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 , among other things , 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 , not treated here . 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 { \mathchoice { \mbox { } } { \mbox { } } { \mbox { } } { \mbox { } } } _ { 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 . Instead , the gravitomagnetic spin octupole precessions are too small to be detectable .