We derived an analytical scaling relation of the coronal X-ray emission with the unsigned surface magnetic flux, Lx ∝ Φm in Eq. (3.13). Previously, this relation had only been derived using observations, without the backing of a theoretical framework. We based our approach on the coronal loop scaling laws of Rosner et al. (1978) (see Eq. (3.9) and
(3.10)), and the idea that the heating of the corona is mainly driven by an upward-directed Poynting flux generated in the photosphere.
The power-law indexmthat we derive in Eq. (3.13) depends on the area of the active region, the heating mechanism, and the wavelength range covered by the respective X-ray instrument, namely, its temperature response function. Each of these factors can be represented by power laws. The active region area impact is constrained observationally (δ = 0.819, Eq. 3.7), the heating mechanism is inspired by basic considerations (βfrom 1 to 2; Eq. 3.4), and the temperature response between 1 and 10 MK is based on atomic data (αin the range of 1 to 3, Table 3.2).
The power-law indicesm we find through our analytical approach are generally in a range between just belowm ≈ 1 and almost 2 (see Table 3.2). This is within the range found by most observations, which are mostly composed of a combination of stellar stud-ies with different instruments (see Table 3.1; a larger value only found by Kochukhov et al. 2020). As such, we consider our simple analytical model approach to be a good first step to build a theoretical foundation for the observed power-law relations between X-ray emission and magnetic field. However, with our simplified model approach it is dif-ficult to distinguish between different heating mechanisms, mainly because the different X-ray instruments have quite different responses to the temperature of the coronal plasma.
4.1 Introduction
The numerical setup, presented in this chapter, models a small part of the solar corona above an active region in a 3D Cartesian box (see Bingert 2009, for more details). The model extends in the vertical direction up to the corona, including also the photosphere.
The motivation of this model is the fieldline braiding mechanism proposed by Parker (1972, 1983). At the lower boundary, we include the vertical magnetic field of an active region together with a description of photospheric velocities mimicking solar granulation.
The shuffling of the footpoints of the photospheric magnetic fieldlines because of the horizontal velocities will generate the necessary energy flux (or Poynting flux) needed to heat the corona to high temperatures of several million Kelvin degrees, similar to the solar corona. Once the system reaches an equilibrium state, the energy input in the corona is balanced by the radiative losses. In this state, the modeled corona can maintain a high temperature indefinitely.
Our computational box is divided into four parts. The lower part of our computational domain is the photosphere, where the active region is located. In the photosphere, the shuffling of the magnetic fieldlines will induce currents in the corona that will dissipate and heat the corona. The chromosphere is the part of the box located above the photo-sphere. Given the highly complicated nature of the chromosphere, it is only treated as a reservoir for energy. We mainly care about the energy transfer in the corona. Therefore, excluding the chromosphere from our analysis will not affect the results. The transition region is a thin layer located between the chromosphere and the corona. In the transi-tion region, the temperature increases steeply, reaching from a few thousand Kelvins to coronal temperatures of one million Kelvin. The high gradients in temperature have to be resolved numerically so that numerical instabilities are avoided and will not affect the simulations. The corona extends for the largest part of our numerical box. For the proper physical description of the corona, we include an optically thin radiative loss function and a Spitzer heat conduction along the magnetic field. This model has been proved success-ful in describing some of the characteristics observed in the solar corona (see e.g. Bingert and Peter 2011; Warnecke and Peter 2019a). Consequently, it can be used for a parameter study and extend the analysis even to other stars.
We aim to run several simulations to test the effect of the surface magnetic flux on the coronal X-ray emission. The reason for that is to explain the observed X-ray emission coming from stellar coronae of stars more active than the Sun. To be able to run several simulations for our analysis, we choose a spatial resolution of 390 km.
Our work is divided into two parts. For the first part, we increase the surface magnetic
flux by increasing the strength of the vertical surface magnetic field of the active region.
In this case, we keep the box size fixed with a volume of 50 x 50 x 50 Mm3. For the simulations we used the supercomputers in Gesellschaft für wissenschaftliche Datenver-arbeitung mbH Göttingen (GWDG) with a 1283 grid points in all directions. The results will be presented in Chap. 5. For the second part, we increase the surface magnetic flux by increasing the surface area covered by the active region. In this case, our numerical box has to increase. The biggest computational box reaches a volume of 200 x 200 x 200 Mm3. The individual active regions have the same surface vertical magnetic field strength fixed at a constant value. The simulations have been carried out at the Max Planck Com-puting and Data Facility (MPCDF) in Münich. The number of grid points scales with the size, but the spatial resolution is constant at 390 km. The results of this work will be presented in Chap. 6. Finally, the different physical parameters have the same values for all simulations so that the corona will be affected solely by the change in the surface magnetic flux.