- Split View
-
Views
-
Cite
Cite
Michael D. Smith, Evolutionary tracks of massive stars during formation, Monthly Notices of the Royal Astronomical Society, Volume 438, Issue 2, 21 February 2014, Pages 1051–1066, https://doi.org/10.1093/mnras/stt2210
- Share Icon Share
Abstract
A model for massive stars is constructed by piecing together evolutionary algorithms for the protostellar structure, the environment, the inflow and the radiation feedback. We investigate specified accretion histories of constant, decelerating and accelerating forms and consider both hot and cold accretion, identified with spherical free-fall and disc accretion, respectively. Diagnostic tools for the interpretation of the phases of massive star formation and testing the evolutionary models are then developed. Evolutionary tracks able to fit Herschel Space Telescope data require the generated stars to be three to four times less massive than in previous interpretations, thus being consistent with clump star formation efficiencies of 10–15 per cent. However, for these cold Herschel clumps, the bolometric temperature is not a good diagnostic to differentiate between accretion models. We also find that neither spherical nor disc accretion can explain the high radio luminosities of many protostars. Nevertheless, we discover a solution in which the extreme ultraviolet flux needed to explain the radio emission is produced if the accretion flow is via free-fall on to hotspots covering less than 10 per cent of the surface area. Moreover, the protostar must be compact, and so has formed through cold accretion. We show that these conclusions are independent of the imposed accretion history. This suggests that massive stars form via gas accretion through discs which, in the phase before the star bloats, download their mass via magnetic flux tubes on to the protostar.
INTRODUCTION
Massive stars are the principal source of heavy elements and ultraviolet (UV) radiation, and a major supplier of wind and supernova energy, within the Universe. Individually, they dominate cluster formation, and collectively, they influence the evolution of galaxies. However, our knowledge of massive stars remains limited (Zinnecker & Yorke 2007). We know they are born in massive and dense clumps embedded within giant molecular clouds (Blitz 1991). We also think we understand fairly well how physical mechanisms conspire to describe the emergence of low- and intermediate-mass stars. But the events which conspire to produce the high-mass counterparts are still controversial (McKee & Ostriker 2007). This is due to a combination of factors including their rarity, large distances, rapid evolution, high extinction and their confusion with associated clusters.
A formation model requires integration of evolutionary models for the internal star, the environment and the feedback. Together, these constitute a ‘protostellar system’ which consists of several distinct components including a hydrostatic core, disc, envelope, wind, jets and bipolar outflow. Recently, with Herschel, Balloon-borne Large Aperture Submillimeter Telescope (BLAST) and Planck following Spitzer (e.g. Netterfield et al. 2009; Elia et al. 2010; Planck Collaboration et al. 2011), we have acquired the quality and quantity of data to explore the early formation stages. Processes that can now be addressed include clump dispersal, cluster formation, mass inflow, stellar accretion, stellar flux and mass outflow.
The aim here is to link the system components and underlying processes together through a model framework that predicts resulting observable correlations. We thus develop a scheme which explores plausible paths from clump to exposed star. It is set up to track how mass is passed between the components, while accounting for direct ejection from the supplying clump and indirect ejection through jets.
The interaction of a gradually emerging star with a massive clump is described in terms of a time sequence going from compact and hot molecular cores to extended H ii regions (Churchwell 2002) as expansion occurs. This has emphasized an issue concerning the time-scales with too many bound hypercompact or the later ultracompact H ii regions being observed. This may be resolved by taking less ideal approaches to accretion and dispersal from and to the clump (Peters et al. 2010), allowing for choking of the cavity through the infall of dense filaments and fragments of the clump.
The very early stages of protostar formation within cold clumps have now been identified in large numbers as infrared dark clouds (IRDCs; Rathborne, Jackson & Simon 2006; Simon et al. 2006). These could be the starless objects from which massive stars will form. However, the earliest signs of accretion have also been signposted by directed outflows of cold molecular gas and, it turns out, most of the IRDCs studied so far host weak 24 μm emission sources and already drive molecular outflows, both strong indicators for active star formation (Rathborne et al. 2010). In addition, the Herschel Space Telescope has helped reveal the full population of early evolutionary stages at the very onset of massive star formation (Beuther et al. 2010; Ragan et al. 2012).
It has been proposed that the large-scale evolution can be split into two phases: an accretion phase and a clean-up phase. Initially, as the protostar gains in mass, its sphere of influence grows, leading to accelerated accretion (McKee & Tan 2003). There is observational support for this as discussed by Davies et al. (2011) which favours either turbulent core (McKee & Tan 2003) or competitive accretion (Bonnell et al. 2001) models. Subsequently, in a distinct second phase, termed the clean-up phase, the accretion has stopped abruptly and the remaining clump material is partly dispersed or integrated into a surrounding cluster (Molinari et al. 2008).
In another scenario, the massive star does not fully form early. Low-mass star formation dominates until the clump has considerably evolved. These stars would remain difficult to detect. It has been remarked that the current initial mass function implies two peaks of star formation with the majority of low-mass stars forming first and high-mass stars forming later (Behrend & Maeder 2001). In competitive accretion, gas is funnelling down to the cloud centre where stars, initially accreting gas with low relative velocity, already have large mass before accreting the late-arriving higher velocity gas (Bonnell & Bate 2006).
Stellar collisions and mergers could supplement gas accretion (Bonnell & Bate 2002). Massive stars could form via coalescence of intermediate-mass stars within very dense systems. Although not considered here, along with other scenarios involving fragmentation, these should lead to alternative predictions.
Do high-mass stars form in a scaled-up analogue of the low-mass formation scenario? Molinari et al. (2008) find a consistent interpretation in favour of an analogy. They analysed wide spectral energy distributions (SEDs) to derive bolometric luminosity (Lbol) and clump mass (Mclump). This leads to the diagnostic diagram of Lbol versus Mclump (we have replaced the term envelope with clump here, employing envelope to refer to the inner part of the clump which accretes directly on to the protostar). The high-mass objects were then shown to occupy a sequence of regions on this diagram in a similar manner to those evolving classes which populate the low-mass regions. Herschel data have recently extended this to include protostars (e.g. Elia et al. 2010).
The SED parameters now available include fluxes derived from the Red MSX Source survey, Spitzer Infrared Array Camera (IRAC) and Multiband Imaging Photometer for Spitzer (MIPS), Stratospheric Observatory for Infrared Astronomy (SOFIA) Faint Object Infrared Camera for the SOFIA Telescope (FORCAST), Herschel Photoconductor Array Camera and Spectrometer (PACS), Herschel Spectral and Photometric Imaging Receiver (SPIRE) and BLAST. Further data are available across the radio, submillimetre and infrared. From these, we can derive clump mass, temperature, luminosity, UV flux, outflow mass and outflow power. So we can now construct several diagrams to employ as diagnostic tools to estimate the evolutionary stages.
The bolometric luminosity and temperature, Tbol, have been used in isolation to test accretion models for a version of the model tuned to low-mass protostars (Froebrich et al. 2006). A similar approach but just using the luminosity function was adopted by Davies et al. (2011) for high-mass stars using Midcourse Space Experiment (MSX) data and radio identifications to distinguish protostars from later stages. In place of an SED derivation of luminosity, the 21 μm flux was utilized with the assumption that the bolometric luminosity can be taken as a reliable proxy for the set of protostars being investigated. Both these approaches led to constraints on the time-scale of young stars and the general form of the accretion process.
A potential test would use the radio luminosity, Lr, produced by free–free emission after extreme UV excitation. The ratio of Lr/Lbol should then provide a measure of the development while Lbol/Mclump provides a distinct measure. Plotted together, such a diagram provides a distance independent distribution of evolutionary phases.
Outflow parameters have also been used to distinguish the phases of low-mass stars where accretion is known to decelerate rather than accelerate with time (Caratti o Garatti et al. 2006). This leads to a decrease in the force of the outflow as the source ages (Curtis et al. 2010). Beuther et al. (2002) showed that bipolar outflows are indeed ubiquitous phenomena in the formation process of massive stars, suggesting similar flow-formation scenarios for all masses, consistent again with scaled-up, but otherwise similar, physical processes – mainly accretion – to their low-mass counterparts. Going further, it was shown that the measured molecular hydrogen outflow luminosity is tightly related to the source bolometric luminosity for low-mass stars (Caratti o Garatti et al. 2006), and this relationship extends to massive objects (Caratti o Garatti et al. 2008). It is clear that this only applies to the youngest protostars in which the bolometric luminosity is dominated by the release of energy through accretion. Those sources associated with jets are very young (well before the main sequence turn-on), while those without detectable jets possess ultracompact H ii regions (Beuther & Shepherd 2005; Ramsay et al. 2011).
The objectives of this first paper are to set up the model framework and consider the effects of radiation feedback. We impose simple accretion rates that slowly vary. Subsequent works will tackle accretion outbursts, the outflow properties and the strength of feedback in self-regulation.
The accelerated-accretion model generates a particular Lbol versus Mclump relation (Molinari et al. 2008). We begin here by exploring how general this is. A constant accretion rate is a common working assumption while there is evidence for both a declining rate as well as a sporadic/episodic rate. These models involve less dramatic cut-offs in the accretion and should generate models with different statistical properties.
A major issue to address later is the existence of two distinct phases. If accelerated accretion is followed by a clean-up phase, we expect the outflow feedback phase to precede the radiative feedback phase. It may prove difficult for accretion phenomena to still dominate once the rapid rise in UV flux has started at late times. Quite remarkably, nature has no problem: all of the sources with infall signatures on to ultracompact H ii regions have corresponding outflow signatures as well (Klaassen, Testi & Beuther 2012). This observation suggests that accretion may continue, consistent with the gravoturbulent model (Schmeja & Klessen 2004). However, both accretion and collimated outflows are probably weak when the star has advanced to its ultracompact H ii stage (Varricatt et al. 2010). An ultimate aim of this study will therefore be to determine the conditions under which the accretion-outflow phase can significantly overlap with the UV phase, associated with the final contraction of a massive star.
METHOD
Model construction: mass movement
The model is constructed upon (1) the extension of the unification scheme for low-mass stars, (2) the strategy and model invoked for high-mass stars, (3) the detailed evolutionary tracks of an accreting massive protostar, (4) the results for a range of potential accretion rates and (5) predicted outcomes for radiative and outflow feedback. All algorithms and graphics are written and processed in idl.
We will consider two important free parameters. The most critical is the fraction, ξ, of the initial clump mass which ends up as part of the star. Even for low-mass stars, it is well known that there must be a much larger obscuring mass than necessary to form the star (Myers et al. 1998). This mass prolongs the embedded phase and extends the late Class 0 and early Class 1 stages. The best estimate for the excess mass was found to be a factor of 2 in the low-mass version of the present scheme on using accretion rates derived from gravoturbulent models (Froebrich et al. 2006).
Also for high-mass protostars, the surrounding bound clumps are estimated to exceed the star's mass by a factor which can exceed 30 (Molinari et al. 2008). We thus anticipate quite low values for ξ. Hence, in this work, we assume that the clump mass is sufficient to form the protostar and the associated stellar cluster (see Section 3.1) in addition to gas expelled directly from the clump. We assume both mass-loss rates to be constant with the same time-scale.
The second parameter is the fractional efficiency ϵ of mass diversion from inflow to outflow, from the disc to the jets. The extended magnetocentrifugal model is expected to be quite efficient and the X-wind model is expected to reach the 30 per cent level (Shu et al. 1988, 1994). In doing so, such magnetocentrifugal mechanisms can carry away the total angular momentum and kinetic energy of the accreting disc material.
The variable jet efficiency was introduced in order to account for evolving outflow properties of low-mass protostars. The Class 0 outflows appear to have a mechanical luminosity of order of the bolometric luminosity of the protostellar core. On the other hand, Class 1 outflows have mechanical luminosities and momentum flow rates up to a factor of 10 lower (Bontemps et al. 1996; Caratti o Garatti et al. 2006). With the same mass outflow efficiency, ζ = 0, this would then require that the jet speed is higher in Class 0 outflows. This is, however, contrary to the observations which associate lower velocities (∼100 km s−1) to the Class 0 outflows. Here, we shall assume a constant outflow mass fraction, taking the case ϵ = 0.3 throughout this work.
Model construction: accretion rates
The mass accretion rate |$\skew4\dot{M}_{\rm acc}(t)$| is the main prescribed parameter. We choose the four forms as shown in Fig. 1 and set out below. Note that the power-law and exponential models both include a significant phase of accelerated accretion prior to the prolonged decline.
Constant rate models
Constant accretion models were favoured following the work of Shu (1977) on singular isothermal spheres. It is clear, however, that the rate must eventually fall (by the pre-main-sequence stage for low-mass stars) as the reservoir becomes exhausted. Here we assume a constant rate until a cut-off time, to, at which the accretion is abruptly halted.
Exponential models
Power-law models
The favoured model for low-mass evolution involves a sharp exponential rise followed by a prolonged power law decrease in time (Smith 1999, 2000). The power law has substantial observational support (Calvet, Hartmann & Strom 2000). The early peak may reach |$\skew4\dot{M}_{\rm acc} = 10^{-4}\,\mathrm{M}_{{\odot }}\,{\rm yr}^{-1}$| for 104 yr, and eventually fall to |$\skew4\dot{M}_{\rm acc} = 10^{-7}\,{\rm M}_{{\odot }}\,{\rm yr}^{-1}$| for 106 yr, corresponding to Class 0 and Class 2 or classical T Tauri stars, respectively.
Accelerated accretion
The turbulent core model is reproduced with n = 1 which leads to |$\skew4\dot{M}_{*} \propto M_{*}^{1/2}$|. This model has been shown to lead to some predictions which are consistent with different sets of data (Molinari et al. 2008; Davies et al. 2011).
Note that observed correlations between accretion rates and mass such as |$\skew4\dot{M}_{\rm acc} \propto M_{*}^{1.8 \pm 0.2}$| (Natta, Testi & Randich 2006) apply to a relatively mature stage of young stars. In contrast, |$\skew4\dot{M}_{\rm acc} \propto M_{*}^{1}$| was uncovered (Barentsen et al. 2011; Spezzi et al. 2012). However, it is no surprise that the results differ according to the precise sample selection criteria.
Growth of the star
The structure of a protostar while it accretes at a constant rate has been calculated by Hosokawa & Omukai (2009) in the ‘hot accretion’ scenario corresponding to spherical free-fall. If, instead of free-fall on to the surface, the gas settles via an accretion disc, the ‘cold accretion’ structure is appropriate (Hosokawa, Yorke & Omukai 2010). In the hot accretion case, the stellar radius swells up to over 100 R⊙ for |$\skew4\dot{M}_{*} > 10^{-3}$| M⊙ yr−1. The accretion may continue until after the arrival on the main sequence, arriving at higher masses for higher accretion rates.
For this work, we have fitted analytical functions to the template figures provided in the above two studies for the radius, R*, and luminosity, L*. The four functions correspond to the four main stages with smooth interpolation between these stages. The stages are (1) the adiabatic accretion, (2) the swelling (or bloating), (3) the Kelvin–Helmholtz contraction and, finally, (4) main-sequence accretion. The resulting functions are shown in Figs 2 and 3. Fig. 3 displays both the accretion and stellar luminosities, and demonstrates that the accretion luminosity dominates until the radial swelling stage which is apparent as a dip in the bolometric luminosity.
The stellar structure depends on the stellar mass, the initial interior state and the accretion rate history. For the case of a constant accretion rate, the above published works provide accurate templates for fiducial cases. However, to employ these figures for time-varying accretion, Davies et al. (2011) took the current accretion rate to look up the radius and luminosity from Hosokawa & Omukai (2009). This method may be a reasonable approximation when the accretion rate continues to increase such that most of the mass has been accumulated within a factor of 2 of the present accretion rate. More accurately for the adiabatic phases, we here calculate how the star has accumulated the mass and entropy over its entire evolution. Thus a mass-averaged accretion rate, proportional to the accumulated entropy, is employed. However, this method would still lead to very large errors when the accretion rate varies by large amounts, decreases considerably or varies rapidly in the post-adiabatic stages.
When the above method yields a Kelvin–Helmholtz time that is comparable or shorter than the accretion time-scale, the adiabatic approximation is invalid. A compromise solution would employ both the current and the mass-averaged accretion rates so as to deliver the correct stellar parameters for the two limiting cases of adiabatic accretion and rapid loss of entropy. This is parametrized with the use of the factor f = tacc/tKH where, as usually defined, |$t_{\rm acc} = M/\skew4\dot{M}$| and tKH = GM2/(RLint), as determined by the mass-averaged accretion rate. We then take this factor to determine the stellar radius and luminosity relative to the mass-averaged values, Ro, Lo, by the additive factors (1 − exp( − f))(R(t) − Ro), where R(t) is the stellar radius calculated from the current rate. While this compromise is more accurate, this would still lead to spurious properties if accretion variations are very rapid.
In this paper, therefore, we will use a second method, where the structure at any time is determined by the mass-averaged accretion rate over the history of the star limited to the past Kelvin–Helmholtz time. We implement an iterative process to determine the radius, luminosity and mass-average accretion rate since these are themselves functions of tKH. Upon testing, we find that the maximum difference between the two methods is typically a few per cent and occurs in the Kelvin–Helmholtz phase. In comparison, simply assuming the current accretion rate and reading the stellar structure from the constant-rate simulations, generated errors of order of 10 per cent in radius throughout the first three evolutionary stages for the slowly varying rates displayed in Fig. 1. Although not ideal, it could still be implemented for most present purposes.
Radiation feedback modelling
The radiation feedback into the environment consists of an ionizing effect and a heating effect on the surrounding envelope. The UV Lyman flux from the protostar and the accreting material ionizes the environment, generating an H ii region. This region can be observed in the radio continuum through free–free emission.
Envelope and bolometric temperature
The observed SEDs of protostars are often complex with multiple peaks. In this work, we restrict the analysis to predicting the bolometric temperature of the optically thick core of the clump given a spherically symmetric model. This provides an indication of how the peak wavelength will change with age. However, this should be only considered indicative since, even in this homogeneous spherically symmetric approximation, we require several other major assertions concerning the gas and dust distribution.
The above Method 1 was devised for the low-mass protostellar case in which the clump is described as a core, and its mass may correspond to that of the enhanced density and temperature which distinguishes it from the embedding molecular cloud. This case leads to linear isotherms on the diagnostic Lbol–Mclump logarithmic diagrams with the bolometric temperature simply proportional to Mclump/Lbol for the case β = 1.5. As shown in the top panel of Fig. 4, the predicted isotherms are quite close together thus requiring a wide range of bolometric temperatures to cover the data for the displayed Herschel clumps.
In the high-mass case, we associate the clump with a size that is only mildly dependent on the clump mass. The observed median half-size is 5 × 104 au (Molinari et al. 2008; Veneziani et al. 2013). Here, we will assume this value as a constant outer radius with larger clumps having the extra mass ‘squeezed in’ as concluded by Beuther et al. (2013). To complete the model for Method 2, the inner radius, Rin, of the clump is taken to be the sublimation radius, i.e. Tin = 1400 K. With these two boundary conditions replacing those of Method 1 (the distribution otherwise the same as described above), this Method 2 yields well-separated isotherms as shown in the lower panel of Fig. 4. We will display isotherms derived from Method 2 in the following.
Disc accretion
At the inner radius, the envelope feeds a circumstellar disc. We assume here, and will test in a following study, the working assumption that the disc is ‘viscous’ and steady, and that the gas spends relatively little time in the disc and so reaches an inner accretion disc at the same rate as with which it is supplied by the envelope. The inner accretion may be non-steady, the gas either being expelled in the jets or accumulated on to the surface of the protostar, later to become the star itself. The disc mass is thus proportional to the accretion rate and the accretion time-scale.
Standard turbulent viscosity is efficient at separating the flux of angular momentum from the mass out to radii of about 100 au for the initial rapid accretion rates from the envelope. Hence massive outer discs could build up until the viscous mechanisms associated with self-gravity are effective. This could lead to the formation of secondary objects (stars, brown dwarfs), and so cut off both the star and jet supply line. Perhaps more likely is that high accretion rates lead to simultaneous binary formation and powerful molecular jets.
We assume here that the inner disc processes the material fast and so remains steady. The outer part of the disc will lag behind. The outer radius of the steady state disc can be found by requiring the disc accretion time-scale tν(R) = R2/ν to be less than the time-scale for changes in the accretion rate |$\skew4\dot{M}_{\rm acc}/\ddot{M}_{\rm acc}$|. This yields a steady disc extent Rs.
RESULTS
Clump mass versus bolometric luminosity
Cloud mass, bolometric luminosity and bolometric temperature are quantities derived from observations. The bolometric temperature is sensitive to the geometry, orientation and uniformity of the clump. Therefore, the relationship between cloud mass and bolometric luminosity is most often employed as a diagnostic tool for the formation of high-mass stars (Molinari et al. 2008; Beltrán et al. 2013) as well as lower mass stars (Reipurth et al. 1993; Saraceno et al. 1996; Smith 2000). More recently, Lbol–Mclump or Lbol/Mclump–Mclump diagrams have been utilized to analyse Herschel Space Telescope data (Elia et al. 2010; Ragan et al. 2012).
To interpret these data points as a time sequence, it is assumed that the clump mass decreases in time as the luminosity increases (Andre, Ward-Thompson & Barsony 1993). Model evolutionary tracks are then easily calculated based on the principles discussed in Section 2 upon choosing an accretion type and rate.
The clump gas is taken to be reduced at a constant rate (excluding the inner envelope which supplies the massive star). However, other relationships between the stellar cluster and most-massive star have been considered and, above all, there is a wide spread in the measured values of A (Weidner, Kroupa & Bonnell 2010) corresponding to an order of magnitude in mass.
Diagrams for models with constant hot accretion are shown in Fig. 5 and for variable rates in Fig. 6, with specific parameters listed in Table 1. Also shown are observational data for far-infrared and infrared sources and the theoretical bolometric temperature isotherms. Note that these isotherms as calculated from equation (27) are in agreement with the bolometric temperatures directly derived in the literature.
The top panel of Fig. 5 displays the tracks for a slow evolution (e.g. Behrend & Maeder 2001) with a simultaneous cloud evolution. This yields a strong dependence between the two parameters with a wide distribution of sources predicted. In recent works, the clump mass is assumed to fall at a slow constant rate with time while the massive star forms abruptly from a compact envelope, which is consistent with both available data and theoretical expectations (McKee & Tan 2002). We therefore discount the slow accretion scenario as a model for clump/cluster evolution although it is relevant in following the possible evolution of an isolated core especially in the low-mass case.
The constant fast accretion case is shown in the lower panel of Fig. 5. It is clear that massive infrared protostars would be rarer in this case (as indicated by the arrowheads placed at regular time intervals). In addition, a much narrower range in clump masses for a given luminosity interval is predicted in the latter case.
The accelerated accretion model generates similar results (top panel of Fig. 6) with two distinct track stages. This was the model explored by Molinari et al. (2008) but we note here that both the constant and power law fall-off models produce very similar tracks. To differentiate between models will require a detailed statistical analysis. The power-law model generates tracks in which there is an extended transition stage between the accretion and clean-up stages (middle panel of Fig. 6). This transition stage occurs at bolometric temperatures between 60 and 80 K.
It should also be remarked that the track direction changes from almost vertically up to down for a short period. This is caused by the swelling of the protostar which reduces the bolometric luminosity by reducing the accretion luminosity. While hardly visible in Fig. 6, this effect becomes prominent in the cold accretion scenario where the expansion phase is stronger. This is shown in Fig. 7 for the power-law model in which it is most obvious. In general, however, we find that cold accretion does not significantly alter the tracks on the Lbol–Mclump plots.
Bolometric temperature
The bolometric isotherms calculated through Method 2 are shown on the Lbol–Mclump plots. The temperatures are broadly consistent with the range of temperatures derived from the SEDs for the Herschel sources. We do not expect more than this given the known sensitivity to geometry and orientation. Indeed, as shown in Fig. 4, there is a strong dependence on the radial density distribution as given by the index β.
However, we can compare the accretion models statistically to determine if they would yield significantly different statistics in terms of numbers of source in any temperature interval. These numbers are provided in Table 2. As also illustrated in Fig. 8, there is a remarkable difference between the accelerated accretion model and the alternative evolutions. In the accelerated accretion model, significantly more time elapses in the low-temperature (<30 K) regime. For the 100 M⊙ case, all the other models investigated (with the exception of slow accretion) spend much less time at low temperatures, by a factor of 2–3. This result is consistent with expectations: accelerated accretion takes time to get off the ground.
Accretion model . | Max. accretion . | Accretion . | Clump . | Max. accretion . | Accretion . | Clump . |
---|---|---|---|---|---|---|
. | rate . | time-scale . | time-scale . | rate . | time-scale . | time-scale . |
. | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . |
Constant – slow | 0.141 | 10.0 | 10 | 0.0014 | 100 | 100 |
Constant – fast | 1.413 | 1.0 | 10 | 0.0141 | 10 | 100 |
Accelerated | 2.717 | 1.0 | 10 | 0.0907 | 3 | 10 |
Power law | 2.861 | 0.2 | 10 | 0.2861 | 0.2 | 10 |
Exponential | 1.980 | 1.0 | 10 | 0.1980 | 1.0 | 10 |
Accretion model . | Max. accretion . | Accretion . | Clump . | Max. accretion . | Accretion . | Clump . |
---|---|---|---|---|---|---|
. | rate . | time-scale . | time-scale . | rate . | time-scale . | time-scale . |
. | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . |
Constant – slow | 0.141 | 10.0 | 10 | 0.0014 | 100 | 100 |
Constant – fast | 1.413 | 1.0 | 10 | 0.0141 | 10 | 100 |
Accelerated | 2.717 | 1.0 | 10 | 0.0907 | 3 | 10 |
Power law | 2.861 | 0.2 | 10 | 0.2861 | 0.2 | 10 |
Exponential | 1.980 | 1.0 | 10 | 0.1980 | 1.0 | 10 |
Accretion model . | Max. accretion . | Accretion . | Clump . | Max. accretion . | Accretion . | Clump . |
---|---|---|---|---|---|---|
. | rate . | time-scale . | time-scale . | rate . | time-scale . | time-scale . |
. | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . |
Constant – slow | 0.141 | 10.0 | 10 | 0.0014 | 100 | 100 |
Constant – fast | 1.413 | 1.0 | 10 | 0.0141 | 10 | 100 |
Accelerated | 2.717 | 1.0 | 10 | 0.0907 | 3 | 10 |
Power law | 2.861 | 0.2 | 10 | 0.2861 | 0.2 | 10 |
Exponential | 1.980 | 1.0 | 10 | 0.1980 | 1.0 | 10 |
Accretion model . | Max. accretion . | Accretion . | Clump . | Max. accretion . | Accretion . | Clump . |
---|---|---|---|---|---|---|
. | rate . | time-scale . | time-scale . | rate . | time-scale . | time-scale . |
. | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . | (10−3 M⊙ yr−1) . | (105 yr) . | (105 yr) . |
Constant – slow | 0.141 | 10.0 | 10 | 0.0014 | 100 | 100 |
Constant – fast | 1.413 | 1.0 | 10 | 0.0141 | 10 | 100 |
Accelerated | 2.717 | 1.0 | 10 | 0.0907 | 3 | 10 |
Power law | 2.861 | 0.2 | 10 | 0.2861 | 0.2 | 10 |
Exponential | 1.980 | 1.0 | 10 | 0.1980 | 1.0 | 10 |
Accretion model . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (×1000 yr) . |
---|---|---|---|---|---|
Hot accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.027 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.058 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.020 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.025 | 0.051 | 0.402 | 0.356 | 845 |
Cold accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.022 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.052 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.017 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.021 | 0.051 | 0.402 | 0.356 | 8459 |
Hi-Gal data (Elia et al. 2010) | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Accretion model . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (×1000 yr) . |
---|---|---|---|---|---|
Hot accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.027 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.058 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.020 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.025 | 0.051 | 0.402 | 0.356 | 845 |
Cold accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.022 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.052 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.017 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.021 | 0.051 | 0.402 | 0.356 | 8459 |
Hi-Gal data (Elia et al. 2010) | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Accretion model . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (×1000 yr) . |
---|---|---|---|---|---|
Hot accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.027 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.058 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.020 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.025 | 0.051 | 0.402 | 0.356 | 845 |
Cold accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.022 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.052 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.017 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.021 | 0.051 | 0.402 | 0.356 | 8459 |
Hi-Gal data (Elia et al. 2010) | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Accretion model . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (×1000 yr) . |
---|---|---|---|---|---|
Hot accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.027 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.058 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.020 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.025 | 0.051 | 0.402 | 0.356 | 845 |
Cold accretion | |||||
Constant – slow | 0.250 | 0.442 | 0.279 | 0.158 | 866 |
Constant – fast | 0.022 | 0.057 | 0.397 | 0.356 | 845 |
Accelerated | 0.052 | 0.080 | 0.373 | 0.356 | 845 |
Power law | 0.017 | 0.049 | 0.446 | 0.321 | 847 |
Exponential | 0.021 | 0.051 | 0.402 | 0.356 | 8459 |
Hi-Gal data (Elia et al. 2010) | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Herschel Hi-Gal data
Data directly comparable to that predicted in Table 2 are available through several Herschel programmes. The distribution of bolometric temperatures in Table 2 along with the data employed here is inconsistent. Over 80 per cent of observed clumps have temperatures below 30 K (bottom lines in Table 2) whereas the models predict a much more even number distribution with temperature. It is clear that only a small fraction of the Herschel cores on the very high mass tracks will go on to form such massive stars.
A resolution to this problem is straightforward: the most massive forming star observed in the clumps is a factor of about 3 smaller than that used in the literature to calculate tracks. In Table 3, we present re-calculated number distributions on the assumption that the clump masses are indeed large but are actually being heated by protostars which will form much lower mass stars. The new tracks are illustrated in Fig. 10. Extremely good correspondences are apparent. Note that the initial clump masses are now 4.7 times larger, corresponding to stars of three times the mass according to equation (34).
Final stellar mass . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (yr) . |
---|---|---|---|---|---|
Constant – fast | |||||
Clump mass: 6683 M⊙ | |||||
Clump radius: 50 000 au | |||||
100 | 0.022 | 0.057 | 0.396 | 0.356 | 845 |
50 | 0.042 | 0.233 | 0.538 | 0.148 | 929 |
30 | 0.077 | 0.581 | 0.294 | 0.081 | 960 |
15 | 0.550 | 0.810 | 0.133 | 0.036 | 1472 |
10 | 0.715 | 0.880 | 0.084 | 0.023 | 988 |
Clump mass: 5740 M⊙ | |||||
Clump radius: 30 000 au | |||||
Stellar mass: 30 M⊙ | |||||
Constant – slow | 0.672 | 0.826 | 0.114 | 0.037 | 2936 |
Constant – fast | 0.473 | 0.778 | 0.156 | 0.043 | 2935 |
Accelerated | 0.473 | 0.777 | 0.156 | 0.043 | 978 |
Power law | 0.497 | 0.781 | 0.153 | 0.042 | 978 |
Exponential | 0.472 | 0.777 | 0.156 | 0.043 | 978 |
Hi-Gal data | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Final stellar mass . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (yr) . |
---|---|---|---|---|---|
Constant – fast | |||||
Clump mass: 6683 M⊙ | |||||
Clump radius: 50 000 au | |||||
100 | 0.022 | 0.057 | 0.396 | 0.356 | 845 |
50 | 0.042 | 0.233 | 0.538 | 0.148 | 929 |
30 | 0.077 | 0.581 | 0.294 | 0.081 | 960 |
15 | 0.550 | 0.810 | 0.133 | 0.036 | 1472 |
10 | 0.715 | 0.880 | 0.084 | 0.023 | 988 |
Clump mass: 5740 M⊙ | |||||
Clump radius: 30 000 au | |||||
Stellar mass: 30 M⊙ | |||||
Constant – slow | 0.672 | 0.826 | 0.114 | 0.037 | 2936 |
Constant – fast | 0.473 | 0.778 | 0.156 | 0.043 | 2935 |
Accelerated | 0.473 | 0.777 | 0.156 | 0.043 | 978 |
Power law | 0.497 | 0.781 | 0.153 | 0.042 | 978 |
Exponential | 0.472 | 0.777 | 0.156 | 0.043 | 978 |
Hi-Gal data | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Final stellar mass . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (yr) . |
---|---|---|---|---|---|
Constant – fast | |||||
Clump mass: 6683 M⊙ | |||||
Clump radius: 50 000 au | |||||
100 | 0.022 | 0.057 | 0.396 | 0.356 | 845 |
50 | 0.042 | 0.233 | 0.538 | 0.148 | 929 |
30 | 0.077 | 0.581 | 0.294 | 0.081 | 960 |
15 | 0.550 | 0.810 | 0.133 | 0.036 | 1472 |
10 | 0.715 | 0.880 | 0.084 | 0.023 | 988 |
Clump mass: 5740 M⊙ | |||||
Clump radius: 30 000 au | |||||
Stellar mass: 30 M⊙ | |||||
Constant – slow | 0.672 | 0.826 | 0.114 | 0.037 | 2936 |
Constant – fast | 0.473 | 0.778 | 0.156 | 0.043 | 2935 |
Accelerated | 0.473 | 0.777 | 0.156 | 0.043 | 978 |
Power law | 0.497 | 0.781 | 0.153 | 0.042 | 978 |
Exponential | 0.472 | 0.777 | 0.156 | 0.043 | 978 |
Hi-Gal data | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
Final stellar mass . | <20 K . | <30 K . | 30–50 K . | 50–70 K . | <100 K time (yr) . |
---|---|---|---|---|---|
Constant – fast | |||||
Clump mass: 6683 M⊙ | |||||
Clump radius: 50 000 au | |||||
100 | 0.022 | 0.057 | 0.396 | 0.356 | 845 |
50 | 0.042 | 0.233 | 0.538 | 0.148 | 929 |
30 | 0.077 | 0.581 | 0.294 | 0.081 | 960 |
15 | 0.550 | 0.810 | 0.133 | 0.036 | 1472 |
10 | 0.715 | 0.880 | 0.084 | 0.023 | 988 |
Clump mass: 5740 M⊙ | |||||
Clump radius: 30 000 au | |||||
Stellar mass: 30 M⊙ | |||||
Constant – slow | 0.672 | 0.826 | 0.114 | 0.037 | 2936 |
Constant – fast | 0.473 | 0.778 | 0.156 | 0.043 | 2935 |
Accelerated | 0.473 | 0.777 | 0.156 | 0.043 | 978 |
Power law | 0.497 | 0.781 | 0.153 | 0.042 | 978 |
Exponential | 0.472 | 0.777 | 0.156 | 0.043 | 978 |
Hi-Gal data | Number | ||||
l = 30° field | 0.463 | 0.832 | 0.128 | 0.022 | 311 |
l = 59° field | 0.417 | 0.846 | 0.142 | 0.011 | 91 |
H-Gal YSOs (Veneziani et al. 2013) | 0.463 | 0.989 | 0.011 | 0 | 284 |
This interpretation of the statistics, in which only 10–15 per cent of the initial clump mass ends up in stars, is independent of the accretion model. As shown on the panels of Fig. 9, it is very difficult to distinguish between the models for the high-mass clumps.
The new interpretation is consistent with the data relating star clusters to the most massive stars as presented by Weidner et al. (2010). Their fig. 3 shows that there is a minimum mass of the most-massive star for a star cluster of a given size which is approximately three times lower than the average stellar mass. This minimum mass would, of course, be the most likely if drawn randomly from a distribution corresponding to the initial mass function. We thus recommend that far-infrared data be interpreted by tracks as shown in Fig. 10.
Fig. 10 also demonstrates that the evolutionary phase of the observed sample of Herschel protostars is more advanced than previously interpreted because the final mass of the stars had been overestimated. The assertion here is that there are far fewer embedded protostars with mass exceeding 30 M⊙.
Radiative feedback: hot accretion
As the protostar matures, the rapid increase in luminosity leads to a high surface temperature and a high number of extreme UV photons, NLy, capable of generating a surrounding source of free–free radio emission as quantified in Section 2.4. With this interpretation, we can compare two distinct indicators of evolution: Mclump/Lbol and NLy/Lbol. Remarkably, both these quantities are, at least in principle, distance independent. It is apparent from Fig. 11 that there is a significant difference between the two extreme models with the accelerated accretion tracks for the most massive stars occupying a wider region whereas the power-law model predicts much less variation.
To compare to available data, we plot in Fig. 13 the Lyman photon flux and bolometric luminosity for a typical hot accretion model, with and without hotspots. The observed data are taken from Sánchez-Monge et al. (2013) and are clearly at variance with this model: there remains a significant number of data point lying above the tracks. This problem was discussed by Lumsden et al. (2013), Urquhart et al. (2013) and Sánchez-Monge et al. (2013) on comparing data to the expected Lyman flux from zero-age main sequence (ZAMS) stars. Sánchez-Monge et al. (2013) speculated that one resolution could be if there was an extra component from the accretion. The lower panel, however, demonstrates that this is not sufficient in the case where the star itself has formed through spherical accretion: the bloated protostar is too large to permit a significant release of extreme UV photons through free-fall on to the surface. This conclusion applies to all accretion models discussed here.
Radiative feedback: cold accretion
Cold accretion generates a young star with a considerably smaller radius through the early phases. Hence cold accretion does indeed generate more Lyman photons earlier as shown in the top panel of Fig. 14 although not greatly different from the hot accretion examples (note the different axial scales).
We now again suppose that the inner accretion is guided by the magnetic field to form accretion hotspots on the surface. As shown in the lower panel of Fig. 14, the behaviour is very different: the accretion luminosity dominates the early UV emission. Moreover, the hotspots are very important Lyman emitters for the stars of mass 10–20 M⊙ and will dominate the radio emission during the early phases of star formation.
Fig. 15 compares the constant accretion model to the ATCA data. While cold accretion does not generate sufficient extreme UV (top panel) if a considerable fraction of the accreting material is funnelled on to accretion hotspots, then the data can be very well interpreted (lower panel). This implies that the star has formed through a disc rather than spherical infall. However, at some stage, the magnetic field becomes sufficiently strong so that material is diverted and funnelled from the inner disc radius to effectively free-fall on to the surface. The star, of course, has previously formed through the cold accretion and so maintains the relatively small radius. The enhanced Lyman flux is a result of the high accretion on to a small growing protostar. The subsequent temporary extreme fall in the Lyman flux occurs as the bolometric luminosity falls and the star expands.
We find that the conclusion that funnelled accretion on to a compact protostar is occurring is independent of the chosen accretion model, as illustrated in Fig. 16 for two extreme accretion types. Good fits to the ATCA data require hotspot surface areas of less than 3–5 per cent for mass fractions of 50 and 75 per cent, respectively. These ranges are consistent with the fractions deduced from observations of young stars (Calvet & Gullbring 1998). However, it should be noted that while we concentrate on explaining the enigmatic high Lyman flux, most observed sources are either consistent with ZAMS or are underluminous. Some of these sources are not consistent with that expected on taking into account the additional low-Lyman flux of the associated stellar cluster (Lumsden et al. 2013; Urquhart et al. 2013). In the present context, these sources can be the result of either (1) distributed accretion over the surface and/or (2) single stars in the early bloating or late Kelvin–Helmholtz contraction phases.
The anomalously high Lyman fluxes only require the formation of intermediate-mass stars. High accretion rates are required to explain some data points. However, as can be seen from Fig. 16, the protostar need only grow up to the beginning of the bloating phase in order to account for the Lyman flux through hotspot accretion.
CONCLUSIONS
A model for massive stars has been constructed by piecing together models for the protostellar structure, the inflow from a large clump and the radiation feedback. The framework requires the accretion rate from the clump to be specified. In this first work we consider a specific subset of possible flows. We consider both hot and cold accretion scenarios, identified as the limiting cases for spherical free-fall and disc accretion, respectively. We assume the fiducial cases presented by Hosokawa & Omukai (2009) in the ‘hot accretion’ scenario and Hosokawa et al. (2010) for the ‘cold accretion’ structure but it should be noted that there is considerable uncertainty, depending on the assumed initial interior state and the physics of the radiation feedback (see also Kuiper & Yorke 2013).
Strongly variable accretion rates have been investigated by Smith et al. (2012) as well as Kuiper & Yorke (2013) by utilizing hydrodynamic simulations. In both these works, the puffy extended nature of the protostars is evident. Here we only consider smooth evolutions and do not consider accretion outbursts or pulsations, or the jet and outflow properties. We also do not consider binary formation, geometry and inclination effects, or the evolution of the size of the H ii region. Hence, this first work sets up the fundamental algorithms and compares results for two recent diagnostic tools.
Models for the formation of massive stars through accretion can be tested by comparing predictions to a range of observational parameters. These include the bolometric and extreme UV luminosities, the envelope and disc mass and the outflow momentum and energy. We here determine possible evolutionary tracks on assuming the variation with time of the accretion rate from a molecular clump on to the star and calculate how the star, envelope and outflow simultaneously evolve. This is achieved by making analytical prescriptions for the components based on current knowledge. We update and extend previous models and confirm previous conclusions that the clump mass must far exceeds the accreted mass, most of it being converted into a surrounding cluster of low-mass objects or dispersed.
We find that accelerated accretion is not favoured on the basis of the Lbol–Mclump diagnostic diagram which does not directly provide a test to differentiate the models. Only a slow accretion model can be distinguished in which the star and clump evolve on the same time-scale, which is not pursued since it seems unlikely given the contrasting sound-crossing time-scales between cores and clumps. This is mainly because the protostar tends to accrete most of its mass within a short time span in all the other models.
Instead, we show that the time spent within each range of bolometric temperature can be closely related to the underlying accretion model. As shown in Table 2, accelerated accretion generates relatively more sources at low bolometric temperatures. Sets of far-infrared Herschel data covering the temperature range from 20 to 70 K should provide some insight. However, modelling and observations of the bolometric temperature both remain problematic especially at the low temperatures which can be dominated by unbound non-stellar and pre-stellar objects in addition to asymptotic giant branch (AGB) stars (Veneziani et al. 2013). However, we find a solution which fits the data in which the initial clump mass is four to five times larger than that necessary to generate the associated star cluster corresponding to the mass of the most massive star. We thus generate revised evolutionary tracks which are consistent with statistics for the bolometric temperature. In these revised models, the star remains deeply embedded throughout its formation and the bolometric temperature distribution is no longer a sensitive diagnostic to differentiate between accretion models.
Accretion models may be better tested through a complete sample of hotter protostars with temperatures in the range 50–100 K. In addition, as with their low-mass counterparts, large periodic accretion variations could dominate the statistics.
The accelerated accretion model has been advanced in the literature because we expect the gravitational sphere of influence of a central object to grow as its mass grows. Here, however, with the large accretion rates often assumed, we take an inner envelope to already exist with sufficient mass to meet the later needs of the star and outflow. In this case, the accretion rate depends on how fast mass can flow in from the envelope, through the disc, rather than how massive the central protostar has become.
Finally, we have investigated Lyman fluxes as deduced observationally from radio fluxes. Observationally, it has been shown that objects of luminosity ∼10 000 L⊙ can possess very high Lyman fluxes, inconsistent with their expected stellar temperatures (Sánchez-Monge et al. 2013). We have shown here that the problem is resolvable if the protostar is relatively compact, formed through cold accretion via a disc. However, the present accretion must involve a funnelling free-fall mechanism on to a fraction of the stellar surface estimated to be less than 10 per cent. The mechanism for this remains unknown but if the evidence that massive protostars are configured in the same way as low-mass stars continues to grow, then accretion via magnetic flux tubes and jets driven by magnetocentrifugal processes are conceivable.
The above explanation of the excess Lyman photon flux requires evolution under the cold accretion scenario. Hot accretion falls far short even with the inclusion of hotspots, as demonstrated in Fig. 13. Cold accretion was defined as accretion on to the photosphere with no back-heating, so that the accreting material has the same entropy as that of the photosphere (Hosokawa et al. 2010). This is a limiting case which may be difficult to realize. It could be generally expected that the rapid mass accretion should be somewhat hot because a fraction of the entropy should be advected into the stellar interior (Hartmann, Zhu & Calvet 2011). Taking larger protostellar radii will reduce the Lyman flux; taking radii 50 per cent higher than that predicted for cold accretion, however, does not significantly alter the qualitative of fit to the ATCA data while doubling the radius has a considerable effect. Accurate results will require the implementation of a full stellar evolution code.
The major objective here has been to construct a consistent model which links the components. In the following works, we will investigate the consequences of mass outflows, accretion variations, maser production, thermal radio jets and H ii regions with the purpose of determining how their evolutions are coordinated.
I wish to thank Riccardo Cesaroni, Davide Elia, Sergio Molinari and Alvaro Sanchez-Monge for their encouragement and comment.