A&A 368, 888-900 (2001)
DOI: 10.1051/0004-6361:20010026

CHANDRA-LETGS X-ray observations of Capella

Temperature, density and abundance diagnostics

R. Mewe1 - A. J. J. Raassen1,2 - J. J. Drake3 - J. S. Kaastra1 - R. L. J. van der Meer1 - D. Porquet4

1 - Space Research Organization Netherlands (SRON), Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands
2 - Astronomical Institute "Anton Pannekoek'', Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
3 - Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138, USA
4 - CEA/DSM/DAPNIA, Service d'Astrophysique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France

Received 12 October 2000 / Accepted 15 December 2000

We report an analysis of the X-ray spectrum of Capella from 6 to 175 Å obtained with the Low Energy Transmission Grating Spectrometer (LETGS) on board of the X-ray space observatory CHANDRA. Many emission line features appear that can be resolved much better as compared to former instruments (EUVE and ASCA). Coronal electron densities ($n_{\rm e}$) and temperatures (T) of brightly emitting regions are constrained by an analysis of ratios of density- and temperature-sensitive lines of helium-like ions and highly ionized iron atoms. Lines emitted by e.g., O VII & VIII, Mg X-XII, Si XII-XIV, Fe IX, X & XV-XXIII are used to derive T. Line ratios in the helium-like triplets of C V, N VI, O VII, Mg XI, and Si XIII yield T in the range 0.5-10 MK, and $n_{\rm e}$ in the range 109-1013 cm-3. The Fe IX/X ratio yields $T \simeq$ 0.9 MK, while lines from Fe XVIII to XXII give $T
\sim$ 6-10 MK. Flux ratios of Fe XX-XXII lines indicate for the electron densities an upper limit in the range $n_{\rm e}
\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...er{\offinterlineskip\halign{\hfil$\scriptscriptstyle ... (2-5) 1012 cm-3. From line ratios of Fe XVII and Fe XVIII we derive constraints on the optical depth $\tau$ of $\sim$1-1.5. An emission measure distribution is derived from Fe line intensities. Results for element abundances (relative to Fe) from a 4-T model are: O and Ne/Fe about solar, N, Mg and Si/Fe $\sim$2$\times$ solar. The results for Tand $n_{\rm e}$ are described in terms of quasi-static coronal loop models and it is shown that the X-ray emission originates from compact structures much smaller than the stellar radii.

Key words: stars: individual: Capella - stars: coronae - stars: late-type - stars: activity - X-rays: stars -
atomic processes vspace-1mm

1 Introduction

The G1 III + G8/K0 III binary Capella with a 104.0233-day orbit at a distance of 13 pc (Strassmeier et al. 1993) is one of the brightest non-degenerate stellar X-ray sources. Observations by Linsky et al. (1998) with the GHRS spectrometer on the Hubble Space Telescope (HST) on Sept. 1995 and on April 1996 indicate that the optically brightest (by $0\hbox{$.\!\!^{\rm m}$ }15$) and more fastly rotating (v sin i=36 kms-1) G1 component (mass 2.6 $M_\odot$, radius 9.2 $R_\odot$) is about 3-10 times brighter in the chromospheric and transition-region UV lines than the slowly rotating (v sin i=3 kms-1) G8/K0 star (2.7 $M_\odot$, 12.2 $R_\odot$). The Sept. 1995 observations showed that the coronae around the G1 and G8 stars were equally bright in the hot (107 K) coronal Fe XXI 1354 Å line, but recent observations with the HST STIS by Johnson et al. (2001) on October 1999 indicate that the G8 corona was at least 5 times fainter than the G1 corona in the Fe XXI line, the latter having changed relative to the Sept. 1995 observations by only 50% or less. These results suggest that the hot corona of the G8 star is quite strongly variable whereas that of the G1 is not.

Due to the coronal structure of Capella its spectrum is rich in emission lines in the soft X-ray region and has been well studied by many earlier instruments with lower spectral resolution (for references see, e.g., Brinkman et al. 2000). The spectra of Capella obtained with the LETGS are of high quality and agree with the pre-flight calculated instrumental spectral resolution (cf. Brinkman et al. 2000). The wavelength region of the LETGS between 6-42 Å contains a number of helium-like line "triplets'' from Si XIII to C V which provide density as well as temperature diagnostics for plasmas with temperatures in the range 1-10 MK and densities in the range 108-1013 cm-3. The long-wavelength region between 90-140 Å contains a series of lines from $2\ell$-$2\ell'$transitions in Fe XVIII-Fe XXIII which provide density and temperature diagnostics for relatively hot ( $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... 5 MK) and dense ( $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... 1012 cm-3) plasmas (e.g., Mewe et al. 1985, 1991; Brickhouse et al. 1995). From EUVE spectra of Capella, Dupree et al. (1993), Schrijver et al. (1995), and Brickhouse (1996) have derived electron densities $n_{\rm e}$ in the range 1012-1013 cm-3 at a temperature $T
\sim 10$ MK. More recently, Brickhouse et al. (2000) have analyzed the coronal temperature and abundance structure of the coronae of Capella based on EUVE and ASCA observations covering a broad wavelength range. Our work presented here covers the ASCA region and part of the EUVE region, though in more detail due to the higher spectral resolution ( $\Delta\lambda$ $\simeq$ 0.06 Å) of the LETG spectrometer. In the present paper, the long-wavelength iron lines have been studied for the purposes of providing further insights into the coronal temperature structure as well as for density diagnostics. Section 2 discusses the observations, the data reduction, the counting rate/photon flux conversion, and the correction for interstellar absorption. Section 3 discusses branching ratios for $n_{\rm e}$-independent lines decaying from the same upper level or within the same ion in relation to theoretical values and the consequences for optical depth and source size. In Sect. 4 the analysis of the temperature diagnostic for helium-like triplets and iron lines is presented and Sect. 5 discusses briefly the determination of the electron density from the iron lines and the helium-like triplets. Section 6 describes the emission measure distribution and Sect. 7 discusses the results in terms of quasi-static loop models.

\par {
...2499fig1c.ps,angle=-90,height=6.7truecm,width=16truecm,clip=} }
\end{figure} Figure 1: CHANDRA-LETGS spectrum of Capella observed on 9 Nov. 1999 in +1 order (top), in -1 order (middle), and detail of -1 order spectrum between 12-25 Å (bottom). Spectra are corrected for background. Several prominent lines are labelled with the emitting ions
Open with DEXTER


Table 1: Strongest non-iron lines in Capella spectrum shown in Fig. 1
$\lambda_{+1}^a$I+1b $\lambda_{-1}^a$I-1bFc $\lambda_{\rm pred}^d$iondIDd

15 $\pm$ 36.1813 $\pm$ 30.76.180Si XIVHLy$\alpha$
6.6643 $\pm$ 56.6443 $\pm$ 62.06.648Si XIIIHer
6.7013 $\pm$ 46.6819 $\pm$ 50.76.688Si XIIIHei
6.7521 $\pm$ 36.7322 $\pm$ 31.06.740Si XIIIHef
8.4333 $\pm$ 48.4136 $\pm$ 42.08.421Mg XIIHLy$\alpha$
9.1844 $\pm$ 49.1650 $\pm$ 53.29.170Mg XIHer
9.2312 $\pm$ 39.229 $\pm$ 30.79.231Mg XIHei
9.3224 $\pm$ 39.3123 $\pm$ 41.69.315Mg XIHef
12.4433 $\pm$ 512.4225 $\pm$ 52.312.430Ni XIXNe5
13.4698 $\pm$ 813.4589 $\pm$ 77.413.448Ne IXHer
14.0739 $\pm$ 814.0640 $\pm$ 83.114.045Ni XIXNe8A
15.2139 $\pm$ 715.2040 $\pm$ 83.015.176O VIIIHLy$\gamma$
18.6420 $\pm$ 418.6218 $\pm$ 31.718.627O VIIHe3A
18.97275 $\pm$ 1118.96309 $\pm$112618.969O VIIIHLy$\alpha$
21.6166 $\pm$ 521.6171 $\pm$ 59.921.602O VIIHer
21.8112 $\pm$ 221.8115 $\pm$ 21.921.804O VIIHei
22.1146 $\pm$ 422.1049 $\pm$ 46.922.101O VIIHef
24.7946 $\pm$ 524.7947 $\pm$ 56.924.781N VIIHLy$\alpha$
27.0014 $\pm$ 427.0117 $\pm$ 42.426.990C VIHLy$\gamma$
28.4416 $\pm$ 428.4314 $\pm$ 32.528.446C VIHLy$\beta$
28.7910 $\pm$ 328.7713 $\pm$ 31.928.787N VIHer
29.105 $\pm$ 229.086 $\pm$ 20.929.084N VIHei
29.549 $\pm$ 229.5211 $\pm$ 21.829.534N VIHef
33.7443 $\pm$ 533.7441 $\pm$ 48.733.736C VIHLy$\alpha$
40.2620 $\pm$ 3(sum+1&-1)5.040.268C VHer
40.735 $\pm$ 3(sum+1&-1)1.940.731C VHei
41.475 $\pm$ 3(sum+1&-1)2.241.472C VHef
43.7822 $\pm$ 343.7836 $\pm$ 53.243.763Si XIBe5A
44.0323 $\pm$ 344.0229 $\pm$ 42.944.021Si XIILi6A
44.1741 $\pm$ 544.1749 $\pm$ 55.144.165Si XIILi6B
45.529 $\pm$ 245.5111 $\pm$ 21.145.520Si XIILi7B
45.6916 $\pm$ 345.6820 $\pm$ 32.145.692Si XIILi7A
49.2114 $\pm$ 349.2014 $\pm$ 72.149.222Si XIBe8
57.8918 $\pm$ 257.8917 $\pm$ 23.457.876Mg XLi5A
gap 63.2921 $\pm$ 22.463.295Mg XLi6B

Notes to Tables 1 and 2
a observed wavelength in Å.
b measured counting rate in 10-4 c/s.
c line photon flux in 10-4 phot/cm2/s, averaged over +1 & -1 order and corrected for interstellar absorption.
d predicted wavelength in Å and line identification (ID); see Phillips et al. (1999) (solar and laboratory measurements between 5 and 20 Å), Mason et al. (1984) (solar observations between 90 and 175 Å), and Mewe et al. (1985, 1995a) (full wavelength range in MEKAL code)
e only total intensity is given because Fe XVIII and O VIII lines are blended


Table 2: Strongest iron lines in Capella spectrum shown in Fig. 1
$\lambda_{+1}^a$I+1b $\lambda_{-1}^a$I-1bFc $\lambda_{\rm pred}^d$iondIDd

142 $\pm$ 812.13142 $\pm$ 81112.124Fe XVIINe4C
     12.134Ne IXHLy$\alpha$
12.2857 $\pm$ 512.2761 $\pm$ 64.712.264Fe XVIINe4D
12.8446 $\pm$ 512.8349 $\pm$ 63.712.831Fe XXN16
13.8390 $\pm$ 713.8185 $\pm$ 76.813.826Fe XVIINe3A
14.23163 $\pm$ 1014.22186 $\pm$ 101414.208Fe XVIIIF14
14.3943 $\pm$ 514.3839 $\pm$ 43.214.374Fe XVIIIF12
14.5636 $\pm$ 512.5433 $\pm$ 52.714.540Fe XVIIIF10
15.03426 $\pm$ 1415.02444 $\pm$ 153415.014Fe XVIINe3C
15.28145 $\pm$ 915.26147 $\pm$ 91115.265Fe XVIINe3D
15.4732 $\pm$ 415.4637 $\pm$ 42.715.474Fe XVIINe3E
15.6441 $\pm$ 415.6253 $\pm$ 63.615.628Fe XVIIIF7
15.8435 $\pm$ 515.8229 $\pm$ 102.515.831Fe XVIIIF6
15.8837 $\pm$ 515.8648 $\pm$ 73.315.873Fe XVIIIF5
16.02107 $\pm$ 716.00127 $\pm$ 79.116.002Fe XVIIIF4e
     16.006O VIIIHLy$\beta^e$
16.09135 $\pm$ 816.07151 $\pm$ 101116.078Fe XVIIIF3
16.78259 $\pm$ 1016.77279 $\pm$ 112016.780Fe XVIINe3F
17.06268 $\pm$ 1817.05446 $\pm$ 263117.055Fe XVIINe3G
17.10309 $\pm$ 1617.09195 $\pm$ 382417.100Fe XVIINeM2
17.6338 $\pm$ 417.6037 $\pm$ 43.617.626Fe XVIIIF1
50.3462 $\pm$ 550.3438 $\pm$ 59.050.350Fe XVINa6A
50.5623 $\pm$ 350.5611 $\pm$ 33.250.555Fe XVINa6B
52.904 $\pm$ 2gap 0.652.911Fe XVMg1
54.1426 $\pm$ 4gap 3.954.142Fe XVINa7B
54.7237 $\pm$ 4gap 5.854.728Fe XVINa7A
gap 62.8718 $\pm$ 32.062.879Fe XVINa8B
gap 63.7136 $\pm$ 44.263.719Fe XVINa8A
66.4016 $\pm$ 366.3631 $\pm$ 43.366.368Fe XVINa9
91.0614 $\pm$ 391.0115 $\pm$ 33.291.020Fe XIXO6F
93.95160 $\pm$ 893.92159 $\pm$ 83793.923Fe XVIIIF4A
97.90597.852 $\pm$ 10.997.880Fe XXIC6C
101.5828 $\pm$ 3101.5527 $\pm$ 37.6101.550Fe XIXO6B
102.2610 $\pm$ 2102.229 $\pm$ 22.6102.220Fe XXIC6D
103.9754 $\pm$ 5103.9554 $\pm$ 515103.937Fe XVIIIF4B
108.3887 $\pm$ 6108.3694 $\pm$ 625108.370Fe XIXO6A
109.9812 $\pm$ 2109.989 $\pm$ 32.9109.970Fe XIXO6E
110.673 $\pm$ 1110.644 $\pm$ 10.9110.630Fe XXN6D
111.729 $\pm$ 3111.707 $\pm$ 12.2111.700Fe XIXO6C
114.413 $\pm$ 2114.414 $\pm$ 11.1114.410Fe XXIIB10A
117.1424 $\pm$ 3117.1622 $\pm$ 36.4117.170Fe XXIIB11
117.508 $\pm$ 3117.534 $\pm$ 21.7117.510Fe XXIC7B
118.7017 $\pm$ 3118.7023 $\pm$ 35.7118.660Fe XXN6C
120.0022 $\pm$ 3120.0124 $\pm$ 46.7120.000Fe XIXO6D
121.8736 $\pm$ 4121.8833 $\pm$ 411121.830Fe XXN6B
128.7622 $\pm$ 3128.7422 $\pm$ 311128.730Fe XXIC6A
132.8737 $\pm$ 4132.8751 $\pm$ 520132.850Fe XXN6A
135.8111 $\pm$ 3135.8212 $\pm$ 35.4135.780Fe XXIIB12A
171.0639 $\pm$ 5offdetector44171.075Fe IXA4
174.516 $\pm$ 2offdetector18174.535Fe XCl4A

2 Observations

Spectra of Capella were obtained in the calibration phase during 6-12 September 1999 (cf. Brinkman et al. 2000) and subsequently on 9 November 1999, using the High Resolution Camera (HRC-S) and the Low Energy Transmission Grating (LETG) on board the X-ray observatory CHANDRA. In the present paper we study the second spectrum (exposure time 85.4 ksec). The HRC-S is equipped with three microchannel plate detectors (10 cm each) in one row. The 0th order diffraction signal is aimed at the central detector, such, that for the +1 and -1 diffraction orders, the wavelength regions that fall in the gaps between the detector plates do not overlap (see Fig. 1). This way the central plate covers nominally the range of -50 (-1 order) to +60 (+1 order) Å, the outer plates -55 to -165 Å and +65 to +175 Å. The dispersion is 1.15 Å/mm. Figure 1 shows the LETGS spectra of Capella in the +1 & -1 order with the gaps between 62-65 Å and 52-56 Å (+1 and -1 order, respectively) and a detail of the -1 order spectrum between 12-25 Å. The spectra were rebinned to intervals of 0.02 Å and corrected for the background. Around the dispersion axis a source box is defined. The edge of the box is a function of the wavelength, small at short wavelengths and wide at long wavelengths, like a bowtie. The background box is defined as the area from 10 to 39.5 arcsec above and under the dispersion axis. The ratio of the source to background area ranges from 1:25 at the short wavelengths to 1:6 at the edge of the detector. For each bin the number of counts is integrated in the cross dispersion direction inside the boxes. The net number of source counts is calculated as follows: $s = {\rm raw} - f \cdot {\rm back}$, ${\rm d}s = \sqrt{{\rm raw} + f^2\cdot {\rm back}}$, $f{\rm = {{area(source)}\over{area(back)}}}$, where ${\rm back = \char93 }$ counts in background box, ${\rm raw = \char93 }$ counts in source box, and $s = \char93 $ net source counts in source box with error ds.

By visual inspection we selected the prominent lines and fitted them with Gaussian profiles. This has yielded values for the wavelength, FWHM, and the number of photon counts (including uncertainties) for each line. By comparing wavelengths and line counts with expected values we have identified all lines given in Tables 1 and 2. To represent the background due to continuum radiation we introduce a constant shift. The intensities of the emission line features in the spectrum are based on integration over the line profile to obtain the total counts. For isolated lines the Gaussian line width turns out to be consistent with the average theoretical instrumental width, which implies that Doppler broadening can be neglected compared to the instrumental width. The measured FWHM of the instrumental profile is about 0.068 Å for wavelengths $\lambda \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...{\offinterlineskip\halign{\hfil$\scriptscriptstyle ... Å (and increasing with $\lambda$ at long wavelengths). This is in good agreement with the results of the pre-flight ground calibration. Blended lines were fitted by multiple Gauss profiles with the proper instrumental width. The LETGS instrumental profile is actually closer to a Lorentzian function than to a Gaussian; we have investigated the effect of using Lorentzian profiles in the analysis but usually found no significant differences in derived line parameters.

Counting rates I and $1\sigma$ uncertainties are given in Tables 1 and 2. The observed counting rates I (10-4 c/s) have been converted to photon fluxes F (10-4 phot/cm2/s) using the preliminaryin-flight effective area model for the spectrometer (for details of the calibration, see van der Meer et al. 2001). The observed flux at Earth was further corrected for interstellar absorption, assuming an interstellar column density $N_{\rm H} = 1.8 ~ 10^{18}$ cm-2 (see Schrijver et al. 1995), using the cross sections of Morrison & McCammon (1983). In Tables 1 and 2 we give the photon flux F averaged over the +1 & -1 order and corrected for interstellar absorption.

In the overlapping wavelength regions ($\lambda$ <35 Å) the obtained fluxes are in agreement with values obtained with the CHANDRA-HETGS by Canizares et al. (2000) and with data from the RGS on board XMM-Newton (see Audard et al. 2001; cf. their Table 1 for a comparison of fluxes). For the long-wavelength region ($\lambda$ >90 Å) we have compared the LETGS fluxes with the EUVE fluxes obtained by Brickhouse et al. (2000). The LETGS/EUVE flux ratio was in the range $\sim$0.5 to 0.8 which is explained by a different activity level of Capella (see e.g., comparison by Linsky et al. 1998).


Table 3: Comparison of branching ratios
$\lambda$ (Å) +1a -1a av.b Nc Md Ce Bf
Si XII              
${45.52\over 45.692}$ 0.57 0.52 0.54(.15) 0.51 0.50 0.51 --
Fe XVI              
${62.87\over 63.719}$ gap 0.50 0.49(.08) 0.48 0.48 0.52 --
Fe XVIII              
${93.92\over 103.94}$ 3.0 2.9 2.5(.2) 2.76 2.43 2.70 2.71
Fe XIX              
${108.37\over 120.0}$ 4.0 4.0 3.5(.4) 3.75 3.51 3.64 3.75
${101.55\over 109.97}$ 2.3 3.0 2.6(.4) 1.98 1.80 1.95 2.00
] ${101.55\over 111.7}$ 3.3 3.8 3.5(1.1) 2.52 2.34 2.51 2.54
${109.97\over 111.7}$ 1.42 1.28 1.4(.4) 1.27 1.27 1.29 1.27
Fe XXI              
${102.22\over 97.88}$ 2.0 4.0 2.8(1.7) 2.42 2.37 2.39 2.42

a measured counting rate ratio in +1 and -1 order of spectrum.
b line photon flux ratio averaged over +1 and -1 order and corrected for interstellar absorption (in parentheses the mean statistical 1$\sigma$ error).
c from ratio of transition probabilities A given by NIST database with uncertainty classification C (< 25%).
d calculated with SPEX/MEKAL code (Mewe et al. 1985, 1995a; Kaastra et al. 1996).
e from recent calculations of transition probabilities by Raassen using Cowan (1981) code.
f from Brickhouse et al. (1995).

3 Comparison of observations with models

This section presents a comparison of observations with model predictions. First, we consider branching ratios for lines decaying from the same upper level and some optical depth effects. Second, we consider some lines decaying from different upper levels.

3.1 Branching ratios

The branching ratio is defined as the ratio of the photon fluxes of two lines decaying from the same upper level. The uncertainty of the branching ratio is given by the uncertainty of the radiative transition probabilities A of the lines which is typically about 25% each (see below). The comparison of the measured and theoretical branching ratios provides a good figure of merit for the quality of the flux measurements as well as a check on the assumption of the optically thin model, i.e., optical line depth $\tau \simeq 0$.

In Table 3, photon flux ratios of two lines decaying from the same upper level are compared to the theoretical branching ratios. The theoretical values are from the NIST compilation, from MEKAL (Mewe et al. 1985, 1995a) as implemented in the SPEX code (Kaastra et al. 1996), from recent calculations at SRON using the Cowan code programs, and from work by Brickhouse et al. (1995). From this table we notice that the theoretical ratios from MEKAL for the Fe XVIII pair and three of the Fe XIX pairs are somewhat lower than those from the three other theories, which are in good agreement with each other. For all except one the theoretical line ratios in Table 3 are within the experimental statistical errors of the Capella line ratios. For some of the ratios the statistical error is also very large, e.g., the ratio 102.22/97.88 for which the deviation between the +1 and -1 order is apparently large, though the flux for 97.88 Å should only be treated as an upper limit. The only disagreement concerns the ratio 101.55/109.97. However, taking into account an uncertainty of 25% for the theoretical values as indicated by the NIST database (classification "C'') all ratios agree with the theoretical values. Brickhouse et al. (1995) observed a deviation for the ratio 101.55/111.7. She suggests a blend for the line at 101.55 Å. Although in our case the ratios related to the 101.55 Å line are about 25% larger than the theoretical value (which is in agreement with some blending) they are all within the combined statistical and NIST error bars.

3.1.1 Optical depth effects

Schrijver et al. (1995) and Mewe et al. (1995b) have investigated the possibility that strong resonance photons are scattered out of the line of sight in late-type stellar coronae. Branching ratios can be used to check the assumption of the optical thin model because effects of resonance scattering would affect the measured branching ratio. From the fact that the intensities of, e.g., the strong resonance lines Fe XVIII $\lambda$93.92 and Fe XIX $\lambda$108.37 are in good agreement with the intensities of other lines sharing the same upper levels, we can derive a constraint on the optical depth taking into account the systematic uncertainties of the theoretical transition probabilities A (typical 25% for each A, hence 35% for the branching ratio) which dominate over the statistical errors (typically 10%). If we rule out a reduction in the resonance line intensity larger than about 30%, then on the basis of a simple "escape-factor'' model (escape factor $P(\tau) \simeq [1 + 0.43 \tau]^{-1}$ for a homogeneous mixture of emitters and absorbers in a slab geometry (e.g., Kaastra & Mewe 1995)) we can put a constraint on the optical depth: $P \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... yields $\tau \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...r{\offinterlineskip\halign{\hfil$\scriptscriptstyle ... for the Fe XVIII $\lambda$93.92 line.

Line transitions of Ne-like Fe XVII yield the strongest emission lines seen in the LETGS spectrum in the range 15-17 Å (cf. Fig. 1 and Table 2). For the very strong Fe XVII $\lambda$15.014 resonance line we can also derive a constraint on the optical depth as follows.

In principle, the ratio of the strong Fe XVII 15.014 Å resonance line (with a large oscillator strength f=2.66) to a presumably optically thin Fe XVII line with a relatively small oscillator strength (e.g., 16.780 Å, f=0.101, or 15.265 Å, f=0.593) yields the optical depth $\tau$in the 15.014 Å line, applying the escape-factor model. Solar physicists have used this technique to derive the density in active regions on the Sun from the optical depth and the estimated plasma dimension $\ell$ (e.g., Saba et al. 1999). Though the 16.780 Å line has the smallest f-value we prefer to use the 15.265 Å line because the 16.780 Å $3{\rm s} \to~2{\rm p}$ line can be affected by radiative cascades.

From the measured Fe XVII 15.014/15.265 photon flux ratio of 2.85$\pm$0.14 we derive for a plasma of 7 MK (as derived from the Fe XVII 15.265/Fe XVIII 16.078 ratio, see Sect. 4.1) an upper limit to the optical depth $\tau$of the Fe XVII 15.014 Å line by comparing to the corresponding line ratios given by experiment (2.8-3.2) (Livermore Electron Beam Ion Trap (EBIT)) or theory (3.3-4.7) (cf. Brown et al. 1998; Bhatia & Doschek 1992). Taking the EBIT ratio we obtain from the condition $P \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... an upper limit of $\tau \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...\offinterlineskip\halign{\hfil$\scriptscriptstyle ..., whereas the theoretical ratio would give a more conservative upper limit of $\tau \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...\offinterlineskip\halign{\hfil$\scriptscriptstyle ... from $P \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ....

3.1.2 Constraints on the source size
The constraints on $\tau$ imply a constraint on the extent of the emitting plasma though the averaging over the stellar surface makes a straightforward interpretation difficult. The optical depth $\tau$ for a Doppler-broadened resonance line can be written as (Mewe et al. 1995b):
$\displaystyle \tau$ = $\displaystyle 1.16\,10^{-14} \left({n_{\rm i}\over n_{\rm el}}\right) A_z \left({n_{\rm H
}\over n_{\rm e}}\right) \lambda f \sqrt{{M\over T}}
n_{\rm e} \ell$  
  $\textstyle \equiv$ $\displaystyle 10^{-19} C_{\rm d} \left({A_z\over A_{z \odot}}\right)
\left({{n_{\rm e}\ell}\over \sqrt{T_6}}\right),$ (1)

where $(n_{\rm i}/n_{\rm el}$) is the ion fraction (from Arnaud & Raymond 1992), $A_z=n_{\rm el}/n_{\rm H}$ the elemental abundance relative to hydrogen with $A_{z \odot}$ the corresponding value for the solar photosphere as given by Anders & Grevesse (1989), $n_{\rm H}/n_{\rm e}\simeq 0.85$ the ratio of hydrogen to electron density (in cm-3), $\lambda$ the wavelength in Å, f the absorption oscillator strength, M the atomic weight, T the temperature (in K or T6in MK), and $\ell$ a characteristic plasma dimension (in cm), and

\begin{displaymath}C_{\rm d} \equiv
98.5 (n_{\rm i}/n_{\rm el}) A_{z \odot} \lambda f \sqrt{M}.
\end{displaymath} (2)

For Fe XVII $\lambda$15.014, Fe XVIII $\lambda$93.92, and Fe XIX $\lambda$108.37 we obtain from Eq. (2) $C_{\rm d} \simeq$ 0.25, 0.05, and 0.04, respectively for a temperature $T_6 \simeq 6.5$ (see Sect. 4). Assuming solar abundances ( $A_z = A_{z \odot}$ = 4.68 10-5 for Fe), the constraints on $\tau$ yield corresponding upper limits for $n_{\rm e} \ell$ of about 2 1020, 5 1020, and 6 1020 cm-2 for these three cases. With a density of typically $n_{\rm e} \approx 3$ 1012 cm-3 (see Sect. 5.1) we derive upper limits to the typical size $\ell$ of the emitting region in the range (1-2) 108 cm, i.e. much smaller than the stellar radii ($\sim$ $10~R_{\odot} \simeq 10^{12}$ cm), but slightly above the typical loop size as derived from the electron density in a comparable temperature region (cf. Table 9, Fe XX-XXI). For the Sun loop sizes typically range from 3 109 cm for compact active region loops to 2 1010 cm for large-scale quiet region loops (see e.g., Mewe 1991, Table 1).

The above analysis, however, is strictly valid only if all flux is concentrated in one single loop. If one observes a collection of loops distributed more or less homogeneously over the stellar surface, the effects of optical depth tend to average out. There may be significant resonance scattering of photons into the line of sight. In this situation it is possible that there is indeed significant optical depth in certain structures, but that this is compensated for by the azimuthal average over the stellar disk whereby some lines of sight pick up photons that are scattered and others lose photons to scattering (e.g., Wood & Raymond 2000). Nevertheless, the above analysis still serves to demonstrate the usefulness of investigating optical depths to provide further clues on the emitting geometry.


Table 4: Line ratios based on different upper levels
$\lambda$ (Å) +1a -1a av.b Mc Bd
Fe XVI          
54.14/54.728 0.70 gap 0.67(.15) 0.56 --
Fe XIX          
91.02/101.55 0.49 0.56 0.42(.07) 0.156 0.241
91.02/108.37 0.16 0.16 0.130(.014) 0.057 0.082
91.02/109.97 1.1 1.7 1.09(.19) 0.287 0.476
108.37/109.97 7.0 10.4 8.4(1.4) 5.00 5.48
108.37/111.70 10.0 13.6 11.4(2.6) 6.34 6.95
Fe XX          
118.66/121.83 0.48 0.68 0.55(.07) 0.517 0.517
118.66/132.85 0.47 0.44 0.28(.03) 0.355 0.386
121.83/132.85 0.97 0.65 0.55(.06) 0.679 0.754
Fe XXII          
117.17/135.78 2.1 1.8 1.17(.2) 0.853 0.991

a measured counting rate ratio in +1 and -1 order of spectrum.
b line photon flux ratio averaged over +1 and -1 order and corrected for interstellar absorption (in parentheses the mean statistical 1$\sigma$ error).
c calculated with SPEX/MEKAL code (Mewe et al. 1985, 1995a; Kaastra et al. 1996).
d Brickhouse et al. (1995).

3.2 Lines decaying from different upper levels

In Table 4 ratios are given for lines that belong to the same Fe ion, but that do not decay from the same upper level. In this case they cannot be compared to the ratios of the transition probabilities, the branching ratio as given by NIST and COWAN. However, comparison with MEKAL and Brickhouse et al. (1995) is still possible, as they take into account the populations of the levels due to collisional excitations and radiative decays. Only density-independent lines are used. In Table 4 all ratios in Fe XIX that include the line at 91.02 Å are far too high. This implies blending by an unknown feature or a theoretical flux value far too low for that line. There is some indication for this, given the considerable difference between the MEKAL and Brickhouse values, suggesting some uncertainties in the calculation of the proper flux value. This situation renders this line at 91.02 Å problematic for density and temperature diagnostics. The ratios including the Fe XX line at 132.85 Å also deviate from the theoretical values. However, 132.85 Å is well known to be blended with a line of Fe XXIII; the relative contributions of Fe XX and Fe XXIII depend on the temperature regime and for Capella are expected to be dominated by Fe XX. The deviant behaviour of the lines at 91.02 Å and 132.85 Å appears again in Sects. 4 and 5 where we discuss the temperature and density diagnostics. For completeness, these lines were included in the tables though the results based on their ratios should be considered as less reliable.

\par\includegraphics[angle=-90, width=8.2cm, clip]{h2499fig2a.ps}...
...ps}\par\includegraphics[angle=-90, width=8.2cm, clip]{h2499fig2c.ps}\end{figure} Figure 2: Temperature dependence of photon flux ratios between Fe XVIII to Fe XXI lines. The observed ratios are indicated
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Table 5: Temperature determination from line ratios for different Fe ions
$\lambda$ (Å) +1a -1a av.b Tc
Fe IX/Fe X        
171.07/174.53 6.36 -- $\sim$2.4(.8) $\sim$0.85
Fe XVII/Fe XVIII        
15.265/16.078 1.07 0.97 1.01(.07) $7.0\pm0.2$
16.780/15.628 6.36 5.30 5.7(.6) $6.9\pm0.2$
Fe XVIII/Fe XVIII        
15.628/93.92 0.25 0.33 0.10(.01) $10.0\pm1.0$
Fe XVIII/Fe XIX        
103.94/108.37 0.63 0.58 0.61(.05) $5.7\pm0.2$
103.94/101.55 1.9 2.0 2.0(.2) $5.4\pm0.2$
93.92/91.02d 11.5 10.6 11.6(1.6) $6.6\pm0.3$
93.92/108.37 1.84 1.70 1.51(.15) $5.9 \pm0.2$
103.94/111.70 6.26 7.72 7.0(1.5) $5.0 \pm0.4$
Fe XIX/Fe XX        
120.00/118.66 1.3 1.0 1.2(.2) $6.0 \pm0.3$
120.00/121.83 0.61 0.70 0.63(.07) $5.9 \pm0.2$
120.00/132.85d 0.59 0.46 0.35(.04) $6.3 \pm0.3$
108.37/118.66 5.0 4.1 4.3(.5) $6.0 \pm0.3$
108.37/121.83 2.4 2.8 2.2(.2) $6.0\pm0.2$
108.37/132.85d 2.4 1.8 1.21(.09) $6.6\pm0.2$
111.70/118.66 0.50 0.31 0.38(.12) 7.0 +0.8-0.6
111.70/121.83 0.24 0.21 0.21(.06) 6.8 +0.9-0.5
101.55/121.83 0.79 0.79 0.73(.07) $6.2\pm0.3$
109.97/121.83 0.35 0.27 0.28(.05) $6.7\pm0.5$
91.02d/121.83 0.39 0.45 0.31(.03) $5.3\pm0.2$
Fe XX/Fe XXI        
118.66/117.51 2.2 5.2 3.3(1.0) 8.4 +1.0-0.7
121.83/117.51 4.6 7.6 6.5(2.0) 8.4 +0.9-0.7
132.85d/117.51 4.8 11.7 11.9(3.9) 7.7 +0.9-0.5
Fe XXI/Fe XXII        
117.51/117.17 0.32 0.20 0.27(.08) 9.7 +1.2-0.8

a measured counting rate ratio in +1 and -1 order of spectrum.
b line photon flux ratio averaged over +1 and -1 order and corrected for interstellar absorption (in parentheses the mean statistical 1$\sigma$ error).
c Temperature (in MK) derived from Brickhouse et al. (1995) but for the 15-16 Å Fe XVII/Fe XVIII line ratios from corrected MEKAL (see text).
d line intensity too high compared to theoretical values (see Table 4 and paragraph 3.2); results less reliable.


Table 6: Temperature determination from the line intensity ratio G=(i+f)/r for different He-like ions and from the ratio of the He- and H-like resonance lines.
$\lambda$ (Å) +1a -1a av.b Tc Td
C V          
${{40.731+41.472}\over 40.268}$ sum   0.80(.26) $1.4\pm{0.5}$ 1.0
N VI          
${{29.084+29.534}\over 28.787}$ 1.44 1.28 1.44(.46) 0.5+0.5-0.2 0.35
O VII          
${{21.804+22.101}\over 21.602}$ 0.88 0.90 0.89(.09) $1.8\pm0.3$ 1.9
Mg XI          
${{9.231+9.315}\over 9.170}$ 0.83 0.67 0.74(.11) 4.6 +1.4-1.0 5.7
Si XIII          
${{6.688+6.740}\over 6.648}$ 0.78 0.94 0.87(.17) 5 +3-2 6
C V/C VI          
${40.268\over 33.736}$ sum   0.59(.24) $1.10\pm0.15$  
N VI/VII          
${28.787\over 24.781}$ 0.213 0.269 0.27(.04) $2.50\pm0.14$  
O VII/VIII          
${21.602\over 18.969}$ 0.239 0.229 0.37(.02) $3.37\pm0.06$  
Mg XI/XII          
${9.170\over 8.421}$ 1.34 1.38 1.61(.18) $6.9\pm0.2$  
Si XIII/XIV          
${6.648\over 6.182}$ 2.79 3.33 2.82(.44) $9.4\pm0.5 $  

a measured counting rate ratio in +1 and -1 order of spectrum.
b line photon flux ratio averaged over +1 and -1 order and corrected for interstellar absorption (in parentheses the mean statistical 1$\sigma$ error).
c Temperature (in MK) derived from SPEX/MEKAL.
d Temperature (in MK) derived from Porquet & Dubau (2000) and Mewe et al. (2001).

4 Temperature diagnostics

4.1 Fe ions

In Fig. 3 and Table 5 theoretical ratios of temperature-sensitive lines of higher charge state ions of iron are shown and compared with the observations (Brickhouse et al. 1995). The line ratios from the lower charge state Fe ions Fe IX and Fe X indicate that these ions are formed in a plasma with a temperature of about 1 MK. For the higher Fe ions Fe XVIII, Fe XIX, and Fe XX temperatures of 6 MK or more are observed, but for ions higher than Fe XX the temperature increases to 8 or 10 MK. These results are consistent with those obtained by Brickhouse et al. (2000) on the basis of EUVE observations.

We notice some deviations including the Fe XIX lines at 108 and 120 Å and those at 101, 109, and 111 Å. The first two both decay from the same upper level (922.908 cm-1), while the three others all decay from the level at 984.760 cm-1. In the ratio between the Fe XIX/Fe XX this results in a lower temperature when the two lines at 108 and 120 Å are involved and a somewhat higher temperature when the three other lines are used. Errors induced in the computed populations of the levels in the theoretical model resulting from inaccurate collisional excitation rates are likely to be the underlying reason. We notice from Table 5 a fairly regular increase of the temperature from 6 MK up to 10 MK going from Fe XVIII to Fe XXII. This favours a multi-temperature model; this comes as no surprise based on the earlier analyses using EUVE and ASCA observations that arrived at similar conclusions (e.g., Dupree et al. 1993; Brickhouse et al. 2000).

The wide dynamic wavelength range of the LETGS allows to study simultaneously the $n = 2 \rightarrow2$ and $n = 3 \rightarrow2$ iron line transitions in the L-shell and to compare the derived temperatures. In the short wavelength range, around 16 Å the observed features correspond to $n = 3 \rightarrow2$ transitions, while in the wavelength range above 80 Å the lines originate from $n = 2 \rightarrow2$ transitions. From the measured photon flux ratio Fe XVII 15.265Å/Fe XVIII $16.078{\rm\AA} =
1.01 \pm 0.07$ we derive a temperature of 7.0 $\pm$ 0.2 MK using MEKAL, but after correcting (enhancing) the Fe XVIII flux by a factor 2.14 as indicated by the results from a benchmark study of the MEKAL code with solar flare spectra by Phillips et al. (1999; their Table A8). The two Fe XVII/Fe XVIII $n = 3 \rightarrow2$ ratios result in higher temperatures than the ratios based on $n = 2 \rightarrow2$ transitions. As a test case the ratio Fe XVIII/Fe XVIII between an $n = 3 \rightarrow2$ and an $n = 2 \rightarrow2$transition was calculated. It results in an even higher temperature of 10 MK. However, this ratio covers a large wavelength range and is more sensitive to flux calibration errors.

Higher temperature regions (T = 10 MK) are present in the coronae of Capella (e.g., Brickhouse et al. 2000), though with emission measures more than an order of magnitude below those of the dominant temperature of several MK. Lines of higher Fe ionization stages (e.g., Fe XXIV) that could constrain higher temperatures are outside the range in which the LETGS has either insufficient spectral resolution or sensitivity to yield accurate measurents of their fluxes.

4.2 He-like ions

The LETGS spectra of Capella contain a number of interesting emission line features in the short wavelength region between 6 and 45 Å that originate from the He-like ions Si XIII, Mg XI, Ne IX, O VII, N VI, and C V. It concerns the resonance line 1s$^{2\ 1}$S0 - 1s2p 1P1(r), the intercombination line 1s$^{2\ 1}$S0 - 1s2p 3P1,2(i)and the forbidden line 1s$^{2\ 1}$S0 - 1s2s 3S1(f). There are combinations of these three transitions that are strongly temperature- and density-sensitive. The resonance line intensity is comparable to the sum of the intensities of the two other lines and increases at higher temperatures, while for high density the intercombination line becomes intense compared to the forbidden line. Of the six ions mentioned above, the Ne IX triplet could not be used because of severe blending of the r and i lines by Fe XIX lines and partly blending of the f line by an Fe XVII line. For C V the i and f lines are rather weak, but for N VI and O VII three clearly separated components were measured. For Mg XI and Si XIII the intercombination line was blended by the resonance line, but a double Gaussian fit enabled us to deconvolve these lines, resulting in measured wavelengths in very good agreement (within about 0.005-0.01 Å) with the theoretical values. In the case of Si, the measurement should be considered less certain due to the lower spectral resolution at these wavelengths.

From the ratio G = (i + f)/r the temperature is determined for each He-like ion and is given in Table 6. We observe a wide range of temperatures from about 0.5 MK for N VI up to 5 MK for the higher stages of ionization (Si XIII). However, the N VI temperature appears somewhat discrepant in comparison to those derived for C and O and we suspect it might be affected by line blending. The temperatures were all derived using an updated version of the plasma radiative loss and line fitting program SPEX developed at SRON and from recent calculations by Porquet & Dubau (2000) and by Mewe et al. (2001) including all recombinations. Another possibility for obtaining the coronal temperature from He-like ions is to use the ratio between the resonance lines of these ions and the resonance lines of the one higher ionized stages, the H-like ions, of the same element. These ratios and the derived temperatures are collected also in Table 6. The single He-like ion ratios result in a lower temperature than the ratios from He-like versus H-like ions. The conclusion is that the H-like ions are formed in another (hotter) area than the He-like ions. Therefore the obtained temperature is a mixture and average of the two areas.

5 Density diagnostics


Table 7: Densities from ratios of $n_{\rm e}$-dependent iron lines
$\lambda$ (Å) +1a -1a av.b $n_{\rm e}^c$


${110.63\over 118.66}$ 0.15 0.16 0.15(.06) $<5\ 10^{12}$
${110.63\over 121.83}$ 0.07 0.11 0.08(.03) $<5\ 10^{12}$
${110.63\over 132.85^d}$ 0.07 0.07 0.04(.02) $<2\ 10^{12}$
Fe XXI        
${ 97.88\over 128.74}$ 0.22 0.10 0.09(.03) $<2\ 10^{12}$
${117.51\over 128.73}$ 0.34 0.20 0.17(.05) $<3\ 10^{12}$
Fe XXII        
${114.41\over 117.17}$ 0.14 0.18 0.17(.10) $<3\ 10^{12}$
${114.41\over 135.78}$ 0.30 0.33 0.19(.10) $<5\ 10^{12}$

a measured counting rate ratio in +1 and -1 order of spectrum.
b line photon flux ratio averaged over +1 and -1 order and corrected for interstellar absorption (in parentheses the mean statistical 1$\sigma$ error).
c $n_{\rm e}$ (in cm-3) derived from Brickhouse et al. (1995).
d line intensity too high compared to theoretical values (see Table 4 and paragraph 3.2); results less reliable.


Table 8: Density determination from the i/f line intensity ratio for different He-like ions
$\lambda$(Å) +1a -1a av.b $n_{\rm e}^c$ $n_{\rm e}^d$
C V          
${40.731\over 41.472}$ sum   0.89(.6) $3\pm 2$ 109 4 109
N VI          
${29.084\over 29.534}$ 0.52 0.53 0.49(.17) $5\pm 3$ 109 6 109
O VII          
${21.804\over 22.101}$ 0.26 0.31 0.28(.05) $<7\ 10^{9}$ 3 109
Mg XI          
${9.231\over 9.315}$ 0.51 0.40 0.45(.13) $<4\ 10^{12}$ 3 1012
Si XIII          
${6.688\over 6.740}$ 0.62 0.82 0.72(.24) $3\pm 2.5$ 1013 4 1013

a measured counting rate ratio in +1 and -1 order of spectrum.
b line photon flux ratio averaged over +1 and -1 order and corrected for interstellar absorption (in parentheses the mean statistical 1$\sigma$ error).
c Electron density (in cm-3) derived from SPEX/MEKAL.
d Electron density (in cm-3) derived from Porquet and Dubau (2000) and Mewe et al. (2001); for C V a radiation temp. of 4585 K is assumed (see Ness et al. 2001).

5.1 Fe ions

From a number of density-sensitive iron lines at wavelengths above 90 Å (e.g., Fe XX to Fe XXII) the electron density can be investigated for plasmas with relatively high temperatures ( $T \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... MK). For the determination of the electron density in the coronal plasma of Capella ratios of density-dependent lines of Fe XX, Fe XXI, and Fe XXII were used. The data used and density values obtained are listed in Table 7. The theoretical density dependence for these lines is taken from the work by Brickhouse et al. (1995). The theoretical flux ratio-$n_{\rm e}$ curves are very flat for low densities and start to deviate from the low-density plasma situation only for values of $n_{\rm e} \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...interlineskip\halign{\hfil$\scriptscriptstyle ... cm-3. This makes the method insensitive for the lower limit. The $n_{\rm e}$ values obtained for the ions Fe XX to Fe XXII, however, are consistently around $\sim$ 1012 cm-3and all with an upper limit of $n_{\rm e} \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...\offinterlineskip\halign{\hfil$\scriptscriptstyle ...- $5\ 10^{12}$ cm-3.

5.2 He-like ions

Two of the three transitions of the He-like ions described above are also strongly density-dependent (e.g., Gabriel & Jordan 1969), namely the ratio of the intercombination line (i) and the forbidden line (f). Line ratios (i/f) were derived from measured values and the corresponding plasma densities were derived; the results are listed in Table 8. The uncertainty of the counting rate of the O VII intercombination line justifies an upper limit only. As for the temperature diagnostics, the obtained values are from SPEX and from Porquet & Dubau (2000) and Mewe et al. (2001).

If we compare the results for i/f and $n_{\rm e}$ with those obtained by Ness et al. (2001) in the case of C, N, and O, we note that the values are slightly different but in agreement within the error bars. Ness et al. (2001) find for O VII a somewhat different value for i/f( $0.255 \pm 0.015$) based on a method in which the total spectrum is fitted with a model for the continuum plus background, whereas we fit the background-subtracted spectrum. Because below about 109 cm-3 the dependence of i/f on $n_{\rm e}$ becomes very flat (Fig. 5 of Ness et al.) this gives a much lower upper limit for $n_{\rm e}$ (only 5 108 cm-3) than we find but the difference is still acceptable within the statistical uncertainties.

From the observations of Capella with the High-Energy Transmission Grating on CHANDRA Canizares et al. (2000) obtain a slightly larger i/f ratio for O VII ( $0.35 \pm 0.05$, but consistent with our ratio of $0.28 \pm 0.05$) implying a higher electron density within the range (8-20) 109 cm-3. Again the difference in i/f may be due to a different treatment of the continuum plus background.

For the lower ions C V, N VI, and O VII we obtain densities typically in the range 109-1010 cm-3 for temperatures between about 0.5 and 2 MK. Such densities are typical of those for active coronal regions on the Sun. In the case of Capella, similar densities for the lower temperature ($\sim$1-2 MK) plasma were suggested by Schrijver et al. (1995) and by Brickhouse (1996) in analyses of EUVE spectra. The higher ions Mg XI and Si XIII associated with higher temperatures ( $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... MK) suggest much higher densities, i.e. 1012-1013 cm-3, that are consistent with the results obtained for the highly ionized Fe ions. Such high densities have been observed on the Sun only during flares, but were derived earlier from EUVE spectra of Capella (cf. Sect. 1). We note, however, that the density obtained for Si XIII is significantly higher than the upper limit found from Mg XI. For the same line complex observed with the CHANDRA HETG (albeit at a different time), Canizares et al. (2000) obtained a ratio i/f of $0.39 \pm
0.03$, which is to be compared with our value of $0.72 \pm 0.24$. It is clear from the recent analysis of HST STIS spectra by Johnson et al. (2001) that the hot 10 MK component of the Capella coronae can be highly variable, and so it is therefore possible that the higher density we derived based on observations made at a different time is real. However, since the spectral resolution of the LETGS is fairly low at the wavelength of the Si He-like complex, rendering line deconvolution more difficult, this density should perhaps be treated with caution at this time.

6 Emission measure modeling

We performed an emission measure (EM) analysis of the Capella LETGS data by comparing measured line fluxes with the theoretical fluxes as calculated by MEKAL for a given emission measure and formation temperature for a particular line, assuming solar abundances (Anders & Grevesse 1989). Figure 4 (top) presents the results of the EM distribution based on the analysis of the iron lines. For all Fe-ions used the error bars are given. These error bars contain two components. When having many lines (and values) for one particular ion, the error is dominated by the statistical error of these values. In case of one single transition or two closely lying values for an ion, the error is dominated by the uncertainty in the flux as given in Tables 1 and 2. The larger error bar for Fe XVI is probably due to the fact that all lines used are near the gap between the detectors where the efficiency declines rapidly and the calibration is uncertain.

\par\includegraphics[angle=-90, width=8cm, clip]{h2499fig3a.ps}\v...
\includegraphics[angle=-90, width=8cm, clip]{h2499fig3b.ps} %\end{figure} Figure 3: EM distribution derived from Fe IX to Fe XXII line intensities for the Capella CHANDRA-LETGS observations (top) compared to the results from a 4-T fitting with SPEX and from observations with EUVE (Brickhouse et al. 2000) and EXOSAT (Lemen et al. 1989) (bottom)
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For comparison we give in the bottom panel of Fig. 3 the results from two other instruments: EXOSAT (Lemen et al. 1989) and EUVE (Brickhouse et al. 2000). These distributions are comparable to our results given the fact that the results are obtained at different times and with different instruments.

Using the SPEX code (Kaastra et al. 1996) we have fitted the Chandra Capella spectrum with a 4-temperature model with variable abundances. The result scaled to the solar abundance of iron is shown in the bottom panel of Fig. 3 and agrees satisfactorily with the emission measure distribution given on top. The derived ratios of and Ne to Fe are close to solar or mildly subsolar, while the N, Mg, and Si ratios are about a factor of 2 larger than compared to that taken from Anders & Grevesse (1989). For Mg and Si these results agree with those from Brickhouse et al. (2000) based on an analysis of EUVE and ASCA spectra, but our Ne/Fe ratio is a factor of 3 larger.


Table 9: Characteristic loop parameters derived from observations
Ion $n_{\rm e}^a$ $T_{\rm m}^b$ Lc $H_{\rm p}^d$ EMe ff fg
C V 0.03 1.0 4.3 1.0 0.74 73 6.5
N VI 0.05 1.4 5.1 1.4 1.10 34 3.0
O VII 0.07h 2.0 7.4 2.0 0.71 7.4 0.68
Mg XI 40h 6.3 0.13 6.2 6.6 1.1-2 1.0-3
Si XIII 300i 10 0.043 9.8 5.3 5.2-4 4.8-5
Fe XX 30h 8.9 0.34 8.7 1.2 1.5-3 1.4-4
Fe XXI 30h 10 0.43 9.8 0.64 6.3-4 5.8-4
Fe XXII 30h 12 0.62 12 0.38 2.5-4 2.3-5

a electron density at loop apex in 1011 cm-3 from SPEX/MEKAL.
b temperature at loop apex in MK taken equal to maximum line formation temperature from SPEX/MEKAL.
c loop semilength in 108 cm for $\Gamma=1$; for $\Gamma=10$ L will be factor $\Gamma^{0.3}\simeq 2$ larger; L scales $\propto T_{\rm m}^2/n_{\rm e}$ and $H_{\rm p} \propto T_{\rm m}$.
d pressure scale height in 1011 cm.
e emission measure in 1052 cm-3.
f filling factor (in %) for $\Gamma$=1; 1.1-2 = 1.1 10-2, etc.
g filling factor (in %) for $\Gamma$=10; 1.0-3 = 1.0 10-3, etc.
h to be considered as upper limit, corresponding loop length and filling factor as lower limit.
i lower limit is 50.

7 Loop modeling

7.1 Scaling laws and loop lengths

Because the two binary components both contribute to the observed X-ray emission (see Sect. 1), any interpretation of the composite spectrum can only be given in terms of average stellar properties. We derive from our observations global properties of the X-ray emitting coronal structures using a standard model of a quasi-static, semi-circular, magnetic loop with constant cross section as developed by Rosner et al. (1978; hereafter RTV). In the RTV loop model the basic loop parameters are related through a simple scaling law:

\begin{displaymath}T_{\rm m}\simeq 1400(pL)^{1/3},
\end{displaymath} (3)

where the apex temperature $T_{\rm m}$ in K, the loop semilength L in cm, and the pressure p in dyne cm-2. Or with $p \simeq 2n_{\rm e} k T_{\rm m}$ we write:

\begin{displaymath}n_{\rm e} L = 1.3\ 10^6 T_{\rm m}^2,
\end{displaymath} (4)

where $n_{\rm e}$ is the apex electron density in cm-3.

Generalizations of the basic RTV model have been developed by Vesecky et al. (1979; hereafter VAU), including cross sections that increase with height in a way representing loop field structures by a magnetic line dipole below the chromosphere. The scaling law is only slightly modified by the VAU results which can be conveniently approximated by introducing into the rhs of Eqs. (3 or 4) an additional multiplication factor $\Gamma^{-0.1}$ or $\Gamma^{0.3}$, respectively. Here $\Gamma$ is the ratio of the loop cross section at the apex to that at the loop footpoint, the expansion factor. It turns out that a large variety of solar loops ranging from small bright points to active-region loops and large-scale structures can be modeled by these scaling laws (e.g., discussion by Mewe 1991).

Applying Eq. (4) for the derived densities (Sect. 5) we obtain estimates for the loop semilengths L (Table 9), taking $T_{\rm m}$ equal to the temperature of maximum line formation (from MEKAL).

7.2 Emission measures and filling factors

For a given density the emission measure places an additional constraint on the emitting volume (i.e. the "filling factor''). From the measured line fluxes and the known emissivities at temperature of maximum line formation we derive the emission measure $EM=\int n_{\rm e}n_{\rm H} {\rm d}V$, where V is the volume contributing to the emission (for cosmic abundances the hydrogen density $n_{\rm H}\simeq 0.85n_{\rm e}$). In Table 9 we give EM for various lines.

Assuming for simplicity that the X-ray corona consists of one class of N identical loops with parameters $T_{\rm m},n_{\rm e},L$, we derive for the emission measure

\begin{displaymath}EM = 0.85 G C(\Gamma) f n_{\rm e}^2 (4\pi R^2) \ell,
\end{displaymath} (5)

where R is the stellar radius $\simeq 10~R_\odot \simeq 7\ 10^{11}$ cm (Strassmeier et al. 1993) averaged for the two components of Capella. The factor G corrects for the fraction of the corona occulted by the stellar disk: G=0.5 for compact loops which we consider here. $C(\Gamma)\simeq (1+\Gamma)/2$ is an empirical correction for $\Gamma>1$ to the RTV loop volume derived from the VAU model. The fractional coverage of the stellar surface with footpoints of N identical X-ray emitting loops with cross section S at the base, the socalled $filling\ factor$, equals $f=2NS/4\pi R^2$. Finally, $\ell$ is the emission scale length, which is limited either by the loop size ( $\ell\approx L$ for $L<H_{\rm p}/2$) (case 1) or by the pressure scale height ( $\ell\approx H_{\rm p}/2$ for $L>H_{\rm p}/2$) (case 2). For case 1 (as $L \ll H_{\rm p}$) we derive from Eq. (6) and using Eq. (4) (with the factor $\Gamma^{0.3}$) for the filling factor

\begin{displaymath}f = 1.45\ 10^{-7} {EM\over {n_{\rm e} R^2 T_m^2 C(\Gamma)\Gamma^{0.3}}}\cdot
\end{displaymath} (6)

Using the measured fluxes for various helium-like lines and some iron lines we present in Table 9 the filling factors calculated with Eq. (7) for two loop geometries: $\Gamma=1$ (RTV loop) and $\Gamma=10$, respectively. The filling factors for the low-temperature regions are about 30% or more for the case $\Gamma=1$, i.e., comparable to those for active RS CVn systems, but much higher than for active regions on the Sun (few %, cf. Mewe 1991). However, loops with large expansion ($\Gamma=10$) should have a filling factor that is closer to that of active regions on the Sun. The hot, compact loops have probably a very small filling factor.

The results appear to show the existence of two main classes of loop structures in the coronae of the binary system Capella. On the one hand, the analysis of the lower ions N VI and O VII indicates temperatures, densities, and loop lengths that are more or less representative of active coronal regions on the Sun (with relatively large filling factors).

On the other hand, the higher He-like ions Mg XI and Si XIII and the highly ionized Fe ions (Fe XX-XXII) indicate the presence of a class of much more compact and dense, hot loops (with very small filling factors). On the Sun temperatures and densities this high - as observed in non-flaring active regions in the coronae of Capella - are seen only during flares. For both classes of loop systems $L \ll H_{\rm p}$ and R, where the pressure scale height $H_{\rm p}$ (in cm) is given by

\begin{displaymath}H_{\rm p}=5\ 10^3T (g/g_{\odot})^{-1},
\end{displaymath} (7)

where $g/g_{\odot}$ is the surface gravity expressed in solar units (for ${\rm Capella} = 0.051$) and T in K.

In all cases the loop sizes are much smaller than the stellar radius so that it is clear that the X-ray emission originates from compact regions that must be supported by strong magnetic fields (e.g., of the order of several hundred Gauss as was suggested by Dupree et al. 1993). Similar conclusions were drawn for other active binaries based on a review of EUVE observations by Drake (1996).

8 Conclusions

Echo the salient features of earlier EUVE analyses, we have established different temperature regimes in the Capella coronal plasmas: from $T \sim 0.5$-2 MK for Fe IX & X, N VI, and O VII, $\sim$5-6. MK for Fe XVIII-XX up to $\sim$9 MK for Fe XXI and Fe XXII. The upper limit to the densities obtained from Fe XX-XXII lines is about 3 1012 cm-3. The density of the plasma obtained from density sensitive lines of He-like triplets N VI and O VII is in the range 109-1010 cm-3 for $T \simeq$ 0.5-2 MK and from Mg XI and Si XIII in the range 1012-1013 cm-3 for $T \simeq$ 5-9 MK.

Abundances derived for various elements are about solar (O, Ne) or twice solar (N, Mg, Si).

The results for the densities and emission measures are interpreted in terms of quasi-static loop models. They appear to show the existence of two main classes of loop structures in the coronae of Capella. The analysis of the lower ions N VI and O VII indicates temperatures, densities, and loop lengths that are more or less representative for active coronal regions on the Sun (with relatively large filling factors). On the other hand, the higher He-like ions Mg XI and Si XIII and the highly ionized Fe ions (Fe XX-XXII) indicate the presence of a class of much more compact and dense, hot loops (with very small filling factors). On the Sun temperatures and densities this high are seen only during flares. For both classes of loops the size is much smaller than the pressure scale height and the stellar radius.

Comparisons between ratios of lines decaying from the same upper level and branching ratios give confidence in the in-flight total area calibration and the detector sensitivity as well as the reliability of the calculated transition probabilities. Based on photon fluxes of lines, sensitive to resonance scattering, we have derived conservative upper limits to the optical depth of the lines Fe XVIII $\lambda$93.92 and Fe XVII $\lambda$15.014 of 1 and 1.5, respectively.


The Space Research Organization Netherlands (SRON) is supported financially by NWO. We thank Nancy Brickhouse for her contribution to the effort to obtain for Capella a long observation time with the LETGS during the initial in-flight calibration of the CHANDRA transmission gratings in the context of the "Emission Line Project'' headed by her and Jeremy Drake. JJD was supported by NASA contract NAS8-39073 to the Chandra X-ray Center during the course of this research. Finally, we wish to thank the referee for helpful comments that allowed us to improve the paper.



Copyright ESO 2001