Patterns, an efficient way to analyse the p-mode frequency content in rapidly rotating stars.

Abstract

The objective of this work is to apply the recent techniques used with the pulsating spectrum of A-F stars [REFS] to study non-pertubative rotating models (SCF) [REFS] and targets. The techniques are the Fourier transform (FT, from now), the autocorrelation function (ACF, from now), the histogram of frequency differences (HFD, from now), and the Echelle diagram (ED, from now). We are looking for the best procedure to obtain a reliable mode identification.

Here, the intro. Yes, I propose here to include a relation of the papers we have published on this topic, including the last accepted one. Then, we should also mention the works by Lignières and Reese that are more related (e.g. those mentioning the periodic patterns predicted by non-perturbative theory)

To test the confidence in the tools and figure out what information we can obtain with each of them, we started the analysis of a set of rotating models. We have computed 54 SCF different models [REFS] with the following characteristics. They have 9 different rotations from \(\Omega/\Omega_{K} = 0\) to \(\Omega/\Omega_{K} = 0.8\). For each rotational velocity, we have two different inclination angles and three different ways of setting the mode amplitudes. In one case, the mode amplitude is given by the mode visibility; in a second case these amplitudes are multiplied by a random number comprised between 1 and 10 (the random number is uniformly distributed in a logarithmic scale); in a third case these amplitudes are multiplied by a random number comprised between 0.1 and 10 (the random number is uniformly distributed in a logarithmic scale).

The most visible modes (in terms of visibility calculations) are the island modes, then the chaotic and then the whispering gallery modes. As rotation increases the number of visible whispering gallery mode decreases while the number of chaotic mode increases (together with the number of island modes).

The other subsets of modes (mainly chaotic modes and whispering gallery modes) have a different organisation that is not fully understood yet. Nevertheless a large separation (but not half the large separation) and a spacing close to \(2\Omega\) (between \(m=-1\) and \(m=1\) modes) should still be present in the chaotic mode subspectrum.

The models contain the following information:

\(\Delta_{n}\) | \(\Delta_{\ell}\) | \(\Delta_{m}\) | \(\Omega\) |

7.1969685 | 3.0115791 | 0.23874654 | 0.0000000 |

7.1296325 | 2.9668959 | 0.14821643 | 0.60528506 |

7.0805956 | 2.5754183 | -0.020148432 | 1.1861165 |

7.0474367 | 1.8731673 | -0.072648819 | 1.7204604 |

6.9380044 | 1.1871432 | -0.20184019 | 2.1902553 |

6.6969111 | 1.0043267 | -0.63984686 | 2.5823731 |

6.2993731 | 1.1582393 | -1.0687608 | 2.8913942 |

6.0308690 | 1.1349926 | -1.0545724 | 3.1175864 |

5.6754183 | 0.84165972 | -0.82021388 | 3.2670515 |

Fourier transform, histogram of frequency differences, Echelle diagrams, autocorrelation function.

The large separation could be determined using the FT. Even chaotic modes can follow a pseudo large frequency separation, at least over a limited range of modes. These may at least reinforce the \(\Delta_{n}\) signal, even if they make mode identification more complicated.

We computed the FT for different subsets of frequencies for each one of the frequency lists of the rotating models following García Hernández et al. (2009) and we was able to derive the correct value of the large separation in almost all the cases. We trust that a peak in the Fourier transform corresponds to the large separation (or its sub-multiple) when it is conserved when increasing the number of frequencies involved in it. The accuracy of this method has to be precisely quantify, because we did not obtain the exacts values of \(\Delta_{n}\). This is why, by the moment, the results are only approximate. These are the values I obtained (only 1 value per rotation rate):

\(\Omega\) | \(\Delta_{n}\) |

0.0 | 7.08 |

0.6 | 6.96 |

1.18 | 7.00 |

1.72 | 6.96 |

2.19 | 6.70 |

2.58 | 6.56 |

2.89 | 6.20 |

3.11 | 5.83 |

3.26 | 5.53 |

Mode identification, even in the synthetic spectra, is not easy. As a first step, we were working on a procedure to find \(\Omega\) and identify the (n, l, m) and (n, l, −m) pairs of modes. The procedure was to select the frequency pairs separated by ∆n, once it is determine by the FT, and search for the (m, −m) pairs. The theory predicts a slightly decreasing of ∆ν with n for these pairs.

Once the large separation is derived, it is convenient to use the frequencies normalised to ∆ν. We are able to see, approximately, the range in radial order. Large separation will appear more clearly for n\(\ge\)4.

Sometimes, an histogram of difference could give an idea of the rotational splitting: \(2\Omega ≈ \Delta_{n}/2\) or \(\Omega ≈ \Delta_{n}/2\). This is because the frequencies cluster around the radial orders when those relations are fulfilled (typically for \(\Omega/\Omega_{K} ≈ 0.3\) or 0.7, see Fig. 2).

Reese et al. 2013 (submitted paper) simulations showed that \(2\Omega\) is the easiest secondary spacings to find. It is found in the autocorrelation (or distribution of all frequency spacings) rather than in the FT of the spectrum because it is due to frequency pairs (m = +1 and m = −1) rather than a frequency comb.

Can we say something about what the best rotation rate, inclination angle and visibility of the mode are to be easily detectable with the FT?

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