Predictions for Observing Protostellar Outflows with ALMA

Abstract

Introduction

Young protostars are observed to launch energetic collimated bipolar mass outflows (Frank et al., 2014). These protostellar outflows play a fundamental role in the star formation process on a variety of scales. On sub-pc scales they entrain and unbind core gas, thus setting the efficiency at which dense gas turns into stars (Matzner et al., 1999; Alves et al., 2007; Machida et al., 2013; (citation not found: Offner14). Interaction between outflows and infalling material may regulate protostellar accretion and, ultimately, terminate it . On sub-pc up to cloud scales, outflows inject substantial energy into their surroundings, potentially providing a means of sustaining cloud turbulence over multiple dynamical times(Nakamura et al., 2007; Carroll et al., 2009; Wang et al., 2010; Arce et al., 2010; Nakamura et al., 2011; (citation not found: hansen11) Nakamura et al., 2014).

The origin of outflows is attributed to the presence of magnetic fields, and a variety of different models have been proposed to explain the launching mechanism (e.g., Arce et al., 2007). Of these, the “disk-wind" model (Blandford et al., 1982; Pelletier et al., 1992), in which the gas is centrifugally accelerated from the accretion disk surface, and the “X-wind" model (Shu et al., 1988), in which gas is accelerated along tightly wound field lines, are most commonly invoked to explain observed outflow signatures. However, investigating the launching mechanism is challenging because launching occurs on scales of a few stellar radii and during times when the protostar is heavily extincted by its natal gas. Consequently, separating outflow gas from accreting core gas, discriminating between models, and determining fundamental outflow properties are nontrivial.

Three main approaches have been applied to studying outflows. First, single-dish molecular line observations have been successful in mapping the extent of outflows and their kinematics on core to cloud scales (citation not found: bourke97) Arce et al., 2010; Dunham et al., 2014). However, outflow gas with velocities comparable to the cloud turbulent velocity can only be extracted with additional assumptions and modeling (e.g., Arce et al., 2001; Dunham et al., 2014), which are difficult to apply to confused, clustered star forming environments (Plunkett et al., 2013). Second, interferometry provide a means of mapping outflows down to 1,000 AU scales scales (Arce et al., 2006; Plunkett et al., 2013), and the Atacama Large Millimeter/submilllimeter Antenna (ALMA) is extending these limits down to sub-AU scales (Arce et al., 2013; Lee et al., 2014). However, interferometry is not suitable for producing large high-resolution maps and it resolves out larger scale structure. Consequently, it is difficult to assemble a complete and multi-scale picture of outflow properties with these observations. Finally, numerical simulations provide a complementary approach that supplies three-dimensional predictions for launching, entrainment and energy injection (Machida et al., 2013; (citation not found: siefried13) (citation not found: Offner14).

The most promising avenue for understanding outflows lies at the intersection of numerical modeling and observations. By performing synthetic observations to model molecular and atomic lines, continuum, and observational effects, simulations can be mapped into the observational domain where they can be compared directly to observations (e.g., Offner et al., 2011; (citation not found: Offner12b) (citation not found: Mairs13). Such direct comparisons are important for assessing the “reality" of the simulations, to interpret observational data and to assess observational uncertainties (Goodman, 2011). In addition to observational instrument limitations, chemistry and radiative transfer introduce additional uncertainties that are difficult to quantify without realistic models (citation not found: beaumont13). Synthetic observations have previously been performed in the context of understanding outflow opening angles (Offner et al., 2011), observed morphology (citation not found: peters13), and impact on spectral energy distributions (citation not found: Offner12a).

The immanent completion of ALMA provides further motivation for predictive synthetic observations. Although ALMA will have unprecedented sensitivity and resolution compared to existing instruments, by nature interferometry resolves out large-scale structure and different configurations will be sensitive to different scales. Atmospheric noise and total observing time may also effect the fidelity of the data. Previous synthetic observations performed by (citation not found: Offner12b) suggest that the superior resolution of full ALMA and the Atacama Compact Array (ACA) will be able to resolve core structure and fragmentation prior to binary formation. (citation not found: peters13) predicts that ALMA will be able to resolve complex outflow velocity structure and helical structure in molecular emission.

In this paper we seek to quantify the accuracy of different ALMA configurations in recovering fundamental gas properties such as mass, line-of-sight momentum, and energy. We use the casa software package to synthetically observe protostellar outflows in the radiation-hydrodynamic simulations of (citation not found: Offner14). By modeling the emission at different times, inclinations, molecular lines, and observing configurations we evaluate how well physical quantities can be measured in the star formation process. In section §\ref{Methods} we describe our methods for modeling and observing outflows. In section §\ref{results} we evaluate the effects of different observational parameters on bulk quantities. We discuss results and summarize conclusions in §\ref{Conclusions}.

Methods

Hydrodynamic Simulation

In order to assess the recovery of information using ALMA, we post-process a self-gravitating radiation-hydrodynamic simulation in which we have complete three-dimensional temperature, density and velocity information. The simulation, th0.1fw0.3, was previously presented in (citation not found: Offner14). We briefly describe the simulation properties here and refer the reader to OA14 for additional details.

The calculation was performed with the orion adaptive mesh refinement (AMR) code (Truelove 1998, Klein 1999). The simulation follows the collapse of an isolated, turbulent low-mass core. It begins with an initially uniform, cold \(4M_\odot\) sphere of radius \(R_c=2\times 10^{17}\)cm, density \(\rho_c=2\times 10^{-19}\) g cm\(^{-3}\) and temperature \(T_c=10\) K . This core is embedded in a warm, diffuse gas with \(\rho=rho_c/100\) and \(T=100 T_c\) K. The dense gas is initialized with a grid of random velocity perturbations such that the initial rms velocity dispersion is 0.5 km s\(^{-1}\).

Additional levels of adaptive mesh refinement (AMR) are inserted as the core collapses under the influence of gravity. The core itself is resolved with a minimum cell size of \(\Delta_{\rm min}\simeq 0.001\) pc, where the maximum level of refinement has \(\Delta_{\rm min}\simeq 26 AU\). Once the central region exceeds the maximum grid resolution (\(\rho_{\rm max}\simeq 6.5 \times 10^{-15}\) g cm\(^{-3}\), (e.g., Truelove et al., 1998)), a “star" forms. This star, which is represented by a Lagrangian sink particle, accretes, radiates and launches a collimated bipolar outflow (Krumholz et al., 2004; (citation not found: Offner09) (citation not found: cunninham11). The rate of mass loss due to the outflow is set to a fixed fraction of the instantaneous accretion rate: \(\dot m_w = f_w \dot m_*\), where \(f_w = 0.3\) is the outflow launching rate given by the X-wind model (Shu et al., 1988). The distribution of outflow momentum is parameterized by a fixed collimation angle, \(\theta_0=0.1\), which is empirically determined to be similar to that of observed outflows (Matzner et al., 1999; Cunningham et al., 2011). Although \(\theta_0\) is constant in time, the outflow injection into the AMR grid occurs on such small scales that the outflow properties such as the opening angle and morphology evolve hydrodynamically; these appear to agree well with observed low-mass outflows (Offner et al., 2011; (citation not found: Offner14).

By the end of the calculation (\(t= 0.5\) Myr), the simulation contains a single star with a mass of \(\sim\)1.45 \(\msun\). Most of the core mass has been ejected from the domain and the remaining gas has a rms mass-weighted velocity dispersion of \(\sim\)1 km s\(^{-1}\). Since OA14 found that the final stellar mass and star formation efficiency did not depend strongly on \(\theta_0\) and \(f_w\), we analyze only a single run. However, we note that different initial core masses, rotations, and magnetic field strengths might produce qualitatively different results (Machida et al., 2013).

Molecular Line Modeling

We use the non-local thermodynamical equilibrium radiative transfer code radmc-3d1 to compute the line emission in \(^{12}\)CO(1-0) and \(^{13}\)CO(1-0). We adopt the Large Velocity Gradient (LVG) approximation (Shetty et al., 2011), which solves for the rotational level populations by solving the equations for local radiative statistical equilllibrium. radmc-3d requires 3D input gas densities, velocities and temperatures, which are produced as outputs by the hydrodynamic simulation. We perform the radiative transfer on a uniform \(256^3\) grid, where we interpolate all the AMR data to the second refinement level (\(\Delta x =0.001\)pc). We include turbulent line broadening on scales at and below the grid resolution by adding a constant microturbulence of 0.05 km s\(^{-1}\). For \(^{12}\)CO we smooth the velocity field by using a doppler parameter of 0.025, such that the velocity field is linearly interpolated between velocity jumps greater that 0.025\(c_s\), where \(c_s\) is the local sound speed. \(^{13}\)CO has a doppler parameter of 0.25.

To obtain the CO abundances from the total gas density, we assume that molecular Hydrogen dominates in all gas cooler than 1,000 K, where \(n_{{\rm H}_2}=\rho/(2.8 m_p)\). We adopt constant CO abundances of [\(^{12}\)CO/H\(_2\)]=\(8.6 \times 10^{-5}\) and [\(^{12}\)CO/\(^{13}\)CO]=62 for gas cooler than 900 K; otherwise the CO abundance is set to zero. Thus, line emission only originates in relatively cold gas in the dense core and gas entrained by the outflow; the warm, low-density ambient material and the hot, outflow gas, which is ionized by construction, do not emit. We adopt the molecular collisional coefficients from Schöier et al. (2005).

Inteferometry Modelling

We convert the RADMC-3D output into skymaps that we used as input for the Common Astronomy Software Applications (CASA) task ’simobserve’. Using simobserve, we placed the model at the distance (450 pc) and the RA and Dec. of HH46/47, an outflow recently observed by ALMA (Arce, source). We add thermal noise to the synthetic observations, assuming 0.5mm of precipitable water vapour and a ground temperature of 269K which reflect good observing conditions. Synthetic observations were made with various ALMA configurations, pointing times and integration times. We deconvolve and clean the CASA output using task ’simanalyze’. Table \ref{tab_runs} lists the CASA parameters and input values for each run we perform.

Interferometry analysis

The choice of settings used to clean the simulated observations is discussed in the appendix.


  1. http://www.ita.uni-heidelberg.de/ dullemond/software/radmc-3d/

A summary of our simulated obervations can be found in table \ref{tab_runs}

\label{tab_runs}

Run Name Molecule Viewing Angle ALMA Configuration Integrations Time Pointing Time Cleaning Iterations Cleaning Threshold
R1 12 00 Compact 7200 10 10000 0.1
R2 12 00 Cycle 1.3 7200 10 10000 0.1
R3 12 00 Full 3 3600 10 10000 0.1
R4 12 00 Full 3 7200 10 10000 0.1
R5 12 45 Compact 7200 10 10000 0.1
R6 12 45 Cycle 1.1 7200 10 10000 0.1
R7 12 45 Full 3 3600 10 10000 0.1
R8 12 45 Full 3 7200 10 10000 0.1
R9 13 45 Compact 7200 10 10000 0.1
R10 13 45 Cycle 1.1 7200 10 10000 0.1
R11 13 45 Full 3 3600 10 10000 0.1
R12 13 45 Full 3 7200 10 10000 0.1
R13 13 00 Cycle 1.1 7200 30 10000 0.01
R14 13 00 Cycle 1.1 7200 30 10000 0.1
R15 13 00 Cycle 1.1 7200 30 10000 1.0
R16 13 00 Cycle 1.3 7200 10 10000 0.1
R17 13 00 Cycle 1.3 7200 30 10000 0.1
R18 13 00 Compact 7200 10 10000 0.1
R19 13 00 Compact 7200 30 10000 0.1
R20 13 00 Compact 7200 100 10000 0.1
R21 13 00 Full 3 3600 10 1000 0.1
R22 13 00 Full 3 3600 10 10000 0.1
R23 13 00 Full 3 3600 10 20000 0.1
R24 13 00 Full 3 300 10 10000 0.1
R25 13 00 Full 3 600 10 10000 0.1
R26 13 00 Full 3 7200 10 10000 0.1

Derivation of Mass, Momentum and Energy

We used the following formulae to determine the mass of the outflow:

\[%\label{eqn:Mass from temp} M(v, \alpha, \delta) = [{\rm H}_2/^{12}{\rm CO}] \mu_m A(\alpha, \delta) F (T_{ex}) T_B(v, \alpha, \delta), \Delta v\]

where \[%\label{eqn:F(T_ex)} F(T_{ex}) = 2.31 * 10^{14} \frac{T_{ex} + 0.92}{1 - e^{-5.53/t_{ex}}} \frac{1}{J(T_{ex}) - J(T_{bg})}\] and assuming that the gas is a black body \[%\label{eqn:rayleigh jean} T_B = \frac{ \Sigma B_v c^2}{2 v^2 k T}\] and \[%\label{eqn:J(T)} J(T) = \frac{hv}{k(e ^ {\frac{hv}{kT}} - 1)}\] \(M\) is the total mass, \([{\rm H}_2/^{12}{\rm CO}]\) the mass ratio between \(H_2\) and \(^{12}CO\) (\(1/(8.6 * 10 ^{-5})\), \(\mu_m\) the mean molecular mass of \(H_2\) (\(2 * 1.6733 * 10 ^{-24}g\)), \(A\) the area of emission (1 pixel), \(T_{ex}\) the assumed temperature of excitation, \(T_{bg}\) the background temperature (\(0K\)), \(T_B\) the temperature calculated using the Rayleigh Jeans law, \(\Delta v\) the velocity resolution, \(k\) the Boltzman constant, \(v\) the frequency of observation (115.3Ghz for \(^{12}CO\), 110.2Ghz for \(^{13}CO\)), \(h\) Planck’s constant.

(Bourke et al. 1997) Every symbol here must be defined. What is v, what is alpha, A, etc? Explain it to yourself a year ago. If that you wouldn’t understand, the reader probably won’t either. State what this equation assumes. Also how do you get \(T_B\) from the datacube I gave you? Give the equation you use. Edit: I think that this has been dealt with, though not 100% sure I have explained everything. Have checked the code and these are the equations. Should I include the fact that mass is the integral/sum over all pixels or is that implied?

To ontain the momentum and energy, we used the calculated mass for each voxel as well as its known velocity. The data had velocity resolution of 0.08 km/s and included mass with \(|v| < 10 km/s\). need to add a couple sentences on how you derive the momentum and energy: “To obtain the momentum...”

Before discussing excitation temperature, need a general paragraph on the basic trends, i.e. what does the raw simulation show, what are the general trends with time. I can write this if you want. Edit: I feel that there is a lot to add to this - this description just came from watching the video a couple of times.

The simulation shows the gas cloud collpasing and the protostar forming at t=0.17Myr. The outflow begins to form immediately and is well extablished by t=0.22Myr with a significant mass of dense, high velocity gas being expelled from the protostar at the poles. However, the outflow dissapates and by t=0.35Myr its structure is no longer visible. Any gas left outside of the protostar is of low density and moving at a low velocity.

Results

Excitation Temperature

Figure \ref{texcit} illustrates the impact of the assumed excitation temperature on the mass estimate. We found that there was a systematic bias to overestimate the mass of the outflow. We attribute this to errors in assuming a single temperature of excitation. We examined the simulation data and found that there was significant temperature variation in the data cube and slight temperature variation over time. We followed the same protocol as observers and assumed a single temperature of excitation for each image. However we used a different temperature for each time. These temperatures are shown in Table \ref{tab_texcit}.

See how I added the table number and caption below. Do this for the other tables. Also explain how you get this temperature. I think you did: \[\bar T = \frac{ \Sigma \rho_i T_i }{\Sigma \rho_i}\] update if this is not what you did - Edit this is slightly complicated as T average was actually really high, even when accounting for mass. However if you looked at a graph of the temperatures you saw two spikes - one somewhere between 100K - 1000K (which was low density gas) and another at  10K. I took the average of this lower spike. Should I call of T_avg high density gas?

\label{tab_texcit} Time (Myr) Temperature (K)
0.2 17.2254077826
0.24 11.2679129249
0.27 11.4728610325
0.3 11.1062020028
0.35 11.6817959508
0.4 11.8538009796
0.5 12.4322632105

Mass with \(|v| > 1\) calculated with various values T\(_{ex}\) versus time using the ^13CO simulated emission. \label{fig2}

Outflow Identification

Throughout this paper we define mass moving at \(|v| > 1\) km/s as part of the outflow. This cutoff was determined by observing the simulation data. We show here that this choice does not impact results.

Figure \ref{fig3} shows that our observations improve as we increase the cutoff, though we found that we still overestimate the mass slightly.

Observed and simulation mass versus time with various cutoff velocities. The simulation mass is taken from density data cubes and so is the true simulation mass. The observed mass was observed with the settings from R26. \label{fig3}

Figure \ref{fig4} shows that while different observational parameteres disagree about the total mass with various cutoffs, they agree very well about the percentage of mass above a certain velocity. All observations miss out on very low velocity gas due to its optical thickness, but detect the correct ratio of higher velocity gas.

Mass percentage versus velocity cutoff for two synthetic observations and the simulation. Observations were made at \(t=0.27\)Myrs with the R26 and R18 configurations. \label{fig4}

Integration time

We analyzed the effect changes to the total integration time had on the quality of the observations. We used data from observations R23-R26, all of which used the same ALMA configuration with 10s pointings.

We found that longer observations obtained accurate results for mass with \(|v| > 1km/s\). However, these observations could not detect much of the low velocity gas. Shorter observations did not provide accurate results. These results are shown in Figure \ref{fig5} and \ref{fig6}

Mass versus time for synthetic \(^{13}\)CO with various integration times. The synthetic observations are performed with the full ALMA configuration, with 10 second pointings. \label{fig5}

Mass \(|v| > 1km/s\) versus time for synthetic \(^{13}\)CO with various integration times. The synthetic observations are performed with the full ALMA configuration, with 10 second pointings. \label{fig6}

We found that longer observations, as shown in \ref{fig7} were able to provide an detailed and accurate map of the velocity of the outflow. Shorter observations were able to determine the general structure of the outflow but lacked detail - \ref{fig8}.

Velocity contour plot of a synthetic \(^{13}\)CO observation at \(t = 0.27\)Myr. The synthetic observation is performed with the full ALMA configuration, over 300s with 10 second pointings. \label{fig7}

Velocity contour plot of a synthetic \(^{13}\)CO observation at \(t = 0.27\)Myr. The synthetic observation is performed with the full ALMA configuration, over 7200s with 10 second pointings. \label{fig8}

Pointing Time

We analyzed the effect changes to the pointing time had on the quality of the observations. We used data from observations R18-R20, all of which used the compact ALMA array with 7200s integrations.

We found that shorter pointings were better, but that we only lost a small amount of information with longer pointings. Thus, if there is any overhead to short pointings, longer pointings with a longer integration time would improve the data. These results are shown in Figure \ref{fig9}

We found that increasing the pointing time had a similarly small effect when observing the velocity.

ALMA Configuration

We simulated observations using the ALMA configurations listed in Table \ref{tab_mass}.

\label{tab_mass} Array Name Dishes Average Dish Separation (m) Longest? Baseline (m) Beam (") & Total Area (m^2)
Cycle 0 Compact & 16 & 56.9 & 126 & 5.23 & 7240
Cycle 1.1 & 32 & 71.4 & 166 & 3.99 & 14500
Cycle 1.3 & 32 & 164 & 443 & 1.49 & 14500
Full 3 & 50 & 93.2 & 260 & 2.54 & 22600

We found that the ALMA configuration had a significant effect on our results.

We found that objects of this size were best observed with ALMA configurations with a large beam. Figure (n?) shows the mass observed by the Cycle 1.1 and 1.3 configurations. Configuration 1.3 with its large baseline and small beam was significantly less effective than configuration 1.1, with all other observational parameters kept constant (R14 and R17).

Mass versus time of synthetic ALMA observations. The observations were performed with the cycle 1.1 and 1.3 configurations and lasted 7200s with 30s pointings.

While it has less of an effect than the beam, increasing the number of antennae and area of the array helps. Despite having a larger beam, the full array in configuration 3 performs better than the compact array - figure (nasdf).

Mass versus time of synthetic ALMA observations. The observations were performed with the Full and Compact configurations and lasted 7200s with 10s pointings.

Configurations with larger areas and beams perform better at detecting the velocity as well. While the 1.3 configuration purports to give a very detailed map of of the structure of the outflow, much of this is noise. The data from the full array is not visibly different from that of the 1.1 configuration. On close examination, it does provide slightly more detail as a result of its increased area and smaller beam.

Velocity contour plot of a synthetic \(^{13}\)CO observation at \(t = 0.27\)Myr. The synthetic observation is performed with the 1.3 ALMA configuration, over 7200s with 30 second pointings.

Velocity contour plot of a synthetic \(^{13}\)CO observation at \(t = 0.27\)Myr. The synthetic observation is performed with the 1.1 ALMA configuration, over 7200s with 30 second pointings.

\(^{12}\)CO compared to \(^{13}\)CO

An important assumption made by observers is that the gas is optically thin. When observing \(^{13}\)CO we found that this is for all but the densest gas, and certainly in the outflow. However, we found that \(^{12}\)CO is optically thick at all velocities.

Figure (asdf) shows the calculated mass versus time for observations in \(^{12}\)CO and \(^{13}\)CO if we assume that both are optically thin. These results make it clear that \(^{12}\)CO is optically thick, even for gas with \(|v| > 1\).

Mass versus time for synthetic \(^{13}\)CO and \(^{12}\)CO observations. The synthetic observations are performed with the full ALMA configuration, 10 second pointings and 7200s total observation time.

Optical Depth

We can use the optically thin \(^{13}\)CO emission to correct the \(^{12}\)CO. Given the brightness temperatures for each line, it is possible to derive the optical depth, \(\tau_{12}\) (citation not found: dunham13):

\[\begin{aligned} \label{eqn:mass} \frac{1 - e^{-\tau_{12}}}{\tau_{12}} = \frac{ T_{\rm mb 12}}{T_{\rm mb 13}} \frac{[ ^{13}{\rm CO}]}{[^{12}{\rm CO}]}\end{aligned}\]

We then correct the \(^{12}\)CO data by calculating the optical depth of each voxel and multiplying the emission by this factor. Figure [] shows the estimated mass after this correction is applied.

Mass versus time for synthetic ^13CO and ^12CO observations. The synthetic observations are performed with the full ALMA configuration, 10 second pointings and 7200s total observation time. The ^12CO is corrected using equation (X)

We found that while the ^12CO observation has some artifacts, it is useful for determining the velocity and structure of the outflow. However, the ^13CO observations are siginificantly better.

Velocity contour plot of a synthetic ^12CO observation at 0.3Myr. The synthetic observation is performed with the full ALMA configuration, over 7200s with 10 second pointings.

Velocity contour plot of a synthetic ^12CO observation at 0.3Myr. The synthetic observation is performed with the full ALMA configuration, over 7200s with 10 second pointings. The ^12CO is corrected using equation (X)

Viewing angle

We also considered how changes in the observation location could affect observations. The simulation was observed from the Z axis and 45 degrees away. 45 degrees is the inclination, not the position in the sky. The two cases are if you were on a sphere and walked over 45 degrees to look at the same object, at the same distance, but from a different view. In RADMC coordinates the views are along the Z axis (\(\phi=0\))–in which the outflow is inclined by about 20degrees– and from \(\phi=\pi/2\) when the outflow is inclined about 40 degrees. Note that because the outflow is not aligned in radmc coordiantes, rotating by 45 degrees along 1 axis doesn’t nicely coorespond to the same degree of rotation along that axis for the simulated outflow.

Mass versus time for synthetic \(^{13}\)CO observations from above the Z axis and between the Z and Y axes. The synthetic observations are performed with the full ALMA configuration, 10 second pointings and 7200s total observation time.

Mass \(|v| > 1km/s \) versus time for synthetic \(^{13}\)CO observations from above the Z axis and between the Z and Y axes. The synthetic observations are performed with the full ALMA configuration, 10 second pointings and 7200s total observation time.

This shows that while we expect slightly different results for the mass moving a $|v| > 1km/s$ in the line of sight of the observation, these results do not differ by too much. {\bf we can't claim this, since we haven't directly checked it: Most observations angles will give similar results. Only a very few will be significantly different. I think you can say, "For optically thin emission, as in the case of $^{13]$CO, we expect the estimated mass should be similar for outflow inclinations between $\sim$10-35 degrees." Note these are when you are not looking at the outflow either totally pole-on or edge on. You could mention here that the projected velocity will be very different for different angles so the momentum and energy estimates could be significantly different.}
Simulation Mass Simulation Mass, |V| > 1 ^13CO CASA Mass ^13CO CASA Mass, |V| > 1 ^12CO CASA Mass (uncorrected) ^12CO CASA Mass (uncorrected), |V| > 1 ^12CO CASA Mass (corrected) ^12CO CASA Mass (corrected), |V| > 1
Mass (\(M_{\odot}\)) 3.216 0.388 2.041 0.492 0.0957 0.0513 2.066 0.498
Momentum (\(M_{\odot} km/s\)) 1.566 0.590 1.376 0.788 0.122 0.104 1.392 0.797
Energy (\(CGS 10^{42}\)) 18.75 13.45 19.75 16.47 2.947 2.840 19.98 16.67
Time (Myr) M_tot(\(M_{\odot}\)) M_|v|>1(\(M_{\odot}\)) M_^13CO(\(M_{\odot}\)) M_^13CO, |v|>1(\(M_{\odot}\)) M_^12CO, uncorr(\(M_{\odot}\)) M_^12CO, |v|>1, uncorr(\(M_{\odot}\)) M_^12CO(\(M_{\odot}\)) M_^12CO,|v|>1(\(M_{\odot}\))
0.2 4.218 0.183 1.697 0.282 0.085 0.031 1.710 0.284
0.24 3.747 0.472 1.844 0.530 0.105 0.064 1.867 0.537
0.27 3.261 0.387 2.040 0.492 0.095 0.051 2.065 0.498
0.3 2.762 0.292 1.718 0.399 0.107 0.042 1.740 0.404
0.35 1.888 0.062 1.089 0.040 0.091 0.026 1.101 0.041
0.4 1.494 0.045 0.836 0.023 0.073 0.014 0.845 0.024
0.5 1.306 0.040 0.736 0.026 0.114 0.024 0.745 0.026

Output time, total simulated outflow mass, simulated outflow mass for \(|v_z|>1\), total mass estimated from \(^{13}\)CO full ALMA configuration 3, mass estimated from \(^{13}\)CO full ALMA configuration 3 for \(|v_z|>1\), total mass estimated from \(^{12}\)CO full ALMA configuration 3 with no opacity correction, mass estimated from \(^{12}\)CO full ALMA configuration 3 for \(|v_z|>1\) with no opacity correction, total mass estimated from \(^{12}\)CO full ALMA configuration 3 with opacity correction, mass estimated from \(^{12}\)CO full ALMA configuration 3 for \(|v_z|>1\) with opacity correction.

\label{tab_momentum}

Time (Myr) M_totV_z(\(M_{\odot}\)kms\(^{-1}\)) MV_|v_z|>1(\(M_{\odot}\)kms\(^{-1}\)) M_^13COV_z(\(M_{\odot}\)kms\(^{-1}\)) M_^13COV_|v_z|>1(\(M_{\odot}\)kms\(^{-1}\)) M_^12COV_z, uncorr (\(M_{\odot}\)kms\(^{-1}\)) M_^12COV_|v_z|>1, uncorr (\(M_{\odot}\)kms\(^{-1}\)) M_^12COV_z, corr (\(M_{\odot}\)kms\(^{-1}\)) M_^12COzV_|v_z|>1, corr (\(M_{\odot}\)kms\(^{-1}\))
0.2 1.402 0.255 1.100 0.598 0.133 0.114 1.109 0.603
0.24 1.853 0.790 1.440 0.916 0.148 0.131 1.458 0.927
0.27 1.565 0.590 1.375 0.787 0.122 0.104 1.392 0.797
0.3 1.332 0.420 1.109 0.590 0.095 0.069 1.123 0.598
0.35 0.758 0.110 0.499 0.103 0.070 0.044 0.504 0.104
0.4 0.525 0.073 0.349 0.094 0.045 0.022 0.353 0.095
0.5 0.426 0.067 0.309 0.099 0.069 0.034 0.313 0.101

Output time, total simulated outflow momentum, simulated outflow momentum for \(|v_z|>1\), total momentum estimated from \(^{13}\)CO full ALMA configuration 3, momentum estimated from \(^{13}\)CO full ALMA configuration 3 for \(|v_z|>1\), total momentum estimated from \(^{12}\)CO full ALMA configuration 3 with no opacity correction, momentum estimated from \(^{12}\)CO full ALMA configuration 3 for \(|v_z|>1\) with no opacity correction, total momentum estimated from \(^{12}\)CO full ALMA configuration 3 with opacity correction, momentum estimated from \(^{12}\)CO full ALMA configuration 3 for \(|v_z|>1\) with opacity correction.’

\label{tab_energy}

Time (Myr) Energy_tot (erg) Energy_|v_z|>1 (erg) Energy_^13CO (erg) Energy_^13CO |v_z| > 1 (erg) Energy_^12CO, uncorr (erg) Energy_^12CO |v_z| > 1, uncorr (erg) Energy_^12CO, corr (erg) Energy_^12CO |v_z| > 1, corr (erg)
0.2 12.43 6.983 22.16 19.58 6.159 6.064 22.34 19.73
0.24 23.12 17.52 22.91 19.92 3.188 3.089 23.19 20.16
0.27 18.75 13.44 19.74 16.47 2.947 2.839 19.98 16.66
0.3 15.65 10.53 13.40 10.48 1.470 1.321 13.56 10.61
0.35 7.146 3.846 6.435 4.310 1.103 0.958 6.510 4.361
0.4 4.204 2.080 6.667 5.520 0.481 0.350 6.745 5.584
0.5 3.679 1.994 6.839 5.891 0.702 0.507 6.914 5.956

Output time, total simulated outflow energy, simulated outflow energy for \(|v_z|>1\), total energy estimated from \(^{13}\)CO full ALMA configuration 3, energy estimated from \(^{13}\)CO full ALMA configuration 3 for \(|v_z|>1\), total energy estimated from \(^{12}\)CO full ALMA configuration 3 with no opacity correction, energy estimated from \(^{12}\)CO full ALMA configuration 3 for \(|v_z|>1\) with no opacity correction, total energy estimated from \(^{12}\)CO full ALMA configuration 3 with opacity correction, energymo estimated from \(^{12}\)CO full ALMA configuration 3 for \(|v_z|>1\) with opacity correction.’ Energy is out by a factor is 10^42.

All observations were taken with the full ALMA 3 congfiguration, with an integration time of 7200s and a pointing time of 10s. These values were calculated at T=0.27Myr.

Conclusions

We have shown that the mass of a protostellar stellar outflow cannot be inferred from observations in ^12CO but that the velocity and structure can. To determine the mass, observations in the optically thin ^13CO are needed, and these observations will also significantly improve the data on the velocity and structure of the outflow.

We have also shown that an appropriatly sized beam is the most important observational property. Observations with too small a beam gave no useful data about the outflow’s mass and very noisy data on its structure and velocity. Once the correct beam size has been chosen, the data can be improved by increasing the number of antennae (or the antennae area), or increasing the total integration time.

We found that while shorter pointings resulted in slightly better results, the difference was far smaller than that from increasing the integration time. If the total integration time can be increased by reducing the overhead of shorter pointings, this would almost certainly be beneficial.

Appendix

There were numerous other factors that went into these observations other than those mentioned in the article. We specified a limit on the number of cleaning iterations performed by Simanalyze, as well as ahreshold for early termination. We found that we never hit the early termination but that changes in the number of iterations did affect our results. These effects are shown in figure \ref{fig1}.

We settled on 10 000 cleaning iterations as it seems to provide a balance between good results, and efficient use to time and processor cycles.

Mass versus time for synthetic \(^{13}\)CO observations with various number of cleaning cycles. The synthetic observations are performed with the full ALMA configuration, with a 3600s integration time and 10s pointings. \label{fig1}

We also considered the effect of changing the pointing time. The CASA documentation suggests that this should have asignificant effect on our results, with shorter pointings giving better results. However, we found that there was very little difference in the quality of data gained by observation using the recommended 10s pointings and 100s pointings. Figure \ref{figAp2} shows that shorter pointings perform slightly better at determining the mass. However, Figures \ref{figAp3} and \ref{figAp4} show that changing the poitning time has no discernable effect on observations of the velocity.

Despite these, we used the CASA recommended 10s pointings for most of our observations but suggest that in real observing to shorten pointing time only if it does not affect other observational parameters.

Mass versus time for synthetic \(^{13}\)CO with various pointing times. The synthetic observations are performed with the compact ALMA configuration, with 7200s observations. \label{figAp2}

Velocity contour plot of a synthetic \(^{13}\)CO observation at \(t = 0.27\)Myr. The synthetic observation is performed with the cycle 0 compact ALMA configuration, over 7200s with 10 second pointings. which ’compact’ configurations is this? (cycle 0) \ref{figAp3}

Velocity contour plot of a synthetic \(^{13}\)CO observation at \(t = 0.27\)Myr. The synthetic observation is performed with the compact ALMA configuration, over 7200s with 100 second pointings. \ref{figAp4}

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