A&A 493, 115-118 (2009)
DOI: 10.1051/0004-6361:200810900
A. C. Raga1 - J. Cantó2 - A. Rodríguez-González1 - A. Esquivel1
1 - Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de Mexico, Ap. Postal 70-543, 04510 Mexico, Mexico
2 - Instituto de Astronomía, Universidad Nacional Autónoma de Mexico, Ap. Postal 70-468, 04510 Mexico, Mexico
Received 2 September 2008 / Accepted 7 October 2008
Abstract
Context. Several curved Herbig-Haro (HH) jet/counterjet systems have been observed. Some of these systems are the result of a relative motion between the outflow source and the surrounding environment.
Aims. In this paper, we consider the case of an HH jet/counterjet system that is immersed in an isotropic stellar wind. This situation is somewhat different from the previously studied HH jet/plane-parallel side wind interaction, because the isotropic wind is divergent, and has a radially dependent density structure.
Methods. We present an analytic model for a jet/wind interaction that is based on the balance between the ram pressure of the wind and the centrifugal pressure along the curved jet/counterjet path. We consider the case of an isothermal jet.
Results. We find that, if we have a jet and a wind with similar velocities and mass loss rates, the jet/counterjet system only shows a shallow curvature, with a total deflection angle of only a few degrees. However, if we have stronger winds, quite large deflections are produced (e.g., for the interaction of an HH jet with the wind from an O star). The deflection angle depends on neither the separation between the jet and wind sources nor on the ejection direction of the jet/counterjet.
Key words: ISM: kinematics and dynamics - ISM: jets and outflows - ISM: Herbig-Haro objects - stars: winds, outflows
Cantó & Raga (1995) presented an analytic model of a jet/sidewind interaction, with the idea of modelling the curvature that would result from having an HH jet/counterjet system ejected from a source in relative motion with respect to the surrounding environment. A comparison between the analytic model and a 3D numerical simulation was presented by Lim & Raga (1998). At the time there was no clear application of the model except for the case of HH30, which shows a general curved structure (see López et al. 1995).
A much more convincing example of a curved, molecular jet resulting from a jet/sidewind interaction was later found by Salas et al. (1998). However, the field of HH jet/sidewind interactions came to life with the paper of Bally & Reipurth (2001), who discovered many curved jet/counterjet systems. Several of these objects are in the outskirts of the Orion Nebula, and are outflows from low mass stars interacting with the expanding H II region (see Bally et al. 2006).
These observations gave rise to papers on modelling specific jet/sidewind interaction systems (Masciadri & Raga 2001; Kajdic & Raga 2007). Also, Ciardi et al. (2008) recently described results from laboratory experiments and 3D simulations of jet/sidewind interactions.
In the present paper, we discuss an extension of the analytic model of Cantó & Raga (1995, who studied the interaction of a jet with a plane-parallel sidewind) to the case of an HH jet/counterjet system immersed in an isotropic stellar wind. This model is relevant for the case of an HH jet interacting with a more isotropic wind from an intermediate mass star (a Herbig Ae/Be star) or from a massive star (an O or B star).
The paper is organized as follows. The model is derived in Sect. 2. The solutions obtained for different parameters are shown in Sect. 3. Applications of the models to different jet/wind interaction systems are discussed in Sect. 4. Finally, the results are summarized in Sect. 5.
We derive an analytic model for the interaction of a steady jet with an isotropic stellar wind. The model is based on the same assumptions as the one of Cantó & Raga (1995), who modeled a jet interacting with a plane-parallel sidewind.
In order to derive the shape of the jet/counterjet system, one assumes that the ram pressure of the wind (impinging on the side of the jet) is balanced by the centrifugal pressure of the material flowing along the curved jet path. Cantó & Raga (1995) found that the modulus of the jet/counterjet velocity does not depend on position (a result of the fact that the force exerted by the external wind on the jet is always perpendicular to the local jet direction). They also found that the path of the jet/counterjet system remains unchanged if one changes the position of the source along the derived path. This is a direct result of the fact that the centrifugal pressure does not depend on the direction along the jet/counterjet locus in which the flow is moving.
Assuming that the cross section of the jet adopts an equilibrium, isothermal ``plasmon'' shape, Cantó & Raga (1995) derived the relation
We assume that Eq. (1) is still valid in the case of the interaction of a jet with a divergent, isotropic stellar wind. This assumption is valid provided that the width of the jet is much smaller than the separation between the jet and the stellar wind source.
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Figure 1:
Schematic diagram showing the interaction of
an HH jet/counterjet system with an isotropic stellar wind. The
position of the jet source is shown with a circle. The stellar wind
source is located at the origin of the coordinate system(s).
Top frame: the locus of the jet and counterjet is shown with the
thick, solid curve. This curve reaches the point of closest
approach to the stellar wind source (i.e., the ``stagnation
point'' of the curve) on the x-axis of the xy-coordinate
system. The locus of the jet/counterjet is given by ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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We now consider the ram pressure
exerted by the wind
on the leading edge of the jet cross section, which is
Assuming a constant velocity, isotropic wind, we have
The path of the jet/counterjet system can then be found by setting
.
Combining Eqs. (1-5) we then obtain the differential equation
Therefore, the total deflection angle of the jet/counterjet
path (see Fig. 1) is
![]() |
Figure 2:
Curves for the jet/counterjet locus for different
values of the dimensionless parameter ![]() ![]() ![]() |
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In Fig. 2 we show the jet/counterjet paths given by Eq. (8)
for different values of the dimensionless parameter (see Eq. (7)). These solutions represent the paths seen in the (x,y) coordinate system which has the x-axis along the direction joining the stellar wind source and the ``stagnation point'' (i.e., the point of
maximum approach betwee the jet/counterjet path and the stellar wind source, see Fig. 1). In Fig. 2, the coordinates are given in units of the stagnation distance
.
In the paths shown (for different values of )
in
Fig. 2, one is free to place the jet/counterjet source in any
position along the trajectories. In this section, we describe
how to choose a solution with an appropriate
and
if one has a jet source at a distance D from the
stellar wind source, ejecting a jet/counterjet system at
an angle
with respect to the direction to the stellar
wind source (see Fig. 1).
![]() |
Figure 3:
The jet/counterjet locus for
![]() ![]() ![]() ![]() |
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Through an inspection of Fig. 1, it is clear that
In Fig. 3 we present a few examples of this kind of solution. The
top panel shows solutions with
and the bottom panel
with
,
for ejection angles
,
15, 30, 45, 60,
75 and
(note that the relation between
and
is given by Eq. (8)). It is clear that regardless of the ejection angle, the solutions always have the total deflection angle
given
by Eq. (10), which only depends on
.
Examples of HH jet/O star wind interactions could be the jets from the innermost proplyds in the Orion Nebula (see, e.g. Henney et al. 2002). The brightest star of the Trapezium, Ori C, is an O7 V star. For such stars, Sternberg et al. (2003) compute
a terminal wind velocity
km s-1 and a mass
loss rate
yr-1.
For a typical jet from a low mass star we take
km s-1and
yr-1. Assuming that the jet has an isothermal sound speed
km s-1, we compute
(see Eq. (7)). From Eq. (10) we then derive a total deviation angle
for the curved jet/counterjet locus.
Larger deviations would of course be obtained for lower velocity
jets and/or for a wind from a brighter O star. For example,
for a jet with
km s-1and
yr-1 interacting
with the wind from an O4 V star
(with
km s-1 and a mass
loss rate
yr-1,
see Sternberg et al. 2003), we obtain
.
Therefore, for this ``weak jet/early O star'' interaction
we obtain a total deflection
.
Therefore, we conclude that in an interaction between an HH jet
and a wind from an O star one can easily obtain locci for
the jet/counterjet systems with substantial deviations from a
straight line. The observed total change in position angle
of the jet/counterjet path directly gives us the dimensionless parameter
of the system (see Eqs. (7) and (10)).
Examples of interactions between a jet and a Herbig Ae/Be wind could be provided by HH jets in clusters of low/intermediate mass stars. Recently, Rodríguez-González et al. (2008) have modeled the interaction between the winds from a cluster of isotropic stellar wind sources. Carroll et al. (2008) have modeled the interaction of collimated jet/counterjet systems from sources in a cluster of young stars. A real cluster of low and intermediate mass young stars would of course have both relatively isotropic winds (from Herbig Ae/Be stars) and well collimated jets (from lower mass stars). An example of such a system of multiple outflow sources might be the embedded cluster in NGC1333 studied by Walawender et al. (2005) and Jorgensen et al. (2006).
Let us then consider the interaction between a jet
(with
km s-1 and
yr-1) and an isotropic wind from a Herbig Ae/Be star, for which we assume
km s-1 and
yr-1 (Finkenzeller & Mundt 1984).
For these parameters, we obtain
,
resulting
in a total deflection angle
.
From this, we conclude
that an HH jet/counterjet system embedded within a wind from
a Herbig Ae/Be star will also show substantial deviations from
a straight path.
We have derived an analytic model for the interaction of an isothermal jet/counterjet system with an isotropic stellar wind. The model has a full, analytic solution which has interesting properties.
We find that the full deflection angle
of the jet/counterjet
locus is only a function of the dimensionless parameter
(where
is the jet Mach number, see Eqs. (7) and (10)).
does not depend on the separation D between the jet and wind sources
nor on the direction
of the jet/counterjet
ejection. However, the distance of maximum approach between the wind source and the jet/counterjet locus (i.e., the ``stagnation distance''
,
see Fig. 1) does depend on
,
D and
(see Eq. (13)).
Let us assume that we observe a jet/wind interaction, and
that we have the relevant radial velocity
and proper motion information necessary to de-project the observed
jet/counterjet locus. From the observed source separation D, stagnation distance
and ejection angle
we can then calculate the dimensionless parameter
(using Eq. (13)). Alternatively, if the
observed jet/counterjet system extends to regions in which it
becomes approximately parallel to the direction from the wind
source, we can measure the full deflection angle
,
and
use it to derive
(using Eq. (7)). We could
combine the determination of
with an observationally
determined
in order to determine the value of
the jet to wind momentum rate ration
.
We estimate that total deflection angles
should be
observed for the interactions between an HH jet and winds from
either an O star or from a Herbig Ae/Be star. Therefore, it is
to be expected that future observations of such interacting systems
should show jet/counterjet systems with substantial curvatures away
from the nearby stellar wind sources.
We end by discussing again the limitations of the present model. In our model, we assume that the jet/stellar wind interaction produces a steady flow in which the jet cross section adopts a ``plasmon'' configuration. This plasmon configuration is not solved correctly, as we use the cross section of Cantó & Raga (1995), who derived the shape resulting from the interaction of a plane-parallel sidewind with a jet. We argue that this cross section will still be approximately correct provided that the separation between the jet locus and the stellar wind source is considerably larger than the jet radius. This of course might not be true for some of the jet locci shown in Fig. 3, in which the jets have close encounters with the stellar wind source.
Another possibly incorrect assumption is that we have a steady flow. Ciardi et al. (2008) find that a jet/sidewind interaction leads to instabilities that fragment the jet flow. Even though such effects clearly lie beyond the simple model described in the present paper, from the work of Ciardi et al. (2008) it is clear that the analytic model of Cantó & Raga (1995, calculated with basically the same assumptions as our present model) does produce a reasonable estimate of the jet path obtained from 3D numerical simulations.
Finally, we should mention that the full jet deflection predicted from our models (see Eq. (10)) is in many cases attained only quite far from the stellar wind source. This full deflection therefore could be not reached by actual jet/stellar wind interaction systems. This could be because the jets still have not propagated far enough, or because the stellar wind is confined to a smaller distance from the wind source (due to the presence of a dense, surrounding ISM). Also, at large distances from the wind source, the jet/wind interaction might have different characteristics as the wind becomes a collisionless plasma, resulting in interactions with richer behaviour than can be described with a fluid model.
Acknowledgements
This work was supported by the DGAPA (UNAM) grant IN108207, the CONACyT grants 46828-F and 61547, and by the ``Macroproyecto de Tecnologías para la Universidad de la Información y la Computación'' (Secretaría de Desarrollo Institucional de la UNAM). We thank Andrea Ciardi (the referee) for helpful comments.