Monthly Notices of the Royal Astronomical Society, 2006
We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potentia... more We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.
An analysis of the observed radial velocities in the system CH Cygni has been conducted in the fr... more An analysis of the observed radial velocities in the system CH Cygni has been conducted in the framework of three-body dynamics. The observations and the fact that this star began symbiotic-like activity in the beginning of the 1960s can be explained in terms of a model consisting of a close binary of total mass D4 that is orbited by a normal white dwarf of mass D1 The 4 binary consists of a M _ M _. M _ red giant of mass D3.5 and a D0.5 dim star of unknown type, which is probably difficult to M _ M _ detect observationally. The orbit of this inner binary has high inclination (more than 40¡) with respect to the orbit of the white dwarf. The dynamical phenomenon known as the Kozai resonance then causes large long-period eccentricity variations in the inner binary. When in the high-eccentricity state, the binary expels gas out of the red giant. Part of this ends up on the white dwarf, causing the activity. In our model, the eccentricity of the binary was low for a long time and has been growing steadily during this century. This explains the only recent symbiotic activity.
We derive the universal solution to the Kepler-Coulomb problem with an additional inverse-square ... more We derive the universal solution to the Kepler-Coulomb problem with an additional inverse-square potential, valid for any type of orbit, and describe three prominent applications in astrodynamics: the relativistic precession of the apsides, the numerical integration of perturbed Kepler-Coulomb problems with a generalized leapfrog, and the averaged motion of earth-orbiting satellites with the J 2 perturbation. The modified orbital elements and Delaunay variables are presented as well.
We discuss the implementation of a new regular algorithm for simulation of the gravitational few-... more We discuss the implementation of a new regular algorithm for simulation of the gravitational few-body problem. The algorithm uses components from earlier methods, including the chain structure, the logarithmic Hamiltonian, and the time-transformed leapfrog. This algorithmic regularization code, AR-CHAIN, can be used for the normal N-body problem, as well as for problems with softened potentials and/or with velocity-dependent external perturbations, including post-Newtonian terms, which we include up to order PN2.5. Arbitrarily extreme mass ratios are allowed. Only linear coordinate transformations are used and thus the algorithm is somewhat simpler than many earlier regularized schemes. We present the results of performance tests which suggest that the new code is either comparable in performance or superior to the existing regularization schemes based on the Kustaanheimo-Stiefel (KS) transformation. This is true even for the two-body problem, independent of eccentricity. An important advantage of the new method is that, contrary to the older KS-CHAIN code, zero masses are allowed. We use our algorithm to integrate the orbits of the S stars around the Milky Way supermassive black hole for one million years, including PN2.5 terms and an intermediate-mass black hole. The three S stars with shortest periods are observed to escape from the system after a few hundred thousand years.
We describe a novel N-body code designed for simulations of the central regions of galaxies conta... more We describe a novel N-body code designed for simulations of the central regions of galaxies containing massive black holes. The code incorporates Mikkola's 'algorithmic' chain regularization scheme including post-Newtonian terms up to PN2.5 order. Stars moving beyond the chain are advanced using a fourth-order integrator with forces computed on a GRAPE board. Performance tests confirm that the hybrid code achieves better energy conservation, in less elapsed time, than the standard scheme and that it reproduces the orbits of stars tightly bound to the black hole with high precision. The hybrid code is applied to two sample problems: the effect of finite-N gravitational fluctuations on the orbits of the S-stars; and inspiral of an intermediate-mass black hole into the galactic center.
Blazar OJ287 exhibits large thermal flares at least twice every 12 years. The times of these flar... more Blazar OJ287 exhibits large thermal flares at least twice every 12 years. The times of these flares have been predicted successfully using the model of a quasi-Keplerian eccentric black hole binary where the secondary impacts the accretion disk of the primary, creating the thermal flares. New measurements of the historical light curve have been combined with the observations of the 2015 November/December flare to identify the impact record since year 1886, and to constrain the orbit of the binary. The orbit solution shows that the binary period, now 12.062 year, is decreasing at the rate of 36 days per century. This corresponds to an energy loss to gravitational waves that is 6.5 +- 4 % less than the rate predicted by the standard quadrupolar gravitational wave (GW) emission. We show that the difference is due to higher order gravitational radiation reaction terms that include the dominant order tail contributions.
We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potentia... more We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.
Monthly Notices of the Royal Astronomical Society, 2006
We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potentia... more We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.
An analysis of the observed radial velocities in the system CH Cygni has been conducted in the fr... more An analysis of the observed radial velocities in the system CH Cygni has been conducted in the framework of three-body dynamics. The observations and the fact that this star began symbiotic-like activity in the beginning of the 1960s can be explained in terms of a model consisting of a close binary of total mass D4 that is orbited by a normal white dwarf of mass D1 The 4 binary consists of a M _ M _. M _ red giant of mass D3.5 and a D0.5 dim star of unknown type, which is probably difficult to M _ M _ detect observationally. The orbit of this inner binary has high inclination (more than 40¡) with respect to the orbit of the white dwarf. The dynamical phenomenon known as the Kozai resonance then causes large long-period eccentricity variations in the inner binary. When in the high-eccentricity state, the binary expels gas out of the red giant. Part of this ends up on the white dwarf, causing the activity. In our model, the eccentricity of the binary was low for a long time and has been growing steadily during this century. This explains the only recent symbiotic activity.
We derive the universal solution to the Kepler-Coulomb problem with an additional inverse-square ... more We derive the universal solution to the Kepler-Coulomb problem with an additional inverse-square potential, valid for any type of orbit, and describe three prominent applications in astrodynamics: the relativistic precession of the apsides, the numerical integration of perturbed Kepler-Coulomb problems with a generalized leapfrog, and the averaged motion of earth-orbiting satellites with the J 2 perturbation. The modified orbital elements and Delaunay variables are presented as well.
We discuss the implementation of a new regular algorithm for simulation of the gravitational few-... more We discuss the implementation of a new regular algorithm for simulation of the gravitational few-body problem. The algorithm uses components from earlier methods, including the chain structure, the logarithmic Hamiltonian, and the time-transformed leapfrog. This algorithmic regularization code, AR-CHAIN, can be used for the normal N-body problem, as well as for problems with softened potentials and/or with velocity-dependent external perturbations, including post-Newtonian terms, which we include up to order PN2.5. Arbitrarily extreme mass ratios are allowed. Only linear coordinate transformations are used and thus the algorithm is somewhat simpler than many earlier regularized schemes. We present the results of performance tests which suggest that the new code is either comparable in performance or superior to the existing regularization schemes based on the Kustaanheimo-Stiefel (KS) transformation. This is true even for the two-body problem, independent of eccentricity. An important advantage of the new method is that, contrary to the older KS-CHAIN code, zero masses are allowed. We use our algorithm to integrate the orbits of the S stars around the Milky Way supermassive black hole for one million years, including PN2.5 terms and an intermediate-mass black hole. The three S stars with shortest periods are observed to escape from the system after a few hundred thousand years.
We describe a novel N-body code designed for simulations of the central regions of galaxies conta... more We describe a novel N-body code designed for simulations of the central regions of galaxies containing massive black holes. The code incorporates Mikkola's 'algorithmic' chain regularization scheme including post-Newtonian terms up to PN2.5 order. Stars moving beyond the chain are advanced using a fourth-order integrator with forces computed on a GRAPE board. Performance tests confirm that the hybrid code achieves better energy conservation, in less elapsed time, than the standard scheme and that it reproduces the orbits of stars tightly bound to the black hole with high precision. The hybrid code is applied to two sample problems: the effect of finite-N gravitational fluctuations on the orbits of the S-stars; and inspiral of an intermediate-mass black hole into the galactic center.
Blazar OJ287 exhibits large thermal flares at least twice every 12 years. The times of these flar... more Blazar OJ287 exhibits large thermal flares at least twice every 12 years. The times of these flares have been predicted successfully using the model of a quasi-Keplerian eccentric black hole binary where the secondary impacts the accretion disk of the primary, creating the thermal flares. New measurements of the historical light curve have been combined with the observations of the 2015 November/December flare to identify the impact record since year 1886, and to constrain the orbit of the binary. The orbit solution shows that the binary period, now 12.062 year, is decreasing at the rate of 36 days per century. This corresponds to an energy loss to gravitational waves that is 6.5 +- 4 % less than the rate predicted by the standard quadrupolar gravitational wave (GW) emission. We show that the difference is due to higher order gravitational radiation reaction terms that include the dominant order tail contributions.
We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potentia... more We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.
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Papers by S. Mikkola