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Spectroscopic Evidence for Gas Infall in GF9-2

We present spectroscopic evidence for infall motion of gas in the natal cloud core harboring an extremely young low-mass protostar GF9-2. We previously discussed that the ongoing collapse of the GF9-2 core has agreement with the Larson-Penston-Hunter (LPH) theoretical solution for the gravitational collapse of a core (Furuya et al.; paper I). To discuss the gas infall on firmer ground, we have carried out On-The-Fly mapping observations of the HCO+ (1--0) line using the Nobeyama 45m telescope equipped with the 25 Beam Array Receiver System. Furthermore, we observed the HCN (1--0) line with the 45m telescope, and the HCO+ (3--2) line with the Caltech Submillimeter Observatory 10.4 m telescope. The optically thick HCO+ and HCN lines show blueskewed profiles whose deepest absorptions are seen at the peak velocity of optically thin lines, i.e., the systemic velocity of the cloud (paper I), indicating the presence of gas infall toward the central protostar. We compared the observed HCO+ line profiles with model ones by solving the radiative transfer in the core under LTE assumption.We found that the core gas has a constant infall velocity of ~0.5 km/s in the central region, leading to a mass accretion rate of 2.5x10^{-5} Msun/yr. Consequently, we confirm that the gas infall in the GF9-2 core is consistent with the LPH solution.


Introduction
Without accurate knowledge of the collapsing process of an isolated dense cloud core, we cannot understand how a low-mass protostar forms through the accretion process. One of the major limiting factors is that the initial conditions of core collapse are not necessarily revealed by observations in detail. There has been a long standing theoretical debate about the gravitational collapse process since the early 1970s. Two extreme models of runaway collapse and quasi-static, inside-out collapse have been developed; the former is originally proposed by the Larson (1969) and Penston (1969) and extended by Hunter (1977), and the latter by Shu (1977). In this paper, we refer to the runaway collapse scenario as the Larson-Penston-Hunter (LPH) solution by following the nomenclature in McKee & Ostriker (2007). The validity of these solutions has been observationally studied by e.g., Ward-Thompson et al. (1994), and Looney, Mundy & Welch (2003).
In this context, we performed a detailed study of the natal cloud core harboring an extremely young low-mass protostar GF 9-2 (Furuya, Kitamura & Shinnaga 2006; hereafter paper I). The protostar is believed not to have generated an extensive molecular outflow, yielding a rare opportunity to investigate core collapse conditions free from the disturbance by the outflow. We discussed that the observed velocity field of the core is consistent with the initial conditions assumed in the LPH solution. The core shows a radial density profile of ρ(r) ∝ r −2 and the entire core has non-thermal line widths of ∆v nth ∼ (2-3)c iso , suggestive of gas infall motions (c iso is isothermal sound velocity). In order to shed light on the physical properties of the infall motions, we have carried out molecular line observations of the GF 9-2 core to detect the blueskewed profiles characterizing infall motion (e.g., Walker et al. 1986;Zhou et al. 1993;Ward-Thompson et al. 1996;Myers et al. 1996;Mardones et al. 1997;Onishi, Mizuno & Fukui 1999;Remijan & Hollis 2006).

Observations
We have carried out On-The-Fly (OTF) mapping observations (Sawada et al. 2008) of the HCO + J =1-0 line [rest frequency (ν rest ) = 89188.526 MHz] using the Nobeyama Radio Observatory (NRO) 1 45 m telescope in 2006 May. We used the 25 Beam Array Receiver System (BEARS), and configured auto-correlators (ACs) as a backend, yielding an effective velocity resolution (∆v res ) of 0.048 km s −1 in the 8 MHz bandwidth mode. At the line frequency, the mean beam size (θ HPBW ) of the 25 beams was 18. ′′ 6, and the mean main-beam efficiency (η mb ) was 0.51. All the spectra were calibrated by the standard chopper wheel method, and were converted into main-beam brightness temperature (T mb ) by dividing by η mb . We adopted the correction factors provided by the observatory for gain differences between the 25 beams. The uncertainty in our intensity calibration is estimated to be ∼ 15%. The OTF mapping was carried out over an area with a size of ∼320 ′′ , employing position switching method. The telescope pointing was checked every 1.2 hrs, and was found to be accurate within 3 ′′ . The data reduction was done using the NOSTAR package. By using a spheroidal function as a gridding convolution function, we produced a data cube with a spatial resolution of 20. ′′ 6 and a pixel size of 5 ′′ . Furthermore, in 2008 June, we have performed a single-point integration of the HCN (1-0) line toward the core center with the 45 m telescope. The beam size of the S80 receiver at the line frequency was 18. ′′ 2 in HPBW, and η mb was 0.44. We used acousto-optical spectrometers (AOSs), AOS-H, providing ∆v res of 0.13 km s −1 .
In 2006 July, 2007 August, and 2008 June, we have used the Caltech Submillimeter Observatory (CSO) 2 10.4 m telescope to observe the HCO + J =3-2 transition (ν rest = 267557.633 MHz). We assumed θ HPBW = 26 ′′ at 267 GHz and estimated η mb of 0.70±0.15 from our measurements of Jupiter. We configured the 50 MHz band width AOS, providing ∆v res of 0.055 km s −1 . The HCO + line was observed toward 3×3 positions with a grid spacing of 30 ′′ centered on the GF 9-2 core center. The pointing accuracy was better than 3 ′′ , and the overall uncertainty in flux calibration was 30 %. Figure 1 shows molecular line spectra toward the GF 9-2 core center in unit of T mb . All the spectra, except the HCO + J = 1 − 0, are the single-dish spectra taken with position switching. The HCO + (1-0) spectrum was produced from the data cube ( §2) by convolving with a Gaussian profile in order to have an effective spatial resolution of 26 ′′ in FWHM, which is equal to the CSO beam size for the HCO + (3-2) line. Clearly, all the spectra show blueskewed profiles. The LSR-velocities of their deepest absorption agree with the systemic velocity (V sys ) of the cloud, V LSR = −2.48 km s −1 , which was estimated from the optically thin H 13 CO + (1-0) and CCS 4 3 − 3 2 lines (paper I) as well as NH 3 inversion lines at 23 GHz (Furuya, Kitamura & Shinnaga 2008).

Results
The velocities of the blueshifted peaks of the HCO + (1-0) and (3-2) lines, V p,blue are estimated to be −2.96 ± 0.02 km s −1 and −2.86 ± 0.01 km s −1 , respectively, from Gaussian fitting. Similarly, we obtained V p,red = −2.29 ± 0.01 km s −1 and −2.25 ± 0.05 km s −1 for the redshifted peaks of the (1-0) and (3-2) transitions, respectively. The obtained V p,red for both the transitions agree with each other, while the V p,blue values differ by 0.1 km s −1 for the two transitions, suggesting that the velocity difference should be attributed to the intrinsic property of the gas rather than uncertainties in ν rest .
The three hyperfine (HF) components of the HCN (1-0) emission, showing blueskewed profiles, have comparable peak intensities, suggestive of large optical depths. We have performed HFS analysis of the blue-and redshifted components separately assuming that infall motions traced by the blue-and redshifted emission are independent, and that each emission can be approximated by a Gaussian profile. Our analysis gave the representative LSRvelocity (V 0 ) of V 0,blue = −2.90 ± 0.01 km s −1 , intrinsic velocity width (∆v int ) of 0.34 ± 0.01 km s −1 , and total optical depth (τ tot ) of 15 ± 2 for the blueshifted components. Similarly, we obtained V 0,red = −2.21 ± 0.03 km s −1 , ∆v int = 0.17 ± 0.02 km s −1 , and τ tot = 12 ± 1.6 for the redshifted components. These τ tot values lead to optical depths (τ ) of 1.6 and 1.3 for the blue-and redshifted F = 0 − 1 components, 8.1 and 6.7 for the F = 2 − 1 components, and 4.9 and 4.0 for the F = 1 − 1 components, respectively, with uncertainty of 20%. The estimated τ for the blue-and redshifted gas of each HF component are comparable to each other, as expected from their intensity ratios, and seem to be consistent with the fact that the absorption dips in the F = 2 − 1 and F = 1 − 1 transitions are deeper than that in the 0 − 1 transition. Figure 2 compares total integrated intensity maps of the H 13 CO + (1-0) (paper I) and HCO + (1-0) emission. Both lines show a similar spatial structure, but the HCO + emission is more extended than the H 13 CO + . Figure 3 represents the HCO + (3-2) and (1-0) mosaic spectra taken with 30 ′′ spacing. Here, the (1-0) spectra were produced from the smoothed cube data in the same manner as for Figure 1. Both the transitions show the blueskewed profiles not only toward the core center, but also at some surrounding positions of, e.g., (∆α, ∆δ) = (0 ′′ , ± 30 ′′ ) and (+30 ′′ , ± 0 ′′ ). Due to the limited S/N, it is difficult to assess whether or not the outer spectra (e.g., in the row of ∆δ = +60 ′′ and the column of ∆α = +60 ′′ ) have blueskewed profiles. The HCO + (1-0) emission show strong blueshifted emission toward the southwest, leaving a possibility that the emission represents another blueshifted component of the gas. In fact, the possibility is supported by Position-Velocity (PV) diagrams of the HCO + (1-0) and H 13 CO + (1-0) emission shown in Figure 4. Clearly, there exists an isolated blueshifted component of the HCO + emission peaked at the offset position of ∼ −120 ′′ (Figure 4a). It should be noted that the velocity structures in the central region of r ∼ < 30 ′′ are fairly consistent with the blueskewed profiles expected for infall motion.

Discussion
Although the nature of the blueshifted HCO + emission to the southwest is not clear, our observations show that the gas motion in the central region (r ∼ < 30 ′′ ) is governed by the infall. Therefore, we focus on the discussion of the spectrum toward the core center. To give more quantitative constraints on the gas infall, we have developed a radiative transfer code to model the H 12 CO + and H 13 CO + spectra for a spherical core dominated by infall motion. The code was originally written for the CO emission from a protoplanetary disk (Omodaka, Kitamura & Kawazoe 1992). In this model, we assumed that local thermodynamic equilibrium (LTE) holds for the line excitation. To calculate the spectra, we assumed that the gas is infalling with a constant velocity (v = V inf ) for r ≤ 30 ′′ , whereas a static state (v = 0) for 30 ′′ < r ≤ 100 ′′ from Figures 2-4. We adopted the mass density radial profile reported in paper I: ρ(r) = 8.4 × 10 23 (r/200 AU) −2 cm −3 . We used a radial temperature profile of T (r) = 1000 (r/1 AU) −p K with p = 1/2, on the basis of the temperature profile at 2 × 10 4 yr after the formation of a protostar in Figure 2 by Masunaga & Inutsuka (2000b). We took a constant fractional abundance for H 13 CO + of (8.5 ± 4.7) × 10 −12 (paper I) all over the core, although this simplification may not be valid in a dynamically collapsing core (e.g., Aikawa et al. 2001). For a comparison with the observed spectra, we smoothed the calculated ones with a Gaussian function with θ HPBW = 26 ′′ and converted the flux density into the T mb (see Omodaka et al. 1992).
Figure 5 compares calculated and observed spectral profiles whose intensity scales are normalized. Here we discuss only the velocity differences between the blue-and redshifted peaks because our calculations adopted the LTE approximation, and because rigorous cal-culations (Masunaga & Inutsuka 2000b) show that the intensities of the model spectra significantly vary with the evolutionary stage of the protostar. We searched for a plausible value of V inf to reproduce the observed line profiles, and found that a velocity of 0.5 km s −1 gives the most plausible H 12 CO + line profiles. In fact, the calculated H 12 CO + spectra have a reasonable consistency with the observed ones within the errors ( Figure 5). We confirmed that the two cases of V inf = 0.3 and 0.7 km s −1 cannot reproduce the observed velocity differences between the blue-and redshifted peaks ( §3). Notice that V inf = 0.5 km s −1 agrees with an estimate from V inf ≈ 1 2 |V blue − V red |/ cos(45 • ) = 0.45 km s −1 in §3. Although the model H 13 CO + line profile has the blue and red shoulders due to the central infalling region, the overall line profile seems agree with the observed one that has a single peak at V sys owing to the outer static gas. As for p, we also tried the two cases of p = 1/3 and 1, but found that both the p values cannot reproduce the observed absorption features around V sys . Here, the p value of 1/3 is expected in the case that the dust opacity coefficient has the frequency dependence of ∝ ν 2 (Doty & Leung 1994). We addressed such a case because the overall properties of the dust in the GF 9-2 core, which is at a very early evolutionary stage, may be similar to those for the interstellar medium. In addition, the deep absorption dips around V sys require the presence of a cold static envelope surrounding the contracting region.
It is known that a density profile of ρ(r) ∝ r − 3 2 is expected for a central free-fall region onto the forming protostar. Our previous observations revealed that the profile of ρ(r) ∝ r −2 continues down to the central r ∼ 600 AU region, although it was impossible to know whether or not ρ(r) ∝ r − 3 2 exists at r ∼ < 600 AU owing to our insufficient angular resolution. Given p = 1/2 and V inf = 0.5 km s −1 , we further calculated the HCO + spectra for the case that the gas is freely falling onto the central protostar of 0.1 M ⊙ (paper I) at r < 600 AU. However, the resultant line profiles were not significantly changed compared with the above results.
Although the LPH solution well describes the on-going collapse in the GF 9-2 core (paper I), we examined a case for the Shu's solution. We assumed that the gas is freely falling toward a 0.1 M ⊙ protostar at r ∼ < 600 AU and is static at r ∼ > 600 AU on the basis of the observed density profile (paper I). The calculated HCO + spectra for the two transitions do not show blueskewed profiles, but double-peaked spectra having an absorption dip at the V sys . This would be a natural consequence by giving the compact free-fall region embedded in the large static core.
Assuming that the infall motion is radial, we can estimate the mass accretion rateṀ acc by 4πr 2 ρV inf . We obtainṀ acc of 2.5×10 −5 M ⊙ yr −1 with uncertainty of a factor ∼ 2. Since theoretical models of protostar formation (e.g., Stahler, Shu & Tamm 1980;Masunaga & Inutsuka 2000a) predict that the protostar radius, R * , should be ∼ 5 R ⊙ for a low-mass protostar, we can calculate an accretion luminosity (L acc ) for the central protostar(s) of L acc = 16L ⊙ G(M * /0.1M ⊙ )[Ṁ acc /(2.5 × 10 −5 M ⊙ yr −1 )](R * /5R ⊙ ) −1 . This seems reasonable compared with the luminosity expected for a protostar at a very early evolutionary stage (see e.g., Figure 7 in Masunaga & Inutsuka 2000a). Consequently, our study confirms that the gas in the GF 9-2 core is infalling onto the protostar(s) in the central region, and that the infalling velocity is consistent with the prediction from the LPH solution. Furthermore, we plan to perform deeper integration of these HCO + lines to examine the presence of the infalling gas in the outer region of the core.
R. S. F. gratefully acknowledges Thomas A. Bell, Thomas G. Phillips, and Ruisheng Peng for their generous help during the CSO observations. The authors sincerely thank Thomas A. Bell for a critical reading of the manuscript at the final stage of preparation. This work is partially supported by Grant-in-Aids from the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 19204020 and No. 20740113).
Facilities: Nobeyama 45 m telescope, Caltech Submillimeter Observatory 10.4 m telescope Fig. 1.-Single-dish spectra of the molecular line emission toward the GF 9-2 core center shown in the main-beam brightness temperature (T mb ) scale. The HCO + (1-0) spectrum is smoothed so as to have the effective resolution of 26 ′′ in FWHM, i.e., the CSO beam size for the HCO + (3-2) line. The RMS noise levels are 73 mK, 61 mK, and 88 mK for the HCO + (1-0), HCO + (3-2), and the HCN lines, respectively. To convert the rest-frequency (ν rest ) of the HCN hyperfine emission to the LSR-velocity (V LSR ), we adopted the systemic velocity (V sys ) of −2.48 km s −1 (paper I), and ν rest = 88630.41406 MHz for F = 1-1, 88631.84375 MHz for F = 2-1, and 88633.93750 MHz for F = 0-1. The vertical dashed line shows the systemic velocity of the cloud, V sys = −2.48 km s −1 (paper I). Fig. 2.-Overlay of the total integrated intensity maps of the H 13 CO + (1-0) emission (white contour; paper I) on that of the HCO + (1-0) emission (grey-scale with thin contour; this work) emission. The contour levels for the HCO + emission have the 3σ intervals starting from the 3σ level where σ = 0.59 K km s −1 . The HCO + emission is integrated over an LSR-velocity range between −3.60 km s −1 and −1.95 km s −1 . The central star marks the peak position of the 3 mm continuum emission (paper I). The hatched circle at the bottom left corner indicates the effective beam size of the HCO + emission, which is set to be equal to the CSO beam size for HCO + (3-2) (θ HPBW = 26 ′′ ; see §2).   -Model (green) and observed (black) spectra of the H 12 CO + (3-2), H 12 CO + (1-0), and H 13 CO + (1-0) emission toward the center of the core. The model spectra are calculated with the most plausible infall velocity (V inf ) of 0.5 km s −1 (see text). To compare with the observed profiles, the model spectra are convolved by a Gaussian function with θ HPBW = 26 ′′ (see §4).