  We consider the origin of the UV bright phase of metal-rich helium-burning
stars, the slow blue phase (SBP), that was predicted by various earlier works.
Based on improved physics including OPAL opacities, we confirm the existence of
the SBP. In addition to our grid of evolutionary tracks, we provide an
analytical understanding of the main characteristics of the SBP phenomenon.
  The SBP is slow because it is a slow evolving helium-shell-burning phase
which is analogous to the early AGB phase. The SBP of a more metal-rich star is
slower than a metal-poor counterpart if their Teff's are the same because a
more metal-rich helium-burning star has a smaller mass than a metal-poor one
and because lifetime increases as mass decreases. Metal-rich helium-burning
stars easily become hot because the luminosity from the hydrogen-burning shell
is extremely sensitive to the mean molecular weight whereas the luminosity from
the helium-burning core is not. We suggest that the SBP phenomenon is a major
cause of the UV upturn phenomenon in giant elliptical galaxies as will be shown
in subsequent papers. The new HB tracks can be retrieved from S.Y.'s web site
http://shemesh.gsfc.nasa.gov/astronomy.html.
