2.4 What is the plasma-β in sunspots?

In the previous Section 2.3 we have argued that the magnetic field vector in sunspots is non-potential. However, in order to establish its degree of non-potentiality it is important to develop this statement further. The way this has been traditionally done is through the study of the plasma-β parameter. The plasma-β is defined as the ratio between the gas pressure and the magnetic pressure:
Pg 8πPg β = --- = ---2-(in cgs units). (22 ) Pm B

In the solar atmosphere, if β > > 1 the dynamics of the system are dominated by the plasma motions, which twist and drag the magnetic field lines while forcing them into highly non-potential configurations. If β < < 1 the opposite situation occurs, that is, the magnetic field is not influenced by the plasma motions. In this case, the magnetic field will evolve into a state of minimum energy which happens to coincide with a potential configuration (see Chapter 3.4 in Priest, 1982). Therefore, many works throughout the literature focus on the plasma-β in order to study the potentiality of the magnetic field. Here, we will employ our results from the inversion of spectropolarimetric data in Section 2.2 to investigate the value of the plasma-β parameter in a sunspot. Figure 16View Image shows the variation of the azimuthally averaged plasma-β (along ellipses in Figure 10View Image) as a function of the normalized radial distance in the sunspot r∕Rs. This figure displays β at four different optical depths, from the deep photosphere τc = 1 (yellow) to the high-photosphere τc = 10−3 (blue). This figure shows that β < < 1 above −2 τc ≤ 10 and, thus, the magnetic field can be considered to be nearly potential (or at least force-free) in these high layers. At τc = 1 (referred to as continuum) β ≥ 1 and, therefore, the magnetic field is non-potential. At the intermediate layer of τc = 0.1 (around 100 kilometers above the continuum) the magnetic field is nearly potential in the umbra, but it cannot be considered this way in the penumbra: r∕Rs > 0.4.

View Image

Figure 16: Similar to Figure 11View Image but for the plasma-β as a function of the normalized radial distance in the sunspot: r∕R s. The different curves refer to different optical depths in the sunspot: τc = 1 (yellow), τc = 0.1 (red), −2 τc = 10 (green), and − 3 τc = 10 (blue). The vertical dashed line at r∕Rs ≃ 0.4 indicates the umbra-penumbra boundary.

In Figure 16View Image the gas pressure was obtained under the assumption of hydrostatic equilibrium (Section 1.3.3), which we know not to be very reliable in sunpots. A more realistic approach was followed by Mathew et al. (2004Jump To The Next Citation Point, and references therein), where an attempt to consider the effect of the magnetic field in the force balance of the sunspot was made. Their results for the deep photosphere (τc = 1) obtained from the inversion of the Fe i line pair at 1564.8 nm are consistent with our Figure 16View Image (obtained from the inversion of the Fe i line pair at 630 nm), with β ≈ 1 close to the continuum everywhere in the sunspot. Similar results were also obtained by Puschmann et al. (2010cJump To The Next Citation Point, see their Figure 4), who performed an even more realistic estimation of the geometrical height scale, considering the three components of the Lorenz force term (j × B; Equation (17View Equation)). In Figure 17View Image we reproduce their results, which further confirm that the β ≈ 1 in the deep photospheric layers of the penumbra.

View Image

Figure 17: Same as Figures 9View Image and 15View Image but for the plasma-β at z = 0 in the inner penumbra of a sunspot. The white contours are the same as in Figure 15View Image: B ρ = 650 (solid white) and B ρ = 450 (dashed white). This sunspot is AR 10953 observed on May 1st, 2007 with Hinode/SP (from Puschmann et al., 2010cJump To The Next Citation Point, reproduced by permission of the AAS).

These results have important consequences for magnetic field extrapolations from the photosphere towards the corona, because they imply that those extrapolations cannot be potential. In addition, as pointed out by Puschmann et al. (2010c) the magnetic field is not force-free because in many regions the current density vector j and the magnetic field vector B are not parallel. Unfortunately, extrapolations cannot deal thus far with non-force-free magnetic field configurations. Considering that it has now become possible to infer the full current density vector j, developing tools to perform non-force free magnetic field extrapolations will be a necessary and important step for future investigations. These results also have important consequences for sunspot’s helioseismology, because of the deep photospheric location of the β = 1 region, which is the region where most of the conversion from sound waves into magneto-acoustic waves takes place.

In the chromosphere of sunspots, the magnetic field strength is about half of the photospheric value (see Figure 4 in Orozco Suarez et al., 2005). Therefore, the magnetic pressure in the chromosphere is only about 25% of the photospheric value. However, the density and gas pressure are at least 2 – 3 orders of magnitude smaller. Thus, the chromosphere of sunspots is clearly a low-β (β < < 1) environment, which in turn means that the magnetic field configuration is nearly potential.

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