  We use the data of the Chandra Orion Ultradeep Project (COUP) to study the
nearly 600 X-ray sources that can be reliably identified with optically well
characterized T Tauri stars (TTS) in the Orion Nebula Cluster. We detect X-ray
emission from more than 97% of the optically visible late-type (spectral types
F to M) cluster stars. This proofs that there is no ``X-ray quiet'' population
of late-type stars with suppressed magnetic activity. All TTS with known
rotation periods lie in the saturated or super-saturated regime of the relation
between activity and Rossby numbers seen for main-sequence (MS) stars, but the
TTS show a much larger scatter in X-ray activity than seen for the MS stars.
Strong near-linear relations between X-ray luminosities, bolometric
luminosities and mass are present. We also find that the fractional X-ray
luminosity rises slowly with mass over the 0.1 - 2 M_sun range. The plasma
temperatures determined from the X-ray spectra of the TTS are much hotter than
in MS stars, but seem to follow a general solar-stellar correlation between
plasma temperature and activity level. The large scatter about the relations
between X-ray activity and stellar parameters seems to be related to the
influence of accretion on the X-ray emission. While the X-ray activity of the
non-accreting TTS is consistent with that of rapidly rotating MS stars, the
accreting stars are less X-ray active (by a factor of ~2-3 on average) and
produce much less well defined correlations than the non-accretors. We discuss
possible reasons for the suppression of X-ray emission by accretion and the
implications of our findings on long-standing questions related to the origin
of the X-ray emission from young stars.
