Protostellar outflow-driven turbulence

0.28 MB·English

by  Christopher D. Matzner

Draftversion February5,2008
PreprinttypesetusingLATEXstyleemulateapjv.04/03/99
PROTOSTELLAR OUTFLOW-DRIVEN TURBULENCE
Christopher D. Matzner
Department of Astronomy & Astrophysics
University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada
Subject headings: ISM: clouds – ISM: kinematics and dynamics – ISM: indivdual (NGC 1333)
stars: formation – stars: winds, outflows – turbulence
Draft version February 5, 2008
7
0
ABSTRACT
0
2 Protostellar outflows crisscross the regions of star cluster formation, stirring turbulence and

n altering the evolution of the forming cluster. We model the stirring of turbulent motions by
a protostellar outflows, building on an observation that the scaling law of supersonic turbulence
J
implies a momentum cascade analogous to the energy cascade in Kolmogorov turbulence. We

1
then generalize this model to account for a diversity of outflow strengths, and for outflow col-

limation, both of which enhance turbulence. For a single value of its coupling coefficient the
1
v model is consistent with turbulence simulations by Li & Nakamura and, plausibly, with obser-
2 vations of the NGC1333 cluster-forming region. Outflow-driven turbulence is strong enough
2
to stall collapse in cluster-forming regions for several crossing times, relieving the mismatch
0
between star formation and turbulentdecay rates. Thepredicted line-width-size scaling implies
1
0 radial density indices between -1 and -2 for regions supported by outflow-driven turbulence,
7 with a tendency for steeper profiles in regions that are more massive or have higher column
0
densities.
/
h 1. INTRODUCTION dense) context of the Sun’s birth cluster.
p
There is growing evidence that star cluster forma-
- Star formation in the Milky Way is known to occur
o tionisaslowprocess,inthesensethatthestarforma-
predominantlywithinstarclusters. Clusterswithdif-
r
tion rate, though quite uncertain, is about an order
t ferentmultiplicites (N )areborninaremarkablyflat
s ⋆
of magnitude slower than the mean free-fall rate of
a distribution, dN˙ /dlnN ∝ N−1. This is flat in the
: cl ⋆ ⋆ the parent gas “clump”. Palla & Stahler (1999) and
v sense that the birth cohort of a given star is equally
Huff & Stahler (2006) argue from the Orion Nebula
Xi likely to be in any decade of N⋆ within the allowed Cluster’s Hertzsprung-Russell diagram that star for-
range,fromalowerlimitofN ∼ 50−100toanupper
r ⋆ mation began there at least 10 Myr ago, as has been
a limit that depends on the area sampled – a few thou-
dramatically confirmed by the detection of lithium
sand, based on surveys out to 2 kpc (Lada & Lada
depletion in some of its members (Palla et al. 2005).
2003), or about half a million (McKee & Williams
The common origin of runaway stars AE Aurigae
1997), based on HII region surveys of the entire in-
and µ Columbae in the cluster’s vicinity 2.6 Myr
ner Galaxy. The number, density, and lifetime of
ago (Blaauw & Morgan 1954) supportsthis assertion
the birth cohort affect the likelihood of exposure to
(Tan et al. 2006). Tan et al. also point to the fact
ionizing radiation and supernovae from the massive
that cluster mass profiles are quite smooth, requir-
clustermembers,aswellasthelikelihoodofclosestel-
ing time for subclusters to merge. Krumholz & Tan
lar encounters, all of which affect the formation and
(2006) compute the star formation rate parameter
evolution of planetary systems.
SFR , which is normalized to the free-fall rate, from
The Solar System, for instance, is thought to ff
molecular andinfraredobservations of dense, cluster-
derive from a cluster with a few thousand mem-
forming clumps, finding SFR ∼ 10−1.7 (with signifi-
bers (Adams & Laughlin 2001) on the basis that a ff
cantuncertainty)forhydrogendensitiesrangingfrom
nearby supernova supplied short-lived radionuclides,
n = 102 to 105.3cm−3. As they point out, this is
and thatpassingstars didnotdisturbtheouter plan- H
consistent both with theories in which star forma-
ets. The orbit of the planetoid Sedna (Brown et al.
tion is regulated by ambipolar diffusion within mag-
2005) appears to require an encounter with another
netized gas (McKee 1989), and with those in which
star well within a thousand AU, an event which is
turbulence suppresses the rate of localized collapse
nearly impossible except in the dense (but not too
(Krumholz & McKee 2005). Regardless of its cause,
1
2
slow cluster formation is in sharp contrast with the flow intensities in §4.1, and evaluate it for the stellar
rapid decay of supersonic turbulence over at most initial mass function (IMF) and outflow collimation
a couple free-fall times (Stone et al. 1998; Mac Low in §4.1.1 and 4.1.2, respectively. The loss of momen-
1999). Therefore, forming star clusters must tap tum from a finite region due to collimated outflows
some source of energy to sustain the observed tur- is considered in §4.2. We compare to available sim-
bulence – and the nature of this feedback may leave ulations in §5 and to observations of NGC1333 in
its imprint on the clusters’ properties. §6, before drawing conclusions about the dynamics
This paper addresses the driving of turbulence by of stellar cluster formation in §7.
protostellar jets and winds, which are thought to
2. DIMENSIONALANALYSIS
emerge from stars of all masses as they form. Since
To begin as generally as possible, let us first con-
outflows driven by these jets and winds are ubiqui-
siderapressurelessmediumofmeandensityρ stirred
tous, outflow-driven turbulence is a pervasive and in- 0
by spherical explosions whose characteristic momen-
evitable consequence of stellar cluster formation.
tum is I, that occur at a rate per unit volume S. We
A second, more sporadic form of feedback comes
choose to describe these by their momentum rather
from the most massive cluster members, whose
thantheirenergy, becauseweexpecttheshockedma-
ionizing radiation is a strong driver of turbulence
terial to be highly radiative. These three parameters
within giant molecular clouds (GMCs; Matzner 2002;
with independent dimensions define scales of mass,
Krumholz et al. 2006). However, cluster-forming
length, and time:
clumps are much denser than the GMCs surrounding
them (∼ 104−5 rather than ∼ 102 H atoms cm−3), ρ4/7I3/7 I1/7 ρ3/7
and high densities damp the dynamical effects of m = 0 , ℓ = , t = 0 . (1)
photoionization. We shall therefore concentrate on S3/7 ρ10/7S1/7 I3/7S4/7
outflow-driven turbulence. (See Tan & McKee 2000 One quantity of interest is the turbulent line-width,
for preliminary cluster-formation models which in- i.e., the characteristic turbulent velocity, measured
clude photoionization.) on scale ℓ:
Norman & Silk (1980) proposed that protostel- ℓ I4/7S3/7
= . (2)
lar outflows sustain the turbulence in GMCs, be- t ρ4/7
fore it was known that star formation is re- 0
The dynamical state of the gas will be highly tur-
stricted to the densest and most massive clumps
bulent and clumpy, but in the absence of other di-
within them. McKee (1989) adopted protostel-
mensional parameters any quantity must obey these
lar outflows as the driving agents in his equi-
scalings. There is one explosion, on average, per ℓ3t.
librium theory of GMCs. Concentrating on the
Intermittent explosions are the dominant feature of
scale of a cluster-forming clump, Matzner (1999)
flow on scales smaller than ℓ, whereas flow on larger
and Matzner & McKee (1999b) incorporated out-
scales reflects many overlapping explosions.
flows into dynamical models for star cluster forma-
To estimate characteristic scales for the cluster-
tion. Recent simulations of cluster formation by
forming regions discussed later, consider I =
Li & Nakamura (2006) provide an important demon-
1039.6gcms−1, S = 10−67.2cm−3s−1, and ρ =
stration that protostellar outflows are dynamically 0
10−19.6gcm−3 – values typical of cluster-forming re-
important in this process.
gions like NGC1333 (§ 6). Then, ℓ = 0.38pc,
The theory of outflow-driven turbulence presented
here improves upon these previous studies by ac- t = 0.34Myr, m = 19M⊙, and ℓ/t = 1.1kms−1.
Outflow-driven turbulence (1) extends well beyond
counting for two of the outflows’ key properties: a
the regions of localized collapse associated with in-
strong collimation toward an outflow axis, and a di-
dividual forming stars, and (2) involves motions sig-
versity in strength associated with the range of pos-
nificantly faster than the thermal sound speed (c ∼
sible stellar masses. These effects conspire to en- s
0.2kms−1). In§4.1 weshallfindthat thesequalities,
hance the outflows’ effect on the dynamics of cluster-
whichdistinguishoutflowsaslikelyagentsofsupport,
formingclumps,relative toamodelinwhichoutflows
are only enhanced by effects like outflow collimation.
are spherical and all have the same strength.
In §2 we establish the basic dimensional scales and 2.1. Gravity, stellar mass, and finite-size regions
dimensionlessratiosthatdefineoutflow-driventurbu-
We now introduce gravity, and restrict our atten-
lence. In §4 we propose models for the energy spec-
tion to a region of finite size R – hence of mass
trum and line-width-size relation of outflow-driven
M = 4πR3ρ /3 if we take ρ to be the mean den-
turbulence, drawing on an insight about the dynami- 0 0
sity. Gravity and finite radius come together nat-
cal nature of supersonic turbulence in strongly radia-
urally, because star formation is known to occur in
tive gas. We extend this model to a diversity of out-
self-gravitating “clumps” of finite mass and radius.
3
Let us also introduce a stellar mass scale M , and outflows cannot provide the dynamical pressure re-
⋆
suppose that the explosion momentum is related to quiredforcloud support. Onemighttherefore expect
M through a characteristic velocity: both parameters to be > 1 in a real cluster-forming
⋆ ∼
environment. More specific predictions are provided
I = M v . (3) in §4.2 where we detail a model for turbulence in a
⋆ c
cloud of finite radius and mass, and in §7 where we
If we consider the stellar mass function fixed, we consider the cloud’s dynamical expansion or contrac-
might take M⋆ ≃ 0.5M⊙; Matzner & McKee (1999a) tion in response to outflow-generated turbulence.
adopt v ≃ 40kms−1 in protostellar outflows, imply-
c
ing I ≃ 1039.6gcms−1. 2.2. Finite sound speed and magnetization
Given M , we may define
⋆ Continuing in thespiritof dimensionalanalysis, let
us allow the gas to have a finite sound speed c ; the
s
32 1/2 G1/2ρ3/2 relevant dimensionless parameter is
S = SFR 0 (4)
ff
3π M
(cid:18) (cid:19) ⋆ c t
s
Π = (7)
c
where SFR is the star formation rate normalized ℓ
ff
to the free fall rate at the mean density. McKee
which is an estimate for the inverse of the turbu-
(1989) estimates this parameter under the assump-
lent Mach number on scale ℓ. Turbulence is super-
tion that ambipolar diffusion determines the rate
sonic and strongly compressive so long as Π ≪ 1;
c
of collapse, and finds SFR ≃ 0.08 in regions
ff for the parameters adopted above, this is true if
well shielded from external far-ultraviolet radiation. SFR ≫ 10−3.2n−1/6(c /0.2kms−1)7/3, where the
The turbulence-regulated star formation theory of ff H4 s
hydrogen density is 104n cm−3. So long as Π ≪ 1
Krumholz & McKee (2005) predicts SFR ≃ 0.034 H4 c
ff we expect outflow-driven turbulence to beinsensitive
in nearby star-froming clumps.
to the precise value of c ; this is supported by the
Although we used M in defining v and SFR , it s
⋆ c ff more detailed calculations given below in §4.1.
isimportanttorealizethatoutflow-driven turbulence
The influence of sound speed on the star formation
is invariant under the transformation M → αM ,
⋆ ⋆ processitselfshouldnotbeunderstated,however. By
SFR → αSFR , and v → α−1v , which leaves S
ff ff c c setting the thermal Jeans scale, finite sound speed
and I unchanged. Therefore M introduces no new
⋆ determines both the normalization of stellar masses
dimensionalscalesofarasturbulenceisconcerned. A
(e.g., Padoan & Nordlund 2002), and the star forma-
corollary isthat,althoughv entersexplicitly intothe
c tionrate(atleastinthemodelofKrumholz & McKee
starformationefficiency(Matzner & McKee2000),it
2005), thereby affecting S and I.
will always appear in the combination SFR v ∝ SI
ff c The dynamics of the gas will certainly be affected
in dimensionless ratios given here.
by magnetic fields (e.g., McKee et al. 1993). These
Self-gravity andfiniteradius,ontheotherhand,in-
introduce at least two additional dimensionless pa-
troduce the dimensional quantities G (Newton’s con-
rameters: the degree of magnetization, as measured
stant) and R, which can be used to form two dimen-
by the global mass-to-flux ratio or the local plasma-β
sionlessratioswithS,I,andρ . Usefulformsinclude
0 parameter; and the ambipolar diffusion rate, as mea-
the radius ratio
sured by the ratio of the neutral-ion collision time to
R
ΠR ≡ ℓ , (5) the free-fall time, say. We expect that, like Πc, these
parameters influence the coefficients that appear be-
the acceleration ratio
low – the coupling factor Λ defined in §4, and the
critical value of α in §7 – but introduce no other
ℓ R IS R 2 3/2 v vir
Π ≡ = = SFR c , (6) qualitative change to the theory. This expectation
acc t2vK2 ρ0 vK2 (cid:18)π(cid:19) ffvK can only be tested by comparing the theory to obser-
vations and simulations.
which measures the acceleration scale SI/ρ in units
0
of the surface gravity v2 /R = GM/R2 (for Kepler 3. OUTFLOW MOMENTUM PER STELLAR MASS,v
K c
speed v ), and the quantity I /I¯ ≃ Π−1/2Π7/2 The outflow momentum per unit stellar mass, v ,
K esc acc R c
which we define in §4.2. was estimated to be 40kms−1 by Matzner & McKee
The scalings in equations (1) and (2) pertain to an (2000) on the basis that the typical protostellar wind
infinite(Π ≫ 1)andnon-self-gravitating(Π ≫ 1) velocity is ∼ 200kms−1 and about one-sixth of the
R acc
medium rather than a realistic clump or cloud. Con- accreted mass is ejected as wind – a value which is
sider the opposite: if Π ≪ 1, then the region will be intermediate among several theoretical predictions.
R
cleared by a typical outflow; and if Π ≪ 1, then In order to check our estimate of v , we combine
acc c
4
the Palla & Stahler (1992) evolutionary models for if ρ and u are the local density and fluid velocity,
accreting protostars with an observational inference u is the center-of-mass velocity, and the integral
cm
by Richer et al. (2000) of the relation between stellar extends over the volume affected by the outflow. A
and outflow properties. For those stars whose lumi- populationofexpandingoutflowsdeliversmomentum
nosity is dominated by accretion, Richer et al. show at a rate SI per unit volume. Turbulence is there-
that the outflow momentum is roughly 0.3v , where foredriven,onscaleℓ,byacharacteristicacceleration
k⋆
v is the Kepler velocity at the surface of the proto- SI/ρ .
k⋆ 0
star. We obtain v from Palla & Stahler’s models, Does a radiative, supersoniccascade transportmo-
k⋆
and apply the Richer et al. rule to all the material mentum to small scales the way an incompressive
accreting onto a star of final mass M in order to cal- cascade transports energy? This seems plausible,
⋆
culate its net outflow momentum I(M ). Dividing based on the dynamics of expanding outflow shells.
⋆
the total momentum by the total mass for a stellar Moreover, this hypothesis is consistent with the well-
population drawn from the Kroupa (2001) IMF with known scaling laws of supersonic turbulence. The
an upper cutoff of M , we find one-dimensional (spherically averaged) spectrum of
⋆u
the velocity field u(x) is defined as
M
⋆u
v ≃ (28,27,22)log (8)
c 10 0.11M⊙ 1 ∞
assuming stars all accrete at (10−5,3 × E(k) = 2π hu(x)·u(x+nr)ie−ikrdr (9)
10−5,10−4)M⊙yr−1,respectively. Theresultislower Z0
for a higher accretion rate, because these stars have where angle brackets indicate an average over time,
less time to contract during accretion. space, and the direction of the unit vector n. Sup-
For an upper cutoff M⋆u = 120M⊙ this exercise posetheturbulencetransportsanacceleration SI/ρ0
gives about twice our fiducial estimate of 40kms−1, from ℓ to small scales. As S has units cm−3s−1 and
as previously noted by Tan & McKee (2002). How- as SI/ρ has units cms−2,
0
ever, theexcess momentum comes frommassive stars
that (1) require an extrapolation of the Richer et al. SI
(2000) scaling, (2) form only sporadically, relative E(k) = A k−2 (k ≫ ℓ−1) (10)
ρ
0
to low-mass objects, and (3) may not form at all,
if the cluster in question is too small to sample the
for some coefficient A which depends on other di-
IMF. For simplicity we hold to the initial estimate
mensionless parameters. Equation (10) matches the
v = 40kms−1 for the purpose of making estimates,
c spectral slope for supersonic turbulence, as deter-
although a more complicated model could be accom-
mined by numerical experiment (Porter et al. 1992).
modated using the theory we present in § 4.1.
Itcoincides withthespectralslopeinBurgers’turbu-
4. VELOCITY SPECTRUM AND LINE-WIDTH-SIZE lence (Kida 1979), and indeed with that of individual
RELATION shocks. Moreover it resembles Larson’s (1981) and
Thewindsthatdriveoutflowspossessahighenergy matches Solomon et al.’s(1987)estimates oftheline-
per unit mass, which reflects the depth of the poten- width-sizerelationinmolecularclouds. Therefore,we
tial well from which they werelaunched. Most of this adopt the central hypothesis that outflow-driven tur-
is lost to radiation at the wind deceleration shock. bulence transports the flux of scalar momentum per
What remains is the kinetic energy of the expanding unit mass (i.e, an acceleration) from the driving scale
shell, but this declines in proportion to the shell ve- to small scales, where it is destroyed by the collision
locity as additional mass is swept up. At a radius of ofoppositely-directedmotions. Theremainderofthis
order ℓ the shell loses coherence and merges with su- paper amounts to an exploration of this hypothesis.
personic turbulent motions – which themselves dissi- Consider the one-dimensional velocity dispersion,
pateenergy,asdescribed,forinstance,byStone et al. or line-width, σ(r). To be specific we define 3σ(r)2/2
(1998), Mac Low et al. (1998), and Mac Low (1999). tobethemeankineticenergyperunitmassofspheres
IncontrasttotheKolmogorov cascadeofincompress- of radiusr, relative totheir centers of mass. Sincefor
ible turbulence, this supersonic cascade is not de- now we consider the gas infinitely cold (an assump-
scribed by a constant energy flux to small scales. tion we drop in §4), σ → 0 as r → 0. On small scales
Ontheotherhand,stronglyradiativeoutflowshells (r ≪ ℓ), σ(r) is determined by dimensional analysis:
conserve momentum in each direction, and therefore
conserve a net scalar momentum I as they expand – SI
σ(r)2 = Λ2 r (r ≪ ℓ) (11)
where by “scalar momentum” we mean ρ
0
ρ|u−ucm|d3x for some coupling coefficient Λ.
Z
5
A lower limit on Λ can be estimated from the fact intermittency of the motions on scale r. Moreover it
that σ(r)2 includes the kinetic energy of decelerat- neglects magnetic forces, which can cause merging at
ing shells. The mean kinetic energy density in shells smaller radii. Here M(r ) refers to the mass en-
m1
traveling faster than v is countered within a radius r of the driving source;
m1
we write
t(v) Iv′ ′ M(r) = 4πφ ρ r3 (16)
S dt (12) m 0
3
2
Z0
where φ > 1 if stars tend to form within extended
m
wheret′(v′)istheageatwhichtheflowdecelerates to overdensities. Ingeneral φ is afunctionof theother
m
v′. Since the shell expands according to dr′ = v′dt′, dimensionlessparameters, includingr/ℓ, butweshall
the integral can be written treat it as a number.
We previously considered the outflow strength I
r(v) I r(v)
′ and rate S as unique values. We generalize now to a
S dr = SI, (13)
Z0 2 2 range of values for I, and consider S(< I) as the cu-
mulative rate of outflows weaker than I. To proceed
which is SIℓ/2at the mergingscale, r(v) = r(σ) = ℓ. we suppose that σ(r)2 – which is proportional to the
The total energy density on this scale is 3σ2(ℓ)/2;
specific kinetic energy of motions on scales smaller
comparing to equation (11), we find
than r – is a linear superposition of the contributions
from outflows of a single intensity.
Λ2 > 1/3. (14)
Therefore, let us reexamine the single-intensity
limit considered in §2. This is given by S(< I) →
This is only approximate, since our estimate of the
S H(I −I ), where H(x) is the Heaviside function,
kinetic energy did not account for the complicated 1 1
and where subscript ‘1’ denotes a single value of I;
dynamics of merging, and since equation (11) is ex-
then dS/dI = S δ(I −I ). The contribution σ2(r)
pected to break down at the merging scale. However, 1 1 1
is ∝ r on scales smaller than r , and is constant on
it guides us to expect Λ2 to be of order unity, and to m1
larger scales; therefore we adopt the idealization
attribute avaluein excess of 1/3 to thepersistenceof
turbulent energy. In fact, the Krumholz et al. (2006)
S I r, r < r
etostiamcahteanogfetuinrbΛu2lenotf d0.r6iv4infogratnhdisdpecearysisctoernrtestpuornbdus- σ12(r)= Λ2 ρ101 (cid:26) rm1, r > rmm11 . (17)
lence, so our best estimate of Λ2 is 1/3+0.64 = 0.97. The contribution of outflows to σ2 on scales larger
1
On scales large enough that no individual outflow
than r should not be interpreted as an inverse cas-
m1
can dominate the motion, the velocity spectrum is cade: it simply reflects the fact that σ2(r) incorpo-
likely to be affected by a cascade from even larger
rates all motions within spheres of size r. Differenti-
scales. Indeed, if outflows are sufficiently weak or the
ating twice,
drivingissufficientlystrong,itispossibleforthiscas-
cadetodominateoveroutflow-drivenmotionsevenon d2σ2 Λ2S I Λ2I dS/dI
scales smaller than ℓ. Inthe equations below we shall dr21 = − ρ1 1δ(r−rm1) = − ρ 1dr /dI.
0 0 m1
include a term to describe this external cascade.
(18)
This equation holds for each value of I in our linear
4.1. Variations of outflow intensity
superposition model, so in general
Uptothispointwehaveconsideredoutflowsofuni-
form intensity. However fluctuations should be com- d2σ2 Λ2I dS
=− . (19)
monplace – due to variations in the mass and accre- dr2 ρ dr
0
tion rate of the driving star, for instance, or due to
In this equation, as in the rest of this section, we
the angle from which a collimated outflow is viewed.
take I tomean thevalueforwhichr = r according
To treat this possibility, consider a single outflow m1
to equation (15), so that I, S, and r are monotonic
whose momentum scale I differs from the character-
1
functions of one another; dS/dr could be replaced
istic value I. It dominates the motion of gas it over-
with (dS/dI)(dI/dr). Keep in mind that the rela-
takes within a “merging” radius r , defined such
m1
tions between these quantities dependon σ(r), which
that
must be determined self-consistently.
M(r )σ(r )= I . (15)
m1 m1 1
Integrating,
This equation relies on the basic assumption that the
one-dimensional velocity dispersion σ(r) can be used dσ2 Λ2 Stot
′ ′
= a + I dS (20)
to gauge themerging radius, withoutreference to the dr ext ρ0 ZS(r)
6
wherea is theacceleration duetoacascade, if any, Thisfactor illustrates theincreaseinσ2 expected due
ext
from larger scales, and S is the total outflow rate to the broadening of the outflow distribution. It de-
tot
per unit volume. Therefore pends on c and a because of the implicit nature
s ext
of equation (24), but to be definite we calculate it
dσ2 Λ2 Stot
lim = aext + IdS, (21) assuming cs = aext =0.
r→0 dr ρ0 Z0 Our expectation that broader distributions of out-
flow strengths lead to stronger turbulent motions is
whichgeneralizes equation(11). Takingσ2 toinclude
verified in Table 1, where E is presented for the out-
thermal motions, we also have
flow distributions discussed below.
limσ2 = c2. (22)
r→0 s
4.1.1. Mass function
Given a total momentum injection rate IdS,
We now consider the two primary sources of vari-
what form for S(< I) maximizes σ(r) on large
R ation in outflow intensity: differences in the stellar
scales? It is obvious from equation (2) that since
mass, in this subsection, and differences with angle
ℓ/t ∝ (IS)3/7I1/7, turbulence is enhanced if momen-
in collimated winds in the next subsection.
tum is emitted in fewer, stronger bursts. (Up to a
Let F(> M /M ) be the fraction of stars that
⋆ ⋆
point: in a finite region, bursts that are too strong
exceed the mean stellar mass M at birth; then
⋆
will escape, and those that are too rare are irrele-
F(0) = 1 and x|dF/dx|dx = 1 if the limits of in-
vant.) Suppose we fix the number of events and thus
tegration include all stellar masses. Assuming I is
the mean outflow strength R
strictly proportional to M as in equation (3), and
⋆
parametrizing the star formation rate using equation
IdS
I¯ ≡ (23) (4), we have
S
R tot
as well as IdS. Then, by the same argument, tur- 32 1/2 G1/2ρ3/2 I
bulence is maximized if all but one outflow is negli- S(> I)= SFRff 0 F > .
R 3π M M v
gible in strength, and the last carries all the momen- (cid:18) (cid:19) ⋆ (cid:18) ⋆ c(cid:19)
(27)
tum. The corresponding σ(r)2 achieves a value enhanced
More generally, we expect that turbulence is en-
by E = 1.85 relative to a single-mass model.
hanced by broadening the range of outflow strengths.
Consider the integration by parts of equation (20)
4.1.2. Outflow collimation
over radius, from 0 to r:
A second and even more significant source of vari-
Λ2 Stot
σ2(r)=c2+ra + r I′dS′ ation in outflow intensity is the collimation of proto-
s ext ρ0 ZS(r) stellarwinds. Matzner & McKee(1999a)arguedthat
Λ2 S(r) magnetic stresses cause both disk winds and X-winds
′ ′ ′
+ r I dS . (24) to approach the same asymptotic structure at large
ρ
0 Z0 distances from the source: the apparent strength at
Thissolvesdifferentialequation(19),butonlyimplic- an angle µ to the outflow axis is
itly, since the function r(I) given by equation (15),
and therefore S(r), depend on σ(r). Equation (24) is Iˆ(µ)= IP(|µ|), P(|µ|) = 1
useful, however, because the first integral disappears ln(2/θ )(1+θ2−µ2)
0 0
at the maximum outflow extent (r = maxr ). The (28)
m1
contribution ofoutflowstoσ2(r)onevenlarger scales where the normalization factor ln(2/θ ) ensures
0
is therefore given by the second integral 1P(|µ|)d|µ| = 1 so long as θ ≪ 1. The soften-
0 0
ing angle θ encompasses precession, internal shocks,
Λ2 Stot R 0
rIdS. (25) fluid and magnetic instabilities, phenomena at the
ρ
Z0 jet-ambient interaction, and anything else that de-
This allows us to define a turbulence enhancement grades collimation. Matzner & McKee (1999a) esti-
factor as the ratio of the outflow-driven turbulent mate θ0 ∼ 10−2 for several well-observed outflows.
pressure (eq. [25]) to what it would be for a mono- This implies a variation of ∼ 104 in outflow strength
lithic model in which I = I¯: with angle, which has a strong effect on the ampli-
tude and character of outflow-driven turbulence. (It
E ≡ 0StotrIdS (26) is possible, however, that a larger value of θ0 applies
R3 I¯4St3ot 2/7 tothelaterstageofoutflowevolution andtothedriv-
(cid:18)4πφm Λρ10/2(cid:19) ing of turbulence.)
7
GivenanoutflowofnetmomentumI,theapparent outflows as they expand. However, outflows take mo-
strength equals Iˆ at angle mentumaswellasmasswiththemwhentheyescape.
Collimation allows this to happen in some directions
1/2 without the entire clump being unbound. Specifi-
1
|µ|= P−1(Iˆ/I) = 1+θ2− , cally, escape occursindirection µ whenIˆ(|µ|) > I ,
" 0 ln(2/θ0)Iˆ/I# where esc
(29)
I = c Mv (31)
where P−1 is the functional inverse of P, so long as esc g esc
this gives 0 ≤ |µ| ≤ 1. For a random orientation if M = M(R) and vesc = (2GM/R)1/2 are the mass
|µ| is evenly distributed within this range; therefore and escape velocity of the region. The factor cg ac-
P−1(Iˆ/I) is the cumulative probability that the ap- counts for gravitational deceleration of an expand-
parent strength is weaker than Iˆ, and dP−1/d(Iˆ/I) ing shell, which saps its momentum. It is close to
unity, however: for density distributions ρ(r) ∝ r−kρ,
isthedistributionofIˆ/I. (Formally, dP−1/d(Iˆ/I)=
Matzner & McKee (2000) found
d|µ|/dP.)
We adopt the basic assumption that turbulence 1/2
9−3k
ρ
driven by collimated outflows is very similar to tur- cg = (32)
8−3kρ!
bulence driven by spherical outflows, if one general-
izes thedistributionS(I)totheapparent distribution (their equation [A13]). This formula holds when out-
Sˆ(Iˆ), which is the convolution of S and P−1 over the flows are driven impulsively – a safe assumption, so
logarithm of their arguments: long as the wind that drives an outflow has a du-
ration similar to the free-fall time of its collapsing,
dSˆ dS dP−1 overdense core.
= ⊗ . (30) Given that outflow intensities exceeding I get
dlogIˆ dlogI dlog Iˆ/I esc
away, the theory of §4.1 is easy to modify: simply
(cid:16) (cid:17) replace S with S(I ), throwing away the remain-
tot esc
The influence of collimation on σ(r) is illustrated ing momentum. It is important to realize that some
in figure 1, which displays numerical integrations of outflows emerge from the clump surface and rain
equation (20). Two points can be drawn from these backonitlater; thereplacementjustsuggestedtreats
results: (1)Afinitethermalsoundspeedisofnoprac- them no differently from those that merge within the
tical consequence so long as ℓ > c t, since solutions clump. Although approximate, this approach cap-
s
converge when σ > c ; (2) σ(r) continues to rise until tures the essential division between capture and es-
s
r ≃ max(r ). (Recall that max(r ) refers to the cape.
m1 m1
value obtained by setting I → maxI.) However the Theeffect ofoutflow eruptionsonthevelocity scale
rise in σ(r) slows to zero as this limit is approached. σ(R) is determined by how much outflow momentum
We expect the coupling coefficient Λ to depend is eliminated in this procedure. It depends, there-
somewhat on the degree of collimation, thanks to fore, on the dimensionless ratio Iesc/I¯ as well as on
complicated phenomenalikecollisions between young the shape of S(I) (see also §2.1). This dependence
outflows (Cunningham et al. 2006), although the is illustrated in figure 2, where a slow dependence on
sense and magnitude of this dependence are un- Iesc/I¯ is apparent. This is natural, because the ap-
known. parent intensity Iˆ can exceed I¯ by a factor of 106.7
in the model plotted – of which 102.7 comes from the
4.2. Finite-size regions range of stellar masses, and 104 arises from collima-
tion. For a default model in which we adopt the
What if the region of interest (a dense “clump”,
Kroupa (2001) IMF, take v to be independent of
say) is too small to catch all the outflow momentum? M , consider winds to be collcimated with θ = 10−2,
The eruption of outflows from protocluster regions is ⋆ 0
and take a = 0, we find
frequently observed, and this process deserves special ext
attention. Matzner & McKee (2000) calculated the σ2t2 c2t2 r −η −1/η
mthaessouetjflecotwios’ndryantaeminicathliesffceactsse.; we wish to consider ℓ2 ≃ ℓs2 +Λ2"K1−2η +K2−2η(cid:18)ℓ(cid:19) # (33)
It is not sufficient to simply evaluate σ(r) at the where the fit parameters K1, K2, and η are functions
clumpsizeR usingtheresultsoftheprevioussection. of Iesc/I¯:
Equation (20) assumes a homogeneous background I −10 −1/10
on scales larger than r, as σ(r) reflects the downward K = 2.33−10 + 0.32+0.40log esc ,
cascade driven by outflows merging on larger scales, 1 " (cid:18) 10 I¯ (cid:19) #
as well as the upward cascade composed by those (34)
8
2.5
2
Kroupa IMF, θ =10−2
0
1.5
)
Λl
( Kroupa IMF, spherical
/
t
σ
1
0.5
0
10−2 10−1 100 101 102
r/l
Fig. 1.— Line-width σ(r) versus size, normalized to the characteristic values in § 2, for outflows driven by stars drawn from
the Kroupa (2001) IMF. If outflows are spherical, the theory of §4.1.1 gives the lower curve; if they are collimated, the theory
of §4.1.2 gives the upper curve. At scales much smaller than the driving scale ℓ, σ ∝r1/2 – in agreement with the Larson’s-law
scalings for molecular clouds (Larson 1981) as reported bySolomon et al. (1987).
Table 1
Turbulence enhancement factors.
Mass function E (isotropic) E (θ = 10−2)
0
Single-mass 1.0 3.04
Kroupa (2001) IMF 1.85 5.38
9
I −6 −1/6 expectchanges innumerical resolution, cloudmagne-
K2 = "1.22−6 +(cid:18)0.55+0.26log10 Ie¯sc(cid:19) # , tbiyzavtaiornia,tiaonnds oinutΛfl.ow collimation to be accompanied
(35)
Li & Nakamura also mention that collimated out-
and
11+15log (I /I¯) flows appear to drive more vigorous turbulence, as
η = 10 esc . (36) we would expect from §4.1.2. Nakamura & Li (2006)
1+10log (I /I¯)
10 esc have modified their previous simulations by collimat-
This fit reproduces the numerical evaluation of σ(r) ingoutflowswithin30◦ halfopeningangle,andreport
to within 5% for Iesc > 6I. The quantities used in thattheenhancementisnotdiscerniblefromthesim-
these formulae are, in convenient form, ulations. In our theory, this modification amounts to
increasingI byafactor of 7.5 whileholdingSI fixed;
I 40kms−1
esc = 770M5/4Σ1/4 , (37) we would expect σ to increase by 7.51/7 = 1.3 (as
I¯ 3 cgs vc in eq. [2]), so long as Λ does not change. Whether
this mild enhancement of turbulent velocity is seen
ℓ = 0.079M5/28Σ−15/28 40kms−1 0.034 1/7 pc, in simulation will require further investigation. In
3 cgs vc SFRff! an equilibrium cloud, as our referee points out, one
(38) must tease this effect from the redistribution of mass
and caused by more vigorous turbulence.
Adeeperinvestigation of thesameclass of modelis
ℓ = 1.24M−1/28Σ3/28 vc 4/7 0.034 3/7 km reported by Nakamura & Li (2007), who report sev-
t 3 cgs 40kms−1 SFR s eral results in line with our predictions. In this work,
(cid:18) (cid:19) (cid:18) ff(cid:19)
(39) outflows posses a 30◦ “jet” component and a uniform
where M(R) = 103M3M⊙ and M(R)/(πR2) = “wind” component. The authors report that the jet
Σcgsgcm−2, and we used kρ = 1.5 to evaluate cg. component is more effective at driving turbulence,
based on the slowdown of star formation observed
5. COMPARISONTO SIMULATION
if jets are strengthened (all else being equal). Their
ThesimulationbyLi & Nakamura(2006)offersthe turbulentpowerspectrumobeysE(k) ∝ k−2 asantic-
closest comparison to the theory presented above. ipated in equation (10), and flattens at large scales –
These authors simulate a self-gravitating, weakly possibly reflecting the flattening of σ(r) (figure 1).
magnetized, initially turbulent cloud, identifying re- Finally, they obtain an equilibrium density profile
gions of localized collapse and replacing them with ρ∝ r−1.5,inlinewiththeargumentwepresentbelow
spherical outflows. Using their fiducial conver- in §7.
sion from code units to physical units, their out- Mac Low (2000) has also simulated turbulence
flow momentum scale is nearly constant at I = driven by collimated sources, but a direct compar-
1039.7gcms−1. The 960 M⊙ of cloud gas is initially ison is not possible because the driving field in his
arranged in a mildly centrally peaked density pro- simulation was steady rather than impulsive.
file of outer radius 0.75 pc, and becomes more cen-
trally condensed over the course of the run. After 0.6
6. COMPARISONTO OBSERVATION:NGC 1333
Myrstarformationbeginsinearnest,andbetween0.9
and 1.2 Myr the stellar fraction increases from 6.5% Located within the Perseus molecular cloud, the
to 13.4% of the cloud mass. Li & Nakamura report NGC 1333 reflection nebula is the site of a vigor-
a final, three-dimensional rms velocity of 1.5kms−1 ous burst of clustered star formation within a dense
corresponding to σ = 0.85kms−1. molecular clump. The current population of at least
If one assumes that the gas has contracted by a 143 young stars is estimated to be 1-2 Myr old
fraction f of its initial radius during the period (Lada et al. 1996), and about ten molecular outflows
R
0.9-1.2 Myr, then ρ ≃ 10−19.4f−3gcm−3 and S ≃ and Herbig-Haro systems emerge from the most re-
0 R
10−66.6f−3cm−3s−1 in that period. Equation (17) cently formed objects. Lada & Lada (2003) estimate
then preRdicts σ = 1.7f Λkms−1, which is consis- a stellar mass of ∼ 79M⊙, based on the limiting K
R
−3/7 magnitude of 14.5. Distance estimates range from
tent with theLi & Nakamura results if Λ= 0.6f .
R D =210 pc to 350 pc, as discussed by Warin et al.
Contraction by f = 1/2 appears reasonable based
R (1996).
on their figure 2, in which case
Ridge et al. (2003) map the NGC 1333 core (and
many others) in 13CO(1-0), C18O(1-0), and C18O(2-
Λ≃ 0.8.
1), probing sequentially smaller and denser regions.
This seems entirely consistent with our expectations They derive masses from a large velocity gradient
from §2 that Λ ∼ 1. Note, however, that one should analysis; combining these with their reported veloc-
10
2.5
∞
2
104
1.5
) 103
Λl
/(
σt
1 100
10
0.5
0
10−3 10−2 10−1 100 101 102
r/l
Fig. 2.— Effect of finite clump radius on turbulence driven by collimated outflows (θ0 = 10−2) driven by stars drawn from
the Kroupa (2001) IMF. The thick curve represents an infinite uniform medium; the other curves are labeled by Iesc/I¯, which
determineshowmuchmomentumislost from afiniteclump. Circles marktheedgeoftheregion forfiducialparametersnH4 =1,
vc =40kms−1, SFRff =0.034, and Λ=1.
ity dispersions, and assuming that gas dominates This is entirely consistent with the observed value
the mass budget, we derive the virial parameter of 1.1kms−1, for precisely the same value of Λ we es-
(Bertoldi & McKee 1992) timated in §5. This result is entirely consistent with
the proposition that turbulence in the star-forming
5Rσ(R)2 220pc region of NGC 1333 has been regenerated by out-
α ≡ = (1.27,1.32,1.19) (40)
vir GM(R) D flows. The cluster outflow inferred by Warin et al.
(1996) could then legitimately be viewed as the mass
for 13CO(1-0), C18O(1-0), and C18O(2-1), respec- ejected by erupting jets, and the inflow detected by
tively. Since α is thought to vary, in strongly Walsh et al. (2006) might represent a return flow of
vir
self-gravitating regions, between 1.11 (for molecular gas ejected below the escape speed.
clouds; Solomon et al. 1987) and 1.34 (for massive, Although the value v = 40kms−1 (which was
c
magnetized molecular cores; McKee & Tan 2003), we justified in § 3 using the observational relations re-
adopt D = 220 pc (close to D = 212 pc adopted by ported by Richer et al. 2000) is in line with the mo-
Quillen et al. 2005). The embedded star cluster then mentum of the NGC 1333 outflow HH 7-11 as es-
haveaneffective radiusR = 0.39pc,scalingthevalue timated by Snell & Edwards (1981), Knee & Sandell
given by Lada & Lada (2003) to the closer distance; (2000) and Quillen et al. (2005) have inferred signif-
the enclosed mass is then 446M⊙, and the escape ve- icantly lower momenta for the NGC 1333 outflows.
locity is 3.1kms−1. The velocity dispersion at the Knee & Sandell estimate ∼ 1M⊙kms−1 per outflow
same radius is σ =1.1kms−1. for ten currently active outflows, and Quillen et al.
Can this velocity dispersion be maintained by the derive a similar momentum scale for the explosions
protocluster outflows? To evaluate the formulae in that open cavities in the gas. Given that the mean
§4.1 and §4.2, we adopt the Kroupa (2001) IMF stellar mass in the Kroupa (2001) IMF is 0.21M⊙,
and assume the current star formation rate equals this momentum scale corresponds to v ∼ 5kms−1.
c
79M⊙ per Myr. Other model parameters are Stot = Intheformalismof§4.2,thislowervaluewouldimply
10−65.8cm−3s−1, ρ = 10−18.9gcm−3, T = 20 K,θ = σ(R) = 0.47Λkms−1.
0 0
10−2, and Iesc = 1041.5gcms−1 (where we have used It is possible that, for strongly collimated out-
kρ = 1.35 to estimate cg = 1.1 in eq. [32]). For ref- flows in a real, magnetized star-forming region like
erence, these values give SFR ≃ 0.034 in the region NGC1333, Λ = 2.3 as required to make this
ff
of interest – virtually identical to the value SFR = lower estimate consistent with the observed value of
ff
0.038 we derive from the Krumholz & McKee (2005) 1.1kms−1. However it is equally probable that the
theory. We take aext = 0 to probe the influence of inference vc ∼ 5kms−1 is an underestimate: com-
outflows alone. pare this to the estimates in §3, and note that the
Using v = 40kms−1 and applying the prescrip- minimum value estimated by Richer et al. (2000, in
c
tion of §4.2 to predict the velocity dispersion at their §III.B) is ∼30kms−1. One solution to this dis-
R = 0.39pc, we find σ(R) = 1.12(Λ/0.8)kms−1.