The Problem of the Height Dependence of Magnetic Fields in Sunspots

To understand the physics of sunspots, it is important to know the properties of their magnetic field, and especially its height stratification plays a substantial role. There are mainly two methods to assess this stratification, but they yield different magnetic gradients in the photospheric layers. Determinations based on the several spectral lines of different formation heights and the slope of their profiles result in gradients of -2 to -3 G km(-1), or even steeper. This is similar for the total magnetic field strength and for the vertical component of the magnetic field. The other option is to determine the horizontal partial derivatives of the magnetic field, and with the condition also the vertical derivative is known. With this method, gradients of -0.5 G km(-1) and even shallower are obtained. Obviously, these results do not agree. If chromospheric spectral lines are included, only shallow gradients around -0.5 G km(-1) are obtained. Shallow gradients are also found from gyro-resonance measurements in the radio wave range 300 - 2000 GHz. Some indirect methods are also considered, but they cannot clarify the total picture. An analysis of a numerical simulation of a sunspot indicates a shallow gradient over a wide height range, but with slightly steeper gradients in deep layers. Several ideas to explain the discrepancy are also discussed. With no doubts cast on Maxwell's equations, the first one is to look at the uncertainties of the formation heights of spectral lines, but a wider range of these heights would require an extension of the solar photosphere that is incompatible with observations and the theory of stellar atmospheres. Submerging and rising magnetic flux might play a role in the outer penumbra, if the resolution is too low to separate them, but it is not likely that this effect acts also in the umbra. A quick investigation assuming a spatial small scale structure of sunspots together with twist and writhe of individual flux tubes shows a reduction of the measured magnetic field strength for spectral lines sensitive to a larger height range. However, sophisticated investigations are required to prove that the explanation for the discrepancy lies here, and the problem of the height gradient of the magnetic field in sunspots is still not solved.