First Results from the Hertzsprung SONG Telescope: Asteroseismology of the G5 Subgiant Star μ Herculis^*

Because of its brightness, the basic parameters for μ Her have been determined in many studies. The most recent publication is the 2016 version of the PASTEL catalog (Soubiran et al. 2016), which also summarizes nearly all literature values.

The reported effective temperature determinations range from 5397¹² to 5650 K, $\mathrm{log}g$ from 3.7 to 4.1 and [Fe/H] values between +0.04 and +0.3. Most of these studies employed "standard" 1D–LTE analysis of high-resolution, high signal-to-noise spectra to determine these parameters and arrived at slightly different conclusions. We do not have a quantitative way to decide which values are the best. We therefore adopted the most recent parameters (Jofré et al. 2015, hereafter J15) and list them in Table 1. To reflect that this choice is a compromise, we assigned larger uncertainties than reported by J15. Specifically, Bruntt et al. (2010) have discussed the accuracy of the determination of stellar temperature, gravity, and [Fe/H] and concluded that realistic error bars for these quantities are 80 K, 0.08 dex, and 0.07 dex, respectively. We have adopted these values here. Finally, J15 also determined $v\,\sin i=1.7$ km s⁻¹, which is in accordance with expectations for an old, slightly evolved low-mass star.

Table 1.  Classical Parameters for μ Her

Parameter	Value	Uncertainty	Reference
${T}_{\mathrm{eff}}$ [K]	5560	80	J15, our uncertainty
$[\mathrm{Fe}/{\rm{H}}]$ [dex]	0.28	0.07	J15, our uncertainty
$\mathrm{log}g$ [dex]	3.98	0.10	J15, our uncertainty
$v\,\sin \,i$ [km s−1]	1.7	0.4	J15
Parallax [mas]	120.33	0.16	van Leeuwen (2007)
${\theta }_{\mathrm{LD}}$[mas]	1.93	0.03	Derived here
$R/{R}_{\odot }$	1.73	0.02	Derived here
$L/{L}_{\odot }$	2.54	0.08	Derived here
MV	3.82	0.03	Derived here
System velocity [km s−1]	−17.07	0.12	SIMBAD
log${R}_{\mathrm{HK}}^{\prime }$	−5.1	0.1	Isaacson & Fischer (2010)

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To estimate the luminosity, we used the measured $V=3.42$ (Bessell 2000), the Hipparcos parallax (120.33 ± 0.16 mas), and assumed ${A}_{V}=0$, which yielded M_V = 3.82. The bolometric correction was calculated using Equation (9) from Torres (2010). We used the Casagrande & VandenBerg (2014) V filter bolometric corrections and found −0086 and −0068 for μ Her and the Sun, respectively. Based on these values, we determined $L=2.54\pm 0.08\,{L}_{\odot }$.

The radius can be determined from angular-diameter measurements. Observations of μ Her were recently made with the Precision Astronomical Visual Observations beam combiner (Ireland et al. 2008) at the Center for High Angular Resolution Astronomy Array (ten Brummelaar et al. 2005). A fit of a uniform-disk model to these observations resulted in a uniform-disk diameter of ${\theta }_{\mathrm{UD}}=1.821\pm 0.018$ mas (I. Karovicova et al. 2017, in preparation). We determined a linear limb-darkening coefficient in the R band (0.60 ± 0.04) by interpolating the model grids of Claret & Bloemen (2011) to the spectroscopic values of T_eff, $\mathrm{log}g$, and [Fe/H]. The subsequent limb-darkened diameter is determined to be ${\theta }_{\mathrm{LD}}=1.93\pm 0.02$ mas. Using the parallax, this translates to a radius of $R=1.73\pm 0.02{R}_{\odot }$.

While angular diameters are often used to determine effective stellar temperatures, we have opted not to do this here because we have found three independent literature values for the bolometric flux that differ by 25%, which makes it problematic to select the correct value (Mozurkewich et al. 2003; Boyajian et al. 2013; Baines et al. 2014). We note that if we adopt the luminosity from photometry, our interferometric radius and neglecting the uncertainty in the parallax, the inferred temperature for μ Her becomes 5540 ± 80 K, which is fully consistent with the adopted spectroscopic temperature.

There are only two reports on the activity level for μ Her, based on the Ca HK lines: Wright et al. (2004) reported $\mathrm{log}\,$R${}_{\mathrm{HK}}=-5.11$ and Isaacson & Fischer (2010) found $\mathrm{log}\,$R${}_{\mathrm{HK}}=-5.08$. Both values are lower than the level found for the Sun, suggesting that μ Her is a rather inactive star, consistent with its evolutionary stage. This is, however, contradicted by the newly released measurements by the Mount Wilson Observatory HK Project.¹³ These measurements of μ Her indicate an abrupt change in the S index from 0.14 to more than 0.3. At this stage, it is impossible to conclude whether this jump is of stellar origin, and only additional data can solve this ambiguity.

Roberts et al. (2016) provided a detailed summary of the quadruple nature of μ Her. All other components of the system are M-type dwarfs. Interestingly, the inner pair (μ Her and one of the M-dwarfs) of the system has an orbital inclination of $63^\circ \pm 5^\circ$ (Roberts et al. 2016), which agrees very well with the inclination of the μ Her rotation axis determined from our seismic measurements (see Section 7). From the analysis of published radial-velocity and astrometric measurements, Roberts et al. (2016) determined an orbital period of ∼100 years and concluded that this pair is currently close to the lower inflection point of the radial-velocity curve. We expect to cover this portion of the orbit with SONG radial-velocity measurements in the coming years.

The power spectrum of the μ Her time series was calculated as a weighted fit of sinusoids, following the algorithms described by Frandsen et al. (1995) and Handberg (2013). We calculated power spectra separately for the 2014 and 2015 series, and then combined them into one power spectrum as a weighted average based on their mean noise levels. The relative weights were 42% and 58% for the 2014 and 2015 data, respectively. The individual and combined power spectra are shown in Figure 3.

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The noise level in the combined power spectrum corresponds to 2 cm s⁻¹ in amplitude at a frequency of 3000 μHz, which translates to 19.4 cm² s⁻² μHz⁻¹ in power density. This is similar to the noise levels in the α Cen A and B time series data (see Bedding et al. 2004; Butler et al. 2004; Kjeldsen et al. 2005). For example, the noise level in amplitude for α Cen B was 1.4 cm s⁻¹ at 7000 μHz, but close to 2 cm s⁻¹ at 3000 μHz. Thus the 1 m SONG telescope and spectrograph has achieved a noise level in μ Her over the 200 nights that is comparable to that achieved with the 8 m VLT and 4 m AAT over nine nights in a star that is seven times brighter.

Extraction of mode frequencies was done in the combined power spectrum. A large number of p-modes are clearly present, especially near the maximum power at 1200 μHz. However, the single-site data result in a complicated spectral window with strong sidelobes. The spectral windows for the 2014 and 2015 data are shown in Figures 4 and 5.

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As a next step, we determined the large frequency separation. Figure 6 shows the autocorrelation of the power spectrum after smoothing with a Gaussian of FWHM 0.5 μHz, for frequency shifts between 0 and 100 μHz. The peaks at 11.6 and 23.1 μHz correspond to 1 and 2 cycles per day, respectively, arising from the daily gaps. We can identify the large frequency separation of μ Her as ${\rm{\Delta }}\nu =64\,\mu \mathrm{Hz}$. This value for ${\rm{\Delta }}\nu$ agrees with the prediction by Bedding et al. (1996), which was based on their estimates of the mass and radius of the star. It is also consistent with the observed value of ${\nu }_{\max }=1200\,\mu \mathrm{Hz}$. Our measurement disagrees with the value of 56.5 μHz determined by Bonanno et al. (2008) based on seven nights of radial-velocity measurements. However, we note that in their Figure 3, which is a comb-response function of their power spectrum (analogous to an autocorrelation), there is a secondary peak close to 64 μHz. The incorrect determination of ${\rm{\Delta }}\nu$ is most likely due to the short time span of the observations and the confusion with the daily sidebands.

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Figure 7 illustrates the observed power spectrum in échelle format, where we see clear vertical ridges corresponding to modes with different degrees. We tested other values of ${\rm{\Delta }}\nu$ and found that none gave the same clear structure of vertical ridges. Due to the single-site nature of the data, the first and second daily sidelobes are prominent. To locate the individual oscillation modes, we constructed a folded power spectrum using the following procedure. We first smoothed the power spectrum by using a Gaussian function with a FWHM of 1 μHz. This was then folded with a spacing of 64 μHz between 976 μHz and 1424 μHz (7 radial orders, centered at 1200 μHz). The resulting folded power spectrum is shown in Figure 8, where the positions of modes of different degrees (l = 0, 1 and 2) can be seen. We used the peaks identified in Figure 8 to estimate the parameters in the asymptotic relation (Tassoul 1980; Scherrer et al. 1983; Christensen-Dalsgaard 1988):

$$\begin{eqnarray}&&\nu (n,l)\approx {\rm{\Delta }}\nu \left(n+\tfrac{1}{2}l+\epsilon \right)-l(l+1){D}_{0}.\end{eqnarray} \tag{ 2 }$$

We found ${\rm{\Delta }}\nu =64.2\,\mu \mathrm{Hz}$, D₀ = 0.80 μHz, and $\epsilon =1.44$. Note that this value of is consistent with expectations for a star with the effective temperature of μ Her (White et al. 2012).

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Using Equation (2), we estimated the expected frequencies of the individual p-modes and identified them in the power spectrum. Thanks to the high data quality, we also detected five l = 3 modes in the range of 1100–1400 μHz, where the S/N is highest. Additionally, one bumped l = 1 mixed mode is apparently present at low frequencies.

In total, we identified 49 probable modes, shown as filled symbols in the right panel of Figure 7, superimposed on a smoothed version of the observed power spectrum. The open symbols show the first and second daily sidelobes on either side of each mode. It is remarkable that none of the sidelobes coincide with other p-modes or their sidelobes. It is extremely fortunate that the single-site spectral window has little impact on our efforts to identify and measure the oscillation modes. Indeed, it seems that its frequency spacings make μ Her an ideal target for single-site observations (see Arentoft et al. 2014, for a discussion of SONG's spectral window and its influence on choice of targets).

We estimated uncertainties in the frequencies based on their S/N using a procedure similar to Kjeldsen et al. (2005, Section 4). These frequencies and their uncertainties were used as input for the Markov Chain Monte-Carlo (MCMC) frequency extraction described in Section 5.4. The identified modes in the central part of the spectrum, together with their daily sidelobes, are shown in Figure 9.

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To determine the frequency of maximum power (${\nu }_{\max }$) and the peak oscillation amplitude (${A}_{\mathrm{osc}}$) for μ Her, we followed the procedure described in Section 3.2 of Kjeldsen et al. (2008). This involves smoothing the power spectrum to estimate the total power in the oscillations in a manner that is insensitive to the spectral window.

We found the following values: ${\nu }_{\max }=1216\pm 11\,\mu \mathrm{Hz}$ and ${A}_{\mathrm{osc}}=38.9\pm 1.2\,\mathrm{cm}\,{{\rm{s}}}^{-1}$. Note that this velocity amplitude corresponds to radial modes and is 2.08 ± 0.10 times the mean solar value (see Kjeldsen et al. 2008 for details).

Interestingly, the oscillation amplitude for μ Her decreased significantly from 2014 to 2015. This is clearly seen in Figure 3. Analyzing the two power spectra separately, as described above, showed the peak amplitude to be $41.6\pm 1.7\,\mathrm{cm}\,{{\rm{s}}}^{-1}$ in 2014 and $36.1\pm 1.5\,\mathrm{cm}\,{{\rm{s}}}^{-1}$ in 2015.

The next step was to measure parameters for the 49 individual modes using an MCMC analysis. In order to use the full time span of the measurements we constructed a full time series using all the available data from the two observing runs. The two time series were concatenated, but the gap between them was reduced to 80 days. This can be justified by the fact that we are searching for stochastic oscillations where the mode lifetime is significantly shorter than 80 days. In this way we ensure that any oscillations from the 2014 data set will have disappeared and do not affect the 2015 data set. This concatenation creates a better window function. The final power density spectrum and corresponding spectral window function were then calculated from the time series specified above, following the prescriptions outlined in Section 4.

5.4.1. MCMC Peakbagging

The fit to the power spectrum was performed using the APT MCMC algorithm (Handberg & Campante 2011) and the preliminary frequencies determined in Section 5.2 were used as starting guesses. We ran four million iterations, which were subsequently thinned to two million, using 10 parallel tempering levels to avoid local maxima solutions. The model limit spectrum that was fitted to the observed power spectrum was defined as

$$\begin{eqnarray}&&{\mathscr{P}}(\nu )=\eta (\nu )\displaystyle \sum _{n,l}\displaystyle \sum _{m=-l}^{l}\displaystyle \frac{{H}_{{nl}}{{\mathscr{E}}}_{{lm}}(i)}{1+\tfrac{4}{{{\rm{\Gamma }}}_{{nl}}^{2}}{(\nu -{\nu }_{{nl}}-m\delta {\nu }_{s})}^{2}}+N(\nu )\,\end{eqnarray} \tag{ 3 }$$

where ${\nu }_{{nl}}$ is the mode frequency, H_nl is the mode height, ${{\rm{\Gamma }}}_{{nl}}$ is the linewidth (which is inversely proportional to the mode lifetime), and $\delta {\nu }_{s}$ is the rotational splitting. The factor $\eta (\nu )\equiv {\mathrm{sinc}}^{2}({\rm{\Delta }}{T}_{\mathrm{int}}\nu )$ is the attenuation of signals arising from the non-zero integration time (${\rm{\Delta }}{T}_{\mathrm{int}}$). The noise model $N(\nu )$ was simply a white-noise profile across the region of interest. In order to limit the number of free parameters, H_nl and ${{\rm{\Gamma }}}_{{nl}}$ were linearly interpolated in frequency between H_n0 and ${{\rm{\Gamma }}}_{n0}$, respectively, and the height was scaled with the visibility of the mode (see Handberg & Campante 2011). The relative heights of rotationally split components within a multiplet were taken as (Gizon & Solanki 2003)

$$\begin{eqnarray}&&{{\mathscr{E}}}_{{lm}}(i)=\displaystyle \frac{(l-| m| )!}{(l+| m| )!}\{{P}_{l}^{| m| }(\cos i){\}}^{2}\,\end{eqnarray} \tag{ 4 }$$

where ${P}_{l}^{m}(x)$ are the associated Legendre functions.

Instead of using the mode height, H_nl, directly as the free parameter in the fit, the mode amplitude was used. This is less correlated with the linewidth, ${{\rm{\Gamma }}}_{{nl}}$, and therefore provides a more stable fit. The conversion from amplitude to height was done following Fletcher et al. (2006), which allows for linewidths becoming comparable to the frequency resolution. Similarly, the projected rotational splitting, ${\nu }_{s}\sin i$, was used as the free parameter instead of the rotational splitting itself, to avoid known correlations.

Uniform priors were set for mode frequencies, the rotational splitting and the inclination angle, whereas modified Jeffreys priors were used for mode heights and linewidths.

In order to account for the single-site window function, the model spectrum, $P(\nu )$, was convolved with the spectral window in each iteration of the MCMC. This has a very significant impact on the computing time, but is essential in order to describe the spread of power to sidelobes due to the non-continuous observations.

From the resulting Markov chain, the final parameters and errors listed in Table 2 were estimated from the full posterior probability distributions as the median values and 68.3% confidence interval. The final mode frequencies (with uncertainties) were corrected for the systemic radial-velocity Doppler shift (${v}_{\mathrm{rad}}=-17.07\pm 0.12$ km s⁻¹) in order to list the frequencies in the rest frame of the star (Davies et al. 2014).

Table 2.  Frequencies (μHz) for Individual Oscillation Modes Extracted from the MCMC Analysis, Listed in échelle Format (see Figure 7)

n	l = 2	l = 0	l = 3	l = 1
24	${1636.68}_{-0.45}^{+0.30}$	⋯	⋯	${1669.55}_{-0.37}^{+0.27}$
23	⋯	⋯	⋯	${1599.67}_{-0.22}^{+0.68}$
22	${1501.34}_{-0.42}^{+0.20}$	${1505.04}_{-0.72}^{+0.21}$	⋯	${1534.83}_{-0.30}^{+0.72}$
21	${1436.28}_{-0.10}^{+0.05}$	${1440.74}_{-0.05}^{+0.02}$	⋯	⋯
20	${1371.87}_{-0.33}^{+0.24}$	${1376.41}_{-0.14}^{+0.10}$	${1398.21}_{-0.43}^{+0.53}$	${1404.15}_{-0.17}^{+0.12}$
19	${1307.24}_{-0.12}^{+0.13}$	${1311.81}_{-0.13}^{+0.08}$	${1334.13}_{-0.25}^{+0.52}$	${1339.85}_{-0.10}^{+0.10}$
18	${1243.05}_{-0.06}^{+0.07}$	${1247.89}_{-0.04}^{+0.04}$	${1268.56}_{-0.27}^{+0.11}$	${1274.93}_{-0.06}^{+0.05}$
17	${1178.69}_{-0.09}^{+0.11}$	${1183.20}_{-0.05}^{+0.05}$	${1203.74}_{-0.26}^{+0.37}$	${1211.12}_{-0.06}^{+0.05}$
16	${1113.04}_{-0.07}^{+0.05}$	${1119.03}_{-0.07}^{+0.06}$	${1139.15}_{-0.18}^{+0.57}$	${1147.38}_{-0.04}^{+0.04}$
15	${1049.94}_{-0.11}^{+0.21}$	${1054.90}_{-0.06}^{+0.06}$	⋯	${1083.81}_{-0.06}^{+0.06}$
14	${986.14}_{-0.09}^{+0.12}$	${991.75}_{-0.14}^{+0.11}$	⋯	${1021.14}_{-0.15}^{+0.13}$
13	${922.93}_{-0.12}^{+0.08}$	${928.64}_{-0.06}^{+0.06}$	⋯	${958.75}_{-0.17}^{+0.17}$
12	${858.51}_{-0.04}^{+0.13}$	${865.34}_{-0.12}^{+0.10}$	⋯	${903.95}_{-0.07}^{+0.08}$
11	${795.49}_{-0.45}^{+0.02}$	${801.42}_{-0.49}^{+0.06}$	⋯	${824.11}_{-0.03}^{+0.26}$
10	${731.32}_{-0.15}^{+0.15}$	${737.47}_{-0.21}^{+0.27}$	⋯	${766.27}_{-0.02}^{+0.03}$
9	${668.56}_{-0.05}^{+0.19}$	${676.76}_{-0.03}^{+0.02}$	⋯	${702.89}_{-0.08}^{+0.08}$

Note. Note that n corresponds to the radial order for the l = 0 modes.

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Finally, we calculated the frequency-separation ratio as defined by Roxburgh & Vorontsov (2003), which are used in the following sections for modeling of the observations:

$$\begin{eqnarray}&&{r}_{01}(n)=\displaystyle \frac{1}{8}\displaystyle \frac{{\nu }_{n-\mathrm{1,0}}-4{\nu }_{n-\mathrm{1,1}}+6{\nu }_{n,0}-4{\nu }_{n,1}+{\nu }_{n+\mathrm{1,0}}}{{\nu }_{n,1}-{\nu }_{n-\mathrm{1,1}}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{r}_{02}(n)=\displaystyle \frac{{\nu }_{n,0}-{\nu }_{n-\mathrm{1,2}}}{{\nu }_{n,1}-{\nu }_{n-\mathrm{1,1}}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{r}_{10}(n)=\displaystyle \frac{-1}{8}\displaystyle \frac{{\nu }_{n-\mathrm{1,1}}-4{\nu }_{n,0}+6{\nu }_{n,1}-4{\nu }_{n+\mathrm{1,0}}+{\nu }_{n+\mathrm{1,1}}}{{\nu }_{n+\mathrm{1,0}}-{\nu }_{n,0}}.\end{eqnarray} \tag{ 7 }$$

These were calculated using the full Markov chains for each frequency coming from the MCMC analysis, yielding the full correlation matrices between all ratios.

From the MCMC analysis, we were also able to constrain the rotational splitting between the different m-components of the l = 1 multiplets. We also measured the stellar inclination angle, based on the relative heights of these m-components (Gizon & Solanki 2003; see Figures 10 and 11). The resulting rotational period is ${P}_{\mathrm{rot}}={52}_{-1}^{+3}$ days and the stellar rotational inclination angle is $i={63}_{-10}^{+9}$ degrees. Both values and their errors were determined as the mode values and 68.3% confidence intervals in Figures 10 and 11.

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We fitted the 49 frequencies in Table 2 and their ratios following procedures described by Silva Aguirre et al. (2015), in this case taking into account the presence of the mixed modes. One fit was applied to the individual frequencies, which were assumed to be statistically independent. A grid of models and oscillation frequencies was calculated using the ASTEC stellar evolution code (Christensen-Dalsgaard 2008a) and the ADIPLS adiabatic pulsation code (Christensen-Dalsgaard 2008b). The evolution modeling used the OPAL equation of state (Rogers & Nayfonov 2002) and OPAL opacities (Iglesias & Rogers 1996), supplemented by the Ferguson et al. (2005) low-temperature opacities. The nuclear reaction rates were obtained from the NACRE compilation (Angulo et al. 1999). Diffusion and settling of helium and heavy elements were not included. Convection was described using the mixing-length formalism (Böhm-Vitense 1958), and convective overshoot was not included. The grid spanned a large range in mass and composition, though was constrained by an assumed Galactic chemical evolution model with ${\rm{\Delta }}Y/{\rm{\Delta }}Z=1.4$, where Y and Z are the abundances of helium and heavy elements, respectively. Models with three values of the mixing-length parameter ${\alpha }_{\mathrm{ML}}$, 1.5, 1.8, and 2.1, were included, where ${\alpha }_{\mathrm{ML}}=1.8$ roughly corresponds to the solar calibration.

To match the observed frequencies, the computed frequencies were corrected for the errors introduced by the treatment of the near-surface layers by applying a fitted scaled solar surface correction (Christensen-Dalsgaard 2012), described in more detail, together with other aspects of this so-called ASTFIT fitting technique, by Silva Aguirre et al. (2015). Briefly, the fit is carried out by minimizing, along each evolution sequence,

$$\begin{eqnarray}&&{\chi }^{2}={\chi }_{\mathrm{spec}}^{2}+{\chi }_{\nu }^{2}.\end{eqnarray} \tag{ 8 }$$

Here, ${\chi }_{\mathrm{spec}}^{2}$ is based on observed values of ${T}_{\mathrm{eff}}$ and [Fe/H] (see Table 1), and

$$\begin{eqnarray}&&{\chi }_{\nu }^{2}=\displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N}{\left(\displaystyle \frac{{\nu }_{i}^{(\mathrm{obs})}-{\nu }_{i}^{(\mathrm{mod})}}{{\sigma }_{i}}\right)}^{2}\end{eqnarray} \tag{ 9 }$$

is based on the observed frequencies ${\nu }_{i}^{(\mathrm{obs})}$ and standard deviations ${\sigma }_{i}$ listed in Table 2. The remaining observed properties were not included in the fit but were used to check the results. The model frequencies, ${\nu }_{i}^{(\mathrm{mod})}$, included the surface correction (see above). An initial minimization was carried out between timesteps in the evolution sequence by assuming that the frequencies scale as ${R}^{-3/2}$. This defined a minimum ${\chi }_{\min }^{2}$ for each evolution track in the grid. The best-fitting models were found by locating the smallest resulting values of ${\chi }_{\min }^{2}$.

Owing to the presence of mixed modes, the ${R}^{-3/2}$ scaling of the frequencies is not universally valid, leading to potential systematic errors in the fits. To correct for this, the fit was refined by computing, in the vicinity of the minima determined by the above scaling procedure, frequencies for a small set of models suitably interpolated between timesteps in the evolution sequence. As shown by Christensen-Dalsgaard & Houdek (2010), this allows us to fully resolve the behavior of the frequencies in the vicinity of an avoided crossing involving mixed modes. In practice, this was applied only to evolution tracks where the ${\chi }_{\min }^{2}$ as determined by the simple procedure was less than twice the minimum among the values of ${\chi }_{\min }^{2}$ so determined.

The stars analyzed by Silva Aguirre et al. (2015) were all on the main sequence and the observed modes were purely acoustic. In contrast, μ Her is a subgiant with clearly identified mixed modes (Figure 12), and the relevant models also have several mixed modes. This complicates the identification of the observed modes with those of the models in the grid. We have applied a relatively simple technique to identify the relevant model modes in cases with mixed modes, taking into account that the present observations show only one nonradial mode of each degree in each interval between two adjacent radial modes. Thus in each radial-mode interval, we chose (with an exception noted below) the frequency of a given degree that minimized the normalized inertia

$$\begin{eqnarray}&&{Q}_{{nl}}=\displaystyle \frac{{E}_{{nl}}}{{\bar{E}}_{0}({\nu }_{{nl}})},\end{eqnarray} \tag{ 10 }$$

where E_nl is the inertia of the mode and ${\bar{E}}_{0}({\nu }_{{nl}})$ is the radial-mode inertia, interpolated logarithmically to the frequency ${\nu }_{{nl}}$ of the given mode. The underlying assumption is that this is the mode most likely to be observed.

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For l = 2 and 3, there was typically a clear minimum of Q_nl among the relevant modes, and the above procedure produced a reasonable fit. For l = 1, however, there may be two modes in a given radial-mode interval with comparable values of Q_nl, and there is a risk that the selected mode does not provide the optimal fit to the observations. To circumvent this problem, the procedure was modified by including in the minimization a suitably weighted measure of the distance to the nearest observed dipolar mode. Although fairly crude, this method yielded a reasonable behavior of the fit along the evolution tracks.

As applied by Silva Aguirre et al. (2015), ASTFIT determined likelihood-weighted averages of the various stellar parameters. In the present case, we have found that ${\chi }_{\nu }^{2}$ (see Equation (9)) is dominated by a few modes, particularly the dipolar mode undergoing avoided crossing, and hence this statistical procedure has little meaning (see also Figure 13). For this preliminary analysis, we therefore simply considered a few examples of optimized fits for representative selected evolution tracks, chosen to yield values of ${T}_{\mathrm{eff}}$ and [Fe/H] within $2\sigma$ of the observed values and ${\chi }_{\nu }^{2}$ near its minimum value. These are listed in Table 3. Two examples, with masses of $1.12\,{M}_{\odot }$ and $1.15\,{M}_{\odot }$, are shown in the échelle diagram in Figure 12. Figure 13 shows the resulting frequency differences for the $1.12\,{M}_{\odot }$ model, compared to the fitted surface function.

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Table 3.  Results of Model Fits

Model	M	R	L	${T}_{\mathrm{eff}}$	[Fe/H]	Age
	$(\,{M}_{\odot })$	$(\,{R}_{\odot })$	$(\,{L}_{\odot })$	(K)	(dex)	(Gyr)
ASTFIT1	1.12	1.71	2.7	5650	0.26	7.6
ASTFIT2	1.15	1.73	2.6	5600	0.30	7.9
BASTA	${1.11}_{-0.01}^{+0.01}$	${1.71}_{-0.02}^{+0.01}$	${2.6}_{-0.1}^{+0.1}$	${5600}_{-50}^{+50}$	${0.21}_{-0.06}^{+0.06}$	${7.8}_{-0.4}^{+0.3}$
From Table 1		1.73 ± 0.02	2.54 ± 0.08	5560 ± 80	0.28 ± 0.07

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To determine the uncertainties in the stellar properties, we also fitted combinations of p-mode dominated frequencies using the BAyesian STellar Algorithm (BASTA, see Silva Aguirre et al. 2015). Briefly, this Bayesian approach relies on a large grid of stellar models to determine the probability density function of a given stellar property based on the fit to a set of observational quantities. In this case, we considered the spectroscopic constraints ${T}_{\mathrm{eff}}$ and [Fe/H] and the frequency-separation ratios r₀₁ and r₁₀ above 1000 μHz (to avoid the impact of the mixed modes in the fit) as the input parameters to be reproduced. We report in Table 3 the median and the 16 and 84 percentiles of the posterior probability density function. The results are in excellent agreement with those obtained with ASTFIT, as well as with the independent radius determination from interferometry.

Using the effective temperature, large frequency separation, and [Fe/H] (see Table 4) as inputs, we also calculated the stellar parameters using the Asteroseismology Made Easy (AME Lundkvist et al. 2014) grid-based method and found the values to be in agreement with those listed in Table 3.

Table 4.  Summary of Results

Parameter	Value	Comment
T${}_{\mathrm{eff}}$[K]	5560 ± 80	J15
$\mathrm{log}g$ [cgs]	3.98 ± 10	J15
$[\mathrm{Fe}/{\rm{H}}]$	0.28 ± 07	J15
$v\,\sin i$ [km s−1]	1.7 ± 0.4	J15
${\theta }_{\mathrm{LD}}$[mas]	1.93 ± 0.03	measured
R $[\,{R}_{\odot }]$	1.73 ± 0.02	angular diameter + parallax
L $[\,{L}_{\odot }]$	2.54 ± 0.08	assuming ${A}_{V}=0$
${\nu }_{\max }$ $[\mu \mathrm{Hz}]$	1216 ± 11	measured
${\rm{\Delta }}\nu$ $[\mu \mathrm{Hz}]$	64.2 ± 0.2	measured
	1.44	measured
i $[^\circ ]$	${63}_{-10}^{+9}$	measured
P${}_{\mathrm{rot}}$ [day]	${52}_{-1}^{+3}$	measured
age [Gyr]	${7.8}_{-0.4}^{+0.3}$	from model
M $[\,{M}_{\odot }]$	1.11 ± 0.01	from model
R $[\,{R}_{\odot }]$	1.72 ± 0.02	from model
L $[\,{L}_{\odot }]$	2.6 ± 0.1	from model
$\mathrm{log}g$ [cgs]	4.01 ± 0.01	from model
${\tau }_{{II}}$ [s]	1938	from model
${\tau }_{c}$ [s]	4488	from model

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Mode linewidths and amplitudes can be used to test models of stellar structure and stability. In particular, comparison between observations and models can be used to calibrate the parameters in the convection model used in the numerical stability analysis. In Figure 14, theoretical estimates of linear damping rates¹⁵ of radial modes are compared to the SONG observations. These computations were performed for global model parameters of the models ASTFIT1 and ASTFIT2 (see Table 3).

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The depth of the (surface) convection zone was calibrated to the values obtained from the seismic models ASTFIT1 and ASTFIT2 described in Section 6.1. The basic stability computations were as in Houdek (2006) using Gough's (1977a, 1977b) nonlocal, time-dependent convection model, but adopted for the stellar atmosphere a temperature—optical depth ($T\mbox{--}\tau$) relation from Trampedach et al. (2014) 3D hydrodynamical simulations. The agreement with the observations is reasonably good.

We also estimated the maximum value of the velocity amplitudes of the acoustic oscillations. Various excitation models have been used in the past to estimate amplitudes of stochastically excited oscillations (Goldreich & Keeley 1977; Balmforth 1992; Samadi & Goupil 2001; Chaplin et al. 2005; Houdek 2006). Here we adapt the scaling relation by Chaplin et al. (2011) for estimating the maximum velocity amplitude. Adopting the global parameters listed in Table 1, we estimate for μ Her a relative maximum velocity amplitude $V/{V}_{\odot }\simeq 1.83$ (${V}_{\odot }$ being the maximum solar velocity amplitude), which is in reasonable agreement with the observed value of 2.08 ± 0.10 discussed in Section 5.3. We should, however, note that the β function in Chaplin et al. (2011) Equation (7) is rather uncertain and will add to the uncertainty from the adopted effective temperature for μ Her. The predicted value of the velocity amplitude will therefore capture the uncertainties in both the observations and the scaling relation.

Abrupt variations in the sound speed, which are called acoustic glitches, produce seismic signatures in the spacing of the observed frequencies. From these seismic signatures, the locations of the abrupt variation (in terms of acoustic depth τ) can be estimated. Figure 15 displays observed second differences ${{\rm{\Delta }}}_{2}{\nu }_{n,l}:= {\nu }_{n-1,l}-2{\nu }_{n,l}+{\nu }_{n+1,l}$ of low-degree ($l=0,1,2$, symbols), together with results of the seismic diagnostic D₂ by Houdek & Gough (2007). This analysis adopts Airy functions for the pulsation eigenfunctions and glitches of both stages of helium ionization. We estimated the acoustic depths of the glitches brought about by the second stage of helium ionization, ${\tau }_{\mathrm{II}}$, and by the abrupt variation of the sound speed at the base of the convection zone, ${\tau }_{{\rm{c}}}$. We found ${\tau }_{\mathrm{II}}\simeq 1938\,{\rm{s}}$ and ${\tau }_{{\rm{c}}}\simeq 4488$ s. The depths of the acoustic glitches obtained directly from the equilibrium structures of the models listed in Table 3, agree with ${\tau }_{\mathrm{II}}$ to within 3% and for ${\tau }_{{\rm{c}}}$ to within 15%. For the present work, we did not perform an error analysis for the acoustic-glitch depths, but plan to conduct a Monte-Carlo error analysis in an upcoming paper.

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