On the Prospects for Detection and Identification of Low-Frequency Oscillation Modes in Rotating B Type Stars
J. Daszynska-Daszkiewicz, W. A. Dziembowski, A. A. Pamyatnykh
aa r X i v : . [ a s t r o - ph ] A p r On the Prospects for Detection and Identificationof Low-Frequency Oscillation Modes in RotatingB Type Stars
J. D a s z y ´n s k a-D a s z k i e w i c z , W. A. D z i e m b o w s k i , and A. A. P a m y a t n y k h , Instytut Astronomiczny, Uniwersytet Wrocławski. ul. Kopernika 11, Polande-mail: [email protected] Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Polande-mail:[email protected] Copernicus Astronomical Center, ul. Bartycka 18, 00-716 Warsaw, Polande-mail: [email protected] Institute of Astronomy, Pyatnitskaya Str. 48, 109017 Moscow, Russia
ABSTRACTWe study how rotation affects observable amplitudes of high-order g- and mixed r/g-modes and examineprospects for their detection and identification. Our formalism, which is described in some detail, relies on anonadiabatic generalization of the traditional approximation. Numerical results are presented for a numberof unstable modes in a model of SPB star, at rotation rates up to 250 km/s. It is shown that rotation has alarge effect on mode visibility in light and in mean radial velocity variations. In most cases, fast rotationimpairs mode detectability of g-modes in light variation, as Townsend (2003b) has already noted, but it helpsdetection in radial velocity variation. The mixed modes, which exist only at sufficiently fast rotation, are alsomore easily seen in radial velocity. The amplitude ratios and phase differences are strongly dependent on theaspect, the rotational velocity and on the mode. The latter dependence is essential for mode identification.
Stars: oscillations – Stars: emission-line, Be – Stars: rotation
Variability with frequencies comparable to rotation frequency has been found in a num-ber of hot (most often Be) stars. Whether such variability is caused by slow modes hasbeen debated for some time (see e.g. , Baade 1982, Balona 1985). However, recent datafrom the MOST on z Oph (Walker et al. et al. b CMi (Saio et al. µ Eri (Jerzykiewicz et al. ⊙ star in the mid of its main sequence evolution. Ignoring effects of rotation, wefind that all dipole modes with period between 1.8 d to 2.8 d are unstable. Rotation pe-riods in this range correspond to equatorial velocities between 100 km/s and 150 km/s,which is not high for such a star.For low frequency modes the surface dependence may be approximately describedin terms of the Hough functions (see e.g. , Lee and Saio 1997, Bildsten et al. e.g. , Daszy´nska-Daszkiewicz, Dziembowski and Pamyat-nykh 2005), however, that in the case of B-type pulsators, it is very important to com-bine photometric and radial velocity data for a unique discrimination of excited modesand constraining stellar parameters. Thus, this work, which may be regarded as anextension of Townsend’s paper, focuses on calculation of radial velocity amplitudes.We adopt an uniform approach in our calculation of all disk-averaged amplitudes andit is different from that used by him. Furthermore, in addition to g-modes, we includer-modes, which become propagative in stellar radiative envelopes once rotation is fastenough (see Savonije 2005, Townsend 2005b, who uses the term mixed gravity-Rossbymodes, Lee 2006).In Section 2, after specifying assumptions adopted in our calculations, we summa-rize formulae for angular dependence of velocity and atmospheric parameters. Expres-sions for the light and disk-averaged radial velocity variations are given in Sections 3and 4, respectively. Sections 5, 6 and 7 present numerical results for selected modesin one representative stellar model considering range of rotation rates but ignoring ef-fects of changes of centrifugal force on model structure. Unstable mode propertiesare briefly described in Section 5. In Section 6, upon adopting an arbitrary normal-ization of linear eigenfunctions, we calculate observable amplitudes of various modes,which may be excited and detected. Prospects of mode identification are discussedin Section 7. Examples of diagnostic diagrams employing amplitude ratios and phasedifferences are shown there. In our study we adopt the standard approximations, which are the linear nonadiabatictheory of stellar oscillation and static plane-parallel atmosphere models. These ap-proximations are well justified in our application. Modes detected in slowly pulsatingB-type stars have indeed very low amplitudes, the oscillation periods are much longerthan the thermal time scale in the atmosphere, and vertical variations of mode am-plitude are small across the whole atmosphere. The constant kinematic accelerationis easily included. Like Townsend (2003b) we adopt the traditional approximation,which allows to separate latitudinal and radial dependencies of the pulsational ampli-tudes. We essentially follow his formalism, expect that we choose the azimuthal andtemporal dependence as Z = exp [ i ( m j − w t )] , which implies m > m < x ( R , q , j ) = e R (cid:18) Q , v ˆ Q sin q , − i v ˜ Q sin q (cid:19) Z ( ) where e is an arbitrary, but small, complex constant, and v = GM w R . The three functions Q , ˆ Q , ˜ Q describe the latitudinal dependence of solutions and arethe Hough functions, which are obtained as solutions of the Laplace’s tidal equations ( D + msµ ) Q = ( s µ − ) ˆ Q , ( )( D − msµ ) ˆ Q = [ l ( − µ ) − m ] Q , ( ) where s = W / w is called the spin parameter, µ = cos q and D ≡ ( − µ ) dd µ . Theequations together with boundary conditions at µ = µ = l . The third function is given by˜ Q = − m Q + sµ ˆ Q . ( ) Bildsten et al. (1996), Lee and Saio (1997), and Townsend (2003a) discussed in greatdetail the l ( s ) dependence and asymptotic properties of the Hough functions. Here,we recall only the essentials.For specified m and s →
0, there are branches with l → ℓ ( ℓ + ) and Q ( q ) → P | m | ℓ (ex-cept for normalization). These branches correspond to g-modes distorted by rotation.For prograde sectorial modes ( m = ℓ ), l slowly decreases with s . For all other g-modes,the function l ( s ) is increasing quite rapidly. We will identify g-mode branches by the ℓ value at s =
0. Thus, like in the case of no rotation, we will use ( ℓ, m ) values as theangular quantum numbers and refer to ℓ as the mode degree. However, now specifi-cation of the angular dependence requires also the value of s . The symmetry aboutthe equatorial plane is determined by the parity of ℓ + | m | . If it is even, the functions Q ( q ) and ˜ Q ( q ) are symmetrical and ˆ Q ( q ) is antisymmetrical. The opposite is true if ℓ + | m | is odd. The branches for which l → − ¥ at s → m <
0, there is one branch crossing zero at s = | m | +
1. If l >
0, the associatedmodes become propagatory in the radiative regions, they may be excited and visible inthe light variations. However, following Lee (2006), we will still call them r -modes.The functions Q ( q ) and ˜ Q ( q ) are antisymmetrical and ˆ Q ( q ) is symmetrical about theequator for these r-modes.Upon replacing ℓ ( ℓ + ) with l ( s ) , the nonadiabatic mode properties may be calcu-lated with a reasonable accuracy using the same code as for non-rotating stars. This isso because for the mode of our interest, horizontal flux losses, which are not correctlydescribed, are small. The latitudinal dependence of perturbed thermodynamical pa-rameters is then given by Q ( q ) . Thus, the bolometric flux perturbation may be writtenas d F bol F bol = e f Q Z , ( ) where f is a complex quantity determined by solution of linear nonadiabatic equationsand it depends on the stellar model and the mode parameters m , l and w .For evaluation of perturbed monochromatic fluxes, we also need the perturbed grav-ity, which, as follows from Eq. (1), is given by d gg = − e (cid:0) + v − (cid:1) Q Z . ( ) The effect of gravity perturbation plays a relatively small role in light variability causedby slow modes but it is easy to include. More important is perturbation of star shapeleading to changes in the projected surface element and the limb-darkening. For bothwe need the normal to stellar surface which, as follows from Eq. (1), is given by d n s = − e(cid:209) H ( Q Z ) = − e (cid:18) , ¶Q ¶q , i m Q sin q (cid:19) Z . ( ) The change of the directed element of the surface is d d S d S = e (cid:18) Q , − ¶Q ¶q , − i m Q sin q (cid:19) Z ( ) where d S = R d µ d j .From Eq. (1), we also obtain the perturbed pulsation velocity field as seen from aninertial system ( j = j + W t ) d v = d x d t = (cid:18) ¶¶ t + W ¶¶j (cid:19) x = e R [ − i wx + W e z × x ] . ( ) Use of the Lagrangian pulsational velocity is adequate for representing velocity at thephotospheric layer because x is nearly constant across the outer layers for the slowmodes considered by us. The most straightforward extension of the expression for the light variation to the caseof rotating stars is through the expansion of the Hough function into the truncatedseries of the associated Legendre functions. This was the way Townsend (2003b)derived his expression. We write the equivalent expression in the form, which is astraightforward generalization of our formula (Daszy´nska-Daszkiewicz, Dziembowskiand Pamyatnykh 2003) derived for the case of modes described by single sphericalharmonic. Now the complex amplitude of the light variation in the x passband may beexpressed as A x ( i ) = e ¥ (cid:229) j = g m ℓ j ( s ) Y m ℓ j ( i , ) h D x ℓ j f + E x ℓ j i ( ) where ℓ j = (cid:26) | m | + ( j − ) even − parity modes | m | + ( j − ) + − parity modesand D x ℓ = − . b x ℓ ¶ log ( F x | b x ℓ | ) ¶ log T eff , E x ℓ = − . b x ℓ (cid:20) ( + ℓ )( − ℓ ) − (cid:0) + v − (cid:1) ¶ log ( F x | b x ℓ | ) ¶ log g (cid:21) , b x ℓ = Z h x ( ˜ µ ) ˜ µP ℓ ( ˜ µ ) d˜ µ . ( ) where h x is the limb-darkening law, adopted in the nonlinear form (Claret 2000). Withthis form Townsend obtained his analytical expressions for b x ℓ . The quantities g m ℓ j ( s ) ,which have to be calculated numerically, are the expansion coefficients of the Q func-tion into the series of the Legendre functions. That is Q ( q ) = ¥ (cid:229) j = g m ℓ j ( s ) P m ℓ j ( q ) . ( ) The expression given by Eq. (9) is quite revealing. At specified e , the light ampli-tudes of low degree modes must decrease with s because of increasing role of higherorder terms which suffer more from disk-averaging. The higher order terms lead to theaspect-dependence of the amplitude ratios.Unfortunately, we could not find a corresponding semi-analytical expression forthe radial velocity and this is why we decided to rely on two-dimensional numericalintegration over the visible hemisphere. We used Eq. (9) to check the accuracy. The total flux in the x passband toward the observer is given by L x = Z S F x h x n obs · d S ( ) where integration is carried over visible part of star surface, S , and n obs is the unitvector toward observer.In the spherical coordinate system with the polar axis parallel to the rotation axis,we have n obs ≡ ( o r , o q , o j ) ( ) where o r ≡ ˜ µ = cos i cos q + sin i sin q cos ( j − j ) , o q = − cos i sin q + sin i cos q cos ( j − j ) , o j = − sin i sin ( j − j ) . The observer’s angular coordinates are ( i , j ) . The first order perturbation of the totalflux is given by d L x = Z S [( d F x h x + F x d h x ) d S e r + F x h x d d S ] · n obs . ( ) Assuming an equilibrium atmosphere, we have from Eqs. (4) and (5) d F x = e (cid:20) a xT f − a xg (cid:0) + v − (cid:1)(cid:21) Q Z where a xT = ¶ log F x ¶ log T eff and a xg = ¶ log F x ¶ log g are the derivatives determined numerically from grids of stellar atmosphere models.For perturbed limb darkening, we take into account perturbations of the coefficientson T eff and g though they are only secondary contributors to light variation. Moreimportant contribution arises from d n s (see Eq. 6). With all terms included, we obtain d h x = e (cid:26)(cid:20) ¶ h x ¶ ln T f − ¶ h x ¶ ln g (cid:0) + v − (cid:1)(cid:21) − ¶ h x ¶ ˜ µ (cid:18) ¶Q ¶q o q + i m Q sin q o j (cid:19)(cid:27) Q Z . The derivatives of Claret’s h x are given in Appendix A1. The expression forperturbed surface element is given in Eq. (7). The integration is carried over the unper-turbed visible hemisphere. Within the linear approximation, the integration boundaryis unchanged. Note also that the horizontal component of the displacement does notenter the expression. The domains of integration over q and j are shown in Fig. 1 andFig. 2, respectively. From Figs. 1 and 2 it follows that the ranges are:0 ≤ q ≤ p + i , i obs Fig. 1. The meridional view of the integration area. i z yx yx Fig. 2. Integration area over the azimuthal angle ( left ) and the edge-on view ( right ). − ( p − a ) ≤ j − j ≤ ( p − a ) where a = arccos [ cot q cot i ] . For each q , we carry the integration over the azimuthalangle, Y = j − j , from − b to + b , where b = p − a = p − arccos [ cot q cot i ] (see Fig. 1and Fig. 2). We use identities b Z − b G ( Y ) Zd Y = b R G ( Y ) cos m Y d Y if G ( Y ) is even2i b R G ( Y ) sin m Y d d Y if G ( Y ) is oddThe final expression for the total light variation can be written in the following form d L x L x = e (cid:20)(cid:18) a xT B + B (cid:19) f + B + B − ( + v − )( a xg B + B ) (cid:21) Z ( ) and Z = exp [ i ( m j − w t )] . In Appendix A2 we give explicit expressions for the two-dimensional integrals, B , which depend on two angular numbers, spin and the aspect.The integrals take into account changes in the limb-darkening resulting from the changeof the normal (Eq. 6) as well as the change due to perturbation of the local temperature(Eq. 4) and gravity (Eq. 5). The two latter are given through derivatives of h x withrespect to log T eff and log g . There are many terms contributing to d L x . However, in ourapplications two are far dominant: the one resulting from temperature perturbation,which is proportional to a xT B , and the other resulting from the surface distortion,which is proportional to B . Adopting the standard sign convention, we write the radial velocity averaged over thestellar disk in the following form: h V rad i = − R S ( v · n obs ) F h x n obs · d S R S F h x n obs · d S ( ) where for the total velocity field we use v = d v + W R sin q e j . ( ) The pulsational component, d v , as results from Eqs. (1) and (8), is given by d v = − i we R Q − s v ˜ Qv (cid:20) ˆ Q sin q − s q sin q ˜ Q (cid:21) − i v (cid:20) ˜ Q sin q − s q sin q ˆ Q (cid:21) + i s Q sin q Z ( ) The contribution of rotation to the mean radial velocity arises from the same pulsationalchanges of photospheric parameters which cause luminosity change and may be calcu-lated in the same way as outlined in Section 3. Clearly, there is a nonzero contributiononly for non-axisymmetric modes.We write our final expression for the perturbed radial velocity in the followingform, d h V rad i = i we R ( C puls + C rot ) Z , ( ) with C puls = (cid:16) B − s B (cid:17) + v (cid:16) B − s B (cid:17) and C rot = − s sin i (cid:20)(cid:18) a xT B + B (cid:19) f + B + B − (cid:0) + v − (cid:1) (cid:0) a xg B + B (cid:1)(cid:21) . Explicit expressions for the B coefficients in terms of the three Hough functions aregiven in Appendix A3.If s ≈ m =
0, the two contributions are of the same order. Similarly to thecase of the luminosity, the dominant contributions to C rot are the ones resulting fromtemperature perturbation, which here is proportional to a xT B , and the other resultingfrom the surface distortion, which is proportional to B . ⊙⊙⊙ Main Sequence Star
To illustrate how visibility of various modes depends on equatorial velocity of rota-tion we choose a model of a 6 M ⊙ Population I star in the mid of its main sequenceevolution. Parameters of the model are given in Table 1. The model is spherically sym-metric, which is consistent with our use of the traditional approximation, but includesaveraged effects of centrifugal force corresponding to uniform rotation with equatorialvelocity of 250 km/s. The mean effects of centrifugal force are still reasonably smalland therefore we used the same model to study effects of the Coriolis force at lowerequatorial velocities.There are many low frequency modes that are unstable in our selected star model.These are mainly g-modes. However, at sufficiently high rotation rate there are alsocertain r-modes, which become propagatory in radiative envelope and may become un-stable. In all cases, the instability is caused by the k -mechanism acting in the metalopacity bump layer. For g-modes, the angular order, ℓ , is defined at the limit of zerorotation where the angular dependence of amplitude is described by individual spher-ical harmonics. The actual mode geometry is determined by the ( ℓ, m ) numbers andthe spin parameter, s . Since at each azimuthal order m < l changes sign, we will identify r-modes by the m value alone. Naturally, wefocus attention on modes suffering least reduction of observable amplitude caused bydisk-averaging. Therefore, we consider g-modes with ℓ ≤ m = − − -1.0-0.50.00.51.01.52.02.5 -1.0-0.50.00.51.01.52.02.50 10 20 30 40 50 60 70 80 90-1.0-0.50.00.51.01.52.02.5 0 10 20 30 40 50 60 70 80 90-1.0-0.50.00.51.01.52.02.5 V rot = 0 km/s V rot = 50 km/s V rot = 150 km/s V rot = 250 km/s r , m =-1 =1, m = +1 =1, m = -1 =1, m =0 Fig. 3. The Hough functions for the selected g-modes with ℓ = m = − Typically, at each degree and azimuthal order, we find instability extending overmany (up to 40) radial orders. For analysis of visibility, we selected the mode charac-terized by the highest normalized growth rate, h , which varies between − ℓ as the first entry. The effect ofrotation on mode surface geometry at specified V rot depends on s which determines l . T a b l e 1
Most unstable low degree modes in the B star model with M = . ⊙ , log T eff = . L / L ⊙ = . ℓ m V rot spin l v f n obs [c/d] n star [c/d] h + + + + − − − − + + + + − − − − − . + + + + − − − − − − − − -2-1012 -2-1012-2-1012 -2-10120 10 20 30 40 50 60 70 80 90-2-1012 0 10 20 30 40 50 60 70 80 90-2-1012 V rot = 0 km/s V rot = 50 km/s V rot = 150 km/s V rot = 250 km/s r , m = -2 =2, m = +1 =2, m = -1 =2, m = +2 =2, m = -2 =2, m =0 Fig. 4. Same as Fig. 3 but for the g-modes with ℓ = m = − The depth-dependence of eigenfunctions is determined primarily by the product lv . Inparticular, the radial orders in this model are given by n ≈ . √ lv . For the selectedmodes they are between 26 and 34. Nonadiabatic parameters, f and h , depend on n butalso on the pulsation frequency in the star reference system, n star .The Hough Q -functions for selected modes are shown in Figs. 3 and 4. We may seethe well-known effect of equatorial amplitude confinement, which increases with V rot .The effect is present in all modes including those with m = ℓ though l ( s ) is a decreasingfunction in these cases. As for the remaining Hough functions, which are important,the confinement is also present. ˜ Q and Q have the same symmetry about the equatorand the symmetry of ˆ Q is opposite. For the modes listed in Table 1, we calculated amplitudes of light variation in the U and V Geneva passbands, A U and A V , with Eqs. (15), as well as that of the radial velocity, A V rad , with Eqs. (19), adopting an arbitrary normalization, e = .
01. Coefficients a T and a g occurring in Eq. (15), were interpolated from the line-blanketed models of stellaratmospheres (Castelli and Kurucz 2004).1Figs. 5–7 show the A V and A V rad in function of the aspect angle. Because of thearbitrary normalization, they cannot be regarded as reliable predictors of expected am-plitudes. We may expect that chances of excitation are related to h but it does notdetermine e . Still plots like these provide important hints for interpretation of the richoscillation spectra such as that of HD 163868 (Walker et al. ℓ = A V on the rotation rate. Theaspect-dependence is qualitatively the same as in the case of no rotation. Rotation givesrise to a departure from the dipolar angular dependence of x r and d F bol but the contri-bution of the surface distortion to light variation remains small even at V rot =
250 km/s.Thus, the A V ( i ) -dependence is a simple reflection of the Q ( q ) -dependence shown inFig. 3. The equatorial confinement leads to a reduction of amplitude upon averagingcontributions from the whole hemisphere. =1, m=0 =1, m=-1=1, m=-1 =1, m=+1=1, m=+1 A V [ mm ag ] =1, m=0 A V r ad [ k m / s ] rot = 0 km/s V rot = 50 km/s V rot =150 km/s V rot =250 km/s A V [ mm ag ] A V r ad [ k m / s ] A V [ mm ag ] inclination A V r ad [ k m / s ] inclination Fig. 5. Amplitudes of light in the V -band of Geneva photometry, A V , and of the radial velocity, A V rad , atindicated equatorial rotational velocities plotted as functions of the aspect angle, for the ℓ = e = .
01 (see Eq. 1).
The pattern of the dependence of radial velocity amplitude on the rotation rate,shown in the right panels of Fig. 5, is more complicated. In all three cases, the dominantcontribution arises from pulsational velocity (the C puls term in Eq. 19). A secondary butsignificant contribution, the C rot term, adds in the case of prograde modes and subtractsin the case of retrograde modes. However, the main reason for the large differencebetween the modes and for the nonmonotonic aspect dependence is connected withproperties of C puls . At high rotation rates the contribution from the advective term in d v is large and causes that all its three components play a role. The large values of A V rad inthe case of ℓ = m =+ A V ( i ) -dependencies for the ℓ = V rot but on average the effectis the same as for ℓ =
1, that is fast rotation decreases chances for photometric detectionof slow modes. Also on average, amplitudes of the ℓ = ℓ =
1. In the right panels of Fig. 6, we may see that radial velocity amplitudes ofprograde modes increase with the rotation rate and that the effect is opposite for theretrograde modes. The role of the C rot term is much more significant than at ℓ = rot = 0 km/s V rot = 50 km/s V rot =150 km/s V rot =250 km/s =2, m= -2=2, m= -2 =2, m=+2=2, m=+2=2, m=0 =2, m=-1=2, m=-1 =2, m=+1=2, m=+1 A V [ mm ag ] =2, m=0 A V r ad [ k m / s ] A V [ mm ag ] A V r ad [ k m / s ] A V [ mm ag ] A V r ad [ k m / s ] A V [ mm ag ] A V r ad [ k m / s ] A V [ mm ag ] inclination A V r ad [ k m / s ] inclination Fig. 6. Same as Fig. 5 but for the ℓ = rot =150 km/s V rot =250 km/s r, m= -1 r, m= -2r, m= -2 A V [ mm ag ] r, m= -1 A V r ad [ k m / s ] inclination A V [ mm ag ] inclination A V r ad [ k m / s ] Fig. 7. Same as Fig. 5 but for the r-modes with | m | ≤ We may see in Fig. 3 that Q ( q ) for the m = − ℓ = , m = Q ’s for the m = − ℓ = , m = + f . Moreover,the effect of distortion, which is significant, cancels a part of the effect of temperatureperturbation. The r-modes are antisymmetric with respect to the equator and are bestseen from the intermediate aspect angles. The radial velocity amplitudes, shown in theright panels of Fig. 7, are much less reduced. Thus, spectroscopy gives a better chancefor detecting r-modes. Rotation has a very profound effect on slow mode visibility and hence on the procedureof mode identification. Unlike in non-rotating stars, the amplitude ratios depend on theazimuthal order, m , and on the aspect, i . Modes with various m differ not only in thesurface geometry but also in their nonadiabatic properties. There are constraints on m following from localization in the frequency spectrum. Modes differing in the sign of m are well separated. As we have seen in Figs. 5–7, the aspect is an important factorin mode selection which has to be taken into account considering possible m and i values. Nonetheless, a unique discrimination of modes is not likely possible withoutemploying amplitude and phase data.Let us begin with photometric data alone. The observables, which do not depend on e , are amplitude ratios and phase differences from multiband photometry. In this case,the potential for mode discrimination rests mainly on the difference in the relative tem-perature and distortion contribution to the total light variation in different passbands.The latter contribution for the ℓ = ℓ = i and V rot is imprecise. A discrimination between the two ℓ = m = −
1, if V rot is between 150 km/s and 175 km/s, and m = −
2, if V rot is 250 km/s.4 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.02.2 r, m= -1 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.02.2 A U / A V V rot = 0 km/s V rot = 50 km/s V rot =150 km/s V rot =250 km/s =1, m= -1 A U / A V -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.02.2 =2, m= -2 U - V [rad] -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.02.2 U - V [rad] =2, m= -1 Fig. 8. Photometric diagnostic diagrams for selected modes based on light amplitudes and phases in the U and V Geneva passbands, for retrograde g-modes and for the r-mode with m = −
1. The arrows indicatedirection of increasing i . The i -ranges are limited by the condition A V >
10 mmag
Any discrimination between these four modes based on frequencies alone wouldnot be possible because all of them are retrograde and occupy the low frequency endof the oscillation spectrum. It is important to remember that the plots refer only tomost unstable modes of each type. Thus, in real life, there is additional uncertainties infrequencies and in the f -values that affect the amplitude ratios and phase differences.With mean radial velocity data, discrimination between the two ℓ = ℓ = m = + m = ℓ with no need for specifying the value of the complex parameter f .Instead, as described in the case of non-rotating stars by Daszy´nska-Daszkiewicz etal. (2005), the combined spectroscopy and photometry data on amplitudes and phasesmay be used to determine f , which becomes an independent seismic observable, andthe mode degree, ℓ . In order to see how the method may be extended to the presentcase, let us rewrite Eqs. (15) and (19) as expressions for the complex amplitudes oflight in the x -band and of radial velocity, respectively. From Eq. (15) we obtain A x ( i ) = D x ( i ) f ˜ e + E x ( i ) ˜ e ( ) where D x ( i ) = − . (cid:18) a xT B + B (cid:19) , E x ( i ) = − . [ B + B − ( + v − )( a xg B + B )] and ˜ e = e exp ( i m j ) . Similarly, from Eq. (19) the first moment of spectral line is equalto M x ( i ) = i w R [ H x ( i ) f ˜ e + G x ( i ) ˜ e ] ( ) -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 A V r ad / A V [ k m / s / m ag ] V rot = 0 km/s V rot = 50 km/s V rot =150 km/s V rot =250 km/s =2, m= -2 Vrad - V [rad] -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 Vrad - V [rad] =2, m= -1 -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 =1, m= -1 A V r ad / A V [ k m / s / m ag ] -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 =1, m= 0 Fig. 9. Diagnostic diagrams for selected modes based on amplitudes and phases in the V Geneva photometricpassband and in the radial velocity, for g-modes with ℓ = where H x ( i ) = − s sin i (cid:18) a xT B + B (cid:19) and G x ( i ) = B − s B + v (cid:16) B − s B (cid:17) − s sin i (cid:2) B + B − (cid:0) + v − (cid:1) (cid:0) a xg B + B (cid:1)(cid:3) . These two equations are counterparts of Eqs. (1) and (2) of Daszy´nska-Daszkiewicz etal. (2005).In the case of negligible rotation, the dependence of amplitudes on m and i hasbeen absorbed in ˜ e and the problem was reduced to finding ˜ e f and ˜ e by the least squareminimization of c ( ℓ ) . The method requires stellar atmosphere parameters, which infact may be improved on the process of pulsation amplitude fitting. An unique deter-mination of ℓ is possible even with imprecise atmosphere parameters if the min ( c )( ℓ ) dependence is strong. In the present case we have two more stellar parameters to im-prove which are i and V rot . Moreover, we have to take into account the m -dependence.Prospects for mode identification depend on the strength of the the min ( c )( ℓ, m ) de-pendence. The plots in Figs. 8 and 9 suggest that it is likely the case. However, itremains to be seen when the method is applied to real data. Our goal was to examine chances for detection and identification of slow oscillationmodes whose frequencies are of the order of angular velocity of rotation. In our cal-culations of expected mode amplitudes, we relied on a nonadiabatic generalization ofthe traditional approximation, similar to that introduced by Townsend (2005a). Thechances for detecting a particular mode depend, in part, on its intrinsic amplitudes,which may be calculated only in the framework of a nonlinear modeling. Our linearmodels give us only a hint which is the growth rate. Such models are expected to6be adequate for describing geometry of the mode which has important impact on modevisibility, its aspect and the angular degree dependence. With the formalism outlined inSections 3 and 4, we calculated the observable amplitudes for selected unstable modesin models of a 6 M ⊙ main sequence star as a function of the rotation rate and the aspectangle. The model may be regarded as representative for SPB variables.Departure from the individual spherical-harmonic dependence which increases withthe rotation rate leads, in most cases, to lower photometric amplitudes. In contrast, theeffect on radial velocity amplitude is most often opposite. However, the light to radialvelocity amplitude ratio changes significantly from mode to mode and depends on theaspect. We showed, in particular, that the mixed r/g-modes are most easily detectable inradial velocity. Considering possible identification of peaks in rich oscillation spectrasuch as of HD 163868 (Walker et al. et al. (2007).The observables yielding numerical constraints on mode geometry and the aspectangle are amplitude ratios and phase differences. We calculated examples of diagramsbased on photometric data in two passbands and on mean radial velocity measurements.Not always photometric data are sufficient for mode discrimination. Radial velocitydata not only help discrimination but also allow to proceed in a less model-dependentmanner. The photospheric value of the complex radial eigenfunction correspondingto the radiative flux may then be determined from data rather than taken from linearnonadiabatic calculations. A comparison of calculated and deduced values yields aconstraint on the model. Acknowledgements.
This work has been supported by the Polish MNiSW grantNo 1 P03D 021 28.
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A1. The Limb-Darkening Law
For calculating the surface integrals in Eq. (15) and Eq. (19), we need to specifythe limb-darkening law. Here, we use the nonlinear Claret (2000) formulae which werewrite in the following form h x ( ˜ µ ) = − (cid:229) k = a xk ( − ˜ µ k / ) − (cid:229) k = kk + a xk . The derivative with respect to ˜ µ can be easily obtained from this formula. The deriva-tives with respect to the effective temperature and gravity are given by: ¶ h x ¶ ln T eff = − (cid:229) k = kk + a xk · (cid:229) k = (cid:20) kk + h x − ( − ˜ µ k / ) (cid:21) ¶ a xk ¶ ln T eff , and ¶ h x ¶ ln g = − (cid:229) k = kk + a xk · (cid:229) k = (cid:20) kk + h x − ( − ˜ µ k / ) (cid:21) ¶ a xk ¶ ln g respectively. A2. Integrals in the Expression for the Light Variation B = p + i Z Q P sin q d q B = p + i Z (cid:18) ¶Q ¶q sin q P + m Q P (cid:19) d q B = p + i Z Q P sin q d q B = p + i Z Q P sin q d q where P = p b Z cos m Y h x o r d Y P = − p b Z cos m Y (cid:18) o r d h x d o r + h x (cid:19) o q d Y P = p b Z sin m Y (cid:18) o r d h x d o r + h x (cid:19) o j d Y P = p b Z cos m Y ¶ h x ¶ ln T eff o r d Y P = p b Z cos m Y ¶ h x ¶ ln g o r d Y . The function Q is one of three Hough functions (see Section 2). The components ofthe unit vector directed to the observer ( o r , o q , o j ) ) are given in Eq. (13). A3. Integrals in the Expression for the Radial Velocity Variation B = p + i Z Q P sin q d q B = p + i Z ( ˆ Q P + ˜ Q P ) d q B = p + i Z Q P sin q d q B = p + i Z (cid:2) ˜ Q P sin q + (cid:0) ˜ Q P + ˆ Q P (cid:1) cos q (cid:3) d q B = p + i Z Q P sin q d q B = p + i Z (cid:18) ¶Q ¶q sin q P + m Q P (cid:19) sin q d q B = p + i Z Q P sin q d q B = p + i Z Q P sin q d q where P = p b Z cos m Y h x o r d Y P = p b Z cos m Y h x o r o q d Y P = p b Z sin m Y h x o r o j d Y P = p b Z sin m Y sin Y h x o r d Y P = − p b Z sin m Y sin Y (cid:18) o r d h x d o r + h x (cid:19) o q d Y P = − p b Z cos m Y sin Y (cid:18) o r d h x d o r + h x (cid:19) o j d Y P = p b Z sin m Y sin Y ¶ h x ¶ ln T eff o r d Y P = p b Z sin m Y sin Y ¶ h x ¶ ln g o r dd