2.3 Indirect diagnostics

We know from the Sun that magnetic regions lead to enhanced emission both in the solar chromosphere and in the corona. Chromospheric and coronal emission can be observed in tracers like Ca ii emission in H & K lines or the Ca triplet, in Hα, in UV, X-ray, or radio emission. If we assume that other stars obey the same relations between magnetic fields and emission processes, we can determine their magnetic fields from observations of these tracers. For most of the indirect tracers, the determination of magnetic field requires: 1) that magnetic fields and their configuration in other stars are not too different from the solar field; and 2) that we correctly identify the mechanism coupling magnetic fields to observable emission. In this review, I will not give a detailed discussion of the results from indirect diagnostics, but rather introduce the general ideas and refer to the original literature.

Intensity contrast

Figure 1View Image gives a clear example of the correspondence between surface brightness and magnetic flux density on the Sun. Relations between these two values were provided, e.g., by Ortiz et al. (2002). They determine contrasts of active region faculae and the network as a function of heliocentric angle and magnetogram signal. Although this information is not available in spatially unresolved observations of other stars, it can be very helpful for the analysis of stellar variability from high quality photometric data, e.g., from the CoRoT or Kepler satellites.

Chromospheric emission

The correspondence between chromospheric Ca ii K emission and magnetic fields for the solar surface was investigated by Schrijver et al. (1989Jump To The Next Citation Point). Figure 10View Image shows that a close relation exists between the field strength and Ca ii emission in solar surface observations:

Ic − 0.13 --------- = 0.008 ⟨fB ⟩0.6, (7 ) IW
with IC the core intensity and IW the intensity in the wings (see Schrijver et al., 1989Jump To The Next Citation Point). Schrijver (1990) found a relation between C iv and magnetic flux density of the form
FC iv ∝ ⟨fB ⟩0.7. (8 )
A similar correspondence was also observed in M stars but using Hα (Reiners and Basri, 2007Jump To The Next Citation Point, 2010Jump To The Next Citation Point). Disk-integrated measurements of stellar chromospheric activity can therefore trace changes in surface activity induced by, e.g., rotation or magnetic cycles (see, e.g., Baliunas et al., 1995; Hall, 2008).
View Image

Figure 10: Ca ii K core wing intensity ratio vs. absolute value of the magnetic flux density from resolved solar surface observations after degradation of the resolution to 14.4” × 14.4” (from Schrijver et al., 1989, reproduced by permission of the AAS).

X-ray emission

X-ray observations are available for the Sun and many stars. A close relation between magnetic flux and X-ray spectral radiance was shown by Pevtsov et al. (2003Jump To The Next Citation Point) (see also Güdel, 2004). The relation holds for solar quiet regions, active regions, and disk-integrated measurements of very active stars and covers more than ten orders of magnitude in both parameters (see Figure 11View Image). The relation is approximated by

L ∝ Φ1.15, (9 ) X
with LX the X-ray spectral radiance and Φ the magnetic flux. The relation is similar to the one found by Saar (2001Jump To The Next Citation Point) for cool stars,
FX ∝ Φ0.95. (10 )
View Image

Figure 11: X-ray spectral radiance vs. total unsigned magnetic flux for solar and stellar observations. Dots: Quiet Sun. Squares: X-ray bright points. Diamonds: Solar active regions. Pluses: Solar disk averages. Crosses: G, K, and M dwarfs. Circles: T Tauri stars. Solid line: Power-law approximation 1.15 LX ∝ Φ (from Pevtsov et al., 2003, reproduced by permission of the AAS).

Radio emission

Emission of radiation at radio wavelengths is indicative of ionized atmospheres that in many cases are related to stellar magnetic activity. Radio emission can be generated by different processes leading to characteristic signatures of radio emission; for an overview see Güdel (2002Jump To The Next Citation Point). The close correlation between radio and X-ray emission (the Güdel–Benz relation; Benz and Güdel, 1994) shows that radio and X-ray emission are generated by the same or at least correlated processes. The relation holds for quiescent and active emission of the Sun and a wide variety of stars. In very low-mass stars or brown dwarfs, however, Berger et al. (2005) showed that this relation is violated with objects that are overluminous at radio wavelengths.

Depending on the emission process, or on the question whether the emitting electrons are relativistic or not, radio emission has characteristic properties that can allow the determination of magnetic fields (see Güdel, 2002). The gyrofrequency, or cyclotron frequency, in a magnetic field is

--eB--- 6 νc = 2πmec ≈ 2.8 × 10 B, (11 )
with the magnetic field strength B in Gauss and νc in Hz. Gyrosynchrotron emission from a power-law electron distribution is proportional to γ−δ (with γ the Lorentz factor) and shows polarization characteristic for the magnetic field. For a chosen angle between the line of sight and the magnetic field, 𝜃 = π∕3, the polarization p can be written (Dulk, 1985)
p ≈ 103.35+0.035δ(ν∕B )−0.51. (12 )
In principle, this equation can be used to determine the magnetic field strength from the fractional polarization of radio emission, but it rests on several assumptions and is very sensitive to the geometry of the emitting regions, which is not known in spatially unresolved stars.

Coherent emission, in particular electron cyclotron maser emission, can be a reliable tracer of the magnetic field strength because it is emitted mostly at the fundamental and the second harmonic of νc. Detection of radio emission at a given frequency indicates the presence of magnetic field corresponding to that frequency, for example the detection of 8.5 GHz radio emission indicates a field of strength B ≥ 3 kG (see, e.g., Hallinan et al., 2008Jump To The Next Citation Point).

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