Normality condition in the ideal resonance problem     

Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1972,6(2):151-166

The Ideal Resonance Problem, defined by the Hamiltonian $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x,\mu \ll 1,$$ has been solved in Garfinkelet al. (1971). As a perturbed simple pendulum, this solution furnishes a convenient and accurate reference orbit for the study of resonance. In order to preserve the penduloid character of the motion, the solution is subject to thenormality condition, which boundsAB" andB' away from zero indeep and inshallow resonance, respectively. For a first-order solution, the paper derives the normality condition in the form $$pi \leqslant max(|\alpha /\alpha _1 |,|\alpha /\alpha _1 |^{2i} ),i = 1,2.$$ Herep _i are known functions of the constant ‘mean element’y', α is the resonance parameter defined by $$\alpha \equiv - {\rm B}'/|4AB\prime \prime |^{1/2} \mu ,$$ and $$\alpha _1 \equiv \mu ^{ - 1/2}$$ defines the conventionaldemarcation point separating the deep and the shallow resonance regions. The results are applied to the problem of the critical inclination of a satellite of an oblate planet. There the normality condition takes the form $$\Lambda _1 (\lambda ) \leqslant e \leqslant \Lambda _2 (\lambda )if|i - tan^{ - 1} 2| \leqslant \lambda e/2(1 + e)$$ withΛ ₁, andΛ ₂ known functions of λ, defined by $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$   相似文献   

A comparison of theoretical Fe xii emission line strengths with EUV observations of a solar active region     

F. P. Keenan  J. G. Doyle  S. S. Tayal  R. J. W. Henry 《Solar physics》1991,135(2):353-360

New theoretical electron-density-sensitive Fe xii emission line ratios $$R_1 = I(3s^2 3p^3 {}^4S_{3/2} - 3s3p^4 {}^4P_{5/2} )/I(3s^2 3p^3 {}^2P_{3/2} - 3s3p^4 D_{5/2} )$$ and $$R_2 = I(3s^2 3p^3 {}^2P_{3/2} - 3s3p^4 {}^2D_{5/2} )/I(3s^2 3p^3 {}^4S_{3/2} - 3s3p^2 P_{3/2} )$$ are derived using R-matrix electron impact excitation rate calculations. We have identified the Fexii \(3s^2 3p^3 {}^4S_{3/2} - 3s3p^4 {}^4P_{5/2} ,{\text{ }}3s^2 3p^3 {}^2P_{3/2} - 3s^3 3p^4 {}^2D_{5/2} ,{\text{ }}3s^2 3p^3 S_{3/2} - 3s^2 3p^3 P_{3/2} \) and \(3s^2 3p^3 {}^4S_{3/2} - 3s^2 3p^3 {}^2P_{1/2}\) transitions in an active region spectrum obtained with the Harvard S-055 spectrometer on board Skylab at wavelengths of 364.0, 382.8, 1241.7, and 1349.4 Å, respectively. Electron densities determined from the observed values of R ₁ (log N _e ? 11.0) and R ₂(log N _e ? 11.4) are significantly larger than the typical active region measurements, but are similar to those derived from some active region spectra observed with the Skylab 2082A instrument, which provides observational support for the atomic data adopted in the line ratio calculations, and also for the identification of the Fe xii transitions in the S-055 spectrum. However the observed value of R ₃ = I(1349.4 Å)/I(1241.7 Å) is approximately a factor of two larger than one would expect from theory which, considering that the 1349.4 Å line lies at the edge of the S-055 wavelength coverage, may reflect errors in the instrument efficiency curve. Another possibility is that the 1349.4 Å transition is blended, probably with Si ii 1350.1 Å.  相似文献   

Some identities for spheroidal harmonics     

R. G. Langebartel 《Astrophysics and Space Science》1991,175(1):51-56

The spheroidal harmonics expressions $$\left[ {P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) - P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ and $$\left[ {\eta ^2 P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) + \xi ^2 P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ , have ξ²+η² as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ²+η² in the form of a linear combination of terms of the formP _2m ^2s (iξ)P _2n ^2s (η)e ^i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.  相似文献   

Accretion flow dynamics during 1999 outburst of XTE J1859+226—modeling of broadband spectra and constraining the source mass     

Anuj Nandi  S. Mandal  H. Sreehari  D. Radhika  Santabrata Das  I. Chattopadhyay  N. Iyer  V. K. Agrawal  R. Aktar 《Astrophysics and Space Science》2018,363(5):90

We examine the dynamical behavior of accretion flow around XTE J1859+226 during the 1999 outburst by analyzing the entire outburst data (～166 days) from RXTE Satellite. Towards this, we study the hysteresis behavior in the hardness intensity diagram (HID) based on the broadband (3–150 keV) spectral modeling, spectral signature of jet ejection and the evolution of Quasi-periodic Oscillation (QPO) frequencies using the two-component advective flow model around a black hole. We compute the flow parameters, namely Keplerian accretion rate (\({\dot{m}}_{d}\)), sub-Keplerian accretion rate (\({\dot{m}}_{h}\)), shock location (\(r_{s}\)) and black hole mass (\(M_{\mathit{bh}}\)) from the spectral modeling and study their evolution along the q-diagram. Subsequently, the kinetic jet power is computed as \(L^{\mathrm{obs}}_{\mathrm{jet}} \sim3\mbox{--}6 \times10^{37}~\mbox{erg}\,\mbox{s}^{-1}\) during one of the observed radio flares which indicates that jet power corresponds to 8–16% mass outflow rate from the disc. This estimate of mass outflow rate is in close agreement with the change in total accretion rate (～14%) required for spectral modeling before and during the flare. Finally, we provide a mass estimate of the source XTE J1859+226 based on the spectral modeling that lies in the range of 5.2–7.9 \(M_{\odot}\) with 90% confidence.  相似文献   

Global solution of the ideal resonance problem     

Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):207-212

If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 $$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$ Herey is the momentum-vectory _k withk=1,2?N, x ₁ is thecritical argument, andx _k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian. The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by 1 $$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$ The solution isglobal in the sense that it is valid for all values of α² in the range 1 $$0 \leqslant \alpha ^2 \leqslant \infty ,$$ which embrances thelibration and thecirculation regimes of the co-ordinatex ₁, associated with α² < 1 and α² > 1, respectively. The solution includes asymptotically the limit α² → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ², and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε^1/2. The singularities that commonly arise within the libration region α² < 1 and on the separatrix α² = 1 of the phase-plane have been suppressed by means of aregularizing function 1 $$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$ introduced into the new Hamiltonian. The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α² < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α² > 1/μ, where the classical solution is valid. Thedemarcation point 1 $$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$ conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 $$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$ It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).  相似文献   

Addendum to: Strength of the Solar Coronal Magnetic Field – A Comparison of Independent Estimates Using Contemporaneous Radio and White-Light Observations     

Anshu Kumari  R. Ramesh  C. Kathiravan  T. J. Wang 《Solar physics》2017,292(12):177

This addendum uses an alternate fit for the electron density distribution \(N(r)\) (see Figure 1) and estimates the coronal magnetic field using the new model. We find that the estimates of the magnetic field are in close agreement using both the models.  相似文献   

Pulsar emission     

G. S. Sahakian 《Astrophysics》1999,42(2):190-200

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Probing the mass and anisotropy of the Milky Way gaseous halo: sight-lines toward Mrk 421 and PKS 2155-304     

A. Gupta  S. Mathur  M. Galeazzi  Y. Krongold 《Astrophysics and Space Science》2014,352(2):775-787

We recently found that the halo of the Milky Way contains a large reservoir of warm-hot gas that accounts for large fraction of the missing baryons from the Galaxy. The average physical properties of this circumgalactic medium (CGM) are determined by combining average absorption and emission measurements along several extragalactic sightlines. However, there is a wide distribution of both, the halo emission measure and the O?vii column density, suggesting that the Galactic warm-hot gaseous halo is anisotropic. We present Suzaku observations of fields close to two sightlines along which we have precise O?vii absorption measurements with Chandra. The column densities along these two sightlines are similar within errors, but we find that the emission measures are different: 0.0025±0.0006 cm^?6?pc near the Mrk 421 direction and 0.0042±0.0008 cm^?6?pc close to the PKS 2155-304 sightline. Therefore the densities and pathlengths in the two directions must be different, providing a suggestive evidence that the warm-hot gas in the CGM of the Milky Way is not distributed uniformly. However, the formal errors on derived parameters are too large to make such a claim. In the Mrk 421 direction we derive the density of \(1.6^{+2.6}_{-0.8} \times 10^{-4}~\mbox{cm}^{-3}\) and pathlength of \(334^{+685}_{-274}~\mbox{kpc}\) . In the PKS 2155-304 direction we measure the gas density of \(3.6^{+4.5}_{-1.8} \times10^{-4}~\mbox{cm}^{-3}\) and path-length of \(109^{+200}_{-82}~\mbox{kpc}\) . Thus the density and pathlength along these sightlines are consistent with each other within errors. The average density and pathlength of the two sightlines are similar to the global averages, so the halo mass is still huge, over 10 billion solar masses. With more such studies, we will be able to better characterize the CGM anisotropy and measure its mass more accurately. We can then compare the observational results with theoretical models and investigate if/how the CGM structure is related to the larger scale environment of the Milky Way. We also show that the Galactic disk makes insignificant contribution to the observed O?vii absorption; a similar conclusion was also reached by Henley and Shelton (2013) about the emission measure. We further argue that any density inhomogeneity in the warm-hot gas, be it from clumping, from the disk, or from a non-constant density gradient, would strengthen our result in that the Galactic halo path-length and the mass would become larger than what we estimate here. As such, our results are conservative and robust.  相似文献   

Diffuse cosmic gamma-ray background in the 28 ke V-4.1 MeV range from Kosmos 461 observations     

E. P. Mazets  S. V. Golenetskii  V. N. Il'inskii  Yu. A. Gur'yan  T. V. Kharitonova 《Astrophysics and Space Science》1975,33(2):347-357

Diffuse cosmic background and atmospheric gamma-radiation in the range 28 keV-4.1 MeV were studied with a scintillation spectrometer on board of the Kosmos 461 satellite. Separation of the cosmic and atmospheric components was made possible through a reliable determination of the geomagnetic dependences of albedo gamma-radiation: The spectrum of diffuse background in the energy range covered cannot be fitted with a common law. At energies below 400 keV the spectrum follows a power-law $$I = (5.6 \pm 0.5) \times 10^{ - 3} E^{ - (2.80 \pm 0.05)} cm^{ - 2} s^{ - 1} sr^{ - 1} MeV^{ - 1} .$$ Starting from 400 keV, this power-law breaks down; the spectrum revealing a clearly pronounced shoulder. Extrapolation of the power-law spectrum to higher energies shows that the gamma-ray component responsible for the change in the shape of the spectrum is quite strong, becoming predominant in the diffuse background in the range 1–100 MeV. The intensity of excess radiation is maximum in the region of 700–800 keV reaching ～1.8×10^?2 cm^?2s^?1sr^?1 MeV^?1. The shape of the high energy component spectrum of the diffuse background constructed using the data of Kosmos 461 and SAS-2 is in agreement with the hypotheses of the cosmological origin of the radiation.  相似文献   

The cosmogonic shadow effect     

Michel Azar  William B. Thompson 《Astrophysics and Space Science》1988,144(1-2):373-406

We consider the Alfvén-Arrhenius fall-down mechanism and describe an approximate model for the infall, capture and distribution of dust particles on a given magnetic field line and their possible neutralization at the ‘2’/3 points, the points at which the field aligned compnents of the gravitational and centrifugal forces are equal and opposite. We find that a small fraction (<10%) of an incoming particle distribution will actually contribute to the above ‘2’/3 fall-down process. We also show that if at the 2/3 points, the ratio of dust to plasma density is $$\frac{{n_D \left( {\tfrac{2}{3}} \right)}}{{n_p \left( {\tfrac{2}{3}} \right)}} > \frac{{10^{ - 3} }}{{r_{g_\mu } T_{eV} }}$$ . (r _gμ=radius of a grain in microns,T=plasma temperature in eV), then the dust particles will lose their charge, decouple from the field line and follow Keplerian orbits in accordance with the Alfvén-Arrhenius mechanism. We then determine the limits on the plasma parameters in order that rotation of a quasi-neutral plasma in thermal equilibrium be possible in the gravitational and dipole field of a rotating central body. The constraints imposed by the above conditions are rather weak, and the plasma parameters can have a wide range of values. For a plasma corotating with an angular velocity Ω～10^?4s^?1, we show that the plasma temperature and density must satisfy $$10^{ - 1}<< T_{(eV)}<< 10^2 ,10T_{eV}^2<< n^p \left( {cm^3 } \right)<< 10^6 $$ .  相似文献   

Gamma-ray burst measurements with A 6.3 m2 array     

K. Beurle  A. Bewick  J. S. Mills  J. J. Quenby 《Astrophysics and Space Science》1981,77(1):201-214

Two flights from Alice Springs, Australia, were achieved in November 1977 and November 1978 with a plastic scintillator -burst detector, effective area 6.3 m², thickness 5 cm, energy response in the range 50 keV to 2 MeV. In 33 hr of good, high altitude data, two bursts were detected, yielding a rate corrected to an isotropic flux of at a size of 8.5×10^–9 erg cm^–2. One event, seen at 22.14 on 15 Nov 1978, was confirmed by spacecraft measurements. The second, too small to be detected by spacecraft, arrived from 0 hr RA, –13.2° Decl. ±12° and possibly comes from a confirmed -burst source location. A galactic origin with a source distribution originating from a relatively thick disk, is favoured by these results.  相似文献