Chromospheric changes in K stars with activity
aa r X i v : . [ a s t r o - ph . S R ] J un Mon. Not. R. Astron. Soc. , 1– ?? (2009) Printed 31 October 2018 (MN L A TEX style file v2.2)
Chromospheric changes in K stars with activity
Mariela C. Vieytes ⋆ , Pablo J. D. Mauas, and Rodrigo F. D´ıaz † Instituto de Astronom´ıa y F´ısica del Espacio,CC. 67 Suc. 28 (1428)Buenos Aires, Argentina
Accepted . Received ; in original form
ABSTRACT
We study the differences in chromospheric structure induced in K stars by stellaractivity, to expand our previous work for G stars, including the Sun as a star. Weselected six stars of spectral type K with 0.82 < B − V < β observed profiles. We also computed in detail the net radiative lossesfor each model to constrain the heating mechanism that can maintain the structurein the atmosphere. We find a strong correlation between these losses and S CaII , theindex generally used as a proxy for activity, as we found for G stars.
Key words: radiative transfer - stars: atmosphere - stars: activity
Solar and stellar chromospheric models have been developedto study the dependency of chromospheric plasma parame-ters with height and temperature.
The best known exam-ples are the models for the solar atmosphere com-puted by E. Avrett and his co-workers, in particularmodel C for the average quit Sun by Vernazza et al.(1981), which was later modified by Fontenla et al.(1993)
In several cases, these models were used to character-ize changes due to activity and spectral type. For example,Kelch et al. (1979) studied a sample of eight main-sequencestars ranging in spectral type from F0 to M0, some of whichwere of similar spectral type and different levels of chromo-spheric activity. They computed the photospheric structurestarting from a radiative equilibrium model for the T eff ofeach star and fitting the Ca II K line wings. The chromo-sphere was built using the emission core of the Ca II K line.To estimate the radiative cooling rate in the K line they usedthe K index (Linsky & Ayres 1978), which is calculated asthe difference between the integrated flux inside the two K minima of the Ca II K line and the corresponding flux forthe model in radiative equilibrium.Their results showed that non-radiative heating is im-portant in the lower photosphere of all the late-type stars ⋆ E-mail:[email protected] † Visiting Astronomer, Complejo Astron´omico El Leoncito oper-ated under agreement between the Consejo Nacional de Investi-gaciones Cient´ıficas y T´ecnicas de la Rep´ublica Argentina and theNational Universities of La Plata, C´ordoba and San Juan. under study. They found that the value of the K indexand the temperature gradient in the lower chromosphere ofthese stars, as a function of T eff , divides active and inac-tive stars, and that the cooling rate in chromospheric linesdecreases with T eff . Regarding the chromospheric structure,they found that the temperature minimum moves outward,to lower values of column mass density, with decreasing mag-netic activity, i.e. with decreasing non radiative heating inthe lower chromosphere.Semi-empirical models of the dM star AD Leo in its qui-escent state and during a flare were built by Mauas & Falchi(1994, 1996). Subsequently, models of two “basal” ( i.e. inac-tive) stars of the same spectral type, Gl588 and Gl628, wereconstructed by Mauas et al. (1997).In a previous paper (Vieytes et al. 2005, hereafter PaperI), we computed chromospheric models for a sample of dwarfstars of spectral type G, including the Sun as a star, usingthe FAL models Fontenla et al. (1993) as a startingpoint . Our purpose was to study the changes in chromo-spheric structure induced by magnetic activity. The starswe modeled were chosen to have similar colors than the Sun,and therefore similar photospheric structures, but differentchromospheric activity levels, probably due to different agesand/or rotation periods. These stars can be considered assolar analogues, since they share several characteristics withthe Sun.To extend our research to cooler stars and to study howthe chromospheric structure changes with spectral type andchromospheric activity, in this paper we perform a studysimilar to the one in Paper I for several dwarfs of spectraltype K, selected with similar colour, i.e. similar photosphericstructure, and with different levels of magnetic activity. c (cid:13) M. Vieytes, P. Mauas and R. D´ıaz
As the base for our sample we selected one of the moststudied K stars, Epsilon Eridani (HD 22049), which is anactive star of spectral type K2 V ( B − V =0.88), with T eff =5110 K. This star has been widely studied because it is oneof the ten nearest stars. It has two planets and a belt ofdust particles around it, which has been compared to theKuiper belt in the Sun. These facts make this stellar systemresemble our own Solar System.Several chromospheric models have been computed forthis star. Kelch (1978) modeled the lower chromosphere tomatch the Ca II K line profile and integrated fluxes of theMg II h and k lines. Using observations of the ultravioletlines of C II, Mg II, Si II and Si III from the IUE satellite,Simon et al. (1980) obtained a model for Epsilon Eridani,which also reproduces hydrogen line profiles not fitted byKelch’s model. The thermal structure of this model has theonset of the transition zone deeper in the chromosphere anda lower temperature in the plateau than Kelch’s model.Another chromospheric model for Epsilon Eridani is theone by Thatcher et al. (1991), who fitted the Ca II K line,the infrared triplet lines of Ca II, the Na D doublet, H α and H β . More recently, Sim & Jordan (2005), using ultra-violet observations from STIS and FUSE, developed a newsemiempirical model for the upper chromosphere and lowertransition region of this star keeping the photosphere andlower chromosphere of Thatcher et al. (1991).Finally, Ness & Jordan (2008) studied the relative ele-ment abundances from the conora and upper transition re-gion of Epsilon Eridani, using observations from Chandra,EUE, FUSE and XMM-Newton.This Paper is arranged as follows: we present our stellarsample and discuss the observational data in §
2. In § § § The largest observational study of chromospheric activity isthe one started in 1966 at the Mount Wilson Observatory,which at present includes more than 2200 stars in the spec-tral range between F and early K. As the indicator of chro-mospheric stellar activity, they use the S
CaII index, whichis the ratio of the fluxes in the H and K line cores and twonearby reference windows 20 ˚A wide (Vaughan et al. 1978).The emission in the cores of these lines increase with in-creasing chromospheric activity, i . e . with increasing surfacemagnetism. In this work we used the same activity indicator.To select the stars in our sample, we require that0 . < B − V < .
90, a colour similar to ε Eri, and thatthe magnetic activity levels are different. All the stars arepart of the library of southern late-type dwarfs published byCincunegui & Mauas (2004, hereafter CM04).The stellar parameters of the stars in our sample arelisted in Table 2. In the third column we list the spectraltype, in the fourth to sixth columns we indicate the colourindex B − V , T eff and the metallicity. In column 7 we showthe mean values of the S CaII index obtained at the CerroTololo InterAmerican Observatory (Henry et al. 1996), andin columns 8 and 9 the maximum and minimum S
CaII ob-
Figure 1. S CaII for each observation for ε Eri in our library. Theopen triangles are the values for the different observations, thesquares indicate the annual averages, and the largest full trianglesshow the two spectra modeled in this paper. tained from our spectra (see Cincunegui et al. 2007 for de-tails on how this index is obtained) and from the modelswe built in this paper . Finally, in the last two columnsof Table 2 we include the observing dates of each spectrumused in the present work.The observations were made at the 2.15 m telescopeof the Complejo Astronomico El Leoncito (CASLEO), lo-cated in San Juan, Argentina. They were obtained with aREOSC spectrograph designed to work between 3500 and7500 ˚A and a 1024 x 1024 pixel TEK CCD as detector. Thespectral resolution ranges from 0.141 to 0.249 ˚A per pixel( R = λ/δλ ≃ CaII index of ε Eri obtainedfrom our observations (open triangles). The two spectramodeled in this paper are represented by full triangles. Thedifference in the Ca II K line flux between the maximum andminimum is 17%. With squares we also present the annualaverage of the S
CaII index. For details on the variability of ε Eri, see Buccino & Mauas (2008).
For each star we built a different chromospheric model, as-suming one-dimensional, plane-parallel atmospheres. We si-multaneously solved the equations of hydrostatic equilib-rium, radiative transfer and statistical equilibrium, using thecomputer code Pandora. A description of this code can befound in Avrett & Loeser (2003).For a given distribution of temperature with height, weself-consistently computed non-LTE populations for 15 lev-els of H, 13 of He I, 6 of He II, 15 of Fe I, 8 of Ca I, 5 of Ca c (cid:13) , 1– ?? hromospheric changes in K stars with activity Table 1.
The stellar sample. Columns 3 to 6 list the stellar parameters (from Perryman et al. 1997, and from Cincunegui& Mauas 2004). The next three columns give the S
CaII measured by Henry et al. (1996) at CTIO and by Cincunegui &Mauas (2004) at CASLEO, both converted to Mount Wilson S
CaII compared whit S
CaII calculated from our models ;and the last two columns list the observing dates for each spectrum we used.HD (Name)
S. type B − V T eff (K) [Fe/H] S CTIO S max CM / S max mod S min CM / S min mod Min Max17925 (V* EP Eri) K1 V 0.86 4956 0.10 0.662 0.792/ ε Eri) K2 V 0.88 5110 -0.14 0.483 0.555/ — — 11/24/04128621 ( α Cen B) K1 V 0.90 5037 0.24 0.209 0.247/ — — 6/27/02 II, 7 of Mg I, 6 of Mg II, 21 of Si I, 8 of Na I and 6 of Al I.The atomic models we used for H and Ca II are described inMauas et al. (1997) and Falchi & Mauas (1998). The Ca IIlines and Ly α were computed using Partial Redistribution, as it has been done in previous chromospheric mod-els (like, for example, the Vernazza et al. (1981) so-lar models) .An important element to include in this kind of model-ing, in particular for the coolest stars, is the effect of bound-bound absorptions due to the numerous atomic and molecu-lar lines present in the stellar atmosphere, referred to as lineblanketing (Falchi & Mauas 1998), which plays a crucial rolein determining both the emergent energy distribution andthe physical structure of the atmosphere. In solar-type starsthe most important effects come from neutral or single ion-ized metals. In even cooler stars, molecular bands, as CN,CO, H O, etc, could dominate. In this paper, line blanket-ing is treated in non LTE, as explained in Falchi & Mauas(1998), assuming the source function is given by S ν = αJ ν + (1 − α ) B ν , (1)where B ν is the Planck function and J ν is the mean intensity. α is the scattering albedo, for which we used the expressiongiven by Anderson (1989) which depends on wavelength,depth and temperature.From the finished model, we computed the emitted pro-files of H β and of the Ca II H and K lines, and modified themodel until we found a satisfactory match with the observedprofiles. As a check of the accuracy of the models, we alsocompared the observed computed profiles of the Mg I b andthe Na I D lines for each model (details of these features canbe found in Mauas et al. 1988 and D´ıaz et al. 2007).For comparison with synthetic profiles, the observationswere converted to the stellar surface through log ( F surf /f earth ) = 0 .
35 + 0 . V + BC ) + 4 log ( T eff ) , (2)where F surf is the stellar surface flux, f earth is the flux ob-served at earth, V is the visual magnitude, BC is the bolo-metric correction given by Johnson et al. (1966), and T eff isthe effective temperature for each star, given in Table 2.Of course, semiempirical models like this one are onlya first approximation to the structure of stellar chromo-spheres, which are neither static nor homogeneous. Regard-ing temporal variations, we took care of picking our ob-servations at times when no flares were present, using themethod explained in Cincunegui et al. (2007). Spatial inho-mogeneities characteristic of magnetically active stars, likestarspots or active regions, cannot be resolved on the stellar surface. The models presented here, however, can be usedas a first step to build two component models as was done,for example, by Mauas & Falchi (1996).Faster temporal variations, like waves, cannot be re-produced with this kind of models, of course. We are alsonot considering possible small-scale spatial inhomogeneitieslike, for example, the chromospheric bifurcation proposedfor the Sun by Ayres (1981), which should be produced byCO cooling. However, on one hand this cooling was prob-ably overestimated (Mauas et al. 1990), and on the otherit is probably too slow compared to atmospheric dynamics(Wedemeyer-Bohm & Steffen 2007). In any case, homoge-neous models provide information on the ”mean” state ofthe stellar atmosphere, where the different components areweighted by their effect on the emitted radiation, in partic-ular on the spectral features under study. ε Eri
Before building the model atmosphere, a set of atmosphericparameters has to be determined. Both the surface gravityand the metallicity are fundamental input parameters in anyatmospheric model, and the effective temperature, althoughis not needed as input, is used in Ec. 2 to calculate the stellarsurface flux needed to analyze the results.In Table 3.1 we summarize several values of these quan-tities that can be found in the literature. Given the astro-physical interest on ε Eri, Drake & Smith (1993) recognizedthe necessity of determining these parameters with high pre-cision and they summarized the methods used to obtainthem until 1993, and the validity of these determinations.To improve these values, they determined the surface grav-ity, metallicity and effective temperature in a self-consistentway, comparing the equivalent widths of several Fe I, Fe IIand Ca I lines with theoretical profiles from different modelatmospheres. The parameters derived by Drake & Smith(1993) were used in the most recent model for ε Eri bySim & Jordan (2005, hereafter SJ05), although they recog-nized that the value of log(g) adopted could be too high(private communication).The difficulty in the calculation of the surface gravityis that it is indirectly determined from the values of massand stellar radius. Since these two parameters can be calcu-lated more precisely for stars in binary systems, we studiedanother star of our sample, α Centauri B (HD 128621), per-taining to the system α Centauri AB. For close systems likethis visual binary, the stellar radii and masses can be derivedwith an error of 1 to 10% (Guenther & Demarque 2000). c (cid:13) , 1– ?? M. Vieytes, P. Mauas and R. D´ıaz
Table 2.
Stellar characteristics for Epsilon Eridani (HD22049) from Cayrel de Strobel et al. (2001) and Table 1 fromDrake & Smith (1993).log(g) T eff [ F e/H ] Reference
According to Cayrel de Strobel et al. (2001), the valuesof log(g) found for α Cen B range from 4.51 to 4.73, withan average value of 4.60. We therefore adopted a value of log ( g ) = 4 .
65 for all the stars in our sample, since thisvalue is contained in the range given by Drake & Smith(1993), considering the error in their calculation ( log ( g ) =4 . ± . This same value of log(g) was adopted byNess & Jordan (2008) in their recent study of the coronaand transition region of ε Eri.Regarding the rest of the stellar parameters, we adopted T eff = 5110 K (Tomkin & Lambert 1999), which is close tothe value by Drake & Smith (1993). We adopted solar metal-licity as a good approximation for ε Eri, as has been done inall the previous models for this star, since it is a young starwhich is probably not metal defficient. This was suggestedby Krishna Swamy (1966), who built a grid of model atmo-spheres for ε Eri with different metallicities to fit the Ca IIK line and found that using solar metallicity results in thebest agreement with observations.In the case of α Cen B, Ayres & Linsky (1976) builttwo models for this star assuming in one case solar metal-licity and in the other an abundance twice as large. Theyconcluded that the computed profiles of the Ca II K linediffer very little and are in both cases consistent with theobservations.
To build the atmospheric models for ε Eri, as a first stepwe computed a photospheric structure capable of reproduc-ing the observed continuum spectrum for this star. Once thephotospheric model was obtained, we changed the chromo-spheric structure to fit the Ca II K and H β lines for bothsituations of interest, i.e. the maximum and minimum levelsof chromospheric activity. This is the first time this sort ofanalysis is made.Figure 2 shows the resulting models, which are pre-sented in column mass for comparison with the best one-component model from SJ05 (their model B). In Figure 3we compare the computed and observed continuum spec- Figure 2.
Models for ε Eri in its minimum (dashed line) andits maximum situation (full line). For comparison, we show themodel B from Sim & Jordan (2005) (dotted line) trum of ε Eri, and in Figure 4 and Figure 5 we show thecomparison of the observed and computed profiles for bothlevels of activity. It is important to note the good agreementof the fit, even better than the one by Thatcher et al. (1991)for all the diagnostic lines and continuum.
On top of the chromosphere, we added a tran-sition region with a similar structure to the solarone. Since we have no observations of lines formedin this region, we could no constrain it further. How-ever, the position at which the transition region be-gins was adjusted to fit the observed emission of theCa II k line.
There are several differences between our model and theone by SJ05. In our model, the photosphere is hotter, thetemperature minimum region is narrower and the chromo-spheric rise has a larger slope. Also, their transition regionis placed deeper in the atmosphere, i.e. , at higher values ofthe column mass.The differences between these models may be caused byseveral factors. As we have already noted, SJ05 used a highervalue of surface gravity which could explain the differences inall the thermal structure. The differences in the photosphericstructure could arise from the fact that we used the completespectra to fit the continuum emission, and SJ05 used thephotospheric model by Thatcher et al. (1991), built to fitonly the Ca II K line wings, which are formed in the higherphotosphere.Another important factor to consider is the moment ofthe activity cycle in which the observations used to buildthe model were taken. In our case, all the lines used as diag-nostics correspond to the same activity level since they wereall observed simultaneously. But in the model by SJ05, thestructure of the higher chromosphere and transition regionwas assembled with the model by Thatcher et al. (1991) forthe lower chromosphere and photosphere, whitout takinginto account that these thermal structures were obtained c (cid:13) , 1– ?? hromospheric changes in K stars with activity Figure 3.
Comparison between the observed (grey) and com-puted continuum (black) for ε Eri. -6 -4 -2 0 2 4 6123456 -4 -2 0 2 40.511.5-4 -2 0 2 412345 -4 -2 0 2 4123456
Figure 4.
Comparison of observed (dashed line) and computedprofiles (full line) for ε Eri in its minimum. using line profiles that correspond to different parts of theactivity cycle. For these reasons, the comparison betweenthese models is only qualitative.Regarding the differences in the atmospheric structurebetween the maximum and the minimum level of activity,the changes occur all along the atmosphere (Figure 2), fromthe temperature minimum to the transition region. The posi-tion of the minimum is the same in both situations, althoughthe temperature increases from 3980 K to 4050 K.
Finally, to check whether our results are affectedby the adopted value of the metallicity, we computed -6 -4 -2 0 2 4 6123456 -4 -2 0 2 40.511.5-4 -2 0 2 412345 -4 -2 0 2 4123456
Figure 5.
Comparison of observed (dashed line) and computedprofiles (full line) for ε Eri in its maximum. the emitted profiles for our models with the metal-licity given by Zhao et al. (2002), which was usedby Ness & Jordan (2008), and we found no signif-icant differences. This result is consistent with theone obtained by Ayres & Linsky (1976).3.3 The other stars
To build the models for the other stars in the sample, weused solar metallicity and the same surface gravity that wasused for ε Eri. The stellar surface flux was computed withEquation 2, using for each star the T eff values shown in Table2. Since we want to study the changes in thermal structureinduced by activity, we made the approximation that all thestars have the same photosphere than ε Eri.The models for the less active stars (HD 128621 and HD26965 in its maximum and minimum activity level, and ε Eriin its minimum) are shown in Figure 6. It can be seen thatall these models have the temperature minimum between 60and 100 km higher, and from 20 to 240 K cooler that ε Eriin its minimum.The temperature in the chromosphere, from thetemperature-minimum region up to 1100 km, increases withactivity, although the largest differences are in the chromo-spheric plateau. These changes with activity are different tothose obtained for G stars (Paper I) with similar activitylevels, because in that case only the temperature minimumregion changed, and the rest of the atmospheric structureremained the same.An important fact which can be seen in Figure 6, isthat the differences in the atmospheric structure for a star inits maximum and minimum activity levels are comparableto the changes seen beetwen two different stars. This factstresses how important it is, when building an atmosphericmodel, the moment at which the observations to be ajusted c (cid:13) , 1– ?? M. Vieytes, P. Mauas and R. D´ıaz
Figure 6.
Models for the less active group. All the models havethe same structure below 150 km. are made, and, in particular, how important it is to usesimultaneous observations of the diagnostic lines.In Figure 13 to 16 we show the observed and computedprofiles for α Cen B (HD 128621) and HD 26965 in its maxi-mum and minimum states. It is important to note the changein scale to compare with Figures 4 and 5, since these twostars are less active than ε Eridani.The models for the most active stars (HD 17925 in bothactivity levels, and HD 22049, HD 37572 and HD 177996in their maximum) are shown in Figure 7. Again the differ-ences in the atmospheric structure for a star in its maximumand minimum activity levels are similar to the changes seenbeetwen two different stars.In Figure 7 it can be seen that for the stars in this groupthe temperature minimum is hotter than for the stars in Fig-ure 6, and this temperature is almost constant as the activitylevel increases, varying only 50 K. The position of this regionis also the same for all these stars. The atmospheric struc-ture changes with activity everywhere in the chromosphere,mainly in the plateau and the rise to the transition region.The observed and synthetic profiles for the most activestars are compared in Figures 17 to 20. Again, it is importantto note the change in the scale for comparison with the lessactive stars and the good fit in all cases.
As was mentioned in Kelch et al. (1979), the ratio of thetemperature in the minimum and the effective temperature( T min / T eff ) gives an indication of the importance of nonra-diative heating in the upper photosphere of stars. In thatpaper, they compare this ratio with T eff to study the trenddue to spectral type.In Figure 8 we plot this ratio versus S CaII , which isan indicator of the level of magnetic activity in the chro-mosphere for all stars independently of spectral type. The
Figure 7.
Models for the more active group. All the models havethe same structure below 150 km.
Figure 8. T min / T eff vs. S CaII computed from the models forK stars (this paper, squares) and for G stars (Paper I, triangles). values of S
CaII were obtained by integration of the syntheticprofiles, and in the figure we include the values obtainedfrom the models for K stars built in this paper and those forG stars constructed in Paper I.In the figure it is possible to observe that there is a sat-uration in T min . In fact, its value increases with activity upto T min / T eff ∼ .
79, and after that it remains almost con-stant even if activity increases further. On the other hand,the computed value of T min / T eff for G stars is larger thanfor K stars with similar activity levels.In Figure 9 (left) we show the position of the tempera-ture minimum region in column mass as a function of S CaII for G (Paper I, triangles) and K stars (squares). For K starsthe temperature minimum occurs deeper than for G stars, c (cid:13) , 1– ?? hromospheric changes in K stars with activity Figure 9.
Position (in column mass) of the temperature minimum region (left) and the transition region (right), as a function of computed S CaII for G (Paper I, triangles) and K stars (squares). and there is a tendency for this region to move inward asactivity grows. In other words, the temperature inversion oc-curs deeper for more active stars, indicating that the energydeposition starts deeper in the atmosphere as the activitylevel increases, for both spectral types. In Figure 9 we alsoshow the position of the transition region (TR), specificallythe height at which the temperature reaches 36000 K. It canbe seen that for G stars the chromosphere is more extendedthan for K stars, and that in both cases the TR moves in-ward as activity increases.To study the energetic requirements to maintain theatmospheric structure, we calculated the total net radiativeloss for each model in the same way as in Paper I. At a givendepth, the radiative cooling rate Φ (ergs cm − sec − ) in agiven spectral feature (line or continuum) can be computedas (Vernazza et al. 1981)Φ = 4 π Z κ ν ( S ν − J ν ) dν , (3)where S ν is the source function and J ν is the mean inten-sity at frequency ν . A positive value of Φ implies a net lossof energy (cooling), and a negative value represents a netenergy absorption.Here, we considered line and continua of H, H-, H-ff,Mg I and II, Fe I, Si I, Ca II, Na I and CO. The totalrates for each star are shown in Figure 10 for the less activemodels, and in Figure 11 for the more active ones. As it isexpected, the amount of non-radiative energy supplied to thechromosphere increases everywhere with magnetic activity.In both figures, it is possible to note a region where thenet cooling rate is negative. This fact was already known forthe Sun (Vernazza et al. 1981), and was later found in PaperI for other G stars, for which negative cooling rates in thetemperature minimum region were also obtained. Within theplane-parallel, homogeneous approximation we are investi-gating, this implies either mechanical energy extraction or, more likely, that the calculations have neglected importantsources of radiative cooling (see Mauas 1993).The main contributions in this zone are H-, Si I, Fe Iand CO, the same than for G stars. It is important to notethat since the temperature for K stars is lower in this re-gion, there could be an important contribution of severalmolecules which we do not consider in our calculations, like,for example, CH, that could act as cooling agents. Consider-ing these contributions could bring our computations closerto energy balance.For the less active models the cooling rate becomes pos-itive at around 300 km, implying that there is mechanicalenergy deposition above this height. For the most activemodels, this energy deposition starts deeper in the atmo-sphere, i.e. the chromosphere starts deeper.Also in the chromosphere, the most important contrib-utors to the cooling rate are the same than for G stars, butthe proportions are different: for ε Eri in its minimum, forexample, Mg II and Ca II contribute with ∼
9% each, whilefor the Sun these contributions are of ∼ ∼
15% in ε Eri, butonly ∼
10% in the Sun. In both cases, almost half of thetotal cooling rate corresponds to line blanketing.Finally, to quantify the total amount of mechanical en-ergy deposited in the chromosphere, we integrated the netradiative cooling rate from the depth in the chromospherewhere the cooling rate becomes positive to the region wherethe temperature reaches 10 K. To compare the results forboth spectral types, we normalized the integrated rate, φ int ,by the surface luminosity ( σT ) The resulting quantity,therefore, gives an idea of the fraction of the totalenergy emitted by the star that goes into heating thechromosphere.
The results are shown in Figure 12, whereit can be seen that there is a unique trend for all stars, in-dependently of spectral type. This fact seems to imply that c (cid:13) , 1– ?? M. Vieytes, P. Mauas and R. D´ıaz
Figure 10.
Logarithm of the total cooling rate for the less activestars. A positive value of log Φ implies a net loss of energy (cool-ing), and a negative value represents a net energy absorption.
Figure 11.
Logarithm of the total cooling rate for the more ac-tive stars. A positive value of log Φ implies a net loss of energy(cooling), and a negative value represents a net energy absorption. the physical processes that supply the energy to sustain theatmospheric structure are independent of spectral type.Given the good corelation between S
CaII and the nor-malized φ int , we fit the data with a polynomial function,given by φ int σT = − .
14 10 − + 1 .
28 10 − S CaII +2 .
80 10 − S − .
80 10 − S . (4)In Figure 12 it can be seen that the fit is very good and,therefore, the energetic requirements of a given star can beestimated from its chromospheric activity level as mesuredby S CaII . Figure 12.
Normalized φ int versus computed S CaII index.
Empty squares represent the K star models from this work and full squares indicate the G star models from Paper I.
One of the main goals of chromospheric modelling is to ac-curately estimate the radiative losses in the chromospherein detail, using only the information that can be obtainedfrom the observations, without any assumption about thephysical processes involved. In this way, these losses can beequaled to the energy requirements that any proposed mech-anism of chromospheric heating should match.
For example, we saw in the previous section thatthe contribution of the different features to the totalcooling rate is not the same for G and K stars. Inparticular, the Ca II, Mg II and Fe I relative con-tributions are not the same for both spectral types.Therefore, it might not be correct to scale the rela-tive contributions computed for the Sun to K stars,as it has been done sometimes (see Cuntz et al.1999, and Rammacher et al. 2005).
Cuntz et al. (1999) computed theoretical two-component models for K dwarfs of different activity levels.They proposed that the energy is deposited in the chro-mosphere by acoustic and magnetic shocks, and found thatthese shocks are stronger and are produced deeper in thechromosphere as the activity of the star increases. This re-sult is in agreement with our calculations, which shows thatthe energy deposition is larger and deposited deeper withincreasing activity.On the other hand, they reproduced the lineal trendbetween the Ca II H and K lines fluxes and the rotationalperiod, although their computed fluxes are smaller than theobservations, which could be due to their sketchy calculationof the radiative cooling rate.
In this paper we present chromospheric models for six Kdwarfs, including ε Eridani, with similar photospheric prop-erties but different magnetic activity levels. In most cases c (cid:13) , 1– ?? hromospheric changes in K stars with activity we computed models for two moments of the activity cyclefor the same star.These models were based on, and reproduced very well,the Ca II H and K and the H β line profiles for all the stars inour sample. The reliability of the stellar atmospheric modelswas checked with other features, the Na I D and Mg I b lines.Also for these lines we found very good agreement betweencomputed and observed profiles.We found that the changes in atmospheric structure inK dwarfs with activity are produced all along the chromo-sphere, from the region of the temperature minimum to thetransition region and mainly in the chromospheric plateau,independently of the activity level of the star. This was notthe case for the G dwarfs modelled in Paper I, since for theless active G stars the changes with activity occur only inthe region of the temperature minimum.The ratio of the minimum and effective temperatures( T min / T eff ) can give an idea of the importance of non-radiative heating in the upper photosphere of stars. Bothfor K and G stars, this value increases with activity up to T min / T eff ∼ .
79, where it saturates, and it remains con-stant even if the activity level increases further. On the otherhand, the computed value of T min / T eff for G stars is largerthan for K stars with similar activity.For both spectral types, the position of the temperatureminimum moves inward as activity increases, implying thatthe chromosphere starts deeper in more active stars. This, inturn, implies that as the activity level increases, the energydeposition occurs deeper in the atmosphere.On the other hand, the transition region is placed athigher column masses for G stars than for K stars, and inboth cases it moves inward as activity increases.Regarding the energetic requirements, the integratedchromospheric radiative losses, normalized to the surfaceluminosity, show a unique trend for G and K dwarfs whenplotted against S CaII , the main proxy of stellar activity. Thismight indicate that the same physical processes are heatingthe stellar chromospheres in both cases. We calculated anempirical relationship between the S
CaII index and the en-ergy deposited in the chromosphere, which can be used toestimate the energetic requirements of a given star knowingits chromospheric activity level.There are significant differences in the contributions ofMg II, Ca II and Fe I to the total net cooling rate in thechromosphere between G and K stars, which implies thatvalues obtained for a given star should not be extrapolatedto another one of a different spectral type. In both casesabout half of the total rate is due to line blanketing.
ACKNOWLEDGMENTS
We would like to thank the director of the CASLEO Ob-servatory, and all the staff of this institution. The CCDand data acquisition system at CASLEO has been partlyfinanced by R.M. Rich through U.S. NSF grant AST-90-15827. We also thank the anonymous referee, whose com-ments help us to improve this paper. This work made exten-sive use of the SIMBAD database, operated at CDS, Stras-bourg, France. -4 -2 0 2 412345 -4 -2 0 2 40.20.40.60.81-4 -2 0 2 412345 -4 -2 0 2 412345
Figure 13.
Comparison of observed (dashed line) and computedprofiles (full line) for α Cen B (HD 128621) in its maximum. -4 -2 0 2 412345 -4 -2 0 2 40.20.40.60.81-4 -2 0 2 412345 -4 -2 0 2 412345
Figure 14.
Comparison of observed (dashed line) and computedprofiles (full line) for α Cen B (HD 128621) in its minimum.
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