of temperature ( MK) and high luminosities ( erg s) and do not match with our model results. The jet associated with Z CMa appears to be intermediate to the two cases discussed above, the distance of the X-ray source from the YSO being AU. In the framework of our model, we can interpret the X-ray knots located further away from the YSO as due to strongly decelerated (and radiatively cooled) plasma blobs at low temperatures. In general, our model provides the first attempt to describe the characteristics of almost all the X-ray emitting HH jets detected so far.

4.4 Location of X-ray sources within the jet

Another important parameter that allows us to compare model predictions with observations is the position of X-ray sources within the jet. In order to infer the position that is the most likely to detect within the jet, we integrate the spatial distribution of X-ray emission derived from the model both along the radial direction and in time. This analysis requires that the computational domain has been already fully perturbed by the first high-speed blob (ejected with the maximum initial velocity) to avoid the initial transient phase. The time interval of the integration therefore varies for the different ejection rates considered: from to yr for yr, from yr to yr for LJ2-M1000, and from to yr for yr, the head of the jet traveling outside the domain after yr in the former case and yr in the latter case. Figure 8 shows the normalized count rate integrated along the jet axis and in time for the LJ0.5-M1000 (solid line in black), the LJ2-M1000 (dashed line in red), and the LJ8-M1000 run (dashed-dotted line in green). Most of the emission is located at the base of the jet, within about AU, with a first bump within about AU from the YSO. The position of observed X-ray sources associated with HH jets is superimposed to the figure for those cases that show emission close to the base of the jet (i.e. HH jets to the right of the dashed line in Fig. 7). From the nearest to the farthest from the protostar: DG Tau (in magenta), HH 154 (in violet), HH 540 (in green), TKH 8 (in cyan), and Z CMa (in red).

Figure 8: Normalized count rate in the band [0.3 - 10] keV integrated over the radial direction and in time along the axis of the jet within AU for the LJ0.5-M1000 (solid line in black), the LJ2-M1000 (dashed line in red), and the LJ8-M1000 (dashed-dotted line in green) run after the computational domain has been fully perturbed by the first blob ejected (time interval = from yr to yr for LJ0.5-M1000, from yr to yr for LJ2-M1000, and from yr to yr for LJ8-M1000). The crosses superimposed indicate the position of observed X-ray sources associated with HH jets for those cases that show emission close to the base of the jet. From the nearest to the farthest from the stellar source: DG Tau (in magenta), HH 154 (in violet), HH 540 (in green), TKH 8 (in cyan), and Z CMa (in red).

The X-ray emission is the largest (and, therefore, more easily detectable) at the base of the jet. In fact, the interactions among blobs and, possibly, among knots involve larger energies close to the jet base than at larger distances and the outgoing knots progressively fade as they propagate away from the stellar source due to radiative cooling. We note that the position of observed X-ray emitting sources associated with HH jets seems to be related to the mass of the central object from which the jet originates. In particular, almost all the LM objects show emission localized preferentially at the base of the jet (see Fig. 8), with the exception of HH 2. On the contrary, the X-ray sources detected in the HM jets HH 80/81 and HH 210 are located at distances AU from stellar source (not reported in the figure).

4.5 Predictions on future X-ray observations of HH 154

Among the known X-ray emitting jets, HH 154 is the only one that allows us to challenge the predictions of our model because of its proximity and brightness. In particular, this is the only object for which it was possible to analyze its morphology and the time variability in the X-ray band to date (Favata et al. 2006). The HH 154 X-ray source has been detected as an almost point-like source in 2001 (Bally et al. 2003) and as a knotty source, consisting of a stationary (over a time baseline of yr) and an elongated source (showing a proper motion of km s) in 2005 (Favata et al. 2006). New future observations of this object promise therefore to add important pieces of information to shed light on the nature of the X-ray emission from jets. Here we use the results of our pulsed jet model to interpret future observations of HH 154.

  • Stationary source. As discussed in Sect. 4.2, a pulsed jet with a random ejection speed can lead to the same configuration (i.e. an apparent stationary source) if the X-ray emission detected at different epochs arises from different knots brightening roughly at the same position. However, the probability to see an almost stationary source is lower than the probability to see single moving knots according to our model, although not rejectable. In this case, therefore, we cannot exclude alternative scenarios to that of a pulsed jet that may explain the stationary source, as suggested by Bally et al. (2003). Among these, the X-rays produced by the central protostar may be reflected into our line of sight by a scattering layer located about 100-200 AU from the parent star (a mechanism similar to that observed in Seyfert 2 galaxies). Alternatively, near the location of jet collimation the dense medium or the magnetic field could act like a nozzle leading to quasi-stationary shocks emitting X-rays (see discussions in Bonito et al. 2007).

  • Previously detected sources show detectable proper motion. The new observations may show that the X-ray sources are located in different positions with respect to previous observations. This may happen because of the proper motion of the knots (not detectable on time-scales of 4 years but measurable on a time-scales of 10 years) or because new knots have emerged, while those observed in 2001 and 2005 have faded down, giving the impression of a motion of the sources. Both these cases are predicted by our model.

  • New sources appeared. Our model predicts that the brightness and the morphology of the X-ray source associated with the pulsed jet can strongly change over time-scales of the order of 10 years. We expect, therefore, that new observations may show a very different morphology of HH 154, with new emerging knots and X-ray features (e.g. stationary shocks) catching the previously existing ones. This is probably the most frequent configuration predicted by the model.

  • Previously detected sources disappeared. Since the last observation of HH 154 with Chandra has been performed in 2005, the time-lapse between this and a new observation will be at least of years, with a total time baseline since the first observation in 2001 of about years. Over this time-scale, the sources can both disappear, as also predicted by our model. In fact, if the X-ray source is not powered by new energetic blobs, its X-ray luminosity drops with a characteristic time-scale of a few years.

5 Conclusions

We investigated the X-ray emission expected to arise from a randomly pulsed jet with the aim to explain the nature of X-ray emission detected in protostellar jets. We also explored the observable X-ray features predicted to form as a consequence of the collisions between blobs and knots within the jet, by exploring the parameter space given by the ejection rate, initial jet Mach number, and initial density contrast between the ambient medium and the jet. Our findings lead to the following conclusions:

  • In all the cases, the interactions of the ejected plasma blobs with the surrounding medium produce X-ray emitting features. The main components contributing to the X-ray emission are: irregular chains of knots; isolated high speed knots; steady knots; reverse shocks; oblique shocks.

  • Light jets produce significant X-ray emission consistent with the levels observed, whereas heavy jets are characterized by very faint or no X-ray emission and emit mostly in the optical band.

  • In the case of light jets (leading to detectable X-ray emission), the X-ray luminosity is mainly determined by the ejection rate of plasma blobs, rather than by the jet Mach number: higher ejection rates are related to more energetic objects, thus leading to higher X-ray luminosities.

  • Our model predicts X-ray luminosity and best-fit temperature in nice agreement with most of observed X-ray emitting jets. In particular the HH 154 jet is well described as a pulsed jet with ejection rate yr. Our model represents the first attempt to describe all the X-ray emitting jets detected so far.

  • We found that most of the X-ray emission is located at the base of the optical jet where the plasma blob collisions are the most energetic and where, therefore, the probability to detect X-ray emission is the highest. This result is consistent with the evidence that almost all the LM objects show emission localized preferentially at the base of the jet.

In conclusion, our model explains why only a small fraction of HH jets has been detected in X-rays. Indeed detectable X-ray emission may arise only under favorite conditions, namely high ejection rates of plasma blobs and high density contrast between the perturbed ambient medium and the ejected blob.

We stress that erroneous interpretations of observations can easily be drawn if the X-ray sources (reverse shocks, stationary knots, etc.) are not monitored frequently, since they show fast variability over time-scales of a few years. On this respect, numerical models can provide strong diagnostic tools to interpret observations. To this end, we provided here model predictions on the morphology and characteristics of the X-ray source associated with HH 154 that may be revealed in future observations.

Acknowledgements.
We would like to thank Joel Kastner and Ettore Flaccomio for providing us the spectral parameters of the COUP source HH 540. R.B., S.O., M.M, and G.P. acknowledge support by the Marie Curie Fellowship Contract No. MTKD-CT-2005-029768. The software used in this work was in part developed by the DOE-supported ASC/Alliances Center for Astrophysical Thermonuclear Flashes at the University of Chicago, using modules for thermal conduction and optically thin radiation constructed at the Osservatorio Astronomico di Palermo. The calculations were performed on the cluster at the SCAN (Sistema di Calcolo per l’Astrofisica Numerica) facility of the INAF – Osservatorio Astronomico di Palermo and at CINECA (Bologna, Italy). This work was partially supported by Ministero Istruzione Università e Ricerca and by INAF.

Footnotes

  1. offprints: R. Bonito
  2. Assumed to be perpendicular to the jet axis
  3. See Bonito et al. (2007) for definitions and details on the EM.
  4. As discussed in Paper I, we use the term ”blob” to indicate the ejected clump of plasma and ”knot” to indicate the observable structure formed along the jet.
  5. For this reason the epoch time in Fig. 5 corresponds to yr, yr, and yr from the beginning of the simulation for run LJ0.5-M1000, LJ2-M1000, and LJ8-M1000, respectively
  6. A catalogue of HH objects has been prepared by B. Reipurth and it can be found in http://casa.colorado.edu/hhcat/

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