Optimizing expansions of the Earth’s gravity potential and its derivatives over ellipsoidal harmonics

A set of spherical harmonics is the most widely used representation of the Earth’s gravity potential. This series converges outside and on the surface of a reference sphere enveloping the Earth. However, the Earth’s surface is better approximated by the reference ellipsoid—a compressed ellipsoid of revolution that covers the entire Earth. The gravity potential can be expanded in a series over ellipsoidal harmonics on the surface of the reference ellipsoid and on the surface of other external confocal ellipsoids of revolution. In contrast to spherical harmonics, depending on the associated Legendre functions of the first kind, ellipsoidal harmonics depend also on the associated Legendre functions of the second kind. The latter contain the very slowly converging hypergeometric Gauss series. The number of series increases with increasing the order of their derivatives. In this work, we derived new series for the gravitational potential of the Earth and its derivatives over ellipsoidal harmonics. Starting from the first order derivative, all the series corresponding to higher order derivatives depend on the same two hypergeometric Gauss series. The latter converges considerably faster than that for the hypergeometric series previously used when computing the gravity potential and its derivatives.     

Applying an Expansion of the Second Derivatives of the Earth’s Gravitational Potential in Modified Spherical Harmonics for Constructing and Estimating Its Models

Second-order derivatives of the Earth’s potential in a local north-oriented coordinate system are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. Based on these relations, recurrent procedures are developed for estimating the geopotential coefficients from the spectrum of each derivative and, conversely, for simulating the spectrum from a known geopotential model. The very simple structure of the expressions for the derivatives is convenient for estimating the coefficients of the geopotential by the least squares method at a certain step of processing satellite gradiometry data. Since the new series are orthogonal, the method with a quadrature formula can be applied, which helps avoid aliasing errors caused by the truncation of the series. The spectral coefficients of the derivatives are estimated using the derived relations for different models on an average orbital sphere of the GOCE satellite and at other altitudes above the Earth’s surface.

     

Laplace series for the level ellipsoid of revolution

The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity \(\varepsilon \) of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities \(\varepsilon \) and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients \(I_n\) of Laplace series representing the outer potential of T are uniquely determined by \(\varepsilon \) and q. In this paper, we have found explicit expressions for Stokes coefficients via \(\varepsilon \) and q, as well as their asymptotics when \(n\rightarrow \infty \). If T does not coincide with a Maclaurin ellipsoid, then \(|I_n|\sim B\varepsilon ^n/n\) with a certain constant B. Let us compare this asymptotics with one of \(I_n\) for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: \(|I_n|\sim \bar{B}\varepsilon ^n/(n^2)\). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.     

The exact transformation from spherical harmonic to ellipsoidal harmonic coefficients for gravitational field modeling

The spherical and ellipsoidal harmonic series of the external gravitational potential for a given mass distribution are equivalent in their mutual region of uniform convergence. In an instructive case, the equality of the two series on the common coordinate surface of an infinitely large sphere reveals the exact correspondence between the spherical and ellipsoidal harmonic coefficients. The transformation between the two sets of coefficients can be accomplished via the numerical methods by Walter (Celest Mech 2:389–397, 1970) and Dechambre and Scheeres (Astron Astrophys 387:1114–1122, 2002), respectively. On the other hand, the harmonic coefficients are defined by the integrals of mass density moments in terms of the respective solid harmonics. This paper presents general algebraic formulas for expressing the solid ellipsoidal harmonics as a linear combination of the corresponding solid spherical harmonics. An exact transformation from spherical to ellipsoidal harmonic coefficients is found by incorporating these connecting expressions into the density integral. A computational procedure is proposed for the transformation. Numerical results based on the nearly ellipsoidal Martian moon, Phobos, are presented for validation of the method.     

The Responses of the Deformable Earth to Different Driving Forces

The spatial and temporal variations of the Earth deformation and the gravitational field are important both in the theoretical research and in the construction of geospatial database. The Earth deforms due to various mechanisms and the deformation further induces changes in the gravitational potential of the Earth, i.e. the deformation-induced additional potential or the Euler gravitational increment. Based on the theory of vector spherical harmonics, we discuss in this paper the Earth deformation and gravitational increment resulting from the tidal force, loading force and the stress of the Earth's surface. We write out the expression for the Euler gravitational potential increment and the relations between different Love numbers. These are all important points in the research on Earth deformation.     

A New Reference Equipotential Surface, and Reference Ellipsoid for the Planet Mars

Using the shape model of Mars GTM090AA in terms of spherical harmonics complete to degree and order 90 and gravitational field model of Mars GGM2BC80 in terms of spherical harmonics complete to degree and order 80, both from Mars Global Surveyor (MGS) mission, the geometry (shape) and gravity potential value of reference equipotential surface of Mars (Areoid) are computed based on a constrained optimization problem. In this paper, the Areoid is defined as a reference equipotential surface, which best fits to the shape of Mars in least squares sense. The estimated gravity potential value of the Areoid from this study, i.e. W ₀ = (12,654,875 ± 69) (m²/s²), is used as one of the four fundamental gravity parameters of Mars namely, {W ₀, GM, ω, J ₂₀}, i.e. {Areoid’s gravity potential, gravitational constant of Mars, angular velocity of Mars, second zonal spherical harmonic of gravitational field expansion of Mars}, to compute a bi-axial reference ellipsoid of Somigliana-Pizzetti type as the hydrostatic approximate figure of Mars. The estimated values of semi-major and semi-minor axis of the computed reference ellipsoid of Mars are (3,395,428 ± 19) (m), and (3,377,678 ± 19) (m), respectively. Finally the computed Areoid is presented with respect to the computed reference ellipsoid.     

The low-eccentricity earth satellite orbit at the critical inclination

A solution to the orbital motion of an Earth satellite at the critical inclination and with near-zero eccentricity is developed by the von Zeipel method to the first order in the eccentricity, and to the first order in the higher gravitational harmonics, using elements which do not degenerate at zero eccentricity.     

Fast computation of satellite gravitational gradient

The computation of the Earth's potential function at high order and degree with the method of reference [1], causes overflow most of the time. The normalized method [2–6] can eliminate the overflows, but leads to formulae much more involved than those in reference [1]; besides, the programming is more complex and the computer time required larger. The method presented in this paper has the following features: each component of the satellite gravitational gradient can be computed; the formulae are short and easy to be programme; the method is much quicker than the normalization method and can be carried out with a micro‐computer, without overflow even in the case of Earth's spherical harmonics of order and degree as high as 1025 or higher. This method satisfies the present demand to compute satellite gravitational gradient with high accuracy. Furthermore, we present formulae for the fast computation, without overflow, of the gravitational gradient corresponding to Earth's spherical harmonics up to order and degree of 3170 × 3170 or higher. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical lunar libration tables

We have developed a theory of the rotation of the Moon, for the purpose of obtaining libration series explicitly dependent upon lunar gravitational field model parameters. A nonlinear model is used in which the rigid Moon, whose motion in space is that of the main problem of lunar theory, and whose gravity potential is represented through its third degree harmonics, is torqued by the Earth and Sun. The analytical series are then obtained as Poisson series. Numerical comparisons with Eckhardt's solution are briefly exposed.     

Enhancement of on-board forecasting accuracy of GEO SC COG motion using the compensative transversal acceleration

A method is suggested for enhancing the on-board forecasting accuracy of the COG motion of a GEO SC with a long time of independent operation. The suggested method consists of introducing so-called compensative transversal acceleration (CTA), along with zonal harmonics into the right sides of the differential equations of SC motion among other disturbances due to the Earth’s gravitational field eccentricity. The CTA compensates the integral effect of the sectoral and tesseral harmonics; its value is constant for a specified point of GEO SC location (standing point) and is calculated on the Earth from numerical integration of differential equations of motion taking into account the complete set of gravitational field harmonics. The CTA value is transmitted on-board of an SC as program command data. The method is implemented in algorithms of on-board forecasting of Electro-L SC motion and can be used to enhance the on-board forecasting accuracy of the COG motion of GEO SCs with a long time of independent operation.     

Determining the orientation and spin period of TOPEX/Poseidon satellite by a photometric method

We present the results of photometric observations of the TOPEX/Poseidon satellite performed during 2008–2016. The satellite become space debris after a failure in January, 2006, in a low Earth orbit. In the Laboratory of Space Research of Uzhhorod National University 73 light curves of the spacecraft were obtained. Standardization of photometric light curves is briefly explained. We have calculated the color indices of reflecting surfaces and the spin rate change. The general tendency of the latter is described by an exponential decay function. The satellite spin periods based on 126 light curves (including 53 light curves from the MMT-9 project operating since 2014) were taken into account. In 2016 the period of its own rotation reached its minimum of 10.6 s.A method to derive the direction of the spin axis of an artificial satellite and the angles of the light scattered by its surface has been developed in the Laboratory of Space Research of Uzhhorod National University. We briefly describe the “Orientation” program used for these purposes. The orientation of the TOPEX/Poseidon satellite in mid-2016 is given. The angle of precession β = 45°–50° and period of precession P _pr = 141.5 s have been defined. The reasons for the identified nature of the satellite’s own rotation have been found. They amount to the perturbation caused by a deviation of the Earth gravity field from a central-symmetric shape and the presence of moving parts on the satellite.     

The representation of the force functions of the Earth's and Moon's gravitation in ellipsoidal coordinates

This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc ₂₀ andc ₂₂.     

A theory of the equilibrium figure and gravitational field of the Galilean satellite Io: The second approximation

We construct a theory of the equilibrium figure and gravitational field of the Galilean satellite Io to within terms of the second order in the small parameter α. We show that to describe all effects of the second approximation, the equation for the figure of the satellite must contain not only the components of the second spherical function, but also the components of the third and fourth spherical functions. The contribution of the third spherical function is determined by the Love number of the third order h₃, whose model value is 1.6582. Measurements of the third-order gravitational moments could reveal the extent to which the hydrostatic equilibrium conditions are satisfied for Io. These conditions are J₃=C₃₂=0 and C₃₁/C₃₃=?6. We have calculated the corrections of the second order of smallness to the gravitational moments J₂ and C₂₂. We conclude that when modeling the internal structure of Io, it is better to use the observed value of k₂ than the moment of inertia derived from k₂. The corrections to the lengths of the semiaxes of the equilibrium figure of Io are all positive and equal to ～64.5, ～26, and ～14 m for the a, b, and c axes, respectively. Our theory allows the parameters of the figure and the fourth-order gravitational moments that differ from zero to be calculated. For the homogeneous model, their values are:\(s_4 = \frac{{885}}{{224}}\alpha ^2 ,s_{42} = - \frac{{75}}{{224}}\alpha ^2 ,s_{44} = \frac{{15}}{{896}}\alpha ^2 ,J_4 = - \frac{{885}}{{224}}\alpha ^2 ,C_{42} = \frac{{75}}{{224}}\alpha ^2 ,C_{44} = \frac{{15}}{{896}}\alpha ^2 \).     

Regional positioning using a low Earth orbit satellite constellation

Global and regional satellite navigation systems are constellations orbiting the Earth and transmitting radio signals for determining position and velocity of users around the globe. The state-of-the-art navigation satellite systems are located in medium Earth orbits and geosynchronous Earth orbits and are characterized by high launching, building and maintenance costs. For applications that require only regional coverage, the continuous and global coverage that existing systems provide may be unnecessary. Thus, a nano-satellites-based regional navigation satellite system in Low Earth Orbit (LEO), with significantly reduced launching, building and maintenance costs, can be considered. Thus, this paper is aimed at developing a LEO constellation optimization and design method, using genetic algorithms and gradient-based optimization. The preliminary results of this study include 268 LEO constellations, aimed at regional navigation in an approximately 1000 km \(\times \) 1000 km area centered at the geographic coordinates [30, 30] degrees. The constellations performance is examined using simulations, and the figures of merit include total coverage time, revisit time, and geometric dilution of precision (GDOP) percentiles. The GDOP is a quantity that determines the positioning solution accuracy and solely depends on the spatial geometry of the satellites. Whereas the optimization method takes into account only the Earth’s second zonal harmonic coefficient, the simulations include the Earth’s gravitational field with zonal and tesseral harmonics up to degree 10 and order 10, Solar radiation pressure, drag, and the lunisolar gravitational perturbation.     

The Phobos gravitational field modeled on the basis of its topography

The parameters of the best-fitting ellipsoid have been derived using the latest spherical harmonics of the Phobos topography (Duxbury, 1989) by solution of non-linear overdetermined inverse problem. The lengths of the equatorial axes of the ellipsoid have been determined (a = 12.9 km, b = 11.4 km). They are nearly the same as established by Duxbury (ibid.) on the basis of the linearized relationship between the squared lengths of ellipsoidal axes and the topography coefficients C ₂₀ and C _22. The length of the polar axis (c = 9.1 km) differs of about 20% from Duxbury's value. Supposing mass homogeneity of Phobos, the Stokes parameters of the external gravitational field have been derived up to those of the sixth degree and order. The large irregularities in the Phobos figure cause the values of the Duxbury's potential coefficients be fairly inaccurate except the harmonics C ₂₀, C ₃₂, S ₄₃ and S ₅₁, i.e. linearized relationship between gravity and topography cannot be applied for Phobos. Finally, positions of the centre of figure and the directions of the principal axes of inertia have been established.     

月球卫星轨道变化的分析解

由于月球自转缓慢及其引力位的特点,使得讨论月球卫星与人造地球卫星轨道变化的方法有所不同。     

New orbits of wide visual double stars

Based on photographic and CCD observations with the Pulkovo 26-inch refractor, radial velocity measurements with the 1.5-m RTT-150 telescope (TUBITAK National Observatory, Turkey), and highly accurate observations published in the WDS catalog, we have obtained the orbits of ten wide visual double stars by the apparent motion parameter method. The orientation of the orbits in the Galactic coordinate system has been determined. For the outer pair of the multiple star HIP 12780 we have calculated a family of orbits with a minimum period P = 4634 yr. Two equivalent solutions with the same period have been obtained for the stars HIP 50 (P = 949 yr) and HIP 66195 (P = 3237 yr). We have unambiguously determined the orbits of six stars: HIP 12777 (P = 3327 yr), HIP 15058 (P = 420 yr), HIP 33287 (P = 1090 yr), HIP 48429 (P = 1066 yr), HIP 69751 (P = 957 yr), and HIP 73846 (P = 1348 yr). The orbit of HIP 55068 is orientated perpendicularly to the plane of the sky, P >1000 yr. The star HIP 48429 is suspected to have an invisible companion.     

On the feasibility of studying the exospheres of Earth-like exoplanets by Lyman-<Emphasis Type="Italic">α</Emphasis> monitoring

Observations of the Earth’s exosphere have unveiled an extended envelope of hydrogen reaching further than 10 Earth radii composed of atoms orbiting around the Earth. This large envelope increases significantly the opacity of the Earth to Lyman α (Lyα) photons coming from the Sun, to the point of making feasible the detection of the Earth’s transit signature from 1.35 pc if pointing with an 8 meter primary mirror space telescope through a clean line of sight (N _H <?10¹⁷ cm^??2), as we show. In this work, we evaluate the potential detectability of Earth analogs orbiting around nearby M-type stars by monitoring the variability of the Lyα flux variability. We show that, in spite of the interstellar, heliospheric and astrospheric absorption, the transit signature in M5 V type stars would be detectable with a dedicated Lyα flux monitor implemented in a 4–8 m class space telescope. Such monitoring programs would enable measuring the robustness of planetary atmospheres under heavy space weather conditions like those produced by M-type stars. A 2-m class telescope, such as the World Space Observatory, would suffice to detect an Earth-like planet orbiting around Proxima Centauri, if there was such a planet or nearby M5 type stars.     

On Estimating the Dissipative Factor of the Martian Interior

An attempt is made to construct a trial Q_μ(l) distribution in the silicate mantle of Mars. With the allowance for the fact that on the P–T plane the Earth’s geotherm is close to the distribution of areotherms, it was concluded that Q_μ(l) should be distributed in the Martian interior topologically close to the Q_μ(l) distribution in the Earth. The initial distribution was specified by the four-layer piecewise-constant distribution from the QML9 model. An important step was to select the power index in the frequency dependence of Q_μ. Based on the laboratory data and on the experience of studying this problem for the Earth, n was specified in the interval 0.1–0.3. It was found that with the conversion of the initial distribution to the orbital period of Phobos around Mars, which is the only constraint for the problem derived from the observations, this distribution agrees reasonably well with the observational data at n = 0.1.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.