aa r X i v : . [ a s t r o - ph . S R ] M a r Perspectives on Low-Mass Star Formation
Shantanu Basu [email protected]
Department of Physics and Astronomy, The University of Western Ontario, London, Canada.
Abstract
I review some recent work on low-mass star formation,with an emphasis on theory, basic principles, and unre-solved questions. Star formation is both a gravitationalfragmentation problem as well as an accretion prob-lem. Molecular cloud structure can be understood as afragmentation process driven by the interplay of turbu-lence, magnetic fields, and gravity (acting on either adynamical or ambipolar-diffusion time scale). This re-sults in a natural way to understand filamentary struc-ture as magnetic ribbons that have an apparent widththat scales differently than the Jeans length. Recentwork also shows that stellar mass accretion througha disk is episodic. We show through numerical simu-lations that bursts of FU Ori type may be clustered,since the clump that accretes to the center is tidallysheared apart in its last stage of infall. Finally, weutilize a simplified model of stellar mass accretion andaccretion termination to derive an analytic form for theinitial mass function that has a lognormal-like body anda power-law tail. This scenario is consistent with an ex-pectation of a larger number of substellar objects thanmay have been previously detected.
1. Introduction
Astronomers often divide the star formation processinto a handful of neat phases (e.g., Shu et al., 1987).We can for example think of star formation as pro-ceeding through: (1) the fragmentation of a molecularcloud, resulting in filaments, cores, and any kind oflocalized collapsing objects; (2) the collapse of thesecores to form star-disk systems; (3) the phase of diskaccretion and evolution, which includes companionand planet formation and jet/outflow launching; and(4) the termination of mass accretion (due to outflows,ejection, competition with other sinks of cloud mass,magnetic support of envelopes, or other means) thatdetermines the final mass of a star. In this paper Ireview some recent work and add commentary on thefirst, third, and fourth phases of star formation as de-fined here.
Proceedings of the Star Formation in Different Environ-ments , ICISE, Quy Nhon, Vietnam, 2016. PSFDE: volume01. Copyright 2016 by the author(s).
2. Fragmentation of Clouds: MagneticRibbon Model
Recent years have seen a renewed interest in filamen-tary structures in molecular clouds, thanks to the
Her-schel Space Observatory observations of dust emissionfrom molecular clouds (Andr´e et al., 2010). These ob-servations reveal a filamentary network of dust emis-sion that is present in both star-forming and non-star-forming clouds. This implies that the initial structureof molecular clouds may be an imprint of initial (likelyturbulent) conditions and exists before self-gravity ofthe cloud can make a major impact and lead to starformation. Furthermore, observations of a handful ofmolecular clouds in the Gould Belt have found thatthe average projected lateral width of the filamentarystructures are clustered around 0.1 pc, even thoughthe mean column density of the filaments vary by atleast two orders of magnitude.A mean width of filaments that clusters around ∼ . a ≃ c s / ( G Σ c ) ∝ Σ − c where c s isthe isothermal sound speed and Σ c is the central col-umn density. How are we to understand this result?On one hand it could be some kind of observationalbias. However, the resolution of Herschel is about 0.01pc at the distance of the Gould Belt clouds so we canrule out beam smearing. It is possible that the averag-ing technique can converge to a narrow range of meanwidths even if the filaments intrinsically have a widerange of widths along their spine (Panopoulou et al.,2017). One may even think of the filaments as an en-semble of collapsing cores that have narrow FWHMand wider regions that are not collapsing. What ex-plains the widths of the non-collapsing regions? This isprobably an imprint of the filament formation process.In Auddy et al. (2016) we proposed that the filamentsare actually magnetic ribbons, part of the scenario ofdynamically-oscillating molecular clouds supported bymagnetic fields due to a large-scale subcritical mass-to-flux ratio. In this scenario molecular clouds are sup-ported by magnetic fields, and the initial structure isinherited from initial conditions. If the initial condi- ow-Mass Star Formation tions are turbulent, perhaps inherited from a largerscale turbulent cascade, then filamentary structure isnaturally generated. These filaments differ from thosein cosmological simulations that are driven purely bythe gravity of the dark matter and collapse on a dy-namical time. Instead, these are quasi-equilibriumfilaments (really ribbons, with preferential flatteningin the direction of the mean magnetic field) that arein approximate force balance between turbulent rampressure and the pressure and tension of the magneticfield. Since the mass-to-flux ratio is subcritical, theribbon cannot collapse in the flux-freezing limit, andrebounds when the magnetic pressure equals the rampressure. If the initial turbulent compression occurson a characteristic scale L , Auddy et al. (2016) showthat the equilibrium lateral thickness of the resultingribbon is L = L " (cid:18) v t v A (cid:19) + 1 − , (1)where v t is the nonlinear flow speed and v A is thebackground Alfv´en speed. This thickness depends on L but is independent of density, unlike the Jeansscale. However, the ribbon is expected to be flattenedpreferentially along the magnetic field direction andhave a thickness H = c s / √ πGρ that is comparableto the Jeans length. In this model we ignore turbu-lent pressure effects in the direction of the magneticfield (Kudoh & Basu, 2003; 2006). Figure 1 showsa schematic picture of the ribbon formation, with aturbulent flow that is in the plane perpendicular toa mean magnetic field along which the cloud is flat-tened. An ensemble of such ribbons will be observedat a random set of viewing angles, each with a uniqueprojected width and line-of-sight column density. Fig-ure 2 shows the result of a synthetic set of projectedwidths generated for a random distribution of viewingangles.The magnetic ribbon scenario can explain why theobserved widths can: (1) cluster around an approxi-mately fixed width, since they are quasi-equilibriumobjects and not on a one way path of collapse; (2)why the widths are not wholly different that the Jeanslength, since one dimension has that width; (3) whythe observed polarization patterns imply a large scalemagnetic field perpendicular to the filament. However,an explanation for a preferred wavelength L for thebackground turbulence, of order 1 pc, remains an openissue.
3. Disk Accretion: Migrating EmbryoModel
Detailed self-consistent simulations of the formation ofcircumstellar disks from their parent cores and theirfurther evolution shows that the mass accretion rate˙ M at small scales near the star can differ considerablyfrom the mass infall rate onto the disk. In a series ofpapers (Vorobyov & Basu, 2005; 2006; 2010; 2015) wehave shown that ˙ M in the disk is episodic and char-acterized by bursts with ≥ − M ⊙ yr − that resultfrom gravitational instability. The instability leadsto fragmentation and the migration of the fragmentsto the protostar. The recurrent gravitational insta-bility occurs while there is significant envelope accre-tion onto the disk, which can temporarily reduce theToomre Q parameter to be less than the critical value.This paradigm has become the standard one for earlydisk accretion, as the mean and median luminosity ofprotostars in star-forming regions are lower by aboutan order of magnitude than predicted from the stan-dard spherical accretion models (Evans et al., 2009).This so-called “luminosity problem” can be resolvedby episodic accretion coupled with a slowly decliningmean accretion rate (Dunham & Vorobyov, 2012).Figure 3 shows the time evolution of ˙ M at both aninner radius (black solid lines) and an outer radius(red dashed lines) characteristic of envelope infall, inthree models with different initial core masses M core as labeled. The main elements of stellar mass accre-tion are evident in these plots. The low mass modelwith M core = 0 . M ⊙ illustrates all the phases. The˙ M near the center rapidly rises to a value ∼ c /G upon the formation of a first hydrostatic core, where c s is the isothermal sound speed. Subsequently thereis a smooth accretion until the formation of a cen-trifugal disk. Although this occurs ∼ yr afterthe formation of the first core, this is a model de-pendent effect. The central sink cell sets the timefor the start of disk accretion, since the centrifugalradius of infalling mass shells must reach the sinkcell radius of 6 AU for a disk to form in the simu-lation. Higher resolution but one-dimensional simula-tions that also include non-ideal magnetohydrodynam-ics (Dapp & Basu, 2010; Dapp et al., 2012) show thata disk actually forms immediately after the formationof a stellar core. All three panels of Figure 3 showsthat the disk accretion has a baseline rate that is wellbelow the rate of infall onto the disk but is punctuatedby bursts that temporarily increase ˙ M by one to twoor more orders of magnitude. This burst mode of ac-cretion continues until the envelope accretion, whichis a declining function of time due to a finite massreservoir, drops to negligible values. The top panel of ow-Mass Star Formation B =B Z L 2H ν t0 ν t0 yxz To observer φθ Figure 1.
Formation of a magnetic ribbon under the influence of ram pressure and the magnetic field. The cloud isflattened along the mean magnetic field direction and the compression is in one dimension in the perpendicular plane.The thick black arrow points to an observer located at a random orientation.
Observed Column Density [cm − ] O b s e r v e d R i bb o n W i d t h [ p c ] Mean L obs θ = 0 ° θ = 90 ° Figure 2.
Observed ribbon width L obs vs. observed column density N obs . Each blue dot corresponds to a magnetic ribbonwith intrinsic column density N and observing angle θ chosen randomly. The black dashed line is the mean ribbon widthfor the entire range of values of N obs . The black dotted line is the width when the ribbon is viewed at θ = 0 ◦ . The bluedotted-dashed line is the width for the side on view θ = 90 ◦ . ow-Mass Star Formation Figure 3 shows that the disk accretion continues afterthis time in a power-law manner. This later evolutionof ˙ M is set by internal transport mechanisms of thedisk, mainly gravitational torques arising from per-sistent nonaxisymmetric structure (Vorobyov & Basu,2007). The dependence ˙ M ∝ t − n with n ≃ M core have not yet settled into the self-similarregime during the times shown, as the envelope in-fall continues for a longer time. The mass dependenceof the rate of decline of the disk accretion rate is animportant observable and merits a careful comparisonwith models. Within the first ∼ Myr of accretion, theobjects with greater mass, accreting from more mas-sive envelopes, will show a more rapid decrease of massaccretion rate than lower mass accretors. This is be-cause the more massive object’s accretion rate is stillset by the envelope accretion rate onto the disk, whilethe lower mass object may have ended its phase ofenvelope-driven accretion and already settled into self-similar disk accretion. At later times, it is possible that˙ M may decline more rapidly for the low mass objects,as inferred in one observational study (Manara et al.,2012), due to photoevaporation of the inner disk dueto x-ray and UV stellar flux (Ercolano et al., 2014).An interesting new prediction based on high-resolutionmodeling (Vorobyov & Basu, 2015) was that a massaccretion burst of FU Ori magnitude could actuallyoccur with a series of secondary bursts. This is due tothe physics of clump infall that leads to the burst. Inthe late stages of infall to the center, the clump can besheared apart and include secondary fragments. Themass infall can occur in the form of a clustered burst.Figure 4 shows the late stages of infall of a clump asit approaches the center. A clustered burst has veryrecently been confirmed in the FU Ori object V346(Kraus et al., 2016).
4. Stellar and Substellar Masses: MLPdistribution
Despite claims that the observed core mass function(CMF) may be the explanation for the stellar (andsubstellar) initial mass function (IMF), there are road-blocks to the idea of a simple one-to-one mapping ofthe CMF to the IMF. The discovery of an ever growingset of substellar objects, down to as low as ≃ − M ⊙ ,for example by (Drass et al., 2016), imply that directcollapse scenario would have to produce extremely lowmass collapsing cores. The mean Jeans mass in amolecular cloud is already as high as ≃ M ⊙ .In the accretion scenario, there is still a CMF but it is not fundamentally important to determining the IMF.The exact shape and mean value of a CMF dependson a variety of physical factors as well as the obser-vational technique and definition of core boundaries.The CMF will inevitably be drawn toward a lognormaldistribution due to the multiple effects of turbulence,magnetic fields, gravity, and temperature fluctuations.However, it need not be the cause of the IMF, whichmay also resemble a lognormal for statistical reasonshaving to do with the Central Limit Theorem.An attractive scenario is to think of star formation asa killed accretion process. Since a protostellar seedor first hydrostatic core starts with a very low mass ≃ − M ⊙ (Larson, 1969), the formation of substellarand stellar mass objects can be thought of as arisingfrom a combination of mass accretion and a mecha-nism for termination of the accretion. Accretion ter-mination can arise from a variety of effects, includingejection from a multiple system of proto-objects in adisk (Basu & Vorobyov, 2012), outflows from the pro-tostar, and sweeping away of molecular cloud gas byfeedback from a nearby high-mass star. If the distribu-tion of accretion stopping times is f ( t ) = δ e − δt , rep-resenting equally likely stopping (ELS) in equal timeintervals, and the mass growth law is exponential, i.e., dm/dt = γm , then the resulting normalized pdf formasses after accretion termination is f ( m ) = α (cid:2) αµ + α σ / (cid:3) m − − α × erfc (cid:20) √ (cid:18) ασ − ln m − µ σ (cid:19)(cid:21) . (2)Here µ and σ are the mean and dispersion of a start-ing lognormal distribution of masses which then un-dergo accretion growth, and α = δ/γ is the dimen-sionless ratio of “death” rate to “growth” rate of pro-tostars. The exponential growth law of protostars isan approximation to a two-stage process where weenvision that the early stage accretion rate is nearlyconstant but later needs to increase rapidly if massivestars are to be formed. This is because of the relativelysmall age spread of stars in young clusters containingstars of widely varying masses (Myers & Fuller, 1993).This modified lognormal power-law (MLP) model hy-pothesizes that the initial distribution of accreting pro-tostellar seed masses is lognormal. However, a lognor-mal of very small dispersion σ approaches a delta func-tion, and we may also think of the initial distributionas being essentially a delta function at the mass of thefirst hydrostatic cores, ≃ − M ⊙ . This interpreta-tion is aided by a newly derived mass function of theOrion Nebula Cluster (Drass et al., 2016). Their fitof a lognornal to the low mass data yields parameters ow-Mass Star Formation Figure 3.
Mass accretion rates versus time for models with three different initial core masses, as labeled. The black solidlines are the accretion rates at the inner sink cell (at radius 6 AU) and the red dashed lines are the envelope infall ratesmeasured at 2000 AU. The arrows in the top panel mark the formation of the first hydrostatic core and the disk. ow-Mass Star Formation
Figure 4.
A clustered burst. The images zoom-in onto a fragment approaching the central star. The gas surface densityvalues are shown in the color bar in log g cm − . During the short time shown the infalling clump is sheared out by tidalforces and the mass infall occurs in multiple bursts. The red circle in the center corresponds to the central sink cell. that imply that the mode of the density function is ≃ . M ⊙ .
5. Magnetic Fields
The SFDE conference had a special session on mag-netic fields and important points about the observabil-ity of magnetic fields were discussed. Why are mag-netic fields important to star formation studies? Themost important potential role of magnetic fields wouldbe to ensure the observed low efficiency of star for-mation. This would primarily require that the mass-to-flux ratio in much of the cloud volume is subcrit-ical. So far there is no equivocal direct evidence ofthis, as magnetic field detections are difficult enoughin dense regions let alone in the low density regionsof clouds. A strong magnetic field could also explainfeatures like the magnetic ribbons described in this pa-per, and the striations that are observed perpendicularto large filaments (Tritsis & Tassis, 2016). They couldalso provide a conduit for rapidly spreading kineticenergy throughout the cloud (Kudoh & Basu, 2003;2006; Wang et al., 2010). Neutral-ion slip must existin molecular clouds since the neutrals cannot feel anyLorentz force but for a collisional interaction with ionsthat requires a relative drift between the two species.Is it worthwhile to look for direct evidence of neutral-ion drift? Observational comparison of individual neu- tral and ionic species line widths can be problematicdue to their differing (and uncertain) critical densi-ties and optical depths. It is even more problematicto interpret any alleged systematic difference in theneutral and ion line widths based on a linear theoryof Alfv´en waves in a uniform non-gravitating medium(Li & Houde, 2008). Since self-gravity is importantin molecular clouds, particularly in clumps and cores,there should be a systematically greater line width ofneutrals than ions due to unresolved gravitationally-driven ambipolar diffusion. This effect is typically onlya fraction of the sound speed on a core scale. How-ever, there is the possibility of high resolution obser-vations of the protostellar collapse zone revealing anambipolar diffusion shock at a typical distance ∼ AU from a young stellar object (Li & McKee, 1996;Contopoulos et al., 1998). In this region, the neutralinfall motions are supersonic and the neutral-ion driftspeed is also supersonic. The infall of neutrals, letalone ions, within an expansion wave (Shu, 1977), hasyet to be characterized observationally and comparedto theoretical models.
6. Final Thoughts
How is the research field of star formation faring? Wehave been encouraged by the organizers to express ouropinions. All areas of human activity follow trends, ow-Mass Star Formation − − . . . − . − − . − − .
50 log( m ) l og ( m f ( m )) lognormalMLP Figure 5.
Comparison of a lognormal density function with the MLP density function. The lognormal has parameters µ = − . σ = 0 . µ = − . σ = 0 .
5, and α = 1. − − − − − m ) l og ( m f ( m )) MLP N = 10 N = 10 N = 10 Figure 6.
The MLP function with parameters µ = − . σ = 1 . α = 1 . N as labeled. All histograms are binned in increments∆ log m = 0 .
2, where m is in units of M ⊙ . The analytic function is plotted as m f ( m ) where f ( m ) is the density function,and the histogram is the fractional number in each bin divided by ∆ ln m . ow-Mass Star Formation and this field is no exception. In the 1980s the hottopic was the singular isothermal sphere, the 1990ssaw the development of models with magnetic fieldsand then turbulence, the 2000s saw a resurgence of in-terest in the Bonnor-Ebert sphere and the core massfunction, and the 2010s have focused a lot on filamen-tary structure. The field may have settled into chas-ing “rabbits” and we appear to be in a quiescent eraof progress while the bursts of activity are happeningin related areas of planetary and substellar objects orthe application of star formation (e.g., star formationrate history and IMF) to the extragalactic and highredshift universe. Population III star formation maybecome a hot topic depending on the findings of the James Webb Space Telescope (JWST).There are still many outlets for research in star forma-tion studies. Although the core subject of molecularcloud structure may have stalled in my opinion, the bigoutstanding question is still to be solved. Can mag-netic fields or turbulence enforce a very low efficiencyof star formation in molecular clouds? If the formerthen we need to find evidence of subcritical mass-to-flux ratio in cloud envelopes. If the latter then we needto identify drivers of turbulence, either on the largescale or through stellar feedback. New high resolutionobservations that reveal ribbon-like structure or stri-ations are interesting in my view insofar as they canhelp distinguish between scenarios, e.g., magnetically-dominated or not.Stellar mass accretion is an interesting area with muchtheoretical and observational development in the lastdecade. The episodic accretion paradigm is now firmlyestablished, and can be further explored with the
Ata-cama Large Millimeter/submillimeter Array (ALMA)disk observations and the upcoming JWST ability toidentify numerous low mass and substellar objects.Even the IMF may be understood as essentially anaccretion process, and accretion models like the MLPcan be constrained by new observations of the substel-lar and low mass IMF.
Acknowledgments
SB thanks Eduard Vorobyov and Takahiro Kudoh, andstudents Sayantan Auddy, Deepakshi Madaan, andPranav Manangath for valuable collaboration and dis-cussions that contributed to this review paper. SB wassupported by a Discovery Grant from NSERC.