© The Authors 2024

1. Introduction

In their paper on the discovery of the six-planet system around K0V-type star HD 110067, Luque et al. (2023) find a rather old age for the host star. They note the relevance of this find as an example of the long-term stability of a planetary system that compares in terms of complexity to our Solar System, which appears to have undergone dramatic changes of state in its first 800 million years or so, according to computational work on orbital stability with the Nice model (Tsiganis et al. 2005). Interestingly, Lammers & Winn (2024) modelled planetary systems with the properties of HD 110067, within the margins of the available data, and found stability beyond 25 Myr for only a few specific cases. They conclude that only a resonance chain between the six planets can maintain stability for much longer timescales (i.e. gigayears). This makes this planetary system a very interesting equivalent to the Solar System. The only question is whether it is comparable to the current Solar System or a past version.

The motivation for this work is this intrinsic age ambiguity because any evolution track matching a K0V star runs parallel to the main sequence (MS) for the first ≈8 Gyr. Here, therefore, we focus on the stellar activity level instead since, according to the Skumanich law (Skumanich 1972), it serves as an indicator of time.

We measured the S-index to characterise the chromospheric activity as it was introduced, together with a set of over 40 calibration stars, by Olin Wilson (see e.g. Baliunas et al. 1995). To obtain a more physical chromospheric activity indicator, we then calculated the excess flux R index, R_HK⁺, which puts the purely activity-related chromospheric emission in the Ca II H&K lines in proportion to the bolometric luminosity. The excess flux R index, R_HK⁺ is formed by the emission of faculae over active regions at the bottom of the chromosphere. Consequently, both the photospheric line core flux and the basal (non-activity-related) chromospheric fluxes have to be subtracted from the total measure in the 1 Å HK line core passbands. Their combined value is then divided by the bolometric flux of the star, while in the S-index, the denominator is given by the local UV fluxes in the adjacent radial velocity (RV) windows of 20 Å (see e.g. Rutten 1984 or Mittag et al. 2012). The conversion can be carried out following established recipes.

However, different stars have different spin-down timescales, and they differ in the ranges of their rotation periods, including the longest observed rotation periods of their least active specimens. This was empirically studied by Mittag et al. (2018). Their work derived empirical convective turnover times for MS stars, grouped into different B − V spans. If, however, rotation is represented by the respective empirical Rossby number, Ro, the activity measured in terms of R_HK⁺ decreases in all B − V groups of their Fig. 2 over a stretch of 0.1–1 in Ro. We therefore used the empirical Rossby number as a universal representation of the rotation and spin-down of MS stars. To account for different spin-down timescales, such as that of KO dwarfs in comparison to the Sun, we then used the parameterised description of Reiners & Mohanty (2012).

In addition, direct comparison with a virtually identical twin that has already been thoroughly studied provides a simpler, albeit more limited, alternative. We chose the K0V standard star σ Dra (HD 185144), for which we have good spectra obtained at the Telescopio Internacional de Guanajuato Robótico Espectroscópico (TIGRE) using the Heidelberg Extended Range Optical Spectrograph (HEROS) facility in Guanajuato, Mexico, a collaboration of the universities of Hamburg, Liege, and Guanajuato (see Schmitt et al. 2014). The spectra of HD 110067 and σ Dra look nearly identical, and even their luminosities (based on Gaia DR3 parallaxes) coincide within only 1% (M_V = 5.89 and 5.88, respectively). Not astonishingly, our spectroscopically derived physical parameters, using a synthetic spectrum comparison method with iSpec (see description by Rosas-Portilla et al. 2022), agree within their uncertainties.

σ Dra is well known for its pronounced magnetic activity cycle, and its Mount Wilson S-index varies around an average of 0.22, with extremes of 0.16 and 0.28 (see Gray et al. 1992; Baliunas et al. 1995). Ramírez et al. (2012) estimated a nominal age of 3 Gyr for this star based on the very weak presence of the lithium doublet at 6707.8 Å. We come back to this point below.

2. Persistent, strong chromospheric emission

FGK MS stars that have evolved into the second half of their central-hydrogen-burning lifetime all have a very modest chromospheric emission in Ca II H&K lines, between calibrated Mount Wilson S-values of 0.15–0.17 (see Schröder et al. 2013). Most of this emission is of a non-active nature.

This contribution to S, unrelated to activity, comprises the photospheric line core and the aforementioned minimal ‘basal’ chromospheric flux, observed in the absence of any active regions in even the least active stars. In the S-distribution of FGK MS stars over B − V, we therefore observe this fairly constant contribution as a sharp lower cutoff or envelope at 0.15 (see e.g. Mittag et al. 2012, Fig. 2 therein).

We used the TIGRE facility in Guanajuato, Mexico, with its HEROS spectrograph, to observe the Ca II H&K emission of HD 110067 and obtain a well-calibrated S-index. To our surprise, this star showed a high and sustained activity level at S = 0.32 with little variation over 6 weeks in early 2024 (see Fig. 1). The sum of the 78 spectra available to us – High Accuracy Radial velocity Planet Searcher in the Northern hemisphere (HARPS-N) spectra obtained at the Telescopio Nazionale Galileo (TNG) on the Canary Islands between 2021 and 2023 (see Fig. 2) – yields a nearly identical picture. Hence, deducting 0.15 for the non-active contribution to S, an activity-related value of 0.17 remains. For the Sun, where the S-value is, on average, between 0.18 and 0.19, that activity-related S-contribution is only 0.035. By comparison, the average S-index of σ Dra’s activity is 0.07.

Fig. 1. Three TIGRE-HEROS spectra of HD 110067 obtained between January and March 2024, summed together to reduce noise. They show a strong chromospheric emission in Ca II K (blue line), here directly compared to the modest emission of virtual twin σ Dra.

Fig. 2. Sum of the 78 HARPS-N spectra available to us, obtained for HD 110067 at the TNG on the Canary Islands from 2021 to 2023 (arbitrary flux over wavelength in Å). It shows a high chromospheric emission, identical to that in Fig. 1.

This striking observation raises the question of whether the large emission of HD 110067 can be understood in the context of a much younger age, or whether are there planet-host star interactions at play. Proof of such mechanism would indicate a stark discrepancy between the age related to the high level of activity observed and any other age indicator, that is, if the age deduced from its activity is much younger than what is derived from any other reasonable method . Hence, we also took a closer look at other, conventional age-deriving methods.

3. Physical parameters from spectroscopic analysis

When synthesising physical parameters, iSpec (see Blanco-Cuaresma 2019 and references therein) uses a set of non-blended, well-known lines of different ion species, most of which are not yet saturated to preserve the respective density information. This package compares those line strengths with a spectral library of atmospheric models. However, crosstalk between different parameters and multiple local χ² minima in the parameter space produces larger uncertainties than what individual solutions for best matches suggest. We therefore employed the well-tested, systematic, and script-driven iterative use of iSpec, as described by Rosas-Portilla et al. (2022), which uses high S/N TIGRE spectra of R = 20 000. Table 1 lists the parameters derived for HD 110067 and its twin, σ Dra.

Complementing iSpec results, the wings of the strongest lines, namely the Balmer series and Ca II K, contain valuable additional information about effective temperature and gravity. We used the sum of the 78 HARPS-N spectra to benefit from their very high spectroscopic resolution (R = 115 000) and optimum S/N (625 around H_α). We studied the line profiles of the strongest lines, including Ca I and iron lines, (i) by comparing them with the spectra of 400 stars with empirically well-known parameters using Specmatch-emp (reducing the HARPS-N spectral resolution to the 55 000 of the Specmatch library), and (ii) via a Spectroscopy Made Easy (SME) analysis (as developed and described by Piskunov & Valenti (2017).

While the set of T_eff, metallicity, and gravity values obtained with Specmatch agrees well with that of iSpec within the uncertainties, our SME line profile analysis demonstrates the same parameter crosstalk often seen in individual iSpec synthesising results. A gap also remains between the T_eff value derived from the SME model and the empirical T_eff derived from the Specmatch comparison with observational data. In absolute terms, therefore, the uncertainties we estimated for our adopted values (bottom line in Table 1) are larger than the nominal ones suggested by any tool. We also obtained an approximate value for the rotation of vsin(i)≈2.5 km/s from these line profile studies.

The other important parameter used to place a star in the Hertzsprung–Russell (HR) diagram and compare it with matching evolution tracks is its luminosity. Using the physical parameters derived by Luque et al. (2023) for HD 110067 (namely, T_eff and R; see their Table 3), we find a value for L of 0.431 L_⊙ (log L = −0.365). Using M_V = 5.89 and a bolometric correction (for B − V = 0.79) of −0.192 (from Flower 1996, their Table 3), and with M_Bol, ⊙ = 4.74, we arrive at log L = −0.38. We adopted log L = −0.37 ± 0.01 for our analysis.

4. Evolution tracks: The problem with K0 dwarfs

As noted above, in the mass range of K0V stars, evolution tracks run almost exactly parallel to the zero-age main sequence for their first ≈8 Gyr (see Fig. 3), which makes it impossible to conclude an age from the HR diagram position alone. However, we can at least state that our spectroscopically derived physical parameters are in good agreement with MESA evolution tracks (Choi et al. 2016) in the mass range 0.78–0.85 for a metallicity of [Fe/H] = 0.16, close to the one we have derived here. To illustrate this age ambiguity: for its luminosity of log L = −0.37, the 0.80 M_⊙ track matches HD 110067 at an age of 6.8 Gyr, while on the 0.85 M_⊙ track this star would only be 0.8 Gyr old.

Fig. 3. HR diagram positions of K0V stars HD 110067 and its virtual twin, σ Dra. They both coincide with e.g. a 0.8 Gyr-old model of a 0.85 M⊙ MESA evolution track and a 6.8-Gyr model on an evolution track with 0.80 M⊙, using here a metallicity of [Fe/H] = −0.16. Solid lines indicate stable, well-modelled phases, while dashed lines indicate transitions, uncertainties, or complex phenomena in the star’s evolution.

If both the mass and luminosity of the star were known independently to a precision of 1%, the age could be assessed with a satisfactory uncertainty of close to 1 Gyr. But despite a precise Gaia parallax, the luminosity remains less certain because of the bolometric correction. And for a precise, independent mass determination, planetary orbits need to be resolved in order to obtain semi-major axes and/or RVs, not only periods. Hence, the dilemma of finding ages for K0 dwarf stars from matching evolution tracks is intrinsic. A more feasible solution could be oscillation analysis from extensive, high-precision photometric monitoring, where available, to provide a very precise value for gravity, which in spectroscopy remains somewhat uncertain.

5. Does lithium resolve the age question?

Ramírez et al. (2012) gave a nominal lithium age for sigma Dra of 3 Gyr, without specifying their equivalent width (EW) measurement. High S/N (∼300) TIGRE spectra give a 1 mÅ upper limit to the lithium 6707.85 Å doublet in σ Dra, in agreement with the upper limit determined by Takeda & Kawanomoto (2005). At the same time, TIGRE spectra provide a marginal lithium detection in HD 110067. And indeed, adding up the 78 HARPS-N spectra (see Fig. 4, R = 115.000) enabled us to obtain a reliable EW measurement of 1.1(±0.2) mÅ – fortunately, the lithium line is adequately separated from the Fe I 6707.4 Å line. From this, we expect HD 110067 to be younger than σ Dra.

Fig. 4. Sum of the 78 HARP-N spectra of HD 110067 with a resolution of R = 115.000. The sum yields a S/N of about 600 in the region of the lithium line at 6707.85 Å, here seen as a shallow indentation of only about 1.1 mA EW to the right of the Fe I line at 6707.4 Å.

In general, however, deriving lithium ages for K0 dwarfs above about 2 Gyr is an almost impossible and highly ambiguous task, since the EWs reach below 10 mÅ and are therefore of the same order as the intrinsic star-to-star variation (see Jeffries et al. 2023). In M 67, at an age of just under 4 Gyr, a K0 dwarf’s lithium is already below the detection limit (see Pace et al. 2012). Hence, in absolute terms, the lithium line EW only tells us that HD 110067 may have reached an age of ≈2 Gyr and is likely younger than M 67 and σ Dra.

6. Estimating an activity age

Since Skumanich (1972), it has been well documented how stellar rotation and magnetic activity decline in lockstep among MS stars (see e.g. Barnes 2007 and references therein, or Mittag et al. 2018). However, the span of rotation periods found varies with stellar mass – or, empirically, with B − V (see Fig. 2 of Barnes 2007). Hence, rotation periods cannot be compared directly between two different kinds of stars. A universal picture only emerges when, in each stellar B − V group, the rotation periods are divided by the respective empirical convection turnover times as derived by Mittag et al. (2018). Then the activity parameter R_HK⁺ declines in a straight line over a logarithmic scale of the Rossby number, which varies from about 0.1 (very active stars) to 1 (the upper limit of the rotation period defined by the least active stars, which serves as an empirical turnover time) in each B − V group of FGK MS stars.

The Rossby number (Ro) is known to give a better (closer) relationship with age compared to the rotation period (see Barnes 2010). More recently, Mittag et al. (2023) confirmed empirically what dynamo theorists have argued for decades: that Ro is the physical parameter that is most closely linked with activity cycle periods, not the rotation period. This raises the problem of how to establish a meaningful convective turnover time. The value derived from stellar models depends critically on where in the convective envelope it is measured and on the specific details of the model. Consequently, we used the approach of Mittag et al. (2018), who defined an empirical convective turnover time, which varies along the F, G, and K MS, as the upper envelope of the observed rotation period distribution. Here, the longest observed period of any given star type is then assigned to Ro = 1 and therefore equals the empirical convective turnover time. At least on a relative scale, this removes the problem of the rotation period giving different relations for different stars.

Following the conventions described by Rutten (1984) and extracting the solely activity-related H&K fluxes by subtracting the photospheric and basal flux components in the S-index (i.e. subtracting 0.15), for HD 110067 we obtain

R_HK⁺ = C_cf × K × 10⁻¹⁴× (S_MWO − 0.15)/σ = 2.4 × 10⁻⁵.

Here, for the original full width at half maximum 1.09 Å triangular Mount Wilson Observatory (MWO) pass window of the HK flux measurements, onto which our S-values are calibrated, K is given by 1.07 × 10⁶ erg/cm² sK⁴, according to the observations of Hall et al. (2007), and σ is the Stefan Boltzmann constant in cgs units (5.67 × 10⁻⁵ erg/cm²s K⁴).

The B − V-dependent correction factor, C_cf, dates back to Middelkoop (1982) and here effectively compensates for how the ratio of bolometric flux (forming the denominator of R_HK⁺) over the UV reference flux (forming the denominator of S) changes with effective temperature from the Sun to any other MS star. It is 0.75 for a K0V star of B − V = 0.79 (according to Eq. (8) of Mittag et al. 2018) and 1.0 for the Sun. Another advantage of converting the S-index directly into R_HK⁺ is the cancellation of the T_eff⁴ terms; additionally, C_cf carries the only small temperature impact on R_HK⁺ when obtained from S.

In its B − V group, HD 110067 would thus have a rotation period of about 25 days and a respective empirical turnover time (the upper limit to the rotation period) of 63 days. From this, we find an empirical Rossby number (Ro) of 0.40. The Sun, by comparison, is much closer to the upper rotation limit of its B − V group according to the Mittag et al. (2018) classification, with an empirical Rossby number of about 0.8.

If seen about equator-on, a rotation period of 25 days would mean a vsin(i)  =  1.7 km/s, close to the lower uncertainty margin of 2.5 km/s obtained from the line profile analysis, which suggests a somewhat shorter period. Likewise, the available Transiting Exoplanet Survey Satellite (TESS) photometry of HD 110067 carries a rotation signature of about 20 days (Luque et al. 2023). Based on this rotation period, we calculate a Rossby number (of our empirical scaling) of 0.32.

Since the spin-down timescale also depends on stellar parameters, the activity age clock runs at different speeds for different stars. Schröder et al. (2013) found, empirically, that a slower activity decline on the FGK MS for lower-mass stars is responsible for their longer evolutionary timescales. This confirms the theoretical work of Reiners & Mohanty (2012), who derived a parameterised description of the spin-down timescales in the non-saturated dynamo domain of

τ_*/τ_⊙ = M ⋅ R² ⋅ (M²/R¹⁶)^1/3.

Here, for simplicity, M and R are in solar units. Based on this, the present-day spin-down timescale of a K0V star of M_* = 0.85 M_⊙ and R_* = 0.8 R_⊙ is 1.4 times longer than that of the Sun.

Based on the fact that the stellar parameters undergo evolutionary changes on the MS, in particular a continuous growth of the radius, Reiners & Mohanty (2012) found that the exponential spin-down of a solar-type star effectively assumes a time dependence very similar to the empirical Skumanich (1972) law Ω_*(t)∝t^−1/2, which translates into the more universal relation Ro(t)² ∝ t. Hence, if we wish to use this relation as a clock, the spin-down timescale for K0V stars, which is 1.4 times longer than that of the Sun, must be entered quadratically alongside the Rossby number. Under the non-saturated dynamo regime, this suggests:

t_*/t_⊙ = (τ_*/τ_⊙)² ⋅ (Ro/Ro_⊙)².

Here, t is the time running from the beginning of the non-saturated dynamo regime, that is, starting around an age of about 0.5 Gyr, according to current estimates of how long the saturated regime lasts. For the Sun, therefore, t_⊙ is 4.1 Gyr.

To reach the same Rossby number as that of the Sun (0.8), a K0V star would have to pass about twice the solar time span in the non-saturated regime. But for a much more active star like HD 110067 with a much lower Rossby number of 0.40, this time decreases to 2.0 Gyr, which brings the estimate of the total age to 2.5 Gyr. For a Rossby number of 0.32 (using the empirical rotation period of ≈20 d), the same calculation yields an activity age of 1.7 Gyr. This alternative value marks the lower uncertainty margin of our approach and illustrates the magnitude of a reasonable error estimate of 30%,  or 0.8 Gyr.

Interestingly, along the same lines, σ Dra has an empirical Rossby number of about 0.65, and we derived a total activity age of 5.5–6 Gyr. This is not inconsistent with its even lower, hardly detectable lithium content. According to the matching evolution track method, this twin star has a mass close to 0.81 M_⊙.

7. Discussion and conclusions

Since it is almost impossible to estimate the age of a K0V star by means of evolution tracks or a low lithium content, and given the enormous interest in the possibly long-lived and complex planetary system of HD 110067, we observed the Ca II H&K emission of this interesting host star. It is surprisingly strong and persistent, and any interference from flare activity can be ruled out.

From S and the respective R_HK⁺, we deduced the empirical rotation period (25 days) and the Rossby number. Based on the latter, the spin-down timescale (in solar units), and the Skumanich law, we obtained an activity age of 2.5 Gyr. This is the same age as that derived for the galactic cluster NGC 6819, where K0V stars show a Kepler data rotation period of about 22 days (see Meibom et al. 2015, their Fig. 2, over B − V = 0.8).

A rotation period of 20 days, as indicated by TESS photometry and our vsin(i) of HD 110067, similarly suggests an age of 1.7 Gyr and highlights the lower uncertainty margin of our activity age assessment. We would like to emphasise that, while much lower than originally proclaimed, an age of this order is fully consistent with evolution tracks. According to the range of matching evolution tracks in Fig. 3, the lithium EW of HD 110067 falls between 0.83 and 0.84 M_⊙. Also, while twin star σ Dra has a much weaker Ca II H&K emission and a narrower lithium EW, a comparison of the two stars is still apt.

Therefore, no hypothetical planet-host star interaction would be required to explain the large Ca II H&K emission of HD 110067. On the other hand, given the ambiguities in the evolutionary age and lithium age, such an interaction cannot be entirely ruled out. But since HD 110067’s rotation period is close to those found for K0V stars in NGC 6819 with the same age as the activity age derived here, we see this as strong evidence that the high levels of observed activity of HD 110067 are genuinely inherent to the star.

Consequently, our activity age approach then also suggests that the level of extreme ultraviolet, and presumably X-ray, irradiation produced by HD 110067 in its habitable zone is comparable to that in the Solar System about 3.3–3.5 Gyr ago. Interestingly, this roughly coincides with the appearance of the first primitive marine life forms on our planet.

Given the various simplifications and uncertainties in this approach and the scatter in the R_HK⁺ over log P diagrams of Mittag et al. (2018), by which we obtained the empirical Rossby number, we consider the error margin of our activity age estimate to be about 30%. Still, for HD 110067, this is significantly better than the huge uncertainty of a lithium or evolution age, as demonstrated above for K0-type MS stars in general. While the lithium abundance of HD 110067 could suggest any age above 1 Gyr, matching evolution tracks agree with any age below about 8 Gyr. Hence, results from automatically matchmaking routines can indeed fool their user.

Acknowledgments

We are grateful to the Universidad de Guanajuato and the University of Hamburg for supporting the operation of TIGRE. We also acknowledge the online access to TNG/HARPN data provided by the Centro Italiano Archivi Astronomici, and to the MESA evolution model library MIST, hosted by the CfA in Harvard, and some travel support by the UG.

References

All Tables

All Figures

	Fig. 1. Three TIGRE-HEROS spectra of HD 110067 obtained between January and March 2024, summed together to reduce noise. They show a strong chromospheric emission in Ca II K (blue line), here directly compared to the modest emission of virtual twin σ Dra.
In the text

	Fig. 2. Sum of the 78 HARPS-N spectra available to us, obtained for HD 110067 at the TNG on the Canary Islands from 2021 to 2023 (arbitrary flux over wavelength in Å). It shows a high chromospheric emission, identical to that in Fig. 1.
In the text

Fig. 3. HR diagram positions of K0V stars HD 110067 and its virtual twin, σ Dra. They both coincide with e.g. a 0.8 Gyr-old model of a 0.85 M⊙ MESA evolution track and a 6.8-Gyr model on an evolution track with 0.80 M⊙, using here a metallicity of [Fe/H] = −0.16. Solid lines indicate stable, well-modelled phases, while dashed lines indicate transitions, uncertainties, or complex phenomena in the star’s evolution.

In the text

	Fig. 4. Sum of the 78 HARP-N spectra of HD 110067 with a resolution of R = 115.000. The sum yields a S/N of about 600 in the region of the lithium line at 6707.85 Å, here seen as a shallow indentation of only about 1.1 mA EW to the right of the Fe I line at 6707.4 Å.
In the text