Issue 
A&A
Volume 626, June 2019



Article Number  A84  
Number of page(s)  20  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201834998  
Published online  18 June 2019 
Analysis and test of the centralbluespot infall hallmark
^{1}
Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos, Universitat de Barcelona, IEECUB,
Martí i Franquès 1,
08028
Barcelona,
Spain
email: robert.estalella@ub.edu
^{2}
Instituto de Astrofísica de Andalucía, CSIC,
Glorieta de la Astronomía, s/n,
18008
Granada,
Spain
Received:
30
December
2018
Accepted:
13
April
2019
Aims. The infall of material onto a protostar, in the case of optically thick line emission, produces an asymmetry in the blue and redwing line emissions. For an angularly resolved emission, this translates in a blue central spot in the firstorder moment (intensity weighted velocity) map.
Methods. An analytical expression for the firstorder moment intensity as a function of the projected distance was derived, for the cases of infinite and finite infall radius. The effect of a finite angular resolution, which requires the numerical convolution with the beam, was also studied.
Results. This method was applied to existing data of several starforming regions, namely G31.41+0.31 HMC, B335, and LDN 1287, obtaining good fits to the firstorder moment intensity maps, and deriving values of the central masses onto which the infall is taking place (G31.41+0.31 HMC: 70–120 M_{⊙}; B335: 0.1 M_{⊙}; Guitar Core of LDN 1287: 4.8 M_{⊙}). The centralbluespot infall hallmark appears to be a robust and reliable indicator of infall.
Key words: ISM: jets and outflows / ISM: individual objects: G31.41+0.31 HMC / ISM: individual objects: B335 / stars: formation / ISM: individual objects: LDN 1287
© ESO 2019
1 Introduction
The early phases of star formation are characterized by infall motions of ambient material onto a central protostellar object. However, obtaining unequivocal observational evidence for these motions constitutes a longstanding problem. Rotation, infall and outflow motions can be present simultaneously in the early phases of star formation, and may produce similar observational features in the line profiles, making an unambiguous interpretation of the observations difficult.
The most common method used so far to search for evidence of infall is based on the socalled “blue asymmetry”. This signature consists in the appearance of two peaks in the spectral line profiles, with the blueshifted peak stronger than the redshifted peak (e.g., Zhou et al. 1994; Klaassen & Wilson 2007; Wu et al. 2007). However, the effect of protostellar infall on molecular line profiles cannot be easily isolated from those of other dynamical processes, resulting in ambiguities (e.g., Purcell et al. 2006; Szymczak et al. 2007). Inverse PCygni profiles, which consist in the detection of absorption at redshifted velocities against a bright background continuum source, have been observed in molecular lines against a bright background HII region (e.g., Keto et al. 1987; Zhang et al. 1998) or against the bright dust continuum emission of the hot protostellar core itself (e.g., Di Francesco et al. 2001; Girart et al. 2009), and have been interpreted as infall motions of the surrounding envelope. Nevertheless, it uniquely indicates that foreground matter is moving toward a hotter source, no matter how or if it is indeed gravitationally bound. See MayenGijon et al. (2014) and MayenGijon (2015) for a thorough review of infall signatures.
Anglada et al. (1991) introduce a more complete signature, the “3D spectral imaging infall signature”, which is based on the spatial distribution of the line emission intensity in the images as a function of the lineofsight (LOS) velocity (i.e., the channel maps). This signature, appropriate for angularly resolved sources, results as an extension of the formalism initially developed by Anglada et al. (1987) for the study of an angularly unresolved infalling core. These signatures are focused on relatively high velocities, in order to avoid confusion with emission from the ambient cloud at low velocities. With the assumption of gravitational infall motions dominating the kinematics over turbulent and thermal motions, and spherical symmetry with an infall velocity increasing inwards, it can be shown that the points of the infalling core with the same LOS velocity form closed surfaces. The equalLOSvelocity surfaces are a nested set of surfaces with the same shape, decreasing in size with increasing absolute value of the LOS velocity, and converging to the position of the core center (see Fig. 1).
Because the equalLOSvelocity surfaces are closed surfaces, a given LOS, in general, intersects the same surface twice. However, if the opacity is high enough, only a narrow layer at the front side of the equalLOSvelocity surface is observable. Hence, the intensity map at a given LOS velocity is an image of the excitation temperature distribution in the side of the equalLOSvelocity surface facing the observer (thick lines in Fig. 1), while the emission from the rear side remains hidden (thin lines in Fig. 1). Thus, the shapes of the blueshifted and redshifted emitting regions^{1} are different. Since the temperature increases inwards, the integrated blueshifted emission comes from a region closer, on average, to the central protostar (and, therefore, hotter) than the corresponding redshifted emission, resulting in asymmetric line profiles, with the blueshifted wing stronger than the redshifted wing (Anglada et al. 1987). The difference is still more remarkable for a pair of blueshifted–redshifted channel maps. Assuming that the maps are centered on the position of the protostar, for the redshifted channel the intensity is slightly lower at the center of the map than at the edges, because toward the center the side of the equalLOSvelocity surface facing the observer is slightly farther away from the center of the core and thus, colder than the rest of the surface. For the blueshifted channel, the intensity increases sharply toward the center of the image since this emission comes from a region very close to the center of the core and, therefore, very hot (see Fig. 2). In addition, since the size of the equalLOSvelocity surfaces decreases with increasing absolute value of the LOS velocity, the emission becomes more compact for increasing absolute value of the LOS velocity (see Fig. 2).
This behavior of the intensity maps for redshifted and blueshifted LOS velocities produces a characteristic signature in the intensityweighted mean velocity (firstorder moment) map, which was pointed out by MayenGijon et al. (2014), in other words, that the central region of the firstorder map appears blueshifted because of the higher weight of the strong blueshifted emission. Additionally, the integrated intensity, (zerothorder moment) peaks toward the central position. At larger distances from the center, the integrated intensity decreases, the blue andredshifted intensities become similar, and the intensityweighted mean velocity approaches the systemic velocity of the cloud. Therefore, the firstorder moment of an infalling envelope is characterized by a compact spot of blueshifted emission toward the position of the zerothordermoment peak. This infall hallmark is designated as the “central blue spot” (MayenGijon et al. 2014; MayenGijon 2015). One of the advantages of the centralbluespot infall hallmark is that its detection does not require a beforehand knowledge of the systemic velocity of the cloud. An accurate knowledge of the systemic velocity is critical in searching for infall through the analysis of asymmetries in the line profiles.
MayenGijon et al. (2014) and MayenGijon (2015) identify both the 3D spectral imaging infall signature and the centralbluespot hallmark in highangular resolution maps of the emission of several NH_{3} transitions toward G31.41+0.31 HMC, and compare the observed emission with the predictions of a spherically symmetric model with full transport of radiation calculation (Osorio et al. 2009).
If there is rotation, the LOS velocity has contributions from both the infalling velocity and the rotation velocity. MayenGijon et al. (2014) and MayenGijon (2015), in the analysis of the NH_{3} data of G31.41+0.31 HMC, discuss qualitatively how rotation affects the channel maps of an infalling core. These authors find that the radial intensity profile of the image for a given LOSvelocity channel is stretched toward the side where rotation has the same sign than the channel velocity, and it is shrunk on the opposite side. Nevertheless, as in the nonrotating case, the images in blueshifted channels present a centrally peaked intensity distribution, while in the redshifted channels they present a flatter intensity distribution. Thus, the rotation signature makes the spatial intensity profiles asymmetric with respect to the central position but it does not mask the 3D spectral imaging infall signature of Anglada et al. (1991).
Regarding the firstorder moment map, MayenGijon (2015) explores how the centralbluespot hallmark of an infalling core is modified by the presence of rotation. He finds that rotation makes the centralbluespot even bluer and moves it off the center toward the half of the core where rotation tends to shift velocities to the blue. Additionally, a dimmer red spot appears symmetrically located on the opposite side of the rotation axis.
In the present paper we studied quantitatively the centralbluespot infall hallmark, restricted to the spherically symmetric case without rotation, taking as a basis the work of Anglada et al. (1987, 1991). Using the same assumptions than in these papers, we derived analytical expressions for the intensity profiles (Sect. 2), the line profiles (Sect. 3), and the firstorder moment of the intensity profile (Sect. 4) as functions of the angular distance from the center. This was done for the case of an infinite infall radius, without considering the effect of a finite angular resolution. The details of the derivation for arbitrary values of the powerlaw indices are given in Appendix A. The effect of a finite spectral resolution is addressed in Sect. 4.2 and Appendix C, while the effect of a finite angular resolution is studied in Sect. 4.3 and Appendix D. The case of a finite infall radius is presented in Sect. 5, where an analytical expression for the firstorder moment is obtained. The details of the derivation for arbitrary values of the powerlaw indices are given in Appendix B. The transformation between reduced units and practical units is described in Sect. 6. The results are applied to several cases (G31.41+0.31 HMC, B335, LDN 1287) in Sect. 7, with the analysis of preexisting data that show the centralbluespot infall hallmark. Finally, the conclusions are given in Sect. 8.
2 Intensity profile
Based on Anglada et al. (1987, 1991), we are assuming an infalling molecular gas core, with infall velocity and temperature given by power laws of the radius (Eq. (1) of Anglada et al. 1987), (1)
where R_{0} is a reference radius. The powerlaw indices of the infall velocity and temperature are taken with a value α = β = 1∕2, that is, freefall velocity, and optically thin dust heating from a central protostar, which are characteristic of the main accretion phase in the Larson collapse model (Larson 1972) or in the Shu (1977) insideout collapse model. In the Appendices A and B the case of arbitrary values of the powerlaw indices is developed.
Let the coordinate z be along the line of sight, positive outwards from the observer, and p the impact parameter, that is, the distance to the center projected on the plane of the sky (see Fig. 1). All coordinates are lengths in units of R_{0}, so that the distance to the center in units of R_{0} is (2)
Let us define the reduced lineofsight (LOS) velocity and temperature (3)
Equations (9) and (10) of Anglada et al. (1991) can be expressed in reduced variables as (4)
For a given v_{z} Eq. (4) gives the parametric equation (with z as parameter) of the intensity profile, t(p). We are assuming that the intensity observed at a given LOS velocity comes from the part of the equalLOSvelocity surface facing the observer. Thus, the bluewing intensity profile (v_{z} < 0) is obtained for 0 < z < z_{m}, while the redwing intensity profile (v_{z} > 0) is obtained for − z^{*} < z < −z_{m}, where z_{m} is the value of z for which p is maximum for a given v_{z}, p = p_{m} (see Fig. 1), and z^{*} is the maximum value of z for a given equalLOSvelocity surface. The value of z^{*} is obtained from Eq. (4) for p = 0, z^{*} = v_{z}^{−2}. The value of z_{m} can be obtained from the derivative of p(z) given in Eq. (4), and the values obtained for z_{m} and p_{m} are (5)
The ratio z_{m}∕p_{m} is independentof v_{z}, , meaning that the points (z_{m}, p_{m}) are aligned along a straight line passing through the center, with a slope of (see Fig. 1).
For any LOSvelocity the emission is confined inside a projected distance p < p_{m}. For blueshifted LOSvelocities (v_{z} < 0) the intensity increases sharply for small projected distances p, while for redshifted LOSvelocity (v_{z} > 0) the intensityis almost flat up to p_{m} (see Figs. 3 and 4). This can be seen in Fig. 2, where we show the intensity maps, for LOS pairs of positive and negative LOS velocities.
For the maximum projected distance p_{m} the redwing intensity is maximum, and equal to the minimum bluewing intensity (see Fig. 3), (6)
The blue wing intensity for small projected distances is very high, and for the adimensional temperature and LOS velocity t ≫ v_{z} we obtain the asymptotic behavior (see Fig. 3) (7)
Fig. 1 Intersection of the surfaces of equal lineofsight (LOS) velocity, v_{z}, for a collapsing protostellar core, with the plane (p, z) (see Sect. 2). The LOSvelocities are different pairs of negative and positive velocities (v_{z} = ±2^{−1∕2}, ±1, ±2^{1∕2}, ±2). The blue lines are the contours for v_{z} < 0, and the red lines for v_{z} > 0. The observer, located to the right, at p = 0, z = −∞, sees the emission coming from the part of the surfaces of equal LOSvelocity facing the observer, traced with thick lines. The dashed lines indicate the position of (z_{m}, p_{m}) for each contour, where p_{m} is the maximum value of p, and z_{m} the corresponding value of z. The square frame is drawn at p = ±1, z = ±1. 

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Fig. 2 Intensity maps for pairs of LOSvelocities, V_{z} = ±0.8 km s^{−1} (top), ± 1.1 km s^{−1} (middle), and ± 1.4 km s^{−1} (bottom), calculated for a a mass M_{*} = 1 M_{⊙}, a distance of 140 pc, and a beam of 0.′′1. The axes are position offsets labeled in arcsec. The intensity scale is the same for all maps. We note the sharp peak at the center of the blueshifted LOSvelocity maps (left), and the flat, slightly concave shape of the redshifted LOSvelocity maps (right), and the more compact emission for high values V_{z}  (bottom) than for low values of V_{z} (top). 

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3 Line profile
In order to calculate the firstorder moment intensity, we need to obtain the line intensity for any value of p. Thus, we want to derive the intensity and LOS velocity for a given projected distance p. From Eq. (4) we can obtain easily the LOS velocity and temperature as functions of p and z, (8)
These equations can be interpreted as the parametric equation (with z as parameter) of the line profile t(v_{z}) for a given projected distance p (see Figs. 5 and 6).
The line of sight with a given p intersects equalLOSvelocity surfaces of decreasing size, down to that corresponding to p_{m} = p. For this equalLOSvelocity surface, the blueshifted velocity is minimum (maximum absolute value) and the redshifted velocity is maximum (see Fig. 1). This occurs at the points with coordinate .
Fig. 3 Intensity profiles t(p) for v_{z} = −1 (top panel, blue) and v_{z} = +1 (bottom panel, red). The different dashed lines indicate the blue and red wing temperature at the maximum value of p, p = p_{m}, and the blue wing asymptotic behavior for p ≪ 1. 

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3.1 Blue wing
The observable intensity at blueshifted velocities (v_{z} < 0) is obtained for . For a given value of p, the minimum blueshifted intensity and minimum velocity (maximum absolute value) are obtainedat the point , (9)
while the maximum blueshifted intensity and maximum velocity are obtained at the point (p, z = 0) (10)
3.2 Red wing
The observable intensity at redshifted velocities (v_{z} > 0) is obtained for . For a given value of p, the minimum redshifted intensity and minimum velocity are obtained at the point (p, z = +∞), (11)
while the maximum redshifted intensity and maximum velocity are obtained at the point , (12)
Fig. 4 Intensity profiles t(p) for pairs of positive and negative velocities, v_{z} = ±1, ±2^{1∕2}, ±2. 

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Fig. 5 Line profile showing the emission at a projected distance p as a function of LOS velocity. The blue wing emission encompasses LOS velocities from − v_{m} to 0, and its peak value (at v_{z} = 0) is t_{2} = p^{−1∕2}. The red wing emission encompasses LOS velocities from 0 to v_{m} = 0.620 p^{−1∕2}, and its peak value (at v_{z} = v_{m}) is t_{1} = 0.760 p^{−1∕2}. 

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Fig. 6 Line profiles for values of p = 1, 2^{1∕2}, 2. 

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4 Firstorder moment for an infinite infall radius
4.1 Dependence on the projected distance
We are interested in calculating the firstorder normalized moment of the line profile as a function of the projected distance, (13)
where μ_{0} and μ_{1} are the zerothorder and unnormalized firstorder moments, (14)
The moment μ_{0} has units of intensity times velocity, while μ_{1} has units of velocity.
From Eq. (8) we can obtain v_{z} as an explicit function of t, (15)
where the bluewing profile is obtained for 3^{−1∕4} p^{−1∕2} < t < p^{−1∕2}, and the redwing profile for 0 < t < 3^{−1∕4} p^{−1∕2}. Since we know theinverse function v_{z}(t), we can evaluate the integrals by integration by parts, (16)
The details of the calculation of these integrals are given in Appendix A. As described inthe appendix, the final results obtained can be summarized as follows. The moments μ_{0}, , and μ_{1} are power laws of the projected distance p, and given by (17)
with H_{0} and H_{1} given by (18)
A plot of the zeroth and firstorder moments can be seen in Fig. 7.
Fig. 7 Log–log plot of the moments μ_{0} and − μ_{1}, as a functionof projected distance p. 

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4.2 Effect of a finite spectral resolution
Let us assume that we are observing the lines with an spectrometer with a finite spectral resolution. The response of the spectrometer can be represented by the convolution of the “real” line profile with the instrumental response, which has a width equal to the spectral resolution. In general, we can assume that instrumental response is symmetric with respect to velocity. As shown in Appendix C, in this case the firstorder moment of the line is not modified by the spectrometer.
4.3 Effect of a finite angular resolution
Let us assume that we are observing with a telescope with a circularly symmetric Gaussian beam with halfpower beamwidth (HPBW) θ_{b}. In the following we will use the beamwidth in units of R_{0}, that is, b ≡ D θ_{b}∕R_{0}, where D is the distanceto the source.
The observed intensity as a function of projected distance will be the 2D convolution of t(p), given by Eq. (4), with the beam. Since the moments μ_{0} and (but not μ_{1}) depend linearly on the line profile t(p) (Eq. (14)), the convolution (a linear operator) can be performed onto μ_{0} and , and after that, obtain the normalized firstorder moment .
The 2D convolution of the powerlaw functions μ_{0} and with a Gaussian beam has to be done numerically. However, it has to be done only once, because the result can be scaled for any value of b. For instance, if the numerical calculation has been done for b = 1 and μ_{1}(p;1) is obtained, then for any value of b we have (19)
The function − μ_{1}(p;1) is shown as a blue solid line in the log–log plot of Fig. 8.
However, since both μ_{0} and are power laws of p, as shown in Appendix D, the value at the origin, p = 0, and the asymptotic expression for large projected distances, p ≫ b, can be obtained analytically. By application of Eqs. (D.8) and (D.10) to μ_{0} we obtain (see Fig. 8) (20)
with a characteristic size (the intersection of the two asymptotes) p_{0} = 0.339 b. For we obtain (21)
with a characteristic size p_{1} = 0.254 b. Finally, for the firstorder moment, , we derive (22)
Fig. 8 Log–log plots of the moments μ_{0} and − μ_{1} (solid lines) for a Gaussian beam with HPBW b = 1 (dotted line), as a function of the projected distance p, and their asymptotic values (dashed lines) for p ≪ b and p ≫ b. 

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5 Firstorder moment for a finite infall radius
Up to now we have assumed that the infall velocity pattern in r^{−1∕2} extends up to an infinite radius. A more realistic approach is to assume a finite radius. For instance, in the insideout collapse model (Shu 1977), the collapse propagates outwards from the center at the speed of sound. The radius of the expansion wave is usually called the infall radius R_{i}, and let us call r_{i} the infall radius in units of R_{0}, that is, r_{i} ≡ R_{i}∕R_{0}. The envelope with a radius greater than the infall radius is approximately static, while the material inside the infall radius is in free fall. Thus, let us assume that the infall occurs only for radii r < r_{i}. The static material will only contribute to the ambientgas lineemission, centered on v_{z} = 0, and will not be taken into account.
5.1 Line profile (finite infall radius)
The effect of having a finite infall radius is that the equalLOSvelocity surfaces are truncated at a radius r = r_{i}. Thus, for the redwing emission, a part of the equalLOSvelocity surface near the center of the core will no longer be hidden from the observer by the part facing the observer. A critical value of p is that for which the sphere of radius r_{i} intersects the equalLOSvelocity surfaces at the points (p = p_{m}, z = z_{m}), that is, , or . As illustrated in Fig. 9, for a given projected distance , the bluewing emission, like the infinite infall radius case, comes from material along the line of sight at radii . However, the redwing emission, unlike the infinite infall radius case, comes from two regions: material along the line of sight at radii , in the part of the equalLOSvelocity surface facing the observer, and material closer to the center, located at − z_{b} < z < 0, which is no longer hidden by the part of the equalLOSvelocity surface facing the observer because this part of the equalLOSvelocity surface would be outside the infalling sphere of radius r_{i}. The material in the part of the equalLOSvelocity surface facing the observer has LOS velocities v_{a} < v_{z} < v_{m}, where v_{a} is the velocity of the material where the line of sight intersects the sphere of radius r_{i}. The material closer to the center has LOS velocities 0 < v_{r} < v_{a} (see Fig. 10).
Let us use the reduced coordinate q ≡ p∕r_{i}. The corresponding critical value of q is (see Fig. 9). For values , only the redwing emission is affected. For the line becomes symmetric because none of the red wing emission at any LOS velocity is hidden by the part of the equalLOSvelocity surface facing the observer, and the moment μ_{1} becomes zero (see Figs. 9 and 10). For q ≥ 1 all the wing emission disappears (μ_{0} = μ_{1} = 0).
5.2 Firstorder moment (finite infall radius)
The details of the derivation of an analytical expression for the moments, for a finite infall radius are given in Appendix B. The moments μ_{0}, , and μ_{1} obtained are no longer power laws of the projected distance p, although the dependence on r_{i} can be separated from the explicit dependence on p using the parameter q = p∕r_{i}. In this way, the resulting expressions are similar to those obtained for the case of an infinite infall radius, (23)
with H_{0} and H_{1} given by (see Fig. B.2) (24)
where q′ is an auxiliary parameter depending on q only (see Fig. B.1), (25)
An example of the moments obtained can be seen in Fig. 11.
The limiting values of the moments for r_{i} →∞, corresponding to q = 0, q′ = 1, coincide with the results derived for an infinite infall radius (Eqs. (17) and (18)), (27)
while the limiting values for , corresponding to , are (28)
Fig. 9 For a given projected distance the wing emission comes from the thick part of the line of sight, for radii less than r_{i}. The emission from the thin part of the line of sight is hidden to the observer.For the blue wing, the observer can observe the emission coming from z = 0 (with v_{z} = 0) to z = z_{m} (with v_{z} = −v_{m}). For the red wing, the observer can observe the emission coming from z = −z_{a} (with r = r_{i}, v_{z} = v_{a}) to z = −z_{m} (with v_{z} = +v_{m}), and, since the corresponding part of the equalLOSvelocity surface facing the observer is missing (it would be outside the infall radius), from z = 0 (with v_{z} = 0) to z = −z_{b} (with v_{z} = v_{a}). The velocity v_{a} is the LOS velocity of the equalLOSvelocity surface that intersects the sphere r = r_{i} at a projected distance p, i.e. v_{a} = v_{z}(p, −z_{a}) = v_{z}(p, −z_{b}). 

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Fig. 10 Line profile for and p = 1. The blue wing is unaffected. For the red wing the emission at velocities v_{z} < v_{a} comes from the hot part of the red equalLOSvelocity surface. For the blue and red wings become symmetric. 

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Fig. 11 Log–log plot of μ_{0} and − μ_{1} as a functionof the projected distance p, for an infall radius r_{i} = 1 (solid lines) and for an infinite infall radius (dashed lines). 

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5.3 Finite angular resolution (finite infall radius)
In this case, μ_{0} and are no longer power laws of p, but have a characteristic scale size given by r_{i}. Thus, unlike the infinite infall radius case, the beam convolution has to be performed for every value of r_{i}, and the result will depend on both r_{i} and b. For r_{i} very large, the results for an infinite infall radius of the last section are reproduced.
6 Practical units
In the previous analysis all lengths, z, p, r_{i}, b, are measured in units of the reference radius R_{0}; velocities in units of V _{0}, the infall velocity at the reference radius R_{0}; and temperatures in units of T_{0}, the temperature at the reference radius R_{0}. The projected distance p in practical units, that is, θ in arcsec, can be obtained from p in units of R_{0} through^{2} (29)
where D is the distance to the source. The value of R_{0} is arbitrary, but the values of R_{0} and V _{0} are related to the central mass of the protostar onto which the accretion is taking place, M_{*}, (30)
6.1 Infinite infall radius
The firstorder moment at the origin (Eq. (22)) in practical units becomes, (32)
Taking into account Eq. (30), we have for θ = 0 (34)
Examples of μ_{1}(θ;θ_{b}) for some values of θ_{b} and M_{*} are shown in Figs. 13 and 14. A value of the central mass can be derived directly from the value of the firstorder moment at the origin, (36)
It may seem surprising that the value of the firstorder moment at the origin (Eq. (34)) does not depend on the temperature T_{0}. This is because μ_{1} is the normalized moment , and the dependence on T_{0} of both and μ_{0} cancels. However, μ_{1} does depend on the temperature gradient, that is, on the powerlaw index of temperature law (β in Eq. (A.36)). For instance, for β = 0 (no temperature gradient), μ_{1} is zero.
Fig. 12 Log–log plot of μ_{0} and − μ_{1} as a functionof the projected distance, for an infall radius r_{i} = 1 and beamwidth b = 0.1 in reduced units (solid lines) and b = 0 (dashed lines). 

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6.2 Finite infall radius
In this case the 2D convolution has to be computed for any value of the infall radius. A possible strategy for computing the firstorder moment is the following.
 1.
Choose avalue for R_{0}, for instance R_{0} = 1 kau, so that the infall radius in reduced coordinates is r_{i} = [R_{i}∕kau].
 2.
For a given value of r_{i}, and a range of values of p, construct the functions μ_{0} = H_{0}(q) p^{−1} and (Eq. (23)) in reduced coordinates (p and r_{i} in units of R_{0}).
 3.
Transform the p coordinate to practical units, (Eq. (29)). Compute the beam 2Dconvolution (Eq. (D.2)) of μ_{0} and , and obtain . See in Fig. 12 an example of the resulting μ_{0} and μ_{1} after beam convolution.
 4.
The resulting firstorder moment is in units of V _{0}. To have it in km s^{−1}, scale μ_{1} by a factor , where M_{*} is the central mass (Eq. (31)).
Examples of μ_{1}(θ;θ_{b}, R_{i}) for some values of θ_{b}, R_{i}, and M_{*} are shown in Figs. 15 and 16.
Let us see now what can be derived from the value of the firstorder moment at the origin. The value of μ_{1} for θ = 0 (Eq. (B.38)) in practical units becomes (37)
where s = R_{i}∕(Dθ_{b}), and J_{0} and J_{1} are given by Eq. (B.37). Taking into account Eq. (30), we have (38)
where J(s) = 1.331 J_{1}(s)∕J_{0}(s). The function J(s), calculated numerically, is shown in Fig. 17. The asymptotic expression for large infall radii, R_{i} ≫ Dθ_{b}, (39)
coincides, asexpected, with the expression derived for an infinite infall radius, Eq. (34). In the case of a poor angular resolution compared with the infall radius (θ_{b} ≫ R_{i}∕D), we obtain (40)
The value at the origin of the firstorder moment in the case of a finite infall radius, μ_{1} (0;θ_{b}, R_{i}), in contrast with the infinite infall radius case, does not provide a unique value of the central mass, unless the infall radius is known. There is a degeneracy between the infall radius and the central mass: different pairs of values of the infall radius and central mass produce the same value of the firstorder moment at the origin. In order to disentangle this degeneracy, it is necessary to fit not only the value at the origin, but also the variation of the observed firstorder moment as a function of the projected distance.
Fig. 13 Firstorder moment μ_{1} for an infinite infall radius, as a function of the projected distance, D θ (in kau), for beamwidths D θ_{b} = 0.1, 0.2, 0.5, 1, 2, 5, and 10 kau, where D is the distance to the source. The vertical axis scales as . The left axis is μ_{1} for a central mass M_{*} = 1 M_{⊙}, while the right axis is for M_{*} = 10 M_{⊙}. 

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Fig. 14 Firstorder moment μ_{1} for an infinite infall radius, as a function of the projected distance in units of the beamwidth, θ∕θ_{b} (adimensional), for central masses M_{*} = 0.1, 0.2, 0.5, 1, 2, 5, and 10 M_{⊙}. The vertical axis scales as , where D is the distance to the source. The left axis is μ_{1} for D θ_{b}= 1 kau, while the right axis is for D θ_{b} = 0.1 kau. 

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Fig. 15 Firstorder moment μ_{1}, as a functionof projected distance in units of the infall radius, D θ∕R_{i} (adimensional), for beamwidths D θ_{b} = 0, 0.1, 0.2, 0.5, 1, and 2 kau, where D is the distance to the source. The vertical axis scales as . The left axis is μ_{1} for a central mass M_{*} = 1 M_{⊙}, while the right axis is for M_{*} = 10 M_{⊙}. 

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Fig. 16 Same as Fig. 15, for D θ_{b} = 0.1 kau, and R_{i} = 0.2, 0.5, 1, 2, 5, 10, and ∞ kau. The horizontal axis is the projected distance, D θ (in kau). Note that the line for R_{i} = 1 kau in this figure coincides with the line for D θ_{b} = 0.1 kau in Fig. 15. 

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Fig. 17 Function J(s) in the expression of μ_{1}(0;θ_{b}, R_{i}), showing its asymptotic values for s ≪ 1 and s ≫ 1. 

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7 Application of the centralbluespot infall hallmark to real data
7.1 G31.41+0.31 HMC
G31.41+0.31 HMC (hereafter G31) is a hot molecular core whose distance was estimated to be 7.9 kpc, but recent determinations (Reid et al., in prep.) give a value of 3.7 kpc for its distance. Infall motions in G31 have been reported by Girart et al. (2009) from inverse PCygni profiles, and by MayenGijon et al. (2014) from VLA observations of the ammonia inversion transitions (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6) showing a central bluespot in the firstorder moment maps.
Here we are analyzing the firstorder moment maps shown in Fig. 2 of MayenGijon et al. (2014) (see Fig. 18). The halfpower beamwidth of the observations were 0.′′ 33 for the (2, 2), (3, 3), and (6, 6) maps, 0.′′ 16 for the (4, 4) map, and 0.′′ 37 for the (5, 5) map. The value of the firstorder moment as a function of the angular distance was obtained for the five maps by averaging the firstorder moment in concentric rings of width 0.′′1 centered onthe average position of the peak of the blue spot, α(J2000) = 18^{h}47^{m}34.^{s}32, δ(J2000) = −01°12′46.′′1. In Fig. 19, we present the firstorder moment profiles obtained. We considered that the velocity far from the center was 97.4 km s^{−1}, the value used by MayenGijon et al. (2014), which is consistent with the values of the systemic velocity quoted for G31, ranging from 96.26 km s^{−1} (Beltrán et al. 2005) to 98.8 km s^{−1} (Cesaroni et al. 1994).The firstorder moment of the different transitions show very similar profiles, with a value of approximately − 3 km s^{−1} for small angular distances, except for the (2, 2) transition, which shows a shallower dip in velocity. This could be due to a lower opacity of the (2, 2) line, and partial blending of the central line with the inner satellite lines, and we will no longer consider this transition. By application of Eq. (36) we see that a value of ~ − 3 km s^{−1} for the firstorder moment means roughly a central mass of the order of 50 M_{⊙}.
In order to obtain a more accurate value of the central mass, the hallmark model was calculated for the beam size of each transition, and fitted to the observed data for different values of the infall radius and the central mass. For an infinite infall radius the best fit was found for a central mass of 44 M_{⊙}, with a residual χ^{2} statistic for ν = 39 degrees of freedom (the total number of rings of all transitions used in the fit, minus 1), χ^{2} = 36.6, which gives a reduced . For finite values of the infall radius we obtained higher values of the central mass. For R_{i} = 20 kau (corresponding to an angular radius of 5.′′ 4 at 3.7 kpc), we obtained a better fit, with a central mass M_{*} = 69M_{⊙} (χ^{2} = 21.2, χ_{r} = 0.74), while for R_{i} = 5 kau (1.′′ 4), a lower limit for the infall radius to be consistent with the size of the NH_{3} maps, the best fit was for a central mass M_{*} = 122 M_{⊙} (χ^{2} = 10.7, χ_{r} = 0.52) (see Fig. 20). In conclusion, the central mass obtained is always greater than ~ 44 M_{⊙}, and the best fit is obtained for values of the infall radius between 5 and 20 kau, with central masses between ~ 70 and ~ 120 M_{⊙}.
Osorio et al. (2009) model and the central core of G31 obtain, assuming a distance of 7.9 kpc, a central star with a mass of ~ 25M_{⊙}, a mass accretion rate of 3 × 10^{−3} M_{⊙} yr^{−1}, and a total luminosity of 2 × 10^{5} L_{⊙}. The luminosity, scaled to the distance of 3.7 kpc adopted here, is 4.4 × 10^{4} L_{⊙}. A single star with a mass equal to the central mass derived here would have a luminosity two orders of magnitude higher. This apparent lack of luminosity can be explained considering, as usually found in highmass star forming regions, that there is not a single highmass star at the center of G31, but a cluster of less massive stars. In the case of G31, recent highangular resolution ALMA continuum observations (Beltrán et al., in prep.) reveal the presence of at least four cores at the center of G31.
If we assume that the stars of the cluster at the center of G31 have masses that follow the Salpeter initial mass function (Salpeter 1955) there would be a few highmass young stellar objects, probably associated with the four cores detected, which could account for most of the total luminosity observed. A higher number of lowmass young stellar objects yet undetected, with little contribution to the overall luminosity, could total a mass of 70–120 M_{⊙} for the cluster.
Fig. 18 Maps of G31 firstorder (color scale) and zerothorder moment (contours) for the ammonia inversion transitions (2, 2) to (6, 6) (MayenGijon et al. 2014). The color scale, the same for all panels, ranges from 94 to 101 km s^{−1}. The contours are in steps of 10% of the peak value for all the maps, except for the (4, 4) transition, for which the steps are 20%. The beam is shown in the lower right corner of each panel. 

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Fig. 19 G31 firstorder moment for the ammonia inversion transitions (2, 2) (black line and open circles), (3, 3) (red line andfilled circles), (4, 4) (blue line and filled squares), (5, 5) (magenta line and open squares), and (6, 6) (green line and filled diamonds) (MayenGijon et al. 2014), as a function of angular distance, measured for rings of width 0.′′ 1 and average radius θ. The error bars are the rms of the velocity inside each ring. The righthand vertical axis shows the velocities obtained from the firstorder moment maps, while the lefthand vertical axis shows the velocity with respect to the systemic velocity of G31, taken as V _{sys} = 97.4 km s^{−1}. 

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Fig. 20 Model (black dashed line) fitted to the G31 firstorder moment, calculated for a central mass of 122 M_{⊙}, an infall radius of 5 kau, and a halfpower beamwidth of ~0.′′33 for the (3, 3), (5, 5), and (6, 6) transitions (top), and 0.′′16 for the (4, 4) transition (bottom). 

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7.2 B335
B335 is an isolated Class 0 protostar with a bolometric luminosity of ~ 1 L_{⊙}, at a distance of 105 pc (Olofsson & Olofsson 2009). Several authors claim the detection of infall (Kurono et al. 2013; Evans et al. 2015). Here we are analyzing two observations, one of H^{13} CO^{+} (J = 1–0) at 87 GHz, with a moderate angular resolution, combining data from the 45 m Nobeyama telescope and the Nobeyama Millimeter Array (Kurono et al. 2013), and the other of ^{13}CO (J = 2–1) at 220 GHz, with very high angular resolution, carried out with the Atacama Large Millimeter/Submillimeter Array (ALMA; Yen et al. 2015).
Regarding the Nobeyama data, the beam obtained for the combined data from both instruments was ~ 5.′′0, and the spectral resolution was 0.108 km s^{−1}. The channel maps of the H^{13} CO^{+} emission were retrieved from Fig. 5 of Kurono et al. (2013), and the data cube obtained was resampled with a cell size of 1″. The zeroth and firstorder moments obtained are shown in Fig. 21. The value of the firstorder moment as a function of the angular distance was obtained by averaging the firstorder moment in concentric rings of width 2″ centered onthe position of the continuum compact source, α(J2000) = 19^{h}37^{m}00.^{s}89, δ(J2000) = +07°34′09.′′6 (Yen et al. 2015), the (0, 0) position in Figs. 21 and 22. The values obtained are shown in Fig. 23 (top panel). The error bars are the rms dispersion of velocities inside each ring, added quadratically to the uncertainty due to the finite spectral resolution, the same for all rings.
The ALMA data had an angular resolution of ~ 0.′′31, and the spectral resolution was 0.166 km s^{−1}. The firstorder moment for all the line, μ_{1}, is shown in the middle panel of Fig. 22. As can be seen, the central blue spot is not centered on the position of the continuum compact source (plus sign), but is offset and extends to the southeast. The angular resolution of the data corresponds to a linear resolution of 33 au at a distance of 105 pc. At this small scale, the kinematics can be dominated by the rotation of the protostellar disk, and infall can be no longer spherically symmetric, as assumed by the hallmark model. In order to check the validity of the model, we also computed the first order moment separately for the redshifted channels, with velocity higher than the systemic velocity, , and for the blueshifted channels (with velocity lower than V _{sys}), . The hallmark model predicts that shows a central redspot and a central blue spot, higher in absolute value than the firstorder moment of the full line. As can be seen in the top and bottom panels of Fig. 22, this is what is observed for and , with quite well centered red and blue spots. The values of μ_{1}, , and as a function of the angular distance were calculated for concentric rings of 0.′′ 1 width, centered on the position of the continuum compact source. The error bars were calculated in the same way as for the Nobeyama data. The moments as a function of the angular distance θ, are shown in Fig. 23 (bottom panel). As can be seen, the three moments follow the same kind of dependence on the projected distance, and we will see that they can be fitted by power laws of index − 1∕2, convolved with the beam, as predicted by the hallmark model. Thus, the gas kinematics for the range of projected distances sampled by ALMA, from 0.′′ 1 to 0.′′ 8 (10–84 au), appears to be dominated by infall.
The hallmark model was calculated and fitted simultaneously to the Nobeyama firstorder moment μ_{1}, and to the ALMA and (but not to μ_{1}). Since the Nobeyama data are very sensitive to the value adopted for the systemic velocity, we fitted both the value of the systemic velocity and the central mass. We tested the infall radius reported by Kurono et al. (2013), 2900 au (corrected from their assumed distance of 150 pc), but a better fit was obtained with a larger infall radius. The best fit was obtained for an infinite infall radius, for a systemic velocity V _{sys} = 8.29 km s^{−1}, and central mass M_{*} = 0.09M_{⊙} (dashed lines in the top and bottom panels of Fig. 23). The goodness of the fit is given by a χ^{2} statistic for ν = 27 degrees of freedom, χ^{2} = 29.9, corresponding to a reduced , an indication that the model fits well the data within the uncertainties. As an additional check, we computed the predicted value of the firstorder moment for the ALMA data of the full line, μ_{1}, (dotted line in the bottom panel of Fig. 23), which is a linear combination of and , that is, . As can be seen in the figure, the values predicted for μ_{1} match well those observed.
We can conclude that the kinematics of the gas in B335, for the linear scales sampled by ALMA and Nobeyama, from ~ 10 to ~ 2500 au, in other words, more than two orders of magnitude, can be explained by a simple model of infall onto a central protostar of ~ 0.1 M_{⊙}. This can be considered as an outstanding result of the centralbluespot infall hallmark model.
Fig. 21 B335 zerothorder (contours) and firstorder moment (color scale) of the H^{13} CO^{+} (J = 1–0) line, obtained from the channel maps of Kurono et al. (2013). Contours are in steps of 511 mJy beam^{−1} km s^{−1}. The color scale at the right border is in m s^{−1}. The (0, 0) position corresponds to α(J2000) = 19^{h}37^{m}00.^{s}89, δ(J2000) = +07°34′09.′′6. The synthesized beam is shown in the lower left corner. 

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Fig. 22 B335 firstorder moment of the ^{13}CO (J = 2–1) line observed with ALMA. Top: moment of the red channels, with velocities higher than the systemic velocity, V _{sys} (color scale: 8.9–10.1 km s^{−1}). Middle: moment of all the channels (color scale: 7.4–9.2 km s^{−1}). Bottom: moment of the blue channels, with velocities lower than V _{sys} (color scale: 6.8–8.0 km s^{−1}). The (0, 0) position, marked with a plus sign, is the same as in Fig. 21. The synthesized beam is shown in the lower left corner of the middle panel. 

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Fig. 23 Same as Fig. 19 for B335. Top: H^{13}CO^{+} (J = 1–0) line observed at Nobeyama (2″ ring width). Bottom: ^{13}CO (J = 2–1) line observed with ALMA (0.′′1 ring width; , red line and symbols; μ_{1}, black line and symbols; , blue line and symbols) The dashed lines in the top and bottom panels show the best simultaneous fit to the Nobeyama moment μ_{1} (beamwidth 5.′′0), and the ALMA moments and (beamwidth 0.′′31). The best fit is obtained for an infinite infall radius, a systemic velocity V _{sys} = 8.29 km s^{−1}, and a centralmass of 0.09 M_{⊙}. The black dotted line in the bottom panel is not a fit, but the value of μ_{1} predicted by the bestfit model for the ALMA firstorder moment. 

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Fig. 24 NH_{3} (1, 1) line central velocity of the Guitar Core in L1287. The central blue spot is blueshifted ~ 0.5 km s^{−1} with respect to the ambient gas. The embedded sources in the center are indicated by dots (mm, Juárez et al. 2019), and crosses (cm, Anglada et al. 1994). The synthesized beam is shown in the lowerright corner of the map. 

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7.3 LDN 1287
LDN 1287 (hereafter L1287) is a molecular cloud located at a distance of 929 pc, associated with an energeticbipolar CO outflow (Yang et al. 1991). The source was mapped with a single dish in NH_{3} (Estalella et al. 1993; Sepúlveda et al. 2011). A cluster of mm sources has been detected at the center of L1287 (Juárez et al. 2019), one of the mm sources being associated with VLA 3 (Anglada et al. 1994), a jetlike cmcontinuum source that appears to be driving the outflow.
Here we are analyzing VLA observations of the NH_{3} (1, 1) and (2, 2) transitions, which show a complicated structure with a complex kinematics (Sepúlveda et al. in prep.). The NH_{3} lines were analyzed by means of the Hyperfine Structure Tool (HfS) (Estalella 2017). After careful inspection of the spectra, three different velocity components were identified, with nonoverlapping velocity ranges (Guitar Core, Blue Filament, and Red Filament). The Guitar Core does not show any sign of interaction with the embedded young stellar objects (no increase in linewidth, nor in rotational temperature at the projected position of the embedded sources). Our results suggest that the Guitar core is a very young protostellar core. Given the poor velocity resolution of the observations, the only way to separate the emission of the Guitar Core from that of the filaments was to fit Gaussian components to the observed spectra. Thus, the asymmetry of the line was inferred from the shift of the central velocity of the line fitted. Nevertheless, a compact spot of ~ 10″ in diameter of blueshifted velocities appears at the center of the Guitar Core (see Fig. 24).
The central velocity of the Guitar Core, obtained from the HfS fits, was averaged in concentric rings 1″ wide, centered on the emission peak, up to a radius of 20″. The velocity profile obtained is shown in Fig. 25. The error bars are the rms dispersion of the velocities averaged in each ring, added quadratically to the error of the average value of central velocity obtained from HfS. The average velocity at large distances from the peak (the systemic velocity of the Guitar Core), − 18.54 km s^{−1}, has been subtracted from the values of the central velocity. The best fit to the (1, 1) and (2, 2) data, for a beamwidth of 3.′′ 48, was obtained for a central massof 4.8 M_{⊙} and a infinite infall radius (dashed line in Fig. 25). The goodness of the fit is indicated by the value of the χ^{2} statistic for ν = 39 degrees of freedom (the total number of rings used in the fit, minus 1), χ^{2} = 42.7, which gives a reduced .
Fig. 25 Same as Fig. 19 for the Guitar Core in L1287, for the NH_{3} (1, 1) (blue circles)and (2, 2) lines (red circles). The rings used were 1″ wide, and a systemic velocity V _{sys} = −18.55 km s^{−1} was adopted. The best fit was obtained for an infinite infall radius, and a central mass M_{*} = 4.4 M_{⊙} (black dashed line). 

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8 Conclusions
The centralbluespot infall hallmark (MayenGijon et al. 2014) was studied quantitatively, taking as a basis the work of Anglada et al. (1987, 1991). The assumptions were that the line emission was optically thick, the gravitational infall motions dominated the kinematics over turbulent and thermal motions, the infall velocity and temperature were powerlaws of radius and increase inwards, with powerlaw indices of − 1∕2, and that the Sobolev approximation was valid. With these assumptions an analytical expression for the firstorder moment as a function of the projected distance was derived, for the cases of infinite and finite infall radius. The effect of a finite angular resolution was also studied, but the convolution with the beam has to be calculated numerically. These results were applied to existing data of several starforming regions (G31, B335, and L1287), obtaining good fits to the firstorder moment maps, and deriving values of the central masses onto which the infall is taking place. The values obtained for the central masses are 70–120 M_{⊙} for G31, 0.1 M_{⊙} for B335, and 4.8 M_{⊙} for the Guitar Core of L1287.
In conclusion, the centralbluespot infall hallmark appears to be a robust and reliable indicator of infall.
Acknowledgements
This work has been partially supported by the Spanish MINECO grants AYA201457369C3 and AYA201784390C2 (cofunded with FEDER funds), by MDM20140369 of ICCUB (Unidad de Excelencia “María de Maeztu”), and through the “Center of Excellence Severo Ochoa” award for the Instituto de Astrofísica de Andalucía (SEV20170709).
Appendix A Calculation of the moments for an infinite infall radius and arbitrary powerlaw indices
A.1 Intensity profile (infinite infall radius)
Let us assume that the infall velocity and temperature in an infalling molecular gas core are given by power laws with arbitrary powerlaw indices, − α and − β, (A.1)
The development made in Sect. 2 for α = β = 1∕2 can be generalized for any positive values of the powerlaw indices, α, β > 0. The projected distance and temperature (Eq. (4)) are now (A.2)
The expressions for z^{*}, z_{m}, p_{m} (Eq. (5)) become (A.3)
and t(p_{m}) (Eq. (6)) is now (A.4)
A.2 Line profile (infinite infall radius)
The equations derived in Sect. 3 can be generalized as follows. The temperature and LOS velocity (Eq. (8)) become (A.5)
The velocity v_{m} and temperatures t_{1} and t_{2} (Eqs. (9), (10), (12)) are now given by (A.6)
A.3 Moments calculation (infinite infall radius)
From Eq. (A.5) we can obtain v_{z} as an explicit function of t, (A.7)
where the bluewing profile (v_{z} < 0) is obtained for t_{1} < t < t_{2}, and the redwingprofile (v_{z} > 0) for 0 < t < t_{1} (see Eq. (A.6)).
In order to calculate the firstorder normalized moment μ_{1} (p) as a function of the projected distance, (A.8)
we need to evaluate the integrals (A.9)
A.4 Zerothorder moment (infinite infall radius)
For the blue wing, the limits of integration are from (t = t_{1}, v_{z} = −v_{m}) to (t = t_{2}, v_{z} = 0). The resulting integral for is (A.10)
where v_{m}, t_{1}, and t_{2} have already been defined in Eq. (A.6). The first term is (A.11)
The integral of the second term has the same dependence on p. This can be seen with the change of variables x = pt^{1∕β}, resulting in (A.12)
It is useful to define the function G_{0}, (A.13)
which, in general, is not analytical, but is convergent for α + β > 0, and the function to integrate is continuum for x > 0. With this notation, the zerothorder moment can be expressed as (A.14)
Similarly, for the red wing the limits of integration are from (t = 0, v_{z} = 0) to (t = t_{1}, v_{z} = v_{m}). The resulting integral for is (A.16)
Using the same notation used for the blue wing, we can obtain (A.17)
is analytical for α + β > 0, and its value is (A.20)
where Γ is the Gamma function.
Thus, the expression for the total zerothorder moment is (A.21)
A.5 Firstorder moment (infinite infall radius)
For the blue wing, using the same limits of integration as for the zerothorder moment, we have (see Fig. 5) (A.23)
and the integral can be evaluated using the same change of variables as for the zerothorder moment, obtaining (A.25)
where we defined the function G_{1} as (A.26)
which is analytical, and convergent for 2α + β > 0, (A.27)
Thus, the unnormalized firstorder moment of the blue wing can be expressed as (A.28)
Similarly, for the red wing we have (A.30)
that can be expressed as (A.31)
Note that B_{1} and R_{1} cancel each other partially, so that (A.33)
Thus, the expression for the total unnormalized firstorder moment is simply (A.34)
The final results, for an infinite infall radius, can be summarized as follows. (A.36)
where Γ is the Gamma function, and (A.37)
which, in general, has to be evaluated numerically.
A.6 Particular case (infinite infall radius) for powerlaw indices 1/2
For the particular case of α = β = 1∕2, the former expressions are simpler. In particular, the integral G_{0} is analytical, (A.38)
For the firstorder moment we have (A.42)
and the value of H_{1} is (A.44)
It can be useful to give the normalized firstorder moment separately for the blue and red wings, (A.45)
which have the same powerlaw dependence on the projected distance as μ_{1}.
In conclusion, for the total zeroth and firstorder moments we have (A.46)
Appendix B Calculation of the moments for a finite infall radius and arbitrary powerlaw indices
The critical value of the reduced coordinate q = p∕r_{i}, for an arbitrary value of the powerlax index α, is . For values , only the redwing emission is affected; for the line becomes symmetric (μ_{1} = 0); for q ≥ 1 all the wing emission disappears (μ_{0} = μ_{1} = 0).
B.1 Line profile (finite infall radius)
In order to calculate the moments of the red wing emission we need the values of v_{a}, t_{a}, and t_{b} as a function of p and q (see Fig. 10). For this, we need the values of z_{a} and z_{b}, the z coordinate of the two intersections of the equalLOSvelocity surface with the lineofsight.
The distance z_{a} is obtained readily from , which gives (B.1)
and, from Eq. (A.6) we have (B.2)
Fig. B.1 Plot of q′ as a functionof q (case α = β = 1∕2), for . Note that , and that q′ = q for . 

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The distance z_{b} requires more work. The equation of the equalLOSvelocity surface of velocity v_{a} (Eq. (A.6)) can be written as (B.3)
For a given p and v_{a}, z_{b} is a root of this equation that satisfies 0 < z_{b} < z_{a}. It is useful to use the variable , so that the equation to solve depends only on q and α, and becomes (B.4)
For rational values of α this equation is a polynomial in x. However, in general, the root has to be found numerically. An iterative algorithm that gives the correct root for is the following (B.5)
It is useful to introduce the parameter q′, which plays a role similar to that of q, (B.6)
so that the temperature t_{b}, corresponding to z_{b}, can be expressed as (B.7)
It can be easily shown that for , the parameter q′ is always bounded by q < q′ < 1, and that for , q′ = q (see Fig. B.1).
B.2 Zerothorder moment (finite infall radius)
The zerothorder moment of the blue wing, , does not depend on q, and we havealready seen that it can be expressed as (B.8)
The zerothorder moment of the red wing, , can be calculated as (B.9)
Using the expressions obtained for v_{a}, t_{a}, and t_{b}, can be expressed as (B.10)
where q′ is given by Eq. (B.6), and G_{0} by Eq. (A.13).
Finally, the total zerothorder moment can be written as (B.12)
B.3 Firstorder moment (finite infall radius)
The firstorder moment for the blue wing does not depend on q, and we have already seen that (B.14)
The firstorder moment of the red wing is the opposite of , except for a deficit of redwing emission (see Fig. 10), so that (B.15)
Taking into account the expressions of v_{a}, t_{a}, and t_{b}, the firstorder moment of the red wing can be written as (B.16)
where q′ is given by Eq. (B.6), and G_{1} by Eq. (A.26). Thus, the total unnormalized firstorder moment is (B.18)
The normalized first order moments for the blue and red wings separately are given by (B.20)
where B_{0}, B_{1}, R_{0} (q), and R_{1} (q) have already been given. Finally, the expression obtained for the normalized firstorder moment of the whole line, , is (B.21)
B.4 Final results (finite infall radius)
Assuming infall velocity and temperature that are power laws of the radius, (B.22)
we have, for a finite infall radius r_{i}, with q = p∕r_{i}, (B.23)
with H_{0}(q) and H_{1}(q) given by (B.24)
where q′ is an auxiliary parameter (B.25)
and x is the root, satisfying 0 < x < q^{−2}, of the equation (B.26)
to be solved numerically (see Eq. (B.5)), and (B.27)
B.5 Particular case (finite infall radius) for powerlaw indices 1/2
For the particular case of α = β = 1∕2, the former expressions are simpler. In particular, as already seen, the integral G_{0} is analytical, and the equation in x to find q′ = x^{−1∕2} is a 3rd degree polynomial, (B.28)
Since wealready know a root corresponding to z_{a}, x = q^{−2}, the polynomial is divisible by (x − q^{−2}), (B.29)
and thus the root x (and q′) can be found analytically (see Fig. B.1), (B.30)
Fig. B.2 Log–log plot of H_{0} and − H_{1}∕H_{0} as a function of q (case α = β = 1∕2), for . 

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Thus, the final results for this particular case are (B.31)
where B_{0} is given by Eq. (A.40), H_{0} and H_{1} (see Fig. B.2) are given by (B.32)
q′ is given by Eq. (B.30), G_{0} is given by Eq. (A.38), and G_{1} results in (B.33)
B.6 Finite angular resolution (finite infall radius) for powerlaw indices 1/2
Although the firstorder moment for a finite infall radius r_{i}, observed with a finite angular resolution, has to be calculated numerically, we can study its value at the origin, for p = 0. The convolution of the moments μ_{0}(p;r_{i}) and with a Gaussian beam of unit area and halfpower beamwidth b, (B.34)
It is useful to use the variable s ≡ r_{i}∕b, so that bothmoments can be expressed as (B.36)
The normalized firstorder moment at the origin will be (B.38)
The asymptotic values of μ_{1}(0;b, r_{i}) for r_{i} →∞ (equivalent to s≫ 1) must coincide with the expression derived with infinite infall radius (Eq. (22)). Effectively, (B.39)
as expected. Let us now evaluate the asymptotic expression for μ_{1} (0;b, r_{i}) for b →∞ (equivalent to s ≪1). Taking into account that H_{0}(q) = 0 for q > 1 and H_{1}(q) = 0 for , (B.41)
where and are constants to be evaluated numerically, resulting in C_{1}∕C_{0} ≃−0.060. Thus, (B.42)
Appendix C Finite spectral resolution and firstorder moment of a line
Let I(v) be the intensity of a line as a function of radial velocity. We are interested in calculating the firstorder moment of the line profile, (C.1)
We will make use of the relationship between the zeroth and firstorder moments of a function f and the values of its Fourier transform F[f] and its derivative F′[f] at the origin (see Bracewell 2000, Chap. 8) (C.3)
Let us assume that the finite spectral resolution of the spectrometer can be represented by the convolution of the real line profile with an instrumental response function W(v), (C.4)
and that the instrumental response is normalized to unit area, and is symmetric with respect to velocity or, more precisely, with zero firstorder moment, (C.5)
In the Fourier domain we have (C.6)
The zerothorder moment calculated from the observed profiles will be (C.7)
where we used that the Fourier transform of a convolution is the product of Fourier transforms (Bracewell 2000). Thus, the zerothordermoment is independent of the spectral resolution.
Let us consider now the unnormalized firstorder moment, (C.8)
By using the relations with the derivative of the Fourier transform, we obtain (C.9)
Thus, is independent of the spectral resolution, and so is the normalized firstorder moment, .
In conclusion. the spectral resolution does not affect the value of the firstorder moment of a line, as long as the spectrometer response can be described as a convolution with an instrumental response symmetric with respect to velocity.
Appendix D 2D Convolution of a powerlaw function with a Gaussian beam
Let F(p) be a 2D powerlaw function of the radial distance (D.1)
that has to be convolved with a Gaussian beam of unit area and halfpower beamwidth b, (D.2)
will have a different behavior for p ≫ b and p ≪ b. On the one hand, for p ≫ b the convolution with the Gaussian beam will not modify noticeably the powerlaw function and the result will not depend on b, (D.4)
On the other hand, for p ≪ b, F_{b} will not depend much on p and will be approximately constant, (D.5)
The integral can be evaluated analytically and the result is (D.7)
For the cases of interest, we have (D.8)
The characteristic radius that separates the two regions of F_{b} can be estimated as the intersection of the two asymptotic values, that is, F(p_{c}) = F_{0}, resulting in (D.9)
For the cases of interest, we have (D.10)
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All Figures
Fig. 1 Intersection of the surfaces of equal lineofsight (LOS) velocity, v_{z}, for a collapsing protostellar core, with the plane (p, z) (see Sect. 2). The LOSvelocities are different pairs of negative and positive velocities (v_{z} = ±2^{−1∕2}, ±1, ±2^{1∕2}, ±2). The blue lines are the contours for v_{z} < 0, and the red lines for v_{z} > 0. The observer, located to the right, at p = 0, z = −∞, sees the emission coming from the part of the surfaces of equal LOSvelocity facing the observer, traced with thick lines. The dashed lines indicate the position of (z_{m}, p_{m}) for each contour, where p_{m} is the maximum value of p, and z_{m} the corresponding value of z. The square frame is drawn at p = ±1, z = ±1. 

Open with DEXTER  
In the text 
Fig. 2 Intensity maps for pairs of LOSvelocities, V_{z} = ±0.8 km s^{−1} (top), ± 1.1 km s^{−1} (middle), and ± 1.4 km s^{−1} (bottom), calculated for a a mass M_{*} = 1 M_{⊙}, a distance of 140 pc, and a beam of 0.′′1. The axes are position offsets labeled in arcsec. The intensity scale is the same for all maps. We note the sharp peak at the center of the blueshifted LOSvelocity maps (left), and the flat, slightly concave shape of the redshifted LOSvelocity maps (right), and the more compact emission for high values V_{z}  (bottom) than for low values of V_{z} (top). 

Open with DEXTER  
In the text 
Fig. 3 Intensity profiles t(p) for v_{z} = −1 (top panel, blue) and v_{z} = +1 (bottom panel, red). The different dashed lines indicate the blue and red wing temperature at the maximum value of p, p = p_{m}, and the blue wing asymptotic behavior for p ≪ 1. 

Open with DEXTER  
In the text 
Fig. 4 Intensity profiles t(p) for pairs of positive and negative velocities, v_{z} = ±1, ±2^{1∕2}, ±2. 

Open with DEXTER  
In the text 
Fig. 5 Line profile showing the emission at a projected distance p as a function of LOS velocity. The blue wing emission encompasses LOS velocities from − v_{m} to 0, and its peak value (at v_{z} = 0) is t_{2} = p^{−1∕2}. The red wing emission encompasses LOS velocities from 0 to v_{m} = 0.620 p^{−1∕2}, and its peak value (at v_{z} = v_{m}) is t_{1} = 0.760 p^{−1∕2}. 

Open with DEXTER  
In the text 
Fig. 6 Line profiles for values of p = 1, 2^{1∕2}, 2. 

Open with DEXTER  
In the text 
Fig. 7 Log–log plot of the moments μ_{0} and − μ_{1}, as a functionof projected distance p. 

Open with DEXTER  
In the text 
Fig. 8 Log–log plots of the moments μ_{0} and − μ_{1} (solid lines) for a Gaussian beam with HPBW b = 1 (dotted line), as a function of the projected distance p, and their asymptotic values (dashed lines) for p ≪ b and p ≫ b. 

Open with DEXTER  
In the text 
Fig. 9 For a given projected distance the wing emission comes from the thick part of the line of sight, for radii less than r_{i}. The emission from the thin part of the line of sight is hidden to the observer.For the blue wing, the observer can observe the emission coming from z = 0 (with v_{z} = 0) to z = z_{m} (with v_{z} = −v_{m}). For the red wing, the observer can observe the emission coming from z = −z_{a} (with r = r_{i}, v_{z} = v_{a}) to z = −z_{m} (with v_{z} = +v_{m}), and, since the corresponding part of the equalLOSvelocity surface facing the observer is missing (it would be outside the infall radius), from z = 0 (with v_{z} = 0) to z = −z_{b} (with v_{z} = v_{a}). The velocity v_{a} is the LOS velocity of the equalLOSvelocity surface that intersects the sphere r = r_{i} at a projected distance p, i.e. v_{a} = v_{z}(p, −z_{a}) = v_{z}(p, −z_{b}). 

Open with DEXTER  
In the text 
Fig. 10 Line profile for and p = 1. The blue wing is unaffected. For the red wing the emission at velocities v_{z} < v_{a} comes from the hot part of the red equalLOSvelocity surface. For the blue and red wings become symmetric. 

Open with DEXTER  
In the text 
Fig. 11 Log–log plot of μ_{0} and − μ_{1} as a functionof the projected distance p, for an infall radius r_{i} = 1 (solid lines) and for an infinite infall radius (dashed lines). 

Open with DEXTER  
In the text 
Fig. 12 Log–log plot of μ_{0} and − μ_{1} as a functionof the projected distance, for an infall radius r_{i} = 1 and beamwidth b = 0.1 in reduced units (solid lines) and b = 0 (dashed lines). 

Open with DEXTER  
In the text 
Fig. 13 Firstorder moment μ_{1} for an infinite infall radius, as a function of the projected distance, D θ (in kau), for beamwidths D θ_{b} = 0.1, 0.2, 0.5, 1, 2, 5, and 10 kau, where D is the distance to the source. The vertical axis scales as . The left axis is μ_{1} for a central mass M_{*} = 1 M_{⊙}, while the right axis is for M_{*} = 10 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 14 Firstorder moment μ_{1} for an infinite infall radius, as a function of the projected distance in units of the beamwidth, θ∕θ_{b} (adimensional), for central masses M_{*} = 0.1, 0.2, 0.5, 1, 2, 5, and 10 M_{⊙}. The vertical axis scales as , where D is the distance to the source. The left axis is μ_{1} for D θ_{b}= 1 kau, while the right axis is for D θ_{b} = 0.1 kau. 

Open with DEXTER  
In the text 
Fig. 15 Firstorder moment μ_{1}, as a functionof projected distance in units of the infall radius, D θ∕R_{i} (adimensional), for beamwidths D θ_{b} = 0, 0.1, 0.2, 0.5, 1, and 2 kau, where D is the distance to the source. The vertical axis scales as . The left axis is μ_{1} for a central mass M_{*} = 1 M_{⊙}, while the right axis is for M_{*} = 10 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 16 Same as Fig. 15, for D θ_{b} = 0.1 kau, and R_{i} = 0.2, 0.5, 1, 2, 5, 10, and ∞ kau. The horizontal axis is the projected distance, D θ (in kau). Note that the line for R_{i} = 1 kau in this figure coincides with the line for D θ_{b} = 0.1 kau in Fig. 15. 

Open with DEXTER  
In the text 
Fig. 17 Function J(s) in the expression of μ_{1}(0;θ_{b}, R_{i}), showing its asymptotic values for s ≪ 1 and s ≫ 1. 

Open with DEXTER  
In the text 
Fig. 18 Maps of G31 firstorder (color scale) and zerothorder moment (contours) for the ammonia inversion transitions (2, 2) to (6, 6) (MayenGijon et al. 2014). The color scale, the same for all panels, ranges from 94 to 101 km s^{−1}. The contours are in steps of 10% of the peak value for all the maps, except for the (4, 4) transition, for which the steps are 20%. The beam is shown in the lower right corner of each panel. 

Open with DEXTER  
In the text 
Fig. 19 G31 firstorder moment for the ammonia inversion transitions (2, 2) (black line and open circles), (3, 3) (red line andfilled circles), (4, 4) (blue line and filled squares), (5, 5) (magenta line and open squares), and (6, 6) (green line and filled diamonds) (MayenGijon et al. 2014), as a function of angular distance, measured for rings of width 0.′′ 1 and average radius θ. The error bars are the rms of the velocity inside each ring. The righthand vertical axis shows the velocities obtained from the firstorder moment maps, while the lefthand vertical axis shows the velocity with respect to the systemic velocity of G31, taken as V _{sys} = 97.4 km s^{−1}. 

Open with DEXTER  
In the text 
Fig. 20 Model (black dashed line) fitted to the G31 firstorder moment, calculated for a central mass of 122 M_{⊙}, an infall radius of 5 kau, and a halfpower beamwidth of ~0.′′33 for the (3, 3), (5, 5), and (6, 6) transitions (top), and 0.′′16 for the (4, 4) transition (bottom). 

Open with DEXTER  
In the text 
Fig. 21 B335 zerothorder (contours) and firstorder moment (color scale) of the H^{13} CO^{+} (J = 1–0) line, obtained from the channel maps of Kurono et al. (2013). Contours are in steps of 511 mJy beam^{−1} km s^{−1}. The color scale at the right border is in m s^{−1}. The (0, 0) position corresponds to α(J2000) = 19^{h}37^{m}00.^{s}89, δ(J2000) = +07°34′09.′′6. The synthesized beam is shown in the lower left corner. 

Open with DEXTER  
In the text 
Fig. 22 B335 firstorder moment of the ^{13}CO (J = 2–1) line observed with ALMA. Top: moment of the red channels, with velocities higher than the systemic velocity, V _{sys} (color scale: 8.9–10.1 km s^{−1}). Middle: moment of all the channels (color scale: 7.4–9.2 km s^{−1}). Bottom: moment of the blue channels, with velocities lower than V _{sys} (color scale: 6.8–8.0 km s^{−1}). The (0, 0) position, marked with a plus sign, is the same as in Fig. 21. The synthesized beam is shown in the lower left corner of the middle panel. 

Open with DEXTER  
In the text 
Fig. 23 Same as Fig. 19 for B335. Top: H^{13}CO^{+} (J = 1–0) line observed at Nobeyama (2″ ring width). Bottom: ^{13}CO (J = 2–1) line observed with ALMA (0.′′1 ring width; , red line and symbols; μ_{1}, black line and symbols; , blue line and symbols) The dashed lines in the top and bottom panels show the best simultaneous fit to the Nobeyama moment μ_{1} (beamwidth 5.′′0), and the ALMA moments and (beamwidth 0.′′31). The best fit is obtained for an infinite infall radius, a systemic velocity V _{sys} = 8.29 km s^{−1}, and a centralmass of 0.09 M_{⊙}. The black dotted line in the bottom panel is not a fit, but the value of μ_{1} predicted by the bestfit model for the ALMA firstorder moment. 

Open with DEXTER  
In the text 
Fig. 24 NH_{3} (1, 1) line central velocity of the Guitar Core in L1287. The central blue spot is blueshifted ~ 0.5 km s^{−1} with respect to the ambient gas. The embedded sources in the center are indicated by dots (mm, Juárez et al. 2019), and crosses (cm, Anglada et al. 1994). The synthesized beam is shown in the lowerright corner of the map. 

Open with DEXTER  
In the text 
Fig. 25 Same as Fig. 19 for the Guitar Core in L1287, for the NH_{3} (1, 1) (blue circles)and (2, 2) lines (red circles). The rings used were 1″ wide, and a systemic velocity V _{sys} = −18.55 km s^{−1} was adopted. The best fit was obtained for an infinite infall radius, and a central mass M_{*} = 4.4 M_{⊙} (black dashed line). 

Open with DEXTER  
In the text 
Fig. B.1 Plot of q′ as a functionof q (case α = β = 1∕2), for . Note that , and that q′ = q for . 

Open with DEXTER  
In the text 
Fig. B.2 Log–log plot of H_{0} and − H_{1}∕H_{0} as a function of q (case α = β = 1∕2), for . 

Open with DEXTER  
In the text 
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