2.5 Sunspots’ thermal brightness and thermal-magnetic relation

The Eddington–Barbier approximation can be employed to relate the observed intensity from any solar structure with a temperature close to the continuum layer: τ = 2∕3. This is done by assuming that the observed intensity is equal to the Planck’s function, and solving for the temperature:
2hc2 hc Iobs ∼ ---5-exp − ----- . (23 ) λ λKT

Variations in the observed intensity can be related to a change in the temperature through:

ΔIobs-∼ dIobs = --hc--. (24 ) ΔT dT λKT 2 (25 )

The observed brightness of a sunspot umbra at visible wavelengths is about 5 – 25% of the observed brightness of the granulation at the same wavelength: I ≈ 0.05 –0.25I umb qs. In the penumbra this number is about 65 – 85% of the granulation brightness: Ipen ≈ 0.65– 0.85Iqs. Assuming that the temperature at τ = 2 ∕3 for the quiet Sun is about 6050 K, the numbers we obtain from Equation (25View Equation) are: Tumb (τ = 2∕3) ≈ 4800 K and Tpen(τ = 2∕3) ≈ 5650 K.

At infrared wavelengths the difference in the brightness between quiet Sun and umbra or penumbra is greatly reduced: Iumb ≈ 0.4 − 0.6Ic,qs and Ipen ≈ 0.7 − 0.9Ic,qs (see Figure 1 in Mathew et al., 2003Jump To The Next Citation Point). This happens as a consequence of the behavior of the Planck’s function B (λ, T), whose ratio for two different temperatures decreases towards larger wavelengths. All numbers mentioned thus far are strongly dependent on the spatial resolution and optical quality of the instruments. For example, large amounts of scattered light tend to reduce the intensity contrast and, therefore, temperature differences between different solar structures.

In Figure 18View Image, we present scatter plots showing the relationship between the sunspot’s thermal brightness and the components of the magnetic field vector. These plots have been adapted from Figure 4 in Mathew et al. (2004Jump To The Next Citation Point). They show T (τ = 1): vs. B (total magnetic field strength; upper-left), vs. ζ (zenith angle – Equation (10View Equation) – upper-right), vs. B z (or our B ρ vertical component of the magnetic field; lower-left), and vs. Br (or our Bh horizontal component of the magnetic field; lower-right). As expected, the thermal brightness anti-correlates with the total field strength B since the latter is larger (see Figures 4View Image and 11View Image) in the darkest part of the sunspot: the umbra. However, the inclination of the magnetic field ζ correlates well with the thermal brightness. Again, this was to be expected (see Figures 7View Image and 11View Image) since the inclination of the magnetic field increases towards the penumbra, which is brighter (see Figures 2View Image and 3View Image). As we will discuss intensively throughout Section 3, these trivial results have important consequences for the energy transport in sunspots.

View Image

Figure 18: Upper-left: scatter plot of the total field strength vs. temperature at τc = 1. Upper-right: inclination of the magnetic field with respect to the vertical direction on the solar surface (ζ; see Equation (10View Equation)) versus temperature at τc = 1. Bottom-left: vertical component of the magnetic field Bz (called B ρ in our Section 2.1) vs. temperature at τc = 1. Bottom-right: horizontal component of the magnetic field Br (called Bh in Section 2.1) versus the temperature at τc = 1. In all these panels circles represent umbral points, whereas crosses and triangles correspond to points in the umbra-penumbra boundary and penumbral points, respectively (from Mathew et al., 2004, reproduced by permission of the ESO).


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