The double-averaging (DA) approximation is widely employed as the standard technique in studying the secular evolution of the hierarchical three-body system. We show that effects stemmed from the ...short-time-scale oscillations ignored by DA can accumulate over long time-scales and lead to significant errors in the long-term evolution of the Lidov–Kozai cycles. In particular, the conditions for having an orbital flip, where the inner orbit switches between prograde and retrograde with respect to the outer orbit and the associated extremely high eccentricities during the switch, can be modified significantly. The failure of DA can arise for a relatively strong perturber where the mass of the tertiary is considerable compared to the total mass of the inner binary. This issue can be relevant for astrophysical systems such as stellar triples, planets in stellar binaries, stellar-mass binaries orbiting massive black holes and moons of the planets perturbed by the Sun. We derive analytical equations for the short-term oscillations of the inner orbit to the leading order for all inclinations, eccentricities and mass ratios. Under the test particle approximation, we derive the ‘corrected double-averaging’ (CDA) equations by incorporating the effects of short-term oscillations into the DA. By comparing to N-body integrations, we show that the CDA equations successfully correct most of the errors of the long-term evolution under the DA approximation for a large range of initial conditions. We provide an implementation of CDA that can be directly added to codes employing DA equations.
ABSTRACT
Resolving the small length-scale of thermonuclear detonation waves (TNDWs) in supernovae is currently not possible in multidimensional full-star simulations. Additionally, multidimensional ...simulations usually use small, oversimplistic reaction networks and adopt an ad hoc transition criterion to nuclear statistical equilibrium (NSE). The errors due to the applied approximations are not well understood. We present here a new accurate and efficient numerical scheme that accelerates the calculations by orders of magnitudes and allows the structure of TNDWs to be resolved. The numerical scheme has two important ingredients: (1) a burning limiter that broadens the width of the TNDW while accurately preserving its internal structure, and (2) an adaptive separation of isotopes into groups that are in nuclear statistical quasi-equilibrium, which resolves the time-consuming burning calculation of reactions that are nearly balanced out. Burning is calculated in situ employing the required large networks without the use of post-processing or pre-describing the conditions behind the TNDW. In particular, the approach to and deviation from NSE are calculated self-consistently. The scheme can be easily implemented in multidimensional codes. We test our scheme against accurate solutions of the structure of TNDWs and against homogeneous expansion from NSE. We show that with resolutions that are typical for multidimensional full-star simulations, we reproduce the accurate thermodynamic trajectory (density, temperature, etc.) to an accuracy that is better than a per cent for the resolved scales (where the burning limiter is not applied), while keeping the error for unresolved scales (broadened by the burning limiter) within a few per cent.
We propose a stringent observational test on the formation of warm Jupiters (gas-giant planets with 10 days <, ~ P <, ~ 100 days) by high-eccentricity (high-e) migration mechanisms. Unlike hot ...Jupiters, the majority of observed warm Jupiters have pericenter distances too large to allow efficient tidal dissipation to induce migration. To access the close pericenter required for migration during a Kozai-Lidov cycle, they must be accompanied by a strong enough perturber to overcome the precession caused by general relativity, placing a strong upper limit on the perturber's separation. For a warm Jupiter at a ~ 0.2 AU, a Jupiter-mass (solar-mass) perturber is required to be <, ~ 3 AU (<, ~ 30 AU) and can be identified observationally. Among warm Jupiters detected by radial velocities (RVs), > ~ 50% (5 out of 9) with large eccentricities (e > ~ 0.4) have known Jovian companions satisfying this necessary condition for high-e migration. In contrast, <, ~ 20% (3 out of 17) of the low-e (e <, ~ 0.2) warm Jupiters have detected additional Jovian companions, suggesting that high-e migration with planetary perturbers may not be the dominant formation channel. Complete, long-term RV follow-ups of the warm-Jupiter population will allow a firm upper limit to be put on the fraction of these planets formed by high-e migration. Transiting warm Jupiters showing spin-orbit misalignments will be interesting to apply our test. If the misalignments are solely due to high-e migration as commonly suggested, we expect that the majority of warm Jupiters with low-e (e <, ~ 0.2) are not misaligned, in contrast with low-e hot Jupiters.
Abstract
The very long term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is investigated analytically for librating Kozai–Lidov ...cycles (KLCs). Klein & Katz presented an analytic solution for the approximate dynamics on a very long timescale developed in the neighborhood of the KLCs' fixed point where the eccentricity vector is close to unity and aligned (or anti-aligned) with the quadrupole axis and for a precession rate equal to the angular frequency of the secular Kozai–Lidov equations around this fixed point. In this paper, we generalize the analytic solution to encompass a wider range of precession rates. We show that the analytic solution approximately describes the quantitative dynamics for systems with librating KLCs for a wide range of initial conditions, including values that are far from the fixed point, which is somewhat unexpected. In particular, using the analytic solution, we map the strikingly rich structures that arise for precession rates similar to the Kozai–Lidov timescale (ratio of a few).
We present a simple analytic model for the structure of non-relativistic and relativistic radiation mediated shocks. At shock velocities {beta} {sub s} {identical_to} v{sub s} /c {approx_gt} 0.1, the ...shock transition region is far from thermal equilibrium since the transition crossing time is too short for the production of a blackbody photon density (by bremsstrahlung emission). In this region, electrons and photons (and positrons) are in Compton (pair) equilibrium at temperatures T{sub s} significantly exceeding the far downstream temperature, T{sub s} >> T{sub d} {approx} 2({epsilon}n{sub u} {h_bar}{sup 3} c {sup 3}){sup 1/4}. T{sub s} {approx_gt} 10keV is reached at shock velocities {beta} {sub s} {approx} 0.2. At higher velocities, {beta} {sub s} {approx_gt} 0.6, the plasma is dominated in the transition region by e {sup {+-}} pairs and 60keV {approx}< T{sub s} {approx}< 200keV. We argue that the spectrum emitted during the breaking out of supernova (SN) shocks from the stellar envelopes (or the surrounding winds) of blue supergiants and Wolf-Rayet stars, which reach {beta} {sub s}>0.1 for reasonable stellar parameters, may include a hard component with photon energies reaching tens or even hundreds of keV. Our breakout analysis is restricted to temperatures T{sub s} {approx}< 50keV corresponding to photon energies h{nu} {approx}< 150keV, where pair creation can be neglected. This may account for the X-ray outburst associated with SN2008D, and possibly for other SN-associated outbursts with spectra not extending beyond few 100keV (e.g., XRF060218/SN2006aj).
Abstract
The very long-term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is studied analytically. This problem describes the ...evolution of a star and a planet that are perturbed either by a (circular and not too inclined) binary star system or by one other star and a second more distant star, as well as a perturbation by one distant star and the host galaxy or a compact-object binary system orbiting a massive black hole in nonspherical nuclear star clusters. Previous numerical experiments have shown that when the precession frequency is comparable to the Kozai–Lidov timescale, long-term evolution emerges that involves extremely high eccentricities with potential applications for a broad scope of astrophysical phenomena, including systems with merging black holes, neutron stars, or white dwarfs. By averaging the secular equations of motion over the Kozai–Lidov cycles (KLCs), we solve the problem analytically in the neighborhood of the KLC fixed point where the eccentricity vector is close to unity and aligned with the quadrupole axis and for a precession rate similar to the Kozai–Lidov timescale. In this regime the dynamics is dominated by a resonance between the perturbation frequency and the precession frequency of the eccentricity vector. While the quantitative evolution of the system is not reproduced by the solution far away from this fixed point, it sheds light on the qualitative behavior.
The spectrum of radiation emitted following shock breakout from a star's surface with a power-law density profile rho is proportional to x super(n) is investigated. Assuming planar geometry, local ...Compton equilibrium, and bremsstrahlung emission as the dominant photon production mechanism, numerical solutions are obtained for the photon number density and temperature profiles as a function of time for hydrogen-helium envelopes. The temperature solutions are determined by the breakout shock velocity v sub(0) and the pre-shock breakout density rho sub(0) and depend weakly on the value of n. Fitting formulae for the peak surface temperature at breakout as a function of v sub(0) and rho sub(0) are provided, with Tpeak approx = 9.44 exp 12.63(nu0/c)1/2 eV, and the time dependence of the surface temperature is tabulated. The time integrated emitted spectrum is a robust prediction of the model, determined by T sub(peak) and v sub(0) alone and insensitive to details of light travel time or slight deviations from spherical symmetry. Adopting commonly assumed progenitor parameters, breakout luminosities of approx = 10 super(45) erg s super(-1) and approx = 10 super(44) erg s super(-1) in the 0.3-10 keV band are expected for blue supergiant (BSG) and red supergiant (RSG)/He-WR progenitors, respectively (is well below the band for RSGs, unless their radius is ~10 super(13) cm). > 30 detections of SN 1987A-like (BSG) breakouts are expected over the lifetime of ROSAT and XMM-Newton. An absence of such detections would imply either that the typical parameters assumed for BSG progenitors are grossly incorrect or that their envelopes are not hydrostatic. The observed spectrum and duration of XRF 080109/SN 2008D are in tension with a non-relativistic breakout from a stellar surface interpretation.
ABSTRACT
Simple expressions for the radius and magnetic field of a system emitting synchrotron self‐absorbed radio and inverse‐Compton X‐rays are derived from first principles which involve ...observable quantities only. These expressions are useful for analysing observations of supernova blast waves interacting with dense circumstellar material at early times.
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