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Mon. Not. R. Astron. Soc. 356, 167–182 (2005) doi:10.1111/j.1365-2966.2004.08431.x
The spin of accreting stars: dependence on magnetic coupling to the disc
Sean Matt?† and Ralph E. Pudritz†
Physics & Astronomy Department, McMaster University, Hamilton ON, Canada L8S 4M1
Accepted 2004 September 22. Received 2004 August 10; in original form 2004 May 30
ABSTRACT
We formulate a general, steady-state model for the torque on a magnetized star from a sur-
rounding accretion disc. For the first time, we include the opening of dipolar magnetic-field
lines due to the differential rotation between the star and disc, so the magnetic topology then
depends on the strength of the magnetic coupling to the disc. This coupling is determined by
theeffectivesliprateofmagnetic-fieldlinesthatpenetratethediffusivedisc.Strongercoupling
(i.e. lower slip rate) leads to a more open topology and thus to a weaker magnetic torque on the
star from the disc. In the expected strong coupling regime, we find that the spin-down torque
on the star is more than an order of magnitude smaller than calculated by previous models.
We also use our general approach to examine the equilibrium (‘disc-locked’) state, in which
the net torque on the star is zero. In this state, we show that the stellar spin rate is roughly
an order of magnitude faster than predicted by previous models. This challenges the idea that
slowly-rotating,accretingprotostarsaredisclocked.Furthermore,whenthefieldissufficiently
open (e.g. for mass accretion rates ?5 × 10−9M?yr−1, for typical accreting protostars), the
star will receive no magnetic spin-down torque from the disc at all. We therefore conclude that
protostars must experience a spin-down torque from a source that has not yet been considered
in the star–disc torque models – possibly from a stellar wind along the open field lines.
Key words: accretion, accretion discs – MHD – stars: formation – stars: magnetic fields –
stars: pre-main-sequence – stars: rotation.
1 INTRODUCTION
Accretion discs are responsible for some of the most energetic and
spectacular phenomena in many classes of astrophysical objects,
including protostars, white dwarfs (cataclysmic variables and in-
termediate polars), neutron stars (binary X-ray pulsars), and black
holes (both stellar mass X-ray transients and supermassive active
galactic nuclei). Gravitational potential energy liberated by the ac-
cretionprocessgivesrisetoexceptionalluminosityexcessesandcan
drive powerful jets and outflows. Accretion on to the central object
can occur only as quickly as angular momentum can be transported
away from the system. Furthermore, the accretion of disc material,
which has high specific angular momentum, spins up the central
object, if the object rotates at less than the break-up rate. It is there-
fore surprising that the central objects (hereinafter ‘stars’) are often
observedtospinfarbelowtheirbreak-uprates,inspiteoflong-lived
accretion. Why does this happen?
There is good evidence that accretion on to magnetized stars oc-
curs along closed magnetospheric field lines that connect the star
?CITA National Fellow.
†E-mail: matt@physics.mcmaster.ca (SM); pudritz@physics.mcmaster.ca
(REP)
to the inner edge of the disc. Theoretical models of this sort have
beensuccessfulinexplainingnumerousobservedfeaturesinaccret-
ing protostars (e.g. Hayashi, Shibata & Matsumoto 1996; Goodson,
B¨ ohm & Winglee 1999; Muzerolle, Calvet & Hartmann 2001), in-
termediate polars (e.g. Patterson 1994), and X-ray pulsars (e.g. Joss
& Rappaport 1984; Aly & Kuijpers 1990; Kato et al. 2001; Kato,
Hayashi & Matsumoto 2004). In some cases, there is even direct
evidence that the stars are magnetized, namely for accreting proto-
stars (Johns-Krull et al. 1999), intermediate polars (Piirola, Hakala
& Coyne 1993), and X-ray pulsars (Makishima et al. 1999).
Magnetic fields can also be effective at transferring angular mo-
mentum away from the star, possibly explaining the observed rota-
tionrates.Torquesonthestarthatareexertedbymagnetic-fieldlines
anchoredtothestarandthatarealsoconnectedtothedischavebeen
calculated by several authors (e.g. Ghosh & Lamb 1979; Cameron
&Campbell1993;Lovelace,Romanova&Bisnovatyi-Kogan1995;
Wang 1995; Yi 1995; Armitage & Clarke 1996, hereinafter AC96;
Rappaport, Fregeau & Spruit 2004). Under certain circumstances,
this torque can counteract the angular momentum deposited by ac-
cretion, leading to a net spin-down of the star (possibly explain-
ing spin-down episodes observed in X-ray pulsars; Ghosh & Lamb
1978; Lovelace et al. 1995) or giving rise to an equilibrium state, in
whichthenettorqueonthestariszero,possiblyexplainingtheslow
spin of some accreting protostars (K¨ onigl 1991, hereinafter K91).
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S. Matt and R. E. Pudritz
In this equilibrium state, the spin rate of the central object depends
on the accretion rate in the disc, and so a system is then considered
to be ‘disc locked’. Because these models for the magnetic star–
disc interaction show that accreting stars can spin more slowly than
the break-up rate, there is a general perception that the presence of
an accretion disc in any system leads to slow rotation rates. This
idea of disc locking has been applied to a variety of problems. As
an example, in systems where the moment of inertia of the star is
changing (e.g. during contraction), some authors have assumed that
disc-locking keeps the star at a constant spin rate (e.g. as applied
to protostars by Bouvier, Forestini & Allain 1997; Sills, Pinson-
neault&Terndrup2000;Barnes,Sofia&Pinsonneault2001;Tinker,
Pinsonneault & Terndrup 2002; Rebull, Wolff & Strom 2004; and
suggested for stellar collision products by Leonard & Livio 1995;
Sills et al. 2001; De et al. 2004).
There is a nagging problem with this physical picture, however,
because the magnetic torque calculations discussed above (with the
exception of Lovelace et al. 1995) assume that the stellar magnetic
field remains largely closed and that field lines connect to a large
portionofthedisc.1Thisassumptionisquestionablebecauseclosed
magnetic structures tend to open when enough energy is added to
them, and these systems possess a natural source of energy in the
form of gravitational potential energy that is released during disc
accretion. This energy release can drive outflows and twist field
lines, thereby adding energy to the magnetic field. Thus, the gen-
eral surplus of energy in accreting systems suggests that associated
magnetic fields should be dominated by open, rather than closed,
topologies. How are low spin rates achieved in this case?
In this paper, we generalize the star–disc interaction model to
include the effect of varying field topology (i.e. connectedness). We
consider the mechanical energy that is added to the field via dif-
ferential rotation between the star and disc as the only mechanism
responsible for opening the field (though our formulation is easily
adaptable for other mechanisms). The time-dependent behaviour of
a dipole stellar field attached to a rotating, conducting disc has been
studied,usingananalyticapproach,byseveralauthors(e.g.Lynden-
Bell&Boily1994;Agapitou&Papaloizou2000;Uzdensky,K¨ onigl
&Litwin2002a,hereinafterUKL).Theyhaveshownthat,asthedif-
ferential twist angle between the star and disc (??) monotonically
increases, the torque exerted by field lines first reaches a maximum
value,thendecreases.Thisoccursbecauseazimuthaltwistingofthe
dipolefieldlinesgeneratesanazimuthalcomponenttothefield,and
the magnetic pressure associated with this component acts to inflate
the field, which then balloons outward at an angle of ∼60◦from the
rotation axis, causally disconnecting the star and disc (see also Aly
1985; Aly & Kuijpers 1990; Newman, Newman & Lovelace 1992;
Lovelace et al. 1995; Bardou & Heyvaerts 1996). Typically, this
inflation or opening of the field occurs when a critical differential
rotationangleof??c≈πhasbeenreached,andtheamountofflux
that opens depends on the strength of the magnetic coupling of the
field to the disc (UKL). This analytic work on the field opening has
been corroborated by time-dependent, numerical magnetohydrody-
namic simulations of the stellar dipole–disc interaction (Hayashi
1Note that the X-wind model of Shu et al. (1994, and subsequent works) is
unique in star–disc interaction theory in the literature, as it assumes that a
systemwillalwaysaccreteverynearitsdisc-lockedstate.Themagnetic-field
geometry employed by the X-wind model is also unique and was designed,
in part, to avoid the problem of field opening due to differential twisting, as
considered in this paper. Therefore, much of our discussion does not apply
to the X-wind.
etal.1996;Goodson,Winglee&B¨ ohm1997;Miller&Stone1997;
Goodson et al. 1999; Kato et al. 2001; Matt et al. 2002; Romanova
et al. 2002; K¨ uker, Henning & R¨ udiger 2003; Kato et al. 2004).
Our primary goal in this paper is to determine the effect of the
topologyofthemagneticfieldonthetorquesinthesteady-state,star–
disc interaction model. In a previous paper (Matt & Pudritz 2004),
we gave a brief outline of this theory and showed that a more open
(i.e. less connected) field topology results in a spin-down torque
on the star that is less than for the closed-field assumption. Con-
sequently, the equilibrium state (with a net zero torque) features a
fasterspinthanpredictedbypreviousmodels,whichcallsintoques-
tion the general belief that accretion discs necessarily lead to slow
rotation. The present paper contains a more detailed presentation of
the theory and our assumptions, and we consider all possible spin
states of the system (not just the equilibrium state). We also extend
our analysis to show that there are at least three different modes
in which a magnetic star–disc system can operate. Our analysis is
applicable to all classes of magnetized objects that accrete from
Keplerian discs. However, because an abundance of observational
dataexistsforaccretingprotostars,inparticularforclassicalTTauri
stars (CTTSs), we adopt a set of fiducial parameters that are appro-
priate for these systems and discuss various aspects of the model in
this context.
Section2containsaformulationofthegeneralmodel.Thespecial
case of a disc-locked system is the topic of Section 3. The final
Section (Section 4) contains a summary of our results and includes
a list of problems with using the disc-locking scenario to explain
CTTS spins, plus a discussion of three possible configurations of
the general system.
2 STAR–DISC INTERACTION MODEL
Magnetic, star–disc interaction models in the literature differ in
their various assumptions, adopted parameter values, and in the
introduction of ‘fudge factors,’ but they are quite similar on the
whole (for a review, see Uzdensky 2004). We formulate a general
modelthatbuildsuponthispreviouswork(mostlyfollowingAC96),
by including the effect of varying magnetic-field topology, via the
introduction of the physical parameters β and γc(defined below).
According to the usual model assumptions, a rotating star is sur-
roundedbyathin,Keplerianaccretiondisc.Theangularmomentum
vector of the disc is aligned with that of the star, which rotates as
a solid body and at a rate that is some fraction of break-up speed,
defined by
?
where ?∗, R∗and M∗are the angular rotation rate, radius and mass
of the star, respectively. Note that f is always within the range zero
to one.2The disc rotates at an angular rate different from that of the
star at all radii, except at the singular corotation radius given by
f ≡ ?∗
R3
∗
GM∗,
(1)
Rco= f−2/3R∗.
For r < Rco, the disc rotates faster than the star, while for r > Rco,
the angular rotation rate of the star is greater than that of the disc.
(2)
2Disc accretion solutions do exist for f > 1 (Paczynski 1991; Popham &
Narayan 1991), in which the star is actually spun down by accretion toward
f =1,evenwithoutanymagnetictorques.However,weonlyconsidercases
with f ? 1 in this paper, because this characterizes the spin of observed
protostellar systems.
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The disc is assumed to be in a steady-state wherein the mass ac-
cretion rate ˙Mais constant in time and at all radii. In a real disc
from which winds are launched,˙Mamay have a weak radial depen-
dence, but we assume this has a negligible effect on the model. A
rotation-axis-aligned dipole magnetic field, anchored into the stel-
lar surface, also connects to the disc. The field is strong enough to
truncate the disc at some inner location Rtfrom where disc material
is subsequently channelled along magnetic-field lines as it accretes
on to the star. In general, the disc may have its own magnetic field
(either generated in a disc dynamo or carried in by the disc from
larger scales). We do not specifically include this field in the model,
though it may be responsible for angular momentum transport in
the disc (providing˙Ma) and may also aid in the connection of the
stellar field to the disc. Within the disc, the kinetic energy of the gas
is much greater than the magnetic energy of the stellar field, but the
region above the star and disc (the corona) is filled with low-density
material, and so the corona is magnetically dominated.
Inthisconfiguration,themagneticfieldconnectsthestaranddisc
by conveying torques between the two. Torques are conveyed on an
Alfv´ enwavecrossingtime,whichismuchshorterthantheKeplerian
orbital time. Everywhere that the magnetic field connects the star to
the disc, except at Rco, the magnetic field is twisted azimuthally by
differential rotation between the two. Inside Rcothe field is twisted
such that field lines ‘lead’ the stellar rotation, so torques from field
lines threading the region r < Rcoact to spin up the star (and spin
down the disc). Conversely, torques from field lines threading r >
Rcoact to spin the star down (and spin the disc up). The accretion
of disc matter on to the star also deposits angular momentum on to
the star.
In order for disc material to accrete, Rtmust be less than Rco
so that accreting material loses angular momentum to the star as
it falls inward. In order for the star with f ? 1 to feel any spin-
downtorquesfromthedisc,thestellarfieldmustconnecttothedisc
beyond Rco.Underthiscondition,thefieldthatconnectsoutside Rco
transfers angular momentum from the star to the disc. To maintain a
steadyaccretionrate,thediscmustthentransportthisexcessangular
momentum outward, resulting in an altered disc structure (Sunyaev
& Shakura 1977a,b; Spruit & Taam 1993; Rappaport et al. 2004).
We will define Routas the outermost radial extent of the magnetic
connection, and the usual assumption is that Rout? Rco. Fig. 1
illustrates the basic picture, and shows the locations of Rt, Rcoand
Routfor a possible configuration of the star–disc interaction model.
The assumption of a dipole field refers to the poloidal component
of the field, Bp= (B2
r- and z-components of the magnetic field, and the closed field only
exists in the region interior to Rout. The dipole field is used for
simplicity and because it has the weakest radial dependence (Bz∝
r+ B2
z)1/2, where Brand Bzare the cylindrical
Rco Routt
R
Disc Star
Figure 1. Magnetic star–disc interaction. The stellar field connects to a
region of the disc, from Rtto Rout, reaching beyond Rcoin this case. The
stellar field dominates the accretion flow on to the star (arrow).
r−3along the equator) of any natural magnetic multipole. In reality,
the twisting of dipole field lines alters the poloidal field, but this
perturbation should be slight in the region where the magnetic field
remainsclosed(asjustifiedbytheworkcitedinSection1,e.g.UKL,
and see also Livio & Pringle 1992).
For discussion throughout this paper, it is often instructive to use
physical units, especially for comparison with observations. For
this purpose, we adopt a set of observationally determined ‘fiducial
parameters’ that are appropriate for CTTSs (e.g. see Johns-Krull &
Gafford 2002):
˙Ma= 5 × 10−8M?yr−1,
M∗= 0.5M?,
R∗ = 2R?,
B∗ = 2 × 103G,
where B∗is the stellar magnetic-field strength at the equator.3How-
ever, our formulation of the problem is applicable to any magnetic
star–disc system.
2.1 Twisting and slipping of magnetic-field lines
The torque exerted by magnetic-field lines threading an annulus of
a disc of radial width dr is given by (e.g. AC96)
dτm= −γµ2
where
r4dr,
(3)
γ ≡ Bφ/Bz.
Here, µ is the dipole moment and Bφis the azimuthal component
of the magnetic field. The radial component, Br, is assumed to be
negligible within the disc, the torque has been vertically integrated
throughthedisc,and Bφreferstothevalueatthediscsurface.Here,
and throughout this paper, we choose the sign of the torque to be
relative to the star such that a positive torque spins the star up, and
consequently spins the disc down (and vice versa for a negative
torque).
Thedifferentialmagnetictorqueofequation(3)dependsnotonly
on µ, but also γ, which is the ‘twist’,4or pitch angle, of the field.
The total (integrated) magnetic torque will also depend on the size
and radial location of the magnetically connected region in the disc.
Whileµisaconstantparameterofthesystem,theradialdependence
of γ depends on the physical coupling of the magnetic field to the
disc.
In general, the coupling is not perfect. Magnetic forces act to
resist the twisting of the field, and so the field will ‘slip’ backward
throughthediscatsomeratevdproportionaltoγ.Theexactslipping
rate depends upon which physical mechanism is at work. In the
literature, there are generally three mechanisms discussed (e.g. see
Wang1995):(1)magneticreconnectioninthedisc,(2)reconnection
outside the disc, and (3) turbulent diffusion of the magnetic field
through the disc. We adopt the latter mechanism, but we further
discuss the other two, below.
The magnetic field slips azimuthally at a speed (e.g. Lovelace
et al. 1995; UKL)
vd=ηt
(4)
hγ = βvkγ,
(5)
3These fiducial parameters are slightly different from those used for figs
2 and 3 of Matt & Pudritz (2004), who considered the specific case of the
CTTS BP Tau.
4This should not be confused with the ‘twist angle’ of the footpoints of the
field, ??, as discussed by UKL, though ?? and γ are intimately related.
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S. Matt and R. E. Pudritz
whereηtistheturbulentmagneticdiffusivity,histhelocaldiscscale
height,vkistheKeplerianorbitalspeed,andwehaveintroducedthe
dimensionless ‘diffusion parameter’ β ≡ ηt(hvk)−1. The variable β
simply parametrizes the coupling of the stellar magnetic field to the
disc such that β ? 1 corresponds to weak coupling, and β ? 1 to
strong coupling.
Generally, β is a scale factor that compares vdwith vk, and we
have chosen this generic formulation so that the system behaviour
is largely independent of any particular disc model – as long as the
disc obeys Keplerian rotation and provides a steady accretion rate.
However, if we temporarily consider a standard α-disc (Shakura &
Sunyaev 1973), we may rewrite our diffusivity parameter β in a
more physically revealing way:
β =α
whereαhasitsusualmeaningandPtistheturbulentPrandtlnumber,
equal to the turbulent viscosity divided by ηt. The disc turbulence
is likely to be driven by the magneto-rotational instability (MRI;
Balbus & Hawley 1991), which follows the general behaviour of an
α-disc. Because both α and h/r typically have weak radial depen-
dences,andthevalueofPtisunknown,weassumethatβ isconstant
in the region of the disc connected to the stellar field.
The value of β is not well constrained (AC96 used β = 1),
but extreme α-disc parameters give an upper limit of β ? 1. For
a more reasonable estimate, note that a thin disc usually means
h/r ? 0.1, and α is typically in the range 0.001 to 0.1 (Sano et al.
2004). So, assuming Ptis of order unity, β ? 10−2. We get a similar
estimate using equation (5) and reasonable guesses for CTTS disc
parameters:
?
However, given the uncertainties and possible variation among dif-
ferent systems, and to assess the effect of the coupling of the field
to the disc, we retain β as a free parameter.
In the disc-connected region, if vdis anywhere less (greater) than
thelocaldifferentialrotationspeedbetweenthestaranddisc,γ will
increase(decrease)onanorbitaltime-scale.Thusthemagneticfield
will quickly achieve a steady-state configuration in which vdequals
the local differential rotation rate (e.g. UKL), which gives
γ = β−1?(r/Rco)3/2− 1?.
ThesolidlineinFig.2showsthequantityβγ asafunctionofradius
(normalizedto Rco)alongthesurfaceofthedisc.Themagnetictwist
is zero at Rco, and is oppositely directed on either side of Rco. Also,
atagivenradius,thetwistwillbelargerforsmallervaluesofβ (and
vice versa).
We have assumed that the field coupling is determined by turbu-
lent diffusion and, when β is constant, we find that γ ∝r1.5(equa-
tion 8, for r ? Rco). Other coupling mechanisms (as discussed
above) result in a different radial dependence of γ. For example,
Wang (1995) showed that if the twist is limited by reconnection in
the disc, γ ∝ r1.6(for r ? Rco), while for reconnection outside the
disc, γ approaches a constant value (for r ? Rco). On the other
hand, Livio & Pringle (1992) and AC96, assumed γ was limited by
reconnection in the stellar corona, and they used the same formula-
tion as equation (8) (with β = 1). In any case, note that the radial
dependence of the differential magnetic torque is dominated by the
fall-off of the dipole magnetic field (which results in ther−4depen-
dence of equation 3). Therefore, the choice of magnetic coupling
mechansim will not much influence our results (AC96). Similarly,
Pt
h
r,
(6)
β ≈ 10−2
ηt
1016cm2s−1
??
h
R?
?−1?
vk
100kms−1
?−1
.
(7)
(8)
0.00.5 1.0
r / Rco
1.5 2.0
-1.0
-0.5
0.0
0.5
1.0
1.5
βγ
Figure 2. The magnetic twist (γ ≡ Bφ/Bz) times the diffusion parameter
β, as a function of radius along the surface of the disc (thick, solid line).
The dotted lines indicate critical radii for a field-opening twist of γc =
± 0.5/β (chosen for illustrative purposes). The locations of Rin, Rcoand
Routare where γ equals −γc, 0 and γc, respectively.
a small radial dependence of β (which we take as constant) will not
introduce a large error.
2.1.1 Maximum twist for dipole field
As discussed in Section 1, several authors have shown that dipole
field lines will transition from a closed to an open topology when a
criticaldifferentialrotationangleof??c≈πhasbeenreached.This
corresponds to a critical field twist of γc≈ 1. Because the twisting
of field lines does not significantly alter the poloidal configuration
of the field lines that remain closed, and as an approximation we
will assume that the opening of field lines is only important for
the determination of the size of the connected region (i.e. to deter-
mine Rout), we include the effect of field line opening in the steady-
state torque theory in the following manner: we will use equation
(8) only where γ < γcand assume that the field will be open ev-
erywhere else (a similar approach was used by Lovelace et al. 1995
and justified by the work of UKL). In other words, wherever equa-
tion (8) predicts γ ? γc, the magnetic connection is assumed to be
severed, so the star and disc are causally diconnected, such that no
torquescanbeconveyedbetweenthetwo.Thesizeoftheconnected
regioninaKepleriandiscisthenlimitedtoafiniteradialextentnear
Rco, where the differential rotation between the star and disc is the
smallest.
Willfieldlines,onceopenedbydifferentialrotation,remainopen?
It has been suggested that such field lines could reconnect in the
current sheet formed during the opening (Aly & Kuijpers 1990;
Uzdensky,K¨ onigl&Litwin2002b).Inorderforthistobeimportant,
thetime-scaleforreconnectionshouldbecomparablewithorshorter
thanthatforfieldlineopening.Itisnotclearwhetherthisisthecase
in these systems (Matt et al. 2002; Uzdensky et al. 2002b), but even
if it is, there are other considerations. First, due to the topology of
the field, reconnection must initially occur between open field lines
at the smallest radii (connecting to the lowest latitude on the star). It
is possible that, if reconnection does occur, only a small amount of
fluxwillbeabletoreconnectbeforethisnewlyconnectedfieldagain
begins to open (as in the simulations of Goodson et al. 1999, and
see Uzdensky et al. 2002b). In this case, the size of the connected
region (in a time-averaged sense) will be only slightly larger than if
the reconnection were never to occur. Secondly, the configuration
oftheopenedfieldisfavourabletolaunchmagnetocentrifugalflows
from the disc (Blandford & Payne 1982). We ignore such outflows
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The spin of accreting stars
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in our model, but in a real system, they could help to maintain an
open magnetic-field configuration. Thus we conclude that, once the
field has opened, reconnection along the current sheet is unlikely to
affect the size of the connected region significantly.
2.1.2 Maximal spin-down torque for maximal twist
It is instructive to look at the maximum possible spin-down torque
in this system. Regardless of any disc model or any magnetic cou-
pling physics, the largest possible magnetic torque on a star that
connects to a disc via a dipole magnetic field occurs when the field
is maximally twisted (γ = γc) at all radii along the surface of the
disc. Spin-down torques on the star only occur along field lines
threading the disc outside Rco. Also, the accretion of mass from a
Keplerian disc always adds angular momentum to the star. There-
fore, the largest net spin-down torque on the star occurs when the
disc is truncated exactly at the corotation radius (Rt= Rco), the
field threads the disc to Rout→ ∞, and the disc does not accrete
(˙Ma= 0). One then integrates equation (3) from Rcoto ∞ to get
τmax= −γc
3
This is the absolute maximum spin-down torque that a star can un-
dergofromadiscthatexistsintheequatorialplaneandtowhichthe
star is connected via a dipole magnetic field. It is even independent
of the rotation profile of the disc, except that the angular rotation
rate of the disc is slower than the star outside some radius Rco. It
is also independent of the angular momentum transport mechanism
within the disc.
To achieve this maximal torque requires that: (a) the field twist
has no radial dependence; and (b) the twist is very nearly equal to
the maximum allowed value of γc. If the coupling of the field to a
Keplerian disc is determined by turbulent diffusion, a constant γ
can only be achieved in the unlikely event that ηtdecreases with
radius to exactly counteract the increase in differential rotation rate.
Alternatively, reconnection in the stellar or disc corona may also
lead to a constant γ (Aly & Kuijpers 1990; Wang 1995, but see
the discussion in Section 2.1.1). However, in either case, it is not
clear why the value of the constant twist would necessarily be near
the maximal value γc(instead of, for example, 0.1 γc). Though this
torque may not be very realistic, it is similar in strength to the spin-
down torque used in previous models (e.g. it is the equivalent to the
solution of AC96 for γc= 1).
µ2
Rco3.
(9)
2.1.3 Determination of Rinand Rout
To derive a more realistic magnetic torque, we adopt equation (8)
for the radial dependence of γ. Following Lovelace et al. (1995),
we assume that this equation is only valid where |γ| ? γc, and that
the field is open everywhere else. Thus, equation (8) predicts that
the outer radius of the magnetically connected region in the disc is
Rout= (1 + βγc)2/3Rco.
There is a corresponding location inside Rco at which the twist
formally exceeds the critical value, given by
(10)
Rin= (1 − βγc)2/3Rco.
ThedottedlinesinFig.2indicatetheseradiiforβγc=0.5,inwhich
case Rin≈ 0.63Rcoand Rout≈ 1.31Rco. It is evident that the field
topology is a function of βγcsuch that more diffusion in the disc
allows for a larger connected region. Note that, if βγc? 1, the field
can remain connected to the disc at any r < Rco(as Rinis then not
defined).
(11)
The typical assumption of a closed magnetic topology, corre-
sponding to Rout → ∞ (e.g. Yi 1995; AC96), is equivalent to
γc→ ∞ – the field is allowed to twist to arbitrarily large values
without opening. In order to consider the effect of varying topology
(i.e. where the field is open beyond some finite Rout), we adopt a
value of γc= 1 (as justified by e.g. UKL). However, we will retain
γcas a parameter in all of our formulae for a comparison between
the two cases (γc→ ∞ and γc= 1) and so that different values of
γcmay be considered by the reader. The combined parameter βγc
appears throughout our formulation. We generally think of this pa-
rameter in two ways. First, when γc= ∞, the stellar field is closed
and connects to the entire disc, and this represents the ‘standard’
star–disc interaction model. Secondly, for the more realistic case of
γc= 1, the field topology is partially open, and Routthen depends
on β.
2.2 Three possible states of the system
The inner edge of the disc is delimited by Rt(discussed in Sec-
tion 2.3.1), and so there are two possible magnetic configurations
for an accreting system, depending on the location of Rtrelative
to Rin. First, if Rt< Rin, the stellar field will be largely open, and
equation (8) is not valid anywhere. The outer radius of the mag-
netically connected region, R
equation 10) will be near the inner edge of the disc (R
will refer to this situation as ‘state 1’ of the system. In state 1, the
star receives no spin-down torques from the disc.
In ‘state 2’, Rin< Rt< Rcoand the star is magnetically con-
nected to the disc from Rtto Rout. State 2 represents the typical
configuration considered in many models and was discussed at the
beginning of this section. Also, systems near their disc-locked state
(Section 3) are always in state 2.
Finally,thereexistsathirdpossible,non-accretingstate,‘state3,’
that occurs when the disc is overpowered by the magnetic field (e.g.
forfastrotation,largeµorsmall˙Ma)andthediscbecomestruncated
outside Rco(e.g. Illarionov & Sunyaev 1975). In state 3, there are
no positive (spin-up) torques on the star, so it can never be in spin
equilibrium (Sunyaev & Shakura 1977b).
Fig. 3 illustrates the basic magnetic configuration of each state.
One can think of this figure, for example, as a sequence (from top to
bottom) of decreasing˙Ma(or increasing µ). A steadily decreasing
˙Mamay represent an evolutionary sequence (e.g.) for protostars as
one goes from class 0 sources to weak-lined T Tauri stars (class 3;
as shown in Reid, Pudritz & Wadsley 2002). As˙Madecreases from
a system in state 1, the disc truncation radius (which is at the inner
edgeofthediscinthefigure.)movesoutwardandeventuallycrosses
the location of Rin(entering state 2) and then Rco(state 3).
The conditions for which a system transitions from an accreting
state to state 3 is unknown (see Rappaport et al. 2004 and Sec-
tion 4.2.3). In the current work, we do not consider state 3, other
than to note that it occurs somewhere below the lower˙Malimit of
accreting systems. Instead, we focus most of our attention on state
2 and discuss state 1, where appropriate.
?
out (which is not then determined by
?
out∼ Rt). We
2.3 Torques between the star and disc
A combination of equations (3) and (8) gives the full radial depen-
dence of the differential magnetic torque in the system,
?
dτm
dr
=
f8/3
β
µ2
R4
∗
r
Rco
?−4?
1 −
?
r
Rco
?3/2?
,
(12)
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172
S. Matt and R. E. Pudritz
co
R
Rco in
R
Rco Rout in
R
State 2
State 3
State 1
Figure 3. Three possible configurations of the magnetic star–disc interac-
tion. There are two possible accreting states: either the stellar field connects
only to the inner edge of the disc (top panel) or it connects to an extend
region, reaching beyond Rco(middle panel). In either case, the stellar field
dominatestheaccretionflow(arrows)ontothestar.Undercertainconditions
(e.g. low˙Ma), accretion on to the central star will cease, defining the third,
non-accreting state (bottom panel).
where,forconvenience,wehaveusedequation(2)toexpressthera-
diusinunitsof Rco.Furthermore,theangularmomentumcarriedby
accretingmaterialthrougheachannulusofaKepleriandisc(ofwidth
dr and vertically integrated) equals dτa = 0.5˙Ma(GM∗/r)−1/2dr
(e.g. Clarke et al. 1995). This can be combined with equation (2) to
give the differential accretion torque, as a function of r/Rco,
?GM∗
The assumption that˙Mais constant at all radii in the disc requires
thatthenetangularmomentumtransportedawayfromeachannulus
in the disc equals dτa. The disc must therefore be structured in such
a way that the differential torques internal to the disc, dτi, satisfy
dτa
dr
=1
2
˙Maf1/3
R∗
?1/2?
r
Rco
?−1/2
.
(13)
dτi≡ dτa− dτm.
These internal torques could result from angular momentum trans-
port via (e.g.) turbulent viscosity (Shakura & Sunyaev 1973), MRI
(Balbus & Hawley 1991) or disc winds (see review by K¨ onigl &
Pudritz2000).Ifoneassumesaparticularangularmomentumtrans-
portmechanisminthedisc,thesolutiontoequation(14)determines
the structure of the disc. As an example, for the case of α viscos-
ity, Rappaport et al. (2004) showed that the disc can respond to
external magnetic torques by increasing its surface density in order
(14)
0.51.0 1.5 2.0 2.53.0
r / Rco
-1
0
1
2
3
dτ / dr [ 1025 dynes ]
internal
accretion
magnetic
Rt
Figure 4. Differential torques in the fiducial CTTS system (see discussion
above Section 2.1) for β = 1 and γc→ ∞, where the field is assumed to
remain connected over the entire disc. The truncation radius, Rt, is where
dτm= dτaand dτi= 0, indicated by the vertical dotted line (Rt≈ 0.91 Rco).
The system is shown in its equilibrium state, where the net torque on the star
is zero, requiring a stellar spin period of 6.0 days (Rco≈ 5.5R∗).
to transport the additional angular momentum outward. A detailed
treatment of the disc adjustment is not necessary here, as we are
presently concerned with torques on the star, and we simply assume
that the disc structures itself such that equation (14) is valid.
Fig. 4 shows the differential torques (dτ/dr) as a function of
r/Rcofor the adopted fiducial parameters. The solid, dash–dotted,
and dashed lines in Fig. 4 represent the differential torques from
equations 12, 13 and 14, respectively. The system shown has β =
1 and γc→ ∞, so that the field is connected to the entire surface
of the disc, and so that the figure represents the closed topology of
several models in the literature (e.g. AC96).
The differential magnetic torque (dτm, solid line in Fig. 4) is
strongest near the star, where the dipole field is strongest, and it
acts to spin up the star (and thus spins down the disc) for r <
Rco. At Rco, dτmgoes to zero, as the field is not twisted (γ = 0)
there. Outside Rco, the magnetic torque becomes stronger again, as
the twist increases, though now acting to spin the star down (and
the disc up). The dipole field strength falls off faster with distance
(Bz∝ r−3) than the magnetic twist increases (γ ∝ r3/2), so dτmhas
a minimum value at r/Rco≈ 1.37 and then goes to zero as r → ∞.
Thedashedlineinthefigure(dτi)givesussomeinformationabout
the disc structure. In this case, the structure will be significantly
different from a case with dτm= 0. For a very large dτmoutside
Rco, the assumption that the disc can counteract it (via an increase
in dτi) must eventually break down, and the system would then be
in state 3 (Section 2.2).
2.3.1 Truncation radius
Inside the corotation radius, the stellar magnetic torque acts to ex-
tract angular momentum from the disc, further enabling accretion
(not hindering accretion, as for r > Rco). The differential magnetic
torque increases rapidly as one moves toward the star, and at some
point, dτm= dτa, the external magnetic torque alone is capable of
maintaining˙Ma. Consequently, dτigoes to zero at the same radius
and formally becomes negative for smaller r (see Fig. 4). Nega-
tive dτiis unrealistic, however, because it would require angular
momentum transfer inward through the disc, from slower spinning
material to faster spinning material, so the Keplerian disc does not
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The spin of accreting stars
173
exist where where dτi? 0. Thus, the disc truncation radius,5Rt, is
where dτi= 0.
At Rt, the stellar magnetic field will quickly spin down the disc
material, forcing it into corotation with the star. Sub-Keplerian ro-
tation leads to a free-fall of disc material on to the surface of the star
in a ‘funnel flow’ along magnetic-field lines (e.g. K91; Romanova
et al. 2002). Whether or not the funnel flow originates exactly from
Rtorfrominsidethatradiusissubjecttodebate(e.g.Aly&Kuijpers
1990). However, for the present discussion of angular momentum
transport, the most important thing is that Rtdefines the location
where the stellar magnetic field completely dominates over the disc
internal stresses, and so all the angular momentum of disc material
at Rtwill end up on the star.
By setting equations (12) equal to (13), we derive a relationship
defining Rt,
?
where
ψ ≡ 2µ2˙M−1
is a dimensionless parameter relating the strength of the magnetic
field to the strength of accretion. This formula was also derived by
Yi (1995), but with different disc parameters in place of our β. For
any given β, f and ψ, there is only one solution to equation (15)
such that Rt< Rco. A real system may deviate slightly from our
simple picture (e.g. of an unperturbed dipole field), leading to an
uncertaintyintheexactvalueof Rt.However,duetothesteepradial
dependence of dτmrelative to dτa, the location of Rtshould not be
significantlyaffected.ForthesystemplottedinFig.4,thesolutionto
equation (15) is Rt/Rco≈ 0.915, represented by the vertical dotted
line in the figure.
For a system in state 1 (with Rt< Rin), equation (15) is not valid
because the field will open (see Section 2.1.3). A substitution of
Rt< Rinin equation (15) indicates that the system will be in state
1 if
Rt
Rco
?−7/2?
1 −
?
Rt
Rco
?3/2?
=β
ψf−7/3,
(15)
a(GM∗)−1/2R−7/2
∗
(16)
f < (1 − βγc)(γcψ)−3/7,
and it will be in state 2 for any larger f. Note that condition (17) can
never be satisfied if βγc? 1 (as Rinis then undefined), and so the
system would then always exist in state 2. For the more probable
case that βγc? 1, state 1 is a possible configuration of any system.
To determine Rtin state 1, instead of using equation (12) for the
differentialmagnetictorqueonemustconsiderthemaximumpossi-
ble dτmin order that the field remains closed. This is determined by
usingγ =−γcinequation(3).Bysettingthisequaltoequation(13),
one finds
Rt= (γcψ)2/7R∗= (2γc)2/7(GM∗)−1/7(˙Ma)−2/7µ4/7.
Notethatthisequationdoesnotdependontherotationrateorradius
ofthestarandhasthesamedependencesonothersystemparameters
as in many previous theoretical works (e.g. Davidson & Ostriker
1973; Ghosh & Lamb 1979; Shu et al. 1994). Because Rt< Rin
in state 1, the field lines will be open inside Rco. Thus, in state 1,
the stellar field connects only to a small portion of the disc near Rt,
from where a funnel flow originates, and all exterior field lines are
open,asshowninthetoppanelofFig.3.State1isdiscussedfurther
in Section 4.2.
(17)
(18)
5Note that Rt is related to the ‘fastness parameter’, ω, in X-ray pulsar
literature: ω = (Rt/Rco)3/2.
-2.0 -1.5 -1.0
log(f)
-0.50.0
0
2
4
6
8
10
12
Rt / R*
β = 2
1
0.5
0.1
Rco / R*
Figure 5. Predicted location of Rt/R∗as a function of log( f ), for the fidu-
cial CTTS system. The dashed lines represent the prediction from equation
(15) for β = 0.1, 0.5, 1.0 and 2.0, as indicated, and all have γc→ ∞ so
that the magnetic field is assumed to remain closed. The solid line is for
β = 0.1, but with γc= 1, so that the field is partially open. The system with
γc= 1 switches to state 1, for log ( f ) ? −1.5, and Rtis then predicted by
equation (18). The dotted line shows the location of Rco(equation 2).
Fig. 5 shows the predicted location of Rtin units of R∗for the
adopted fiducial parameters, as a function of the logarithm of the
spin rate f. For reference, the dotted line shows the location of Rco
(equation 2). The dashed lines show Rtfor systems with γc→
∞ and β = 0.1, 0.5, 1.0 and 2.0, as indicated in the figure. The
solid line shows the β = 0.1 case, but with a more realistic value of
γc=1.Notethat Rtisalwayslessthan Rco,andthatbothdecreaseas
f increases.Forstrongerfieldcoupling(smaller β),thefieldismore
strongly twisted, and so Rtis closer to Rco.
All the dashed lines in Fig. 5 represent systems with γc→ ∞, in
which the field remains closed for arbitrarily large magnetic twist.
Thus, these systems are always in state 2, and the dashed lines are
everywhere given by equation (15). On the other hand, the solid
line represents a system with γc= 1, so it is in state 2 only when
f ? 0.03 (equation 17). For smaller f, it is in state 1, and Rtis then
determined by equation (18). Because equation (18) is independent
of f, state 1 is represented by the constant value of Rt≈ 9.1R∗in
the figure. Real systems will probably have γc= 1, in which case
the value of Rt≈ 9.1R∗represents an upper limit for the fiducial
CTTS system, regardless of β.
2.3.2 Accretion torque
We assume that accreted disc material is quickly integrated into the
structure of the star and the accreted angular momentum is redis-
tributed into the stellar rotation profile. So there is a torque on the
steadily accreting star that is given by τa=˙Ma[ld(Rt) − l∗], where
ld(Rt) is the specific angular momentum of the disc material at Rt
and l∗is that of the star. Combined with equation (1), and assuming
solid body rotation of the star, this becomes
τa=˙Ma(GM∗R∗)1/2?(Rt/R∗)1/2− k2f?,
where k is the normalized radius of gyration (k2≈ 0.2 for a fully
convective star; AC96). This formula is valid for a system in either
state 1 or state 2.
The term in square brackets in equation (19) is dimensionless
and compares the accreting angular momentum (first term) with
how much the star already has (second term). Note that the first
(19)
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term will always be greater than or equal to one, while the second
term has a maximum value of k2(when f = 1). Thus, the second
term is usually negligible (particularly when f ? 1).
2.3.3 Magnetic torque
When in state 2 (i.e. when Rt> Rin), the stellar field connects to a
significant portion of the disc, and one can integrate equation (12)
over the connected region, from Rtto Rout, to obtain the total mag-
netic torque on the star,
?2(Rco/Rout)3/2− (Rco/Rout)3
τm=
1
3β
−2(Rco/Rt)3/2+ (Rco/Rt)3?.
This torque is independent of the detailed structure of the Keplerian
disc. Also, equation (20) includes the dependence of the magnetic
torque on the field topolgy via the variable Rout. For example, the
spin-down torque (found by setting Rt= Rco) exerted by field lines
connected out to Rout≈ 2.4Rcois one half of the spin-down torque
for Rout→ ∞. Similarly, for Rout≈ 7.2Rcoor Rout≈ 34Rco, the
spin-down torque is 90 per cent or 99 per cent (respectively) of the
spin-downtorquefor Rout→∞.Itisevidentthat,evenwhen Routis
large,mostofthespin-downtorquecomesfromfieldlinesconnected
nottoofarfrom Rco.Thisissimplybecausethedifferentialmagnetic
torque (equation 12) becomes very weak far from the star. Thus the
typical assumption of Rout→ ∞ is not significantly effected by the
fact that real discs have finite radial extents, so long as they reach
to several times Rco.
Above, we have taken Routas arbitrary, but our goal in this paper
is to consider the opening of the field from differential rotation, so
Routis then given by equation (10), and the preferred formulation
of the magnetic torque becomes
?2(1 + βγc)−1− (1 + βγc)−2
µ2
Rco3
(20)
τm=
1
3β
µ2
R3
co
−2(Rco/Rt)3/2+ (Rco/Rt)3?.
(21)
This is exactly the solution found by AC96 for the special case
of β = 1 and γc→ ∞ (so that Rout→ ∞), but our formulation
includestheeffectoffieldopeningviadifferentialrotation,inwhich
case the field topology depends on the diffusion parameter β. The
total magnetic torque on the star can be either positive or negative,
depending on the size of the connected region inside Rco, compared
with the connected region outside Rco. In the next section (Section
2.4), we shall show that, for reasonable values of β and γc, Routis
verycloseto Rco,andthespin-downtorqueissignificantlyaffected.
2.4 Effect of opened field
Fig. 6 shows the differential torques in the fiducial CTTS system
with β = 1, as in Fig. 4. However, unlike Fig. 4, here we show
the system for γc= 1, so that the field is open beyond Rout≈ 1.59
Rco. The star now rotates significantly faster, with a period of 3.3 d
( f ≈ 0.14) and Rt≈ 0.974Rco. A comparison between Figs 4 and
6 illustrates the effects of varying field topology on the differential
torques in the star–disc system.
TherearesomeinterestingdifferencesbetweenFigs4and6.Most
notably, the differential magnetic torque in Fig. 6 abruptly goes to
zero at the location of Rout, due to the assumption that there is no
torque on the star from the disc where the field lines are open. It
is evident that a more open topology results in a smaller connected
region, which leads to a net (integrated) spin-down torque that is
0.51.0 1.52.0 2.53.0
r / Rco
-5
0
5
10
dτ / dr [ 1025 dynes ]
internal
accretion
magnetic
Rt
Figure 6. Same as Fig. 4, except that the equilibrium spin period is 3.3 d
(Rco≈ 3.7R∗), and γc= 1, so that the magnetic field is not connected for
r > Rout≈ 1.6Rco. Also, Rt≈ 0.97Rco.
smaller than for the completely closed topology. Thus, a more open
topology results in a faster equilibrium spin rate, as can be seen by
comparing the stellar spin rate of 6.0 d for Fig. 4 with 3.3 d for
Fig. 6.
To quantify the effect of a more open topology on the magnetic
torqueandtodeterminethedependenceonβ,wefirstconsideronly
the portion of the magnetic torque that acts to spin down the star
by setting Rt= Rcoin equation (21). For normalization, we use the
maximum spin-down torque, τmax(equation 9). We define the ratio
of the true spin-down torque to this maximum torque as χ ≡ τm(Rt
= Rco)/τmax, which is given by
χ = (βγc)−1?1 + (1 + βγc)−2− 2(1 + βγc)−1?
and only depends on the parameter βγc. It is immediately evident
that for βγc= 1, χ = 1/4, so the spin-down torque is four times
less than that used by AC96, when one considers a more realistic
magnetic topology.
Fig. 7 illustrates the dependence of the torque ratio χ on
βγc. It decreases as (βγc)−1for β ? 1 and increases as βγcfor
β ? 1 (as revealed by Taylor expansion of equation 22). The lim-
iting behaviour of χ is indicated by the two dotted lines in the
figure. This behaviour can be understood as a competition between
two effects: in the strong magnetic coupling limit (β ? 1), the
(22)
-2 -1012
log(βγc)
-2.0
-1.5
-1.0
-0.5
0.0
log(χ)
Figure 7.
(equation 22), as a function of log(βγc). Dotted lines represent χ = βγcand
χ = (βγc)−1.
Logarithm of the ratio of real to maximal spin-down torque
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The spin of accreting stars
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-2.0-1.5 -1.0
log(f)
-0.50.0
-2
-1
0
1
2
3
τa + τm [ 10 37 ergs ]
(β, γc) = (1, ∞)
(1, 1)
(0.1, 1)
(0.01, 1)
Figure 8. Net torque, τa+ τm(equations 19 and 21), on the star as a
function of the logarithm of the fractional spin rate, for the fiducial CTTS
system. The four solid lines represent (β, γc) = (1, ∞), (1, 1), (0.1, 1) and
(0.01, 1), as indicated, corresponding to different field topologies, ranging
from completely closed to more open. A system will be in spin equilibrium
when the net torque equals zero (dotted line).
field topology becomes more open for smaller βγc, reducing the
spin-down torque. In the weak coupling limit (β ? 1), the topology
is largely closed, but the twisting of the field lines is smaller for
larger β, which reduces the differential magnetic torque at all radii
(dτm∝ β−1). These two effects conspire to give a maximal value of
χ for the special case of βγc= 1. For the more probable value of
β = 10−2, the spin-down torque is two orders of magnitude lower
than that used by AC96. While it may at first seem surprising that
strong magnetic coupling leads to weaker spin-down torques on the
star,furtherreflectionrevealsthatthisisnecessarilytrue,asstronger
coupling leads to stronger twisting, which further disconnects the
star from the disc.
Finally, equations (19) and (21) can be used to calculate the net
torqueonthestar.Fig.8showsthisnettorqueasafunctionoflog( f )
for the fiducial CTTS parameters and for different values of β and
γc. For each case, we calculate the net torque as follows: first, for a
givenvalueofβ andγc,andforeachvalueoff,weuseequation(17)
to determine whether the system is in state 1 or 2. We then find the
location of the truncation radius, using equation (15) if the system
is in state 2, or equation (18) if in state 1. Finally, we calculate the
integrated torques τa(equation 19) and τm(equation 21, if in state
2; τm= 0, if in state 1). As discussed in Section 2.3.1, only cases
with βγc< 1 can be in state 1, so only those cases show a transition
at log f ≈ −1.45 in Fig. 8. The figure also shows the effect of
the field topology on the net torque on the star from the disc. It is
evident that, when the magnetic field is partially open (γc= 1), the
net torque is larger than for the case where the field is everywhere
closed (γc= ∞). The spin rate at which the net torque on the star
is zero indicates the equilibrium spin state, which is the topic of the
next section.
3 THE DISC-LOCKED STATE
The general theory presented in Section 2 enables one to calculate
the net torque (τa+ τm) on the star for any accreting system with
known M∗, R∗,µ,˙Maand ?∗(one must also adopt values for γc
and β). The system is stable, in that a positive torque spins the star
up, and a faster spin reduces the total torque. Conversely, a negative
torque spins the star down, and the torque increases (becoming less
negative) for slower spin. Therefore, in a system where the other
parameters are relatively constant, the spin rate of the star naturally
adjusts to an equilibrium state in which τa+ τm= 0, known as the
‘disc-locked’state(e.g.K91;Cameron&Campbell1993;Shuetal.
1994; AC96). Because the only torques that spin the star down orig-
inate along field lines that connect to the disc outside Rco, systems
in their equilibrium state must be in state 2 (Rt> Rin, see Fig. 3). In
thissection,weshowthatboth Rtand?∗inthedisc-lockedstateare
significantly affected by the field topology and thus have a strong
dependence on the magnetic diffusion parameter β.
3.1 Truncation radius in the disc-locked state
The disc-locked state is defined by the condition τm= −τa. Thus,
by combining equations (19) and (21), and using equations (2) and
(15) to eliminate f, this condition can be rearranged to be
K(βγc) − (Rco/Rt)3/2
(Rco/Rt)3/2
eq eq
where
eq
?1 − (Rco/Rt)3/2
? = 7,
(23)
K(βγc) ≡ 2(1 + βγc)−1− (1 + βγc)−2,
and the subscript ‘eq’ refers to the value in the disc-locked state.
In deriving equation (23), we have ignored the term proportional
to k2f in equation (19), as justified in the discussion following
that equation. The function K(βγc) characterizes the topology of
the field in the sense that, when βγcvaries between 0 and ∞, K
variesbetween1(completelyopenfield)and0(completelyclosed).
Equation (23) has exactly one solution such that (Rt/Rco)eq< 1
(which is the only physical solution) for any given βγc> 0.
The location of Rtfor accreting systems is, in principle, an ob-
servable parameter. For example, Kenyon, Yi & Hartmann (1996)
used a magnetic accretion model to predict infrared excesses in
CTTSs and then to determine the value of Rt/Rcofor a sample of
starsintheTaurus-Aurigamolecularcloud.Thevalueof(Rt/Rco)eq
predicted by equation (23) represents the value for a system that is
disc locked.
The solid line in Fig. 9 shows (Rt/Rco)eq as a function of
log(βγc). When a closed-field topology is assumed (γc → ∞),
(Rt/Rco)eq≈ 0.915. However, for the more reasonable value of
γc= 1, (Rt/Rco)eqincreases (approaching unity) as the magnetic
(24)
-2-2 -1 -1001122
log(βγc) log(βγc)
0.80 0.80
0.850.85
0.900.90
0.950.95
1.00 1.00
1.051.05
Rt / Rco
Rt / Rco
State 1
State 2
State 3, no accretion
Rin / Rco
(Rt / Rco)eq
spin down
spin up
Figure 9. Location of Rt/Rcoin the disc-locked state (solid line; equa-
tion 23), as function of log(βγc). The dashed lines represent Rin/Rcoand
the location where Rt/Rco= 1. Any system with Rt> Rcois in state 3
(dark-grey shaded region), while any system with Rt< Rinis in state 1
(light grey). Furthermore, if Rt/Rcois anywhere below (above) the solid
line, the net torque from the disc will spin the star up (down).
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S. Matt and R. E. Pudritz
coupling becomes stronger (smaller β). This confirms the conclu-
sion of Wang (1995, and see Cameron & Campbell 1993; Yi 1994,
1995) that any disc-locked system will have (Rt/Rco)eq? 0.9, and
we find that the effect of a more open field topology is to increase
this value significantly.
The figure also indicates the three possible states of the system
(discussed further in Section 4.2), determined by the location of Rt
relativeto Rinand Rco(dashedlines).Asystemthatisdisc-lockedis
alwaysinstate2.Ifasystemisobservedwith Rt/Rcolargerthanthe
solidlineinthefigure,thestarshouldbespinningdown.Conversely,
if Rt/Rcois smaller than the solid line in the figure, the net torque
from accretion and from field lines connecting the star and disc will
act to spin the star up. It is interesting that Kenyon et al. (1996)
found typical values of Rt/Rcoin the range 0.6 to 0.8 for the stars
in their sample. If true, these stars cannot be in spin equilibrium,
unlesstheyfeelsignificantspin-downtorquesotherthanthosefrom
field lines connecting them to their discs. Furthermore, if βγc< 0.3
is appropriate, the stars in their sample should exist in state 1.
3.2 Stellar spin rate in equilibrium
Now that we can calculate (Rt/Rco)eqvia equation (23), we rewrite
equation(15)tofindthefractionalspinrateofthestarinequilibrium:
feq= C(β,γc)(2/ψ)3/7,
where
?
(25)
C(β,γc) ≡
2
β
?Rco
Rt
?2
eq
??Rco
Rt
?3/2
eq
− 1
??−3/7
.
(26)
Because (Rco/Rt)eqdepends only on βγc(via equations 23 and 24),
the dimensionless function C(β, γc) depends only on β and γc. We
canalsocombineequations(1),(16)and(25)tofindtheequilibrium
angular spin rate of the star:
?eq
∗= C(β,γc)˙M3/7
This equation has the same dependence of ?eq
equation (3) of K91, and as in the theory of Shu et al. (1994) and
Ostriker&Shu(1995).Theonlydifferenceisthevalueofthefactor
C used in the various theories. K91 used C ≈ 1.15, and Ostriker
& Shu (1995)6found C ≈ 1.13. However, our formulation of the
problem allows us to determine the effect of the field topology on
the equilibrium spin rate, via the function C(β, γc), for arbitrary
values of the diffusion parameter β.
Fig. 10 reveals the dependence of C(β, γc) on β for the two
values of γcwe have considered throughout. For γc→ ∞, C(β) ≈
1.59β3/7, which is represented by the dotted line in the figure. The
solid line shows C(β, γc) for the more realistic value of γc= 1 and
illustrates the effect of a reduced magnetic connection to the disc.
For comparison, the dashed line shows the spin rate factor used by
K91,whichalsoroughlyrepresentsthetypicalfactorsoforderunity
used in most star–disc interaction models.
ThedottedlineinFig.10representstheassumptionthatthemag-
netic field everywhere connects to the disc, regardless of the field
twist. In that case, the magnetic torque increases with with decreas-
ingβ,asthefieldthenbecomeshighlytwisted,andsotheprediction
is that ?eq
a
(GM∗)5/7µ−6/7.
(27)
∗ on˙Ma, M∗and µ as
∗ ∝ β3/7. However, when one considers that dipole field
6While it is interesting that our equation (27) resembles the formulation of
Ostriker & Shu (1995), their assumed magnetic-field geometry is different
from ours, so the comparison of C values should not be taken too seriously.
-2-1012
log(β)
0
2
4
6
8
10
12
C(β)
Figure 10. Spin rate factor C(β, γc) (equation 27) as a function of log(β).
The dotted line is C(β) for γc→ ∞ and represents the assumption of a
completelyclosedmagnetictopology.Thesolidlineshowsthemorerealistic
case with a partially open topology (γc= 1). The dashed lines represents
C ≈ 1.15 from K91. The figure is from Matt & Pudritz (2004).
lines will become open when largely twisted, the torque has a maxi-
mumvalueforβ =1anddecreasesforanyotherβ (seeSection2.4).
Correspondingly,thesolidlineinFig.10hasaminimumvalueforβ
=1.Thisminimumvaluerepresentsthe‘bestcase’fordisclocking,
and even at this location C is a factor of 1.8 larger than the value
for γc→ ∞ and 2.5 times larger than used by K91. For the more
probable value of β = 10−2, the predicted equilibrium spin rate of
the star is more than an order of magnitude faster than predicted by
any other model. Note that the spin rates of the system plotted in
Figs 4 and 6 were chosen to be in equilibrium, and a comparison
between the two figures shows the effect of field topology for the
β = 1 ‘best case’.
Matt & Pudritz (2004) applied the analysis presented in this sec-
tion to the CTTS BP Tau, which is one of the few stars for which all
of the relevant system parameters are known (or well-constrained).
They argued that the existence of slowly rotating, accreting stars,
such as BP Tau, cannot be explained by a disc-locking scenario. To
further illustrate the effect of field topology on the predictions of
disc-locking,andtoapplytheanalysistoallCTTSs,wehaveplotted
Fig. 11. The figure shows the predicted spin period for a wide range
of observable parameters. The spin period is given by 2 π/?eq
we have used the relationship µ = B∗R3
The solid lines are for γc→ ∞ and β = 1, and so they represent
the ‘standard prediction’ by previous models. The broken lines take
into account that some of the field should be open (γc= 1) for the
three different values of β = 1, 0.1 and 0.01. Note that β = 0.1
predicts the same period as for β = 10 (owing to the approximate
symmetry of the solid line in Fig. 10), and β = 0.01 corresponds
to β = 100. It is evident that, even in the ‘best case’ (β = 1) for
the disc-locking scenario, the effect of a more open topology is to
reducetheequilibriumspinperiodbyafactoroftwo,comparedwith
the closed-field assumption. Given the uncertainties in some of the
observed parameters, it may not yet be possible to constrain the
predicted period to within a factor of two. In particular, a difference
of a factor of two in the predicted spin period could result from an
error of a factor of 5.0, 2.2, 2.6 or 1.3 in the observed parameters
˙Ma, B∗, M∗or R∗, respectively. However, for the more probable
valueofβ =10−2(dottedlinesinFig.11),thepredictedspinperiod
isanorderofmagnitudelowerthanthe‘standardprediction’,which
cannot be reconciled by observational errors.
∗, and
∗.
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The spin of accreting stars
177
-9 -8-7 -6
log(M
•
a) [ M / yr ]
0
5
10
15
20
Period (days)
0.00.51.0 1.52.02.5
B* [ 103 Gauss ]
0
2
4
6
8
10
Period (days)
0.00.5 1.01.52.0
M* [ M ]
0
2
4
6
8
10
Period (days)
β , γc
1.00, ∞
1.00, 1
0.10, 1
0.01, 1
1234
R* [ R ]
0
5
10
15
20
Period (days)
Figure 11. Equilibrium spin period as a function of observable parameters˙Ma(upper left), B∗(upper right), M∗(lower left) and R∗(lower right). In each
panel, all parameters are held fixed at the fiducial value except for that which is plotted along the abscissa. The vertical dotted line in each panel shows the
fiducial value that is held fixed in the other panels. The solid line in each panel represents the assumption that the magnetic-field topology is completely closed
(γc→ ∞) and β = 1. All of the broken lines represent a partially open topology (γc= 1) for β = 1 (dash–dotted lines), 0.1 (dashed lines), and the more
probable value of 0.01 (dotted lines).
3.3 Time to reach equilibrium
Itisimportanttodeterminehowquicklythestarwillspinupordown
to reach the equilibrium state. Rather than fully solving the time-
dependentproblem,whichshouldalsoincludethespin-upduetothe
contraction of the protostar (e.g. Yi 1994), one typically estimates a
characteristic spin-down time using the angular momentum of the
star,L∗,dividedbythenettorqueonthestar.Tobemoreprecise,and
for arbitrary spin rates, one should replace L∗with the difference
between the current L∗and the value of L∗for the equilibrium spin
rate.Assumingsolidbodyrotationofthestar,thecharacteristictime
to reach spin equilibrium is then
??eq
which corresponds to a spin-up (down) time for a star currently
spinning slower (faster) than ?eq
time-scale for other system parameters to change (e.g. compared
with the lifetime of the disc), the star is unlikely ever to be in a spin
equilibrium state.
Fig. 12 shows tspinfor the adopted fiducial parameters, as a func-
tion of the spin rate fraction f and for different values of β and γc. It
isevidentfromthefigurethattspingenerallydecreasesforincreasing
f, because faster spin means Rcois closer to the star so the magnetic
torque will be stronger (whether it spins the star up or down). There
is an exception to this when the star spins much slower than feq.
For example, for the case with (β, γc) = (0.01, 1), when f ? 0.2,
tspindecreases with decreasing f. This can be understood, as then
tspin= M∗k2R2
∗
∗ − ?∗
τa+ τm
?
,
(28)
∗. If tspinis long compared with the
0.00.2 0.40.6 0.81.0
f
0
1
2
3
4
5
tspin (105 yrs)
(β, γc) = (0.01, 1)
(0.1, 1)
(1, 1)
(1, ∞)
Figure 12. Time to reach equilibrium, tspin(equation 28), as a function of
the spin rate fraction f, for the fiducial CTTS system. The four solid lines
represent (β, γc) = (1, ∞), (1, 1), (0.1, 1) and (0.01, 1), as indicated. The
vertical dotted lines indicate the equilibrium spin rate for each case.
Rt/Rcodecreases rapidly with decreasing f (see Fig. 5), leading to
much stronger magnetic spin-up torques. Note also that the cases
with (β, γc) = (0.01, 1) and (0.1, 1) are in state 1 for f ? 0.036 and
0.033, respectively, in which τm= 0.
Forthe‘standard’predictionwith(β,γc)=(1,∞),tspinisalways
less than 5 × 104yr, which is much shorter than the expected disc
lifetimeofmorethan106yr(Muzerolleetal.2000).However,when
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one considers the effect of a more open field topology (γc= 1), the
magnetictorqueisreduced,andsotspinislongerandincreaseswhen
β decreases. For the probable case with (β, γc) = (0.01, 1), tspin∼
4 × 105yr. This is still relatively short compared with the expected
disc lifetime. Therefore, we agree with previous authors (e.g. K91;
AC96) that systems such as those considered above should exist
near their equilibrium spin states throughout most of their accretion
lifetimes, but only if β ? 0.01. However, the effect of a more open
field topology is that the equilibrium spin rate is much faster than
previously predicted.
We note in passing that the characteristic time tspin we have
calculated here is significantly shorter than recently calculated by
Hartmann(2002).Hartmann’sestimateassumesthattheupperlimit
to the spin-down torque on the star is equal to ˙Ma(GM∗Rout)1/2.
However, as discussed in Section 2.3, this is the torque necessary to
provideasteadyaccretionrateof˙Mathroughtheradius Rout.Inorder
for the disc to exert a spin-down torque on the star (via the magnetic
connection),itmustprovideatorqueinadditionto˙Ma(GM∗Rout)1/2,
requiring the disc to have a different structure from that in the ab-
sence of a stellar field (see Rappaport et al. 2004). It is not yet clear
to what extent the disc can be restructured (before accretion will
cease), and so Hartmann’s estimate does not represent a true limit.
4 DISCUSSION AND CONCLUSIONS
We extended the standard picture of the interaction of a magnetized
star with a steady-state accretion disc. Our more comprehensive
formulation of this problem allows us to determine the location of
the disc truncation radius Rtand calculate the torque on the star
for a system with arbitrary values of ˙Ma,?∗, M∗, R∗, B∗and β,
which parametrizes the coupling of the magnetic field to the disc.
We consider only two sources for the torques: (a) torque from the
angularmomentumdepositedbyaccretionofdiscmaterialfrom Rt;
and (b) torques exerted by field lines connecting the star to the disc
over the region from Rtto Rout.
Ourmodelresemblesseveralpreviousstudies(e.g.AC96),except
that we have now determined the dependence of the torques on the
magnetic coupling to the disc. Specifically, the differential rotation
between the star and disc results in a largely open topology (e.g.
UKL), so the size of the region of the disc that is magnetically
connected to the star is smaller (i.e. Routis smaller). Thus, when
one considers this effect, the magnetic spin-down torque on the star
is less than if one assumes the field remains everywhere closed. The
strongest spin-down torque occurs for intermediate magnetic-field
coupling (β = 1), in which case the spin-down torque is a factor of
four less than for the closed-field assumption. For strong magnetic
coupling (β ? 1), as expected near the inner edge of an accretion
disc, Routis very close to Rco, and the spin-down torque then is
proportional to β. For the probable value of β = 0.01, the spin-
down torque is 100 times less than for the closed-field assumption!
Furthermore, the possibility that field lines may open inside Rco
characterizes a new mode (state 1) in the system, in which the star
feels no spin-down torques from field lines connected to the disc.
Three possible system states are summarized in Section 4.2.
We also considered the disc-locked state of the system, in which
thenettorqueonthestariszero.Amoreopenfieldtopologyleadsto
anequilibriumstatethathasahigherstellarspinrate.Thetimefora
given system to reach spin equilibrium is also longer when the field
is more open. These results apply to any system in which accretion
occurs on to a magnetized central object. In the general case, not all
the system parameters are observationally known. Thus, one often
assumes that a particular system is disc-locked, and then ‘tunes’ the
unknown system parameter(s) to satisfy equation (27). We found
that equation (27) contains the function C(β, γc), which is plotted
in Fig. 10 (solid line) as a function of β. It is evident that when the
coupling of the field to the disc is strong (β ? 1) or weak (β ? 1),
the ‘tunable’ system parameters will be significantly different from
those in the usual assumption that C(β, γc) is a constant near unity.
In particular, systems that are thought to be disc-locked will require
a larger µ, larger ?eq
using previous formulations of equation (27). In the next section,
we discuss the implications of these results, and recent results from
the literature, for disc-locking in CTTSs.
∗, smaller M∗, or smaller˙Mathan calculated
4.1 Problems with disc locking for CTTSs
ObservationalsupportfordisclockinginCTTSsisstillcontroversial
(in particualar, see Stassun et al. 1999; Herbst et al. 2000; Stassun
et al. 2001; Herbst et al. 2002), so we have taken another look at
the problem from a theoretical standpoint. We find that spin-down
torques on a CTTS are significantly reduced for strong coupling of
the stellar field to the disc (small β), resulting in equilibrium spin
periods as low as a few days or less, for a wide range of system
parameters (see Fig. 11). Small values of β are probable for CTTSs
(see Section 2.1), although β is an uncertain parameter to calculate
fromfirstprinciplesandmayevenvaryfromsystemtosystem.Also,
as discussed in Section 2.1.2, there may exist special circumstances
that allow the field to remain connected, but whether these circum-
stances can exist in CTTS systems remains to be shown. Therefore,
whileouranalysisoftheproblem(inSection3)doesnotcompletely
ruleoutthepossibilityofdisclockinginallsystems(particularlyfor
fastrotators),itsignificantlyreducesthelikelihoodthatdisc-locking
can explain the existence of accreting stars spinning at ∼10 per cent
(e.g. Bouvier et al. 1993) of break-up speed. There are several other
issues in recent literature that, when combined with our analysis,
castadditionaldoubtontheapplicabilityofdisc-lockingtotheslow
rotators.7
The most notable observational challenge to the disc-locking
model is the apparent lack of strong dipole fields on CTTSs (first
suggested by Safier 1998). It is generally accepted that CTTSs have
field strengths of a few kilogauss. This was predicted by K91, and
subsequent observations (Basri, Marcy & Valenti 1992; Guenther
et al. 1999; Johns-Krull, Valenti & Koresko 1999; Johns-Krull &
Valenti 2000; Johns-Krull et al. 2001) have indeed found a mean
field on the surface of the central stars of typically 2 kG. Thus
far in our analysis, we have adopted a field strength of 2 kG to
represent a typical CTTS system and guide our discussion. How-
ever, stringent measurements of the mean line of sight field have
been carried out for three CTTSs: BP Tau (Johns-Krull et al. 1999),
TW Hya (Johns-Krull & Valenti 2001), and T Tau (Smirnov et al.
2003, 2004), and all measurements give an upper limit of roughly
200Gforthestrengthofthedipolecomponentofthestellarmagnetic
field. The measured mean fields of 2 kG thus represent a field that
is disordered or characterized by multipoles of higher order than a
dipole(Johns-Krulletal.1999).Suchhigh-orderfields,evenifvery
strong on the stellar surface, decrease in strength too quickly with
7Owing to the the unique field geometry of the X-wind, in which all the
stellar field lines are squeezed into the inner edge of the disc (Shu et al.
1994), that model avoids the problem of field opening due to differential
twisting, as considered in this paper. However, the issues discussed after the
first paragraph in this section apply to all disc-locking models, including the
X-wind.
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The spin of accreting stars
179
increasing radius to exert significant spin-down torques, especially
forslowrotatorsinwhich Rcoisatseveralstellarradii.Furthermore,
a 200-G dipole field cannot exert a significant spin-down torque on
a CTTS, even if the field connects to the disc everywhere outside
Rco.Thisisevident,forexample,intheupperrightpanelofFig.11,
which indicates that the equilibrium spin period for a star with such
a field is less than one day. For the cases with γc= 1 and β ? 0.1,
there is no equilibrium possible, as the magnetic spin-down torques
are not strong enough to counteract the angular momentum added
by accretion, even for maximal stellar spin ( f = 1). Also, the time
to reach equilibrium (for cases in which equilibrium is possible, as
discussed in Section 3.3) increases by an order of magnitude when
B∗= 200 G, compared with 2 kG.
Second is the issue pointed out by Wang (1995) and discussed in
Section 3.1 that stars in their disc-locked state must have Rt/Rco?
0.9, for a wide range of possible assumptions in the model. Interest-
ingly, Kenyon et al. (1996) concluded that, for their CTTS sample,
the typical value of Rt/Rcowas in the range 0.6 to 0.8, well below
the disc-locked value. If true, these stars cannot be disc-locked, be-
cause the sum of the accretion torque and the torque carried by field
lines connected to the disc will be positive (spinning the stars up;
see Sections 2.3.3 and 3.1). Thus, these stars can only be in spin
equilibrium if they feel significant spin-down torques other than
those from field lines connecting them to their discs. This conclu-
sion does not depend on whether or not the stellar field can open, as
the calculation of Rtin equilibrium also does not.
Finally, CTTSs may drive stellar winds (Safier 1998), and out-
flowsfromthediscareknowntoexplainseveralaspectsofobserved
protostellar outflows (e.g. K¨ onigl & Pudritz 2000). Winds escape
from regions with open magnetic-field lines, or they can themselves
open the field (e.g. as in the solar wind; Parker 1958), disconnect-
ing the star and disc. Safier (1998) concluded that CTTSs’ winds
should open all stellar field lines beyond roughly 3R∗. Furthermore,
a recent measurement of rotation in the jet from the CTTS DG Tau
(Bacciotti et al. 2002, and see Testi et al. 2002) suggests that the
low-velocitycomponent(∼70kms−1)originatesinthediscfromas
closeas0.3aufromthestar(Andersonetal.2003).Thisisprobably
an upper limit (Pesenti et al. 2004), and the more tightly collimated,
high-velocity component (∼220 km s−1; Pyo et al. 2003) must then
originate from well within this radius in the disc (Anderson et al.
2003). Theoretical disc-wind models (e.g. K¨ onigl & Pudritz 2000)
predict jet speeds of the order of the Keplerian velocity from where
the wind is launched, and observed protostellar jets typically travel
with speeds of a few hundred km s−1(e.g. Reipurth & Bally 2001).
Considering a star with M∗= 0.5 M?and R∗= 2 R?, and assum-
ingvk=100kms−1atthelaunchpoint,thisrequiresthediscwinds
to originate from near GM∗/v2
prevent the star from being magnetically connected to the disc be-
yondtheinnermostlocationofthewindlaunchingpoint(effectively
giving an upper limit to Rout), and these winds carry angular mo-
mentum from the disc, not from the star. As calculated in this paper,
a reduced size of the connected region leads to a reduced spin-down
torqueonthestar,regardlessofthecauseofthefieldopening.Proto-
stellaroutflowsthusprovideanindependentandstringentconstraint
on disc-locking models.
We conclude that spin-down torques exerted by field lines con-
necting the star to the disc outside Rcoare likely to be much weaker
than usually assumed. Therefore, the existence of slowly rotating
( f ? 0.1, or perhaps higher) CTTSs probably cannot be explained
by a disc-locking scenario. Either these stars are all in the process
of spinning up, or the stars feel torques other than those related to
a magnetic connnection to the accretion disc. Given that the typical
k≈ 4.8R∗. These ionized disc winds
spin-uptimesforthesesystemsareshort(seeSection3.3),thelatter
possibility appears the most likely.
4.2 Three states of the system
We identified three possible configurations of the system, deter-
mined by the location of Rtrelative to the two key radii Rin(equa-
tion 11) and Rco. There are two accreting configurations, which we
call states 1 and 2, and one non-accreting configuration, state 3.
Here, we summarize the conditions that determine and characterize
each state.
Fig. 3 illustrates the basic magnetic configuration of the three
states, which may even represent an evolutionary sequence for a
system with (e.g.) an evolving˙Ma(see Section 2.2). Fig. 9 shows
the location of Rin/Rco(curved, dashed line) for various values
of βγc, and the horizontal dashed line represents the location of
Rco. For a system with a given value of βγcand for which Rtis
determined, the figure indicates which state the system will be in
and whether the star should be spinning up or down.
4.2.1 State 1: Rt< Rin(or R?
out< Rco)
This state is a direct consequence of the opening of field lines via
the differential rotation between the star and disc. When Rtis suf-
ficiently less than Rco (i.e. when Rt < Rin), the magnetic field
becomes highly twisted there, opening the field inside Rcoand re-
sulting in a highly open field topology. Note that state 1, in this
context, is only possible for βγc< 1, because otherwise Rinis un-
defined (see Fig. 9 and discussion in Section 2.1.3).
As illustrated in the top panel of Fig. 3 and discussed in Sec-
tion 2.3.1, the stellar field in state 1 connects only to a very small,
innermost region of the disc near Rt, and all exterior field lines are
open. The size of the small connected region is likely to be deter-
minedbydissipativeprocesseswithinthedisc,butwedonotattempt
to calculate this here. The location of Rtis determined by equation
(18) (and see Fig. 5), and accretion on to the star occurs from there.
A star in this state will always feel a net positive torque from the
disc, because no field connects outside Rco. Specifically, the star is
spunupbytheaccretiontorque(τa;equation19),andequation(21)
for the magnetic torque is not applicable. In fact, because Rtin-
creases with stellar field strength, and as τa∝ Rt1/2, the presence
of a stellar field actually increases the spin-up torque on the star,
relative to non-magnetic accretion. In any case, a system in state 1
cannot be in spin equilibrium, unless it receives torques other than
those considered in this paper.
Isstate1aprobable,orevencommon,configurationforaccreting
systems? Fig. 9 indicates that, when the magnetic coupling to the
disc is strong, the range of possible values of Rtfor which a system
canbeinstate2issignificantlyreduced.Forexample,ifβγc=0.01,
any system with Rt< 0.993Rcoshould be in state 1. The specific
conditionsunderwhichanygivensystemwillbeinstate1isgivenby
equation (17). For illustrative purposes, we can solve this equation
(and use equation 16) for the mass accretion rate. Assuming γc=
1, β = 0.01 and f = 0.1, and using the fiducial CTTS values (see
discussion above Section 2.1), one finds that a system should be in
state 1 if
?γc
?
˙Ma> 5.4 × 10−7
1
??1 − βγc
?2?
0.99
?5/2?
?−7/3?
f
0.1
M∗
0.5M?
?7/3
?−1/2
×
B∗
2kG
R∗
2R?
M?yr−1.
(29)
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This threshold value of ˙Mais an order of magnitude larger than
the fiducial value, suggesting that CTTS slow rotators will most
commonly exist in state 2.
However, equation (29) assumes a magnetic-field strength of
2 kG, and, as discussed in Section 4.1, the stars probably have
surface dipole field strengths of less than 200 G. This considera-
tion decreases the threshold value of˙Maby at least two orders of
magnitude, suggesting that typical CTTS systems may exist in state
1. We can also look at this from the standpoint of stellar spin, using
equation (17), which indicates that a system with the adopted fidu-
cial parameters (but with B∗= 200 G) will be in state 1 if it spins
more slowly than 26 per cent of break-up speed. (As discussed
in Section 2.3.1, this also corresponds to an upper limit of Rt?
2.4 R∗.) Thus, if the dipole fields are indeed weak, it is more prob-
able that slow rotators, and even some fast rotators, will be in state
1. Furthermore, the conclusion of Kenyon et al. (1996), that Rt/Rco
typicallyrangesfrom0.6to0.8,suggeststhatCTTSswillbeinstate
1, as long as βγc< 0.3 (see Section 3.1).
We have thus far considered the opening of field lines via the
differential rotation, so the existence of state 1 requires that βγcis
significantly less than unity. Given the significant uncertainty in the
value of β in real systems, it is still not clear whether or not state
1 should be common. From the standpoint of torques on the star,
the most important feature of state 1 is that the stellar field never
connects to the disc outside Rco. Thus, for the following discussion,
we will generalize the definition of state 1 to include any magnetic
configuration in which the star does not connect outside Rco.
State 1 is characterized by a large amount of open stellar field,
so it is natural to consider the effects of a stellar wind in the mag-
netically open region. As discussed in Section 4.1, a wind can even
be responsible for opening the field (which does not depend on our
parameters β and γc). Thus, if a wind (or any other process) keeps
thestellarfieldopenbeyondsomeradius R?
system will be in state 1. For example, Safier (1998) concluded that
stellar winds from CTTSs could result in R?
means that any system rotating more slowly than 19 per cent of
break-up speed will have R?
There is empirical evidence for systems in state 1 from some
numerical simulations of the star–disc interaction, which usually
represent systems with β ? 1. In the simulations of Goodson &
Winglee (1999) and von Rekowski & Brandenburg (2004), as an
example, after the initial state, the stellar field never connects to the
disc outside Rco, even immediately following reconnection events.
These authors report that the only significant spin-down torques on
the star come from the open field regions (though a stellar wind was
not properly included), rather than along field lines connecting the
stars to their discs (but also see Romanova et al. 2002).
It seems that state 1 is a probable configuration for accreting
stars, particularly among slow rotators. Because, in this state, the
net torque from the interaction with the accretion disc only acts to
spin up the star, stars with long-lived accretion phases must some-
howridthemselvesofthisexcessangularmomentum.Stellarwinds
can exert spin-down torques on the star, and if these torques are sig-
nificant(e.g.Tout&Pringle1992),theequilibriumspinratemaybe
simply determined by a balance between this torque and τa. In this
situation,state1couldactuallyrepresenttheexpectedconfiguration
for accreting systems in spin equilibrium.
out,andif R?
out< Rco,the
out? 3R∗. If true, this
out< Rco(equation 1) and be in state 1.
4.2.2 State 2: Rin< Rt< Rco
In this state, the stellar field connects to a finite region of the disc
between Rtand Rout,asillustratedinthemiddlepanelofFig.3.This
represents the typical configuration in many star–disc interaction
models,exceptthatthedeterminationof Routvariesbetweenmodels.
The location of Rtis determined by equation (15) (and see Fig. 5),
and accretion on to the star occurs from there.
The star is spun up by the accretion torque (equation 19) and
magnetic torques (equation 21) from field lines connected to the
region of the disc between Rtand Rcoand spun down by mag-
netic torques from field lines connected between Rco and Rout.
Therefore, a system can exist in an equilibrium, disc-locked state,
in state 2, in which the net torque on the star is zero, and the
spin rate of the star then correlates with accretion parameters (see
Section 3).
When one considers that the differential rotation determines Rout
(via equation 10), both ?eq
Also, as shown in Fig. 9, the range of (non-equilibrium) values of
Rtthat exist in state 2 becomes narrower as β decreases. For the
strong coupling case of βγc= 0.01, a system can only be in state 2
if 0.993 < Rt/Rco< 1.0. So for strong coupling, it is unlikely that a
givensystemwillexistinstate2unlessitisverynearitsdisc-locked
state, which then requires a fast stellar spin.
Anotherintriguingeffectofalargelyopenfieldtopologyisthat,in
orderforasystemtobedisc-locked,thedifferentialmagnetictorque
in the disc dτmmust be stronger (compared with the completely
closed assumption) in order to make up for the decreased size of
the connected region (compare Figs 4 and 6). When the connected
regionisverysmall(i.e.forsmallβ),dτmisverylarge,whichought
tohaveasignificanteffectonthediscstructurethere.Theremaybea
physical limit beyond which the disc cannot respond and accretion
will cease (see discussion of state 3, below). This could possibly
leadtoatime-dependentprocess(e.g.Spruit&Taam1993),perhaps
analogous to the simulations of (e.g.) Goodson et al. (1999, though
their stellar field does not connect outside Rco), and in which there
may still be a time-averaged net torque of zero.
∗ and (Rt/Rco)eqare larger for smaller β.
4.2.3 State 3: Rt> Rco
Because no accretion on to the star occurs, state 3 may characterize
the non-accreting, weak-line T Tauri phase of pre-main-sequence
stellar evolution. Also, as the disc does not extend inside Rco, the
star feels no spin-up torques, only spin-down torques, and so it
cannot exist in spin equilibrium (Sunyaev & Shakura 1977b). A
possible magnetic-field configuration of this state is illustrated in
the bottom panel of Fig. 3.
It is not yet clear under which conditions a system will be in
this state, though the relative values of differential torques (or
stresses)inthediscarecertainlyimportant.Someauthors(e.g.Wang
1995; Clarke et al. 1995) have speculated that state 3 occurs when
| dτm/dτa| becomes greater than one (Cameron & Campbell 1993,
suggested a value of 2) anywhere outside Rco.Inthis work, we have
assumed that the disc will be structured such that equation (14) is
satisfied (see Section 2.3 and Rappaport et al. 2004). However, this
assumption must eventually break down for large enough f, large
enough µ, or small enough˙Ma.
The general question of what conditions govern a system in
state 3, to our knowledge, remains an unanswered astrophysical
problem. It is not clear what will determine the location of Rtin
state 3 (as neither of equations 15 or 18 is then valid). In addition
to magnetic torques, outflows and/or radiation from the star may be
important (Johnstone 1995). Understanding this state is probably
relevant to understanding the transition from classical to weak-line
T Tauri phases, and it may even have further implications for gas
giant planet formation/migration (Lin, Bodenheimer & Richardson
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The spin of accreting stars
181
1996;Trilling,Lunine&Benz2002)andfortheultimatedissipation
of the gas disc.
4.3 Conclusions
We have considered that the opening of magnetic-field lines ex-
pected from differential rotation in the star–disc interaction results
in a largely open field topology. This significantly alters the torque
that a star receives from its accretion disc, compared with previous
models that assume a closed field. Our main conclusions from this
work are the following.
(1) This more open field topology results in a weaker spin-down
torque felt by the star from the disc (Section 2.4). The strongest
possible torque occurs for intermediate magnetic coupling to the
disc. Stronger coupling, as expected near the inner edge of the disc,
resultsinaspin-downtorquethatismorethananorderofmagnitude
below the torque found for the closed-field assumption.
(2) In the disc-locked, spin equilibrium state, this results in a
stellarspinratethatismuchfasterthanpredictedbypreviousmodels
(Section 3.2).
(3) We have identified and discussed three possible magnetic-
fieldconfigurationsinmagneticstar–discsystems(Section4.2).The
three configurations could represent, for example, an evolutionary
sequence for a system with a gradually decreasing mass accretion
rate(ore.g.agraduallyincreasingstellarspinrate).Ourconclusions
about each state, in the context of T Tauri stars, are the following.
(i) Assuming strong magnetic coupling to the disc, slowly ro-
tating CTTSs should be in state 1 if ˙Ma ? 5 × 10−9M?yr−1
(equation 29 for B∗= 200 G). Because typical accretion rates
are higher than this, state 1 may represent a common configura-
tion in these systems. In this state, the star feels no spin-down
torques from the disc. However, if spin-down torques from (e.g.)
a stellar wind are significant, stars may be in spin equilibrium in
state 1, though they should not then be considered ‘disc locked’.
(ii) State 2 is the typical configuration assumed in star–disc
interaction models. For strong magnetic coupling in the disc, or
if stellar or disc winds are important, we find that a system can
only be in state 2 under special circumstances. In particular, the
accretion rate must be lower than for state 1.
(iii) State 3 probably represents the non-accreting, weak-line
T Tauri phase. Given that the accretion disc can restructure it-
self in response to (e.g.) external magnetic torques, it it not yet
clear when a system will transition into this state. This important
evolutionary phase requires more theoretical study.
(4) These considerations, and additional issues from the litera-
ture, suggest that slowly rotating CTTSs probably cannot be ex-
plained by a disc-locking scenario (Section 4.1).
(5) IfslowlyrotatingCTTSsareinspinequilibrium,thenanother
spin-down torque must be active in the system. We suggest that this
might arise from magnetized stellar winds.
ACKNOWLEDGMENTS
Thisworkhasbeensignificantlyinfluencedbyopencommunication
lines with Dmitri Uzdensky, Keivan Stassun and Arieh K¨ onigl, and
discussion with Cathie Clarke and Bob Mathieu – we are grateful
for their contributions. This research was supported by the National
Science and Engineering Research Council (NSERC) of Canada,
McMaster University, and the Canadian Institute for Theoretical
Astrophysics through a CITA National Fellowship awarded to REP.
REFERENCES
Agapitou V., Papaloizou J. C. B., 2000, MNRAS, 317, 273
Aly J. J., 1985, A&A, 143, 19
Aly J. J., Kuijpers J., 1990, A&A, 227, 473
Anderson J. M., Li Z., Krasnopolsky R., Blandford R. D., 2003, ApJ, 590,
L107
Armitage P. J., Clarke C. J., 1996, MNRAS, 280, 458 (AC96)
Bacciotti F., Ray T. P., Mundt R., Eisl¨ offel J., Solf J., 2002, ApJ, 222
Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214
Bardou A., Heyvaerts J., 1996, A&A, 307, 1009
Barnes S., Sofia S., Pinsonneault M., 2001, ApJ, 548, 1071
Basri G., Marcy G. W., Valenti J. A., 1992, ApJ, 390, 622
Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883
Bouvier J., Cabrit S., Fernandez M., Martin E. L., Matthews J. M., 1993,
A&A, 272, 176
Bouvier J., Forestini M., Allain S., 1997, A&A, 326, 1023
Cameron A. C., Campbell C. G., 1993, A&A, 274, 309
Clarke C. J., Armitage P. J., Smith K. W., Pringle J. E., 1995, MNRAS, 273,
639
Davidson K., Ostriker J. P., 1973, ApJ, 179, 585
De Marco O., Lanz T., Ouellette J. A., Zurek D., Shara M. M., 2004, ApJ,
606, L151
Ghosh P., Lamb F. K., 1978, ApJ, 223, L83
Ghosh P., Lamb F. K., 1979, ApJ, 234, 296
Goodson A. P., Winglee R. M., 1999, ApJ, 524, 159
Goodson A. P., Winglee R. M., B¨ ohm K. H., 1997, ApJ, 489, 199
Goodson A. P., B¨ ohm K., Winglee R. M., 1999, ApJ, 524, 142
Guenther E. W., Lehmann H., Emerson J. P., Staude J., 1999, A&A, 341,
768
Hartmann L., 2002, ApJ, 566, L29
Hayashi M. R., Shibata K., Matsumoto R., 1996, ApJ, 468, L37
Herbst W., Rhode K. L., Hillenbrand L. A., Curran G., 2000, AJ, 119, 261
HerbstW.,Bailer-JonesC.A.L.,MundtR.,MeisenheimerK.,Wackermann
R., 2002, A&A, 396, 513
Illarionov A. F., Sunyaev R. A., 1975, A&A, 39, 185
Johns-Krull C. M., Gafford A. D., 2002, ApJ, 573, 685
Johns-KrullC.M.,ValentiJ.A.,2000,inASPConf.Ser.198,StellarClusters
andAssociations:Convection,Rotation,andDynamosMeasurementsof
Stellar Magnetic Fields. Astron. Soc. Pac., San Francisco, p. 371
Johns-Krull C. M., Valenti J. A., 2001, in ASP Conf. Ser. 244, Young Stars
NearEarth:ProgressandProspects–TheMagneticFieldofTWHydrae.
Astron. Soc. Pac., San Francisco, p. 147
Johns-Krull C. M., Valenti J. A., Hatzes A. P., Kanaan A., 1999, ApJ, 510,
L41
Johns-Krull C. M., Valenti J. A., Koresko C., 1999, ApJ, 516, 900
Johns-KrullC.M.,ValentiJ.A.,SaarS.H.,HatzesA.P.,2001,inASPConf.
Ser. 223, Cool Stars, Stellar Systems and the Sun: Eleventh Cambridge
Workshop. Astron. Soc. Pac., San Francisco, p. 521
Johnstone D. I., 1995, PhD thesis
Joss P. C., Rappaport S. A., 1984, ARA&A, 22, 537
Kato Y., Hayashi M. R., Miyaji S., Matsumoto R., 2001, Adv. Space Res.,
28, 505
Kato Y., Hayashi M. R., Matsumoto R., 2004, ApJ, 600, 338
Kenyon S. J., Yi I., Hartmann L., 1996, ApJ, 462, 439
K¨ onigl A., 1991, ApJ, 370, L39 (K91)
K¨ onigl A., Pudritz R. E., 2000, in Mannings V., Boss A. P., Russell S. S.,
eds, Protostars and Planets IV. Univ. of Arizona Press, Tucson, p. 759
K¨ uker M., Henning T., R¨ udiger G., 2003, ApJ, 589, 397
Leonard P. J. T., Livio M., 1995, ApJ, 447, L121
Lin D. N. C., Bodenheimer P., Richardson D. C., 1996, Nat, 380, 606
Livio M., Pringle J. E., 1992, MNRAS, 259, 23P
Lovelace R. V. E., Romanova M. M., Bisnovatyi-Kogan G. S., 1995,
MNRAS, 275, 244
Lynden-Bell D., Boily C., 1994, MNRAS, 267, 146
Makishima K., Mihara T., Nagase F., Tanaka Y., 1999, ApJ, 525, 978
Matt S., Pudritz R. E., 2004, ApJ, 607, L43
Matt S., Goodson A. P., Winglee R. M., B¨ ohm K., 2002, ApJ, 574, 232
C ?2004 RAS, MNRAS 356, 167–182
by guest on December 21, 2015
http://mnras.oxfordjournals.org/
Downloaded from
Page 16
182
S. Matt and R. E. Pudritz
Miller K. A., Stone J. M., 1997, ApJ, 489, 890
Muzerolle J., Calvet N., Brice˜ no C., Hartmann L., Hillenbrand L., 2000,
ApJ, 535, L47
Muzerolle J., Calvet N., Hartmann L., 2001, ApJ, 550, 944
Newman W. I., Newman A. L., Lovelace R. V. E., 1992, ApJ, 392, 622
Ostriker E. C., Shu F. H., 1995, ApJ, 447, 813
Paczynski B., 1991, ApJ, 370, 597
Parker E. N., 1958, ApJ, 128, 664
Patterson J., 1994, PASP, 106, 209
Pesenti N., Dougados C., Cabrit S., Ferreira J., Casse F., Garcia P., O’Brien
D., 2004, A&A, 416, L9
Piirola V., Hakala P., Coyne G. V., 1993, ApJ, 410, L107
Popham R., Narayan R., 1991, ApJ, 370, 604
Pyo T. et al., 2003, ApJ, 590, 340
Rappaport S. A., Fregeau J. M., Spruit H., 2004, ApJ, 606, 436
Rebull L. M., Wolff S. C., Strom S. E., 2004, AJ, 127, 1029
Reipurth B., Bally J., 2001, ARA&A, 39, 403
Reid M. A., Pudritz R. E., Wadsley J., 2002, ApJ, 570, 231
RomanovaM.M.,UstyugovaG.V.,KoldobaA.V.,LovelaceR.V.E.,2002,
ApJ, 578, 420
Safier P. N., 1998, ApJ, 494, 336
Sano T., Inutsuka S., Turner N. J., Stone J. M., 2004, ApJ, 605, 321
Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
ShuF.,NajitaJ.,OstrikerE.,WilkinF.,RudenS.,LizanoS.,1994,ApJ,429,
781
Sills A., Pinsonneault M. H., Terndrup D. M., 2000, ApJ, 534, 335
Sills A., Faber J. A., Lombardi J. C., Rasio F. A., Warren A. R., 2001, ApJ,
548, 323
Smirnov D. A., Fabrika S. N., Lamzin S. A., Valyavin G. G., 2003, A&A,
401, 1057
SmirnovD.A.,LamzinS.A.,FabrikaS.N.,ChuntonovG.A.,2004,Astron.
Lett., 30, 456
Spruit H. C., Taam R. E., 1993, ApJ, 402, 593
Stassun K. G., Mathieu R. D., Mazeh T., Vrba F. J., 1999, AJ, 117, 2941
Stassun K. G., Mathieu R. D., Vrba F. J., Mazeh T., Henden A., 2001, AJ,
121, 1003
Sunyaev R. A., Shakura N. I., 1977a, Pis’ma Astron. Zh., 3, 262
Sunyaev R. A., Shakura N. I., 1977b, Pis’ma Astron. Zh., 3, 216
Testi L., Bacciotti F., Sargent A. I., Ray T. P., Eisl¨ offel J., 2002, A&A, 394,
L31
Tinker J., Pinsonneault M., Terndrup D., 2002, ApJ, 564, 877
Tout C. A., Pringle J. E., 1992, MNRAS, 256, 269
Trilling D. E., Lunine J. I., Benz W., 2002, A&A, 394, 241
Uzdensky D. A., 2004, in Gomez de Castro A. I., Heyer M., Vazquez-
SemadeniE.,ReboloR.,TaggerM.,PudritzR.E.,eds,Astrophys.Space
Sci. Library Vol. 291, Magnetic Fields and Star Formation: Theory ver-
sus Observations. Kluwer, Dordrecht, in press (astro-ph/0310104)
Uzdensky D. A., K¨ onigl A., Litwin C., 2002a, ApJ, 565, 1191 (UKL)
Uzdensky D. A., K¨ onigl A., Litwin C., 2002b, ApJ, 565, 1205
von Rekowski B., Brandenburg A., 2004, A&A, 420, 17
Wang Y.-M., 1995, ApJ, 449, L153
Yi I., 1994, ApJ, 428, 760
Yi I., 1995, ApJ, 442, 768
This paper has been typeset from a TEX/LATEX file prepared by the author.
C ?2004 RAS, MNRAS 356, 167–182
by guest on December 21, 2015
http://mnras.oxfordjournals.org/
Downloaded from