Kernel density estimation of the PDF of the time τ_c of the first secular collision from 1042 integrations of the forced inner system over 100 Gyr (blue line in the lower panel). The estimate of the CDF is shown as a blue line in the upper panel. The blue regions represent the pointwise confidence intervals at the 98% level from bootstrap. In red we show the PDF estimate over 5 Gyr from the 10 560 orbital solutions of Sect. 7. The dashed black lines stand for the analytical PDF (57) with T₁ = 27.6 Gyr and α = 0.9 and the corresponding CDF.