The residual permanent magnetic dipole moment of the moon

The residual dipole moment of the outer spherical shell of the Moon, magnetized in the field of an internal dipole is calculated for the case when the permeability of the shell differs from unity. It is shown that, using an average value of surface magnetization from returned lunar crystalline rock samples and a global figure for the lunar permeability of 1.012, that a residual moment of the order of 10¹⁵ to 10¹⁶ Am² is expected. This value is some two or three orders of magnitude lower than the moment for a shell magnetized in an external uniform field and is of the same order as the upper limit of the residual moment detected by Russellet al. (1974). At present the magnetic data and the thermal state of the Moon are not known with sufficient accuracy to distinguish between a crust magnetized in an internal dipole field of constant polarity and a crust magnetized in the dipole field of a self-reversing core dynamo. Refined measurements of the relevant parameters together with the theory presented in this paper could enable these two possibilities to be distinguished.     

The proximity of Mercury's spin to Cassini state 1 from adiabatic invariance

In determining Mercury's core structure from its rotational properties, the value of the normalized moment of inertia, C/MR², from the location of Cassini 1 is crucial. If Mercury's spin axis occupies Cassini state 1, its position defines the location of the state, where the axis is fixed in the frame precessing with the orbit. Although tidal and core-mantle dissipation drive the spin to the Cassini state with a time scale O(¹⁰⁵) years, the spin might still be displaced from the Cassini state if the variations in the orbital elements induced by planetary perturbations, which change the position of the Cassini state, cause the spin to lag behind as it attempts to follow the state. After being brought to the state by dissipative processes, the spin axis is expected to follow the Cassini state for orbit variations with time scales long compared to the 1000 year precession period of the spin about the Cassini state because the solid angle swept out by the spin axis as it precesses is an adiabatic invariant. Short period variations in the orbital elements of small amplitude should cause displacements that are commensurate with the amplitudes of the short period terms. The exception would be if there are forcing terms in the perturbations that are nearly resonant with the 1000 year precession period. The precision of the radar and eventual spacecraft measurements of the position of Mercury's spin axis warrants a check on the likely proximity of the spin axis to the Cassini state. How confident should we be that the spin axis position defines the Cassini state sufficiently well for a precise determination of C/MR²? By following simultaneously the spin position and the Cassini state position during long time scale orbital variations over past 3 million years [Quinn, T.R., Tremaine, S., Duncan, M., 1991. Astron. J. 101, 2287-2305] and short time scale variations for 20,000 years [JPL Ephemeris DE 408; Standish, E.M., private communication, 2005], we show that the spin axis will remain within one arcsec of the Cassini state after it is brought there by dissipative torques. In this process the spin is located in the orbit frame of reference, which in turn is referenced to the inertial ecliptic plane of J2000. There are no perturbations with periods resonant with the precession period that could cause large separations. We thus expect Mercury's spin to occupy Cassini state 1 well within the uncertainties for both radar and spacecraft measurements, with correspondingly tight constraints on C/MR² and the extent of Mercury's molten core. Two unlikely caveats for this conclusion are: (1) an excitation of a free spin precession by an unknown mechanism or (2) a displacement by a dissipative core mantle interaction that exceeds the measurement uncertainties.     

The decay of the mean solar magnetic field

We have analyzed the effects that differential rotation and a hypothetical meridional flow would have on the evolution of the Sun's mean line-of-sight magnetic field as seen from Earth. By winding the large-scale field into strips of alternating positive and negative polarity, differential rotation causes the mean-field amplitude to decay and the mean-field rotation period to acquire the value corresponding to the latitude of the surviving unwound magnetic flux. For a latitudinally broad two-sector initial field such as a horizontal dipole, the decay is rapid for about 5 rotations and slow with a t ^–1/2 dependence thereafter. If a poleward meridional flow is present, it will accelerate the decay by carrying the residual flux to high latitudes where the line-of-sight components are small. The resulting decay is exponential with an e-folding time of 0.75 yr (10 rotations) for an assumed 15 m s^–1 peak meridional flow speed.E.O. Hulburt Center for Space Research.Laboratory for Computational Physics.     

Possible dynamical evolution of the rotation of Venus since formation

The past evolution of the rotation of Venus has been studied by a numerical integration method using the hypothesis that only solar tidal torques and core-mantle coupling have been active since formation. It is found quite conceivable that Venus had originally a rotation similar to the other planets and has evolved in 4.5×10⁹ years from a rapid and direct rotation (12-hour spin period and nearly zero obliquity) to the present slow retrograde one.While the solid tidal torque may be quite efficient in despinning the planet, a thermally driven atmospheric tidal torque has the capability to drive the obliquity from 0° towards 180° and to stabilize the spin axis in the latter position. The effect of a liquid core is discussed and it is shown that core-mantle friction hastens the latter part of the evolution and makes even stronger the state of equilibrium at 180°. The model assumes a nearly stable balance between solid and atmospheric tides at the current rotation rate interpreting the present 243 day spin period as being very close to the limiting value.A large family of solutions allowing for the evolution, in a few billions years, of a rapid prograde rotation to the present state have been found. Noticeably different histories of evolution are observed when the initial conditions and the values of the physical parameters are slightly modified, but generally the principal trend is maintained.The proposed evolutionary explanation of the current rotation of Venus has led us to place constraints on the solid bodyQ and on the magnitude of the atmospheric tidal torque. While the constraints seem rather severe in the absence of core-mantle friction (aQ15 at the annual frequency is required, and a dominant diurnal thermal response in the atmosphere is needed), for a large range of values of the core's viscosity, the liquid core effect allows us to relax somewhat these constraints: a solid bodyQ of the order 40 can then be allowed. ThisQ value implies that a semi-diurnal ground pressure oscillation of 2 mb is needed in the atmosphere in order for a stable balance to occur between the solid and atmospheric tides at the current rotation rate. No model of atmospheric tides on Venus has been attempted in this study, however the value of 2 mb agrees well with that predicted by the model given in Dobrovolskis (1978).     

Spectral Constraints for Millisecond Pulsars Due to General Relativistic Frame Dragging

We develop a numerical code for simulating the magnetospheres of millisecond pulsars, which are expected to have unscreened electric potentials due to the lack of magnetic pair production. We incorporate General Relativistic (GR) expressions for the electric field and charge density and include curvature radiation (CR) due to primary electrons accelerated above the stellar surface, whereas inverse Compton scattering (ICS) of thermal X-ray photons by these electrons are neglected as a second-order effect. We apply the model to PSR J0437-4715, a prime candidate for testing the GR-Electrodynamic theory, and find that the curvature radiation spectrum cuts off at energies below 15 GeV, which are well below the threshold of the H.E.S.S. telescope, whereas Classical Electrodynamics predict a much higher cutoff near 100 GeV, which should be visible for H.E.S.S., if standard assumed Classical Electrodynamics apply. GR theory also predicts a relatively narrow pulse (2φ _L ∼ 0.2 phase width) centered on the magnetic axis, which sets the beaming solid angle to ∼0.5 sr per polar cap (PC) for a magnetic inclination angle of 35^∘ relative to the spin axis, given an observer which sweeps close to the magnetic axis. We also find that EGRET observations above 100 MeV of this pulsar constrain the polar magnetic field strength to B _pc < 4× 10⁸ G for a pulsar radius of 10 km and moment of inertia of 10⁴⁵ g cm². The field strength constraint becomes even tighter for a larger radius and moment of inertia. Furthermore, a reanalysis of the full EGRET data set of this pulsar, assuming the predicted pulse shape and position, should lead to even tighter constraints on neutron star and GR parameters, up to the point where the GR-derived potential and polar cap current may be questioned.     

The moon’s permanent magnetic field: a cratered-shell model

As part of our study of the larger-scale remanent magnetic field of the Moon, we have examined the effects of cratering in an otherwise spherically symmetrical shell magnetized by a concentric dipolar magnetic fieldH _o to an intensity of magnetizationc H _o, wherec is a constant. In our initial model, we assume that the material excavated from the craters is distributed with random orientation and thus does not contribute to the remanent dipole momentM _g . We further assume that the mare fill does not contribute significantly toM _g . We choose the magnetizing dipole momentM _o and the constantc such that the magnitude of the productcH _o ≃ 3 × 10⁻⁴Г at the outer surface of the shell in the equatorial plane of the dipole. This value of the intensity of remanent magnetization was chosen to be within the range 10⁻⁷−10⁻³Г’; the intensities of thermo-remanent magnetization exhibited by Apollo samples. Finally, we use the locations and diameters of the 10 largest craters on the Moon and the depth-to-diameter ratios of Pike’s formulation to model approximately the excavation of the magnetized shell. The remanent dipole momentM _g was calculated for each of three orthogonal orientations of the magnetizing dipoleM _o. The three magnitudes ofM _g fall in the range 4 × 10¹⁸−1 × 10¹⁹Г cm³ which is close to the upper limit of 10¹⁹Г cm³ estimated forM _g from the field measurements obtained with the Apollo subsatellites. Further, the distribution of the craters is such as to produce a significant transverse component ofM _g with acute angles between the spin axis andM _g in the range 51°–77°.     

Strong and Weak Damping of Slow MHD Standing Waves in Hot Coronal Loops

Slow magnetohydrodynamic (MHD) standing wave oscillations in hot coronal loops for both strong (i.e. τ_d/P∼ 1) and weak (i.e. τ_d/P≥ 2) damping are investigated taking account of viscosity, thermal conductivity and optically thin radiation. The individual effect of the dissipative terms is not sufficient to explain the observed damping. However, the combined effect of these dissipative terms is sufficient to explain the observed strong damping, as well as weak damping seen by SUMER. We find that, the ratio of decay time (τ_d) and period (P) of wave, i.e., τ_d/P (which defines the modes of damping, whether it is strong or weak) is density dependent. By varying density from 10⁸ to 10¹⁰ cm⁻³ at a fixed temperature in the temperature range 6 – 10 MK, observed by SUMER, we get two sets of damping: one for which τ _d/P∼ 1 corresponds to strong damping that occurs at lower density and another that occurs at higher density for which τ_d/P ≥ 2 corresponds to weak damping. Contrary to strong-damped oscillations, the effect of optically thin radiation provides some additional dissipation apart from thermal conductivity and viscosity in weak-damped oscillations. We have, therefore, derived a resultant dispersion relation including the effect of optically thin radiation. Solutions of this dispersion relation illustrate how damping time varies with physical parameters of loops in both strong and weak damping cases.     

Extreme sensitivity of the YORP effect to small-scale topography

Radiation recoil (YORP) torques are shown to be extremely sensitive to small-scale surface topography, using numerical simulations. Starting from a set of “base objects” representative of the near-Earth object population, random realizations of three types of small-scale topography are added: Gaussian surface fluctuations, craters, and boulders. For each, the expected relative errors in the spin and obliquity components of the YORP torque caused by the observationally unresolved small-scale topography are computed. Gaussian power, at angular scales below an observational limit, produces expected errors of order 100% if observations constrain the surface to a spherical harmonic order l?10. For errors under 10%, the surface must be constrained to at least l=20. A single crater with diameter roughly half the object's mean radius, placed at random locations, results in expected errors of several tens of percent. The errors scale with crater diameter D as D² for D>0.3 and as D³ for D<0.3 mean radii. Objects that are identical except for the location of a single large crater can differ by factors of several in YORP torque, while being photometrically indistinguishable at the level of hundredths of a magnitude. Boulders placed randomly on identical base objects create torque errors roughly 3 times larger than do craters of the same diameter, with errors scaling as the square of the boulder diameter. A single boulder comparable to Yoshinodai on 25143 Itokawa, moved by as little as twice its own diameter, can alter the magnitude of the torque by factors of several, and change the sign of its spin component at all obliquities. Most of the total torque error produced by multiple unresolved craters is contributed by the handful of largest craters; but both large and small boulders contribute comparably to the total boulder-induced error. A YORP torque prediction derived from groundbased data can be expected to be in error by of order 100% due to unresolved topography. Small surface changes caused by slow spin-up or spin-down may have significant stochastic effects on the spin evolution of small bodies. For rotation periods between roughly 2 and 10 h, these unpredictable changes may reverse the sign of the YORP torque. Objects in this spin regime may random-walk up and down in spin rate before the rubble-pile limit is exceeded and fissioning or loss of surface objects occurs. Similar behavior may be expected at rotation rates approaching the limiting values for tensile-strength dominated objects.     

Spin-up and spin-down in intermediate polars and X-ray pulsars

The accretion torque theory of Wang (1987) is applied to the spin periods of intermediate polars and X-ray pulsars. It is shown that, if the primaries are spinning near their equilibrium values, the range of positive and negative values ofP/P observed in intermediate polars is readily understood in terms of relatively small variations of mass transfer rateM of the kind generated either by long term (10³ yr) cycling ofM suggested by the nova hibernation model, or by the short term (10 yr) variations caused by magnetic cycling in the secondary stars. The variations in spin rates of the disc-fed X-ray pulsars Her X-1 and Cen X-3 can also be accounted for by the latter process.     

The simulation of the magnetic field and spin period evolution of accreting neutron stars

By means of the Monte Carlo method, we simulate the evolutionary distribution of accreting neutron stars (NSs) in the magnetic field versus spin period (B‐P) diagram where the accretion induced magnetic‐field decay model is exploited. The simulated results show that by mass accretion the B‐P distribution of the accreting NS would evolve along the equilibrium period line to a region with low field and short period. The B‐P distributions of the simulated accreting NSs are consistent with those of the observed millisecond pulsars (MSPs) after accretion of ∼ 0.1–0.2 M⊙. We also test the effects of the initial magnetic field and the spin period on the evolved B‐P distribution of the accreting NSs. It is shown that the evolved distributions of the simulated samples are independent of the selection of the initial condition when the NS magnetic field decays to a value less than ∼10¹⁰ G. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radiative lifetimes of the second negative system of O2+

The dipole moment of the A²Π_u?x²Π_g transition of O⁺₂ is calculated as a function of internuclear distance using ab initio methods. The band absorption oscillator strengths and band transition probabilities of the second negative system are derived and the resulting lifetimes are compared with experimental data. The high-lying v″ levels of the ground state may decay into low-lying v′ levels of the excited state. The corresponding radiative lifetimes are calculated.     

Nonlinear excitation of small-scale Alfvén waves by fast waves and plasma heating in the solar atmosphere

We study a nonlinear mechanism for the excitation of kinetic Alfvén waves (KAWs) by fast magneto-acoustic waves (FWs) in the solar atmosphere. Our focus is on the excitation of KAWs that have very small wavelengths in the direction perpendicular to the background magnetic field. Because of their small perpendicular length scales, these waves are very efficient in the energy exchange with plasmas and other waves. We show that the nonlinear coupling of the energy of the finite-amplitude FWs to the small-scale KAWs can be much faster than other dissipation mechanisms for fast wave, such as electron viscous damping, Landau damping, and modulational instability. The nonlinear damping of the FWs due to decay FW = KAW + KAW places a limit on the amplitude of the magnetic field in the fast waves in the solar corona and solar-wind at the level B/B ₀∼10⁻². In turn, the nonlinearly excited small-scale KAWs undergo strong dissipation due to resistive or Landau damping and can provide coronal and solar-wind heating. The transient coronal heating observed by Yohkoh and SOHO may be produced by the kinetic Alfvén waves that are excited by parametric decay of fast waves propagating from the reconnection sites.     

Tidal disruption of dissipative planetesimals

Tidal disruption is a potentially important process for the accumulation of the planets from planetesimals. The fact that stable equilibria do not exist for circular orbits inside the Roche limit has often been hypothesized to mean that any object that passes within the Roche limit is totally disrupted. We have disproven this hypothesis by solving the dynamic problem of the tidal disruption of a dissipative planetestimal during a close encounter with a protoplanet. The solution consists of a numerical integration of the three-dimensional, nonlinear equations of motion, including an approximate treatment of viscous dissipation in the solid regions of the planetesimal. The numerical methods have been extensively tested on a series of one-, two- (Jeans), and three-(Roche) dimensional test problems involving the equilibrium of a body subjected to tidal forces. The results may be scaled to planetesimals of arbitrary size, providing that the scaled equation of state applied. The calculations show that a strongly dissipative planetesimal which passes by the Earth on a parabolic orbit with a perigee within the Roche limit (≈3R_Earth) is not tidally disrupted (even for grazing incidence), and loses no more than a few percent of its mass. This result applies to bodies of radius R which have a kinematic viscosity ν ? 10¹²(R/1000km)² cm²sec^?1. Less dissipative planetesimals (ν ≈ 10¹³(R/1000 km)² cm²sec^?1) may lose up to about 20% of their mass. There are two coupled reasons why this result differs from previous hypotheses: (1) in a dynamic encounter, there is insufficient time to disrupt the planetesimal, and (2) even in circular orbit, the small velocities in the solid region imply that many orbital periods are necessary to completely disrupt the planetesimal. Hence solid and partially molten planetesimals will not experience substantial tidal disruption; completely molten bodies may be sufficiently inviscid to undergo tidal disruption.     

Particle and field environment of Uranus

In 1985 the spin axis of Uranus points within 10° of the Sun and the planet's position is very near the solar apex direction. A Uranus mission with an encounter near 1985 might expect to measure the unusual particle and field configuration of a “pole-on” magnetosphere and also properties of the interstellar medium. We give here estimates of the particle and field environment of Uranus based on extrapolation of solar wind data from 1 AU and on scaling relations for an Earth-type magnetosphere. Since the magnetic moment of Uranus is unknown, all magnetospheric parameters are derived as a function of the dipole strength. The onset of special magnetospheric properties are identified as the dipole moment increases from small to large values. A fairly complete set of magnetospheric parameters is given for a specific dipole moment to illustrate the case of a large moment.     

Fitting X-ray Afterglow Light curves of Gamma-ray Bursts by Using the Magnetar Energy Injection Model

The central compact object for some gamma-ray bursts (GRBs) may be a strongly magnetized millisecond pulsar. It can inject energy to the outer shock of the GRB by through the magnetic dipole radiation, and therefore causes the shallow decay of the early afterglow. Recently, from a large number of GRB X-ray afterglows observed by Swift/XRT(X-ray telescope), it is revealed that many of them exhibit the shallow decay about 10²∼10⁴ s after the burst prompt emission. We have fitted the X-ray afterglow light curves of 11 GRBs by using the energy injection model of a magnetar with the rotation period in the millisecond order of magnitude. The obtained result shows the validity and universality of the magnetar energy injection model in explaining the shallow decay of afterglows, and simultaneously provides some constraints on the magnetic field strength and rotation period of the central magnetar.     

Stability criterion for a light binary attracted by a heavy body

Dynamical behaviour of a small binary with equal components, each of mass m, is considered under attraction of a heavy body of mass M. Differential equations of the general three-body problem are integrated numerically using the code by S. J. Aarseth (Aarseth, Zare 1974) for mass ratios m/M within 10⁻¹¹–10⁻⁴ range. The direct and retrograde orbits of light bodies about each other are considered which lie either in the plane of moving their center of mass or in the plane perpendicular to it. It is shown numerically that the critical separation between the binary components which leads to disruption of binary is proportional to (m/M)^1/3. The criterion can be used for studying (in the first approximation) the motion of double stars and binary asteroids or computing the parameters of magnetic monopol and antimonopol pairs.     

Rotational modeling of Hyperion

Saturn’s moon, Hyperion, is subject to strongly-varying solid body torques from its primary and lacks a stable spin state resonant with its orbital frequency. In fact, its rotation is chaotic, with a Lyapunov timescale on the order of 100 days. In 2005, Cassini made three close passes of Hyperion at intervals of 40 and 67 days, when the moon was imaged extensively and the spin state could be measured. Curiously, the spin axis was observed at the same location within the body, within errors, during all three fly-bys—~ 30° from the long axis of the moon and rotating between 4.2 and 4.5 times faster than the synchronous rate. Our dynamical modeling predicts that the rotation axis should be precessing within the body, with a period of ~ 16 days. If the spin axis retains its orientation during all three fly-bys, then this puts a strong constraint on the in-body precessional period, and thus the moments of inertia. However, the location of the principal axes in our model are derived from the shape model of Hyperion, assuming a uniform composition. This may not be a valid assumption, as Hyperion has significant void space, as shown by its density of 544± 50  kg m⁻³ (Thomas et al. in Nature 448:50, 2007). This paper will examine both a rotation model with principal axes fixed by the shape model, and one with offsets from the shape model. We favor the latter interpretation, which produces a best-fit with principal axes offset of ~ 30° from the shape model, placing the A axis at the spin axis in 2005, but returns a lower reduced χ ² than the best-fit fixed-axes model.     

Precession of the isolated neutron star PSR B1828−11

Stairs, Lyne & Shemar have found that the arrival-time residuals from PSR B1828−11 vary periodically with a period ≈500 d. This behaviour can be accounted for by precession of the radio pulsar, an interpretation that is reinforced by the detection of variations in its pulse profile on the same time-scale. Here, we model the period residuals from PSR B1828−11 in terms of precession of a triaxial rigid body. We include two contributions to the residuals: (i) the geometric effect, which arises because the times at which the pulsar emission beam points towards the observer varies with precession phase; and (ii) the spin-down contribution, which arises from any dependence of the spin-down torque acting on the pulsar on the angle between its spin     and magnetic     axes. We use the data to probe numerous properties of the pulsar, most notably its shape, and the dependence of its spin-down torque on     , for which we assume the sum of a spin-aligned component (with a weight  1 − a   ) and a dipolar component perpendicular to the magnetic beam axis (weight a ), rather than the vacuum dipole torque  ( a = 1)  . We find that a variety of shapes are consistent with the residuals, with a slight statistical preference for a prolate star. Moreover, a range of torque possibilities fit the data equally well, with no strong preference for the vacuum model. In the case of a prolate star, we find evidence for an angle-dependent spin-down torque. Our results show that the combination of geometrical and spin-down effects associated with precession can account for the principal features of the timing behaviour of PSR B1828−11, without fine tuning of the parameters.     

The ultraviolet extinction by hollow spherical particles of graphite

Using the discrete dipole approximation, we have calculated the extinction efficiency of hollow spherical particles of graphite as a possible constituent of interstellar grains. The particles had a shell structure with the basal plane perpendicular to the radius. The calculations were made on the particles having the outer radiusR ₀=10 and 5nm in the wave number region from 0.8 to 8.0 m^–1 using the anisotropic optical constants. It was found that the hollow particles with the inner radiusR ₁0.65R ₀ yield an extinction feature at 4.6 m^–1, which fits fairly well to one observed in the interstellar extinction.