The star formation timescale of elliptical galaxies -- Fitting [Mg/Fe]
  and total metallicity simultaneously
  The alpha element to iron peak element ratio, for example [Mg/Fe], is a
commonly applied indicator of the galaxy star formation timescale (SFT) since
the two groups of elements are mainly produced by different types of supernovae
that explode over different timescales. However, it is insufficient to consider
only [Mg/Fe] when estimating the SFT. The [Mg/Fe] yield of a stellar population
depends on its metallicity. Therefore, it is possible for galaxies with
different SFTs and at the same time different total metallicity to have the
same [Mg/Fe]. This effect has not been properly taken into consideration in
previous studies. In this study, we assume the galaxy-wide stellar initial mass
function (gwIMF) to be canonical and invariant. We demonstrate that our
computation code reproduces the SFT estimations of previous studies where only
the [Mg/Fe] observational constraint is applied. We then demonstrate that once
both metallicity and [Mg/Fe] observations are considered, a more severe
"downsizing relation" is required. This means that either low-mass ellipticals
have longer SFTs (> 4 Gyr for galaxies with mass below $10^{10}$ M$_\odot$) or
massive ellipticals have shorter SFTs ($\approx 200$ Myr for galaxies more
massive than $10^{11}$ M$_\odot$) than previously thought. This modification
increases the difficulty in reconciling such SFTs with other observational
constraints. We show that applying different stellar yield modifications does
not relieve this formation timescale problem. The quite unrealistically short
SFT required by [Mg/Fe] and total metallicity would be prolonged if a variable
stellar gwIMF were assumed. Since a systematically varying gwIMF has been
suggested by various observations this could present a natural solution to this
problem.
