  Eccentric exoplanets offer an opportunity to study the response of an
atmosphere to changing thermal forcing and the robustness of the super-rotating
equatorial jet seen on tidally locked hot Jupiters. However, the atmospheric
dynamics on eccentric planets strongly depend on the planetary rotation period,
which is difficult to constrain observationally. The ringing phenomenon,
whereby the observed emission increases and decreases after the periastron
passage as the flash-heated hemisphere rotates into and out of view, can
provide a tight constraint on rotation. We studied five highly eccentric
transiting exoplanets HAT-P-2 b, HD 80606 b, TOI-3362 b, TOI-4127 b and HD
17156 b to find which displays strong ringing signals that are sufficiently
strong for the James Webb Space Telescope (JWST) to detect. We implemented the
treatment of eccentricity and non-synchronous rotation in the non-grey climate
model expeRT/MITgcm and generated synthetic light curves. We find four
detectable ringing peaks on HD 80606 b and some undetectable ringing on
TOI-4127 b and HD 17156 b. The lack of clouds, photo-chemistry and obliquity in
our models may have led us to overestimate the amplitude of the ringing
however. The strength of the ringing signal is mostly determined by the
eccentricity, planetary rotation period, planet-to-star radius ratio and
apparent magnitude of the system. We searched for more exoplanets that could
show ringing but found no candidates as promising as HD 80606 b. We recommend
prioritising HD 80606 b as a target for ringing with JWST. A baseline of five
days after the periastron passage would capture three ringing peaks, which is
sufficient to tightly constrain the planetary rotation period. An extension to
seven days would add a fourth peak, which would allow us to verify the rotation
period.
