A HERO for General Relativity

Abstract

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 $1047\mathrm{km}$ and a slow, 21-h 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 ≃4–12 $\mathrm{cm}{\mathrm{yr}}^{-1}$, 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 $0.7\mathrm{cm}{\mathrm{yr}}^{-1}$. 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 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 $G{J}_2/c^2$ 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 ${\mathsf{\sigma }}_{J_{\ell }}$ 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. Instead, the gravitomagnetic spin octupole precessions are too small to be detectable.

1. Introduction

The (slow) motion of a test particle moving in spacetime (weakly) deformed by the mass-energy content of an isolated, axially symmetric rotating body of mass M, angular momentum $\mathit{S}$, polar and equatorial radii $R_{\mathrm{p}},R_{\mathrm{e}}$, ellipticity $\epsilon =\sqrt{1-R_{\mathrm{p}}^2/R_{\mathrm{e}}^2}$, dimensionless quadrupole mass moment $J_2$ exhibits several post-Newtonian (pN) features. Some of them have never been put to the test so far because of their smallness; they are the gravitoelectric and gravitomagnetic effects associated with the asphericity of the central body induced by its mass quadrupole and spin octupole moments, respectively [1,2,3,4,5,6].

Instead, the pN orbital effects which have been extensively tested so far in several terrestrial and astronomical scenarios are the gravitoelectric and gravitomagnetic precessions due to the mass monopole and spin dipole moments, respectively. The former is responsible for the time-honored, previously anomalous perihelion precession of Mercury [7], whose explanation by Reference [8] was the first empirical success of his newly born theory of gravitation. It was later repeatedly measured with radar measurements of Mercury itself [9,10], of other inner planets [11,12], and of the asteroid Icarus [13,14] as well. Binary pulsars [15] and Earth’s artificial satellites [16,17] have also been used. The latter is the so-called Lense-Thirring effect [18,19,20,21,22], which is currently under scrutiny in the Earth’s surrounding with the geodetic satellites of the LAGEOS family; see, for example, Renzetti [23], and Lucchesi et al. [24], and references therein. Another gravitomagnetic effect—the Pugh-Schiff rates of change of orbiting gyroscopes [25,26]—was successfully tested in the field of the Earth with the dedicated Gravity Probe B (GP-B) spaceborne mission a few years ago [27,28] to a $19\%$ accuracy level, despite the originally expected error level of the order of $1\%$ [29].

By assuming the validity of general relativity, its mass quadrupole and spin octupole accelerations are, to the first post-Newtonian (pN) order,

$$\begin{array}{cc}\hfill {\mathit{A}}^{\mathrm{pN}M{J}_2}& =\frac{\mu J_2{R}_{\mathrm{e}}^2}{c^2{r}^4}\left\{\frac{3}{2}\left[\left(5{\xi }^2-1\right)\widehat{\mathit{r}}-2\xi \widehat{\mathit{S}}\right]\left({\mathrm{v}}^2-\frac{4\mu }{r}\right)-6\left[\left(5{\xi }^2-1\right){\mathrm{v}}_r-2\xi {\mathrm{v}}_S\right]\mathbf{v}-\frac{2\mu }{r}\left(3{\xi }^2-1\right)\widehat{\mathit{r}}\right\},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\mathit{A}}^{\mathrm{pN}S\epsilon }& =\frac{3GS{R}_{\mathrm{e}}^2{\epsilon }^2}{7c^2{r}^5}\mathbf{v}\times \left\{5\xi \left[7{\xi }^2-3\right]\widehat{\mathit{r}}+3\left[1-5{\xi }^2\right]\widehat{\mathit{S}}\right\},\hfill \end{array}$$

(2)

where G is the Newtonian constant of gravitation, $\mu \doteq GM$ is the gravitational parameter of the primary, c is the speed of light in vacuum, $\widehat{\mathit{S}}$ is the unit vector of the rotational axis,

$$\xi \doteq \widehat{\mathit{S}}·\widehat{\mathit{r}}$$

(3)

is the cosine of the angle between the directions of the body’s angular momentum and the orbiter’s position vector $\mathit{r}$, and

$$\begin{array}{cc}\hfill {\mathrm{v}}_r& \doteq \mathbf{v}·\widehat{\mathit{r}},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathrm{v}}_S& \doteq \mathbf{v}·\widehat{\mathit{S}}\hfill \end{array}$$

(5)

are the components of the particle’s velocity $\mathbf{v}$ along the radial direction and the primary’s spin respectively. The averaged rates of change of the semimajor axis a, the eccentricity e, the inclination I, the longitude of the ascending node $\Omega$ and the argument of pericenter $\omega$ induced by Equations (1) and (2) were analytically calculated for a general orientation of $\widehat{\mathit{S}}$ in space by Iorio [30,31]; previous derivations of the gravitoelectric mass quadrupole effects in the particular case of an equatorial coordinate system with its reference z axis aligned with $\widehat{\mathit{S}}$ can be found in Brumberg [1], Soffel [3], Will [4], Soffel et al. [6].

In this paper, we will preliminarily explore the perspectives of measuring, for the first time, some consequences of Equations (1) and (2) by suitably designing a dedicated drag-free satellite-based mission around the Earth encompassing a highly eccentric geocentric orbit exploiting the frozen perigee configuration; we provisionally name it as “HERO” (Highly Eccentric Relativity Orbiter). For some embryonal thoughts about the possibility of using an Earth’s spacecraft to measure the pN gravitoelectric effects proportional to $G{J}_2/c^2$, see Iorio [30,32]; for deeper investigations concerning a possible probe around Jupiter to measure them and the pN gravitomagnetic signature proportional to $GS{\epsilon }^2/c^2$, see Iorio [31,32]. About the propagation of the electromagnetic waves in the deformed spacetime of an oblate body and the perspectives of measuring the resulting deflection due to Jupiter with astrometric techniques, see, e.g., Abbas, Bucciarelli & Lattanzi [33], Crosta & Mignard [34], Kopeikin & Makarov [35], Le Poncin-Lafitte & Teyssandier [36], and references therein. Other more or less similar mission concepts can be found, for example, in Angélil et al. [37], Schärer et al. [38,39]. We will show that the size of the secular rate of a predicted by Equation (1) falls within the recently reached experimental accuracy in measuring phenomenologically such a kind of an effect with the existing passive LARES satellite [24]. Be that as it may, we will show that, as a by-product, other general relativistic features of motion could also be measured with high accuracy, at least as far as the systematic error due to the current formal level of mismodeling in the competing classical precessions due to the zonal harmonic coefficients $J_{\ell },\ell =2,3,4,\dots$ of the multipolar expansion of the Earth’s gravity potential is concerned. To this aim, it is crucial to assess the level of possible cancellation of the non-gravitational perturbations by the drag-free technology. Its evaluation is outside the scope of the present paper. However, we will look in detail at the atmospheric drag, which is one of the major disturbing non-conservative accelerations inducing relevant competing signatures, especially on a. For the sake of simplicity, we will assume a spherical shape for a passive spacecraft.

The high eccentricity1 of the suggested orbit of HERO would allow also for accurate tests of the gravitational redshift provided that the spacecraft is endowed with accurate atomic clocks; see Table A2 and Table A6 for the expected sizes of it. For recent tests of such an effect performed with the H-maser clocks carried onboard the satellites GSAT0201 (5-Doresa) and GSAT0202 (6-Milena) of the Galileo constellation by exploiting their fortuitous rather high eccentricity due to their erroneous orbital injection, see Delva et al. [40], Herrmann et al. [41]. It is interesting to recall that the formerly proposed Space-Time Explorer and QUantum Equivalence Space Test (STE-QUEST) space mission [42] was pre-selected by the European Space Agency (ESA) for the Cosmic Vision M3 launch opportunity; although it was not finally selected due to budgetary and technological reasons, its science case was highly rated. HERO may somehow inherit (part of) its legacy.

Finally, several long-range modified models of gravity [43] imply spherically symmetric modifications of the Newtonian inverse-square law which induce net secular precessions of the pericenter and the mean anomaly at epoch. They would represent further valuable goals for HERO.

The paper is organized as follows. Section 2 is the main body of the paper; it contains a discussion of two possible orbital configurations along with the magnitude of the various pN effects and the size of the corresponding systematic errors due to the current level of mismodeling in the multipolar zonal coefficients of the geopotential. It also deals with the linear combination approach which could be implemented in order to reduce the bias due to the latter ones. Section 3 contains a summary of our findings and offer our conclusions. Appendix A contains the tables and the figures. Appendix B displays the analytically calculated orbital precessions due to the zonal harmonics of the geopotential up to degree $\ell =8$. Appendix C deals in detail with the impact of the atmospheric drag on all the orbital elements of a spherical, passive geodetic satellite in a highly elliptical orbit.

2. Two Different Orbital Configurations for HERO

In Table A2 and Table A6, two different orbital configurations are proposed. They imply highly eccentric orbits, characterized by values of the eccentricity as large as $e=0.45$ and $e=0.82$, respectively, and the critical inclination $I_{\mathrm{crit}}=arcsin\left(2/\sqrt{5}\right)$ which allows one to keep the argument of perigee $\omega$ essentially fixed over any reasonable time span for an actual data analysis and the longitude of the ascending node $\Omega$ circulating at a relatively high pace. Their orbital periods are $P_{\mathrm{b}}=4.3\mathrm{h}$ and $P_{\mathrm{b}}=21.3\mathrm{h}$, with perigee heights of $h_{\min }=1047\mathrm{km}$ and $h_{\min }=642\mathrm{km}$, respectively.

One of the most interesting relativistic features of motion is, perhaps, the relatively large value of the expected semimajor axis increase $⟨\dot{a}⟩$ induced by the pN gravitoelectric quadrupolar acceleration of Equation (1); let us recall that it is [1,6,32]

$$⟨\frac{\mathrm{d}a}{\mathrm{d}t}⟩=\frac{9a{n}_{\mathrm{b}}^3{R}_{\mathrm{e}}^2{J}_2{e}^2\left(6+e^2\right){sin}^{2}Isin2\omega }{8c^2{\left(1-e^2\right)}^4},$$

(6)

where $n_{\mathrm{b}}=\sqrt{\mu /a^3}=2/P_{\mathrm{b}}$ is the Keplerian mean motion. The need of a highly eccentric orbit is apparent from Equation (6). The frozen perigee would make the resulting integrated pN shift a secular trend. According to Table A3 and Table A7, its predicted rates for the orbital geometries considered in Table A2 and Table A6 are $⟨\dot{a}⟩=3.8\mathrm{cm}{\mathrm{yr}}^{-1}$ and $⟨\dot{a}⟩=11.6\mathrm{cm}{\mathrm{yr}}^{-1}$, respectively. Suffice it to say that for the existing passive satellites of the LAGEOS family, whose orbits are essentially circular, secular decay rates have been measured over the last decades with an experimental accuracy of ${}_{⟨\dot{a}⟩}\simeq 3\mathrm{cm}{\mathrm{yr}}^{-1}$ for LAGEOS [44,45,46], and ${}_{⟨\dot{a}⟩}=0.7\mathrm{cm}{\mathrm{yr}}^{-1}$ for LARES [24]. It is arguable that an active mechanism of compensation of the non-gravitational accelerations would allow the increase of such accuracies, allowing, perhaps, to measure the pN quadrupolar effect at a $\simeq 1-10\%$ level, depending on the orbital configuration adopted. As far as possible competing effects of gravitational origin are concerned, neither the static and time-dependent parts of the geopotential nor 3rd-body lunisolar attractions induce nonvanishing averaged perturbations on a. Thus, the reduction of the non-conservative accelerations is of the utmost importance. Among them, a prominent role is played by the atmospheric drag, treated in detail in Appendix C, because it causes a net long-term averaged decay rate of a. By modeling the spacecraft as a LARES-like cannonball geodetic satellite for the sake of simplicity, it can be shown that, in the case of the orbital configuration of Table A2, the average acceleration due to the neutral drag only amounts to ${⟨A⟩}_{\mathrm{drag}}=(2.3-0.8)\times {10}^{-11}\mathrm{m}{\mathrm{s}}^{-2}$ over one orbital period, so that the resulting effect on a would be as large as ${⟨\dot{a}⟩}_{\mathrm{drag}}=-(5.1-2.0)\mathrm{m}{\mathrm{yr}}^{-1}$; see Table A5, and Figure A1 and Figure A2. For the more eccentric orbital configuration of Table A6, we have ${⟨A⟩}_{\mathrm{drag}}=(1.22-7.3)\times {10}^{-11}\mathrm{m}{\mathrm{s}}^{-2}$ and ${⟨\dot{a}⟩}_{\mathrm{drag}}=-(27.6-164.6)\mathrm{m}{\mathrm{yr}}^{-1}$, as shown by Table A9, and Figure A3 and Figure A4. An inspection of Figure A1 and Figure A3 reveals that, as expected, most of the disturbing effect occurs around the perigee passage, i.e., for $f\simeq 0$. This may help in suitably calibrating the counteracting action of the drag-free mechanism.

Table A3 and Table A6 show that all the other Keplerian orbital elements also exhibit non-zero secular rates of change due to Equation (1). This is a potentially important feature since they could be linearly combined, as suggested by2 Shapiro [9] in a different context, in order to decouple the pN effect(s) of interest from the disturbing mismodeled Newtonian secular precessions induced by the Earth’s zonal multipoles. Indeed, contrary to a, the other Keplerian orbital elements do exhibit long-term averaged precessions due to the classical zonal harmonics $J_{\ell },\ell =2,3,4,\dots$ of the geopotential; they are analytically calculated in Appendix B up to degree $\ell =8$. Depending on the specific orbital geometry, they can be both secular and harmonic, or entirely3 secular. To this aim, we complete the set of the pN orbital effects by analytically calculating the averaged rate of the mean anomaly at epoch due to the pN accelerations considered. It turns out that Equations (1) and (2) induce

$$\begin{array}{cc}\hfill ⟨\dot{\eta }⟩& =\frac{\mu n_{\mathrm{b}}{R}^2{J}_2\left[-\left(80+73e^2\right)\left(1+3cos2I\right)-84\left(1+2e^2\right){sin}^{2}Icos2\omega \right]}{32c^2{a}^3{\left(1-e^2\right)}^{5/2}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill ⟨\dot{\eta }⟩& =\frac{9GS{R}^2{\epsilon }^2\left[5cos3I+cosI\left(3+10{sin}^{2}Icos2\omega \right)\right]}{56a^5{c}^2{\left(1-e^2\right)}^2},\hfill \end{array}$$

(8)

respectively, while the gravitoelectric mass monopole acceleration yields

$$⟨\dot{\eta }⟩=\frac{\mu n_{\mathrm{b}}\left[-15+6\sqrt{1-e^2}+\left(9-7\sqrt{1-e^2}\right)\zeta \right]}{c^{2}a\sqrt{1-e^2}},$$

(9)

where $\zeta \doteq Mm/{\left(M+m\right)}^2$, and m is the satellite’s mass. Instead, it turns out that there is no net Lense-Thirring effect on $\eta$. To the benefit of the reader, we review the linear combination approach, which is a generalization of that proposed explicitly for the first time by Ciufolini et al. [47] to test the pN Lense-Thirring effect in the gravitomagnetic field of the Earth with the artificial satellites of the LAGEOS family. It should be noted that, actually, it is quite general, being not necessarily limited just to the pN spin dipole case. By looking at N orbital elements4 ${\kappa }^{\left(i\right)}$ experiencing classical long-term precessions due to the zonals of the geopotential, the following N linear combinations can be written down

$${\mathsf{\mu }}_{\mathrm{pN}}{⟨\dot{\kappa }⟩}_{\mathrm{pN}}^{\left(i\right)}+\sum _{s=2}^N\left(\frac{\partial {⟨\dot{\kappa }⟩}_{J_s}^{\left(i\right)}}{\partial J_s}\right)\delta J_s,i=1,2,\dots N.$$

(10)

They involve the pN averaged precessions ${⟨\dot{\kappa }⟩}_{\mathrm{pN}}^{\left(i\right)}$ as predicted by general relativity and scaled by a multiplicative parameter ${\mathsf{\mu }}_{\mathrm{pN}}$, and the errors in the computed secular node precessions due to the uncertainties in the first $N-1$ zonals $J_s,s=2,3,\dots N$, assumed as mismodeled through $\delta J_s,s=2,3,\dots N$. In the following, we will use the shorthand

$${\dot{\kappa }}_{.\ell }\doteq \frac{\partial {⟨\dot{\kappa }⟩}_{J_{\ell }}}{\partial J_{\ell }}$$

(11)

for the partial derivative of the classical averaged precession ${⟨\dot{\kappa }⟩}_{J_{\ell }}$ with respect to the generic even zonal $J_{\ell }$ of degree ℓ; see Appendix B. Then, the N combinations of Equation (10) are posed equal to the experimental residuals $\delta {\dot{\kappa }}^{\left(i\right)},i=1,2,\dots N$ of each of the N orbital elements considered. In principle, such residuals account for the purposely unmodelled pN effect, the mismodelling of the static and time-varying parts of the geopotential, and the non-gravitational forces. Thus, one gets

$$\delta {\dot{\kappa }}^{\left(i\right)}={\mathsf{\mu }}_{\mathrm{pN}}{⟨\dot{\kappa }⟩}_{\mathrm{pN}}^{\left(i\right)}+\sum _{s=2}^N{\dot{\kappa }}_{.s}^{\left(i\right)}\delta J_s,i=1,2,\dots N.$$

(12)

If we look at the pN scaling parameter5 ${\mathsf{\mu }}_{\mathrm{pN}}$ and the mismodeling in the first $N-1$ zonals $\delta J_s,s=2,3,\dots N$ as unknowns, we can interpret Equation (12) as an inhomogenous linear system of N algebraic equations in the N unknowns

$$\underset{N}{\underbrace{{\mathsf{\mu }}_{\mathrm{pN}},\delta J_2,\delta J_3\dots \delta J_N}},$$

(13)

whose coefficients are

$${⟨\dot{\kappa }⟩}_{\mathrm{pN}}^{\left(i\right)},{\dot{\kappa }}_{.s}^{\left(i\right)},i=1,2,\dots N,s=2,3,\dots N,$$

(14)

while the constant terms are the N orbital residuals

$$\delta {\dot{\kappa }}^{\left(i\right)},i=1,2,\dots N.$$

(15)

It turns out that, after some algebraic manipulations, the dimensionless pN scaling parameter, which is 1 in general relativity, can be expressed as

$${\mathsf{\mu }}_{\mathrm{pN}}=\frac{{\mathcal{C}}_{\delta }}{{\mathcal{C}}_{\mathrm{pN}}}.$$

(16)

In Equation (16), the combination of the N orbital residuals

$${\mathcal{C}}_{\delta }\doteq \delta {\dot{\kappa }}^{\left(1\right)}+\sum _{j=1}^{N-1}{c}_j\delta {\dot{\kappa }}^{(j+1)}$$

(17)

is, by construction, independent of the first $N-1$ zonals, being impacted by the other ones of degree $\ell >N$ along with the non-gravitational perturbations and other possible orbital perturbations which cannot be reduced to the same formal expressions of the first $N-1$ zonal rates. Instead,

$${\mathcal{C}}_{\mathrm{pN}}\doteq {⟨\dot{\kappa }⟩}_{\mathrm{pN}}^{\left(1\right)}+\sum _{j=1}^{N-1}{c}_j{⟨\dot{\kappa }⟩}_{\mathrm{pN}}^{(j+1)}$$

(18)

combines the N pN orbital precessions as predicted by general relativity. The dimensionless coefficients $c_j,j=1,2,\dots N-1$ in Equations (17) and (18) depend only on some of the orbital parameters of the satellite(s) involved in such a way that, by construction, ${\mathcal{C}}_{\delta }=0$ if Equation (17) is calculated by posing

$$\delta {\dot{\kappa }}^{\left(i\right)}={\dot{\kappa }}_{.\ell }^{\left(i\right)}\delta J_{\ell },i=1,2,\dots N$$

(19)

for any of the first $N-1$ zonals, independently of the value assumed for its uncertainty $\delta J_{\ell }$.

As far as HERO is concerned, the linear combination of the four experimental residuals $\delta \Omega ,\delta \eta ,\delta e,\delta \omega$ of the satellite’s node, mean anomaly at epoch, eccentricity and perigee suitably designed to cancel out the secular precessions due to the first three zonal harmonics $J_2,J_3,J_4$ of the geopotential is

$${\mathcal{C}}_{\delta }=\delta \Omega +c_1\delta \eta +c_2\delta e+c_3\delta \omega .$$

(20)

The coefficients $c_1,c_2,c_3$ turn out to be

$$\begin{array}{cc}\hfill c_1& =\frac{{\dot{\Omega }}_{.2}{\dot{e}}_{.3}{\dot{\omega }}_{.4}-{\dot{e}}_{.2}{\dot{\Omega }}_{.3}{\dot{\omega }}_{.4}-{\dot{\Omega }}_{.2}{\dot{\omega }}_{.3}{\dot{e}}_{.4}+{\dot{\omega }}_{.2}{\dot{\Omega }}_{.3}{\dot{e}}_{.4}+{\dot{e}}_{.2}{\dot{\omega }}_{.3}{\dot{\Omega }}_{.4}-{\dot{\omega }}_{.2}{\dot{e}}_{.3}{\dot{\Omega }}_{.4}}{{\dot{e}}_{.2}{\dot{\eta }}_{.3}{\dot{\omega }}_{.4}-{\dot{\eta }}_{.2}{\dot{e}}_{.3}{\dot{\omega }}_{.4}-{\dot{e}}_{.2}{\dot{\omega }}_{.3}{\dot{\eta }}_{.4}+{\dot{\omega }}_{.2}{\dot{e}}_{.3}{\dot{\eta }}_{.4}+{\dot{\eta }}_{.2}{\dot{\omega }}_{.3}{\dot{e}}_{.4}-{\dot{\omega }}_{.2}{\dot{\eta }}_{.3}{\dot{e}}_{.4}},\hfill \\ \hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill c_2& =\frac{-{\dot{\Omega }}_{.2}{\dot{\eta }}_{.3}{\dot{\omega }}_{.4}+{\dot{\eta }}_{.2}{\dot{\Omega }}_{.3}{\dot{\omega }}_{.4}+{\dot{\Omega }}_{.2}{\dot{\omega }}_{.3}{\dot{\eta }}_{.4}-{\dot{\omega }}_{.2}{\dot{\Omega }}_{.3}{\dot{\eta }}_{.4}-{\dot{\eta }}_{.2}{\dot{\omega }}_{.3}{\dot{\Omega }}_{.4}+{\dot{\omega }}_{.2}{\dot{\eta }}_{.3}{\dot{\Omega }}_{.4}}{{\dot{e}}_{.2}{\dot{\eta }}_{.3}{\dot{\omega }}_{.4}-{\dot{\eta }}_{.2}{\dot{e}}_{.3}{\dot{\omega }}_{.4}-{\dot{e}}_{.2}{\dot{\omega }}_{.3}{\dot{\eta }}_{.4}+{\dot{\omega }}_{.2}{\dot{e}}_{.3}{\dot{\eta }}_{.4}+{\dot{\eta }}_{.2}{\dot{\omega }}_{.3}{\dot{e}}_{.4}-{\dot{\omega }}_{.2}{\dot{\eta }}_{.3}{\dot{e}}_{.4}},\hfill \\ \hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill c_3& =\frac{-{\dot{\Omega }}_{.2}{\dot{e}}_{.3}{\dot{\eta }}_{.4}+{\dot{e}}_{.2}{\dot{\Omega }}_{.3}{\dot{\eta }}_{.4}+{\dot{\Omega }}_{.2}{\dot{\eta }}_{.3}{\dot{e}}_{.4}-{\dot{\eta }}_{.2}{\dot{\Omega }}_{.3}{\dot{e}}_{.4}-{\dot{e}}_{.2}{\dot{\eta }}_{.3}{\dot{\Omega }}_{.4}+{\dot{\eta }}_{.2}{\dot{e}}_{.3}{\dot{\Omega }}_{.4}}{{\dot{e}}_{.2}{\dot{\eta }}_{.3}{\dot{\omega }}_{.4}-{\dot{\eta }}_{.2}{\dot{e}}_{.3}{\dot{\omega }}_{.4}-{\dot{e}}_{.2}{\dot{\omega }}_{.3}{\dot{\eta }}_{.4}+{\dot{\omega }}_{.2}{\dot{e}}_{.3}{\dot{\eta }}_{.4}+{\dot{\eta }}_{.2}{\dot{\omega }}_{.3}{\dot{e}}_{.4}-{\dot{\omega }}_{.2}{\dot{\eta }}_{.3}{\dot{e}}_{.4}}.\hfill \end{array}$$

(23)

Their numerical values, computed with the formulas of Appendix B for the orbital configurations of Table A2 and Table A6, are listed in Table A4 and Table A8, respectively. In them, the combined mismodeled classical precessions due to the uncancelled zonals, calculated by assuming the formal, statistical sigmas ${\mathsf{\sigma }}_{J_{\ell }},\ell =5,6,\dots$ of the recent global gravity field solution Tongji-Grace02s [48] as a measure of their uncertainties $\delta J_{\ell },\ell =5,6,\dots$, are reported as well. However, caution is in order since the realistic level of mismodeling in the geopotential’s coefficients is usually larger than the mere formal errors released in the models produced by various institutions and publicly available on the Internet at http://icgem.gfz-potsdam.de/tom_longtime. A correct evaluation of the actual uncertainties in the zonal harmonics require great care by suitably comparing different global gravity field models; we will not deal with such a task here. From an inspection of Table A4 and Table A8, it can be noted that the (formal) impact of the uncancelled zonals on the combined pN mass quadrupole effect ($G{J}_2/c^2$) is at the $\simeq 0.6-0.07\%$ level for the proposed orbital configurations of Table A2 and Table A6. If the pN spin dipole Lense-Thirring effect ($GS/c^2$) is considered, the systematic error due to the mismodeling in $J_{\ell },\ell >4$ is about $\simeq 0.1-0.03\%$. The pN mass monopole combined precessions ($GM/c^2$) are affected at the $\simeq (20-5)\times {10}^{-7}$ relative level. Instead, it turns out that the the combined mismodelled classical precessions are at the same level of the pN spin octupole trends ($GS{\epsilon }^2/c^2$). It may not be unrealistic to expect that, when the forthcoming global gravity field models based on the analysis of the entire long data records of the dedicated GRACE and GOCE missions will be finally available, the current merely formal level of uncertainties in the geopotential’s zonal harmonics may be considered as realistic. Moreover, in the next years, the mission GRACE-FO (GFO) [49], launched in May 2018, will also contribute to the production of new global Earth’s gravity field models of increased quality.

On the other hand, the size of the coefficients $c_1,c_2,c_3$ amplifies the impact of any non-gravitational perturbations that may affect the spacecraft; thus, they should be effectively counteracted by some active drag-free apparatus. In particular, the coefficient $c_2$ of the eccentricity is ≃30–40; Table A5 and Table A9 show that the expected secular decrease rate of e due to the atmospheric drag is rather large. Thus, some trade-off may be required among the need of reducing the systematic error of gravitational origin and the actual performance of the drag-free mechanism by looking, e.g., at different linear combinations. It may be interesting to note the case of the mean anomaly at epoch. Indeed, in the case of the high perigee orbital configuration of Table A2, Table A3 and Table A5 tell us that the neutral atmospheric drag would represent just ≃1–2% of the predicted pN $G{J}_2/c^2$ precession on $\eta$. On the other hand, the present-day formal mismodeling in the classical $J_2$-induced rate is about $19\%$ of it. If the pN Schwarzschild-like effect is considered, the formal bias due to $J_2$ is at the $\simeq {10}^{-5}$ level, while the impact of the atmospheric drag is as little as $\simeq {10}^{-6}$. The neutral atmospheric drag has a larger impact on the pN precessions of $\eta$ in the case of the low perigee configuration of Table A6, as shown by Table A7 and Table A9.

3. Summary and Overview

The HERO concept—meant as a hopefully drag-free spacecraft moving in a highly eccentric orbit in a frozen perigee configuration that aimed to perform several tests of relativistic gravity in the Earth’s spacetime—represents, in principle, a promising opportunity to measure a general relativistic effect for the first time, which has never received the same attention as the more well-known Schwarzschild and Lense-Thirring precessions. The post-Newtonian gravitoelectric orbital shifts due to the mass quadrupole moment of the Earth. Indeed, the systematic uncertainty in the combined satellite’s precessions due to the formal, statistical errors in the competing Newtonian mass multipoles of the geopotential—as per one of the most recent global gravity field models—is currently below the per cent level for both the orbital configurations proposed. A unique feature of such a post-Newtonian effect is also that the semimajor axis a undergoes a long-term variation which, for a frozen perigee configuration, resembles a secular trend of the order of ≃4–11 $\mathrm{cm}{\mathrm{yr}}^{-1}$, depending on the orbital geometry chosen. At present, the secular decay of the semimajor axis of the existing passive geodetic satellite LARES has been measured to an accuracy better than $1\mathrm{cm}{\mathrm{yr}}^{-1}$ at 2σ level. As far as the traditional Lense-Thirring and Schwarzschild-like post-Newtonian precessions, the formal systematic bias due to the present-day mismodeling in the classical Earth’s zonal harmonics is currently $\lesssim 0.1\%$ and $\lesssim 0.0002\%$, respectively, if a suitable linear combination of some of the orbital elements of HERO is adopted. However, it must be stressed that the actual uncertainties in the zonal multipoles of the terrestrial gravity field may usually be (much) worse than the sigmas released in the various global gravity solutions. Nonetheless, it cannot be ruled out that, if and when HERO will fly, our knowledge of the Earth’s gravity field will have reached such levels that today’s only formal uncertainties can finally be considered as truly realistic. In addition to the post-Newtonian accelerations, HERO may perform an accurate test of the gravitational red-shift in view of its high eccentricity. Also several models of modified gravity, which generally affect the perigee and the mean anomaly at epoch with secular precessions, could be fruitfully put to the test. A crucial aspect is represented by the level of compensation of the non-gravitational perturbation which will be practically attainable with some drag-free apparatus; suffice it to say that the nominal size of the competing secular decrease of the semimajor axis due to, for example, the neutral atmospheric drag reaching the ≃ 5–160 $\mathrm{m}{\mathrm{yr}}^{-1}$ level if a passive, cannonball satellite is considered. The variability of atmospheric density is such that, in order to achieve drag-free technology, one would have to measure atmospheric density at the satellite in real-time and transmit the data back to Earth for inclusion and analysis within a drag model. Adding such complexities will, however, add to the actual drag as antennas, batteries, solar panels and so forth, and will increase drag and accompanying errors. Perhaps, as the orbit will have to be determined with Satellite Laser Ranging (SLR), one could design and build a type of corner cube reflector where some aspect of the returned SLR pulse is modulated by the atmospheric pressure signal. SLR systems with kHz pulse rates will be useful here. The investigation of such delicate issues deserve dedicated and detailed analyses.

Appendix B.1. The Eccentricity

$$\begin{array}{cc}\hfill {\dot{e}}_{.2}& =0,\hfill \end{array}$$

(A2)

$$\begin{array}{cc}\hfill {\dot{e}}_{.3}& =-\frac{3n_{\mathrm{b}}{R}_{\mathrm{e}}^3\left(3+5cos2I\right)sinIcos\omega }{16a^3{\left(1-e^2\right)}^2},\hfill \end{array}$$

(A3)

$$\begin{array}{cc}\hfill {\dot{e}}_{.4}& =-\frac{15e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^4\left(5+7cos2I\right){sin}^{2}Isin2\omega }{64a^4{\left(1-e^2\right)}^3},\hfill \end{array}$$

(A4)

$$\begin{array}{cc}\hfill {\dot{e}}_{.5}& =\frac{15n_{\mathrm{b}}{R}_{\mathrm{e}}^5}{2048a^5{\left(1-e^2\right)}^4}\times \hfill \\ & \times \left\{28e^2\left(7+9cos2I\right){sin}^{3}Icos3\omega +\left(4+3e^2\right)\left[2sinI+7\left(sin3I+3sin5I\right)\right]cos\omega \right\},\hfill \end{array}$$

(A5)

$$\begin{array}{cc}\hfill {\dot{e}}_{.6}& =\frac{105e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^6{sin}^{2}I}{8192a^6{\left(1-e^2\right)}^5}\times \hfill \\ & \times \left[5\left(2+e^2\right)\left(35+60cos2I+33cos4I\right)sin2\omega +12e^2\left(9+11cos2I\right){sin}^{2}Isin4\omega \right],\hfill \end{array}$$

(A6)

$$\begin{array}{cc}\hfill {\dot{e}}_{.7}& =\frac{21n_{\mathrm{b}}{R}_{\mathrm{e}}^7}{524288a^7{\left(1-e^2\right)}^6}\times \hfill \\ \hfill & \times \left\{60e^2{sin}^{3}I\left[-3\left(8+3e^2\right)\left(189+308cos2I+143cos4I\right)cos3\omega -\right.\right.\hfill \\ \hfill & \left.-44e^2\left(11+13cos2I\right){sin}^{2}Icos5\omega \right]-\hfill \\ & \left.-5\left[8+5e^2\left(4+e^2\right)\right]\left(25sinI+81sin3I+165sin5I+429sin7I\right)cos\omega \right\},\hfill \end{array}$$

(A7)

$$\begin{array}{cc}\hfill {\dot{e}}_{.8}& =-\frac{63e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^8{sin}^{2}I}{2097152a^8{\left(1-e^2\right)}^7}\times \hfill \\ \hfill & \times \left\{35\left(48+80e^2+15e^4\right)\left(210+385cos2I+286cos4I+143cos6I\right)sin2\omega +\right.\hfill \\ \hfill & +88e^2{sin}^{2}I\left[7\left(10+3e^2\right)\left(99+156cos2I+65cos4I\right)sin4\omega +\right.\hfill \\ & \left.\left.+26e^2\left(13+15cos2I\right){sin}^{2}Isin6\omega \right]\right\}.\hfill \end{array}$$

(A8)

Appendix B.2. The Inclination

$$\begin{array}{cc}\hfill {\dot{I}}_{.2}& =0,\hfill \end{array}$$

(A9)

$$\begin{array}{cc}\hfill {\dot{I}}_{.3}& =\frac{3e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^{3}cosI\left(3+5cos2I\right)cos\omega }{16a^3{\left(1-e^2\right)}^3},\hfill \end{array}$$

(A10)

$$\begin{array}{cc}\hfill {\dot{I}}_{.4}& =\frac{15e^2{n}_{\mathrm{b}}{R}_{\mathrm{e}}^{4}cosI\left(5+7cos2I\right)sinIsin2\omega }{64a^4{\left(1-e^2\right)}^4},\hfill \end{array}$$

(A11)

$$\begin{array}{cc}\hfill {\dot{I}}_{.5}& =-\frac{15e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^{5}cotI}{2048a^5{\left(1-e^2\right)}^5}\times \hfill \\ & \times \left\{28e^2\left(7+9cos2I\right){sin}^{3}Icos3\omega +\left(4+3e^2\right)\left[2sinI+7\left(sin3I+3sin5I\right)\right]cos\omega \right\},\hfill \end{array}$$

(A12)

$$\begin{array}{cc}\hfill {\dot{I}}_{.6}& =\frac{5n_{\mathrm{b}}{R}_{\mathrm{e}}^{6}cotI}{65536a^6{\left(1-e^2\right)}^6}\times \hfill \\ \hfill & \times \left[-840e^2\left(2+e^2\right)\left(35+60cos2I+33cos4I\right){sin}^{2}Isin2\omega -\right.\hfill \\ & \left.-2016e^4\left(9+11cos2I\right){sin}^{4}Isin4\omega \right],\hfill \end{array}$$

(A13)

$$\begin{array}{cc}\hfill {\dot{I}}_{.7}& =-\frac{21e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^{7}cotI}{524288a^7{\left(1-e^2\right)}^7}\times \hfill \\ \hfill & \times \left\{60e^2{sin}^{3}I\left[-3\left(8+3e^2\right)\left(189+308cos2I+143cos4I\right)cos3\omega -\right.\right.\hfill \\ \hfill & \left.-44e^2\left(11+13cos2I\right){sin}^{2}Icos5\omega \right]-\hfill \\ & \left.-5\left[8+5e^2\left(4+e^2\right)\right]\left(25sinI+81sin3I+165sin5I+429sin7I\right)cos\omega \right\},\hfill \end{array}$$

(A14)

$$\begin{array}{cc}\hfill {\dot{I}}_{.8}& =\frac{63e^2{n}_{\mathrm{b}}{R}_{\mathrm{e}}^{8}sinIcosI}{2097152a^8{\left(1-e^2\right)}^8}\times \hfill \\ \hfill & \times \left\{35\left(48+80e^2+15e^4\right)\left(210+385cos2I+286cos4I+143cos6I\right)sin2\omega +\right.\hfill \\ \hfill & +88e^2{sin}^{2}I\left[7\left(10+3e^2\right)\left(99+156cos2I+65cos4I\right)sin4\omega +\right.\hfill \\ & \left.\left.+26e^2\left(13+15cos2I\right){sin}^{2}Isin6\omega \right]\right\}.\hfill \end{array}$$

(A15)

Appendix B.3. The Longitude of the Ascending Node

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.2}& =-\frac{3n_{\mathrm{b}}{R}_{\mathrm{e}}^{2}cosI}{2a^2{\left(1-e^2\right)}^2},\hfill \end{array}$$

(A16)

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.3}& =\frac{3e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^3\left(cosI+15cos3I\right)cscIsin\omega }{32a^3{\left(1-e^2\right)}^3},\hfill \end{array}$$

(A17)

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.4}& =\frac{15n_{\mathrm{b}}{R}_{\mathrm{e}}^4\left[\left(2+3e^2\right)\left(9cosI+7cos3I\right)-2e^2\left(5cosI+7cos3I\right)cos2\omega \right]}{128a^4{\left(1-e^2\right)}^4},\hfill \end{array}$$

(A18)

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.5}& =-\frac{15e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^5}{2048a^5{\left(1-e^2\right)}^5}\times \hfill \\ & \times \left\{\left(4+3e^2\right)\left[2cosI+21\left(cos3I+5cos5I\right)\right]cscIsin\omega +7e^2\left(2sin2I+15sin4I\right)sin3\omega \right\},\hfill \end{array}$$

(A19)

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.6}& =\frac{105n_{\mathrm{b}}{R}_{\mathrm{e}}^6}{16384a^6{\left(1-e^2\right)}^6}\times \hfill \\ \hfill & \times \left[-\left(8+40e^2+15e^4\right)\left(50cosI+45cos3I+33cos5I\right)+\right.\hfill \\ \hfill & +5e^2\left(2+e^2\right)\left(70cosI+87cos3I+99cos5I\right)cos2\omega +\hfill \\ & \left.+6e^4\left(47cosI+33cos3I\right){sin}^{2}Icos4\omega \right],\hfill \end{array}$$

(A20)

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.7}& =-\frac{21e{n}_{\mathrm{b}}{R}_{\mathrm{e}}^{7}cscI}{524288a^7{\left(1-e^2\right)}^7}\times \hfill \\ \hfill & \times \left\{-5\left[8+5e^2\left(4+e^2\right)\right]\left(25cosI+243cos3I+825cos5I+3003cos7I\right)sin\omega -\right.\hfill \\ \hfill & -30e^2\left(8+3e^2\right)\left(1442cosI+1397cos3I+1001cos5I\right){sin}^{2}Isin3\omega -\hfill \\ & \left.-264e^4\left(149cosI+91cos3I\right){sin}^{4}Isin5\omega \right\},\hfill \end{array}$$

(A21)

$$\begin{array}{cc}\hfill {\dot{\Omega }}_{.8}& =\frac{63n_{\mathrm{b}}{R}_{\mathrm{e}}^8}{2097152a^8{\left(1-e^2\right)}^8}\times \hfill \\ \hfill & \times \left(5\left\{16+7e^2\left[24+5e^2\left(6+e^2\right)\right]\right\}\left[1225cosI+11\left(105cos3I+91cos5I+65cos7I\right)\right]-\right.\hfill \\ \hfill & -70e^2\left(48+80e^2+15e^4\right)\left\{105cosI+11\left[11cos3I+13\left(cos5I+cos7I\right)\right]\right\}cos2\omega -\hfill \\ \hfill & -616e^4\left(10+3e^2\right)\left(138cosI+117cos3I+65cos5I\right){sin}^{2}Icos4\omega -\hfill \\ & \left.-9152e^{6}cosI\left(2+5cos2I\right){sin}^{4}Icos6\omega \right).\hfill \end{array}$$

(A22)

Appendix B.4. The Argument of Perigee

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.2}& =\frac{3n_{\mathrm{b}}{R}_{\mathrm{e}}^2\left(3+5cos2I\right)}{8a^2{\left(1-e^2\right)}^2},\hfill \end{array}$$

(A23)

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.3}& =-\frac{3n_{\mathrm{b}}{R}_{\mathrm{e}}^3}{64a^{3}e{\left(1-e^2\right)}^3}\left[-1-3e^2-4cos2I+5\left(1+7e^2\right)cos4I\right]cscIsin\omega ,\hfill \end{array}$$

(A24)

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.4}& =\frac{15n_{\mathrm{b}}{R}_{\mathrm{e}}^4}{1024a^4{\left(1-e^2\right)}^4}\times \hfill \\ \hfill & \times \left\{-27\left(4+5e^2\right)+4cos2I\left[-52-63e^2+2\left(-2+7e^2\right)cos2\omega \right]+2\left(-6+5e^2\right)cos2\omega +\right.\hfill \\ & \left.+7cos4I\left[-28-27e^2+2\left(2+9e^2\right)cos2\omega \right]\right\},\hfill \end{array}$$

(A25)

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.5}& =-\frac{15n_{\mathrm{b}}{R}_{\mathrm{e}}^{5}sinI}{4096a^{5}e{\left(1-e^2\right)}^5}\times \hfill \\ \hfill & \times \left\{\left[8+74e^2+30e^4+\left(20+113e^2+21e^4\right)cos2I+14\left(4+5e^2-9e^4\right)cos4I-\right.\right.\hfill \\ \hfill & \left.-21\left(4+61e^2+33e^4\right)cos6I\right]{csc}^{2}Isin\omega -\hfill \\ & \left.-14e^2\left[-5+7e^2+4\left(-1+6e^2\right)cos2I+\left(9+33e^2\right)cos4I\right]sin3\omega \right\},\hfill \end{array}$$

(A26)

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.6}& =\frac{105n_{\mathrm{b}}{R}_{\mathrm{e}}^6}{65536a^6{\left(1-e^2\right)}^6}\times \hfill \\ \hfill & \times \left(5\left\{\left(472+1,940e^2+675e^4\right)cos2I+\right.\right.\hfill \\ \hfill & \left.+3e^2\left[2\left(292+99e^2\right)cos4I+11\left(44+13e^2\right)cos6I\right]\right\}-\hfill \\ \hfill & -5\left[10e^2\left(6+7e^2\right)+\left(-68+254e^2+195e^4\right)cos2I+6\left(-4+102e^2+55e^4\right)cos4I+\right.\hfill \\ \hfill & \left.+33\left(4+34e^2+13e^4\right)cos6I\right]cos2\omega +\hfill \\ \hfill & +2\left[1128cos4I+1188cos6I+25\left(24+100e^2+35e^4+4cos2\omega \right)\right]-\hfill \\ & \left.-6e^2\left[-28+45e^2+4\left(-4+33e^2\right)cos2I+11\left(4+13e^2\right)cos4I\right]{sin}^{2}Icos4\omega \right),\hfill \end{array}$$

(A27)

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.7}& =-\frac{21n_{\mathrm{b}}{R}_{\mathrm{e}}^7}{524288e{a}^7{\left(1-e^2\right)}^7}\times \hfill \\ \hfill & \times \left(-5\left(8+156e^2+225e^4+40e^6\right)\left(25sinI+81sin3I+165sin5I+429sin7I\right)sin\omega -\right.\hfill \\ \hfill & -60e^2\left(24+95e^2+24e^4\right)\left(189+308cos2I+143cos4I\right){sin}^{3}Isin3\omega +\hfill \\ \hfill & +10e^{2}cosI\left\{\left[8+5e^2\left(4+e^2\right)\right]\left(-1198+2421cos2I-2178cos4I+\right.\right.\hfill \\ \hfill & \left.+3003cos6I\right)cotIsin\omega +\hfill \\ \hfill & \left.+3e^2\left(8+3e^2\right)\left(523+396cos2I+1001cos4I\right)sin2Isin3\omega \right\}+\hfill \\ \hfill & +528e^6{cos}^{2}I\left(29+91cos2I\right){sin}^{3}Isin5\omega -\hfill \\ & \left.-528e^4\left(5+8e^2\right)\left(11+13cos2I\right){sin}^{5}Isin5\omega \right),\hfill \end{array}$$

(A28)

$$\begin{array}{cc}\hfill {\dot{\omega }}_{.8}& =\frac{63n_{\mathrm{b}}{R}_{\mathrm{e}}^8}{33554432a^8{\left(1-e^2\right)}^8}\times \hfill \\ \hfill & \times \left(5\left\{-1225\left[192+35e^2\left(48+56e^2+9e^4\right)\right]-\right.\right.\hfill \\ \hfill & -280\left[1664+7e^2\left(2064+2400e^2+385e^4\right)\right]cos2I-\hfill \\ \hfill & -308\left[1472+7e^2\left(1776+2040e^2+325e^4\right)\right]cos4I-\hfill \\ \hfill & -3432\left[128+7e^2\left(144+160e^2+25e^4\right)\right]cos6I-\hfill \\ \hfill & \left.-715\left[704+7e^2\left(624+600e^2+85e^4\right)\right]cos8I\right\}+\hfill \\ \hfill & +70\left[35\left(-96+208e^2+950e^4+225e^6\right)+\right.\hfill \\ \hfill & +16\left(-384+1648e^2+5160e^4+1155e^6\right)cos2I+\hfill \\ \hfill & +44\left(-96+1360e^2+2870e^4+585e^6\right)cos4I+\hfill \\ \hfill & +2288e^2\left(48+80e^2+15e^4\right)cos6I+\hfill \\ \hfill & \left.+143\left(96+1328e^2+1610e^4+255e^6\right)cos8I\right]cos2\omega +\hfill \\ \hfill & +616e^2\left[6\left(-280+944e^2+363e^4\right)+\left(-1960+14128e^2+4797e^4\right)cos2I+\right.\hfill \\ \hfill & \left.+26\left(40+688e^2+195e^4\right)cos4I+65\left(40+208e^2+51e^4\right)cos6I\right]{sin}^{2}Icos4\omega +\hfill \\ & \left.+4576e^4\left[-22+39e^2+4\left(-2+25e^2\right)cos2I+5\left(6+17e^2\right)cos4I\right]{sin}^{4}Icos6\omega \right).\hfill \end{array}$$

(A29)

Appendix B.5. The Mean Anomaly at Epoch

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.2}& =\frac{3n_{\mathrm{b}}{R}_{\mathrm{e}}^2\left(1+3cos2I\right)}{8a^2{\left(1-e^2\right)}^{3/2}},\hfill \end{array}$$

(A30)

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.3}& =\frac{3n_{\mathrm{b}}{R}_{\mathrm{e}}^3}{16e{a}^3{\left(1-e^2\right)}^{5/2}}\left[\left(-1+4e^2\right)\left(3+5cos2I\right)sinIsin\omega \right],\hfill \end{array}$$

(A31)

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.4}& =-\frac{15n_{\mathrm{b}}{R}_{\mathrm{e}}^4}{1024a^4{\left(1-e^2\right)}^{7/2}}\times \hfill \\ & \times \left[3e^2\left(9+20cos2I+35cos4I\right)+8\left(-2+5e^2\right)\left(5+7cos2I\right){sin}^{2}Icos2\omega \right],\hfill \end{array}$$

(A32)

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.5}& =\frac{15n_{\mathrm{b}}{R}_{\mathrm{e}}^5}{2048e{a}^5{\left(1-e^2\right)}^{9/2}}\times \hfill \\ \hfill & \times \left\{-\left(-4+7e^2+18e^4\right)\left[2sinI+7\left(sin3I+3sin5I\right)\right]sin\omega -\right.\hfill \\ & \left.-28e^2\left(-1+2e^2\right)\left(7+9cos2I\right){sin}^{3}Isin3\omega \right\},\hfill \end{array}$$

(A33)

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.6}& =-\frac{35n_{\mathrm{b}}{R}_{\mathrm{e}}^6}{65536a^6{\left(1-e^2\right)}^{11/2}}\times \hfill \\ \hfill & \times \left[-\left(-8+20e^2+15e^4\right)\left(50+105cos2I+126cos4I+231cos6I\right)-\right.\hfill \\ \hfill & -60\left(-4+6e^2+7e^4\right)\left(35+60cos2I+33cos4I\right){sin}^{2}Icos2\omega -\hfill \\ & \left.-72e^2\left(-4+7e^2\right)\left(9+11cos2I\right){sin}^{4}Icos4\omega \right],\hfill \end{array}$$

(A34)

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.7}& =\frac{21n_{\mathrm{b}}{R}_{\mathrm{e}}^7}{524288e{a}^7{\left(1-e^2\right)}^{13/2}}\times \hfill \\ \hfill & \times \left[5\left(-8-28e^2+95e^4+40e^6\right)\left(25sinI+81sin3I+165sin5I+429sin7I\right)sin\omega +\right.\hfill \\ \hfill & +180e^2\left(-8+11e^2+8e^4\right)\left(189+308cos2I+143cos4I\right){sin}^{3}Isin3\omega +\hfill \\ & \left.+528e^4\left(-5+8e^2\right)\left(11+13cos2I\right){sin}^{5}Isin5\omega \right],\hfill \end{array}$$

(A35)

$$\begin{array}{cc}\hfill {\dot{\eta }}_{.8}& =-\frac{63n_{\mathrm{b}}{R}_{\mathrm{e}}^8}{33554432a^8{\left(1-e^2\right)}^{15/2}}\times \hfill \\ \hfill & \times \left\{5\left[-32+35e^4\left(4+e^2\right)\right]\left(1225+2520cos2I+2772cos4I+3432cos6I+\right.\right.\hfill \\ \hfill & \left.+6435cos8I\right)+280\left(-96-80e^2+470e^4+135e^6\right)\left(210+385cos2I+286cos4I+\right.\hfill \\ \hfill & \left.+143cos6I\right){sin}^{2}Icos2\omega +\hfill \\ \hfill & +2464e^2\left(-40+52e^2+27e^4\right)\left(99+156cos2I+65cos4I\right){sin}^{4}Icos4\omega +\hfill \\ & \left.+18304e^4\left(-2+3e^2\right)\left(13+15cos2I\right){sin}^{6}Icos6\omega \right\}.\hfill \end{array}$$

(A36)