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1.
Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m 1,..., m n we denote by N(m 1, ..., m n, k) the number of central configurations' of the n-body problem in k modulus dilatations and rotations. If m n 1,..., m n, k) is finite, then we give a bound of N(m 1,..., m n, k) which only depends of n and k.  相似文献   

2.
The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS 1 andS 2 changing the eccentricity of a solution. The family of symmetries,S 1, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S 2, is related to a possibleaffinity of the configurations space. Via a symmetry of theS 2 family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS 1.S 2, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC 2 H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:
  1. The two families of symmetriesS 1 andS 2 are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
  2. TheS 1 family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
  3. TheS 2 family of symmetries may regularize multiple collisions among central solutions of the same type.
Therefore any central solution, via a symmetryS 1 orS 2 orS 1.S 2, is equivalent to a central circular solution of the same type. That is a form of regularization.  相似文献   

3.
New stacked central configurations for the planar 5-body problem   总被引：1，自引：0，他引：1  
A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m 1, m 2, m 3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m 4, m 5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.  相似文献   

4.
Europa's surface is chemically altered by radiolysis from energetic charged particle bombardment. It has been suggested that hydrated sulfuric acid (H2SO4·nH2O) is a major surface species and is part of a radiolytic sulfur cycle, where a dynamic equilibrium exists between continuous production and destruction of sulfur polymers Sx, sulfur dioxide SO2, hydrogen sulfide H2S, and H2SO4·nH2O. We measured the rate of sulfate anion production for cyclo-octal sulfur grains in frozen water at temperatures, energies, and dose rates appropriate for Europa using energetic electrons. The measured rate is GMixture(SO42−)=fSulfur (r0/r)βG1 molecules (100 eV)−1, where fSulfur is the sulfur weight fraction, r is the grain radius, r0=50 μm, β≈1.9, and G1=0.4±0.1. Equilibrium column densities N are derived for Europa's surface and follow the ordering N(H2SO4) » N(S)>N(SO2)>N(H2S). The lifetime of a sulfur atom on Europa's surface for radiolysis to H2SO4 is τ(−S)=120(r/r0)β years. Rapid radiolytic processing hides the identity of the original source of the sulfurous material, but Iogenic plasma ion implantation and an acidic or salty ocean are candidate sources. Sulfate salts, if present, would be decomposed in <3800 years and be rapidly assimilated into the sulfur cycle.  相似文献   

5.
Ices in the solar system are observed on the surface of planets, satellites, comets and asteroids where they are continuously subordinate at particle fluxes (cosmic ions, solar wind and charged particles caught in the magnetosphere of the planets) that deeply modify their physical and structural properties. Each incoming ion destroys molecular bonds producing fragments that, by recombination, form new molecules also different from the original ones. Moreover, if the incoming ion is reactive (H+, On+, Sn+, etc.), it can concur to the formation of new molecules.Those effects can be studied by laboratory experiments where, with some limitation, it is possible to reproduce the astrophysical environments of planetary ices.In this work, we describe some experiments of 15-100 keV H+ and He+ implantation in pure sulfur dioxide (SO2) at 16 and 80 K and carbon dioxide (CO2) at 16 K ices aimed to search for the formation of new molecules. Among other results we confirm that carbonic acid (H2CO3) is formed after H-implantation in CO2, vice versa H-implantation in SO2 at both temperatures does not produce measurable quantity of sulfurous acid (H2SO3). The results are discussed in the light of their relevance to the chemistry of some solar system objects, particularly of Io, the innermost of Jupiter's Galilean satellites, that exhibits a surface very rich in frost SO2 and it is continuously bombarded with H+ ions caught in Jupiter's magnetosphere.  相似文献   

6.
Observations of the Composite InfraRed Spectrometer (CIRS) during the entire nominal Cassini mission (2004-2008) provide us with an accurate global view of composition and temperature in the middle atmosphere of Titan (between 100 and 500 km). We investigated limb spectra acquired at resolution at nine different latitudes between 56°S and 80°N, with a better sampling in the northern hemisphere where molecular abundances and temperature present strong latitudinal variations. From this limb data acquired between February 2005 and May 2008, we retrieved the vertical mixing ratio profiles of C2H2, C2H4, C2H6, C3H8, CH3C2H, C4H2, C6H6, HCN, HC3N and CO2. We present here for the first time, the latitudinal variations of the C2H6, C3H8, CO2, C2H4 and C6H6 vertical mixing ratios profiles. Some molecules, such as C2H6 or C3H8 present little variations above their condensation level. The other molecules (except CO2) show a significant enhancement of their mixing ratios poleward of 50°N. C2H4 is the only molecule whose mixing ratio decreases with height at latitudes below 46°N. Regions depleted in C2H2, HCN and C4H2 are observed around 400 km (0.01 mbar) and 55°N. We also inferred a region enriched in CO2 located between 30 and 40°N in the 2-0.7 mbar pressure range. At 80°N, almost all molecules studied here present a local minimum of their mixing ratio profiles near 300 km (∼0.07 mbar), which is in contradiction with Global Circulation Models that predict constant-with-height vertical profiles due to subsidence at the north pole.  相似文献   

7.
We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric.  相似文献   

8.
The number of equivalence classes of central configurations (abbr. c.c.) in the planar 4-body problem with three arbitrary and a fourth small mass is investigated. These c.c. are derived according to their generic origin in the 3-body problem. It is shown that each 3-body collinear c.c. generates exactly 2 non-collinear c.c. (besides 4 collinear ones) of 4 bodies with smallm 40; and that any 3-body equilateral triangle c.c. generates exactly 8 or 9 or 10 (depending onm 1,m 2,m 3) planar 4-body c.c. withm 4=0. Further, every one of these c.c. can be continued uniquely to sufficiently smallm 4>0 except when there are just 9; then exactly one of them is degenerate, and we conjecture that it is not continuable tom 4>0.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.  相似文献   

9.
We report 12.6-cm-wavelength radar observations of Europa, Ganymede, and Callisto made at the Arecibo Observatory in November 1977 and February 1979. When combined with previous observations, our results establish firmly the distinguishing radar properties of these satellites: (i) high geometric albedos, α; (ii) circular polarization ratios, μC, which anomalously exceed unity; (iii) linear polarization ratios, μL, which are approximately 0.5; and (iv) diffuse scattering which varies as cosnθ, where θ is angle of incidence and 1 ? n ? 2. We tabulate weighted-mean values of α, μC, μL, and n derived from observations between 1975 and 1979. The values of μC for Ganymede and Europa are nearly identical and significantly larger than that for Callisto. The values of n for Ganymede and Callisto are nearly identical and significantly smaller than that for Europa. Although significant albedo and/or polarization features are common in the radar spectra, the fractional rms fluctuation in disk-integrated properties is only ～10%. No time variation in the radar properties has been evident during 1976–1979.  相似文献   

10.
We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z 0 values, where z 0 lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.  相似文献   

11.
In this work, we have considered the flat FRW model of the universe in (n+2)-dimensions filled with the dark matter and the magnetic field. We present the Hubble parameter in terms of the observable parameters Ω m0 and H 0 with the redshift z and the other parameters like B 0, ω, μ 0, δ, n, w m . The natures of magnetic field B, deceleration parameter q and $\operatorname{Om}$ diagnostic have also been analyzed for accelerating expansion of the universe. From Stern data set (12 points), we have obtained the bounds of the arbitrary parameters by minimizing the χ 2 test. The best-fit values of the parameters are obtained by 66 %, 90 % and 99 % confidence levels. Now to find the bounds of the parameters (B 0,ω) and to draw the statistical confidence contour, we fixed four parameters μ 0, δ, n, w m . Here the parameter n determines the higher dimensions and we perform comparative study between three cases: 4D (n=2), 5D (n=3) and 6D (n=4) respectively. Next due to joint analysis with BAO observation, we have also obtained the bounds of the parameters (B 0,ω) by fixing other parameters μ 0, δ, n, w m for 4D, 5D and 6D. The best fit of distance modulus for our theoretical model and the Supernova Type Ia Union2 sample are drawn for different dimensions.  相似文献   

12.
Sang J. Kim  T.R. Geballe  A. Jung  Y.C. Minh 《Icarus》2010,208(2):837-849
We present latitudinally-resolved high-resolution (R = 37,000) pole-to-pole spectra of Jupiter in various narrow longitudinal ranges, in spectral intervals covering roughly half of the spectral range 2.86-3.53 μm. We have analyzed the data with the aid of synthetic spectra generated from a model jovian atmosphere that included lines of CH4, CH3D, NH3, C2H2, C2H6, PH3, and HCN, as well as clouds and haze. Numerous spectral features of many of these molecular species are present and are individually identified for the first time, as are many lines of and a few unidentified spectral features. In both polar regions the 2.86-3.10-μm continuum is more than 10 times weaker than in spectra at lower latitudes, implying that in this wavelength range the single-scattering albedos of polar haze particles are very low. In contrast, the 3.24-3.53 μm the weak polar and equatorial continua are of comparable intensity. We derive vertical distributions of NH3, C2H2 and C2H6, and find that the mixing ratios of NH3 and C2H6 show little variation between equatorial and polar regions. However, the mixing ratios of C2H2 in the northern and southern polar regions are ∼6 and ∼3 times, respectively, less than those in the equatorial regions. The derived mixing ratio curves of C2H2 and C2H6 extend up to the 10−6 bar level, a significantly higher altitude than most previous results in the literature. Further ground-based observations covering other longitudes are needed to test if these mixing ratios are representative values for the equatorial and polar regions.  相似文献   

13.
Ethane (C2H6), methylacetylene (CH3C2H or C3H4) and diacetylene (C4H2) have been discovered in Spitzer 10-20 μm spectra of Uranus, with 0.1-mbar volume mixing ratios of (1.0±0.1)×10−8, (2.5±0.3)×10−10, and (1.6±0.2)×10−10, respectively. These hydrocarbons complement previously detected methane (CH4) and acetylene (C2H2). Carbon dioxide (CO2) was also detected at the 7-σ level with a 0.1-mbar volume mixing ratio of (4±0.5)×10−11. Although the reactions producing hydrocarbons in the atmospheres of giant planets start from radicals, the methyl radical (CH3) was not found in the spectra, implying much lower abundances than in the atmospheres of Saturn or Neptune where it has been detected. This finding underlines the fact that Uranus' atmosphere occupies a special position among the giant planets, and our results shed light on the chemical reactions happening in the absence of a substantial internal energy source.  相似文献   

14.
Exploratory models of the collapse of spherical self-gravitating clouds are studied in relation to the problem of the formation of first generation star-systems. The masses which were considered are in the range of 83 to 5.2×1010 M ⊙. For simplicity, the assumed composition includes hydrogen only, which could be in the form of H, H2, H+ or H?. Since the physical conditions that might have prevailed in a primeval nebula are not well known, rather simple initial conditions were chosen: The gas starts from rest and has initially a uniform temperature. We consider the case of rather cool (T 0～100 K) neutral clouds with different initial ionization degrees. Some of the initial density-distributions here considered are uniform while others are decreasing from the center outwards. The assumed initial values for the densities are ～10?24 g cm?3, except for one of the models, for which it is ～10?26 g cm?3. Several atomic processes within the gas, including physical-chemical reactions and the evaluation of radiative emission coefficients are considered. A system of differential equations is set up in order to evaluate the concentrationsn H,n H 2,n H +,n H ? andn e as a function of time. The treatment makes possible the study of the cooling and heating properties of the gas. Furthermore, the dynamical, thermal and chemical evolution of the cloud can be followed during the collapse. The computations apply only to the optically thin stages. The models show the importance of a correct evaluation of the chemical reactions and dissipative mechanisms, which cannot be ignored in a realistic treatment of the collapse of self-gravitating clouds. The influence of the initial conditions on the dynamical and thermal properties during evolution are also analysed.  相似文献   

15.
The reaction between dicarbon (C2) and acetylene was recently suggested as a possible competitive reaction in the atmospheres of Titan, Saturn and Uranus by rate constant measurements at very low temperatures [see Canosa, A., Páramo, A., Le Picard, S.D., Sims, I.R., 2007. An experimental study of the reaction kinetics of C2(X1Σg+) with hydrocarbons (CH4, C2H2, C2H4, C2H6 and C3H8) over the temperature range 24-300 K: implications for the atmospheres of Titan and the Giant Planets. Icarus 187, 558-568]. We have investigated the reaction of the two low lying electron states of C2 and acetylene by the crossed molecular beam (CMB) technique with mass spectrometric detection. C4H, already identified as a primary product in previous CMB experiments, is confirmed as such, even though the mechanism of formation is inferred to be partly different with respect to the previous study. An experimental setup has been devised to characterize the internal population of C2 and refine the interpretation of the scattering results. The implications for the modelling of the atmospheres of Giant Planets and Titan, as well as cometary comae and the interstellar medium, are discussed.  相似文献   

16.
E.L. Gibb  M.J. Mumma  M.A. DiSanti 《Icarus》2003,165(2):391-406
We detected CH4 in eight Oort cloud comets using high-dispersion (λ/Δλ∼2×104) infrared spectra acquired with CSHELL at NASA's IRTF and NIRSPEC at the W.M. Keck Observatory. The observed comets were C/1995 O1 (Hale-Bopp), C/1996 B2 (Hyakutake), C/1999 H1 (Lee), C/1999 T1 (McNaught-Hartley), C/1999 S4 (LINEAR), C/2000 WM1 (LINEAR), C/2001 A2 (LINEAR), and 153/P Ikeya-Zhang (C/2002 C1). We detected the R0 and R1 lines of the ν3 vibrational band of CH4 near 3.3 μm in each comet, with the exception of McNaught-Hartley where only the R0 line was measured. In order to obtain production rates, a fluorescence model has been developed for this band of CH4. We report g-factors for the R0 and R1 transitions at several rotational temperatures typically found in comet comae and relevant to our observations. Using g-factors appropriate to Trot as determined from HCN, CO and/or H2O and C2H6, CH4 production rates and mixing ratios are presented. Abundances of CH4/H2O are compared among our existing sample of comets, in the context of establishing their place of origin. In addition, CH4 is compared to native CO, another hypervolatile species, and no correlation is found among the comets observed.  相似文献   

17.
A numerical model is utilized to investigate the temperature (T) and solar zenith angle (χ) control of D-region positive ion chemistry between 75 and 90 km. It is assumed that NO? is the precursor ion in a chain which involves three-body formation of the intermediary cluster ions NO+(H2O)m?1(X) (m = 1–3), where X can be N2,O2, H2O, or CO2, switching reactions which convert these weakly bound clusters to hydrates of NO+ and reaction of the third hydrate of NO+ with H2O to initiate the chain to form H+(H2O)n (n = 1–7). Zonal mean and tidal temperatures from rocket observations and theory are synthesized to obtain the best available estimate of mean latitudinal, seasonal and local time variations of temperature in this height region. Relative compositions of NO+(H2O)m and H+(H2O)n are found to vary widely over the complete range of realistic conditions; however, the relative ion populations are entirely explicable in terms of the effects of χ and T on the relative life-times of the intermediary ion clusters with respect to recombination, switching and thermal decomposition. For instance, as χ increases (and electron production decreases) beyond 60° for a given temperature, the recombination times of the intermediate ion cluster species lengthen with respect to the formation time of the H+ water clusters, causing the relative H+ water cluster population to increase and thus raise the level where the cluster ion and NO+ concentrations are equal from about 85 km (normal midday) to 90 km. For a given χ the concentrations of NO+H2O and H+(H2O)4 increase (decrease) for temperatures less than (greater than) 190 and 205 K, respectively. The transition occurs when the temperature becomes sufficiently high that the lifetimes of intermediary ion clusters with respect to thermal decomposition become less than their lifetimes with respect to H2O switching (which ultimately leads to the third hydrate of NO+ and entry into the water chain). At this point, the formation time of H+(H2O)4 becomes long compared with its lifetime with respect to thermal decomposition and its relative concentration decreases also. Implications of these results with respect to studies of the D-region are discussed.  相似文献   

18.
We investigated the chemical evolution in IC 63 nebula, a photo-dominated region (PDR). The chemical structure and the ionization state depend directly on the intensity of the incident UV radiation. The electron density is also affected by the incident UV radiation. It decreases gradually with the increase of the depth in the cloud varying from 5.9×10-5 at the surface to 9.6×10-9 in the core. Ionic carbon(C+) dominates the electron density in the outer region while ionic metals and other ions (H+, CH2D+, and HCO+) are the most dominant in the deepest region. Our results at A V = 6.7 mag are in good agreement with observations except in the case of H2S, where the calculated value is lower than the observed value by about two orders of magnitude.  相似文献   

19.
The coefficients of correlation between spectroscopic data published by Yamashita (1967) and others for carbon stars and the statistical population indices calculated for these stars at the Toru Observatory are calculated (Table II). The intensity estimates of Cai 4227 Å, Nai D lines, the C13/C12 ratio, then=n(Li/Ca) index, as well as CN and probably C2 bands are higher in population I carbon stars. The CH(G) band and probably hydrogen (H, H, H) lines as well as Baii 4554 and 4934 Å lines are stronger in population II carbon stars. The photoelectric colour indices, corrected for interstellar reddening do not show significant population effects. They can be used as spectral type equivalents. For a population criterion the CH/CN intensity ratio is proposed.  相似文献   

20.
In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle.  相似文献   

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