Can Fu Orionis Outbursts Regulate the Rotation

CAN FU ORIONIS OUTBURSTS REGULATE THE

ROTATION RATES OF T TAURI STARS?

Rob ert Popham

Harvard-Smithsonian Center for Astrophysics

ABSTRACT

We prop ose that FU Orionis outbursts may play an imp ortant role in

maintaining the slow rotation of classical disk-accreting T Tauri stars. Current

estimates for the frequency and duration of FU Orionis outbursts and the mass

accretion rates of T Tauri and FU Orionis stars suggest that more mass maybe

accreted during the outbursts than during the T Tauri phases. If this is the case,

then the outbursts should also dominate the accretion of angular momentum.

During the outbursts, the accretion rate is so high that the magnetic eld of

the star should not disrupt the disk, and the disk will extend all the wayinto

the stellar surface. Standard thin disk mo dels then predict that the star should

accrete large amounts of angular momentum, which will pro duce a secular

spinup of the star.

We present b oundary layer solutions for FU Orionis parameters which show

that the rotation rate of the accreting material reaches a maximum value which

is far less than the Keplerian rotation rate at the stellar surface. This is due to

the imp ortance of radial pressure supp ort at these high mass accretion rates.

As a result, the rate at which the star accretes angular momentum from the

disk drops rapidly as the stellar rotation rate increases. In fact, the angular

momentum accretion rate drops b elow zero for stellar rotation rates which are

always substantially b elow breakup, but dep end on the mass accretion rate and

on the adopted de nition of the stellar radius. When the angular momentum

accretion rate drops close to zero, the star will stop spinning up. Faster stellar

rotation rates will pro duce negative angular momentum accretion which will

spin the star down. Therefore, FU Orionis outbursts can keep the stellar

rotation rate close to some equilibrium value for which the angular momentum

accretion rate is small. We show that this equilibrium rotation rate maybe

similar to the observed rotation rates of T Tauri stars; thus we prop ose that FU

Orionis outbursts may b e resp onsible for the observed slow rotation of T Tauri

stars. This mechanism is indep endent of whether the disk is disrupted by the

stellar magnetic eld during the T Tauri phase.

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Subject headings: accretion, accretion disks|stars: formation|stars:

pre-main-sequence|stars: rotation

1. Intro duction

The FU Orionis stars are a class of accreting pre-main sequence stars which are

observed to b e T Tauri stars undergoing large outbursts (see Hartmann, Kenyon, &

Hartigan 1993 for a review). During the outbursts, these systems brighten by 5 6

magnitudes over a p erio d of 1 10 years. They reach luminosities of a few hundred

L , with sp ectra indicating maximum temp eratures 6500 7000 K. The outbursts last

for 100 years, although this numb er is p o orly known, since no outbursting systems have

b een observed to return to their pre-outburst state. Hartmann & Kenyon (1985, 1987)

showed that FU Orionis stars contain accretion disks by demonstrating that sp ectra of these

systems contain double-p eaked absorption lines. Simple steady accretion disk mo dels of two

FU Orionis systems, FU Orionis and V1057 Cygni, were constructed by Kenyon, Hartmann,

& Hewett (1988, hereafter KHH). The mo dels which b est tted the observed broad-band

4 1

sp ectral energy distributions of these systems had accretion rates M 10 M yr , and

central stars with masses M 0:3 1M and radii R 4R . The presence of a disk

in these systems suggests that their outbursts may result from a disk instability similar to

those b elieved to cause similar outbursts in dwarf novae (Clarke, Lin, & Papaloizou 1989;

Clarke, Lin, & Pringle 1990; Bell & Lin 1994; Bell et al. 1995).

Disk accretion also has imp ortant implications for the rotational evolution of the

accreting star. In the standard picture of disk accretion (Shakura & Sunyaev 1973), the

star accretes angular momentum from the disk. The angular momentum accretion rate is

simply the Keplerian sp eci c angular momentum at the stellar surface multiplied by the

mass accretion rate. This quickly spins up the accreting star. Thus, in the standard picture,

ifTTauri or FU Orionis stars have accreted even a small fraction of their mass through a

disk, they should b e spinning rapidly. The rotation rates of FU Orionis stars are not well

known, since the accretion luminosity is so high that it overwhelms the stellar luminosity.

The rotation rates of T Tauri stars are more easily observable and have b een studied in

some detail (Bouvier et al. 1993, 1995). They tend to b e ab out a tenth of breakup sp eed,

on average, with rotation p erio ds usually around 6{8 days. Since the angular momentum

added by disk accretion should rapidly spin T Tauri stars up to breakup sp eed, some

authors havehyp othesized that the spinup of T Tauri stars is controlled by the interaction

between the disk and a strong stellar magnetic eld (Konigl 1991; Cameron & Campb ell

1993; Hartmann 1994; Shu et al. 1994).

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This mechanism for maintaining the spin rate of a star near some equilibrium value

was rst explored in detail in order to explain the spin b ehavior of some accreting neutron

stars (Ghosh, Lamb, & Pethick 1977; Ghosh & Lamb 1979a,b). In this mo del, the stellar

magnetic eld disrupts the disk at some distance from the stellar surface. The lo cation

of this disruption radius is determined by the strength of the magnetic eld and the mass

accretion rate; a stronger magnetic eld disrupts the disk farther away from the star, while

a larger mass accretion rate moves the disruption radius closer to the stellar surface. The

Keplerian rotation sp eed at the disruption radius determines approximately how fast the

star will rotate. In order to pro duce the rotation sp eeds of most T Tauri stars, the disk

would have to b e disrupted around 5 stellar radii from the stellar surface, and for the

accretion rates commonly ascrib ed to T Tauri stars, this would require a stellar magnetic

eld 1 kG (Konigl 1991).

The magnetic hyp othesis for explaining the slow rotation of T Tauri stars do es not

include the e ects of angular momentum accretion during FU Orionis outbursts. Event

statistics of these outbursts, in which the numb er of observed events is compared to the

number of young stars in the observable volume, suggest that each star probably undergo es

multiple outbursts (Herbig 1977; Hartmann & Kenyon 1985). Estimates for the time

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between outbursts vary widely from 10 10 years (e.g. Bell & Lin 1994; Kenyon 1995).

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If the accretion rate during this p erio d is the typical T Tauri rate of 10 M yr , then

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10 10 M will b e accreted during the T Tauri phase. During the intervening FU

4 1 2

Orionis outbursts, the accretion rate is 10 M yr for 100 years, so 10 M

should b e accreted during the outburst. This suggests that as an accreting pre-main

sequence star evolves through multiple cycles of FU Orionis outbursts and T Tauri quiescent

phases, most of the mass accreted by the star may b e added during the outbursts. If this

is the case, then we should exp ect that the outbursts will also dominate the accretion of

angular momentum.

During the FU Orionis outbursts, the stellar magnetic eld should have little e ect on

the accretion disk. As noted ab ove, in order to pro duce the observed rotation rates of T

Tauri stars, the eld should disrupt the disk at a radius of ab out 5 stellar radii. When such

a star exp eriences an FU Orionis outburst, the mass accretion rate increases by a factor of

order 1000. This should b e sucienttooverwhelm the magnetic eld, and allow the disk

to reach all the way in to the stellar surface. In the standard picture of accretion disks,

since the star accretes a large amount of mass during the FU Orionis outburst, it should

also accrete a large amount of angular momentum, which will spin the star up substantially.

This will in turn require that the star loses large amounts of angular momentum during the

TTauri phase in order to stay at the observed slow rotation rate. If the star accretes less

mass in the T Tauri phase than it do es in the FU Orionis outburst, it will b e unable to lose

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sucient angular momentum to spin backdown, and over time, the star will continue to

spin up toward breakup sp eed.

If the disk reaches the stellar surface, and the star is rotating slowly, a viscous b oundary

layer forms b etween the rapidly rotating disk material and the star. The b oundary layer

controls the ow of angular momentum b etween the disk and the star. Also, if the central

star is not rotating, the b oundary layer should pro duce as much luminosity as the disk.

Clearly,ifwe wish to understand the angular momentum transfer and to compare disk

mo dels to observed sp ectra of FU Orionis systems, it is essential to include the b oundary

layer in our disk mo dels.

Our initial investigation of the b oundary layer structure examined the question of

whether a star continues to accrete mass and angular momentum when it has b een spun

up to breakup sp eed (Popham & Narayan 1991, hereafter PN91). We found that as the

star approaches the breakup rotation rate, the amount of angular momentum accreted

p er unit of mass accreted drops dramatically, and can even b ecome negative, so that the

star continues to accrete mass while losing angular momentum to the disk. Thus, there

is some equilibrium rotation rate close to breakup for which the star accretes no angular

momentum, so that it can continue to accrete mass without spinning up or down. This

study used a very simple disk mo del with a p olytropic relation b etween the disk pressure

and density,soitwas not clear that similar conclusions would hold when a more realistic

treatment of the disk was used. More recently,wehave explored the structure of the

b oundary layer region for several typ es of accreting stars, including T Tauri and FU Orionis

stars (Popham et al. 1993, hereafter PNHK) and cataclysmic variables (Narayan&Popham

1993; Popham & Narayan 1995), using a more realistic mo del which includes the energy

balance and radiative transfer.

In this pap er, we examine the angular momentum transfer b etween the star and the

disk for b oundary layer solutions calculated for parameters corresp onding to FU Orionis

systems. The b oundary layer mo del we use is similar to the one used in our studies of

cataclysmic variables (Narayan & Popham 1993; Popham & Narayan 1995) and in our

previous study of pre-main sequence stars (PNHK). We nd that many of the conclusions

reached by PN91 continue to apply when our more detailed mo del is applied to FU Orionis

systems. When the stellar rotation rate is plotted against the height of the disk at the

stellar surface, the solutions fall on multiple solution branches. We nd that the angular

momentum accretion rate drops as the stellar rotation rate increases, and that solutions

exist for a wide range of angular momentum accretion rates, including negativevalues.

There is an equilibrium stellar rotation rate for which the star can accrete mass without

spinning up or down. The ma jor di erence from the PN91 results is that for these FU

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Orionis-typ e solutions, the equilibrium stellar rotation rate is well b elow breakup.

On the basis of these solutions, we prop ose that FU Orionis outbursts playan

imp ortant role in the spin evolution of accreting pre-main sequence stars. The large amount

of mass accreted during an outburst can change the stellar rotation rate substantially.If

more mass is accreted during the FU Orionis outburst phases than during the intervening T

Tauri phases, then the stellar rotation rate will stay close to the equilibrium rate established

during the FU Orionis phase. Although this equilibrium rate is rather uncertain, we will

show that for reasonable choices of parameters, it is comparable to the observed rotation

rates of T Tauri stars. Thus, angular momentum transfer during the FU Orionis outbursts

can explain the slow rotation of T Tauri stars without invoking strong stellar magnetic

elds. If T Tauri stars haveweak elds which allow the accretion disk to extend down to

the surface of the star and spin up the star, negative angular momentum accretion during

the FU Orionis outburst phases can keep the star spinning slowly. Alternatively,ifTTauri

stars have strong magnetic elds which disrupt the disk, the FU Orionis phases will still

dominate the spin evolution as long as they dominate the mass accretion.

In x2, we discuss the mo del used to calculate our b oundary layer and disk solutions

for FU Orionis parameters, and the input parameters for our solutions. We describ e a

typical solution in detail in x3. We demonstrate the presence of multiple solution branches,

and showhow the character of the solutions varies along those branches. We also showa

solution which accretes mass without accreting angular momentum. Finally,we showhow

variations in the mass accretion rate a ect the solutions and solution branches. In x4, we

describ e the choice of an additional condition which allows us to nd a relation b etween

the stellar rotation rate and the angular momentum accretion rate. We discuss the spin

evolution of T Tauri/FU Orionis systems and the implications of our results in x5.

2. Disk and Boundary Layer Mo del

2.1. Disk Equations

The mo del we use to study the b oundary layer uses a version of the \slim disk"

equations develop ed byPaczynski and collab orato rs (Paczynski & Bisnovatyi-Kogan 1981;

Muchotrzeb & Paczynski 1982). This mo del is similar to the mo dels used in previous pap ers

(Narayan & Popham 1993; PNHK; Popham & Narayan 1995), so we will only describ e it

qualitatively here, and refer the reader to those pap ers for more details.

The mo del assumes a steady state, so that the mass and angular momentum accretion

_ _

rates M and J are constant with radius. We use the slim disk versions of the angular

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momentum, radial momentum, and energy equations. Thus, we disp ense with some of the

simplifying assumptions made in the standard thin disk treatment (Shakura & Sunyaev

1973). We include pressure supp ort and acceleration in the radial momentum equation, so

that the rotational velo city is not required to equal the Keplerian value .We allow

_ _

the angular momentum accretion rate J to deviate from the standard value M (R )R

_ _

. Finally,we drop the assumption that the energy by a factor j , so that J = j M (R )R

dissipated by viscosity is radiated lo cally at the same radius in the disk. Instead, we include

terms for the radial transp ort of energy by radiation and by the accreting material itself.

The radiative transfer scheme we use is the one describ ed in PNHK.

The mo di ed angular momentum, radial momentum, and energy equations, along with

the radiative transfer equations, provide a set of di erential equations which can b e solved

with appropriate b oundary conditions. The most imp ortant of these are the conditions on

the angular velo city and the radial radiative ux. The angular velo city must match the

stellar rotation rate at R , and the Keplerian value (R )atR=R = 100R .At

K out out

R, there is an outward radiative ux characterized by an e ective temp erature T , which

34 1

we take to b e 5000 K. This pro duces a luminosityof10 ergs s entering the b oundary

36 1

layer from the star, which is a small fraction of the accretion luminosity L 10 ergs s .

acc

The equations are solved using a relaxation metho d.

2.2. Solution Parameters

Within the framework of the disk and b oundary layer mo del describ ed ab ove and in

the aforementioned pap ers, a large numb er of solutions can b e found. These solutions are

distinguished byvarious choices of input parameters. For our mo dels, these include the

_ _

mass accretion rate M , the angular momentum accretion rate J , the stellar mass, radius,

and rotation rate M , R , and , and the viscosity parameter . In the case of FU Orionis

systems, none of these parameters are directly observable, so they must b e estimated by

using the b est current mo dels to interpret the available observations.

2.2.1. M; M ; andR

These are the standard accretion disk parameters which determine the disk luminosity

and temp erature. Observed sp ectra of FU Orionis and V1057 Cygni were tted with simple

accretion disk mo dels by KHH. Although these mo dels did not include the b oundary layer,

they allow us to estimate the basic disk parameters as follows. First, the infrared ux is

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pro duced primarily in regions of the disk which are fairly far from the stellar surface, so that

the ux pro duced is insensitive to the value of R .Wehave calculated simple blackbody

sp ectra of our disk and b oundary layer solutions for various choices of M and M , and

compared them to photometric data of V1057 Cygni. Fits to infrared photometry suggest

1 5

yr . Our sp ectra and ts will b e presented in that the combination MM ' 5 10 M

a future pap er (Popham et al. 1995).

Another constraintisprovided by sp ectra of FU Orionis and V1057 Cygni, which

show that the maximum e ective temp erature reached in the inner disk is around 7000 K

(KHH; Kenyon et al. 1989). In the standard Shakura & Sunyaev (1973) mo del, the disk

1=4 3

) at R =49=36R . reaches a maximum e ective temp erature T =0:287(GM M=R

max

3=4

,soitprovides a reasonable way of estimating R ;however, the Shakura- T / R

max

Sunyaev formulation do es not include any luminosity from the b oundary layer, so the true

1 5

yr ,we temp erature of the inner disk will b e higher than T .For MM =510 M

max

nd T ' 6500 K for R =310 cm=4:31 R . This value for R is somewhat larger

max

than the observational estimates for T Tauri star radii, such as those tabulated by Bouvier

et al. (1995), where R ' 1:5 2:5R .

This combination of parameters gives an accretion luminosity L '

acc

36 1

1:4 10 ergs s ' 350L , which agrees with the b olometric luminosity estimates

for most of the FU Orionis systems listed by Bell & Lin (1994). In the solutions presented

in this pap er, wekeep M =0:5M constant, and vary M and R so that the accretion

4:0 4:15

luminosity remains nearly constant. The three sets of solutions have M =10 ;10 ,

4:3 1 11 11 11

and 10 M yr , and R =410 ; 3 10 , and 2:25 10 cm, resp ectively.

2.2.2.

The viscosity parameter is more dicult to determine. The most common wayto

do so is to try to estimate the viscous timescale in the disk by observing the disk outburst

b ehavior. This assumes that the FU Orionis outbursts are pro duced by disk instabiliti es,

or at least by some disturbance which dies away on the viscous timescale. Disk instability

3 4

mo dels of FU Orionis outbursts have found that small values of 10 10 were

required to repro duce the long timescales of observed outbursts (Clarke, Lin & Pringle

1990; Bell & Lin (1994); Bell et al. 1995). Unfortunately, these values of lead to problems

with the evolutionary timescales and gravitational stability of the disk (Bell et al. 1995).

Wehave therefore adopted an intermediate value of =0:01 whichwe use for all of our mo dels.

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2.2.3. and j

These parameters are usually ignored in accretion disk studies, since in the standard

Shakura & Sunyaev-typ e mo del, the accreting star is assumed to b e nonrotating, and the

angular momentum accretion rate is just given by the Keplerian sp eci c angular momentum

at the stellar surface multiplied by the mass accretion rate. Our study treats the b oundary

layer region explicitly, so these assumptions are no longer necessary.We treat b oth the

stellar rotation rate and the angular momentum accretion rate as free parameters for

purp oses of nding our solutions. Later, when we discuss the spin evolution of accreting

pre-main-sequence stars in x4, we will attempt to de ne a relation which sp eci es the

angular momentum accretion rate as a function of the stellar rotation rate.

3. Solutions and Solution Branches

3.1. ATypical Solution

4:15 1

We b egin by discussing an individual b oundary layer solution with M =10 M yr

11 11

and R =310 cm, with j =0:90 and H (R )=10 cm. These parameters are typical

of most of the solutions presented here. Note that wehave sp eci ed the disk heightat R

rather than ; the reason for this is discussed b elow. Figure 1 shows the variation of the

rotational velo city , the radial velo city v , the e ective temp erature T , and the disk

R ef f

height H in the region R =13 R .

The rotational velo city reaches a maximum around R ' 5 10 cm ' 5=3 R ,so

the dynamical b oundary layer is ab out 2=3 R wide. Note that the maximum is only

ab out 70% of at that radius. This means that even at the outer edge of the dynamical

2 2

b oundary layer, R is only ab out 50% of the gravitational term R; the other half of

the supp ort against gravity is provided by the pressure gradient.

The radial velo city v shows a similar pro le to . It reaches a maximum of

4 1 11

cm s at R ' 6 10 cm ' 2 R .Thus, the accreting material moves radially 3 10

through the b oundary layer region in ab out 1 year. In the b oundary layer, the radial

velo city drops rapidly, reaching 2000 cm s at the stellar surface.

The e ective temp erature T p eaks closer to the stellar surface than either or v .

ef f R

In fact, the maximum T 8300 K is reached at R =3:810 cm, in the middle of the

ef f

dynamical b oundary layer. The radiative energy released in the dynamical b oundary layer

propagates out to larger radii, but most of it has b een radiated from the disk surface inside

2 R .Thus, the thermal b oundary layer (the region of elevated e ective temp erature) is

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only slightly wider than the dynamical b oundary layer.

The disk height H is fairly large: the ratio H=R is close to 1/3 throughout the inner

region. Note that in earlier studies of the b oundary layer structure at lower accretion

rates, we generally found that H increased rapidly in the pressure-gradient supp orted

region inside the dynamical b oundary layer. Here, as noted ab ove, the accreting material

is already substantially pressure-gradient supp orted at the outer edge of the dynamical

b oundary layer, and no rapid increase in H is seen at the innermost radii.

3.2. Solution Branches

One goal of this pap er is to examine the variation in the b oundary layer structure as

the various system parameters discussed in x2.2 change. We are particularly interested in

the spin history of accreting pre-main sequence stars, so we fo cus on the variations pro duced

bychanges in the rotation rate of the star and the angular momentum accretion rate j .

Accordingly,we rst varied while keeping all of the other parameters constant, including

M , M , R , , and j . This pro duced an intriguing result. We found that for some of our

solutions, as we increased , the disk height H H (R ) increased, while other solutions

showed the opp osite b ehavior: H decreased as the star spun faster.

We continued to systematically increase and decrease , and plotted the values of H

in the resulting solutions as a function of . This revealed that the tracks traced out in

the H plane connected together into a single curve, like the one shown in Figure 2.

This curve has a Z shap e; multiple solutions can b e found for a single value of . These

solutions have di erentvalues of the disk height H , and they are distinct solutions, despite

the fact that they share the same values of all of the input parameters M; M ;R ; ; ; and

j . In fact, by sp ecifying H rather than ,we can nd a unique solution. The Z-shap ed

curve has three branches; on the middle branch, H increases with increasing , while on

the upp er and lower branches, H decreases with increasing . The existence of multiple

solution branches for a single value of j agrees with the ndings of our earlier work (PN91),

which used a much simpler disk formulation.

3.2.1. Variations Along the Branches

The solutions along a Z-shap ed curve show a wide range of variation in the structure

of the b oundary layer. In Figure 3 wehave plotted 10 solutions with H ranging from

10 11

8 10 cm to 1:25 10 cm. The p ositions of these 10 solutions in the H plane are

{10{

marked on the Z-shap ed curve shown in Fig. 2 The solutions in the upp er left have large

values of H and small values of corresp onding to a slowly rotating star. These solutions

havea very broad b oundary layer, as shown in Figure 3. The p eak values of and v sit

out at R ' 4R . The p eak e ective temp erature, o ccurs much closer to the star, but the

region of elevated e ective temp erature extends out to several stellar radii. The disk height

H is approximately prop ortional to R, with H=R 0:4, as shown in Fig. 3c.

Solutions with smaller values of H , but still lying on the upp er branch, have

progressively narrower b oundary layers. The last solution shown on the upp er branch, with

H =1:05 10 cm, has at R<2R (Fig. 3). Meanwhile, increases gradually,

max

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reaching a value of 3:5 10 s at the transition from the upp er to the middle branch.

5 1

Note that this value of is less than 10% of (R ) ' 5 10 s .

As H continues to decrease along the middle branch, the b oundary layer continues

to shrink in radial extent. The changes in the solutions along the middle branch with

decreasing H seem to continue smo othly the changes along the upp er branch, with one

exception: drops backdown toward zero. The H =910 cm solution, near the

transition from the middle to the lower branch, has at R ' 4 10 cm ' 4=3 R (Fig.

max

3), so the radial extent of the b oundary layer has decreased substantially. This solution has

7 1

' 5 10 s ' 0:01 (R ).

Finally, along the lower branch, the trends noted ab ove for the upp er and middle

branches continue. The b oundary layer b ecomes quite narrow, with at R ' 1:15R in

max

the H =810 cm solution. The value of increases rapidly, and the solutions havea

large angular velo city gradient d =dR at R . This concentrated energy release pro duces a

strong p eak in T near R , but T b egins to drop as H continues to decrease, b ecause

ef f ef f

the large values of for these solutions means that a large p ortion of the rotational energy

of the accreting material is retained rather than b eing dissipated in the b oundary layer.

The ratio H=R varies more with R for the lower branch solutions than for solutions on the

other branches; the H =810 cm solution has H=R =0:24 at R = R but H=R ' 1=3

at R =3R .

3.3. Angular Momentum Accretion Rate

Thus far wehavekept the angular momentum accretion rate j constant and showed

that a family of solutions with a given value of j traces out a Z-shap ed curve in the H

plane. Nowwe show the e ects of varying the angular momentum accretion rate j .For each

value of j , and using the same M , M , R , and as b efore, we can follow the variations in

{11{

as wevary H . These variations are shown in Figure 4.

As we increase j ,we nd a similar Z-shap ed curve in the H plane, but the curve

is displaced to the left, i.e. to smaller values of .For j =0:91, the transition b etween the

lower and middle branches has moved to negativevalues of . The middle-to-upp er branch

transition also moves left, but by a smaller amount. The Z-shap ed curves for adjacent

values of j nest inside one another, and they continue to b e displaced to smaller as j

continues to increase.

If we decrease j , the curves are displaced to larger , but they also b egin to change

shap e dramatically.Atj=0:89, the middle branch has b ecome much shorter and steep er,

so that it extends over a very narrow range in .Atj=0:84, the middle branch has

vanished completely, leaving a smo oth, monotonic curve with H decreasing as increases.

If we continue to decrease j ,we continue to nd solutions, even for j 0. In these

solutions, the star accretes mass while losing angular momentum. An example of a j =0

solution is shown in Figure 5. The angular velo city continues to rise all the wayintothe

stellar surface. Nonetheless, is far b elow (R), so the inner region of the disk is

almost completely pressure-supp orted. There is no maximum in , so in essence there is no

b oundary layer at all. The disk is fairly thick, with H=R ' 0:4. We can nd solutions for

negativevalues of j , and even for large negativevalues suchas j =10. As j drops, the

disk continues to thicken. Wehave included the curves for j = 0 and j = 1 in the H

plane in Figure 4. The curves are rather similar to the j =0:84 curve discussed ab ove; H

decreases monotonically with increasing .

3.4. Variations with M

Having describ ed the app earance of the family of curves in the H plane pro duced

by a range of values of j ,we can examine the variations in this family of curves as the

mass accretion rate changes. As discussed in x2.2, we also vary R along with M so as to

keep the accretion luminosity approximately constant, so that we examine 3 combinations

4:0 4:15 4:3 1 11 11

_ _

of M and R : M =10 ;10 , and 10 M yr , and R =410 ; 3 10 , and

2:25 10 cm, resp ectively.

4:0 1

In Figure 6, we show the families of curves in the H plane for M =10 M yr ,

11 4:3 1 11

R =410 cm and for M =10 M yr , R =2:25 10 cm. These can b e compared

4:15 1 11

to Fig. 4, which used intermediate values of M =10 M yr , R =310 cm.

The shap es of the curves and their variation with j are quite similar for the 3 di erent

accretion rates. The ma jor di erence is that the family of curves is displaced to larger

{12{

_ _

as M decreases, and to smaller as M increases. Thus, at the lowest accretion rate,

4:3 1

M =10 M yr , the transition from Z-shap ed to monotonic curves o ccurs around

5 1 4:15 1

' 1:5 10 s ' 0:2 (R ). This can b e compared to the M =10 M yr

6 1

case, where the same transition happ ens around ' 5 10 s ' 0:1 (R ). At

4:0 1

M =10 M yr , all of the Z-shap ed curves havemoved to < 0, corresp onding to a

star rotating in the opp osite sense to the disk.

4. Spin Evolution of Pre-Main-Sequence Stars

4.1. Angular Momentum Accretion

In order to study the spin evolution of a disk-accreting star, we need to knowhowmuch

angular momentum is accreted by the star as a function of its rotation rate. The answer to

this question will determine the spin evolution of the star. Of course, the spin evolution

may b e complicated byvariations in the mass accretion rate or the mass and radius of the

star. These considerations are imp ortant for T Tauri/FU Orionis stars, and we will discuss

them later.

To b egin with, however, we concentrate on the variation of j as a function of for a

star with constant M , M , and R . The general sense of this variation is clear from Figs.

4 and 6, which showhowj varies with and H .Ifwe assume for the moment that H

stays constant, then j decreases as increases. At small values of , where the constant-j

curves are Z-shap ed, j decreases gradually. As the star spins up, the constant-j curves

b ecome monotonic, and j decreases more rapidly. This suggests that the spin evolution of

the star pro ceeds as follows: as long as j remains p ositive, the star continues to spin up.

Eventually, j decreases to zero, and the star reaches an equilibrium rotation rate where it

neither spins up nor spins down.

The ma jor uncertainty in this picture is in H : for a given value of , j also dep ends

strongly on the value of H .Thus, in some regions of the H plane, if H were to

decrease rapidly enough as increases, j would increase. As we discussed in x3, we can

nd solutions by sp ecifying j and either or H . In nature, one would exp ect that only

the stellar rotation rate is an intrinsic parameter of the accreting system, and that

b oth j and H should b e determined by the physics of the accretion ow. In order to

repro duce this situation within our mo del, we need an additional condition which gives H

as a function of . Such a condition will de ne a track in the H plane, and the

variation of j as a function of along this track provides a basis for understanding the

spin evolution of accreting stars.

{13{

4.2. Need for an Additional Boundary Condition

The ma jor reason for the uncertaintyin H is the lack of a clear de nition of the stellar

radius R . In the solutions presented thus far, wehave sp eci ed a value for R , but wehave

not sp eci ed a b oundary condition which ensures that the sp eci ed R represents the true

stellar radius in anyphysical sense. As a result, we can obtain a wide range of solutions

whichhavevery di erentphysical characteristics at R , as shown in Fig. 3. Thus, we

need an additional b oundary condition, a de nition for R which marks the lo cation of the

transition from the b oundary layer to the star, based on the changes in the accretion ow.

This is a dicult task, for several reasons. First, our mo del is set up in a way that

sp eci cally tries to avoid making distinctions b etween the disk, b oundary layer and star.

All of these regions are assumed to ob ey the same equations, with the same physical

parameters. Second, the distinction b etween star and disk is much more dicult to make

at large accretion rates. This can b e seen clearly in Fig. 1. It is quite dicult to separate

the accretion owinto di erent regions; , T and H all vary smo othly over large length

ef f

scales. This can b e contrasted with the clear di erences b etween the disk, b oundary layer,

and star which are visible in mo dels for lower accretion rates, like those for cataclysmic

variables (Popham & Narayan 1995) and T Tauri stars (PNHK). In those mo dels,

dropp ed rapidly in the b oundary layer, and then stayed almost constant in the star; the

transition b etween the two regions was quite rapid. Similarly, H b egan to rise rapidly with

decreasing R inside the b oundary layer. In the current solutions, the pro le shows only

weak indications of leveling o , and H continues to decrease approximately linearly with R.

This makes it quite dicult to select a radius which represents a transition from b oundary

layer to star.

Another p ersp ective on this issue comes from examining the range of solutions in Fig.

3, all of which use a single value of j . One way to lo cate the stellar radius is to select a

solution from this set for which the b oundary layer{star transition app ears to o ccur at the

inner edge of our computational grid, where R = R . It seems clear that the solutions with

the smallest values of H are not appropriate. They havevery large values of d =dR at R ,

and it seems clear that if the solution were to continue in to R

drop precipitously.Thus, it app ears that in these solutions, R falls in the middle of the

b oundary layer, and the b oundary layer{star transition would o ccur at R

other hand, the solutions with the largest values of H havevery broad regions, extending

out to a few times R , in which decreases quite gradually. It app ears that these solutions

have reached the b oundary layer{star transition at R>R , and the material then simply

continues to settle as it ows in slowly to R . The intermediate solutions with H ' 10 cm

seem to have the b est characteristics. Both and v have dropp ed substantially but are R

{14{

leveling o around R .

4.3. Choice of a Boundary Condition

Wehave many p ossible choices for b oundary conditions to sp ecify R . One p ossibili ty

would b e to try to match the shap e of the disk surface to the shap e of the surface of a star

rotating at a sp eed ;however, as wehave discussed, the disk height H do es not vary in a

way that resembles the transition from a disk to a star, so this would b e dicult. The b est

variables to use for a b oundary condition are probably the angular and radial velo cities

and v . These variables show directly where the b oundary layer is lo cated, and compared

to other variables like the central and e ective temp eratures of the disk, they dep end far

less on the details of radiative transfer, opacities, etc.

Of the twovelo cities, we b elieve that the radial velo city provides a sup erior measure

of the lo cation of the b oundary layer{star transition. This is b ecause the angular velo city

mayormay not drop as the accreting material approaches the stellar surface; if the star is

rapidly rotating, may drop only slightly,ormay continue to increase, as wesawinthe

solutions with small and negativevalues of j in x3.3. In such cases, it b ecomes very dicult

to distinguish the disk from the star on the basis of the pro le. The radial velo city, on the

other hand, always reaches a maximum value and then drops substantially as it approaches

the stellar surface. Put another way, do es not have to b e small, but v do es.

The simplest condition we can put on v is simply to select a value for v , and de ne

R R;

R as the radius where v = v . This can then b e used to determine the variation of

R R;

H and j with . One simple way to do this is to keep j constant and vary H ,soasto

move along one of the constant-j curves shown in Figs. 4 and 6. Fig. 3 shows that v

varies monotonically along a constant-j curve, such that v decreases with increasing H .

Thus, byvarying H ,we can nd a solution for which v = v at R = R . Corresp onding

R R;

solutions can b e found for other values of j , and the values of and H for these solutions

will then pro duce the desired relations for H ( ) and j ( ).

4.4. Variation of j with

In order to illustrate how j varies with for this b oundary condition, wecho ose a

1 4

typical value of v = 1000 cm s . This value is a factor of 2 10 smaller than the

free-fall velo city, and it is 5-15% of the maximum radial velo city reached in the resulting

solutions. The variation of H ( ) for the solutions with v = v at R =310 cm

R R;

{15{

is shown as a dashed line in Fig. 4. Note that H is nearly constant for the full range

of , with H ' 1:07 1:11 10 cm, so that H =R ' 0:36. There is some variation

11 11

of H ; it decreases gradually from ab out 1:088 10 cm at = 0 to 1:075 10 cm

5 1 11

at ' 1:3 10 s , and then rises more rapidly, reaching ab out 1:108 10 cm at

5 1

' 1:83 10 s .

More imp ortantly, j decreases with increasing ; as the star spins up, it accretes less

angular momentum. Figure 7a shows the variation of j as a function of ; j decreases

gradually at rst, and then drops more rapidly. The angular momentum accretion rate

5 1 5 1

reaches zero at ' 1:5 10 s , and rapidly drops to j = 1at '1:810 s .

Note that these values of are only ab out one third of the Keplerian angular velo cityat

5 1

this radius, (R ) ' 5 10 s .Thus, the star stops accreting angular momentum

when it is still rotating well b elow breakup.

At other mass accretion rates, the variation of H and j with increasing is quite

similar: H stays approximately constant and j decreases gradually at rst, then more

rapidly. The ma jor di erence is that j decreases more rapidly at higher mass accretion

rates, and reaches j = 0 at a smaller value of . This is illustrated in Figure 7b and

4:0 1 4:3 1

_ _

c, where we plot j ( ) for the M =10 M yr and M =10 M yr solutions.

Wehave used the same criterion, v = 1000 cm s , to de ne the stellar radius. At

4:0 1 6 1

M =10 M yr , j = 0 for ' 6:6 10 s , corresp onding to a rotation p erio d of

11 days. Also note that for this higher accretion rate, j is substantially smaller than one

4:3 1 5 1

even at =0.At M =10 M yr , (j =0)'310 s , which corresp onds

toamuch shorter rotation p erio d of 2:4days. Recall that as M changes, wehave also

varied R by a similar amount in order to keep the accretion luminosity constant. Thus

the variation in (j = 0) as a fraction of the breakup rotation rate (R ) is not as

dramatic as the variation in (j = 0) itself; we nd (j =0)= (R )'0:2;0:3;0:4 for

4:0 4:15 4:3 1

M =10 ;10 ; 10 M yr , resp ectively.

5. Discussion

5.1. Spin Evolution of T Tauri/FU Orionis Stars

Wehave found that b oundary layer solutions for parameters corresp onding to FU

Orionis outbursts have small and negative angular momentum accretion rates for fairly low

stellar rotation rates. This suggests that angular momentum transfer during the FU Orionis

outbursts can keep the rotation rate of T Tauri/FU Orionis stars near to an equilibrium

rotation rate ' (j = 0). If the star is rotating faster than when it

;eq ;F U ;eq ;F U

{16{

exp eriences an FU Orionis outburst, it will lose angular momentum in the outburst phase

and spin backdown to . If, on the other hand, the star is rotating slowly when an

;eq ;F U

outburst o ccurs, it will have j 1 in the outburst phase and will spin up toward .

;eq ;F U

5 4 1

Wehave found that for accretion rates M =510 10 M yr , j = 0 for spin rates

5 1

(j =0)'0:66 3:04 10 s ' , corresp onding to rotation p erio ds of 2:4 11

;eq ;F U

days. These are similar to the observed rotation p erio ds of T Tauri stars (Bouvier et al.

1993, 1995); however, (j = 0) is sensitive to the value assumed for v , as discussed

b elow.

During the T Tauri phase, the stellar rotation rate will tend to movetoward a di erent

equilibrium value . If the stellar magnetic eld is to o weak to disrupt the disk, then

;eq ;T T

the star will spin up with j 1until it reaches ' (R ). If the stellar eld do es

;eq ;T T K

disrupt the disk, then will b e determined by the radius where the disruption o ccurs,

;eq ;T T

and may b e smaller or larger than . Therefore j may b e p ositive or negative, with

;eq ;F U

1=2

a magnitude jj j (R =R ) (Ghosh, Lamb,&Pethick 1977), where R is the radius at

d d

which the eld disrupts the disk. Mo dels of magnetically disrupted disks in T Tauri stars

suggest R 4 5R , whichwould give j 2.

The spin evolution of a T Tauri/FU Orionis star will dep end on , , and

;eq ;F U ;eq ;T T

on the amounts of mass accreted in typical FU Orionis and T Tauri phases, M and

M .IfM > M , as suggested by current estimates of the frequency and duration

TT FU TT

of FU Orionis outbursts, the stellar rotation rate will movetoward during each

;eq ;T T

TTauri phase, but will return to during each FU Orionis phase.

;eq ;F U

How large will the excursions in b e? A rigidly rotating star has angular momentum

_ _ _

J = I , where I is the star's moment of inertia, so J = I + I .Ifwe assume that the

_ _

moment of inertia changes slowly, so that J ' I , then the star spins up and down at a

_ _

rate = (R )= (j=k)(M=M), where wehave de ned I kM R . Since k is substantially

less than 1, the increase in as a fraction of the breakup rotation rate (R ) is several

times larger than the fractional increase in the stellar mass M .Ifwe assume j =1, k =0:2,

then the accretion of M =0:01 M onto a 0:5M star would increase by0:1 (R ).

If the typical change in during the T Tauri phase is larger than the di erence b etween

the equilibrium spin rates and , then will reach during eachT

;eq ;F U ;eq ;T T ;eq ;T T

Tauri phase, and remain there for the duration of the phase.

It is imp ortant to note that even though the FU Orionis phases may pro duce more

mass accretion than the T Tauri phases, M > M ,aTTauri/FU Orionis star

FU TT

will still sp end most of its lifetime with moving away from , due to the short

;eq ;F U

duration of the FU Orionis phases. This means that the mean value of during the T

Tauri phase will b e o set somewhat from . The sign of this o set will dep end on

;eq ;F U

{17{

whether the star spins up or down during the T Tauri phase, i.e. on whether is

;eq ;T T

smaller or larger than . Also, the return to the T Tauri state after an FU Orionis

;eq ;F U

outburst may b e gradual, with the mass accretion rate declining slowly over a p erio d of

a few decades. At mass accretion rates intermediate b etween T Tauri and FU Orionis

rates, the star should spin up, since the magnetic eld may still b e to o weak to disrupt the

disk, while the mechanism describ ed in this pap er will only op erate at a large value of the

equilibrium rotation rate. Despite these deviations, the high accretion rate of mass and

angular momentum in the FU Orionis phase will return to quite rapidly, and as

;eq ;F U

long as M > M , will continue to b e close to .

FU TT ;eq ;F U

Note also that in general the equilibrium spin rate will di er slightly from the

;eq

______

value of which gives j = 0. Since J = I + I ,we will have = 0 for J = I .

This simply means that in equilibrium, J must b e sucient to comp ensate for changes in

the star's moment of inertia. If the moment of inertia increases in resp onse to accretion,

j>0at . If the moment of inertia decreases, as it might in an accreting white dwarf,

;eq

then j ( ) < 0. For instance, if the star's radius of gyration stays constant, so that

;eq

_ _

I=I = M=M, then j ( )= k = (R ), where again I kM R .If = (R )isin

;eq ;eq K ;eq K

the range 0.2-0.4 as suggested by our results, and k ' 0:2, then j ( ) ' 0:04 0:08, and

;eq

will b e only slightly smaller than (j = 0).

;eq

5.2. Solutions with Negative Angular Momentum Accretion

One of the most imp ortant results of this pap er is that it establishes the existence of

disk and b oundary layer solutions with small and negative angular momentum accretion

rates. In these solutions, the central star accretes mass but loses angular momentum. This

is quite di erent from the standard thin disk formulation, in which the angular momentum

_ _

accretion rate is always J = M (R )R ,or j = 1 in our units. The key to understanding

this result is that the pro les of our solutions can b e very di erent from the purely

Keplerian assumed in the thin disk case. Most imp ortantly, do es not reach a maximum

and then drop down to the stellar rotation rate . Instead, continues to increase all the

way in to the stellar surface at R = R . The fact that is monotonic means that there is no

radius R at which d =dR = 0, where the viscous torque would vanish, and the angular

momentum accretion rate would simply b e the amount of angular momentum carried in

_ _ _

the accreting material at that radius, J = M R . Since for a steady ow J must b e

max

constant with radius, this would constrain J to b e p ositive. In our low-j and negative-j

solutions, there is no maximum in , so that the viscous torque carries angular momentum

outward at all radii. If the torque exceeds the rate at which angular momentum is carried

{18{

in by the accreting material, the net angular momentum accretion rate will b e negative.

These solutions are very similar to the ones we found in an earlier pap er (PN91).

In that pap er, we calculated the structure of the b oundary layer using a simpli ed disk

mo del with a p olytropic pressure-density relation. We found that our solutions formed two

branches in the H plane for large values of j 1. We did not nd the lower branch,

but seems likely that it exists for these solutions as well. The PN91 solution branches

formed sharp corners where they met, unlike the rounded transitions b etween branches

seen in the current solutions. This di erence probably arises from the very di erent disk

thicknesses in the two studies; the PN91 solutions generally had H=R < 0:01. Wewere also

able to nd solutions with small and negativevalues of j ; for these values only a single

branchwas present in the H plane. Thus, despite the use of a di erent disk mo del,

the qualitative b ehavior of these solutions is very close to that of the solutions describ ed in

this pap er.

One imp ortant question is why the solutions presented here reach negative angular

momentum accretion rates when the stellar rotation rate is still well b elow the breakup

rate (R ). For the PN91 solutions, we de ned the stellar radius as the p oint where

H=R =0:1 in the accretion ow, and found that with this de nition, j was very close to 1

for most , but dropp ed precipitously when ' 0:915 (R ). Wehave seen that the

current solutions reach j = 0 for 0:2 0:5 (R ). As discussed ab ove, solutions with

small or negativevalues of j must have d =dR < 0 for all R.Thus, if ' close to R ,

the star must b e rotating close to breakup, ' (R ), in order to allow a negative-j

solution. If is much less than (R ), will have to decrease close to R , and there will

b e a maximum in , so that j ' 1.

In the FU Orionis solutions presented here, two factors combine to pro duce small and

negativevalues of j when (R ). The rst is the large radial width of the b oundary

layer. In the solutions with j 1, p eaks fairly far from R , so that the radial width of

the b oundary layer is comparable to R .At the radius where p eaks, is substantially

smaller than (R ); for instance, at R =2R , '0:35 (R ). As the star spins up,

K K K

only needs to reach this value to make d =dR < 0everywhere and allow j to b e small or

negative. Second, pressure supp ort plays an imp ortant role, and is substantially smaller

than . This allows solutions with d =dR < 0everywhere for even smaller values of .

_ _

This also explains whywe reach j = 0 at smaller for larger values of M ;as M increases,

pressure supp ort plays a larger role, the disk b ecomes thicker, and is a smaller fraction of

It is also worth noting that the total accretion luminosity increases substantially as j

{19{

decreases. In general, the accretion luminosityvaries according to the expression

# "

GM M 1

L ' 1 j +

acc

R (R ) 2 (R )

K K

(Popham & Narayan 1995). Thus, for large negativevalues of j , the accretion luminosity

can b e substantially larger than the standard value GM M=R . If the star spins up

substantially during the T Tauri phase, then when an FU Orionis outbursts o ccurs, j

may reach fairly large negativevalues. For instance, Fig. 4 shows that j = 1 for

5 1

' 1:83 10 s , which is only ab out a 6% increase in = (R )over the p oint

where j = 0. A 10% variation from = (R ) ' 0:3 to 0.4, as discussed ab ove, would give

j 2. This would give an accretion luminosityof1:88GM M=R , whichwould make

the outburst even brighter than one would exp ect from the increase in M . This additional

luminositywould come from the rotational energy of the star, which is released as the star

spins down.

Disk solutions with negative angular momentum accretion rates may b e imp ortantin

other typ es of accreting systems. In systems such as cataclysmic variables, the accretion

rates are generally much smaller than those of FU Orionis systems. This pro duces a

narrower b oundary layer, and pressure supp ort plays a smaller role, so wewould exp ect

that negative angular momentum accretion rates are only found for ' (R ). We are

currently completing a study of negative-j solutions for disks around cataclysmic variables

8 1

with M =10 M yr which con rms these exp ectations (Popham 1995). In some other

systems, suchasemb edded pre-main-sequence stars, or some X-ray binaries, the accretion

rates may b e large enough to pro duce solutions like the ones presented here, where negative

angular momentum accretion rates are reached when the star is still spinning relatively

slowly.

5.3. Assumptions

The results describ ed ab ove dep end on a numb er of assumptions. One very imp ortant

assumption is that FU Orionis outbursts o ccur in classical T Tauri stars; i.e., that the T

Tauri and FU Orionis phases o ccur during the same ep o ch of pre-main sequence stellar

evolution. Mo dels of FU Orionis outbursts based on disk instabiliti es generally assume

that the outbursts arise in T Tauri star disks, so that the FU Orionis and T Tauri phases

corresp ond to the high and low states observed in other typ es of accreting stars, suchas

dwarf novae. The observational connection b etweenTTauri and FU Orionis stars is rather

tenuous, since only one of the current FU Orionis systems was observed sp ectroscopically

b efore it wentinto outburst, and no known FU Orionis systems haveyet returned to their

{20{

pre-outburst levels. The one system for which a sp ectrum was taken prior to outburst,

V1057 Cygni, resembled a typical T Tauri star (Herbig 1977). However, some of the

observed characteristics of FU Orionis systems, such as their asso ciation with re ection

nebulae and out ows, and their large far-infrared excesses, suggest that FU Orionis

outbursts may arise in younger emb edded sources rather than in classical T Tauri stars

(Kenyon 1995). If this is the case, then the mechanism describ ed here should serve to limit

the stellar rotation rate during the emb edded phase of pre-main-sequence stellar evolution.

The rotation rate would then presumably b e controlled by the stellar magnetic eld during

theTTauri phase.

Wemust also make some assumption ab out the lo cation of the stellar radius, whichis

dicult to de ne in our mo del (see x4 for a discussion of this problem). Wehavechosen a

particular value of v to de ne R , in order to provide an example of the variation of H

and j as a function of . Unfortunately, the value of (j = 0) is fairly sensitivetothe

choice of v .Wehave illustrated this in Fig. 4, where wehave plotted lines corresp onding

1 1

to v = 500 cm s and v = 2000 cm s .For these values of v , j reaches zero

R; R; R;

5 1

at ' 0:94 and 2:19 10 s , resp ectively. This means that for a factor of 4 variation

in v , (j =0)varies by more than a factor of 2. Thus, even when we sp ecify all the

parameters of an FU Orionis system, we still have a substantial uncertaintyin (j =0)

resulting from the uncertainty in the de nition of the stellar radius. As discussed in x4.5,

(j = 0) also varies dramatically with M . Therefore we can only say that for what

we b elieve to b e reasonable choices of v and M , the resulting values of (j =0)are

comparable to T Tauri rotation rates.

Our results are not strongly dep endent on the magnetic eld of the star, as long as the

eld is not strong enough to disrupt the accretion ow in the high-M FU Orionis phase.

This would require a eld substantially larger than the 1 kG needed to disrupt the disk

2=7 4=7

during the T Tauri phase. The disruption radius varies as R / M B (Pringle &

1=2

Rees 1972), so to maintain the same R requires that B vary as M . Since FU Orionis

accretion rates are a factor of 1000 higher than T Tauri rates, B would havetobeabout

30 times larger to disrupt the disk, whichwould require B 3 10 G. A eld this strong

would disrupt T Tauri disks at R 20R , which can probably b e ruled out by observations.

Wehave used stellar radii of 2,25, 3, and 4 10 cm, corresp onding to 2.88, 4.31,

and 5.75 R , in our calculations. These are larger than the radii generally found for T

Tauri stars, which are in the range 1.5{2.5 R . These larger radii app ear to b e required by

observations: smaller stellar radii would pro duce maximum temp eratures in the inner disk

which are substantially larger than those observed. This suggests that the accreting star

may expand during the FU Orionis outburst.

{21{

Prialnik & Livio (1985) calculated the evolution of a fully convective 0.2 M star during

3 1 10 1

the accretion of 2:5 10 M of material at rates ranging from 10 to 10 M yr .

They also varied the fraction f of the accretion energy carried into the star with the

accreting material; f ranged from 0.001 to 0.5. They found that for the accretion rates

appropriate to FU Orionis outbursts, the star expanded stably if f 0:03. If f 0:1,

the star expanded unstably. In our FU Orionis solutions, the central temp erature of the

5 6

disk at R is 1 3 10 K. The virial temp erature for these parameters is 10 K,

so a reasonable fraction f 0:1 0:3 of the accretion energy is advected into the star.

This suggests that rapid stellar expansion may result from the high-M accretion during

an FU Orionis outburst, but it is dicult to knowhow the mass of the star, the presence

of rotation, and the departure from spherical symmetry would a ect the Prialnik & Livio

results. This expansion, and the subsequent contraction during the T Tauri phase, could

a ect the spin evolution of the star, and other asp ects of the star's interaction with the

disk, and clearly deserves further study.

Our mo dels also assume that the disk and b oundary layer are in a steady state. This

is clearly not true for FU Orionis systems, since they are exp eriencing outbursts. The

2 1 2

viscous timescale is t ' R = = (H=R) .For our solutions with =0:01,

v isc

5 1

H=R ' 0:3, this gives t ' 1000 . In the inner disk, wehave 'few 10 s ,so

v isc K

t ' few 10 s ' 1 yr. This is comparable to the rise times of the fastest-rising systems,

v isc

FU Ori and V1057 Cyg, and much shorter than their decline times. Thus FU Orionis

systems should b e reasonably well-mo deled by steady state solutions for this value of .

5.4. Comparison with Observations

Recent measurements of p erio dic photometric variations in T Tauri stars have clearly

established that the classical disk-accreting T Tauri stars spin more slowly than weak-line

TTauri stars, which app ear to lack accretion disks (Bouvier et al. 1993, 1995; Edwards et

al. 1993). This has generally b een interpreted to mean that the spinup of T Tauri stars

is controlled by a stellar magnetic eld strong enough to disrupt the disk. If FU Orionis

outbursts regulate the spin evolution of T Tauri stars in the way describ ed in this pap er,

it would also account for this observation. In b oth cases, disk accretion maintains the

slow rotation sp eed of classical T Tauri stars. If FU Orionis outbursts cease late in the

classical T Tauri stage, then the more rapid rotation of the weak-line T Tauri stars could

b e pro duced by accretion at the end of the classical T Tauri stage.

Observations of the spin evolution of T Tauri stars could b e used to test the ideas

discussed in this pap er. The most useful data would b e measurements of the spin p erio d

{22{

b efore and after an FU Orionis outburst. If the star spins down during an outburst, it would

o er clear evidence that outbursts are regulating the rotation rates of T Tauri stars. If the

star spins up, the result is less clear. If the amount of spinup is less than exp ected based on

the amount of mass accreted during the outburst, it would suggest that the star is reaching

the equilibrium rotation rate at some p oint during the outburst. In this case the

;eq ;F U

outbursts would still b e limiting the rotation rate of the star. If, on the other hand, the

star spins up as much as exp ected, it would suggest that the rotation rate must b e limited

by pro cesses taking place during the T Tauri phase. One complication in interpreting the

observations is the p ossibil ity that signi cant amounts of angular momentum mightbe

carried o in the strong out ows which o ccur during outbursts.

Observations of variations in the rotation rates of T Tauri stars whichhave not

exp erienced observed outbursts could also b e useful. Over a short timescale, variations

in M could pro duce irregular p erio d variations. Nonetheless, if FU Orionis outbursts are

controlling the rotation rates of these stars, then their spin p erio ds should vary secularly

during the T Tauri phase as the rotation rate moves from toward .If

;eq ;F U ;eq ;T T

instead the rotation rates of T Tauri stars remain constantover a long timescale, it would

suggest that has stabilized at , whichwould indicate that the spin evolution is

;eq ;T T

b eing controlled by angular momentum transfer in the T Tauri phase.

The main sources of data which can b e used to further constrain the solutions

presented in this pap er are the sp ectra of FU Orionis systems. These sp ectra yield valuable

information on the disk temp eratures and rotational velo cities (Hartmann & Kenyon 1985,

1987; KHH). The solutions presented in this pap er were selected to have luminosities and

temp eratures in approximate agreement with those derived from observations. Because our

main aim in this pap er has b een to understand the spinup and angular momentum transfer

in FU Orionis systems, wehave not attempted to match our solutions with observations

in any detail. We defer this to a subsequent pap er (Popham et al. 1995), where we plan

to compare the sp ectra and rotational velo cities pro duced by our disk and b oundary layer

solutions to those observed in FU Orionis systems.

Iwould like to thank Ramesh Narayan, Scott Kenyon, and Lee Hartmann for helpful

discussions. I would also like to thank the organizers of the conference on Circumstellar

Disks, Out ows and Star Formation, held in Cozumel, Mexico, where an early version of

this work was presented. This work was supp orted by NASA grantNAG5-2837 at the

Center for Astrophysics and grants NASA NAGW-1583, NSF AST 93-15133, and NSF

PHY 91-00283 at the University of Illinois.

{23{

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{24{

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This preprintwas prepared with the AAS L T X macros v4.0. E

{25{

Fig. 1.| A b oundary layer solution for typical FU Orionis parameters. The solution shown

4:15 1 11 11

here has M =10 M yr , M =0:5M , R =310 cm, j =0:90, H =10 cm, and

=10 . The four panels show the angular velo city , the radial velo city v , the e ective

temp erature T , and the disk height H in the range from R =3910 cm = 1 3R .

ef f

The dashed line shows the Keplerian angular velo city .

Fig. 2.| Variation of the rotation rate and the disk height H at R = R =310 cm,

4:15 1 11

for solutions with j =0:90, M =10 M yr , M =0:5M , R =310 cm, and

=10 . The solutions form a Z-shap ed curve in the H plane, with three solution

branches.

Fig. 3.| Same as Fig. 1, but for the 10 solutions marked along the Z-shap ed curve in Fig.

2, with H =0:80; 0:85; 0:90; :::1:25 10 cm. The b oundary layer width increases rapidly

as H increases, and angular velo city, radial velo city, and e ective temp erature all decrease.

Fig. 4.| Same as Fig. 2, but for various values of j as lab eled. All of the solutions on all

4:15 1 11 2

curves have M =10 M yr , M =0:5M , R =310 cm, and =10 .As and

H increase, j decreases, and the curves change from Z-shap ed to monotonic. The dotted line

1 1

marks the solutions with v = 1000 cm s , the upp er dashed line v = 500 cm s ,

R; R;

the lower dashed line v = 2000 cm s .

Fig. 5.| A b oundary layer solution with j = 0, where the star accretes mass but no angular

4:15 1 11

momentum. This solution has M =10 M yr , M =0:5M , R =310 cm,

11 2 1

H ' 1:08 10 cm, and =10 . Note that v = 1000 cm s , as discussed in x4.4; if

5 1

this condition is used to de ne R , then j = 0 for ' 1:55 10 s , corresp onding to a

rotation p erio d of 4.7 days.

_ _

Fig. 6.| Same as Fig. 4, but for two other choices of M and R : (a) has M =

4:0 1 11 4:3 1

10 M yr and R =410 cm = 5:75 R , and (b) has M =10 M yr and

R =2:25 10 cm = 3:23 R . Note that as M increases, the Z-shap e of the curves

b ecomes less pronounced and only app ears at lower values of . The dotted lines mark the

solutions with v = 1000 cm s .

Fig. 7.| Variation of the angular momentum accretion rate j with the stellar rotation rate

4:15 1

, for a set of solutions with a given value of v at R : (a) is for M =10 M yr ,

11 1

R =310 cm, and shows j ( ) for v = -500, -1000, and -2000 cm s ; (b) is

4:0 1 11 1

for M =10 M yr , R =410 cm, and v = 1000 cm s ; and (c) is for

4:3 1 11 1

M =10 M yr , R =2:25 10 cm, and v = 1000 cm s . In all cases, j drops

with increasing and reaches j = 0 for well b elow (R ).