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THE ASTROPHYSICAL JOURNAL, 511 : 896È903, 1999 February 1 ( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A. ON THE THERMAL STABILITY OF IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS PAOLA DÏALESSIO,1 JORGE CANTO ,1 LEE HARTMANN,2 NURIA CALVET,2 AND SUSANA LIZANO1 Received 1998 April 16 ; accepted 1998 September 3 ABSTRACT The dusty disks of many young stars are probably heated mostly by absorption of light from the central star. This stellar irradiation can control the vertical structure of the disk, particularly in the outer regions. Because the irradiation heating is sensitive to the disk structureÈthe disk vertical thickness and the tilt of the disk photosphere relative to the starÈthe possibility of an unstable feedback is present. To study this problem, we present calculations of the evolution of perturbations in vertically isothermal disks. We Ðnd that such disks are generally stable. In outer disk regions of T Tauri stars, linear analysis indicates that the radiative cooling time is so short that temperature perturbations will be damped faster than the disk structure can respond. Using our results for steady ““ alpha ÏÏ viscosity disks, we estimate that this is true for distances larger than 2 AU (M0 /10~8 M yr~1)7@9(a/0.01)~7@9 for typical T Tauri _ stars. Inside this radius, if the disk surface tilt (““ Ñaring ÏÏ) is still signiÐcant, numerical Ðnite-amplitude calculations show that temperature perturbations will travel inward as they damp. We Ðnd that disk self-shadowing has a small e†ect on the results because the perturbation is damped on a timescale shorter than the time in which the shadowed disk region can respond. Our results help justify steady, smooth treatments of the e†ects of irradiation on the disks of young stellar objects. Subject headings : accretion, accretion disks È circumstellar matter È instabilities È stars : preÈmain-sequence 1. INTRODUCTION the heating produced by irradiation exceeds that resulting from viscous dissipation of accretion energy near the disk midplane, the outer layers of the disk would be hotter than the interior, owing to nongray opacities and to the oblique entry of starlight into the disk (see also Malbet & Bertout 1991). Irradiation-produced temperature inversions may explain the 2.2 km CO overtone band emission seen in some objects (Calvet et al. 1991), as well as silicate emission features in the mid-infrared range (Calvet & Hartmann 1992 ; Chiang & Goldreich 1997). Since irradiation heating can play such an important role in determining the vertical structure of T Tauri disks, it is important to investigate the stability of such disks to ripples in their surfaces. Tilting the disk photosphere away from the disk plane, toward the central star, increases the irradiation heating, which increases the disk temperature. In turn, this increases the local disk scale height, which might yield an even larger tilt with the consequent instability. In this paper we consider the problem of the stability of nonaccreting, irradiated disks, approximating the disk as having a well-deÐned photospheric level where it absorbs the central starÏs light and assuming vertical isothermality. We show that such disks are stable to rippling perturbations of their surfaces and argue that this result probably applies to realistic protostellar disks. In ° 2 we consider small perturbations adopting disk parameters typical of our detailed structure calculations for T Tauri disks (DÏAlessio et al. 1998). In ° 3 we consider the problem in more generality, which allows us to account for the di†ering slope of the surface in inner and outer disk regions. In ° 4 we consider Ðnite-amplitude perturbations, and in ° 5 we present a discussion of the results. It has become increasingly clear that dusty preÈmainsequence disks are signiÐcantly heated by absorption of light from the central T Tauri star (““ irradiation ÏÏ), and that this heating plays an important role in establishing the physical disk structure. The Ðrst clue to this came from observations of the infrared spectral energy distributions of T Tauri disks (Rydgren & Zak 1987), which implied disk surface temperatures T that declined with increasing radius R more slowly than the T P R~3@4 predicted by simple optically thick, Ñat disk models, whether powered by steady accretion (Lynden-Bell & Pringle 1974) or irradiation (Adams , Lada, & Shu 1987). Kenyon & Hartmann (1987) showed that the required disk-temperature distributions could be explained by irradiation if preÈmain-sequence disks are ““ Ñared,ÏÏ i.e., their surfaces curve away from the midplane, which allows the disk to absorb more light from the central star. Since this Ñaring increases the local disk temperature, it increases the local disk scale height ; in outer disk regions, irradiation will dominate disk heating, and thus a self-consistent determination of disk structure requires the simultaneous solution of irradiation and vertical hydrostatic equilibrium (DÏAlessio et al. 1998). The Ñared disk model has been conÐrmed by Hubble Space T elescope images of the young stellar object HH 30 (Burrows et al. 1996), which clearly show the curved disk surfaces in scattered light. A second clue to the importance of disk irradiation was provided by observations of emission features in disk spectra (Carr 1989 ; Cohen & Witteborn 1985). Calvet et al. (1991) and Calvet & Hartmann (1992) pointed out that if 2. 1 Instituto de Astronom• a, UNAM, Ap. Postal 70-264, Cd. Universitaria, 04510 Mexico D.F., Mexico ; dalessio=astroscu.unam.mx, lizano=astrosmo.unam.mx. 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 ; hartmann=cfa.harvard.edu, ncalvet= cfa.harvard.edu. DISK SURFACE STABILITY : OUTER DISK We begin with an analysis modeled after the work by Cunningham (1976), who investigated the problem of stability of disks around black holes. We assume that the disk is vertically isothermal and optically thick to the stellar irra896 IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS diation. We further assume that the disk has a well-deÐned photosphere (Kenyon & Hartmann 1987), i.e., that the stellar photons are absorbed in a relatively narrow layer in the disk. Finally, we assume that the disk vertical structure responds instantaneously to a perturbation in the irradiated Ñux (caused by changes in the position of the disk photosphere). To Ðx ideas in an appropriate context, we Ðrst consider small perturbations and adopt speciÐc results for the background state of the disk motivated by our detailed structure results for typical disk models in their outer regions, where Ñaring is important (e.g., DÏAlessio et al. 1998). The irradiation Ñux is given by L k F \ * k \ pT 4 , irr 4nR2 * r2 (1) where L is the stellar luminosity, T is the stellar e†ective * * temperature, r is the cylindrical radial coordinate R in units of the stellar radius R , and k is the cosine of the mean * stellar radiation and the normal angle between the incident to the disk photosphere. For the outer, Ñared disk regions, the last quantity can be approximated by A B A B h d ln h 1 dh h [ \ [1 , k^ 2r d ln r 2 dr r h \ h T 1@2r3@2 . (3) 1 The unperturbed, steady state, disk temperature T (r) is given by the balance between heating by irradiation0 and radiative cooling, given by pT 4 \ F , (4) 0 irr,0 where the subindex 0 corresponds to the unperturbed quantities. Equation (4), along with equations (1), (2), and (3), is satisÐed by a power-law temperature distribution, T (r) \ T r~3@7 . (5) 0 1 Here T and h are scaling coefficients determined by the 1 disk 1parameters. particular Now we perturb the temperature of the disk, maintaining vertical isothermality and assuming the perturbation is small and axisymmetric. We take T \ T (r)(1 ] v) , (6) 0 where v > 1. The perturbation in temperature introduces an imbalance between heating and cooling, which results in evolution of the perturbation, described by LT \ F [ pT 4 , irr Lt C AB D M0 ) 1 1@2 1[ , (8) 3nac2 r s where ) is the Keplerian angular velocity ) \ (GM /R3)1@2, * M is the stellar mass, G is the gravitational constant, a is * the viscosity parameter (Shakura & Sunyaev 1973), and c is s the sound speed. We assume that the timescale for the perturbation evolution is smaller than the viscous timescale, and that the speciÐc heat is constant in time. For a powerlaw (unperturbed) temperature radial distribution T (r) \ 0 T r~q and for r ? 1, the speciÐc heat C(r) \ k&/m is also a 1 power law in radius and can be written as C(r) \ C rq~3@2. 1 Here k is the Boltzmann constant, m is the mean molecular mass, and the constant C is given by 1 GM 1@2 M0 * . (9) C \ 1 3naT R3 * 1 Assuming that the disk is always in hydrostatic equilibrium, the temperature perturbation results in a perturbation in the gas scale height, &\ A B A B v , h \ h r3@2T 1@2(1 ] v)1@2 B h r3@2T 1@2 1 ] 1 0 1 0 2 (2) where we are assuming k > 1. The height of the photosphere h is taken to be proportional to the gas density scale height (Kenyon & Hartmann 1987 ; Miyake & Nakagawa 1995). Thus, the photospheric height (in units of stellar radii) can be written as C(r) (Pringle 1981) 897 and A B 2 r Lv d ln h . [1 \ ] 7 2 Lr d ln r (10) (11) Since the speciÐc heat is constant in time, we have LT Lv B C(r)T . (12) 0 Lt Lt Substituting equations (10) and (11) into equation (1) and using the thermal equilibrium equation (4), one obtains that the irradiation Ñux is given by C(r) A BA B 2 r Lv ] . 7 2 Lr On the other hand, the radiative cooling is v 7 F \ pT 4 1 ] irr 2 0 2 (13) pT 4 B pT 4(1 ] 4v) . (14) 0 Substituting equations (12), (13), and (14) into equation (7), the evolution of the perturbation is given by 4 C(r) Lv Lv ] 2v [ r \ 0 , (15) 7 pT 3 Lt Lr 0 where we have retained only terms linear in the perturbation. Following Cunningham (1976), we search for a solution of the form v(r, t) \ r2w(t, f) , (16) Lv Lw \ r2 , Lt Lt (17) Lv Lw \ 2rw ] r2 bbrb~1 . Lr Lf (18) where f \ brb. Then (7) where we are neglecting the radial transport of energy. In equation (7), C(r) is the speciÐc heat per surface unit of the disk, which is proportional to the surface mass density. For a standard a viscous disk with a constant mass accretion rate M0 , the mass surface density can be written as and 898 DÏALESSIO ET AL. & Nakagawa 1995 ; Canto, DÏAlessio, & Lizano 1995) Substituting, one Ðnds 4 CT r2 Lw Lw 0 ] 2r2w [ 2r2w [ bbrb`2 \0 . 7 pT 4 Lt Lf 0 Then 4 CT 0 \ bbrb . (21) 7 pT 4 0 The speciÐc heat is proportional to the surface mass density, which we assume does not change while the perturbation evolves. For an unperturbed temperature given by equation (5) and using equation (8), the values of the constants are b \ 3/14 and b \ (8 )C T /pT 4, and f is given by 3 1 1 1 8 C f\ , (22) 3 pT 3 0 which is proportional to the cooling timescale, given by the energy content of the disk (per unit surface) CT , divided by 0 the rate at which this energy is lost (per unit surface), pT 4 0 (the thermal timescale ; Pringle 1981). Equation (19) then becomes Lw Lw . \ Lf Lt (23) Every function w \ w(f ] t) is solution of this equation ; then v \ r2w(f ] t). The velocity of the perturbation is the velocity corresponding to a constant value of the argument (f ] t), i.e., K R dR \[ . (24) bf dt f`t/const The negative sign indicates that the perturbation propagates inward, i.e., toward the central star. This happens because the perturbation in temperature locally changes the slope of the disk surface, which increases the amount of irradiation Ñux intercepted by the disk between the position of the perturbation and the adjacent inner annulus. At the same time, the slope between the perturbation position and the adjacent outer annulus becomes smaller than the unperturbed disk slope, which decreases the irradiation Ñux there. Thus, the perturbation will move toward smaller radii. The perturbation is damped on a timescale CA B v t \[ \ [v d (dv/dt) D Lv Lv ~1 ]v Lt Lr fb 2 C(r)T 0, \ (25) 2 7 pT 4 0 calculated following the perturbation moving through the disk, which is proportional to the cooling timescale. The damping time also can be written as t \ R/2v ; the perturbation disappears in roughly half the dcharacteristic time it would take to reach the star. Now we can check the validity of our assumptions for the particular case of accretion disks around low-mass young stars. The disk temperature can be written as (e.g., Miyake \ AB j 2@7 T (r) \ T r~3@7 , 0 * 7 (19) 4 CT Lw Lw 0 \ bbrb , (20) 7 pT 4 Lt Lf 0 and since w is arbitrary, we choose the constants b and b such that v\ Vol. 511 (26) where j, for typical T Tauri stars parameters (e.g., Gullbring et al. 1998), is given by A B A BA B A B A B A kT R 1@2 * * mGM * g m ~1@2 \ 0.056 3 2m H T 1@2 R 1@2 M ~1@2 * * * , (27) ] 4000 K 2R 0.5 M _ _ where the quantity g is the ratio between the photospheric height and the disk scale height and m is the mean molecular mass. The photospheric height (in units of stellar radius) is h \ j(T /T )1@2r3@2. Assuming*a typical mass accretion rate and central star properties for T Tauri stars and a typical viscosity parameter a, the damping time can be written as j\g B A BA B A B A B A B AB A B M 15@14 R ~29@14 * * 0.5 M 2R _ _ a ~1 T ~32@7 g ~8@7 * ] 0.01 4000 K 3 M0 t \ 0.16r3@14 d 10~8 m 4@7 yr , (28) 2m H where the mass accretion rate is in M yr~1. The viscous _ timescale can be written as (Pringle 1981) ] R2) t \ a~1 vis c2 s A B A B A B A B AB A B M 9@14 R 5@14 * * 0.5 M 2R _ _ a ~1 T ~8@7 g ~2@7 * ] 0.01 4000 K 3 \ 233r13@14 m 8@7 yr . (29) 2m H This is comfortably longer than the damping timescale for any reasonable parameters, so our assumption of constant C(r) (constant &) is justiÐed, since & can only change on the viscous timescale (Pringle 1981). On the other hand, the assumption that the vertical structure responds quickly to the change in irradiation is not justiÐed at all points in the disk. The vertical disk structure can respond to changes in temperature on a timescale H/c , where c is the sound speed in the disk. Since H/R \ c /v s , s orb where vs is the local Keplerian velocity, the hydrostatic orb equilibrium timescale for response is t \ )~1 (Pringle h 1981). Numerically, ] A B A B R 3@2 M ~1@2 1 * * yr . t \ \ 2 ] 10~4r3@2 h ) 2R 0.5 M _ _ (30) No. 2, 1999 IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS FIG. 1.ÈComparison between the damping time of a small perturbation in temperature calculated with the general treatment described in ° 3 (solid line) and calculated with an approximated power-law distribution for the temperature of a Ñared disk, valid for distances larger than the stellar radius, discussed in ° 2 (dot-dashed line), with di†erent timescales of the disk : the viscous timescale (dotted line) and the hydrostatic timescale (dashed line). The disk has M0 \ 10~8 M yr~1, a \ 0.01, and the central star has M \ 0.5 M , R \ 2 R , and T_ \ 4000 K. * _ * _ * This timescale increases with r faster than the damping time. This means that there is an outer region where the damping time of the perturbation is shorter than the hydrostatic equilibrium timescale, and the disk has not enough time to change its height because of the change in temperature (given by the perturbation). The radius beyond which this happens is A B A A B A A B B A B B AB M0 7@9 M 11@9 R ~16@9 * * R B 1.7 h 10~8 0.5 M 2R _ _ a ~7@9 T ~32@9 g ~8@9 * ] 0.01 4000 K 3 m 4@9 AU . (31) 2m H For R Z R , there is no coupling between diskh temperature perturbation and height, and our treatment is not valid. Nevertheless, one expects that, since the disk height is not a†ected by the temperature perturbation, the irradiation Ñux remains constant and the only term that changes with time in the right-hand side of equation (7) is the disk radiative cooling (increased relative to the steady state cooling because of the perturbation itself). As long as the disk cooling is larger than the irradiation heating, its temperature will decrease with time until it reaches the steady state solution. (In this case, there is no reason for the perturbation to move toward the star.) From equation (7) with a constant irradiation Ñux, this damping happens in the cooling timescale, given by t B C(r)T /pT 4, which is 0 pertursimilar to the damping timescaled of the 0moving bation given by equation (50). Therefore, a perturbation in temperature in R [ R is damped locally before the disk h ] 899 can readjust its height to be in hydrostatic equilibrium with its new temperature distribution. From equation (31) we can see that the larger the value of M0 /a, the bigger the radius where t B t and the larger the d h region of the disk where temperature and height are coupled. But for steady disks, larger M0 implies larger viscous dissipation Ñux and disk optical depth, and the assumption of vertical isothermality breaks down, at least for the inner and optically thick regions of the disk. In this case, equation (7) has to be modiÐed to account for the viscous Ñux and also for the fact that the midplane temperature (which is the relevant temperature determining the gas scale height) is di†erent from the e†ective temperature (which gives the radiative cooling of the disk). When viscous dissipation is the main heating source of the disk, a perturbation in height has a small e†ect on the disk heating and one expects that a perturbation in temperature dies out locally (as long as the disk mass accretion rate corresponds to a thermally stable branch of the S curve ; Hartmann & Kenyon 1996, and references therein). In this case, one expects that the perturbation will damp in a thermal timescale, given by t B C(r)T /F , where F is the viscous Ñux vis vis (Pringle 1981). d Figure 1 shows a comparison of the relevant timescales for a disk with M0 \ 10~8 M yr~1 and a \ 0.01. Note that t \ t only in the inner disk._However, in these regions the d has h an almost Ñat surface (Kenyon & Hartmann 1987), disk and there it is not valid to assume equation (2). In this case we must modify our treatment, as discussed in the following section. 3. GENERAL TREATMENT FOR SMALL PERTURBATIONS As in the previous section, the disk is assumed vertically isothermal and optically thick to the stellar irradiation. The mean cosine between the stellar incident radiation and the normal to the disk surface has a more general form than equation (2), to account for those regions of the disk with a photospheric height small compared with the stellar radius. Then, k is given by (Ruden & Pollack 1991) k\ 1 2 CA B D dh h 4 [ ] , 3nr dr r (32) where as in the previous section we assume k > 1. The temperature in the unperturbed steady state disk, given by equation (4), can be found from A BC pT 4 * 2r2 A BD 4 dh h 0[ 0 ] 3nr dr r \ pT 4 , 0 (33) where h is given by equation (3), evaluated at the unper0 turbed temperature T . 0 We perturb the disk temperature as in equation (6), assuming that this perturbation is reÑected instantaneously in the photospheric height and h B h (1 ] v/2). Neglecting nonlinear terms in v, the right-hand0side of equation (7), using equation (1), is F [ pT 4 B irr A BC A B A B D pT 4 * 2r2 ] v 4 ] 1] 2 3nr h dv dh h 0[ 0 ] 0 [ pT 4(1 ] 4v) , 0 2 dr dr r (34) 900 DÏALESSIO ET AL. which can be reduced using the steady state solution C A B D pT 4 Lv pT 4 dh h *h * 0 [ 0 [ 4pT 4 v ] . F [ pT 4 B 0 irr 4r2 0 Lr 4r2 dr r Vol. 511 Since the amplitude of the perturbation is small, we expect that the solution for v is independent of v itself, so that (35) The equation for the evolution of the temperature perturbation can be written as Lv Lv , D(r) \ A(r)v ] B(r) Lt Lr (36) df D(r) \ , dr B(r) (45) A(r) d ln a . \[ dr B(r) (46) Then f(r) \ with D(r) \ A(r) \ A A C(r)q 0, pT 3 * (37) B h 1 dh 0 [ 0 [ 4q4 0 r 4r2 dr B h d ln q 0 [ 4q4 , \ 0 1] 0 8r3 d ln r (38) h B(r) \ 0 , 4r2 P D(r) dr , B(r) CP (47) D a(r) \ exp [ A(r)/B(r)dr . (48) If / A(r)/B(r)dr decreases with r, the amplitude of the perturbation a(r) increases with r (i.e., the amplitude decreases when the perturbation moves toward the star). Since f(r) is a positive quantity the perturbation moves inward and its (phase) velocity is given by the velocity dR/dt, which corresponds to the argument z \ (f ] t) \ constant, i.e., (39) v\ R * . (df/dr) (49) T q \ 0. (40) 0 T * From the result discussed in ° 2, we look for a solution of the form As we have mentioned before, C(r)q P r3@2. Then v P 0 distance to the q1@2r, and the velocity decreases when the 0 central star decreases. The damping time is given by v(r, t) \ a(r)w(z) , R B(r) D(r) df d ln a \[ * \[ . t \ d dr dr vA(r) A(r) (41) where a(r) is the amplitude of the perturbation as a function of distance to the central star, which does not depend explicitly on time. The function w(z) is related to the initial shape of the perturbation, and its argument z depends explicitly on time and position, z \ t ] f(r). The quantity f(r) is related to the velocity of propagation of the perturbation : the radius where the function w(z) is maximum changes in time maintaining z constant. In those regions of the disk where the Ñaring is important, we expect that a(r) B r2, as we have discussed in ° 2. Then Lv dw Lz dw \ a(r) \ a(r) , Lt dz Lt dz dw df da . w(z) ] a(r) dz dr dr dw \ A(r)a(r)w(z) dz ] B(r) C From equation (33), the unperturbed temperature can be approximated by two di†erent power laws, one for the outer and Ñared regions of the disk and one for the inner regions, where the photospheric height is smaller than the stellar radius. For the outer disk, T P r~3@7, q4 B h /7r3, 0 of and equation (48) gives a(r) \ r2, and0 the phase 0velocity the perturbation is v B 7RpT 4/4C(r)T , as before (° 2). On 0 the other hand, for the inner0Ñat regions of the disk, the unperturbed temperature is given by q B (2/3n)1@4r~3@4 and the photospheric height is h B h 0r9@8, where h \ 2 h (2T /n)1@8. Then the amplitude of0 the 2perturbation trav1 * eling through those regions is a B exp [[Er~9@8] , (43) where E B 9.5/nh , and we have assumed that h [ 1. 0 Figure 1 shows2 the damping time for the perturbation, calculated using equation (50) for the unperturbed temperature obtained by numerically integrating equation (33). For the plotted models, the damping time calculated assuming a completely Ñared disk (see ° 2) is a good approximation for R Z 1 AU, but for R [ 1AU, t decreases more d equations (2) rapidly as distance decreases. One expects that and (32) give the same result for radii where Therefore, the equation for the evolution of the perturbation is D(r)a(r) A D da df dw . w(z) ] a(r) dr dr dz (50) (42) Lv da dw Lz \ w(z) ] a(r) Lr dr dz Lr \ N (44) B dh h 4 [ ? . dr r 3nr (51) (52) We estimate the radius r beyond which the disk can be f considered Ñared for the calculation of the irradiation Ñux, No. 2, 1999 IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS taking the Ñat disk irradiation Ñux equal to a factor f times the Ñared disk irradiation Ñux. This gives AB A B j ~8@9 2 7@9 , r B f ~7@9 f 7 3n (a) 0.4 (53) 0.2 and for the typical parameters we have been using and taking f \ 0.5, r B 40. For 0.5 \ f \ 2, 12 \ r \ 40, so f f this can be considered the transition region between the Ñat and Ñared disk description. For smaller distances, r \ 12, the slope of the scale height is small and the disk is Ñat. When this happens, the perturbation in the disk height has a negligible e†ect on the irradiation Ñux (which can be approximated by the irradiation Ñux of a Ñat disk, i.e., F B 2pT 4/3nr3). The disk height and temperature are irr *because a change in height does not a†ect the decoupled heating of the disk. When the perturbation reaches this region, its amplitude decreases with distance exponentially (as is shown by eq. [51]), and it cannot move radially any longer. 0 NUMERICAL EVOLUTION 4. C(r) A B LT LT [ pT 4 , \ F r, T , irr Lr Lt (54) where we have written explicitly the spatial derivative of T in the argument of the irradiation Ñux. We assume that the disk is always in hydrostatic equilibrium. The shadowing is introduced by taking F (r) \ 0 for r [ r , if h(r)/r \ h(r )/r . irr has a local maximum, t t The radius r is where h(r)/r and tit is t given by the radius where the slope (L ln h/L ln r) \ 1 and rt of the L2h/L2r \ 0. At r \ r a straight line from the center t star is tangent to the disk surface. We calculate numerically the unperturbed steady state temperature distribution integrating equation (33). At a time t \ 0, this disk is perturbed by C D C D a(r )r2 (r [ r )2 0 0 exp [ , (55) 2p2 r2 0 v where r is the perturbation initial position, a(r) \ r2a(r )/r2 0 0 0 is its amplitude, and p is its initial width. We use the Macv Cormack predictor-corrector method (MacCormack & Paullay 1974) to integrate equation (54). The time step used in the integration is dt \ 1.25 ] 10~5 yr, and the spatial grid consists of 300 points, logarithmically distributed, between r \ 1.36 and 2000. The disk has M0 \ 10~8 M yr~1, a \ 0.01, and the central star has M \ 0.5 M _, * _ R \ 2 R , and T \ 4000 K. *Figure _ * 2 shows the evolution of a perturbation initiated at r \ 200, with p \ 0.1r , and a(r ) \ 0.15 and 0.5. The 0 position of the v perturbation 0 initial is 0close to the outermost radius at which the hydrostatic timescale is short enough to assume the disk is in hydrostatic equilibrium. The elapsed time is t \ 1 yr, and each curve corresponds to a time step *t \ 0.1 yr. The Ðgure shows the decreasing amplitude of the perturbation while it moves toward the star. The position of the maximum of v is independent of its initial amplitude, as can be seen comparing Figures 2a and 2b. For comparison, Figure 2c shows the evolution of the highv\ (b) 0.4 0.2 0 (c) 0.4 0.2 0 0 In this section, we calculate numerically the evolution of an arbitrary size temperature perturbation, including the e†ect of the shadowing of the disk caused by the perturbation. The equation we integrate is equation (7), i.e., 901 100 200 300 400 r FIG. 2.È(a) Perturbation as a function of position for di†erent times (*t \ 0.1 yr). Initially, the perturbation is a modiÐed Gaussian, centered in r \ 200, with a width equal to p \ 0.1r (dotted line). The disk has a mass 0 p a viscosity 0 accretion rate M0 \ 10~8 M yr~1, parameter a \ 0.01, and it _ surrounds a star with M \ 0.5 M , R \ 2 R , and T \ 4000 K. The * _is v(r * ) \ 0.5. _ (b) The* evolution of the initial amplitude of the perturbation 0 (a), but an initial amplitude perturbation for the same parameters as v(r ) \ 0.15. (c) The evolution of the perturbation with amplitude v(r ) \ 0 0.5,0 calculated assuming the irradiation Ñux is constant in time and equal to the irradiation Ñux of the unperturbed disk (see text). amplitude perturbation if the irradiation Ñux remains constant. This is an example of what happens in those regions where the hydrostatic timescale is larger than the temperature perturbation damping time. We do not include the e†ect of the shadow in this last case, since the perturbation in temperature has no e†ect on the shape of the disk. Comparing this result with the evolution of a perturbation with the same initial amplitude but a changing surface shape, we see that the moving perturbation decreases faster than the static one. For instance, after t \ 1 yr the moving perturbation has a value of the maximum of v smaller by a factor D0.6 with respect to the static one because when the perturbation moves toward the star, the radiative cooling rate increases more than the irradiation heating rate. Figure 2 shows that behind the perturbation there is a region with v \ 0, i.e., with a temperature lower than the unperturbed steady state temperature, which corresponds to the shadow. The external boundary of the shadow moves toward smaller radii, while the perturbation evolves. This is shown by Figure 3, where h/r is plotted as a function of distance for di†erent time steps. From Figure 2 we see that the e†ect of the shadow, as expected, is larger for the larger initial amplitude. The minimum value of v (related to the shadow) is D[0.15, around r \ 250, for the perturbation with a(r ) \ 0.5 initiated at r \ 200. The temperature at the 0 region decreases with time, with a thermal timeshadowed scale proportional to the local damping time. As we have shown in ° 2 and ° 3, this timescale increases with distance (see Fig. 1). Therefore, the bigger the shadow, the longer the time its outer regions need to decrease their temperature because of the decreasing heating. But the timescale in 902 DÏALESSIO ET AL. FIG. 3.ÈRatio between the photospheric height and the radial coordinate h(r)/r as a function of r for a perturbation with a(r ) \ 0.5, at r \ 500. 0 0 t\2 The time t \ 0 is plotted with a dotted line. The last plotted time was yr and *t \ 5 ] 10~2 yr. Disk parameters are the same as in Fig. 2. The shadow is produced at those outer regions where h/r is smaller than the maximum value of h/r at the bump. which the perturbation evolves is smaller than the timescale in which the shadows evolve (because the perturbation is always located at a smaller radius than the shadow it produces). Figure 3 shows that the perturbation is damped fast enough to prevent a large e†ect of the shadow on the disk structure. We calculate the radius r where the function v/r2 is maximum. According to ° 2, max for a small perturbation in the FIG. 4.È(a) Amplitude (v/r2) as a function of the radius r . Each curve maxwith a di†erent initial amplitude max and posicorresponds to a perturbation tion : a(r ) \ 0.5, r \ 200 (solid line), a(r ) \ 0.5, r \ 500 (dotted line), 0 0 a(r ) \ 0.15, r \ 1000 (dashed line), a(r )0\ 0.15, r0 \ 5000 (dot-dashed 0 0 0 of the maximum of line). In each case, we take p \ 0.1r . (b) 0The position 0 perturbations plotted in (a). v/r2 as a function of time, for vthe same Vol. 511 Ñared region of the disk, w(f ] t) \ v/r2. Therefore, r is max the position of the maximum of w(f ] t) and w(r ) \ const max in those regions where the assumptions used in ° 2 are valid. Figure 4 shows w(r ) \ (v/r2) as a function of r , max rmax for perturbations withmaxinitial amplitudes of 0.15 and 0.5, initiated at r \ 200, 500, and 1000. In each case there is a 0 range of r for which the amplitude of the perturbation is max approximately proportional to r2 as we expect from equation (16) (see ° 2). For the large amplitude perturbations, at larger distances (or shorter times) the perturbation is not small enough to be described by the linear equations discussed in °° 2 and 3, and its amplitude decreases with position (or time) faster than the linear prediction. There is an inner region around r D 100 where the irradiation max changes because the disk is almost Ñat and the damping time decreases faster than the linear prediction (see Fig. 1). When the distance between the perturbation and the central star is similar to the width of the perturbation, the presence of the inner boundary is reÑected in the sudden decrease of w(r ) toward smaller radii. max 5. CONCLUSIONS We have considered the evolution of a temperature perturbation in a vertically isothermal disk in vertical hydrostatic equilibrium, irradiated by the central star. We Ðnd that there are three main regions whose details depend on the speciÐc parameters of the disk and the star : 1. The outer disk, for R Z 2 AU for typical parameters, where Ñaring is important in determining the disk unperturbed temperature, but where the hydrostatic timescale is longer than the perturbation damping time. In this region the disk temperature and scale height are decoupled : the disk has not enough time to reach the hydrostatic equilibrium corresponding to the new temperature before radiative cooling damps out the perturbation. The perturbation is damped in a cooling timescale (D3.7t ), between D0.7 d and a typical and 10 yr, for R between 2 and 1000 AU T Tauri accretion disk. 2. The intermediate disk, for 0.1 Z R Z 2 AU, where the temperature distribution is changing from the ““ Ñat ÏÏ to the ““ Ñared ÏÏ characteristic temperature distributions, but where the damping time of the perturbation is longer than the hydrostatic timescale. In this region the disk is always in hydrostatic equilibrium and the temperature and scale height are coupled. The perturbation in temperature (and photospheric height) moves toward the star, with decreasing amplitude, in timescales between 0.001 and 0.1 yr, for typical parameters. 3. The inner disk, for R [ 0.1 AU, where the disk photospheric height is smaller or similar to the stellar radius and the temperature distribution is similar to that of a Ñat irradiated disk. The perturbation moves toward the central star and its amplitude decreases exponentially. For a disk in hydrostatic equilibrium, a shadow caused by a bump at the disk surface produces a variation in the disk-temperature structure that evolves more slowly than the bump itself. The consequence of this is that the original perturbation disappears before the shadow can produce a large e†ect on the disk structure and after a time similar to the perturbation damping time the disk returns to its original, unperturbed structure. No. 2, 1999 IRRADIATION-DOMINATED PREÈMAIN-SEQUENCE DISKS We have neglected some e†ects that could be present in a more realistic disk model. For instance, as discussed in ° 1, disks where irradiation dominates may have temperature inversions. Nevertheless, we do not see why the modest change in the local scale height should produce qualitatively di†erent results. Similarly, we have ignored radial transport of energy and accretion viscous heating. These two e†ects should help stabilize the disk further because they tend to smooth the temperature distribution. Since the rippling of the disk arising from temperature inhomogeneities dies out in a time shorter than the viscous timescale, we conclude that the irradiated disk is stable (Cunningham 1976) and, as in Blair et al. (1984), that a 903 smooth surface can be assumed to calculate the Ñux and incidence direction of the stellar irradiation in our detailed modeling of the disk vertical structure (e.g., DÏAlessio et al. 1998). We are grateful to John Raymond for making us aware of the Cunningham (1976) paper and to Alejandro Raga for fruitful discussions. This work was supported in part by Instituto de Astronom• a, UNAM, Mexico, DGAPAUNAM, ConaCyT, and NASA grant NAG5-4282. 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