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Appendix A: Magnetic flux and magnetic helicity evolution

The plots in Figs. A.1 and A.2 report the evolution of total, positive, and negative magnetic flux expressed in 10²² Mx (first row), helicity injection rate dH/dt in 10⁴² Mx² h^-1 (second row), accumulated helicity injection in 10⁴² Mx² (third row), ratio (fourth row), positive (H₊) and negative (H_-) accumulated helicity injection (fifth row) for NOAA 8948, 9114+9115 and 9393+9394, belonging to the Class I and for NOAA 10162, 10229 and 10365, belonging to Class II.

NOAA 8948 (Fig. A.1, leftt panel): the total, positive, and negative magnetic flux are slightly decreasing during the period analyzed. However, if we limit our attention to the time interval preceding the CME, we note a slight increase of both positive and negative magnetic fluxes. The helicity injection rate dH/dt (second row) is prevalently positive, with the exception of the last part of the analyzed period. The accumulated helicity injection (third row) is initially characterized by increasing positive values and in the last part of the analyzed period by a decreasing trend. The ratio (fourth row) shows an increasing trend till the time of the corresponding increase in H_- (see the graph in the fifth row), but not correlated with the CME occurrence. The graph in the fifth row shows that the main contribution to the helicity injection is due to the increase of H₊, while there is an increase of H_- only during the last part of the analyzed period. The CME had a velocity of 409 km s^-1, an acceleration of 2.5 m s^-2, and was associated to an M3.1 flare.

NOAA 9114+9115 (Fig. A.1, middle panel): the total magnetic flux is slightly increasing during the analyzed period and in particular just before the CME occurrence; the helicity injection rate (second row), which is always characterized by negative values, shows an evident increasing trend before the CME occurrence (till the value of  Mx²) and a phase characterized by lower values (till  Mx²) after it. The accumulated helicity injection (third row) shows a different trend before and after the CME occurrence. This behavior is also evident in the curve showing the ratio . The graph reported in the fifth row indicates that the contribution to is only due to H_-. Note that the CME had a velocity of 720 km s^-1, an acceleration of 2.8 m s^-2 and it was associated with a C2.3 flare and an erupting filament.

NOAA 9393+9394 (Fig. A.1, right panel): the total magnetic flux shows an increase during the analyzed period (in this case, the positive flux increases, whereas the negative flux decreases); the helicity injection rate (second row) is predominantly characterized by negative values. The curve showing the accumulated helicity injection indicates that there is a continuous increase of helicity, always characterized by negative values, also after the CME. Interestingly, the trend shows a slight variation a few hours before the CME. A similar trend is shown by the values (fourth row). The graph reported in the fifth row indicates that the main contribution to is due to H_-, and that H₊ starts to increase some hours before the CME occurrence. In this case, the CME, associated to an X1.7 flare, had a velocity of 942 km s^-1 and an acceleration of 3.5 m s^-2.

	Figure A.1: Same as in Fig. 3 for ARs 8948, 9114+9115 and 9393+9394, belonging to Class I (gradual CMEs).
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NOAA 10162 (Fig. A.2, left panel): the total magnetic flux is almost constant during the analyzed period (this is due to a smooth increase of the positive flux and to a contemporary decrease of the negative flux); the helicity injection rate shows a very spiky behavior, generally characterized by positive values. The accumulated helicity shows an almost monotonic increase, with a small plateau (reflecting the dH/dt value) at the CME occurrence. The ratio shows an increasing trend with a similar plateau in coincidence with the CME occurrence. H₊ has a trend similar to H and , while H_- remains constant (assuming values very close to zero). In this case, the CME had a velocity of 870 km s^-1, a deceleration of 23.4 m s^-2 and was associated with an erupting filament.

NOAA 10229 (Fig. A.2, middle panel): in this active region the total magnetic flux decreases during the analyzed period (the negative flux shows a stronger decrease than the positive flux), whereas the helicity injection rate generally shows negative values, with some abrupt changes to positive values. The accumulated helicity, and correspondingly the ratio are monotonically increasing. In this case H_- (see the graph in the fifth row) shows a continuous increase, while H₊ has only a slight increase just before the CME occurrence and again some hours later. The velocity and deceleration of the CME were 1092 km s^-1 and 36.2 m s^-2, respectively, and the associated flare was of M2.7 class.

NOAA 10365 (Fig. A.2, right panel): in the first row we can see that the total magnetic flux shows an evident increase during the analyzed period. However, a more careful inspection of this graph indicates that there is a clear increase before the first CME and almost no increase before the second CME. The helicity injection rate dH/dt is initially characterized by positive values and does not show significant variation after the first CME, while it changes, assuming also negative values, after the second CME. This behavior is clearly shown in the accumulated helicity (third row) and even more evidently in the trend, especially as far as the second event is concerned. Similarly, H₊ (see the graph in the fifth row), which increases during the entire analyzed period, shows a small plateau after the second CME, and later starts to increase again, while H_-, which was almost zero until this event, starts to increase. The first CME had a velocity of 509 km s^-1, a deceleration of 14.6 m s^-2, and was associated with an M1.6 flare, while the second CME presented a velocity of 964 km s^-1, a deceleration of 9.6 m s^-2, and was associated with an X1.3 flare.

	Figure A.2: Same as in Fig. 3 for ARs 10162, 10229 and 10365, belonging to Class II (impulsive CMEs).
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