Magnetic Activity of F-, G-, and K-type Stars in the LAMOST-Kepler Field
Jinghua Zhang, Shaolan Bi, Yaguang Li, Jie Jiang, Tanda Li, Han He, Jie Yu, Shourya Khanna, Zhishuai Ge, Kang Liu, Zhijia Tian, Yaqian Wu, Xianfei Zhang
MMagnetic Activity of F-, G-, and K-type Stars in the LAMOST – Kepler
Field
Jinghua Zhang , Shaolan Bi , Yaguang Li , Jie Jiang , Tanda Li , Han He , Jie Yu , Shourya Khanna , Zhishuai Ge ,Kang Liu , Zhijia Tian , Yaqian Wu , and Xianfei Zhang Department of Astronomy, Beijing Normal University, Beijing 100875, People ’ s Republic of China; [email protected], [email protected] School of Space and Environment, Beihang University, Beijing 100083, People ’ s Republic of China Sydney Institute for Astronomy ( SIfA ) , School of Physics, University of Sydney, Sydney, NSW 2006, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark CAS Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, People ’ s Republic of China Max Planck Institute for Solar System Research, D-37077 Göttingen, Germany Key Laboratory of Beam Technology of Ministry of Education, Beijing Radiation Center, Beijing 100875, People ’ s Republic of China School of Physics and Astronomy, Yunnan University, Kunming 650091, People ’ s Republic of China Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, People ’ s Republic of China Received 2019 April 30; revised 2019 December 2; accepted 2019 December 10; published 2020 February 20
Abstract
Monitoring chromospheric and photospheric indexes of magnetic activity can provide valuable information,especially the interaction between different parts of the atmosphere and their response to magnetic fi elds. Weextract chromospheric indexes, S and + R HK , for 59,816 stars from LAMOST spectra in the LAMOST – Kepler program, and photospheric index, R eff , for 5575 stars from Kepler light curves. The log R eff shows positivecorrelation with log + R HK . We estimate the power-law indexes between R eff and + R HK for F-, G-, and K-type stars,respectively. We also con fi rm the dependence of both chromospheric and photospheric activity on stellar rotation.Ca II H and K emissions and photospheric variations generally decrease with increasing rotation periods for starswith rotation periods exceeding a few days. The power-law indexes in exponential decay regimes show differentcharacteristics in the two activity – rotation relations. The updated largest sample including the activity proxies andreported rotation periods provides more information to understand the magnetic activity for cool stars. Uni fi ed Astronomy Thesaurus concepts: Stellar activity ( ) ; Stellar atmospheres ( ) ; Stellar chromospheres ( ) ; Stellar magnetic fi elds ( ) Supporting material: machine-readable table
1. Introduction
The study of stellar activity in terms of different emissionfeatures originating from different parts of atmospheres hasdemonstrated the close connection between stellar activity,rotation, magnetic fi eld, and inner stellar dynamo ( Bab-cock 1958; Middelkoop 1982; Noyes et al. 1984 ) . Themagnetic fi elds generated through magnetohydrodynamicprocesses control the structure and the energetic balance ofstellar atmospheric plasma, bringing a variety of phenomena,e.g., inhomogeneous dark spots and bright faculae from thephotosphere, emissions in the line cores from the chromosphereand the transition region, and thermal X-rays and eruptive fl ares from the coronal region ( Catalano et al. 1999 ) .The chromospheric Ca II H and K emissions are well knownto correlate well with the stellar actual magnetic fl ux and thusare the most commonly used indicator of chromosphericactivity for cool stars ( Saar & Schrijver 1987; Chatzistergoset al. 2019 ) . The Mount Wilson program measured thechromospheric Ca II H and K emissions of more than 1000stars for over four decades ( Wilson 1963; Duncan et al. 1991;Baliunas et al. 1995 ) . These emissions were quanti fi ed as theMount Wilson S value ( S MW ; Duncan et al. 1991 ) . It was usedto monitor long-term stellar chromospheric activity ( Baliunaset al. 1995; Lockwood et al. 2007 ) . However, there are somephotospheric fl ux contributions in the wings of the H and K lines so that the values of S MW do not depend on chromospheresolely ( Linsky & Ayres 1978 ) . Noyes et al. ( ) derived the chromospheric emission fraction parameter ¢ R HK , which wasconverted from S by removing an empirically determinedphotospheric contribution R phot . Schrijver ( ) introducedthe concept of basal components in the chromospheric fl uxemissions from an atmosphere heated by acoustic waves andshocks. The basal fl ux characterizes stars with minimal activitylevels and depends sensitively on effective temperature ( Schrijver et al. 1989; Rutten & Uitenbroek 1991 ) . Using the S index from large samples with different luminosity classes,Mittag et al. ( ) parameterized the empirical basal chromo-spheric fl ux. They derived a new activity indicator + R HK tocharacterize the pure activity-related Ca II H and K line surface fl ux for stars of different spectral types.The unprecedented quality of the continuous four-yearphotometric observations carried out by the Kepler spacemission have extended our understanding of photosphericactivity to a large number of fi eld stars ( Borucki et al. 2010 ) . Itsbroad optical photometry measures the variability caused bystarspots or faculae zone rotating into and out of visibility asthe star rotates. Hence, the range or amplitude of light-curve fl uctuation is commonly used as a proxy for stellar photo-spheric activity ( e.g., Basri et al. 2013; García et al. 2014 ) .Basri et al. ( ) used the range between the 5th and 95thpercentile of fl ux as a proxy for photometric variability. Thismethod could underestimate the variability of very active stars ( García et al. 2014 ) . Instead, García et al. ( ) de fi ned a newindex of photometric variability ( S ph ) as the mean value of thelight-curve fl uctuations over subseries of length 5 × P rot ,where P rot is the rotation period of the star. Since the fl uctuation amplitude is not generally uniform for a given The Astrophysical Journal Supplement Series, ( ))
The study of stellar activity in terms of different emissionfeatures originating from different parts of atmospheres hasdemonstrated the close connection between stellar activity,rotation, magnetic fi eld, and inner stellar dynamo ( Bab-cock 1958; Middelkoop 1982; Noyes et al. 1984 ) . Themagnetic fi elds generated through magnetohydrodynamicprocesses control the structure and the energetic balance ofstellar atmospheric plasma, bringing a variety of phenomena,e.g., inhomogeneous dark spots and bright faculae from thephotosphere, emissions in the line cores from the chromosphereand the transition region, and thermal X-rays and eruptive fl ares from the coronal region ( Catalano et al. 1999 ) .The chromospheric Ca II H and K emissions are well knownto correlate well with the stellar actual magnetic fl ux and thusare the most commonly used indicator of chromosphericactivity for cool stars ( Saar & Schrijver 1987; Chatzistergoset al. 2019 ) . The Mount Wilson program measured thechromospheric Ca II H and K emissions of more than 1000stars for over four decades ( Wilson 1963; Duncan et al. 1991;Baliunas et al. 1995 ) . These emissions were quanti fi ed as theMount Wilson S value ( S MW ; Duncan et al. 1991 ) . It was usedto monitor long-term stellar chromospheric activity ( Baliunaset al. 1995; Lockwood et al. 2007 ) . However, there are somephotospheric fl ux contributions in the wings of the H and K lines so that the values of S MW do not depend on chromospheresolely ( Linsky & Ayres 1978 ) . Noyes et al. ( ) derived the chromospheric emission fraction parameter ¢ R HK , which wasconverted from S by removing an empirically determinedphotospheric contribution R phot . Schrijver ( ) introducedthe concept of basal components in the chromospheric fl uxemissions from an atmosphere heated by acoustic waves andshocks. The basal fl ux characterizes stars with minimal activitylevels and depends sensitively on effective temperature ( Schrijver et al. 1989; Rutten & Uitenbroek 1991 ) . Using the S index from large samples with different luminosity classes,Mittag et al. ( ) parameterized the empirical basal chromo-spheric fl ux. They derived a new activity indicator + R HK tocharacterize the pure activity-related Ca II H and K line surface fl ux for stars of different spectral types.The unprecedented quality of the continuous four-yearphotometric observations carried out by the Kepler spacemission have extended our understanding of photosphericactivity to a large number of fi eld stars ( Borucki et al. 2010 ) . Itsbroad optical photometry measures the variability caused bystarspots or faculae zone rotating into and out of visibility asthe star rotates. Hence, the range or amplitude of light-curve fl uctuation is commonly used as a proxy for stellar photo-spheric activity ( e.g., Basri et al. 2013; García et al. 2014 ) .Basri et al. ( ) used the range between the 5th and 95thpercentile of fl ux as a proxy for photometric variability. Thismethod could underestimate the variability of very active stars ( García et al. 2014 ) . Instead, García et al. ( ) de fi ned a newindex of photometric variability ( S ph ) as the mean value of thelight-curve fl uctuations over subseries of length 5 × P rot ,where P rot is the rotation period of the star. Since the fl uctuation amplitude is not generally uniform for a given The Astrophysical Journal Supplement Series, ( )) , 2020 March https: // doi.org / / / ab6165 © 2020. The American Astronomical Society. All rights reserved. Corresponding author. ( R eff ) was de fi ned byHe et al. ( ) to represent the effective range of the light-curve fl uctuation amplitude.It might be expected that more rapid rotation would lead toeither an increased number of spots and / or larger spots overall,similar to the patterns between rotation and chromosphericactivities. Therefore, the amplitude of light-curve fl uctuationand the chromospheric activities tend to be related to eachother. Notsu et al. ( ) investigated the connection betweenmean stellar brightness variation and the residual fl ux in theinfrared Ca II line core, and showed that the two quantities arestrongly correlated. Karoff et al. ( ) con fi rmed thecorrelation for 1400 G-type stars and found that suchcorrelation is absent for stars with activity levels lower thanthe Sun. There is extensive research on the relations betweendifferent activity proxies for the solar case ( e.g., Bennett et al.1984; Cappelli et al. 1989; Schrijver et al. 1989 ) . Schrijveret al. ( ) originally derived a power-law index ofapproximately 0.6 between the Ca II K-line core excess fl uxdensity and the absolute value of the magnetic fl ux density forsolar active regions. A power-law exponent of 0.2 wassuggested by Rezaei et al. ( ) for locations in a quiet Sunand higher values of 0.4 – ( ) interpreted the below-unity power-law exponent asthe geometric expansion model of magnetic fl ux tubes. Thisqualitative picture was con fi rmed by Solanki et al. ( ) byapplying a two-dimensional magnetostatic model. Recently,Barczynski et al. ( ) used the original geometric expansionmodel to explain the relations between emissions of solarchromospheric or transition regions and the magnetic fi elds.Whether correlations between photospheric and chromosphericactivity exist as seen in a large range of spectral types is still anopen question. The Large Sky Area Multi-Object FibreSpectroscopic Telescope ( LAMOST ) spectroscopic survey ( Cui et al. 2012; Zhao et al. 2012 ) has collected millions ofstellar spectra with a resolution of 1800 in a broad wavelengthrange of 3700 – Å . This wealth of data provides a goldenopportunity to answer this question.Stellar rotation plays an important role in the dynamo andaffects magnetic activity. The rotational spin-down due to theloss of stellar angular momentum weakens the ef fi ciency of thedynamo, leading to a decreasing magnetic activity ( Parker 1955;Skumanich 1972 ) . The relationships between stellar rotationperiod and the magnetic activity levels in terms of chromo-spheric and coronal proxies have been studied. The relationsshow different con fi gurations, with fast rotators falling in thesaturated activity regime, and slow rotators falling in theexponential decay regime. ( e.g., Kraft 1967; Noyes et al. 1984;Pizzolato et al. 2003; Wright et al. 2011 ) . In addition, therelations between rotation period and photospheric activityproxies have been investigated ( McQuillan et al. 2014 ) . Withthe help of LAMOST and the Kepler mission, we can study therelations among stellar rotation periods, the chromosphericactivity proxies, and the photospheric activity proxies in a largesample comprehensively.In this paper, we construct a large sample of cool stars withphotospheric and chromospheric activity proxies in theLAMOST – Kepler fi eld. Section 2 describes methods of dataanalysis. We explore the relations between the activity indexesin Section 3. In Section 4, we investigate the activity – rotationrelationship in detail. We summarize the conclusions inSection 5.
2. Data Analysis
The LAMOST – Kepler project was initiated to use theLAMOST spectroscopic survey to perform spectroscopicfollow-up observations for the targets in the fi eld of the Kepler mission ( De Cat et al. 2015 ) . By 2016 June, this project hadcollected more than 180,000 optical spectra covering3700 – Å in low-resolution R ∼ ( LSS – GAC ) fl ux calibration pipeline ( Liu et al.2014; Yuan et al. 2014 ) . We use stellar atmospheric parameters ( T eff , g log , and [ Fe / H ] ) from the LAMOST Stellar ParameterPipeline at Peking University ( LSP3; Xiang et al. 2015 ) . Forthe spectra with signal-to-noise ratios ( S / Ns ) higher than 50,stellar parameters from LSP3 have typical uncertainties of 100K for T eff , 0.1 dex for g log , and 0.1 dex for [ Fe / H ] ( Xianget al. 2017 ) .We aim at targets of dwarfs, removing giant stars with anempirical T eff − g log relation determined by Ciardi et al. ( ) .Binaries labeled by Berger et al. ( ) were also excluded. Toselect F- to K-type stars, we use T eff in the range of 3800 – / Ns at the blue end of the spectra are higherthan 10. With these constraints, we gathered 86,689 spectra for59,816 stars.For photospheric activity analysis, we selected the samplestars from catalog in McQuillan et al. ( ) . Note that thiscatalog is biased to stars that produce measurable rotationalcurves, which leaves out photometrically quiet or long-periodstars. This catalog provides the updated largest sample set withrotation period. The period is acquired through rotationalmodulation due to the existence of inhomogeneous spots andfaculae that lead to fl uctuating light curves. We cross match59,816 stars with the catalog of McQuillan et al. ( ) andobtain 5575 targets with both photometric observational dataand spectroscopic observational data. S Index
The chromospheric activity level is typically quanti fi edthrough the classical S index, i.e., the ratio of the fl ux in thecore of the Ca II H and K lines to the nearby continuouswindows ( Vaughan et al. 1978 ) . Figure 1 shows examples ofspectra for typical FGK-type stars at different activity levels.The spectra of these stars have been normalized in the spectralrange of 3900 Å – ∼ Å . Following Karoff et al. ( ) , we computed the fl ux ratio S as the emission inthe Ca II H and K lines relative to the continuum, a = ++ S H KR V · ( ) where H and K are the fl uxes integrated in 1.09 Å FWHMtriangular windows centered on the line cores of 3968 and3934 Å . R and V are the fl uxes integrated in 20 Å rectangularwindows centered on 4001 Å and 3901 Å . The normalizationfactor α = ( ) . The factorof 8 is the ratio of exposure time between HK and RV channelsof the Mount Wilson HKP-2 spectrophotometer. For stars withmultiple observations, the S values were calculated by theweighted mean values of these multiple spectra.2 The Astrophysical Journal Supplement Series, ( ))
The chromospheric activity level is typically quanti fi edthrough the classical S index, i.e., the ratio of the fl ux in thecore of the Ca II H and K lines to the nearby continuouswindows ( Vaughan et al. 1978 ) . Figure 1 shows examples ofspectra for typical FGK-type stars at different activity levels.The spectra of these stars have been normalized in the spectralrange of 3900 Å – ∼ Å . Following Karoff et al. ( ) , we computed the fl ux ratio S as the emission inthe Ca II H and K lines relative to the continuum, a = ++ S H KR V · ( ) where H and K are the fl uxes integrated in 1.09 Å FWHMtriangular windows centered on the line cores of 3968 and3934 Å . R and V are the fl uxes integrated in 20 Å rectangularwindows centered on 4001 Å and 3901 Å . The normalizationfactor α = ( ) . The factorof 8 is the ratio of exposure time between HK and RV channelsof the Mount Wilson HKP-2 spectrophotometer. For stars withmultiple observations, the S values were calculated by theweighted mean values of these multiple spectra.2 The Astrophysical Journal Supplement Series, ( )) , 2020 March Zhang et al. o estimate the uncertainties of S indexes, we applied asimilar procedure as in K16. We calculated the relation of thestandard deviation of different S measurements as a function ofthe mean S / N at the spectra ’ s blue end for stars with multipleobservations as s = - + S log 1.5 log S N 1.0 ( ) ( ) . The uncer-tainties given by this relation are considered the causes ofstellar intrinsic chromospheric activity variation. We alsoconsidered the random errors. We used a Monte Carloapproach to estimate them. We added Gaussian noises to theoriginal spectrum to generate a simulated spectrum. The S value was then calculated for the simulated spectrum. This wasdone 1000 times, and the standard deviation of the 1000 S values was adopted as the uncertainty. We fi nally took thestandard error of the uncertainties given by the two proceduresas the S measurement uncertainty for each star.K16 calculated the S index for ∼ – Kepler fi eld. In Figure 2 we compare our S indexvalues with those of K16 for stars in common. There is a goodagreement between the two sets of S values. The differencemay arise from the fact that we used spectra processed with theLSP3 pipeline, while K16 used spectra processed with theLAMOST Stellar Parameter pipeline ( LASP; Luo et al. 2015 ) .We also compared the S index distribution with that of Isaacson& Fischer ( ) and found agreement for moderately activestars but a lack of high-activity stars in the LAMOST sample.Given the low-resolution power of LAMOST, the wavelengthranges of the HK emissions are not easy to identify. Thus, it ispossible to take some fl uxes outside the veritable emissionwindows in the S measurements, which would lead to differentresults between high-resolution spectra and low-resolutionspectra. K16 found a similar lack of high-activity stars in theLAMOST sample as well. + R HK Index
The quantity S is sensitive to the integrated emission overthese windows and the photospheric radiation transmitted by H and K instrumental passbands, both of which are colordependent ( Middelkoop 1982 ) . As mentioned in Section 1,Mittag et al. ( ) de fi ned a new proxy, + R HK ,which is converted from the S index by eliminating thephotospheric contributions ( Noyes et al. 1984 ) and the so-called basal chromospheric fl ux ( Schrijver 1987 ) . Using M13 ʼ s method, we calculated the index + R HK following s s = - - = + + R T T , 2
HK HK HK HK HK ,phot ,basaleff4 eff4 ( ) where T eff is the effective temperature, and σ is the Stefan-Boltzmann constant. Here the HK surface fl ux is derived fromthe RV continuum fl ux and the S index through a = S HK RV · ( ) ( Middelkoop 1982 ) . The photospheric fl ux in the HK bands HK ,phot , the basal chromospheric fl ux HK ,basal , and the continuum fl ux RV were calculated from the B − V color index ( Ramírez & Meléndez 2005 ) . The uncertain-ties of + R HK were estimated by considering the uncertainties ofthe S index. Note that both HK ,phot and HK ,basal are virtuallygiven by the empirical formula that is based on stellar color. Figure 1.
Representative spectra of different chromospheric emission levels in F-, G-, and K-type stars. A vertical offset of 0.4 is applied between each spectrum forclarity. The spectra lines from bottom to top in each panel are the stars with different emission levels. The cores of Ca II H and K emission lines are indicated by reddashed lines in each panel. Figure 2.
Comparison of the S index with those of Karoff et al. ( ) for starsin common. The solid line shows the line of equality; the dashed line shows aleast-squares method fi tting of the results. In the lower panel, the dispersionsfor the S index are plotted on the Y -axis. The Astrophysical Journal Supplement Series, ( ))
Comparison of the S index with those of Karoff et al. ( ) for starsin common. The solid line shows the line of equality; the dashed line shows aleast-squares method fi tting of the results. In the lower panel, the dispersionsfor the S index are plotted on the Y -axis. The Astrophysical Journal Supplement Series, ( )) , 2020 March Zhang et al. he color range of the sample selected in M13 is < - < B V . For early F-type stars ( - < B V ) ,the + R HK values calculated by Equation ( ) are not reasonable.Figure 3 shows the distribution of + R log HK with T eff of oursample. The histogram of + R log HK is in the right panel. Thevalues of + R log HK for most stars are in the range of − ∼− + R log HK are mostly lowerthan ∼− The photometric data were obtained by the
Kepler missionand were acquired in the long-cadence mode ( ) . We used data processed by the PresearchData Conditioning module ( Smith et al. 2012; Stumpe et al.2012 ) of the Kepler data analysis pipeline.We used R eff to represent the photospheric activity ( He et al.2015, 2018 ) . For a given light curve F t , t =
0, 1, 2, 3, ..., N – N points, we fi rst obtain the relative fl ux: = - ~~ f F FF , 3 t t ( ) where ~ F is the median of F t . Then, we used a Fourier-basedlow-pass fi lter to remove high-frequency variations present in f t . The nature of these variations could be outliers, fl are spikes, oscillation signals, and granulation-driven fl ickers ( Cranmeret al. 2014; Kallinger et al. 2014 ) . The cutoff frequency is givenby the empirical relation, = f P upper 10.3 rot , which takes intoaccount that different stars may have different noise levels and fl uctuation properties ( He et al. 2015; Mehrabi et al. 2017 ) .Finally, we obtained the pure gradual variation component, f G .The effective fl uctuation range of the light curve is given by = R f eff rms · ( · ) ( ) where f rms is the rms value of f G ( García et al. 2010; Chaplinet al. 2011 ) . The factor in Equation ( ) is given tointroduce a corrected value of the fl uctuation range; see Heet al. ( ) for a detailed illustration. For each star, wecalculated R eff quarter by quarter and then took the averagevalue as the evaluated proxy. The uncertainty of R eff was takenby the standard error of R eff values in all quarters. Table 1 lists T eff , log g , [ Fe / H ] , S , + R log HK , and R eff of our sample.In Figure 4, we plot the distributions of R log eff with T eff .The histogram of R log eff is plotted in the right panel. Thedifference of photospheric activity levels is almost 3 orders ofmagnitude between the most active stars and the inactive stars.The photospheric activity levels ( R log eff ) for most stars are inthe range of − ∼− R log eff values of eachtemperature bin shown by the red line increase with decreasing T eff . When T eff < R log eff values are around Figure 3.
Distribution of + R log HK with T eff of all the dwarfs that have the chromospheric observations. Panel ( b ) shows the histogram of + R log HK . Table 1
Sample Entries of Deduced Activity Proxies of the 5575 StarsKIC T eff log g [ Fe / H ] S log + R HK R eff ± ± ± ± − ± ± ± ± ± ± − ± ± ± ± − ± ± − ± ± ± ± ± ± − ± ± ± ± − ± ± − ± ± L L L L L L L
Note.
Columns 2 – ( ) . Stars without + R log HK are labeled with NaN. ( This table is available in its entirety in machine-readable form. ) The Astrophysical Journal Supplement Series, ( ))
Columns 2 – ( ) . Stars without + R log HK are labeled with NaN. ( This table is available in its entirety in machine-readable form. ) The Astrophysical Journal Supplement Series, ( )) , 2020 March Zhang et al. ∼− ( ) , who took a similar photospheric activity index toinvestigate the correlation between the size of starspots and thestellar effective temperature.
3. Relations between Chromospheric and PhotosphericActivities
We compared R eff with + R HK for F-, G-, and K-type stars inFigure 5. The numbers of F-, G-, and K-type stars are 1444,2997, and 1134, respectively.As shown in Figure 5, for F-, G-, and K-type stars, the + R log HK values show positive correlations with the R log eff values. We use Spearman ’ s rank order correlation coef fi cient ( r s ) to test the potential connections between the two quantities.The r s values for the F-, G-, and K-type stars are 0.49, 0.50, and0.53, respectively, indicating that there is a relation between R eff and + R HK . The scatter is probably due to the physicaldifferences between measuring photometric intensity contrastsand chromospheric emissions. The different observational timelengths between photometric and spectroscopic observationsmay be another reason. For each star, the R eff was obtainedfrom continuous observation in 4 yr, while the + R HK wasobtained from a few observational records. To estimate theserelations, we performed an orthogonal regression method ( Isobe et al. 1990; Feigelson & Baru 1992 ) in terms of = + + R c k R log log , 5 HK eff · ( ) where c is a scaling parameter and k is a power-law index. The fi tted power-law indexes are listed in Table 2 with fi ttedrelations shown in Figure 5 by red lines. The power-lawexponents for F- and G-type stars are similar, while for K-typestars the exponent is larger than that of earlier type stars. The Figure 4.
Distribution of R log eff with T eff of our sample. The average R log eff values of each temperature bin are shown by the red line. The sample stars are dividedinto seven bins by T eff . Panel ( b ) shows the histogram of R log eff . Figure 5.
Relations between logarithmic R eff and + R HK index for F-type ( left ) , G-type ( middle ) , and K-type ( right ) stars. The red lines represent the power-lawapproximation. The residuals are plotted on the Y -axis in the respective lower panels. The Astrophysical Journal Supplement Series, ( ))
Relations between logarithmic R eff and + R HK index for F-type ( left ) , G-type ( middle ) , and K-type ( right ) stars. The red lines represent the power-lawapproximation. The residuals are plotted on the Y -axis in the respective lower panels. The Astrophysical Journal Supplement Series, ( )) , 2020 March Zhang et al. eason for this difference may be also related to the lack ofhigh-activity stars at the cool end in our sample.The relationship between chromospheric activity and photo-spheric variability is also investigated on the decadal timescaleof the solar activity cycle ( Radick et al. 1998; Lockwood et al.2007 ) . Lockwood et al. ( ) showed that on a year-to-yeartimescale, the young active stars become fainter as their Ca II H and K emission increases, and older less active stars tend toshow a pattern of direct correlation. The correlation studied inthis work is based on the stellar rotational timescale. The resultsindicate that on rotational timescales, the photosphericvariability always shows positive correlation with the Ca II H and K emissions for cool stars. On the other hand, the relationsbetween the chromosphere activity proxies and the activityproxies in transition region or in the coronal region have alsobeen studied in the stellar case ( e.g., Oranje 1986; Schrijveret al. 1992; Güdel 2004 ) . In this work, we additionally give therelation between activity proxies in the photosphere and in thechromosphere. Our results and the previous results providerelated magnetic information from the photosphere to coronalregions for cool stars.
4. Activity – Rotation Relations
We studied the relations between stellar rotation periods andchromospheric emission as well as photospheric variation forF-, G-, and K-type stars. Figure 6 and Figure 7 show + R HK versus P rot and R eff versus P rot . The histograms of rotationperiods for each subset are shown in the respective upperpanels. We found that when the rotation periods exceed about1, 3, and 6 days for F-, G-, and K-type stars, the probabilities ofthe density distribution become signi fi cant. Beyond that, thecorresponding rotation periods at the density peaks are near 6,18, and 25 days, which become longer from F- to K-type stars. Compared with K-type stars, the activity – rotation relations aremuch more dispersive for F- and G-type stars, especially in the R eff − P rot relation. This phenomenon is also observed in the fl are and X-ray band ( Huiqin & Jifeng 2019; Pizzolato et al.2003 ) . For stars with rotation periods exceeding about 1, 3, and6 days, Ca II H and K emissions and photospheric variationsdecrease signi fi cantly with increasing rotation periods, formingan exponential decay regime.We parameterized the relations for F-, G-, and K-type starsthat possess P rot larger than 1, 3, and 6 days, individually. We fi tted the activity index ( + R HK and R eff ) versus P rot with a powerlaw as the form: b = + R c P log log , 6 i rot · ( ) where R i is + R HK or R eff , c is the scaling factor, and β is thepower-law index ( Wright et al. 2011 ) . The parameters weredetermined by the Ordinary Least Squares bisector ( Isobe et al.1990; Feigelson & Baru 1992 ) . The fi tted results withrespective errors are listed in Table 3. In + R HK − P rot relations,the absolute values of β increase from F- to K-type stars, whichmeans the slope of the exponential decay regime becomessteeper. In R eff − P rot relations, the β value of F-type stars is lessthan that of G- and K-type stars. Beyond that, the slopes in R eff − P rot relations are all steeper than those in + R HK − P rot relations, which implies that the dependence of the photometricintensity contrasts on stellar rotation is different from thedependence of the chromosphric emissions on rotation.The similar decreasing trends of activity proxies withincreasing rotation periods have also been shown in previousworks ( Pizzolato et al. 2003; Wright et al. 2011; Wright &Drake 2016 ) . The explanation for this trend might be based onthe αω -type dynamo theory ( Noyes et al. 1984; Charbon-neau 2014 ) . That is, the observed decrease in proxies of stellaractivity driven by the stellar magnetic dynamo could beattributed to the rotational spin-down of the star, which isdriven by mass loss through a magnetized stellar wind ( Skumanich 1972 ) . As the number of the fast rotators arefew, and they show the severe dispersion in the acitivy – rotationrelations, the saturation region is not as signi fi cant as thatshown in the X-ray band. The results suggest that for slowrotators, the relations between rotation period and the Ca II H Table 2
Power-law Indexes Fit for Different Type Stars in This WorkIndex F-type G-type K-type k ± ± ± c − ± − ± − ± Figure 6.
Relations between the P rot and + R HK indexes for F-type ( left ) , G-type ( middle ) , and K-type ( right ) stars. The top section of each panel shows the histogram of P log rot . The red solid lines represent the fi tted relations. The red dashed lines represent the lower limits of rotation periods for the fi tting. The Astrophysical Journal Supplement Series, ( ))
Relations between the P rot and + R HK indexes for F-type ( left ) , G-type ( middle ) , and K-type ( right ) stars. The top section of each panel shows the histogram of P log rot . The red solid lines represent the fi tted relations. The red dashed lines represent the lower limits of rotation periods for the fi tting. The Astrophysical Journal Supplement Series, ( )) , 2020 March Zhang et al. nd K emissions as well as the photospheric variations couldalso be important probes of the physical dynamo process.
5. Conclusion
We constructed two updated largest catalogs with stellaractivity proxies in the LAMOST – Kepler program. Onecontains 59,816 F-, G-, and K-type stars with the chromo-spheric activity proxies S and + R HK . Another one includes 5575stars with the photospheric activity proxy R eff and the rotationperiods. We studied the relations between the activity proxies,as well as the relations between activity proxies and rotationperiods.The + R log HK shows positive correlation with R log eff . Thereexists a power-law relation between + R HK and R eff . The power-law indexes fi tted for F- and G-type stars are 0.20 and 0.18,while the value fi tted for K-type stars is 0.46.Our analysis con fi rmed the relations between the two activityindices ( + R HK and R eff ) and rotation period. For stars withrotation periods exceeding a few days, Ca II H and K emissionsand photospheric variations generally decrease with increasingrotation period. The absolute values of the power-law index β increase in the + R HK − P rot relations from F- to K-type stars,while it does not show a similar trend in R eff − P rot relations.Our results indicated that the relations between rotation periodand the Ca II H and K emissions as well as the photosphericvariations could also be important probes of the physicaldynamo process.The authors would like to thank the referees for theconstructive criticism and useful advice, which helped usgreatly improve the paper. The authors acknowledge supportfrom the Joint Research Fund in Astronomy ( U1631236 ) undercooperative agreement between the National Natural ScienceFoundation of China ( NSFC ) and Chinese Academy of Sciences ( CAS ) , and grants 11522325, and 11873023 fromthe National Natural Science Foundation of China, and theFundamental Research Funds for the Central Universities andYouth Scholars Program of Beijing Normal University. T.L.acknowledges funding from an Australian Research CouncilDP grant DP150104667, the Danish National ResearchFoundation ( grant DNRF106 ) . H.H. acknowledges the supportof the Astronomical Big Data Joint Research Center,cofounded by the National Astronomical Observatories,Chinese Academy of Sciences and the Alibaba Cloud. Thispaper includes data collected by the Kepler
Discovery Mission,whose funding is provided by NASA ’ s Science MissionDirectorate. Guoshoujing Telescope ( the Large Sky AreaMulti-Object Fiber Spectroscopic Telescope, LAMOST ) is aNational Major Scienti fi c Project built by the ChineseAcademy of Sciences. Funding for the project has beenprovided by the National Development and Reform Commis-sion. LAMOST is operated and managed by the NationalAstronomical Observatories, Chinese Academy of Sciences. ORCID iDs
Shaolan Bi https: // orcid.org / // orcid.org / // orcid.org / // orcid.org / References
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