© The Authors 2024

1. Introduction

Asteroseismology is the study of stellar oscillations, which allows for detailed inferences regarding the interior structure of stars; a property that conventional techniques such as photometry and spectroscopy fail to obtain due to the obscuring stellar opacity. In-depth analyses using individual frequencies are routinely carried out for the Sun and other solar-type stars along the main sequence (MS; see e.g. Houdek & Gough 2009; Silva Aguirre et al. 2017; Winther et al. 2023). Here, the stars behave according to the predictions of the asymptotic theory (see Sect. 2) and are stochastically excited by convection. We say that they exhibit solar-like oscillations. As the stars evolve past the MS turn-off and onto the sub-giant branch (SGB), individual oscillations frequencies can still be employed. Here, the stars begin to develop features that deviate from the asymptotic theory, but only modestly, such that it can be handled accordingly by our oscillation codes. The deviations can provide sensitive information on the interior structure, yielding constraints on the characterisation of the SGB stars (e.g., Christensen-Dalsgaard et al. 1995; Metcalfe et al. 2010; Grundahl et al. 2017; Stokholm et al. 2019). When we wish to extend the considerations to the more evolved counterparts, the red giants, however, our efforts are obstructed by the nature of their oscillation spectra.

Asteroseismology and the inferences that are possible from the individual frequencies could provide valuable insights into the nature of red giants – aiding in their characterisation and understanding. The crucial criteria for making these inferences, however, is the possibility to link observed oscillations to the oscillation modes of the stellar models. On the observational side, the asteroseismic data available for giants have been revolutionised by space missions such as Convetion, Rotation and planetary Transits (CoRoT; Baglin et al. 2009) and Kepler (Gilliland et al. 2010). The observed asteroseismic parameters for the expansive collection of available time series have been extensively derived, and as the oscillation amplitudes roughly scale as the ratio between the stellar luminosity and mass, L/M, as summarised by Kjeldsen & Bedding (1995), evolved stars have readily apparent oscillation patterns. In contrast, the oscillatory properties of the red-giant stellar models impose the constraints on our investigations (e.g., Hekker & Christensen-Dalsgaard 2017).

A stellar model represents a snapshot in time of the interior stellar profile produced by a stellar evolution code. Various resulting stellar tracks and isochrone compilations exist (e.g. Pietrinferni et al. 2004; Dotter et al. 2008; Dotter 2016; Hidalgo et al. 2018), and the modelling of stars on the red-giant branch (RGB) hinges on connecting such compilations to observations (Salaris et al. 2002). As a part of the Aarhus red-giant challenge, the numerical accuracy and consistency of an ensemble of evolution codes used for red-giant modelling was evaluated (Silva Aguirre et al. 2020). The connection between asteroseismology and the stellar evolution codes with the stellar structures they produce, however, is not a trivial problem. The subsequent paper by Christensen-Dalsgaard et al. (2020) discussed the complex relation between evolved stellar structures and oscillation properties and employed the same oscillation code as we do here.

Previous asteroseismic characterisation and modelling of red giants has therefore relied on the global parameters (Mosser et al. 2011; Hekker et al. 2011; Wang et al. 2023, and references therein). These are the large frequency separation Δν, the frequency of maximum oscillatory power ν_max and the dipolar period spacing ΔΠ₁. In practice, Δν and ν_max can be obtained from the scaling relations (Kjeldsen & Bedding 1995) (see Eqs. (6) and (7)), which makes their retrieval readily available for the stellar models for a comparison with observed values. We note that in recent years, some concern has arisen about the applicability of the scaling relations, particularly for ν_max, for evolved stars with sub-solar metallicity (Epstein et al. 2014; Viani et al. 2017). Notably, ΔΠ₁ has been successfully used as a constraint on the stellar interior, after it was shown by Bedding et al. (2011) to clearly distinguish helium-burning red-clump and first-ascent red-giant stars.

In the effort to constrain the stellar interior using asteroseismology, it is possible to go one step further than the period spacing by considering the gravity phase offset ϵ_g in the asymptotic relation for gravity modes. The ϵ_g term can assist in constraining the determination of ΔΠ₁ (Buysschaert et al. 2016) as well as identifying and studying the borders of the internal cavity for the gravity modes (Pinçon et al. 2019). Further diagnostic power is provided by the coupling factor q of mixed modes (see Sect. 2), which provides insights into the intermediate regions between the stellar envelope and core. The coupling factor was also shown to vary with stellar mass and evolutionary stage (Mosser et al. 2017).

Fitting the red-giant models directly to the entire spectrum of observed individual frequencies has so far been unfeasible, and the potential of such investigations is left unexplored. However, the properties of the individual red-giant oscillations provide a unique gateway for observational constraints throughout the stellar interior. Realising this link between the observed and modelled individual frequencies by circumventing the computational cost of obtaining the oscillations for red-giant models is the focus of this paper.

In the following, we outline the oscillatory behaviour of red giants in Sect. 2, emphasising the features they develop that both complicate the numerical computations and offer constraints for the stellar interior profiles, before we discuss the observability of the pulsations in real stars. The necessary considerations and requirements for computing giant-star oscillation spectra are then introduced in Sect. 3. The fundamental idea and approach of this work in realising the computations is outlined in Sect. 4, along with a brief overview of the complications we encountered. The retrieval of proper observable frequency intervals for a given stellar model is covered in Sect. 5, before we validate the method on a stellar grid covering a wide parameter space in Sect. 6. Section 7 presents the resulting outcomes of the truncated scanning method for a representative model. Lastly, in Sect. 8 we discuss further avenues for improvement of the concepts and methods we presented, before we conclude.

2. Oscillations of red giants

In order to understand the reason behind the troublesome oscillation properties of red-giant models, we return to the fundamental asymptotic description for stellar oscillations. Under the assumption of spherical symmetry, neglecting the effects of rotation, the stellar oscillation modes may be described at co-latitude θ and longitude ϕ by spherical harmonics, Y_l^m(θ, ϕ). In interpreting the spherical harmonics we may describe the modes by their mode degree l as the total number of nodes on the stellar surface, and the azimuthal order m counting the number of nodal lines crossing the equator. In spherically symmetric stars the pulsations become independent of m. The final addition for the mode characterisation is the radial order n, defining the number and properties of nodes in the radial direction (see, e.g., Aerts et al. 2010). For purely acoustic modes, or acoustic resonances for mixed modes (see Eq. (5) and discussion thereof), this results in the asymptotic relation, in the first-order approximation, describing a given oscillation frequency of a given spherical degree l and acoustic (pressure) mode order n_p as

$$\begin{array}{c}\hfill {\nu }_{\mathit{nl}}\simeq \mathrm{\Delta }\nu (n_{\mathrm{p}}+\frac{l}{2}+ϵ).\end{array}$$(1)

Here, Δν is the aforementioned large frequency separation describing the separation between modes of identical degree l but consecutive order n_p, and ϵ is a constant close to unity called the phase term (White et al. 2011).

Red-giant stars still exhibit solar-like oscillations, but display additional features in their oscillation patterns (for extensive reviews see, e.g., Chaplin & Miglio 2013; Hekker & Christensen-Dalsgaard 2017). This occurs because red-giant stars have evolved to form a dense degenerate core, which causes the local gravitational acceleration, g, to reach very high values in the deep interior. By extension, this holds for the buoyancy frequency N that governs the propagation of the gravity modes. It is defined as (Aerts et al. 2010, Eq. (3.73))

$$\begin{array}{c}\hfill N^2=g(\frac{1}{{\mathrm{\Gamma }}_1}\frac{\mathrm{d}lnp}{\mathrm{d}r}-\frac{\mathrm{d}ln\rho }{\mathrm{d}r}).\end{array}$$(2)

Here, r is the radial distance from the centre of the star, p and ρ denote pressure and density, respectively, and Γ₁ = (∂lnp/∂lnρ)_ad is the first adiabatic exponent. In convectively stable regions N² is positive and g-modes can exist at frequencies 0 < ν ≤ N² – within their trapping region. Meanwhile, as the radius of the star has increased, this, with a contribution from the decreasing temperature in the exterior, results in a reduction in the sound speed c, which leads to a fall in the Lamb frequency S_l (Lamb 1932), defined as

$$\begin{array}{c}\hfill S_l^2=\frac{l(l+1)c^2}{r^2}·\end{array}$$(3)

This means that we can describe the trapping region, or acoustic cavity, of a given p-mode as being between the characteristic acoustic frequency and the acoustic cut-off frequency, which governs the wave behaviour near the surface layers, as S_l ≤ ν_nl ≤ ν_ac. Figure 1 illustrates the characteristic frequencies as cyclic frequencies N/2π and S_l/2π within the interior of an evolved solar stellar model, the parameters of which are presented in Table 1. This model was used to generate the representative figures throughout this work and is referred to simply as model ℳ.

Fig. 1. Characteristic frequencies for model ℳ with the parameters given in Table 1, expressed in terms of cyclic frequencies N/2π (solid), S1/2π and S2/2π (both dashed, S2 residing further from the interior). The horizontal dotted line indicates the frequency of maximum power νmax. The gravity-mode cavity is indicated by the red section with its upper boundary at r1, and the pressure-mode cavity by the blue section (with the lower boundary at r2) for l = 1. A prominent buoyancy glitch can be seen at r/R ≈ 0.075.

The rise in N and fall in S_l for giant stars ultimately causes the g- and p-mode cavities to exist in identical frequency ranges near the observable region in frequency, near ν_max, as shown in Fig. 1. This fact leads to the phenomenon of mixed-mode oscillations once the separation between the cavities, the evanescent region (marked by r₁ and r₂), becomes sufficiently small for a coupling between the outer acoustic modes and inner gravity modes to occur (Jiang & Christensen-Dalsgaard 2014). In a sense, the g-modes can now leave an imprint on the p-modes which in turn propagates to the exterior of the star. Mixed modes exist for all non-radial modes in giants, although only the dipole and quadrupole modes have observable amplitudes in disk-integrated light. The radial modes with l = 0 are unable to display mixed-mode characteristics, because no gravity modes of l = 0 exist to which the radial p-modes might couple. The radial modes of giants thus display a purely acoustic behaviour, and are simple in their numerical computation. In the subsequent work, they are therefore trivially determined from the complete stellar models without the need for further treatment.

The existence of the mixed modes for non-radial pulsations is the source of the complication we wish to circumvent. A single exterior pressure mode with acoustic mode order n_p may couple to a multitude of interior gravity modes, each with an associated mode order n_g. This leads to dense oscillation spectra for giants with thousands of theoretical oscillation modes potentially being excited, which is computationally expensive to evaluate for each individual stellar model. However, the mixed modes display a sensitivity both to the outer stellar layers through their p-mode component and to the internal structure by the influence from the g-mode counterpart, leading to valuable information in modelling efforts (Hjørringgaard et al. 2017; Mosser et al. 2018).

For the purpose of characterising the effect of the evolutionary stage on the oscillatory nature along the RGB, the global parameters Δν, ν_max and ΔΠ₁ can provide valuable insights. The number of possible theoretical stellar oscillations becomes a challenge for stars evolved beyond Δν ≈ 15 μHz, that is, well into the RGB evolution, but prior to the location of the RGB luminosity bump (Khan et al. 2018). This can be seen from the parameter K, which estimates the number of calculations that must be carried out for each Δν-wide frequency range in an RGB model to obtain all theoretical dipole modes (Eq. (22) of Mosser et al. 2015),

$$\begin{array}{c}\hfill K=\frac{\mathrm{\Delta }\nu }{\mathrm{\Delta }{\mathrm{\Pi }}_1{\nu }_{\max }^2}·\end{array}$$(4)

Figure 2 illustrates this fact for the solar stellar track of model ℳ, covering an evolutionary region from the base of the RGB (BRGB) and onward, colour-coded by the value of K. The figure shows three classification regions of the RGB: the lower RGB (LRGB), middle RGB (MRGB) and the tip of the RGB (TRGB). The LRGB represents the initial region of interest, when the method proposed here starts to become relevant. The MRGB comprises the largest region of interest for the method and contains the RGB bump, and the TRGB is the last and most densely population region in terms of oscillation modes. The total number of calculations required becomes even larger with the inclusion of the quadrupole (l = 2) modes and their period spacing ΔΠ₂. In this context, Fig. 2 and K illustrates the rapidly increasing computational difficulties along the RGB.

Fig. 2. Evolutionary diagram for a solar stellar track with M = 1 M⊙ and [Fe/H] = 0 dex, plotting the large frequency separation Δν and luminosity L as a function of effective temperature Teff. The colour bar indicates the number of dipole modes in a Δν-wide frequency range around νmax, according to Eq. (4). The three classifications from top to bottom denote the tip (TRGB), middle (MRGB) and lower RGB (LRGB).

Our method demonstrates a consistent way of isolating the frequency regions within the dense oscillation spectra of giants that are observable. This reduces the computation time substantially by allowing for a large fraction of the theoretical oscillations in each stellar model to be disregarded entirely. To isolate the most observable modes, we consider the mode inertia

$$\begin{array}{c}\hfill E=\frac{{\int }_V\rho {|\delta r|}^2\mathrm{d}V}{{M|\delta r|}_{\mathit{phot}}^2}·\end{array}$$(5)

Here, δ⃗r denotes the displacement vector, and phot the photospheric value of the vector, where the integral is performed over the volume V of the star. An intuitive interpretation of the mode inertia is that the lower the inertia of a mode, the easier it is excited into oscillation, as a smaller fraction of the stellar mass is involved in the pulsation (Chaplin & Miglio 2013). The approach is therefore to isolate the mixed modes of each acoustic mode order n_p carrying the lowest inertia for a given stellar model. Crucially, the identification of these mixed modes must be realised without the need for calculating the entire pulsation spectrum. This approach and Eq. (5) can be represented in a plot of the mode inertia, as displayed in Fig. 3. It is apparent that the radial modes do not display mixed-mode nature. For non-radial modes, the inertia shows acoustic resonances, characterised by an acoustic mode order n_p at frequencies approximately satisfying Eq. (1). Isolating the most observable mixed modes thus entails identifying the modes in these inertia valleys while excluding the many modes exhibiting higher inertia seen at the peaks.

Fig. 3. Inertia plot of model ℳ. The red upwards triangles and blue downwards triangles denote the dipole and quadrupole modes, respectively. The radial modes are depicted as orange diamonds. The plotted modes are connected by solid lines to guide the eye.

In characterising the stellar models considered during the development of this work, the large frequency separation and frequency of maximum oscillatory power were consistently used. Both parameters were obtained under the assumption of an isothermal atmosphere (Chaplin & Miglio 2013) such that

$$\begin{array}{cc}\hfill \mathrm{\Delta }\nu & \simeq {\left(\frac{M}{M_{\odot }}\right)}^{0.5}{\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot }}\right)}^{3.0}{\left(\frac{L}{L_{\odot }}\right)}^{0.75}\mathrm{\Delta }{\nu }_{\odot },\hfill \end{array}$$(6)

$$\begin{array}{cc}\hfill {\nu }_{\max }& \simeq \left(\frac{M}{M_{\odot }}\right){\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot }}\right)}^{3.5}\left(\frac{L}{L_{\odot }}\right){\nu }_{\max ,\odot },\hfill \end{array}$$(7)

where the stellar parameters carry their usual interpretation. This formulation is used throughout this work whenever Δν or ν_max are referenced.

2.1. Observability of red-giant oscillations

The previous section outlined the nature of red-giant oscillations and the situation we face in the stellar models. We now wish to clarify the phenomena related to the observability of the oscillations. These aspects were investigated both from a theoretical and observational perspective in the past, offering insight into potential constraints to impose on our method.

From the theoretical side, Dupret et al. (2009) investigated three RGB models in detail to examine their oscillation spectra. They found that non-radial modes should be observable throughout the RGB, but becoming limited to the purely exterior p-modes beyond the RGB bump. The main cause of this is the degree of radiative damping that affects the lifetime of the oscillation, which becomes increasingly significant along the RGB. This interpretation was seconded by Grosjean et al. (2014), who similarly found finding that mixed-mode oscillations become undetectable for evolved red giants. They argued that it is both an effect of the increased radiative damping and a decrease in the coupling strength q.

Observations from Kepler were interpreted by Bedding et al. (2010) and later by Stello et al. (2014). They confirmed the above theoretical view, observing non-radial mixed modes for less evolved red giants. As the star becomes more evolved, however, the mixed modes gradually disappear. Yet, Stello et al. (2014) still recovered the acoustic resonances of non-radial oscillations for stars with Δν ≲ 1 μHz when using the entire four-year Kepler time series, indicating the importance and application of asteroseismology throughout the RGB (see their Fig. 4 for reference).

To incorporate these considerations and constraints, we followed the approach by Mosser et al. (2018), who used the properties of the ζ function. The ζ function is defined as the fraction of mode inertia from the interior relative to the total mode inertia, and it can be reformulated to depend on K and the mode coupling q (Hekker & Christensen-Dalsgaard 2017). Mosser et al. (2018) used the full width at half maximum of the ζ function to derive a formula for the observable region of mixed modes around an acoustic resonance, which expressed in frequency becomes

$$\begin{array}{c}\hfill \delta {\nu }_{\mathrm{obs}}(\nu )=\frac{2q(\nu )}{\pi }\sqrt{1+\frac{1}{Kq(\nu )}},\end{array}$$(8)

where q(ν) is the coupling constant of a given frequency, and K is defined by Eq. (4) for the respective period spacing of the mode. The estimation of q for any given frequency within a stellar model was described by Shibahashi (1979),

$$\begin{array}{c}\hfill q=\frac{1}{4}exp(-2{\displaystyle {\int }_{r_1}^{r_2}}{|\kappa |}^{1/2}dr).\end{array}$$(9)

Here, r₁ and r₂ denote the lower and upper radial coordinate of the evanescent region extent, as indicated in Fig. 1. The radial wave vector κ is calculated following Shibahashi (1979) as

$$\begin{array}{c}\hfill \kappa =\frac{{\omega }^2}{c^2}(\frac{S_l^2}{{\omega }^2}-1)(\frac{N^2}{{\omega }^2}-1),\end{array}$$(10)

where ω = 2πν is the angular frequency, c is the sound speed in the deep interior and the characteristic frequencies N and S_l are defined in Eqs. (2) and (3), respectively. The expression for q in Eq. (9), by construction has an upper limit at 1/4. This effectively results in a limitation on the dimensions of q when used in Eq. (8).

The use of Eq. (8) enables us to overlay realistic constraints on the observability of the model frequencies retrieved. Its use and application within this method are further outlined in Sect. 5.

2.2. Numerical scheme

The oscillation code employed in this work is the Aarhus adiabatic pulsation package (ADIPLS; Christensen-Dalsgaard 2008). The code solves the adiabatic oscillation equations separately from the centre (or a truncation point; see Sect. 4) and the surface, identifying eigenfrequencies by requiring a continuous match of the two solutions. For radial modes, or in truncated models, the matching point is close to the surface of the model, while for non-radial oscillations in complete models, it is typically located at the innermost maximum in the buoyancy frequency. The eigenfrequencies are determined by scanning in frequency and locating zeros in a matching determinant. The step in the scan is roughly based on the expected asymptotic behaviour of the frequencies. For truncated models, we solved the second-order set of equations in the Cowling approximation, neglecting the perturbation to the gravitational potential, and obtained the frequency correction resulting from the potential perturbation through a perturbation analysis (Christensen-Dalsgaard 1982, Eq. (D1)); for complete models, the full set of equations was solved. The equations were solved with a fourth-order Runge-Kutta technique (Cash & Moore 1980), using linear interpolation to obtain model values at points intermediate between the mesh points.

In complete models, the central boundary conditions were obtained from an expansion to second order around the singular central point. In truncated models, the truncation point was assumed to be in the evanescent region between the p- and g-mode trapping regions (see also Appendix A), and the inner boundary condition was set to obtain the solution that decreases exponentially towards the interior. The dynamic surface boundary condition assumed a continuous match to an isothermal atmosphere.

3. Calculating giant-star oscillations

Numerically calculating accurate oscillations for red-giant models that reflect the true pulsations of real stars involves changing the computational approach considerably. We modified ADIPLS here, but the method can be extended to other pulsation codes. The process of implementing the necessary changes to a pulsation code in order to achieve the appropriate settings for complete red giant models, thereby producing a resulting scan that incorporates the proper frequency interval and mode sampling for a given stellar model, is briefly presented in this section.

The properties of a mode are characterised by its order n = n_p − n_g, where n_p and n_g are the orders associated with the acoustic and gravity-mode cavities, as introduced in Sect. 2. Depending on the evolutionary state of the star, the relevant range in n progresses to increasingly negative values as n_g rises. Defining a suitable interval in n in which to search for modes must therefore be tied to the evolutionary state of a given star. In order to ensure that this interval encapsulates the entire range of possible observable modes, a conservative lower boundary was set. The lower boundary was defaulted to n = −2000 and the upper to n = 50. The lower boundary was gradually altered according to the steady evolution in ν_max through a simple step-function, always opting for the conservative choice and ensuring proper sampling. As we approach the TRGB, the lower boundary goes as low as n = −20 000, while the upper boundary remained unchanged at all times. It is readily apparent that such a large interval in n for highly evolved red-giant stars results in a plethora of possible theoretical oscillation frequencies for calculation.

In order to resolve such dense oscillation spectra, the number of numerical mesh points, 𝒩, must be increased accordingly in order to trace the structure of these evolved stars. Providing a suitable 𝒩 aims to ensure that the accuracy of the numerically derived modes remains higher than the typical uncertainty on the observed frequencies of giant stars – which is of the order of 10⁻⁴ μHz (Corsaro et al. 2015; de Montellano et al. 2018). However, a strict treatment for the required 𝒩 as a function of the evolutionary state of a given stellar model has not yet been obtained. Within this work, the number of mesh points was therefore doubled at specific points along the RGB. The requirement to increase the number of mesh points 𝒩 is one of the main components that significantly increases the computation time of the modes for giant stars, in combination with the density of the oscillation spectra themselves.

The settings outlined above do provide suitable giant-star oscillation spectra, but at a considerable computational cost and with a dense mode spectrum associated with each stellar model. Figure 4 shows an example of such an expensive full calculation for model ℳ in a replicated échelle diagram (Bedding 2012). The smear of overlapping oscillations consist of a combination of l = 1, 2 modes that take on mixed-mode characteristics, and it visualises the large host of theoretical modes present. Each line of points corresponds to a specific radial mode order n_p0. Even for such a model below the RGB bump, the calculation time necessary is upwards of 5 minutes¹, and it rapidly increases as Δν decreases. For post-RGB-bump models, the dense nature of the modes seen at the lower frequencies in Fig. 4 will cover the entire frequency range for both the dipole and quadrupole modes. The calculation time necessary for such models reach a couple of hours per model near the TRGB.

Fig. 4. Replicated échelle diagram of a full calculation of model ℳ. The orange diamonds, red upwards and blue downwards triangles depict the l = 0, l = 1 and l = 2 modes, respectively, for a model below the RGB bump. For each plotted mode, a scaling in size related to the inverse of its inertia is applied.

4. Isolating p-mode-like frequencies

The most likely oscillation frequencies to be observable are the mixed modes that are the most p-mode-like; these are the so-called (p − m) modes (Mosser et al. 2015). The reason is that the acoustic cavity of the p-modes resides in the exterior of the star where the density ρ is lower, which causes the modes to obtain a lower mode inertia (Christensen-Dalsgaard et al. 2020). These modes were theoretically isolated through a truncation in the interior of the input model. A truncation in other words cuts off part of the model, makes it inaccessible for the pulsation code, and thereby removes its possible influence on the oscillation modes. This framework requires an inner truncation point to be given in an evanescent region. If this is satisfied, we can isolate the acoustic resonances as those that exhibit eigenfunctions that decrease exponentially towards the interior (Aerts et al. 2010, Chap. 3). This requirement is directly satisfied by the implemented truncation algorithm and is illustrated in Fig. 5. For details of the truncation algorithm, we refer to Appendix A.

Fig. 5. Propagation diagram for model ℳ, displaying a zoomed-in view of Fig. 1 near the bottom of the convective region. The buoyancy frequency N/2π and Lamb frequency Sl/2π for l = 1, 2 are shown as the full and dashed (S2 being the higher) profiles, respectively. The horizontal dotted line indicates a representative high frequency νhigh of the model, with its g-mode cavity marked in red. The horizontal dot-dashed line shows a representative lower frequency νlow. The vertical dotted and dot-dashed grey lines indicate suitable respective truncation points in r/R. The vertical solid grey line represents a truncation point at the bottom of the convective region. The buoyancy glitch is marked as the hatched area.

Figure 5 displays a propagation diagram for model ℳ, providing a zoomed-in perspective of Fig. 1. It depicts the interior structure of N and S_l near the bottom of the convective region, where the g-mode cavity has its boundary. The truncation point obtained for a representative frequency just above ν_max of the model is marked by the vertical dotted line, below which the model could subsequently be truncated. However, when including a tentative lower frequency ν_low marked by the horizontal dot-dashed line, a problem arises. The shaded grey area marks a possible g-mode cavity for ν_low if the truncation point marked by the vertical dotted line is employed, resulting in the mode obtaining g-mode characteristics. In such cases, the implemented frequency-dependent truncation scheme therefore finds a more suitable truncation for ν_low, indicated by the vertical dot-dashed line, at a higher value of r/R. This variable choice of truncation point ensures that we always truncate within an evanescent region and at a sufficiently large separation from N to avoid the influence of g-mode characteristics on a given mode. Subsequently, we are thus able to isolate the acoustic resonances by considering the modes whose eigenfunction decreases exponentially towards the interior. The further details and considerations of Fig. 5 are discussed in Sect. 4.2.

4.1. Frequency shift by truncation

The main complication incurred by the model truncation is a frequency shift in the dipole and quadrupole modes. The shift occurs because the boundary conditions for the numerical scheme outlined in Sect. 2.2 are unchanged, although the extent of the model was altered by the truncation. In other words, the inner boundary conditions do not fully represent the behaviour of the modes in a full model, resulting in a small shift in the retrieved frequencies. When the recovered acoustic resonances are to be used as centre points of the observable regions, we need to correct for this shift. To do this in a coherent way becomes crucial for the success of the method, as otherwise the final frequencies fail to properly estimate the lowest inertia modes of a full model calculation.

For this purpose, the validation grid to be presented in Sect. 6 was employed (see Appendix D for details). For all models across the nine tracks in the grid, both the full model frequencies calculated with the settings outlined in Sect. 3 and the truncated frequencies are at hand. The specifics of deriving the frequency-shift correction can be found in Appendix B. We concluded that the frequency shift becomes relevant for the dipole modes and has a slight (and predictably consistent) effect on the three lowest acoustic-mode-order quadrupole resonances. Furthermore, we found that the uncertainty on the corrected resonances to estimate the full model resonances was negligible within the framework of this method for all but the lowest acoustic-mode-order dipole resonance. Proper recovery of the lowest acoustic dipole resonance is thus problematic. This complication is discussed and solved in Sect. 5.

4.2. Buoyancy glitch

In pre-RGB-bump models, the phenomenon of a buoyancy glitch occurs. The glitch is a real physical effect left behind by the first dredge-up, and produces a discontinuity in composition and hence density in the affected models (Christensen-Dalsgaard 2015; Lindsay et al. 2022). This can cause problems in the application of the truncated scanning method. The variable truncation algorithm initially neglects the glitch and assumes a smooth decrease in N with increasing r/R in the stellar interior. An example of a prominent buoyancy glitch is shown in the propagation diagram in Fig. 5 as the hatched area. This may give rise to a mode that essentially corresponds to the interface mode at a density discontinuity (Dziembowski & Pamyatnykh 1991; Bildsten & Cumming 1998), with a frequency that increases with degree. This frequency may be in the range of frequencies searched for acoustic resonances.

In the variable truncation algorithm, modes in the truncated model are tested for being associated with the glitch. A tentative glitch effect on a given mode calculated with the variable truncation algorithm will reveal itself through an oscillating eigenfunction in the glitch region. If a node in the eigenfunction is encountered within the glitch, the given truncated mode is recalculated with the truncation point set at the bottom of the convective region, where the buoyancy frequency from Eq. (2) becomes imaginary. This effectively means placing the truncation point on the exterior side of the glitch, as illustrated by the vertical solid line in Fig. 5, thus eliminating the possible glitch effects.

Placing the truncation point at the bottom of the convective region may be problematic due to the proximity of S_l. Recalling that we must ensure a truncation within an evanescent region, we check for this tentative situation. If the truncation at the bottom of the convective region is invalid due to the evanescent criteria, we reverse to the previous truncation on the interior of the glitch. In such cases, the possible interference with the glitch on the mode cannot be avoided.

We note that the induced effect of the buoyancy glitch on stellar pulsations is still an active research area. This affects not just our truncated models, but also the full pulsation calculations of red-giant models in general. As described by Cunha et al. (2024), the modelling of the glitch and its assumed shape in the stellar interior imprints various features onto the pulsations. It is beyond the scope of this work to investigate these effects further, and we chose to model the glitch as either a δ function or a resolved Gaussian profile, depending on the resolution of the glitch in the given stellar model.

In summary, the tentative glitch interference on the truncated models affect a minuscule number of models when the method is applied to entire stellar grids. Furthermore, the framework to be described in Sect. 5 absorbs the interference such that no model in the validation grid of Sect. 6 exhibits an incomplete oscillations spectrum.

5. Scanning in observable intervals

With the acoustic resonances estimated, the diagnostic potential of the mixed modes now has to be realised and included. The aim is to place appropriate scanning intervals around the recovered acoustic resonances for both the dipole and quadrupole modes, thereby yielding a span of theoretical mixed-mode frequencies for each acoustic mode order n_p that is centred on the feasibly observable regions. Crucially, these scanning regions are computed using the full non-truncated model, and no interference from a truncation is therefore present in the final frequencies we obtain. Thus, the resulting oscillation frequencies are affected by any tentative structural features present, for instance the buoyancy glitch. The placement of suitable scanning intervals is the focus of this section.

A complication was described in Sect. 4.1, namely the inability of the generalised frequency correction to estimate the acoustic resonance for the lowest-order dipole mode (see Fig. B.4 and discussion thereof). An attempt to place a scanning interval here risks missing the bottom of the inertia valley, while also being the most computationally expensive dipole interval to calculate. However, a redeeming aspect is that the lowest acoustic resonance that theoretically exists is well outside of the observable frequency range. For example, for model ℳ, the lowest resonance lies near 20 μHz. This is significantly below the observable region for stars exhibiting a Δν ≈ 8 μHz, having a value of ν_max ≈ 80 μHz. Within the framework of the truncated scanning method, the lowest-order dipole acoustic resonance has therefore excluded from the calculation.

For models evolved beyond Δν ≲ 5 μHz, the condition of numerical precision imposes a limit on the number of nodes in the eigenfunction relative to the maximum number 𝒩_max of mesh points in the computation, and hence a lower limit on the frequency of quadrupolar modes. This clearly does not affect the calculation for the truncated model. Thus, the lowest-order quadrupolar acoustic resonance has to be discarded before proceeding to the scanning interval treatment.

In a sense, we wish to fill out the bottom of the inertia valleys depicted in Fig. 3 for every acoustic resonance in a given model. For this purpose, we employ Eq. (8) repeatedly for every acoustic resonance recovered from the truncated run, except for the two exceptions mentioned above. Equation (8) is applied twice: once with wide intervals, ensuring that we capture the acoustic resonances of the full model, then again to sort and trim the intervals based on the recovered resonances.

5.1. Initial and trimmed intervals

Importantly, we cannot risk missing the bottom of each inertia valley in the pulsation calculations. This may occur for example for quadrupole frequencies of low mode order where the observable width determined from Eq. (8) becomes so small (it can fall below 1 × 10⁻⁴ μHz) that only the resonance itself is observable. The risk of missing the full model resonance (hereafter true resonance) based on the truncated resonances thus becomes significant. To err on the side of caution, the observable region estimated by Eq. (8) is amplified by a gradually increasing scaling factor for the LRGB, MRGB and TRGB for both the dipole and quadrupole cases. The classification of the evolutionary regions and the amplification factors are listed in Table 2. The evolutionary region that is of key importance for future investigations is represented by the MRGB classification. It effectively covers an extended region around the RGB bump and represents the region in which our method starts to offer significant computational improvement (see Sect. 7).

As a final safeguard, a minimum width of 2 and 1% of the Δν value of the model is enforced for the dipole and quadrupole modes, respectively. This ensures that we recover the true resonance in the scan, even for complex model interiors for which the procedure of Sect. 2.1 struggles, and also in situations in which the observable width becomes incredibly narrow. We note that the Δν safeguard is frequently employed for the quadrupole frequencies of models in the MRGB and TRGB.

As previously mentioned, the frequency shift on the quadrupole modes only affects the three lowest acoustic mode order resonances, and always with a consistent small shift to a slightly higher frequency. To account for this, we therefore apply twice the width, either amplified or set by Δν, to the lower boundary of the scanning interval of these three modes.

The calculations then yield an initial frequency list for a given stellar model. The recovered intervals are too wide when considering the limitations on the observability discussed in Sect. 2.1, and a final assessment is therefore made. Within each obtained interval, we locate the resonance as the minima in mode inertia. Employing these true acoustic resonances in Eq. (8) provides the actual observational regions of the complete stellar model. Because of the asymmetry in the frequency spacing, that is, two potential modes/resonances of comparable inertia existing at the bottom of an inertia valley, of which the minimisation routine chooses one, as well as the applicational uncertainties and a tentative metallicity dependence of Eq. (8), we amplify the observational width obtained from Eq. (8) by a factor of 2. The initial wide intervals are subsequently trimmed to these new widths.

5.2. LRGB stars

For stars exhibiting a Δν ≳ 8 μHz, the mode coupling q can become very strong and may reach the upper limit of 1/4. Additionally, as the radiative damping is as yet inefficient, this means that mixed modes with a considerable shift in frequency become observable. At higher mode orders, this effectively means that the amplified observational width determined for the initial scanning intervals crosses the inertia bump and overlaps. This is unproblematic, because the number of modes at the inertia peaks at higher frequency for the LRGB members is modest. In this situation, the intervals are simply joined into a complete scan covering the relevant region in frequency. A visual example similar to Fig. 6 is shown in Fig. C.1 for an LRGB star with Δν = 14.86 μHz, illustrating the reduced number of modes at higher acoustic mode order at the inertia peaks, and how significantly displaced observable mixed modes are recovered.

Fig. 6. Mode inertia for the model ℳ, depicting the modes from a full model (transparent points) and truncated scanning method (opaque points) calculation, abbreviated FMC and TSM, respectively. All plotted points are connected by a solid grey line to guide the eye. The inertia valleys containing observable p-mode-like oscillations for both the dipole (red) and quadrupole (blue) modes are recovered. The lowest-order dipole mode has been removed from consideration, as can be clearly seen at ν ≈ 20 μHz. Note that the recovered radial modes are identical for an FMC and TSM calculation.

5.3. TRGB stars

Highly evolved red giants also deserve special mention. Here, the coupling strength has become very low in combination with efficient radiative damping, such that only the true resonances or mixed modes very close to it are observable. However, due to the nature of red-giant oscillations outlined in Sect. 2, an immense number of theoretical modes exist. Figure C.2 shows an example of a TRGB star with Δν = 1.04 μHz. Despite the observational limitations, we still recover a small number of mixed modes around the true resonances. They are displaced very little in frequency, however, and would likely overlap in observations to provide a widened frequency peak around the acoustic resonance, as found by Stello et al. (2014).

6. Validation

In order to validate the performance of the truncated scanning method, we applied it to a set of stellar tracks covering a large parameter space. A Cartesian grid of nine tracks was calculated with masses M = 0.7, 1.3 and 1.9 M_⊙ and metallicities [Fe/H]= − 1.5, −0.5 and 0.5 dex with GARSTEC (Weiss & Schlattl 2008). For brevity, the details of the input to and settings within GARSTEC can be found in Appendix D. For each of the tracks, stellar models were recorded from Δν ≈ 20 μHz until the helium flash at Δν ≈ 0.8 μHz, providing an appropriate sampling for testing purposes of ∼100 models per track along this evolutionary region.

The complete truncated scanning method framework described in Sects. 3–5 was applied to all models in the grid. Additionally, the computationally expensive full model calculations were also obtained for the entire grid to provide grounds for a proper validation of the truncated scanning method. An equivalent to Fig. 6 was inspected for every model, enabling visual assessment and verification of the performance of the method.

The truncated scanning method performed as intended for models across the large parameter space covered by the validation grid. For a limited number of models, the glitch interference causes the obtained observational regions to take peculiar shapes, as briefly mentioned in 4.2, but nonetheless localised the observable regions with the acoustic resonances. Generally, the wide applicability of our method was verified for solar-like stars on the RGB and confidently provides an avenue for obtaining the oscillations of red-giant models.

7. Results

Employing the truncated scanning method we can compute the realistically observable theoretical mixed-mode oscillations of red-giant models at significantly reduced computational cost. The result for model ℳ is displayed in a replicated échelle diagram in Fig. 7, which is similar to the full model version shown previously in Fig. 4. Figure 7 shows the resulting scanning intervals in a more intuitive way in connection to observational asteroseismology. The observable regions that are determined return the modes adjacent to the acoustic resonance of each mode order, distributing the modes along the approximately vertical ridges usually found in échelle diagrams of observed pulsations. The determined dipole mixed modes reside in the region with the lowest inertia for the model, as indicated by the size of the plotted mode markers. For model ℳ, the observable quadrupole frequencies number just the acoustic resonance in each mode order, except for the highest frequencies. These singular quadrupole resonances were previously unattainable and hidden among the multitude of theoretical frequencies, but are now recovered by the truncated scanning method. The resulting theoretical mode spectrum obtained for a given model is thus situated as desired in the regions that are crucial for investigations that fit observed stellar data.

Fig. 7. Replicated échelle diagram of model ℳ obtained after application of the truncated scanning method to calculate red-giant oscillations. The orange diamonds, red upwards and blue downwards triangles depict the l = 0, l = 1 and l = 2 modes, respectively. A scaling in size related to the inverse of its inertia is applied to each plotted mode. The observable mixed-mode regions surrounding the acoustic resonances have been calculated and plotted for the dipole and quadrupole modes. For contrast to the initial situation, compare to Fig. 4.

It is important to acknowledge that due to the complications with the variable truncation scheme and glitch interference outlined in Sect. 4, a small number of stellar models (primarily in the LRGB region) will inevitably fail in their computation. Such models must be recalculated in their full form with the settings outlined in Sect. 3. However, for the few affected and less evolved models, the computation remains sufficiently efficient to carry out.

Prior to this work, the only option at hand was to calculate the entire frequency spectra of evolved stars. As became clear in previous discussions, these incredibly dense spectra, and hence, the large number of nodes in the mode eigenfunctions that require a significant increase in the number 𝒩 of numerical mesh points, results in excessive computational cost. Briefly outlining the obtained computational efficiency puts into perspective the importance and applicability of the truncated scanning method. The full model calculation in Fig. 4 takes approximately 5 minutes, while the results obtained from the truncated scanning method in Fig. 7 were computed in 16 seconds. This computational difference only becomes increasingly more significant during the ascent of the RGB. For highly evolved red-giant stars near the TRGB, a full model calculation of a single stellar model takes hours, while the truncated scanning method can perform the calculation in less than 10 minutes per model.

Figure 8 illustrates the obtained efficiency for a number of models from the solar origin track of model ℳ, selected in each of the classification intervals from Table 2. Prior to the method being applied at Δν = 15 μHz, the computation time is identical, but it varies significantly once the truncated scanning method is employed. The calculation time for the truncated scanning method varies slightly along the LRGB and MRGB due to the dynamic interval boundaries that depend on the specific model, based on the considerations of Sects. 2.1 and 5.

Fig. 8. Computational efficiency of the truncated scanning method. The computation time for a given model is plotted against its Δν value for a full calculation and for the truncated scanning method as solid and dashed lines, respectively. The time is displayed on a logarithmic scale, and the overplotted lines aid in the visualisation. The vertical dashed lines indicate the classification transition points from Table 2, as shown.

Until the TRGB at Δν < 2 μHz, the calculation time with the truncated scanning method remains below ∼2 minutes, which is an impressive improvement of the previous situation. Furthermore, it speaks to the applicability and performance of the derived method. For the TRGB models, the computation time exceeds 2–3 hours for the full calculations, but it remains below ∼8 minutes at all times for the truncated scanning method. The appearance of a reduced slope for the full calculations near the TRGB is due to the logarithmic scale of the y-axis. It does not reflect an actual difference in the scaling of the computation time between results produced by a full calculation and the truncated scanning method.

8. Discussion and conclusion

The truncated scanning method contributes a novel approach to deriving the oscillation spectra of evolved giants efficiently, by limiting the derivation to the observable regions. However, as with any scientific method in its infancy, there is room for future improvement to the method outlined herein.

The simplicity of the method is reduced by the need for a frequency-shift correction of the dipole modes. The shift in frequency impacted all recovered acoustic resonances since the truncation of the model and the applied inner boundary condition did not fully reflect the properties of modes in the full model. While this frequency shift is small in magnitude, it would be preferable to avoid it. In seeking out a way to avoid the frequency shift, the entire procedure could be redefined to avoid the necessity of a truncation altogether. Through reformulation of the equations behind the stellar oscillations, the contributions that lead to g-mode behaviour might be discarded following the ideas proposed in Ong & Basu (2020). Further studies are required, but accuracy and simplicity might be improved when the need for a truncation and the accompanied frequency shift from the truncated scanning method is removed. The potential of this approach is the subject of further testing and development efforts.

The induced signatures on the pulsations caused by the buoyancy glitch cause problems for a very limited number of models in the evolutionary region in which the glitch is dominant (when the convective region is receding). We stress that the effect of the glitch is not unique to the approach in the truncated scanning method, but is a general effect imposed on all oscillation calculations of the affected red-giant models. Constraining the glitch signatures and understanding their nature is an important avenue for further research and improvement for red-giant pulsations.

The results of this paper serve as a novel method for efficiently calculating stellar oscillations for stellar models of giant stars. We obtained the desired results of a much faster computation that maintains reliability in the returned theoretical frequencies spaced across the observational regions for any given red-giant model. Realising these computations unlocks the possibility of asteroseismic modelling utilising individual frequencies of red-giant stars, which could serve as an independent modelling procedure to the existing reliance on the global parameters.

On a larger scale, the truncated scanning method moves investigations of individual frequencies by fitting to stellar grids situated on the RGB into the realm of possibility. The calculation time of all pulsation modes for an entire grid of evolved stellar tracks was previously infeasible both in terms of computational resources and time invested for a certain research project, but has now become achievable in a matter of days as opposed to months or even years. In this context, the application of the truncated scanning method for an entire stellar grid in connection with asteroseismic modelling of red giants with individual frequencies is the subject of an upcoming paper.

Data availability

The code products and required software that produces ADIPLS input files with the truncated scanning method settings are available upon request to the first author. Various versions of the validation grid with the stellar models and associated frequency files used for the representative figures, along with the plotting code, can be found here: https://www.erda.au.dk/archives/a6d160238869305894269d9a74068333/published-archive.html.

Acknowledgments

JRL wishes to thank the members of SAC for comments and discussions regarding the paper. The authors thank the anonymous referees for the constructive criticism, which aided in improving the quality of the work in this paper. This work was supported by a research grant (42101) from VILLUM FONDEN. Funding for the Stellar Astrophysics Centre was provided by The Danish National Research Foundation (grant agreement no.: DNRF106). MSL acknowledges support from The Independent Research Fund Denmark’s Inge Lehmann program (grant agreement no.: 1131-00014B). The numerical results presented in this work were obtained at the Centre for Scientific Computing, Aarhus https://phys.au.dk/forskning/faciliteter/cscaa/.

References

Appendix A: Frequency-dependent truncation algorithm

The choice of truncation point is determined through the following equation from JWKB theory:

$$\begin{array}{c}\hfill k^2=\frac{l(l+1)}{x^2}(1-\frac{{\stackrel{\sim }{N}}^2}{{\sigma }^2})(1-\frac{{\sigma }^2}{{\stackrel{\sim }{S}}_l^2})\end{array}$$(A.1)

Here, ${\sigma }^2=\frac{R^3}{\mathit{GM}}{\omega }^2$ is the squared dimensionless frequency with ω = 2πν describing any angular oscillation frequency of a given mode and x = r/R the radial coordinate. The dimensionless characteristic frequencies are denoted as $\stackrel{\sim }{N}$ and ${\stackrel{\sim }{S}}_l$. The modes have clear acoustic behaviour through the Lamb frequency ${\stackrel{\sim }{S}}_l$, i.e. p-mode behaviour, as long as the term $\stackrel{\sim }{N}/{\sigma }^2\ll 1$. However, as $\stackrel{\sim }{N}$ grows large for giant stars $\stackrel{\sim }{N}/{\sigma }^2\simeq 1$ is reached, resulting in gravity-dominated mixed modes – so-called (p − g) modes (Mosser et al. 2015) – arising, all exhibiting higher mode inertia. If we wish to isolate the observable p-mode-like oscillations, we therefore need to truncate the model according to the behaviour of $\stackrel{\sim }{N}/{\sigma }^2$. Note that this fraction is frequency dependent. Hence, the approach is to implement a variable truncation depending on frequency.

Instead of directly using $\stackrel{\sim }{N}/{\sigma }^2$ to define the truncation, the term relating to the Lamb frequency ${\sigma }^2/{\stackrel{\sim }{S}}_l^2$ was also considered to provide an upper boundary condition. The aim is to find a truncation point x_tr as the point where both ${\stackrel{\sim }{N}}^2/{\sigma }^2$ and ${\sigma }^2/{\stackrel{\sim }{S}}_l^2$ are well below 1. A balance between minimising x in relation to

$$\begin{array}{c}\hfill \frac{{\stackrel{\sim }{N}}^2(x_{\mathrm{tr}}^{(1)})}{{\sigma }^2}\le ϵ,\end{array}$$(A.2)

in order to obtain $x_{\mathrm{tr}}^{(1)}$, and concurrently maximising x in relation to

$$\begin{array}{c}\hfill \frac{{\sigma }^2}{{\stackrel{\sim }{S}}_l^2(x_{\mathrm{tr}}^{(2)})}\ge \zeta ,\end{array}$$(A.3)

to estimate $x_{\mathrm{tr}}^{(2)}$ is performed. Currently, we hard-coded the values of ϵ = ζ = 0.5. In the case where $x_{\mathrm{tr}}^{(1)}\le x_{\mathrm{tr}}^{(2)}$, the truncation point can directly be set to $x_{\mathrm{tr}}=x_{\mathrm{tr}}^{(1)}$ and a successful truncation is obtained ensuring that

$$\begin{array}{c}\hfill \frac{{\stackrel{\sim }{N}}^2}{{\sigma }^2}\le ϵ,\frac{{\sigma }^2}{{\stackrel{\sim }{S}}_l^2}\ge \zeta .\end{array}$$(A.4)

In the case where the evanescent zone is narrow, the above conditions may not be satisfied resulting in $x_{\mathrm{tr}}^{(1)}\ge x_{\mathrm{tr}}^{(2)}$. A last-ditch effort to set $x_{\mathrm{tr}}=x_{\mathrm{tr}}^{(3)}$ is evaluated by requiring

$$\begin{array}{c}\hfill \frac{{\stackrel{\sim }{N}}^2(x_{\mathrm{tr}}^{(3)})}{{\sigma }^2}=\frac{{\sigma }^2}{{\stackrel{\sim }{S}}_l^2(x_{\mathrm{tr}}^{(3)})}.\end{array}$$(A.5)

For the dipole modes where N and S_l are in closer proximity, this criterion is often employed and sets $x_{\mathrm{tr}}=x_{\mathrm{tr}}^{(3)}$. However, if this does not provide a realistic estimate, as there is no guarantee that k² > 0 at $x_{\mathrm{tr}}^{(3)}$ in Eq. A.1, the scan is stopped and an error message to the standard error is produced.

Appendix B: Generalised frequency correction

Here we consider the diagnostic consequences of the incurred frequency shift mentioned in Sect. 4.1, and subsequently the derivation of a generalised model for its correction. For the purpose of the derivation and subsequent testing, the validation grid described in Sect. 6 and Appendix D was employed.

Figure B.1 illustrates the situation in a relative parameter space for model ℳ, as the fractional difference between the truncated frequencies ν_T and full frequencies ν_F recovered at minimum inertia for each acoustic resonance as a function of the truncated frequencies ν_T. One immediately notices a key difference between the effect on the l = 1 and l = 2 modes. The frequency shift is almost entirely insignificant for the quadrupole modes, while the contrary is true for certain dipole modes. This is due to the nature of the modes in the stellar interior. The dipole modes have an inner turning point for their p-mode cavity deeper in the interior than the quadrupole modes (see Fig. 1 or 5). As such, the l = 1 modes in a sense ‘feel’ the effect of the truncation to a larger extent than do those with l = 2. The necessary frequency shift for the dipole modes, as depicted in Fig. B.1, is dominant for the lower-order modes and significantly smaller for the higher-order modes.

Fig. B.1. Visual representation of the frequency shift incurred on the truncated frequencies for model ℳ. The fractional frequency shift (νF − νT)/νT between the full (νF) and truncated (νT) frequencies is plotted against νT for the dipole and quadrupole modes, shown by the red and blue lines, respectively. The grey profile shows the resulting generalised correction for model ℳ overplotted, for which its value is subsequently subtracted from all dipole acoustic resonances to provide the red triangles. The uncertainty on the correction for each dipole mode is indicated, as well as the minimal size of the scanning interval boundary as the shaded beams in the respective colour.

The necessary correction for the quadrupole modes takes a much simpler form. The effect of the small shift seen will generally be absorbed by the inclusion of a frequency scanning interval (see Sect. 5). Only for the three lowest acoustic-mode-order resonances is the shift considerable enough to warrant further consideration. However, the shift is consistent and predictable, resulting in a small shift to larger frequencies. This will be accounted for by the treatment in Sect. 5. The approach presented in the following therefore focuses entirely on the dipole modes of the models.

For all the models in the validation grid it is possible to do an exact fit to the dipole shift trend. However, this is only possible since we have obtained the expensive full model frequencies. The aim is therefore to derive a fitting function that could be generalised and applied widely when the full model calculations are omitted. The fitting function decided upon is of the form

$$\begin{array}{c}\hfill f=\frac{1}{{(\nu -a)}^b},a,b\in \mathbb{R}.\end{array}$$(B.1)

The approach is to fit Eq. B.1 to the dipole shift of all models in the validation grid, and subsequently investigate how each of the fit parameters varies with the value of Δν for the given model. All fitting efforts at this stage utilise a non-linear Neyman least-squares fitting algorithm (Neyman 1949).

The procedure for deriving the general correction based on each model in the validation grid is as follows:

1. Fit the complete fitting function in Eq. B.1 to every model, each with an associated Δν value. This obtains a range of values for the fit parameters a and b as a function of Δν.

2. Determine the relation between a and Δν (Fig. B.2), chosen as a linear fit of the form f_a = a₁Δν + a₂.

3. Determine the relation between b and Δν (Fig. B.3), fitted according to f_b = b₁/(Δν)^b₂.

For points 2 and 3, we bin the respective fit parameters in bins equally distributed in log-space. The median value of the fit parameters within each bin is calculated and the 5 − σ confidence interval used as a measure of the uncertainty. Subsequently we fit the associated function using the iminuit package² with the MINUIT algorithm (James & Roos 1975), which is a fitting tool for rigorous statistical analyses that incorporates the covariances in its uncertainty estimates.

The resulting generalised model for the frequency-shift correction is then defined as

$$\begin{array}{c}\hfill f_{\mathrm{corr}}(\nu )=\frac{1}{{(\nu -f_a\left(\mathrm{\Delta }\nu \right))}^{f_b(\mathrm{\Delta }\nu )}},\end{array}$$(B.2)

where the value of Δν is in μHz. The two correlation functions f_a and f_b are given by the fits in Figures B.2 and B.3:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_a(\mathrm{\Delta }\nu )& =(1.778\pm 0.071)\mathrm{\Delta }\nu -(2.201\pm 0.191)\hfill \\ \hfill f_b(\mathrm{\Delta }\nu )& =\frac{(2.749\pm 0.052)}{\mathrm{\Delta }{\nu }^{(0.292\pm 0.014)}}.\hfill \end{array}\end{array}$$

Note that for f_b(Δν), the fitted function displays only a slightly asymptotic behaviour as Δν approaches zero. This is desirable, as the risk of b being set to unreasonably high values for highly evolved stars is avoided. The uncertainties on the parameters in f_a and f_b are propagated when employing Eq. B.2 for model ℳ and overplotted on the corrected dipole frequencies in Fig. B.1. The uncertainties on the corrected frequencies are negligible in comparison to the later employed minimum boundary for the scanning intervals (see Sect. 5), for all but the lowest acoustic-mode-order resonance. For this mode, the correction takes its largest amplitude and is the most uncertain. This complication was also discussed Sect. 5.

To summarise, we note that the application of the derived generalised frequency correction does not require the full model frequencies, just the recovered acoustic resonances and the Δν of the model. As such, it can be applied to the dipole modes after a given truncated calculation to correct and shift the recovered acoustic resonances. Figure B.4 shows the generalised correction applied to the lower frequency range of model ℳ, visualised through a plot of the mode inertia. One can clearly see how the acoustic resonances have been shifted to be coincident with the inertia valleys as desired – a larger correction being necessary at lower frequency – by both an exact fit to model ℳ and the generalised correction. The previously noted difficulty of the lowest acoustic-mode-order resonance can also be seen in Fig. B.4. The minimum interval for the subsequent scanning is indicated by the shaded bands, and shows how the minima in inertia are widely covered for all but this one problematic resonance.

Fig. B.2. Fit parameter a obtained from all fits to Eq. B.1 for all models investigated, plotted against Δν. The resulting linear fit to the binned data has been overlaid, with the resulting parameters a1 and a2 indicated.

Fig. B.3. Fit parameter b obtained from all fits to Eq. B.1 for all models investigated, plotted against Δν. The fit to the binned data using a power function with the indicated parameters b1 and b2 has been overplotted, showing the determined trend of b with Δν.

Fig. B.4. Application of the generalised frequency-shift correction for the lower frequency dipole modes of model ℳ visualised in an inertia diagram. The frequencies of the uncorrected resonance frequencies (dotted lines), corrected resonance frequencies from an exact fit to model ℳ (dashed lines) and the generalised model-corrected resonance frequencies (solid lines) are shown. Note that the exact and generalised corrections almost overlap in all acoustic resonances. The minimal size of the subsequently imposed scanning interval is indicated by the surrounding shaded bands.

Appendix C: Scanning intervals for LRGB and TRGB stars

Figure C.1 and C.2 display the mode inertia for an LRGB and TRGB model, respectively. They show the results of the truncated scanning method utilising the altered dimensions of the scanning intervals for the two classification regions mentioned and discussed in Sect. 5.2 and 5.3, respectively. The differences of the scanning interval dimensions and resulting observable regions for the LRGB and TRGB evolutionary regions, is visualised in the figures.

Fig. C.1. Mode inertia for an unevolved RGB star with Δν = 14.86 μHz, depicting the modes from a full model (transparent points) and truncated scanning method (opaque points) calculation, abbreviated FMC and TSM, respectively. The observable intervals are recovered for both the dipole (red) and quadrupole (blue) oscillation modes.

Fig. C.2. Mode inertia for a highly evolved star with Δν = 1.04 μHz, depicting the modes from a full model (transparent points) and truncated scanning method (opaque points) calculation, abbreviated FMC and TSM, respectively. The observable regions for both the dipole (red) and quadrupole (blue) modes are recovered, depicting very few observable modes.

Appendix D: GARSTEC settings for the validation grid

All stellar evolution models employed in this work are computed using the Garching Stellar Evolution Code, GARSTEC (Weiss & Schlattl 2008). The choice for equations of state are the ones from OPAL (Rogers et al. 1996; Rogers & Nayfonov 2002) and the work of Mihalas-Hummer-Däppen (Daeppen et al. 1988; Hummer & Mihalas 1988; Mihalas et al. 1988, 1990). We treat atomic diffusion following the prescription by Thoul et al. (1994). Different choices for opacities are used. For high temperatures we employ the ones from OPAL (Rogers & Iglesias 1992; Iglesias & Rogers 1996), and for low temperatures the opacities from Ferguson et al. (2005). The nuclear reaction rate cross-sections are from NACRE (Angulo et al. 1999), with the exception of the reactions ¹⁴N(p, γ)¹⁵O and ¹²C(α, γ)¹⁶O which are from Formicola et al. (2004) and Hammer et al. (2005), respectively. The stellar abundances used are from Asplund et al. (2009).

The coverage of the validation grid is shown in Fig. C.3, which illustrates the Cartesian grid of 9 tracks presented in Sect. 6. It was constructed to widely span the parameter space of red giants and therefore have tracks at extreme combinations of mass and metallicity. Each track was evolved until the He flash. The track that evolved the furthest reached a Δν = 0.810 μHz, while the least evolved reached Δν = 0.877 μHz. In both cases, this is far into the TRGB region that we consider.

Fig. C.3. Simple plot showing the spanned parameter space of the validation grid in the two varied parameters, mass and [Fe/H].

All Tables

All Figures

Fig. 1. Characteristic frequencies for model ℳ with the parameters given in Table 1, expressed in terms of cyclic frequencies N/2π (solid), S1/2π and S2/2π (both dashed, S2 residing further from the interior). The horizontal dotted line indicates the frequency of maximum power νmax. The gravity-mode cavity is indicated by the red section with its upper boundary at r1, and the pressure-mode cavity by the blue section (with the lower boundary at r2) for l = 1. A prominent buoyancy glitch can be seen at r/R ≈ 0.075.

In the text

Fig. 2. Evolutionary diagram for a solar stellar track with M = 1 M⊙ and [Fe/H] = 0 dex, plotting the large frequency separation Δν and luminosity L as a function of effective temperature Teff. The colour bar indicates the number of dipole modes in a Δν-wide frequency range around νmax, according to Eq. (4). The three classifications from top to bottom denote the tip (TRGB), middle (MRGB) and lower RGB (LRGB).

In the text

	Fig. 3. Inertia plot of model ℳ. The red upwards triangles and blue downwards triangles denote the dipole and quadrupole modes, respectively. The radial modes are depicted as orange diamonds. The plotted modes are connected by solid lines to guide the eye.
In the text

	Fig. 4. Replicated échelle diagram of a full calculation of model ℳ. The orange diamonds, red upwards and blue downwards triangles depict the l = 0, l = 1 and l = 2 modes, respectively, for a model below the RGB bump. For each plotted mode, a scaling in size related to the inverse of its inertia is applied.
In the text

Fig. 5. Propagation diagram for model ℳ, displaying a zoomed-in view of Fig. 1 near the bottom of the convective region. The buoyancy frequency N/2π and Lamb frequency Sl/2π for l = 1, 2 are shown as the full and dashed (S2 being the higher) profiles, respectively. The horizontal dotted line indicates a representative high frequency νhigh of the model, with its g-mode cavity marked in red. The horizontal dot-dashed line shows a representative lower frequency νlow. The vertical dotted and dot-dashed grey lines indicate suitable respective truncation points in r/R. The vertical solid grey line represents a truncation point at the bottom of the convective region. The buoyancy glitch is marked as the hatched area.

In the text

Fig. 6. Mode inertia for the model ℳ, depicting the modes from a full model (transparent points) and truncated scanning method (opaque points) calculation, abbreviated FMC and TSM, respectively. All plotted points are connected by a solid grey line to guide the eye. The inertia valleys containing observable p-mode-like oscillations for both the dipole (red) and quadrupole (blue) modes are recovered. The lowest-order dipole mode has been removed from consideration, as can be clearly seen at ν ≈ 20 μHz. Note that the recovered radial modes are identical for an FMC and TSM calculation.

In the text

Fig. 7. Replicated échelle diagram of model ℳ obtained after application of the truncated scanning method to calculate red-giant oscillations. The orange diamonds, red upwards and blue downwards triangles depict the l = 0, l = 1 and l = 2 modes, respectively. A scaling in size related to the inverse of its inertia is applied to each plotted mode. The observable mixed-mode regions surrounding the acoustic resonances have been calculated and plotted for the dipole and quadrupole modes. For contrast to the initial situation, compare to Fig. 4.

In the text

Fig. 8. Computational efficiency of the truncated scanning method. The computation time for a given model is plotted against its Δν value for a full calculation and for the truncated scanning method as solid and dashed lines, respectively. The time is displayed on a logarithmic scale, and the overplotted lines aid in the visualisation. The vertical dashed lines indicate the classification transition points from Table 2, as shown.

In the text

Fig. B.1. Visual representation of the frequency shift incurred on the truncated frequencies for model ℳ. The fractional frequency shift (νF − νT)/νT between the full (νF) and truncated (νT) frequencies is plotted against νT for the dipole and quadrupole modes, shown by the red and blue lines, respectively. The grey profile shows the resulting generalised correction for model ℳ overplotted, for which its value is subsequently subtracted from all dipole acoustic resonances to provide the red triangles. The uncertainty on the correction for each dipole mode is indicated, as well as the minimal size of the scanning interval boundary as the shaded beams in the respective colour.

In the text

	Fig. B.2. Fit parameter a obtained from all fits to Eq. B.1 for all models investigated, plotted against Δν. The resulting linear fit to the binned data has been overlaid, with the resulting parameters a1 and a2 indicated.
In the text

	Fig. B.3. Fit parameter b obtained from all fits to Eq. B.1 for all models investigated, plotted against Δν. The fit to the binned data using a power function with the indicated parameters b1 and b2 has been overplotted, showing the determined trend of b with Δν.
In the text

Fig. B.4. Application of the generalised frequency-shift correction for the lower frequency dipole modes of model ℳ visualised in an inertia diagram. The frequencies of the uncorrected resonance frequencies (dotted lines), corrected resonance frequencies from an exact fit to model ℳ (dashed lines) and the generalised model-corrected resonance frequencies (solid lines) are shown. Note that the exact and generalised corrections almost overlap in all acoustic resonances. The minimal size of the subsequently imposed scanning interval is indicated by the surrounding shaded bands.

In the text

	Fig. C.1. Mode inertia for an unevolved RGB star with Δν = 14.86 μHz, depicting the modes from a full model (transparent points) and truncated scanning method (opaque points) calculation, abbreviated FMC and TSM, respectively. The observable intervals are recovered for both the dipole (red) and quadrupole (blue) oscillation modes.
In the text

	Fig. C.2. Mode inertia for a highly evolved star with Δν = 1.04 μHz, depicting the modes from a full model (transparent points) and truncated scanning method (opaque points) calculation, abbreviated FMC and TSM, respectively. The observable regions for both the dipole (red) and quadrupole (blue) modes are recovered, depicting very few observable modes.
In the text

	Fig. C.3. Simple plot showing the spanned parameter space of the validation grid in the two varied parameters, mass and [Fe/H].
In the text