1. INTRODUCTION 2. METHOD OF SOLUTION

Transcription

1 THE ASTOPHYSICAL JOUNAL, 557:311È319, 1 August 1 ( 1. The American Astronomical Society. All rights rerved. Printed in U.S.A. PULSATIONAL STABILITY OF g-modes IN SLOWLY PULSATING B STAS UMIN LEE1 Institute of Astronomy, University of Cambridge, Madingley oad, CB3 HA, UK eceived 1 January 9; accepted 1 March 1 ABSTACT Low-frequency g-mod excited by the opacity bump mechanism in main-sequence stars with the mass M D M are believed to be rponsible for the periodic variability observed in slowly pulsating B (SPB) stars. We calculate fully nonadiabatic g-mod with a low-degree l in an SPB star model to examine the pulsational stability by taking into account the e ects of the Coriolis force and assuming a uniform rotation. We Ðnd that the slow rotation of the star contribut to the excitation of the high radial order retrograde g-mod with low l. This contribution to the excitation is most signiðcant for the g-mod with l \ 1. At rapid rotation, many g-mod with low l in the model are found to be pulsationally unstable even in the inertial regime in which o u o \ ), where u denot the oscillation frequency observed in the corotating frame and ) is the angular rotation frequency. As the rotation rate increas, mode coupling caused by the Coriolis force between g-mod with di erent valu of l becom inñuential on the pulsational stability. We Ðnd that some of the g-mod in the inertial regime are stabilized by mode coupling at rapid rotation and that this stabilization e ect is most signiðcant for the retrograde g-mod with l [ o m o. Subject headings: instabiliti È stars: oscillations È stars: rotation 1. INTODUCTION Since the introduction of the new generation of opacity data (Iglias, ogers, & Wilson 199; Iglias & ogers 199), it has been clear that low-frequency g-mod in main-sequence stars having M D M are excited by the opacity bump mechanism operating at temperatur of T D ] 15 K (see, e.g., Gautschy & Saio 1993; Dziembowski, Moskalik, & Pamyatnykh 1993). This excitation of g-mod in the main-sequence stars is believed to be rponsible for the periodic variability found in the socalled slowly pulsating B (SPB) stars (Waelkens 1991). Although SPB stars are not necsarily rapid rotators (Waelkens 1991; Waelkens et al. 199), the slow rotation of the star can be important in determining the modal properti of high radial order g-mod when they have the oscillation frequency u, observed in the corotating frame, comparable to or ls than the rotation frequency ). From the observational point of view, when Balona (199) and Balona & Koen (199) found no SPB stars in the open clusters NGC 393 and NGC 755, in which many b Cephei variabl, which are also thought to be excited by the same opacity bump mechanism, were identiðed, Balona & Koen (199) conjectured that rapid rotation supprs the g-mode pulsations in SPB stars. This conjecture appears to be consistent with the Ðnding by Waelkens (1991) and Waelkens et al. (199) that SPB stars tend to be slow rotators. We think it important to examine the e ect of rotation on the pulsational stability of g-mod in SPB stars. The e ect of rotation on the pulsational stability of g-mod in main-sequence stars has been invtigated by several authors. Employing a perturbative technique in which the e ect of the Coriolis force was taken into account to the Ðrst order of ), Carroll & Hansen (19) carried out a fully nonadiabatic pulsational stability analysis of b Cephei models against nonradial mod with a low-degree l, 1 On leave of absence from the Astronomical Institute, Tohoku University, Sendai, Miyagi 9-57, Japan; lee=astr.tohoku.ac.jp. 311 and they suggted that slow rotation contribut to the dtabilization of the prograde mod. Using a nonperturbative method of calculation of nonradial oscillations in rotating stars, Lee & Bara e (1995) argued that the centrifugal force plays no signiðcant role in determining the pulsational stability of g-mod in SPB models. ather recently, under the quasi-adiabatic and the traditional approximations, Ushomirsky & Bildsten (199) examined the e ect of the Coriolis force on the pulsational stability of g-mod having the oscillation frequency o u o [ ) in an SPB model and showed that rapid rotation stabiliz some of the g-mod that are unstable at ) \ and excit some of the g-mod that are stable at ) \. In this paper, carrying out a fully nonadiabatic pulsational stability analysis for g-mod in main-sequence models with the mass comparable to SPB stars, we examine the e ect of rotation on the pulsational stability, particularly for g-mod in the inertial regime in which o u o [ ). Since the centrifugal force plays no signiðcant role in determining the modal properti of low-frequency g-mod (see, e.g., Lee & Bara e 1995), in this paper we invtigate the e ects of the Coriolis force on the g-mod neglecting the centrifugal force. In the method of calculation employed in this paper is brieñy reviewed. Section 3 is for numerical rults, and is for discussion and conclusion.. METHOD OF SOLUTION The method for solving fully nonadiabatic and nonradial oscillations in uniformly rotating stars is the same as that employed by Lee & Saio (197b). The governing equations for small-amplitude oscillations in uniformly rotating stars are derived by linearizing Ñuid dynamic equations, assuming that the time dependence of the perturbations is given by eipt, with p being the oscillation frequency in an inertial frame. If the equilibrium conðguration of the star is axisymmetric, we can employ a seri expansion in terms of spherical harmonic functions Y m(h, /) with di erent valu l of l for a given m to reprent the perturbations (see, e.g.,

2 31 LEE Vol. 557 Lee & Saio 19). In this expansion scheme, the displacement vector n(r, h, /, t) is expanded as = m \ r ; Slj (r)y m eipt, (1) r lj ljz@ = C m \ r ; Hlj (r) LY m lj h Lh ] T lj{ (r) 1 LY m l{j D eipt, sin h L/ lj,ljz@m@ = C 1 LY m m \ r ; Hlj (r) lj Õ sin h L/ [ T lj{ (r) LY md l{j eipt, Lh lj,ljz@m@ where l \ o m o ] ( j [ 1) and l@ \ l ] 1 for even mod and l \ j o m o ] j [ 1 and l@ \ j l [ j 1 for odd mod for integers j j \ 1,,... The Eulerian j j prsure perturbation p@(r, h, /, t), for example, is given by = p@ \ ; (r)y m eipt. () lj ljz@ Substituting the expansions into linearized Ñuid dynamic equations, we obtain simultaneous linear ordinary di erential equations for the expansion coefficients (see Lee & Saio 197b). Here, we ignore the e ects of the centrifugal force and assume g/g \ 1, where g is the e ective gravity. In eff eff the case of uniformly rotating stars, it is convenient to use the oscillation frequency u p ] m) observed in the corotating frame of the star, where ) denot the angular rotation velocity. We solve the simultaneous di erential equations as an eigenvalue problem for u by imposing appropriate boundary conditions at the center and the surface of the star (see Lee & Bara e 1995 for the boundary conditions). Since the time dependence of the perturbations is given by eipt with p \ p ] ip, pulsationally stable I (unstable) mod have positive (negative) p \ u. I I In this paper, prograde and retrograde mod are deðned in the corotating frame of the rotating star. The surface wave pattern of a prograde mode is traveling in the same direction as that of the stellar rotation in the corotating frame. Note that for positive m, prograde (retrograde) mod have negative (positive) u. For numerical computation, we truncate the inðnite seri expansions (eqs. [1]È[]) by retaining a Ðnite number of the expansion terms with l (and l@) from j \ 1toj \ j j j max and discarding all the terms with j [ j. How many max expansion terms must be retained in the seri expansion depends on the oscillation mod we are interted in. For g-mod with l Domo and o )/u o [ 1, retaining the Ðrst several terms is in most cas sufficient in the sense that the eigenfrequency and eigenfunction calculated are insensitive to further increasing the number of the expansion terms. For low-frequency g-mod with l Domo and o )/u o Z 1, however, it is necsary to retain more than several expansion terms to obtain a reasonable convergence of the eigenfrequency and the eigenfunction. In this paper, we assume j \ 1, which is sufficient for most of the g-mod with l max Domo calculated here. When we look for a g-mode associated with a given l, we count the numbers of nod of the expansion coefficients S (r) and p@ (r) for mode identiðcation. We also calculate lj/l lj/l the partial oscillation energy A, deðned as P lj A \ o u o dr or lj ][Sp S ] l (l ] 1)Hp H ] l@(l@ ] 1)T p T lj lj j j lj lj j j lj{ lj { ], (5) to watch that the ratio A /A, with A \ jmax A, is lj/l among the dominant ratios, where the asterisk indicat j/1 lj the complex conjugate of the quantity. 3. PULSATIONAL STABILITY OF g-modes WITH LOW l We carried out a fully nonadiabatic pulsational stability analysis for a zero-age main-sequence model of M \ M, the physical parameters of which are log (L /L ) \.3, log T \.1, / \.39, X \.7, and Z \. The model eff was calculated using a standard stellar evolution code implemented with the OPAL opacity with a Grevse and Noels solar composition (Iglias & ogers 199). For the stability analysis, we applied the Cowling approximation, neglecting the Euler perturbation of the gravitational potential. This approximation is reasonable for high radial order g-mod. At ) \, many g-mod with a lowdegree l in the model are excited by the opacity bump mechanism (see, e.g., Gautschy & Saio 1993). A plot of sgn u versus for g-mod with low l is given in I log o u I o u Figure 1, where u \ u/(gm/3)1@ u/u, and the cross, Ðlled squar, and open squar are for c l \ 1,, and 3, rpectively. The radial order k of the g-mod shown in Figure 1 rang between k \ and k \ for l \ 1, between k \ 3 and k \ 7 for l \, and between k \ 5 and k \ 9 for l \ 3. Since o u pulsationally unstable I o \ 1, (stable) mod have a positive (negative) sgn u I log o u I o. For convenience, we introduce, for a given l, the maximum and minimum radial orders k and k, rpectively, max min between which pulsationally unstable g-mod are found. In the case of )1 )/u \, we Ðnd (k, k ) \ (9, 1) for c min max ω FIG. 1.ÈThe sgn u for g-mod in the main-sequence I log o u I o M model plotted vs. u for the case of )1 \, where the cross, Ðlled squar, and open squar are for l \ 1,, and 3, rpectively, and the radial order k of the g-mod rang between k \ and k \ for l \ 1, between k \ 3 and k \ 7 for l \, and between k \ 5 and k \ 9 for l \ 3. The frequenci are normalized as u \ u/u and )1 \ )/u with u \ (GM/3)1@, where G is the gravitational constant c and M and c denote c the mass and radius of the model. Mod with positive sgn u are I log o u I o pulsationally unstable

3 No. 1, 1 g-modes IN SLOWLY PULSATING B STAS 313 l \ 1, (9, ) for l \, and (9, 3) for l \ 3, and all the g-mod between k and k are unstable in this case. For the cas of )1 \.1 min and.3, max we have tabulated (k, k ) in Tabl 1 and, rpectively. Note that not all the min g-mod max between k and k are unstable when )1 D. It may be useful to min keep in mind max that the rotation velocity at the equator of the model is given by v \ ) \ 5)1 km s~1. At small rotation rat, we may eq expand the complex eigenfrequency u of a g-mode in terms of )1 or l )/u as u \ u ] md 1 (l))1 ] D (m, l))1 ]O()1 3) u ] O()1 3), ex () or u \ u [1 ] 1 mc 1 (l)l ] 1 C3 (m, l)l]o(l3)], (7) where u is the eigenfrequency of the mode at )1 \ and D (l), D (m, l), C (l), and C (m, l), evaluated at )1 \, are dimensionls 1 complex 1 coefficients that depend on m, l, and the radial order k. Note that D \ C and D \ u In 1 1 C3 /. equation (), we have deðned u as the extrapolation to the ex order of )1. The deðnition of the coefficient C3 given in the equation (7) is the same as that of C given in Saio (191) except that the coefficient C3 calculated here do not contain the contribution from the centrifugal force. For a given mode, the coefficients D and D in this paper are obtained by calculating the mode 1 at three di erent rotation speeds: )1 \, 1, and [1. We conðrmed that the complex coefficient D thus obtained for g-mod with low l is in good agreement 1 with that calculated using the perturbation method employed in Carroll & Hansen (19). For adiabatic g-mod in a polytropic model with the index N \ 3, the contribution to the coefficient C from the Coriolis force dominat that from the centrifugal force (Saio 191). For nonadiabatic g-mod with l \ 1 in the M model, using the method of calculation given in Lee & Bara e (1995), we found that C /C3 for the B.9 g mode, that the ratio C /C3 quickly approach 1 as the radial order k increas, and that C /C3 except for a I I D 1 few g-mod near the frequenci at which C3 chang sign. In Figure, the real and imaginary parts of I the coefficients D and D are plotted as a function of u for the g-mod in D 1.3 (a) D 3 (c) D 1I - (b) D I - (d) x x ω ω FIG..ÈComplex coefficients D and D in the expansion (eq. []) are plotted as a function of u for the g-mod given in Fig. 1, where the solid, dashed, 1 and dash-dotted lin are for l \ 1,, and 3, rpectively. In panel a, the frequency u of the g-mod is indicated by the short vertical line.

4 31 LEE Vol. 557 Figure 1, where the solid, dotted, and dash-dotted lin are for l \ 1,, and 3, rpectively. As is well known, the real part D is given in a good approximation by D D 1/l(l ] 1) for 1 g-mod with k? 1. The imaginary parts 1 D and D are negative with large magnitud for the high 1I radial order I g-mod. This suggts that slow rotation contribut to the pulsational instability for the high radial order retrograde g-mod in the model. For the prograde g-mod, however, since D contribut to the stabilization and D 1I I to the dtabilization for large k, the stabilization is not necsarily e ective. Note that o D o can be much larger than o u only for the high radial 1I order g-mod with I o l \ 1 and o D o [ o u for the g-mod with l \ and 3. 1I I o For the case of )1 \.1, sgn u is plotted versus I log o u I o u for the g-mod with l \ m \ 1,, and 3 (upper panel) and those with l \ m ] 1 \ and l \ m ] \ 3 (lower panel) in Figure 3, where the cross, Ðlled squar, and open squar are for l \ 1,, and 3, rpectively, and the L=m=1 L=m= L=m= L= & m=1 L=3 & m= ω FIG. 3.ÈThe sgn u for g-mod with l \ m \ 1,, and 3 (upper panel) and those with l \ m ] 1 \ and l \ m ] \ 3(lower panel) in the I log o u I o M model plotted vs. u for the case of )1 \.1, where the cross, Ðlled squar, and open squar are for l \ 1,, and 3, rpectively, and the radial order k of the g-mod spans the same range as that of the g-mod plotted in Fig. 1 for each value of l. For positive valu of m, prograde (retrograde) mod, observed in the corotating frame of the star, have negative (positive) u.

5 No. 1, 1 g-modes IN SLOWLY PULSATING B STAS 315 ADIAL ODES TABLE 1 (k, k ) FO UNSTABLE g-modes AT )1 \.1 min max POGADE ETOGADE DEGEE m \ 1 m \ m \ 3 m \ 1 m \ m \ 3 l \ 1... (9,15) (9,19) l \... (,) (,)... (9,) (9,)... l \ 3... (9,) (9,3) (9,3) (9,) (9,) (9,) radial order k of the g-mod spans the same range as that given in Figure 1 for each value of l. Since we use positive m, prograde (retrograde) mod have negative (positive) u. The apparent asymmetry found in the frequency spectrum u between the prograde and retrograde mod is attributable to the fact that the frequency u of a g-mode with a radial order k is given by u B u on the retrograde side and by ] m)1 D u 1 B [ (u on the prograde side. Since is [ m)1 D 1 ) D smaller for larger l, the g-mod with l \ and 3 1 show L=m= L=m+1= ω FIG..ÈThe sgn u for g-mod with l \ m \ 1(upper panel) and those with l \ m ] 1 \ (lower panel) in the model plotted as a I log o u I o M function of u for )1 \.1, where the Ðlled squar and the open squar denote the extrapolated frequency u (see eq. []) and the eigenfrequency u, ex rpectively. The radial order k of the g-mod spans the same range as that of the g-mod plotted in Fig. 1 for each value of l.

6 31 LEE Vol L=m=1 L=m= L=m= L= & m=1 L=3 & m= ω FIG. 5.ÈSame as in Fig. 3, but for the case of )1 \.3 ADIAL ODES TABLE (k, k ) FO UNSTABLE g-modes AT )1 \.3 min max POGADE ETOGADE DEGEE m \ 1 m \ m \ 3 m \ 1 m \ m \ 3 l \ 1... (,15) (11, ) l \... (7,3) (,)... (7,) (,)... l \ 3... (7,1) (9,) (,3) (,) (9,) (1, ) weaker asymmetry in the frequency spectrum u between the prograde and retrograde sid than those with l \ 1. As shown by Table 1, for both prograde and retrograde mod, the radial orders k and k for unstable g-mod are almost the same as min those found max in the case of )1 \, except for k \ for the prograde g-mod with l \ m ] 1 \. We may min notice that the radial orders k for the retrograde max g-mod appear to be slightly larger than those in the case of )1 \. This slight shift of k is consistent with the fact max

7 No. 1, 1 g-modes IN SLOWLY PULSATING B STAS A L /A. L=m= A L /A. L=m+1= ω FIG..Èatios A /A for the g-mod with l \ m \ 1(upper panel) and those with l \ m ] 1 \ (lower panel) in the M model at )1 \.3 plotted as a lj function of u where the cross, Ðlled squar, and open squar are for the ratios with m ], and m ], rpectively, in the upper panel, and, A /A l lj j \ m, for the ratios A /Awith l \ m ] 1, m ] 3, and m ] 5, rpectively, in the lower panel. lj j that D and D are negative for the high radial order g-mod 1I at )1 \. I The usefulns of equation () in timating the complex oscillation frequency u of g-mod in slowly rotating stars can be examined by comparing u with the eigenfrequency u calculated at a Ðnite value of )1. In Figure we plotted the complex frequenci u and u for g-mod with l \ m \ 1 (upper panel) and l \ m ] 1 \ (lower panel), where the Ðlled squar and open squar are used to reprent u and u, rpectively, and the radial order k of the g-mod spans the same range as that in Figure 1 for each l. The Ðgure shows that the agreement between u and u is good for o l o [ 1 but becom poor as o l o increas, where l )/u. For the retrograde g-mod with o l o Z 1, for example, Im(u indicat instability, but the imaginary ex ) part of the eigenfrequency u shows that they are pulsa- tionally stable. We sometim Ðnd signiðcant deviation of u from u in the regime of o l o \ 1. The deviation is caused by mode coupling with g-mod with di erent valu of l, which is more likely to happen for g-mod with l [ o m o.

8 31 LEE Vol. 557 For example, we found that the prograde g mode with l \ m ] 1 \ having u is coupled kmin/ with the \[.59 g mode with l \ m ] 3 \, which is pulsationally unstable 1 at )1 \. For the case of )1 \.3, sgn u is plotted versus I log o u I o u for the g-mod with l \ m \ 1,, and 3 (upper panel) and those with l \ m ] 1 \ and l \ m ] \ 3 (lower panel) in Figure 5, where the cross, Ðlled squar, and open squar are used to denote the g-mod with l \ 1,, and 3, rpectively, and the radial order k of the g-mod spans the same range as that in Figure 1 for each value of l. Many pulsationally unstable g-mod are found in the inertial regime with o l o Z 1, in which equation () is not expected to give a reasonable timate of the complex eigenfrequency u. As shown by Table, the radial orders k and k for the unstable g-mod are, loosely speaking, min almost max the same as those found in the cas of )1 \ and.1 except for the retrograde g-mod with l \ m \ 1, for which k shifts to a substantially larger value compared to that in max the case of )1 \. For each value of l, several retrograde g-mod between k and k, mainly in the inertial regime, are pulsationally min stable max and they are in most cas coupled with g-mod with harmonic degre di erent from l. We note that the mode coupling e ect on the stability is more signiðcant for the retrograde mod than for the prograde mod. This is because the h dependence of the eigenfunction of a retrograde g-mode with o l o Z 1, which is approximately reprented by a Hough function (see, e.g., Lee & Saio 1997), obtains an extra node compared to that of the prograde g-mode with the same o l o in the inertial regime of the oscillation frequency. We also note that the e ect of mode coupling between g-mod with di erent valu of l on the stability is more signiðcant for the g-mod with l [ o m o than for those with l \ o m o. In fact, as shown by Figure 5, we Ðnd that the numbers of the unstable retrograde g-mod for l [ o m o are substantially smaller than those for l ] o m o in the case of )1 \.3. In Figure, we plotted the ratios A /A as a function of u for lj the g-mod with l \ m \ 1 (upper panel) and l \ m ] 1 \ (lower panel) at )1 \.3, where the cross, Ðlled squar, and open squar are for the ratios A /A with l \ m, m ], and m ] in the upper panel and lj for the j ratios with l \ m ] 1, m ] 3, and m ] 5 in the lower panel. j Mode coupling occurs frequently in the inertial regime, particularly on the retrograde side.. DISCUSSION AND CONCLUSION When the frequency of an envelope g-mode in a rotating star becom close to that of an inertial mode in the convective core in which the Schwarzschild discriminant A is thought to almost vanish, the eigenfunction of the g-mode behav like an inertial mode in the core. For the M main-sequence model, in which the outer boundary of the convective core is at r/ \.197, we can show an example of this phenomenon in Figure 7, where the real parts of the expansion coefficients S (solid line), H (dashed line), and it (dash-dotted line) l/ of the retrograde l/ g mode with l {/3 l \ m \ at )1 \.3 are given versus the fractional 3 radius r/, and the normalization given by e (S ) \ 1 is applied at the surface. The g mode has u and the ratio 3 \.3317 u /)B1.15. It is instructive to compare the expansion coefficients in the convective core with those of the inertial mode i (l [ o m o \ ) with m \ with the ratio u/)b1.1 1 in an insentropic polytropic model with the index N \ 1 Amplitud FIG. 7.Èeal parts of the expansion coefficients S (solid line), H (dashed line), and it (dash-dotted line) of the retrograde l/ g mode l/ (u with l \ l {/3 m \ in the main-sequence model 3 at \.3317) M )1 \.3 given vs. the fractional radius r/, where we have employed the normalization given by e (S ) \ 1 at the surface. The outer boundary of the convective core is at r/ \.197. (Yoshida & Lee ). The remblance between the expansion coefficients in the two cas is obvious. Savonije & Papaloizou (1997) have also discussed this ronance phenomenon between a g-mode in the envelope and an inertial mode in the convective core in the context of dynamical tid in a rotating main-sequence star of M in a binary system. The oscillation frequency of a g-mode in a uniformly rotating star may be asymptotically given by (see, e.g., Lee & Saio 197a) PS (k ] a)n \ j (l) N dr nm u r, () where k is an integer reprenting the radial order, a is a phase factor weakly dependent on u, N is the Brunt-Va isa la frequency, and the integration should be done in the propagation region of the g-mode. The symbol j denot the eigenvalue of LaplaceÏs tidal equation and is nm a function of the ratio l )/u for given integral indic n and m, for which we use the convention employed in Lee & Saio (1997). For n º, we have j (l) º in [O \l\]o, and j (l) B ( o m o ] n)( o m o ] nm n ] 1) ] ml when o l o>1. In the limit nm of o l o ], we may identify the oscillation mod associated with j as those with a spherical harmonic func- tion Y m(h, /) with nm the indic l \ o m o ] n and m. Note that, if we di erentiate l equation (7) with rpect to l at l \ for a given radial order k, we obtain C D 1/l(l ] 1). For a given value of o l o D, j with n º is 1 larger on the retrograde side than on the prograde nm side (see, e.g., Lee & Saio 1997). This means that at a given rotation speed ), the radial orders k of g-mod associated with j whose frequenci nm r/.3.5

9 No. 1, 1 g-modes IN SLOWLY PULSATING B STAS 319 satisfy l ¹ o l o ¹ l and fall in the interval u \ )/l ¹ o u o ¹ u 1 \ )/l are higher on the retrograde side than on the prograde 1 side. 1 In other words, for a given value of the radial order k, the frequency of a g-mode associated with j is higher for retrograde mod than for prograde mod, which nm is clearly seen in Figure 5. Equation () can give a good timation of the complex eigenfrequency u for a g-mode that satisð o )/u o [ 1 if mode coupling do not play any signiðcant role. From equation (), it is also obvious that slow rotation can be inñuential in determining the pulsational stability of a g-mode only when o D o for the mode is much greater than o u In the case of the 1I main-sequence model, I o. M o D o largely exceeds o u o for high radial order (k Z 1I 15) g-mod with l \ 1 at I )1 \, which leads to the shift of the radial order k to a larger value, for example, for the retrograde g-mod max with l \ m \ 1 at )1 \.1. It may be useful to calculate the coefficients D and D for mainsequence models with di erent mass 1 comparable with SPB stars. We have calculated the coefficients for two zeroage main-sequence models of M \ 5 and M \ 3 M with X \.7 and Z \, where log (L /L ) \.7 and log T \.5 for the M \ 5 M model and eff log (L /L ) \ 1.9 and log T \.1 for the M \ 3 M eff model. For the two main-sequence models, we eventually found almost the same behavior of the coefficients D and 1 D as functions of u as that found for the M model. We found that o D o is much larger than ou only for the 1I I o high radial order g-mod with l \ 1 and o D o [ o u for 1I I o the g-mod with l º. This means that slow rotation can be e ective in dtabilizing the high radial order retrograde g-mod with l \ 1. In fact, although no g-mod with l \ 1 are pulsationally unstable in the 5 M model at )1 \, we found by computing complex eigenfrequenci u at )1 D that a few high-k retrograde g-mod with l \ m \ 1 become pulsationally unstable, for example, at )1 \.1. This dtabilization is not e ective, however, for high radial order g-mod with l º because o D o [ o u for them. 1I I o In fact, in the 3 M model, g-mod with l \ are found to be pulsationally stable at )1 \, and they remain stable, for example, at )1 \.1. In this paper we have examined the e ects of the Coriolis force on the pulsational stability of g-mod with low l in a main-sequence model with M \ M, in which many g-mod with low l are excited by the opacity bump mechanism in the absence of rotation. We found that the slow rotation of the star contribut to the excitation of the high radial order g-mod with low l. This excitation is most signiðcant for the g-mod with l \ 1. At rapid rotation, we found many pulsationally unstable g-mod with low l even in the inertial regime in which o u o \ ). As the rotation speed increas, mode coupling caused by the Coriolis term between g-mod with di erent valu of l becom signiðcant, and this is most signiðcant for retrograde g-mod with l [ o m o. As a rult of mode coupling, some of the g-mod between k and k become pulsationally stable, which in fact leads to min a substantial max reduction in the numbers of the unstable retrograde g-mod with l \ m ] 1 \ and l \ m ] \ 3 in the M model. Some of the rults discussed in this paper are consistent with those obtained by Ushomirsky & Bildsten (199), who employed the quasiadiabatic and traditional approximations for the analysis. Under the approximations, however, the e ects of mode coupling on the stability cannot be properly taken into account. In this sense, the stability analysis carried out in this paper is more thorough than that of Ushomirsky & Bildsten (199). Since the centrifugal force and the Coriolis force can not be e ective in stabilizing low-frequency g-mod in SPB stars as shown by Lee & Bara e (1995) and by the prent paper, the rotation of the star may not be a key to solving the problem raised by Balona (199) and Balona & Koen (199) so long as uniform rotation is assumed. I am most grateful to Profsor D. Gough for his hospitality during my stay in Institute of Astronomy, University of Cambridge. Balona, L. 199, MNAS, 7, 1 Balona, L., & Koen, C. 199, MNAS, 7, 171 Carroll, B. W., & Hansen, C. J. 19, ApJ, 3, 35 Dziembowski, W. A., Moskalik, P., & Pamyatnykh, A. A. 1993, MNAS, 5, 5 Gautschy, A., & Saio, H. 1993, MNAS,, 13 Iglias, C. A., & ogers, F. J. 199, ApJ,, 93 Iglias, C. A., ogers, F. J., & Wilson, B. G. 199, ApJ, 397, 717 Lee, U., & Bara e, I. 1995, A&A, 31, 19 Lee, U., & Saio, H. 19, MNAS, 1, 35 EFEENCES Lee, U., & Saio, H. 197a, MNAS,, 513 ÈÈÈ. 197b, MNAS, 5, 3 ÈÈÈ. 1997, ApJ, 91, 39 Saio, H. 191, ApJ,, 99 Savonije, G. J., & Papaloizou, J. C. B. 1997, MNAS, 91, 33 Ushomirsky, G., & Bildsten, L. 199, ApJ, 97, L11 Waelkens, C. 1991, A&A,, 53 Waelkens, C., Aerts, C., Ktens, E., Grenon, M., & Eyer, L. 199, A&A, 33, 15 Yoshida, S., & Lee, U., ApJ, 59, 997