On the Extension of Adams–Bashforth–Moulton Methods for Numerical Integration of Delay Differential Equations and Application to the Moon’s Orbit

Abstract

One of the problems arising in modern celestial mechanics is the need of precise numerical integration of dynamical equations of motion of the Moon. The action of tidal forces is modeled with a time delay and the motion of the Moon is therefore described by a functional differential equation (FDE) called delay differential equation (DDE).Numerical integration of the orbit is normally being performed in both directions (forwards and backwards in time) starting from some epoch (moment in time). While the theory of normal forwards-in-time numerical integration of DDEs is developed and well-known, integrating a DDE backwards in time is equivalent to solving a different kind of FDE called advanced differential equation, where the derivative of the function depends on not yet known future states of the function.We examine a modification of Adams-Bashforth-Moulton method allowing to perform integration of the Moon's DDE forwards and backwards in time and the results of such integration.

Estimates of the change rate of solar mass and gravitational constant based on the dynamics of the Solar System

“…The Lense-Thirring effect is especially important for the determination of the time-varying value of GM ⊙ because it allows us to build a much more correct orbit of Mercury, which is fit to the ranging observations of the MESSENGER spacecraft. An important technical achievement was also made with the new multistep integrator (Aksim & Pavlov 2020), which is capable of handling delay differential equations in a manner that does not decrease the performance. Because the planets and the Moon are integrated together and because the lunar equations contain delay (Pavlov et al 2016), the new integrator has allowed us to build the ephemeris twice as fast.…”

Penyelesaian Model Epidemi Sir Menggunakan Metode Runge-Kutta Orde Empat Dan Metode Adams-Bashforth-Moulton

“…Pemodelan matematika telah berperan penting dalam kehidupan manusia [1]. Banyak masalah terkait bidang ilmu lain yang dapat diselesaikan menggunakan model matematika, misalnya masalah terkait fisika [2,3], biologi [4,5], dan komputasi [6][7][8][9]. Ketika masalah nyata dinyatakan ke dalam model matematika, model tersebut perlu diselesaikan sehingga penyelesaian model matematika menjadi pendekatan atas penyelesaian masalah nyata.…”

Role of lunar laser ranging in realization of terrestrial, lunar, and ephemeris reference frames

“…For numerical integration, an implementation of Adams-Bashforth-Moulton method, modified to handle delay differential equations [1], was used. Timedelayed terms appears in the differential equations of lunar rotation due to the nature of the tidal dissipation.…”

“…Mars orbiters which have an accuracy of 55 cm at best [32]. Laser ranging to the distance of Mars or Venus has not ever been 1 A collection of materials on LLR and relativity is available at http://www.issibern.ch/teams/lunarlaser 2 Another EOP series that provides both terrestrial and celestial poles is JPL EOP2: https://eop2-external.jpl.nasa.gov. This relatively new product was unknown to author at the time of writing.…”

“…Some of them are dedicated to fundamental properties of spacetime (see e.g. [25,78,79,70,24]) 1 , while some others have provided new information about tides on the Earth [76], and tides and the internal structure of the Moon [75,41,51]. In addition to the theoretical results, LLR has allowed to build a high-precision geocentric ephemeris of the Moon, featuring both orbital and rotational motion [17,58,70].…”