Starless Cloud Core L1517B in Envelope Expansion with Core Collapse
aa r X i v : . [ a s t r o - ph . S R ] A ug Received 2010...; accepted 2011...; published...
Preprint typeset using L A TEX style emulateapj v. 8/13/10
STARLESS CLOUD CORE L1517B IN ENVELOPE EXPANSION WITH CORE COLLAPSE
Tian-Ming Fu ,Yang Gao , Yu-Qing Lou Received 2010...; accepted 2011...; published...
ABSTRACTVarious spectral emission lines from star-forming molecular cloud core L1517B manifest red asym-metric double-peaked profiles with stronger red peaks and weaker blue peaks, in contrast to the oft-observed blue-skewed molecular spectral line profiles with blue peaks stronger than red peaks. Invokinga spherically symmetric general polytropic hydrodynamic shock model for the envelope expansion withcore collapse (EECC) phase, we show the radial flow velocity, mass density and temperature struc-tures of self-similar evolution for L1517B in a dynamically consistent manner. By prescribing simpleradial profiles of abundance distribution for pertinent molecules, we perform molecular excitation andradiative transfer calculations using the publicly available RATRAN code set for the spherically sym-metric case. Emphatically, spectral profiles of line emissions from the same molecules but for differentline transitions as well as spectra of closely pertinent isotopologues strongly constrain the self-similarhydrodynamics of a cloud core with prescribed abundances. Our computational results show that theEECC model reproduces molecular spectral line profiles in sensible agreement with observational dataof Institut de Radioastronomie Millim´etrique (IRAM), Five College Radio Astronomical Observatory(FCRAO) and Effelsberg 100 m telescopes for L1517B. We also report spatially resolved observationsof optically thick line HCO + (1 −
0) using the Purple Mountain Observatory (PMO) 13.7 m telescope atDelingha in China and the relevant fitting results. Hyperfine line structures of NH and N H + transi-tions are also fitted to consistently reveal the dynamics of central core collapse. As a consistent modelcheck, radial profiles of 1.2 mm and 850 µ m dust continua observed by IRAM 30 m telescope andthe Submillimeter Common-User Bolometer Array (SCUBA), respectively, are also fitted numericallyusing the same EECC model that produces the molecular line profiles. L1517B is likely undergoingan EECC shock phase. For future observational tests, we also predict several molecular line profileswith spatial distributions, radial profile of sub-millimeter continuum at wavelength 450 µ m, as well asthe radial profiles of the column density and visual extinction for L1517B. Subject headings: dust, extinction — hydrodynamics — ISM: clouds — line: profiles — radio contin-uum: ISM — shock waves INTRODUCTION
Observations over past two decades provide effec-tive diagnostics to examine dynamic properties ofmolecular clouds where star formations are takingplace in relatively early phases. These observa-tions, including molecular spectral line profiles (e.g.Benson & Myers 1989; Tafalla et al. 2002; Aguti et al.2007), radial profiles of (sub)millimeter radio continua(e.g. Motte, Andr´e & Neri 1998; Shirley et al. 2000;Stanke et al. 2006; Nutter & Ward-Thompson 2007),and infrared dust extinctions (e.g. Alves, Lada & Lada2001), can be utilized separately and/or in combinationto reveal several key hydrodynamic cloud features, suchas core collapses, envelope expansions, travelling shocksand turbulence in the early evolution of star-formingcloud cores.On a relatively independent track, the theoret-ical model framework based on self-similar hydro-dynamic collapses has achieved substantial develop-ment since late 1960s (e.g. Bodenheimer & Sweigart Department of Physics and Tsinghua Center for Astrophysics(THCA), Tsinghua University, Beijing 100084, China. Center for Combustion Energy and Department of ThermalEngineering, Tsinghua University, Beijing 100084, China. Department of Astronomy and Astrophysics, the Universityof Chicago, 5460 South Ellis Avenue, Chicago, IL 60637, USA. National Astronomical Observatories, Chinese Academy ofScience, A20, Datun Road, Beijing 100012, China. + (1 − + (3 − CO(2 − ),H CO(2 − ), CS(2 −
1) and CS(3 −
2) from thiscandidate source clearly show red asymmetric spec-tral line profiles (e.g. Tafalla et al. 2004, 2006), whichlikely reveal the existence of an EECC shock dy-namic phase. Moreover, millimeter and sub-millimetercontinuum mappings serve as independent constraintson radial density and temperature profiles of L1517B(e.g. Tafalla et al. 2004; Kirk, Ward-Thompson & Andr´e2005) for the same EECC shock model used for fittingmolecular line profiles.Our motivation is to demonstrate specifically that thetheoretical explanation of red profiles based on generalpolytropic EECC solutions with collapses, expansionsand shocks (Gao & Lou 2010), appears grossly consis-tent with available observations for starless cloud coreL1517B. To further constrain and verify the EECC shockmodel, we present the result of a 5 × ′′ steps)spatially resolved HCO + (1 −
0) molecular line profile ob-servation using the 13.7 m telescope of Purple MountainObservatory (PMO) at Delingha in Qinghai Province ofChina. We also report the model fitting results of thesespatially resolved spectra on the basis of the same self-similar EECC shock model and show that the EECCshock dynamic phase will influence molecular line pro-files observed at different lines of sight (LOSs) from thecloud, rather than simply affects the central spectral lineprofiles alone. We also predict spatially resolved profilesfor several other molecular emission lines, sub-millimetercontinuum at 450 µ m, and the column density distribu-tion of L1517B for future observational tests.This paper is structured as follows. The backgroundinformation and our motivation are introduced here inSection 1. In Section 2, we specify model parametersand physical properties for L1517B using the frameworkof general polytropic EECC hydrodynamic shock model(Wang & Lou 2008) without magnetic fields. Model fit-tings of central spectra from several molecular line tran-sitions as well as spatially resolved spectral line profilesfor the HCO + (1 −
0) transition of L1517B are shown andanalyzed in Section 3. We also venture to predict otherspectral line profiles with spatial resolutions. In Sec-tion 4, we show millimeter and sub-millimeter continuaas further proof-tests to the same EECC shock dynamicmodel. Column number density and visual extinctionsare predicted in Section 4.3 and we conclude in Section5 with speculations. Details of analysis are summarizedin Appendices A and B for a convenient reference. PROPERTIES OF STARLESS CLOUD CORE L1517B
Earlier Studies on Cloud Core L1517B
The candidate source L1517B as a star-forming cloudcore is located in the Taurus complex, at an estimateddistance of ∼
140 pc from us (Elias 1978). Accordingto the radio continuum mappings and molecular line in-tensity mappings (Tafalla et al. 2002, 2006), cloud coreL1517B has a radius of ∼ ′′ (corresponding to ∼ . × AU at a distance of ∼
140 pc) and a center iden-ECC for Star-Forming Cloud Core L1517B 3tified at the right ascension α J2000 = 04 h m . s andthe declination δ J2000 = 30 ◦ ′ ′′ . Millimeter and sub-millimeter radio continuum data of L1517B have beenacquired at 1.2 mm wavelength using the Institut de Ra-dioastronomie Millim´etrique (IRAM) 30m telescope byTafalla et al. (2002, 2004), and at wavelengths of 850 µ mand 450 µ m using the Sub-millimeter Common-UserBolometer Array (SCUBA) on the James Clerk MaxwellTelescope (JCMT) by Kirk, Ward-Thompson & Andr´e(2005). Model analyses of the 1.2 mm continuum havebeen done in two ways in the literature. One involvesa presumed form of the number density radial profile n / [1 + ( r/r ) α ] (where n = 2 . × cm − is the cen-tral number density, r is the radius, r = 35 ′′ is thereference radius and α = 2 . T = 9 . ∼ . ∼ . × cm − in the inner region.Information has also been drawn from the 850 µ m con-tinuum observation of L1517B (e.g. Kirk et al. 2005), in-cluding a temperature estimate of ∼
10 K, a full width athalf maximum (FWHM) diameter of ∼ . − .
028 pc(by a 2D Gaussian fit to the 850 µ m data), a central vol-ume number density of ∼ × cm − and a total cloudmass of ∼ . M ⊙ within the 150-arcsec aperture. Ra-dio continuum mappings on shorter wavelengths (850 µ mand 450 µ m in Kirk, Ward-Thompson & Andr´e 2005) re-veal more substructures (thus not that spherically sym-metric as compared to the mapping of 1.2 mm continuumemissions) and may be used to constrain theoretical mod-els (see subsection 4.2). Polarization observations for 850 µ m and 450 µ m emission continua are performed usingSCUBA by Kirk, Ward-Thompson & Crutcher (2006),from which a magnetic field strength of ∼ µ G in thiscloud core is inferred. For the moment, magnetic field isnot included in our current model formulation.Various molecular transition lines from L1517B havebeen observed using the IRAM 30 m telescope byTafalla et al. (2004, 2006), and spectral line profile fit-tings based on a static gas sphere close to the centerwith an empirical density profile, a constant tempera-ture and an ad hoc outer ( r > AU) flow velocity gra-dient are performed therein. They also argued for otherscenarios, such as rotation and pure contraction with cer-tain cooling mechanisms near the center to generate suchasymmetric profiles (e.g. Tafalla et al. 2004). There areinconsistencies in such empirical approach from the the-oretical perspective as also discussed in Gao, Lou & Wu(2009). Nevertheless, these model fittings might providegross information for L1517B. In Tafalla et al. (2004),they declare the existence of internal motions of the or-der of ∼ . − in addition to turbulence, which weshall see in the following analysis might be related tothe cloud systematic infall and expansion motions pre-dicted by our general polytropic EECC shock hydrody-namic model. Another important result is the findingof molecular abundance drops towards the cloud center TABLE 1Six independent physical scaling parameters anddimensionless self-similar EECC model parameters
Scaling k / (km s − . ) t (yrs)Value 66 . . × Parameters γ a A B b x Values 1.2 10.02 4.58 1 . a γ is the polytropic index of the EoS of a general polytropic gas. b Two integration constants A and B are mass and velocity pa-rameters in the asymptotic solution for x → ∞ [see eqns. (A11) inAppendix A] adopted as the ‘boundary condition’. c x is the dimensionless upstream location of an outgoing shock. (e.g. Tafalla et al. 2006), which contributes to molecularline profile structures and will also be adopted in spectralline profile fittings of our model analysis (see subsection3.4 for details). Self-similar EECC Shock Model Parameters
We adopt the general polytropic self-similar EECChydrodynamic shock model of Wang & Lou (2008) tosimultaneously fit both the (sub)millimeter radio con-tinuum emissions and several available molecular lineprofiles, especially those profiles with red skewed peaksand with spatial resolutions. The general polytropicmodel can semi-analytically and numerically describeself-similar behaviors of the hydrodynamic evolution,i.e., a time-evolving molecular cloud preserves its basicstructural profiles (e.g. the density, temperature, veloc-ity and pressure profiles remain similar to their initialprofiles), of a quasi-spherical polytropic gas under self-gravity with the specific entropy conserved along stream-lines (see Appendix A for basic nonlinear hydrodynamicpartial differential equations (PDEs), the self-similartransformation, notations, and definitions therein). Agas thermal temperature of ∼
10 K (e.g. Tafalla et al.2002; Kirk, Ward-Thompson & Andr´e 2005) and a typi-cal cloud radius of ∼ . − .
028 pc Kirk et al. (2005)are considered for setting pertinent physical scalings inour self-similar EECC shock model. As L1517B appearsin an early evolutionary phase of protostar formation, wemay estimate a characteristic infall age of about ∼ × yrs for this cloud core according to Kirk et al. (2005)and Myers (2005). We also adopt an outer radius of ∼ . × AU for our cloud core model of L1517B, byreferring to the observed 150 ′′ radius (e.g. Tafalla et al.2002) at an estimated distance of ∼
140 pc (e.g. Elias1978).Using these empirical information of L1517B, wechoose the self-similar model parameters and the phys-ical scalings of time t as well as sound parameter k aslisted in Tables 1 and 2, of which the sound parameter k is derived from the following empirical length scale k / t n ∼ = 10 AU . (1)The number density scale of L1517B cloud core for hy- To explore the origin of “red profiles”, we actually reduce thenumber of free model parameters in our data fittings by requiring aconstant specific entropy everywhere (i.e. n + γ = 2 is introduced),corresponding to the conventional polytropic cloud core. This con-straint is not necessary in the theoretical framework of our generalpolytropic model (Wang & Lou 2008). Fu, Gao and Lou
TABLE 2Derived physical scaling k and dimensionless self-similarEECC hydrodynamic shock model parameters Scaling k / (km s − . ) a Value 66 . n q b m v α x v α − .
17 0 .
70 1 .
85 0 .
73 1 . a The downstream sound parameter k = ( x /x ) k . b Parameter q ≡ n + γ − / (3 n −
2) is determined by the poly-tropic index γ and the scaling index n = 2 − γ adopted in theself-similar EECC hydrodynamic shock model; this corresponds toa constant specific entropy everywhere. c The reduced central point mass m is obtained by solving theself-similar hydrodynamic equations numerically and by matchingthe central free-fall solution (see Appendix A for details). d Parameters x (regarded as independent in Table 1), x , v , v , α and α are the dimensionless reduced upstream and down-stream shock locations, velocities and densities respectively, andare obtained from hydrodynamic shock jump conditions [see eqns.(B2) − (B4)] in the self-similar shock model in Appendix B. drogen molecule H is taken as [eq (A6) in Appendix A]14 πGµm H t ∼ = 10 cm − , (2)where we adopt the mean molecular weight µ = 2 .
29 asused by Harvey et al. (2003). This number density scaleis one order of magnitude smaller than the empirical val-ues for the central number density ∼ − × cm − (e.g. Tafalla et al. 2002, Kirk et al. 2005), as the densityradial profile of our dynamic model increases rapidly to-wards the core center (see Fig. 1).With these specified scalings, physical variables includ-ing radius, radial flow velocity, number density and ther-mal temperature of L1517B can be expressed in termsof the dimensionless self-similar variable x and reduceddependent variables of x using eqns (A5) − (A7), viz. r = 1 . × x AU , (3) r = 1 . × x AU , (4) u = 0 . v ( x ) km s − , (5) u = 0 . v ( x ) km s − , (6) N = N = 3 . × α ( x ) cm − , (7) T = 7 . α ( x ) γ − m ( x ) q K , (8) T = 7 . α ( x ) γ − m ( x ) q K . (9)The enclosed mass and the central mass accretion rate[see eqns. (A6) and (A8) in Appendix A] are M = 0 . m ( x )(3 n − M ⊙ , (10) M = 0 . m ( x )(3 n − M ⊙ , (11)and ˙ M = 1 . × − m M ⊙ yr − , (12)where subscripts 1 and 2 refer to physical variables onthe immediate upstream and downstream sides of theoutgoing shock front, respectively. As required by themass conservation across a shock surface, we confirm that M | x = x = M | x = x = 2 . M ⊙ by a direct numerical Radial flow velocity T ( K ) R (10 AU)
L1517B
Temperature
Dynamic parametersx =1.87A=10.02, B=4.58 R shock =1.98x10 AU Expansion region - u ( k m s - ) Collapse region R inf =1.68x10 AU Number density N ( c m - ) shock waveshock waveshock wave M tot =3.89 M , M =0.034 M Fig. 1.—
Structures of cloud core L1517B. From top to bottomare radial profiles of radial flow velocity in unit of km s − (positivevalues for radial infall), number density in unit of 10 cm − andtemperature in unit of Kelvin (K), respectively. The abscissa isradius R in unit of 10 AU in a logarithmic scale. Infall radius R inf (= 1 . × AU) and shock radius R sh (= 1 . × AU) ex-pand at speeds of ∼ .
021 km s − and ∼ .
25 km s − , respectively.Variables exhibit discontinuities across the shock front. Other pa-rameters for this general polytropic self-similar EECC shock solu-tion are summarized in Table 3. The radial density profile of ourdynamic polytropic sphere with expanding envelope and free-fallcollapsing core thus consists of two parts connected by an expand-ing stagnation surface, of which the inner part can be described as ρ ∝ r − / [eq. (A12)] and the outer part as ρ ∝ r − /n [eq. (A11)](see the middle panel here). This broken power law radial densityprofile is a typical feature of starless cores (e.g. Caselli et al. 2002). computation with model parameters. To simplify physi-cal units of variables, we have already substituted valuesof k, t and n as listed in Tables 1 and 2 into eq. (A8).All dynamic model parameters in Table 1 are chosenin the procedure of data fitting of molecular spectrallines and dust emission continua from observations asdescribed in Sections 3 and 4. In particular, red profiles,e.g. HCO + (1 −
0) and HCO + (3 −
2) transitions, appearsuggestive of an expanding envelope in L1517B; this mo-tivates us to invoke a self-similar polytropic EECC shocksolution. Meanwhile, in order to fit the molecular spec-tral line profiles of these emissions, we introduce an out-going shock in the EECC model to avoid the outer en-velope of this molecular cloud expanding too fast. Thisshock emerges as an expanding flow rushes into the enve-lope (see the top panel of Fig. 1). We have systematicallyexplored a wide range of model parameters (includingpolytropic index γ , asymptotic behavior characterizedby mass and velocity parameters A and B , and locationor speed of the outgoing shock) to identify the best-fitEECC shock model. The finally chosen EECC shockmodel with parameters in Table 1 does self-consistentlyand simultaneously fit (sub)millimeter emission continuaand molecular spectral line profiles of L1517B. Velocity,density and temperature radial profiles at the presentepoch of the star-forming L1517B described by the cho-sen polytropic EECC shock model are displayed in Fig.1. By the very dynamic nature, these radial profiles inFig. 1 qualitatively differ from those of both fitting mod-els adopted by Tafalla et al. (2004, 2006) and Kirk et al.ECC for Star-Forming Cloud Core L1517B 5 TABLE 3Physical properties of star-forming cloud core L1517B derived from our polytropic hydrodynamic EECC shock model
Variables M M tot a ˙ M ( M ⊙ yr − ) b R inf (AU) R sh (AU) c u (km s − ) u (km s − ) d Values 0 . M ⊙ . M ⊙ . × − . × . × − .
03 0 . Note . — These physical parameters are grossly consistent with those estimated from SCUBA observations by(Kirk, Ward-Thompson & Andr´e 2005), except for the conspicuous feature of the variable temperature profile obtained self-consistentlyfrom our self-similar EECC hydrodynamic shock model, which differs from the static and isothermal assumptions in earlier models.(a) M and M tot are the central point mass and the total cloud mass within a radius R = 2 . × AU.(b) ˙ M denotes the central mass accretion rate which decreases with time in our EECC shock model (see equation A8).(c) R inf is the expanding boundary separating the core collapse and envelope expansion regions and R sh stands for the shock radius.(d) u and u are the upstream and downstream radial velocities across the outgoing shock front, with negative values for local inflows (2005). In principle, all these model fits should be con-strained simultaneously by other available observations.As fitted by the underlying hydrodynamic shockmodel, L1517B appears to involve a self-similar core col-lapse with an envelope expansion at a typical outflowingspeed of u exp ∼ . − in the radial range of 1 . × AU ≤ R ≤ . × AU, and a concurrent core collapsewithin the infall radius R inf ∼ = 1 . × AU with atypical infall speed of u inf ∼ . − . These infalland expansion speeds are in order-of-magnitude agree-ment with the ∼ . − internal speed estimatedin Tafalla et al. (2004). There is an outgoing shock at R sh ∼ = 1 . × AU with a travelling speed of ∼ .
25 kms − . The central point mass which represents the mass ofthe pre-protostar is M ∼ . M ⊙ and the total cloudmass within a radius of 21000 AU is M tot ∼ . M ⊙ .Thus, there is not yet a star (in the sense of thermalnuclear burning) at the center of L1517B since the in-ferred mass of the pre-protostar is much smaller thanthe threshold mass value ( ∼ . M ⊙ ) to initiate nuclearreaction. Besides, according to the standard equation forthe accretion luminosity L acc = GM ⋆ ˙ M ⋆ /R ⋆ (e.g. equa-tion 2 in Kenyon & Hartmann 1995), we can estimatean accretion luminosity of about ∼ . L ⊙ for L1517B,which is likely below the current infrared (IR) detectionlimit. Moreover, according to Caselli et al. (2002), star-less cores are less massive (for example, a statistical meanmass is estimated as h M tot i≃ M ⊙ , while the total massof L1517B inferred from our model is ∼ . M ⊙ ) thancloud cores with nuclear burning stars (e.g., h M tot i≃ M ⊙ in contrast). Therefore, L1517B can be consis-tently considered starless, rather than a core with a star,as noted before. Our inferred total cloud mass is twicethat derived from dust continuum flux densities by Kirket al. (2005), but should still be classified as low-mass( M ≤ M ⊙ , cf. McKee & Ostriker 2007) star-formingcloud cores. The ratio of pre-protostar mass to the cloudmass is very low, at M /M tot = 0 . M = 4 . × − M ⊙ yr − , giv-ing an evolution timescale of t E ∼ M / ˙ M = 7 . × yr. From the polytropic model with n <
1, the massaccretion rate decreases with increasing time (see eq.A8), which appears to be a typical feature in low-massstar formation systems (e.g. Schmeja & Klessen 2004;Evans et al. 2009). Thus the realistic infall age of thecloud should be less than the derived evolution timescale t E . Derived physical properties of the pre-protostellarcore L1517B are summarized in Table 3 for a convenientreference. MOLECULAR SPECTRAL LINE PROFILES ∼ vdtak/ratran/ratran.html),which deals with both LRT and non-local thermalequilibrium (non-LTE) excitations of molecular energylevels based on Monte Carlo method. This Monte Carlocode can handle both optically thin and thick lines,barring very large optical depths (e.g. > ∼ u , number density N and gas temperature T are allderived self-consistently from this EECC model. Thedust temperature T d in the cloud core is assumed tobe equal to the gas temperature T as expected forhigh-density cloud cores where gas molecules and dustshave sufficiently frequent collisional exchanges (e.g.Goldsmith & Langer 1978; Goldsmith 2001). Whenrunning the RATRAN code, we divide the sphericalcloud core into 16 uneven shells with enough accuracyfor calculations based on the principle that physicalquantities in each shell do not vary significantly. Weinput the number density N , gas and dust temperature T , radial flow velocity u derived self-consistently fromour EECC shock hydrodynamic model for radiativetransfer calculations. Besides, the temperature of back-ground radiation is the cosmic microwave background(CMB) temperature T bg = 2 .
73 K. All molecular linetransition data are obtained from the Leiden Atomic andMolecular Database (LAMDA, see Sch¨oier et al. 2005),and various molecular abundance ratios with respect toH molecules are presented in Table 4 and more detailsof abundance profiles can be found in subsection 3.4. Red Skewed Double-Peak Line Profiles
Numerical results (based on our EECC model) of cen-tral molecular spectral line profiles in comparison withrelevant observational data from Tafalla et al. (2004,2006) are displayed in Figs. 2, 3 and 4. Emission lines Actually, 12 and 14 uneven shells have also been tested sepa-rately in running the RATRAN code and one can sense the gradualconvergence to the final results with little variance.
Fu, Gao and Lou
HCO + (1-0) HCO + (3-2) T m b ( K ) V LSR (km s -1 ) H CO(2 -1 ) H CO(2 -1 ) T m b ( K ) V LSR (km s -1 ) CS(2-1)
CS(3-2)
Fig. 2.—
Central molecular spectral line profiles of L1517Bwith salient red skewed characteristics. The ordinate stands forthe brightness temperature in Kelvin and the abscissa representsthe LOS velocity component. Histograms are observational datataken from Tafalla et al. (2004, 2006) while solid curves representRATRAN LRT computational results based on our EECC shockmodel. Relevant abundances with respect to H are listed in Ta-ble 4. Our spectral profile fitting for the molecular transition lineHCO + (1 −
0) shows slightly stronger red peak than observed. How-ever, spatially resolved observation of this line transition revealsthat the profiles in some spatial positions [see e.g. ( − ′′ , − ′′ )in Fig. 5] exhibit somewhat stronger red peaks than our calculationin the central position due to non-spherical effects in L1517B. Formolecular line transition CS(2 − of HCO + (1 −
0) and H CO + (1 −
0) are averaged over abeam area of about 50 ′′ × ′′ to match with Five Col-lege Radio Astronomical Observatory (FCRAO) 13.7 mtelescope observations in 2001 April (e.g. Tafalla et al.2006). Spectral profiles for the following eight moleculartransition lines, namely HCO + (3 − CO(2 − ),H CO(2 − ), CS(2 − − + (3 − −
12) and SO(34 − ′′ × ′′ to match with the dataacquired using the IRAM 30 m telescope in Spain be-tween 1999 October and 2002 November (Tafalla et al.2002, 2004, 2006). Noting that the observational dataof the molecular transition line CS(2 −
1) publishedin Tafalla et al. (2004) actually differ from those of
TABLE 4Rest-frame molecular line transition frequencies andmolecular abundances in reference to H molecules Molecular Line Frequency AbundanceTransitions (GHz) X HCO + (J= 1 −
0) 89.188523 1 . × − HCO + (J= 3 −
2) 267.557619 1 . × − H CO(J K − K +1 = 2 − ) 140.839502 5 . × − H CO(J K − K +1 = 2 − ) 150.498334 5 . × − CS(J= 2 −
1) 97.980953 2 . × − CS(J= 3 −
2) 146.969026 2 . × − H CO + (J= 1 −
0) 86.754288 7 . × −
11 (a)
DCO + (J= 3 −
2) 216.112582 4 . × −
10 (a)
SO(NJ= 23 −
12) 99.299890 1 . × − SO(NJ= 34 −
23) 138.178670 1 . × − N H + (JF F= 101 − . × −
10 (b) N H + (JF F= 344 − . × −
10 (b) NH (JKF F= 1110 . − .
5) 23.694501 2 . × − NH (JKF F= 2210 . − .
5) 23.722680 2 . × − Note . — Frequencies of molecular transitions are obtained fromthe LAMDA database (e.g. Sch¨oier et al. 2005). a Molecules with central depletion hole. X = X for r > r hole ,and X = 10 − X for r < r hole . For simplification and for keepinga minimum number of parameters in our model construction, wehave assumed the same radius of abundance hole, r hole = 5 . × AU, for all molecular species in the fitting procedure. b Constant abundance without depletion hole. c The central abundance enhancement with Para- X (NH ) = X (NH )[ n ( r ) /n ], where n ( r ) is the H number density at ra-dius r . We adopt n = 2 . × cm − which is the same asTafalla et al. (2004) for comparisons. Tafalla et al. (2002), we here take the observational dataof Tafalla et al. (2004) in our model analysis. Follow-ing Tafalla et al. (2004), we produce the N H + (1 − H + (3 −
2) spectra by averaging over a beamarea of ∼ ′′ × ′′ and ∼ ′′ × ′′ , respectively.As the NH ( J, K ) = (1 ,
1) and (2 ,
2) lines were ob-served simultaneously using the 100 m telescope of theMax Planck Institute for Radio Astronomy (MPIRA) atEffelsberg near Bonn between 1998 October and 2001May (Tafalla et al. 2002), we made an average over a ∼ ′′ × ′′ beam area to simulate relevant results.A constant intrinsic line broadening with Doppler b-parameter (defined as 1/e the half-width of a line profilein executing the RATRAN code) ∼ .
08 km s − (i.e.with a FWHM of ∼ .
13 km s − for a Gaussian line pro-file) and a cloud receding velocity of u cloud ∼ .
73 kms − are used for all the data fitting procedure.Optical depths play consequential roles in producingasymmetric spectroscopic signatures in molecular line In our fittings of spectral profiles from emission lines of CS,we note that the observed intensity of the red peak of CS(2 − −
1) and CS(3 −
2) transition lines. Wethen checked the observations of Tafalla et al. (2004) and foundthat Tafalla et al. (2004) have made a correction of the relevantdata, and the new data appears more consistent with our modelfitting calculations. The difference between the CS data of Tafallaet al. (2002) and Tafalla et al. (2004) may be caused by differentradio telescopes between FCRAO 14m observations (Tafalla et al.2002) and IRAM 30m observations (Tafalla et al. 2004). Both the thermal (∆ v T ) and turbulent (∆ v t ) components thatcontribute to the line broadening have been included in RATRANcode calculations. ∆ v T is estimated by ∼ ( k B T/m ) / with T consistently obtained from each shell and m is the mean parti-cle weight, and ∆ v t = e × (Doppler b-parameter). The total linebroadening ∆ v = [(∆ v T ) + (∆ v t ) ] / . ECC for Star-Forming Cloud Core L1517B 7profiles. Optically thick molecular line transitions ofHCO + and CS display deeper self-absorption dips thanoptically thin emissions from molecular line transitionsof H CO, which have almost no central dips and showonly stronger red shoulders (see Fig. 2). Moreover, theHCO + (3 −
2) line transition manifests less apparent self-absorption dip in comparison with HCO + (1 −
0) tran-sition. This contrast is likely due to lower energy levelpopulations on J=3 and J=2 in such a cold interstellarmedium (ISM) environment of T ∼
10 K, leading to alower optical depth and source function of the J= 3 − + (Gao & Lou 2010).Emissions from the same molecule but for differentenergy level transitions strongly constrain the under-lying polytropic hydrodynamic cloud core model, be-cause all the dynamic and thermal parameters as wellas molecular abundance profiles should remain identi-cal in the LRT calculations using the RATRAN code.Fig. 2 shows the spectral profiles of HCO + , H COand CS each for two distinct transitions from centralL1517B. These molecular line spectral profiles presentexplicit red skewed double-peak signatures (i.e. redprofiles), consistent with the plausible existence of self-similar EECC shock phase (Lou & Shen 2004; Shen &Lou 2004; Thompson & White 2004; Gao & Lou 2010). Such a cloud envelope expansion with core collapse mayarise from either large-scale cloud radial oscillations (e.g.Aguti et al. 2007) or molecular outflows driven by pro-tostar embedded in the core (e.g. Thompson & White2004). It has been claimed that the presence of molec-ular outflows is always associated with the evidenceof non-Gaussian CO line wings (e.g. Thompson et al.2004). However, spectra for C O(1 − O(2 − O(1 −
0) and C O(2 −
1) towards L1517B (see figure8 of Tafalla et al. 2004) do not manifest such evidenceof outflows. Besides, no spatially separated “molecular”outflow lobes has yet been detected in this star formingcloud core. We thus propose that the global envelopeexpansion with central core collapse revealed in L1517Bmay originate from damped acoustic radial pulsationson large spatial and temporal scales in molecular cloudcores (Lou & Shen 2004; Lada et al. 2003; Keto et al.2006; Gao & Lou 2010; Lou & Gao 2011). Other Relevant Molecular Transition Lines
As consistent checks and further constraints, wealso compute central spectra of isotopologues, namelyH CO + and DCO + , of the formyl cation HCO + andtwo distinct line transitions of sulfur monoxide SO usingthe same EECC shock model in comparison with obser-vation data of Tafalla et al. (2006). Fittings of isotopo-logue emissions with constant abundance ratios to eachother provide an effective way to investigate the in-fluence of abundance patterns on spectral profiles. Asalready noted in subsection 3.1, emissions from identi-cal molecules with different energy level transitions serveas effective evidences to verify our EECC shock model. A schematic explanation for the origin of red profiles in molec-ular cloud cores can be found in Lou & Gao (2011). More explanations for the coexistence of expansion and col-lapse are elaborated in introduction and discussion sections. We simply multiply a constant ratio to the molecular abun-dance of HCO + for its isotopologues in our model profile fittings. In extensive numerical explorations, we adopt molecularabundances and isotopic ratios which are somewhat dif-ferent from those used in Tafalla et al. (2006), e.g. theSO abundance with respect to hydrogen molecule H dif-fers from that adopted by Tafalla et al. (2006) due to thevariation in the chosen radius of central depletion hole.There is no conspicuous appearance of red skeweddouble-peak profiles in molecular line emissions in Fig.3; instead, single-peak lines are observed in these opti-cally thin lines as expected. As discussed in subsection3.1, the absence of red asymmetries and self-absorptiondips in spectral line profiles is attributed to the decreaseof molecular level populations (which in turn leads tothe optical depth decrease). This point is highlightedby comparing the spectral line profile of the opticallythin H CO + (1 −
0) in Fig. 3 with that of the opti-cally thick HCO + (1 −
0) in Fig. 2 [n.b. the opticaldepth of H CO + (1 −
0) is ∼ /
20 that of HCO + (1 − + and SO, tend to be broaderin line profile widths. Actually, model lines of thesemolecules could be narrowed by reducing their depletionhole sizes . For illustrations, we have decreased the de-pletion hole radius r hole from 5 . × AU to 2 . × AUfor both DCO + and SO. Meanwhile, their abundancesare reduced to 1 . × − and 5 × − , respectively, andthen both molecular lines become narrower (see dashedcurves in Fig. 3). Physical explanations for the reduc-tions of depletion holes are as follows: The difference be-tween the abundance hole radius of the DCO + and thatof the HCO + results from a central increase in the deu-terium fractionation caused by the CO depletion, whichpartly compensates the DCO + freeze out at the innercore (e.g. Tafalla et al. 2006). The non-Carbon-bearingmolecule SO does not suffer from the depletion of Carboncomponents, so that its depletion hole may differ fromthose of Carbon-bearing molecules. Here, SO providescomplementary information to emissions from Carbon-bearing molecules for the protostar-forming cloud core. Molecules Without Central Depletion Holes
Molecules with central depletion holes offer importantdiagnostics to probe the dynamic structure of the outermolecular envelope of L1517B. Meanwhile, moleculartracers of the central core, such as N H + and NH , offeressential information to examine the collapsing core, asthese molecules do not freeze out onto dust grains at typ-ical core densities and they are present in the gas phasethroughout the core (e.g. Tafalla et al. 2004). As shownin Table 4, a constant N H + abundance and a variableabundance profile with a central enhancement for NH have been adopted in our RATRAN calculations. The reduction of the depletion hole (still larger than the col-lapsing core) would bring extra emission contributions from thedenser and hotter inner region, so we need to decrease the totalabundance of these molecular species to better fit observations.Due to the lower abundance, emissions from the outer expandingenvelope which mainly contribute to the edges of the line spectraare weakened, and this leads to narrower spectral line profiles.
Fu, Gao and Lou H CO + (1-0) SO(23-12) T m b ( K ) V LSR (km s -1 ) SO(34-23)
DCO + (3-2) Fig. 3.—
Molecular line profiles of central spectra for four transi-tions H CO + (1 − + (3 − −
12) and SO(34 − r hole =5 . × AU for all species. Constant isotopic ratios of C/ Cand H/D (see Table 4 for details) are adopted in our model fittinganalysis. Dashed curves displayed in the DCO + and SO line pro-files correspond to the simulation results for a smaller depletionhole with r hole = 2 . × AU, and lower molecular abundancesas described in the text. The absence of red asymmetry and self-absorption dip in these molecular spectral line profiles is attributedto the decrease of optical depths because of the lower energy levelpopulations for these molecular line transitions. T m b ( K ) V LSR (km s -1 ) NH (1,1) NH (2,2) N2H+(1-0)
N2H+(3-2)
Fig. 4.—
Molecular line profiles from the central part of L1517Bwithout depletion holes. Multi-peaks present in both moleculartransitions are caused by hyperfine (hf) splittings rather than bythe cloud core dynamic collapse. Histograms are observational datataken from Tafalla et al. (2002, 2004). As shown in Table 4, aconstant N H + (top panels) abundance and a central abundanceenhancement pattern for NH (bottom panels) have been adoptedin order to fit the observational data. Unlike other molecular transitions discussed in this pa-per, rotational levels of N H + are split into multiplehyperfine (hf) components by the two nitrogen atoms(e.g. Caselli et al. 2002; Tafalla et al. 2002). Such hfsplittings make the radiative transfer computation muchmore complicated. Fortunately, available N H + molec-ular data file containing hf structures is provided inLAMDA, which simplifies our calculations considerably.In all, 15 different transitions of N H + (with 7 distinctfrequencies due to the overlap of some transitions) for J = 1 − J = 3 − J = 1 − J = 3 − ∼ . − has been predicted by our EECC model,this region is very small as compared to the entire col-lapsing core ( ∼ /
27 in volume) and is almost negligiblecompared to the entire cloud ( ∼ / ( J, K ) = (1 , ,
2) inversion lines arise from para-NH (e.g.Tafalla et al. 2002), so we only focus on this particularspecies in our modelling. Like N H + , ammonia NH alsohas hf splittings due to the interaction between the elec-trical quadrupole moment of the nitrogen nucleus and theelectric field of electrons. Apart from this major con-tribution, other weaker interactions, including the I N · J (where I N and J are the nitrogen spin and the to-tal angular momentum of ammonia, respectively) and I · J (where I is the sum of hydrogen spins) magneticinteractions, as well as H-N and H-H spin-spin inter-actions, further split the transition components on theorder of ∼ by assuming that the sublevels are popu-lated according to their statistical weights. Once pop-ulations for the hf sublevels are derived, we take the Information about hf sub-levels are taken from the NASA-JPLmolecular database. Only the electrical quadrupole hf energy levelstructure has been considered in the data files. Therefore, 5 distinctcomponents of the transition frequency for both (
J, K ) = (1 ,
1) and(2 ,
2) are taken into account in our radiative transfer modelling.
ECC for Star-Forming Cloud Core L1517B 9full hf structures into account and predict the emergent(
J, K ) = (1 ,
1) and (2 ,
2) spectra (as shown in Fig. 4)by integrating the radiative transfer equation along theLOS and summing up each hf component . The validityof this approach was discussed by Tafalla et al. (2002).We briefly comment here: according to quantum statis-tics, the population ratio between two different levels isgiven by N /N = ( g /g ) × e − ∆ E/ ( k B T ) , where g i arethe statistical weights (degeneracies) for level i ( i =1, 2)and ∆ E = E − E is the energy difference between thetwo energy levels. As long as ∆ E ≪ k B T , i.e. the energydifference is far less than the energy scale involving theexcitation temperature (which is usually satisfied whenwe consider the population of hf sub-levels for moleculesin starless cores), we can then approximately take thepopulation ratio as simply g /g . The excitation tem-perature and the kinetic temperature can be equal underthermodynamic equilibrium, but this is not necessarily soin general.Ammonia NH spectral data are usually used toprobe the central gas kinetic temperature of star-formingmolecular cores (e.g. Tafalla et al. 2004). The databaseof dense molecular cores mapped in the ( J, K ) = (1 , ,
2) transition lines of NH was compiled and pre-sented by Jijina, Myers & Adams (1999), who concludethat the temperature distribution of molecular cores inTaurus complex is very narrow around ∼
10 K. In ourpolytropic EECC shock model, however, the gas temper-ature gradually rises up towards the center and reaches ∼
15 K in the collapsing core of L1517B as shown inFig. 1. According to Tafalla et al. (2004) and Ho &Townes (1983), the gas temperature of ∼
10 K inferredfrom NH3 observations relies on the presumed (static)density profile, which is an empirical asymptotic power-law envelope with a “flat” central region. It appearsthat such a constant ∼
10 K temperature reasonablyfits the gas kinetic temperature for L1517B derived fromNH data analysis (e.g. figure 4 of Tafalla et al. 2004).We note, however, that the gas kinetic temperature thusdetermined is inferred directly from the rotational tem-perature T of NH (e.g. Walmsley & Ungerechts 1983;Tafalla et al. 2004) which relates to the relative bright-ness temperature/population between the ( J, K ) = (2 , ,
1) radiative transitions (e.g. Ho & Townes 1983);therefore, the gas kinetic temperature thus derived onlyrepresents an “averaged” temperature along the LOS,not necessarily indicating the actual temperature at acertain spatial point. In other words, it is possible that aradially variable temperature profile could also reproducethe NH data as shown by our dynamic model fittinganalysis (see also Galli et al. 2002 for variable tempera-ture profiles in molecular cloud cores). Noting that thetemperature profile of our model reaches ∼
15 K onlyin the very inner region ( ∼ )while remaining around ∼
10 K over a wide radial rangein the outer portion (see Fig. 1), we would expect thatour radially variable temperature profile may mimic the We compute the Einstein A coefficients for different hf linetransitions based on equation (9) of Pickett et al. (1998). The gas kinetic temperature for L1517B derived from NH observations starts from ∼
900 AU (see figure 4 of Tafalla et al.2004); therefore, the “hot” region in our model occupies the innercore and contributes only a very small part to the LOS averagedgas kinetic temperature. constant LOS temperature distribution of the NH anal-ysis for L1517B. Besides, as the temperature distributionof L1517B in figure 4 of Tafalla et al. (2004) representsLOS averages spatially smoothed with 40”, these aver-ages over the telescope beam resolution would furtherflatten the temperature distribution and make the wholeprofile closer to a roughly constant ∼
10 K distribution.Moreover, as our EECC model shown in Fig. 1 as awhole reasonably reproduces both numerous molecularline emissions (including those of NH and N H + ) and(sub)millimeter dust continuum observations of L1517B(see Table 4 and Section 4 for details), we conclude thatthis hydrodynamic self-similar EECC model is a viablephysical description of cloud core L1517B. Our temper-ature variation would only introduce minor corrections,because the inner “hot” region is fairly small comparedwith the entire molecular cloud ( R inf < . R out ). Inother words, a variable temperature profile might ap-pear somewhat flattened in this type of inferences byLOS NH observations. Noting that our radial temper-ature profile drops below ∼
10 K in the outer envelopeportion, it is possible to produce a 10 K “averaged” rota-tional temperature from our EECC model. Jijina et al.(1999) have adopted the T - T K (here T K is the gas ki-netic temperature) from Walmsley & Ungerechts (1983)to convert the inferred rotational temperature into gastemperature, which may cause further uncertainty in thedetermination of cloud temperature since the T - T K re-lation of Walmsley & Ungerechts (1983) may not be thataccurate (e.g. Tafalla et al. 2004). Influence of Molecular Abundance Distributions
The spatial distribution of molecular abundances playsa crucial role in producing molecular spectral line profilesand thus has received considerable attention in the liter-ature (e.g. Herbst & Klemperer 1973; Rawlings & Yates2001; Tsamis et al. 2008). Also, the depletion of molecu-lar species by adhesion onto cold dust grain surfaces hasbeen discovered in central regions of star-forming cloudcores (e.g. Tafalla et al. 2002, 2004; Walmsley et al.2004). We have selected molecular abundance ratios (rel-ative to the number of H molecules) by referring toTafalla et al. (2004, 2006) yet with variations to probemolecular distributions in L1517B. Central molecular de-pletion holes with very much lower abundance ratios[i.e. ∼ − of the ratio in the outer cloud layers asin Tafalla et al. (e.g. 2006)] are essential in reproducingabsolute intensities and relative strengths of ‘blue’ and‘red’ peaks in molecular spectral line profiles. It is pos-sible to treat the radii of abundance holes, which canvary for different molecular species in general, as extraindependent parameters as in Tafalla et al. (2006). How-ever, in order to focus on exploring the dynamic processof L1517B and to reduce the number of independent pa-rameters, we have assumed a common size of depletionhole for all molecules, except for NH and N H + molecu-lar transitions. Our chosen abundance distributions andthe radius of the depletion hole r hole used in the datafitting are summarized in Table 4 for reference. Basedon LRT RATRAN calculations, intensities of both ‘blue’and ‘red’ peaks would decrease with the increase of r hole ;this is also intuitively sensible. In fact, the intensity ofthe stronger ‘red’ peak appears to be much more sensi-tive to the variation of r hole than that of the weaker ‘blue’0 Fu, Gao and Lou Fig. 5.—
We show here the grid-map for the observed molecular transition HCO + (1 −
0) spectra with a 50 ′′ step for L1517B. The emissionintensities are expressed in main beam temperature T mb (using Kelvin as the unit) with elevation corrections f ( θ ) = 1 − . ± .
02) cos θ where θ is the elevation angle of the Delingha telescope, and the efficiency of the Delingha 13.7 m telescope is η mb = 0 .
62. This 5 × α J2000 = 04 h m . s , δ J2000 = 30 ◦ ′ ′′ based on the central position of the 1.2 mm continuum source of L1517Bas observed by Tafalla et al. (2002). The LOS velocity resolution is ∆ v = 0 .
04 km s − , and the root-mean-square error in the antennatemperature T ∗ A is σ rms = 0 .
083 K. The legend panel to the upper right corner illustrates the central panel as an example. The spectralprofiles are not perfectly spherically symmetric with the eastern spectra slightly shifting towards the blue. The difference between theeastern and western spectra might be due to the rotation of L1517B about its north-south axis. However, double-peaked red profiles areevident in HCO + (1 −
0) lines across the cloud globule and an average over the same distance from the center reveals stronger red peaks inall radial positions (see Fig. 8). In this sense, the presence of red profiles is indicative of cloud core expansive motions (see also Aguti et al.2007). peak does. This may allow us to estimate the radius ofthe depletion hole in L1517B by carefully calibrating the‘blue’ to ‘red’ peak intensity ratios in pertinent molecularspectral line profiles.
Spatially Resolved Spectral Line Profiles
Central spectral profile model fittings of various molec-ular line transitions provide an important and effectiveapproach to probe both dynamical and thermal struc-tures of pre-protostellar cores. However, only presentingsignatures from the cloud center, central spectral pro-file fittings of molecular lines might still have certainambiguity and uncertainty in determining all pertinentphysical properties of star-forming cloud cores. Mappingmolecular transition line profiles with high enough spa-tial resolutions, on the other hand, would further con-strain the large-scale physical conditions within molec-ular clouds (e.g. Aguti et al. 2007; Gao & Lou 2010;Lou & Gao 2011). Here, we report a position-switchobservation (5 × + (1 −
0) line transition, which is the most prominentdiagnostics for dynamic motions within L1517B, usingthe 13.7 m telescope at Delingha in Qinghai Province ofChina and present our model fitting results of spatiallyresolved spectral profiles based on the same underlyingEECC shock model. Meanwhile, we predict other perti-nent red skewed spectral profiles of molecular lines withspatial resolutions for future observational tests.
Data Acquisition and Comparison
We recently observed L1517B for HCO + (1 −
0) at89.186769 GHz using the 13.7 m millimeter-wave radiotelescope of PMO at Delingha during April 12 −
22, 2010.The telescope is ∼ ∼
90 GHzwas used in the observation together with a FFTS digitalspectrometer with 16384 channels and a working band-width of 200 MHz, which gives a velocity resolution of0.04 km s − at the frequency of 89 GHz. The position-switch mode was chosen, with the pointing and track-ECC for Star-Forming Cloud Core L1517B 11 T m b ( K ) V LSR (km s -1 ) FCRAO 2001 Delingha 2010
Fig. 6.—
A direct comparison between the central spectra, whichhave been generated from the average of typically 5 spectra withina 50 ′′ radius from the core center, of the molecular transitionHCO + (1 −
0) from L1517B observed by Delingha 13.7 m millimeter-wave radio telescope in mid-April 2010 (solid histogram) and byFCRAO 13.7 m telescope in April 2001 (dashed histogram). Theordinate stands for the main beam brightness temperature T mb converted using the beam efficiencies (0.62 for Delingha telescopeand 0.55 for FCRAO telescope respectively) from the respective an-tenna temperatures T ∗ A . The Delingha observation has a velocityresolution of 0.04 km s − as compared with the velocity resolutionbetween 0.03 km s − and 0.07 km s − of the FCRAO observation,which used the QUARRY array receiver in frequency switchingmode together with the facility correlator. With longer integrationtimes for the observation of each spatially resolved regions, the re-sult from the Delingha observation has relatively higher signal tonoise ratio (S/N) compared with the FCRAO data. ing accuracy within the range of 2 ′′ − ′′ . The telescopebeam size is approximately 50 ′′ at this frequency, and weadopted a mapping step size of 50 ′′ in observation. Thetypical system temperature T sys is ∼
250 K during ourobservations. The standard chopper wheel calibrationwas used during the observation runs to get the antennatemperature T ∗ A , which has been corrected for the atmo-spheric absorption and telescope elevations. The beamefficiency of 0.62 is used to convert the telescope antennatemperatures T ∗ A into main beam brightness tempera-tures T mb . The net integration time for most positionsis 60 minutes, and is 80 minutes for the central position,resulting a root-mean-square (rms) noise of σ rms = 0 . T ∗ A .The spectral profile data are processed with the stan-dard CLASS software. Linear baselines are removedfrom all spectra and an elevation correction f ( θ ) =1 − . ± .
02) cos θ ( θ is the elevation angle) is adjusted.We also made an average of five spectra within a50 ′′ radius from the cloud core center (i.e. the centralspectrum together with four spectra that are 50 ′′ awayfrom the center) to directly compare with the result ofTafalla et al. (2006) obtained by the FCRAO 13.7 m tele-scope in April 2001. From the data comparison of Fig.6, we see that these two observational results are gen-erally consistent with each other with the notable ex-ception of integrated intensities around the dip betweenthe two peaks. This difference may be tolerated withinuncertainties of the observations and might be due toslight discrepancies in telescope calibrations or velocity T m b ( K ) V LSR (km/s) (0",-20")(0",20")(20",0") (-20",0")(0",0")
Fig. 7.—
The HCO + (1 −
0) grid-map of four spectral profilesat a distance of ∼ ′′ around the central spectral profile plot forL1517B (see Fig. 8 for the average of these four spectra). Emissionintensities are expressed in the main beam brightness temperature T mb . The net integration time for each spectral profile with theimpact parameter b= 20 ′′ is 50 minutes, and is 80 minutes forthe spectral profile at the center. While these 20 ′′ spacing spectraoverlap previous observations with the beam size of 50 ′′ , it is stillvaluable to examine global dynamic properties of L1517B fromthese spectra, as the effect of possible rotation about the north-south axis in the inner cloud region appears weaker than in theouter region. Compared with Fig. 5, apparent red profiles in nearlyall these inner spectra (n.b. the red asymmetry is not particularlyobvious in the eastern spectrum) strongly indicate that this redasymmetry is an intrinsic characteristic of L1517B caused by itsinternal motion rather than by rotation. resolutions. This comparison of data from two indepen-dent observations 9 years apart confirm the validity ofboth observational data from FCRAO 2001 April andDelingha 2010 April.To examine whether these “red-profiles” are caused byrotation, we have observed four spectral profiles at a dis-tance of 20 ′′ from the center (Fig. 7) since the rotationaleffect is much weaker in the inner region than in the outerregion. Conspicuous red asymmetric profile characteris-tics in nearly all these spectra exclude the possibility thata pure rotation produces these “red-profiles”. Model Fittings of Observational Data
Simultaneous comparisons between model results andobservational data for molecular line profiles with dis-tinct impact parameters b (distance of LOS from thecore center) is another powerful way to examine theoverall dynamical and thermal structures of star-formingcloud cores. As our model is spherically symmetric whilethe spatially resolved observation (Fig. 5) exhibits non-spherical effects, we make an average over grids of datawith the same radial distance from the center. We carriedout observations of four spectra spaced by 20 ′′ from thecore center under the same observational conditions (yetwith 50 minute integration time each) to check EECCshock model (Fig. 7). All the dynamic parameters as wellas the abundance distribution and the intrinsic broaden-ing in these simulations are consistent with those used in2 Fu, Gao and Loufitting the central spectrum of HCO + (1 −
0) observed bythe FCRAO 13.7m telescope (see Section 2).From the model fitting results shown in Fig. 8, we seethat our computed molecular line profiles based on thegeneral polytropic EECC shock hydrodynamic model co-incide with the observational data. In comparison withobservations, molecular line profiles produced from ournumerical model calculations show somewhat strongerred peaks and weaker blue peaks at the very center (Fig.8). This likely originates from the spatial asymmetry ofthe spectral line profiles, which might be caused by thepossible rotation of L1517B about its north-south axis.The small blue shifts in the eastern spectra (see Fig. 5)will reduce the intensity of the red peaks and enhance theblue peaks when we make an average of the observationaldata over the same radial distance. Meanwhile, these av-eraged spectra exhibit increasing broadening towards theouter envelope while the broadening of our numerical re-sults remain invariant. This phenomenon might be possi-bly attributed to the incremental influence of turbulencewith increasing radius, which is typical in star-formingclouds (e.g. McKee & Tan 2003). In order to reduce thenumber of free parameters and to better understand thedynamics of L1517B, we simply assumed that the intrin-sic broadenings (which are characterized by the Dopplerb − parameter in the RATRAN code) take the same valueeverywhere in the numerical model fitting calculations.All these averaged spectral profiles present evident redprofile characteristics, which are indicative of cloud coreexpansions. Our fitting results reveal that L1517B ismost likely undergoing a core collapse and expansion pro-cess as described by the polytropic EECC shock model. Several Model Predictions for Cloud Core L1517B
As predictions for future observations of L1517B, wecompute spatially resolved relevant molecular line pro-files shown in Fig. 9 for molecular line transitions withred asymmetries in their central spectra.Our model simulation results in Fig. 9 show a gradualdisappearance of red asymmetries in spectral line pro-files and an explicit decrease in overall magnitudes asthe LOS departs away from the cloud core center. Thisvariation trend is attributed to the diminution in bothcloud density and temperature, as well as optical depth(Gao & Lou 2010). Therefore, it would help to assess theinfluence of optical depths on molecular line emissions bymaking comparisons between model results and observa-tions for molecular transitions with spatial resolution. MILLIMETER AND SUB-MILLIMETER CONTINUA OFSTAR-FORMING CLOUD CORE L1517B
While model fittings to observed molecular spectralline profiles provide an important means to infer phys-ical properties of cloud cores, molecular line emissionsonly present under certain conditions and some moleculesmay be frozen onto dust grain surfaces in certain re-gions (e.g. Walmsley et al. 2004). Meanwhile, the ra-dial profiles of flux intensities at (sub)millimeter wave-lengths from dust grain emissions are generally indepen-dent of chemical processes, thus can serve as an indepen-dent observational diagnostics to examine and constraindensity and temperature distributions in star-formingclouds (see e.g. Bacmann et al. 2000; Shirley et al. 2000;Andr´e et al. 2004). Moreover, as the dust density is al- T m b ( K ) V LSR (km s -1 ) Fig. 8.—
Model fittings of spatially resolved spectral lineHCO + (1 −
0) observations for L1517B. Histograms are Delinghaobservational data produced by the average of line spectra withthe same distance from the core center. A telescope efficiency of η mb = 0 .
62 was used to convert the antenna temperature T ∗ A intothe main beam brightness temperature T mb . Solid curves presentLRT fitting simulations averaged over a beam area of 50 ′′ × ′′ based on the same underlying polytropic EECC shock hydrody-namic model and molecular abundance pattern. Spectra corre-spond to the impact parameter b (distance of LOS from the corecenter) being 0 ′′ , 20 ′′ , 50 ′′ , 71 ′′ , 100 ′′ , 112 ′′ and 141 ′′ , respectively.The same intrinsic broadening with Doppler b − parameter ∼ . − and receding velocity of the cloud core u cloud ∼ .
73 kms − as being used previously are adopted in our data fitting pro-cedure.
40 arcsec
72 arcsec HC O + ( − )
80 arcsec H C O ( − ) B r i gh t n ess T e m p e r a t u r e ( K ) H C O ( − ) C S ( − ) Velocity (km/s) C S ( − )
60 arcsec
Fig. 9.—
Red-skewed molecular spectral profiles with spatial res-olutions for L1517B. Emissions from HCO + (3 − CO(2 − ), H CO(2 − ), CS(2 −
1) and CS(3 −
2) are averagedover a radius of ∼ ′′ to simulate the observation of IRAM 30m telescope. We present emission spectral profiles with the chosenimpact parameter b being 0 ′′ , 40 ′′ , 60 ′′ , 72 ′′ and 80 ′′ respectively toreveal the variation of the relative difference in strengths betweenred and blue peaks as well as the overall magnitudes. ECC for Star-Forming Cloud Core L1517B 13most independent of depletion holes of molecular abun-dances, millimeter and (sub)millimeter continua wouldoffer an additional probe to distinguish different models.For optically thin dust emissions, the integral form ofthe specific intensity from a spherically symmetric cloudglobule along a LOS with an impact parameter b is givenby I ν ( b ) = 2 Z R out b B ν [ T d ( r )] κ ν ( r ) ρ ( r ) r ( r − b ) / dr , (13)(e.g. Adams 1991) where R out is the outer radius of acloud globule, and T d ( r ), ρ ( r ), κ ν ( r ) and B ν ( T ) are thedust temperature, cloud mass density, specific dust opac-ity and the Planck function, respectively. Radio Continuum Emissions at 1.2 mm
The 1.2 mm wavelength radio continuum mapping ofL1517B is centrally concentrated and appears grosslyspherical (see figure 1 of Tafalla et al. 2004), so the spher-ical symmetry is a reasonable first-order approximationfor L1517B. We use the same EECC shock hydrodynamicmodel described in Section 2 to fit the radial profile of1.2 mm radio continuum emission data (see figure 2 inTafalla et al. 2004). Tafalla et al. (2004) acquired the1.2 mm radio continuum mapping from the IRAM 30m telescope with a beam size of ∼ ′′ . Therefore, weintegrate the intensities from model calculations withinthis beam size. Parameters of the EECC shock modelare shown in Tables 1 and 2, and our theoretical modelfitting to the observed radial profile of radio continuumis displayed in Fig. 10. In the model computations, thedust temperature is assumed the same as the gas temper-ature adopted for previous molecular spectral line profilecalculations shown in the Section 4. We utilize the dustopacity model proposed by Ossenkopf & Henning (1994)with emissions and absorptions included but with scat-ters ignored. A typical gas to dust mass ratio of 100 to 1is assumed in our cloud model calculations as usual. Nu-merical calculations are carried out using the RATRANcode with the spherical cloud divided into 256 sphericaluneven shells. We have also doubled the number of shells(i.e. 512) in our calculations (see the dashed curve in thetop panel of Fig. 10) and find that, except for a tiny in-crement at small radii, the result generally coincides withthat produced by 256 shells. Therefore, our adoption of256 shells provides sufficient computational accuracy formodeling the continuum emissions. Though the densityand temperature of the cloud increase rapidly near thecenter, they do not lead to a noticeable central peak or aprompt increase in the dust continuum since our calcula-tion results have been convolved with a telescope beamsize of ∼ ′′ which smooths the singularities of densityand temperature profiles at very small radii. Continuum Emissions for 850 µ m and 450 µ m High-quality sub-millimeter continuum observationsat wavelengths of 850 µ m and 450 µ m reveal morestructural complexities of pre-protostellar cores (e.g.Holland et al. 1999; Shirley et al. 2000; Dye et al. 2008).As a consistent check, we compute radial profiles for 850 µ m and 450 µ m flux intensities using the same EECCmodel and compare them with the observational data ofSCUBA by Kirk et al. (2005). The same opacity model for κ ν (Ossenkopf & Henning 1994) and the gas-to-dustmass ratio of 100 to 1 are adopted in these model calcu-lations.In order to be compatible with the JCMT beamFWHM of ∼ . µ m and ∼ . µ m, our numerical calculations areconvolved with a telescope beam size of ∼ ′′ × ′′ for 850 µ m and of ∼ ′′ × ′′ for 450 µ m, respectively.The fitting result of 850 µ m normalized flux density isshown in Fig. 10 and the model calculation of the 450 µ m normalized flux density radial profile is shown in Fig.11. We find a reasonable agreement of the model resultswith observations of 850 µ m continuum inside 10 AU;outside this radius, the quality of data becomes poorwith considerable error bars. The 450 µ m flux densityprofile is grossly consistent in intensity with the obser-vational data (see figure 3 of Kirk et al. 2005) at cor-responding radius, with several substructures being ig-nored. As we have not included the influence of the cen-tral point source in our calculations, peak flux intensi-ties of both 850 µ m ( ∼
155 mJy beam − ) and 450 µ m( ∼
790 mJy beam − ) are somewhat lower than thoseobserved, namely S peak850 = 170 ±
12 mJy beam − and S peak450 = 920 ±
120 mJy beam − (see table 1 of Kirk et al.2005). Since the radio observations of 1.2 mm continuumand 850 µ m data were carried out by two different tele-scopes (i.e. the IRAM for 1.2 mm and the SCUBA for850 µ m) and our model computational result agrees wellwith the 1.2 mm continuum of dust emissions (see Fig.10), the systematic difference of our fitting for the 850 µ m continuum profile may also be related to differencesin calibrations between the two telescopes, which couldbe checked by future direct radio observations of 850 µ mcontinuum using the IRAM 30m telescope. Column Number Densities and Dust Extinctions
The near-infrared dust extinction provides an indepen-dent diagnostics to trace the column density in molecu-lar clouds without involving the dust temperature dis-tributions (Alves, Lada & Lada 2001). It serves as animportant constraint to the density radial distributionof our general polytropic EECC shock hydrodynamicmodel, as a complementary yet independent diagnosticsto various molecular line profiles and dust radio contin-uum emission measurements (Lou & Gao 2011) . More-over, column densities in the low-density outer regionof molecular cloud, where (sub)millimeter dust contin-uum emissions may not be easy to detect, can still betraced by near-infrared dust extinction measurements(Kandori et al. 2005). As an essential and necessarycheck, we thus compute the column density radial pro-file of H molecules by simply integrating volume densi-ties from the same EECC shock hydrodynamic model.Since the column density radial profile is sensitive tothe outer radius R out of a cloud globule, our modelfitting to the observed radial profile would identify aproper outer radius of the cloud core at R out = 2 . × AU. We then invoke the empirical conversion relation N (H ) /A V = 9 . × cm − mag − (e.g. Kandori et al. Because the temperature profile is not isothermal (as oftenassumed in the literature) in our EECC model, the column den-sity radial distribution cannot be directly mapped out by dust(sub)millimeter continuum emissions.
R (arcsec) N o r m a li z e d F l u x R (AU)
850 m continuum
L1517B F ( m Jy / b ea m ) Fig. 10.—
Radial profiles of the 1.2 mm (top) and 850 µ m (bot-tom) radio continuum emissions from L1517B. For the 1.2 mmcontinuum, the intensity scale is in mJy per 11 ′′ beam size andthe abscissa is the angular distance from the cloud core center inarcsec. Solid squares are data from figure 2 of Tafalla et al. (2004).The solid and dashed curves present the simulation fitting resultsbased on the density and temperature profiles of our EECC shockhydrodynamic model by dividing the cloud into 256 and 512 shells,respectively. For 850 µ m, we show the normalized radial profile forthe flux intensity (per 15 ′′ beam size) with the peak flux density S peak850 ∼ = 155 mJy beam − . Data points from SCUBA observa-tions (Kirk et al. 2005) are shown here in a logarithmic scale athalf beam spacings with 1 σ error bars ( ∼
17 mJy beam − ). Theabscissa stands for radius from the center in unit of AU on a loga-rithmic scale. The scatter of data points, in both the 1.2 mm and850 µ m continua, at the most outer part of the core ( R > ′′ ,where the inconsistency between our simulations and observationsemerge) is highly likely due to the asymmetry of the core (see figure1 of Tafalla et al. 2004). No conspicuous shock discontinuities areseen for either 1.2 mm or 850 µ m continua, since this shock locatesat the outer envelope ( ∼
140 AU) where both the temperature anddensity change slightly due to the shock wave. N (H ) is the H molecule column number density and A V is the visual extinction, to predict the column den-sity and visual extinction for cloud core L1517B usingour EECC shock model which can reasonably fit othercurrently available observational data.In the column density calculation, we convolve numer-ical integration results with a typical beam resolutionof ∼ ′′ . As a comparison and prediction, results ofboth the H molecule column density and the visual ex-tinction A V derived from the EECC shock model areshown in the lower panel of Fig. 11. For example, thevalue of central column density predicted by our EECCshock model ( ∼ . × cm − ) is close to yet a bitlower than the model result of Kirk et al. (2005) [i.e. N (H ) c = 4 × cm − with a typical error range of ± ∼
30 percent (see table 4 of Kirk et al. 2005). Theircentral “column density”, however, instead of being di-rectly obtained from infrared dust extinction measure-ments, is actually derived from the 850 µ m radio con-tinuum profile modelled by a static isothermal ( T = 10K ) Bonnor-Ebert sphere and a constant dust opacity κ ≡ .
01 cm g − . However, with those estimatedparameters, such an isothermal Bonnor-Ebert sphere forL1517B turns out to be dynamically unstable. Besides, R (AU) A V ( m a g ) N o r m a li z e d F l u x L1517B
450 m continuum N ( H ) ( c m - ) Column density & Visual extinction N ( H ) /A V =9.4 10 cm -2 mag -1 R (arcsec)
Fig. 11.—
Radial profile for the 450 µ m radio continuum emis-sions (top); and column densities for hydrogen molecules N (H )and corresponding magnitudes of visual extinction A V (bottom)predicted for observations of L1517B using the EECC shock model.For the 450 µ m continuum, here we show the normalized radialprofile for the flux density (per 8 ′′ beam size) with the peak fluxintensity S peak450 ∼ = 790 mJy beam − . Results are shown in a log-log scale with the abscissa being the radial distance in AU. For thecolumn density and visual extinction profiles, the ordinate on theleft axis marks the column number density distribution N (H ) inunit of cm − and the ordinate on the right axis indicates the mag-nitudes of visual extinction A V . The abscissa is the radial angulardistance from the cloud core center in arcsec. These results areconvolved with a typical observational resolution of ∼ ′′ . their central “column density” is model dependent with-out being constrained by other available observations ofL1517B, so we propose here to test and examine theirprediction as well as ours for the column density radialprofile of L1517B by dust extinction observations. CONCLUSIONS AND DISCUSSION
In this paper, we have systematically investigated thephysical and chemical (abundance) properties of the star-less cloud core L1517B from three complementary ob-servational aspects, viz., molecular spectral line profiles(some of them are spatially resolved), (sub)millimetercontinuum emissions and the column density radial pro-file through near-infrared measurements of dust extinc-tion. We invoke a self-similar polytropic EECC shockmodel with six independent model parameters involved,viz. the upstream sound parameter k , time scale t , poly-tropic index γ , coefficients of asymptotic solution condi-tions A and B , and the shock location x , as summarizedin Table 1 to present reasonable data fittings to variousasymmetric ‘red profiles’ together with optically thin sin-gle peak spectral profiles, as well as to 1.2 mm and 850 µ m continuum radial profiles from observations (Kirk etal. 2005; Tafalla et al. 2004, 2006). In addition, we haverecently completed a 5 × ′′ spacing) spatiallyresolved spectral profile observation of the HCO + (1 − M ∼ . M ⊙ (incapable of thermal nuclearburnings) and a current central mass accretion rate of˙ M ∼ . × − M ⊙ yr − . (2) The total mass ofthe cloud core is estimated as M tot ∼ . M ⊙ reveal-ing that L1517B belongs to a low-mass cloud core (e.g.Tafalla et al. 2002; McKee & Ostriker 2007). (3) Thecloud core L1517B has an expanding envelope with atypical outgoing speed of u exp ∼ . − and a corecollapsing at u inf ∼ . − . (4) There is also anexpanding shock at radius R sh ∼ = 1 . × AU with anoutgoing radial speed of ∼ .
25 km s − . This outgoingshock is initiated when an expanding outflow runs intoa slowly infalling gas in our dynamic EECC model sce-nario. More importantly, we demonstrate that the poly-tropic EECC shock dynamic phase of L1517B with a setof sensible parameters can give rise to the manifestationof various observed molecular spectral line profiles.In contrast to the results of Tafalla et al. (2004, 2006)and Kirk et al. (2005) (see section 2.1 for details),our general polytropic EECC shock hydrodynamic modelsimply depends on physical properties of L1517B in-side the observed outer boundary of ∼ ′′ and self-consistently produces thermal as well as dynamic prop-erties of the cloud core in the framework of self-similarhydrodynamics to fit the molecular spectral line profiles,(sub)millimeter continuum radial profiles and to predictthe dust extinction property and the column density ra-dial profile in the plane of sky.The uniqueness of our model fitting is generally sup-ported by the theoretical work done by Gao & Lou(2010) and Lou & Gao (2011), which discussed the rela-tionship between different sets of model parameters andthe corresponding molecular line spectral profiles. In-stead of other possibilities that might cause red-skewedmolecular spectral line profiles as noted in the introduc-tion of Gao & Lou (2010), we conclude that the ‘red’asymmetries present in several molecular central spec-tra of L1517B are most plausibly produced by the co-existence of core collapse and envelope expansion mo-tions in the cloud. Other scenarios, such as rotationand pure contraction without expansion were suggestedby Tafalla et al. (2004). However, as discussed in Sec-tion 3.5.1, spatially-resolved molecular spectra aroundthe core center reveal that this asymmetry is most likelyan intrinsic property of the cloud dynamics, rather thanbeing caused by a rotation. Besides, as the temperature profile in our model increases towards the center (a typi-cal characteristics for most molecular clouds), it is impos-sible to produce molecular line profiles of red asymmetryfrom a pure contraction (see eq. B4 of Gao & Lou 2010).As indicated by Broderick et al. (2002), there exist ra-dial breathing modes generated from nonlinear evolutionof acoustic radial pulsations on large spatial and tempo-ral scales, with long crossing times in protostar-formingmolecular cloud cores. We advanced the following sce-nario for the possible emergence of the EECC phase ofL1517B. The molecular cloud is undergoing an envelopeexpanding phase of the breathing mode when the cen-tral contracting region collapses due to nonlinear insta-bilities. This central infalling dynamics is promoted byself-gravity and the collapsed region expands towards theenvelope, similar to the EWCS in Shu (1977). Eventu-ally, all gas materials would continue to fall towards thecenter to form a proto-stellar core. This scenario is con-sistent with both theoretical (e.g. Stahler & Yen 2010)and observational (e.g. Lee & Myers 2011) pictures con-cerning the evolution of internal motions in starless cores.We find in our model analysis that a comparison be-tween optically thin and thick molecular spectral lineprofiles (or between different energy level transitions ofthe same molecule) serves as an effective and sensiblemethod to study the influence of optical depth varia-tions. In addition, analyses of the same molecular linetransition along LOS with distinct impact parameter b from the center may further provide an effective ap-proach to examine effects of both optical depths and ra-dial variations of cloud physical properties (see Section3.5). We have prescribed the abundance patterns of dif-ferent molecules with central depletion holes, which hasbeen widely invoked for molecular line profiles from star-forming clouds as radiative diagnosis (e.g. Tafalla et al.2002, 2004; Walmsley et al. 2004). Intensities of ‘red’peaks appear to be much more sensitive to the size of adepletion hole, as compared to ‘blue’ peaks. This mayallow us to assess molecular depletions by comparing therelative differences between intensities of ‘red’ and ‘blue’peaks in their molecular spectral line profiles.We thank the anonymous referee for suggestions to im-prove the quality of the manuscript. We would like tothank the hospitality of PMO Delingha Observatory andthe 13.7 m telescope staff for their support during theobservation. D.R. Lu and Y. Sun are acknowledged forassistance in the data processing. We thank J. Yang ofPMO for advice and suggestions. This research was sup-ported in part by the National Undergraduate InnovationResearch Program 091000344 for two consecutive yearsat Tsinghua University from the Ministry of Education. We show the velocity profile from our model in Fig. 1; thesetypical values only imply the characteristic magnitudes of expand- ing and collapsing flow velocities, rather than implying the outgo-ing/infalling region has an uniform velocity of this typical value.
APPENDIX
GENERAL POLYTROPIC SELF-SIMILAR HYDRODYNAMIC MODEL
In spherical polar coordinates ( r, θ, φ ), general polytropic nonlinear hydrodynamic partial differential equations(PDEs) for a spherically symmetric molecular cloud under the self-gravity and gas pressure force are given by ∂ρ∂t + 1 r ∂∂r ( r ρu ) = 0 , (A1) ∂M∂t + u ∂M∂r = 0 , ∂M∂r = 4 πr ρ , (A2) ρ (cid:18) ∂u∂t + u ∂u∂r (cid:19) = − ∂p∂r − GM ρr , (A3) (cid:18) ∂∂t + u ∂∂r (cid:19)(cid:18) ln pρ γ (cid:19) = 0 , (A4)where the mass density ρ , the radial bulk flow velocity u , the thermal gas pressure p and the enclosed mass M withinradius r at time t depend on r and t in general; G = 6 . × − dyne cm g − is the gravitational constant and γ isthe polytropic index. PDEs (A1) and (A2) are the two complementary forms of the mass conservation. PDE (A3) isthe radial momentum conservation of a molecular cloud under the gas pressure force and the self-gravity but in theabsence of random magnetic fields (Wang & Lou 2007, 2008). PDE (A4) requires the specific entropy conservationalong streamlines corresponding to a general polytropic EoS p = K ( r, t ) ρ γ with K ( r, t ) formally related to the specificentropy that varies in both time t and radius r in general.To derive an important subset of nonlinear self-similar solutions, these hydrodynamic PDEs can be cast into a setof coupled nonlinear ordinary differential equations (ODEs) with the following transformation (Wang & Lou 2008), r = k / t n x , (the independent self − similar variable x defined) (A5) u = k / t n − v ( x ) , ρ = α ( x )4 πGt , p = kt n − β ( x )4 πG , M = k / t n − m ( x )(3 n − G , (A6)where x is the independent self-similar variable combining r and t in a proper manner, and v ( x ), α ( x ), β ( x ) and m ( x ) are the dimensionless reduced radial flow speed, mass density, thermal pressure and enclosed mass, respectively.Two constants k and n are the sound parameter and scaling index which consistently make x , v ( x ), α ( x ), β ( x ) and m ( x ) dimensionless and control the spatial and temporal scalings of physical variables in a hydrodynamic cloud. Thethermal gas temperature is given by the ideal gas law T ≡ µm H k B pρ = µm H k B kt (2 n − α ( x ) γ − m ( x ) q , (A7)where k B , m H and µ are Boltzmann’s constant, the hydrogen atomic mass and the mean molecular weight, respectively.For a finite dm ( x ) /dx as x → + in the central free-fall solution (e.g. Wang & Lou 2008), the last expression in equation(A6) gives the central mass accretion rate as ˙ M = k / t n − m /G (A8)with m being the reduced central enclosed point mass. For n smaller or larger than 1, this central mass accretionrate decreases or increases with increasing t , respectively. For n = 1, the central mass accretion rate is constant.Substituting self-similar transformation (A5) − (A6) into PDEs (A1) − (A4), we obtain two coupled nonlinear ODEs( nx − v ) α ′ − αv ′ = − x − v ) α/x , (A9) β ′ /α − ( nx − v ) v ′ = (1 − n ) v − ( nx − v ) α/ (3 n − , (A10)with β = α γ m q for γ = 4 /
3, where parameter q ≡ n + γ − / (3 n − x → + ∞ , we have the asymptotic self-similar solution to the leading orders as α = Ax − /n + · · · , v = Bx − /n − (cid:20) B (cid:18) − n (cid:19) + nA (3 n −
2) + (2 n − n q − A − n +3 nq/ (cid:21) x − /n + · · · , (A11)(see Lou & Shi 2011 in preparation for more specific comments on the B term) where A and B are two integrationconstants, referred to as the mass and velocity parameters, respectively.In the other limit of x → + and to the leading order, we have the asymptotic central free-fall solution α ( x ) = (cid:20) (3 n − m x (cid:21) / , v ( x ) = − (cid:20) m (3 n − x (cid:21) / , (A12)ECC for Star-Forming Cloud Core L1517B 17where the constant m is the reduced enclosed point mass, representing the dimensionless proto-stellar mass.By solving coupled nonlinear ODEs (A9) and (A10) with analytic asymptotic solutions (A11) and (A12) as “boundaryconditions”, and by taking proper care of the sonic critical curve, we can derive the radial profiles of velocity, densityand thermal temperature simultaneously and self-consistently [see Wang & Lou (2008) for more details]. HYDRODYNAMIC SHOCK CONDITIONS IN THE SELF-SIMILAR FORM
Self-similar shocks may appear in dynamic molecular clouds and can be constructed within the framework of ageneral polytropic hydrodynamic model. For a hydrodynamic shock, we apply the three conservation laws in the shockcomoving framework, namely conservations of mass, radial momentum and energy, across the shock front[ ρ ( u s − u )] = 0 , [ p + ρ ( u s − u ) ] = 0 , (cid:20) ρ ( u s − u ) γp ( u s − u )( γ − (cid:21) = 0 , (B1)where u , u s , ρ and p represent the flow velocity, shock front speed, gas mass density and thermal gas pressure,respectively. We use a pair of square brackets outside each expression embraced to denote the difference betweenthe upstream (marked by subscript ‘1’) and downstream (marked by subscript ‘2’) quantities, as has been doneconventionally for shock analyses (Landau & Lifshitz 1959).As the parameter k in the self-similarity transformation equation (A5) is related to the sound speed which are gener-ally different in the upstream and downstream sides of a shock, there are two parallel sets of self-similar transformationin the two flow regions across a shock. We set k = λ k with the dimensionless ratio λ representing this difference.The relation x = λx is then required for consistency. With this similarity scaling relation, hydrodynamic shock jumpconditions (B1) can be readily cast into the following self-similar form (Wang & Lou 2008), namely α ( nx − v ) = λα ( nx − v ) , (B2) α − n +3 nq/ x q ( nx − v ) q + α ( nx − v ) = λ [ α − n +3 nq/ x q ( nx − v ) q + α ( nx − v ) ] , (B3)( nx − v ) + 2 γx q ( γ − α − n +3 nq/ ( nx − v ) q = λ (cid:20) ( nx − v ) + 2 γx q ( γ − α − n +3 nq/ ( nx − v ) q (cid:21) . (B4)These three self-similar shock conditions (B2) − (B4) can be solved explicitly and more relevant details can be foundin Appendix D of Wang & Lou (2008) in the absence of random magnetic fields. REFERENCESAdams, F. C. 1991, ApJ, 382, 544Aguti, E. D., Lada, C. J., Bergin, E. A., Alves, J. F., &Birkinshaw, M. 2007, ApJ, 665, 457Alves, J., Lada, C. J., & Lada, E. A. 2001, Nature, 409, 159Andr´e, P., Bouwman, J., Belloche, A., & Hennebelle, P. 2004,ApSS, 292, 325Bacmann, A., Andr´e, P., Puget, J. L., Abergel, A., Bontemps, S.,& Ward-Thompson, D. 2000, A&A, 361, 555Benson, P. J., & Myers, P. C. 1989, ApJS, 71, 89Bodenheimer, P., & Sweigart, A. 1968, ApJ, 152, 515Bonnor, W. 1956, MNRAS, 116, 351Broderick, A. E., Keto, E., Lada, C. J., & Narayan, R. 2007, ApJ,671, 1832Caselli, P., Benson, P. J., Myers, P. C., & Tafalla, M. 2002, ApJ,572, 238Dye, S., et al. 2008, MNRAS, 386, 1107Dyson, J. E., & Williams, D. A. 1997, The Physics of theInterstellar Medium. 2nd ed., IOP Publishing Ltd., Bristol andPhiladelphiaEbert, R. 1955, Z. f Astr., 37, 217Elias, J. H. 1978, ApJ, 224, 857Evans, N. J. II, et al. 2009, ApJS, 181, 321Fuller, G. A., Williams, S. J., & Sridharan T. K. 2005, A&A, 424,949Galli, D., Walmsley, M., & Concalves, J. 2002, A&A, 394, 275Gao, Y., Lou, Y.-Q., & Wu, K. 2009, MNRAS, 400, 887Gao, Y., & Lou, Y.-Q. 2010, MNRAS, 403, 1919Goldsmith, P. F. 2001, ApJ, 557, 736Goldsmith, P. F., & Langer, W. D. 1978, ApJ, 222, 881Gregersen, E. M., Evans, N. J. II, Mardones, D., & Myers, P. C.2000, ApJ, 533, 440Gregersen, E. M., Evans, N. J. II, Zhou, S., & Choi, M. 1997,ApJ, 484, 256Harvey, D. W. A., Wilner, D. J., & Myers, P. C. 2003, ApJ, 583,809Herbst, E., & Klemperer, W. 1973, ApJ, 185, 505 Ho, P. T. P., & Townes, C. H. 1983, ARA&A, 21, 239Hogerheijde, M. R., & van der Tak, F. F. S. 2000, A&A, 362, 697Holland, W. S., et al. 1999, MNRAS, 303, 659Hunter, C. 1977, ApJ, 218, 834Jijina, J., Myers, P. C., & Adams, F. C. 1999, ApJS, 125, 161Jørgensen, J. K., et al. 2004, A&A, 415, 1021Kandori, R., et al. 2005, AJ, 130, 2166Kenyon, S. J., & Hartmann, L. W. 1995, ApJS, 101, 117Keto, E., Broderick, A. E., Lada, C. J., & Narayan, R. 2006, ApJ,652, 1366Kirk, J. M., Ward-Thompson, D., & Andr´e, P. 2005, MNRAS,360, 1506Kirk, J. M., Ward-Thompson, D., & Crutcher, R. M. 2006,MNRAS, 369, 1445Lada, C. J., Bergin, E. A., Alves, J. F., & Huard, T. L. 2003,ApJ, 586, 286Landau, L. D., & Lifshitz, E. M. 1959, Fluid Mechanics, SpringerLarson, R. B. 1969, MNRAS, 145, 271Larson, R. B. 1981, MNRAS, 194, 809Lee, C. F., Myers, P. C., & Tafalla, M. 1999, ApJ, 526, 788Lee, C. W., & Myers, P. C. ApJ, in press, 2011arXiv1104.2950LLou, Y.-Q., & Shen, Y. 2004, MNRAS, 348, 717Lou, Y.-Q., & Gao, Y. 2011, MNRAS, 412, 1755Mardones, D., Myers, P. C., Tafalla, M., & Wilner, D. J. 1997,ApJ, 489, 719McKee, C. F., & Tan, J. C. 2003, ApJ, 585, 850McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565Motte, F., & Andr´e, P. 2001, A&A, 365, 440Motte, F., Andr´e, P., & Neri, R. 1998, A&A, 336, 150Myers, P. C. 2005, ApJ, 623, 280Myers, P. C., Evans, N. J. II, & Ohashi, N. 2000, in Mannings V.,Boss A. P., Russell S. S., eds, Protostars and Planets IV.University of Arizona Press, Tucson. p. 217Nutter, D., & Ward-Thompson, D. 2007, MNRAS, 374, 1413Ossenkopf, V., & Henning, Th. 1994, A&A, 291, 9438 Fu, Gao and Lou