1990) who found differences of 10 −3 when they applied the variational principle to their calculations. An explicit formulation of the variational principle in spheroidal geometry is given in Appendix C.

Influence of the parameters from the numerical method
A final test consists in modifying different input parameters and seeing the effect it has on the results. We have therefore applied this test to a few modes which are representative of all the modes that have been calculated. The parameters that were modified are the radial resolution N r (which is the same for the equilibrium model and the pulsation mode), the harmonic resolution of the model L mod, the harmonic resolution of the pulsation mode L max, the shift σ used in the Arnoldi-Chebyshev algorithm 5, and ǫ which controls the relative error of the enthalpy in the equilibrium model. Table 3 lists the values used for the different parameters and the induced frequency variations. For a given parameter, we used the frequency obtained at highest resolution (or lowest value for ǫ) as a reference. In most cases, we obtained a rough plateau at the different levels given in Table 3. In some cases however, there was a definite decrease of the error. For instance, for those modes in which it was tested, the error was roughly proportional to ǫ. Also, for high frequency modes, the error strongly decreased as N r increased, as could be expected for high radial orders. In the table, we put the lower/final values of the error for both of these parameters. The information on the shift is slightly different. The line "Values" gives the amplitude of the variation on the value of the shift. The next two lines contain the standard deviation of the results.
The results on ǫ are not representative of the calculated frequencies. As was pointed out in Sect. 3.1, the number iterations was usually less than optimal because of a large value of ǫ (10 −8 instead of 10 −10 ), thus resulting in a decreased accuracy. The relative error on low frequency modes is 10 −8 and that of high frequency modes 10 −7. Table 3. Frequencies variations in terms of different parameters. Lmax is the harmonic resolution of the pulsation modes, L mod the harmonic resolution of the models and Nr the radial resolution. The line "Values" gives the different values that were used for the resolutions, ǫ and the width giving the variation of the shift. The two following lines give the order of magnitude of the induced frequency variations (in units of √ 4πGρc)... 60 2 − 5 × 10 −4 10 −8...10 −10 Low frequency modes < 10 −15 10 −10 10 −10 10 −13 10 −10 High frequency modes 10 −14 10 −9 10 −10 10 −11 10 −9

Discussion
Overall, the main source of error in the present calculations is the uncertainties on the equilibrium model. This is because we chose a convergence criteria which was sure to be met, but which lead to a number of iterations less than optimal. This therefore leads to a global accuracy of 7 digits after the decimal point (in units of √ 4πGρ c ), the last digit being uncertain. Table 3 however shows that these calculations could potentially be made more accurate. The present accuracy is nonetheless largely sufficient for the requirements of COROT, which will be at most 5 significant digits.

Results
We now proceed to present the results themselves. We followed acoustic adiabatic pulsation modes (with Γ 1 = 5/3) from a zero rotation rate to 0.59Ω K (where Ω K = GM/R 3 eq is the Keplerian break-up rotation rate), using the same procedure as in Paper I. This involves identifying the frequencies at Ω = 0, following their evolution while progressively increasing the rotation rate, and working through a number of avoided crossings. The underlying polytropic models have an index N = 3 which gives a polytropic exponent γ = 4/3. The modes that were calculated are: ℓ = 0 to 3, n = 1 to 10 and m = −ℓ to ℓ both with and without the Coriolis force.

Comparison with perturbative methods
In this section, we compare complete and perturbative calculations so as to determine the range of validity of perturbative methods.

Perturbative coefficients
In order to compare perturbative calculations with complete ones, it proved necessary to compute our own perturbative coefficients, since we were unable to find perturbative coefficients for polytropic models with a sufficient accuracy in the literature. Instead of using the traditional method of perturbing the fluid equations and finding corrections of various orders on the frequencies (see Soufi et al. (1998) for a complete description), we did a series of complete calculations for small rotation rates ((Ω ⋆ ) i = 0, 10 −6, 10 −5, 10 −4, 10 −3, 0.002, 0.004... 0.018) and applied a least squares fit to the results. In order to increase the accuracy of such calculations, we made use of Eq. (55) and separated even and odd powers of the rotation rate: ω n, l, m + ω n, l, −m 2 = ω 0 n, l, m + ω 2 n, l, m Ω 2 + ω 4 n, l, m Ω 4 + O(Ω 6 ), ω n, l, m − ω n, l, −m 2 = ω 1 n, l, m Ω + ω 3 n, l, m Ω 3 + ω 5 n, l, m Ω 5 + O(Ω 7 ).
By fitting (ω n, l, m +ω n, l, −m )/2 and (ω n, l, m −ω n, l, −m )/2 the number of unknowns is reduced to three and the residues are smaller. It is necessary to include the fourth and fifth powers of the rotation rate so as to ensure that the second and third order coefficients are reasonably accurate. The results are given in Table 4 for frequencies and rotation rates in units of GM/R 3 pol. From these coefficients the frequencies are given through the following formula: The form of the second degree coefficients was obtained from Saio (1981) and that of the third degree coefficients from Goupil (private communication). In order to express these results in units of Ω K instead, one can use the following perturbative formula: where Ω pol K = GM/R 3 pol and A ≃ 0.77166. In order to estimate the accuracy of these perturbative coefficients, there are a number of tests that can be done. First of all, the zeroth order coefficients are simply the pulsation frequencies without rotation but are treated as unknowns in the least squares development. The frequencies without rotation are recovered in the least squares fit to an accuracy of at least 5.4 × 10 −8 in the units of Table 4. The first order coefficient C, can be calculated via integrals based on the zeroth order solution (Ledoux, 1951). This alternate way of calculating the coefficients agrees to within 1.4 × 10 −9 (this does not necessarily mean that the coefficients are accurate to that precision but does show a high degree of internal coherence). For the second and third degree coefficients, we checked to see if they satisfied the forms given in Eq. (66); the number of significant digits in Table 4 has been adjusted accordingly. These forms were a constraint only on the ℓ = 2 and 3 second order coefficients and on the ℓ = 3 third order coefficients. Another test we did consisted in applying the least squares fit to a subset of the results used in the first fit and seeing whether the coefficients were altered. This test indicates roughly the same accuracy as the other tests.

Comparison
Based on these coefficients, it is possible to calculate perturbative frequencies which can then be compared to the complete calculations, thereby establishing a domain of validity for perturbative methods. In Fig. 4, we show two such domains for 3 rd order methods, one for each of COROT's error bars (0.6 µHz for the 20 runs and 0.08 µHz for the 150 day runs). The underlying polytropic models have a fixed mass of 1.9 M ⊙ and a fixed polar radius of 2.3 R ⊙, both of which are typical of δ Scuti pulsators. When the distance between the perturbative frequency and the complete one exceeds COROT's error bars, the frequency is shown in black. Otherwise, it is shown in grey. From these figures, it is clear that complete methods are required beyond v · sin i = 75 km.s −1 for COROT's 20 day programs and v · sin i = 50 km.s −1 for COROT's 150 day programs.
It is important to bear in mind that the domain of validity obtained for perturbative methods depends on the choice of rotational variable used in the development. In order to illustrate this, suppose we develop a frequency in terms of two different rotational parameters X and Y : ω = a 0 +a 1 X +a 2 X 2 +a 3 X 3 +O( When the relationship between X and Y is more complex than a simple proportionality, the neglected terms, O(X 4 ) and O(Y 4 ), are not the same. As a result, a 3 rd order development in terms of X or Y will give different values for ω, thus modifying the corresponding domain of validity. Therefore, we decided to compute the domain of validity associated with the variables Ω/Ω K and Ω/(GM/R 3 pol ) 1/2 to see if there was a substantial difference between the two. For individual frequencies, there can be large differences, but when all the frequencies are considered, the global result is roughly the same.
In Fig. 5