  HERO (Highly Eccentric Relativity Orbiter) is a space-based mission concept
aimed to perform several tests of post-Newtonian gravity around the Earth with
a preferably drag-free spacecraft moving along a highly elliptical path fixed
in its plane undergoing a relatively fast secular precession. We considered two
possible scenarios: a fast, 4-hr orbit with high perigee height of
$1,047\,\mathrm{km}$, and a slow, 21-hr path with a low perigee height of
$642\,\mathrm{km}$. HERO may detect, for the first time, the post-Newtonian
orbital effects induced by the mass quadrupole moment $J_2$ of the Earth which
affects the semimajor axis $a$ via a secular trend of $\simeq
4-12\,\mathrm{cm\,yr}^{-1}$, depending on the orbital configuration. Recently,
the secular decay of the semimajor axis of the passive satellite LARES was
measured with an error as little as $0.7\,\mathrm{cm\,yr}^{-1}$. Also the
post-Newtonian spin dipole (Lense-Thirring) and mass monopole (Schwarzschild)
effects could be tested to a high accuracy depending on the level of
compensation of the non-gravitational perturbations. Moreover, the large
eccentricity of the orbit would allow to constrain several long-range modified
models of gravity and to accurately measure the gravitational red-shift as
well. Each of the six Keplerian orbital elements could be individually
monitored to extract the $GJ_2/c^2$ signature, or they could be suitably
combined in order to disentangle the post-Newtonian effect(s) of interest from
the competing mismodeled Newtonian secular precessions induced by the zonal
harmonic multipoles $J_\ell$ of the geopotential. In the latter case, the
systematic uncertainty due to the current formal errors $\sigma_{J_\ell}$ of a
recent global Earth's gravity field model are better than $1\%$ for all the
post-Newtonian effects considered, with a peak of $\simeq 10^{-7}$ for the
Schwarzschild-like shifts. [Abridged]
