A&A 471, 873-884 (2007)
DOI: 10.1051/0004-6361:20066921
T. P. Downes1,2 - S. Cabrit3
1 - School of Mathematical Sciences, Dublin City University,
Dublin 9, Ireland
2 - National Centre for Plasma Science and Technology,
Dublin City University, Dublin 9, Ireland
3 - LERMA, Observatoire de Paris,
UMR 8112 du CNRS, 61 Av. de l'Observatoire, 75014 Paris, France
Received 11 December 2006 / Accepted 4 July 2007
Abstract
Context. The momentum, age and momentum injection rate (thrust) of molecular outflows are key parameters in theories of star formation. Systematic biases in these quantities as inferred from CO line observations are introduced through simplified calculations. These biases were quantified for radially expanding flows. However, recent studies suggest that the youngest outflows may be better described by jet-driven bowshocks, where additional biases are expected.
Aims. We investigate quantitatively the biases in momentum, age, and thrust estimates in the case of young jet-driven molecular outflows, and propose more accurate methods of determining these quantities.
Methods. We use long-duration (1500 yr) high resolution numerical simulations in concert with the standard observational methods of inferring the relevant quantities to quantify the systematic biases in these calculations introduced, in particular, by dissociation, erroneous inclusion of transverse momentum, and hidden material at cloud velocity. Jet/ambient density contrasts of 0.1-1 are considered, leading to bow speeds of 60-135 km s-1.
Results. When mass-weighted velocities are used, lifetimes are overestimated by typically an order of magnitude. The molecular thrust is then underestimated by similar amounts. Using the maximum velocity in CO profiles gives better results, if empirical corrections for inclination are applied. We propose a new method of calculating the lifetime of an outflow which dramatically improves estimates of age and molecular thrust independent of inclination. Our results are applicable to younger flows which have not broken out of their parent cloud.
Conclusions. Published correlations between the molecular flow thrust and the source bolometric luminosity obtained with the maximum CO velocity method should remain valid. However, dissociation at the bow head may cause the observable thrust to underestimate the total flow thrust by a factor of up to 2-4, depending on the bow propagation speed and the magnetic field strength. Detailed evaluation of this effect would greatly help to better constrain the efficiency of the ejection mechanism in protostars.
Key words: hydrodynamics - shock waves - ISM: jets and outflows - ISM: molecules
The cumulative momentum and the momentum injection rate ("thrust'') in swept-up molecular outflows from young stars are both key parameters for theories of star formation: the cumulative outflow momentum is a measure of the feed-back from star formation on cloud turbulence, and its value per unit stellar mass is an essential parameter for theories of self-regulated star formation (e.g. Norman & Silk 1980). The outflow thrust (i.e. the ratio of flow momentum to age) gives a measure of the rate of momentum injection by the protostellar wind, and its ratio to the source luminosity sets a key constraint for theoretical ejection models (cf. Lada 1985; Richer et al. 2000).
To evaluate accurately the above properties, however, it is necessary
to have a reasonably good measure of both the total outflow momentum,
and of the time over which this momentum was deposited ("flow
age''). Systematic biases in these quantities were examined for simple
models of radially expanding outflows with a power-law velocity field
by Cabrit & Bertout (1990). The method found as the most accurate was
then used to derive updated correlations between the outflow thrust
and the source luminosity,
:
very high values of wind
thrust
were inferred for low-luminosity
outflow sources, strongly favoring magneto-centrifugally driven winds
from accretion disks (Cabrit & Bertout 1992). This analysis
was extended to higher luminosity sources (e.g. Richer et al. 2000; Beuther et al. 2002).
Since this early work, there has been mounting evidence that the youngest molecular flows, driven by Class 0 protostars, show a kinematic pattern in better agreement with a jet-driven bowshock than with a radially expanding shell (Masson & Chernin 1993; Raga & Cabrit 1993; Smith et al. 1997; Cabrit et al. 1997; Gueth & Guilloteau 1999; Downes & Ray 1999; Lee et al. 2001; Downes & Cabrit 2003; Arce et al. 2006). This situation will introduce new biases in observational derivations of the flow momentum and age, for several reasons:
In this work, we evaluate quantitatively the above biases in momentum, age and thrust estimates in jet-driven molecular outflows using long-duration high resolution numerical simulations reaching timescales of 1500 years. Both a density-matched and a strongly underdense jet (1:10) are considered, to study the effect of the jet/ambient density contrast. We include a treatment of molecular dissociation, to estimate the amount of "unseen'' momentum in atomic gas, as well as a simplified treatment of NLTE CO emission, to gauge additional errors made in the conversion from line intensity to mass assuming LTE. We then apply the methods most widely used by observers to derive the outflow mass, momentum, age, and thrust, and quantify the various errors introduced in these quantities.
Our numerical model is presented in Sect. 2. In Sect. 3 we describe the errors introduced by molecular dissociation, the assumption of constant CO excitation temperature, the erroneous inclusion of transverse momentum, inclination effects, and hidden outflow material at ambient velocities. We then examine the accuracy of inferred timescales and of thrust estimates using the usual methods and introduce a new, more robust method for calculating the age of young jet-driven outflows. We also discuss the limits of applicability of our results. Section 4 outlines our main conclusions.
In this section we describe the numerical method and initial conditions used to perform the simulations in this work.
The code is a modified version of that used in Keegan & Downes (2005) to perform long-duration simulations of YSO jets. The main modification is the inclusion of a more accurate H2 cooling function of Le Bourlot, Pineau des Forêts & Flower (1999) valid from T=100 K to 104 K. The hydrodynamic equations are solved using a second order Godunov scheme (e.g. Falle 1991) in which a nonlinear Riemann solver is used for strong rarefactions, while a linear Riemann solver is used for all other flux calculations in order to reduce computational overhead. The simulations presented here were run in cylindrical symmetry.
The code itself is parallelised efficiently and load-balanced in order to ensure that only the active region of the computational domain is being integrated (see Keegan & Downes 2005). This yields a speed-up of a factor of around 4 over traditional domain decomposition which, while not as good as one would expect from adaptive mesh refinement which typically gives a factor of 10, is still significant.
We considered a typical protostellar jet mean velocity of 215 km s-1 (e.g. Mundt 1987). Superimposed on the jet velocity was a spectrum of sine waves with total amplitude 127 km s-1 and periods of 5, 10, 20 and 50 yrs. The jet temperature was taken to be 1000 K while the ambient temperature in each case was taken to be 100 K. The time-averaged jet Mach number was therefore about 93. Note that, while the ambient temperature is rather high compared with that expected in molecular clouds, reducing the temperature to 10 K does not change the results presented here. This is due to the fact that the jet is hypersonic with respect to the ambient medium in either case.
The molecular fractions in the jet and ambient medium were taken to be
the same, with
while overall the jet
and ambient gases were assumed to be of solar abundances. The total
jet number density was set to 100 cm-3. For the jet radius, we
adopted a value of
cm = 333 AU, consistent
with an extrapolation of typical atomic jet widths (see e.g. Reipurth et al. 2000) to the propagation distances
0.1 pc
covered in our simulations. The total injected z-momentum over 1500 yr is then
km s-1 and the
time-averaged jet thrust is
km s-1 yr-1.
Two simulations were run for the purposes of this work, with different
values of the mass density contrast
:
a
density-matched case (
)
where the ambient density was 100 cm-3,
and an underdense jet case (
)
where the ambient density was
set to 1000 cm-3. In each case a resolution of
cm in
both the z and r directions was used. Hence the jet diameter was
well resolved, being 100 pixels across. Our relatively low densities
were chosen to ensure that all shock fronts are also well resolved, an
essential condition for an accurate treatment of H2 dissociation
and cooling. The full grid size of the
simulation was
(0.11 pc
0.03 pc) while for the
simulation it was
(0.21 pc
0.016 pc).
The time-averaged advance speed of the bow head is 60 km s-1 for
and
135 km s-1 for the
case, hence
substantial dissociation occurs in both cases, though in different amounts.
These relatively high advance speeds are in line with the proper motions of
150-260 km s-1 observed at the tip of the HH47 redshifted outflow lobe
(Eisloeffel & Mundt 1994; Micono et al. 1998), the
only measurements of this kind available so far (see
Sect. 3.7).
We examine our simulations (both
and
)
at an
age of t=1500 yr. As noted in Keegan & Downes (2005) this
ensures that the system has overcome initial transients, and
characteristics such as the mass-velocity relations have reached a
quasi-steady state. As in Downes & Cabrit (2003), we derive
the observable properties of the simulated outflows using the NLTE
emission formula of McKee et al. (1982) for the CO(2-1)
line, assuming optically thin emission. Intensity-velocity
distributions for swept-up material integrated over the flow are
constructed for various view angles, and used to derive "observed''
outflow mass, momentum, kinetic energy, and age in the same way as
usually done by observers. These values are then compared with the
actual parameters of our simulated outflow.
For obvious symmetry reasons, we conduct our analysis only on a single lobe of the outflow (that tilted towards the observer). We also exclude any molecular material originating from the jet, as in reality this contribution would depend on the atomic/molecular fraction in Class 0 jets, which is ill-known at present. Finally, we consider that the outflow is young enough that the whole lobe still lies within the confines of the parent molecular cloud, so that its full (projected) length can be measured in CO. This, and the neglect of any wide-angle wind component, mean that our results may not be applicable to evolved flows around Class I sources (see discussion in Sect. 3.7).
The sources of errors that we will examine are the following:
In observed flows, the assumption of optically thin CO emission
introduces an additional error in mass and momentum determinations. We
could not quantify this effect in our simulations, as our low adopted
densities lead to small swept-up masses (a few
,
see
Tables 1, 2) where the optically thin assumption is in fact well
justified. However, Cabrit & Bertout (1990) showed that an empirical
correction for CO optical depth using the 13CO to 12CO
intensity ratio is usually adequate for typical outflow
opacities. Since such an optical-depth correction is now routinely
applied in recent observational studies, this source of error is not
considered further in our analysis.
In the jet-driven molecular outflow model, the outflow is accelerated by the bowshock of the driving stellar jet. Downes & Ray (1999) pointed out that there is the possibility of dissociating a substantial fraction of the swept-up gas during this process.
Downes & Cabrit (2003) further showed that molecular
dissociation plays a crucial role in producing a break in slope around
20 km s-1 in the m(v) relation of jet-driven flows (contrary to the
interpretation of Birks, Fuller & Gibb (2006) who erroneously
reported that this work attributed the break to the excitation
temperature of the line being observed). The break in m(v)appears as soon as one considers molecular swept-up material only, as
illustrated in Fig. 1 for our
simulation. Line
excitation effects only help to further steepen the I(v) relation
at higher velocities than the break, yielding CO(2-1) intensity-velocity
relations in very good agreement with observations (Bachiller &
Tafalla 1999; Downes & Cabrit 2003).
Here we examine the effect of molecular dissociation on the observable swept-up mass, axial momentum along the jet z-axis, and kinetic energy. Tables 1 and 2 contain a summary of the results of our calculations. The "Total'' property refers to the ambient material swept up by the bowshock in both atomic and molecular form, the "H2'' property is only for molecular swept up material. The numbers in parentheses are the proportion of the actual total for that property.
Table 1:
Inclination-independent errors for the
underdense jet simulation. The total swept-up mass, the molecular swept-up
mass, the mass inferred assuming T=10 K, and the mass inferred
assuming T=100 K for the simulation at t=1500 yrs. The numbers
in parentheses represent the fraction of the actual mass in the outflow.
Also shown are the same quantities for momentum and kinetic energy.
Table 2:
Inclination-independent errors for the
density-matched
simulation. Content as in Table 1.
We can see from Tables 1 and 2 that
only about 60% of the swept up mass is actually molecular, compared to
almost 95% of the unshocked ambient medium (see Sect. 2.2). The result is common between the
and
simulations.
The effect is more substantial for the axial momentum: Only 45%
at ,
and 26% at
of the z-momentum in the
swept-up outflow is molecular. The results we find here, for our much
longer simulations, are in excellent agreement with the results of
Downes & Ray (1999). Their model G corresponds most closely
to our
simulation. For this model they found that the
proportion of swept-up momentum residing in molecules was about 0.25and we find this figure to be 0.26 from our
simulation. Hence this number does not seem to depend strongly on
the flow age.
The fraction residing in molecular material is even smaller for
kinetic energy than for momentum, with the molecular kinetic
energy being only 31% ()
to 14% (
)
of the total
kinetic energy in the swept-up outflow.
![]() |
Figure 1:
Plots of the mass-velocity relations for all swept-up
material (top curve), and for molecular swept-up material only
(short-dash) for our ![]() |
Open with DEXTER |
We can easily understand why the momentum and kinetic energy estimates
are progressively worse than the mass estimates as follows. We know
(e.g. Smith et al. 1997; Downes & Ray 1999; Lee et
al. 2000; Downes & Cabrit 2003) that the swept-up
mass in a jet-driven flow varies with velocity as
with
-2. This is confirmed for our
simulations in Fig. 1.
Then the integrated mass, M, for the outflow is given by
In contrast, since
,
the largest contribution to P comes from
material at the highest velocity. This is precisely the material which
will have passed through the strongest shocks, and hence will have the lowest
fraction of molecules (again, see Fig. 1). It is clear,
then, that we can expect a larger discrepancy between momentum estimates
based on molecular material and the true momentum in the outflow than
encountered when making mass estimates in the same way. The same argument
holds for the energy estimates, leading to the expectation, verified in our
simulations, that the kinetic energy in molecular gas is an even more severe
underestimate of the total energy than is the case for momentum.
We also find that the effect of dissociation depends on .
Underestimation of the total momentum and energy due to
molecular dissociation is twice as severe in the
case
compared with
(see Tables 1 and 2).
This may be attributed to the twice faster
propagation speed of the bow head in the density-matched
case, which induces more dissociation at the highest flow velocities
than in the under-dense
case, where the bow propagates at
only 60 km s-1. Since the molecular momentum and
energy are weighted towards high speed material (see above), this differential
effect due to the variation in bow speed is clearly noticeable for these
quantities. There is little differential effect of
for the molecular
mass which is dominated by low speed material.
We conclude that molecular dissociation may cause a large fraction of
the outflow momentum and kinetic energy to be unobservable in CO. The
magnitude of this effect depends on the bow advance speed,
controlled by the jet/ambient density contrast. It would be much
reduced if
is less than 0.01, leading to bow speeds
20 km s-1 (cf. Sect. 3.7). It could also be lowered by the
presence of transverse magnetic fields allowing non-dissociative
C-shocks at velocities
25 km s-1 (Le Bourlot et al. 2002). In the following subsections we will thus examine
errors introduced in the properties of the observable molecular
gas only, and reintroduce the extra effect of molecular dissociation
in our final discussion.
The bottom two rows in Tables 1 and 2
entitled "Inferred'' give the properties which would be inferred by
CO(2-1) observations of our simulated outflows, assuming CO is
optically thin and in LTE at a constant temperature of 10 K or 100 K
across the entire outflow, while the lowest two curves in Fig. 1 plot the mass-velocity relations obtained with these
assumed temperatures. These values (and the inferred
relations in Fig. 1) are calculated using
The assumption of T=100 K gives masses 2.2 times larger than
assuming T=10 K. This results from the depletion of the
J=2 level
at temperatures above 20 K, as higher energy levels become
accessible to the molecule. This "partition function'' effect causes
the J=2-1 line emissivity per molecule (expression between
parentheses in Eq. (4)) to eventually become
inversely proportional to T above 50 K, so that dm(I(v))increases with T.
As can be seen in Tables 1 and 2,
an assumption of T=10 K gives a reasonable estimate of the molecular mass and momentum, while kinetic energy tends to be too low (by
a factor of 1.5 for
,
and 3.5 for
). This can be
understood from Fig. 1, which shows that m(v) is correctly
estimated up to 30 km s-1, but is underestimated for the higher
velocity, warmer material that dominates the kinetic energy integral
(see previous section). In contrast, the T=100 K assumption gives
an overestimate of all molecular parameters (with the exception of
molecular kinetic energy for
,
which is still significantly
too low).
The assumption most commonly used by observers is that K.
We have seen that this introduces only a small error in the molecular swept-up momentum for both values of
.
Therefore,
the main inclination-independent error on the momentum in outflows
will remain that resulting from the neglect of dissociated material
(see previous section). On the other hand, molecular kinetic energy is
significantly underestimated by this assumption when
(by a
factor of 3.5). Using a specific excitation temperature for
high-velocity gas (determined, for example, from high-J CO line ratios;
cf. Hatchell Fuller & Ladd 1999) is needed to
alleviate this problem. In any case, the error factor is only half of
that introduced from the neglect of dissociated material, which
contains most of the kinetic energy of swept-up gas in our
simulations.
Here we investigate the error in the molecular swept-up momentum introduced from taking into account transverse momentum. As noted in Sect. 1, transverse motions in jet-driven outflows result from the "energy-driven'' sideways thermal expansion of hot gas in the immediate post-shock zone behind the working surface (see, e.g., Ostriker et al. 2001). They do not correspond to actual linear z-momentum injected by the jet. Taking transverse momentum into account might, therefore, lead to an overestimate of the momentum necessary to drive the outflow. Such a bias has not been investigated to date. We will examine the influence of the viewing angle on this error, and the effect of applying a global inclination correction.
In a lobe which is driven only by momentum injection in the axial
z-direction, the net vector contribution of transverse momentum
to the total momentum in the system, ,
is, by symmetry, zero:
![]() |
(2) |
When the flow is inclined by an angle
to the plane
of the sky, the line of sight velocities include contribution from
both the z motions and the transverse motions. The latter may
introduce a mix of forward and backward (blueshifted and
redshifted) momentum towards a given lobe of the flow. To extract
only the z-momentum component (projected onto the line of sight) one
must perform a proper vector summation along the line of sight
so that transverse momentum from the front and back sides of the
bowshock cancel out by symmetry, i.e. one should count positively the
momentum towards the observer, and negatively the momentum going away:
![]() |
(3) |
A quantitative illustration of both of these effects as a function of
inclination to the plane of the sky, ,
is presented in Fig. 2 for the
simulation, and in Fig. 3 for the
case. All quantities are
normalised to the actual molecular z-momentum in the flow.
![]() |
Figure 2:
Plots of the approaching and receding line-of-sight
molecular momentum present in the blueshifted lobe of the ![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 3:
As Fig. 2 but for the ![]() |
Open with DEXTER |
The behavior of
with inclination (star
symbols in Figs. 2 and
3) is found to depend upon
.
It increases at
smaller
(more edge-on flows) when
,
but decreases
when
.
The data here suggest that the
outflow
has more sideways expansion than the
outflow. This is to
be expected given that the ambient density in the
outflow
is higher and hence the cooling is stronger. It is well known that
this leads to narrower bowshocks with post-shock motions oriented more
closely to the axis of the driving jet. Despite this difference, for
both values of
,
we find that
remains within a factor of 2 of the actual molecular z-momentum in
the outflow, independent of the inclination angle of the flow: by
coincidence, the erroneous inclusion of transverse momentum
compensates almost exactly for the reduction in line-of-sight z-momentum
due to projection effects.
In contrast, the inclination-corrected absolute momentum calculated
from Eq. (9) (top curves in Figs. 2 and 3) systematically
overestimates the molecular z-momentum in the flow, as expected,
even going to infinity as the inclination angle goes to zero. Another
source of error, not quantified here, is the error on
itself.
The flow inclination is difficult to evaluate accurately without a
proper kinematic model, unless there are proper motion measurements
for the associated jet, but the latter are rarely available in deeply
embedded outflow sources.
As a result, at all values of inclination,
gives a much better estimate of Pz than the
inclination-corrected prescription of Eq. (9).
Compared with the correct expression of Eq. (7), it
also has the advantage of not requiring an accurate inclination
value. Therefore, based on our simulations, this would seem like the
most robust method to evaluate the injected molecular momentum
from observations of jet-driven flows
.
Unlike momentum, the kinetic energy in an outflow is not a conserved quantity - a significant fraction of it is lost through radiative cooling, particularly in post-shock regions. Indeed, the high level of far-infrared line emission from outflows found by the ISO satellite supports the idea that they are driven by highly-radiative, momentum-conserving shocks (e.g. Giannini et al. 2001). Hence, kinetic energy estimates are not as useful in constraining the physical mechanism powering the jet. However, the kinetic energy in molecular swept-up gas will probe the energy mediated through non-dissociative shocks, and will be relevant to compare with the cooling budget of H2, CO, and H2O observed with ISO, Spitzer, and the forthcoming Herschel observatory. We thus include a brief discussion of the projection effects here, as most observers provide estimates of the molecular kinetic energy in an outflow.
Energy estimates do not pose the same problems as momentum estimates
since energy is a scalar quantity. In this case it is correct to add
all the kinetic energy in the flow lobe together (i.e. both the energy in
forward and backward motions within a single lobe). A problem which remains
is that there is always some amount of energy which is not in
line-of-sight motion, hence using the integral in
Eq. (3) (e.g. Lada & Fich 1996; Shepherd et al. 2000) will always give a lower limit. One may consider
applying a mean correction factor of
to take into
account inclination effects on the energy. However, this will be
correct only for z-motions, and not for transverse motions.
Figures 4 and 5 illustrate the
resulting error on molecular kinetic energy for the
and
simulation respectively by plotting the line-of-sight kinetic
energy (sum of the red-shifted and blue-shifted components) as a
function of inclination, as well as the same quantity after global
correction by
.
![]() |
Figure 4:
Plots of the total line-of-sight energy for the ![]() ![]() |
Open with DEXTER |
![]() |
Figure 5:
As Fig. 4 but for the ![]() |
Open with DEXTER |
Table 3:
Masses and momenta of the outflows calculated from the
relation using the velocity ranges 2-40 km s-1 and
7-40 km s-1 with
a bin width of 1 km s-1. Also shown are
and
,
the latter chosen as the velocity at which
is 1/100 that at
(see text).
In contrast to what we found for the absolute line-of-sight momentum
(
), the line-of-sight energy in molecular
gas always severely underestimates the true molecular kinetic energy. In
the
case, the underestimate is a factor of 4 at the median
inclination of 30
,
and up to a factor of 10 in the edge-on case.
Even when the flow is viewed pole-on, the energy estimate is
slightly too low by 20%. This is due to the presence of kinetic
energy in transverse motions perpendicular to the line of sight. The
inclination correction
works rather well for
greater than about
,
with errors of less than a factor of 2.
However, the corrected values become very much larger than the true value
for
.
The picture is slightly different for the
case; the
differences arising from the higher transverse motions in this
simulation due to the lower ambient density (see the previous
section). In this case the variation with inclination angle is
lessened, with an underestimate of about a factor of 3 at all
angles. As a result, the inclination correction
works well only at median inclinations
,
but
over/underestimates the energy significantly at smaller/larger
angles.
Overall, it appears difficult to estimate the molecular kinetic energy of swept-up gas to better than a factor of 3 even if inclination corrections are applied.
In actual observations, CO emission at low velocities is often dominated by ambient molecular cloud material moving turbulently along the line of sight. To avoid contamination of outflow quantities by ambient gas, a low velocity cut-off around 1-2 km s-1 is typically used, while sometimes a cut-off as high as 7 km s-1 is necessary (see, e.g. Lada & Fich 1996).
The effect of truncating the minimum velocity at which material is
considered to be in the flow is illustrated in Table 3 for our jet-driven simulations. The
masses and line-of-sight momenta of the outflows are calculated from
the
relations at t=1500 yr, with cut-off
velocities
of 2 and 7 km s-1. They are normalised to the true
outflow mass and z-momentum in molecular form (from Tables 1
and 2). Momentum is estimated as
,
the prescription found as the most
robust against projection effects in Sect. 3.3.
In the typical case of
km s-1, ignoring low velocity
material leads to an underestimate in the molecular mass of typically
a factor of 2-3. However, the effect is not as marked for the molecular
momentum, which is more heavily weighted towards high velocities (see
Sect. 3.1, in particular Eqs. (1) and (2)). The errors on this quantity are
very similar to those found previously without a low-velocity cut-off
(see Figs. 2 and 3).
They remain less than a factor of 2 in most cases, hence it does not appear
necessary to correct the absolute line-of-sight momentum for hidden
outflow material at cloud velocities when
is small.
In the extreme case where
km s-1, molecular mass
becomes severely underestimated (less than 10% is recovered), while
momentum is always too low by a typical factor of 2-3, even reaching
10-100 in the
outflow at
(a result
of the strongly forward-directed motions in this simulation). Hence
we expect that
will significantly
underestimate the molecular momentum when such large values of
are used, which is, fortunately, very rare.
It is worth noting that, while one expects inclination effects in these
calculations due to the dependence of the m(v) relation on ,
their effect is small, leading to only roughly a change of a factor of
two in the mass and momentum values, except for the
km s-1,
case. The origin of the dependence on
arises
because, depending on the inclination angle used, and the velocity range
used, material which at some angles is measured as moving (and therefore
contributing to the mass and momentum of the outflow) is, at other angles,
counted as being static. The specific details of the bowshock and the
velocity ranges determine how significant an effect this is.
In this section we discuss three methods which have been commonly used to estimate the age and thrust of molecular outflows from observations.
One of the most commonly used ways of doing this is to calculate the
intensity-weighted absolute velocity averaged over the entire
flow lobe
A third "
'' method discussed by Lada (1985),
Cabrit & Bertout (1992), and Beuther et al. (2002) uses
We now compare the results of each of these calculations with the
actual age and thrust of the simulated outflows. We will then propose
a fourth, "perpendicular'' method to estimate these quantities,
which appears more robust for the ages and density contrasts covered
by our simulations. Note that we compare the inferred thrusts with
the molecular axial thrust only, to see which method best
retrieves the properties of the emitting gas. To calculate errors on
the total (molecular+atomic) thrust in the simulated outflow,
one should multiply the relevant numbers in Tables 4
to 7 by 0.45 (for
;
see
Table 1) or 0.26 (for
;
see Table 2).
Table 4:
Inferred ages and thrusts of the outflow using the global
method and using the velocity ranges 2-40 km s-1 and 7-40 km s-1.
The quantities are normalized to the true age and molecular
thrust of the outflow. Errors on the total
(molecular+atomic) thrust may be obtained by
multiplying the relevant numbers by 0.45 (
)
or 0.26 (
).
We have taken
to be the projected distance from the
the point of injection to the furthest point from
this in the simulated bowshock.
Values of
and
for our simulations are listed
in the third and fourth column of Table 3.
Note that the lengths calculated here are similar to the length of
young Class 0 outflows such as HH211 (Gueth & Guilloteau 1999).
We then calculate
and the flow
thrust using Eqs. (11) and (12). Table 4 presents the ages and thrusts calculated for
and 7 km s-1 (see Sect. 3.5).
The values of
in
Table 3 are in good agreement with typical
observed values in CO outflows. At the same time, they are clearly
much smaller than the mean advance speeds of the bow head in our
simulations, which are 60-135 km s-1 for
-1. This occurs
because the mass-weighted velocity in a bowshock is more
representative of the bow wing motions than of the propagation speed
of the head of the bowshock (which contains very little mass). It
is then clear that estimating the age of a jet-driven outflow using
this method will be very misleading: as shown by
Table 4, the ages will be always significantly
overestimated, unless the flow is very close to the line of sight. The
error on age can be as large as a factor of 35, with a median error
(at
)
of a factor of between 10 (for
)
and 30 (for
). Correcting the age for inclination by
multiplying by
would only help significantly for flows close to the plane of
the sky. In the extreme case where
km s-1, the error on
age is decreased due to the resulting higher mean velocity.
As a result of the overestimate in the age, the molecular thrust is
very significantly underestimated. The typical error is again quite
large, with the median estimate being a factor of 20 too low for both
values of .
Similar errors are found for a low-velocity cut-off
at
km s-1, the smaller amount of observable
momentum roughly compensating for the shorter ages.
It is interesting to note that the only instance where global
age estimates are actually quite accurate is when the flow is closely
aligned to the line of sight. This is because the "length'' of the
flow lobe on the sky is then the same as its half-width, which
expands at a velocity close to
in our
simulations (cf. Sect. 3.6.4). The thrust is then much
better estimated as well. However, a pole-on inclination is
statistically very unlikely.
Table 5:
Inferred ages and thrusts of the outflow using the local
method of Lada & Fich (1996) with the velocity ranges 2-40 km s-1 and
7-40 km s-1. The quantities are normalized to the true
age and molecular thrust of the outflow. Errors on the total
(molecular+atomic) thrust may be obtained by
multiplying the relevant numbers by 0.45 (
)
or 0.26 (
).
Table 6:
Inferred ages and thrusts of the outflow using the
method of Cabrit & Bertout (1992) and using the velocity ranges
2-40 km s-1 and 7-40 km s-1. The quantities are normalized to the
true age and molecular thrust of the outflow. Errors on the total
(molecular+atomic) thrust may be obtained by
multiplying the relevant numbers by 0.45 (
)
or 0.26 (
).
Table 7:
Inferred ages and thrusts of the outflow using the
perpendicular method and using the velocity ranges 2-40 km s-1 and 7-40 km s-1. The quantities are normalized to the true
age and molecular thrust of the outflow. Errors on the total
(molecular+atomic) thrust may be obtained by
multiplying the relevant numbers by 0.45 (
)
or 0.26 (
).
![]() |
Figure 6:
Frequency distribution of the intensity-weighted local dynamical
imescales for both the ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
Figure 6 contains histograms of the frequency
distributions of the "local'' timescales calculated from Eq. (13) in both the
and
simulations for
an angle to the plane of the sky of
and
.
These
simulated results show a fairly peaked distribution, as observed by
Lada & Fich (1996) (their Fig. 5). The peaks move to smaller
timescales as the flow is viewed more pole-on, as expected from
projection effects.
We have performed these calculations for a number of different spatial
resolutions on the sky and find that the results are not strongly
dependent on this. The results shown here are for a resolution of
1016 cm (
pc) on the sky. Table
5 lists the inferred
timescales (i.e. the peak of the histogram) and the resulting thrusts.
The local method is found to perform better than the global method.
This occurs because most points in the flow have
,
yielding shorter timescales
(cf. Eq. (13)). However, the ages are still
generally overestimated, with a median error of 7.5-13.5 for a
typical cut-off velocity
= 2 km s-1. The errors are again strongly dependent on the inclination angle
of the flow, and are reduced only if the flow is close to the line of
sight (or if
is large). Once again we are left with a
method which strongly overestimates the age of the outflow, by an order
of magnitude typically.
The molecular thrust calculated from Eq. (14)
is also better than that obtained with the global method, but is still
significantly underestimated (except again in pole-on flows). The median
error on molecular thrust is an underestimate by a factor of 5-7 for
= 2 km s-1, and by a factor of 10-15 for
= 7 km s-1. Correcting
for inclination (assuming
is perfectly known)
would reduce the median errors by a factor of around 3.5.
For other inclinations, the associated correction would not seem to improve the
situation significantly, leading to an overestimate for flows very
close to the plane of the sky and an underestimate for flows viewed
almost pole-on.
Values of
for our simulations were measured at 1/100 of
the CO intensity level at
,
and are listed in
Table 3. The resulting range
6-35 km s-1 is typical of observed
values reported
for "standard'' low-velocity CO outflows, excluding localized
extremely high-velocity features (e.g. Cabrit
& Bertout 1992; Beuther et al. 2002). Table 6 contains
the inferred ages and thrusts using Eqs. (15) and (16) and the values of
from
Table 3. The thrusts were corrected for
velocity gradients and projection effects by inter/extrapolating
linearly as a function of
the mean correction
factors
listed in Table 1 of Cabrit
& Bertout (1992). The adopted correction factors (in Log10) are
thus 0.45, -0.15, -0.35, and -0.65 for
,
30
,
60
,
and 90
.
Since
is closer than
to the true
bow advance speed, this method of age estimate is a clear improvement
over the local and global methods. The median error is an
overestimate by a factor of 2-8 only for
= 2 km s-1, and even
less when
= 7 km s-1. However there is still a marked
inclination dependence, and the age becomes underestimated when the
flow is close to the line of sight, due to foreshortening in
.
The molecular thrusts calculated using this method are also better
than for the two previous methods. For
= 2 km s-1, the median
error is 0.5-0.7. This is reassuring, given that the adopted
inclination corrections were "calibrated'' on highly simplified
kinematic outflow models (Cabrit & Bertout 1990). Because of this
mismatch, the variation of error with inclination angle is not
completely compensated for, but it is reduced (a factor of 2-8,
compared to a factor of 20-100 if the correction were not applied). As
a result, the error on molecular thrust is less than a factor of 2,
except in the
flow viewed edge-on. When
= 7 km s-1,
the thrust underestimate is more significant (median factor of 5-8),
though again not as severe as with the previous two methods.
The main limitation of this method remains the need for a reliable
inclination estimate. Care must also be exercised, when using
Eq. (16), to exclude from
any faint
extremely high-velocity CO "bullets'' at 50-200 km s-1
closely associated with jet working surface(s) (Bachiller & Tafalla
1999): while such high values of
would give much
better age estimates (being close to the true advance speed of the bow
head), they would lead to greatly overestimating the momentum and
thrust in the bulk of the swept-up gas (respectively
and
in this method).
It is clear from observations of molecular outflows that a very large
amount of the material in the outflow is moving rather slowly. If the
jet-driven model of molecular outflows is accepted then this slow
moving material will be along the wings of the bowshock, and will
have predominantly transverse motions. The characteristic length
appropriate to measure the age is then the half-width of the lobe - i.e. the
perpendicular distance from the jet axis,
,
rather than the
distance from the jet source.
Consider an element of fluid in the ambient medium close to the axis
of the jet. As the bowshock, driven by the jet head, sweeps up this
element of fluid, a thermal pressure gradient perpendicular to the jet
axis is set up. This drives the fluid element away from the jet axis
(see Sect. 1). Once the element has propagated a
distance significantly greater than the jet radius away from the jet
axis, it begins to coast with constant momentum as it no longer feels
a significant pressure gradient. Its movement away from the jet axis,
as part of the the bow wing, thus becomes progressively slower as it
sweeps up more ambient gas along its path. In this case we must
account for the fact that the velocity of expansion measured at a
given time is lower than the expansion velocity at any previous time.
Hence, the bow is younger than one would calculate from its current
expansion speed and width. As demonstrated by Masson & Chernin
(1993), Wilkin (1996), and Ostriker et al. (2001), the bowshock wings expand asymptotically as
,
hence the transverse expansion speed varies with
width as
.
One then
finds that the time taken to expand to the current half-width
R is
t | = | ![]() |
|
= | ![]() |
||
= | ![]() |
(14) |
Table 7 shows the inferred age of the outflow,
,
and the corresponding molecular thrust,
,
normalized to their actual values. We can see that this
"perpendicular'' method performs extremely well when compared with
any of the previous methods, when
km s-1. For example,
when
,
all other methods overestimate the age by a
factor of 10 or more, while this method is correct to within 12-17%.
The median error is also smaller for this method, with the estimated
age being only 9%-36% too low. The largest error, which occurs
for an outflow aligned with the line of sight, results in an
underestimate of less than a factor of 2.5. The thrusts are also
generally more accurate than any other method, especially in highly
inclined flows, since timescales are now more accurately determined.
In the extreme case where the molecular cloud shows a lot of
broadening, and the cutoff is at
= 7 km s-1,
underestimates the age by a factor of 3 (
)
to 5
(
), independent of inclination. However, the molecular thrust is
still better estimated than with any of the other methods, as the
shorter flow age partly compensates for the smaller amount of
observable momentum at
.
A further attractive feature of this method is that the inclination
dependence of the estimated parameters is relatively minor, as the
transverse radius
does not change with inclination, and
and
generally do not
vary by more than a factor of 2 (see
Table 3). Hence no correction for the flow
inclination (which in itself is an uncertain quantity) is required,
unlike in the
method.
The only uncertainty in age in this method is thus introduced by the
low-velocity cut-off
,
which affects
through the power-law shape of m(v). However, the thrust appears much less
affected.
Our conclusion that dynamical timescales overestimate the age of
jet-driven flows appears to contradict the statistics of Class I
outflow surveys, which suggest that dynamical ages underestimate the
true outflow lifetimes by an order of magnitude (Parker et al. 1991). There is a simple explanation for this
discrepancy: We have assumed in our calculations that the flow is
young enough that the full length of the bowshock is observable in CO. On the other hand, the discovery of giant, parsec-scale optical
outflows from many Class I sources has revealed that their jets have
broken well out of their parent molecular cloud (see Reipurth and
Bally 2001, for a review). The CO lobes are thus
severely truncated compared to the full flow extent, and in this
case CO dynamical timescales may indeed be shorter than the
source age. This fact resolves the apparent contradiction noted by Parker et al. (1991) without the need to invoke a static CO cavity. It
should be noted that the sources of Parker et al. (1991) were
selected to be in isolated dark globules, all much less than 1 pc in size,
hence one expects the jet to have left the cloud in
yr, and
all the associated CO flows to be truncated.
One might think that our new "perpendicular'' method for estimating the
flow age should remain applicable even in Class I sources as it relies
only on the transverse lobe width, which is unaffected by
truncation. However, the sideways expansion of bow wings decays as
,
hence at an age of 105 yr it will
fall well below the measurable limit
,
while the
intensity-weighted
will remain greater than
by
construction. The assumption of the "perpendicular'' method that
may thus no longer be valid for
Class I jet-driven flows. Another (reverse) problem is that some
Class I flows exhibit unexpectedly high expansion speeds, with in a few cases
perpendicular timescales
104 yr much shorter than the
source age, e.g. in L43-RNO91 (Lee & Ho 2005). This phenomenon
probably requires an additional uncollimated wind component at late
evolutionary stages, not included in the present study.
Therefore, our conclusions about age determinations are not applicable to Class I flows and remain limited to the youngest jet-driven outflows from Class 0 sources.
The situation is slightly different regarding thrust estimates
in Class I flows: in a truncated CO flow, the relevant time to compute
the thrust injected into the molecular lobe will not be the full
lifetime of the jet, but only the time it took for the jet to
propagate over the CO lobe length
before leaving the
cloud. We would thus expect the thrust of truncated, jet-driven CO outflows to suffer similar underestimates as found here for younger
flows. An alternative method to estimate thrust in truncated CO outflows has been proposed by Bontemps et al. (1996),
involving the local momentum flux across a section of the lobe. The accuracy
of this alternative method will be investigated in a forthcoming paper.
Another limitation of the scope of our results is that they are
applicable only to outflows that propagate at relatively high speed of
60-135 km s-1 in the ambient medium. To our knowledge, the only
reported measurements of proper motions at the tip of a CO molecular
lobe are those obtained in optical and H2 lines in the HH47
counterflow (Eisloeffel & Mundt 1994; Micono et al. 1998). They indicate high proper motions of 150-260 km s-1 at the bow apex, consistent with the values of
adopted in
the present work. A much lower jet/ambient density contrast of 0.001-0.01 has been invoked by Richer et al. (1992) in the
Orion B outflow to alleviate requirements on jet thrust. In such a
regime, where the bowshock propagates at only 10-20 km s-1, the
effect of dissociation would be negligible. However, such a low
propagation speed implies a true age of
yr for
the Orion B flow (Richer et al. 1992), which appears
too long given the Class 0 status of the driving source FIR5 (André
et al. 2000). Proper motion measurements
of CO cavity tips other than HH47 would be highly desirable
to verify which range of
and propagation speed
is relevant over the observable phase of Class 0 jet-driven flows.
We have investigated the accuracy of observational methods for inferring the physical parameters of young, purely jet-driven molecular outflows, dealing in detail with many of the usual assumptions made when making these calculations. The most important conclusions are the following:
Acknowledgements
This work was partly funded by the CosmoGrid project, funded under the Programme for Research in Third Level Institutions (PRTLI) administered by the Irish Higher Education Authority under the National Development Plan and with partial support from the European Regional Development Fund. The present work was also supported in part by the European Community's Marie Curie Actions - Human Resource and Mobility within the JETSET (Jet Simulations, Experiments and Theory) network under contract MRTN-CT-2004 005592.