  The Earth's magnetosphere extension is controlled by the solar activity level
via solar wind properties. Understanding such a relation in the Solar System is
important for predicting also the condition of exoplanetary magnetospheres near
Sun-like stars. We use measurements of a chromospheric proxy, the Ca II K
index, and solar wind OMNI parameters to connect the solar activity variations,
on the decennial time scales, to the solar wind properties. The data span over
the time interval 1965-2021, which almost entirely covers the last 5 solar
cycles. Using both cross-correlation and mutual information analysis, a
3.2-year lag of the solar wind speed with respect to the Ca II K index is
found. Analogously, a 3.6-year lag is found once considering the dynamic
pressure. A correlation between the solar wind dynamic pressure and the solar
UV emission is found and used to derive the Earth's magnetopause standoff
distance. Moreover, the advantage of using a chromospheric proxy, such as the
Ca II K index, opens the possibility to extend the relation found for the Sun
to Sun-like stars, by linking stellar variability to stellar wind properties.
The model is applied to a sample of Sun-like stars as a case study, where we
assume the presence of an Earth-like exoplanet at 1 AU. Finally, we compare our
results with previous estimates of the magnetosphere extension for the same set
of Sun-like stars.
