A Summary of the Different Classes of Stellar Pulsators A summary of all the classes of pulsating stars and their main properties as described in Chapter 2 is given in the tables below. This list originated from a combination of observational discoveries, measured stellar properties, and theoretical developments. Observers who found a new type of pulsator either named it after the prototype or gave the class a name according to the ob- served characteristics of the oscillations. Several pulsators, or even groups of pulsators, were afterwards found to originate from the same physical mech- anism and were thus merged into one and the same class. We sort this out here in Tables A.1 and A.2 in order to avoid further confusion on pulsating star nomenclature. The effective temperature and luminosity indicated in Tables A.1 and A.2 should be taken as rough indications only of the borders of instability strips. Often the theory is not sufficiently refined to consider these boundaries as final. Moreover, there is overlap between various classes where so-called hy- brid pulsators, whose oscillations are excited in two different layers and/or by two different mechanisms, occur. Finally, new discoveries are being made frequently, which then drive new theoretical developments possibly leading to new instability regions. The results from the future observing facilities as described in Chapter 8 will surely lead to new classes and/or subclasses with lower amplitudes compared to what is presently achievable. In the tables below, F stands for fundamental radial mode, FO for first radial overtone and S for strange mode oscillations. Several classes undergo (quasi-)periodicities with different time scales due to outbursts, cyclic variabil- ity, rotational modulation, differential rotation, activity, binarity, etc., besides oscillations. We list here only the period and amplitude ranges for the oscil- latory behaviour. Exceptions to the listed ranges are possible. The various names for each class as they occur in the literature are listed; the left column was the authors’ choice, dominantly based on papers in the literature which discuss the physical cause of the oscillations. For a full description of the variability characteristics, we refer to Chapter 2 and the numerous references listed there. 679 Table A.1. The names and basic properties of main sequence, giant and compact pulsators. Class name Other names Mode Period Amplitudes Amplitudes log Teff (K) log L/L adopted here in the literature Type Ranges (brightness) (velocity) approx. range approx. range Solar-like main sequence p 3to10min few ppm < 50 cm s−1 [3.70, 3.82] [−0.5, 1.0] pulsator red giants p few hours few 10 ppm few m s−1 [3.65, 3.70] [−0.5, 2.0] γ Dor slowly pulsating F g 8h to5d < 50 mmag < 5kms−1 [3.83, 3.90] [0.7, 1.1] δ Sct SX Phe (Pop. II) p 15 min to 8 h <0.3 mag < 10 km s−1 [3.82, 3.95] [0.6, 2.0] roAp — p 5to22min < 10mmag < 10 km s−1 [3.82, 3.93] [0.8, 1.5] SPB 53 Per g 0.5 to 5 d < 50 mmag < 15 km s−1 [4.05, 4.35] [2.0, 4.0] β Cep β CMa, ζ Oph (SpT O) p&g 1to12h(p) < 0.1mag < 20 km s−1 [4.25, 4.50] [3.2, 5.0] 53 Per few days (g) < 0.01 mag < 10 km s−1 pulsating Be λ Eri, SPBe p&g 0.1 to 5 d < 20 mmag < 20 km s−1 [4.05, 4.50] [2.0, 5.0] pre-MS pulsator pulsating T Tauri p 1to8h < 5 mmag pulsating Herbig Ae/Be p 1to8h < 5 mmag pulsating T Tauri g 8h to5d < 5 mmag p-mode sdBV EC14026, V361 Hya p 80 to 800 s < 0.1mag [4.20, 4.50] [1.2, 2.2] g-mode sdBV PG1716+426, g 0.5 to 3 h < 0.01 mag [4.40, 4.60] [1.2, 2.6] Betsy star, lpsdBV PNNV ZZ Lep, [WCE] g 5h to5d < 0.3mag GW Vir DOV, PG1159 g 5to80min < 0.2mag [4.80, 5.10] [1.5, 3.5] DBV V777 Her g 2to16min < 0.2mag [4.40, 4.60] [−1.0, 0.7] DAV ZZ Ceti g 0.5 to 25 min < 0.2mag [3.95, 4.15] [−2.6, −2.2] Table A.2. The names and basic properties of supergiant pulsators. Class name Other names Mode Period Amplitudes Amplitudes log Teff (K) log L/L adopted here in literature Type Ranges (brightness) (velocity) approx. range approx. range RR Lyr RRab F ∼ 0.5d < 1.5mag < 30 km s−1 [3.78, 3.88] [1.4, 1.7] RRc FO ∼ 0.3d < 0.5mag < 10 km s−1 RRd F+FO 0.3 to 0.5 d < 0.2mag < 10 km s−1 Type II Cepheid WVir F 10 to 30 d < 1mag < 30 km s−1 [3.70, 3.90] [2.0, 4.0] BL Her F 1to5d < 1mag < 30 km s−1 RV Tauri RVa, RVb F? 30 to 150 d < 3mag < 30 km s−1 [3.60, 3.90] [3.2, 4.2] Type I Cepheid — F 1to50d < 1mag < 30 km s−1 [3.55, 3.85] [2.0, 5.5] s-Cepheid FO < 20 d < 0.1mag < 10 km s−1 Mira SRa, SRb l =0 > 80 d < 8mag [3.45, 3.75] [2.5, 4.0] SRc l =0 > 80 d < 1mag SRd l =0 < 80 d < 1mag PVSG (SpT A) α Cyg g,S? 10 to 100 d < 0.3mag < 20 km s−1 [3.45, 4.00] [2.5, 4.5] PVSG (SpT B) SPBsg g 1to10d < 0.3mag < 20 km s−1 [4.00, 4.50] [4.3, 5.8] LBV SDor g,S 2to40d < 0.1mag [3.80, 4.20] [5.5, 6.5] WR WC, WN g,S 1h to 5 d < 0.2mag [4.40, 4.70] [4.5, 6.0] HdC RCrB g? 40 to 100 d < 0.05 mag < 10 km s−1 [3.50, 4.20] [3.5, 4.5] eHe PV Tel S ∼20 d V652 Her g,S? ∼0.1 d V2076 Oph g,S? 0.5 to 8 d B Properties of Legendre Functions and Spherical Harmonics B.1 Properties of Legendre Functions We provide here some basic properties of the Legendre functions, which are essential to the understanding of nonradial oscillations. The following expres- sions are mainly taken from Abramowitz & Stegun (1964) and Whittaker & Watson (1927). The Legendre function are solutions of the differential equation: d2P m dP m m2 (1 − x2) l − 2x l + l(l +1)− P m =0, (B.1) dx2 dx 1 − x2 l or, equivalently, d dP m m2 (1 − x2) l + l(l +1)− P m =0. (B.2) dx dx 1 − x2 l In the special case of m = 0, we are dealing with the Legendre polynomials: 0 Pl(x)=Pl (x). Explicit expressions for the first few cases are: P0(x)=1, P1(x)=x, 1 − − 2 1/2 P1 (x)= (1 x ) , (B.3) 2 P2(x)=1/2(3x − 1) , 1 − − 2 1/2 P2 (x)= 3(1 x ) , 2 − 2 P2 (x)=3(1 x ) . General expressions, valid for m>0, are: 1 dl(x2 − 1)l P (x)= , (B.4) l 2ll! dxl 683 684 B Properties of Legendre Functions and Spherical Harmonics dmP (x) P m(x)=(−1)m(1 − x2)m/2 l , (B.5) l dxm (2m)!2−m P m(x)=(−1)m (1 − x2)m/2 . (B.6) m m! Note that Eq. (B.4)showsthatPl(x) is a polynomial of degree l.Also, Eq. (B.6)showsthat (2m)!2−m P m(cos θ)=(−1)m sinm θ (B.7) m m! The Legendre function for negative azimuthal order is obtained from the one for positive m as (l − m)! P −m(cos θ)= P m(cos θ) . (B.8) l (l + m)! l The Legendre functions are easily computed from the following recursion re- lations: − m m − m (l m +1)Pl+1(x)=(2l +1)xPl (x) (l + m)Pl−1(x) , (B.9) P m+1(x)= (B.10) l " # − 2 −1/2 − m − m (1 x ) (l m)xPl (x) (l + m)Pl−1(x) , m 2 dPl m m (1 − x ) = lxP (x) − (l + m)P − (x) , (B.11) dx l l 1 dP dP − x l − l 1 = lP (x) . (B.12) dx dx l The following integral expression defines the orthogonality and normaliza- tion of the Legendre functions: 1 m m (n + m)! Pl (x)Pl (x)dx = δll − . (B.13) −1 (l +1/2)(l m)! The Legendre functions have the following asymptotic expansion, for fixed m ≥ 0andlargel: Γ (l + m +1) π −1/2 1 π mπ P m(cos θ)= sin θ cos l + θ − + l Γ (l +3/2) 2 2 4 2 +O(l−1) (B.14) m For arbitrary, large l, m an approximate asymptotic representation of Pl is m 2 − 2 −1/4 Pl (cos θ) Alm cos Θlm cos θ cos[Ψlm(θ)] , cos θ ∈ [− cos Θlm, cos Θlm] , (B.15) for a suitable amplitude Alm and a rapidly varying phase function Ψlm;here B.2 Properties of Spherical Harmonics 685 −1 Θlm =sin [m/ l(l +1)] (B.16) (e.g. Gough & Thompson 1990).1 Thus the Legendre function is confined between turning-point latitudes ±(π/2−Θlm); in particular, as also illustrated in Fig. B.1,modeswithm ±l are confined very near the equator.