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Appendix A: Equations in spheroidal coordinates

In this section, we derive explicit expressions for the pulsation equations and the mechanical boundary condition, based on the coordinate system described in Sect. 2.4. However, before giving these expressions, it is useful to recall a few definitions. The natural covariant basis, denoted (E_ζ, E_θ, E_φ), is defined via the relation E_i = ∂_ir, where i stands for ζ, θ or φ, and r = re_r: (A.1)Here (e_r, e_θ, e_φ) is the usual spherical basis associated with the spherical coordinates (r  θ, φ). The associated dual (contravariant) basis is defined such that : (A.2)The vector E^ζ is perpendicular to surfaces of constant ζ value, including the stellar surface. As in Reese et al. (2006), we derive an alternate basis from (E_ζ, E_θ, E_φ) as follows: (A.3)In the spherical limit, the alternate basis converges to the spherical basis. The Lagrangian displacement is decomposed over the alternate basis as follows: (A.4)These components are related to the spherical components (see Eq. (20)) as follows: (A.5)where superscripts are used with the alternate components, and subscripts with the spherical components. Based on the alternate components, the dot product ξ·g_eff becomes: (A.6)where we have introduced the following quantities: (A.7)

A.1. Pulsation equations

We now give explicit expressions for the pulsation equations in spheroidal coordinates. The continuity equation is: (A.8)Euler’s equation takes on the following form: where s = rsinθ is the distance to the rotation axis. Poisson’s equation becomes: (A.12)where As was pointed out in Sect. 2.2, the relative Lagrangian density perturbation, δρ/ρ₀, can be eliminated in favour of the relative Lagrangian pressure perturbation, δp/P₀, thanks to the adiabatic relation, Eq. (6).

A.2. Mechanical boundary condition

As explained in Sect. 2.5, the mechanical boundary condition is obtained by calculating the dot product between E^ζ and Euler’s equation, and cancelling out the vertical gradient of δp/P₀. Furthermore, the quantity ∂_ζξ^ζ is eliminated through the continuity equation, and the terms and vanish at the surface. In spheroidal components, one obtains: (A.15)

Appendix B: Lagrangian perturbation to the effective gravity

As was explained in Sect. 3.2, the Lagrangian perturbation to the effective gravity, δg_eff, is deduced from the vectorial Lagrangian perturbation to the effective gravity, δg_eff, via the relation δg_eff = −n·δg_eff, where n is the outward normal at the surface. Furthermore, δg_eff includes the Lagrangian perturbation to the gradient of the gravitational potential and the acceleration of a particle tied to the surface, resulting from the oscillatory motions. After adding and subtracting ξ·∇(sΩ²e_s) in order to introduce the equilibrium effective gravity, a vectorial expression is obtained in Eq. (34), and is reproduced here for convenience: (B.1)In what follows, we will go through the above equation one term at a time in order to obtain explicit expressions for the dot product of n with each one.

The Eulerian perturbation to gravity is obtained through tensor analysis: (B.2)where we have used the relation . Furthermore, we have used Einstein’s summation convention on repeated indices.

Before dealing with the next term, it is useful to introduce the contravariant components of the Lagrangian displacement, which we distinguish from the components given in Eq. (A.4) by placing a tilde over the top: (B.3)where (E_ζ, E_θ, E_φ) is given in Eq. (A.1). We also introduce the covariant components of the effective gravity: (B.4)From Eq. (31), it is straightforward to see that . Furthermore, and at the stellar surface.

In tensorial notation, the term n·{ξ·∇g_eff} becomes: (B.5)where g^ij = Eⁱ·E^j denotes the contravariant components of the metric tensor, and the Christoffel coefficients. Equation (B.5) can be simplified if we use the following relation: (B.6)which is a modified form of Poisson’s equation. The result is: (B.7)where and where we have made use of the following simplifications: on the surface, g^ij = g^ji and . The above expression can then be re-expressed in terms of ξ_r, ξ_θ and g_eff to yield: (B.10)The term ∇·(sΩ²e_s) takes on the following expression for a general rotation profile, Ω ≡ Ω(ζ,θ): (B.11)For a cylindrical rotation profile, Ω ≡ Ω(s), it becomes: (B.12)The term (ω + mΩ)²n·ξ is given by: (B.13)The Coriolis term is given by: (B.14)The last two terms are more conveniently treated together. They take on the following expression for a general rotation profile: (B.15)If the rotation profile is cylindrical, they become: (B.16)Combining all of these equations together, and remembering the minus sign, yields Eq. (35).

Appendix C: Cancelling of simplified disk-integration factors

Given the simplified form of the disk-integration factors given in Eq. (45) (see Sect. 3.7), it turns out that some of these cancel out regardless of inclination. To see this, one needs to start with an explicit form for Eq. (45): (C.1)where ψ is a suitably chosen phase, ⟨δT⟩ is given in Eq. (45), and f(θ) corresponds to the visibility curve (i.e. the border between the visible and hidden side of the star). The function f obeys the following symmetry: f(θ) + f(π − θ) = π. We have made use of Eqs. (16) and (22) in obtaining an explicit expression for e_obs.·dS. The integral in Eq. (C.1) is then split into two equal halves and the second half is modified according to the variable changes θ′ = π − θ and φ′ = φ − π: (C.2)where we have made use of the relation f(θ) + f(π − θ) = π. The two halves can be combined to give a single integral over the entire stellar surface only if . If we assume this is the case, we can then see under conditions the integral vanishes. We specifically look at the integration over φ, which now is between the bounds 0 and 2π. Remembering that if |m| ≠ 1 and that if m ≠ 0, we deduce the second condition for cancelling the disk-integration factor, i.e. |m| ≥ 2.

Appendix D: Normalisation of multi-colour visibilities

In order to find a normalisation which minimises the distances between a set of multi-colour visibilities, we start with the cost function given in Eq. (47) and include an additional term so as to enforce the constraint . Without loss of generality, we work with the normalised components, , instead of the original ones: (D.1)where Λ represents a Lagrange multiplier. Setting the derivatives, , ∂J/∂W_i, and ∂J/∂Λ, to zero leads to the following system: where we’ve used vectorial notation for conciseness and where A·B represents the dot product . Using the normalised components allows us to simplify Eq. (D.2) to . When combined with Eq. (D.3), this yields: (D.5)This last equation is in fact an eigenvalue problem where N − Λ is the eigenvalue and W the eigenvector. In order to determine which eigensolution yields the minimal value for J, we develop the cost function as follows: (D.6)where we’ve used the simplifications , , and W·W = 1. In order to simplify the term , we calculate the dot product between W and Eq. (D.3): (D.7)Hence, (D.8)Therefore, the minimal value of Λ (and hence the maximal value of N − Λ) corresponds to the minimal value of J. The vector W is therefore the principal component of the vector set and can be found either via a singular value decomposition of the associated matrix, or more simply through a power iteration.

© ESO, 2013