Perturbation growth in accreting filaments

0.28 MB·

by  Seamus D. Clarke

Mon.Not.R.Astron.Soc.000,1–7(2002) Printed26February2016 (MNLATEXstylefilev2.2)
Perturbation growth in accreting filaments
S. D. Clarke1∗, A. P. Whitworth1 and D. A. Hubber2 3
6
1 1School of Physicsand Astronomy, Cardiff University,Cardiff, CF24 3AA, UK
0 2UniversityObservatory Munich, Ludwig-Maximilians-University Munich, Scheinerstr.1, D-81679 Munich, Germany
2 3Excellence Cluster Universe, Boltzmannstr. 2, D-85748 Garching, Germany

b
e
F

5
2 ABSTRACT

] We use smoothed particle hydrodynamic simulations to investigate the growthof
A
perturbations in infinitely long filaments as they form and grow by accretion. The
G growth of these perturbations leads to filament fragmentation and the formation of
. cores. Most previous work on this subject has been confined to the growth and frag-
h
mentationofequilibriumfilamentsandhasfoundthatthereexists apreferentialfrag-
p
mentation length scale which is roughly 4 times the filament’s diameter. Our results
-
o show a more complicated dispersion relation with a series of peaks linking perturba-
r tion wavelength and growth rate. These are due to gravo-acoustic oscillations along
t
s the longitudinal axis during the sub-critical phase of growth. The positions of the
a peaks in growth rate have a strong dependence on both the mass accretion rate onto
[
the filament and the temperature of the gas. When seeded with a multi-wavelength

2 density power spectrum there exists a clear preferred core separation equal to the
v largest peak in the dispersion relation. Our results allow one to estimate a minimum
1 age for a filament which is breaking up into regularly spaced fragments, as well as an
5 averageaccretion rate. We apply the model to observations of filaments in Taurus by
6 Tafalla & Hacar (2015) and find accretion rates consistent with those estimated by
7
Palmeirim et al. (2013).
0
. Key words: ISM: clouds - ISM: kinematics and dynamics - ISM: structure - stars:
2
0 formation
6
1
:
v
i 1 INTRODUCTION tational constant and T is the gas temperature (Ostriker
X
1964).
Theprevalenceoffilamentarystructuresacrossawiderange
r TheHerschelGouldBeltSurveyhasshownthatthema-
a ofscales(Schneider& Elmegreen1979;Lada, Alves & Lada
jorityofstar-formingcoresarefoundwithinfilamentswhich
1999;Myers2009;Andr´eet al.2010;Palmeirim et al. 2013;
havesuper-criticallinedensities,whilesub-criticalfilaments
Beutheret al. 2015) has lead to several papers study-
are sterile (Andr´eet al. 2010). It has also been shown the-
ing their structure, stability, fragmentation and collapse
oretically that due to their geometry, filaments are apt to
(Ostriker 1964; Larson 1985; Inutsuka& Miyama 1992;
fragment; small-scale perturbations can readily collapse lo-
Burkert & Hartmann2004;Pon, Johnstone & Heitsch2011;
cally before global longitudinal collapse overwhelms them
Fischera & Martin2012;Heitsch2013;Clarke & Whitworth
(Pon, Johnstone & Heitsch 2011). This suggest a paradigm
2015).
in which filaments are formed inside molecular clouds, and
Ithasbeenshownthatafilament’slinedensity(defined
thedensestofthesefilamentsthenbecomesuper-criticaland
asthemassperunitlength)determinesthefilament’sradial
go on to fragment into cores.
stability:filamentsbelow acritical linedensitywill not and
Inutsuka& Miyama (1992) have analysed how small-
cannot bemade tocollapse radially, those with line density
scaleperturbationsgrowandcollapseinanequilibriumfila-
abovethecritical valuewill. Foran isothermal filament the
ment;theyderiveadispersion relation linkingperturbation
critical line density is
wavelength with perturbation growth rate. They find that
µ = 2a2o ≈ 16.7M pc−1 T , (1) perturbations are unstable when their wavelength is larger
CRIT G ⊙ 10K thantwicethefilament’sdiameter,andthereexistsafastest
(cid:18) (cid:19)
growingmodeatapproximately4timesthefilament’sdiam-
where ao is the isothermal sound speed, G is the gravi-
eter.Whenthelinedensityofanisothermalfilamentexceeds
the critical value by a small amount, the perturbations do
∗E-mail:[email protected] not have time to grow before global radial collapse takes
2 S. D. Clarke, A. P. Whitworth and D. A. Hubber
2 NUMERICAL SETUP
-1.0
Thesimulationspresentedinthispaperareperformedusing
0 the Smoothed Particle Hydrodynamics (SPH) code GAN-
DALF(Hubberetal.inprep).Thesimulationsinvokeboth
0.6 0.6 self-gravity and hydrodynamics, with the barotropic equa-
-2m] -1.2 -2m] tion of state
y [g c y [g c T(ρ)=T 1+ ρ 2/3 . (2)
ensit 0.4 pc] 0.4 -1 ensit O (cid:18)ρBARO(cid:19) !
n d z [ n d Here T is the initial temperature and ρ =
olum -1.4 olum 10−14 gcOm−3isthecriticaldensityatwhichthegaBsAcRhOanges
c c
from beingapproximately isothermal toapproximately adi-
abatic. We present two sets of simulations, one with T =
0.2 0.2 10KandtheotherwithT =40K;thisresultsinisotherOmal
-2 sound speeds of ∼ 0.19kmO/s and ∼ 0.37km/s respectively,
assumingsolarmetallicity.Grad-hSPH(Price & Monaghan
-1.6 2004) is implemented, with η=1.2, so that a typical parti-
cle has ∼56 neighbours. Sink particles are implemented as
-0.05 0 0.05 -0.05 0 0.05 described in Hubber,Walch & Whitworth (2013) using the
x [pc] x [pc] sink creation density,ρ =10−12gcm−3.
SINK
Figure 1. The column density projected onto the x-z plane for The computational domain is open in the x and y di-
the fiducial case (i.e. A = 0.2, λ = 0.2pc, T = 40K and µ˙ = rections, but is periodic in z, the long axis of the filament
70M⊙pc−1Myr−1) at two different times. OOn the left, at t = (see Wunsch et al. in prep, for the implementation of the
0.15Myr,thefilamenthasformedonthez axis,itissub-critical modified Ewald field). This in effect allows us to study the
and confined by the ram pressure of the accreting gas. On the
perturbationgrowthinaninfinitelylongfilament,andhence
right,att= 0.55Myr,thefilamenthasbecomesupercriticaland
toignore thecomplicating effectsofglobal longitudinal col-
iscontractingradially;atthesametimetheseededperturbations
lapse (Clarke & Whitworth 2015).
havebecomesitesoflocalcollapse.Createdusingthevisualisation
To generate the initial conditions, a cylindrical settled
toolSPLASH(Price2007).
glassofparticleswithauniformdensityisstretchedsothat
it reproducesthe density profile,
ρ r 2πz
ρ(r,z) = O O 1+Asin . (3)
r λ
(cid:18) (cid:18) (cid:19)(cid:19)
Here ρ is the density at r = 0.1pc, A is the amplitude
O O
place; in this case, it is thought that fragmentation occurs of theperturbation and λ is theperturbation’s wavelength.
at the point when isothermality breaks and radial collapse The initial velocity field is,
is halted.
Though perturbationsinequilibriumandsuper-critical v=−v rˆ. (4)
filaments have been studied before (e.g. Larson 1985; O
Inutsuka& Miyama 1992, 1997; Freundlich,Jog & Combes Combined, these density and velocity profiles result in a
2014),non-equilibriumfilamentshavebeenneglected.How- setup which can be characterised by A, λ and the influx
ever, it is unlikely that when a filament first forms it is of mass perunit length, µ˙ =2πρ r v .
O O O
in equilibrium or has a super-critical line density; it is far The resulting perturbed cylinder of particles has a ra-
more likely that a filament will be sub-critical when it first dius of r = 1pc, and a length, L = mλ, where m is
max
forms and mass then accretes on to it until it becomes un- the largest integer that satisfies L 6 1pc. This provides a
stable and fragments. As filaments and perturbations form sufficientlylargecomputationaldomaintoallowustostudy
together and co-evolve it is important to understand how a wide range of plausible perturbation wavelengths, while
density perturbations behave during the sub-critical phase, maintaining good resolution.
especiallysoifoneattemptstolinkthedensityperturbation Westressthatthereisnofilamentpresentattheoutset.
power spectrum to the core mass function (Inutsuka 2001; Att=0allthematerialisflowingradiallyinwards,towards
Roy et al. 2015). the z-axis. As soon as the simulation starts, an accretion
In this paper, we present numerical simulations of ini- shock forms on the z-axis and then propagates outwards;
tially sub-critical perturbed accreting filaments, in order the filament is the dense material inside this shock. Our
to investigate the dispersion relation between perturbation initial setup should be seen as a simple approximation to
wavelengthandperturbationgrowthrate,andtocompareto the formation of a filament, a locally convergent flow in a
therelationfoundbyInutsuka& Miyama(1992)forequilib- globally turbulentfield,whichisabouttocreateafilament.
rium filaments. In section 2, we detail the simulation setup Weconsiderperturbationswhichareinitiallysmall,tak-
and the initial conditions used; in section 3, we present the ingA=0.1or0.2.Theperturbationwavelengthλisvaried
results of these simulations; in section 4, we discuss their between0.05pcand0.5pc.Thelowerlimitisduetoresolu-
significanceandcomparetopreviouswork;andinsection5, tion concerns, and theupper limit to ensure that at least 2
we summarize ourconclusions. wavelengthsfitwithinthecomputationaldomain.Themass
Perturbation growth in accreting filaments 3
1.90
A=0.2 1.15
A=0.1
1.85
1.10
1−Myr] 1.80 wthrate
Growthrate[ 11..7705 Normalisedgro1.05
1.00 µ˙=70M pc−1Myr−1
1.65 µ˙=100M⊙ pc−1Myr−1
⊙
Stretchedµ˙=100M pc−1Myr−1
⊙
1.600.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.905.05 0.10 0.15 0.20 0.25 0.30 0.35
Perturbationwavelength[pc] Perturbationwavelength[pc]
Figure 2. The dispersion relation between perturbation wave- Figure 3. The dispersion relation between perturbation wave-
length and perturbation growth rate for T = 40K, µ˙ = length and normalised growth rate for the fiducial case
70M pc−1Myr−1 andA=0.2(thefiducialcOase)andalsowith (µ˙ = 70M pc−1Myr−1), and for the case with µ˙ =
⊙ ⊙
A=0.1.Therenolongerexistsasinglelocalmaximumat4times 100M pc−1Myr−1. The normalised growth rate is defined as
⊙
thefilament’sdiameter,insteadthereexistsaseriesofpeaksand gˆλ =gλ/g0.05pc. The fiducial case (solid blue line) and the case
troughs. The initial amplitude of the perturbations does not af- with a higher accretion rate (dashed red line) are out of phase,
fecttheshapeofthedispersionrelation,ratheritonlyaffectsthe the peaks in the dispersion relation do not line up. The higher
magnitudeofthegrowthrate. accretion rate has caused the relation to be squeezed in the x-
direction. The green dotted line is the result of stretching the
highaccretionlineassumingthedispersionrelationtakestheform
flux per unit length is varied between ∼10M⊙pc−1Myr−1 gλ=f(λ/τCRIT),whereτCRIT isthetimeatwhichthefilament
and ∼100M pc−1Myr−1, in line with the accretion rates becomes supercritical. The peaks in the stretched dispersion re-
⊙ lationnowlineupwiththosefromthefiducialcase.
estimated observationally by Palmeirim et al. (2013) and
Kirk et al. (2013). We define the critical time as the time
at which the filament reaches the critical line density and
becomes radially unstable, τ ∼µ /µ˙.
CRIT CRIT 1.16
Afterthefirstsinkforms,weonlyfollowthesimulation
for a further 0.01 Myr, since we are principally interested 1.14
in the dynamics that lead to instability, rather than the 1.12
subsequentcollapse.
Thesimulationsarerunwith3millionparticlesperpar- rate1.10
sec,givingamassresolutionofbetween1.5 ×10−3M and wth1.08
t4he×s1m0a−l3lMwa⊙v.elSeuncghthhipgehrtruersboalutitoionns aisrenewceelslsarersyoltvoede⊙nesvuerne malisedgro1.06
at low densities. The artificial viscosity needed to capture Nor1.04
shocks is known to overly dampen short wavelength oscil-
lations on theorder of thesmoothing length. Therefore, we 1.02 TO=40K
haveinvokedtimedependentartificialviscosityasdescribed 1.00 TSOtre=tc1h0eKdT =10K
O
in Morris & Monaghan (1997) to further lessen the effect 0.908.05 0.10 0.15 0.20 0.25 0.30 0.35
of artificial viscosity on short wavelength oscillations. Tests Perturbationwavelength[pc]
withdifferentnumbersofparticlesshowthatthesimulations
haveconverged at 3 million particles per parsec. Figure 4. The dispersion relation between perturbation wave-
length and normalised growth rate for the fiducial case (T =
O
40K), and for the case with T =10K. The normalised growth
O
rate is defined as gˆλ = gλ/g0.05pc. The fiducial case (solid blue
3 RESULTS line) and the case with the lower temperature (dashed red line)
areoutof phase. Thelowertemperature has caused the relation
The perturbations seeded in the initial density profile sur- to be squeezed in the x-direction. The green dotted line is the
vivetheapproximatelycylindricalaccretionshockbounding result of stretching the T = 10K line assuming the dispersion
the filament, and form a radially sub-critical perturbed fil- relation takes the form gλO=f′(λ/aO), where aO is the isother-
ament of typical width ∼0.1pc (figure 1a). The rest of the mal sound speed. The peaks inthe stretched dispersionrelation
gas continues to accrete onto the filament until it reaches nowlineupwiththosefromthefiducialcase.
the critical line density and the perturbed sections collapse
(figure1b).
Wetakethetimeat whichthefirst sinkparticleforms,
τ , as a proxy for the perturbation growth time. Sink
SINK
particles are used in numerical simulations to reduce com-
4 S. D. Clarke, A. P. Whitworth and D. A. Hubber
Figure 5. The normalised volume-density profile, ρ/hρi, along Figure6.Thelongitudinaldensity(bluesolidline)andvelocity
thez-axisforthefiducialcaseattwodifferenttimes,t=0.05Myr (greendashedline)profilesatt=0.45Myrforthesimulationwith
(bluesolidline)andt=0.30Myr(greendashedline).Astanding λ = 0.2pc, T = 40K, µ˙ = 70M pc−1Myr−1 and A = 0.2.
O ⊙
gravo-acoustic wave issetupalongthe filament’s length; the lo- This is the time at which the perturbations are just becoming
cationsoftheinitialdensitypeaksaretheanti-nodesofthewave. self-gravitating and the filament is close to becoming globally
Thedensitypeaks andtrough switchafter0.25Myrhas passed, super-critical. The red dotted line is the v = 0km/s line, which
i.e.halfanoscillationperiod. we include to help the reader see the converging and diverging
regionsalongthefilament.Thevelocityfieldisoutofphasewith
the density field, and the main converging flows are positioned
where the troughs of the density profile are. Because of this the
majority of the gas is moving away from the density peaks just
putational cost; they are used to replace dense, collaps-
ascollapseisabouttobegin.Therearesmallconvergingflowsat
ing bound regions that will inevitably become stars. Thus
the central density peaks because these peaks have just become
the earlier a sink particle forms the faster the perturbation
self-gravitating.
became unstable. Specifically we define the perturbation
growth rate as g =1/τ . Figure 2 shows thedispersion
λ SINK
relation linking perturbation wavelength with perturbation
4 DISCUSSION
growth rate for T = 40K, µ˙ = 70M pc−1Myr−1 and
O ⊙
A = 0.2 (the fiducial case) and also with A = 0.1. It is ev- The primary characteristic of the sub-critical phase is that
ident that the inclusion of the non-equilibrium sub-critical thermalpressureforcesaregreaterthangravitationalforces.
phase has dramatically changed the relationship from the Thus, during this phase the evolution of the perturbations
onederivedbyInutsuka& Miyama(1992).Therenolonger within the filament is dominated by acoustic oscillations,
existsasinglelocalmaximumat4timesthefilament’sdiam- and standing waves are set up along its length. Figure 5
eter.Insteadthedispersionrelationappearstohavetwofea- shows the volume-density profile along the z-axis at two
tures: longer wavelength perturbations tend to grow faster, different times. The density peaks oscillate with a period
and superimposed on this there is a series of peaks and λ/a ∼0.50Myr. Thus, between 0.05 Myr and 0.30 Myr, a
O
troughs. half period has elapsed, and the density peaks and troughs
Varyingthemass accretion rate wefindthat theshape haveswitched.
of the dispersion relation is unchanged, but it is rescaled It is the presence of these acoustic density oscillations
in the x-direction (figure 3). As the mass accretion rate is thatgivesrisetotheoscillations seeninthedispersion rela-
increased,andthetimetakenforthefilamenttobecomesu- tion (figure 2). The depressed growth rate at certain wave-
percritical is decreased, the dispersion relation is squeezed lengths is due to the fact that in these cases the time at
and the peaks in the growth rate move to shorter wave- whichradialcollapsebegins,τ ,coincideswiththepoint
CRIT
lengths. Let us consider that there exists a function, f(x), in the oscillation at which the peak of the density pertur-
which transforms theparameters, λ,a and µ˙,intotheob- bation is being dispersed and reformed at a trough (figure
O
served dispersion relation g . Figure 3 shows that the dis- 6).Thuswhenthegasmakingupthedensitypeakbecomes
λ
persion relation takes the form g = f(λµ˙) = f(λ/τ ), self-gravitating it is moving outwards. This has two conse-
λ CRIT
where τ is the time at which the filament becomes su- quences:thegasmustfirstbedeceleratedandturnedaround
CRIT
percritical. beforetheperturbationcollapseslongitudinallyaswellasra-
Whentheinitialtemperatureofthegas, andbyexten- dially,andtheamplitudeofthedensitypeakisdecreasedas
sion the isothermal sound speed, is varied we see the same masshasmovedintowhatoncewasatrough.Thesetwoef-
typeof behaviouraswhen themass accretion rateis varied fectscausethecollapseoftheperturbationtoproceedmore
(figure 4). As the temperature of the gas is decreased the slowlyoncethefilamenthasbecomesuper-critical,resulting
dispersion relation is squeezed and the peaks in the growth in a significantly lower growth rate. Conversely, the peaks
ratemovetoshorter wavelengths.Wefindthatthestretch- arecausedbythelongitudinalmotionsconvergingonaden-
ing goes as T−1/2 or as a−1, so we can write thedispersion sity peak at thetime of radial collapse, when theperturba-
O O
relation as g =F(λ/a τ ). tion is reaching its peak amplitude.
λ O CRIT
Perturbation growth in accreting filaments 5
4.8×10−19 4.0×10−16
4.6 3.5
3.0
4.4
3−m] 3−m] 2.5
nsity[gc 4.2 nsity[gc 2.0
de 4.0 de
ume ume 1.5
Vol 3.8 Vol
1.0
3.6 0.5
3.40.0 0.2 0.4 0.6 0.8 1.0 0.00.0 0.2 0.4 0.6 0.8 1.0
z[pc] z[pc]
Figure 7. The longitudinal density profile for one of the α = −1.6 simulations at t = 0.05Myr (left) and t = 0.56Myr (right). The
densitypeaks (cores)att=0.56Myr,areroughlyperiodicallyspaced,havinganaverageseparationof0.24pc.
One can write an equation to predict the positions of dentonthesoundspeeditisnotclearyetiftheywillsurvive
the peaks in the dispersion relation by considering when in the presence of turbulence. However, it has been shown
the oscillation period and the time at which the filament that filaments are decoupled from the supersonic medium
becomessupercritical,τ ,areinphase.Forthetwotobe surroundingthemandappeartoonlycontainsub-sonicmo-
CRIT
in phaserequires tions(Hacar & Tafalla2011),whichmayleadtoonlyasmall
correction to the model presented here.
Asnotedabove,thedispersionrelation(figure2)shows
τ ∼nτ , (5)
CRIT HALFOSCILLATION that there is an upwards trend in growth rate with in-
where n is a positive integer and creasing wavelength, longer wavelength perturbations grow
faster than shorter wavelength perturbations. This can
λ
τ = . (6) be explained in terms of the Jeans length. The Jeans
HALFOSCILLATION 2aO length is applicable because the perturbations are roughly
Hence spherical as they approach instability, as also observed by
Inutsuka& Miyama(1997).Foraperturbationtobeunsta-
2τ a
λ ∼ CRIT O. (7) ble and collapse it’s length must be greater than the Jeans
DOMn n length, λ = a (π/Gρ)1/2. Smaller wavelength perturba-
J O
At T = 40K the isothermal sound speed, a , is tions must therefore reach higher densities before they be-
O O
0.37km/s, and taking τ as 0.45 Myr we expect thedis- come unstable, and this delays their growth.
CRIT
persionrelationtoshowpeaksatλ =0.33,0.17,0.11,...
PEAK
corresponding to n = 1,2,3,.... We see the n 6 6 peaks
in figure 2 where equation 7 predicts. The dominant wave-
length, the one that grows most quickly, is the n = 1 peak
4.1 Multi-wavelength perturbations
(figure2).
This dominant wavelength, λ , is defined by a reso- We now apply our analysis to filaments which are seeded
DOM
nance between the timescale, τ on which the line den- with perturbations at multiple wavelengths, in order to in-
CRIT
sity of the filament approaches the critical value for radial vestigate whether the dispersion relation continues to hold
collapse, and the timescale on which the longitudinal os- true. We use initial conditions informed by the work of
cillations of the perturbation complete a half oscillation, Roy et al.(2015),whofindthatthepowerspectrumofden-
τ , so that both motions act to enhance the sity perturbations in interstellar filaments is well described
HALFOSCILLATION
perturbation - i.e. both radial and longitudinal flows are bya power law with an index of α=−1.6 ± 0.3.
converging. We have performed a set of simulations whose initial
Fromequation7wecanseethatifthetemperature,and longitudinaldensityprofileischaracterisedbysuchapower
hencethesoundspeed,a ,arekeptconstant,andthemass law.Weperformtenrealisationswithanindexofα=−1.3,
O
influx per unit length, µ˙, is increased, the dominant wave- tenwithα=−1.6andtenwithα=−1.9.Soastocompare
lengthdecreasesbecauseτ hasbeenreduced(seefigure the results to the dispersion relation (figure 2) in section
CRIT
3).Converselyifτ iskeptconstant,andthetemperature 3, we take T = 40K, the computational domain is set to
and sound speedCaRrIeTdecreased, the dominant wavelength 1pc, µ˙ =70MO pc−1Myr−1, the maximum amplitude per-
⊙
again decreases, butnowbecausethetimescale tocomplete turbationhas A=0.2, thesameradial density andvelocity
half alongitudinal oscillation hasbeen increased (see figure profiles are used, and the k = 1 mode has an amplitude
4). of 0. We set the amplitude of the k = 1 mode to zero as
Astheoscillationsareacousticinnatureandaredepen- it corresponds to a wavelength of 1pc, beyond the range
6 S. D. Clarke, A. P. Whitworth and D. A. Hubber
of perturbation wavelengths we considered in the previous
section.
Figure7showsthelongitudinaldensityprofileforoneof α=−1.6
the α=−1.6 simulations at t=0.05Myr and t=0.56Myr 10 αα==−−11..93
(just before the first sink forms). The density peaks are
8
roughly periodic at t=0.56Myr despitethere being no ob-
viousindication ofitin theinitial conditions.Thissuggests bin
that there exists a preferential length scale for fragmenta- per 6
tion. mber
We determine the core separations for each set of sim- Nu 4
ulationsandplotthehistograms infigure8.Separationdis-
tances below 0.05pc have been removed; in a number of
2
simulations afew coresform veryclosetogether, withsepa-
rations ∼0.02pc, and these clusters of cores are then sepa-
rated bymuch greater distances. 00.15 0.20 0.25 0.30 0.35 0.40 0.45
Changingtheindexofthepowerdoesnotappeartoaf- Corespacing[pc]
fectthedistributionofcoreseparations,allthreearesharply
peakedat∼0.3pc.Asthevalueofαdoesnothaveastrong Figure 8.Histograms showing the distribution of core spacings
inaset of 30simulations.The bluelineshows the core spacings
effect on the distribution we combine the data from all 30
for the ten simulations where α = −1.6, the green line shows
simulations, this leads to a sample of 114 spacings with a
the core spacings from the ten simulations where α=−1.3 and
mean of 0.296 pcand a standard deviation of 0.070 pc.
theredlineshowsthecorespacingforthetensimulationswhere
The peak in core separations corresponds to the n=1 α = −1.9. Spacings below 0.05pc were removed; in a number
peakinthedispersion relation(figure2);eventhoughthere of simulations a few cores form very close together, ∼ 0.02pc,
isinitiallygreaterpowerinothermodes,itisthewavelength theseclustersarethenseparatedbymuchgreaterdistances.The
with thefastest growth ratewhichdeterminesthefragmen- change inpower index does not affect the distribution, all three
tation length scale. distributions arestrongly peaked at ∼0.3pc. Consideringall 30
Oursimulationssuggestthatonecouldestimatealower simulations together, a sample of 114 spacings, the distribution
hasameanof0.296pcandastandarddeviationof0.070pc
agelimitforafilamentwhichisfragmentingperiodicallyby
measuringtheaveragecoreseparation distance.Equation 7
can bere-written as,
and sets up standing gravo-acoustic oscillations, leading to
aseries ofoscillations inthedispersion relation linkingper-
λ
τ >τ ≃ CORE. (8) turbation growth rate and perturbation wavelength.
AGE CRIT 2a
O The fastest growing mode in an evolving filament is
This is a lower limit on the filament’s age, as we no longer simply linked to the diameter of a filament
do not know how much time has elapsed between when (as shown to be the case for equilibrium filaments by
the cores formed and when the filament is observed. We Inutsuka& Miyama 1992). Instead it is dependent on the
can also estimate the average accretion rate that the fila- temperature and the accretion rate onto the filament, be-
ment experienced during its assembly, µ˙ = µ /τ = cause the fastest growing wavelength is the one for which
CRIT CRIT
2a µ /λ .Afilamentwhichhascloselyspacedcores there is a resonance between the timescale on which the
O CRIT CORE
is likely to have experienced a very high accretion rate and filament becomes super-critical and the period of the lon-
vice versa. gitudinal oscillations. We have shown that this dependence
Tafalla & Hacar (2015) find that there exists a pref- holds when filaments are seeded with a multi-wavelength
erential core separation of ∼ 0.2pc in the sub-filaments density power spectrum. Moreover, the results are insensi-
making up the L1495/B213 complex in Taurus. Taking tive to the exact power law index in the range of indices
a gas temperature of 10 K and assuming solar metallic- (−1.36α6−1.9) observed by Roy et al. (2015).
ity the sound speed is 0.19km/s. This results in a mini- These results allow observers to constrain the age of
mum age for these sub-filaments of 0.53Myr. The critical a filament which is breaking up into regularly spaced frag-
line density at this temperature is 16.7M pc−1. There- ments (e.g. Tafalla & Hacar 2015; Beuther et al. 2015) as
⊙
fore the average accretion rate during the filament’s for- well as its accretion history.
mation was ∼ 32M pc−1Myr−1, in agreement with the
⊙
accretion rate inferred observationally by Palmeirim et al.
(2013), 27−50M pc−1Myr−1.
⊙ 6 ACKNOWLEDGMENTS
SDC gratefully acknowledges the support of a STFC post-
graduate studentship. APW gratefully acknowledges the
5 CONCLUSIONS
support of a consolidated grant (ST/K00926/1) from the
Filamentfragmentationisacomplexphenomenon.Wehave UKScience&TechnologyFacilitiesCouncil.DAHacknowl-
shown that the sub-critical accretion phase in a filament’s edges the support of the DFG cluster of excellence “Origin
evolutionstronglyinfluencesthegrowthoftheperturbations and Structure of the Universe”. We also thank the anony-
which lead to fragmentation. During the sub-critical accre- mous referee for their very useful comments which helped
tionphasethethermalpressuretermdominatesovergravity to improve this paper. This work was performed using the
Perturbation growth in accreting filaments 7
computationalfacilities oftheAdvancedResearchComput-
ing at Cardiff (ARCCA) Division, Cardiff University.
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0.15 0.20 0.25 0.30 0.35 0.40 0.45
Core spacing [pc]