Variations in the observed intensity can be related to a change in the temperature through:
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: . In the penumbra this number is about 65 – 85% of the granulation brightness: . Assuming that the temperature at for the quiet Sun is about 6050 K, the numbers we obtain from Equation (25) are: and .
At infrared wavelengths the difference in the brightness between quiet Sun and umbra or penumbra is greatly reduced: and (see Figure 1 in Mathew et al., 2003). This happens as a consequence of the behavior of the Planck’s function , 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 18, 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. (2004). They show : vs. (total magnetic field strength; upper-left), vs. (zenith angle – Equation (10) – upper-right), vs. (or our vertical component of the magnetic field; lower-left), and vs. (or our horizontal component of the magnetic field; lower-right). As expected, the thermal brightness anti-correlates with the total field strength since the latter is larger (see Figures 4 and 11) in the darkest part of the sunspot: the umbra. However, the inclination of the magnetic field correlates well with the thermal brightness. Again, this was to be expected (see Figures 7 and 11) since the inclination of the magnetic field increases towards the penumbra, which is brighter (see Figures 2 and 3). As we will discuss intensively throughout Section 3, these trivial results have important consequences for the energy transport in sunspots.
Living Rev. Solar Phys. 8, (2011), 4
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