The core/envelope asymmetry in p-mode pulsating stars

Abstract

It has been shown that there is a potential ambiguity in the asteroseismic determination of the location of internal structures in a pulsating star (Montgomery et al. 2003). We show how, in the case of high-order non-radial acoustic modes, we can possibly remove this ambiguity by considering modes of different degree. To support our conclusions we have investigated the seismic signatures of sharp density variations in the structure of quasi-homogeneous models. Aliasing It is known that a sharp variation in the equilibrium structure of a star gives rise to a periodic component in the frequencies of oscillation (see for example Monteiro et al. 2000). A way to isolate these components, in high order modes, is to consider deviations from asymptotic expressions for period (frequency) spacings in g-mode (p-mode) pulsators. Montgomery et al. (2003) reported that in the case of white dwarfs, where only high order gravity modes have been detected, there is a potential ambiguity in determining where in the stellar interior the variation that generates the periodic signal is located. With the aim of extending the analysis to acoustic modes, we present how we could possibly remove such an ambiguity by considering modes of different degree. A general form for the periodic signal generated by a sharp variation located at an acoustic depth τd could be approximated by δν ' A(ν) sin (2π ν 2τd + φ) (1) where A(ν) is a slowly decreasing function of frequency which depends on the characteristics of the sharp variation (Monteiro et al., 2000). When looking for such a periodic signal in the frequencies of an acoustic oscillation spectrum it is clear that the signal can be evaluated only in a discrete set of frequencies νn,`, solutions of the oscillation equations. Let us consider modes of some degree ` and a periodic signal as in Eq. (1). Having defined the acoustic radius as θ(r) = ∫ r 0 dr c and remembering the simple first order asymptotic relation (Tassoul, 1980) νn ' ∆ν ( n+ φ ) (2) and T ≡ τ(0) ≡ θ(R) ' 1/(2∆ν) (3) 36 The core/envelope asymmetry in p-mode pulsating stars it is straightforward to show that δν (Eq. (1)) can be also written as δν = A(ν) sin (2π ν 2θd + φ ) (4) This means that we cannot distinguish whether a variation is located at an acoustic depth τd or θd = T − τd (see Mazumdar & Antia (2001) and Montgomery et al. (2003) in the case of g modes). Figure 1: (a) If the discontinuity is located near the surface, the periodic signal can be described as slowly changing and independent from ` with a period ∼ 1/(2τd) (continuous line) or as a signal with a short period (1/(2θd)) with a phase which depends on ` being even or odd (dashed and dotted lines). The signal evaluated at the discrete frequencies in Eq. (5) is represented by asterisks (` = 0) and diamonds (` = 1). The values on the axes are arbitrary. (b) The discontinuity is located near the center of the star. Since we would like to include in our treatment modes of different degree ` we generalized the previous argument considering, instead of Eq. (2), an asymptotic expression which includes the dependence on ` (Tassoul, 1980):