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TSI Modellings

Over the last two solar cycles, three TSI composites are available (RMIB, ACRIM and PMOD) showing a common 11 year cyclic TSI variation, but disagreeing on possible long term variations of the solar minimum TSI over the solar cycles. Both ACRIM and PMOD composites indicate a decreasing TSI between the minima in 1996 and 2009. This result is not confirmed by the RMIB composite for which no significant changes are detected between the minima. It is worth notice that such a stability of the minima was already predicted two years ago through an empirical model.


Previous estimations of the occurrence of the end of solar cycle 23 turned out to be inaccurate. (Mekaoui & Dewitte 2008) we developed an empirical TSI model based on proxies of solar activity. According to this approach, it was concluded that both cycle 23 TSI minima will have the same level. We use the parameters of the RMIB model (Mekaoui & Dewitte 2008) with the newly updated proxy indices. We then compare the results with the RMIB composite extended to 2009.

2. RMIB composite

The RMIB composite is built using the data from five instruments covering cycle 23 . Figure 1 (top) shows the updated results of three of them: TIM, PMO6 and DIARAD. The RMIB composite (Figure 1 (bottom)) is finally obtained by averaging the partial VIRGO composite and the AET (ACRIM II- ERBS-TIM) composite. These two partial composites are represented in Figure 2(top).

3.Index updates

The key ingredients of the modelling approach are composed of short and long-term indices of solar variability. Figures 3 and 4 show their updated values. Of particular interest, the Mg ii index is based on the ratio of two narrow spectral bands. It was therefore thought that this index is relatively insensitive to ageing degradations. Figure 4 displays two different interpretations of the evolution of the Mg ii index during the declining phase of cycle 23.

4.Models and residuals
4.1.Optical model update

We apply Equation 2 with the parameters of the RMIB composite in Table 1:

                  T SIi (t) = a + b Σred (t) + c M gII(t)      (2)


                                     Table 1 : The three regression models


                T SI(t) = 1366.85 + 1.725 × 10−3 Σred (t) + 89.40 M gII(t)

Since two versions of the Mg ii index are available, two TSI models are obtained (i.e. one using the updated red curve of Figure 4 and one using the blue curve). Results for cycle 23 are displayed in Figure 5 (top panel). The RMIB composite (black curve) is within the modelling uncertainties up to 2005 (Mekaoui & Dewitte 2008).

Figure 1: Top: RMIB components update. An arbitrary offset is applied to ACRIM II, ERBS and TIM (AET) components for a better visualization. Bottom: the RMIB composite covering solar cycle 23.

Figure 2: Top: partial VIRGO and AET composites. Bottom: 121-day running mean
of the difference beteween VIRGO and AET partial composites.From 2005 onward, the measurements extend between the two models: the red curve is the updated model using the Mg ii component displayed in red in Figure 4. The turquoise curve is the model using the component in blue. Residuals (measurements minus model) are shown for each model (middle panel in Figure 5). The 2σ error is around 0.34 W.m−2 . We notice that both models do not lead a flat residuals during the ending phase of solar cycle 23.

Figure 3: Top: updated MPSI and MWSI indices from Mount Wilson Observatory (red
curve) and data used for computing the RMIB model (Mekaoui & Dewitte.2008) (black curve).
Bottom: updated Σred index from the San Fernando Observatory (Preminger, 2009) and
previous measurements (black curve).

Figure 4: The black curve displays the Mg ii core to wing ratio as function of time (Mekaoui & Dewitte 2008). These data were used to compute the RMIB model . The red curve represents the update of this index up to 2008. Unfortunately no data are available after 2008. The blue curve shows another interpretation of the long-term evolution the Mg ii core to wing ratio. Compared to the red curve, this version has a steeper decrease during the declining phase of cycle 23.

4.2 Magnetic model determination

Alternatively to the use of the Mg ii component. The magnetic model uses the MPSI
and the MWSI indices according to :
                                  T SI(t) = a + b M P SI(t) + c M W SI(t) (1)
Where a, b and c are the coefficients to be determined. Using this approach we find
for the RMIB composite: a = 1366.268, b = 0.786193 and c = -1.94507 with σ=0.192
W.m−2 and R2 =0.86. The curve in magenta (Figure 5, top panel) shows the 121-
running mean of the model described by Equation 1. The modelled values are
surprisingly close to the measurements displayed by the black curve. The residuals
are shown in the lower panel of Figure 5 (see the curve in magenta). A 39-day
moving average is also plotted. We conclude that this modelling approach is better
suited for long-term TSI modelling. Nevertheless, the residuals depict an offset of the
order of -0.2 W.m−2 during the 1996-1997 period. This difference can be explained
by the accelerated degradation of the PMO6V radiometer during the beginning of
the mission .

Figure 5: Top panel: measured and modelled TSI during cycle 23. A 121-day moving average is applied. The black curve is the updated RMIB composite (Mekaoui & Dewitte 2008). The red curve is the model update. This model uses the Mg ii component (red curve in Figure 4). Alternatively, the curve in turquoise is the model update using the other version of the Mg ii (blue curve in Figure 4). The curve in magenta is the model based on Equation 1. The residuals of each model is displayed in the middle and lower panel. Middle panel: measured minus modeled TSI value for the two model update according to interpretation of the evolution of the Mg ii component. Lower panel: residuals obtained from the MPSI-MWSI model (curve in magenta). All curves contain a 39-day moving average.

5. Conclusion

Despite the disagreement of the two available Mg ii components. The RMIB model
succeeded in reproducing the 11 year TSI modulation with an uncertainty of ± 0.34
W.m−2 ( 2σ error). The level of the minima in 1996 and 2009 are not significantly
different. However, it should be noticed that cycle 23 lasts now for 13 years which
is relatively long, but not unusual. Cycle 20 (1964-1977) had the same duration.