Position and velocity perturbations in the orbital frame in terms of classical element perturbations

Casotto, S.

doi:10.1007/bf00692510

Cited by 46 publications

(36 citation statements)

References 6 publications

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“…If the motion of the binary is affected by some relatively small pK acceleration A, either Newtonian or pN in nature, its impact on the RV of the visible component A can be calculated perturbatively as follows. Casotto (1993) analytically worked out the instantaneous changes ∆v ρ , ∆v τ , ∆v ν of the radial, transverse and out-of-plane components v ρ , v τ , v ν of the velocity vector v of the relative motion of a test particle about its primary: they are…”

Section: Outline Of the Proposed Methodsmentioning

confidence: 99%

Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the star S2 in 2018

Iorio

2017

Monthly Notices of the Royal Astronomical Society

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One of the directly measured quantities used in monitoring the orbits of many of the S stars revolving around the Supermassive Black Hole (SMBH) in the Galactic Center (GC) is their radial velocity (RV) V obtained with near-infrared spectroscopy. Here, we devise a general approach to calculate both the instantaneous variations ∆V (t) and the net shifts per revolution ∆V induced on such an observable by some post-Keplerian (pK) accelerations. In particular, we look at the general relativistic Schwarzschild (gravitoelectric) and Lense-Thirring (gravitomagnetic frame-dragging) effects, and the mass quadrupole. It turns out that we may be on the verge of measuring the Schwarzschild-type 1pN static component of the SMBH's field with the star S2 for which RV measurements accurate to about ≃ 30 − 50 km s −1 dating back to t 0 = 2003.271 are currently available, and whose orbital period amounts to P b = 16 yr. Indeed, while its expected general relativistic RV net shift per orbit amounts to just ∆V GE = −11.6 km s −1 , it should reach a peak value as large as ∆V GE max (t max ) = 551 km s −1 at t max = 2018.35. Current uncertainties in the S2 and SMBH's estimated parameters yield a variation range from 505 km s −1 (2018.79) to 575 km s −1 (2018.45). The periastron shift ∆ω GE of S2 over the same time span will not be larger than 0.2 deg, while the current accuracy in estimating such an orbital element for this star is of the order of 0.6 deg. The frame-dragging and quadrupole-induced RV shifts are far smaller for S2, amounting to, at most, 0.19 km s −1 , 0.0039 km s −1 , respectively.

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Section: Outline Of the Proposed Methodsmentioning

confidence: 99%

Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the star S2 in 2018

Iorio

2017

Monthly Notices of the Royal Astronomical Society

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“…The matrix of partial derivatives T xq , though complicated, can be found in Kaula (1966) and Casotto (1993). By substituting the quasi-linear, linear and second-order perturbations of x t andẋ t into (20), we can obtain the corresponding quasi-linear, linear and second-order perturbations of the six orbital elements, respectively.…”

Section: Perturbations Of Orbital Elementsmentioning

confidence: 99%

“…Based on the partial derivatives of the disturbing potential with respect to the orbital elements given by Groves (1960) and following the method of Kozai (1959), Kaula (1961aKaula ( , 1966) developed a linear perturbation theory and connected space geodetic measurements with the geopotential coefficients. Kaula linear perturbation was then used to derive the radial, transverse and normal satellite position perturbation by Rosborough and Tapley (1987), to derive the Cartesian coordinate and velocity perturbations through the transformation between the six orbital elements and the coordinate and velocity by Casotto (1993), and to derive the perturbations of the non-singular elements for the eccentricity by Deleflie et al (2006). The linear perturbation theory was also extended by Cheng (2002) to analyze satellite-to-satellite tracking (SST) data.…”

Section: Introductionmentioning

confidence: 99%

Position and velocity perturbations for the determination of geopotential from space geodetic measurements

2008

Celestial Mech Dyn Astr

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Although space geodetic observing systems have been advanced recently to such a revolutionary level that low Earth Orbiting (LEO) satellites can now be tracked almost continuously and at the unprecedented high accuracy, none of the three basic methods for mapping the Earth's gravity field, namely, Kaula linear perturbation, the numerical integration method and the orbit energy-based method, could meet the demand of these challenging data. Some theoretical effort has been made in order to establish comparable mathematical modellings for these measurements, notably by Mayer-Gürr et al. (J Geod 78:462-480, 2005). Although the numerical integration method has been routinely used to produce models of the Earth's gravity field, for example, from recent satellite gravity missions CHAMP and GRACE, the modelling error of the method increases with the increase of the length of an arc. In order to best exploit the almost continuity and unprecedented high accuracy provided by modern space observing technology for the determination of the Earth's gravity field, we propose using measured orbits as approximate values and derive the corresponding coordinate and velocity perturbations. The perturbations derived are quasi-linear, linear and of second-order approximation. Unlike conventional perturbation techniques which are only valid in the vicinity of reference mean values, our coordinate and velocity perturbations are mathematically valid uniformly through a whole orbital arc of any length. In particular, the derived coordinate and velocity perturbations are free of singularity due to the critical inclination and resonance inherent in the solution of artificial satellite motion by using various types of orbital elements. We then transform the coordinate and velocity perturbations into those of the six Keplerian orbital elements. For completeness, we also briefly outline how to use the derived coordinate and velocity perturbations to establish observation equations of space geodetic measurements for the determination of geopotential.

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“…The instantaneous changes of the R − T − N components of the test particle's position vector r can be worked out from the following general expression (Casotto 1993)…”

Section: Analytical Calculation Of the Orbital Effectsmentioning

confidence: 99%

Orbital effects of spatial variations of fundamental coupling constants

Iorio¹

2011

Monthly Notices of the Royal Astronomical Society

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We deal with the effects induced on the orbit of a test particle revolving around a central body by putative spatial variations of fundamental coupling constants $\zeta$. In particular, we assume a dipole gradient for $\zeta(\bds r)/\bar{\zeta}$ along a generic direction $\bds{\hat{k}}$ in space. We analytically work out the long-term variations of all the six standard Keplerian orbital elements parameterizing the orbit of a test particle in a gravitationally bound two-body system. It turns out that, apart from the semi-major axis $a$, the eccentricity $e$, the inclination $I$, the longitude of the ascending node $\Omega$, the longitude of pericenter $\pi$ and the mean anomaly $\mathcal{M}$ undergo non-zero long-term changes. By using the usual decomposition along the radial ($R$), transverse ($T$) and normal ($N$) directions, we also analytically work out the long-term changes $\Delta R,\Delta T,\Delta N$ and $\Delta v_R,\Delta v_T,\Delta v_N$ experienced by the position and the velocity vectors $\bds r$ and $\bds v$ of the test particle. It turns out that, apart from $\Delta N$, all the other five shifts do not vanish over one full orbital revolution. In the calculation we do not use \textit{a-priori} simplifying assumptions concerning $e$ and $I$. Thus, our results are valid for a generic orbital geometry; moreover, they hold for any gradient direction (abridged).Comment: Latex2e, 20 pages, 1 figure, 7 tables. Version accepted by Monthly Notices of the Royal Astronomical Society (MNRAS). Error in the caption of Table 5 corrected. References update

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Position and velocity perturbations in the orbital frame in terms of classical element perturbations

Cited by 46 publications

References 6 publications

Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the star S2 in 2018

Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the star S2 in 2018

Position and velocity perturbations for the determination of geopotential from space geodetic measurements

Orbital effects of spatial variations of fundamental coupling constants

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