Estimates of Active Region Area Coverage through Simultaneous Measurements of He I $\lambda\lambda$ 5876 and 10830 Lines

TL;DR

This study measures the active region coverage in solar-type stars using helium triplet lines, revealing that most stars have less than full coverage and that activity differences are mainly due to the area coverage of active regions.

Contribution

It introduces a method to estimate active region filling factors in stars using simultaneous helium line measurements and chromospheric models, highlighting the role of area coverage in stellar activity.

Findings

Most stars have filling factors less than one.

Discrepancies in K-type and F-type stars are due to model limitations and measurement uncertainties.

Active region heating rates are similar across different stars.

Abstract

Simultaneous, high-quality measurements of the neutral helium triplet features at 5876~\AA\ and 10830~\AA, respectively, in a sample of solar-type stars are presented. The observations were made with ESO telescopes at the La Silla Paranal Observatory under program ID 088.D-0028(A) and MPG Utility Run for FEROS 088.A-9029(A). The equivalent widths of these features combined with chromospheric models are utilized to infer the fractional area coverage, or filling factor, of magnetic regions outside of spots. We find that the majority of the sample is characterized by filling factors less than unity. However, discrepancies occur among the coolest K-type and warmest and most rapidly rotating F-type dwarf stars. We discuss these apparently anomalous results and find that in the case of K-type stars they are an artifact of the application of chromospheric models best suited to the Sun than to stars with significantly lower $T_\mathrm{eff}$. The case of the F-type rapid rotators can be explained with the measurement uncertainties of the equivalent widths, but they may also be due to a non-magnetic heating component in their atmospheres. With the exceptions noted above, preliminary results suggest that the average heating rates in the active regions are the same from one star to the other, differing in the spatially integrated, observed level of activity due to the area coverage. Hence, differences in activity in this sample are mainly due to the filling factor of active regions.

Tables

Table 1. Table 1: Journal of Observations. The UT date for all start times is 2011 December 7. Label ‘std’ in the notes indicate telluric standards. Table  1 is published in its entirety in the machine readable format. A portion is shown here for guidance regarding its form and content.

HD	Sp. typeaaFrom the Bright Star Catalog (Hoffleit & Jaschek, 1991) or the Simbad data base	$B-V$aaFrom the Bright Star Catalog (Hoffleit & Jaschek, 1991) or the Simbad data base	UT start time		Nod. pos.	Notes
			FEROS	CRIRES
HD 49933	F3V	0.36	06:22:47
			06:25:23
			06:27:59
HD 29992	F3IV	0.37	03:19:05	03:40:28	A
			03:20:41	03:42:28	B
			03:22:19
HD 37495	F5V	0.46	05:07:55	05:10:26	A	No AO
			05:10:01	05:15:42	B	No AO
			05:12:07
HD 27861	A1V	0.08	06:00:11	00:55:58	A	std
			06:02:23	00:58:43	B	std
			06:04:35			std
HD 18331	A1V	0.09	02:55:09			std
			02:57:31			std
			02:59:53			std

Table 2. Table 2: Linear fit coefficients for stellar blends near He I λ 𝜆 \lambda 5876

Line\@alignment@align	$A_{B-V}$		$B_{B-V}$	${\sigma }_{B-V}$
	mÅ		mÅ	mÅ
Unid. $\lambda 5875.778$	0\@alignment@align.7	.	0\@alignment@align.4
Unid. $\lambda 5875.15$	-0\@alignment@align.4	4.7	1\@alignment@align.1
Unid. $\lambda 5876.1$	0\@alignment@align.9	1.8	0\@alignment@align.7
Fe I $\lambda 5876.296$	-1\@alignment@align.8	12.9	1\@alignment@align.8
Cr I $\lambda 5876.556$	-1\@alignment@align.3	9.4	1\@alignment@align.3

Table 3. Table 3: Helium Equivalent Width Measurements and Data for Program Stars: B V ≤ 0.5 B V 0.5 \bv\leq 0.5

HD	$B-V$	$v\sin i$aaFrom Ammler-von Eiff & Reiners (2012) when available, otherwise from Głȩbocki & Gnaciński (2005).	$\log L_X$bbFrom Hünsch et al. (1999), except $L_X$ for HD 49933 which is from Hünsch et al. (1998).	$W_{\mathrm{fit}}$(He I $\lambda$5876)	$W_{\mathrm{corr}}$(He I $\lambda$5876)	$W_{\mathrm{bl}}$($\lambda$5876)	$W$(He I $\lambda$10830)	$W$(He I $\lambda$10829.1)	$f$(C)	$f$(C-np)	$f_{\min }$
		km s-1	erg s-1	mÅ	mÅ	mÅ	mÅ	mÅ
HD 49933	0.36	9.9	29.50	27.6(1.4)	27.6(1.4)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 29992	0.37	97.5	28.83	27.1(5.5)	18.4(6.1)	8.7(2.6)	364.3(9.6)	$\cdots$	$\cdots$	$\cdots$	0.88
HD 219693	0.39	19.9	28.98	30.1(1.8)	26.4(2.3)	3.7(1.4)	259.6(5.8)	38.5(5.1)	0.66(0.04)	0.72(0.03)	0.59
HD 32743	0.39	22.7	29.02	21.3(2.2)	18.3(2.5)	3.0(1.3)	184.2(4.2)	28.6(3.7)	0.44(0.04)	0.50(0.03)	0.39
HD 104731	0.41	15.9	28.49	14.1(1.7)	10.3(2.2)	3.8(1.4)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 3302	0.41	17.8	29.40	33.2(2.0)	30.1(2.4)	3.1(1.3)	305.0(6.0)	48.7(5.3)	0.83(0.05)	0.89(0.04)	0.72
HD 77370	0.42	60.4	29.07	23.2(4.6)	16.6(4.9)	6.6(1.9)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 68456	0.43	8.8	29.15	33.1(1.3)	31.4(1.5)	1.7(0.7)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 30652	0.44	17.3	29.03	20.8(2.0)	17.4(2.4)	3.3(1.3)	187.7(11.0)	31.6(9.1)	0.47(0.07)	0.52(0.06)	0.40
HD 88697	0.44	19.8	29.49	38.1(2.1)	34.1(2.5)	4.0(1.4)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 76653	0.45	10.3	29.33	28.8(1.4)	27.1(1.6)	1.7(0.7)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 201647	0.45	22.4	28.92	18.5(2.1)	14.5(2.5)	4.1(1.4)	174.7(9.1)	24.6(8.1)	0.45(0.07)	0.50(0.07)	0.36
HD 79940	0.45	117.2	28.79	28.5(6.9)	17.4(7.4)	11.0(2.6)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 37495	0.46	27.2	29.31	25.9(2.2)	21.7(2.6)	4.1(1.4)	290.2(8.6)	$\cdots$	0.89(0.07)	0.92(0.05)	0.68
HD 219482	0.47	7.5	29.42	35.9(1.3)	35.9(1.3)	$\cdots$	318.5(4.3)	62.6(3.6)	0.85(0.03)	0.91(0.02)	0.75
HD 33262	0.47	15.4	28.71	34.9(1.6)	33.1(1.8)	1.7(0.7)	317.2(4.3)	56.3(3.7)	0.86(0.04)	0.92(0.03)	0.75
HD 189245	0.49	72.6	29.90	76.9(5.3)	64.7(5.9)	12.2(2.6)	409.6(4.1)	$\cdots$	$\cdots$	$\cdots$	1.00
HD 199260	0.50	13.7	29.18	31.4(1.8)	29.5(2.1)	1.9(1.1)	294.3(5.7)	51.5(4.9)	0.81(0.05)	0.87(0.04)	0.69

Table 4. Table 4: Helium Equivalent Width Measurements and Data for Program Stars: B V > 0.5 B V 0.5 \bv>0.5

HD	$B-V$	$v\sin i$aaFrom Ammler-von Eiff & Reiners (2012) when available, otherwise from Głȩbocki & Gnaciński (2005) or (HD 48189A) from Schröder et al. (2009).	$\log L_X$bbFrom Hünsch et al. (1999).	$W_{\mathrm{fit}}$(He I $\lambda$5876)	$W_{\mathrm{corr}}$(He I $\lambda$5876)	$W_{\mathrm{bl}}$($\lambda$5876)	$W$(He I $\lambda$10830)	$W$(He I $\lambda$10829.1)	$f$(C)	$f$(C-np)	$f_{\min }$
		km s-1	erg s-1	mÅ	mÅ	mÅ	mÅ	mÅ
HD 41700	0.52	14.7	29.63	45.6(1.6)	41.1(2.1)	4.5(1.4)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 100563	0.53	13.5	29.13	26.2(1.6)	21.6(2.1)	4.6(1.4)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 17051	0.57	4.2	28.83	24.2(1.1)	24.2(1.1)	$\cdots$	239.0(1.4)	47.7(1.1)	0.66(0.03)	0.71(0.03)	0.54
HD 88742	0.59	2.7	28.58	15.8(1.0)	15.8(1.0)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 48189A	0.61	15.5	29.99	38.3(1.6)	35.8(2.0)	2.5(1.1)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 30495	0.64	2.9	28.83	19.8(1.0)	19.8(1.0)	$\cdots$	223.6(5.7)	41.4(4.6)	0.72(0.08)	0.76(0.07)	0.50
HD 16417	0.66	2.5	27.81	4.5(1.2)	4.5(1.2)	$\cdots$	71.7(9.0)	12.1(7.3)	0.23(0.18)	0.31(0.20)	0.09
HD 1835	0.66	7.0	28.99	33.2(1.2)	33.2(1.2)	$\cdots$	323.7(5.8)	67.0(4.8)	0.92(0.04)	0.96(0.03)	0.77
HD 20630	0.67	4.5	28.89	27.4(1.1)	27.4(1.1)	$\cdots$	279.0(4.8)	55.6(3.7)	0.82(0.05)	0.87(0.04)	0.65
HD 76151	0.67	2.4	28.33	12.6(1.0)	12.6(1.0)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 42807	0.68	3.6	28.68	24.1(1.1)	24.1(1.1)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 43162	0.68	5.5	29.13	31.5(1.2)	31.5(1.2)	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
HD 10700	0.72	1.8	26.30	3.2(1.2)	3.2(1.2)	$\cdots$	51.6(3.0)	$\cdots$	0.12(0.14)	0.19(0.15)	0.03
HD 17925	0.86	4.8	29.08	28.1(1.1)	28.1(1.1)	$\cdots$	317.6(20.9)	70.0(16.0)	$\cdots$	$\cdots$	0.75
HD 22049	0.88	1.9	28.32	18.1(1.0)	18.1(1.0)	$\cdots$	257.8(3.6)	51.3(3.0)	$\cdots$	$\cdots$	0.59
HD 5133	0.94	1.8	27.79	12.6(1.0)	12.6(1.0)	$\cdots$	229.8(11.7)	42.3(9.6)	$\cdots$	$\cdots$	0.51

Equations

$$W_{\mathrm{obs}}=(1-f)W_{\mathrm{q}}+f{W}_{\mathrm{a}}.$$

$$f\ge \frac{W_{\mathrm{obs}}-W_{\mathrm{q}}}{W_{\max }-W_{\mathrm{q}}},$$

Full text

stars: activity — stars: magnetic field — stars: solar-type — techniques: spectroscopic

††slugcomment: Draft version ††facilities: ESO VLT (CRIRES), ESO MPG (FEROS), NSF McMath-Pierce (FTS), NSF SOLIS (VSM), NASA SDO (AIA)††software: MIDAS, ESOREX, IDL

1 Introduction

While sunspots are the most visible manifestations of magnetic flux emergence resulting from dynamo processes, magnetic flux concentrations outside of spots in active regions form a significant fraction of the total (unsigned) magnetic flux budget of the Sun. Likewise, the total and spectral solar irradiance as functions of time cannot be modeled by considering the contribution of sunspots only (for a recent review, see Yeo et al., 2014).

Determining the distribution, or at least the fractional area coverage of magnetic active regions, is relevant to both dynamo theory and to models of chromospheric and coronal heating. With regard to the latter, flux-calibrated chromospheric emission line profiles yield the surface-averaged emission that represents a lower limit to the intrinsic emission in localized active regions. A more accurate knowledge of the actual radiative losses resulting from chromospheric heating would provide a further constraint for the development of models based on, for example, local plasma heating by Joule dissipation associated with an Alfvén wave field (Tu & Song, 2013). While information on the spatial distribution of magnetically active regions on stellar surfaces can be obtained in some special cases (mostly rapid rotators through Doppler imaging), such measurements have always been elusive in more solar-like stars.

A census of the solar magnetic flux in its various forms can be performed directly with the distinct advantage of spatially resolved observations. In the case of stars, however, we generally rely on radiative proxies to infer the properties of magnetic flux on the spatially-unresolved stellar surface. The analog of the solar cycle in late-type stars is typically seen through the modulation of chromospheric radiative emissions, such as in the deep cores of the Ca II resonance lines, that are spatially associated with sites of emergent magnetic fields. (Skumanich et al., 1975; Wilson, 1978; Baliunas et al., 1995). The amplitude modulation of this magnetic flux is widely regarded as a property of non-linear dynamo processes of which $\alpha-\omega$ kinematic dynamos are a particular class of mean-field dynamo models (Tobias, 1997).

The high-quality photometric data from the space missions CoRoT (Baglin et al., 2009) and Kepler (Koch et al., 2010) have yielded new insight on the rotation and magnetic properties of solar-type stars by providing rotation periods for thousands of main-sequence stars (McQuillan et al., 2014; Nielsen et al., 2013; Reinhold et al., 2013; Buzasi et al., 2016) as well as new photospheric proxies of magnetic activity based on the periodicity and amplitude of the light-curve modulation (He et al., 2015).

In order to provide a broader parameter space for the advancement of stellar and solar dynamo models, we further develop herein a method for the measurement of active region area coverages on solar-type stars (Giampapa, 1985; Andretta & Giampapa, 1995 (AG95). In particular, we extend our previous work through the results of simultaneously acquired, high-resolution spectroscopic observations of the He I triplet lines at 5876 Å and 10830 Å, respectively.

Solar observations, such as those in Fig. 1, demonstrate that these lines are ideal tracers of magnetic regions outside of cool spots, appearing in absorption in active regions and only weakly in quiet network elements and the (non-magnetic) photosphere333See Harvey et al. (1975) for an analogous figure that includes more rare spectroheliograms in He I $\lambda$5876 (D3) obtained simultaneously with solar X-ray images from $Skylab$..

Thus, the measured absorption equivalent width is proportional to active region area coverage, or filling factor (the two terms are used interchangeably in this paper), at the height of formation and the time of observation. A more precise estimate of the active region filling factor can be obtained through examination of the joint response of the He I triplet lines to chromospheric heating combined with a model-dependent calibration of the strengths that can be attained by these features in plage-like regions on the stellar surface. Recall that plages are the chromospheric counterpart of faculae, which are localized bright regions in the solar photosphere associated with concentrations of magnetic fields and characterized by reduced opacity, thus allowing us to see the deeper, hotter (and, hence, brighter) walls of the facular area. The overlying plage is distinguished by relatively bright Ca II H & K emission in full-disk spectroheliograms.

We discuss in Sec. 2 our approach to the calibration of the joint response of the helium lines to atmospheric heating along with the results of our model calculations. The observations and their reduction are presented in Secs. 3 and 4. A discussion of the inferred active region filling factors is given in Sec. 5. In Sec. 6 we present our conclusions and our anticipated directions for further research.

2 Model Approach

Andretta & Giampapa (1995 (AG95, hereafter AG95) developed a technique to address this problem by demonstrating that the non-linear response of the two main triplet He I lines (at 10830 Å and 5876 Å) to chromospheric heating can be exploited to infer the fractional area coverage by active regions in solar-like stars. Their approach was based on a two-component representation of the strength of activity diagnostics, where the observed equivalent width of a line, $W_{\mathrm{obs}}$, can be written in terms of the contributions from the quiescent atmosphere, $W_{\mathrm{q}}$, and the active (plage-like) atmosphere, $W_{\mathrm{a}}$, via a filling factor $f$, where

[TABLE]

Further details on the derivation of the above equation can be found in AG95. We only note here that the difference in continuum intensities between the quiescent and active atmosphere is assumed to be negligible. At this level of approximation, this is a valid assumption in the Sun and, by analogy, in solar-type stars.

In the two-component model described above, the quantities $W_{\mathrm{q}}$ and $W_{\mathrm{a}}$ represent the average values in the quiescent atmosphere and in active regions, respectively. In both regions, the observed line strengths can of course vary on smaller scales. In the quiet Sun, for example, chromospheric line strengths are typically distributed in a characteristic spatial pattern called the “supergranular network”. The main assumption of Eq. 1 is therefore that the average value of the strength of the activity diagnostics, $W_{\mathrm{a}}$, is the same in all active regions on the stellar surface independent of their area, i.e. small and large active regions are equally “bright” when imaged in the chosen activity diagnostics. Fig. 1 shows that this assumption is plausible, but for most activity diagnostics this assumption can also be quantitatively verified (e.g. Andretta & Del Zanna, 2014).

In this approach, the filling factor, i.e. the fractional area covered by active regions, is therefore one of the fundamental parameters discriminating between stars with different observed activity levels. In the favorable case of a negligible contribution from the quiescent atmosphere, the filling factor is simply $f=W_{\mathrm{obs}}/W_{\mathrm{a}}$. But even in this case the filling factor cannot be determined unless the intrinsic line strength in stellar active regions is known. AG95 showed that this ambiguity can be resolved by observing two lines with different, non-linear dependences on the atmospheric activity.

In order to apply the method described by AG95 to a pair of activity diagnostics such as the main He I triplet lines, the dependence of the intrinsic strength of both lines on the active region heating needs to be computed, $W_{\mathrm{a}}(p)$, where $p$ is a parameter, or set of parameters characterizing the active regions in the formation layer of the diagnostics under consideration. In AG95, the activity parameter is the mass loading, or column density, $m$, in g cm*-2* at the top of the chromosphere or, equivalently, the increase of total chromospheric pressure relative to the quiescent state, $P/P_{\mathrm{q}}$. A more complete set of activity parameters can be considered, as described below.

In comparison with other diagnostics, the helium triplet lines are especially suitable for this approach since they respond to chromospheric heating, which is parameterized in model computations by higher chromospheric pressure in the line formation region, by increasing their absorption equivalent widths in a non-linear fashion (see AG95, their Fig. 2). At sufficient densities, such as those that may occur in a flare, collisional control eventually overcomes scattering processes and the triplet lines are driven into emission. Thus, as atmospheric heating increases, all He I triplet lines go deeper into absorption, reaching a maximum in their equivalent widths before going eventually into emission.

This general behavior of the helium lines is in qualitative agreement with observations, as illustrated in Fig. 2, where we see varying strengths in the triplet lines at different locations in a solar active region, presumably in response to different degrees of chromospheric heating.

We note that this behavior is very much reminiscent of H$\alpha$ line formation in the chromospheres of M dwarf stars (Cram & Mullan, 1979; Giampapa et al., 1982). Following the approach of Giampapa (1985), the existence of a maximum in intrinsic line strength in active regions, $W_{\mathrm{max}}$, can be exploited to derive a lower limit to the filling factor:

[TABLE]

or, if $W_{\mathrm{q}}\approx 0$, $f\geq W_{\mathrm{obs}}/W_{\mathrm{max}}$.

The key point of the method described in AG95 is however that each line attains its maximum equivalent width at different amounts of atmospheric heating. This leads to a strongly non-linear joint response of the line strengths. Thus, simultaneous observations of two lines can in principle allow an unambiguous determination of the filling factor $f$. This behavior is illustrated in Fig. 2, where the main component of the $\lambda$10830 line seems to reach a level of “saturation” in its equivalent width while both the D3 line and the minor component of the IR triplet line at 10829 Å continue to increase in their strength.

The essence of the method is illustrated in Fig. 3, which displays theoretical diagrams as calculated by AG95 of the joint response of the triplet lines in equivalent width to chromospheric heating (dot-dashed lines), together with the set of calculations adopted here and described in Sec. 2.1 below. The locus $f=1$ defines a region (highlighted in solid color in the figure for the reference calculations) where all measurements should fall; already in AG95 it was shown that observations of solar-like stars do indeed fall in this allowed region. We also note that to infer the filling factor it is not necessary to have a detailed knowledge of the specific activity state of the stellar plage-like regions; only the joint dependence of the two spectral diagnostics. Nevertheless, the values of the activity parameter $p$ that best match the observations can still be derived together with $f$ by inverting Eq. 1 for the selected activity diagnostic pair.

The effect of measurement errors is also shown in Fig. 3. Two hypothetical joint measurements of $\lambda$5876 and $\lambda$10830 with a 10% ($1\sigma$) uncertainty are shown in the left-hand panel of that figure. The corresponding bi-dimensional probability distributions are shown as filled gray contours. In the right-hand panel, the probability distribution transformed by the inversion of Eq. 1 for both lines is shown in the ($P/P_{\mathrm{q}}$,$f$) plane, for both sets of theoretical calculations of $W_{\mathrm{a}}(P/P_{\mathrm{q}})$ we have considered and that are discussed in Sec. 2.1 below. The mean value and the standard deviation of the transformed distributions are shown as error bars. In particular, the mean and the standard deviation of the transformed probability distribution for the filling factor is the value and its error we will attach to the actual measurements described in the remainder of this work.

In addition to the general properties of their joint response to chromospheric heating, the helium triplet lines exhibit several desirable features:

• •

they are purely chromospheric lines: the photospheric contribution to these lines is negligible in solar-like stars;

• •

their strength in the quiescent chromosphere is small: the observed lines in spectra of solar-like stars arise almost entirely in active regions: $W_{\mathrm{q}}(D_{3})\approx 0$ and $W_{\mathrm{q}}(\lambda 10830)\approx 40$ mÅ (the latter value is inferred from full-disk measurements during the minima of solar activity: Harvey & Livingston 1994; Livingston et al. 2010);

• •

they both belong to the same atom: therefore the effect of the elemental abundance is largely factored out;

• •

the transitions giving rise to the two lines share one atomic level ($1s2p$ ${}^{3}\!P$) in so-called orthohelium: thus their differential behavior is relatively insensitive to the details of interactions with other atomic levels;

• •

they form essentially in the same zone of the chromosphere, regardless of the details of the formation mechanism (Andretta et al., 1995; Andretta & Jones, 1997 (AJ97). Hence, they probe exactly the same regions of the stellar atmosphere.

2.1 Reference calculations

In AG95 the atmospheric activity level is parameterized by the column density, $m$, in g cm*-2* at the top of the chromosphere. This formulation has the advantage that the total chromospheric pressure, $P$, is simply given by $P=gm$, where $g$ is the stellar surface gravity. Implicit in this relation is the assumption that the chromosphere is thin with respect to the stellar radius. In a parallel study, Andretta & Jones (1997 (AJ97, hereafter AJ97) carried out a more extensive analysis of the parameters determining the formation of the helium spectrum in the Sun.

The reference quiescent model adopted in both AG95 and AJ97 is the VAL C model of the quiet Sun (Vernazza et al., 1981). In AJ97, two modified versions of the model, termed VAL C-np and VAL C-nt, were also considered, which differ from the VAL C model only in the thickness of the transition region, $\Delta h(\mathrm{TR})$. The C-nt series, i.e. the series starting from the VAL C-nt model was used in AJ97 only to discuss some specific radiative transfer aspects of the line formation; we will not consider this series of models here.

Both AG95 and AJ97 included in their analysis the effect on the helium ionization balance of coronal EUV back-illumination integrated in the range $\lambda<$ 500 Å ($J_{\mathrm{cor}}$). In AG95 the EUV back-illumination was suitably scaled to account for increased coronal emission in active regions. The same scaling was applied by Andretta (1994 (A94, hereafter A94) also for the C-np series of atmospheric models. Hence, the pair $\{m/m_{\mathrm{q}},\Delta h(\mathrm{TR})\}$ constitutes the fundamental set of parameters, $p$, determining both the structure of the atmosphere and the coronal back-illumination, that, in turn, are used to compute the theoretical equivalent widths of the He I triplet lines to be used in Eq. 1, $W_{\mathrm{a}}(p)$.

In AJ97 it was shown that the C-np series, i.e. the series of atmospheres with a reduced temperature plateau at $\approx 2\times 10^{4}$ K, better matches the observed properties of the solar He I spectrum than the C and C-nt series, from the extreme UV (EUV) to the IR. That finding is consistent with the structure of solar plages derived from semiempirical models (e.g.: Fontenla et al., 1993, 2006). We therefore adopt the C-np series as our reference set of models, considering however the scaling of EUV back-illumination as computed by A94 and AG95. We nevertheless also take into consideration the C series of models, i.e. the series based on the VAL C model, as in AG95, for comparison. Figure 4 shows the series of models computed in A94 (panels a and b) which were then employed by AG95, AJ97, and in the present work, together with the corresponding joint response of the triplet lines as functions of the parameter $P/P_{\mathrm{q}}$ (panel c).

The effect of the choice of the series of models on the determination of the filling factors is illustrated in Fig. 3. For the two examples shown, the differences introduced by the different theoretical calculations of $W_{\mathrm{a}}(P/P_{\mathrm{q}})$ are larger than the uncertainties due to measurements errors, if the latter are of the order of 10% or less.

We note, however, that the maximum equivalent width attained by the He I $\lambda$10830 line is very similar in the two series of models: $\approx 370$ mÅ for the C-np series, and $\approx 410$ mÅ for the C series. From an observational point of view, Sanz-Forcada & Dupree (2008) noted that data for cool dwarfs and subdwarfs tend to be below those theoretical limits, with very few exceptions in very active binaries. A similar result is obtained by inspecting the data of Zarro & Zirin (1986).

Regarding the quiescent value for the He I $\lambda$10830 line adopted here ($W_{\mathrm{q}}=40$ mÅ), we note that data published in Zarro & Zirin (1986) for low activity stars for which some He I $\lambda$10830 absorption could be detected tend to cluster around $\sim 50$ mÅ (see their figures 1a and 1b). Values reported by Takeda & Takada-Hidai (2011) have a median of $35$ mÅ, while values reported by Smith et al. (2012a) are around $\sim 30$ mÅ. These last two papers present mostly data for low-metallicity, low-gravity stars whose atmospheres could significantly differ from those of solar-like stars as far as the relevant regions contributing to the formation of the He I optical lines are concerned (photosphere, transition region, corona.) Furthermore, the relatively modest variations in the minimum detected $\lambda$10830 equivalent width have a little effect on the values obtained with Eq. 2, since $W_{\mathrm{max}}$ is about an order of magnitude larger than $W_{\mathrm{q}}$.

In conclusion, the lower limits of the filling factors derived from Eq. 2 are practically insensitive to the details of the adopted models. On the other hand, the D3 line never attains its maximum in the grid of models considered by AG95 and AJ97 and therefore a similar approach based on D3 measurements alone is not feasible in solar-like stars.

Concerning the dependence of the joint response of the He I triplet lines on stellar effective temperature, the calculations of AG95 for an F star with $T_{\mathrm{eff}}=6500$ K show a slightly lower slope of the initial linear part of the curve compared to the case of the Sun, considered as a typical G-type star. This behavior can be understood given that at low activity levels the line formation is dominated by scattering of photospheric radiation (see discussion in A94 and AJ97). The relevant photoexciting radiation determining the slope of the linear part of the joint response of the $\lambda$10830 and $\lambda$5876 lines is the photospheric radiation field at 10830 Å. Following the same argument, we expect calculations for chromospheres illuminated by the photosphere of a K-type star to show a slightly steeper joint response of the two triplet lines at low activity levels. The effect of the photospheric radiation on the joint response of the two He I triplet lines is shown in Fig. 4 for the optically thin case.

Finally, we note that a number of theoretical and observational studies on the formation of the helium spectrum in the Sun have appeared since A94, AG95, and AJ97 (e.g.: MacPherson & Jordan, 1999; Andretta et al., 2000; Smith & Jordan, 2002; Smith, 2003; Andretta et al., 2003; Pietarila & Judge, 2004; Judge & Pietarila, 2004; Mauas et al., 2005; Andretta et al., 2008). Most of those studies, however, were focused on the formation of the EUV lines and continua, while the mechanism responsible for the formation of the optical subordinate lines has attracted comparatively less attention, with some recent exceptions such as Leenaarts et al. (2016). In any case, we remark that in this investigation we are merely utilizing those earlier calculations, and that updating the models is beyond the scope of this paper.

3 Observations

Given the potential effects of variability due to magnetic activity on the strengths of the triplet features, combined with a method based on observations of the joint behavior of these diagnostics, our observational approach was to obtain spectra of the D3 line in the visible and the Near-Infrared (NIR) $\lambda$10830 line on the same night, respectively. Obtaining simultaneous spectra of the two lines is a challenge even in the case of the Sun, but it is nevertheless feasible, as the spectra of Fig. 2 demonstrate. In addition to those FTS spectra, to our knowledge only Muglach & Schmidt (2001) have been successful in obtaining simultaneous observations in the two lines. On the other hand, we could not find analogous observations of solar-like stars in the literature, although both the He I $\lambda$5876 and the $\lambda$10830 lines have been extensively studied in the context of stellar activity, as we briefly recap in the following.

Guided by He I D3 spectra obtained for solar plages (Landman, 1981), extensive stellar observations of D3 as an activity diagnostic utilizing digital detectors with peak sensitivities in the visible soon followed. Lambert & O’Brien (1983) reported the detection of rotational modulation of D3 in selected main sequence stars. Wolff & Heasley (1984) conducted a survey of D3 in a sample of G and K stars followed by a survey focused on main-sequence stars (Wolff et al., 1985). These investigations were soon followed by focused studies addressing specific questions. Examples include the determination of the effective temperature on the main sequence corresponding to the onset of chromospheric activity associated with outer envelope convection (Wolff et al., 1986; Wolff & Heasley, 1987; García-López et al., 1993); the correlation of D3 absorption strength with rotation as well as its empirical relationship with other diagnostics of magnetic-field related activity such as X-ray emission (Saar et al., 1997); and, evidence for cycle-like variability in D3 seen in multi-year stellar programs involving high precision radial velocity monitoring (Santos et al., 2010). While a stronger feature, for a long time studies of the He I $\lambda$10830 line in late-type, dwarf stars have been more limited due to the lack of sensitivity of available detectors in this spectral region though probes of chromospheric structure based on $\lambda$10830 have been carried out in recent years with large-aperture telescopes (e.g.: Takeda & Takada-Hidai, 2011; Smith et al., 2012b). To our knowledge, there are no near-simultaneous observations of both D3 and $\lambda$10830 in stars displaying solar-like activity, while such data exist for T Tauri stars (e.g.: Dupree et al., 2012).

The primary challenges in the utilization of the helium triplet lines are that (a) they are intrinsically weak and (b) they are blended with terrestrial water lines. These issues are best addressed with very high quality spectra (in terms of S/N ratio and resolution) acquired at very dry sites to mitigate the effects of terrestrial water vapor contamination. Even when these requirements are met, the presence of blends with nearby atomic lines in the stellar spectrum due to rotational smearing (which is typically larger in more active late-type stars) introduces an additional source of error in the estimates of equivalent width. In addition to the difficulties of observing each line individually, their wide wavelength separation adds further challenges in obtaining simultaneous spectra with the same spectrograph.

In view of these considerations, we utilized the 8.2-m Very Large Telescope (VLT) and the CRyogenic high-resolution InfraRed Echelle Spectrograph (CRIRES) at the European Southern Observatory (ESO) at Cerro Paranal to obtain the NIR $\lambda$10830 spectra on the night of UT 2011 December 6 – 7. The D3 spectroscopic observations were carried out on the same night using the Fiber Extended-range Optical Spectrograph (FEROS), mounted at the 2.2m Max-Planck Gesellschaft/European Southern Observatory (MPG/ESO) telescope at La Silla (Chile), during MPG guaranteed time.

3.1 Target selection

The principal selection criteria for the stellar sample included visually bright (V $<$ 7) F, G, and K dwarfs that are detected X-ray sources in the ROSAT All-Sky Bright Source Catalogue (Voges et al., 1999) or listed in the Gliese-Jahreiss Catalogue of Nearby Stars (Gliese & Jahreiss, 1991). The application of the large-aperture VLT to bright objects served the dual objectives of efficiently obtaining spectra of the highest quality for a large number of targets in a single allocated night. The target selection criteria are clearly biased toward active stars since it was our intention to obtain spectra with detectable helium triplet lines in order to further develop our analysis methods as opposed to carrying out a survey at this time according to some completeness criteria.

The time differences between FEROS and CRIRES spectra of the same target are below 30 minutes, with the exception of HD 17051 and HD 33262 for which the time difference is about 1 hour. A journal of the observations is presented in Table 1. Note that the CRIRES observations at $\lambda$10830 terminated earlier due to the onset of adverse weather at the Cerro Paranal site, resulting in fewer targets observed than at the La Silla site where the D3 spectra were obtained with FEROS and the ESO 2.2-m telescope. The details of the observations are given below.

3.2 FEROS observations and data reduction

FEROS is a bench-mounted, thermally controlled instrument, fed by two fibers providing simultaneous spectra of either the object and wavelength calibration or the object and sky. It is designed to achieve high resolution ($R$=48,000), high efficiency ($\approx$ 20%), and to provide an almost complete spectral coverage from  3500 to 9200 Å spread over 39 echelle orders (Kaufer et al., 2000). The entrance aperture of the fiber has a projected diameter on the sky of 2.0″. As the cross-disperser is a prism, the spectral orders are strongly curved on the CCD. The detector is an EEV 2k4k CCD.

A total of 134 FEROS spectra of our targets (including telluric standards) were acquired using the object-sky mode to avoid contamination by Th-Ar lines, as for our purposes it was preferable to analyze clean spectra rather than attaining the highest radial velocity precision. The integration times of individual exposures ranged between 12 and 420 s to obtain a signal-to-noise ratio (SNR) greater than 200 for stars brighter than V=7.

The data were reduced using a modified version of the FEROS Data Reduction System (DRS) pipeline, implemented within ESO-MIDAS444Munich Image Data Analysis System (vers. 09SEPpl1.2) under context FEROS, which yields a wavelength-calibrated, normalized, one-dimensional spectrum. The details of the reduction steps are given by Schisano et al. (2009).

For each target, a triplet of consecutive spectra was obtained, with the exception of HD 88697. Pixels with unusually high values were identified by comparing the three spectra, and then flagged as missing. The triplet of spectra was then averaged to obtain a single spectrum with greater SNR. The estimated SNR in reduced spectra reach values of the order of 1000, with a mean value around 650.

3.3 CRIRES observations and data reduction

The He I $\lambda$10830 spectroscopic observations were carried out in visitor mode on the same night as the FEROS D3 observations, using CRIRES (Käufl et al., 2004, 2006), mounted at Unit Telescope 1 (Antu) of the VLT array at Cerro Paranal. The entrance slit width was set to 0.2″ to attain a nominal resolving power of $R=10^{5}$. The CRIRES science spectra are recorded on an array of four 1024$\times$512 Aladdin III detectors. The grating position (#52) was chosen so that the He I $\lambda$10830 line was recorded on detector #3. We verified that spectra on that detector were free of significant ghost features.

Each star was observed at two nodding offset positions along the slit, A and B, with jitter. The total exposure times of each science frame (without overhead) range from 2 to 10 s to obtain a SNR exceeding 200 for the target stars. Almost all the science frames were obtained with the Adaptive Optics (AO) system on to optimize the SNR; only the last few spectra of the observing run were obtained with the AO off, because of the increasingly deteriorating seeing due to the onset of adverse weather which eventually caused early termination of the run.

Data reduction of each CRIRES frame was performed using the ESOREX pipeline for CRIRES. Science frames and flat-field frames were corrected for non-linearity and 1D spectra were extracted from the combined flat-fielded frames with an optimal extraction algorithm. The wavelength solution is based on the Th-Ar calibration frames provided by ESO. The wavelength solution was then refined in the vicinity of the $\lambda$10830 line by matching the average positions of the strongest H2O lines observed in the range 10772–10868 Å with the wavelengths given by Breckinridge & Hall (1973).

The estimated SNR in reduced spectra are in the range 200 – 700.

4 Spectral analysis

After the data were reduced following standard instrument pipelines, we analyzed the 1D spectra around the wavelengths of interest (5876 Å and 10830 Å) using the Interactive Data Language (IDL). The spectra in these regions are shown in Fig. 5. In both wavelength ranges, it was necessary to correct the object spectra for contamination by terrestrial lines by using the spectra of A stars obtained in the same observing run. Contamination by stellar blends also needed to be taken into account. In some cases, further uncertainties in the procedure are introduced by rotational smearing. Since the characteristics of telluric lines and stellar blends are different in the two wavelength ranges, the procedures for measuring the equivalent widths of the He I lines are slightly different, as described in detail in the following two sections. Examples of corrected spectra and of fitted line profiles according to the procedures described in the following sections are shown in Fig. 6. A summary of the results together with relevant stellar parameters is given in Tables 3 and 4. In particular, we adopt the $v\sin i$ values given by Ammler-von Eiff & Reiners (2012), when available, or by Głȩbocki & Gnaciński (2005), with the exception of HD 48189A, for which the value given by Schröder et al. (2009) was adopted.

4.1 Measuring the D3 line

The He I $\lambda$5876 multiplet consists of 6 fine-structure lines arising from the transitions between levels $1s\,2p\>^{3}$P and $1s\,3d\>^{3}$D. The rest wavelength of the strongest component is at 5875.615 Å; four of the other components are separated at most by 25 mÅ from this component, while a sixth component, whose strength accounts for 1/9th of the total (exact value in LS coupling), is at 0.351 Å on the red side, giving a slightly asymmetric shape to the line in high-quality spectra.

The main difficulty in analyzing this line is that it is blended with a group of telluric H2O lines at wavelengths of 5875.444 Å, 5875.596 Å, 5875.769 Å, 5876.124 Å, and 5876.449 Å, as listed by Moore et al. (1966). The procedure we adopted to correct for telluric blends relies on the observed spectra of telluric standards. Since the geometrical air mass inferred from the time of the observations is normally a poor indicator of the water vapor column mass for each spectrum, we chose instead to use as proxies the strongest H2O lines in the range 5855–5930 Å which appear to be not blended with stellar lines in all the spectra. We found that the best proxy for this purpose is the sum of the equivalent widths of the H2O lines at 5919.6 Å, 5920.6 Å, and 5925.0 Å. From the measured value of this proxy in each object spectrum, we derived the corresponding telluric spectrum by interpolating the spectra of the telluric standard pixel-by-pixel as a function of the proxy. This procedure of course ignores the details of the excitation of the individual H2O lines and of all the other telluric lines. We estimated the error introduced by the telluric correction procedure on the $\lambda$5876 line at its rest wavelength and for $v\sin i$ = 0 to be 1 mÅ rms or less; the error increases to 2.7 and 5.0 mÅ for profiles rotationally broadened by 40 and 80 km s*-1* respectively. Assuming the telluric correction to be proportional to the line broadening, we estimated its error correction as $\sigma_{\mathrm{atm}}=0.89+0.047\times v\sin i$, with $v\sin i$ in km s*-1* and $\sigma_{\mathrm{atm}}$ in mÅ.

After the telluric spectrum was removed, we fitted the D3 line profile with a composite profile of 6 Gaussians, each representing one of the fine structure components of the multiplet. The wavelength and Gaussian width of the reference component $\lambda 5875.615$ were allowed to vary, while the wavelength separations and relative Gaussian widths of the other components were kept constant. The relative strengths were also kept constant at values proportional to the relative $gf$ values. Thus, this composite profile for the $\lambda$5876 multiplet is still determined by only three free parameters (multiplet equivalent width, position and width of the main component) as in standard, single-Gaussian fitting procedures. The multiple-Gaussian profile thus constructed was then broadened by the $v\sin i$ value given in literature for each target (see Tables 3 and 4).

The solar line list compilation by Moore et al. (1966) reports the presence of at least three photospheric lines potentially affecting the measurement of the $\lambda$5876 equivalent width: Fe I $\lambda$5876.30, Cr I $\lambda$5876.45, and an unidentified line at 5874.778 Å. In several spectra, particularly in cooler stars, we also noticed at least two other unidentified lines at 5875.15 Å and 5876.1 Å (see Fig. 5). All these lines can affect the determination of the equivalent width of the D3 line for $v\sin i>30$ km s*-1*.

In nearly all the spectra, we included some or all of those lines in the fit procedure as rotationally broadened Gaussians, adopting the $v\sin i$ value given in literature. In spectra with significant rotational broadening we constrained the wavelengths and, in some cases, the Gaussian widths of the fitting functions within reasonable bounds to prevent unphysical results.

We found that these blends tend to increase towards cooler spectral types, thus confirming their stellar origin. Only the very weak 5874.778 Å blend does not exhibit an obvious trend with $B-V$. Table 4.1 reports the coefficients of the linear fit of the equivalent widths of these stellar blends, measured in mÅ, as functions of $B-V$ in the form: $A_{\mathrm{B-V}}+B_{\mathrm{B-V}}\times B-V$, together with the standard deviation from the fit, $\sigma_{\mathrm{B-V}}$. For those stars with high $v\sin i$ for which some or all of those blends could not be measured, the above relation can be used to estimate their contribution to the He I $\lambda$5876 equivalent width. In particular, we found that the equivalent widths of the Fe I $\lambda$5876.30 and Cr I $\lambda$5876.45 lines increase with $B-V$ from $\sim 2$ mÅ at $B-V$=0.4 to about $\sim 12$ and $\sim 8$ mÅ at $B-V$=1, respectively.