Solar rotation and the inclination of the rotation axis

The solar rotation and the inclination of the rotation axis can in principle be measured from the splitting of the azimuthal components of the non-radial modes (` ≥ 1) and their mode height ratio respectively (for details, see Gizon and Solanki 2003). In this analysis I assumed the splitting to be independent of the azimuthal ordermand the radial ordern, i.e. rigid body rotation.

Figure 2.19 shows the solar power spectrum and expectation value for a 120 day block of VIRGO data for modes with angular degree ` ≤ 2 averaged over 12 radial orders (cf. Figure 2.9). In this particular example, I obtained a rotational splitting Ω/2π = 0.40µHz and an inclination angle i = 76<sup>◦</sup>. Due to the solar mode linewidth ofΓ&1µHz and thus 2πΓ/Ω> 2, the azimuthal components of the non-radial modes are not resolved. However, the average line profile of the non-radial modes is broader than the average line profile of the radial` = 0 modes. This broadening can be attributed to the contribution of the|m|> 0 components to the overall line profile. I conclude that the global fit allows us to detect the signature of rotation in the oscillation power spectrum of a slowly rotating star like the Sun. Finally, Figure 2.20 demonstrates the overall good per-formance of the global fit. The figure shows the ratio of the observed power spectrum and the expectation value determined with the global fit, i.e. the ratio of the black and the red lines in Figure 2.19. This ratio is distributed around one and does not show a significant bias.

The estimates for Ω/2π and i for the particular 120 day block of VIRGO data dis-cussed above are derived from a global fit with 100 random initial guesses. The distribu-tion of these guesses and the corresponding fits are shown in Figure 2.21. The best fit of the rotational splitting,Ω/2π = 0.40µHz, seems to be slightly underestimated consider-ing the distribution of the fit results. For the inclination angle,i, the distribution of the fit results does not even show a clear peak. On the other hand, the distribution of the

pro-of 120 day VIRGO time series. The 1σ error is defined such that 68% of all fits fall within the interval given by the subscripts/exponents. For reference some approximate intervals for the expected parameters are listed in thethird column. The rotational splitting corresponds to surface rotation rates between latitudes 0^◦ ≤ λ ≤ 60^◦. The inclination of the solar rotation axis is inclined from the normal to the ecliptic plane by∼7^◦. The given interval considers that the actual inclination during a particular observation block depends on the ephemerides of the Sun.

Parameters for the Sun Global fit (this work) Reference solar values Rotational splittingΩ/2π[µHz] 0.52⁺₋^0.12_0.08 ∈[0.37,0.45]

Inclinationi[^◦] 56±11 ∈[83,90]

sini 0.82⁺_−0.12^0.09 ∈[0.993,1]

Ω/2π sini[µHz] 0.424⁺₋^0.038_0.036 ∈[0.37,0.45]

jected splitting,Ω/2π sini, shows a distinct peak with the best fit being very close to the center of this distribution. This confirms the results of Ballot et al. (2006, 2008): it is very difficult to measure the rotation and the inclination of the rotation axis reliably for slowly rotating stars. However, the projected splitting,Ω/2πsini, can be measured precisely and unbiased.

The results for the rotational splittingΩ/2π, the inclinationi, andΩ/2π siniobtained from global fits of all 35 blocks of VIRGO data are summarized in Table 2.5. The numbers correspond to the median and the 1σerror of all 35 blocks. The table gives also reference values for the individual parameters. The reference interval forΩ/2π corresponds to a surface rotation rate at latitudes 0^◦ ≤ θ ≤ 60^◦. The reference interval for the inclination considers that the solar rotation axis is inclined with respect to the normal of the ecliptic by∼ 7^◦ (Carrington 1863). The actual inclination for a particular block of VIRGO data depends on the ephemerides of the Sun. The estimates onΩ/2πandiobtained in this anal-ysis are significantly biased with respect to the reference values, i.e. the rotational split-ting,Ω/2π = 0.52⁺_−0.08^0.12µHz, is overestimated and the inclination angle, i = 56^◦±11^◦, is underestimated. On the other hand, the projected splitting,Ω/2π sini= 0.424⁺_−0.036^0.038µHz, is in very good agreement with the expected solar reference value.

Figure 2.22 shows the estimates onΩ/2πand sinifor all 35 blocks of VIRGO data.

Both parameters are correlated and equally distributed around a constantΩ/2π sini. The color map represents the shape of the joint PDF (or the log-likelihood function) in the Ω/2π-siniplane. The colors correspond to the value of the joint PDF function obtained from global fits for various pairs of variates, (Ω/2π, sini). For each point, the value of the joint PDF is averaged over all 35 blocks of VIRGO data. The estimates forΩ/2πand ifor all 35 blocks are located around the maximum of the joint PDF. The joint PDF is very flat around its maximum (see bottom panel of Figure 2.22). So, it is very difficult

Figure 2.19: Solar power spectrum and the corresponding global fit for modes with an-gular degree` ≤ 2 averaged over 12 radial orders for one 120 day block of VIRGO data (cf. Figure 2.9). The observed power is shown in black, the expectation value of the power in red. The vertical tick marks indicate the central frequencies of the azimuthal components which are separated by the rotational splittingΩ/2π. The rotational splitting is assumed to be independent of the radial order and the azimuthal order. Theblue and green profiles at the bottom of each panel show the individual odd and even azimuthal components of the corresponding averaged line profile.

Figure 2.20: Ratio of the observed solar power spectrum and the expectation value of the power from Figure 2.19 for modes with ` ≤ 2. The observed power and the fit are averaged over 12 radial orders. Note that there is no systematic bias for any of the three cases indicating the reliability of the global fit.

Figure 2.21: Distribution of 100 initial guesses (thick, grey) and the corresponding fits (fine, black) of the rotational splittingΩ/2π(top) and the inclination angle iof the rota-tion axis (middle) for one 120 day block of VIRGO data. The bottom panel shows the projected splitting, Ω/2π sini, composed of the distributions in the panels above. The vertical dashed linesindicate the parameters of the best fit.

that the rotational splitting can be constrained between 0.30µHz ≤ Ω/2π ≤ 1.05µHz corresponding to a rotation period of 11 days≤ Prot ≤39 days. The inclination angle of the rotation axis can be constrained between 0.4≤ sini≤1.0 (or 24^◦≤ i≤90^◦). The two parameters are correlated and their product isΩ/2π sini = 0.424⁺_−0.036^0.038µHz within a 1σ error bar.

Given the reliable estimate ofΩ/2π sinifrom the analysis of the time series, the in-dividual estimates onΩ/2π andimay be further constrained when complementary mea-surements are taken into account. Figure 2.23 shows a 120 day VIRGO time series and the corresponding power spectrum at low frequencies (data courtesy of C. Fröhlich). There are two peaks atν=0.40µHz andν=0.59µHz. These peaks may be attributed to active regions dragged by solar rotation with a period ofProt = 29 days andProt = 20 days re-spectively. There are also two possible harmonics at∼1µHz and∼ 1.4µHz. I emphasize that this is just one example. For various blocks of data, the exact position of the features in the low frequency power spectrum may vary slightly. Combining the estimate on the solar surface rotation with the seismic estimate onΩ/2π siniallows us to constrain the inclination angle of the solar rotation axis (see intersection of the corresponding lines in Figure 2.22). Thus, I obtain 0.66 ≤ sini ≤ 1.0 or (41^◦ ≤ i≤ 90^◦). Note that in this case the uncertainty on the inclination angle of the rotation axis is dominated by the precision of the rotation measurement, i.e. a more precise constraint on the solar rotation allows one to estimate of the inclination angle more precisely.

2.4 Discussion: Is the global fit good enough for