Extension of the solution of Kepler's equation to high eccentricities

Fernandes, Sandro da Silva

doi:10.1007/bf00691979

Cited by 7 publications

(5 citation statements)

References 2 publications

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“…The series of eq. (3) converges for all values of e < 1 like a geometric series of ratio e exp √ 1 − e 2 / 1 + √ 1 − e 2 (Wintner 1941;da Silva Fernandes 1994). During an orbital revolution E spans an angular interval of 2π, in such a way that the timing τ does not remain constant, as it happened if the pulsar was not perturbed by its companions, but exhibits a time-dependent, harmonic variation ∆τ (E) which reveals the existence of other bodies in the system.…”

Section: The Keplerian Change In the Times Of Arrival Of The Pulsarmentioning

confidence: 99%

Phenomenological constraints on accretion of non-annihilating dark matter on the PSR B1257+12 pulsar from orbital dynamics of its planets

Iorio¹

2010

J. Cosmol. Astropart. Phys.

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We analytically compute the effects that a pulsar's mass variation, whatever its physical origin may be, has on the standard Keplerian changes ∆τ Kep in the times of arrival of its pulses due to potential test particle companions, and on their orbital dynamics over long time scales. We apply our results to the planetary system of the PSR B1257+12 pulsar, located in the Galaxy at ∼ 600 pc from us, to phenomenologically constrain a putative accretion of non-annihilating dark matter on the hosting neutron star. By comparing our prediction for ∆τṀ /M to the root-mean-square accuracy of the timing residuals δ(∆τ ) = 3.0 µs we find for the mass variation rateṀ /M ≤ 1.3 × 10 −6 yr −1 . Actually, considerations related to the pulsar's lifetime, of the order of ∆t ∼ 0.8 Gyr, and to the currently accepted picture of the formation of its planets point toward a tighter constrain on the mass accretion rate, i.e.Ṁ/M ≤ 10 −9 yr −1 . Otherwise, the planets would have formed at about 300 − 700 au from PSR B1257+12, i.e. too far with respect to the expected extension of 1 − 2 au of the part of the protoplanetary disk containing the solid constituents from which they likely originated. In fact, an even smaller upper limit,Ṁ /M ≤ 10 −11 yr −1 , would likely be more realistic to avoid certain technical inconsistencies with the quality of the fit of the timing data, performed by keeping the standard value M = 1.4M ⊙ fixed for the neutron star's mass. Anyway, the entire pulsar data set should be re-processed by explicitly modeling the mass variation rate and solving for it. Model-dependent theoretical predictions for the pulsar's mass accretion, in the framework of the mirror matter scenario, yield a mass increment rate of about 10 −16 yr −1 for a value of the density of mirror matter ρ dm as large as 10 −17 g cm −3 = 5.6 × 10 6 GeV cm −3 . Such a rate corresponds to a fractional mass variation of ∆M/M ∼ 10 −7 over the pulsar's lifetime. It would imply a formation of a black hole from the accreted dark matter inner core for values of the dark matter particle's mass m dm larger than 3 × 10 3 Gev, which are, thus, excluded since PSR B1257+12 is actually not such a kind of compact object. Instead, by assuming ρ dm ∼ 10 −24 g cm −3 = 0.56 GeV cm −3 , the mass accretion rate would beṀ /M ∼ 10 −23 yr −1 , with a fractional mass variation of the order of ∆M/M ∼ 10 −14 . It rules out m dm ≥ 8 × 10 6 Gev. Extreme values ρ dm = 1.8 × 10 −13 g cm −3 = 10 11 GeV cm −3 for non-annihilating dark matter in central spike may yield the constraintṀ /M ≤ 10 −11 yr −1 ; over ∆t = 0.8 Gyr, it rules out m dm ≥ 12 Gev. Subject headings: gravitation−dark matter−planetary systems−pulsars: general−pulsars: individual, (PSR B1257+12)−extrasolar planets (dm) Ch would be smaller than in the usual case. Indeed, in terms of the Planck

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Section: The Keplerian Change In the Times Of Arrival Of The Pulsarmentioning

confidence: 99%

Phenomenological constraints on accretion of non-annihilating dark matter on the PSR B1257+12 pulsar from orbital dynamics of its planets

Iorio¹

2010

J. Cosmol. Astropart. Phys.

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“…It converges for all e < 1 like a geometric series with ratio r = [e/(1 + √ 1 − e 2 )] exp( √ 1 − e 2 )[14]. See also http://mathworld.wolfram.com/KeplersEquation.html.…”

mentioning

confidence: 99%

Solar system constraints on a Rindler-type extra-acceleration from modified gravity at large distances

Iorio¹

2011

J. Cosmol. Astropart. Phys.

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We analytically work out the orbital effects caused by a Rindlertype extra-acceleration A Rin which naturally arises in some recent models of modified gravity at large distances. In particular, we focus on the perturbations induced by it on the two-body range ρ and range-ratė ρ which are commonly used in satellite and planetary investigations as primary observable quantities. The constraints obtained for A Rin by comparing our calculations with the currently available range and range-rate residuals for some of the major bodies of the solar system, obtained without explicitly modeling A Rin , are 1 − 2 × 10 −13 m s . The constraints inferred from the planets' range and range-rate residuals are confirmed also by the latest empirical determinations of the corrections ∆̟ to the usual Newtonian/Einsteinian secular precessions of the planetary longitudes of perihelia ̟: moreover, the Earth yields A Rin ≤ 7 × 10 −16 m s −2 . Another approach which could be followed consists of taking into account A Rin in re-processing all the available data sets with accordingly modified dynamical models, and estimating a dedicated solve-for parameter explicitly accounting for it. Anyway, such a method is time-consuming. A preliminary analysis likely performed in such a way by a different author yields A ≤ 8×10 −14 m s −2 at Mars' distance and A ≤ 1×10 −14 m s −2 at Saturn's distance. The method adopted here can be easily and straightforwardly extended to other long-range modified models of gravity as well.

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“…In the case of the elliptic KE, the result of Equation ( 8) was used in Ref. [19] to derive an expansion in the basis sin nM. Equations ( 7) and (8), taken together with Equations ( 1) and (3), can be used for the iterative computation of all the higher order derivatives entering Equation (2) for e = 1.…”

Section: Methodsmentioning

confidence: 99%

“…[2]). The solution for 0 < e < 1 has been written as an expansion in powers of e [17], or as an expansion in the basis functions sin(nM) with coefficients proportional to the values J n (e) of Bessel functions [18,19]. Levi-Civita [20,21] This article describes a class of solutions of KEs, Equation (1), in terms of bivariate infinite series in powers of both e and M,…”

Section: Introductionmentioning

confidence: 99%

Bivariate Infinite Series Solution of Kepler’s Equations

Tommasini

2021

Mathematics

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A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure for the analytical computation of all the higher-order partial derivatives of the eccentric anomaly with respect to the eccentricity e and mean anomaly M in a given base point (ec,Mc) of the (e,M) plane. Explicit examples of such bivariate infinite series are provided, corresponding to different choices of (ec,Mc), and their convergence is studied numerically. In particular, the polynomials that are obtained by truncating the infinite series up to the fifth degree reach high levels of accuracy in significantly large regions of the parameter space (e,M). Besides their theoretical interest, these series can be used for designing 2-D spline numerical algorithms for efficiently solving Kepler’s equations for all values of the eccentricity and mean anomaly.

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Extension of the solution of Kepler's equation to high eccentricities

Cited by 7 publications

References 2 publications

Phenomenological constraints on accretion of non-annihilating dark matter on the PSR B1257+12 pulsar from orbital dynamics of its planets

Phenomenological constraints on accretion of non-annihilating dark matter on the PSR B1257+12 pulsar from orbital dynamics of its planets

Solar system constraints on a Rindler-type extra-acceleration from modified gravity at large distances

Bivariate Infinite Series Solution of Kepler’s Equations

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