Synthetic coronal emission and temporal evolution

Coronal loops are considered to play an important role in the heating of the solar and stellar corona. Coronal loops in our Sun have been directly observed. However for other stars, this is not possible. Observations of stellar flares have shown that typical stellar loops can vary in length, and they can reach up to solar radius (Getman et al. 2008).

Because of the high plasma temperature located in coronal loops, we observe them in the extreme ultraviolet (EUV) and X-ray regime. To synthesize the EUV emission from our numerical model we use the 171 Å channel of the Atmospheric Imaging Assembly (AIA;

Lemen et al. 2012) onboard the Solar Dynamic Observatory (SDO; Pesnell et al. 2012).

As in Chap. 5, for the X-ray emission we use the Al-poly filter of the X-ray telescope (XRT; Golub et al. 2007b) onboard the Hinode (Kosugi et al. 2007).

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Figure 6.1: Synthetic EUV and X-ray emission for the runs R50, R100 and R200. The first row shows the synthesized EUV emission integrated in the y-direction, as it would have been observed from the AIA instrument for the 171 Å channel. The second and third row show the integrated synthetic X-ray emission in theyandzdirection respectively, as it would have been observed by the Al-poly filter of the XRT onboard Hinode. Both the EUV and X-ray emission are averaged in time during the relaxed phase. Each plot is scaled with the maximum EUV or X-ray emission respectively. For the larger box size coronal loops increase in length.

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Figure 6.2: Time evolution of coronal EUV and X-ray emission. a) Total integrated X-ray emission over time for all the runs seen in Table 6.1. b) Total integrated EUV emission corresponding to the AIA 171 channel over time for all the runs. The vertical dashed lines indicate the relaxed phase used for the temporal averages.

As we discussed in Chap. 3 and Chap. 5, the optically thin radiation is proportional to electrons density squared times the temperature response functionR(T). The temperature response function can be calculated both for the 171 Å channel and the Al-poly of XRT, using the routines of the Chiantidatabase (Dere et al. 1997, 2019). After calculating the X-ray and EUV radiation at each grid point, we integrate along the y-direction to get a side view. That can be considered as a near-the-limb observation. Similarly, we integrate along thez-direction to get a top view. That would correspond to a disk center observation.

Both the side and top view as synthesized from our model is shown in Fig. 6.1 for the runs R50, R100, R200. Each plot depicted in Fig. 6.1 is scaled with the maximum value of the EUV or X-ray emission. For the larger numerical boxes, the EUV and X-ray emission increases indicating a connection to the amount of surface magnetic flux each run host at the bottom boundary. As we approach the upper part of the corona, the magnetic field is weak. Hence, we observe a diffused emission. All three runs show bright loops in EUV and X-ray. The difference lies in the length of each loop. The average length of coronal loops in the R200 is on the order of 80 Mm that is significantly larger than the average length of 20 Mm found in R50.

As a next step, we study the temporal variation of the coronal emission. That is an important step since the quantities studied in scaling relations might vary significantly in time, especially during the initial stages of the simulations. Hence, to properly analyze our data, we have to choose an appropriate time frame. The time range where these quantities

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Figure 6.3: Horizontally averaged profile of temperature and density. a) Horizontal aver-age temperature as a function of height for each run. b) Horizontal density as a function of height for each run. The colors show the different runs. Both quantities are also averaged in time over the relaxed phase (see Sect. 6.3.1).

show the least temporal variation, compared to an average value, can be considered as the relaxed phase of our simulations. In Fig. 6.2 it is illustrated the temporal evolution of the coronal X-ray emission and the EUV emission for all the runs listed in Table 6.1. The X-ray and EUV emission in the corona is integrated into the whole computational box.

One important aspect we notice here is the connection between magnetic activity and coronal emission. The larger box size, hosting a higher amount of surface magnetic flux, shows a much brighter X-ray and EUV corona than the smaller box sizes. The increase is non-linear, and the implications are discussed in Sect. 6.3.6.

The initial phase determines the cooling phase of our simulations. The duration of that phase depends on the strength of the coronal activity. For the R25 (see black solid line Fig. 6.2) run, since it hosts the least amount of surface magnetic flux, the coronal temperature is too low to sustain a hot corona. Therefore, the cooling time will be much shorter than the other more active runs. The temporal evolution of the coronal emission shows a minimal time variability for the range from t=220 min to t=300 min indicated by the dashed vertical dashed lines. We consider, for that specific time frame, the simulations to have reached a relaxed phase. The analysis following next will be concentrated at the relaxed phase, hence for the time 220 min to 300 min.

6.3.2 Coronal temperature and density

The temperature and density stratification over height provide insight into the ability of our model to create a hot corona. In this section, we show horizontally averaged profiles of temperatureT and densityρ. Both quantities are also averaged in time over the relaxed phase. The time interval is chosen appropriately so that the horizontally averaged quanti-ties show minimal variability compared to an average value (see Sect. 6.3.1 and Fig. 6.2).

By increasing the size of the box, we allow energy to be deposited in a larger volume.

That will lead to an overall increase in the coronal temperature. The average stratification ofT and ρare depicted in Fig. 6.3. Almost all the runs can self-consistently form a hot corona (see Fig. 6.3a). Therefore, the size of the numerical box affects the average coronal temperature. The values for temperature shown in Fig. 6.3a are average values, and thus

peak temperature in the coronal part of our computational domain is much higher. The smallest box size, R25, does not manage to reach a high coronal temperature. That is to be expected considering the low surface magnetic flux of this specific run. On the other hand, the averaged densityρ(see Fig. 6.3b) shows only a slight increase mainly at the low corona, but there is no clear distinction between the runs as in the temperature profile.

To understand the reasons behind this particular behavior in temperature and density, we have to go back to the RTV scaling laws. As we mentioned in Chap. 2 the RTV scaling laws are an outcome of a 1D hydrostatic coronal loop model. They provide a simple way to relate the coronal temperature and density with the pressure and loop length. Alterna-tively, it can be written in such a way that to relate the temperatureT and densityρwith the heating rateHand the loop lengthL(see e.g. Zhuleku et al. 2020),

T ∝ H^2/7L^4/7

n∝ H^4/7L^1/7. (6.1)

Since the strength of the magnetic field is kept constant, so is the amount of energy per unit volume in the corona. In other words, the volumetric heating rateHdoes not depend on the size of the numerical box, and it will be the same for each run. Consequently, the difference in temperature and density is solely because of the change in the loop length L. As a result, if the loop length doubles in size, the temperature increases by a factor of 2^4/7 ' 1.5. On the other hand, the difference in density will only be 2^1/7 ' 1.1. That explains why the change in density is insignificant for each run.