Accretion-Powered Stellar Winds II: Numerical Solutions for Stellar Wind Torques

0.46 MB·English

by  Sean Matt

Draftversion February2,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
ACCRETION-POWERED STELLAR WINDS II:
NUMERICAL SOLUTIONS FOR STELLAR WIND TORQUES
Sean Matt1 and Ralph E. Pudritz2
1DepartmentofAstronomy,UniversityofVirginia,P.O.Box400325,Charlottesville,VA22904-4325; [email protected]
2PhysicsandAstronomyDepartment, McMasterUniversity,Hamilton,ONL8S4M1,Canada;[email protected]
Draft versionFebruary 2, 2008
ABSTRACT
In order to explain the slow rotation observed in a large fraction of accreting pre-main-sequence
8
stars (CTTSs), we explore the role of stellar winds in torquing down the stars. For this mechanism
0
to be effective, the stellar winds need to have relatively high outflow rates, and thus would likely be
0
powered by the accretion process itself. Here, we use numerical magnetohydrodynamical simulations
2
to compute detailed 2-dimensional (axisymmetric) stellar wind solutions, in order to determine the

n spin down torque on the star. We discuss wind driving mechanisms and then adopt a Parker-like
a (thermal pressure driven) wind, modified by rotation, magnetic fields, and enhanced mass loss rate
J
(relative to the sun). We explore a range of parameters relevant for CTTSs, including variations

8 in the stellar mass, radius, spin rate, surface magnetic field strength, the mass loss rate, and wind
1
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
h] spinning down a “typical” CTTS, for a mass loss rate of 10−9M⊙ yr−1. The winds are wide-angle,
∼
p self-collimated flows,as expected of magnetic rotatorwinds with moderately fast rotation. The cases
- with quadrupolar field produce a much weaker torque than for a dipole with the same surface field
o
strength, demonstrating that magnetic geometryplays a fundamental role in determining the torque.
r
t Caseswithvaryingwindaccelerationrateshowmuchsmallervariationsin the torquesuggestingthat
s
the details of the wind driving are less important. We use our computed results to fit a semi-analytic
a
[ formula for the effective Alfv´enradius in the wind, as wellas the torque. This allowsfor considerable
predictive power, and is an improvement over existing approximations.

3
Subject headings: accretion, accretion disks — MHD — stars: magnetic fields — stars: pre-main-
v
sequence — stars: rotation — stars: winds, outflows
6
3
4 1. INTRODUCTION 2004, for a compilation) show that approximately half
0
of the stars are rotating rapidly and do seem to spin up
. For more than half a century, the spin rates and the
1 as expected as they approach zero-age main sequence
angular momentum evolution of stars have been topics
0 (Vogel & Kuhi 1981; Bouvier et al. 1997; Rebull et al.
of vigorous study. We know that stellar winds are re-
8 2004; Herbst et al. 2007). However, the surprise is that
sponsible for the spinning down of late-type (later than
0 the other half of TTSs exhibit much slower rotation
F2) main sequence stars (Parker 1958; Schatzman 1962;
v: Kraft 1967; Skumanich 1972; Soderblom 1983; Kawaler rates (∼ 1∼0% of breakup speed) at all ages. Recent
i 1988; MacGregor & Brenner 1991; Barnes & Sofia 1996; studies have shown a correlation between slow rotation
X
and the presence of an a accretion disk (see especially
Bouvier et al. 1997). There is still progress to be made
r Cieza & Baliber2007),thoughthisideahasbeencontro-
on main sequence star spins (Barnes 2003), but per-
a
versialinthepast(e.g.,Stassun et al.1999;Herbst et al.
haps the largestopen questions remain at the pre-main-
2000; Stassun et al. 2001; Herbst et al. 2002). This is
sequence phase, which determines the “initial condi-
still an open issue, but it is clear than an efficient angu-
tions” for the spin histories of stars.
larmomentumlossorregulationmechanismisoperating
By the time intermediate/low mass (. 2M⊙) pre-
for the slow rotators.
main-sequence stars become optically visible (T Tauri
stars; TTSs), they already have ages around 105 – 106 Although alternative ideas have been proposed since
(K¨onigl1991;Shu et al.1994,seeMatt & Pudritz2007b
yrs. A large fraction of TTSs (called classical TTSs;
for a history), Hartmann & Stauffer (1989) offered the
CTTSs) are observed to actively accrete material from
a disk at a rate within a wide range of 10−8M⊙ yr−1 first potential explanation for the slow rotators, namely
∼ that massive stellar winds may be responsible for
(e.g.,Johns-Krull & Gafford2002). At this rate,the an-
carrying off substantial angular momentum (see also
gular momentum accreted from the orbiting disk should
Tout & Pringle 1992). In Matt & Pudritz (2005a, here-
spin up the stars to a substantial fraction of breakup
after Paper I), we extended this idea to consider the ef-
speed in a short amount of time (comparable to their
fects of the magnetic interaction between the star and
ages). The fact that the stars are also still contracting
disk, and we used a 1-dimensional scaling from the so-
(e.g.,Rebull et al.2002),andthattheypresumablywere
lar wind angular momentum loss to estimate the torque
accreting at much higher rates before they became opti-
for TTSs. The scaling suggested that, for an obser-
callyvisible,furtheraddstoexpectationoffastrotation.
vationally constrained dipole magnetic field strength of
LargedatasetsforthespinsofTTSsinstarformation
200G (e.g., Johns-Krull et al. 1999; Bouvier et al. 2007;
regions and clusters of different ages (see Rebull et al.
2 Matt & Pudritz
Johns-Krull 2007a,b; Smirnov et al. 2003a; Yang et al. state solutions of winds from spinning magnetized stars.
2007), it mightindeed be possible for the stellar wind to We carry out a parameter study to provide a range
extract enough angular momentum to explain the slow of possible solutions that are expected to charaterize
rotators. For stellar winds to balance the accreted an- accretion-powered stellar winds. Where possible, we
gular momentum, the wind outflow rate needs to be a compare our results to analytic magnetohydrodynamic
substantial fractionof the accretion rate. In Paper I, we (MHD) stellar wind theory. In a companion paper,
suggestedthatthis is possible,ifa fractionofthe energy we will use these solutions to compare the stellar wind
liberated by the accretion process actually powers the torques and wind driving power with the torque and en-
stellar wind. ergy deposition expected to arise from the interactionof
The pre-main-sequence phase is, in fact, marked by the star with its accretion disk.
powerful outflows (Reipurth & Bally 2001). In the most In the following section ( 2.1), we give a brief intro-
§
powerfulsources,due to the largelinear momenta of the duction to basic stellar wind theory. This provides the
outflows (K¨onigl & Pudritz 2000), the X-ray luminosi- motivation for using a numerical approach and sets the
ties (Decampli 1981), and possible detection of rotation stage for comparing our numerical results with the an-
(Bacciotti et al.2002;Anderson et al.2003;Coffey et al. alytic theory. Section 2.2 contains a discussion of our
2004; Ferreira et al. 2006; Coffey et al. 2007), it appears adopted wind driving mechanism. We describe our nu-
that most of the flow arises from the accretion disk, merical method for obtaining solutions in section 3, and
rather than the star. It is not clear what fraction of presenttheresultsinsection4. Section5containsasemi-
the total outflow may actually originate from the star, analytic formulation for the torque and a comparison to
andthushowpowerfularethestellarwindscomparedto previous theory.
main-sequence phase winds or compared to their accre-
2. MAGNETIZEDSTELLARWINDS:NEEDED
tion rates.
BACKGROUND
Thereissomeobservationalevidenceforpowerfulstel-
2.1. Magnetic Stellar Wind Theory
lar winds from CTTSs, as distinguished from inner disk
winds. In particular, Edwards et al. (2003, 2006) ob- StandardMHDwindtheory(i.e.,magneticrotatorthe-
servedtheHeI10830˚Alinein39CTTSsandsawseveral ory), following Weber & Davis (1967), characterizes a
cases with a broad, deep, blue-shifted absorption, indi- steady-state flow of plasma along a magnetic field line
cating outflow velocities of typically a few hundred, and thatisanchoredtoarotatingobject,whichwewillhere-
up to 400 km s−1. They concluded that this feature after take to be a star. One of the key results is that
∼
is best interpreted as arising in an optically thick stellar the angular momentum outflow rate per unit mass loss
wind (see also Dupree et al. 2005). They also suggested is given very simply as (see, e.g., Weber & Davis 1967;
the winds may be accretion-powered,since the wind sig- Mestel 1968; Michel 1969)
natures are most prevalent in the stars with highest ac-
cretion rates and absent in non-accreting systems. Sub- l=Ω∗rA2, (1)
sequentmodelingoftheHeI10830˚AlinebyKwan et al. where Ω∗ is the angular rotation rate of the star, and
(2007)indicatesthatapproximatelyhalfoftheseCTTSs r is the cylindrical radius at which the outflow speed
A
show evidence for a powerful stellar wind. Furthermore, equals the local magnetic Alfv´en speed,
Kurosawaet al. (2006) modeled the Hα emission line in
B
p
these systems and suggested that a stellar wind compo- vA , (2)
≡ √4πρ
nentcouldmostnaturallyexplaintheprofilesobservedin
7%ofthestarsinasamplecompiledbyReipurth et al. where ρ is the local mass density and B is the strength
p
∼
(1996). of the poloidal magnetic field, B = (B2 +B2)1/2, in
p r z
There already exists some theoretical work on stel- cylindrical (r, φ, z) coordinates. Equation (1) indicates
lar winds, specifically from pre-main-sequence stars, that the quantity of angular momentum carried in the
with a focus on the wind driving mechanism (Decampli wind is as if the wind material is corotating out to r
A
1981; Hartmann et al. 1982, 1990) or the collimation and conserves its angular momentum thereafter. Thus,
of the winds (Fendt et al. 1995; Fendt & Camenzind r is often referred to the magnetic “lever arm.” In re-
A
1996). These do not discuss the expected angu- ality,the azimuthalvelocityof the wind, v , is a smooth
φ
lar momentum outflow rates, however. The works (i.e.,differentiable)functionofradius,andthedifference
that do calculate stellar wind torques for pre-main- between v r and l at all radii equals the torque trans-
φ
sequence stars (Hartmann & MacGregor 1982; Mestel mitted by azimuthally twisted magnetic field lines.
1984; Hartmann & Stauffer 1989; Tout & Pringle 1992; By integrating the mass flux times l over any surface
Paatz & Camenzind 1996, Paper I) are either based on enclosingthestar,oneobtainsanexpressionforthetotal
a 1-dimensional formulation and/or have made a priori angular momentum outflow rate and, by Newton’s third
simplifying assumptions regarding the stellar magnetic law, the torque on the star:
field structure, wind flow speed, and latitudinal depen-
dence of the wind. Calculating the stellar wind torque τw =−M˙wΩ∗ rA2 , (3)
reliably is a complex, multi-dimensional problem, and where M˙ is the integrated win(cid:10)d m(cid:11)ass loss rate. Since
more work is needed to develop the stellar wind theory w
the value of r will generically not be the same along
further. each field line,Aequation (3) defines the quantity r2 ,
The primary goal of this paper therefore, is to take A
whichisthe mass-loss-weightedaverageofr2 (sugg(cid:10)este(cid:11)d
the nextmajor stepin developingthe accretion-powered A
by Washimi & Shibata 1993). Hereafter, we will simply
stellarwindpicturebyrigourouslycomputingthesteady-
refer to this average as r r2 1/2.
A ≡ A
(cid:10) (cid:11)
Stellar Wind Torques 3
The difficulty now lies in calculating r . The lever found evidence for a stellar wind with a coronal tem-
A
arm length clearly depends on the stellar surface field perature in the CTTS TW Hya (though this conclusion
strength(B∗),stellarradius(R∗),andM˙w becausethese has been challenged by Johns-Krull& Herczeg 2007).
directly affect Alfv´en condition. But it also depends on The assumption of thermal pressure driving is a sim-
the flow speed and field structure, which are not possi- plification, even for the solar wind. It is known that a
ble to determine a priori in the wind. The flow speed is major factor in driving the solar wind is Alfv´en wave
influenced by the thermal energy in the wind as well as momentum and energy deposition. Two important re-
rotation. In addition, there exist two different regimes cent studies have done self-consistent analyses of the
(Belcher & MacGregor 1976): the fast magnetic rotator combined problemof both solar wind heating and accel-
regime, where the flow speed is mostly determined by eration (Suzuki & Inutsuka 2006; Cranmer et al. 2007).
magnetorotationaleffects;andtheslowmagneticrotator, Thefirstpapershowsthatlowfrequency,transversemo-
where the flow speed is solely determined by the wind tionsofopenfieldlinesatthephotosphereleadsto tran-
driving. The field structure in the wind, even though sonic solar winds for superradial expansion of the wind
the geometry may be known at the stellar surface, is de- cross-section. If the amplitude of these transverse pho-
termined by the self-consistent interaction between the tospheric motions exceeds 0.7 km s−1, fast winds are
wind and rotating magnetic field and thus is a function produced and the dissipation of wave energy heats the
of all parameters. Therefore one can only calculate r atmosphere to a million degrees. The results are sensi-
A
by making a priori assumptions about the field struc- tive to the amplitude of the velocity perturbations, and
ture and/or flow speed (Weber & Davis 1967; Mestel the simulations show that the solar wind vitually dis-
1968;Okamoto1974;Mestel1984;Mestel & Spruit1987; appears for amplitudes 0.3 km s−1. These numerical
≤
Kawaler1988)orbyusingiterativetechniques(ornumer- simulations also show that Alfv´en wave pressure dom-
ical simulations; Pneuman & Kopp 1971; Sakurai 1985; inates the gas pressure in the solar acceleration region
Washimi & Shibata 1993; Keppens & Goedbloed 2000; (1.5R⊙ R 10R⊙). The second paper shows sim-
≤ ≤
Matt & Balick 2004). ilar results. This work shows that there are three key
All of these methods are complementary. The analyt- parameters that control wind heating and acceleration:
ical work, in which the field structure is guessed, pro- the flux of acoustic power injected at the photosphere,
duces a predictive formulationof the stellar wind torque the Alfv´en wave amplitude there, and the Alfv´en wave
(e.g.,Kawaler1988). However,usuallytheformulationof correlationlength(characterizingwavedampingthrough
the field structure introduces more parameters (such as turbulence) at the photosphere.
apowerlawindexforthe magneticfield),sothatalmost Our primary goal here is to evaluate the angular mo-
any result can be obtained by adjusting these. Further- mentum transported away from the star by the stellar
more, the field structure in the analytic models has no wind. Thus, in this work,we do not discuss the thermo-
explicitdependenceon(e.g.)Ω∗,whichisexhibitedinnu- dynamic propertiesofthe wind andinsteadfocus on the
merical simulations (e.g., Matt & Balick 2004). The nu- angular momentum transport. Fortunately, this torque
mericalsimulationtechnique hasthe advantageofcalcu- doesnotmuchdepend onwhatdrivesthe wind. Rather,
latingthefieldstructureandflowspeedself-consistently. the torque depends primarily on the stellar magnetic
However, a single simulation does not predict the de- field, rotation rate, radius, M˙ , and the wind velocity.
w
pendence of r on parameters, and to date, not enough As long as “something” accelerates the wind to speeds
A
parameterspacehasbeenexplored. Thus,todate,there similar to what we see in our simulations, the torque we
existsnoformulationforthestellarwindtorquethatcon- calculate will be approximately correct.
vincingly applies over a wide range of conditions (e.g., We expect that the Alfv´en waves in accreting TTS
over a range of B∗, M˙w, and Ω∗). winds will have a significant, if not dominant contribu-
In this paper we will use 2-dimensional (axisymmet- tion to both the accelerationand heating of their winds.
ric) MHD simulations to calculate the torque and cor- These waves will be launched along the open field lines
responding value of r . This will allow us to check the that originate from the TTS photosphere at latitudes
A
estimate for r of Paper I (and previous works). In ad- comparabletothosethatharbourfieldlinescarryingthe
A
dition,we willcarryoutaparameterstudy to determine accretionflowontothestar. Theirregularaccretionflow
the dependence of the stellar wind torque on parame- shouldgenerate very large (i.e., much largerthan acous-
ters, over a range of conditions appropriate for TTSs, tic motions in the solar photosphere) acoustical trans-
and compare with the predictions of analytic theory. verse motions in the TTS photosphere as it impinges
uponthestar. Theselargeamplitudeperturbations,gen-
erated by the accretion flow itself, may be the ultimate
2.2. Wind Driving Mechanism
driver for the Alfv´en wave flux that drives our proposed
It is not known what drives winds from TTSs. These accretion-poweredstellar wind.
stars have active coronae (Feigelson & Montmerle 1999; Notethatthedrivingforcecanbeparameterizedasbe-
Stassun et al. 2004; Favata et al. 2005), and it thus ingproportionalto ∇ξ(whereξisthewaveenergyden-
−
seemsareasonableassumptionthattheyalsodrivesolar- sity;Decampli1981). This has the same functionalform
like coronalwinds in whichthermal pressureplays a sig- as the thermal pressure force ( ∇P) used in our sim-
−
nificantroleinthewindacceleration. Basedonacalcula- ulations. Several authors (e.g., Hartmann & MacGregor
tionfromBisnovatyi-Kogan& Lamzin(1977), Decampli 1980; Decampli 1981; Holzer et al. 1983; Suzuki 2007)
(1981) concluded that, in order for the wind emission to computed velocity profiles for cool ( 104 K) Alfv´en
∼
be consistent with the X-ray observations, the mass loss wave-driven winds. These works exhibit wind velocity
rate of a T Tauri star coronal wind must be less than profiles that are similar to what is expected from ther-
10−9M⊙ yr−1. Furthermore, Dupree et al. (2005)
∼
4 Matt & Pudritz
mal pressure driving of hotter winds. Therefore, we can the computational grid with a spherically symmetric,
think of thermal pressure driving as a proxy for some isothermal Parker wind solution (Parker 1958), plus
other driving mechanism. Also, it will be important to force-free dipole (and sometimes quadrupole) magnetic
have these solutions to compare with future work that field. When the simulation begins, the wind solution
includes different driving mechanisms. changes from the initial state due to the presence of the
In this paper, we restrict ourselves to mass loss rates magnetic field, the rotation of the star, and the poly-
aofPMa˙rwke<r-2li×ke1(0P−a9rMke⊙r y1r9−518.)Acosrjounsatlifiwedinadbdorviev,inwgemadeocpht- rtruonpiucnteiqlutahteiosnystoefmstraetleax(ePs in∝toρaγ)s.teaTdyh-estsaitmeu(lwatitiohnins
anism, modified by magnetic fields, stellar rotation, and a small tolerance) MHD wind solution. The code uses
an enhanced mass loss rate (relative to the sun). As nested computational grids so that the wind can be eas-
the nature (e.g., temperature) of TTS stellar winds is ily followed to large distances (several tens to hundreds
not well-known, our detailed solutions of coronal winds of R∗).
will enable us to look at the expected radiative prop- This method results in a steady-state solution for the
erties, a posteriori, allowing for further constraints on wind that is determined solely by the boundary condi-
real systems. We will show in a forthcoming paper (and tions held fixed at the base of the stellar corona (the
see Matt & Pudritz 2007a) that the expected emission “stellar surface”). In order to capture the appropri-
from the simulated winds presented here rules out ther- ate physics within the framework of a finite difference
mal pressure driving at a substantially lower mass loss scheme,weemployafour-layerboundaryforthestar,on
rate than the limit of Decampli (1981). which the various physical quantities are set as follows.
WeconsiderthesphericallocationR=30,inunitsofthe
3. NUMERICALSIMULATIONMETHOD grid spacing, to be the surface of the star. For all grid-
pointssuchthatR 34.5,the poloidalvelocityis forced
We calculate solutions of steady-state winds from iso-
≤
to be parallel with the poloidal magnetic field (v B ,
latedstars(no accretiondisk),using the finite-difference p p
k
where the poloidal component is defined as the vector
MHD code of Matt & Balick (2004), and the reader will
find further details there1 (and references therein). As- component in the r-z plane). Where R 33.5, ρ and
≤
P are held constant (in time) at their initial values. For
suming axisymmetry and using a cylindrical (r, φ, z)
R 32.5, v is held at zero, while v is held at coro-
coordinate system, the code employs a two-step Lax- p φ
≤
tation with the star. For R 31.5, B field is held at
Wendroff scheme (Richtmyer & Morton 1967) to solve p
≤
its initial, dipolar value, while B is set so that there is
the following time-dependent, ideal MHD equations: φ
no poloidal electric current at that layer (which gives it
∂ρ= ∇ (ρv), (4) a dependence on the conditions in the next outer layer,
∂t − · 31.5<R 32.5).
≤
∂(ρv) These boundary conditions properly capture the be-
= ρ(v ∇)v v[∇ (ρv)] havior of a wind acceleratedfrom the surface of a rotat-
∂t − · − ·
ing magnetized star, as follows. There is a layer on the
∇P GM∗ρ Rˆ + 1(J B), (5) stellarboundary(R>32.5)outsideofwhichthevelocity
− − (r2+z2) c × notfixed,butisallowedtovaryintime. Inthis way,the
∂e= ∇ [v(e+P)] GM∗ρ Rˆ v+J E(,6) wind speed and direction is not specified, but is deter-
∂t − · −(cid:20)(r2+z2) (cid:21)· · mined by the code in response to all of the forces. By
holding P fixed at its initial value for all R 33.5, we
∂B ≤
= c(∇ E), (7) constrainthepressuregradientforce(thermaldriving)at
∂t − × the base of the wind to be constant in time. Also, hold-
and uses ing the density fixed at R 33.5 allows the region from
≤
where the wind flows to be instantly replenished with
1
E= (v B), (8) plasma. Thus,thebaseofthewindmaintainsaconstant
−c × temperature and density, regardless of how fast or slow
c
J= (∇ B), (9) the wind flows away from that region. The existence of
4π × a layer in which v = 0 and B can evolve (namely, at
p p
1 P 31.5<R 32.5)allowsB (andv )toreachavaluethat
e= ρv2+ , (10) ≤ p p
2 γ 1 isself-consistentlydeterminedbythebalanceofmagnetic
− and inertial forces. We set the poloidal velocity parallel
whereρisthedensity,v thevelocity,P thegaspressure,
to the poloidalmagnetic fieldfor the nexttwoouter lay-
G Newton’s gravitationalconstant, M∗ the stellar mass, ers,toensureasmoothtransitionfromtheregionofpure
Rthesphericaldistancefromthecenterofthestar(R2 =
dipole field and zero velocity to a that with a perturbed
r2 +z2), e the internal energy density, B the magnetic
field and outflow. Setting B so that the poloidal elec-
field, J the volume current, E the electric field, c the φ
tric current is zero inside some radius ensures that the
speed of light, and γ the ratio of specific heats.
field behaves as if anchored in a rotating conductor (the
To obtain steady-state wind solutions, we follow the
surface of the star). Also, this ensures that B evolves
φ
method of Matt & Balick (2004), which is also simi-
appropriatelyoutsidetheanchoredlayeraccordingtothe
lar to that employed by Washimi & Shibata (1993) and
interaction with the wind plasma.
Keppens & Goedbloed (1999). It involves initializing
The key physical parameters can be represented by
the characteristic speeds of the input physics, namely
1 Matt&Balick (2004) ran cases with isotropic hydrodynamic
the soundspeedatthe base ofthe corona,c , the escape
variables at the base of the wind and also cases with enhanced s
polarwinds. Hereweonlyconsidertheisotropiccase. speedfromthesurfaceofthestar,vesc,therotationspeed
Stellar Wind Torques 5
ofthestar,andtheAlfv´enspeedatthebaseofthewind.
We specify the ratio of c /v as our parameter, rather TABLE 1
s esc
Fiducial StellarWind
thanthesoundspeedalone. Thisseemsthemostreason-
Parameters
able,sincethetemperatureofathermallydrivenwindis
regulated somewhat by the interplay between the ther- Parameter Value
mal energy input and the expansion of the corona (the
wind) against gravity. To first order, a hotter wind ex- M∗ 0.5M⊙
R∗ 2.0R⊙
pandsmorerapidlyagainstgravityallowinglesstime for B∗ (dipole) 200G
the gas to heat, and a cooler wind expands more slowly, f 0.1
allowingmoretimetoheat. Oncethe valueofthestellar M˙wa 1.9×10−9M⊙ yr−1
mass and radius is specified, the ratio of c /v deter- cs/vesc 0.222
s esc γ 1.05
minesthetemperatureheldfixedonthestellarboundary,
asdescribedabove. Thewindplasmaischaracterizedby a In order to treat M˙w as aparam-
a polytropic equation of state, and so γ is also a param- eter in the simulations, our method
eter. We parameterize the stellar rotation rate as the is to adjust the mass density at the
base of the wind until the desired
fraction of breakup speed,
M˙w isachieved inthesteady-state.
f Ω∗R∗3/2(GM∗)−1/2. (11)
≡
The Alfv´en speed is determined by the magnetic field
strength and coronal density. Rather than taking the
Alfv´en speed as a key parameter, we specify the field
1122
strength at the equator of the star (B∗) as our parame-
ter, in order to connect the simulations as much as pos-
sible to observationally constrained quantities. For the
same reason,we specify M˙w as a parameter,ratherthan 99
the coronal density. In the simulations, we must spec-
ify the base density, ρ∗, to be held fixed on the stellar ))**
boundary, and the value of M˙ in the steady-state wind RR
w ((
is not solely determined by ρ∗. For example, the rota- z z 66
tionofthestarcanenhanceM˙ viamagneto-centrifugal
w
flinging, and a strong magnetic field can decrease M˙
w
by inhibiting flow from a region near the equator that
33
remains magnetically closed(the “dead zone”). In other
words, M˙ is not an a priori tunable parameter; rather,
w
itisaresultofthesimulations. Therefore,totreatM˙ as
w
00
our tunable parameter, we adopt an iterative approach.
Thisentailsfirstrunningagivensimulationwithaguess 00 33 66 99 1122
for ρ∗, checking the resulting value of M˙w, and then ad- rr ((RR**))
justingρ∗ andrerunningthesimulation. Weiterateuntil
the desired value of M˙ is achieved (within a tolerance Fig. 1.—Fiducialcase: greyscaleoflogdensity,velocityvectors,
w
of 2%). This typically required 2 to 4 iterations, so the andmagneticfieldlinesillustratethestructureofthesteady-state
windsolution(seetable1). ThedashedlinerepresentstheAlfv´en
ability to treat M˙w as a chosen parameter comes at a surface,where thewindspeed equals the local Alfv´enspeed. The
substantial cost. rotationaxisisvertical,andthelongestvectorcorrespondsto160
kms−1. Blackcorrespondstoadensityabove5.3×10−13 gcm−3
4. STELLARWIND SOLUTIONS andwhitetoadensitybelow2.6×10−16 gcm−3.
4.1. The Fiducial Case
In a thermally-driven wind, the coronal sound speed
We start by presenting the results of our stellar wind should be comparable to the escape speed, and we use
simulation for parameters with values that represent a c /v = 0.222 as our fiducial value. This value gives
s esc
“typical” T Tauri star and follow the fiducial values of wind speeds that are appropriate in the solar case. The
Paper I and Matt & Pudritz (2005b). Table 1 lists the choice of polytropic index γ is also important. At large
fiducial parameters. We consider a low mass pre-main- distances ( AU) from the sun, the solar wind plasma
sequence star, with a surface escape speed of vesc 309 is well char∼acterized by an effective γ between approxi-
km s−1. A dipole magnetic field strength of 200 G≈auss mately 1.5 and 5/3 (Feldman et al. 1998; Krasnopolsky
is consistent with 3σ upper limits (Johns-Krull et al. 2000). However, in the region where the wind is ac-
1999; Smirnov et al. 2004, 2003b) or marginal detection celerated (within a few solar radii), thermal conduction
(Smirnov et al.2003a;Yang et al.2007)ofthelongitudi- and other heating and cooling effects play a role (e.g.,
nalmagneticfieldmeasuredforCTTSs. Weseekprimar- Cranmer et al.2007),resultinginaneffectiveγ closerto
ily to understand the slow rotators,for which a rotation unity (isothermal). Our fiducial value of γ = 1.05 was
rate of 10% of breakup is appropriate. In Paper I, we used by Washimi & Shibata (1993) and Matt & Balick
estimatedthatanaccretion-poweredstellarwindforaT (2004) for solar-like winds. This nearly isothermal value
Tau star might have M˙w 1.9 10−9M⊙ yr−1, so we of γ approximates the thermodynamics of a gas with a
≈ ×
use this as our fiducial value. true value of γ =5/3 that is heated as it expands.
6 Matt & Pudritz
TABLE 2
StellarWindTorques andLeverArm Lengths
1122
Case ρ∗ M˙w τw ˙rA2¸1/2
(10−11cmg3) (10−9Myr⊙) (1036 erg) (R∗)
fiducial 3.67 1.89 1.77 6.97
f =0.004 7.62 1.86 0.0972 8.33 99
f =0.2 1.36 1.87 2.82 6.26
f =0.05 6.01 1.88 1.06 7.65 ))
**
B∗ =400G 3.67 1.86 3.27 9.55 RR
1B∗kG=q2ukadG. 23..9627 11..8972 11.33.78 61.91.73 z (z ( 66
2kGquad. 4.38 1.93 2.11 7.53
lowM˙w 0.377 0.187 0.500 11.8
verylowM˙w 0.0755 0.0378 0.204 16.7
R∗ =1.5R⊙ 5.71 1.86 1.10 5.96 33
R∗ =3R⊙ 1.99 1.89 3.43 8.75
M∗ =0.25M⊙ 5.06 1.91 1.47 7.52
M∗ =1M⊙ 2.59 1.88 2.11 6.42
cs/vesc =0.245 0.773 1.87 1.59 6.64 00
cs/vesc =0.192 55.4 1.89 1.91 7.23
γ =1.10 11.1 1.87 2.19 7.79 00 33 66 99 1122
rr ((RR ))
**
Figure 1 shows the result of our fiducial case sim- Fig. 2.— Same as figure 1, but for the f = 0.004 (similar to
solar) case. The maximum velocity vector corresponds to 95 km
ulation, which illustrates the steady-state wind solu- s−1.
tion. The non-spherical shape of the Alfv´en sur-
4.2. Parameter Study
face (which eventually crosses the rotation axis at
larger radii than shown) is mainly due to magnetorota- To establish the dependence of rA on parameters and
tional effects in the wind (see Washimi & Shibata 1993; to calculate wind solutions that are applicable to a wide
Matt & Balick 2004). This demonstrates that the fidu- range of conditions that are observed or often assumed
cial T Tauri star wind exists in the thermo-centrifugal for T Tauri stars, we carried out a limited parameter
regimewherethermalandmagnetocentrifugaleffectsare study with our simulations. The results are listed in
of similar importance for accelerating the wind (Sakurai table2,andwebriefly discusseachcasebelow. The first
1985; Washimi & Shibata 1993). These winds are self- columnin the table lists the value of the parameter that
collimated, while still exhibiting substantial flow at all ischangedrelativetothe fiducialcase. Foreachcase,all
latitudes. other parameters are identical to the fiducial case. Note
From the simulation, we calculate M˙ and the total thatsince we considerM˙ asa keyparameter,the value
w w
angular momentum outflow rate, τw, as described by of ρ∗ varies from case to case.
Matt & Balick (2004). Then, using equation (3), we
calculate the effective lever arm length, r r2 1/2. 4.2.1. Spin Rate
These results are listed in the first row of tAab≡le 2(cid:10),Aw(cid:11)here As with the fiducialcase,allbut one simulationin our
wealsolistthecoronalbasedensityρ∗thatweiteratively parameter study lie in a regime that is near the bound-
chose to give the desired value of M˙ . ary between slow and fast magnetic rotators. The one
w
The wind base density of 10−11 gm cm−3 is 5 or- exception is a case with a fractional rotation rate equal
ders of magnitude larger tha∼n required for simple solar to the solar value of f = 0.004, which represents a slow
wind models (e.g., Washimi & Shibata 1993). This is magneticrotator. Figure2illustratesthestructureofthe
expected, since the fiducial M˙ is 5 orders of magnitude steady-state wind solution for this case in the same for-
w mat as the previous figure. A comparison between the
higherthanthesolarvalue,andthewindspeedsarecom-
two figures reveals that rotation indeed influences the
parable.
detailed structure of the velocity field and the magnetic
The fiducial stellar wind torque of 1.8 1036 ergs is
≈ × field in the wind, which manifests itself as a difference
capableofbalancingthespinuptorquefromaccretionat
arateof4.4 10−9M⊙ yr−1. Thebasicconclusionhereis in the shape of the Alfv´en surface. All else being equal,
× theeffectoffasterrotationistoreducetheeffectivelever
thatthe stellarwind torqueforthe fiducialcaseis ofthe
arm length, as evident in table 2 (see also Sakurai 1985;
right magnitude to be important for spinning down the
Washimi & Shibata 1993). Although the qualitative ef-
star, as required by the accretion-powered stellar wind
fectofrotationontheshapeoftheAlfv´ensurfacewasan-
scenario. We chose our fiducial parameters to compare
ticipated in analytic theory (e.g., Belcher & MacGregor
with the estimate of Paper I that2 rA/R∗ ≈ 12.2. We 1976), this effectis notproperlyincluded inanyexisting
can see that their estimate, based on scaling of 1D wind
analyticformulationforcalculatingthe torque. Thisisa
theory from solar values, was a 75% overestimate of r .
A primary reason that numerical simulations are required
We will identify the reasons for this in section 4.2.1.
to convincingly calculate the self-consistent wind solu-
tion, especially when considering winds that exist near
2 Paper I actually quotes a value of rA/R∗ = 15, but our defi- the boundary between slow and fast magnetic rotators.
nitiondiffersslightlyhere(compareeq.3herewitheq.2ofPaper
I),sotheleverarmlengthcorrespondsto12.2R∗ here. The effect of rotation on rA was not considered in the
Stellar Wind Torques 7
1122 1122
99 99
)) ))
** **
RR RR
(( ((
z z 66 z z 66
33 33
00 00
00 33 66 99 1122 00 33 66 99 1122
rr ((RR )) rr ((RR ))
** **
Fig. 3.— Same as figure 1, but for the B∗ =400 Gauss dipole Fig. 4.— Same as figure 1, but for the 2 kG quadrupole case.
case. Themaximumvelocityvector correspondsto190kms−1. Themaximumvelocityvectorcorresponds to170kms−1.
estimate of PaperI and accountsfor approximately30% ometries). Given the small number of measurements, it
of their overestimate of r . isstillrelevanttoconsiderstrongerdipolefieldstrengths.
A
As described in Matt & Balick (2004), a given sim- Thus we have run cases with B∗ = 400 and 2 kG.
ulation can be scaled to other systems with the same Figure3 illustrates the wind solutionforthe casewith
characteristic velocity ratios, so that the resulting value B∗ = 400 G, and the results of both cases are listed in
of rA/R∗ is valid for a family of solutions. The sim- table 2. It is clear that the strength of the field has a
ulation with f = 0.004 scales to a solution very sim- strong influence on the stellar wind torque.
ilar to the solar wind with R∗ = 1R⊙, M∗ = 1M⊙,
M˙w = 1.3 10−14M⊙ yr−1, B∗ = 1.5 G, and all speeds 4.2.3. Surface Field Geometry
are increas×ed by a factor of 2. Thus, for these parame- The fact that T Tauri stars have a mean field of
ters, this simulation predicts a lever arm length of 8.33 B∗ 2kGwithamuchweakerdipolecomponent,indi-
| |∼
R⊙ for the case of the solar wind, and τw = 6.8 1029 catesthatthestellarsurfacefieldisdominatedbyhigher
erg. This torque is consistent with the numerical×results order multipole fields. Therefore, it may be important
ofWashimi & Sakurai(1993),butafactorofafewtimes for future work to include much more structured fields
smallerthanobservationallydeterminedvalues(Li1999). thanweconsiderhere. Tobegintoquantifytheeffectsof
To obtain the observed solar torque, corresponding to higherorderfields,werantwocasesthatwereinitialized
rA =12.2R⊙, the simulation would require (e.g.) a sub- with a quadrupolar field of the form
stantiallystrongermagneticfieldthan1G.Thiswasalso
suggested by Li (1999), and our simulations corroborate BR=B∗(3cos2θ 1) R∗ 4
thatsuggestion. As the estimate inPaperI assumedthe − (cid:18) R (cid:19)
canonicalvalueofB∗ ≈1Gforthesun,thisaccountsfor R∗ 4
most of the discrepancy between our simulation results Bθ=2B∗cosθsinθ (12)
(cid:18) R (cid:19)
and the Paper I estimate of r .
A
TocapturearangeofspinsappropriatefortheTTauri in spherical coordinates.
star “slow rotators,”we also rancases with spin rates of We ran cases with B∗ equal to 1 kG and 2 kG, listed
twice and half of the fiducial spin rate. The results of as “1 kG quad.” and “2 kG quad.” respectively in ta-
these simulations are listed in the 3rd and 4th row of ble 2. Figure 4 illustrates the wind solution for the 2
table 2. kG quadrupole case. It is clear from the figure that,
compared to the cases with a dipole field, the shape of
4.2.2. Dipole Field Strength
Alfv´en surface is quite different. Also, for a given value
Measurements for the mean B∗ exist for a number of B∗, the effective lever arm length is much shorter for
| |
of T Tauri stars (e.g., Johns-Krull 2007b). These re- the quadrupole case.
sults show a remarkably consistent field strength for all It is evident from table 2 that the stellar wind torque
stars of around 2 kG. Measurements of the longitudi- fromastarwitha200Gdipolefieldiscomparabletothat
nal field (which limits the global, dipole component) ex- with a 1–2 kG quadrupole field. Thus, the stellar wind
ist only for a handful of accreting stars (Bouvier et al. torque is very sensitive to surface field geometry. How-
2007; Johns-Krull 2007a). These measurements suggest ever,sincethemeasuredsurfacefieldstrengthsof 2kG
∼
the dipole component is no greater than 200 G (though likely include contributions from even higher multipoles
larger values would be allowed for special viewing ge- than a quadrupole, it seems that the dipole component
8 Matt & Pudritz
will generally dominate near the Alfv´en surface. So the mechanism. In a Parker wind, the temperature (param-
strengthofthedipolecomponentshouldgenerallybethe eterized by c /v ) and the cooling/heating of the gas
s esc
most important for determining the torque. as it flows (parameterized by γ) are the key physical
properties determining the velocity profile in the wind.
4.2.4. Mass Loss Rate A hotter wind accelerates more rapidly and achieves a
higherspeed than a coolerwind. A windwith a largerγ
Wehavethusfarconsideredquitemassivewinds,moti-
(closer to 5/3) cools more rapidly as it expands, and so
vatedbyrecentsuggestionsintheliteratureforaccretion
thebulkoftheaccelerationtakesplaceclosertothestar.
poweredwinds. However,the value of M˙ is veryuncer-
w Similarly,ifthewindisinsteadacceleratedbysomething
tain and is likely to exhibit a wide range in values from
other than thermal pressure, the velocity profile may be
one object to the next. In addition, it would be inter-
altered.
esting to predict what torques may be expected for the
In order to quantify the effect of varying the accelera-
winds from the non-accreting, weak line TTSs. We ex-
tion in the wind, within the framework of the pressure-
pect these stars to have solar-like winds that are quite
driving mechanism used here, we have run three more
enhanced relative to their main sequence counterparts,
simulations. The results of these are listed in the last 3
yetprobablylesspowerfulthanwindsfromtheaccreting
rows of table 2. In order, these represent winds that are
stars.
hotter, colder, or with less heating (i.e., more adiabatic
Unfortunately, our method is limited to cases with
cooling)thanthefiducialcase. Therelativelylargeeffect
lever arms that are not too long, since longer lever arms
of c /v and γ on the wind speed near the stellar sur-
requires a larger Alv´en speed on the stellar surface. A s esc
face is evidentby the verydifferentvalues ofρ∗ required
large Alfv´en speed increases the time for the simulation
to keep M˙ fixed, listed in table 2. The wind velocity
to run and also increases the error in the solution (e.g., w
at the base of the corona, for fixed M˙ , varies as the
byincreasingtheeffectivediffusionrateinourcode). For w
thispracticalreason,wewerelimitedtorunningonlytwo inverse of the variation in ρ∗. So these cases represent
cases with lower M˙ covering a range in M˙ of a factor large differences in the wind acceleration rate. The ef-
w w fect on the torque is relatively small, but is not entirely
of 50. These are listed in the 9th and 10th rows of table
negligible. It will be important for future work to deter-
2.
mine the wind torques for different driving mechanisms.
Thepreliminaryconclusiontobedrawnfromthisworkis
4.2.5. Stellar Radius
thatthewindvelocityprofile,andthereforewinddriving
T Tauri stars contract as they age, so stars of a given mechanism, does not have a large effect on the torque.
mass exhibit a range of radii during this phase. Thus,
itisimportanttoconsiderheredifferentcombinationsof 5. SEMI-ANALYTICFITFORTHEEFFECTIVEALFVE´N
stellar mass and radius. Table 2 contains results from RADIUS
two cases with R∗ = 1.5R⊙ and R∗ = 3R⊙. Note that Insection2.1,wepointedoutthatnoreliableformula-
changing R∗ changes vesc, so these cases have a different tionexistsforpredictingtheAlfv´enradius(andtherefore
coronal temperature and Ω∗, in order that cs/vesc and torque) in a stellar wind from fundamental parameters.
f are constant. From the values in the table, it is evi- However our parameter study, even though somewhat
dent that the stellar wind torque is very sensitive to R∗. limited, can be used to provide a numerically based ap-
The reason for this is twofold. First, since B∗ is fixed, proach to this question. We will use the result future
a largerstellar radius correspondsto a largerdipole mo- work,anditalsowillbeofgeneralinterestforotherstel-
ment(µ B∗R∗3),whichiscapableofconveyingalarger lar wind studies.
≡
torque. Second, a largerstellar radius decreasesthe sur- In a 1-dimensional theory, one can assume that the
face gravity, and so the influence of the magnetic field magnetic field strength approximately follows a single
relative to gravity is increased (i.e., vA/vesc increases). powerlawoftheformB =B∗(R∗/R)n. Thenthecondi-
Thus, (rA/R∗)2 increases with R∗, and though Ω∗ de- tion that the wind speed equals the Alfv´en speed at rA
creases (to keep f fixed), the quantity Ω∗R∗2 increases, gives (e.g., Kawaler 1988; Tout & Pringle 1992)
so the net torque increases.
r 2n−2 B2R2
4.2.6. Stellar Mass (cid:18)RA∗(cid:19) = M˙w∗vr∗A, (13)
Cases with half and twice the fiducial stellar mass are
where v is the wind speed atthe Alfv´enradius. There
also listed in table 2. As with the cases of different R∗, are a nurmAber of problems. First, the true magnetic field
notethatchangesinM∗ changevesc,sowehaveadjusted strength in a wind does not follow a single power law
the coronal temperature and Ω∗ to keep the parameters (e.g., Mestel & Spruit 1987). Second, the Alfv´en surface
listed in table 1 fixed. As with the case of varying R∗, is neither a sphere nor a cylinder and a spherical model
a change in v changes the relative importance of the
esc is quite misleading. Third, and perhaps most vexing, is
magnetic field with the gravity. Thus, rA/R∗ is larger the fact that v has different values at different points
for a smaller M∗. However, since we have fixed f, a along the Alfvr´eAn surface and cannot be determined a
smaller M∗ means a smaller Ω∗ so that the net stellar priori. Finally, there is no explicit dependence of r on
wind torque decreases. A
the spinrateor drivingpropertiesofthe wind, whichwe
also know to be false.
4.2.7. Wind Acceleration
There is, however, a more general way of scaling the
The increase of the wind speed with distance from Alfv´enradiusthatissuggestedbybasictheory. Another
the star depends on the details of the wind acceleration clue is that since the numerical simulations are carried
Stellar Wind Torques 9
andc asourfixedparameters,therewouldbealotmore
s
scatter of the diamonds aroundthe line in figure 5. Fur-
20 thermore, our simulations self-consistently capture the
Fiducial interactionbetween the stellar wind, magnetic field, and
Spin, f rotation, without resorting to assumptions about (e.g.)
Quadrupole the extent ofthe dead zone,the dependence ofmagnetic
Wind Accel. field strength with radius, or latitudinal variations in
Other wind quantities. Therefore, our semi-analytic formula-
tion appears to be an improvement over existing theory.
*
R Equation(14)doeshavesomelimitations. Neither the
/ A 10 previousanalyticformulationsnorourownsemi-analytic
r approach properly includes the effects of varying stellar
rotation or wind driving (as evident in figure 5). As an
illustrativeexample,inasmallerparameterstudywitha
rotationratecomparabletothesolarrate(notpresented
here), we found that K 3.0 and m 0.19. Also, note
6 that a line connecting th≈e two points≈with a quadrupole
field suggests K 1.7 and m 0.15, for these cases.
≈ ≈
Thus, the indicies K and m are quite sensitive to the
100 1000 10000 field geometry and have a smaller (but non-negligible)
B 2 R 2 / ( M • v ) sensitivity to changes in the stellar spin rate and the
* * w esc
wind accelerationrate/mechanism. We leave the precise
determinationofthesensitivityofr totheseparameters
Fig. 5.—Effectiveleverarmlengthinthestellarwindversusthe A
for future work.
quantityinbracketsinequation(14). Shownaretheresultsofour
entire parameter study including the fiducial case (filled circle); We can now combine equations (3) and (14) to get a
cases with different spin rates (pluses); cases with a quadrupole formula for the stellar wind torque,
fialelldoth(terriacnagseless()d;icaamseosndws)it,hredpirffeesreennttincgst/hvoessceworitγhd(isffqeuraernets)v;alauneds τ = K2 f v1−2m M˙ 1−2m R1+4m B4m, (15)
of B∗, R∗, M˙w, or M∗. The line represents the best fit to the w √2 esc w ∗ ∗
fiducial and “other” cases, given by equation (14) with K ≈2.11
andm≈0.223 though we know this does not properly contain the de-
pendence (e.g.) on f. This equation is essentially the
out in normalized units, they are scalable to any system
same as that derived by Kawaler (1988), except for the
withwithsimilarcharacteristicspeedsonthestellarsur-
value of the the dimensionless constant out front and of
face (e.g., Matt & Balick 2004). This suggests that we
the expected value of the exponent parameter m. The
canreplacev withthestellarsurfaceescapespeedand
rA constant is not so crucial, and usually this can be cali-
calculate the Alfv´en radius using
brated to the solar wind torque for a predictive theory
r B2R2 m (thoughwehavenotdonethishere). Ontheotherhand,
A =K ∗ ∗ , (14) thevalue ofm isoffargreaterimportanceforpredicting
R∗ (cid:18)M˙wvesc(cid:19) the torque for a range of parameters.
where K and m are dimensionless constants. The In particular, authors typically have chosen a power
quantity inside the bracket measures the effective mag- law such that the stellar wind torque is nearly or
netization of the wind, is similar to that used by completely independent of M˙ (effectively, m =
w
ud-Doula & Owocki (2002, see their eq. 7), and arises 0.5), which results in τ B2 (e.g., Kawaler
w ∗
∝
naturally in disk wind theory (e.g., equation 2.27 of 1988; Pinsonneault et al. 1989; Barnes & Sofia 1996;
Pelletier & Pudritz 1992). It is also a quantity that can Bouvier et al.1997). Ourbasicunderstandingofthe ob-
be fixed by observations of stellar properties and wind served Skumanich-style (Skumanich 1972) spin down of
parameters. mainsequencestars(Ω∗ t−1/2),aswellastheexpected
Infigure5,weplotrA/R∗ asafunctionofthequantity dependence ofmagnetic∝fieldstrengthwith rotationrate
in brackets in equation (14) on a log–log scale, for all 17 (Belcher & MacGregor 1976), appears to rely on this
ofoursimulations. Sinceweknowequation(14)doesnot or a similar formulation. Our fit value of m 0.223
properlyincludetheeffectsofstellarrotationorthewind gives approximately τ B0.9 (this was also fo≈und by
w ∗
drivingmechanism,wecalculatethe bestfitK andmto Washimi & Shibata 1993∝), which is substantially differ-
the fiducial case and only those cases with variations on ent.
B∗, R∗, M˙w, and M∗. The fit, giving K 2.11 and Inordertounderstandthedifferencebetweenourvalue
≈
m 0.223, is plotted as a line in the figure. ofthe exponentandthatusedby others,it isinstructive
≈
Itisremarkablethatthisfitmatchesalloftherelevant to consider the power law formulation of the magnetic
simulations(filledcircleanddiamondsinthefigure)toan fieldusedtoderiveequation(13). Itisgenerallyexpected
accuracy of less than one percent. This is at the level of (e.g., Mestel 1984; Mestel & Spruit 1987; Kawaler 1988)
precisionofthenumericalmethod(Matt & Balick2004). that when the surface magnetic field is dipolar, as we
Remember that we have taken c /v and f as our pa- areconsideringhere,the effectivepowerlawindexofthe
s esc
rameters,sothatcaseswithadifferentvalueofv (i.e., magneticfieldintheflowwillliesomewherebetweenthe
esc
thosewithdifferentR∗ andM∗)actuallyalsohavediffer- value for a dipole (n =3) and that for a split monopole
ent wind temperatures (i.e., c ) and stellar angular spin (n = 2). This has been the primary justification for the
s
rates (Ω∗) than the fiducial case. If we had chosen Ω∗ powerlawsusedintheliterature. Indeedoursimulations
10 Matt & Pudritz
display the expected behaviorof exhibiting a dipolar ge- simulations;e.g.,Gardiner et al.2003;Shang et al.
ometry near the star and an approximately monopolar 2006).
geometry far from the star (e.g., figure 1). However, by 3. The cases with quadrupole fields resulted in a
comparing equations (13) and (14), we see that our fit torquethatismuchweakerthancaseswithadipole
valueofm 0.223seemstoimplyamagneticpowerlaw field of the same surface field strength. Specifi-
≈
of n 3.2. cally, we find that a 200 G dipole field exerts the
≈
It is important to realize that the divergence of the same stellar wind torque upon a star as a 1–2 kG
magnetic field in the flow, captured by the power law quadrupole. This illustrates the very strong effect
index n, is not the only important effect, and this is of magnetic geometry on the stellar wind torque.
whytheformulationofequation(13)ismisleading. Here
are two reasons. First, using 1-dimensional reasoning, 4. We ran cases where the mass loss rate and other
in an accelerating wind, the behavior of vrA mitigates parameters were fixed, but the thermal wind driv-
the response of rA to the parameters. For example, for ing parameterswere varied. For largevariations in
an increase in B∗, the Alfv´en radius will become larger, the wind acceleration, the torque changed by less
but since the flow is accelerating, vrA will also increase. than a factor of 2. This suggests that the details
Kawaler(1988)made the approximationthatvrA equals of the velocity profile are not of fundamental im-
the escape speed at rA. In this case, vrA decreases with portance,andoursolutionsshouldbe areasonable
radius,givingtheoppositeeffectofanacceleratingwind. approximation for winds with other wind driving
Similarly,theapproximationsofMestel(1984)thatvrAis mechanisms. However, it will still be important
constantfor a slow rotatorandproportionalto Ω∗rA for for future work to compare our torque results to
a fast rotator do not well-approximate the acceleration stellar wind solutions that use alternative driving
exhibited in the winds we simulated. The second reason mechanisms.
forthesurprisinglyweakdependenceofr onparameters
A
is in the amount of open magnetic flux that participates
5. Our determination of the torque allowedus to cal-
inthe flow,whichagainis notincluded inthe derivation
culatethe Alfv´enradius(viaeq.3), whichisafun-
of equation (13), and which again mitigates the effect of
damental quantity in MHD wind theory. We com-
parameters on rA. For example, for an increase in B∗, pared our numerical solutions to previous analytic
a smaller area on the stellar surface will have open flux
work and obtained a semi-analytic formulation for
(e.g., compare figures 1 and 3, and see Mestel & Spruit
1987),sorA willnotincreaseasmuchasexpectedinthe 1rA4)/,Rt∗ha∝t [wBe∗2lRl-d∗2e/s(cMr˙iwbvesescm)]amn,ywoitfhoumr ≈sim0u.2la2ti(oenqs.
magnetic power law formulation.
with dipole fields. This formulation appears to be
Infuturework,itwillbeimportanttoextendequation
an improvement over existing work, and the ex-
(14) to include the effects of rotation, etc. Furthermore,
ponent m is significantly smaller than usually as-
much work is needed to explore the full consequences
sumed.
(e.g.,formainsequencestars)ofthesignificantlysmaller
exponent we find, compared to many previous works.
We will continue to develop the theory of accretion-
6. SUMMARYANDCONCLUSIONS poweredstellarwindsinforthcomingwork. Inacompan-
Using 2D (axisymmetric) MHD simulations, we com- ionpaper(thethirdinourseries),wecomparethestellar
puted steady-state, stellar wind solutions for a parame- wind torques computed here to the torques expected to
terrangeappropriateforTTauristars. Wecarriedouta arise from the interaction between the star and an ac-
parameterstudy including variationsofthe stellarmass, cretion disk. We find spin-equilibrium (net zero torque)
radius,surfacemagneticfieldstrength,androtationrate, solutions and test the suggestion of Paper I. In a later
aswellasmasslossrate,windaccelerationrate,andtwo paper,we will use the stellar wind solutions of this work
different magnetic geometries (dipole and quadrupole). to compute emission properties of TTS coronal winds.
Our solutions enabled us to determine the angular mo-
mentumcarriedinthewind,anditsdependenceonmany
Initial results of various aspects of this workwere pre-
of the parameters of the system. Our main conclusions
sented at a number of meetings, and we wish to thank
can be summarized as follows:
many participants who provided interesting questions,
1. For fiducial parameters, the torque is of the same discussion, and ideas including: Gibor Basri, Sylvie
order ( 1036 erg) as estimated in Paper I. There- Cabrit,SteveCranmer,AndreaDupree,SuzanEdwards,
fore, if∼the stellar winds of TTSs have similar pa- ChristianFendt,WillFischer,Shu-ichiroInutsuka,Chris
rameterstothoseconsideredhere,theyshouldhave Johns-Krull,MarinaRomanova,FrankShu,KeivanStas-
a significant influence on the stellar spin. sun, Asif ud-Doula, Jeff Valenti, and others. We also
thank the referee, Ruben Krasnopolsky, for his useful
2. The stellar winds are in the regime of moderately suggestionsforimprovingthepaper. SMissupportedby
fastmagneticrotatorwinds. Theyproducejets, as theUniversityofVirginiathroughaLevinson/VITAFel-
well as a wide-angle flow (see, e.g., Matt & Balick lowship partially funded by The Frank Levinson Family
2004), which should interact with, and be modi- Foundation through the Peninsula Community Founda-
fied by, surrounding material (not included in our tion. REP is supported by a grant from NSERC.
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