A generalized true anomaly-type parametrization, convenient to describe both
bound and open orbits of a two-body system in general relativity is introduced.
A complete description of the time evolution of both the radial and of the
angular equations of a binary system taking into account the first order
post-newtonian (1PN) is given. The gravitational radiation field emitted by the
system is computed in the 1PN approximation including higher multipole moments
beyond the standard quadrupole term. The gravitational waveforms in the time
domain are explicitly given up to the 1PN order for unbound orbits, but the
results are also illustrated on binaries on elliptic orbits with special
attention given to the effects of eccentricity.Comment: 27 pages, 10 figures, to appear in Phys. Rev.
The rotation of the bodies and the eccentricity of the orbit have significant effects on the emitted gravitational radiation of binary systems. This work focuses on the evaluation of the gravitational wave polarization states for spinning compact binaries. We consider binaries on eccentric orbits and the spin-orbit interaction up to the 1.5 post-Newtonian order in a way which is independent of the parameterization of the orbit. The equations of motion for angular variables are included. The formal expressions of the polarization states are given with the inclusion of higher order corrections to the waveform.
By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.
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