Most of these methods are applicable to various types of stars, but results are overall ambiguous. I will not review all aspects of stellar coronal structure (see Güdel, 2004) but concentrate on results that have been obtained – at least in part – specifically for young solar analogs.
In the simplest approach, let us assume that the observed X-ray luminosity LX is produced by an ensemble of identical magnetic coronal loops with characteristic half-length L, surface filling factor f, and an apex temperature T used for the entire loop; then, on using the Rosner–Tucker–Vaiana (RTV, Rosner et al. 1978) loop scaling law and identifying LX = V where is the volumetric heating rate (in erg cm–3 s–1), we obtain
This relation can only hold if L is smaller than the pressure scale height. Based on this expression, the luminous, hot plasma component in magnetically active stars seems to invariably require either very large, moderate-pressure loops with a large filling factor, or solar-sized high-pressure compact loops with a very small (1%) filling factor (Giampapa et al., 1985; Stern et al., 1986; Schrijver et al., 1989; Giampapa et al., 1996; Güdel et al., 1997b; Preibisch, 1997; Sciortino et al., 1999).
Schrijver et al. (1984) modeled T and the emission measure, EM (= nenHV, where ne and nH are the electron and hydrogen number densities, and V is the volume) of a sample of coronal sources based on RTV loop models and found several trends: i) inactive MS stars such as the Sun are covered to a large fraction by large-scale, cool (2 MK) loops of modest size (0.1 ); ii) moderately active dwarfs are dominated by very compact, high-density, hot ( 20 MK) loops that require high heating rates (up to 20 times more than for solar compact active region loops); iii) the most active stars may additionally form rather extended loops with heights similar to .
Similar studies of loop models varied in their results, although for magnetically active stars, most authors reported results that require hot, compact, high-pressure loops (up to several 100 dyn cm–2), somewhat reminiscent of flaring loops (Stern et al., 1986; Giampapa et al., 1996; Maggio and Peres, 1997; Ventura et al., 1998; Sciortino et al., 1999). The hypothesis that the hottest plasma in magnetically active stellar coronae does not form in static loops but in flaring active regions will be encountered again – see Section 5.8.
If the Sun were entirely covered with active regions, the X-ray luminosity would amount to only (2 – 3) × 1029 erg s–1 (Vaiana and Rosner, 1978; Wood et al., 1994), short of LX of the most active solar analogs by one order of magnitude. But the X-ray emission measure distributions of such active stars show excessive amounts of plasma around 10 – 20 MK (Güdel et al., 1997b), which incidentally is the typical range of solar flare temperatures. This again led to the suggestion that the high-T emission measure is in fact due to the superposition of a multitude of temporally unresolved flares (see Section 5.8).
Comprehensive surveys of stellar coronal electron density (ne) measurements based on X-ray spectroscopic line-flux ratios were presented by Ness et al. (2004) and Testa et al. (2004), including a sample of active solar analogs. These studies concluded that the surface filling factor (derived from the emission measure, the measured ne, and a realistic coronal scale height) of magnetic loops containing cool X-ray emitting material increases from inactive to moderately active stars but then “saturates” at levels of about ten percent. In the most active stars, hot coronal loops are added, with a sharply increasing filling factor, thus probably filling the volume left between the cooler coronal magnetic loops. Observations of rotational modulation in very active solar analogs suggests, however, that the coronal volume filling remains significantly below 100% (see Section 4.2.3 below).
Inhomogeneous coronae may reveal signatures in light curves as the star rotates, although success of this method has been moderate given that coronal features evolve on time scales shorter than one rotation (e.g., owing to flares). Two young, near-ZAMS solar analogs have shown clear signatures of X-ray rotational modulation (EK Dra, Güdel et al. 1995c, and 47 Cas B, Güdel et al. 1995a), pointing to a filling factor below unity. This is unexpected because such stars are in the empirical X-ray saturation regime that has often been suggested to be due to complete filling of the surface with X-ray bright coronal magnetic loops (see Section 5.5 below). EK Dra also showed evidence for radio rotational modulation. The depth and length of the modulation (Figure 4a) constrains the X-ray coronal height, and also the electron densities to ne 4 × 1010 cm–3, in agreement with spectroscopic measurements (Ness et al., 2004). This leads to the conclusion that much of the emitting material is concentrated in large “active regions”.
Because the X-ray luminosity in very rapidly rotating, “supersaturated” stars (Section 5.5.1) is below the empirical maximum, rotational modulation would give important structural information on the state of such coronae. A deep modulation in the young solar analog VXR45 (Figure 4b) suggests that extreme activity in these stars is again not due to complete coverage of the surface with active regions (Marino et al. 2003a).
Most binary systems that produce coronal eclipses are close binaries and the components are therefore in no ways similar to the young Sun. A remarkable exception is the relatively wide binary CrB. Its X-ray active, young (age of few 100 Myr) solar analog of spectral type G5 V is totally eclipsed once every 17 days by the optical primary, an A0 V star that is perfectly X-ray dark. Other parameters are ideal as well, such as the non-central eclipse, the eclipse time-scale of a few hours, and the relatively slow rotation period of the secondary. Eclipse observations obtained by ROSAT (Schmitt and Kürster, 1993) and by XMM-Newton (Güdel et al., 2003a) were used to reconstruct projected 2-D images of the X-ray structure. They consistently reveal patches of active regions across the face of the G star; not much material is found significantly beyond its limb (Figure 5). The structures tend to be of modest size ( 5 × 109 cm), with large, X-ray faint areas in between, although the star’s luminosity exceeds that of the active Sun by a factor of 30. These observations imply moderately high densities in the emitting active regions, and both studies mentioned above yielded average electron densities in the brightest active regions of a few 1010 cm–3. The picture of active coronae consisting of features similar to solar active regions thus seems to hold also for intermediately active, young solar analogs.
Information on coronal structure can also be derived indirectly from surface Zeeman–Doppler images as developed for and applied to the stellar case by Jardine et al. (2002a), Jardine et al. (2002b), Hussain et al. (2002), and Hussain et al. (2007) and further references therein. Jardine et al. (2002a) and Jardine et al. (2002b) explored potential field extrapolation, while Hussain et al. (2002) extended the models to include some form of currents in force-free fields.
Such models also require specification of the base thermal pressure of the plasma with respect to the local magnetic pressure, and some cutoff of the corona at locations where the thermal pressure might open up the coronal field lines. They can successfully recover, at least qualitatively, the total X-ray emission measure, the average electron density, and the low level of rotational modulation observed on very active, young stars such as AB Dor. The X-ray rotational modulation is to a large extent suppressed by the highly complex coronal structure, involving both very large magnetic features and more compact loops anchored predominantly at polar latitudes.
Schrijver and Aschwanden (2002) used the model of surface magnetic-field development devised by Schrijver (2001) and Schrijver and Title (2001) (Section 4.1 above) to extrapolate surface magnetic fields into the corona, assuming a potential-field approach. The resulting magnetic structure is illustrated in Figure 6 for a solar-activity star, and two examples with a 10fold and a 30fold magnetic injection rate (the latter corresponding to a solar analog with an age of a few 100 Myr). Near cycle minimum, the coronae of the active stars are dominated by a large-scale dipolar field although the low-activity example also shows compact active regions, more so than the active stars. At activity maximum, the active stars show magnetic-field arcades between the polar rings of opposite polarity, weakening the contribution from the global dipole components. Not all of these loops will be X-ray bright, however; the loop brightness depends on its heating rate, which in turn depends on the mode of coronal heating of a given magnetic loop (see Schrijver and Aschwanden 2002 for further details).
Despite numerous, complementary approaches to the study of coronal structure, the results appear to be inconclusive. Compact active regions are inferred from (and required by) the presence of rotational modulation, and field extrapolation and numerical models suggest compact regions as well. Extended, global fields are more difficult to infer, and this is the consequence of two observational biases rather than implying the absence of such structures. First, most observational methods (e.g., X-ray eclipse and rotational modulation) are insensitive to large structures. Second, the pressure scale height of typical coronal plasma is less than one stellar radius for solar analogs. The dependence of X-ray emission will therefore bias X-ray loop detection toward low heights. There is, nevertheless, evidence for large-scale magnetic fields on young, active stars, from two directions: surface field extrapolation based on observed spot features or based on numerical models of flux transport (Section 4.2.5) imply the presence of structures resembling global dipole components; and spatially resolved radio observations of active stars indeed do show very extended magnetic structures (predominantly above the polar regions; Benz et al. 1998; Mutel et al. 1998), although such evidence has not yet been demonstrated for young solar analogs. In summary, then, observational evidence and model simulations point to the presence of both compact active regions and extended magnetic structures in the young Sun. The relative proportions are likely to relate to the distribution of magnetic field on the stellar surface, as in the models presented by Schrijver and Aschwanden (2002).
There is clear evidence that the surface magnetic filling factor increases with increasing activity (and decreasing rotation period), both from observations (e.g., Montesinos and Jordan, 1993) and theoretical and modeling studies (Montesinos and Jordan, 1993; Fawzy et al., 2002); in contrast, the photospheric magnetic field strength is thought to be primarily restricted by pressure equilibrium with respect to the ambient gas pressure, i.e., the field strength is primarily dependent on spectral type, although a weak activity dependence is also present (e.g., Montesinos and Jordan, 1993). The higher filling factors lead to less expansion of photospheric/chromospheric flux tubes because the tubes merge with adjacent tubes (Cuntz et al., 1999). Therefore, toward more active stars, coronal magnetic fields interact progressively more frequently due to their denser packing. A higher rate of large flares is a consequence. Since flares enhance the electron density along with the temperature, stars with a higher activity level should reveal a predominance of hotter structures (Güdel et al., 1997b). Such a trend is observationally well supported; as further described in Section 5.5.2 below, coronae at higher activity levels are systematically hotter.
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