Correlations in Chromospheric and Coronal Activity Indicators of Giant Stars

As noted above, we have cross-matched the Hünsch et al. (1998b) sample of ROSAT-detected giant stars with FUV sources in the GALEX GR6/7 catalog. We further cross-matched this sample with Gaia Data Release 2 data to find those stars with green and blue photometry, amounting to 191 giants. These stars are listed in Table 2 with their HD identifiers, GALEX FUV GR6/7 magnitude, Gaia green and blue magnitudes, and ROSAT X-ray luminosities. Johnson BV photometry is also available for these stars from a variety of sources including the paper of Hünsch et al. (1998b) and various Hipparcos catalogs.

Uncertainties in the Gaia photometry are derived from data in the Gaia catalog. Magnitude errors are not directly listed within the catalog because they are asymmetric. The asymmetry does not impact the overall conclusions of this work. As such, we used the formula

$$\begin{eqnarray}&&{\sigma }_{\mathrm{mag}}^{2}={\left(\displaystyle \frac{1.086{\sigma }_{F}}{F}\right)}^{2}+{\sigma }_{\mathrm{zp}}^{2}\end{eqnarray} \tag{ 3 }$$

to calculate the standard deviation of Gaia magnitudes, where F is the observed flux and ${\sigma }_{\mathrm{zp}}^{2}$ is the zero-point error (Jordi 2018). The calculated errors in the Gaia magnitudes are listed in Table 2. Most are comparatively small with respect to the plot scales that are used in Figures 2–5. The Vega magnitude zero-point errors estimated by the Gaia Collaboration are σ_G =0.0018 mag, ${\sigma }_{{G}_{{BP}}}=0.0014$ mag, and ${\sigma }_{{G}_{{RP}}}=0.0019$ mag.

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The stars in Table 2 have quite bright Gaia magnitudes. They fall in a magnitude range where the later eDR3 photometry constitutes a significant improvement over DR2 (Riello et al. 2021). One star is particularly commented upon below for which there is a noted difference between DR2 and eDR3 photometry.

Errors associated with GALEX FUV magnitudes were looked at in two ways. The GR6/7 data release lists formal uncertainties in FUV magnitude for the stars in our GALEX-ROSAT cross-matched sample. These have been entered into the column labeled (FUV) in Table 2. The average value of (FUV) among all stars in Table 2 is _av(FUV) = 0.07 mag, while the standard deviation is 0.07 mag. We take this average to be representative of the 1σ uncertainty in FUV mag for the giants in Table 2.

In a second approach uncertainties were considered by applying small number statistics to data for any star with multiple GALEX FUV magnitude measurements listed in the GR6/7 Data Release. The FUV error analysis was based on Table 1 of Keeping (1962), in which for a given small sample of measurements of a quantity, the standard deviation is computed as the ratio of the range among the measurements and a constant value given in the table as a function of sample size. These uncertainty estimates, denoted σ(FUV), range from 0.01 to 0.48 mag. Many stars in the GALEX-ROSAT cross-match sample, however, do not have estimates of this FUV magnitude error, as there were only 78 giants with multiple FUV measurements listed in GR6/7. As such, the average value, _av(FUV), is used for error bars shown in certain figures accompanying this section. These uncertainties _av(FUV) tend to be smaller than those computed on the basis of the small-sample statistics approach, which may be some indication that there is intrinsic variability among the FUV magnitudes of red giant stars. Any such variability would introduce scatter into our attempts to correlate the FUV excess against soft X-ray activity.

We are interested in exploring GALEX FUV photometry as an indicator of stellar activity in red giants. In order to facilitate this effort we have formulated two combined FUV–optical colors: (FUV − G_BP ) from a combination of GALEX FUV and Gaia photometry, and (FUV − B) from combining GALEX and Johnson photometry. It is this second color that was calibrated as an activity and age indicator for FGK main-sequence stars by Crandall et al. (2020). In the next subsections we attempt to place the GALEX-ROSAT cross-matched giants into the context of various color–magnitude diagrams, following which trends between measures of FUV and X-ray activity are searched for.

Figure 2 presents a photometric HR diagram of the crossed-matched ROSAT-GALEX sample of giant stars in which Gaia eDR3 (Gaia Collaboration et al. 2018a) observations are used to provide magnitude and color data. Herein, M_G is the Gaia green absolute magnitude obtained using Gaia parallaxes while the photospheric color plotted is (G_BP − G_RP ). The sample can be divided into two main groups with a gap around (G_BP − G_RP ) = 0.8 mag. To the left of this gap are stars belonging to a main-sequence population, or else are very slightly post-main sequence, whereas the red giants are to the right. This figure is similar to that of Figure 10 in the Gaia Collaboration's paper on the HR diagram (Gaia Collaboration et al. 2018b), although our sample is much smaller. The HR diagram from the Gaia Collaboration is particularly useful due to it being extensively populated by stars in the CHeB phase of evolution. A comparison with Figure 2 reveals that a large fraction of the stars in our ROSAT-GALEX giant sample are also likely to be in this stage of evolution. This is in contrast to stars ascending the RGB for the first or second time.

The stars plotted in Figure 2 are also shown in a different version of the HR diagram in Figure 3 in which M_G is plotted versus the (G_BP − G) color. We will employ this particular Gaia color below as it is most analogous to the Johnson (B − V) color that we will also use in several following sections.

As noted above, the red giant sample discussed in Section 4.2 contains CHeB stars. We have identified candidates for CHeB stars in Figure 3 through a comparison with Modules for Experiments in Stellar Astrophysics (MESA) evolutionary tracks (Paxton et al. 2011, 2013, 2015; Choi et al. 2016; Dotter 2016). The CHeB stars were expected to fall within 0.4 ⪅ (G_BP − G) ⪅ 0.8 mag and −1.0 ⪅ M_G ⪅ 1.0 mag. We plotted a range of MESA models and concluded that CHeB stars are bound by the 0.75 M_⊙ and 3.0 M_⊙ evolutionary tracks, as represented by the blue and purple lines in Figure 3, respectively. The red dots between these tracks in Figure 3 are identified as candidate CHeB stars.

Chromospheric emission is greatly dominated by photospheric flux at most optical wavelengths. However, FUV broadband photometry is sensitive to chromospheric and transition-region flux, including that from emission lines. Thus, a color such as (FUV − G_BP ) is expected to be sensitive to stellar activity. Figure 4 shows a two-color plot of (FUV − G_BP ) versus (G_BP − G) for the ROSAT-GALEX red giants selected for this work. The red symbols denote the Population I equivalent of horizontal branch stars (from Figure 3) that are likely burning helium in their cores. There is a notable range of (FUV − G_BP ) color among these CHeB stars, suggestive of a range in chromospheric activity levels.

In Figure 4 there is a group of bluer stars with a clear separation of activity levels. These stars have been plotted along with MESA evolutionary tracks in Figure 5 to identify their evolutionary phase. Both Figures 4 and 5 depict this set of stars with cyan points. This population is comprised of late-main-sequence and post-main-sequence stars, which have relatively blue FUV colors. The dark blue point in the figure corresponds to the most massive star of this group, and appears to be a subgiant which falls near a pre-RGB segment of the 2.5 M_⊙ evolutionary track (although its luminosity is comparable to that of redder RGB stars). As shown in Figure 4, this subgiant is relatively inactive based on the (FUV − G_BP ) color, indicating that it has evolved and spun down. Both late-main-sequence stars and the proposed subgiant are blueward of the evolutionary tracks marking the RGB region of the HR diagram.

Figure 6 shows the distribution of (B − V) measurements for the subgiant and late-main-sequence stars in our cross-matched ROSAT-GALEX sample. The majority of these stars have (B − V) ≤ 0.55 mag. As such, the photospheric flux likely impacts the utility of their FUV magnitude as an indicator of chromospheric activity. We note a color discrepancy for the subgiant star (dark blue in Figure 4). This star, HD 148374, has a Johnson color of (B − V) = 0.97 mag, but a Gaia DR2 color of (G_BP − G) = 0.31 mag. However, Gaia eDR3 observations give a more consistent color of (G_BP − G) = 0.16 for the subgiant.

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Crandall et al. (2020) calibrated a relationship involving FUV magnitude as an indicator of activity versus stellar age for dwarf stars. Their calibration was also dependent on Johnson (B − V) color. Dwarf stars with (B − V) ≤ 0.55 mag likely have significant photospheric contamination of the GALEX FUV band and so their calibration took an approach of defining an FUV-excess parameter from which photospheric contributions were empirically subtracted. A similar approach is taken here for giant stars.

To assist with quantifying chromospheric effects on FUV–optical colors of giant stars, the left panel of Figure 7 shows a two-color plot for a large sample of giant stars drawn from the rotational velocity study by Massarotti et al. (2008). Both GALEX FUV magnitudes and Johnson B and V magnitudes from the Hipparcos Input Catalogue were compiled for the Massarotti et al. (2008) sample. The stars observed by Massarotti et al. (2008) have a more extended coverage at redder (B − V) color than the ROSAT-GALEX cross-matched sample, which is plotted separately in the right panel of Figure 7.

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To compare FUV and X-ray emission as activity indicators we first introduce a FUV-excess parameter, Q_FUV, based on the Massarotti et al. (2008) sample (see similar excess parameters defined in Findeisen et al. 2011, Smith et al. 2017, Crandall et al. 2020, and Dixon et al. 2020). Here Q_FUV is defined as

$$\begin{eqnarray}&&{Q}_{\mathrm{FUV}}=(\mathrm{FUV}-B)-u,\end{eqnarray} \tag{ 4 }$$

where B is the Johnson blue magnitude and u corresponds to an empirically chosen relation between (FUV − B) and (B − V) for red giants having minimal levels of chromospheric activity. This FUV-excess parameter is defined in such a way that the greater the FUV flux of a star the more negative is the value of Q_FUV. The minimum-activity locus of u versus (B − V) adopted here is plotted in Figure 7 as a red line, and it is intended to represent the boundary of minimum FUV activity for a given (B − V) color. The locus for u (mag) was estimated to fit the quadratic function

$$\begin{eqnarray}&&u=-7.36{\left(B-V\right)}^{2}+16.62(B-V)+4.84(\mathrm{mag}).\end{eqnarray} \tag{ 5 }$$

The giant sample of Massarotti et al. (2008) has been utilized here for defining the function u rather than the sample of Hünsch et al. (1998b), since the former is expected to be less biased with respect to stellar chromospheric and coronal activity. By contrast, the Hünsch et al. (1998b) sample is comprised of stars selected on the basis of being detected in an all-sky soft X-ray survey. As such, it might be weighted toward more active red giants, whereas we have sought to define the locus u on the basis of it being an upper envelope in Figure 7 to giants with the lowest levels of GALEX FUV-band emission. The Massarotti et al. (2008) sample has fewer stars in the color range 0.8 < (B − V) < 0.9 than across the redder domain of 0.9 < (B − V) < 1.2. As such, the locus used here for u may have more inherent uncertainty for giants with (B − V) < 0.90.

We have used Johnson BV photometry rather than Gaia photometry to define the FUV-excess parameter Q_FUV such that our following results for giant stars can be readily compared with the work of Crandall et al. (2020) for main-sequence stars. In the next subsection the baseline of Equation (5) is applied to the X-ray sample of giants of Hünsch et al. (1998b). Uncertainties in values of Q are expected to be dominated by uncertainties in the GALEX FUV magnitudes, since ground-based photoelectric B − V photometry often has uncertainties on the order of 0.01–0.02 mag. Since the B − V colors employed in this work come from several sources it is not straightforward to obtain a precise B − V uncertainty for the stars in Table 2.

Uncertainties in both (FUV − B) and Q_FUV can be addressed more specifically. An uncertainty σ_(FUV−B) in the color (FUV − B) can be estimated from uncertainties σ_FUV and σ_B in the FUV and B magnitudes, respectively, as follows:

$$\begin{eqnarray}&&{\sigma }_{(\mathrm{FUV}-B)}^{2}={\sigma }_{\mathrm{FUV}}^{2}+{\sigma }_{B}^{2}.\end{eqnarray} \tag{ 6 }$$

If we take the uncertainty σ_FUV for a star in Table 2 to be given by the value of (FUV) from that table, and assuming σ_B = 0.02 mag, then estimates of σ_(FUV−B) can be obtained for stars in Table 2. An average uncertainty can be had by setting σ_FUV = _av(FUV) = 0.070 mag and, σ_B = 0.020 mag, from which follows an average σ_(FUV−B) = 0.073 mag. The uncertainty in the (FUV − B) color is therefore going to be typically dominated by measurement uncertainty in the GALEX FUV magnitude.

Error estimation for the Q_FUV excess parameter follows from

$$\begin{eqnarray}&&{\sigma }_{Q}^{2}={\sigma }_{(\mathrm{FUV}-B)}^{2}+{\sigma }_{u}^{2},\end{eqnarray} \tag{ 7 }$$

where σ_(FUV−B) is calculated as in the previous paragraph. Since u is a function of (B − V), the term σ_u in Equation (7) is given by

$$\begin{eqnarray}&&{\sigma }_{u}=\left(\displaystyle \frac{{du}}{d[B-V]}\right){\sigma }_{(B-V)},\end{eqnarray} \tag{ 8 }$$

with du/d[B − V] = − 14.72(B − V) + 16.62 from Equation (5). If we adopt σ_(B−V) = 0.02 mag, then for a giant star with (B − V) = 1.0 mag it follows that σ_u = 0.04 mag. An average value for σ_Q can be calculated upon adopting σ_(FUV−B) = 0.07 mag, as above, leading to an estimate of σ_Q = 0.08 mag. This estimate shows that the uncertainty in Q is dominated by uncertainty in the (FUV − B) color, from which it follows that σ_Q for the typical star in Table 2 is again dominated by uncertainty in the FUV magnitude.

We next compare the chromospheric activity parameter Q_FUV based on GALEX observations to coronal activity using the ROSAT sample of giants published by Hünsch et al. (1998b). Figure 8 shows the FUV-excess parameter Q_FUV versus X-ray-to-bolometric luminosity ratio, $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$, for the cross-matched sample. Not included are stars from Figure 5 with 0.0 ≤ (B − V) ≤ 0.6 mag, as this is a range in which we were unable to define the minimum chromospheric activity locus, u, in Figure 7. This limit excludes all of the late-main-sequence stars (cyan points), and reduces the sample size to one subgiant, 59 CHeB, and 79 other giant stars.

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Correlations between Q_FUV and $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ in Figure 8 were constrained for three samples: all giants in the figure (green line), CHeB stars (red line), and other giants (black line). These fits take the form of

$$\begin{eqnarray}&&{Q}_{\mathrm{FUV}}=-0.74\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})-6.09(\mathrm{mag})\end{eqnarray} \tag{ 9 }$$

for the CHeB giants,

$$\begin{eqnarray}&&{Q}_{\mathrm{FUV}}=-1.22\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})-9.47(\mathrm{mag})\end{eqnarray} \tag{ 10 }$$

for the non-CHeB giants, and

$$\begin{eqnarray}&&{Q}_{\mathrm{FUV}}=-0.94\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})-7.52(\mathrm{mag})\end{eqnarray} \tag{ 11 }$$

for all giants in the figure. The rms scatter in Q_FUV about the fits to CHeB giants, non-CHeB giants, and the full sample in Figure 8 are 1.28, 1.76, and 1.60 mag, respectively. This scatter is considerable compared to the observational uncertainty in the FUV magnitude, which in turn dominates the uncertainty in Q_FUV. The fit parameters, the rms scatter about Q_FUV, as well as the Spearman correlation coefficient, ρ, and coefficient of determination are listed in Table 3. We note that these fits have significant errors due to the scatter in Q_FUV. There is also little correlation between Q_FUV and $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$, as is evident in the poor Spearman coefficients.

Table 2. GALEX Data for a ROSAT Giant Star Sample

Name	GALEX		GALEX	Gaia	Gaia	Gaia	Gaia	ROSAT
	FUV	Q	(FUV)	G	σG	GBP	${\sigma }_{{G}_{\mathrm{BP}}}$	$\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$
	(mag)		(mag)	(mag)	(mag)	(mag)	(mag)
HD 1227	20.08	−0.85	0.10	5.87	0.00	6.36	0.00	−6.09
HD 1522	18.38	−0.56	0.05	3.08	0.01	3.94	0.01	−6.94
HD 1737	18.28	−2.00	0.06	4.87	0.00	5.43	0.00	−5.24
HD 2630	14.65	−1.85	0.01	6.04	0.00	6.26	0.00	−5.35
HD 4482	18.04	−2.46	0.08	5.22	0.00	5.76	0.00	−5.48
HD 4502	15.21	−4.19	0.02	3.68	0.00	4.39	0.01	−4.81
HD 4737	18.84	−2.16	0.11	6.03	0.00	6.50	0.00	−5.32
HD 5516	17.78	−1.54	0.06	4.05	0.01	4.64	0.01	−5.31
HD 6245	18.48	−1.62	0.07	5.11	0.00	5.62	0.00	−5.81
HD 6559	19.15	−2.29	0.11	5.82	0.00	6.39	0.00	−5.31
HD 6763	13.40	−2.10	0.00	5.40	0.00	5.61	0.00	−5.66
HD 6793	17.38	−2.64	0.04	5.09	0.00	5.59	0.00	−4.51
HD 7672	16.38	−3.77	0.01	5.24	0.00	5.78	0.01	−3.9
HD 8634	14.98	−2.26	0.01	6.06	0.00	6.32	0.00	−5.19
HD 8829	12.92	−2.85	0.01	5.41	0.00	5.61	0.00	−5.77
HD 8921	19.89	−1.52	0.13	5.72	0.00	6.45	0.01	−6.11
HD 9774	18.85	−1.40	0.13	4.98	0.00	5.53	0.00	−6.23
HD 10072	18.04	−1.63	0.08	4.73	0.00	5.25	0.00	−5.34
HD 10308	15.98	−1.36	0.02	6.09	0.00	6.34	0.00	−4.6
HD 10588	18.59	7.41	0.09	6.07	0.00	6.55	0.00	−5.1
HD 11025	14.86	−5.73	0.02	5.40	0.00	5.91	0.00	−5.22
HD 11559	17.99	−1.53	0.05	4.30	0.00	4.85	0.00	−5.65
HD 11937	16.83	−1.36	0.03	3.38	0.00	4.01	0.01	−4.92
HD 12055	13.65	−5.82	0.01	4.53	0.00	5.06	0.00	−5.25
HD 12173	13.25	2.18	0.01	6.16	0.00	6.24	0.00	−5.79
HD 12641	16.89	−3.69	0.01	5.70	0.00	6.18	0.00	−5.28
HD 15889	19.56	−1.89	0.16	5.98	0.00	6.53	0.00	−5.15
HD 15920	18.63	−1.18	0.07	4.88	0.00	5.40	0.00	−5.7
HD 16058	18.60	−0.56	0.11	4.41	0.00	5.67	0.01	−5.91
HD 16161	18.66	−0.79	0.10	4.58	0.01	5.11	0.00	−5.99
HD 16246	15.15	−2.18	0.01	6.38	0.00	6.62	0.00	−4.28
HD 16327	15.57	−2.10	0.02	6.07	0.00	6.33	0.00	−5.14
HD 17006	18.86	−1.89	0.07	5.86	0.00	6.33	0.00	−4.98
HD 17824	18.28	−1.25	0.07	4.45	0.00	5.02	0.00	−5.49
HD 18953	18.38	−1.84	0.03	5.06	0.00	5.57	0.00	−4.82
HD 19926	13.95	−6.90	0.01	5.11	0.00	5.80	0.00	−5.37
HD 20313	14.09	−0.94	0.02	5.59	0.00	5.77	0.00	−5.93
HD 21024	14.92	−1.77	0.02	5.39	0.00	5.65	0.00	−5.45
HD 22231	20.56	−0.44	0.22	5.34	0.00	5.93	0.00	−5.87
HD 23838	14.91	−4.73	0.01	5.42	0.00	5.86	0.00	−5.4
HD 24497	11.83	−9.04	0.00	5.90	0.00	6.43	0.00	−5.52
HD 26076	20.82	−0.34	0.22	5.76	0.00	6.29	0.00	−5.05
HD 26575	19.69	−2.02	0.11	6.12	0.00	6.69	0.00	−5.51
HD 27022	17.72	−1.83	0.05	5.02	0.00	5.48	0.00	−5.0
HD 28525	14.60	−5.54	0.02	5.42	0.00	5.90	0.00	−5.09
HD 31553	18.65	−2.47	0.09	5.41	0.00	6.06	0.00	−5.58
HD 31910	15.83	−3.02	0.02	3.69	0.00	4.29	0.01	−6.02
HD 34172	19.11	−1.60	0.12	5.57	0.00	6.07	0.00	−5.99
HD 34658	14.37	−1.80	0.01	5.21	0.00	5.46	0.00	−5.77
HD 37434	17.27	−4.22	0.04	5.76	0.00	6.40	0.00	−4.93
HD 37763	20.59	0.05	0.30	4.81	0.00	5.44	0.00	−6.63
HD 38645	19.25	−1.89	0.09	5.97	0.00	6.46	0.00	−5.25
HD 39070	16.92	−3.21	0.03	5.32	0.00	5.81	0.01	−5.64
HD 39523	18.18	−1.65	0.06	4.13	0.00	4.78	0.00	−5.63
HD 39743	17.88	−3.66	0.04	6.28	0.00	6.83	0.01	−4.19
HD 46730	13.81	−2.33	0.01	6.19	0.00	6.39	0.00	−6.09
HD 47442	14.99	−4.81	0.01	4.02	0.00	4.71	0.00	−6.49
HD 47703	16.54	−1.63	0.02	6.33	0.00	6.62	0.00	−5.54
HD 50337	12.12	−7.10	0.01	4.10	0.00	4.66	0.00	−6.22
HD 50522	15.10	−3.75	0.01	4.19	0.01	4.58	0.00	−5.87
HD 51266	20.77	−0.56	0.29	5.98	0.00	6.49	0.00	−5.54
HD 54719	19.62	−0.14	0.12	3.95	0.01	4.69	0.01	−6.68
HD 57727	18.77	−1.00	0.07	4.73	0.00	5.27	0.00	−5.61
HD 62141	19.60	−1.47	0.11	5.99	0.00	6.49	0.00	−5.69
HD 62264	16.32	−5.03	0.02	5.99	0.00	6.43	0.00	−5.74
HD 62898	19.09	−0.24	0.14	4.34	0.00	5.42	0.00	−6.40
HD 68290	17.93	−1.72	0.05	4.41	0.00	4.95	0.00	−5.53
HD 69148	18.43	−1.98	0.09	5.48	0.00	5.96	0.00	−5.88
HD 71152	13.98	−2.77	0.01	6.88	0.00	7.04	0.00	−4.36
HD 71369	17.01	−0.80	0.05	3.03	0.00	3.77	0.01	−6.44
HD 71433	16.64	−1.87	0.04	6.47	0.00	6.76	0.00	−5.02
HD 73596	14.64	3.70	0.01	6.10	0.00	6.33	0.00	−5.89
HD 74485	18.97	−2.06	0.06	5.87	0.00	6.36	0.00	−5.30
HD 77996	18.74	−1.61	0.06	4.58	0.00	5.26	0.00	−5.58
HD 78235	18.30	−1.83	0.05	5.15	0.00	5.66	0.00	−5.02
HD 78668	18.65	−2.02	0.08	5.50	0.00	6.01	0.00	−5.06
HD 79193	13.39	−1.08	0.01	6.05	0.00	6.19	0.00	−5.59
HD 79940	14.27	−1.63	0.01	4.45	0.00	4.76	0.00	−6.31
HD 80710	16.73	−4.73	0.03	5.71	0.00	6.41	0.00	−5.95
HD 81799	18.18	−1.87	0.08	4.32	0.00	4.97	0.00	−5.66
HD 81873	19.79	−1.12	0.08	5.39	0.00	5.97	0.00	−5.78
HD 82210	16.56	−2.05	0.03	4.28	0.00	4.76	0.00	−4.46
HD 82635	17.06	−2.31	0.04	4.23	0.00	4.79	0.00	−5.10
HD 83108	15.07	−2.36	0.02	6.37	0.00	6.60	0.00	−5.29
HD 84441	15.08	−2.13	0.02	2.60	0.00	3.73	0.07	−7.00
HD 85206	18.29	−3.05	0.12	5.53	0.00	6.22	0.00	−5.72
HD 85396	19.24	−0.90	0.10	5.15	0.00	5.67	0.00	−5.62
HD 85945	18.06	−2.57	0.07	5.71	0.00	6.20	0.00	−4.46
HD 87682	19.51	−1.60	0.17	5.94	0.00	6.43	0.00	−5.26
HD 88639	18.17	−2.37	0.07	5.82	0.00	6.30	0.01	−4.85
HD 88786	19.52	8.22	0.15	6.24	0.00	6.70	0.00	−5.28
HD 90071	13.64	−2.23	0.01	6.16	0.00	6.35	0.00	−5.25
HD 91135	16.66	−2.06	0.03	6.38	0.00	6.67	0.00	−5.94
HD 93813	18.12	−0.36	0.06	2.79	0.03	3.71	0.03	−7.00
HD 95314	20.56	−0.03	0.27	5.24	0.00	6.15	0.00	−5.79
HD 98233	20.48	−1.24	0.26	6.43	0.00	6.93	0.00	−5.28
HD 99564	15.64	−2.00	0.02	5.80	0.00	6.08	0.01	−5.15
HD 99967	19.50	−2.20	0.14	5.91	0.00	6.61	0.01	−6.12
HD 100418	16.42	−2.39	0.03	5.90	0.00	6.22	0.00	−5.27
HD 101107	13.64	−1.80	0.01	5.44	0.00	5.67	0.00	−6.04
HD 101112	20.51	−0.94	0.27	5.90	0.00	6.46	0.00	−5.91
HD 101132	13.77	−1.98	0.01	5.53	0.00	5.75	0.00	−5.42
HD 101154	20.19	−1.37	0.20	5.95	0.00	6.55	0.00	−5.45
HD 102070	17.59	−2.15	0.08	4.37	0.00	4.96	0.00	−6.19
HD 103484	18.12	−2.36	0.07	5.28	0.00	5.81	0.00	−5.23
HD 104438	19.43	−1.29	0.16	5.26	0.00	5.85	0.00	−6.11
HD 106677	16.46	−5.19	0.04	5.86	0.00	6.49	0.01	−3.96
HD 108225	19.42	−0.57	0.09	4.72	0.00	5.26	0.00	−6.15
HD 109272	19.60	−0.53	0.16	5.33	0.00	5.81	0.00	−5.94
HD 109379	16.08	−1.26	0.03	2.25	0.00	7.51	0.41	−7.44
HD 112989	16.63	−3.65	0.04	4.42	0.00	5.12	0.00	−5.96
HD 113049	19.05	−2.03	0.14	5.70	0.00	6.26	0.00	−5.27
HD 113226	17.25	−0.48	0.05	2.45	0.00	3.72	0.08	−6.47
HD 114474	19.60	−0.88	0.11	4.90	0.00	5.50	0.01	−5.97
HD 115337	15.63	−5.52	0.01	6.06	0.00	6.53	0.01	−5.63
HD 115659	17.14	−0.68	0.05	2.62	0.00	3.52	0.03	−6.02
HD 117566	17.46	−2.36	0.04	5.54	0.00	5.97	0.00	−4.73
HD 119458	16.43	−4.05	0.03	5.75	0.00	6.21	0.00	−6.12
HD 120048	18.54	−2.30	0.12	5.68	0.00	6.19	0.00	−5.05
HD 120064	14.38	−3.27	0.01	5.82	0.00	6.12	0.00	−5.03
HD 122744	19.42	−1.74	0.11	6.01	0.00	6.49	0.00	−5.67
HD 129312	17.52	−2.44	0.05	4.53	0.01	5.10	0.00	−5.71
HD 130529	18.23	−2.76	0.11	5.22	0.00	5.98	0.00	−5.44
HD 132813	15.72	−3.13	0.02	3.09	0.02	4.95	0.02	−6.67
HD 133208	17.72	−0.78	0.05	3.14	0.00	3.86	0.01	−6.76
HD 136138	13.95	−6.75	0.01	5.39	0.00	5.94	0.00	−5.11
HD 136407	14.15	−2.97	0.01	6.02	0.00	6.25	0.00	−5.59
HD 139906	17.90	−2.34	0.07	5.60	0.00	6.06	0.00	−5.81
HD 141714	16.83	−2.03	0.05	4.33	0.00	4.80	0.00	−4.91
HD 144208	11.40	−6.83	0.00	5.60	0.00	5.97	0.00	−5.53
HD 147266	19.35	−1.59	0.16	5.77	0.00	6.28	0.00	−5.63
HD 147675	16.80	−1.87	0.04	3.52	0.00	4.16	0.01	−5.27
HD 148374	19.57	−1.07	0.15	5.76	0.01	6.07	0.02	−5.87
HD 148387	17.10	−0.42	0.05	2.48	0.02	3.22	0.02	−7.62
HD 150450	18.78	−0.59	0.09	3.94	0.00	5.15	0.00	−6.97
HD 150682	14.42	−2.22	0.01	5.80	0.00	6.03	0.00	−5.31
HD 150997	17.02	−1.33	0.04	3.12	0.00	3.88	0.01	−6.58
HD 151087	13.54	−1.78	0.01	5.94	0.00	6.12	0.00	−5.28
HD 151900	14.67	−2.60	0.02	6.19	0.00	6.43	0.00	−5.22
HD 153956	20.44	−0.97	0.16	5.69	0.00	6.31	0.00	−6.03
HD 154619	19.50	−1.47	0.15	6.13	0.00	6.61	0.00	−5.71
HD 155103	13.04	−1.94	0.01	5.28	0.00	5.50	0.00	−6.21
HD 156015	13.63	3.40	0.01	5.19	0.01	5.63	0.02	−4.85
HD 156266	19.37	−0.72	0.14	4.39	0.01	4.99	0.00	−6.54
HD 156971	14.00	−2.31	0.01	6.37	0.00	6.56	0.00	−5.60
HD 159181	14.39	−3.44	0.01	2.38	0.00	3.35	0.03	−5.90
HD 161814	19.62	−1.26	0.14	5.49	0.00	6.03	0.00	−5.47
HD 163217	18.27	−2.27	0.09	4.76	0.00	5.44	0.00	−5.94
HD 165462	19.45	−2.14	0.14	5.97	0.00	6.57	0.00	−5.32
HD 166208	17.66	−2.12	0.05	4.71	0.00	5.23	0.00	−5.45
HD 168322	20.71	−0.47	0.28	5.84	0.00	6.37	0.00	−5.80
HD 169836	19.81	−0.99	0.14	5.46	0.00	5.99	0.00	−5.10
HD 175824	14.98	−1.85	0.01	5.71	0.00	5.98	0.00	−5.19
HD 176598	18.43	−2.13	0.10	5.35	0.00	5.88	0.00	−5.30
HD 180006	18.81	−1.44	0.09	4.82	0.00	5.38	0.00	−6.22
HD 181597	21.16	−0.49	0.31	5.99	0.00	6.59	0.00	−5.83
HD 184398	13.11	−8.63	0.01	6.00	0.00	6.66	0.01	−5.02
HD 184492	18.16	−2.31	0.06	4.71	0.00	5.37	0.00	−5.47
HD 187372	19.34	−0.72	0.06	5.31	0.00	6.42	0.00	−6.23
HD 189831	19.61	−0.21	0.12	4.20	0.00	5.04	0.00	−6.80
HD 190252	19.24	−1.73	0.09	6.08	0.00	6.55	0.00	−5.37
HD 196385	13.63	−2.59	0.01	6.29	0.00	6.48	0.00	−5.65
HD 196574	18.71	−0.54	0.06	4.01	0.00	4.56	0.00	−6.22
HD 199253	19.46	−1.05	0.11	4.84	0.00	5.46	0.00	−5.88
HD 199442	21.32	−0.11	0.24	5.70	0.00	6.33	0.00	−5.26
HD 199532	14.40	−2.46	0.01	4.93	0.00	5.26	0.00	−4.45
HD 199665	19.32	−1.02	0.10	5.24	0.00	5.75	0.00	−5.70
HD 199951	17.28	−2.08	0.04	4.37	0.00	4.90	0.00	−5.02
HD 200763	20.12	−0.37	0.15	4.85	0.00	5.46	0.00	−5.95
HD 202951	19.86	0.02	0.08	5.19	0.00	6.25	0.01	−5.81
HD 203387	16.74	−2.28	0.03	3.97	0.00	4.52	0.00	−4.94
HD 204960	19.95	−0.82	0.13	5.26	0.00	5.83	0.00	−6.04
HD 207964	14.35	−2.14	0.01	5.81	0.00	6.03	0.00	−4.92
HD 209278	12.27	−5.05	0.00	7.06	0.00	7.12	0.01	−4.81
HD 210960	14.94	−4.61	0.01	5.27	0.00	5.80	0.00	−5.95
HD 211416	16.40	−1.57	0.04	2.22	0.00	3.57	0.03	−7.08
HD 212132	13.77	−2.08	0.01	5.52	0.00	5.74	0.00	−5.16
HD 212271	18.67	−1.93	0.05	5.26	0.00	5.78	0.00	−4.90
HD 214987	19.45	−1.66	0.09	5.80	0.00	6.32	0.00	−5.36
HD 214995	19.38	−1.85	0.06	5.58	0.00	6.19	0.00	−5.40
HD 215545	13.99	−2.03	0.01	6.54	0.00	6.70	0.00	−5.19
HD 216489	15.94	−5.04	0.02	5.66	0.01	6.28	0.02	−4.02
HD 216718	16.36	−4.01	0.03	5.53	0.01	5.97	0.00	−5.82
HD 216756	14.34	−2.51	0.01	5.79	0.00	6.02	0.00	−5.84
HD 218356	14.79	−5.21	0.02	4.24	0.00	5.03	0.00	−5.23
HD 218527	18.37	−1.81	0.07	5.13	0.00	5.66	0.00	−5.51
HD 218658	14.10	−4.54	0.01	4.20	0.00	4.66	0.01	−5.29
HD 218670	17.44	−1.61	0.04	3.52	0.00	4.16	0.01	−5.43
HD 219916	15.76	−3.44	0.03	4.56	0.00	5.10	0.00	−6.34
HD 220657	14.61	−2.64	0.01	4.20	0.00	4.60	0.00	−4.90
HD 223011	12.45	−1.93	0.00	6.27	0.00	6.39	0.00	−5.19
HD 223346	16.01	−1.62	0.01	6.34	0.00	6.60	0.00	−5.47
HD 223460	16.45	−3.61	0.02	5.62	0.00	6.09	0.00	−4.30

A machine-readable version of the table is available.

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Table 3. Fits for FUV-excess Parameter Q_FUV vs. $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$

Sample a	Nb	ac	bc	ρd	(r2) e	rms f
CHeB giants	59	−0.74	−6.09	−0.42	0.07	1.28
Non-CHeB giants	79	−1.22	−9.47	−0.49	0.19	1.76
All giants and subgiants	139	−0.94	−7.52	−0.43	0.12	1.60

Notes.

^a Sample of giants. ^b Number of stars within each sample. ^c Fit parameters of Q_FUV = $a\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ + b. ^d Spearman coefficient. ^e Coefficient of determination. ^f rms variation of Q_FUV about each fit.

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Figure 8 shows that while there is a range of atmospheric activity for giants, as indicated by both FUV-excess and X-ray observations, there is significant scatter between the two indicators. We investigated binarity as a possible contribution to this phenomenon. In Figure 9 we show the X-ray luminosity ratio versus Q_FUV while focusing on certain populations of stars: no variable stars, RS CVn, visual, symbiotic, or spectroscopic binaries (top left); no spectroscopic, RS CVn, or symbiotic binaries (top right); only spectroscopic, RS CVn, and symbiotic binaries (bottom left); and all RS CVn, visual, symbiotic, and spectroscopic binaries (bottom right). Binarity for each star was found in the SIMBAD online data system. There is a reduced amount of scatter in Q_FUV for the population that does not include any binaries or variable stars (top-left panel of Figure 9), with high activity levels of Q_FUV < −3.0 mag appearing to be less frequent among single giant stars than among binaries.

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In addition to the least-squares fits discussed above, we have experimented with fitting an upper limit to the value of Q_FUV as a function of normalized X-ray luminosity. Green lines on the bottom panels of Figure 9 take the equations

$$\begin{eqnarray}&&{Q}_{\mathrm{FUV}}=-1.42\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})-9.14\mathrm{mag}\end{eqnarray} \tag{ 12 }$$

for the sample of spectroscopic, RS CVn, and symbiotic binaries, and

$$\begin{eqnarray}&&{Q}_{\mathrm{FUV}}=-1.30\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})-8.55\mathrm{mag}\end{eqnarray} \tag{ 13 }$$

for the sample with all binaries. These Q_FUV boundary equations were determined by fitting to the giants with the most positive value of Q_FUV at a given (L_X/L_bol). They indicate that for giants in binary systems there is a lower limit to the FUV excess that is a function of X-ray luminosity. The dashed green line in the top-left panel is the upper boundary to Q_FUV for all binaries transposed from the bottom-right panel. The nonbinaries in the upper panels of Figure 9 do not follow the same boundary to Q_FUV as the binaries. The limiting value of Q_FUV shows a more apparent correlation with X-ray brightness among active binary stars (bottom panels of Figure 9) than among nonbinary giants. At a given value of Q_FUV ∼ −2.0 mag there is a range of a factor of 10 or more in soft X-ray luminosity in all panels. This is similar to the scatter in (L_X/L_bol) seen at a given (F_{Mg II} /F_bol) in Figure 1. In summary, the single giants show much less evidence of a correlation between Q_FUV and X-ray luminosity than binary stars, although there is substantial scatter in these relationships for all groupings of giants in Figure 9. The trend shown by the green line in this figure may pertain to physical situations in which stellar activity is controlled by an internal dynamo that is enhanced within giants whose rotation is "spun up" by virtue of membership in a binary.

There are five stars with Q_FUV < −4.0 mag in the top-left panel of Figure 9 which are not binaries or variables: HD 24497, HD 28525, HD 47442, HD 11025, and HD 12055. These giants are quite active as indicated by their Q_FUV values, and for three of them the X-ray luminosity is also relatively high. As they are single stars, their activity levels are not impacted by companions. Additionally, they are not pulsating. It was of interest to investigate their GALEX FUV measurements further. The left panel of Figure 10 shows Q_FUV versus GALEX FUV magnitude for the population of stars in the top-left panel of Figure 9. The five stars with Q_FUV < −4 mag have the brightest FUV apparent magnitudes in the sample, indicating that their GALEX observations would not likely have enhanced uncertainties from being faint sources. A plot of Q_FUV versus Johnson (B − V) color for this population is shown in the right panel of Figure 10, from which it is apparent that these five stars are not different in effective temperature from the CHeB and giant peers. The nature of these sources is unclear to us. It is possible that these are observational artifacts or a confusion of two unresolved sources in the GALEX FUV images. One possible explanation for the very negative Q_FUV values for the five stars might be a white dwarf companion, but there is no indication from the SIMBAD database to suggest such might be the case.

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Saturation in the GALEX detectors can lead to an underestimation of flux for sources brighter than both near-ultraviolet (NUV) and FUV m_AB ∼ 15 (Morrissey et al. 2007). As such, we have replotted the top-left panel of Figure 9 upon excluding the sources with FUV < 15 mag. The result is shown in Figure 11. The fit between FUV-excess color and $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ is shown in green with an rms scatter about the fit of 0.64 mag. The Spearman coefficient, ρ, for the population is −0.53 and r² = 0.27. The correlation shown by the fit in Figure 11 is not a strong one.

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CHeB stars in our GALEX-ROSAT sample exhibit a range in Q_FUV excess. A number of the chromospheric and transition-region emission lines radiated by G- and K-type stars within the wavelength range of the GALEX FUV band have been studied by Ayres et al. (1981). In their Figures 2 and 3, they demonstrated that K dwarfs and giants can have weaker emission lines than G dwarfs and giants, respectively. Thus, despite the fact that a K giant will have a lower photospheric brightness in the FUV band than a G giant, the contrast between the chromospheric and photospheric contributions may be reduced for the K giant. This may help account for why the greatest range in FUV excesses is observed herein among giants of similar temperatures to the CHeB stars than among redder and cooler giants (e.g., Figure 7). The giants of the Hyades clusters are CHeB stars that are effectively coeval yet are known to exhibit a range of stellar activity. They may provide some context for the spread in FUV-band activity found among the field CHeB stars. As such, we turn to this small ensemble of stars in the next section.

The FUV-excess parameter, Q_FUV, has been introduced in the above work as a measure of the FUV excess of a giant star relative to the least-active giants at a given B − V color. It is dependent on the minimum-activity locus u(B − V), as defined in Equation (5), and as such is influenced by errors in the FUV magnitude, the (B − V) color, and the degree to which the equation for u as a function of (B − V) provides a good match to a locus for minimum-activity giants. The appropriate locus for u might depend on the evolutionary state of a red giant, i.e., shell hydrogen burning versus core helium burning. Furthermore, stars with among the largest FUV magnitude errors in Table 2 typically have values of Q_FUV relatively close to zero. This is a consequence of those stars that are faintest in the FUV band having relatively large values of FUV magnitude, and therefore the Q_FUV index. Consequently, the equation for the envelope function u is sensitive to a subset of stars with among the largest errors in FUV mag. This is a factor of potential error in the derivation of the u function. Were some of the above error terms to affect the placement of u systematically as a function of (B − V), then a (B − V)-dependent scatter would be introduced into Figures 8 through 11. Since the average value of (FUV) for the stars in Table 2 is 0.07 mag, while for stars in Table 5 it is 0.18 mag, possible 3σ offsets in the apparent u locus of 0.5 mag, or more, relative to the intrinsic stellar locus might be present, particularly outside the color range 0.90 ≤ (B − V) ≤ 1.20, where the left panel of Figure 7 is less well populated.

Given such potential uncertainties in the employed equation for u as a function of (B − V), we have also taken an approach to the data in which no FUV-excess parameter such as Q_FUV is involved, but rather the (FUV − B) color is compared directly with the X-ray-to-bolometric luminosity ratio, $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$, among giants within limited (B − V) color ranges. Figure 12 shows (FUV − B) plotted versus $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ for giants grouped into three B − V bins. Stars that have been identified as possible CHeB giants are plotted with open circles to distinguish them from stars (filled circles) in other giant-branch phases of evolution (such as first-ascent giants or asymptotic giant branch stars).

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Within each color bin in Figure 12 there is a trend for the (FUV − B) color to be bluer, on average, among the giants with higher X-ray-to-bolometric luminosity ratios. This tendency is perhaps more clearly evinced among the redder giants as compared to the color range that includes the CHeB stars. However, all three panels of Figure 12 are characterized by significant scatter about any possible correlation. Stars with FUV magnitude errors greater than 0.15 mag, as given in the GR6/7 data release, are designated with crosses in Figure 12. Excluding these stars from consideration does not alter the conclusions or tighten any of the correlations.

A large sample of red giants with rotational velocities was needed in order to have a reasonable number of stars when cross-matched with the GALEX catalog. The sample used in this analysis was derived from Massarotti et al. (2008), who made rotational velocity measurements for 761 giants within 100 pc of the Sun. Massarotti et al. (2008) determined projected rotational $v\sin i$ velocities with spectra acquired with CfA Digital Speedometers (Latham 1992) on the Wyeth Reflector at Oak Ridge Observatory and the Tillinghast Reflector and MMT at the Whipple Observatory. Stellar spectra were cross-correlated with synthetic spectrum templates from Kurucz (1992) to derive rotation speeds. We cross-matched the Massarotti et al. (2008) catalog with the GALEX GR6/7 source archive as described in Section 2, and the Hipparcos catalog, which resulted in a sample of 267 giants with rotational velocities as well as FUV and Johnson B magnitudes. Three of these stars have quoted rotational velocities of $v\sin i\gt 32$ km s⁻¹. There are zero stars with rotational velocities between $v\sin i\sim 15$ km s⁻¹ and these three stars. With such a large gap, it is not enlightening to include the three fast-rotating stars in our attempt to correlate Q_FUV and rotational velocity. These three stars are HIP 3693 (RS CVn, K1 giant; Koevari et al. 2007), HIP 75127 (K1 giant; Houk & Swift 1999), and HIP 112748 (G8 giant; Keenan & McNeil 1989). We were then left with a sample of 264 giant stars. The great majority of these stars have projected rotation speeds less than 6 km s⁻¹.

Stars in the cross-matched GALEX-rotation speed sample are listed in the Appendix as Table 5, along with FUV and NUV magnitudes, plus errors (FUV) and (NUV), all from the GR6/7 data release, as well as the projected rotation speed from Massarotti et al. (2008). The average value of (FUV) for the stars in Table 5 is _av(FUV) = 0.18 mag, with a standard deviation of 0.12 mag. In the case of the NUV magnitude the uncertainties are much smaller, with _av(NUV) = 0.01 mag and a standard deviation of 0.01 mag.

Figure 13 shows GALEX FUV-excess parameter Q_FUV versus Massarotti et al. (2008) rotational velocities. Values of Q_FUV were determined by Equations (4) and (5). We do not see a clear correlation between FUV radiation and rotation for stars with $v\sin i\leqslant 6$ km s⁻¹ in this figure. A possible explanation for the scatter could be a spread in the angle of inclination of the stellar rotation axes. Such variations in $\sin i$ may obscure any trends. Stars bright in the FUV may have intrinsically high rotation, but if the rotation axis is inclined close to the line of sight to a star the full rotation would not be observed. The lack of correlation in Figure 13 would then be the result of a projection effect. In such a case correlations would not be expected between $v\sin i$ and other chromospheric activity indicators among this sample of giants.

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We further investigated the giants in Figure 13 with very negative Q_FUV values, those with Q_FUV ≤ −4.0. Eleven stars fall in that regime, with eight of them being binaries: HIP 7097, HIP 10661, HIP 23221, HIP 52085, HIP 53240, HIP 62886, HIP 74896, and HIP 78481. HIP 68904 is identified as a pulsating star (Strassmeier et al. 2011), which may impact observations of activity levels. The remaining two stars, HIP 79882 and HIP 94490, designated with "+" symbols in Figure 13, are not binaries but high-proper-motion giants.

NUV emission has been shown to correlate with rotational velocities for red giant stars by Dixon et al. (2020). In their work they compiled a set of 133 red giants that were observed by the Sloan Digital Sky Survey APOGEE project and also have GALEX NUV magnitudes. Similar to the analysis in Dixon et al. (2020), we have defined an NUV-excess parameter, described here as Q_NUV. This excess is defined by

$$\begin{eqnarray}&&{Q}_{\mathrm{NUV}}=(\mathrm{NUV}-B)-u,\end{eqnarray} \tag{ 14 }$$

where NUV is the GALEX-observed NUV magnitude, B is the Johnson magnitude, and u is now a reference value relative to which an NUV excess is defined. Figure 14 shows a two-color diagram of the Massarotti et al. (2008) sample of giants with GALEX NUV observations. Values of the B − V color were obtained from the Hipparcos Catalogue. The red line in this figure defines u, and it is likely an effective temperature trend. It is not a locus purely defined by chromospheric activity, as the NUV passband contains a greater amount of photospheric light for FGK stars than does the FUV passband. The locus corresponds to the equation

$$\begin{eqnarray}&&u=-4.30{\left(B-V\right)}^{2}+14.91(B-V)-3.73\,\mathrm{mag}.\end{eqnarray} \tag{ 15 }$$

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We show the NUV excess, Q_NUV, versus rotational velocity, $v\sin i$, in Figure 15. A clear correlation is not evident. By contrast, Dixon et al. (2020) did find a correlation between their NUV-excess parameter and $v\sin i$ for red giants in their Figure 7 using a sample that contained a larger number of stars with $v\sin i\gt 7$–10 km s⁻¹. ² Below this limit the relationship is less evident in both our work and theirs.

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There is a complication to interpreting Figure 15 because of the significant contribution of stellar photospheric light to the NUV bandpass. It was shown by Mohammed et al. (2019) that the NUV magnitude of giant stars in the core-helium-burning phase is sensitive to metallicity. Although most of the giants in Figure 15 are Population I stars, there may be differences in metallicity among them.

Relationships have been sought among red giant stars between two indicators of chromospheric and transition-region activity and coronal soft X-ray emission. In Section 3 a total of 450 giants and subgiants from Hünsch et al. (1998b) has been cross-matched with Pérez Martínez et al. (2011) to obtain a sample of giants with measured Mg ii h and k and ROSAT X-ray emission flux. There is a correlation between $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ and $\mathrm{log}({F}_{\mathrm{Mg}\,{\rm\small{II}}}/{F}_{\mathrm{bol}})$ for this cross-matched sample. Relatively high-activity binary stars contribute significantly to this correlation. Nonetheless, there is a large range of up to 1 dex in $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ for giants with comparable $\mathrm{log}({F}_{\mathrm{Mg}\,{\rm\small{II}}}/{F}_{\mathrm{bol}})$ and (B − V) color, which is particularly notable among lower-activity giants with the weaker λ2800 Mg II emission line strengths.

In Section 4 we have presented a comparative study of FUV excess and soft X-ray emission for a sample of giants from a cross-match between the Hünsch et al. (1998b) ROSAT X-ray catalog and the GALEX all-sky survey. Within this sample 66 stars were identified as being in the CHeB phase of evolution via a comparison between their HR diagram and MESA evolutionary tracks. These 66 giants have measurements of both GALEX FUV and Gaia G_BP magnitudes, and they exhibit a wide range in the (FUV − G_BP ) color: 8.00 ≤(FUV − G_BP ) ≤ 14.63 mag. As FUV magnitudes have been demonstrated as a chromospheric activity indicator, it follows that there is a varied range of activity levels for the CHeB stars. One interpretation of this scatter among the CHeB stars might be that those with the greatest FUV brightness are in a phase of enhanced activity after having decreased in radius and spun up following the helium core flash, causing resurgence of a rotation-induced magnetic dynamo. Since stars spend an extended time in the CHeB phase of evolution, if they subsequently spin down during this phase, a range in activities may result.

Not only is there a range of stellar activity in the FUV among the CHeB stars and other giants, there is also considerable scatter in X-ray luminosity. We attempted to constrain a relationship between an FUV-excess parameter Q_FUV and $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$ for giants with colors (B − V) > 0.62 mag and (FUV − G_BP ) > 11 mag. The correlations do not have a high level of significance, and there is a scatter of a factor ∼10 or more in normalized soft X-ray luminosity for a given Q_FUV. Among the ∼60 binary giants in the lower-right panel of Figure 9 there is a trend for the minimum FUV excess to correlate with $\mathrm{log}({L}_{{\rm{X}}}/{L}_{\mathrm{bol}})$. Additionally, late-main-sequence stars are clearly separated from red giants in a (FUV − G_BP ) versus optical two-color diagram.

We endeavored to constrain correlations between rotational velocities and GALEX UV emission for a sample of giants from Massarotti et al. (2008). However, a clear correlation was not evident between either FUV or NUV excess and $v\sin i$. Scatter in any intrinsic relation between UV excess and surface rotation speed may be obscured by a range in rotation axis inclinations of the giants in the Massarotti et al. (2008) sample. Also, the range in $v\sin i$ among our sample is limited to only ∼7 km s⁻¹.

Combining the results of the present paper with those of Smith (2018) and Smith & Cochrane (2020), the following conclusions are offered as to the behavior of the GALEX FUV-band brightness of red giant stars. ³ (i) Smith (2018) found that the GALEX FUV-band brightness of red giants correlates with the strength of the chromospheric Mg ii λ2800 h and k emission lines, albeit with scatter and above some lower limit of FUV excess, and thus concluded that flux in the FUV bandpass is tracing conditions in the chromospheres of red giants. (ii) The Mg ii λ2800 emission lines in turn are found in this paper to show a correlation with soft X-ray luminosity, but again the overall correlation has considerable scatter in it, and is most notably seen when giants in active binary systems are considered. At the lowest levels of λ2800 hk emission measured there is a spread of a factor of ∼10 in Figure 1 in the soft X-ray luminosity ratio (L_X/L_bol). (iii) Any correlation between FUV-bandpass excess and (L_X/L_bol) is obscured by large scatter and the results of Section 4 herein reveal only weak trends. When active binary giants are considered a correlation is seen between soft X-ray luminosity and the minimum FUV excess at a given (L_X/L_bol). (iv) FUV excesses among red giants do not obviously correlate with stellar age as discerned by Smith & Cochrane (2020) from a comparison between red giants in various kinematic moving groups.

To the above summaries can be added the fact that correlations for giant stars have been discerned among $v\sin i$ and each of the GALEX NUV excess (Dixon et al. 2020), Mg ii λ2800 emission (Smith 2018), and soft X-ray luminosity (Gondoin 1999), although such trends often require the presence of giants with $v\sin i\gt 6$ km s⁻¹ to be clearly discernible. The implication of such observations seems to be that among some giant stars the level of chromospheric and coronal activity may be linked to rotation and, by analogy with main-sequence stars, with a magnetic dynamo. However, as suggested by the results of the present paper in many red giants, perhaps mostly those of low rotation speeds and low levels of activity, the tracers of chromospheric and coronal activity start to become decoupled. One possible explanation might be that in more slowly rotating giants the coronal activity is diminishing and non-dynamo-related processes might be contributing to chromospheric heating. This could be particularly the case for stars near the upper part of their first or second ascents of the RGB if pulsation is starting to set in and acoustic heating of the chromosphere is becoming dominant.

We conclude this discussion by noting that from the results of the present paper the GALEX FUV excess of red giants may not be as useful a chromospheric indicator as the Mg ii λ2800 h and k emission lines, given that the former does not appear to correlate as well with coronal X-ray emission as do the latter, e.g., compare Figure 1 with Figures 8 or 14. This suggests a greater utility in identifying activity levels with the Mg emission lines as compared to GALEX broadband FUV flux. A benefit of using the Mg ii emission lines is that they arise from a single species in a single ionization state, and thus probe a specific region of the chromosphere. By comparison, when utilizing broadband FUV flux as an activity indicator we are sampling a combined sum of multiple emission lines from a variety of species in different ionization states, along with a photospheric background. The various chromospheric emission lines within the GALEX FUV band will not all vary in strength in the same way as a function of temperature, and not all species responsible for the emission lines will reach their maximum populations at the same stellar effective temperature. Thus, there is a complication of having within the FUV band a heterogeneous combination of emission lines each varying with temperature in different ways. This variance may act to obscure the resolution with which GALEX FUV brightness can trace activity within the chromosphere and transition region.