Turbulence decay and cloud core relaxation in molecular clouds
aa r X i v : . [ a s t r o - ph . S R ] D ec Turbulence decay and cloud core relaxation in molecular clouds
Yang Gao , , Haitao Xu Chung K. Law , Center for Combustion Energy, Tsinghua University, Beijing 100084, China Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Department of Mechanical and Aerospace Engineering, Princeton University,Princeton, New Jersey 08544, USA Max Planck Institute for Dynamics and Self-Organization (MPIDS),37077 G¨ottingen, GermanyReceived ; accepted 2 –
ABSTRACT
The turbulent motion within molecular clouds is a key factor controlling starformation. Turbulence supports molecular cloud cores from evolving to gravita-tional collapse and hence sets a lower bound on the size of molecular cloud coresin which star formation can occur. On the other hand, without a continuous ex-ternal energy source maintaining the turbulence, such as in molecular clouds, theturbulence decays with an energy dissipation time comparable to the dynamictimescale of clouds, which could change the size limits obtained from Jean’s crite-rion by assuming constant turbulence intensities. Here we adopt scaling relationsof physical variables in decaying turbulence to analyze its specific effects on theformation of stars. We find that the decay of turbulence provides an additionalapproach for Jeans’ criterion to be achieved, after which gravitational infall gov-erns the motion of the cloud core. This epoch of turbulence decay is defined ascloud core relaxation . The existence of cloud core relaxation provides a morecomplete understanding in the competition between turbulence and gravity onthe dynamics of molecular cloud cores and star formation.
Subject headings: hydrodynamics: turbulence — ISM: molecular clouds — star:formation
1. Introduction
Turbulent motion, magnetic field and gravity are governing factors on thedynamics of star formation in molecular clouds (Pudritz 2002; Ward-Thompson 2002;McKee & Ostriker 2007). Turbulence and its effects on molecular clouds and star formationhave been studied since the pioneer work of Chandrasekhar (1949, 1951a,b); a recentreview on this topic is given in Mac Low & Klessen (2004). Broadly speaking, turbulencehas two competing effects on star formation. On one hand, large-scale turbulence is themain driving mechanism that creates dense cloud cores (Chandrasekhar 1951a; Larson1981; Kritsuk et al. 2013), which incubate star formation. On the other hand, turbulentmotion within the cloud cores provides additional support against gravitational collapse(Chandrasekhar 1951b; Bonazzola et al. 1987; L´eorat et al. 1990), which hinders starformation. A full understanding of the dynamics of start formation therefore requires acomplete analysis of the effects of turbulence.In this work, we examine the resistance to gravitational collapse due to the in-cloud-core turbulence. Most of the previous analyses of this effect were based on the theory ofcompressible or incompressible, fully developed and statistically stationary turbulence, i.e.,turbulent flows maintained by continuous external energy supplies (McKee & Ostriker2007). Most of the energy sources that drive the turbulent motion in a molecular cloud,on the other hand, are neither uniform nor consistent (Mac Low & Klessen 2004), whichmeans that the turbulence in any particular cloud core cannot be continuously maintainedand the turbulent flow will gradually slow down, i.e., the turbulent energy will decay.Additionally, numerical simulations showed that the energy decaying time of a typicalturbulent flow in molecular clouds is smaller or comparable to the dynamic timescale of starformation (Stone et al. 1998; Mac Low et al. 1998). These two factors above suggest thatstar formation in cloud-cores actually occurs in an environment of decaying turbulence. 4 –Therefore, the effect of turbulent energy decay should be taken into account when analyzingstar formation in cloud-cores, which is what we address here.This paper is organized as follows: turbulence driving mechanisms in molecular cloudsare first reviewed, followed by presentation of the scaling law of decaying turbulence, withan analysis of its effect on Jeans’ criterion. The epoch of cloud-core relaxation is thenproposed, and results of our scaling analysis are discussed and concluded.
2. Driving mechanisms of turbulence in molecular clouds
Currently, a general consensus on star formation is that large-scale ( ∼ −
100 pc)turbulence leads to the clustering of dense regions and subsequently the formation of stars.The energy dissipation time of the large-scale turbulence is of the order of ∼ yr, which iscomparable to the free fall timescale of a cloud (Stone et al. 1998; Ossenkopf & Mac Low2002; Offner et al. 2008).Various energy sources that could trigger turbulent flows in molecular clouds have beenproposed and then corresponding driving scales of turbulence have been numerically studied(Genzel et al. 1998; Mac Low & Klessen 2004; Joung & Mac Low 2006; Brunt et al.2009). The magneto-rotational instability (MRI) is an efficient mechanism that coupleslarge-scale galactic rotation with turbulent motions in star forming clouds, whose energyinput into turbulent motion is, however, about two orders of magnitude less than theobserved value. Similarly, turbulent motion due to gravitational instability (GI) is notenergetic enough to drive star formation. These two instabilities (MRI and GI) thereforelikely serve as basic driving mechanisms that contribute to only a small portion of theobserved turbulence in molecular clouds. Protostellar jets and outflows are good sources oflocal turbulence drivers that can affect their surrounding cloud environment but are too 5 –small in dimension to account for the large-scale turbulence observed. Massive stars canaffect the cloud environment significantly by intense radiation, but only a small portionof the radiation energy is converted to turbulent motion. On the other hand, althoughwinds from massive stars can be more energetic, the population of massive star is too smallto make a major contribution, especially when compared to that of supernova explosiondiscussed below.It is suggested by Mac Low & Klessen (2004) that supernova explosion is the dominantturbulence driving mechanism, with sufficient energy input rates to trigger the turbulenceobserved in molecular clouds. Following their analysis, we assume the supernova explosionrate in the Galaxy (100 pc star formation scale height and 15 kpc in radius) to be (50 yr) − ,which then gives an estimate that the supernova explosion rate in a typical star formingcloud with a diameter of 100 pc to be ∼ (10 − yr) − . This result means that there is atypical time lag of 10 − yr between two successive supernova explosions in a molecularcloud, which is comparable or slightly larger than the star formation time of several 10 yr.As a result of a supernova explosion, shock waves sweep the gas and dust in the molecularcloud, cluster them into dense cores through particle collisions, and initiate the turbulentmotion in these cores as well as in the entire cloud (Joung & Mac Low 2006). To give ageneral picture of the molecular clouds and the dense cloud cores discussed here, the typicallength scale of cloud cores is l core ∼ . l ∼
100 pc; and the core density is of the order ρ core ∼ × cm − , which is muchlarger than the density of diffuse regions of a molecular cloud ρ cloud ∼
10 cm − (both arenumber densities of molecules). After the shock waves pass by, the decay laws governs theevolution of turbulence in the molecular cloud. 6 –
3. Turbulence decay in molecular clouds and cloud cores
The decay of turbulent energy, and the associated variation of energy spectra inturbulent flows without an external maintaining force is a classical problem in fluidturbulence research. It can be dated back to the classical paper by von K´arm´an & Howarth(1938) and has been investigated by Kolmogorov (1941); Batchelor & Townsend (1948a,b);Heisenberg (1948), among many others. Recent studies based on terrestrial fluid exper-iments improved the understanding of turbulence decay properties (Kurian & Fransson2009; Krogstad & Davidson 2010, 2011); while in astrophysical investigations, numericalsimulations of star forming clouds confirmed these scaling relations for decaying turbulenceand expanded the results to flows with magnetic fields (Biskamp & M¨uller 1999; Mac Low1999; Cho et al. 2003).In this work we adopt the following decay law for incompressible turbulence, given asEq. (A1) in Krogstad & Davidson (2011), E ( t ) E = u ( t ) u = (cid:16) An u tl (cid:17) − n , (1)where E ( t ) = u ( t ) / t either after the start of thedecaying, or the termination of external forcing, which corresponds to the passing of theshock wave in the case of energy supply by the supernova explosion. Furthermore, E , u and l are the initial values, i.e., at t = 0, of turbulent energy, fluctuating velocity andintegral scale, respectively, and A is a dimensionless number, typically between 1 / / A = 1 / n has not been uniquely determined. Currently availabledata and theories suggest that it should be between 1 and 2 (Biskamp & M¨uller 1999;Krogstad & Davidson 2011), with many recent experimental evidences supporting thatit is close to 1 . ∼ (1 + t/t ) − n law and the slow down ofthe turbulent speed follows ∼ (1 + t/t ) − n/ accordingly, where t = l /u is the turbulencedecay time scale.For a (giant) molecular cloud with typical length scale l ∼
100 pc and mass M ∼ M ⊙ , if all the energy released through a typical supernova explosion( ∼ E SN = 10 erg) is converted to the turbulent energy of the cloud, the turbulentfluctuating velocity in the molecular cloud is u = p E SN /M = 30 km / s. Thecorresponding characteristic energy decay time of the turbulence in the cloud is t = l /u = 3 × yr, which is consistent with the simulation result of Stone et al.(1998). Then for a typical dense cloud core of l core ∼ . u core = u ( l core /l ) / = 3 km / s once it is formed, assuming the Kolmogorovscaling for incompressible turbulence in the inertial range (cf. Kolmogorov 1941) u ∝ l / . (2)Accordingly, the decay time scale for turbulence in such a cloud core is t core = l core /u core =3 × yr, which is much smaller than the decay time of the turbulent motion in the wholecloud. Regarding the above estimates, it is noted that the initial length scale of the cloudcore l core could vary for different cores, resulting in different core masses and turbulentspeeds; while the initial local density of the cloud core ρ core could also be different. The cases in which 10% and 1% of the supernova explosion energy is converted to theturbulent energy are considered in Section 5. Turbulence in molecular clouds are actually compressible thus the spectra index couldbe different from the Kolmogorov scaling law. Discussions on the effect of compressibleturbulence can be found in the last section of this paper. 8 –Although bounded in the gravitational potential of the large cloud, these densecores form their own local potential fields and can be relatively isolated from other cores(Ward-Thompson et al. 2007). Due to the density difference between the dense cores andcloud diffuse regions, ρ core /ρ cloud ∼ × , turbulent motion in the diffuse regions around acloud core can hardly generate strong fluctuations inside the core as a result of mass fluxconservation ( ρu = const. ). This means that the turbulent motion within the dense cores isessentially not affected by turbulence of the diffuse cloud after the initial fluctuation, so thedense cores experience the decay of turbulence in a relatively isolated sense. It has alreadybeen noticed that local star-formation behaviors are different for dense cores of differentproperties (see e.g., McKee & Ostriker 2007): 1) cores that have sufficiently high internalturbulent energy compared to their self-gravity potential will re-disperse and cannot formstars; and 2) cores whose turbulent energy are low enough compared to the gravitationalenergy will collapse under gravity and form star(s). What we will show in the followingsection is that the decay of turbulent motions in the cores may allow some type 1) densecores to eventually evolve to gravitational collapse.
4. Jeans’ criterion in cloud cores with decaying turbulence
Jeans’ criterion for a turbulent cloud core to be gravitationally unstable to perturbationsof wave number k was derived in Chandrasekhar (1951b); Bonazzola et al. (1987): k < πρ c Gc + u c , (3) As an estimate, once cloud cores are formed, a turbulent speed of u cloud = 30 km/s inthe cloud can only result in u core = 0 .
006 km/s flows in the dense core, which is much smallerthan the inside-core turbulent motion of several km/s. 9 –where c is the speed of sound, u c the turbulent speed of the cloud core and G thegravitational constant. Hereafter variables with subscript c represent properties of cloudcores. Equation (3) shows that the gravitational instability is a long-wave instability. Ina cloud core, the core diameter l c limits the longest wave to k = 2 π/λ ∼ π/l c , which isthe first unstable mode when a cloud core becomes unstable due to changes in conditions.If we take a typical cloud-core temperature of ∼
10 K, corresponding to a sound speed of c = 0 . l c > πρ c G h c + 13 u c (cid:16) An u c l c t (cid:17) − n i , (4)in which u c denotes core turbulent speed at the beginning of turbulence decay. Accordingto the turbulence spectra of Eq. (2), the core turbulent speed is related to the core size by u c = u ( l c /l ) / , where u and l denote initial turbulent speed and size of the cloud, thenJeans’ criterion (4) can be further expressed as l c > πρ c G h c + 13 u (cid:16) l c l (cid:17) / (cid:16) An u l / c l / t (cid:17) − n i . (5)Equation (5) shows that the resistance to gravitational collapse in a cloud core has twocontributions: the thermal and turbulent parts, which are respectively the first and secondterms on the right hand side (RHS) of Eq. (5). For initially gravitationally stable cloudcores, the decay of the turbulent motion will diminish the second term and could causethe criterion being satisfied at a later time, hence providing an additional approach for thecloud core to become gravitationally unstable.Therefore the existence of turbulence decay transforms Jeans’ criterion, Eq. (5), intotwo criteria: For a cloud core to be unstable at the beginning of the decaying turbulence( t = 0), the core diameter has to be greater than l c = s πρ c G h c + 13 u (cid:16) l c l (cid:17) / i . (6) 10 –On the other hand, if there is long enough time ( t → + ∞ ) for the turbulent motion todecay, the core diameter has only to be greater than l ′ c = r πρ c G c (7)for it to be gravitational unstable and to collapse at some later time. These two criteria canbe inferred as the critical diameter for core collapse with initial turbulence ( l c ) and theminimum diameter required for core collapse without turbulence ( l ′ c ), respectively. Notethat the two criteria (6) and (7) do not depend on n , the index of the turbulence decay rate,being only affected by the sound speed c and the initial turbulent speed u c = u ( l c /l ) / in cloud cores. Based on these two criteria, three types of cloud-core evolution exist. 1)Cloud cores with diameters l c > l c are gravitationally unstable and will collapse to formstar(s). 2) Cloud cores with diameters in the range l ′ c < l c < l c will not collapse initially,but can evolve to be gravitationally unstable after a period of turbulence decay (turbulentspeed decreases) and eventually collapse to proceed in further star formation. 3) Very smallcloud cores with diameters l c < l ′ c that are stable and do not collapse.
5. Cloud core relaxation
When considering the time needed for a turbulent core to decay to being gravitationallyunstable, it is more informative to re-write the criterion into the following form: t > nl c Au ( l c /l ) / h(cid:16) u ( l c /l ) / l ρ c G/π − c (cid:17) /n − i . (8)For typical cloud cores with c = 0 . ρ c = 5 × cm − , if the initial turbulentspeed of the (giant) cloud of diameter l = 100 pc is u = 30 km/s as estimated in Section 2,the time needed for a core of length scale l c to evolve to be gravitationally unstable can beeasily obtained from Eq. (8) and is illustrated in Fig. 1 (solid line), in which the decay index 11 –is taken as n = 1 .
2. Figure 1 shows that for cloud cores with sizes l c > l c = 3 . t = 0yr, which means that these cores directly collapse as a result of gravity; while for those withsizes l c < l ′ c = 0 . t = + ∞ yr, which means they will never experience gravitationalcollapse. In between the two, cloud cores of sizes l ′ c < l c < l c become gravitationallyunstable after a period of turbulence decay and collapse to form stars. The epoch after theformation of these cores but before the initiation of gravitational collapse can be defined asthe relaxation of cloud cores, during which the turbulent intensity within the cloud coresdecreases. As inferred from Fig. 1 (solid line), the relaxation time for the cloud coreswith sizes between 0 . to 10 yr, which is comparable to thefree fall time for star formation, t ff = [3 π/ (32 Gρ c )] / = 4 × yr (McKee & Ostriker2007). In addition, cloud cores with smaller sizes need longer relaxation times to becomegravitationally unstable.As the total energy released in a supernova explosion may not be fully convertedto the turbulent energy of a star forming cloud, the initial turbulent speed of the entirecloud could be less than 30 km/s as in previous calculation. Observations also suggestthat the turbulent speeds in clouds of ∼
100 pc diameter are typically less than or around10 km/s (e.g., Larson 1981; McKee & Ostriker 2007). Assuming 10% or 1% of thesupernova explosion energy is converted to the turbulent energy of the molecular cloud,the corresponding turbulent speed is u ∼
10 km/s and u ∼ l c , the minimum diameter of a cloud core that can directly evolve to be gravitationalunstable without experiencing turbulence decay, becomes much smaller when the initialturbulent speed decreases; while for cloud cores smaller than l c , the relaxation takes arelatively shorter time (several 10 yr) than in more turbulent clouds. The plots in Fig. 1clearly indicate that the decay of turbulence leads to the existence of a relaxation epoch for 12 –cloud cores with diamater l ′ c < l < l c before they experience gravitational collapse. Evenwhen Jeans’ criterion has been satisfied and gravitational collapse begins, the decay ofturbulent motion will also continue and the turbulent speed decreases until another drivingprocess, such as star winds from nearby, newly formed massive stars, happens. It is alsoto be noted that turbulence can be enhanced as a result of the adiabatic heating in thecompression of a cloud core (Robertson & Goldreich 2012; Murray & Chang 2014). Thisprocess cloud be considered as a self-driven mechanism of turbulence in cores as well, whichmay delay their gravitational collapses as a consequence.Supernova driven turbulence has been presumed in the above analyses, while othersources reviewed in Section 2 will also generate fluid turbulent motions. Although insmaller scales and not energetic enough to be the main energy source for star formations(Mac Low & Klessen 2004), these mechanisms may serve as more frequent energy inputs inlocal star formations. In this sense, the core relaxation discussed above may be interruptedby these local turbulence drives. Also note that magnetic field is not included in theanalyses; the existence of which may lead to different turbulent energy spectra and mayslow the decay of turbulence (e.g., Biskamp & M¨uller 1999; McKee & Ostriker 2007).Consideration of magnetic effects in future works may quantitatively change the turbulencedecay and core relaxation behaviors discussed here.
6. Conclusion and discussions
Based on the scaling laws of decaying turbulence, Jeans’ criterion on the stability ofcloud cores specifies two critical core sizes: l c , if turbulence exists in the core, and l ′ c ( < l c ), when only the thermal effect is considered. For cloud cores with large enoughsizes, ( l c > l c ), they can be gravitationally unstable once formed. For smaller cores thatdo not satisfy Jeans’ criterion at their formation but have sizes between the two criteria 13 –( l ′ c < l c < l c ), they can evolve to be gravitationally unstable through the relaxationof turbulent energy. For cores with even smaller sizes ( l c < l ′ c ), they can never becomeunstable to gravity even with an infinite long epoch of relaxation. The process of turbulencedecay before gravitational collapse is defined as the relaxation of cloud cores, which lasts fora period of 10 to 10 yr for typical conditions in star forming clouds. The existence of corerelaxation provides an additional approach for cloud cores to evolve to be gravitationallyunstable thus collapse.Typical values of cloud core turbulent speed, length scale, density and temperature, aswell as supernova rate and the (giant) cloud diameter are used here for an intuitive pictureof the core relaxation; these values could vary from one star-forming cloud core to anotherby as large as even one or two orders of magnitude.It is also noted that self-similar scaling laws of decay (Krogstad & Davidson 2011)and the Kolmogorov spectra of incompressible turbulence (Kolmogorov 1941) are adoptedhere, with the analytical results apply in the “inertial range” where energy transfers fromlarger to smaller scales with negligible influences from driving or viscosity. Furthermore,although the existence of shock waves and the magnetic field in realistic molecular cloudsmay alter the turbulence spectra, the energy decay rates of turbulence for compressibleand incompressible clouds with or without magnetic fields are quite comparable as foundin numerical simulations (see the reviews and discussions in Mac Low & Klessen 2004;McKee & Ostriker 2007). Nevertheless, the effects of compressibility, magnetic field as wellas the anisotropy of turbulence on its decay properties, and consequently on the cloud-corerelaxation needs to be further investigated.This work was supported by the Center for Combustion Energy at Tsinghua Universityand by the National Science Foundation of China grant 51206088. YG acknowledgesadditional support from the Tsinghua-Santander Program for young faculty performing 14 –research abroad. HX acknowledges the support from the Max Planck Society and theGerman Science Foundation (DFG) through the project A7 of the Collaborative ResearchCenter (CRC) 973 “AstroFIT”. 15 – REFERENCES
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Protostars and Planets V , (ed., Reipurth, B., et al.,University of Arizona Press, p.33-46),This manuscript was prepared with the AAS L A TEX macros v5.2. 18 – l c (pc) t r e l a x ( y r) u =30 km/s, l =100 pcu =10 km/s, l =100 pcu =3 km/s, l =100 pcl c0 =3.0 pc(Giant) Molecular cloud propertiesRelaxation time for cloud cores withdifferent radiusl c0 ’=0.1 pct ff =0.4 × yr Fig. 1.— Relaxation time needed for a cloud core embodied in a (giant) molecular cloudof diameter l = 100 pc containing typical initial turbulent motions of u = 30, 10 and 3km/s to become gravitationally unstable. The abscissa is the scale of cloud core and theordinate is the relaxation time needed before gravitational collapse. For small cloud coreswith l c < l ′ c , the relaxation time is infinity and the core will always relax and can notbecome gravitationally unstable; for large cloud cores with l c > l c , the relaxation time iszero and the core immediately goes to gravitational collapse once formed; for cloud corewith diameter intermediate of the two, it needs a time t relax for the turbulence to decay andeventually become gravitationally unstable. The horizontal dash-dot line denotes the freefall time t ffff