ORE Open Research Exeter
TITLE
Accretion-powered stellar winds. II. Numerical solutions for stellar wind torques
AUTHORS
Matt, Sean P.; Pudritz, Ralph E.
JOURNAL
Astrophysical Journal
DEPOSITED IN ORE
25 January 2016
This version available at
http://hdl.handle.net/10871/19348
COPYRIGHT AND REUSE
Open Research Exeter makes this work available in accordance with publisher policies.
A NOTE ON VERSIONS
The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of
publication
ACCRETION-POWERED STELLAR WINDS. II. NUMERICAL SOLUTIONS
FOR STELLAR WIND TORQUES
Sean Matt
1
and Ralph E. Pudritz
2
Received 2007 August 14; accepted 2008 January 17
ABSTRACT
In order to explain the slow rotation observed in a large fraction of accreting pre-main-sequence stars (CTTSs), we
explore the role of stellar winds in torquing down the stars. For this mechanism to be effective, the stellar winds need
to have relatively high outflow rates, and thus would likely be powered by the accretion process itself. Here, we use
numerical magnetohydrodynamical simulations to compute detailed two-dimensional (axisymmetric) stellar wind
solutions, in order to determine the spin-down torque on the star. We discuss wind driving mechanisms and then adopt
a Parker-like (thermal pressure driven) wind, modified by rotation, magnetic fields, and enhanced mass-loss rate (rela-
tive to the Sun). We explore a range of parameters relevant for CTTSs, including variations in the stellar mass, radius,
spin rate, surface magnetic field strength, mass-loss rate, and wind acceleration rate. We also consider both dipole and
quadrupole magnetic field geometries. Our simulations indicate that the stellar wind torque is of sufficient magnitude
to be important for spinning down a ‘‘typical’’ CTTS, for a mass-loss rate of 10
9
M
yr
1
. The winds are wide-
angle, self-collimated flows, as expected of magnetic rotator winds with moderately fast rotation. The cases with
quadrupolar field produce a much weaker torque than for a dipole with the same surface field strength, demonstrating
that magnetic geometry plays a fundamental role in determining the torque. Cases with varying wind acceleration rate
show much smaller variations in the torque, suggesting that the details of the wind driving are less important. We use
our computed results to fit a semianalytic formula for the effective Alfve
´
n radius in the wind, as well as the torque.
This allows for considerable predictive power, and is an improvement over existing approximations.
Subject headinggs: accretion, accretion disks — MHD — stars: magnetic fields — stars: pre–main-sequence —
stars: rotation — stars: winds , outflows
1. INTRODUCTION
For more than half a century, the spin rates and the angular
momentum evolution of stars have been topics of vigorous study.
We know that stellar winds are responsible for the spinning down
of late-type ( later than F2) main-sequence stars (Parker 1958;
Schatzman 1962; Kraft 1967; Skumanich 1972; Soderblom 1983;
Kawaler 1988; MacGregor & Brenner 1991; Barnes & Sofia
1996; Bouvier et al. 1997). There is still progress to be made on
main-sequence star spins (Barnes 2003), but perhaps the largest
open questions remain at the pre-main-sequence phase, which
determines the ‘‘initial conditions’’ for the spin histories of stars.
By the time intermediate/low-mass (P2 M
) pre-main-sequ ence
stars become optically visible ( T Tauri stars; TTSs), they already
have ages arou nd 10
5
–10
6
yr. A large fraction of TTSs (called
classical TTSs; CTTSs) are observed to actively accrete material
from a disk at a rate within a wide range of 10
8
M
yr
1
(e.g.,
Johns-Krull & Gafford 2002). At this rate, the angular momen-
tum accreted from the orbiting disk should spin up the stars to a
substantial fraction of breakup speed in a short amount of time
(comparable to their ages). The fact that the stars are also still
contracting (e.g., Rebull et al. 2002), and that they presumably
were accreting at much higher rates before they became optically
visible, further adds to the expectation of fast rotation.
Large data sets for the spins of TTSs in star formation regions
and clusters of different ages (see Rebull et al. [2004] for a com-
pilation) show that approximately half of the stars are rotating
rapidly a nd do seem to spin up as expected as they approach
zero-age main sequence (Vogel & Kuhi 1981; Bouvier et al. 1997;
Rebull et al. 2004; Herbst et al. 2007). However, the surprise is
that the other approximately half of TTSs exhibit much slower
rotation rates ( 10% of breakup speed ) at all ages. Recent stud-
ies have shown a correlation between slow rotation and the pres-
ence of an accretion disk (see especially Cieza & Baliber 2007),
although this idea has been controversial in the past (e.g., Stassun
et al. 1999, 2001; Herbst et al. 2000, 2002). This is still an open
issue, but it is clear than an efficient angular momentum loss or
regulation mechanism is operating for the slow rotators.
Although alternative ideas have been proposed since (Ko¨nigl
1991; Shu et al. 1994; see Matt & Pudritz [2008a] for a history),
Hartmann & Stauffer (1989) offered the first potential explana-
tion for the slow rotators, namely that massive stellar winds may
be responsible for carrying off substantial angular momentum
(see also Tout & Pringle 1992). In Matt & Pudritz (2005a; here-
after Paper I ), we extended this idea to consider the effects of the
magnetic interaction between the star and disk, and we used a
one-dimensional scaling from the solar wind angular momentum
loss to estimate the torque for TTSs. The scaling suggested that,
for an observationally constrained dipole magnetic field strength
of 200 G (e. g., Jo hns-Krull et al. 1999; Bouvier et a l. 200 7;
Johns-Krull 2007a, 2007b; Smirnov et al. 2003a; Yang et al.
2007), it might indeed be possible for the stellar wind to extract
enough angular momentum to explain the slow rotators. For stel-
lar winds to balance the accreted angular momentum, the wind
outflow rate needs to be a substantial fraction of the accretion
rate. In Paper I, we suggested that this is possible, if a fraction of
the energy liberated by the accretion process actually powers the
stellar wind.
The pre-main-sequence phase is, in fact, marked by powerful
outflows (Reipurth & Bally 2001). In the most powerful sources,
due to the large linear momenta of the outflows (Ko¨nigl & Pudritz
2000), the X-ray l uminosities (Decampli 1981), and possible
1
Department of Astronomy, University of Virginia, P.O. Box 400325,
Charlottesville, VA 22904-4325; seanmatt@virginia.edu.
2
Physics and Astronomy Department, McMaster University, Hamilton, ON
L8S 4M1, Canada; pudritz@physics.mcmaster.ca.
1109
The Astrophysical Journal, 678:1109–1118, 2008 May 10
# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.
detection of rotation ( Bacciotti et al. 2002; Anderson et al. 2003;
Coffey et al. 2004, 2007; Ferreira et al. 2006), it appears that most
of the flow arises from the accretion disk rather than the star. It is
not clear what fraction of the total outflow may actually originate
from the star, and thus how powerful the stellar winds are com-
pared to main-sequence phase winds or to their accretion rates.
There is some observational evidence for powerful stellar
winds from CTTSs, as distinguished from inner disk winds. In
particular, Edwards et al. (2003, 2006) observed the He i k10830
line in 39 CTTSs and saw several cases with a broad, deep, blue-
shifted absorption, indicating outflow velocities of typically a
few hundred, and up to 400 km s
1
. They concluded that this
feature is best interpreted as arising in an optically thick stellar
wind (see also Dupree et al. 2005). They also suggested that the
winds may be accretion powered, since the wind signatures are
most prevalent in the stars with highest accretion rates and absent
in nonaccreting systems. Subsequent modeling of the He i k10830
line by Kwan et al. (2007) indicates that approximately half of
these CTTSs show evidence for a powerful stellar wind. Further-
more, Kurosawa et al. (2006) modeled the H emission line in
these systems and suggested that a stellar wind component could
most naturally explain the profiles observed in 7% of the stars
in a sample compiled by Reipurth et al. (1996).
There already exists some theoretical work on stellar winds,
specifically from pre-main-sequence stars, with a focus on the
wind driving mechanism ( Decampli 1981; Hartmann et al. 1982,
1990) or the collimation of the winds (Fendt et al. 1995; Fendt
& Camenzind 1996; Bogovalov & Tsinganos 2001; Sauty et al.
2002). These do not discuss the expected angular momentum
outflow rates, however. The works that do calculate stellar wind
torques for pre-main-sequence stars (Hartmann & MacGregor
1982; Mestel 1984; Hartmann & Stauffer 1989; Tout & Pringle
1992; Paatz & Camenzind 1996; Paper I) are either based on a
one-dimensional formulation and/or have made a priori simplifying
assumptions regarding the stellar magnetic field structure, wind flow
speed, and latitudinal dependence of the wind. Calculating the
stellar wind torque reliably is a complex, multidimensional problem,
andmoreworkisneededtodevelopthestellarwindtheoryfurther.
The primary goal of this paper therefore, is to take the next
major step in developing the accretion-powered stellar wind pic-
ture by rigorously computing the steady state solutions of winds
from spinning magnetized stars. We carry out a parameter study
to provide a range of possible solutions that are expected to char-
acterize accretion-powered stellar winds. Where possible, we
compare our results to analytic magnetohydrodynamic (MHD)
stellar wind theory. In a companion paper, we will use these so-
lutions to compare the stellar wind torques and wind driving
power with the torque and energy deposition expected to arise
from the interaction of the star with its accretion disk.
In the following section (x2.1), we give a brief introduction to
basic stellar wind theory. This provides the motivation for using
a numerical approach and sets the stage for comparing our nu-
merical results with the analytic theory. Section 2.2 contains a
discussion of our adopted wind driving mechanism. We describe
our numerical method for obtaining solutions in x 3, and present
the results in x 4. Section 5 contains a semianalytic formulation
for the torque and a comparison to previous theory.
2. MAGNETIZED STELLAR WINDS:
NEEDED BACKGROUND
2.1. Ma
gnetic Stellar Wind Theory
Standard MHD win d theory (i.e., magnetic rotator theory),
following Weber & Davis (1967), characterizes a steady state
flow of plasma along a magnetic field line that is anchored to a
rotating object, which we will hereafter take to be a star. One of
the key results is that the angular momentum outflow rate per
unit mass loss is given very simply as (see, e.g., Weber & Davis
1967; Mestel 1968; Michel 1969)
l ¼
r
2
A
; ð1Þ
where
is the angular rotation rate of the star, and r
A
is the
cylindrical radius at which the outflow speed equals the local
magnetic Alfve
´
n speed,
v
A
B
p
ffiffiffiffiffiffiffiffi
4
p
; ð2Þ
where is the local mass density and B
p
is the strength of the
poloidal magnetic field, B
p
¼ (B
2
r
þ B
2
z
)
1
=
2
,incylindrical(r;;z)
coordinates. Equation (1) indicates that the quantity of angular
momentum carried in the wind is as if the wind material is co-
rotating out to r
A
and conserves its angular momentum there-
after. Thus, r
A
is often referred to as the magnetic ‘‘lever arm.’’ In
reality, the azim uthal velocity of the wind, v
, is a smooth (i.e.,
differentiable) function of radius, and the difference between v
r
and l at all radii equals the torque transmitted b y azimuthally
twisted magnetic field lines.
By integrating the mass flux times l over any surface enclosing
the star, one obtains an expression for the total angular momen-
tum outflow rate and , by Newt on’s t hird law, the torque on the
star:
w
¼
˙
M
w
r
2
A
; ð3Þ
where
˙
M
w
is the integrated wind mass-loss rate. Since the value
of r
A
will generally not be the same along each field line, equa-
tion (3) defines the quantity hr
2
A
i, which is the mass-loss-weighted
average of r
2
A
(suggested by Washimi & Shibata 1993). Hereafter,
we will simply refer to this average a s r
A
hr
2
A
i
1
=
2
.
The difficulty now lies in calculating r
A
. The lever arm length
clearly depends on the stellar surface field strength (B
), stellar
radius (R
), and
˙
M
w
because these directly affect Alfve
´
n condi-
tion. But it also depends on the flow speed and field structure,
which are not possible to determine a priori in the wind. The flow
speed is influenced by the thermal energy in the wind as well as
rotation. There exist two different regimes (Belcher & MacGregor
1976): the fast magnetic rotator regime, where the flow speed is
mostly determined by magnetorotational effects; and the slow
magnetic rotator, where the flow speed is solely determined by
the wind driving. The field structure in the wind, even though the
geometry may be known at the stellar surface, is determined b y
the self-consistent interaction between the wind and rotating
magnetic field and thus is a function of all parameters. There-
fore one can only calculate r
A
by making a priori assumptions
about the field structure and/or flow speed (Weber & Davis
1967; Mestel 1968, 1984; Okamoto 1974; Mestel & Spruit 1987;
Kawaler 1988) or by using iterative techniques (or numerical
simulations; Pneuman & Kopp 1971; Sakurai 1985; Washimi
& Shibata 1993; Keppens & Goedbloed 2000; Matt & Balick
2004).
All of these methods are complementary. The analytical work,
in which the field structure is guessed, produces a predictive for-
mulation of the stellar wind torque (e.g., Kawaler 1988). How-
ever, the formulation of the field structure usually i ntroduc es
more parameters (such as a power-law index for the magnetic
field), so that almost any resul t can be obtained by adjusting
MATT & PUDRITZ1110 Vol. 678
these. Furthermore, the field structure in the analytic models has
no explicit dependence on (for example)
, which is exhibited
in numerical simulations (e.g., Matt & Balick 2004). The numer-
ical simulation technique has the advantage of calculating the
field structure and flow speed self-consistently. However, a single
simulation does not predict the dependence of r
A
on parameters,
and to date, not enough parameter space has been explored. Thus,
to date, there exists no formulation for the stellar wind torque
that convincingly applies over a wide range of conditions (e.g., over
a range of B
;
˙
M
w
, and
).
In this paper we will use two-dimensional (axisymmetric) MHD
simulations to calculate the torque and corresponding value of
r
A
. This will allow us to check the estimate for r
A
of Paper I (and
previous works). In addition, we will carry out a parameter study
to determine the dependence of the stellar wind torque on pa-
rameters, over a range of conditions appropriate for TTSs, and
compare this with the predictions of analytic theory.
2.2. Wind Dri
ving Mechanism
It is not known what drives winds from TTSs. These stars have
active coronae (Feigelson & Montmerle 1999; Stassun et al. 2004;
Favata et al. 2005), and it thus seems a reason able assump -
tion that they also drive solar-like coronal winds in which ther-
mal pressure plays a significant role in the wind acceleration.
Based on a calculation from Bisnovatyi-Kogan & Lamzin (1977),
Decampli (1981) concluded that, in order for the wind emission
to be consistent with the X-ray observations, the mass-loss rate
of a TTS coronal wind must be less than 10
9
M
yr
1
. Fur-
thermore, Dupree et al. (2005) found evidence for a stellar wind
with a coronal temperature in the CTTS TW Hya (although this
conclusion has been challenged by Johns-Krull & Herczeg 2007).
The assumption of thermal pressure driving is a simplifica-
tion, even for the solar wind. It is known that a major factor in
driving the solar wind is Alfve
´
n wave momentum and energy de-
position. Two important recent studies have done self-consistent
analyses of the combined problem of both solar wind heating
and acceleration (Suzuki & Inutsuka 2006; Cranmer et al. 2007).
The first paper shows that low-frequency, transverse motions of
open field lines at the photosphere leads to transonic solar winds
for superradial expansion of the wind cross section. If the ampli-
tude of these transverse photospheric motions exceeds 0.7 km s
1
,
fast winds are produced and the dissipation of wave energy heats
the atmosphere to a million degrees. The results are sensitive to
the amplitude of the velocity perturbations, and the simulations
show that the s olar wind virtually d isappears for amplitudes
0.3 km s
1
. These numerical simulations also show that Alfve
´
n
wave pressure dominates the gas pressure in the solar accelera-
tion region (1:5 R
R 10 R
). The second paper shows sim-
ilar results. This work shows that there are three key parameters
that control wind heating and acceleration: the flux of acoustic
power injected at the photosphere, the Alfve
´
n wave amplitude
there, and the Alfve
´
n wave c orrelation length (characterizing
wave damping through turbulence) at the photosphere.
Our primary goal here is to evaluate the angular momentum
transported away from the star by the stellar wind. Thus, in this
work, we do not discuss the thermodynamic properties of the
wind and instead focus on the angular momentum transport. For-
tunately, this torque does not much depend on what drives the
wind. Rather, the torque depends primarily on the stellar mag-
netic field, rotation rate, radius,
˙
M
w
, and the wind velocity. As
long as ‘‘something’’ accelerates the wind to speeds similar to what
we see in our simulations, the torque we calculat e w ill be ap-
proximately correct.
We expect that the Alfve
´
n waves in accreting TTS winds will
have a significant, if not dominant, contribution to both the accel-
eration and heating of their winds. These waves will be launched
along the open field lines that originate from the TTS photo-
sphere at latitudes comparable to those that harbor field lines
carrying the accretion flow onto the star. The irregular accretion
flow should generate very large (i.e., much larger than acoustic
motions in the solar photosphere) acoustical transverse motions
in the TTS photosphere as it impinges on the star. These large
amplitude perturbations, generated by the accretion flow itself,
may be the ultimate driver for the Alfve
´
n wave flux that drives
our proposed accretion-powered stellar wind.
Note that the driving force can be parameterized as being pro-
portional to : (where is the wave energy density; Decampli
1981). This has the same functional form as the thermal pres-
sure force (: P) used in our simulations. Several authors (e.g.,
Hartmann & MacGregor 1980; Decampli 1981; Holzer et al. 1983;
Suzuki 2007) computed velocity profiles for cool (10
4
K) Alfve
´
n
wave-driven winds. These works exhibit wind velocity profiles
that are similar to what is expected from thermal pressure driving
of hotter winds. Therefore, we can think of thermal pressure driv-
ing as a proxy for some other driving mechanism. Also, it will be
important to have these solutions to compare with future work
that includes different driving mechanisms.
In this paper, we restrict ourselves to mass-loss rates of
˙
M
w
<
2 ; 10
9
M
yr
1
. As justified above, we adopt a Parker-like
(Parker 1958) coronal wind driving mechanism, modified by
magnetic fields, stellar rotation, and an enhanced mass-loss rate
(relative to the Sun). As the nature (e.g., temperature) of TTS
stellar winds is not well known, our detailed solutions of coronal
winds will enable us to look at the expected radiative properties,
a posteriori, allowing for further constraints on real systems. We
will show in a forthcoming paper (and see Matt & Pudritz 2007)
that the expected emission from the simulated winds presented
here rules out thermal pressure driving at a substantially lower
mass-loss rate than the limit of Decampli (1981).
3. NUMERICAL SIMULATION METHOD
We calculate solutions of steady state winds from isolated stars
(no accretion disk), using the finite-difference MHD code of Matt
& Balick (2004); the reader will find further details there
3
(and
references therein). Assuming axisymmetry and using a cylin-
drical (r;;z) coordinate system, the code employs a two-step
Lax-Wendroff scheme ( Richtmyer & Morton 1967) to solve the
following time-dependent, ideal MHD equations:
@
@t
¼: ( v); ð4Þ
@( v)
@t
¼(v :)v v : (v)½
:P
GM
r
2
þ z
2
ðÞ
ˆ
R þ
1
c
(J ; B); ð5Þ
@e
@t
¼: ½v(e þ P)
GM
r
2
þ z
2
ðÞ
ˆ
R
v þ J E; ð6Þ
@B
@t
¼c(:;E ); ð7Þ
3
Matt & Bal ick (2004) ran cases with isotropic hydrodynamic variables at
the base of the wind and also cases with enhanced polar winds. Here we only
consider the isotropic case.
NUMERICAL SOLU TIONS FOR STELLAR WIND TORQUES 1111No. 2, 2008
and uses
E ¼
1
c
(v ; B); ð8Þ
J ¼
c
4
(:;B); ð9Þ
e ¼
1
2
v
2
þ
P
1
; ð10Þ
where is the density, v the velocity, P the gas pressure, G
Newton’s gravitational constant, M
the stellar mass, R the spher-
ical distance from the center of the star (R
2
¼ r
2
þ z
2
), e the in-
ternal energy density, B the magnetic field, J the volume current,
E the electric field, c the speed of light, and the ratio of specific
heats.
To obtain steady state wind solutions, we follow the method
of Matt & Balick (2004), which is also similar to that employed
by Washimi & Shibata (1993) and Keppens & Goedbloed (1999).
It involves initializing the computational grid with a spherically
symmetric, isothermal Parker wind solution ( Parker 1958), plus
force-free dipole ( and sometimes quadrupole) m agneti c field.
When the simulation begins, the wind solution changes from the
initial state due to the presence of the magnetic field, the rota-
tion of the star, and the polytropic equation of state (P /
).
The simulations run until the system relaxes into a steady state
(within a small tolerance) MHD wind solution. The code uses
nested computational grids so that the wind can be easily fol-
lowed to large distances (several tens to hundreds of R
).
This method results in a steady state solution for the wind that
is determined solely by the boundary conditions held fixed at the
base of the stellar corona (the ‘‘stellar surface’’). In order to
capture the appropriate physics within the framework of a finite
difference scheme, we employ a four-layer boundary for the star,
on which the various physical quantities are set as follows. We
consider the spherical location R ¼ 30, in units of the grid spac-
ing, to be the surface of the star. For all grid points such that
R 34:5, the poloidal velocity is forced to be parallel with the
poloidal magnetic field (v
p
k B
p
, where the poloidal component
is defined as the vector component in the r-z plane). Where R
33:5, and P are held constant (in time) at their initial values. For
R 32:5; v
p
is held at zero, while v
is held at corotation with
the star. For R 31:5; B
p
field is held at its initial, dipolar value,
while B
is set so that there is no poloidal electric current at that
layer (which gives it a dependence on the conditions in the next
outer layer, 31:5 < R 32:5).
These boundary conditions properly capture the behavior of a
wind accelerated from the surface of a rotating magnetized star,
as follows. There is a layer on the stellar boundary (R > 32: 5),
outside of which the velocity not fixed, but is allowed to vary in
time. In this way, the wind speed and direction is not specified,
but is determined by the code in response to all of the forces. By
holding P fixed at its initial value for all R 33:5, we constrain
the pressure gradient force (thermal driving) at the base of the
wind to be constant in time. In addition, holding the density fixed
at R 33:5 allows the region from which the wind flows to be
instantly replenished with plasma. Thus, the base of the wind
maintains a constant temperature and density, regardless of how
fast or slow the wind flows away from that region. The exis-
tence of a layer in which v
p
¼ 0 and B
p
can evolve (namely, at
31:5 < R 32:5) allows B
p
(and v
p
) to reach a value that is self-
consistently determined by the balance of magnetic and inertial
forces. We set the poloidal velocity parallel to the poloidal mag-
netic field for the next two outer layers, to ensure a smooth tran-
sition from the region of pure dipole field and zero velocity to
that with a perturbed field and outflow. Setting B
so that the
poloidal electric current is zero inside some radius ensures that
the field behaves as if anchored in a rotating conductor (the sur-
face of the star). Also, this ensures that B
evolves appropriately
outside the anchored layer according to the interaction with the
wind plasma.
The key physical parameters can be represented by the charac-
teristic speeds of the input physics, namely the sound speed at
the base of the corona, c
s
, the escape speed from the surface of
the star, v
esc
, the rotation speed of the star, and the Alfve
´
n speed
at the base of the wind. We specify the ratio of c
s
/v
esc
as our
parameter, rather than the sound speed alone. This seems the
most reasonable, since the temperature of a thermally driven
wind is regulated somewhat by the interplay between the thermal
energy input and the expansion of the corona (the wind) against
gravity. To first order, a hotter wind expands more rapidly against
gravity, allowing less time for the gas to heat, and a cooler wind
expands more slowly, allowing more time to heat. Once the value
of the stellar mass and radius is specified, the ratio of c
s
/v
esc
determines the temperature held fixed on the stellar boundary,
as described above. The wind plasma is characterized by a poly-
tropic equation of state, and so is also a parameter. We pa-
rameterize the stellar rotation rate as the fract ion of breakup
speed,
f
R
3=2
(GM
)
1=2
: ð11Þ
The Alfve
´
n speed is determined by the magnetic field strength
and coronal density. Rather than taking the Alfve
´
n speed as a key
parameter, we specify the field strength at the equator of the star
(B
) as our parameter, in order to connect the simulations as
much as possible to observationally constrained quantities. For
the same reason, we specify
˙
M
w
as a parameter, rather than the
coronal density. In the simulations, we must specify the base
density,
, to be held fixed on the stellar boundary, and the value
of
˙
M
w
in the steady state wind is not solely determined by
. For
example, the rotation of the star can enhance
˙
M
w
via magneto-
centrifugal flinging, and a strong magnetic field can decrease
˙
M
w
by inhibiting flow from a region near the equator that remains
magnetically closed (the ‘‘dead zone’’). In other words,
˙
M
w
is not
an a priori tunable parameter; rather, it is a result of the simu-
lations. Therefore, to treat
˙
M
w
as our tunable parameter, we adopt
an iterative approach. This entails first running a given simula-
tion with a guess for
, checking the resulting value of
˙
M
w
, and
then adjusting
and rerunning the simulation. We iterate until
the desired value of
˙
M
w
is achieved (within a tolerance of 2%).
This typically required 2–4 iterations, so the ability to treat
˙
M
w
as
a chosen parameter comes at a substantial cost.
4. STELLAR WIND SOLUTIONS
4.1. The Fiducial Case
We start by presenting the results of our stellar wind simula-
tion for parameters with values that represent a ‘‘typical’’ TTS
and follow the fiducial values of Paper I and Matt & Pudritz
(2005b). Table 1 lists the fiducial parameters. We consider a low-
mass pre-main-sequence star, with a surface escape speed of
v
esc
309 km s
1
. A dipole magnetic field strength of 200 G is
consistent with 3 upper limits (Johns-Krull et al. 1999; Smirnov
et al. 2004, 2003b) or marginal detection (Smirnov et al. 2003a;
Yang et al. 2007) of the longitudinal magnetic field measured for
CTTSs. We seek primarily to understand the slow rotators, for
which a rotation rate of 10% of breakup is appropriate. In Paper I,
MATT & PUDRITZ1112 Vol. 678
