The spin-induced quadrupole moment of a rapidly rotating star changes the orbital dynamics in a binary system, giving rise to advance (or regression) of periastron and precession of the orbital plane. We show that these effects are important in the recently discovered radio pulsar/main sequence star binary system PSR J0045$-$7319, and can reliably account for the observed peculiar timing residuals. Precise measurements of the apsidal motion and orbital plane precession can yield valuable information on the internal structure and rotation of the star. The detection of orbital precession implies that the spin of the companion star is not aligned with the orbital angular momentum, and suggests that the supernova gave the pulsar a kick out of the original orbital plane. Tidal excitation of g-mode oscillations in the PSR J0045$-$7319 system induces an orbital period change of order $|\Delta P_{\rm orb}/P_{\rm orb}|\sim 10^{-6}$ at each periastron passage, but the secular trend depends on the radiative damping time of the g-modes. We also discuss the spin-orbit coupling effects for the accreting X-ray pulsars and the other known radio pulsar/main sequence binary, PSR B1259$-$63.

Spin-orbit interaction in neutron star/main sequence binaries and implications for pulsar timing

We have carried out high-precision timing measurements of the binary millisecond pulsar PSR J2051$-$0827 with the Effelsberg 100-m radio telescope of the Max-Planck-Institut f\"ur Radioastronomie and with the Lovell 76-m radio telescope at Jodrell Bank. The 6.5-yrs radio timing measurements have revealed a significant secular variation of the projected semi-major axis of the pulsar at a rate of $\dot x\equiv d(a_{\rm 1} \sin i)/dt = (-0.23\pm 0.03)\times 10^{-12}$, which is probably caused by the Newtonian spin-orbit coupling in this binary system leading to a precession of the orbital plane. The required misalignment of the spin and orbital angular momenta of the companion are evidence for an asymmetric supernova explosion. We have also confirmed that the orbital period is currently decreasing at a rate of $\dot P_{\rm b}=(-15.5 \pm 0.8)\times 10^{-12}$s s$^{-1}$ and have measured second and third orbital period derivatives $d^2P_{\rm b}/dt^2=(+2.1 \pm 0.3)\times 10^{-20} {\rm s^{-1}}$ and $d^3P_{\rm b}/dt^3 =(3.6 \pm 0.6)\times 10^{-28} {\rm s^{-2}}$, which indicate a quasi-cyclic orbital period variation similar to those found in another eclipsing pulsar system, PSR B1957+20. The observed variation of the orbital parameters constrains the maximal value of the companion radius to $R_{\rm c max} \sim 0.06 R_{\odot}$ and implies that the companion is underfilling its Roche lobe by 50 %. The derived variation in the quadrupole moment of the companion is probably caused by tidal dissipation similar to the mechanism proposed for PSR B1957+20. We conclude that the companion is at least partially non-degenerate, convective and magnetically active.

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Orbital variability of the PSR J2051-0827 Binary System

We discuss the main characteristics of the orbital pe- riod modulation in close binaries with late-type components. We focus on the various physical scenarios proposed to explain this phenomenon and, in particular, Hall's (1989) suggestion that it may be connected with magnetic activity. Starting from the work of Applegate (1992) and Lanza et al. (1998a), we develop an integral approach to evaluate the gravitational quadrupole mo- ment of an active star and its variations, which we consider to be an important driver of the observed orbital period changes. The method applies the tensor virial theorem after Chandrasekhar (1961) and directly relates the variation of the quadrupole mo- ment with the changes of kinetic and magnetic energy of the stellar hydromagnetic dynamo. Particular effort has been ap- plied in minimizing the number of free parameters entering the problem. A sample of 46 close binaries with period changes of alter- nate signs has been studied by our method. The amplitude of the quadrupole moment change appears to decrease with increasing angular velocity, implying that the time-variable part of the ki- netic energy of rotation varies as T = T/ 0:930:10 , with a correlation coefficient of 0.83. The length of the cycle of the or- bital period modulation seems to be correlated with the angular velocity as Pmod / 0:360:10 , but with a smaller correlation coefficient of 0.62. These results support the suggestion that a distributed non-linear dynamo is at work in the convective en- velopes of very active stars and that it strongly affects the differ- ential rotation. We also discuss the energy budget of the process responsible for the quadrupole moment variation and find that, on average, only 10% of the energy required to maintain the differential rotation may be lost by dissipation in the turbulent convective envelope during a cycle of the orbital period change. The problems of the magnetic field geometry and stability and the relationship between the length of the activity cycle, as de- termined by the change of the area of the starspots and the orbital period modulation, respectively, are also addressed.

http://aa.springer.de/papers/9349003/2300887.pdf

Orbital period modulation and quadrupole moment changes in magnetically active close binaries

The regularization of a new problem, namely the three-body problem, using 'similar' coordinate system is proposed. For this purpose we use the relation of 'similarity', which has been introduced as an equivalence relation in a previous paper (see \cite{rom11}). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The 'similar' polar angle's definition is introduced, in order to analyze the regularization's geometrical transformation. The effect of Levi-Civita's transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.

Regularization of the circular restricted three-body problem using 'similar' coordinate systems

https://commons.erau.edu/cgi/viewcontent.cgi?article=2287&context=publication

J. Sahade

Papers

Dynamics of Close Binary Systems; Double Stars; and Interacting Binary Stars