The Two-dimensional Internal Rotation of KIC 11145123

Kurtz et al. (2014) first measured the frequencies of the 76 detected modes (45 g-modes and 31 p-modes; see Figure 1 of Kurtz et al. 2014) by a nonlinear least-squares method. The estimated errors are as small as the order of 10⁻⁶ cycles per day owing to the coherence of the pulsations. It is evident that the rotational splitting occurs (see Figure 3 of Kurtz et al. 2014), and the values of the rotational shifts in frequencies, or "generalized rotational splittings" (Goupil 2011), are simply computed as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}{\omega }_{{nlm}}=\displaystyle \frac{{\omega }_{{nl},m}-{\omega }_{{nl},-m}}{2},\end{eqnarray} \tag{ 1 }$$

where n, l, and m are the radial order, the spherical degree, and the azimuthal order, respectively. We have used the rotational shifts and the corresponding errors estimated by Kurtz et al. (2014) in the following analyses.

We also see approximately constant g-mode period spacing, which is expected from asymptotic behavior of high-order g-modes (Tassoul 1980), and the observed value is ΔP_g =0.0245 days. Kurtz et al. (2014) fitted the observed mean g-mode period spacing to produce their best model of the star (see Section 2.2). For more information on the individual frequencies, amplitudes, phases, and estimated errors, see Kurtz et al. (2014).

We have chosen Kurtz et al.'s model as a reference model. The model was calculated using Modules for Experiments in Stellar Astrophysics (MESA, version 4298; Paxton et al. 2013). The initial hydrogen content X_init, the initial helium content Y_init, and the initial metallicity Z_init are 0.65, 0.34, and 0.01, respectively. The initial mass is 1.46 M_⊙. The model is found to be at the terminal-age main-sequence (TAMS) stage, where the star has exhausted almost all the hydrogen in the core (Figure 1). There is a sharp feature in the Brunt–Väisälä frequency, and the location of the peak corresponds to the chemical composition gradient left by the receding convective core during the evolution (Figure 1). For more details, see Section 3 in Kurtz et al. (2014).

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Kurtz et al. (2014) also calculated linear adiabatic oscillation of the model based on the pulsation codes developed by Saio & Cox (1980) and Takata (2012), and they conducted mode identification of the detected modes by comparing the modeled frequencies with the observed frequencies (Table 1). In our analyses, we rely on their identification and have used the eigenfunctions they computed, in calculating the 2D rotational splitting kernel that relates the rotational shifts to internal rotation (see Section 3).

As a final remark on the model, we would like to point out that neither several p-mode frequencies nor the atmospheric parameters such as T_eff and log g are reproduced well with the current model. This could be partly because the star is probably a blue straggler, having experienced some interactions with other stars during its evolution; modeling the star as an ordinary A star may not be sufficient. In fact, Takada-Hidai et al. (2017) have carried out spectroscopic analyses of the star with Subaru/HDS and found that it is spectroscopically a blue straggler. We will tackle the problem based on nonstandard modeling of the star in our forthcoming paper, and we assume that the current model is good enough to explain KIC 11145123 in the rest of this paper. Actually, we have confirmed that the results of rotation estimation are not dramatically changed even though we used other models such as those produced by Takada-Hidai et al. (2017) and Y. Hatta, et al. (2019, in preparation).

When the internal rotation is slow enough and we can treat it as a small perturbation, perturbation theory tells us that the rotational shift, Δω_i, with i representing the mode indices (n, l, m), is related to the internal rotation profile Ω(x, μ) by the splitting kernel K_i(x, μ) as follows:

$$\begin{eqnarray}&&{d}_{i}=\int \int {K}_{i}(x,\mu ){\rm{\Omega }}(x,\mu ){dxd}\mu +{e}_{i},\,\,\,\,i=1,\,\ldots ,\,M,\end{eqnarray} \tag{ 2 }$$

where x and μ are the fractional radius and cosine of the colatitude, respectively. We have replaced Δω_i/m with d_i, and M is the number of the observed rotational shifts used in the rotation inversion, namely, 23 in the case of this study. Observational errors e_i are included in expression (2). Relation (2) is correct to first order. The splitting kernels can be calculated once we have solved the linear adiabatic oscillation of the equilibrium model (e.g., Sekii 1993). Note that relation (2) shows that the rotational shifts can be interpreted as averaged values of the internal angular velocity weighted by the splitting kernel. In other words, the splitting kernels give us information about regions where the eigenfrequencies are most influenced by the internal rotation there (Figures 2–4).

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We have this form of equation for each observed rotational shift, and thus we can estimate the internal rotation Ω(x, μ).

What we have to do is to estimate the internal rotation profile Ω(x, μ) based on the set of linear integral constraints (2).

The basic point of the optimally localized averaging (OLA) method is that we express an estimate at a certain target point (x₀, μ₀) as a linear combination of data d_i,

$$\begin{eqnarray}&&\begin{array}{l}\hat{{\rm{\Omega }}}({x}_{0},{\mu }_{0})=\displaystyle \sum _{i=1}^{M}{c}_{i}({x}_{0},{\mu }_{0}){d}_{i}\\ =\int \int D(x,\mu ;{x}_{0},{\mu }_{0}){\rm{\Omega }}(x,\mu ){dxd}\mu +\displaystyle \sum _{i=1}^{M}{c}_{i}({x}_{0},{\mu }_{0}){e}_{i},\end{array}\end{eqnarray} \tag{ 3 }$$

where D(x, μ; x₀, μ₀) is the averaging kernel, defined as

$$\begin{eqnarray}&&D(x,\mu ;{x}_{0},{\mu }_{0})\equiv \displaystyle \sum _{i=1}^{M}{c}_{i}({x}_{0},{\mu }_{0}){K}_{i}(x,\mu ).\end{eqnarray} \tag{ 4 }$$

We can obtain a good estimate of the internal rotation at the target point by making the averaging kernel as localized as possible around the target point. For the purpose of localization, it is common that we minimize the following quantity:

$$\begin{eqnarray}&&S=\int \int D{\left(x,\mu ;{x}_{0},{\mu }_{0}\right)}^{2}\left\{{\left(x-{x}_{0}\right)}^{2}+{x}^{2}{\left(\mu -{\mu }_{0}\right)}^{2}\right\}{dxd}\mu ,\end{eqnarray} \tag{ 5 }$$

with the unimodular condition of D(x, μ; x₀, μ₀)

$$\begin{eqnarray}&&{S}_{u}\equiv \int \int D(x,\mu ;{x}_{0},{\mu }_{0}){dxd}\mu =1.\end{eqnarray} \tag{ 6 }$$

Minimizing S forces the averaging kernel to be large near the target point and to be small far from it (Backus & Gilbert 1967). On the other hand, we would like to keep the estimated errors small at the same time. Thus, what we actually minimize is

$$\begin{eqnarray}&&{S}_{\alpha }=S+\alpha \displaystyle \sum _{i,j=1}^{M}{c}_{i}{c}_{j}{E}_{{ij}}+2{c}_{M+1}(1-{S}_{u}),\end{eqnarray} \tag{ 7 }$$

where α is a trade-off parameter that determines the balance between the first term in expression (7), resolution, and the second term, error magnification. The unimodular condition of the averaging kernel is expressed by using a Lagrangian multiplier c_M+1 introduced as the (M + 1)th coefficient. The error covariance matrix is denoted by E_ij. In this study, we assume that the observational errors are independent of each other, in other words, ${E}_{{ij}}={e}_{i}^{2}{\delta }_{{ij}}$.

The minimization condition can be rewritten using matrices, and we obtain the inversion coefficients c_i(x₀, μ₀) by inverting the linear matrix equation. We finally obtain the estimate for the target point (x₀, μ₀) by substituting c_i for the first line in expression (3).

Because the averaging kernel is just a linear combination of the splitting kernels, where we can localize the averaging kernel is determined by the shapes of the splitting kernels. We present three examples of the splitting kernels (Figures 2–4), and as we see, there are three regions where the splitting kernels have good sensitivity: the outer region at middle latitude (Figure 2), the outer region at low latitude (Figure 3), and the deep radiative region at low latitude (Figure 4). We have chosen the three regions as the target points, namely, (x, μ) = (0.05, 0.00), (0.95, 0.00), and (0.95, 0.70), in the following inversion via the OLA method. Note that when we selected the target points far from these regions we were not able to localize the averaging kernels very well, leading to unreliable estimates.

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Before we show the estimates of the internal rotation, we briefly explain how we have determined the value of the trade-off parameter α (see Equation (7)); we must select α in order to obtain estimates. For this purpose, we have repeated the rotation inversion 40 times for different values of α and then quantified the width of averaging kernels S, which corresponds to the first term in expression (7), and error magnification V, which corresponds to the second term divided by α in the same expression (7). We have plotted the trade-off curves (e.g., Figure 5), and we have chosen several values of α with which both the width of the averaging kernel S and error magnification V are sufficiently suppressed. After the selection of the candidates for a final value of α, we looked at each of the candidate's averaging kernels, and we excluded values of α with which the localization of the averaging kernel is not well achieved. Eventually, we select one or two values of α and obtain the corresponding estimates. The same procedures have been conducted for each target point. We describe the estimates thus obtained in the following subsections.

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4.1.1. The Estimate for the Target Point ($x,\mu$) = (0.05, 0.00)

For this target point, we present two equally reasonable estimates (8) and (9). The first estimate is obtained for α = 10⁸, and the estimate and its standard deviation are

$$\begin{eqnarray}&&\hat{{\rm{\Omega }}}(0.05,0.00)=(0.9940\pm 0.0003){{\rm{\Omega }}}_{100},\end{eqnarray} \tag{ 8 }$$

where Ω₁₀₀ = 2π × 0.01 day⁻¹. The averaging kernel is localized well around the core region at low latitude (Figure 6), suggesting the high reliability of the estimate. The second estimate is obtained for α = 10¹⁰, and the estimate and its standard deviation are

$$\begin{eqnarray}&&\hat{{\rm{\Omega }}}(0.05,0.00)=(0.9492\pm 0.0001)\,{{\rm{\Omega }}}_{100}.\end{eqnarray} \tag{ 9 }$$

The averaging kernel is, again, localized well around the core region at low latitude (Figure 7).

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Though we have achieved the localization of the averaging kernels reasonably well in both cases, there is a significant difference between estimates (8) and (9). This difference can be attributed to the trade-off relation between resolution and error magnification. With the larger trade-off parameter α = 10¹⁰, suppressing error magnification is emphasized (see Equation (7)), leading to the smaller estimated standard deviation in estimate (9) than in estimate (8). Meanwhile, localizing the averaging kernel is prioritized in the case of α = 10⁸, and thus the corresponding averaging kernel behaves better than that with α = 10¹⁰; when we carefully look into the averaging kernel with α = 10¹⁰, we find that it has an oscillatory component between x = 0.05 and x = 0.075 compared with that with α = 10⁸. Therefore, we can qualitatively explain the behavior of the estimates.

For more quantitative hints, we take a look at inversion coefficients determined by rotation inversion with α = 10⁸ and those with α = 10¹⁰ (Figure 8). It is apparent that the averaging kernel for α = 10¹⁰ is mainly composed of the g-mode splitting kernels. On the other hand, we have also found that the averaging kernel for α = 10⁸ is composed not only of the g-mode splitting kernels but also of the mixed-mode splitting kernels. Thus, the distributions of the averaging kernels are slightly different. One possible explanation is as follows: there is a shear of the angular velocities around the core region, and thus small differences between the two averaging kernels lead to the substantial difference in the two estimates. To test the possibility of such a fast-rotating core is one of the purposes in our three-zone modeling introduced in Section 5.1.

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4.1.2. The Estimate for the Target Point ($x,\mu$) = (0.95, 0.00)

We, again, show two equally reasonable estimates as follows:

$$\begin{eqnarray}&&\hat{{\rm{\Omega }}}(0.95,0.00)=(0.9511\pm 0.0005){{\rm{\Omega }}}_{100},\end{eqnarray} \tag{ 10 }$$

which has been obtained with α = 10⁸, and

$$\begin{eqnarray}&&\hat{{\rm{\Omega }}}(0.95,0.00)=(0.9706\pm 0.0002){{\rm{\Omega }}}_{100},\end{eqnarray} \tag{ 11 }$$

which has been obtained with α = 10¹⁰.

The averaging kernels for both estimates have peaks in the outer envelope (Figures 9 and 10), and both of them are apparently localized around the target point. Nevertheless, we see that the averaging kernel for α = 10⁸ has a higher peak than for α = 10¹⁰ because more emphasis is put on resolving the averaging kernel in the case of α = 10⁸. On the other hand, estimate (10) has a larger standard deviation than estimate (11) has since suppressing error magnification is less focused on in the case of α = 10⁸ than in the case of α = 10¹⁰. Thus, we clearly see the trade-off relation.

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In order to see the origins of the significant difference in the two estimates, we assessed the inversion coefficients (Figure 11) as in the previous subsection. In the case of α = 10⁸, the mixed-mode (l = 2) splitting kernels are mainly used for the averaging kernel. However, in the case of α = 10¹⁰, the p-mode (l = 1) splitting kernels are mostly used for the averaging kernel. The above difference should lead to the two equally reasonable estimates. We would like to discuss in more detail the cause of the two reasonable estimates in Section 5.3.

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4.1.3. The Estimate for the Target Point ($x,\mu$) = (0.95, 0.70)

Here one estimate is obtained with α = 10⁸ as follows:

$$\begin{eqnarray}&&\hat{{\rm{\Omega }}}(0.95,0.70)=(0.9564\pm 0.0006)\,{{\rm{\Omega }}}_{100}.\end{eqnarray} \tag{ 12 }$$

The weak dependence of the estimate on the trade-off parameter α is due to the small number of splitting kernels that have sensitivity in high-latitude regions; we do not have any other choice. In contrast to the case with the smaller value of α (Figure 12), we have failed in localizing the averaging kernel (Figure 13) when we increased the importance of error magnification by increasing α. By comparing estimate (12) with estimates (10) and (11), we find the possibility that there is latitudinally differential rotation in the outer envelope. We would like to expand our discussion on the existence of the latitudinally differential rotation in Section 5.4.

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We have carried out the three-zone modeling in order to test the possibility of a velocity shear around the core region (see Section 4.1.1), which has been suggested from the results obtained by the OLA method. In our three-zone modeling, we assume a constant angular velocity for each of three regions, namely, the innermost region (0 < x < x_a, 0 < μ < 1; region 1), the inner region (x_a < x < x_b, 0 < μ < 1; region 2), and the outer region (x_b < x < 1, 0 < μ < 1; region 3). The positions of boundaries, denoted by x_a and x_b, are treated as free parameters.

The linear integral Equation (2) can be expressed in a much simpler form as follows:

$$\begin{eqnarray}&&{d}_{i}=\displaystyle \sum _{j=1}^{3}{K}_{{ij}}{{\rm{\Omega }}}_{j}+{e}_{i},\,\,\,\,i=1,\,\ldots ,\,M,\,\,\,\,j=1,2,3,\end{eqnarray} \tag{ 13 }$$

where K_ij is defined as

$$\begin{eqnarray}&&{K}_{{ij}}\equiv \int {\int }_{j}{K}_{i}(x,\mu ){dxd}\mu ,\end{eqnarray} \tag{ 14 }$$

and the integration is carried out over region j.

Equation (13) can be rewritten in the following form:

$$\begin{eqnarray}&&{\boldsymbol{d}}={\boldsymbol{K}}{\boldsymbol{\Omega }}+{\boldsymbol{e}},\end{eqnarray} \tag{ 15 }$$

where ${\boldsymbol{K}}$ is a 23 × 3 matrix. The data, the angular velocities, and the errors are denoted by ${\boldsymbol{d}}$, ${\boldsymbol{\Omega }}$, and ${\boldsymbol{e}}$, respectively. The linear inverse problem (15) was relatively easy to solve in a least-squares sense because the rank of the observation matrix ${\boldsymbol{K}}$ is found to be full. We have repeated the inversion, changing the values of the three free parameters in the ranges as follows: 0.034 < x_a < 0.048 and 0.19 < x_b < 0.95.

The estimates of the internal rotation that makes the residual of the inversion minimum is as follows:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{1}=(5.57\pm 0.03){{\rm{\Omega }}}_{100},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{2}=(0.9348\pm 0.0001){{\rm{\Omega }}}_{100},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{3}=(1.0930\pm 0.0006){{\rm{\Omega }}}_{100},\end{eqnarray} \tag{ 18 }$$

which are obtained when x_a is 0.046 and x_b is 0.905 (see Figure 14). Interestingly, we have obtained a result that suggests that the innermost region rotates about six times faster than the other parts of the star. Moreover, the inner boundary x_a is almost identical to the edge of the convective core. This result indeed indicates the existence of the shear of the angular velocities, as suggested in the previous section. We also see that the deep radiative region (region 2) rotates slightly slower than the outer region (region 3), which is the same trend as indicated by the result of the two-zone modeling in Kurtz et al. (2014).

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The apparent discrepancy between estimates (8) and (9) can be fully explained provided that the fast-rotating core inferred from the three-zone modeling is valid. The key to understanding the discrepancy is the averaging kernel for each case. Figure 8 shows the inversion coefficients c_i for each case (α = 10⁸ and α = 10¹⁰), from which we see the contribution of each splitting kernel to the corresponding averaging kernel. As we have already mentioned in Section 4.1, the distributions of the averaging kernels are slightly different.

When we compare slices of the g-mode splitting kernels with those of the mixed modes, we find a difference between them (Figure 15), particularly for the mixed mode with (n, l) =(2, 2). The g-mode splitting kernels do not have sensitivity inside the convective core since g-modes are not propagative there. However, the mixed-mode splitting kernel with (n, l) = (2, 2) has, albeit rather small, sensitivity well below the edge of the convective core (x = 0.045), which has been inferred to rotate about 6 times faster than the other parts of the star.

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The discussion in the previous paragraph enables us to explain the reason why there exist two possible estimates. In the case of α = 10¹⁰, the g-mode splitting kernels are mainly used to obtain the averaging kernel, and because they do not have sensitivity inside the convective core, there is no contribution of the fast-rotating core to the slower estimate (9). On the other hand, in the case of α = 10⁸, the mixed-mode splitting kernel with (n, l) = (2, 2) is used, and because it has sensitivity inside the convective core, there is a contribution of the fast-rotating core to the faster estimate (8). Accordingly, we have selected estimate (9) as a final estimate of the internal rotation of the deep radiative region of the star. It is thus evident that the deep radiative region rotates slower than the envelope does because estimate (9) is significantly smaller than estimates (10)–(12).

In this section, we attempt to explain the origin of the seemingly contradictory estimates (10) and (11). We first look at the inversion coefficients of rotation inversion with the target point (x, μ) = (0.95, 0.00) (Figure 11). As we have already mentioned in Section 4.1, the p- or mixed-mode splitting kernels are mainly used for the averaging kernels in both cases (α = 10⁸ and α = 10¹⁰). Then, we repeated rotation inversion excluding one certain p- or mixed-mode rotational shift after another from the inversion procedure to figure out which mode is the most influential on estimation. What we found is that estimates that are hardly dependent on the trade-off parameter α can be obtained when we exclude the p-mode with (n, l) = (3, 1).

It is reasonable because the rotational shift of the mode is much smaller (∼0.0085 day⁻¹) than the other p-mode rotational shift (∼0.0097 day⁻¹), and the splitting kernel of the mode with (n, l) = (3, 1) has the highest peak near the target point x = 0.95 (Figure 16). Therefore, to increase the resolution of the averaging kernel that targets at (x, μ) = (0.95, 0.00), the splitting kernel with (n, l) = (3, 1) is likely to be used, leading to smaller estimates such as estimate (10).

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However, since the radial structures of p- or mixed-mode splitting kernels (Figure 16) are rather complex compared with those of the g-modes (Figure 15), it is difficult to describe the reason why the mode with (n, l) = (3, 1) is so influential on the estimation; the situation is not as simple as what we discuss in Section 5.2. We find an implication that the outer region between 0.83 < x < 0.9, where the splitting kernel with (n, l) = (3, 1) has greater sensitivity than the other kernels have, might rotate much slower than the other outer regions. This is because the latitudinal dependence of the mode with (n, l) = (3, 1) is completely the same as that of the mode with (n, l) = (2, 1), and thus the difference in the rotational shifts must derive from the difference in radial sensitivity of each splitting kernel. Note that we cannot decide whether estimate (10) is better than the other estimate (11) or not owing to the difficulty in interpreting the rotational shift of the mode with (n, l) = (3, 1).

So far, we have not been able to select one estimate from the two equally reasonable estimates (10) and (11) for the target point (x, μ) = (0.95, 0.00). Thus, we cannot tell whether the high-latitude region rotates faster than the low-latitude region does or not because the estimates are ordered as follows: (10) < (12) < (11). Actually, we probably obtain innumerable estimates by changing the value of α between 10⁸ and 10¹⁰; thus, it is possible that there exist estimates based on which we do not find any latitudinal dependence of the internal rotation in the outer envelope.

In order to check whether latitudinally differential rotation does exist or not, we focus on the three quintuplets that are identified as l = 2 (see Table 1), and we directly compute so-called "a-coefficients" of the three multiplets. The a-coefficients are coefficients that are used in an expanded expression of a frequency shift as follows:

$$\begin{eqnarray}&&{\omega }_{{nlm}}-{\omega }_{{nl}0}=\displaystyle \sum _{k}{a}_{k}(n,l){{ \mathcal P }}_{k}(m;l),\end{eqnarray} \tag{ 19 }$$

where k is the order of the expanding polynomial ${{ \mathcal P }}_{k}(m;l)$ (Ritzwoller & Lavely 1991). In the present paper, we followed the formulation of Schou et al. (1994). It is easily shown that the odd-order terms ${a}_{2k+1}$ correspond to rotational splitting by substituting expression (19) into definition (1). In particular, the sign of the a₃ coefficient represents the latitudinal dependence of the internal rotation; the positive (negative) value corresponds to the faster internal rotation in the low-latitude (high-latitude) region. From the observed rotational shifts, we computed three a₃ coefficients for the quintuplets (n, l) =(−1, 2), (0, 2), (2, 2) with the propagated standard deviations as below:

$$\begin{eqnarray*}&&{a}_{3}^{\mathrm{obs}}(-1,2)=(-6.7\pm 3.3)\times {10}^{-6}\ {{\rm{day}}}^{-1}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{a}_{3}^{\mathrm{obs}}(0,2)=(0.15\pm 1.3)\times {10}^{-6}\ {{\rm{day}}}^{-1}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{a}_{3}^{\mathrm{obs}}(2,2)=(-1.8\pm 8.6)\times {10}^{-6}\ {{\rm{day}}}^{-1}.\end{eqnarray*}$$

As we see, ${a}_{3}^{\mathrm{obs}}(-1,2)$ is smaller than zero with more than 2σ significance, suggesting that the high-latitude region rotates faster than the low-latitude region does. Nevertheless, both ${a}_{3}^{\mathrm{obs}}(0,2)$ and ${a}_{3}^{\mathrm{obs}}(2,2)$ are zero within the error bars. For assessing whether there is latitudinal dependence of the internal rotation or not based on all three a₃ coefficients, we carried out a simple statistical test. The null hypothesis we would like to reject is that a₃(n, l) = 0 for any multiplet, and we assume that the probability density function of each observed a₃ coefficient is distributed as Gaussian with the mean and the standard deviation equal to zero and its observational estimate, respectively. Then, we calculated the probability that we measure an a₃ coefficient whose absolute value is larger than that of the actually observed one, ${a}_{3}^{\mathrm{obs}}({\text{}}n,l)$, as follows:

$$\begin{eqnarray*}&&p(n,l)={\int }_{\mathrm{out}}\displaystyle \frac{1}{\sqrt{2\pi }\sigma (n,l)}\exp \left(-\displaystyle \frac{1}{2}\displaystyle \frac{{\left(x-0\right)}^{2}}{\sigma {\left(n,l\right)}^{2}}\right){dx},\end{eqnarray*}$$

where x is a dummy variable and σ(n, l) is an observational estimate of the standard deviation of ${a}_{3}^{\mathrm{obs}}({\text{}}n,l)$. The integration is carried out over the region $-\infty \lt x\lt -{a}_{3}^{\mathrm{obs}}({\text{}}n,l)$ or ${a}_{3}^{\mathrm{obs}}({\text{}}n,l)\lt x\lt \infty$. Note that ${a}_{3}^{\mathrm{obs}}({\text{}}n,l)$ represents the actual observed value here. Thus, the probability that we measure a set of a₃ coefficients {a₃(−1, 2), a₃(0, 2), a₃(2, 2)}, which satisfies the conditions

$$\begin{eqnarray*}&&\begin{array}{l}| {a}_{3}(-1,2)| \geqslant | {a}_{3}^{\mathrm{obs}}(-1,2)| \\ | {a}_{3}(0,2)| \geqslant | {a}_{3}^{\mathrm{obs}}(0,2)| \\ | {a}_{3}(2,2)| \geqslant | {a}_{3}^{\mathrm{obs}}(2,2)| ,\end{array}\end{eqnarray*}$$

is simply calculated as

$$\begin{eqnarray*}&&p=p(-1,2)\times p(0,2)\times p(2,2),\end{eqnarray*}$$

and we found that the value of p is 0.033. This is more than 2σ significance but less than 3σ significance, and thus we conclude that it is marginal to reject the null hypothesis and to claim that ${a}_{3}\ne 0$. Nevertheless, if we admit the existence of the latitudinally differential rotation, it is implied that the high-latitude region rotates faster than the low-latitude region does since the observed a₃ coefficients are mostly negative. This feature is sometimes called antisolar differential rotation (e.g., Brun et al. 2017). Therefore, we consider estimate (10) as a better choice than estimate (11); the former estimate is shown to be supported by the less model-dependent inference.

Aerts et al. (2017) summarized the studies on the internal rotation of main-sequence A- to F-type stars, and they concluded that almost all the stars investigated so far exhibited nearly rigid rotation. This is different from the result we have obtained through the present study; there can be a strong velocity shear between the convective core and the radiative region above. Our result, however, is actually compatible with their view because all the studies in Aerts et al. (2017) have estimated the "core" rotation based on g-modes, with which we cannot extract the information on the convective core. Thus, what they have inferred is the internal rotation of the "deep radiative" region above the convective core. Meanwhile, our estimation of "core" rotation is based on a mixed mode that has sensitivity inside the convective core. We have succeeded in extracting information on the convective core, which enables us to reveal the fast-core rotation of KIC 11145123. Furthermore, our results show that the radiative region of the star rotates nearly rigidly (see, e.g., estimates (17) and (18)). This result is consistent with the current understanding of the internal rotation of A–F stars.

From the theoretical point of view, there have been a series of numerical simulations of the dynamo mechanism inside the convective core of A-type stars (e.g., Browning et al. 2004; Brun et al. 2005; Featherstone et al. 2009), where internal differential rotation has been also calculated. In particular, though they focused more on the magnetism of A stars, one of their studies (Browning et al. 2004) reproduced an internal rotation profile in which the convective core rotates a few times faster than the radiative region above, which is similar to what we have found in this study. The fast-core rotation in an A star might be theoretically feasible.

Finally, we would like to discuss the model dependence of the inferred fast-core rotation. As we have shown in Section 5.2, our inference of the fast-core rotation is strongly dependent on sensitivity of the mixed-mode splitting kernel with (n, l) = (2, 2). However, we have found that the fast-core rotation is inferred even when we exclude the mode (2, 2) in the three-zone modeling inversion, suggesting that the other mixed modes with l = 2 also sense inside the convective core. Thus, the point is whether the modes identified as mixed modes are really mixed modes or not. When we look at the propagation diagram (Figure 17), we see that the Brunt–Väisälä frequency should be smaller in order for the three l = 2 modes to be identified as not mixed modes because we need a broad evanescent region between the g-mode cavity and the p-mode cavity. But the Brunt–Väisälä frequency is well constrained by the observed value of the mean g-mode period spacing (Kurtz et al. 2014), and there is little room for modifying it. Thus, it is likely that the modes in the frequency range are mixed modes, leading to the robustness of the inferred fast-core rotation.

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We would like to provide some physical considerations concerning the inferred internal rotation. One is the internal rotation of the radiative region of the star where the envelope rotates slightly faster than the deep radiative region does. This trend was first found by Kurtz et al. (2014), and we have confirmed it in a 2D way (see Section 5.2). It is apparently peculiar that the envelope rotates faster than the deep radiative region does since the angular momentum redistribution by viscosity alone cannot produce such an internal rotation profile. Rogers (2015) has carried out 2D numerical simulations of angular momentum transfer by internal gravity waves and discussed that it is possible that such a mechanism makes the outer envelope rotate faster than the inner region, which is the case for KIC 11145123. Benomar et al. (2015), however, did not find any evidence for such a reversed trend of stellar rotation via the measurements of radial differential rotation of solar-like stars. Then, another possible hypothesis is that the star has experienced some interactions such as mass accretion or stellar collision with other stars during the evolution and has obtained angular momentum from the outside, leading to the spin-up of the outer envelope. Actually, Takada-Hidai et al. (2017) have shown that the star is spectroscopically a blue straggler, which is thought to be born via interactions with other stars; their results support the above explanation. If this is really the case, our results could be a constraint on the theory of stellar interactions in a binary system.

The other is a strong shear of the angular velocity at the boundary between the convective core and the radiative region above. It is usually expected that such a velocity shear causes instabilities between the two boundaries and leads to an extra mixing there. Interestingly enough, fine-tuning of the model of the star suggests that the atomic diffusion processes in the deep radiative region should be much weaker than usually thought (Y. Hatta, et al. 2019, in preparation) in typical stellar evolution codes; the extra mixing caused by the shear could counteract atomic diffusion processes. This mechanism is rather similar to that occurring at the solar tachocline.