# Can High-velocity Protostellar Jets Help to Drive Low-velocity Outflow?

Masahiro N. Machida1
1 Department of Earth and Planetary Sciences, Faculty of Sciences, Kyushu
University, Fukuoka, Fukuoka 819-0395, Japan

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Using three-dimensional magnetohydrodynamics simulations, the driving of
protostellar jets is investigated in different star-forming cores with the
parameters of magnetic field strength and mass accretion rate. Powerful high-
velocity jets appear in strongly magnetized clouds when the mass accretion
rate onto the protostellar system is lower than $\dot{M}\lesssim 10^{-3}\,{\rm
M}_{\odot}$ yr-1. On the other hand, even at this mass accretion rate range,
no jets appear for magnetic fields of prestellar clouds as weak as
$\mu_{0}\gtrsim 5$–$10$, where $\mu_{0}$ is the mass-to-flux ratio normalized
by the critical value $(2\pi G^{1/2})^{-1}$. For $\dot{M}\gtrsim 10^{-3}\,{\rm
M}_{\odot}$ yr-1, although jets usually appear just after protostar formation
independent of the magnetic field strength, they soon weaken and finally
disappear. Thus, they cannot help drive the low-velocity outflow when there is
no low-velocity flow just before protostar formation. As a result, no
significant mass ejection occurs during the early mass accretion phase either
when the prestellar cloud is weaky magnetized or when the mass accretion rate
is very high. Thus, protostars formed in such environments would trace
different evolutionary paths from the normal star formation process.

###### keywords:

MHD – stars: formation – stars: protostars – stars: magnetic field – stars:
winds, outflows – protoplanetary disks

††pubyear: 2020††pagerange: Can High-velocity Protostellar Jets Help to Drive
Low-velocity Outflow?–LABEL:lastpage

##  1 Introduction

Protostellar outflows (low-velocity wide-angle outflows and high-velocity
collimated jets) are ubiquitously observed in various star-forming regions (Wu
et al., 2004). They are considered to universally appear in the star formation
process. Outflows are powered by accreting matter, in which the gravitational
energy of the accreting matter is converted into the outflow kinetic energy
through the magnetic field (or via magnetic energy; Uchida & Shibata 1985;
Blandford & Payne 1982; Tomisaka 2002). Past studies have shown that
protostellar outflow is necessary for removing the angular momentum of star-
forming clouds (Tomisaka, 2000; Joos, Hennebelle, & Ciardi, 2012; Masson et
al., 2016; Hirano et al., 2020). In addition, the star formation efficiency
should be related to the mass ejection rate of the outflow, as a large
fraction of the cloud mass is ejected by the protostellar outflow (Matzner &
McKee, 2000; Arce et al., 2007; Machida & Hosokawa, 2013). Thus, the
protostellar outflow plays significant roles in the star formation process
(Tanaka et al., 2017, 2018). Nevertheless, some objects (or protostars) have
been observed to show no sign of outflow (e.g. Tokuda et al., 2018; Aso et
al., 2019).

In previous studies (Matsushita et al., 2017, 2018; Machida & Hosokawa, 2020,
hereafter Papers I, II and III), we investigated the driving condition of low-
velocity outflows in various star-forming cloud cores. After performing long-
term simulations, we concluded that low-velocity outflow fails to appear or is
delayed when the magnetic field of the star-forming core is weak and/or when
the mass accretion rate onto the central region is high (see also Wurster,
Price, & Bate 2016 and Lewis & Bate 2017). We called these cases “failed or
delayed outflows” and showed that they correspond to the objects that were
observed to show no protostellar outflow (Aso et al., 2019). For the failed
and delayed outflow cases, a high ram pressure of the infalling envelope
suppresses the outflow driving (see, Papers I and III). In addition, for the
delayed outflow case, the outflow begins to evolve only after a large part of
the infalling envelope dissipates, because the ram pressure of the envelope
weakens as the density of the infalling envelope becomes low (Paper III).
However, there has been criticism about the numerical settings regarding the
failed and delayed outflow cases. In our previous studies (Papers I–III), we
did not resolve the protostar itself. Instead of resolving the protostar, we
used sink cells which cover both the inner disk region and protostar. Although
the sink cell method can accelerate the time evolution of star-forming clouds
in the simulation, the region around the protostar cannot be resolved (Machida
et al., 2010). Thus, we could not simulate a high-velocity outflow component
(i.e. high-velocity jet) driven near the protostar. Note that high-velocity
components can be seen in the high-spatial resolution simulations of low-mass
star formation (Tomisaka, 2002; Banerjee & Pudritz, 2006; Machida, Inutsuka, &
Matsumoto, 2006; Tomida et al., 2013; Bate, Tricco, & Price, 2014; Machida,
2014; Lewis & Bate, 2017; Vaytet et al., 2018; Machida & Basu, 2019). Note
also that few studies focused on the outflow in the high-mass star formation
simulations with sink cells, in which the flows driven near the protostar were
not spatially resolved (Seifried et al., 2011, 2012; Kölligan & Kuiper, 2018).

In both observational and theoretical works, it is well known that two types
of flow appear in the star formation process (e.g. Arce et al., 2007; Machida,
Inutsuka, & Matsumoto, 2008). A high-velocity collimated flow (so-called high-
velocity jets) appears near the protostar, while a low-velocity wide-angle
flow (so-called low-velocity outflow) appears in the region far from the
protostar or in the outer edge of the disk (see review by Inutsuka, 2012).
Note that no flow appears in the intermediate disk region because the magnetic
field is not well coupled with neutrals (Machida, Inutsuka, & Matsumoto, 2007;
Xu & Kunz, 2021). In our previous studies (Papers I–III), we did not resolve
the high-velocity jets. Thus, if the protostar and the disk inner region are
spatially resolved, high-velocity jets possibly appear and may help to drive a
wide-angle flow (or low-velocity outflow). In other words, even if the low-
velocity outflow does not appear in a simulation using sink cells, it may
appear with help from high-velocity jets in high-resolution simulations
without sink cells. In addition, there is a long-standing debate about driving
wide-angle low-velocity outflows. There are two conflicting models or
scenarios for driving low-velocity outflow, the entrainment and direct-driven
scenarios. The low-velocity outflow is entrained by the high-velocity jets in
the former model (Arce et al., 2007), while the low-velocity outflow is driven
directly from the outer disk region in the latter (Matsushita et al., 2019).
Although both recent observations and numerical simulations support the direct
driven scenario (Hennebelle & Fromang, 2008; Machida, 2014; Bjerkeli et al.,
2016; Hirota et al., 2017; Alves et al., 2017; Tabone et al., 2018; de Valon
et al., 2020; Lee et al., 2021; Marchand et al., 2020; Wurster, Bate, &
Bonnell, 2021), entrainment may not be completely rejected. It may be possible
for the high-velocity jets to entrain a part of the infalling gas and produce
the low-velocity component.

In Paper III using simulations with sink cells, I showed that the low-velocity
outflow fails to appear or appears in a late stage in weakly magnetized clouds
when the mass accretion rate onto the central region is high. In this study, I
investigate whether the high-velocity flow can appear and contribute to
driving the low-velocity outflow with almost the same settings as in Paper III
but without using sink cells, in which the protostar is spatially resolved
with a spatial resolution of $7.9\times 10^{-3}$ au.

The structure of this paper is as follows. The numerical settings and methods
are described in §2 and the calculation results are presented in §3. I discuss
the effect of the high-velocity jets on the low-velocity outflow in §4. A
summary is presented in §5.

Model | $f$ | $\mu_{0}$ |  $B_{0}$ [G] |  $\Omega_{0}$ [s-1] | $\alpha_{0}$ | $\beta_{0}$ | $\gamma_{0}$ |  $M_{\rm cl}$ [${\rm M}_{\odot}$] | Jet | Outflow
---|---|---|---|---|---|---|---|---|---|---
AM3 | 1.4 | 3 | $1.0\times 10^{-5}$ | $3.2\times 10^{-14}$ | 0.5 | 0.02 | 0.083 | 11 | S | S
AM5 | 1.4 | 5 | $6.2\times 10^{-6}$ | $3.2\times 10^{-14}$ | 0.5 | 0.02 | 0.030 | 11 | N | D
AM10 | 1.4 | 10 | $3.1\times 10^{-6}$ | $3.2\times 10^{-14}$ | 0.5 | 0.02 | $7.5\times 10^{-3}$ | 11 | N | D
AM20 | 1.4 | 20 | $1.6\times 10^{-6}$ | $3.2\times 10^{-14}$ | 0.5 | 0.02 | $1.9\times 10^{-3}$ | 11 | N | D
BM3 | 3.4 | 3 | $2.5\times 10^{-5}$ | $5.0\times 10^{-14}$ | 0.2 | 0.02 | 0.083 | 28 | S | S
BM5 | 3.4 | 5 | $1.5\times 10^{-5}$ | $5.0\times 10^{-14}$ | 0.2 | 0.02 | 0.030 | 28 | N | D
BM10 | 3.4 | 10 | $7.5\times 10^{-6}$ | $5.0\times 10^{-14}$ | 0.2 | 0.02 | $7.5\times 10^{-3}$ | 28 | N | F
BM20 | 3.4 | 20 | $3.7\times 10^{-6}$ | $5.0\times 10^{-14}$ | 0.2 | 0.02 | $1.9\times 10^{-3}$ | 28 | N | F
CM3 | 8.4 | 3 | $6.2\times 10^{-5}$ | $7.9\times 10^{-14}$ | 0.08 | 0.02 | 0.083 | 68 | S | S
CM5 | 8.4 | 5 | $3.7\times 10^{-5}$ | $7.9\times 10^{-14}$ | 0.08 | 0.02 | 0.030 | 68 | W | F
CM10 | 8.4 | 10 | $1.9\times 10^{-5}$ | $7.9\times 10^{-14}$ | 0.08 | 0.02 | $7.5\times 10^{-3}$ | 68 | W | F
CM20 | 8.4 | 20 | $6.2\times 10^{-6}$ | $7.9\times 10^{-14}$ | 0.08 | 0.02 | $1.9\times 10^{-3}$ | 68 | W | F
DM3 | 16.8 | 3 | $1.2\times 10^{-4}$ | $1.1\times 10^{-13}$ | 0.04 | 0.02 | 0.083 | 132 | S | S
DM5 | 16.8 | 5 | $7.4\times 10^{-5}$ | $1.1\times 10^{-13}$ | 0.04 | 0.02 | 0.030 | 132 | W | F
DM10 | 16.8 | 10 | $3.7\times 10^{-5}$ | $1.1\times 10^{-13}$ | 0.04 | 0.02 | $7.5\times 10^{-3}$ | 132 | W | F
DM20 | 16.8 | 20 | $1.8\times 10^{-5}$ | $1.1\times 10^{-13}$ | 0.04 | 0.02 | $1.9\times 10^{-3}$ | 132 | W | F
EM3 | 33.6 | 3 | $2.4\times 10^{-4}$ | $1.5\times 10^{-13}$ | 0.02 | 0.02 | 0.030 | 272 | N | S
EM5 | 33.6 | 5 | $1.5\times 10^{-4}$ | $1.5\times 10^{-13}$ | 0.02 | 0.02 | 0.083 | 272 | W | D
EM10 | 33.6 | 10 | $7.4\times 10^{-5}$ | $1.5\times 10^{-13}$ | 0.02 | 0.02 | $7.5\times 10^{-3}$ | 272 | N | F
FM3 | 67.2 | 3 | $5.0\times 10^{-4}$ | $2.2\times 10^{-13}$ | 0.01 | 0.02 | 0.083 | 545 | N | D
FM5 | 67.2 | 5 | $3.0\times 10^{-4}$ | $2.2\times 10^{-13}$ | 0.01 | 0.02 | 0.030 | 545 | N | D
FM10 | 67.2 | 10 | $1.5\times 10^{-5}$ | $2.2\times 10^{-13}$ | 0.01 | 0.02 | $7.5\times 10^{-3}$ | 545 | W | D
Table 1:  Model name, initial cloud parameters and calculation results. Column
1 gives the model name. Columns 2 and 3 give the parameters $f$ and $\mu_{0}$.
Columns 4 and 5 give the magnetic field strength $B_{0}$ and angular velocity
$\Omega_{0}$ for the initial state. Columns 6–8 give the ratios of the thermal
$\alpha_{0}$, rotational $\beta_{0}$ and magnetic $\gamma_{0}$ energies to the
gravitational energy of the initial cloud. Column 9 gives the initial cloud
mass. Column 10 describes the calculation results of this study, in which ‘S’,
‘W’ and ‘N’ means that a strong, weak and no jet appears, respectively. Column
11 describes the calculation results of Paper III, in which ‘S’, ‘D’ and ‘F’
mean that an outflow successfully appears, a delayed outflow appears and no
(or failed) outflow appears, respectively.

##  2 Initial Conditions and Numerical Settings

The initial conditions and numerical settings are almost the same as in Papers
I–III. Thus, I only briefly explain them in this section. As the initial
state, I adopt a spherical cloud core with a Bonnor–Ebert (B.E.) density
profile with a central density of $n_{\rm c,0}=10^{5}\,{\rm cm^{-3}}$ and an
isothermal temperature of $T_{\rm iso,0}=20$ K. The initial cloud has twice
the critical B.E. radius ($R_{\rm cl}=4.1\times 10^{4}$ au). The cloud density
is enhanced by a factor of $f$ to promote gravitational collapse, in which $f$
is the density enhancement factor (Papers I and III) and is related to the
cloud (thermal) stability $\alpha_{0}$ (ratio of thermal to gravitational
energy). Therefore, the initial cloud has a central density of $n_{\rm
cl}=f\times 10^{5}\,{\rm cm^{-3}}$ ($=f\times n_{\rm c,0}$). A uniform density
$n_{\rm ISM}=1.25\times 10^{-2}\,n_{\rm cl}$ is set outside the initial cloud
($r>R_{\rm cl}$).

A uniform magnetic field $B_{0}$ is imposed over the whole computational
domain. The magnetic field direction is set to be parallel to the $z$-axis. A
rigid rotation $\Omega_{0}$ is adopted within the cloud ($r<R_{\rm cl}$) and
the rotation axis is set to be inclined by $\theta_{0}$ toward the $x$-axis
from the $z$-axis. Thus, the initial magnetic field is misaligned with the
initial rotation axis. On the other hand, an initial magnetic field parallel
to the initial rotation axis was adopted in Papers I–III. Recently, we have
shown that the aligned cases are not very realistic (Hirano & Machida, 2019;
Hirano et al., 2020; Machida, Hirano, & Kitta, 2020), because the disk normal,
the propagation direction of the outflow and jets and the angular momentum
vector of the disk and protostar are well aligned, and the angular momentum in
such systems is transported only in specific directions (for details, see
Ciardi & Hennebelle, 2010; Joos, Hennebelle, & Ciardi, 2012; Hirano et al.,
2020). Thus, the misaligned case should be adopted as a general (or realistic)
case. In this study, to limit the number of models, referring to the results
in Machida, Hirano, & Kitta (2020), I fixed the angle $\theta_{0}$ to be
$\theta_{0}=20^{\circ}$, which can yield strong outflow and jets in the low-
mass star-formation process (Lewis, Bate, & Price, 2015; Machida, Hirano, &
Kitta, 2020).

As the initial state, a total of 22 different prestellar clouds are prepared
(Table 1), in which the density enhancement factor $f$ (or $\alpha_{0}$) and
the normalized mass-to-flux ratio $\mu_{0}$ (or magnetic field strength
$B_{0}$) are parameters. The parameter $\alpha_{0}$ (or $f$) controls the mass
accretion rate onto the central region $\dot{M}$, in which the mass accretion
rate is proportional to $\alpha_{0}^{-3/2}$ (for details, see Paper I). I will
detail the mass accretion rate for some of the models in §3.1. In this series
of studies, the mass accretion rate parameter (or $\alpha_{0}$) determines
whether star in formation is low- or high-mass. The normalized mass-to-flux
ratio is defined as

$\mu_{0}\equiv\left(\frac{M_{\rm cl}}{\Phi_{\rm
cl}}\right)/\left(\frac{1}{2\pi G^{1/2}}\right),$ (1)

where $\Phi_{\rm cl}$ is the magnetic flux of the initial cloud. The initial
magnetic field strength $B_{0}$ is adjusted so that the mass-to-flux ratio
becomes $\mu_{0}=3$, 5, 10 and 20 in each initial cloud. The rotation rate
$\Omega_{0}$ is determined to give $\beta_{0}=0.02$ in each model, where
$\beta_{0}$ is the ratio of the rotational to gravitational energy. The model
name and cloud parameters are described in Table 1 111 I do not include models
EM20 and FM20 to further limit the number of models. Since these models have a
high mass accretion rate and high infall velocity, a very long calculation
time is necessary to complete the calculation. . Except for the angle
$\theta_{0}$ between the initial magnetic field direction and the rotation
axis, the initial clouds are completely the same as in Paper III.

The numerical settings are almost the same as in Papers I–III. The basic
equations (resistive MHD equations) are described in equations (1)–(4) of
Machida & Basu (2019) and equation (1) of Machida (2014). The nested grid code
is used to cover both prestellar core and protostar (for details of the code,
see Machida et al., 2004, 2005a, 2010; Machida & Hosokawa, 2013). At the
beginning of the calculation, five levels, $l$, of nested grids are prepared.
Each grid is composed of (i, j, k) = (64, 64, 64). The initial prestellar
cloud is set in the fifth level of the grid ($l=5$). The first level of the
grid has a box size of $L(l=1)=6.6\times 10^{5}$ au and a cell width of
$h(l=1)=1.0\times 10^{4}$ au. Both the box size and cell width halve with each
increment of grid level $l$. A new finer grid is generated to satisfy the
Truelove condition, in which the Jeans length is resolved with at least 16
cells (Truelove et al., 1997). The maximum grid level is set to $l_{\rm
max}=21$, which has a box size of $L(l=l_{\rm max})=0.51$ au and a cell width
of $h(l=l_{\rm max})=7.9\times 10^{-3}$ au.

For the purpose of reference, I describe the difference in the numerical
settings between this study and Papers I–III. In Papers I–III, the central
region is covered by the sink cells with a spatial resolution of $h=0.62$ au
and a sink radius of $r_{\rm sink}=1$ au, while in this study the protostar is
resolved with $h=7.9\times 10^{-3}$ au without using sink cells. The detailed
settings can be confirmed in Machida (2014) and Machida & Nakamura (2015). A
small time step is required with a high spatial resolution to ensure the
Courant, Friedrichs and Lewy condition. Thus, the simulation could be executed
for about 10,000 yr after protostar formation in Paper III, while only for
$\sim 500$ yr after protostar formation in this study.

##  3 Results

The evolution of outflows and jets for the models listed in Table 1 are
presented in this section. As described in §2, two parameters, the mass-to-
flux ratio $\mu_{0}$ (corresponding to magnetic field strength) and thermal
stability $\alpha_{0}$ (corresponding to mass accretion rate), are used to
characterize the models. This section first focusses on the evolution of jets
for models with strong magnetic fields (AM3, BM3, CM3, DM3, EM3, FM3). Then,
the jet driving for all the models in Table 1 is shown.

In this paper, as shown in previous studies (Tomisaka, 2002; Banerjee &
Pudritz, 2006; Machida, 2014; Machida & Basu, 2019), two components, the low-
velocity outflow and high-velocity jet, appear from different disk regions.
However, it is very difficult to distinguish them. Thus, I simply call all the
outflowing gas components including the low-velocity outflow and high-velocity
jet the ‘outflow’ in this section. In addition, I describe the high-velocity
outflow component as ‘the high-velocity jet (or high-velocity component)’, and
the low-velocity outflow as ‘the low-velocity outflow (or low-velocity
component)’, though there is no clear velocity boundary between them. It
should be noted that, in Papers I–III, since the high-velocity component was
not resolved and does not appear, the low-velocity flow (i.e. all the
outflowing gas) was referred to as the outflow.

Figure 1:  Density (color) and velocity (arrows) distributions at different
epochs on the $z=0$ (left panels), $y=0$ (middle panels) and $x=0$ (right
panels) planes for DM3 (see also DM3.avi). The elapsed time $t_{\rm ps}$ after
protostar formation and that $t$ after the beginning of the cloud collapse are
described in each left panel.

###  3.1 Outflow Evolution for Strongly Magnetic Field Models

Firstly, the cloud evolutions for DM3, EM3 and FM3 are shown. Figure 1 shows
the time sequence of the central region around the protostar for DM3 with the
density and velocity distributions on the $z=0$ (left), $y=0$ (middle) and
$x=0$ (right) planes plotted. DM3 has the parameters $\mu_{0}=3$ and
$\alpha_{0}=0.04$. We can confirm a rotationally supported disk with a size of
$\sim 10$ au at the very early accretion phase (Fig. 1a). At the same epoch,
the outflow extends to $\sim 20$ au (Figs. 1b and c). The size growth of the
rotationally supported disk can be confirmed in the density distribution on
the $y=0$ plane (nearly edge-on view) in Figures 1b, e, h and k, in which the
disk-like structure gradually increases in size. Although the direction of the
disk normal changes over time (Hirano & Machida, 2019), the rotation motion in
the disk is roughly confirmed on the $z=0$ plane (Figs. 1a, d, g, j).

Figure 1 also shows that strong mass ejection frequently occurs as the disk
grows. Shell-like structures seen in Figures 1f, h and i are caused by
episodic mass ejection (Hirano & Machida, 2019), in which each shell roughly
corresponds to each episode of mass ejection. Although the direction of the
outflow changes with time, the strong mass ejection lasts until the end of the
simulation, as also seen in Machida, Hirano, & Kitta (2020). The maximum speed
of the outflowing gas exceeds $\sim 100\,{\rm km\,s^{-1}}$. Violent mass
ejection by the high-velocity jet can be seen in Figure 1h, i, k and l.

Figure 2:  Density (color) and velocity (arrows) distributions at different
epochs on the $z=0$ (left panels) and $y=0$ (middle and right panels) planes
for EM3 (see also EM3.avi). Note that the box scale of the left and middle
panels is different from that of the right panels. The elapsed time $t_{\rm
ps}$ after protostar formation and that $t$ after the beginning of the cloud
collapse are described in each left panel.  Figure 3:  As Fig. 1 but for FM3
(see also FM3.avi).  Figure 4:  Ram and magnetic pressures on the $y=0$ plane
for FM3. The ram pressure is plotted outside the outflow and the magnetic
pressure is plotted inside the outflow. Both the ram ($P_{\rm ram}/P_{\rm
th,0}$) and magnetic ($P_{\rm mag}/P_{\rm th,0}$) pressures are normalized by
the initial thermal pressure at the center of the cloud $P_{\rm th,0}$. The
boundary between the outflow ($v_{r}>0$) and infalling envelope ($v_{r}<0$) is
plotted by the black solid curve. The arrows represent the velocity at each
point. The elapsed time after protostar formation $t_{\rm ps}$ and that after
the cloud begins to collapse $t$ are described in each panel.

Figure 2 shows the time sequence of the region around the protostar for EM3
with parameters $\mu_{0}=3$ and $\alpha_{0}=0.02$. For this model, the low-
velocity outflow appears before protostar formation and creates a clear
cocoon-like structure, as seen in Figure 2c and f. A high-velocity jet begins
to appear after protostar formation (Fig. 2b). Then, strong mass ejection
occurs around the protostar at $t_{\rm ps}\sim 100$ yr (Fig. 2e), where
$t_{\rm ps}$ is the elapsed time after protostar formation. On the other hand,
Figures 2c and f indicate that the low-velocity outflow weakens and the
cocoon-like structure gradually shrinks. Figure 2f shows that the high-
velocity jet propagates into the shrinking cocoon-like structure formed by the
low-velocity outflow. The high-velocity jet also gradually loses activity for
$t_{\rm ps}\gtrsim 150$ yr (Figs. 2h and i). Finally, the strong mass ejection
stops and both the low-velocity outflow and high-velocity jet disappear, as
seen in Figure 2k and l. The regions above and below the protostar and disk
are disturbed by the outflow and have complicated structures just after
protostar formation (Figs. 2c and f). On the other hand, the density
distribution is rather smooth and no particular structure is seen in these
regions at $t_{\rm ps}\simeq 500$ yr (Fig. 2l). However, a very weak jet
appears in the region very close to the protostar (Fig. 2k).

Figure 3 shows the time sequence of the region around the protostar for FM3
which has the parameters $\mu_{0}=3$ and $\alpha_{0}=0.01$. For this model, no
low-velocity outflow appears before protostar formation (Papers I and III). A
cocoon-like structure can be confirmed just after protostar formation in DM3
(Fig. 1c) and EM3 (Fig. 2c), while no such structure is seen in FM3 (Fig. 3c).
After protostar formation, a high-velocity jet also begins to appear for this
model, as seen in Figures 3e and f. However, the jet rapidly weakens and
disappears at $t_{\rm ps}\sim 160$ yr (Fig. 3h). Then, the jet sometimes
appears near the protostar, but does not grow sufficiently by the end of the
simulation. On the other hand, the disk-like structure grows with time (see
the middle panels of Fig. 2). At the end of the simulation, we cannot see any
sign of mass ejection (Figs. 3k and l).

For FM3, the ram (outside the jet) and magnetic (inside the jet) pressures
when the jet is decaying (or shrinking) are plotted in Figure 4. The figure
indicates that the ram pressure in the envelope (or outside the jet) is larger
than the magnetic pressure inside the jet. In addition, the jet velocity
rapidly decreases near the boundary between the jet and infalling envelope.
Thus, it is expected that a high ram pressure suppresses the growth of the jet
at this epoch, which is also seen in the case of decaying low-velocity outflow
(Paper I and III).

Figure 5:  Three dimensional view of DM3 (upper panels) and FM3 (lower panels)
at four different epochs. In each panel, the blue lines are magnetic field
lines and the central orange region corresponds to the high density region
(iso-density surface of $10^{12}{\rm cm^{-3}}$). The outflowing region with
$v_{r}>0$ is plotted by the nested color surfaces.

Figure 5 shows the time sequence of the structure in a three-dimensional view
for DM3 (upper panels) and FM3 (lower panels). It should be noted that since
the time sequence for EM3 is essentially the same as FM3, EM3 is omitted from
Figure 5. In DM3, we can confirm a tiny disk-like structure at the center and
hourglass shaped magnetic field lines in the whole region at the early mass
accretion stages (Figs. 5a and b). For this model, the outflow gradually grows
with time and the magnetic field lines around the protostar are disturbed by
the outflow (Figs. 5c and d). In Figure 5d, the high-velocity component (jet)
colored yellow and orange is enclosed by the low-velocity component (outflow)
colored blue. The high-velocity jet, which is embedded in the low-velocity
component, has a knotty structure that is caused by the episodic mass
ejection. At the end of the simulation, the outflow structure has a size of
$>1000$ au.

On the other hand, in FM3 (Fig. 5 lower panels), although an hourglass
configuration of the magnetic field lines can be confirmed, no large scale
outflow (either low-velocity outflow or high-velocity jet) appears. In Figure
5h, the magnetic field lines around the protostar are complicated, while we
cannot confirm the structure of (large scale) outflow. Figure 5 shows the
apparent structural difference between DM3 (upper panels) and FM3 (lower
panels).

Figure 6:  Density (color) and velocity (arrows) distributions on the $y=0$
plane at $t_{\rm ps}\sim 500$ yr for AM3 (a), BM3 (b), CM3 (c), DM3 (d), EM3
(e) and FM3 (f). The elapsed time $t_{\rm ps}$ after protostar formation and
that $t$ after the beginning of cloud collapse are described in the upper part
of each panel.

Figure 6 shows snapshots at $t_{\rm ps}\sim 500$ yr after protostar formation
for all the models having $\mu_{0}=3$. In the figure, the high-velocity jet
and low-velocity outflow are sustained for at least $\gtrsim 500$ yr in AM3,
BM3, CM3 and DM3, while no mass ejection occurs in EM3 and FM3 at the same
epoch. In AM3, BM3, CM3 and DM3, although the high-velocity jet extends
roughly from northeast to southwest, there is a slight difference in
directions between the low-velocity outflow and the high-velocity jet.
Especially, the jet axis is somewhat inclined from the outflow axis in CM3
(Fig. 3c), which is also seen in a recent observation (Matsushita et al.,
2019). The misalignment between the high- and low-velocity outflows is caused
by the warp of the disk (for details, see Hirano & Machida, 2019; Machida,
Hirano, & Kitta, 2020). For EM3 (Fig. 6e) and FM3 (Fig. 6f), the disk normal
is also somewhat inclined from the $z$-axis because the initial rotation axis
is set to be inclined from the $z$-axis toward the $x$-axis. Figure 6
indicates that the structure around the protostar differs considerably
depending on the parameter $\alpha_{0}$ or the mass accretion rate (for
details, see below).

Figure 7:  Outflow length against the elapsed time after protostar formation
for AM3, BM3, CM3, DM3, EM3 and FM3.  Figure 8:  Histogram of outflow mass
$\Delta M_{\rm out}$ (left axis) and momentum flux $\Delta F_{\rm out}$ (right
axis) against the outflow velocity $v_{\rm out}$ for AM3, BM3, CM3, DM3, EM3
and FM3. The model name, parameters ($\alpha_{0}$ and $\mu_{0}$) and elapsed
time ($t_{\rm ps}$) are described in each panel.  Figure 9:  Density (color)
and velocity (arrows) distributions on the $y=0$ plane for FM3 at the same
epoch as in Fig. 3k. The elapsed time $t_{\rm ps}$ after protostar formation
and that $t$ after the beginning of the cloud collapse are described.

The time evolution of the outflow length for the models having $\mu_{0}=3$ is
plotted in Figure 7, in which the outflowing gas is identified with
$v_{r}>v_{\rm thr}$ and $v_{\rm thr}=1\,{\rm km\,s^{-1}}$ is adopted. I
confirmed that the outflow physical quantities do not significantly depend on
$v_{\rm thr}$ in the range of $0<v_{\rm thr}<3\,{\rm km\,s^{-1}}$. The figure
indicates that the size of the outflow increases with time and reaches $\sim
700$–$1000$ au for about 600 yr after protostar formation in AM3, BM3, CM3 and
DM3. On the other hand, in EM3 and FM3, the outflow does not grow after
protostar formation. In these models, the outflow length repeatedly increases
and decreases within the range of $\lesssim 200$ au, indicating that the high-
velocity jet and low-velocity outflow appearing episodically around the
protostar lose their power before the outflow grows sufficiently. The
suppression of the outflow due to the ram pressure of the infalling envelope
was elaborately investigated in Paper III (see also Fig. 4).

To confirm the velocity component of the outflow, we plot the mass $\Delta
M_{\rm out}(v_{\rm out})$ and momentum flux $\Delta F_{\rm out}(v_{\rm out})$
of the outflowing gas at each velocity component (or velocity bin), estimated
as

$\Delta M_{\rm out}(v_{\rm out})=\int_{v_{\rm out}-\Delta v}^{v_{\rm
out}+\Delta v}\rho\,dV,$ (2) $\Delta F_{\rm out}(v_{\rm out})=\frac{\Delta
P(v_{\rm out})}{t_{\rm ps}},$ (3)

where $\Delta v=2.5\,{\rm km\,s^{-1}}$ was adopted, against the outflow
velocity at $t_{\rm ps}\simeq 500-600$ yr in Figure 8. In equation (3), the
outflow momentum $\Delta P_{\rm out}(v_{\rm out})$ at each velocity component
is calculated as

$\Delta P_{\rm out}(v_{\rm out})=\int_{v_{\rm out}-\Delta v}^{v_{\rm
out}+\Delta v}\rho\,|\mbox{\boldmath$v$}|\,dV.$ (4)

Note that, in equations (2)–(4), the quantities are integrated only in the
outflowing region of $v_{r}>v_{\rm thr}$. In AM3, BM3, CM3 and DM3, although
the low-velocity components dominate the high-velocity components, there
exists a non-negligible amount of outflowing mass and momentum flux in the
high-velocity components. For these models, although the distributions of
$\Delta M_{\rm out}$ and $\Delta F_{\rm out}$ show time variability, neither
the low- or high-velocity components disappear at any epoch. A similar
distribution to these models can be confirmed in the low-mass star formation
case (Saiki & Machida, 2020).

On the other hand, no high-velocity component exists in EM3 and FM3 in Figure
8. In EM3, the mass and momentum flux of the outflowing gas rapidly decrease
around $v_{\rm out}\simeq 30\,{\rm km\,s^{-1}}$ and disappear in the range
$v_{\rm out}>80\,{\rm km\,s^{-1}}$. In FM3, there is no high-velocity
component for $v_{\rm out}>40\,{\rm km\,s^{-1}}$. Thus, it is clear that no
high-velocity jets appear in EM3 and FM3. In these models, it seems that there
are low-velocity outflow components. However, in fact, the low-velocity
components can be identified only around the region very close to the
protostar and there is no evolved low-velocity outflow by the end of the
simulation, as seen in Figures 2 and 3. To confirm the region around the
protostar, a close-up view of Figure 3k is plotted in Figure 9. We cannot
clearly identify the outflow in a large scale (Figs. 3k and 6f), while we can
confirm the outflow around the protostar in a small scale (Fig. 9). A major
component of the outflow stays within $\sim$2 au and does not grow by the end
of the simulation for FM3 (and EM3).

Figure 10:  Protostellar mass (a), inflow rate (b) and ratio of outflow to
inflow rate (c) against the elapsed time after protostar formation for
AM3–FM3.

Figure 10 shows the protostellar mass (Fig. 10a), the mass infall rate (Fig.
10b) and the ratio of outflow to inflow rate (Fig. 10c) against the elapsed
time after protostar formation. The protostellar mass is estimated as being
the region with $n>n_{\rm ps}$, where $n_{\rm ps}=10^{18}{\rm cm^{-3}}$ is
adopted (Machida & Basu, 2019). As seen in Figure 10a, the protostellar mass
increases as $\alpha_{0}$ decreases, because the mass accretion rate or infall
rate depends on $\alpha_{0}$, as explained in Paper I. The infall rate is
estimated by integrating over the gas with $v_{r}<0$ passing through a
spherical surface with a radius of 1 au as

$\dot{M}_{\rm in}=\int^{v_{r}<0}\rho\,v_{r}\,dS.$ (5)

In the analytical estimate, the mass accretion (or infall) rate is
proportional to $\alpha_{0}^{-3/2}$ (for details, see Paper I). Thus, it is
natural that the mass accretion rate for models with small $\alpha_{0}$ is
higher than that for the model with large $\alpha_{0}$.

We can see a significant difference (or gap) between DM3 and EM3 in Figures
10a and b caused by the lack of outflow in EM3 and FM3. Figure 10c shows the
ratio of outflow to inflow rate, in which the outflow rate,

$\dot{M}_{\rm out}=\int^{v_{r}>0}\rho\,v_{r}\,dS,$ (6)

is estimated using the same spherical surface as used in estimating the inflow
rate ($\dot{M}_{\rm in}$). In the panel, we can also see a significant
difference between the models with and without outflow. As shown in Figures 6
and 7, the outflow grows in AM3, BM3, CM3 and DM3, while it is not evolved in
EM3 and FM3.

For AM3, BM3, CM3 and DM3, the outflow rate is comparable or larger than the
infall rate, as seen in Paper I. The infall and outflow rates are estimated at
a radius of $1$ au and the protostar has a size of 0.01–0.2 au. Thus, a part
of the infalling gas is swept by the high-velocity jet driven near the
protostar. As a result, the mass of the outflowing gas can dominate that of
the infalling gas (see also Papers I and II). Since not all the infalling gas
is ejected by the high-velocity jet, the protostellar mass gradually increases
even when strong mass ejection occurs around the protostar (AM3, BM3, CM3 and
DM3).

On the other hand, the ratio $(\dot{M}_{\rm out}/\dot{M}_{\rm in})$ rapidly
decreases in EM3 and FM3 (Fig. 10c). Since the mass inflow rate is high in
such models, the decrease of the ratio indicates that the mass outflow rate
rapidly decreases. Thus, the infalling gas is not significantly disrupted by
the high-velocity jet in these models. Therefore, a high infall rate (Fig.
10b) and high-mass protostar tends to be realized (Fig. 10c) in these models.
Figure 10a indicates that the outflow affects the mass growth of the
protostar.

Figure 11:  Outflow length against the elapsed time after protostar formation
for all models. The model name and parameters $\alpha_{0}$ and $\mu_{0}$ are
described in each panel.

###  3.2 Parameter Dependence

While we considered the cloud evolution for the models with an initially
strong magnetic field with $\mu_{0}=3$ in §3.1, in this subsection, the cloud
evolution for all models is described. Figure 11 shows the outflow length
against the elapsed time after protostar formation for all the models listed
in Table 1. In the figure, for the models with $\alpha_{0}=0.5$ (Fig. 11a),
0.2 (Fig. 11b) and 0.08 (Fig. 11c), the outflow length continues to increase
only when the initial magnetic field is as strong as $\mu_{0}=3$. Figures
11a–c also show that the outflow lengths for the models with $\mu_{0}\geq 5$
stay within $\sim 200$ au by the end of the simulation 222 A sharp drop of
AM20 at $t_{\rm ps}\sim 200$ yr seen in Fig. 11a is due to the outflow
identification criteria. As described in §3.1, I defined the outflow as the
outflowing gas having $v_{r}\geq 1\,{\rm km\,s^{-1}}$. The head of the outflow
gradually decelerates, while the mass ejection episodically occurs. Thus, the
inner outflowing gas is identified as the head of the outflow after the
forward outflow decelerates to $v_{r}<1\,{\rm km\,s^{-1}}$, which causes the
sharp drop of the outflow length. . Thus, neither the low-velocity outflow nor
high-velocity jet sufficiently evolve in these models. These models
($\alpha_{0}=0.5$, 0.2 and 0.08) have a relatively low mass accretion rate of
$\dot{M}\lesssim 5\times 10^{-5}{\rm M}_{\odot}$ yr-1 (Fig. 10b).

The models with $\alpha_{0}=0.04$ (Fig. 11d) are an exception. As seen in
Figure 10b, the model with $\alpha_{0}=0.04$, DM3, has an intermediate mass
accretion rate of $\sim 10^{-4}\,{\rm M}_{\odot}$ yr-1. In Figure 11d,
although the outflow growth rates are different among the models, the outflows
continue to evolve up to the end of the simulation in all the models (DM3,
MD5, DM10 and DM20).

For the models with $\alpha_{0}=0.02$ and 0.01 (EM3, EM5, EM10, FM3, FM5,
FM10), which have a high mass accretion rate ($>10^{-4}\,{\rm M}_{\odot}$
yr-1), the low-velocity outflow and high-velocity jet are not very active. In
Figures 11e and f, the outflow lengths for these models oscillate within 300
au. Thus, almost none of the models plotted in these panels show a successful
growth of the outflow. Among the models, although EM5 and FM10 show continuous
growth of the outflow, their lengths are much smaller than those of the
successful growth models (AM3, BM3, CM3, DM3).

Figure 12:  Outflow momentum for all models against the elapsed time after
protostar formation. The model name and parameters $\alpha_{0}$ and $\mu_{0}$
are described in each panel.

Figure 12 shows the outflow momentum $P_{\rm out}$, calculated as

$P_{\rm out}=\int^{v_{r}>v_{\rm thr}}\rho|\mbox{\boldmath$v$}|dV,$ (7)

for all model. The trend of the outflow momentum seen in Figure 12 is similar
to that in Figure 11. In Figure 12a–c (models with $\alpha_{0}=0.5$, 0.2 and
0.08), the outflow momenta for the models with $\mu_{0}=3$ are much larger
than those for models with $\mu_{0}\geq 5$. For the models with
$\alpha_{0}=0.04$ (DM3, DM5, DM10, DM20; Fig. 12d), the difference in the
outflow momenta among the models is not very large, because the outflow in
these models continues to grow up to the end of the simulation, as shown in
Figure 11. However, the outflow momentum is larger in the model with a
stronger magnetic field than that with a weaker magnetic field at the end of
the simulation.

For the models with high mass accretion ($\alpha_{0}=0.02$ and 0.01; Figs. 12e
and f), the outflow momenta for the models with the strongest magnetic field
(EM3, FM3) are much larger than those with weak magnetic fields (EM5, EM10,
FM5, FM10). However, Figures 11e and f indicate that the outflow does not
sufficiently grow and the outflowing gas stays near the protostar for the
models with $\mu_{0}=3$ (EM3, FM3). Thus, the large momenta for these models
is due to the dense gas staying around the protostar (see Figs 2, 3 and 9).

In addition, Figures 11e and f also indicate that the outflow can grow in
models EM5 and FM10. However, the outflow momenta for models EM5 and FM10 are
about one order of magnitude smaller than those for the models with
$\mu_{0}=3$ (EM3, FM3). Thus, it is considered that only a very weak outflow
can evolve in such models.

Figure 13:  Three dimensional structure at the end of the simulation plotted
on the parameter plane ($\mu_{0}$ and $\alpha_{0}$). In each panel, the
magnetic field lines (blue stream lines), outflowing component (green, yellow
and red iso-velocity surfaces) and disk-like structure (orange iso-density
surface) are plotted. Strong (red), weak (orange) and no (white) jet models
are represented by the background color.

Figure 13 shows the three dimensional structure at the end of the simulation
for all the models plotted on the $\mu_{0}$–$\alpha_{0}$ parameter plane. The
box size (260 au) is the same in each panel. As described in §2, the parameter
$\mu_{0}$ corresponds to the magnetic field strength of the prestellar cloud,
while the parameter $\alpha_{0}$ controls the mass accretion rate onto the
central region. In the figure, the models are classified into three categories
as strong, weak and no jet cases (or models), with red, orange and white
backgrounds, respectively. Although it is difficult to determine the jet
strength (strong or weak jet), I used the momentum (Fig. 12) and length growth
(Fig. 11) of the outflowing gas as indicators of the jet strength. Note that
the physical quantities of the outflowing gas such as outflow mass, momentum
and kinetic energy increase as the mass accretion rate increases or the
parameter $\alpha_{0}$ decreases (for details, see Papers I and II). Thus, I
only compared the outflow physical quantities (Figs. 11 and 12) for the models
with the same $\alpha_{0}$.

In Figure 13, the strong jet models (red background) are distributed only in
the lower left, indicating that a strong (high-velocity) jet appears only when
both conditions of strong magnetic field and low-mass accretion rate are
realized. A clear bipolar jet structure can been seen in the models in red
panels. In the models located in the lower center and right (panels without
background color), although the magnetic field lines are slightly entangled
around the center, we cannot confirm a clear jet structure.

We can confirm a jet (or outflow) structure in all the models with an
intermediate accretion rate ($\alpha_{0}=0.04$ and 0.08 models) in Figure 13.
A clear bipolar structure can be seen in the models with $\mu_{0}=3$. On the
other hand, the structures of the jet (or outflow) are highly complicated or
small for the models with $\mu_{0}\geq 5$. For the high mass accretion cases
with $\alpha_{0}=0.01$ and 0.02, the models with the strongest magnetic fields
($\mu_{0}=3$) do not show a jet structure. Meanwhile, a bipolar jet-like
structure can be seen in the models with ($\alpha_{0}$, $\mu_{0}$) = (0.02, 5)
and (0.01, 10).

In summary, the models with a strong magnetic field and/or a low-mass
accretion rate always show a clear jet structure. The models with a high-mass
accretion rate and a weak magnetic field do not always show a bipolar jet and
outflow structure. Even if an outflow appears in such models, the momentum and
size of the outflow are very small.

##  4 Discussion

In the star forming regions, protostellar jets are observed as proof of star
formation. Jet and outflow driving in the star formation process has been
investigated in core collapse simulations. However, in many simulations, the
jet driving was investigated only in a limited parameter range (Tomida et al.,
2013). Clouds with strong magnetic fields and low-mass accretion rates are
usually chosen as the initial condition of core collapse simulations to
investigate a high-velocity flow driven by the protostar (Tomida et al., 2013;
Machida, 2014; Machida & Basu, 2019)333 It should be noted that, for the low-
mass star formation case, Bate, Tricco, & Price (2014) investigated the jet
driving in the clouds with different magnetic field strengths ($\mu_{0}=5$,
10, 20, 100) resolving the protostar. Although they calculated the jet driving
only for $\sim 2-3$ yr, the high-velocity jet extends to $\sim 3$ au from the
protostar in the models with $\mu_{0}=5$, 10, and 20. Note that, in their
study, no jet appears in the model with $\mu_{0}=100$. I confirmed that, in my
simulations, the jet appears and is sustained at least in the very early phase
for the model with $\mu_{0}=3$, 5, 10 and 20 (Fig. 11). . The difference in
the initial condition may cause different outcomes (Vaytet et al., 2018;
Wurster, Bate, & Price, 2018). Thus, we need to perform core collapse
simulations with various initial conditions to investigate whether
protostellar jets universally appear in the star formation process. This study
focuses on jet driving with different star-forming environments. In other
words, I especially included jet driving in clouds with weak magnetic fields
and a high mass accretion rates in the investigation. Another purpose of this
study is to investigate whether a jet driven near the protostar can help to
drive a low-velocity and wide-angle outflow, based on the finding in Paper III
that the outflow fails to be driven with a weak magnetic field and/or a high
mass accretion rate.

Figure 14:  Outcomes of this study and Paper III plotted on the parameter
plane ($\mu_{0}$ and $\alpha_{0}$). The model name is described in each panel.
In each panel, the outcomes of this study (i.e. jet driving) and that of Paper
III are represented by color in the upper left and the lower right,
respectively. Color is used to indicate strong jets (upper left) or strong
outflow (lower right) (red), no jet (this study, upper left) or no outflow
(Paper III, lower right) (gray), a weak jet having a low momentum in this
study (orange) and an outflow in a delayed fashion in Paper III (purple).

Figure 14 shows the outcomes of this study and Paper III. In each panel, the
outcome of this study is presented in the upper left, while that of Paper III
is in the lower right. The model names used in this study are the same as
those in Paper III when the parameters $\alpha_{0}$ and $\mu_{0}$ are the
same. The initial condition is almost the same between the two studies. The
difference in the initial condition is the angle between the magnetic and
angular momentum vectors of the prestellar cloud $\theta_{0}$, as described in
§2. The angle $\theta_{0}=0$ was adopted in Paper III, while
$\theta_{0}=20^{\circ}$ is used in this study. Thus, the initial magnetic
field and angular momentum vector are aligned in Paper III, while they are
misaligned in this study. In addition, the integration time $t_{\rm end}$ and
spatial resolution $h$ are different between this study ($t_{\rm end}\sim 500$
yr and $h=7.9\times 10^{-3}$ au ) and Paper III ($t_{\rm end}\sim 10^{4}$ yr
and $h=0.62$ au). Hence, it is difficult to fairly compare the outcomes
between them. However, we cannot realize a long-term cloud evolution with a
high-spatial resolution. Thus, it would be valuable to overview the results
between them in order to further understand the driving mechanism of low- and
high-velocity flows. In this section, I discuss the possibility of jet and
outflow driving in the clouds with different magnetic field strengths and mass
accretion rates using the results in Paper III and this study.

In §3, all the outflowing gas observed in this study is referred to as
outflow, with the outflow being composed of both high-velocity (or high-
velocity jet) and low-velocity (low-velocity outflow) components. Here, I
redefine the terms ‘jet’ and ‘outflow’, used only in the following paragraphs
in this section, to distinguish the terms used for the outflow between this
study and Paper III. In this section, I refer to the outflow as the outgoing
flow observed in Paper III, while I call the jet the outgoing flows observed
in this study. In addition, referring to Figures 11–13, I call a jet with a
large momentum a strong jet and one with small momentum a weak jet.
Furthermore, in this study, since the low-velocity component appears with the
high-velocity jet (or high-velocity component), as seen in Figure 8, I call
the low-velocity component of a jet the low-velocity outflow (or low-velocity
component) and the high-velocity flow the high-velocity jet (high-velocity
component). As seen in Figure 8, there is no clear boundary between a high-
velocity jet and low-velocity outflow. Thus, it is difficult to separate low-
velocity outflow from a high-velocity jet. The low-velocity outflow may be
directly driven by the disk outer edge, as explain in Papers I–III.
Alternatively, it may be entrained by the high-velocity component when the
high-velocity jet appears. In the present study, I do not consider the driving
mechanism of the low-velocity components appearing in this study, as this is
outside the scope of this study.

First, I focus on the models with $\alpha_{0}=0.2$ and 0.5 (the fifth and
sixth lines in Fig. 14; models AM3, AM5, AM10, AM20, BM3, BM5, BM10, BM20),
which have a low mass accretion rate (Fig. 10) and would form low mass stars.
Figure 14 shows that both outflow and jets appear only in AM3 and BM3, which
have the strongest magnetic fields ($\mu_{0}=3$). For these models, the
outflow appears without resolving the jet driving region (Paper III). The jet
also appears as described in §3. Thus, it is considered that a high-velocity
jet and low-velocity outflow are driven at different radii in the
circumstellar disk, as expected from past core collapse simulations (Tomida et
al., 2013; Machida & Basu, 2019). On the other hand, when the magnetic field
is not strong ($\mu_{0}\geq 5$; AM5, AM10, AM20, BM5, BM10, BM20), jets do not
appear. In Figure 14, ‘Delayed Outflow’ refers to the case where a (weak)
outflow begins to grow only in the late accretion phase, just before the
infalling envelope dissipates (Paper III). Also, for BM10 and BM20, neither
jets (this study) nor outflow (Paper III) appear. Thus, for these models, the
outflow and jet would not play a significant role by the end of the mass
accretion phase. Note that since the jet driving was calculated only for $\sim
500$ yr in the main accretion phase with a high spatial resolution, I cannot
clearly state that the jet driving is completely suppressed by the end of the
mass accretion phase. However, since no jet appears, it cannot help to drive
the low-velocity outflow, at least, in the early main accretion phase.

Next, I describe the models with $\alpha_{0}=0.04$ and 0.08 (the third and
fourth lines in Fig. 14; CM3, CM5, CM10, CM20, DM3, DM5, DM10, DM20), which
have moderate mass accretion rates (Fig. 11) and would form intermediate mass
stars. Even for these models, both the jet and outflow appear only when the
magnetic field is as strong as $\mu_{0}=3$. Thus, as described above, the jet
and outflow are expected to be driven at different radii of the circumstellar
disk. The difference between low-mass ($\alpha_{0}=0.2$ and 0.5) and
intermediate-mass ($\alpha=0.04$ and 0.08) stars can be seen in the models
with weak magnetic fields. When the magnetic field is as weak as $\mu\geq 5$,
neither jets nor outflow appear in the early accretion phase for the low-mass
star formation cases. On the other hand, a weak jet appears for all models
with $\mu_{0}\geq 5$ (CM5, CM10, CM20, DM5, DM10, DM20) for intermediate star
formation cases. As shown in Figure 8, in addition to the high-velocity
component, the low-velocity component exists when a jet appears (Fig. 8).
Thus, a part of the outflowing gas would be entrained by the high-velocity
jet. Alternatively, the jet may alleviate the ram pressure barrier which
suppresses the growth of the low-velocity outflow and promotes outflow
directly driven by the outer disk region. However, the momentum of the
outflowing gas in these weak jet models is not as large as that in strong jet
models (Fig. 12). Thus, the contribution of the jet to the low-velocity
outflow is limited.

Finally, I describe the models with $\alpha_{0}=0.01$ and 0.02 (the first and
second lines in Fig. 14; EM3, EM5, EM10, FM3, DM5, FM10) which have high mass
accretion rates (Fig. 11) and would form high mass stars. No strong jets
appear in these models, while only EM3 shows an outflow after protostar
formation. Model EM3 is a special case of all the models listed in Table 1,
because only this model shows a low-velocity outflow (Paper III) without
showing a jet (this study). In Paper III, the outflow begins to grow about
1,000 yr after protostar formation in model EM3. Thus, the protostellar system
drives only the low-velocity outflow without the high-velocity component,
which may explain the observation of Hirota et al. (2017).

For EM5 and FM10, a weak jet appears in the early mass accretion phase (this
study), while the outflow appears in the later accretion phase (Paper III).
Meanwhile, for FM3 and FM5, the outflow appears in the late accretion phase
(Paper III), while no jet appears (this study). For EM10, neither jet nor
outflow appear in either study. Thus, it is expected that, for these models
(EM5, EM10, FM3, FM5, FM10), jets and outflow do not significantly affect the
star formation process throughout the main accretion phase. In addition, in
the models with $\alpha_{0}=0.01$ and 0.02, jets would not greatly contribute
to driving the low-velocity outflow because of a lack of strong jets.

The protostellar outflow is considered to have a significant impact on star
formation. However, Figure 14 indicates that there are various star formation
scenarios (with and without jets and outflow) dependent on the initial
prestellar cloud’s magnetic field strength and intrinsic mass accretion rate.

Finally, I again emphasize the caveat in the above discussion. In Papers
I-III, I prepared the aligned models, in which the rotation axis of the
prestellar cloud is aligned with the global magnetic field. Since such a
setting is not very realistic, I prepared the misaligned models in this study
(see §2). In addition, although the protostar was spatially resolved, I could
not follow the long-term evolution of the jet. For further understanding of
the jet and outflow driving, future studies should focus on the long-term
evolution of the jet resolving the protostar with the parameters $\mu_{0}$,
$\alpha_{0}$ and $\theta_{0}$.

##  5 Summary

In this study, I investigated jet driving in clouds with different magnetic
field strengths and intrinsic mass accretion rates using numerical
simulations. I found that a high-velocity jet appears near the protostar and
is sustained for a long time only when the magnetic field strength of the
prestellar cloud is as strong as $\mu_{0}\leq 3$ and the mass accretion rate
onto the central region is as low as $\dot{M}<10^{-4}\,{\rm M}_{\odot}$ yr-1.
When strongly magnetized clouds ($\mu_{0}=3$) have a mass accretion rate
higher than $\dot{M}>10^{-4}\,{\rm M}_{\odot}$ yr-1, a high-velocity jet can
be driven near the protostar in the early mass accretion phase. However, the
jet does not grow sufficiently and quickly disappears. On the other hand, when
the magnetic field of the star-forming cloud is weak, a jet sometimes appears
for a mass accretion rate higher than $\dot{M}>10^{-5}\,{\rm M}_{\odot}$ yr-1.
In this case, a low-velocity component also appears in conjunction with the
high-velocity jet. However, the outflow momentum in weakly magnetized clouds
is much lower than that in strongly magnetized clouds. Thus, a high-velocity
jet does not significantly contribute to driving a low-velocity outflow. This
study indicates that a protostellar outflow cannot significantly contribute to
the star formation process when the magnetic field strength of the star-
forming cloud is weak.

## Acknowledgements

This research used the computational resources of the HPCI system provided by
the Cyber Science Center at Tohoku University and the Cybermedia Center at
Osaka University (Project ID: hp200004, hp210004). Simulations reported in
this paper were also performed by 2019 and 2020 Koubo Kadai on Earth Simulator
(NEC SX-ACE) at JAMSTEC. The present study was supported by JSPS KAKENHI Grant
(JP17H06360, JP17K0538faa7, JP17KK0096, JP21H00046, JP21K03617: MNM).

## DATA AVAILABILITY

The data underlying this article are available in the article and in its
online supplementary material.

## References

  * Alves et al. (2017) Alves F. O., Girart J. M., Caselli P., Franco G. A. P., Zhao B., Vlemmings W. H. T., Evans M. G., et al., 2017, A&A, 603, L3. doi:10.1051/0004-6361/201731077
  * Aso et al. (2019) Aso Y., Hirano N., Aikawa Y., Machida M. N., Ohashi N., Saito M., Takakuwa S., et al., 2019, ApJ, 887, 209. doi:10.3847/1538-4357/ab5284
  * Arce et al. (2007) Arce H. G., Shepherd D., Gueth F., Lee C.-F., Bachiller R., Rosen A., Beuther H., 2007, prpl.conf, 245
  * Bate, Tricco, & Price (2014) Bate M. R., Tricco T. S., Price D. J., 2014, MNRAS, 437, 77. doi:10.1093/mnras/stt1865
  * Bjerkeli et al. (2016) Bjerkeli P., van der Wiel M. H. D., Harsono D., Ramsey J. P., Jørgensen J. K., 2016, Natur, 540, 406. doi:10.1038/nature20600
  * Banerjee & Pudritz (2006) Banerjee R., Pudritz R. E., 2006, ApJ, 641, 949. doi:10.1086/500496
  * Blandford & Payne (1982) Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883. doi:10.1093/mnras/199.4.883
  * Ciardi & Hennebelle (2010) Ciardi A., Hennebelle P., 2010, MNRAS, 409, L39. doi:10.1111/j.1745-3933.2010.00942.x
  * de Valon et al. (2020) de Valon A., Dougados C., Cabrit S., Louvet F., Zapata L. A., Mardones D., 2020, A&A, 634, L12. doi:10.1051/0004-6361/201936950
  * Hennebelle & Fromang (2008) Hennebelle P., Fromang S., 2008, A&A, 477, 9. doi:10.1051/0004-6361:20078309
  * Hirano & Machida (2019) Hirano S., Machida M. N., 2019, MNRAS, 485, 4667. doi:10.1093/mnras/stz740
  * Hirano et al. (2020) Hirano S., Tsukamoto Y., Basu S., Machida M. N., 2020, ApJ, 898, 118. doi:10.3847/1538-4357/ab9f9d
  * Hirota et al. (2017) Hirota T., Machida M. N., Matsushita Y., Motogi K., Matsumoto N., Kim M. K., Burns R. A., et al., 2017, NatAs, 1, 0146. doi:10.1038/s41550-017-0146
  * Inutsuka (2012) Inutsuka S., 2012, PTEP, 2012, 01A307. doi:10.1093/ptep/pts024
  * Joos, Hennebelle, & Ciardi (2012) Joos M., Hennebelle P., Ciardi A., 2012, A&A, 543, A128. doi:10.1051/0004-6361/201118730
  * Kölligan & Kuiper (2018) Kölligan A., Kuiper R., 2018, A&A, 620, A182. doi:10.1051/0004-6361/201833686
  * Lee et al. (2021) Lee C.-F., Tabone B., Cabrit S., Codella C., Podio L., Ferreira J., Jacquemin-Ide J., 2021, ApJL, 907, L41. doi:10.3847/2041-8213/abda38
  * Lewis, Bate, & Price (2015) Lewis B. T., Bate M. R., Price D. J., 2015, MNRAS, 451, 288. doi:10.1093/mnras/stv957
  * Lewis & Bate (2017) Lewis B. T., Bate M. R., 2017, MNRAS, 467, 3324. doi:10.1093/mnras/stx271
  * Marchand et al. (2020) Marchand P., Tomida K., Tanaka K. E. I., Commerçon B., Chabrier G., 2020, ApJ, 900, 180. doi:10.3847/1538-4357/abad99
  * Machida et al. (2004) Machida, M. N., Tomisaka, K., & Matsumoto, T. 2004, MNRAS, 348, L1
  * Machida et al. (2005a) Machida, M. N., Matsumoto, T., Tomisaka, K., & Hanawa, T. 2005, MNRAS, 362, 369
  * Machida, Inutsuka, & Matsumoto (2006) Machida M. N., Inutsuka S., Matsumoto T., 2006, ApJL, 647, L151. doi:10.1086/507179
  * Machida, Inutsuka, & Matsumoto (2007) Machida M. N., Inutsuka S., Matsumoto T., 2007, ApJ, 670, 1198. doi:10.1086/521779
  * Machida, Inutsuka, & Matsumoto (2008) Machida M. N., Inutsuka S., Matsumoto T., 2008, ApJ, 676, 1088. doi:10.1086/528364
  * Machida et al. (2010) Machida, M. N., Inutsuka, S., & Matsumoto, T. 2010, ApJ, 724, 1006
  * Machida & Hosokawa (2013) Machida M. N., Hosokawa T., 2013, MNRAS, 431, 1719. doi:10.1093/mnras/stt291
  * Machida (2014) Machida M. N., 2014, ApJL, 796, L17. doi:10.1088/2041-8205/796/1/L17
  * Machida & Nakamura (2015) Machida M. N., Nakamura T., 2015, MNRAS, 448, 1405. doi:10.1093/mnras/stu2633
  * Machida & Basu (2019) Machida M. N., Basu S., 2019, ApJ, 876, 149. doi:10.3847/1538-4357/ab18a7
  * Machida & Hosokawa (2020) Machida M. N., Hosokawa T., 2020a, MNRAS, 499, 4490. doi:10.1093/mnras/staa3139
  * Machida, Hirano, & Kitta (2020) Machida M. N., Hirano S., Kitta H., 2020b, MNRAS, 491, 2180. doi:10.1093/mnras/stz3159
  * Matsushita et al. (2017) Matsushita Y., Machida M. N., Sakurai Y., Hosokawa T., 2017, MNRAS, 470, 1026. doi:10.1093/mnras/stx893
  * Matsushita et al. (2018) Matsushita Y., Sakurai Y., Hosokawa T., Machida M. N., 2018, MNRAS, 475, 391. doi:10.1093/mnras/stx3070
  * Matsushita et al. (2019) Matsushita Y., Takahashi S., Machida M. N., Tomisaka K., 2019, ApJ, 871, 221. doi:10.3847/1538-4357/aaf1b6
  * Masson et al. (2016) Masson J., Chabrier G., Hennebelle P., Vaytet N., Commerçon B., 2016, A&A, 587, A32. doi:10.1051/0004-6361/201526371
  * Matzner & McKee (2000) Matzner C. D., McKee C. F., 2000, ApJ, 545, 364. doi:10.1086/317785
  * Saiki & Machida (2020) Saiki Y., Machida M. N., 2020, ApJL, 897, L22. doi:10.3847/2041-8213/ab9d86
  * Seifried et al. (2011) Seifried D., Banerjee R., Klessen R. S., Duffin D., Pudritz R. E., 2011, MNRAS, 417, 1054. doi:10.1111/j.1365-2966.2011.19320.x
  * Seifried et al. (2012) Seifried, D., Pudritz, R. E., Banerjee, R., Duffin, D., & Klessen, R. S. 2012, MNRAS, 422, 347
  * Tabone et al. (2018) Tabone B., Raga A., Cabrit S., Pineau des Forêts G., 2018, A&A, 614, A119. doi:10.1051/0004-6361/201732031
  * Tanaka et al. (2017) Tanaka, K. E. I., Tan, J. C., & Zhang, Y. 2017, ApJ, 835, 32
  * Tanaka et al. (2018) Tanaka, K. E. I., Tan, J. C., Zhang, Y., et al. 2018, ApJ, 861, 68
  * Tokuda et al. (2018) Tokuda K., Onishi T., Saigo K., Matsumoto T., Inoue T., Inutsuka S., Fukui Y., et al., 2018, ApJ, 862, 8. doi:10.3847/1538-4357/aac898
  * Tomida et al. (2013) Tomida K., Tomisaka K., Matsumoto T., Hori Y., Okuzumi S., Machida M. N., Saigo K., 2013, ApJ, 763, 6. doi:10.1088/0004-637X/763/1/6
  * Tomisaka (2000) Tomisaka K., 2000, ApJL, 528, L41. doi:10.1086/312417
  * Tomisaka (2002) Tomisaka K., 2002, ApJ, 575, 306. doi:10.1086/341133
  * Truelove et al. (1997) Truelove, J. K., Klein, R. I., McKee, C. F., et al. 1997, ApJ, 489, L179
  * Uchida & Shibata (1985) Uchida Y., Shibata K., 1985, PASJ, 37, 515
  * Vaytet et al. (2018) Vaytet N., Commerçon B., Masson J., González M., Chabrier G., 2018, A&A, 615, A5. doi:10.1051/0004-6361/201732075
  * Wurster, Price, & Bate (2016) Wurster J., Price D. J., Bate M. R., 2016, MNRAS, 457, 1037. doi:10.1093/mnras/stw013
  * Wurster, Bate, & Price (2018) Wurster J., Bate M. R., Price D. J., 2018, MNRAS, 475, 1859. doi:10.1093/mnras/stx3339
  * Wurster, Bate, & Bonnell (2021) Wurster J., Bate M. R., Bonnell I. A., 2021, MNRAS.tmp. doi:10.1093/mnras/stab2296
  * Xu & Kunz (2021) Xu W., Kunz M. W., 2021, MNRAS, 502, 4911. doi:10.1093/mnras/stab314
  * Wu et al. (2004) Wu Y., Wei Y., Zhao M., Shi Y., Yu W., Qin S., Huang M., 2004, A&A, 426, 503. doi:10.1051/0004-6361:20035767