© The Authors 2024

1 Introduction

The advent of the Atacama Large Millimetre Array (ALMA) has revolutionized the study of the interstellar medium (ISM) with its unprecedented high sensitivity and resolution. A decade of Herschel observations have highlighted the complex filamentary structure of molecular clouds at parsec scales (André et al. 2014). Within these filaments, recent ALMA observations have unravelled an unexpected physical and kinetic complexity at subparsec scales (e.g., Peretto et al. 2013; Hacar et al. 2018; Chen et al. 2019; Li et al. 2021; Sato et al. 2023; Cunningham et al. 2023), illustrating the picture of a hierarchical ISM (Hacar et al. 2024).

Resolving the sub-pc filamentary structure of star-forming regions located at kpc distances can only be performed using high-resolution interferometric observations. However, interferometric-only observations are strongly affected by intrinsic spatial filtering effects (e.g. Ossenkopf-Okada et al. 2016). Interferometry in the radio and sub-millimeter regimes relies on the aperture synthesis technique (Jennison 1958). The limited observing time leads to a sampling of the Fourier (u,v) plane which is always discrete, irregular and incomplete. The resulting interferometric visibilities do not contain information at all spatial scales but only of those Fourier components sampled by the specific baselines (distance between two antennas) available during the observation. By construction, the spatial sensitivity achieved using the aperture synthesis technique is limited by the shortest distance (baselines) between antennas within the array, roughly corresponding to the antenna diameter in modern interferometers such as ALMA. The lack of these short baselines is commonly known as short-spacing problem and critically affects the recovery of extended emission at large angular scales, compared to the interferometric beam size (Wilner & Welch 1994; Kurono et al. 2009).

The solution to the short-spacing problem is to add additional observations sampling large-scales, usually provided by a single-dish (SD) telescope, to be combined with the interferometric data (Stanimirovic 2002). This technique, known as data combination, allows us to preserve the extremely high angular resolution obtained with the interferometer together with the large-scale sensitivity given by the SD data. Several methodologies and algorithms have been developed over recent decades to perform these types of data combination, both in the Fourier and image planes (see Plunkett et al. 2023, for a recent summary of these techniques and their implementation).

In interferometric observations, the large-scale sensitivity is evaluated through the maximum recoverable scale (MRS). According to Wilner & Welch (1994), the MRS is defined as the angular size for an input Gaussian visibility distribution at which the ratio between the observed source peak brightness in absence of the short-spacing data and its true peak brightness is 1/e. In the case of ALMA observations, the MRS is defined as the largest angular scale to which the instrument is still sensitive. The ALMA Technical Handbook provides as empirical formula MRS ~ 0.983λ/L_min, where λ is the observed wavelength and L_min is the shortest baseline of the interferometer¹ (Cortes et al. 2023). The MRS plays a crucial role in the observational setup: both the choice of the interferometric arrays to use and of the array configuration depend on the largest angular scale to observe and on its comparison with the MRS. Thus, a detailed analysis of this parameter is needed in order to investigate the impact of the short-spacing effect on the quality of the observations.

As shown in different Galactic surveys (see e.g., André et al. 2010; Molinari et al. 2010; Arzoumanian et al. 2011), both filaments and cores are typically embedded in large amounts of cloud gas, showing a shallow emission profile with large contributions from the bright diffuse emission extending over several parsecs in size. The recovery of this extended emission determines key observational properties such as the radial profile, source size, and total mass of cores and filaments, and, therefore, becomes essential to interpret the physical characteristics of the ISM structure. This cloud emission at large angular scales can easily exceed the MRS of most interferometers. The absence of large-scale information in interferometric-only observations leads to an incomplete and misleading representation of the true sky emission and poses a critical challenge for both continuum and molecular line observations.

This work is part of the Emergence of high-mass stars in complex fiber networks (EMERGE) project (see Hacar et al. 2024, hereafter Paper I)². The EMERGE project will survey the internal gas organization in a series of high-mass star-forming regions using high-resolution ALMA observations combined with additional high-sensitivity SD data. In this paper, the second of its series (Paper II), we investigate the accuracy of the ALMA observations used to characterize our EMERGE survey (see also Socci et al. 2024, hereafter Paper III). Beyond the combination techniques explored by Plunkett et al. (2023), this work systematically quantifies the effect of the interferometric filtering on the recovery of different emission properties (flux, angular size, and radial dependence) used to derive the physical characteristics (mass, size, stability, evolutionary state, etc.) of relevant ISM structures such as cores and filaments. Since the lack of short-spacing information becomes apparent in resolved targets, the results of our work are most likely applicable to the study of nearby regions within 1 kpc such as Taurus and Orion typically observed using the most compact ALMA configurations (e.g., C43-1; see Paper I). Nonetheless, these results might be applicable to other Galactic targets if observed at high spatial resolution using more extended ALMA configurations.

We have evaluated the current use of data combination in ALMA observations in recent Cycles (Sect. 2). We produce and analyze synthetic ALMA observations (Sect. 3) to evaluate the ability of interferometric observations recovering fundamental emission properties of idealized cores (Sect. 4 and Appendix A), and different filamentary geometries (Sect. 5). We test different data combination methods using the three ALMA arrays and evaluated the Point Spread Function (PSF) obtained form the observations (Sect. 6). We also investigate the combination of ALMA 12 m only with a large SD such as the IRAM-30m telescope (Sect. 7) and discuss the sensitivity requirements for data combination (Sect. 7). Finally, we compare the results obtained with and without data combination, and in between different data combination techniques, with real observations (Sect. 8).

2 Data combination with ALMA

In order to improve its sensitivity at all angular scales, the ALMA observatory includes three arrays sampling different baseline distances: the ALMA 12 m main array (composed by up to fifty 12 m antennas arranged in distinct array configurations with baselines between 15 meters and 16 kilometers and sensitive to small angular scales), the ALMA Compact array (ACA)-7 m array (twelve 7 m antennas with baselines between 9 and 45 m, sampling the intermediate scales), and the Total Power Array (TP Array, comprising four SD 12 m telescopes). Alternatively, the ALMA interferometric observations (12 m-alone or 12 m + 7 m) can be combined with a different SD, always considering the u-v coverage of the different arrays should ideally overlap for a reliable data combination.

Although the short spacing problem and the necessity of data combination have been well known in the community, even today the number of ISM projects aiming for combined 12 m + 7 m + TP array observations, and 7 m + TP data in the case of ACA standalone observations, is lower than expected. To illustrate this issue, we queried the ALMA Science Archive using the ALminer toolkit (Ahmadi & Hacar 2023). We searched for individual Group Observing Unit Sets (GOUS) in ALMA Cycles 6-9 (years 2018-2022) and selected those completed projects within the ISM category requesting mosaic observations. Our conservative selection of mosaics ensures that these observations are aiming for targets with extended emission larger than the ALMA 12 m (or 7 m) primary beam, and, therefore, beyond the interferometric MRS, which most likely require data combination.

We show the percentage of GOUS with 12 m, 7 m, and TP data per year in Fig. 1. Even in the case of large 12 m mosaics obtained with the ALMA main array (left panel), the number of interferometric-only observations (12 m and/or 7 m data; green triangles) accounts for >60% of the observed GOUS over these four Cycles, including >40% of GOUS requesting 12 m-only observations (blue squares) and ~25-30% of GOUS including 12 m+7 m data (orange triangles). In contrast, the number of GOUS requesting 12 m+7 m+TP (red circles) in the last years has been consistently below ~50% indicating that the severity of these effects has not been completely internalized. The situation is slightly better in the case of ACA standalone observations (right panel) showing an apparent increasing trend of ISM projects using ACA mosaics requesting 7 m+TP GOUS in the last years (red circles), although ~30% of them still requested 7 m-only data by 2022 (cyan triangles). The results shown from this sub-sample can potentially affected a much larger number of ALMA projects using single 12 and 7 m pointing observations in crowded galactic environments. Accordingly, the aim of this study is to quantify these effects to raise awareness in the community.

Fig. 1 Observed GOUS with ALMA mosaics for the ISM category over ALMA Cycles 6-9 (years 2018-2022). Left panel: percentage of GOUS observed with ALMA obtained with the 12 m (blue squares), 12 m + 7 m (orange triangles), and the 12 m + 7 m + TP (red circles) arrays, respectively. The green triangles represents all the GOUS requested without SD observations (i.e., 12 m and/or 12 m+7 m). Right panel: percentage of GOUS observed with ACA in standalone mode including 7 m-only (cyan triangles) and 7 m+TP data (red circles). Fluctuation of 10–20% are expected from Poisson statistics given the typical number of GOUS considered per cycle (~40 per year).

3 ALMA simulations

To quantify the impact of the spatial filtering effects on different ISM structures we create a series of synthetic observations of idealized Gaussian (cores) and cylinders (filaments) using the Common Astronomy Software Application (CASA, CASA Team et al. 2022). All these synthetic objects are placed at the center of a 6.25 arcmin wide field located at coordinates RA = 12h00m00.0s and Dec = −23d00m00.0s, the optimal position in the sky that can be observed by ALMA in terms of source elevation and overall sensitivity (although not necessarily in terms of u-v coverage).

We produced models characterized by a single frequency channel to avoid taking into consideration velocity effects. We selected as the central frequency 100 GHz, close to the frequency of the N₂H⁺ (1−0) molecular line, the most used dense gas tracers for cores and filaments (Bergin & Tafalla 2007), and representative of standard the continuum ALMA observations at 3 mm (Band 3). We produce simulations of the three different ALMA arrays separately. We perform our observations using the most compact configuration offered in ALMA 12 m array, C43-1, providing an angular resolution θ_res(12 m) of 3.38 arcsec and MRS θ_MRS(l2 m) of 28.5 arcsec. At these same frequencies, the ACA-7 m array provides a θ_res(7 m) and θ_MRS(7 m) of 12.5 arcsec and 66.7 arcsec, respectively, while the TP antennas the θ_MRS(TP) goes down to 62.9 arcsec³. Since the primary beam (PB) of the ALMA antennas (~58 arcsec for a 12 m antenna) is smaller than the full-width-half-maximum (FWHM) of the largest target (FWHM = 100 arcsec), we observe our targets using mosaics. We run our simulations in CASA v.6.5.2 (CASA Team et al. 2022) using the task simobserve to simulate the visibility file and tclean for the imaging⁴. We adopted the same observational setup for all the synthetic datasets.

3.1 Generation of visibilities with simobserve

As first step in our simulations, and for each of our science targets, we generated their corresponding ALMA visibilities using the task simobserve. Taking a model image in FITS format as input, simobserve simulates the expected visibility dataset (MeasurementSet) obtained for a specific observational setup described by the following parameters: mapsize, angular size of the mosaic map to simulate; maptype, position of the pointings for the mosaic observation; pointingspacing, spacing in between pointings; integration, integration time for each pointing; totaltime, total time of observation or number of repetitions.

For the 12 m array simulations, we mapped the central 3.5 arcmin region of our fields as indicated in the ALMA Cycle 10 Technical Handbook (Cortes et al. 2023). We covered this area using a 67-pointings mosaic following a Nyquist-sampled, hexagonal pointing pattern. Each pointing in our mosaic was observed in two iterations (repetitions), 30 sec each, for a total of 1 min integration per pointing, typical for this type of ALMA observations.

For ACA-7 m observations, the mapped area should be larger than the one observed with the 12 m array by at least half of the PB; therefore, we set it to 4.5 arcmin in size. We thus obtained a 33-pointings mosaic, selecting as integration time 30 s and 14 repetitions. We repeated the observations for three consecutive days over transit mimicking a standard ALMA observing schedule. Our choice of the integration time per pointing reproduces the expected time ratio C43-1 : ACA-7 m = 1 : 7 following the official recommendations to match the sensitivity of the different ALMA arrays (Cortes et al. 2023).

3.2 Generation of images with tclean

In a second step, we image each simulated visibility dataset using the CASA task tclean. This task is based on the CLEAN algorithm (Högbom 1974), the default method for obtaining an image from an interferometric observation, deconvolving it from the instrumental point spread function (PSF). We set mfs as spectral definition mode (continuum imaging with only one output image channel) and selected a standard Briggs weighting with a robust parameter equal to 0.5. We set the pixel size as 0.5 arc-sec and 2 arcsec (~1/6 · θ_res) for the 12 and the ACA-7 m arrays, respectively. To select the cleaning threshold, first we produced a dirty image and from it we estimated its emission peak, then setting the threshold at 10% of the value and the number of maximum iterations at 10⁸. Afterwards, we corrected the images for the PB attenuation using the task impbcor setting a cutoff value of 0.85. We independently imaged each of the 12 and the 7 m arrays, then we produced combined images using different combination techniques (see below). Finally, we convolved the 12 and ACA-7 m array images to obtain a symmetric synthesized beam using the task imsmooth. The final beam sizes are 4.5 arcsec and 16 arcsec for the 12 and ACA-7 m array, respectively.

Fig. 2 Schematic view of our simulation process.

3.3 Simulation of SD images with imsmooth

As third and last step, aimed at simulating SD observations to be combined with the ALMA 12 and ACA-7 m array, we used the task imsmooth, which convolves the synthetic image at the required resolution. We produced simulations of TP observations at a resolution θ_res(TP) of 62.9 arcsec.

3.4 Data combination methods

To overcome the short-spacing problem, we tested different data combination methods already implemented in CASA (see Plunkett et al. 2023, and references therein). We show the workflow followed by our simulations in Fig. 2. For convenience, we briefly describe the different methods here.

We started by combining the two interferometric visibilities (ALMA 12 m + ACA-7 m array), merging small and intermediate spatial scales. We used the CASA task concat, which concatenates several visibility datasets, applying weights of 1 and 0.116 to the simulated 12 and 7 m visibilities, respectively, to take into account the different dish diameters (this is only nedeed for simulated data, see Plunkett et al. 2023). We obtained a single interferometric MeasurementSet that we later imaged (joint-deconvolution) using tclean. We refer to this combined interferometric image as 12 m, hereafter.

Secondly, we explore the Feather method (Cotton 2017) to combine the ALMA 12 m + 7 m interferometric datasets with the corresponding TP map. The Feather method is implemented in CASA as the feather task. This algorithm works in the image plane: it converts both interferometric (high-res) and SD (low-res) images to the visibility plane, combines them in the Fourier domain, and transforms them back into the Feather image. We set the sdfactor, a parameter used to adjust the flux scale of the SD image, to 1.0 (usually constrained between 1.0 and 1.2). We refer to this image as ALMA Feather.

Thirdly, we also tested the Model-Assisted CLEAN plus Feather Method (MACF, Hacar et al. 2018; Plunkett et al. 2023). This technique is a variant of Feather, which is actually used also here to combine the interferometric dataset with the SD image. As main difference, the MACF method first introduces the SD image as source model during the cleaning process of the interferometric visibilities (using the startmodel parameter), informing tclean about the location and brightness of the extended emission sampled by the SD (for a more detailed discussion, see Plunkett et al. 2023). We refer to this image as ALMA MACF.

3.5 Source brightness

The integrated sky brightness (i.e., flux density), F_v, in continuum is determined by the total dust column density, N(H₂), proxied by the H₂ column density assuming a gas-dust coupling, of the target source. Following Kauffmann et al. (2008), the conversion between the dust column density and flux density at a given frequency, v, is described by: $$F_v=N\left({\text{H}}_2\right){\text{Ω}}_A{\mu }_{{\text{H}}_2}{m}_H{k}_v{B}_v\left(T_d\right),$$(1)

where ${\mu }_{{\text{H}}_2}=2.8$ is the mean molecular weight per hydrogen molecule, ${\text{Ω}}_A=2.655\times {10}^{-11}sr{\left(\frac{\text{Δ}{x}_{\text{pix }}}{arcsec}\right)}^2$ the solid angle for a pixel size of ∆x_pix = 0.25 arcsec in our models, and $$B_v\left(T_d\right)=1.475\times {10}^{-23}\frac{W}{m^{2}Hzsr}{\left(\frac{v}{\text{GHz}}\right)}^3\times \frac{1}{e^{{}^{0.0048\left(\frac{v}{\text{GHz}}\right)\left(\frac{10K}{T_d}\right)}}-1}$$(2)

is the Planck function given a dust temperature, T_d. As for the dust emissivity, we adopted a standard k_v = 0.01 cm²g⁻¹ at 300 GHz (1 mm), which translates into k_v = 0.002 cm²g⁻¹ at 100 GHz (3 mm) assuming a frequency dependence v^β with β = 1.6 typical for filaments and cores in Orion (Sadavoy et al. 2016; Mason et al. 2020).

By exploring a series of simplified source geometries and array combinations (Sects. 4–7), we aim to isolate the effects of the interferometer response and quantify its effects on the quality of the observations. In order to simulate realistic ISM conditions, we adopted a source peak intensity of I₀ = 7.5 × 10⁻³ mJy pix⁻¹ in all our synthetic models. According to Eq. (1), this flux density corresponds to an equivalent dust column density of N(H₂) ~ 1.5 × 10²³ cm⁻² (or A_V ~ 150 mag) at v = 100 GHz and assuming a constant T_dust = 15 K, similarly to the maximum peak column densities found in cores and dense filaments in active star-forming regions such as Orion (e.g., Hacar et al. 2018). While undoubtedly bright, these high peak fluxes guarantee a bright emission in our synthetic observations in order to investigate the impact of different interferometric filtering effects.

3.6 Thermal noise

To clearly identify interferometric artefacts such as sidelobes it is convenient to run most of our simulations without adding thermal noise (noise-free). However (and with an aim to investigate these effects under realistic observing conditions), we also produced (noisy) simulations, including atmospheric thermal noise assuming standard weather conditions for Band 3 with precipitated water vapor (PWV) of 1.8 mm (5th PWV octile according to the ALMA Handbook; Cortes et al. 2023). We included noise in our ALMA simulations (Sect. 3.1) by setting the parameter thermalnoise as tsys-atm in the task simobserve. Likewise, we added an additional thermal noise to our SD data using the radiometer formula with similar T_sys values as the interferomet-ric data and the corresponding integration times (see Sect. 7).

4 Observing isolated cores: Single Gaussian analysis

Dense cores are the sites of stellar birth (Benson & Myers 1989). They can be identified as compact objects within molecular clouds in (sub-)millimetre continuum emission (André et al. 2014). Cores are characterized by a roundish shape with a typical diameter of 0.03–0.2 pc (Bergin & Tafalla 2007), that is, ~ 13–90 arcsec at the Orion distance (~420 pc; see Menten et al. 2007). Resolved at the SD resolution in nearby clouds such as Taurus (~140 pc), dense cores are routinely studied in continuum showing typical radii of 20–75 arcsec using SD bolometers (e.g., Tafalla et al. 2002). Cores exemplify the simplest geometry to be recovered by the interferometer and so, they are the first targets in our study.

Despite their favourable conditions, previous ALMA observations of nearby star-forming regions reveal the difficulties of interferometers recovering the full emission profiles of dense cores. Continuum interferometric-only observations in local molecular clouds (~ 130–200 pc) such as Chamaeleon (Cycle 1, 12 m-alone; Dunham et al. 2016), Ophiuchus (Cycle 2, 12 malone Kirk et al. 2017), or Taurus (Cycle 4–6, ACA 7 m-alone Tokuda et al. 2020) result in a few core detections with compact radii of several times the θ_res value (~5 arcsec for the 12 m data and ~ 15–20 arcsec in the case of 7 m data). These core radii are significantly smaller than the expected (deconvolved) sizes estimated from the corresponding SD maps in these fields (see Fig. 4 in Kirk et al. 2017). Despite the high sensitivity of their maps, most of these studies also report a high number of non-detections (e.g., 54 non-detections out of the 73 fields observed by Dunham et al. 2016).

In contrast, the combination of 12 m+7 m data shows a significant improvement of the amount of extended emission recovered (e.g., Tokuda et al. 2016) and a systematically increased core radius in dedicated studies of centrally condensed cores in Taurus (see Fig. 1 in Caselli et al. 2019, for an example). Similar improvements after data combination are also seen in ALMA surveys of dense cores in Orion despite their smaller angular size (e.g., Ohashi et al. 2018; Dutta et al. 2020). Characterizing the effects of interferometric filtering is therefore of paramount importance to assess the accuracy of the core properties (mass, size, and column density) derived from these observations.

The column density profiles of cores have been described using different power-law models (Alves et al. 2001; Tafalla et al. 2002). For simplicity, however, we decided to model our cores as 2D Gaussians of different sizes (defined by their full width half maximum, FWHM) described by $$G(x,y)=I_0\exp \left(-\frac{{\left(x-x_0\right)}^2}{2{\sigma }_x^2}-\frac{{\left(y-y_0\right)}^2}{2{\sigma }_y^2}\right),$$(3)

where I₀ corresponds to the core peak intensity, x₀ and y₀ its central coordinates, and σ_x, σ_y are its two spatial dispersions which can be expressed in terms of the core FWHM as ${\sigma }_x={\sigma }_y=\frac{FWHM}{2\sqrt{2\text{\hspace{0.17em}ln\hspace{0.17em}}2}}$. We produced a sample of several different synthetic cores with FWHM = 5, 10, 15, 20, 30, 40, 50, and 70 arcsec.

We produced noise-free ALMA simulations of all our cores using different arrays and combinations. In Fig. 3, we show the results for cores with different increasing FWHM (form left to right) of 5 arcsec (similar to θ_res(12 m)), 15 arcsec, 30 arcsec (θ_MRS(l2 m)), and 70 arcsec, along with different combinations (from top to bottom) all convolved to the same resolution and shown within the same flux range to facilitate their comparison. The top row (panel A) displays the synthetic reference images used as input.

We show the ALMA 12 m C43-1-only simulations on the second row of Fig. 3 (panel B). The interferometer filtering effect is clearly visible in these images as the core size grows. At 5 arcsec (~θ_res(12 m), first panel), the interferometer is able to reproduce the source with high fidelity. However, as the FWHM of the core increases, the interferometer is not able to recover the total emission of the source anymore (the object becomes fainter) and, in addition, negative sidelobes surrounding the emission become progressively more and more prominent. Interestingly, the flux losses and negative sidelobes become already apparent at FWHM ~ 15 arcsec, which is at scales of <θ_MRS(12 m). We note that the most significant effect is seen at 70 arcsec, where the core is no longer visible due to the heavy filtering at large scales.

The recovery of the shape and flux of the emitting source improves with the data combination. We show our ALMA 12 m + 7 m array simulations on the third row of Fig. 3 (panel C). The overall fraction of emission recovered at intermediate angular scales increases for FWHM values between 15 and 30 arcsec. Also, a faint emission is now visible in case of FWHM = 70 arcsec.

However, we notice that the negative sidelobes are still prominent and present a less circular pattern due to the addition of the 7 m data. The ALMA TP contribution is added in the lower rows of Fig. 3. We show the ALMA Feather and ALMA MACF simulations in panels D and E rows, respectively. The large scale sensitivity allows us to recover the emission even at 70 arcsec (last column) as to reduce the image artifacts stemming from PSF sidelobes.

As representative example of the observed core behavior, we display the radial profile extracted from a horizontal cut in the center of a FWHM = 30 arcsec core in Fig. 4. In the left panel, we show the profile in linear scale compared to the reference (input) peak value (I₀), while on the right, we give it in logarithmic scale with values relative to the peak of each profile (I_peak). As shown by the ALMA 12 m-alone profile (blue line), the value recovered at the center is less than 50% of the one in the reference (left panel). The radial profile of the ALMA 12 m-alone data also decreases faster than the one in the reference (see also right panel) and shows clear negative sidelobes at around 25 arcsec, namely, it is not a scaled-down version of the reference profile. On the other hand, the ALMA 12 m + 7 m array (orange line) recovers higher flux values (~80%) and accurately reproduces the shape of the profile up to ~15 arcsec; beyond that, it slightly diverges decreasing faster than the reference one. Compared to the interferometric images, addition of the SD data in the ALMA Feather and MACF (pink and green lines, respectively) significantly improves the flux recovery ≥90% across the entire profile with only small (<10%) deviations at large radii. The quantitative comparison of the different radial profiles illustrates how the lack of the short-spacing information could potentially affect the emission properties recovered by interferometers, such as ALMA, even for simple sources such as our idealized Gaussian cores.

Fig. 3 Noise-free simulations of isolated cores with FWHM0 = 5, 15, 30, and 70 arcsec (left to right). The first row shows the synthetic reference images used as input for the simulations (panel A). The rows below display the results for different methods of data combination. ALMA 12 m-alone (panel B), ALMA 12 m + 7 m (panel C), ALMA Feather (panel D), ALMA MACF (panel E), and ALMA 12 m + IRAM-30 m MACF (panel F), shown form top to bottom. The size of each image is 2.8 arcmin × 2.8 arcmin.

4.1 Quantifying the effects of the short-spacing information in simple geometries

We further quantify the impact of the intrinsic spatial filtering of the interferometer and of data combination on the recovery of the true properties of the emitting source based on the characterization of the above radial profiles, this time for all our cores with different angular sizes. In Fig. 5, we measure the fraction of recovered flux (top panel), peak flux (I; middle), final FWHM (bottom) with respect to the input model (reference) for the different interferometric (inverted triangles) and combined (circles) reductions.

We first compare the fraction of total flux recovered by our different reductions in Fig. 5 (top) as function of the original core FWHM₀. Even for objects with a FWHM₀ ~ θ_res(12 m) (grey dotted line), the ALMA 12 m-alone (blue) is not able to recover the total emission of the source (already ~90%) and, as the size of the object grows, the fraction of recovered emission decreases dramatically. Around the θ_MRS(12 m) (grey dashed line in Fig. 5), the ALMA 12 m-alone is only able to recover only up to 20% of the emission, a smaller value compared to the expected 1/e definition by Wilner & Welch (1994). The ALMA 12 m-alone is clearly affected by the spatial filtering effect, in particular by the lack short-spacing information, not being able to recover emission at large angular scale (half of the total emission lost in cores with FWHM ~ 20 arcsec < θ_MRS(12 m)).

The ALMA 12 m + 7 m profile (shown in orange in Fig. 5) shows a similar decreasing trend, although much shallower. Adding data sensitivity to intermediate scales helps recover a much larger fraction of the core emission (>70% up to sizes of ~ θ_MRS(12 m)). Still, the flux recovery continues to decrease for cores with larger radii down to scales comparable to θ_MRS(7 m), where ALMA 12 m + 7 m array is losing up to 90% of the core emission. Combining ALMA 12 m plus 7 m data has improved the fraction of emission recovered especially at intermediate scales, but the interferometers-only approach is still affected by significant spatial filtering effects beyond 30 arcsec.

In contrast, the addition of the SD information shown by ALMA Feather and MACF profiles (displayed in pink and green in Fig. 5, respectively) always recovers a fraction >80% independent of the core size. Both profiles overestimate the fraction of recovered emission at small scales (FWHM ≤ 20 arcsec) up to a 10% factor and show a decreasing trend towards larger sizes. Overall, the addition of large scale contributions improves the recovery of the emission of objects at all scales (up to 90% at θ_MRS(12 m)), allowing us to estimate the true flux of the sources within a ~20% uncertainty. The ALMA Feather and MACF show similar but not identical profiles. However their differences are always below 10%, a level of discrepancy that would probably not be detected if the observational and instrumental noise would be added.

We further fit the resulting radial profile with a Gaussian function to estimate the intensity peak and the FWHM, similar to observations. We show the normalized intensity peak recovered for cores of different sizes in Fig. 5 (middle). The behaviour of different methods of data combination is similar to the recovered emission along the entire profile. We estimate the peak values recovered by the ALMA 12 m-alone (blue) at θ_MRS(12 m) to be around 40% (>20%, the fraction of recovered emission at the same scale). The two results are compatible since the fraction of recovered emission takes into account also the negative sidelobes that are not included in the fitting process, resulting in lower values when accounting for the emission along the entire profile. However, even if the effects are less severe for the intensity peak, short-spacing issues are still present and visible, especially at large scales. Similar trends are shown by the ALMA 12 m + 7 m (orange), the ALMA Feather (pink) and MACF (green) with respect to the fraction of recovered emission. The lack of short-spacing information remains noticeable in our interferometric-only (12 m + 7 m) dataset, and it is only mitigated by the combination with the ALMA TP (ALMA Feather and MACF).

To investigate the effects on the recovered source size, we also display FWHM values estimated from the Gaussian fit in respect to the reference value FWHM₀ in Fig. 5 (bottom). In case of a perfect recovery, the observed FWHM of our cores is expected to follow the initial FWHM₀ convolved with the beam resolution (added in quadrature), as shown by the grey solid line in Fig. 5. While the values recovered by the ALMA 12 m-alone (blue) for marginally resolved cores with FWHM₀ ~ 5–10 arcsec follow the expected theoretical prediction, the observed FWHM is systematically underestimated for larger FWHM₀ and saturates to a constant value at FWHM₀ > 30 arcsec. In the case of cores with sizes on the order of θ_MRS(12 m), the estimated FWHM is around 15 arcsec, almost a factor of 2 lower than FWHM₀. As shown by the ALMA 12 m-alone profile, the spatial filtering is not only acting on the flux recovery, but also on the estimate of the object FWHM. The lack of intermediate and large scale observations is critically affecting the recovered FWHM, systematically producing smaller cores.

In comparison, the ALMA 12 m + 7 m results (marked in orange in Fig. 5) are in better agreement with the theoretical predictions up to FWHM₀ ~ 30 arcsec. Still, this method again underestimates the FWHM of cores with FWHM₀ ≳ 40 arcsec, up to a factor of 50% at 70 arcsec (the 7 m array ≳_MRS(7 m)), limited again by the filtering effect at large scales. As already seen for the fraction of recovered emission, the addition of short-spacing information in the ALMA Feather and ALMA MACF methods (pink and green dots, respectively) allows for a better estimate of the true FWHM to be obtained. The resulting FWHM values closely reproduce the expected theoretical growth (grey line) within ~5% error.

Fig. 4 Radial profile extracted from a horizontal cut in the center of a FWHM = 30 arcsec Gaussian core image normalized with respect to the reference image peak (shown by the grey shaded area) displayed in linear scale (left). Same radial profile normalized with respect to its peak value (the reference profile is shown by the grey shaded area) displayed in log-scale (right). The data combination methods used are marked in different colors: ALMA 12 m-alone (blue), ALMA 12 m + 7 m (orange), ALMA Feather (pink), ALMA MACF (green) and ALMA 12 m + IRAM-30 m MACF (black).

Fig. 5 Reconstructed properties of isolated Gaussian cores. Top panel: percentage of emission recovered along the entire radial profile vs. the FWHM0. Different colors are used for the different data combination methods used: blue for the ALMA 12 m-alone, orange for the 12 m + 7 m, pink for the ALMA Feather, green for the ALMA MACF, and black for the ALMA 12 m + IRAM-30 m MACF. The grey dotted and dashed lines show θres(12 m) and θMRS(12 m), respectively. Middle panel: intensity peak estimated from the Gaussian fit vs. FWHM0. Bottom panel: FWHM estimated from the Gaussian fit vs. FWHM0. The grey solid line shows the theoretical expectation. For comparison, the results for the radial profile shown in Fig. 4 with FWHM0 = 30 arcsec correspond to the values close to the θMRS(12 m).

4.2 Additional biases in realistic interferometric observations

The above issues found in interferometric-only datasets could potentially hamper the detection of cores in real observations. Figure 6 shows the recovered emission of cores with FWHM = 5, 15, 30, and 70 arcsec in our ALMA 12 m-alone (panel A) and 12 m+7 m (panel B) datasets, this time using our noisy simulations with PWV = 1.8 mm (see Sect. 3.6). The reduction of the peak intensity in cores of increasing FWHM in interferometric-only observations (see Fig. 5) effectively reduces their signal-to-noise ratio (S/N). Eventually buried within the noise, these effects can severely affect the completeness of dense core surveys biasing their detection rates towards compact, unresolved sources (i.e., cores with FWHM₀ ≲ θ_MRS(12m)).

More subtly, the observational signatures of the interfero-metric filtering could go unnoticed in real observations. While clearly visible in our previous noise-free images (see Figs. 3 B and C for comparison) negative sidelobes can also be hidden within noisy images giving the false impression of a non-detection in a high sensitivity dataset. A comparison between the real core radius (yellow dotted circles) and the emission contour corresponding with a S /N = 3 in our images (cyan contours) demonstrate how hidden negative emission features can inadvertently alter the recovered FWHM of partially resolved sources if these effects were ignored.

Despite their simplicity, our synthetic experiments illustrate how spatial filtering effects can critically bias the derived core properties in case of interferometric-only observations. ALMA 12 m-alone datasets are expected to produce artificially lowintensity, narrow cores in continuum (Fig. 5) explaining many of the observational properties and non-detections obtained in previous ALMA studies (see above). Only the addition of the short-spacing information (at least 12 m+7 m although ideally 12 m+7 m+TP) can guarantee a reliable detection rate and estimate of the actual core emission properties (i.e., total flux, peak flux, and FWHM) with an accuracy better than 10% (e.g., see Caselli et al. 2019).

The above interferometric biases can be particularly severe in the case of ALMA observations of resolved targets in nearby clouds (e.g., Dunham et al. 2016). However, similar effects are expected in more distant targets when observed with extended ALMA configurations if FWHM₀ > θ_MRS(12m). These observational biases should be considered when characterizing physical properties derived from the observed fluxes such as the core mass, peak column density, and size. Given the broad range of column densities and sizes reported for cores in nearby star-forming regions (e.g., Könyves et al. 2020), a careful consideration of these biases is essential when deriving statistical distributions, such as the core mass function, using interferomet-ric observations (see Appendix A for a discussion). In case of doubt, observers should use the full ALMA 12 m+7 m+TP array capabilities when targeting these sources.

5 Observing isolated filaments: effects on elongated geometries

Herschel far-IR surveys (André et al. 2010; Molinari et al. 2010) have revealed the presence of a network of filamentary structures permeating the ISM. The analysis of recent continuum maps, provides the first systematic and homogeneous measurements of key physical properties such as the filament mass, radial profile, and width (Arzoumanian et al. 2011, 2019; Palmeirim et al. 2013; Könyves et al. 2015). Using Herschel observations down to 18 arcsec resolutions, filaments appear to be described by a characteristic typical width of ~0.1 pc (Arzoumanian et al. 2011, 2019). However, molecular line ALMA observations at resolutions of 4.5 arcsec unravelled an unexpected physical and kinetic complexity of filamentary networks of the ISM (e.g., Peretto et al. 2013; Hacar et al. 2018; Shimajiri et al. 2019; Chen et al. 2019). Following Hacar et al. (2018), the filamentary structure identified by Herschel in the two Orion Molecular Clouds OMC-1 and OMC-2 appears to be a collection of small-scale filaments, the so-called fibers, characterized by a narrower width of ~0.03 pc detected using ALMA observations of dense tracers such as N₂H⁺ (1–0). Since interferometric resolutions are needed to resolve filaments up to ~0.03 pc (or 14 arcsec at the distance of Orion), we aim to quantify in this section the impact on the spatial filtering effects on the analysis of these fine gas sub-structures.

Fig. 6 Interferometric ALMA 12 m-alone (panel A) and 12 m+7 m (panel B) simulations of isolated cores with FWHM0 = 5, 15, 30 and 70 arcsec (from left to right) including thermal noise with PWV = 1.8 mm. In each image, we highlight the emission ≥3σ (cyan contours) compared to the core FWHM0 (yellow dotted circles).

5.1 Single Gaussian filament analysis

For simplicity, we first simulated the observation of a simple object elongated towards one direction. We generated noise-free, synthetic observations of an infinitely long filament with a 1D-Gaussian radial profile along the x-axis (similar to the profile of the core, see Sect. 4) as follows: $$G(x)=I_0\exp \left(-\frac{{\left(x-x_0\right)}^2}{2{\sigma }^2}\right),$$(4)

with I₀ as the peak intensity, x₀ position of the peak, x the impact parameter and $\sigma =\frac{FWH{M}_0}{2\sqrt{2\text{\hspace{0.17em}ln\hspace{0.17em}}2}}$, and a constant profile along the y-axis. We modeled Gaussian filaments with FWHM = 5, 10, 15, 20, 30, 40, 50, and 70 arcsec.

In Fig. 7, we show the results of four representative filaments with FWHM of 5 arcsec (similar to θ_res(12 m)), 15 arcsec, 30 arcsec (~θ_MRS(12 m)), and 70 arcsec (panel A) observed using different interferometric observations (panels B-C) and data combination methods (panels D-E), all display within the same intensity range. Qualitatively, the results are similar to those found for cores (Fig. 3) in which the inclusion of first 7 m data (panel C) and later TP observations (e.g., ALMA Feather) systematically improves the recovery of the true sky emission with respect to the 12 m-alone simulations (panel B). Compared to our previous core simulations, however, the simulation of filamentary structures appear to produce larger filtering effects and more prominent sidelobes.

We also display the radial profile extracted from an horizontal cut in the center of a FWHM = 30 arcsec Gaussian filament in Fig. 8, both in linear scale and absolute units (left panel) as well as in relative units and in logarithmic scale with respect to the peak of each profile (right panel). The results obtained for elongated structures are noticeably more dramatic with respect to cores of similar FWHM (see Fig. 4). As shown by the ALMA 12 m-alone profile (blue line in Fig. 8), the recovered peak is only around 20% (a factor of 2 lower than in the cores) and there is a negative sidelobe of comparable intensity to the positive emission. The ALMA 12m + 7m array (orange line in Fig. 4) is recovering 40% of the peak value (80% for the core), still showing a negative sidelobe around 30 arcsec. Only the ALMA Feather and MACF (pink and green lines in Fig. 8, respectively) profiles are closer to the reference one although not even these combination methods are able to recover the 100% of the peak flux in filaments. The ALMA Feather and MACF profiles also show deviations from the reference Gaussian profile at large radii (clearly visible around 35 arcsec in the right panel of Fig. 8). The more severe filtering effects seen on these filaments with respect to Gaussians of similar FWHM suggest that the long dimension in elongated emission features introduce additional artefacts affecting the flux recovery at all scales.

5.2 Effects on elongated geometries

To quantify the emission properties of our filaments, we extracted a radial profile from a cut along the x axis of each image, as done for cores. First, we estimated the total flux recovery integrating the total emission of each filament along their radial entire cut for filaments with different FWHM₀ values in Fig. 9 (top). Even for the narrowest filaments (FWHM₀ ~ 5 arc-sec), the interferometer-only observations (blue solid line) is only able to recover ~70% of the total emission of the source and, as the FWHM₀ grows, the fraction of recovered emission decreases also faster than in cores (see Fig. 5). Around FWHM₀ ~ θ_MRS(12 m) (grey dashed line), the ALMA 12 m-alone is losing almost 95% of the emission (note that the object is barely visible in Fig. 7). The ALMA 12m-alone is clearly affected by the spatial filtering effect at all scales, given it is not able to recover the total emission even at smaller sizes. Adding intermediate-scale data (ALMA 12 m + 7 m; orange solid line) contributes to recover a larger fraction of emission at all cases. However, the improvement is much smaller than for the cores, only recovering ~20% of the emission around θ_MRS(12 m). Even in this case it is clear how combining small and intermediate scales is not enough to fully recover these elongated structures. Only the ALMA Feather and MACF (in pink and green lines, respectively) show constant profiles always recovering a fraction ~100% of the total emission despite the filament size. These two profiles show the effects of combining interferometric data with SD observations, allowing for nearly the total flux to be recovered, even at scales where the interferometric contribution is almost zero.

We continue fitting the extracted radial profile with a Gaussian function to estimate the intensity peak and the observed FWHM. The intensity peak normalized by the reference one I/I₀ is shown in Fig. 9 (middle). The behaviour of different data combinations is similar to what was observed for the recovered emission along the entire profile (see above). Both ALMA 12 m-alone (blue solid line) and ALMA 12 m + 7 m (orange solid line) data satisfactorily recover the peak emission (>95%) in the case of unresolved filaments (FWHM = 5 < θ_res(12 m)). On the other hand, both methods fail in the case of broader filaments as seen by their monotonically decreasing performance at larger FWHM₀. As expected, the filtering effects are more severe in the case of ALMA 12 m-alone, although these issues are clearly visible in both datasets. In comparison, the ALMA Feather (pink solid line) and ALMA MACF (green solid line) reductions show significant and consistent improvements at all scales. Still, the peak recovery at large FWHM values drops to ~70–80% in both cases around 40 arcsec, growing again towards larger scales. Although secondary, we also notice that the ALMA MACF overestimates the peak intensity by few percent in the case of filaments with FWHM₀ ≲ 15 arcsec. While the effects of the spatial filtering are still visible in all interferometric datasets, these issues are heavily reduced by the combination with the short-spacing information provided by the ALMA TP data⁵.

Finally, we display the FWHM recovered from our Gaussian fits compared to the inputs values in our simulations in Fig. 9 (lower panel). Only the values recovered by the ALMA 12 m-alone (blue triangles) for a source FWHM₀ ≤ 15 arcsec lie on the expected prediction, while for larger objects the value of the recovered FWHM is systematically underestimated and becomes close to constant above FWHM₀ ~ 30 arcsec. At θ_MRS(12m) the estimated FWHM is around 10 arcsec, almost a factor of 3 lower than the reference value. The ALMA 12 m + 7 m points (orange dots) recover a slightly larger FWHM value closer to the theoretical prediction, but where at FWHM₀ ~ θ_MRS(l2m), the FWHM is already underestimated by 30%. The spatial filtering is thus affecting both the flux recovery and the object size. Filaments observed with only the interferometer appear to be artificially narrower than expected.

In summary, filtering effects in elongated (filament-like) structures appear to be more severe than in the case of cores of similar FWHM₀ because the overall fraction of power at larger scales is much higher for filaments than for circular cores. These spatial contributions can significantly affect the observational (integrated intensity, peak intensity, and FWHM) and physical properties (i.e. total column density, peak column density, and radial distribution, respectively) derived in interferometric studies (e.g., see Fig. 1 in Wong et al. 2022).

Fig. 7 Noise-free simulations of isolated Gaussian filaments with FWHM0 = 5, 15, 30, and 70 arcsec (left to right). The first row shows the synthetic reference images used as input for the simulations (panel A). The rows below display the results for different methods of data combination. ALMA 12 m-alone (panel B), ALMA 12m + 7m (panel C), ALMA Feather (panel D), ALMA MACF (panel E), and ALMA 12 m + IRAM-30m MACF (panel F), shown from top to bottom. The size of each image is 2.8 arcmin × 2.8 arcmin.

Fig. 8 Radial profile extracted through a horizontal cut in the center of a FWHM0 = 30 arcsec Gaussian filament image normalized with respect to the reference image peak (shown by the grey shaded area) displayed in linear scale (left). Same radial profile this time normalized with respect to the corresponding observed peak value in each map Ipeak displayed in log-scale (right). In both panels the reference profile is shown by the grey shaded area. The results of the distinct data combination methods are indicated with different colors: ALMA 12 m-alone (blue), ALMA 12m + 7 m (orange), ALMA Feather (pink), ALMA MACF (green), and ALMA 12 m + IRAM-30 m MACF (black).

5.3 Finite filaments

Compared to the arbitrarily elongated structures simulated in Sect. 5, the real filaments have finite lengths of L. To investigate how this finite dimension is reproduced by interferometric observations, we decided to simulate a series of elongated objects with different aspect ratio (AR = L/FWHM).

We modeled these new targets as elongated 2D Gaussians of different lengths, described via Eq. (3), where we independently vary σ_x and σ_y describing the characteristic FWHM and length, L, of the object, respectively. For simplicity, we fixed the FWHM of our synthetic structures to 15 arcsec (σ_x) and vary only their length L (σ_y). Our choice for a constant FWHM =15 arcsec (similar to θ_res(12 m)) is justified as the characteristic width in which our previous core and filament models still recover most of the target properties but start diverging in their results (see Figs 3, and 7). We produced ALMA simulations for different AR (or L), ranging from a core-like structure (AR = 1) to an infinite-like filament (AR = ∞), using different arrays and combinations (see Fig. B.1).

Similarly to our previous analysis, we extracted the main radial properties of our simulations from a cut along the x axis of each image and display these results in Fig. 10. As expected, both the ALMA 12 m-alone (blue lines) and ALMA 12 m+7 m (orange lines) observations produce worse results than those datasets including TP information (i.e., Feather or MACF shown by pink and green lines, respectively) in terms of flux recovery (upper panel), peak intensity (mid panel), and recovered FWHM (lower panel). More interesting, Fig. 10 smoothly connects the results obtained in the two limiting cases explored in previous sections. Most variations in terms of flux, peak intensity, and FWHM already occur in mildly elongated core-like structures with AR ~ 2 – 3, while no further losses appear in more filamentary structures above AR > 5.

Although secondary with respect to the FWHM (see above), our results demonstrate how changes in the target’s longest dimension, L, can aggravate the interferometric filtering even in structure with small AR. This is particularly relevant since dense cores usually show prolate geometries with AR ~ 1.5–2 (Myers et al. 1991) inducing an additional source of uncertainty (~20–40% extra losses) to previous observational biases (e.g., Sect. 4.2).

Fig. 9 Reconstructed properties of isolated Gaussian filaments. Top: Percentage of emission recovered along the entire radial profile vs. FWHM0 Different colors are used for the different data combination methods used: blue for the ALMA 12 m-alone, orange for the 12 m + 7 m, pink for the ALMA Feather, green for the ALMA MACF, and black for the ALMA 12 m + IRAM-30 m MACF. The grey dotted and dashed lines show θres(12m) and θMRS(12m), respectively. Middle: Intensity peak estimated from the Gaussian fit vs. FWHM0. Bottom: FWHM estimated from the Gaussian fit vs. FWHM0. The grey solid line shows the theoretical expectation. For comparison, the results for the radial profile shown in Fig. 8 with FWHM0 = 30 arcsec correspond to values close to θMRS(12 m).

5.4 Filaments with Plummer-like profiles

Since filaments observed with Herschel show a characteristic Plummer radial profile (Arzoumanian et al. 2011), we decided to reproduce these more realistic radial profiles in our analysis of filaments. We modeled the radial profile as constant along the y-axis and a Plummer-like profile along the x-axis, described as: $${N\left({\text{H}}_2\right)|}_p(r)=A_p\frac{{\rho }_c{R}_{\text{flat }}}{{\left[1+{\left(r/R_{\text{flat }}\right)}^2\right]}^{\frac{p-1}{2}}},$$(5)

where $A_p=\frac{1}{\cos i}{{{\displaystyle \int }}^{\text{}}}_{-\inf }^{+\inf }\frac{\text{d}u}{{\left(1+u^2\right)}^{p/2}},{\rho }_c$ is the density at the center, R_flat is the characteristic radius of the flat inner portion of the profile, and i is the inclination angle assumed to be equal to 0 for simplicity (see Arzoumanian et al. 2011).

After investigating how the spatial filtering affects the flux recovery and the FWHM estimate in the case of Gaussian filaments, in Sects. 5.1–5.3, we decided to focus on the effects on the profile slope. Thus, we produced two noise-free, Plummer-like filaments with p = 2.5 and 4 as representative values for shallow and steep filaments, respectively (see Hacar et al. 2024), both with radii of R_flat = 30 arcsec and constant peak flux value equivalent to I₀ = A_p × ρ × R_flit = 7.5 × 10⁻³ mJy pix⁻¹. Compared to Sect. 5, the selected p values describe the power-law dependence of a steep, Ostriker-like filament, similar to a Gaussian (p = 4; Ostriker 1964), and along with the much shallower Plummer-like variations similar to those reported by Herschel (p = 2.5, see Arzoumanian et al. 2011). Accurately recovering these filament profiles, whose differences are only noticeable at large radii (r ≫ FWHM), is essential to determine the evolutionary state of these structures (Pineda et al. 2023).

In Fig. 11, we display the radial profiles extracted from a horizontal cut in the two filaments with p = 2.5 (top) and p = 4 (bottom). We show both the normalized intensity I/I₀ (in linear scale; left panels). The observation of these Plummer-like profiles retrieves worse results than Gaussian-like filaments of similar angular size (Sect. 5.1). These new shallower profiles enhance the effective filtering reducing the observed peak intensity in our ALMA 12 m-alone (blue line) and ALMA 12 m + 7 m profile (orange) simulations, particularly in the case of p = 2.5, where we recovered only 5% of the original peak. In addition to their poor performance, we note that interferometric-only observations may also introduce significant sidelobes with intensities comparable to the main filament peak (see secondary peaks in ALMA 12 m-alone and ALMA 12 m + 7 m datasets. On the other hand, only combination methods such as ALMA Feather (pink line) and ALMA MACF (green line) get closer to the true emission profiles of these filaments, albeit not in a complete satisfactory way (see Sect. 6 for a further discussion).

More importantly and as highlighted in the normalized I/I_peak plots in Fig. 11 (right), the previously reported flux looses have a large impact on the resulting radial profiles measured in these filaments. In order to better evaluate the profile’s slope, different dashed red lines show the expected Plummer dependence for the respective p = 2.5 or 4 values (solid red line) and a representative steeper profile with p = 5 (dashed red line). The recovered ALMA 12 m-alone profiles are much sharper than the original ones showing power-law dependencies significantly steeper than p = 5. In addition to their large flux differences, filtering severely impacts the resulting radial profiles producing artificially sharp filaments in all interferometric-only observations (both ALMA 12 m-alone and ALMA 12 m + 7 m). Recent results illustrate the impact of these interferometric filtering effects in ALMA observations of different filamentary structures in the ISM (Klaassen et al. 2020; Yamagishi et al. 2021; Díaz-González et al. 2023; Hacar et al. 2024; Tachihara et al. 2024). These observational biases should be considered in future ALMA studies aiming to characterize the radial dependence of the ISM filaments at high spatial resolution (e.g., Paper III).

Fig. 10 Reconstructed properties of noise-free isolated objects with FWHM0 = 15 arcsec and aspect ratio AR = 1 (core), 2, 5, and ∞ (infinite filament). From top to bottom: Fraction of recovered flux; intensity peak estimated from the Gaussian fit; and FWHM estimated from the Gaussian fit. Different colors are used for the different data combination methods used: ALMA 12 m-alone (blue), 12m + 7m (orange), ALMA Feather (pink), ALMA MACF (green), and ALMA 12 m + IRAM-30 m MACF (black). The grey dotted and dashed lines show θres(12 m) and θMRS(12 m), respectively. The grey solid line in the lower panel shows the theoretical expectation (FWHM0 = 15 arcsec).

Fig. 11 Radial profile extracted through a horizontal cut in the center of a p = 2.5 (top panels) and p = 4 (bottom) Rflat = 30 arcsec Plummer profile. Left: profile is normalized with respect to the reference peak and displayed in normal scale. Right: profile normalized with respect to its peak value displayed in log-scale. The data combination methods use are marked in different colors: ALMA 12 m-alone (blue), ALMA 12 m + 7 m (orange), ALMA Feather (pink), ALMA MACF (green), and ALMA 12 m + IRAM-30 m MACF (black). The reference profile is marked by the gray shadowed area and ideal profiles for different p-values are displayed by red dashed lines.

Fig. 12 uv-coverage of one of the cores simulated in Sect. 3. Different colors represent the uv-sampling obtained by the different ALMA arrays: (blue) ALMA 12 m, (red) ACA 7 m array, and (orange) TP. We also highlight the uv-area covered by the IRAM-30 m SD telescope (grey area) and its overlap with the ALMA 12 m array (black dashed area).

6 PSF analysis

In addition to the source geometry, the recovery of the true sky emission distribution is limited by the corresponding PSF in interferometric observations. The interferometer PSF is the inverse Fourier transform of the u-v components sampled during the observation. The number of antennas, its configuration, and the integration time during an observation determine the u-v coverage and, therefore, the resulting PSF. Considering an ideal case with an uniformly sampled and homogeneous u-v coverage, the PSF would appear as a 2D Gaussian with a FWHM value defining the interferometer resolution. However, since the u-v sampling is always discrete and non homogeneous, realistic PSF can largely depart from the idealized Gaussian functions. Combined with the missing short-spacing information (central hole in the uv-plane), observations carried out with a limited u-v coverage can lead into a PSF with prominent sidelobes that can severely hamper the deconvolution process.

In Fig. 12, we show the u-v components measured during the observations of one of our Gaussian cores (see Sect. 4). Since we adopted the same observational setup for all simulations (see Sect. 3), the u-v coverage and the resulting PSF are similar in all the cases presented so far. Thanks to large number of antennas, the ALMA 12 m array (blue) presents a dense instantaneous (snapshot) coverage that populates the u-v space regularly and homogeneously down to baselines of ~14m (i.e., the shortest baseline in this setup; see Sect. 3). In contrast, the more limited number of ACA 7 m antennas (red) create sparser u-v components usually distributed in a highly non-symmetric pattern (elongated toward the v-direction in our simulations) also leaving large u-v holes between points (e.g., see (u,v)~(20,30) meters). Given their different observational setups, we therefore expect strong departures between the PSF of the 12 and 7 m arrays.

In Fig. 13 we show the image (top) and a representative radial cut of the interferometric PSF (bottom) obtained in our ALMA 12 m-alone (left), 7 m-alone (middle) and 12m + 7m (right) simulations. In Fig. 14 we compare the aforementioned PSF radial profiles. The ALMA 12 m-alone PSF (top left panel) shows a quasi-Gaussian shape (b_maj = 4 arcsec, b_min = 3.5 arcsec) surrounded by a concentric, regular and symmetric sidelobe pattern. In comparison, the ALMA 7 m-alone PSF (top middle panel in Fig. 13) presents a more elongated quasi-Gaussian shape (b_maj = 14.5 arcsec, b_min = 10.3 arcsec) surrounded by an irregular and asymmetric pattern of brighter sidelobes. We extracted a radial profile at the center of the ALMA 12 m PSF image. We plotted it in log-scale (bottom panels) and compared them with their corresponding ideal Gaussian PSF such as θ = b_maj (gray dashed area). As expected, the ALMA 7 m-alone PSF is significantly worse than the ALMA 12 m-alone PSF, showing an earlier departure from gaussianity as well as brighter sidelobes (~15% vs. ≤ 2% of the PSF peak).

Naively, we would expect the combination of ALMA 12 m + 7 m baselines to improve the overall PSF shape. As shown in the right panels in Fig. 13, the 12 m+7 m combined PSF shows a well-behaved Gaussian-like dependence at small radii with a reduced influence of the sidelobes seen in the 7 m-alone data. However, the addition of 7 m data also introduces a strong and unexpected, but prominent sidelobe at scales around 6 arcsec, and around the overlapping baselines shown in Fig. 12, with an estimated peak intensity around 7%.

The observed ALMA 12 m + 7 m PSF can lead into the redistribution of the emission during the deconvolution process of bright extended sources by placing additional flux on the side-lobes. This effect might affect the results of our ALMA Feather and ALMA MACF simulations, and likely explains their discrepancies observed in the flux peak and FWHM recovery with respect to the expected sky emission (also around 10–20%; see Figs. 5 and 9).

Deconvolution algorithms such as CLEAN efficiently mitigate most of the above PSF issues in the case of simple source geometries. These improvements are however more limited when targeting fields with complex emission features such as those explored in our EMERGE survey (see Paper I). Depending on the source structure, relatively minor sidelobes (~10%) could lead into complex dirty images from which CLEAN can no longer recover. While a full analysis of these PSF effects is outside the scope of this paper, optimizing the PSF quality appears crucial for achieving interferometric images with accurate flux measurements better than 20% in ISM studies.

7 Data combination with a large SD: ALMA + IRAM-30 m observations

The analysis presented in this work demonstrates how the inclusion of the short-spacing information using data combination systematically improves the quality of the data, thereby allowing us to retrieve the true properties of the both cores (Sect. 4) and filaments (Sect. 5) observed with ALMA. However, the results of the ALMA Feather and MACF methods combining the three ALMA arrays are still not ideal. Indeed, differences in the recovered flux peak (underestimated) and FWHM (overestimated) are clearly visible in Fig. 11 (among others), this discrepancy can be partially attributed to the resulting PSF (see Sect. 6). Thus, we wanted to explore whether the ALMA 12 m-alone would benefit more from the combination with a larger SD compared to the ALMA one. Such improvements are expected by the more homogeneous uv-coverage obtained by a large SD with respect to a small interferometer, such as the ALMA (ACA) 7 m array (see Plunkett et al. 2023).

We simulate the observations of a SD telescope larger than the ALMA TP (12 m antennas) to be combined with the ALMA 12 m array, replacing the ACA-7 m and the TP. We chose to reproduce the 30-meter Institute de Radioastronomie Millimetric telescope (IRAM-30 m), commonly used in star-formation studies also included as part of our EMERGE survey (see also Sect. 8). To simulate an IRAM-30 m observation, we convolved the synthetic image at the resolution of 25 arcsec (~ resolution of IRAM-30 m telescope at 100 GHz) using the CASA task imsmooth and combined it with ALMA 12 m array observations using the MACF method (see Sect. 3), hereafter referred to as ALMA 12 m + IRAM-30 m observations.

In agreement to our predictions, the ALMA 12 m + IRAM-30 m MACF appears as the best data combination method. A quantitative analysis of their results indicate that the ALMA 12 m + IRAM-30 m MACF observations accurately reproduce the total flux, peak intensities, and FWHM of both Gaussian cores (Fig. 5) and filaments (Figs. 9 and 11) better than any previous combination within the ALMA arrays (ALMA Feather or ALMA MACF) and within less than ~10% with respect to the actual model values. Considering Plummer-like filaments (Fig. 11), arguably the most challenging targets for interferometers given their shallow profiles (p = 2), the ALMA + IRAM-30 m MACF profile is the only method closely reproducing the synthetic filament. Even if the ALMA Feather and MACF reproduce the true slope in profiles with both p = 2.5 and p = 4 up to ~30 arcsec as accurately as the ALMA + IRAM-30 m MACF, the latter is still the only one recovering the intensity peak at the center.

The reason for these improvements is two-fold. First, a large SD provides a high sensitivity across a broader range of scales. Second (and with more relevance with respect to ALMA), the use of a SD provides a uniform uv-coverage of the short-spacing baselines that (unlike the ACA) does not introduce additional sidelobes that perturb the analysis (see Sect. 6).

Overall, the analysis performed in this work suggests the combination of ALMA 12 m with a larger SD (e.g., IRAM-30 m) allows us to achieve better results compared to using ALMA ACA-7 m array + TP, recovering the true properties of the sources despite of their size and shape. The combination with a large SD appears essential to achieve images with high fidelity even with densely populated interferometers such as ALMA (see also Hacar et al. 2024). An enhanced performance is expected for future (50m-class) SD telescopes such as the Atacama Large Aperture Submillimetre Telescope (AtLAST) telescope (Klaassen et al. 2019), particularly in the case of ISM studies (Klaassen et al. 2024).

Fig. 13 Analysis of the interferometer PSF of different ALMA array configurations. From left to right: ALMA 12 m-alone , 7 m-alone, and ALMA 12 m + 7 m. We display the PSF intensity image for each simulation (top) and an intensity profile extracted along a horizontal cut in log-scale (bottom). In the bottom panels, the colored lines show the simulated PSF profiles (see dotted yellow lines in mid-panels) obtained for the ALMA 12 m-alone (blue), 7 m-alone (red) and 12 m + 7 m (orange), while the expected Gaussian PSF profile at the correspondent resolution is marked by the shaded grey area.

Fig. 14 Comparison of the interferometer PSF intensity profile extracted from an horizontal cut for the ALMA 12 m-alone (blue), 7 m-alone (red) and 12m + 7 m (orange). The resolution of the 12 m array (dotted line), and the MRS of both the 12 m (dashed line) and 7 m (dot-dashed line) arrays are indicated by vertical lines.

On the SD sensitivity for data combination

When combining data between interferometric arrays and SD, a good standard of practice is to match the sensitivity for SD data to the one of the interferometer within the range of overlapping baselines (see Mason & Brogan 2013). This requirement translates in longer integration times per pointing for the SD compared to the interferometric observations. Given the large telescopes timespans required (e.g., 1:11.9 for the 12m:TP data in C43-1; Cortes et al. 2023), it is interesting to test whether this theoretical requirement can be actually relaxed.

In order to pursue this goal, we make use of our synthetic datasets including thermal noise (Sect. 3). Following Mason & Brogan (2013), we determined the expected ratio of integration time per pointing (τ) necessary to match the sensitivity between the ALMA 12 m array (int) and the IRAM-30 m telescope (SD) as follows⁶: $$\frac{{\tau }_{\text{SD}}}{{\tau }_{\text{int }}}={\left(\frac{D_{\text{int }}}{D_{\text{SD}}}\right)}^4\frac{2N_{\text{bas }}}{N_{\text{SD}}},$$(6)

with D the diameters of the antennas, N_SD number of SD antennas and N_bas number of overlapping baselines (i.e.,the number of baselines falling within the SD diameter). Considering the ALMA 12 m array in configuration C43-1 in Cycle 9 and IRAM-30 m, τ_SD ~ 4τ_int. Given the integration time ratio, we calculated the equivalent (white) noise in our IRAM-30 m simulations following the radiometer equation. We adopt a standard Gaussian core with FWHM₀ = 30 arcsec as illustrative example were data combination becomes essential (i.e., ~60% of flux losses in the 12 m-alone observations; Sect. 4). We combined the ALMA 12 m-alone data and the IRAM-30 m data using the MACF method. We simulate these observations setting the SD noise as 1, 5, and 10 times the above theoretical predictions.

Different panels in Fig. 15 display the results of the IRAM-30 m (panel A), ALMA 12 m-alone (panel B), and ALMA 12 m + IRAM-30 m MACF combination (panel C), respectively. Two features become apparent in these images. First, the increase of the SD noise rapidly degrades the IRAM-30 m images (top) which also translates in the systematic increase of the noise in the combined MACF images (bottom). Second, and more important, a moderate increase of the SD noise level (up to five times) does not affect the recovery of the extended emission unless the SD image is heavily corrupted (e.g., ten times the expected noise). We illustrate these ALMA 12m + IRAM-30 m MACF source profiles shown in Fig. 16. Even when the noise significantly increases, the recovered characteristics of the target source (i.e., flux peak, FWHM, and radial profile) remain stable (within 10–20%).

The above results could be explained by different contributions of the interferometric and SD data during combination. Our ALMA interferometric observations determine the sensitivity at most spatial scales given the amount of ALMA 12m baselines (including several overlapping with the scales traced by the SD data) and drive the final noise level in the combined maps. On the other hand, SD data mostly provide information of the low frequency (spatial) components. These (Fourier) components are still recognizable in images with high frequency noise levels (Fig. 15A) and therefore can still efficiently contribute to the recovery of extended emission (Fig. 15C)⁷.

Our analysis might have interesting consequences for future ALMA observations. Since the image noise σ_RMS ∝ τ^−1/2, the SD observing times could be reduced without significantly compromising the final outcome of data combination. Also, the addition of short-spacing information appear to have a positive effect on the final data products, even if the original SD data are of lower quality than expected.

8 Comparisons with real ALMA observations

We compare our synthetic simulations with real ALMA observations to validate our results. As target for these comparisons we selected the OMC-3 star-forming region in Orion A, part of the EMERGE Early ALMA Survey (see Paper I). OMC-3 shows a prominent and complex substructure of filaments and cores becoming an ideal testbed for our simulations. The dense gas content of OMC-3 has been investigated using N₂H⁺ (1–0)(Band 3) observations carried out with the ALMA 12 m-array (alone) in its most compact configuration (C43-1) during ALMA Cycle 7, achieving a native resolution of θ_beam ~ 3.5 arc-sec (proj. ID: 2019.1.00641.S; PI: Hacar). The ALMA 12 m array observations have been combined with additional IRAM-30m N₂H⁺ (1–0) large-scale maps (proj. IDs: 032-13, 120-20), using both the Feather and MACF methods and are referred to as ALMA 12 m + IRAM-30 m Feather and MACF, respectively. The observations, the data reduction and imaging procedures, and the data combination technique used are presented in Hacar et al. (2024). Interestingly, the noise level of our IRAM-30m observations (Hacar et al. 2017) are higher than those expected according to theoretical expectations (see Sect. 7).

We show the integrated intensity maps of all interferometric-only (ALMA 12 m-alone) and combined datasets (ALMA 12m + IRAM-30m Feather and MACF) in different panels in Fig. 17. The comparison between these maps clearly demonstrates how the ALMA 12 m-alone one (left panel in Fig. 17) is not able to recover the diffuse emission, showing narrower and fainter filamentary structures surrounded by negative emission. This is in agreement with the predictions of our simulations.

We have characterized the emission in these maps from the analysis of the 23 small-scale filaments (aka fibers; marked by segments on the Fig. 17, middle) identified in this region (see Paper III for additional details). To compare the results on these data with the simulated ones, we extract several radial cuts perpendicular to each fiber axis and analyse their radial profiles. Figure 18 shows a statistical comparison of the flux (top), peak flux (I₀, middle), and FWHM (bottom) recovered in these images. Without a prior knowledge of the true emission distribution, and given the results in Sect. 7, we assume the ALMA 12m + IRAM-30 m MACF observations as reference and compare other results normalized against it. Black lines in each panel show the prediction obtained from the simulations on Gaussian cores (AR = 1; dotted lines) and filaments (AR = ∞, dashed lines), respectively. These new observational results are in close agreement with those in our simulations. The ALMA 12 m-alone image (blue triangles) clearly depart from the expected values roughly following the trends identified in our simplified filament test cases (Sect. 5). On the other hand, only after data combination, the ALMA 12 m + IRAM-30 m Feather data (yellow dots) is able to recover the actual properties of filaments of different sizes FWHM₀.

Despite being overall in agreement with our filamentary predictions (dashed grey lines), our ALMA 12 m-alone results (blue triangles) show a significant scatter of a factor of ~2. This effect might be partially explained by the different aspect ratio (AR; Sect. 5.3) of the structures identified in OMC-3 (see Paper III), which would move these points upwards in these diagrams closer to the distribution expected for cores (dotted grey lines). The expected AR variations appear to be partially responsible of the observed variations in terms of flux recovery (top panel) and FWHM (bottom). Additional discrepancies are nonetheless expected given the simplicity of our synthetic toy models compared to the complex emission features seen in real ISM observations (see Paper I).

We remark here that all simulated and observed datasets are carried out using the ALMA 12 m array in its most compact configuration (C43-1) in Band 3 (3mm). Although this presents significant issues, we note that these low-frequency observations in compact configurations represent the most favourable scenario to recover the extended emission seen in complex star-forming regions such as OMC-3. More severe issues are expected to affect observations at higher frequencies and/or in more extended configurations (e.g., Díaz-González et al. 2023). Additional work is needed to quantify these effects using different ALMA baselines and bands. However our study is sufficiently generic to assert that any observation of objects with extended emission will benefit from combination with an observation with a large SD.

Fig. 15 Simulations of isolated cores with FWHM0 = 30 arcsec. From left to right: noise-free simulations, and simulations with noise levels equal to one, five, and ten times the theoretical noise (assuming PWV = 1.8 mm). From top to bottom: IRAM-30m simulations (panel A), ALMA 12 m-alone (panel B), ALMA 12 m + IRAM-30 m MACF (panel C).

Fig. 16 Radial profile extracted from an horizontal cut in the center of a FWHM0 = 30 arcsec Gaussian cores shown in Fig. 15 C.

Fig. 17 N2H+ (1–0) integrated intensity maps of OMC-3 using three different data combination methods. From left to right: ALMA 12m-alone map; ALMA 12 m array + IRAM-30 m combined using the Feather with the identified filament axes over-plotted (yellow; see also Paper III); and ALMA 12 m array + IRAM-30 m combined using the MACF method.

9 Summary and conclusions

Our aim in this work is to investigate and quantify the impact of the ALMA instrumental response when characterizing the physical properties and gas organization of the ISM. With cores and filaments at different scales, observations of star-forming regions are strongly affected by the interferometric filtering effect and especially by the short-spacing problem (Sect. 1). Although the scientific community is aware of these interferometric issues, the analysis of the results in the ALMA Science Archive suggest that data combination still has not yet been fully implemented in recent ISM observations. To explore the effects of the short-spacing information on the characterization of the physical structure of the ISM with ALMA, we investigated a series of CASA simulations (Sect. 3) and quantified the effects of data combination recovering the emission properties (total flux, peak flux, radial profile, and FWHM) of different core-like (Sect. 4) and filamentary structures (Sect. 5). We explored how the interferometric PSF (Sect. 6) and the use of large SD telescopes (Sect. 7) can affect these analyses. Finally, we also compared our synthetic observations with real ALMA data (Sect. 8).

We summarize our main results as follows:

• 1.

  We explored and compared targets showing different profiles (Gaussian and Plummer-like) and sizes defined by their FWHM, similar to observations. Part of the target emission profile found at larger radii can easily exceed the maximum recoverable scale θ_MRS of the interferometer producing significant filtering effects. As result, interferometric-only observations alone are not able to reproduce the properties of cores and filaments, even when their typical FWHM is below the MRS.

• 2.

  The interferometric intrinsic filtering affects the quality of the observations in two ways: first, it produces large flux losses (~70–80% around the θ_MRS(12m) for Gaussian filaments), leading to strongly underestimated column densities and masses; and secondly, it systematically underestimates the FWHM of the object (by a factor of 2 around the θ_MRS(12 m) for Gaussian filaments). As a result, sources observed with interferometers appear narrower and fainter than what they actually are.

• 3.

  The effect of the ALMA instrumental response depends on the geometry of the source. We demonstrated the filtering effects of the interferometer are much more severe on elongated objects with a large aspect ratio, AR (filamentary), than in more symmetric Gaussian (i.e., core-like with most changes occurring in structures with AR = 2 – 3).

• 4.

  The use of any technique of data combination allows us to get closer to the real physical properties. Among the data combination techniques explored in this work, the MACF procedure seems to give better results compared to feathering, although these differences are within 10%. Observers should use the full ALMA capabilities and apply for 12 m+7 m+TP observations.

• 5.

  The combination of ALMA 12 m array with a large SD (e.g., IRAM-30 m) produces quantitatively better results than the ALMA (ACA) 7m + TP data, especially in the case of filamentary regions. The interferometric dataset will benefit from the combination even if the SD does not perfectly match the expected sensitivity of the ALMA 12 m array observations.

While we have explored in the case of ISM studies using compact ALMA configurations, these filtering effects are intrinsic to all interferometric observations. Additional studies are needed to quantify these effects in other science cases and ALMA configurations: however, our study is sufficiently generic to state that any observation of objects with extended structures exceeding ~0.5 the nominal MRS of the interferometer in size will benefit from being combined with an observation with a large SD telescope. This work demonstrates that data combination as a necessary technique to recover the sky emission in ISM studies exploring complex star-forming regions, especially for nearby molecular clouds within 1 kpc distance such as Taurus and Orion. We will adopt these data techniques as standard procedure in future works of this EMERGE series (e.g., see Paper III).

Fig. 18 Properties of real filaments (total flux, peak flux, and FWHM) identified in the N2H+ (1−0) integrated intensity maps of OMC-3 (see Fig. 17). All panels display the values obtained from the ALMA 12m-alone (yellow dots) and ALMA 12m+ IRAM-30m Feather (blue triangles) compared to those obtained form the combination of the ALMA 12 m array + IRAM-30 m using the MACF (flux, I0, and FWHMo) method assumed as reference. A perfect agreement between these methods is indicated by a grey line. Predictions form simulations are displayed by the grey dashed line. From top to bottom: Percentage of emission recovered along the entire radial profile; intensity peak FWHM estimated from the Gaussian fit; and FWHM estimated from the Gaussian fit. In all cases, the values are represented as function of the reference filament FWHM0 derived from the ALMA 12 m array + IRAM-30 m MACF map.

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 851435). This paper makes use of the following ALMA data: ADS/JAO.ALMA#2019.1.00641.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work is based on IRAM-30 m telescope observations carried out under project numbers 032-13, 120-20. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).

Appendix A Observing the core mass function

Fig. A.1 Analysis of the core mass function (CMF). The synthetic Salpeter-like distribution used as input is marked by the grey-shadowed histogram. The blue and green line histograms describe the CMF recovered using the ALMA 12 m-alone and the ALMA MACF, respectively. The black dashed line displays the expected Salpeter dependence (α = 1.35), while the dotted line is the one that better describes the much shallower, top-heavy slope (α = 1.18) observed by the interferometer-only

Fig. A.2 Analysis of the FWHM distribution recovered in the CMF. Blue and green line histograms describe the FWHM recovered using the ALMA 12 m-alone and the ALMA MACF, respectively.

The CMF describes the mass distributions of cores in molecular clouds prior the formation of stars. It is a crucial diagnostic for investigating the origin of the initial mass function (IMF; see Salpeter 1955) and to test different theoretical models for star formation (e.g., Kroupa 2001; Chabrier 2003). The mass dependence of the CMF can be described by a power-law dependence, such as: $$\frac{dN}{d\log M}\propto M^{-\alpha }.$$(A.1)

Observational low-resolution studies in nearby low-mass, star-forming regions in the last two decades have revealed strong similarities between the slopes of CMF and IMF distributions (both with α = 1.35; Salpeter 1955), where the former is typically shifted a factor of 3 with respect to the latter, suggesting the IMF shape may be inherited from the CMF (e.g., Motte et al. 1998; Testi & Sargent 1998; Alves et al. 2007; Könyves et al. 2015). On the other hand, high-resolution ALMA observations of high-mass, star-forming regions report significantly shallower CMF slopes α < 1, known as top-heavy (Motte et al. 2018; Kong 2019), challenging the direct relation between CMF and IMF.

Most of these recent ALMA studies are based on interferometric-only observations. Among others, the largest core survey to date has been obtained by the ALMA-IMF Large Program (Motte et al. 2022), a survey of 15 nearby massive protoclusters up to a resolution of ~ 0.01 pc mapped in dust continuum at 1.3 and 3 mm. Approximately 700 cores have been detected in this ALMA-IMF program presented in Motte et al. (2022) and identified as Gaussian sources in their continuum maps (see also Pouteau et al. 2023). ALMA-IMF only uses ALMA 12 m-alone data (ALMA 7 m data exist, wherea SD data not available for continuum), while combination was no possible due to issues of inconsistent quality across the sample (see Ginsburg et al. 2022)). Given the filtering effects seen in those Gaussian sources explored in Sect. 4, it is crucial to assess the reliability of the CMF observed in these (and similar) ALMA studies.

We aim to characterize the ALMA instrumental response on a standard CMF observed using the most compact configuration (C43-1) of the ALMA main array at a distance of Orion (D=414 pc). Our choices for both distance and telescope configuration maximize the interferometric recovery and are therefore meant to show the most favourable case for this type of ALMA studies (e.g., Dutta et al. 2020). We created a synthetic CMF drawing 500 core mass values following Eq. A.1 with a standard Salpeter-like slope of α = 1.35 within a mass range between [0.5,10] M_⊙, typical for dense cores, and assign them random FWHM values from a uniform distribution between 5 (similar to θ_beam(12 m)) and 30 arcsec (~ θ_MRS(12 m)), which correspond to physical sizes of 0.01 and 0.06 pc at the selected distance. Given the mass, M, and the FWHM (in terms of spatial dispersion $\sigma =\frac{FWHM}{2\sqrt{2\text{\hspace{0.17em}ln\hspace{0.17em}}2}}$) values for each core, we determined the associated H₂ column density (peak intensity of our cores) following: $$N\left(H_2\right)=\frac{M}{m_H{\mu }_{H_2}2\pi {\sigma }^2},$$(A.2)

and convert them into their corresponding flux densities, F_v following Eq. 1 and assuming a homogeneous dust temperature of T = 10K.

For each individual core we simulate their corresponding ALMA 12 m, 7 m, and TP observations following the same procedure already presented in Sect. 3. We then extracted the individual peak intensity, FWHM, and masses of each target and investigate the resulting combined CMF after applying different data combination techniques.

We display the distinct recovered CMFs (line histograms) and compared them with our input distribution (grey histogram) in Fig. A.1. The black dotted and dashed lines included in the plot do not represent a fit of the distributions: rather, these are merely representative slopes that can describe the data to better compare them. The CMF observed using only the ALMA 12 m-alone (blue histogram) shows a shallower distribution (α = −1.18) shifted towards lower masses with respect to the input one (grey shaded area with α = −1.35). This result can be explained as a combination of the flux losses and the FWHM deviations reported for individual Gaussians in Sect. 4, leading to the systematic underestimation of all core masses in large statistical samples producing apparent top-heavy CMF.

As primary driver of the above deviations, we illustrate the distribution of FWHM in Fig. A.2 (changes in flux peak are minor and thus not shown here). The distribution recovered by the ALMA 12 m-alone (blue histogram) is truncated at ~ 18 arc-sec with respect to the original (grey histogram) and the one recovered by the ALMA MACF method (green histogram). In agreement with Fig. 5 (bottom panel), the recovered FWHM of cores ≳ 20 arcsec is systematically narrower than expected. Moreover, these interferometric effects could also effectively reduce the detection rates in ALMA core surveys since many targets could be shifted below the mass completeness threshold of these observations (e.g., ~ 1 M⊙ in Pouteau et al. 2023). Despite their simplicity, our mock observations demonstrate how observational artefacts such as interferometric filtering can critically bias the estimates of the CMF slope even in the most compact ALMA configurations.

Our findings reinforce the conclusions drawn by Padoan et al. (2023), using more realistic models of clouds and radiative transfer calculations. These authors found that core masses inferred from ALMA observations and, thus, the observed CMF, are highly unreliable due to the combination of projection, filtering, and temperature effects. As result, Padoan et al. (2023) recovered a shallower power-law tail but shifted instead towards higher masses which the authors attribute to the additional effects of background subtraction. Our simulations demonstrate how most of these observational biases are already present under the idealized conditions and can be directly attributed to pure interferometric filtering.

Compared to the above interferometric-only results, we display the CMF recovered by our ALMA MACF observations in Fig. A.1 (green distribution). The simulations illustrate the improvement on the overall flux (and therefore mass) recovery in our sample. The resulting CMF obtained using the ALMA MACF combination reproduces (within the noise) the expected Salpeter-like slope across the entire dynamic range of masses of interest. Our results demonstrate how the combination of zero-spacing information is crucial for obtaining reliable estimates of the core masses in star-forming regions observed with interferometers even when these cores are partially unresolved.

Appendix B Elongated geometries: synthetic observations

Fig. B.1 Noise-free simulations of isolated objects with FWHM0=15 arcsec and different aspect ratios AR. From left to right: AR=1 (core), 2, 5, and ∞ (infinite filament). From top to bottom: Reference model (panel A), and simulated ALMA 12 m-alone (panel B), ALMA 12m + 7m (panel C), ALMA Feather (panel D), ALMA MACF (panel E), and ALMA 12 m + IRAM-30 m MACF (panel F) observations, respectively.

References

All Figures

Fig. 1 Observed GOUS with ALMA mosaics for the ISM category over ALMA Cycles 6-9 (years 2018-2022). Left panel: percentage of GOUS observed with ALMA obtained with the 12 m (blue squares), 12 m + 7 m (orange triangles), and the 12 m + 7 m + TP (red circles) arrays, respectively. The green triangles represents all the GOUS requested without SD observations (i.e., 12 m and/or 12 m+7 m). Right panel: percentage of GOUS observed with ACA in standalone mode including 7 m-only (cyan triangles) and 7 m+TP data (red circles). Fluctuation of 10–20% are expected from Poisson statistics given the typical number of GOUS considered per cycle (~40 per year).

In the text

	Fig. 2 Schematic view of our simulation process.
In the text

Fig. 3 Noise-free simulations of isolated cores with FWHM0 = 5, 15, 30, and 70 arcsec (left to right). The first row shows the synthetic reference images used as input for the simulations (panel A). The rows below display the results for different methods of data combination. ALMA 12 m-alone (panel B), ALMA 12 m + 7 m (panel C), ALMA Feather (panel D), ALMA MACF (panel E), and ALMA 12 m + IRAM-30 m MACF (panel F), shown form top to bottom. The size of each image is 2.8 arcmin × 2.8 arcmin.

In the text

Fig. 4 Radial profile extracted from a horizontal cut in the center of a FWHM = 30 arcsec Gaussian core image normalized with respect to the reference image peak (shown by the grey shaded area) displayed in linear scale (left). Same radial profile normalized with respect to its peak value (the reference profile is shown by the grey shaded area) displayed in log-scale (right). The data combination methods used are marked in different colors: ALMA 12 m-alone (blue), ALMA 12 m + 7 m (orange), ALMA Feather (pink), ALMA MACF (green) and ALMA 12 m + IRAM-30 m MACF (black).

In the text

Fig. 5 Reconstructed properties of isolated Gaussian cores. Top panel: percentage of emission recovered along the entire radial profile vs. the FWHM0. Different colors are used for the different data combination methods used: blue for the ALMA 12 m-alone, orange for the 12 m + 7 m, pink for the ALMA Feather, green for the ALMA MACF, and black for the ALMA 12 m + IRAM-30 m MACF. The grey dotted and dashed lines show θres(12 m) and θMRS(12 m), respectively. Middle panel: intensity peak estimated from the Gaussian fit vs. FWHM0. Bottom panel: FWHM estimated from the Gaussian fit vs. FWHM0. The grey solid line shows the theoretical expectation. For comparison, the results for the radial profile shown in Fig. 4 with FWHM0 = 30 arcsec correspond to the values close to the θMRS(12 m).

In the text

	Fig. 6 Interferometric ALMA 12 m-alone (panel A) and 12 m+7 m (panel B) simulations of isolated cores with FWHM0 = 5, 15, 30 and 70 arcsec (from left to right) including thermal noise with PWV = 1.8 mm. In each image, we highlight the emission ≥3σ (cyan contours) compared to the core FWHM0 (yellow dotted circles).
In the text

Fig. 7 Noise-free simulations of isolated Gaussian filaments with FWHM0 = 5, 15, 30, and 70 arcsec (left to right). The first row shows the synthetic reference images used as input for the simulations (panel A). The rows below display the results for different methods of data combination. ALMA 12 m-alone (panel B), ALMA 12m + 7m (panel C), ALMA Feather (panel D), ALMA MACF (panel E), and ALMA 12 m + IRAM-30m MACF (panel F), shown from top to bottom. The size of each image is 2.8 arcmin × 2.8 arcmin.

In the text

Fig. 8 Radial profile extracted through a horizontal cut in the center of a FWHM0 = 30 arcsec Gaussian filament image normalized with respect to the reference image peak (shown by the grey shaded area) displayed in linear scale (left). Same radial profile this time normalized with respect to the corresponding observed peak value in each map Ipeak displayed in log-scale (right). In both panels the reference profile is shown by the grey shaded area. The results of the distinct data combination methods are indicated with different colors: ALMA 12 m-alone (blue), ALMA 12m + 7 m (orange), ALMA Feather (pink), ALMA MACF (green), and ALMA 12 m + IRAM-30 m MACF (black).

In the text

Fig. 9 Reconstructed properties of isolated Gaussian filaments. Top: Percentage of emission recovered along the entire radial profile vs. FWHM0 Different colors are used for the different data combination methods used: blue for the ALMA 12 m-alone, orange for the 12 m + 7 m, pink for the ALMA Feather, green for the ALMA MACF, and black for the ALMA 12 m + IRAM-30 m MACF. The grey dotted and dashed lines show θres(12m) and θMRS(12m), respectively. Middle: Intensity peak estimated from the Gaussian fit vs. FWHM0. Bottom: FWHM estimated from the Gaussian fit vs. FWHM0. The grey solid line shows the theoretical expectation. For comparison, the results for the radial profile shown in Fig. 8 with FWHM0 = 30 arcsec correspond to values close to θMRS(12 m).

In the text

Fig. 10 Reconstructed properties of noise-free isolated objects with FWHM0 = 15 arcsec and aspect ratio AR = 1 (core), 2, 5, and ∞ (infinite filament). From top to bottom: Fraction of recovered flux; intensity peak estimated from the Gaussian fit; and FWHM estimated from the Gaussian fit. Different colors are used for the different data combination methods used: ALMA 12 m-alone (blue), 12m + 7m (orange), ALMA Feather (pink), ALMA MACF (green), and ALMA 12 m + IRAM-30 m MACF (black). The grey dotted and dashed lines show θres(12 m) and θMRS(12 m), respectively. The grey solid line in the lower panel shows the theoretical expectation (FWHM0 = 15 arcsec).

In the text

Fig. 11 Radial profile extracted through a horizontal cut in the center of a p = 2.5 (top panels) and p = 4 (bottom) Rflat = 30 arcsec Plummer profile. Left: profile is normalized with respect to the reference peak and displayed in normal scale. Right: profile normalized with respect to its peak value displayed in log-scale. The data combination methods use are marked in different colors: ALMA 12 m-alone (blue), ALMA 12 m + 7 m (orange), ALMA Feather (pink), ALMA MACF (green), and ALMA 12 m + IRAM-30 m MACF (black). The reference profile is marked by the gray shadowed area and ideal profiles for different p-values are displayed by red dashed lines.

In the text

	Fig. 12 uv-coverage of one of the cores simulated in Sect. 3. Different colors represent the uv-sampling obtained by the different ALMA arrays: (blue) ALMA 12 m, (red) ACA 7 m array, and (orange) TP. We also highlight the uv-area covered by the IRAM-30 m SD telescope (grey area) and its overlap with the ALMA 12 m array (black dashed area).
In the text

Fig. 13 Analysis of the interferometer PSF of different ALMA array configurations. From left to right: ALMA 12 m-alone , 7 m-alone, and ALMA 12 m + 7 m. We display the PSF intensity image for each simulation (top) and an intensity profile extracted along a horizontal cut in log-scale (bottom). In the bottom panels, the colored lines show the simulated PSF profiles (see dotted yellow lines in mid-panels) obtained for the ALMA 12 m-alone (blue), 7 m-alone (red) and 12 m + 7 m (orange), while the expected Gaussian PSF profile at the correspondent resolution is marked by the shaded grey area.

In the text

	Fig. 14 Comparison of the interferometer PSF intensity profile extracted from an horizontal cut for the ALMA 12 m-alone (blue), 7 m-alone (red) and 12m + 7 m (orange). The resolution of the 12 m array (dotted line), and the MRS of both the 12 m (dashed line) and 7 m (dot-dashed line) arrays are indicated by vertical lines.
In the text

	Fig. 15 Simulations of isolated cores with FWHM0 = 30 arcsec. From left to right: noise-free simulations, and simulations with noise levels equal to one, five, and ten times the theoretical noise (assuming PWV = 1.8 mm). From top to bottom: IRAM-30m simulations (panel A), ALMA 12 m-alone (panel B), ALMA 12 m + IRAM-30 m MACF (panel C).
In the text

	Fig. 16 Radial profile extracted from an horizontal cut in the center of a FWHM0 = 30 arcsec Gaussian cores shown in Fig. 15 C.
In the text

	Fig. 17 N2H+ (1–0) integrated intensity maps of OMC-3 using three different data combination methods. From left to right: ALMA 12m-alone map; ALMA 12 m array + IRAM-30 m combined using the Feather with the identified filament axes over-plotted (yellow; see also Paper III); and ALMA 12 m array + IRAM-30 m combined using the MACF method.
In the text

Fig. 18 Properties of real filaments (total flux, peak flux, and FWHM) identified in the N2H+ (1−0) integrated intensity maps of OMC-3 (see Fig. 17). All panels display the values obtained from the ALMA 12m-alone (yellow dots) and ALMA 12m+ IRAM-30m Feather (blue triangles) compared to those obtained form the combination of the ALMA 12 m array + IRAM-30 m using the MACF (flux, I0, and FWHMo) method assumed as reference. A perfect agreement between these methods is indicated by a grey line. Predictions form simulations are displayed by the grey dashed line. From top to bottom: Percentage of emission recovered along the entire radial profile; intensity peak FWHM estimated from the Gaussian fit; and FWHM estimated from the Gaussian fit. In all cases, the values are represented as function of the reference filament FWHM0 derived from the ALMA 12 m array + IRAM-30 m MACF map.

In the text

Fig. A.1 Analysis of the core mass function (CMF). The synthetic Salpeter-like distribution used as input is marked by the grey-shadowed histogram. The blue and green line histograms describe the CMF recovered using the ALMA 12 m-alone and the ALMA MACF, respectively. The black dashed line displays the expected Salpeter dependence (α = 1.35), while the dotted line is the one that better describes the much shallower, top-heavy slope (α = 1.18) observed by the interferometer-only

In the text

	Fig. A.2 Analysis of the FWHM distribution recovered in the CMF. Blue and green line histograms describe the FWHM recovered using the ALMA 12 m-alone and the ALMA MACF, respectively.
In the text

	Fig. B.1 Noise-free simulations of isolated objects with FWHM0=15 arcsec and different aspect ratios AR. From left to right: AR=1 (core), 2, 5, and ∞ (infinite filament). From top to bottom: Reference model (panel A), and simulated ALMA 12 m-alone (panel B), ALMA 12m + 7m (panel C), ALMA Feather (panel D), ALMA MACF (panel E), and ALMA 12 m + IRAM-30 m MACF (panel F) observations, respectively.
In the text