We have analyzed high-resolution X-ray spectra of a sample of 22 active stars
observed with the High Energy Transmission Grating Spectrometer on {\em
Chandra} in order to investigate their coronal plasma density. Densities where
investigated using the lines of the He-like ions O VII, Mg XI, and Si XIII.
While Si XIII lines in all stars of the sample are compatible with the
low-density limit, Mg XI lines betray the presence of high plasma densities ($>
10^{12}$ cm$^{-3}$) for most of the sources with higher X-ray luminosity ($>
10^{30}$ erg/s); stars with higher $L_X$ and $L_X/L_{bol}$ tend to have higher
densities at high temperatures. Ratios of O VII lines yield much lower
densities of a few $10^{10}$ cm$^{-3}$, indicating that the ``hot'' and
``cool'' plasma resides in physically different structures. Our findings imply
remarkably compact coronal structures, especially for the hotter plasma
emitting the Mg XI lines characterized by coronal surface filling factor,
$f_{MgXI}$, ranging from $10^{-4}$ to $10^{-1}$, while we find $f_{OVII}$
values from a few $10^{-3}$ up to $\sim 1$ for the cooler plasma emitting the O
VII lines. We find that $f_{OVII}$ approaches unity at the same stellar surface
X-ray flux level as solar active regions, suggesting that these stars become
completely covered by active regions. At the same surface flux level,
$f_{MgXI}$ is seen to increase more sharply with increasing surface flux. These
results appear to support earlier suggestions that hot $10^7$ K plasma in
active coronae arises from flaring activity, and that this flaring activity
increases markedly once the stellar surface becomes covered with active
regions.Comment: 53 pages, 19 figures, accepted for publication in Astrophysical
Journal. A version of the paper with higher quality figures is available from
http://www.astropa.unipa.it/Library/preprint.htm