The Mass Profile and Shape of Bars in the Spitzer Survey of Stellar
  Structure in Galaxies (S4G): Search for an Age Indicator for Bars
  We have measured the radial light profiles and global shapes of bars using
two-dimensional 3.6 $\mu m $ image decompositions for 144 face-on barred
galaxies from the Spitzer Survey of Stellar Structure in Galaxies (S4G). The
bar surface brightness profile is correlated with the stellar mass and
bulge-to-total (B/T) ratio of their host galaxies. Bars in massive and
bulge-dominated galaxies (B/T$>$0.2) show a flat profile, while bars in less
massive, disk-dominated galaxies (B/T$\sim$0) show an exponential, disk-like
profile with a wider spread in the radial profile than in the bulge-dominated
galaxies. The global two-dimensional shapes of bars, however, are
rectangular/boxy, independent of the bulge or disk properties. We speculate
that because bars are formed out of disk, bars initially have an exponential
(disk-like) profile which evolves over time, trapping more stars into the boxy
bar orbits. This leads bars to become stronger and have flatter profiles. The
narrow spread of bar radial profiles in more massive disks suggests that these
bars formed earlier (z$>$1), while the disk-like profiles and a larger spread
in the radial profile in less massive systems imply a later and more gradual
evolution, consistent with the cosmological evolution of bars inferred from
observational studies. Therefore, we expect that the flatness of the bar
profile can be used as a dynamical age indicator of the bar to measure the time
elapsed since the bar formation. We argue that cosmic gas accretion is required
to explain our results on bar profile and the presence of gas within the bar
region.
