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1.
The Henon-Heiles mappingx=x+a(y–y 3), y=y–a(x–x3) has been studied, with the aim of finding where the unstable regions of the (x, y) plane are. When this mapping is put into the normal form, it is found to be a typical twist mapping. The criteria of Moser (1971) are used to obtain an upper limit to the size of a stable region around the origin, and this limit decreases to zero as the value of the parameter a increases toward 2.0. However, direct calculation fora=1.99 shows that there is a fairly large region insidex=0.412,y=0, from which escape from near the outer boundary requires at least 160 mappings. The region of high stability thus appears to be much larger than any region of absolute stability predicted by the KAM theorem.A general survey has been made of instability regions for the parameter valuea=1.0, this survey having been carried out to the extent which is allowed by a computer with 18-decimal-place accuracy. First, for all thex-axis fixed points (of the above mapping) deemed to be representative and significant, both the locations and variational matrix traces have been calculated. (The latter show whether the fixed point is elliptic or hyperbolic.) Ifn is the number of mappings andk is the number of circuits around the origin, then the listing (Table IV) is for fractionsk/n between 1/6 and 1/22, inclusive. (This covers the range 0x<0.96, withx=0 the fixed point forn=6,k=1).Escape toward infinity can be rapid, with less than 200 mappings necessary to reach the vicinity of then=1 fixed points (atx=±1,y=0 andx=0,y=±1) from outer regions of the (x, y) plane, such as for |x|>0.93,y=0. In this case, the unstable regions may be tongues encircling the origin. However, as the distance from the origin is decreased, the tongues can be replaced by exceedingly fine threads rapidly becoming less than say 10–16 in thickness. Such a thread issues fromx=0.905468199,y=0 and requires of the order of 40 000 mappings to escape. It does so by spiralling about the origin and penetrating through several series of loops associated with various fixed points at successively greater (absolute) values ofx(y=0). The region between this thread and the origin is therefore highly stable. Practical stability of a region may be regarded as attained when the region is interior to a series of loops for which the trace of the variational matrix is close to 2.0. This occurs forn=53,k=4, with fixed point atx=0.819786,y=0 and Trace=2.0000 0004.If an invariant curve does in fact exist, then one must be able to show that the outward spiralling from a given series of loops is brought to a halt at some stage. This does not occur in the region where direct computation is possible, as we show in this article, and it remains to be seen under what conditions it can take place.  相似文献   

2.
We show that the Hénon-Heiles system with Hamiltonian H=\frac12(y12+y22)+\frac12(ax12+bx22)+\frac13dx23+cx12x2{H=\frac12(y_1^2+y_2^2)+\frac12(ax_1^2+bx_2^2)+\frac13dx_2^3+cx_1^2x_2} is integrable in Liouvillian sense (i.e., the existence of an additional first integral) if and only if c = 0; or \frac dc=1, a=b; or \frac dc=6, a, b{\frac dc=1, a=b; {\rm or}\, \frac dc=6, a, b} arbitrary; or \frac dc=16, b=16a{\frac dc=16, b=16a}. Therefore, we get a complete classification of the Hénon-Heiles system in sense of integrability and non-integrability.  相似文献   

3.
The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form $$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$ The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ 2 is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ?B/?x≡B (1) atx=x 0 characterizes the resonant nature of the problem. With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded.  相似文献   

4.
A simple, semi-analytic method is developed for obtaining the orbits of galaxies undergoing fast collisions in which the galaxies are represented by Plummer models. The results are found to agree fairly well with those of N-body simulations.A simple formula for obtaining the angle of deflection is deduced. The maximum angle of deflection is 180° forV p/V esc(p)=1.00, about 36° forV p/V esc(p)=1.50, and about 18° forV p/V esc(p)=2.00, whereV p is the velocity at closest approachp, andV esc(p) is the parabolic velocity of escape atp. The angle of deflection of a pair of colliding elliptical galaxies without halos is about twice that for a pair of galaxies with halos for the same relative velocity at infinite separation.  相似文献   

5.
If a dynamical system ofN degrees of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 , \mu<< 1.$$ Herey is the momentum-vectory k withk=1, 2,...,N, andx 1 is thecritical argument. A first-orderglobal solution,x 1(t) andy 1(t), for theactive variables of the problem, has been given in Garfinkelet al. (1971). Sincex k fork>1 are ignorable coordinates, it follows that $$y_\kappa = const., k > 1.$$ The solution is completed here by the construction of the functionsx k(t) fork>1, derivable from the new HamiltonianF′(y′) and the generatorS(x, y′) of the von Zeipel canonical transformation used in the cited paper. The solution is subject to thenormality condition, derived in a previous paper fork=1, and extended here to 2≤k≤N. It is shown that the condition is satisfied in the problem of the critical inclination provided it is satisfied fork=1.  相似文献   

6.
A gap in the distribution of a parameter is simply the absence of the parameter for the values corresponding to the gap. The gap in the emission line redshift (z) of QSOs thus represents absence of QSOs with emission line redshift values corresponding to the gap region. Gaps in emission line redshifts of QSOs have been analysed statistically with updated data consisting of 1549 values. The study indicates: (i) There is a critical redshiftz c =2.4, which separates two distinct phases in the creation of QSOs. Forz>z c , the creation appears to have been a slow process. Atz?z c there was a triggering action which produced a burst of QSOs simultaneously. Forz c , the rate of production of QSOs have been fast. (ii) The distribution of gaps atz c ; appear to be consequence of periodicities, provided the periodicities involved are perfect and the redshift values are accurate. (iii) The distribution of gaps atz>z c are not random, but follow a definite trend.  相似文献   

7.
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8.
The fractional convective flux πF c (x c /πF) is computed for the effective level x c = logτ c = 0.125, using bi-dimensional co-spectra for relative continuum-brightness fluctuations ΔI and radial velocity fluctuations ΔV measured for the C i 5052.16 spectral line. A more uncertain flux for x Fe ≈ - 0.9 is obtained for the Fe i 5049.83 line. Since the results (Figure 1) incorporate current uncertainties in RMS ΔI , RMS ΔV and RMS ΔT (x), where ΔT are photospheric temperature fluctuations, they must be considered qualitative until these uncertainties are appreciably reduced. The requirement that the fractional convective flux < 1, places restrictions on these uncertainties which suggest that current RMS ΔT (x)'s are too large. The results confirm the importance of overshoot at the top of the solar hydrogen convection zone and suggest a non-negligible fractional convective flux throughout the lower photosphere. Qualitatively, they do not agree with the predictions of the generally-used, local, mixing-length theory or those of Parsons' (1969) modified mixing-length theory. However, qualitative agreement with the predictions of the non-local, generalized mixing-length theory of Spiegel (1963) and with the non-local theory of Ulrich (1970) cannot be considered as observational confirmation of these theories.  相似文献   

9.
The transport of thermal radiation has been considered within a finite slab which absorb and scatter anisotropically. The problem involves the space-dependent single-scattering albedow(x). Two approximations are taken forw(x). In the first it is represented in exponential form asw(x)=w 0 exp(–x/s), wherew 0 ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = r=0 R d r * p r (x/a), whered r * are known expansion coefficients anda is the half optical thickness of the slab. Analytic expressions for the forward, backward radiation intensities and fluxes are given in each approximation. The solution of the linear transport equation is performed on the basis of integral Fourier transforms.  相似文献   

10.
A family of well behaved perfect fluid balls has been derived starting with the metric potential g 44=B(1+Cr 2) n for all positive integral values of n. For n≥4, the members of this family are seen to satisfy the various physical conditions e.g. c 2 ρ≥p≥0,dp/dr<0,dρ/dr<0, along with the velocity of sound \((\sqrt{dp/c^{2}d\rho} )< 1\) and the adiabatic index ((p+c 2 ρ)/p)(dp/(c 2 dρ))>1. Also the pressure, energy density, velocity of sound and ratio of pressure and energy density are of monotonically decreasing towards the pressure free interface (r=a). The fluid balls join smoothly with the Schwarzschild exterior model at r=a. The well behaved perfect fluid balls so obtained are utilised to construct the superdense star models with their surface density 2×1014  gm/cm3. We have found that the maximum mass of the fluid balls corresponding to various values of n are decreasing with the increasing values of n. Over all maximum mass for the whole family turns out to be 4.1848M Θ and the corresponding radius as 19.4144 km while the red shift at the centre and red shift at surface as Z 0=1.6459 and Z a =0.6538 respectively this all happens for n=4. It is interesting to note that for higher values of n viz n≥170, the physical data start merging with that of Kuchowicz superdense star models and hence the family of fluid models tends to the Kuchowicz fluid models as n→∞. Consequently the maximum mass of the family of solution can not be less than 1.6096 M Θ which is the maximum mass occupied by the Kuchowicz superdense ball. Hence each member of the family for n≥4 provides the astrophysical objects like White dwarfs, Quark star, typical neutron star.  相似文献   

11.
The velocity distribution of hydrogen atoms in the terrestrial exosphere was measured as a function of radial distance (up to 7 Earth radii, ER) with the help of a Lyman-α hydrogen absorption cell, flown in 1968 on board the OGO-5 satellite. This paper contains the final analysis of the measurements. As a basis of comparison, the theory for the calculation of projected velocity distribution along a line of sight is established for the theoretical exospheric model of Chamberlain (1963). Self-absorption of Lyman-α photons along a line of sight is included to derive Lyman-α line profiles emerging from the geocorona. The effect of the hydrogen absorption cell, measured by the reduction factor R(p) is predicted as a function of impact parameter p of the line of sight, for various values of the parameters of a Chamberlain's model, nc (density of exobase level), Tc (temperature at the exobase level), and rcs (satellite critical radius). This predicted reduction factor R(p) is compared to the measured Rm(p), with the following findings: the Ly-α line width decreases with radial distance, as expected from the “evaporation and escape” theory of Chamberlain; the measured temperature Tc = 1080 K is in very good agreement with the exospheric temperature prediction from satellite drag data. An upper limit of 8 × 104at. cm?3 is imposed on nc, regardless of photometric absolute calibration. A good fit to data requires the presence of atoms in satellite orbits, distributed in a different fashion than that described by the concept of satellite critical radius. Lyman-alpha radiation pressure is thought to be the cause of this departure from the exospheric theory of Chamberlain (1963), otherwise perfectly confirmed.The same scientific rationale will be applied to exospheric hydrogen of the planets Mars and Venus in subsequent papers.  相似文献   

12.
According to the general results of a previous work (Caimmi, 1980; hereafter referred to as Paper I), solutions to EC equation, which expresses a necessary and sufficient condition for equilibrium of Emden-Chandresekhar axisymmetric, solid-body rotating polytropes (EC polytropes), are taken into consideration, of the type $$\vartheta (\xi ,\mu ) = A_0 \vartheta _0 (\upsilon ,\xi ) + \sum\limits_l^\infty {_l {\rm A}_{2l} (\upsilon )\vartheta _{2l} (\xi )P_{2l} (\mu ),} $$ with ? 2l later defined as the EC associated function of degree 2l. Thus the EC equation, involving (?, μ), is found to be equivalent to the infinite set of EC associated equations, involving ? 2l (μ). We approximate g (?, μ) by neglecting all terms of degree higher than 2 which appear in the above expression, and then search power series solutions to EC associated equations of degree 0 and 2, corresponding to any choice ofn (polytropic index, related to density distribution) andv (related to rotational distorsion). To this aim, we extend the methods used by Seidov and Kuzakhmedov (1977), and Mohan and Al-Bayaty (1980), to construct power series of the type outlined above, related to solid-body rotating configurations and originating both inside and outside the radial boundary (defined as the first zero of ?0(μ)=0). The corresponding expressions of ?0 and ?2 may serve to derive an approximate expression of, and future work becomes possible concerning the determination of some physical parameters (such as volume, mass, potential energy, angular momentum) related to any choice ofn andv. Computations have been performed forn=k/4 (0≤k≤20, i.e. 0≤n≤5) andv=0,v≈v R/2,v≈v R, withv R lowest value ofv leading to balance between gravitation and centrifugal force at the equator of the system. An upper limit to the error, ε*(μ), done in computing ? 2l , ?? 2l , and ?? 2l at any point ? for a given choice ofn andv, is estimated, ranging from large values (ε*=1E-2) forn close enough to 0 and ? close enough or outside the radial boundary, to low values (ε*=1E-10) forn far enough from 0 and no constraint on ?. Comparison between results of this paper and the accurate results by Linnell (1977, 1981) obtained using a different approach and available forn=2,v=0, andn=3,v=0, lead to a fair agreement (up to (1E?5?1E?6). It is apparent that the method followed here continues to hold when the first EC associated functions up to degree 2l are taken into account, leading — at least in way of principle — to a more refined approximation to the EC function; this would only make the related calculations much more complicated.  相似文献   

13.
A second-order libration solution of theIdeal Resonance Problem is construeted using a Lie-series perturbation technique. The Ideal Resonance Problem is characterized by the equations $$\begin{gathered} - F = B(x) + 2\mu ^2 A(x)sin^2 y, \hfill \\ \dot x = - Fy,\dot y = Fx, \hfill \\ \end{gathered} $$ together with the property thatB x vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initial-value problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished.  相似文献   

14.
The effect of X-ray irradiation on the line-driven stellar wind in X-ray binary systems is studied. The product of the X-ray luminosityL x and the densityn in the wind is a measure to distinguish between the optically thin and thick wind regimes. For an X-ray source of the radiation temperature of 10 keV, the critical value(L x n) c , is of the order of 1037 erg s–1 cm–3; hence, most of wind-fed X-ray sources lie in the optically thick wind regime because ofL x n>(L x n) c . Then the wind structure is determined not only by the parameter =(L x /nR 2),R the distance from the X-ray source, but also by the optical thickness due to helium. The formation of fully ionized helium region depends onL x andn in a way different from that of the Strömgren sphere. In such an optically thick wind, the region behind the He II ionization boundary is little affected by X-ray irradiation and the trace elements remain to be responsible for wind acceleration. Thus, its location is important for the structure of the wind and the interpretation of various phenomena in objects such as wind-fed X-ray pulsars.  相似文献   

15.
《Icarus》1987,72(3):635-646
The occultation of a bright (K∼6) infrared star by Neptune revealed a central flash at two stations and provided accurate measurements of the limb position at these and several additional stations. We have fitted this data ensemble with a general model of an oblate atmosphere to deduce the oblateness e and equatorial radius a0 of Neptune at the 1-μbar pressure level, and the position angle pn of the projected spin axis. The results are e=0.0209±0.0014, a0=25269±10 km, pn=20.1°±1°. Parameters derived from fitting to the limb data alone are in excellent agreement with parameters derived from fitting to central flash data alone (E. Lellouch, W.B. Hubbard, B. Sicardy, F. Vilas, and P. Bouchet, 1986, Nature 324, 227–231), and the principal remaining source of uncertainty appears to be the Neptune-centered declination of the Earth at the time of occultation. As an alternative to the methane absorption model proposed by Lellouch et al., we explain an observed reduction in the central flash intensity by a decrease in temperature from 150 to 135°K as the pressure rises from 1 to 400 μbar. Implications of the oblateness results for Neptune interior models are briefly discussed.  相似文献   

16.
Measurements of the Lyman α airglow intensity were made between June 1969 and June 1970 by a u.v. photometer experiment on the OGO-6 satellite. The data for the zenith intensity at altitudes between 400 and 1100 km were fitted to theoretical airglow models to derive atomic hydrogen density nc at a reference altitude, taken to be 500 km. nc was determined for each of 286 orbits throughout the year. The mean exospheric temperature T∞(J) during this period varied from 900 to 1300 K according to the Jacchia model. The solar Lyman α flux at line center F0 was also determined over each 90-min orbit in the model-fitting procedure. F0 was found to be correlated with sunspot number, in agreement with previous results. A nearly-exact linear relationship was found for F0, when averaged over ‘bins’ which are 20 sunspot numbers in width. nc was found to be inversely correlated with T∞(J); however the dependence is not that predicted by steady-state models whose only escape mechanism is Jeans evaporative escape. Unless the total atmospheric loss rate depends upon 27-day changes in the solar EUV, which is unlikely, an additional upper atmospheric loss is required in order that the total loss remain constant with T∞(J). This extra loss may be largely due to charge-exchange reactions in the exosphere, wherein energetic protons are converted to fast hydrogen atoms, as suggested previously by a number of authors. An additional result is suggested by the apparent spherical symmetry of the inferred density, namely that the familiar diurnal variation of hydrogen is absent at the high latitudes preferentially sampled by the OGO-6 data.  相似文献   

17.
The results of calculations of graphite grain formation in the atmospheres of R CrB stars are given. The parameters for the models wereM=1M ⊙,M bol=?6 mag. The effective temperature ranged from 5300K to 8300K. The chemical composition corresponded to the hydrogen-deficient carbon rich mixture:X=0,Y=0.9,Z c=0.1. The results obtained show the existence of a critical mass loss rate which is ranged fromM *≈10?6 M ⊙yr?1 forT eff=5300 K toM *≈10?5 M ⊙ yr?1 forT eff=8300 K. As soon as the rate of mass loss exceedsM * by 3–5 times the degree of condensation of carbon changes from 0 to 0.7. The finite radii of grains are about from 0.01 μm to 0.6 μm depending on the density near the condensation point, the velocity of matter outflow, and the stellar effective temperature. The duration of grain growth should amount to some dozens of days. It is supposed that the most probable explanation of dust-shell formation around R CrB stars is graphite condensation behind a shock wave arising from nonlinear stellar pulsation.  相似文献   

18.
Jun-Ichi Sakai 《Solar physics》1983,84(1-2):109-118
Transverse amplitude modulations of fast magnetosonic waves propagating perpendicular to the background magnetic field are shown to be unstable on a time scale τ ～- λ∥/V aφ, if the wave amplitude φ exceeds a critical value, φ c = C s/V a. The slow modes generated by the modulational instability under gravity can propagate along the magnetic field with the characteristic velocity, V ph = g/2k ∥ V aφ. The applications of this modulational instability and slow-mode generation mechanism to a solar plasma are discussed.  相似文献   

19.
An attempt has been made to study the effects of the opacity mechanisms on pulsational instability of some theoretical models having different masses and in different evolutionary stages. Different combinations of the opacity exponentsn, s, and helium-hydrogen ratio,B, have been considered. Based on these combinations, Gamma-mechanism and Kappa-mechanism have been compared. Gamma-mechanism (n=0,s=0) is found to be totally ineffective forB=0.01 in producing a driving force. Even a high value ofB (=0.38) is also not suitable for instability. The combination ofn=0.7,s=2.1, andB=0.25 has been found to give the largest negative dissipation for a 15.6M ⊙ star in hydrogen-burning phase although the same does not hold for helium-burning phase. The conditions in which these excitation mechanisms could be most effective have been discussed.  相似文献   

20.
Radiation from an optically thick, tenuous, isothermal and magnetized plasma is considered under conditions typical for X-ray pulsars, in the approximation of coupled diffusion of normal modes. The spectra are calculated of the fluxes and specific intensities of outgoing radiation, their dependences on the plasma densityN, temperatureT and magnetic fieldB are analysed with due regard to the vacuum polarization by a strong magnetic field. Simple analytical expressions are obtained in the limiting cases for the fluxes and intensities. It is shown that atE B »E a (E B =11.6B 12 keV,E a ?0.1N 22 1/2 T 1 ?3/4 keV,B 12=B/1012 G,N 22=N/1022 cm?3,T 1=T/10 keV) the magnetic field strongly intensifies the flux and changes its spectrum in the regionE a ?E ?E B . AtE ?T the spectrum of the energy flux is almost flat in the region \(\sqrt {E_a E_B } \lesssim E \lesssim E_B \) . For homogeneous plasma without Comptonization the cyclotron line atE?=E B appears in emission, though in many other cases it may appear in absorption. The vacuum polarization may produce the ‘vacuum feature’ atE?E W ?13N 22 1/2 B 12 ?1 keV, which, as a rule, appears in absorption. The intensity spectra vary noticeably with the direction of radiation, in particular, at some directions nearB, the spectra become harder than in other directions. Quantization of the magnetic field (E B >T) strongly increases the plasma luminosity (∝E B /T for homogeneous plasma). The results obtained explain a number of basic features in the observed X-ray pulsar spectra.  相似文献   

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