observed stellar rotation periods versus for a number of stars (Noyes et al., 1984 [34]) versus their age inferred according to (102).

The circled symbol is the Sun and the solid curve is described by (100), the expression for the time dependence of solar rotation. We observe quite some scatter round the curve which is due to the spread in initial angular momentum of the stars and uncertainties in their ages. Equation (100) appears to describe the trend for deceleration with time remarkably well. The two other curves are for different exponents in the denominator of (100). The dashed curve with slower rotation in the past (the upper curve for young stars) corresponds to a Skumanich scaling (Skumanich, 1972 [43]) with exponent 1/2, while the dash-dotted curve for faster rotation in the past (the lower curve for young stars) shows the behavior for an exponent of unity. Obviously, the solid curve does the best job of the three curves.

We observe that the exponent 2/3 in (100) is the same as the time exponent in (102). This is not by construction and probably not a coincidence either. It implies that the relationship between chromospheric activity and rotation rate is linear. Because dynamo action which is responsible for generating the solar (and stellar) magnetic field is intimately connected to rotation (differential rotation, to be precise) we do expect a causal relationship. Given the complexity of dynamo physics, the simplicity of this empirical relation is indeed quite remarkable.

Fig. 34. Sunspot number versus chromospheric activity index S for present-day solar activity (left panel) and an extrapolation to possible activity levels in the past (right panel). Dashed lines show the 95% confidence levels for the expectation value for sunspot number based on the fit in the left panel.

Fig. 34. Sunspot number versus chromospheric activity index S for present-day solar activity (left panel) and an extrapolation to possible activity levels in the past (right panel). Dashed lines show the 95% confidence levels for the expectation value for sunspot number based on the fit in the left panel.

We can now proceed to estimate solar activity in the past. We relate sunspot number to chromospheric activity index S in Fig. 34. The left-hand panel shows yearly averaged sunspot numbers (derived from smoothed monthly averages) plotted versus solar chromospheric activity index S. The two quantities are well correlated, the solid line shows the best linear fit and the two dashed lines the 95% confidence level limits in the expectation value for sun spot number for a given value of S. Another solid curve connects subsequent data points ranging from 1974 to 1992. The data are the same as those used in Fig. 31. The right-hand panel shows an extrapolation to large values for S of the relation found in the left-hand panel. From Fig. 33 we can read off that the Sun rotated about 2.5 times faster when it was about 1 Gy old. Because of the linear relation between activity and rotation we know that solar activity must have been about 2.5 times higher at that time. Thus a maximum value for S would lie around Smax « 0.5. This corresponds to an expected yearly sunspot number of about 1400. As attractive as it may appear, this inferred strong enhancement of solar activity in the past is probably misleading. The fit in Fig. 34 would require negative sunspot numbers for S < 0.16. This is certainly not true. Assuming that the peak for low values of S in Fig. 32 is indeed due to stars undergoing Maunder-type activity minima, we see that we need to have a small, non-negative number of sunspots for values of S ~ 0.14. To estimate solar activity in the past we now assume that Smin ~ 0.145 corresponds to the minimal possible activity, and that the difference S — Smin is time dependent. This gives us a maximum value for S, Smax ~ 0.14 + Smin ~ 0.285. This would imply a yearly average of about 500 sunspots when the Sun was 1 Gy old. If the minimum activity index were S ~ 0.16 this number would shrink to about 320. Simply scaling sunspot number by a factor of 2.5 for solar rotation would imply a sunspot number of about 380 for the early Sun. In the coming discussion we will assume a sunspot number of 400 when the Sun was 1 Gy old.

From Fig. 21 we infer a daily coronal mass ejection rate of about 6 CMEs per day at the Sun. From Fig. 22 we see that, for average activity, only about 6% of all CMEs result in sudden commencements which in turn indicate a CME-driven shock. Thus we could reckon with a maximum incidence rate of about 10 CME-driven interplanetary shocks per month 3.5 Gy ago. This is not that much more than the present-day CME incidence rate.


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