  The relationship between the compositions of giant planets and their host
stars is of fundamental interest in understanding planet formation. The solar
system giant planets are enhanced above solar composition in metals, both in
their visible atmospheres and bulk compositions. A key question is whether the
metal enrichment of giant exoplanets is correlated with that of their host
stars. Thorngren et al. (2016) showed that in cool (Teq < 1000 K) giant
exoplanets, the total heavy-element mass increases with total Mp and the heavy
element enrichment relative to the parent star decreases with total Mp. In
their work, the host star metallicity was derived from literature [Fe/H]
measurements. Here we conduct a more detailed and uniform study to determine
whether different host star metals (C, O, Mg, Si, Fe, and Ni) correlate with
the bulk metallicity of their planets, using correlation tests and Bayesian
linear fits. We present new host star abundances of 19 cool giant planet
systems, and combine these with existing host star data for a total of 22 cool
giant planet systems (24 planets). Surprisingly, we find no clear correlation
between stellar metallicity and planetary residual metallicity (the relative
amount of metal versus that expected from the planet mass alone), which is in
conflict with common predictions from formation models. We also find a
potential correlation between residual planet metals and stellar
volatile-to-refractory element ratios. These results provide intriguing new
relationships between giant planet and host star compositions for future
modeling studies of planet formation.
