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Rossby waves (r-modes) in rapidly rotating neutron stars are unstable because of the emission of gravitational radiation. As a result, the stellar rotational energy is converted into both gravitational waves and r-mode energy. The saturation level for the r-mode energy is a fundamental parameter needed to determine how fast the neutron star spins down, as well as whether gravitational waves will be detectable. In this paper we study saturation by nonlinear transfer of energy to the sea of stellar '' inertial '' oscillation modes that arise in rotating stars with negligible buoyancy and elastic restoring forces. We present detailed calculations of stellar inertial modes in the WKB limit, their linear damping by bulk and shear viscosity, and the nonlinear coupling forces among these modes. The saturation amplitude is derived in the extreme limits of strong or weak driving by radiation reaction, as compared to the damping rate of low-order inertial modes. In the weak driving case, energy can be stably transferred to a small number of modes, which damp the energy as heat or neutrinos. In the strong driving case, we show that a turbulent cascade develops, with a constant flux of energy to large wavenumbers and small frequencies where it is damped by shear viscosity. We find that the saturation energy is extremely small, at least 4 orders of magnitude smaller than that found by previous investigators. We show that the large saturation energy found in the simulations of Lindblom and coworkers is an artifact of their unphysically large radiation reaction force. In most physical situations of interest, for either nascent, rapidly rotating neutron stars or neutron stars being spun up by accretion in low-mass X-ray binaries (LMXBs), the strong driving limit is appropriate and the saturation energy is roughly E r mode =ð0:5Mr 2 Ã 2 Þ ' 0:1 gr = ' 10 À6 ð spin =10 3 HzÞ 5 , where M and r * are the stellar mass and radius, respectively, gr is the driving rate by gravitational radiation, is the angular velocity of the star, and spin is the spin frequency. At such a low saturation amplitude, the characteristic time for the star to exit the region of r-mode instability is e10 3 -10 4 yr, depending sensitively on the instability curve. Although our saturation amplitude is smaller than that found by previous investigators, it is still sufficiently large to explain the observed period clustering in LMXBs. We find that the r-mode signal from both newly born neutron stars and LMXBs in the spin-down phase of Levin's limit cycle will be detectable by enhanced LIGO detectors out to $100-200 kpc.

We develop the formalism required to study the nonlinear interaction of modes in rotating Newtonian stars, assuming that the mode amplitudes are only mildly nonlinear. The formalism is simpler than previous treatments of mode-mode interactions for spherical stars, and simplifies and corrects previous treatments for rotating stars. At linear order, we elucidate and extend slightly a formalism due to Schutz, show how to decompose a general motion of a rotating star into a sum over modes, and obtain uncoupled equations of motion for the mode amplitudes under the influence of an external force. Nonlinear effects are added perturbatively via three-mode couplings, which suffices for moderate amplitude modal excitations; the formalism is easy to extend to higher order couplings. We describe a new, efficient way to compute the modal coupling coefficients, to zeroth order in the stellar rotation rate, using spin-weighted spherical harmonics. The formalism is general enough to allow computation of the initial trends in the evolution of the spin frequency and differential rotation of the background star.We apply this formalism to derive some properties of the coupling coefficients relevant to the nonlinear interactions of unstable r-modes in neutron stars, postponing numerical integrations of the coupled equations of motion to a later paper. First, we clarify some aspects of the expansion in stellar rotation frequency Ω that is often used to compute approximate mode functions. We show that in zero-buoyancy stars, the rotational modes (those modes whose frequencies vanish as Ω → 0) are orthogonal to zeroth order in Ω. From an astrophysical viewpoint, the most interesting result of this paper is that many couplings of r−modes to other rotational modes are small: either they vanish altogether because of various selection rules, or they vanish to lowest order in Ω or in compressibility. In particular, in zero-buoyancy stars, the coupling of three r−modes is forbidden entirely and the coupling of two r-modes to one hybrid, or r-g rotational mode vanishes to zeroth order in rotation frequency. The coupling of any three rotational modes vanishes to zeroth order in compressibility and in Ω. In nonzero-buoyancy stars, coupling of the r-modes to each other vanishes to zeroth order in Ω . Couplings to regular modes (those modes whose frequencies are finite in the limit Ω → 0), such as f −modes, are not zero, but since the natural frequencies of these modes are relatively large in the slow rotation limit compared to those of the r-modes, energy transfer to those modes is not expected to be efficient.

We consider radio bursts that originate from extragalactic neutron stars (NSs) by addressing three questions about source distances. What are the physical limitations on coherent radiation at GHz frequencies? Do they permit detection at cosmological distances? How many bursts per NS are needed to produce the inferred burst rate ∼ 10 3 -10 4 sky −1 day −1 ? The burst rate is comparable to the NS formation rate in a Hubble volume, requiring only one per NS if they are bright enough. However, radiation physics causes us to favor a closer population. More bursts per NS are then required but repeats in 10 to 100 yr could still be negligible. Bursts are modeled as sub-ns, coherent shot pulses superposed incoherently to produce msduration ∼ 1 Jy amplitudes; each shot-pulse can be much weaker than the burst amplitude, placing less restrictive requirements on the emission process. Nonetheless, single shot pulses are similar to the extreme, unresolved (< 0.4 ns) MJy shot pulse seen from the Crab pulsar, which is consistent with coherent curvature radiation emitted near the light cylinder by an almost neutral clump with net charge ∼ ±10 21 e and total energy 10 23 ergs. Bursts from Gpc distances require incoherent superposition of ∼ 10 12 d 2 Gpc shot pulses or a total energy 10 35 d 2 Gpc erg. The energy reservoir near the light cylinder limits the detection distance to few × 100 Mpc for a fluence ∼ 1 Jy ms unless conditions are more extreme than for the Crab pulsar. Similarly, extreme single pulses from ordinary pulsars and magnetars could be detectable from throughout the Local Group and perhaps farther. Contributions to dispersion measures from galaxy clusters will be significant for some of the bursts. We discuss tests for the signatures of bursts associated with extragalactic NSs.

Plasma lenses in the host galaxies of fast radio bursts (FRBs) can strongly modulate FRB amplitudes for a wide range of distances, including the ∼ Gpc distance of the repeater FRB121102. To produce caustics, the lens' dispersion-measure depth (DM ), scale size (a), and distance from the source (d sl ) must satisfy DM d sl /a 2 0.65 pc 2 AU −2 cm −3 . Caustics produce strong magnifications ( 10 2 ) on short time scales (∼ hours to days and perhaps shorter) along with narrow, epoch dependent spectral peaks (0.1 to 1 GHz). However, strong suppression also occurs in long-duration (∼ months) troughs. When bursts are multiply imaged, they will arrive differentially by < 1 µs to tens of ms and they will show different apparent dispersion measures, δDM apparent ∼ 1 pc cm −3 . Arrival time perturbations may mask any underlying periodicity with period 1 s. When arrival times differ by less than the burst width, interference effects in dynamic spectra will be seen. Strong lensing requires source sizes smaller than (Fresnel scale) 2 /a, which can be satisfied by compact objects such as neutron star magnetospheres but not by AGNs. Much of the phenomenology of the repeating fast radio burst source FRB121102 can be accounted for with such lenses. The overall picture can be tested by obtaining wideband spectra of bursts (from < 1 to 10 GHz and possibly higher), which can also be used to characterize the plasma environment near FRB sources. A rich variety of phenomena is expected from an ensemble of lenses near the FRB source. We discuss constraints on densities, magnetic fields, and locations of plasma lenses related to requirements for lensing to occur.

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