| Summary: | 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-h orbit with high perigee height of <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1047</mn> <mspace width="0.166667em"></mspace> <mi>km</mi> </mrow> </semantics> </math> </inline-formula> and a slow, 21-h path with a low perigee height of <inline-formula> <math display="inline"> <semantics> <mrow> <mn>642</mn> <mspace width="0.166667em"></mspace> <mi>km</mi> </mrow> </semantics> </math> </inline-formula>. HERO may detect, for the first time, the post-Newtonian orbital effects induced by the mass quadrupole moment <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> of the Earth which, among other things, affects the semimajor axis <i>a</i> via a secular trend of ≃4−12 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>cm</mi> <mspace width="0.166667em"></mspace> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>.</mo> <mn>7</mn> <mspace width="0.166667em"></mspace> <mi>cm</mi> <mspace width="0.166667em"></mspace> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula>. 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 one to constrain several long-range modified models of gravity and accurately measure the gravitational red-shift as well. Each of the six Keplerian orbital elements could be individually monitored to extract the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo>/</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mi>ℓ</mi> </msub> </semantics> </math> </inline-formula> of the geopotential. In the latter case, the systematic uncertainty due to the current formal errors <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">σ</mi> <msub> <mi>J</mi> <mi>ℓ</mi> </msub> </msub> </semantics> </math> </inline-formula> of a recent global Earth’s gravity field model are better than <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>%</mo> </mrow> </semantics> </math> </inline-formula> for all the post-Newtonian effects considered, with a peak of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>≃</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula> for the Schwarzschild-like shifts. Instead, the gravitomagnetic spin octupole precessions are too small to be detectable.
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