3.6 Surface entropy jump

Radiation losses and transport near the solar surface not only produces the low entropy, high density fluid that gravity pulls down to drive the convective motions, but also controls what we observe on the Sun. The surface occurs where photons can escape, so neither the diffusion approximation nor the optically thin approximations are valid. The radiation heating/cooling must be obtained by solving the transfer equation for the radiation. Since the temperature varies both in the vertical and horizontal directions, transfer in three-dimensions is needed. Current computers are still not fast enough to solve the full non-LTE (or even LTE) transfer problem at the large number of frequencies needed to cover the continuum and line spectrum for each of the thousands of time steps needed for a reasonable simulation time sequence, so the binning method mentioned in Section 2.4.1 needs to be used.

Photons escape typically from one mean free path into the Sun. The dominant source of opacity at the surface of the Sun is due to the H ion, whose electron is bound by only 0.75 eV and is easily detached by photons in the near infrared and shorter wavelengths (< 1.64 μm) and by collisions. As a result, the H opacity is very temperature sensitive, 10 κ ≈ T. Because of this great sensitivity to the plasma temperature, photons escape from the cool intergranular lanes at larger depths than from the hot granule centers (cf. Pecker, 1996) and the optical depth unity surface is corrugated with an rms height variation of ≈ 35 km in magnetic field free regions and extending as deep as ≈ 350 km in strong magnetic field (≥ 1.5 kG) concentrations (Wilson depression; see Figure 10View Image). This produces a hilly appearance of granules when viewed toward the limb (Figure 11Watch/download MovieCarlsson et al., 2004Jump To The Next Citation Point). Another consequence is that we observe a much smaller temperature variation across the surface than is actually present at a given geometrical level (Figure 12View ImageStein and Nordlund, 1998Jump To The Next Citation Point). We do not observe the hot plasma because at high temperatures we only see photons that escape from higher elevations, which are cooler, since in the photosphere the mean temperature decreases outward.

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Figure 10: Height of τ = 1 vs vertical velocity (downflows are positive) in the quiet Sun. The rms variation in the τ = 1 surface is ≈ 35 km.

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Figure 11: mov-Movie (48052 KB) The emergent intensity from a magneto-convection simulation showing changed appearance as one approaches the limb (movie by Mats Carlsson, Oslo).

Significant absorption and emission occurs in spectral lines above the level from which continuum photons escape. Spectral lines typically cool the plasma where their optical depth is of order unity and heat the plasma where their optical depth is large and they block the escape of radiation; this process is known as “spectral line blanketing” (Mihalas, 1978). Ascending plasma from granule interiors would otherwise adiabatically cool as it expands going from the visible surface to the temperature minimum (∼ 4 scale heights) by a factor of > 3, or from ∼ 6000 K to < 2000 K. This does not happen because the ascending gas is heated by absorbing radiation in optically thick lines and UV continua. Such absorption blocks some of the radiation from escaping producing a greenhouse effect called “back warming”. The photons absorbed above the τ = 1 500 surface reheat the photospheric plasma which then radiates both upward and downward heating the surface above the temperature it would have if all the radiation could escape to space. Realistic horizontal transport in the presence of evacuated magnetic flux concentrations also requires inclusion of optically thick line opacity, because otherwise the flux concentrations become optically thin and do not absorb photons from the hot granule walls and so do not show up as bright points at disk center and faculae toward the limb (Steiner and Stenflo, 1990Vögler, 2004).

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