Are Type Ia Supernovae Standard Candles?

The use of standard candles for distance measurements is wide spread. Yet, we currently do not know a pure standard candle in astronomy. The concept of standard candles involves not only the secure establishment of a unique luminosity but also a clear observational distinction of the objects as a class. Even Type Ia supernovae, whose maximum luminosity shows amongst the smallest scatter known, need to be normalised to provide accurate distances. Without this normalisation the cosmological claims based on supernovae would not be possible. With a careful normalisation Type Ia supernovae are the best known distance indicators for cosmology to date. This is most easily shown by the small dispersion around the expansion line in the Hubble diagram. Problems with the empirical normalisation remain and a theoretical understanding of this normalisation is missing. This has direct ramifications on systematic uncertainties when deriving cosmological implications from Type Ia supernovae. Improving the understanding of supernova physics is now the prime task to sharpen this tool of observational cosmology. Once the explosion mechanism is revealed a serious discussion of possible evolutionary effects in Type Ia supernovae can start.     

Extension of the critical inclination

The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J ₂ is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J ₂ and J ₄ is analyzed. Besides, when given certain values of J ₂ and J ₄, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.     

On the transverse vibrations of a revolving tether

We analyse the transverse vibrations of a tether, modelled as an inextensible cable, and revolving at an average rate equal to the orbital rate. The reference motion is a revolving rigid tether. During this motion the force in the tether (time and location dependent) remains, in a first approximation, aligned with the tether axis. Separation of variables for the vibrations about this motion gives a Legendre equation for the spatial dependency of the deformations and Hill's equations for time dependency of the in- and out-of-plane deformations. The boundary conditions on the Legendre equation generate a series of admissible values of the separation constant that become equidistant. The two Hill's equations generate a series of intervals, contracting to equidistant critical values, where the solutions are unbounded. The admissible values of the separation constant must avoid these intervals. Asymptotic expressions for the separation constant and the critical values are given. The first and second in-plane deformation mode arc unstable for zero end masses. By increasing the ratio of the concentrated over the distributed mass the deformation modes can be stabilised and the values of the separation constant can be made a multiple of the distribution of the critical points. Introducing unequal tip masses does not affect this result.     

Three-dimensional bifurcations of periodic solutions around the triangular equilibrium points of the restricted three-body problem

Vertically critical, planar periodic solutions around the triangular equilibrium points of the Restricted Three-Body Problem are found to exist for values of the mass parameter in the interval [0.03, 0.5]. Four series of such solutions are computed. The families of three-dimensional periodic solutions that branch off these critical orbits are computed for µ = 0.3 and are continued till their end. All orbits of these families are unstable.     

The effect of radiative transfer on shear-flow instability in the atmospheres on mars and venus

The results of two theoretical investigations concerning the destabilizing effects of radiative transfer on stably stratified shear flows are applied to the CO₂ atmospheres Mars and Venus. It is found that radiatively modified critical Richardson numbers remain below plausible atmospheric values throughout the stratospheres of both planets. Above certain altitudes, however, in the upper stratospheres of these planets (≈50 km on Mars and ≈100 km on Venus), critical Richardson numbers begin to increase significantly above the nonradiating critical value. This trend continues until, in the lower thermosphere, critical Richardson numbers eventually surpass atmospheric values. This effect could lead to observably greater turbulent mixing in the upper atmospheres of Mars and Venus than might be expected from terrestrial observation and from nonradiating theoretical calculations.     

黑洞吸积盘内边缘半径的演化与Blandford-Znajek过程

分别在薄盘和厚盘两种情况下,详细讨论了Ｂｌａｎｄｆｏｒｄ－Ｚｎａｊｅｋ过程对黑洞吸积盘内边缘半径演化的影响．结果表明,中心黑洞的无量纲角动量演化到某一临界值时;黑洞吸积盘的内边缘半径的变化率将由负交正．尽管盘的内边缘半径的极小值出现在Ｋｅｒｒ黑洞的能层的外边界上,但内边缘半径仍不可避免地会演化到Ｋｅｒｒ黑洞的能层中去．     

Bifurcation points in the planar problem of three bodies

An essential parameter in the planar problem of three bodies is the product of the square of the angular momentum and of the total energy (c ² H). The role of this parameter, which may be called abifurcation parameter, in establishing regions of possible motions has been shown by Marchal and Saari (1975) and Zare (1976a). There exist critical values of this parameter below which exchange between bodies cannot occur. These critical values may be calledbifurcation points.This paper gives an analytical criterion to obtain these bifurcation points for any given masses of the participating bodies.     

A New Critical Value Concerning the Genealogy of Long Period Families at L4 in the Restricted Three-body Problem

We found another critical mass ratio value μ between μ4 and μ5 concerning the genealogy of the long period family around the equilateral equilibrium point L4 in the re-stricted three-body problem. This value has not been pointed out before. We used numerical computations to show how the long period family evolves around this critical value. The case is similar to that of the critical values between μ2 and μ4, with slight difference in evolution details.     

On critical values concerning the evolution of the long period families

In a previous paper, we proposed another special critical value concerning the evolution of the long period family around the equilateral equilibrium points, besides the two values given by Henrard. Are there any other special critical values? After studying the stability curves of the long period family carefully, we gave a negative answer. During the study, we found an interesting family of periodic orbits which we called the homo family. We studied the evolution of this family following the increase of μ. With these findings, we were able to explain the origin of the four branches of periodic families emanating from L4 and the stability results of the equilateral equilibrium points.     

On the stability of mean-field models of the solar convection zone

The stability of the solutions of the mean-field theories of turbulent media is questioned. It is done here for the model equations for the solar convection zone which have been used, in particular, to explain the differential rotation. We present an approximation valid for axisymmetric, short-wave disturbances. A critical local Rayleigh number can be defined - involving eddy diffusivities - above which the stratification becomes unstable. For mixing-length models of the solar convection zone we always find sub-critical Rayleigh numbers. One must be careful, however, with other theoretical models. Those we have considered do not reach sufficiently high surface pressure values so that there the associated Rayleigh numbers exceed their critical limits. In the outermost layers in such models, therefore, the solutions could really be unstable.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

三类密近双星自转同步性参量的理论计算和统计分析

利用双星自转同步性理论给出了６９个三类密近双星系统中９３个子星的临界同步自转参量和临界自转周期．并把利用临界自转同步参量所计算的临界自转周期与由气体星自转不稳定理论所计算的临界自转周期做了比较，其结果是两者均属同一量级．     

The propagation of density waves in a Galaxy

We examine the behaviour of the solutions of the Lin-Shu dispersion relation in terms of the stability parameterQ and we find that a modification of the critical values of the dispersion speeds does not change drastically this behaviour. On the contrary the solutions of the Lynden Bell-Kalnajs dispersion relation behave in a more complicated way. The use of this dispersion relation imposes a different definition of the critical values of the dispersion speeds which are in general larger than the corresponding values of the Lin-Shu case.     

An analysis of the critical inclination problem using singularity theory

We use techniques from the theory of singularities of mappings to discuss the critical points of the Brouwer Hamiltonian M_{H, L} on the reduced phase space P_{H, L} near the parameter values H, L where the critical circle of the first order terms of M_{H, L} appears.     

The Galactic Thick Disk – Status

The status of the Galactic thick disk is reviewed. Consideration of the recent literature suggests that its vertical scale height and normalisation with respect to the thin disk remain uncertain to within a factor two, with values reported in the ranges 750–1500 pc, and 0.02–0.13, respectively. The bulk of the thick disk has kinematics (σ_U, σ_V, σ_W) = (65, 54, 38 km s^-1), and lags the thin disk by some 40 km s^-1; differences of opinion exists as to whether kinematics change with distance from the Galactic plane. The bulk of the thick disk has [Fe/H] ∼ −0.6, with little or no evidence for a vertical gradient. The question of gradients is critical for an understanding of thick disk cosmogony and needs closer attention. The reality of the so-called metal-weak thick disk (material having disklike kinematics and [Fe/H] ≤ −1.0) is also considered. The case for such material seems to be steadily growing: in the range −1.6 ≤ [Fe/H] ≤ −1.0, recent estimates suggest ρ_MWTD/ρ_Halo ∼ 0.1-0.3. While many workers regard the thick disk as a discrete entity, the caveat is made that this is a sufficient condition, but not one necessarily required by the observations. Best practice requires that both the discrete model and the alternative extended configuration be compared with observational data to examine the relative likelihood of their relevance. Recent theoretical advances are also discussed, together with the need for in situ measurements of the thick disk away from the Galactic plane. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the development of our knowledge of the motion of the moon around its centre of mass

This paper presents historical stages of the development of methods concerning observation and theory of the Moon's rotation around its centre of mass from the beginning of the 17th century up to the present day. The accuracy and usefulness of these methods is estimated and a critical attitude is assumed towards the results concerning the values of constants of the Moon's physical libration.     

Evolution of the velocity field in model potentials

We consider the evolution of the structure of the velocity field formed by the loops of the trajectory in a generalized Hénon-Heiles model potential. Box-shaped orbits alternating with periodic trajectories are shown to dominate at low values of the energy integral. Signatures of shell-and tube-shaped structures appear as the energy increases. Tube-shaped orbits are associated with stable periodic trajectories with small resonance ratios. Zones of stochastic orbits appear at values of the energy integral close to its critical value, which corresponds to the opening of the zero-velocity contour.     

Internal structure of reconnecting current sheets and the emerging flux model for solar flares

We present a steady-state model for reconnecting current sheets, which relates the central values of temperature, density and pressure within the sheet to the prescribed external values of these parameters as well as the magnetic field strength and inflow velocity (or reconnection rate). The simplifying feature of our model is the assumption of quasi-one-dimensionality so that only variations across the sheet at the centre of symmetry are considered in detail. The dimensions of the sheet, the spatial profiles and their variation with the prescribed dimensionless parameters are obtained from the model. We also obtain the conditions on the dimensionless parameters for the existence of a steady state. A beta-limitation is discovered, such that steady reconnection is impossible when the plasma beta is too small. Also, the sheet dimensions may be an order of magnitude larger than previously thought. Finally, these general results are applied to the emerging flux model for solar flares. A state of thermal nonequilibrium ensues when the current sheet between the emerging and ambient flux reaches a critical height. The effect of the beta-limitation is to make this critical height decrease with increasing magnetic field strength.Now at A.W.R.E., Aldermaston, Berks., England.     

Temperature domains in the ionosphere

The non-linear stationary temperature waves (domains) is analysed. The exact analytical solutions of the non-linear equation of the heat conductivity determine the region of existence of such domains and the critical values of plasma parameters, correspond to the increase of the periodical temperature profiles in the plasma. A stationary source of heating (photo-electrons or electric fields) may stimulate the existence of domains, when the power of the source reaches a critical value. Conditions in the F-region of the ionosphere near the equator favour the increase of the domains.     

The propagation of gravity waves through a critical layer for conditions of moderate wind shear

Solutions of the linearised hydrodynamic equations for a viscous atmosphere using (i) a full-wave integration procedure and (ii) a simplified analytical approach are used to examine the attenuation of gravity waves passing through a critical layer, where the horizontal phase velocity is equal to that of the mean wind. Particular attenuation is paid to the variation of this attenuation with values of Richardson number, Ri, greater than unity. The two sets of results are in good agreement with the predictions of Booker and Bretherton (1967) for an inviscid fluid for values of Ri up to about 4. However, a marked discrepancy from these predicted values is found for larger values of Ri, the present results indicating substantially smaller attenuation. Further calculations suggest that the wave-amplitude attenuation factor predicted by the inviscid model is approached asymptotically in the limit of vanishingly small viscosity and thermal conductivity coefficients.The inclusion of viscosity and thermal conduction gives rise to three characteristic modes of propagation for each direction of energy flow, in place of the single mode occurring in the inviscid case. The importance of energy exchange between these modes in the propagation through the critical layer is demonstrated.