On X-ray Optical Depth in the Coronae of Active Stars
Paola Testa, Jeremy J. Drake, Giovanni Peres, David P. Huenemoerder
aa r X i v : . [ a s t r o - ph ] J un On X-ray Optical Depth in the Coronae of Active Stars
Paola Testa , Jeremy J. Drake , Giovanni Peres , David P. Huenemoerder ABSTRACT
We have investigated the optical thickness of the coronal plasma throughthe analysis of high-resolution X-ray spectra of a large sample of active starsobserved with the High Energy Transmission Grating Spectrometer on
Chandra .In particular, we probed for the presence of significant resonant scattering in thestrong Lyman series lines arising from hydrogen-like oxygen and neon ions. Theactive RS CVn-type binaries II Peg and IM Peg and the single M dwarf EV Lacshow significant optical depth. For these active coronae, the Ly α /Ly β ratiosare significantly depleted as compared with theoretical predictions and with thesame ratios observed in similar active stars. Interpreting these decrements interms of resonance scattering of line photons out of the line-of-sight, we are ableto derive an estimate for the typical size of coronal structures, and from thesewe also derive estimates of coronal filling factors. For all three sources we findthat the both the photon path length as a fraction of the stellar radius, and theimplied surface filling factors are very small and amount to a few percent at most.The measured Ly α /Ly β ratios are in good agreement with APED theoreticalpredictions, thus indicating negligible optical depth, for the other sources in oursample. We discuss the implications for coronal structuring and heating fluxrequirements. For the stellar sample as a whole, the data suggest increasingquenching of Ly α relative to Ly β as function of both L X / L bol and the density-sensitive Mg xi forbidden to intercombination line ratio, as might generally beexpected. Subject headings:
Radiative transfer — X-rays: stars — stars:coronae — stars:late-type Massachusetts Institute of Technology, Kavli Institute for Astrophysics and Space Research, 70 Vassarstreet, Cambridge, MA 02139, USA; [email protected] Smithsonian Astrophysical Observatory, MS 3, 60 Garden Street, Cambridge, MA 02138, USA Dipartimento di Scienze Fisiche & Astronomiche, Sezione di Astronomia, Universit`a di Palermo Piazzadel Parlamento 1, 90134 Palermo, Italy
1. Introduction
A fundamental issue in the physics of stellar outer atmospheres concerns the relationshipbetween magnetic activity on stars with a wide range of physical parameters and solarmagnetic activity. How directly and how far does the solar analogy apply to other stars,and do any of the underlying physical processes differ? The X-ray luminosities of late-typestars can span several decades (e.g., Vaiana et al. 1981), and these hot coronae are foundon such a wide range of spectral types that the extrapolation of the now well-studied solarcorona to the extremes of stellar activity is by no means obvious and could be inappropriate.Indeed, “scaled up Sun” scenarios, in which a stellar surface is covered with bright solar-likeactive regions, only realise X-ray luminosities 100 times that of the typical active Sun (e.g.,Drake et al. 2000). The most active stars, with X-ray luminosities of up to 10,000 times thesolar X-ray luminosity, must have coronae which are structured differently in some way.Since coronal structures can be imaged presently only on the Sun, the structuring ofother stellar coronae is generally investigated through the application of techniques suchas the study of lightcurves during flares (e.g., Schmitt & Favata 1999; Favata et al. 2000a;Maggio et al. 2000; Reale et al. 2004; Testa et al. 2007), rotational modulation (e.g., Brickhouse et al.2001; Marino et al. 2003; Huenemoerder et al. 2006), and study of density properties to-gether with information on emission measure (e.g., Testa et al. 2004a, hereafter TDP04;Ness et al. 2004). Most of these analyses indicate that the emitting plasma is rather com-pact (scale height ≤ . R ⋆ ) and localized at high latitude (see e.g., Schmitt & Favata 1999;Brickhouse et al. 2001; Testa et al. 2004b, TDP04); however, the presence of extended coro-nal plasma has also been claimed on some stars based on UV and X-ray Doppler studies(e.g. Chung et al. 2004; Redfield et al. 2003).The search for signs of quenching in strong lines through resonance scattering representsa further technique that offers a potentially powerful diagnostic of the sizes of X-ray emittingregions; the escape probability of a photon emitted by a resonance line in a low densityhomogeneous plasma is in fact dependent on the line-of-sight path length through the plasmaregion. Significant scattering optical depth can be combined with density measurements toobtain an estimate of photon path length within the emitting plasma.Several existing studies have explored optical depths of both solar and stellar coronalemission lines. Studies of solar X-ray spectra have aimed at probing the optical depth inthe strong ( gf = 2 .
66) 2 p d P − p S resonance line of Fe xvii at 15.01 ˚A as comparedto nearby weaker Fe xvii lines, though with controversial results concerning whether opticaldepth effects were seen or not (Phillips et al. 1996, 1997; Schmelz et al. 1997; Saba et al.1999). In particular, Saba et al. (1999) review recent observational findings on the opacityinferred from the study of the bright iron resonance line at 15.01 ˚A and on the center- 3 –to-limb behaviour. Among other issues, Saba et al. (1999) address the discrepancy theyfind in the derived direction and magnitude of the center-to-limb trend (also in agreementwith Schmelz et al. 1997), as compared to the findings of Phillips et al. (1996) who findthat the effect of resonant scattering is decreasing from the disk center toward the solarlimb, a trend irreconcilable and totally opposite to that found by Saba et al. (1999) andSchmelz et al. (1997). Brickhouse & Schmelz (2006) have recently reanalized solar X-rayspectra and suggest that previously ignored blends might explain the departure of measuredratios from theoretical calculations.Resonant scattering in stellar coronae has been investigated by Phillips et al. (2001)and Ness et al. (2003) through the analysis of the same transition observed at high resolu-tion ( λ/ ∆ λ up to ∼ Chandra and XMM-
Newton . Both stellar studies of Fe xvii transitions fail to find evidence for significant deviation from the optically thin regime, andin particular the large survey of stellar spectra analyzed by Ness et al. (2003) show thatno firm results can be obtained from Fe lines. One exception is the suggestion of resonancescattering in Fe xvii § into the line of sight renders optical depthmeasurements themselves only lower limits to the true scattering depth.In a previous Letter (Testa et al. 2004b, hereafter Paper I) we presented results obtainedusing a different approach to the study of coronal optical depth, through the analysis ofNe and O Ly α to Ly β line strength ratios as observed by the High Energy TransmissionGrating (HETG) on board the Chandra
X-ray Observatory. Significant depletion of Ly α lines to resonance scattering were seen in the spectra of the RS CVn-type binaries II Pegand IM Peg. In this paper we follow up on that exploratory study and examine the Neand O Lyman series lines in a large number of active stars (namely, the same sample forwhich the plasma density was analysed in TDP04) in order to survey the X-ray optical depthproperties of active stellar coronae.We discuss in § xvii lines. The observations are briefly described in §
3. Ourtechniques of line flux measurement and spectral analysis are described in §
4. The resultsare presented in §
5. We combine the results of this study with our earlier density estimatesand discuss these in the context of coronal structure on active stars in §
6; we draw ourconclusions in §
7. 4 –
2. Resonant scattering in Ne and O Lyman series lines.
The Fe xvii soft X-ray complex at ∼ p S -2 p d P xvii–xx lines. There is thusstill some considerable difficulty in reconciling theoretical and observed line strength ratios.Recently, Brickhouse & Schmelz (2006) have found good agreement of solar observed ratioswith new theoretical calculations (Chen & Pradhan 2005), and suggest that center-to-limbobserved trends (Phillips et al. 1996; Schmelz et al. 1997; Saba et al. 1999) are due to chancerather than to optical depth effects.An additional problem in using Fe xvii lines as diagnostics of optical depth is thatthis element has been found to be depleted in the coronae of active stars by factors of upto 10 (e.g., Drake et al. 2001; Huenemoerder et al. 2001; Drake 2003; Audard et al. 2003)as compared with a solar or local cosmic composition. The ratio of the line-center opticaldepths, τ i /τ j , of two lines i and j is given by τ i τ j = f i λ i φ i A i √ m i f j λ j φ j A j √ m j (1)where f is the oscillator strength, λ is the wavelength, φ is the fractional population of theion in question, A is the element abundance and m the ion mass. The line optical depth isdirectly proportional to abundance and, in the case of stellar coronae, where only the verystrongest spectral lines might be expected to undergo any significant resonance scattering,any abundance depletions also reduce the sensitivity of lines as optical depth indicators. Forcoronal abundances typically found in RS CVn systems (see e.g. reviews by Drake 2003;Audard et al. 2003), we expect resonant lines from the more abundant ions like oxygen andneon to be more sensitive to resonant scattering processes than the resonant Fe xvii line.This is illustrated in Figure 1, where we show the relative optical depths for the O viii andNe x Ly α lines and the Fe xvii ∼ .
01 ˚A resonance line for both a representative solarchemical composition (Grevesse & Sauval 1998) and for a chemical composition typicallyfound for active stars. For the latter, we assumed Ne and Fe abundances 0.3 dex higher and0.5 dex lower, respectively, than the corresponding solar values; we note that in several activecoronae abundance anomalies even more pronounced have been found (e.g., Brinkman et al.2001; Huenemoerder et al. 2001). 5 –Fig. 1.— The relative sensitivity of Ly α lines of Ne x , O viii , and the strong Fe xvii viii , are shown for two sets of abundances.Dashed lines correspond to the solar photospheric abundance mixture of Grevesse & Sauval(1998); filled profiles correspond to coronal abundances typically found in RS CVn’s (seetext). Because of the typically higher Ne abundance and lower Fe abundance found in thecoronae of RS CVn systems, the expected resonant scattering effects are greater for O viii and Ne x Ly α than for the Fe xvii . 6 –As discussed in Paper I, the effect of resonance scattering of Ly α and Ly β photons canbe diagnosed by comparison of the measured Ly α /Ly β ratio with respect to the theoreticalratio. In principle, both n = 2 → α and n = 3 → β lines can be affected by resonantscattering: when a large optical depth is reached in Ly α , an enhanced population of the n = 2level can lead to a potentially confusing enhancement in Ly β (and Ba α ) through collisionalexcitation of the n = 2 → β essentially remains optically thinand should make a reliable comparison with which to diagnose optical depth in Ly α .
3. Observations
Chandra
High Energy Transmission Grating Spectrometer (see Canizares et al. 2000,2005 for a description of the instrumentation) observations of 22 cool stars covering a widerange activity level were analysed. The stellar parameters and particulars of the observa-tions were discussed in detail in a companion paper that addressed the coronal densities ofactive stars (TDP04). To the sample analysed in TDP04 we added a set of observations ofIM Peg obtained subsequent to the first three segments analysed in Paper I and TDP04; theparameters of these additional observations are listed in Table 1. 7 –Table 1. Parameters of the HETG observations of IM Peg.
Obs ID Start date and time t exp L Xa L Xb [ks] [erg s − ] [erg s − ]2527 2002-07-01, 15:39:08 24.6 2.95 × × × × × × × × × × × × × × × × relative to the HEG range: 1.5-15 ˚A b relative to the MEG range: 2-24 ˚A hetgs bandpasses; see TDP04)span more than five orders of magnitude, from the relatively weak emission of ProximaCentauri with a few 10 erg s − , up to the very high luminosity ( ∼ × erg s − ) of thegiant HD 223460.
4. Analysis
The data used here were obtained from the
Chandra
Data Archive and have beenreprocessed using standard CIAO v3.2.1 tools and analysis threads. Effective areas were cal-culated using standard CIAO procedures, which include an appropriate observation-specificcorrection for the time-dependent ACIS contamination layer. Positive and negative spectralorders were summed, keeping heg and meg spectra separate. For sources observed in sev-eral different segments, we combined the different observations and analysed the coaddedspectra. For IM Peg, as noted above, a number of observations are available with which toprobe optical depth properties at different times and orbital phases; IM Peg will be discussedin more detail in § IDL software (Kashyap & Drake 2000)using the technique of spectral fitting described in TDP04. We measured the spectral lineintensities of the Ne x Ly α (2 p P / , / − s S / ) and Ly β (3 p P / , / − s S / ) transitionsfrom both heg and meg spectra, and the O viii lines from meg spectra alone (since thelatter lie outside the heg wavelength range).In order to take into account the mild dependence of the Ly α /Ly β ratio on plasmatemperature, we also measured the intensities of the Ne ix and O vii resonance line, r (1 s p P − s S ), providing us with an estimate of a representative temperature throughthe Ly α / r ratio. There are two potential problems with this approach. Firstly, where reso-nant scattering is relevant, both Ly α and r transitions can be depleted, and in such a casethe Ly α / r ratio might deviate from its expected theoretical behaviour. Secondly, the coro-nae in which these X-ray lines are formed are expected to be characterised by continuous http://cxc.harvard.edu/cda Interactive Data Language, Research Systems Inc. α / r ratio and ofthe Ly β /Ly α ratio for Ne lines (O lines show analogous behaviour). The Ly α / r is much moretemperature-sensitive than Ly β /Ly α , and the relatively small deviations from the optically-thin case that might be expected in stellar coronae can only incur small errors in temperaturethat will have a negligible impact on the predicted Ly β /Ly α ratio. 10 –Fig. 2.— Temperature sensitivity (in the isothermal case) of the Ly α / r ratio compared to theLy β /Ly α ratio, for neon lines. The corresponding oxygen lines show analogous behaviour. 11 –The accurate temperature determination from the Ly α / r ratio strictly holds only forisothermal plasma, whereas stellar coronae are characterised by a thermal distribution of theplasma, so that this diagnostic would give us a temperature weighted by the emission mea-sure distribution (DEM). By computing line ratios for non-isothermal plasma models, andinterpreting as if isothermal, we can estimate the errors incurred by an isothermal assump-tion. We considered two different sets of DEMs: those derived from actual observationsof some of the stars in our sample, and simple DEM models in which the emission mea-sure is proportional to T / (as expected for simple hydrostatic loop models; Rosner et al.1978), or proportional to T / (as observed in some stars such as 31 Com, Scelsi et al.2004, and reproduced by some hydrodynamic loop models, Testa et al. 2005). For thesemodels we used peak temperatures varying from 10 . K to 10 . K. Observed DEMs wereculled from the literature for the following sources: AB Dor (Sanz-Forcada et al. 2003b),HD 223460 (Testa et al. 2007), 31 Com, β Cet, µ Vel (Garc´ıa-Alvarez et al. 2006), ER Vul,TZ CrB, ξ UMa (Sanz-Forcada et al. 2003a), 44 Boo (Brickhouse & Dupree 1998), UX Ari(Sanz-Forcada et al. 2002), II Peg (Huenemoerder et al. 2001), λ And (Sanz-Forcada et al.2002), AR Lac (Huenemoerder et al. 2003), HR 1099 (Drake et al. 2001).Given the line ratio computed for each DEM model, we inverted the isothermal relationto obtain T (Ly α / r ). Using that T , we then obtained the theoretical isothermal Ly α /Ly β ratio (Ly α /Ly β [ T (Ly α / r )]) and compared to the synthetic ratio (Ly α /Ly β [DEM]). Wefind that the isothermal assumption is a good predictor of the ratio for both theoreticalDEMs and for representative stellar models: Figure 3 shows that the isothermal and DEMratios are in very good agreement, with differences between the two generally amounting toa few percent. We conclude that the Ly α / r temperatures are quite adequate to assess theappropriate expected Ly α /Ly β ratio, even if the real plasma temperature distributions ofthe coronae in our study are far from isothermality. 12 –Fig. 3.— Left panel:
DEM models (DEM ∝ T / , T / ) used to test the validity of theisothermal approximation. Right panel:
The Ne x Ly α /Ly β ratios obtained for model (leftpanel) and observed DEMs (see text; black empty symbols) compared with the Ly α /Ly β ratios expected for isothermal temperatures diagnosed from the DEM Ne Ly α / r ratios. 13 – x Ly α and O viii Ly β Ne x Ly α and O viii Ly β lines are affected by blending of iron lines unresolved at the hetgs resolution level. Specifically, significant blending is expected for O viii Ly β by anFe xviii line (2 s p ( P )3 s P / − s p P / , λ = 16 . x Ly α line by a nearby Fe xvii transition (2 s p ( P )4 d P − s p S , λ = 12 . O viii Ly β — In order to estimate the often significant contribution of Fe xviii to theO viii Ly β spectral feature, we used the same method described in Paper I, which was alsosubsequently adopted by Ness & Schmitt (2005) in an analysis of spectra of the classicalT Tauri star TW Hya. We estimated the intensity of the 16.004˚A Fe xviii line by scalingthe observed intensity of the slightly stronger neighbouring unblended Fe xviii s p ( P )3 s P / − s p P / ) transition. Gu (2003b) has recently pointed out thesimilar behaviour of Fe L-shell lines originating from 3 s and 3 p upper levels in respectto the indirect excitation processes of radiative recombination, dielectronic recombination,and resonance excitation. The 16.004 ˚A and 16.071 ˚A transitions originate from similar 3 s upper levels and their ratio should therefore not deviate greatly from current theoreticalpredictions. The APED (v.1.3.1) database (Smith et al. 2001) lists their theoretical ratio as0.76 at the temperature of the Fe xviii population peak ( ∼ . xviii in our target stars.We investigated this scaling factor empirically by examining the departure of the re-sulting deblended Ly α /Ly β ratios from the theoretical value as a function of the ratio of theobserved Fe xviii β line strengths. If the deblending scaling factor is correct,there should be no residual slope in the resulting data points; a positive slope would insteadindicate an over-correction for the blend and a negative slope an under-correction. We foundthat a slope of zero was obtained for a scaling factor of ∼ . viii Ly α /Ly β ratios (relative to the theoretical ratio) before ( left ) and after( right ) the correction for the contamination of the Fe xviii line blending with the O Ly β line, plotted vs. the relative intensity of the Fe xviii line at ∼ β be-fore the deblending process. The effect of the blending Fe xviii line is clear from the left plot. The right plot shows the effectiveness of the deblending procedure which eliminatesthe strong correlation between the departure from the theoretical Ly α /Ly β value and theFe xviii /OLy β ratio before the deblending. 15 – Ne x Ly α — In analogy with the procedure used for the O viii Ly β we searched for isolatedand strong Fe xvii lines to estimate the amount of blending from Fe xvii λ = 12 . x Ly α line. A potentially good candidate would be the nearby 12.266˚A Fe xvii line(2 s p ( P )4 d D − s p S ), which has a similar intensity and behavior as a functionof temperature; however this transition is rather weak in most of the observed spectra, andcan be affected itself by blending with a close Fe xxi transition (12.284˚A) barely resolved by hetg .We therefore chose to use a larger set of Fe xvii lines including stronger transitions.The intensity of the Fe xvii line blending with the Ne x Ly α is estimated by scaling theobserved intensity of other Fe xvii lines (12.266, 15.014, 15.261, 16.780, 17.051, 17.096˚A) bythe ratio expected from Landi & Gu (2006) (see also Gu 2003a; Landi et al. 2006), assuminglog T = 6 .
8. These calculations include indirect processes involving the neighboring chargestates, and are quite successful at reproducing the relative intensity of these Fe xvii lines.Previous calculations (including those adopted in APED v.1.3.1) fail here, with, e.g., theratio of the 15˚A line to the 17˚A lines being overestimated by about a factor 2. The detailedcomparison of the observed Fe xvii line ratios with the predictions of different theoreticalcalculations are addressed in a paper in preparation.The measured line fluxes after application of deblending corrections, together with thestatistical errors, are listed in Table 2.
5. Results
The observed Ly α /Ly β ratios and the temperature derived from the Ly α / r ratios arelisted in Table 3 and illustrated in Figure 5. Measurements obtained from the meg spectragenerally result in smaller statistical errors due to the higher S/N; however, we find a generalagreement of heg and meg results . The main differences of the present work with respect tothe analysis of Paper I, where results were presented for 4 sources (II Peg, IM Peg, HR 1099,and AR Lac), are that the line fluxes have been remeasured in the reprocessed data, Ne x The issue of some systematic discrepancies between heg and meg measurements has been addressedbriefly in TDP04, and it is thoroughly discussed in http://space.mit.edu/ASC/calib/heg meg/, where sys-tematic discrepancies of 8-10% are found in the 10-12˚A range ( meg fluxes being lower than corresponding heg fluxes at the same wavelength) therefore not significantly affecting the Ly α /Ly β ratios. We note thatour measured heg fluxes are in agreement with meg measurements within 2 σ for the large majority of cases;the only significant exceptions are the Ne x Ly α of TZ CrB and of HR 1099 (3 , σ respectively) whose heg fluxes are ∼
10% larger than the corresponding meg fluxes in agreement with the comparisons presented inhttp://space.mit.edu/ASC/calib/heg meg/.
16 –Ly α fluxes have been corrected for blending, and the deblending procedure for O viii Ly β hasbeen slightly refined (see previous section). Also, IM Peg fluxes listed here refer to the total( ∼
192 ks) hetgs spectrum including ObsID 2530-2534 not yet publicly available when wecarried out the analysis presented in Paper I (where we used the first three 25 ks pointings;see § − photons cm − s − ). Source Ne O FeLy α a Ly β r Ly α Ly β b r Fe xvii c Fe xviii ±
50 356 ±
15 51 ±
10 44 ± ±
15 990 ±
40 134 ±
20 243 ±
40 21 ± ± ±
20 - 6 . ± . ± ±
28 38 ±
14 100 ±
70 6 ± ± ±
22 262 ±
10 30 ± ± ± ±
19 137 ±
11 330 ±
30 24 ± ± ±
50 682 ±
24 96 ±
16 86 ± ±
27 1400 ±
50 200 ±
22 280 ±
50 41 ± ± ±
19 95 ± ± . ± . ±
14 265 ±
30 38 ±
12 114 ±
40 5 . ± ± ±
20 137 ±
11 22 ± . ± . ± ±
22 27 ±
11 48 ±
24 10 ± ± ± . ± . ± ±
16 15 ± ±
30 8 . ± ± β Cet 419 ±
24 408 ±
10 53 ±
10 48 ± ±
10 764 ±
29 62 ±
23 140 ±
40 116 ±
21 426 ± ± . ± . ± ±
22 16 ± ±
30 11 . ± ± µ Vel 173 ±
21 159 ±
15 16 ±
13 18 ± ± ±
30 45 ±
16 46 ±
27 62 ±
12 163 ± ±
50 832 ±
40 136 ±
23 120 ± ±
20 1318 ±
50 166 ±
30 190 ±
60 63 ±
14 258 ± ±
22 201 ±
12 26 ± . ± . ± ±
29 39 ±
12 66 ±
40 24 ± ±
944 Boo 572 ±
60 537 ±
24 75 ±
11 68 ± ±
20 1450 ±
60 122 ±
22 266 ±
60 53 ±
11 151 ± ±
30 1043 ±
24 145 ±
14 127 ± ±
20 2110 ±
50 178 ±
26 300 ±
50 131 ±
25 445 ± ±
40 767 ±
20 126 ±
17 107 ± ±
18 926 ±
50 147 ±
21 158 ±
40 14 . ± ± ξ UMa 419 ±
27 394 ±
13 42 ± ± ±
16 1510 ±
60 169 ±
22 496 ±
60 83 ±
17 185 ± ±
70 1200 ±
30 183 ±
24 191 ±
10 356 ±
20 1950 ±
70 350 ±
30 250 ±
50 18 ± ± λ And 557 ±
30 556 ±
14 80 ±
11 84 ± ±
14 984 ±
40 113 ±
17 110 ±
30 24 . ± ± ±
29 312 ± ±
12 40 ± ±
15 468 ±
50 51 ±
21 96 ±
50 25 ± ± ±
40 621 ±
17 104 ±
15 94 ± ±
20 888 ±
50 94 ±
23 130 ±
50 34 ± ± ±
40 1730 ±
18 255 ±
14 230 ± ±
18 2680 ±
50 367 ±
21 410 ±
40 58 ±
12 246 ± ±
25 383 ±
10 62 ± ± ± ±
30 75 ±
12 50 ±
20 10 . ± . ± a Ne Ly α fluxes after the deblending with the Fe xvii line at 12.124˚A. b O Ly β fluxes after the deblending with the Fe xviii line at 16.004˚A. c Fe xvii line (at 12.124˚A) flux estimated by scaling the measured fluxes of the other Fe xvii lines.
18 –Fig. 5.— Ly α /Ly β , both from heg (filled symbols) and from meg (empty symbols) vs.temperature, as derived from the ratio of the Ly α and the He-like resonance line. Thesolid line mark the theoretical ratio from APED, expected for isothermal plasma at thecorresponding temperature. As indicated on the right plot, different symbols are used fordifferent classes of sources. Ly α /Ly β ratios from heg are shifted in T by 2% to separatethem from the meg measurements. 19 –Table 3. Ly α /Ly β photon ratios and plasma temperature, with 1 σ errors. Source Ne O T a Ly α /Ly β ∆[ σ ] b T a Ly α /Ly β ∆[ σ ] b [10 K] HEG MEG HEG MEG [10 K] MEG MEGAU Mic 5 . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . ... 10 . ± . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +1 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . ... 7 . ± . . +0 . − . . ± . β Cet 6 . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . ... 11 . ± . . +0 . − . . ± . µ Vel 4 . +0 . − . . ± . . ± . . +1 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . ξ UMa 4 . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . λ And 6 . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . . +0 . − . . ± . . ± . . +0 . − . . ± . a T derived from the Ly α / r line ratio diagnostics. b Discrepancy between Ly α /Ly β measured and theoretical value in units of σ .
20 –The observed Ne x Ly α /Ly β ratios follow closely the APED theoretical predictions asa function of temperature. The values for the O viii ratios (right panel of Fig. 5) showslightly more scatter, with some departures below theory and some 1-2 σ excursions above.We note that an enhancement of the Ly α /Ly β ratio is possible for the particular geometryin which an emitting region is optically-thick in some directions but is optically-thin in theline-of-sight of the observer (see also the radiative transfer study of Kerr et al. 2004). Sucha situation is disfavoured by the isotropic nature of the scattering process in which any lineenhancements through scattering are diluted, roughly speaking, by the ratio of solid anglesof optically-thin to optically-thick lines-of-sight. In the case of loop geometries, such a ratiois much larger than unity. We therefore interpret these small upward 1-2 σ excursions asnormal statistical fluctuations.In the case of the Ne x lines, the meg Ly α /Ly β ratio for II Peg is ∼ σ lower thanthe expected value; the ratio derived from heg is consistent with the meg measurementbut does not depart significantly from theory. The only sources showing O viii Ly α /Ly β significantly ( > σ ) lower than the theoretical ratio are II Peg and EV Lac. We note thatthe low Ly α /Ly β ratios for IM Peg discussed in Paper I were found in the spectra obtainedin the first three pointings while here we analyze the whole set of observations; in § α /Ly β ratios for IM Peg.One other interesting case is the T Tauri star, TW Hya, that shows very high density, n e & cm − , at the temperature of O vii lines (e.g., Kastner et al. 2002). In this case, thehigh densities are generally believed to arise from an accretion shock, rather than coronalloops; significant scattering effects could in principle provide interesting constraints on thedimensions of the shock region. Unfortunately, for this source the relatively low S/N resultsin large error bars in the Ly α /Ly β ratios that do not provide any useful constraints (as wasalso noted in the analysis of Fe xvii lines by Ness & Schmitt 2005). The different
Chandra hetg observations of the RS CVn system IM Peg provide anopportunity for studying the optical depth and its possible variation with time. This systemhas been analyzed in great detail at optical wavelengths by Berdyugina et al. (1999, 2000).These works depict a scenario with stellar spots concentrated mainly close to the polarregions, similar to those found for many other active systems (e.g., Schuessler et al. 1996;Hatzes et al. 1996; Vogt et al. 1999; Hussain et al. 2002; Berdyugina et al. 1998). 21 –
Chandra observed IM Peg eight times over ∼ P rot = 24 .
39 d, P orb = 24 .
65 d, Strassmeier et al. 1993). In Paper I we analyzed thefirst three 25 ks hetgs observations of IM Peg, publicly available at that time, and we foundremarkably small photon path lengths based on optical depth effects. The compact emissionregions implied might be associated with active regions revealed by optical Doppler imagingstudies. Any significant net line photon loss to resonant scattering out of the line-of-sightwould be expected to be sensitive to the orientations of the emitting structures. Modulationof the Ly α /Ly β ratio with time could provide important new structure and morphologydiagnostics. Such regions might also be expected to change on relatively short timescales asa result of flaring activity and consequent re-alignment of their defining magnetic fields. 22 –Table 4. Ly α /Ly β photon ratios with 1 σ errors. Obs. ID Ne x O viii HEG MEG MEG2527+2528 5 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
23 –Fig. 6.— Lightcurve of IM Peg
Chandra observations; both heg ( filled squares ) and meg ( empty squares ) L X are presented. heg values are shifted by 10 s on the time axis withrespect to the corresponding meg measurements. The phases corresponding to the absolutetimes are indicated at the top of the plot. 24 –Fig. 7.— Ne x Ly α, β spectral region for the three chosen time segments of the observationof the RS CVn system IM Peg, as indicated in the plots. Both heg ( upper panel ) and meg ( lower panel ) spectra are presented, normalized to the intensity of the Ly β line at10.24 ˚A. For better readability the spectra corresponding to the second (2529+2530+2531)and the third (2532+2533+2534) segment are shifted in wavelength with respect to the first(2527+2528) by +0.03 ˚A and +0.06 ˚A respectively. Note the pronounced difference in Ly α and Ly β relative intensity in the three spectra especially between the first and the thirdspectra. 25 –Here we present an analysis of the complete dataset, exploring possible temporal vari-ability of the optical depth of the coronal plasma. Unfortunately, the single IM Peg exposuresprovided counts sufficient for reliable intensity measurement only for the strongest lines inthe spectrum, such as Ne x Ly α . In order to probe secular change in the Ly α /Ly β ratio,we therefore measured lines from spectra coadded from contiguous sets of two or three ob-servations that sample different phases of the stellar period: ObsIDs 2527 and 2528; 2529,2530, and 2531; and 2532, 2533, and 2534. The spectra in the Ne and O Ly α, β regions forthese three portions of the observation are shown in Figure 7 and 8. The measured Ly α /Ly β ratios are listed in Table 4. 26 –Fig. 8.— O viii Ly α, β spectral region of meg spectra of IM Peg for the three chosentime segment of the observation, in the same format of the plots in Figure 7. The shift inwavelength of second and third spectra is of +0.06 ˚A with respect to the preceding spectrum.A clear trend in the Ly α /Ly β line ratio is present. 27 –The Ly α /Ly β ratios, while statistically all consistent with one another, are suggestiveof different conditions in the different portions of the observation. Both Ne and O Ly α /Ly β ratios in heg and meg spectra are lower than the corresponding APED theoretical valuein the first segment, as already discussed in Paper I and noted earlier. The rest of theobservation is characterized by Ly α /Ly β values compatible with theory. The measuredratios are illustrated as a function of the hardness ratio ( HR = ( H − S ) / ( H + S ), where H is the flux integrated in the 2-9˚A wavelength band, and S is the flux integrated in the9-25˚A band) in the different segments of the observation in Figure 9. There is no obviouscorrelation of Ly α /Ly β with spectral hardness that might suggest, e.g., that the formerchanges as a result of plasma temperature changes. These results show that it might beworthwhile to explore, when the data quality allows it, temporal variability of optical depthproperties in stellar coronae, expected to some extent on the basis of their dependence onthe line-of-sight and coronal geometry other than on the physical conditions of the emittingplasma. 28 –Fig. 9.— Ly α /Ly β ratios measured from IM Peg spectra for the three time segments selectedfor the analysis (2528+2528, 2529+2530+2531, and 2532+2533+2534; upper panel ), togetherwith the hardness ratio, HR ( lower panel ). In the upper panel the horizontal lines representthe APED values expected for the Ly α /Ly β ratio of O ( dashed line ) for a temperature of5 MK, and of Ne ( dotted line ) for a temperature of 9 MK. For better readability Ly α /Ly β ratios of Ne x from heg , and of O viii are shifted with respect to meg Ne x Ly α /Ly β ratiosby +0.05 and +0.1 respectively on the phase axis. 29 – The data presenting more reliable evidence for optical thickness are the spectra of IM Pegfrom the first two observations, where Ly α of both oxygen and neon appear to be depletedwith respect to the corresponding Ly β transition (deviation of ∼ . σ and ∼ σ respec-tively), and the spectrum of II Peg characterized by an O viii Ly α /Ly β ratio ∼ σ lowerthan the theoretical value. As pointed out in Paper I, the discrepancies between observedand theoretical Ly α /Ly β line strength ratios in IM Peg and II Peg cannot be explainedby photoelectric absorption along the line-of-sight on the basis of their measured H columndensities (Mewe et al. 1997; Mitrou et al. 1997). It is worth discussing whether these transi-tions, and in particular their ratios, might be affected by non-equilibrium conditions that arepossibly relevant for these active stars undergoing frequent flaring activity. Non-equilibriumeffects might be responsible for changes in the O vii G ratio (( f + i ) /r ) observed for EV Lacbetween quiescent and flare phases (Mitra-Kraev 2006). It is well-known that the G ratio issensitive to deviations from coronal equilibrium, because recombination processes contributesignificantly to f and i but are almost negligible for r (e.g. Pradhan 1985). Even though re-combination processes have non-negligible effects on Ly α and Ly β (10-15%), they contributeto a very similar extent to the two transitions, and therefore their ratio is not expected tochange significantly under mildly non-equilibrium conditions. Furthermore, we note that forthe typical conditions of the coronal plasma in these active stars the expected timescalesof these effects are only of the order of hundreds of seconds (e.g., Golub et al. 1989), i.e.very short with respect to our integration times of several tens of kiloseconds. We thereforeinterpreted the discrepant ratios in terms of a relative depletion of the Ly α line flux due toresonance scattering processes.Another source apparently showing significant effects of resonant scattering is EV Lac,whose O viii Ly α /Ly β ratio is more than 3 σ lower than the corresponding theoretical value.This result is supported to some extent by the findings of Ness et al. (2003): in their surveyof coronal optical depth properties based on the analysis of Fe xvii transitions, EV Lac isthe only source deviating significantly from the typical values found for all other coronae.However, Ness et al. (2003) did not consider this result robust due to discrepancies between heg and meg measurements.For the stars with Ly α /Ly β departing from the theoretical values we can derive anestimate of the photon path length using the Kaastra & Mewe (1995) approximation to theescape probability formalism of Kastner & Kastner (1990), as described in § α /Ly β can be expressed in terms of the line center optical depths τ k , f-values 30 – f k , and photon pathlength ℓ , as( I Lyα /I Lyβ ) obs ( I Lyα /I Lyβ ) th = 13 (cid:20)
21 + 0 . C ( ℓ ) f α + 11 + 0 . C ( ℓ ) f α / (cid:21) · (1 + 0 . C ( ℓ ) f β ) (2)where C ( ℓ ) · f k = τ k and the line center optical depth is given by τ = 1 . · − · n i n el A Z n H n e λf r MT n e ℓ (3)where n i /n el is the ion fraction (from Mazzotta et al. 1998), A Z is the element abundance, n H /n e ∼ . f is the oscillator strength, M the atomic weight, T the temperature, and n e the electron density. The Ly α and Ly β f-values are f α = 0 . f β = 0 . n e , derived from the diagnostics of the He-like tripletsTDP04. The coronal abundances assumed for II Peg and IM Peg are discussed in Paper I;for EV Lac we assume O/H=8.40, as derived by Favata et al. (2000b), expressed on theusual spectroscopic logarithmic scale in which X/H= log[ n (X) /n (H)] + 12, where n (X) is thenumber density of element X.In Table 5 we list the values obtained for the path length estimates, ℓ τ , and for com-parison, we list the stellar radii and loop lengths expected for a standard hydrostatic loopmodel (e.g. Rosner et al. 1978, RTV hereafter) corresponding to the observed temperaturesand densities. 31 –Table 5. Path length derived from measured Ly α /Ly β ratios. Source ℓ τ L RTV a ℓ τ /R ⋆ b f s c E S d [cm] [cm] [erg cm − s − ]IM Peg O viii . · . · ∼ . ∼ . · Ne x [HEG] 2 . · . · ∼ . ∼ . · Ne x [MEG] 1 . · . · ∼ . ∼ . · II Peg O viii . · · ∼ . ∼ . · EV Lac O viii . · . · ∼ . ∼ . · Loop length from RTV scaling laws: L RTV ∼ T / [(1 . × ) · p ]. b Path length as fraction of the stellar radius. c Coronal surface filling factor, defined as f s = A/A ⋆ = ( V /ℓ τ ) /A ⋆ = [ EM/ ( n ℓ τ )] /A ⋆ . d Estimate of surface heating flux (see text for details).
32 –
6. Discussion
This work present a detailed and extensive study of the O viii and Ne x Ly α /Ly β ratiosin a sample of active stars, paying careful attention to the presence of blends and to thevalidity of theoretical line ratio predictions. Our analysis shows that optical depth effectsare generally negligible in the disk-integrated X-ray spectra emitted by stellar coronae overa wide range of activity, in line with previous studies based on Fe xvii lines (e.g., Ness et al.2003; Audard et al. 2004). We argued in § into the line of sight thatcould enhance the observed line strength are not included. For instance, in the case of aspherically symmetric corona in which the line optical depth were significant, scattering intoand out of the line of sight would be balanced and line strengths not affected (see e.g.,Wood & Raymond 2000). Strictly, then, the photon path, ℓ , entering into the equations in § lower limit to the true photon path length. This is potentiallyimportant. Phillips et al. (2001), for example, used the lack of appreciable depletion of theFe xvii resonant line at ∼ . upper limit to the lower limit , therefore not providing a constraint. Indeed, the lack ofevidence for line quenching through scattering in most of our sample stars does not implythat scattering is not a significant source term, but merely that the scattering geometry issuch that there is no net loss or gain of photons in the line-of-sight.Instead, a positive detection of resonance scattering has interesting implications andpoints to a non-uniform “aspect ratio” of the dominant coronal emitting regions: for any netline depletion to occur, an emitting structure must generally be more elongated along the lineof sight than in the perpendicular direction. In the context of a corona comprising plasmacontained by magnetic loops, there are different ways in which this can be interpreted: (1)scattering loss results from the structure and viewing angle of the loops themselves; (2)scattering loss arises because of a particular conglomeration of loops viewed at a particularangle. Each case leads to fundamentally different interpretations of observed scattering. Inthe former case, a loop with random orientation must generally be viewed from the top, in 33 –which direction the photon path length through the plasma is largest. Photon loss throughresonance scattering then implies that loops are preferentially placed on the stellar surfacefacing the observer (a perfect alignment of loops seen edge-on on the stellar limb couldproduce a similar effect, though such a chance alignment is unlikely). In the latter case, thecoronal structures or active regions must be placed preferentially on the stellar limb. With the proviso that a scattering-derived path length is a lower limit to the trueemitting region size, we note that the photon path lengths implied by the analysis of lineratios ( § larger than expected from RTV loops.Under the assumption that scattering within individual loops arises when they areviewed from the top, we can interpret the path length estimates, ℓ τ , in terms of the coronalscale height. With knowledge of the total emitting volume we can also derive an estimate ofthe surface filling factors. Emitting volumes can be estimated using the emission measuresimplied by the different lines, V = EM/n , combined with electron densities, n e , derivedin TDP04. The surface filling factors, f s , are then given by f s = A/A ⋆ = ( V /ℓ τ ) /A ⋆ . Thederived filling factors, listed in Table 5, are very small, and especially so for the hotter plasma.They are also an order of magnitude smaller than those previously derived in TDP04 basedon RTV loops—a direct consequence of the optical depth scale height being commensuratelylarger than those suggested by simple quasi-static loop models.The small filling factors we find here have interesting implications for the surface heatingflux requirements. A very rough estimate of the heating required to sustain the observedphysical conditions of the plasma can be obtained from the RTV scaling laws. While ourestimated loop lengths are significantly longer than suggested by RTV models, these relationsshould still suffice for the purposes of estimation. The volumetric heating per unit time, E ,is given by E ∼ · p / L − / , where p is the plasma pressure and L the loop length. Byassuming L = ℓ τ we find for the the surface flux values of order of 10 erg cm − s − forO viii lines, and several 10 erg cm − s − implied by the Ne x lines in IM Peg (Table 5).These compare with typical surface heating rates for the cores of solar active regions of afew 10 erg cm − s − (e.g., Withbroe & Noyes 1977).In the comparison of ℓ τ with L RTV we assumed to some extent that ℓ τ is a reasonableestimate for the coronal scale height (i.e. for the loop length). However, an alternative 34 –interpretation is possible in which the actual coronal loops are larger and ℓ τ represents onlyan estimate of the length of the part of the loop containing plasma at the characteristictemperature of the quenched lines. In such a scenario, the values found for ℓ τ can suggestthe presence of cooler loops with maximum temperature around 3 MK, similar to solar loops,coexistent with loops with much higher maximum temperature ( &
10 MK). In these hotterloops, the plasma emitting the Ne x lines ( T & l , we can use the equationsfor the conductive flux: F c = k c T / · dT /dl ∼ k c T / · ∆ T / ∆ l ; (4)and from RTV relations for a radiative power loss function P ( T ): F c / ∆ l ∼ n P ( T ) . (5)Assuming T is the temperature of maximum formation of the line k , T k max , and ∆ T is thewidth in temperature of the line emissivity curve ( ∼ . l : ∆ l ∼ · cm for O viii ∆ l ∼ · cm for Ne x The resulting values of ∆ l are rather close to L RT V (see Table 5), obtained assuming T k max as maximum temperature of the loop, and are still much smaller than ℓ τ . We can concludethen, that the values derived for the path length do not seem to agree with the hypothesisof standard uniformly heated quasi-static loop models. The maximum path length, l max , through a spherically-symmetric corona of height h ona star with radius R ⋆ and surface filling factor f s (assuming this filling factor does not varysignificantly with height) is l max = 2 f s p h + 2 hR ⋆ . (6)Considering typical parameters found for stellar coronae, we can estimate from the aboveequation whether or not we expect significant photon scattering in stellar coronae, regardlessof whether we see a net photon loss. From Eq. 3 we can estimate the optical depth at the limbfor Ne and O Ly α lines. Assuming the temperature of maximum formation of the line, i.e.about 6 MK and 3 MK for Ne and O respectively, and typical density of about 10 cm − and 35 –10 cm − respectively (see e.g., TDP04, Ness et al. 2004), we derive τ (Ne) ∼ . × − · l max and τ (O) ∼ . × − · l max .The height derived for coronal structures with different techniques (see § . R ⋆ /
2; on the Sun a typical coronal height is closer to R ⋆ /
10. Assuming h = R ⋆ /
10, and R ⋆ = R ⊙ we obtain l max ∼ f s · . R ⋆ ∼ f s · × cm.Under these assumptions, the values of the optical depth at the limb, for a sphericallysymmetric corona, are therefore τ (Ne) ∼ f s and τ (O) ∼ f s (or τ (Ne) ∼ f s and τ (O) ∼ f s if we assume h = R ⋆ /
2, and R ⋆ = R ⊙ ). For the typical filling factors lowerthan a few percent (e.g., TDP04) we obtain optical depth τ . While only three of our stellar sample present a significant case for resonance scattering,it is possible that trends of the departure from the theoretical Ly α /Ly β ratio with funda-mental stellar parameters can be found. We have examined the departures of observed fromtheoretical Ly α /Ly β ratios as a function of X-ray surface flux, surface flux in the Ly α line, L X / L bol , filling factors, and plasma density. These comparisons are suggestive of correla-tions between quenching of Ly α photons and both L X / L bol and the density-sensitive ratioof strengths of the forbidden and intercombination lines, f /i , of Mg xi .These correlations are illustrated in Figure 10 where we show the measured to theoreticalratios of O viii Ly α /Ly β of our sample as a function of L X / L bol ( top ) and Mg xi f /i ( bottom ).Error-weighted linear fits to the data in these figures yield slopes of − . ± .
04 and 0 . ± .
06, respectively. The sources presenting evidence of optical depth effects are the starscharacterised by the highest activity level ( L X / L bol ) and the highest plasma densities (i.e.lowest f/i) in our sample. Such correlations are what are expected based on Eqn. 3. Opticaldepth is proportional to the product of electron density and typical path length within anemitting region, n e ℓ . For a given fixed volume emission measure, n e V ∼ n e l , the opticaldepth varies as n / e or ℓ − / , and so increases with increasing plasma density. In the caseof L X / L bol , an increase in L X can arise through either ℓ or n e , such that any increase in L X might typically be expected to lead to greater scattering optical depth. While only a smallhandful of our measurements present truly significant detections of resonance scattering, theevidence for trends of increasing τ with L X / L bol and Mg xi f /i as is expected adds confidenceto the interpretation of the Ly α /Ly β ratios in these terms. 36 –Fig. 10.— Ratios of measured to theoretical O viii Ly α /Ly β ratios plotted vs. L X / L bol ( top )and vs. the Mg xi forbidden to intercombination line ratio (from TDP04; bottom ), which isa diagnostic of plasma density (low f/i ratios correspond to high density). 37 –
7. Conclusions
We have investigated the optical thickness of stellar coronae through the analysis ofLy α and Ly β lines of hydrogen-like oxygen and neon ions, in Chandra - hetg spectra of alarge sample of active stars. Our study indicates that most stellar coronae are characterisedby negligible visible signs of optical depth, in agreement with the results of previous studiesbased on Fe xvii lines. This indicates that coronae are either in general optically-thin, orthat for cases in which optical depths reach of order unity or higher the geometry does notstrongly favour lines-of-sight showing net Ly α photon loss.We do find evidence of significant optical depth in the O viii Lyman lines of the RS CVnbinary II Peg, and of the single M dwarf EV Lac; the RS CVn binary IM Peg also showsdepletion of both Ne and O Ly α /Ly β ratios as compared with theoretical predictions, andour analysis indicates that it is a transient effect present only in part of the observations.The detection of significant optical depth allows to derive an estimate for the photon pathlength and therefore for the typical height of the corona. The size of coronal structuresderived for all three sources is of the order of a few percent of the stellar radius at most,implying very small coronal filling factors and high surface heating fluxes. We searched forcorrelation with basic stellar parameters and coronal properties and we find that the sourcespresenting evidence of significant optical depth are at the high end of activity level, with L X / L bol at the saturation limit, and high densities in their hot plasma, as revealed by theMg xi He-like triplet lines.For the stellar sample as a whole, we also find evidence of increasing quenching ofLy α relative to Ly β as function of both L X / L bol and the density-sensitive Mg xi forbiddento intercombination line ratio. Such a trend is expected in the scenario in which opticaldepths are significant but generally small: viewing geometry rarely favours large net photonenhacements, but for favourable lines-of-sight photon depletion is expected to increase withboth increasing L X and increasing plasma density.PT and DH were supported by SAO contract SV3-73016 to MIT for support of the Chandra X-ray Center , which is operated by SAO for and on behalf of NASA under contractNAS8-03060. JJD was supported by NASA contract NAS8-39073 to the
Chandra X-rayCenter during the course of this research. GP acknowledges support from Agenzia SpazialeItaliana and italian Ministero dell’Universit`a e della Ricerca. 38 –
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