Fourier analysis of the light curves of eclipsing variables,XXIII

The method of evaluating the photometric perturbationsB _2m of eclipsing variables in the frequency domain, developed by Kopal (1959, 1975e, 1978) for an interpretation of mutual eclipses in systems whose components are distorted by axial rotation and mutual tidal action. The aim of the present paper has been to establish explicit expressions for the photometric perturbationB _2m in such systems, regardless of the kind of eclipses and non-integral values ofm. Recently, Kopal (1978) introduced two different kinds of integrals with respect to associated α-functions andI-integrals which have been expressed in terms of certain general types of series that can be easily programmed for automatic computation within seconds of real time on highspeed computers. Following a brief introduction (Section 1) in which the need of this new approach will be expounded, in Section 3 we shall deduce the integral $$\int_0^{\theta \prime } {\tfrac{{\alpha _n^\prime }}{\delta }} d(sin^{2m} \theta )$$ in terms of a certain general type of series and also β-function, which should enable us to evaluate explicit expressions forf _* ^(h) ,f ₁ ^(h) ,f ₂ ^(h) as well asB _2m .     

Heat and mass transfer of an oscillatory flow with Hall current

In the first part of these notes new expressions—simpler than any previously obtained—are presented in integral form for the derivatives of the α _n ⁰ -functions (required for an interpretation of the observed light changes of eclipsing variables) with respect to the fractional radiir _{1, 2} and projected separation δ of their centres in terms of the modified Bessel functionsK _{0, 1} (x) of the second kind; and utilized for establishing new asymptotic formulae for the computation of ‘boundary integrals’ of the formJ _?1 ⁰ ,n(μ). In the second part of this paper, by a resort to bi-polar coordinates, we shall establish a new type of expansions for the α _n ⁰ -functions valid for any type of eclipses, and converging faster than the expansions of the cross-correlation integral of the form (1) for α _n ⁰ that have so far been established.     

Fourier analysis of light curves of eclipsing variable stars

The photometric perturbationsB _h ^(l) arising from both tidal and rotational distortion of a close eclipsing binary have been given in two previous papers (Livaniou, 1977; Rovithis-Livaniou, 1977). The aim of the present paper will be to find the eclipse perturbationsB _2m =B _{2m, tid} +B _{2m, rot} of a close binary exhibiting partial eclipses. This will be done giving the suitable combinations of theB _h ^(l) 's and will make easier the application to real stars. After a very brief introduction, Section 2 gives both theB _{2m, tid} andB _{2m, rot} for uniformly bright discs; while in Sections 3 and 4 they are given for linear and quadratic limb-darkening, respectively. Finally, Section 5 gives a brief discussion of the results.     

Sur un formalisme explicitant la loi des distances des planetes

The well-known Titius-Bode law (T-B) giving distances of planets from the Sun was improved by Basano and Hughes (1979) who found: $$a_n = 0.285 \times 1.523^n ;$$ a _n being the semi-major axis expressed in astronomical units, of then-th planet. The integern is equal to 1 for Mercury, 2 for Venus etc. The new law (B-H) is more natural than the (T-B) one, because the valuen=?∞ for Mercury is avoided. Furthermore, it accounts for distances of all planets, including Neptune and Pluto. It is striking to note that this law:

1. does not depend on physical parameters of planets (mass, density, temperature, spin, number of satellites and their nature etc.).
2. shows integers suggesting an unknown, obscure wave process in the formation of the solar system.

In this paper, we try to find a formalism accounting for the B-H law. It is based on the turbulence, assumed to be responsible of accretion of matter within the primeval nebula. We consider the function $$\psi ^2 (r,t) = |u^2 (r,t) - u_0^2 |$$ , whereu ²(r, t) stands for the turbulence, i.e., the mean-square deviation velocities of particles at the pointr and the timet; andu ₀ ² is the value of turbulence for which the accretion process of matter is optimum. It is obvious that Ψ²(r _n,t₀) = 0 forr _n=0.285×1.523 ⁿ at the birth timet ₀ of proto-planets. Under these conditions, it is easily found that $$\psi ^2 (r,t_0 ) = \frac{{A^2 }}{r}\sin ^2 [\alpha log r - \Phi (t_0 )]$$ With α=7.47 and Φ(t ₀)=217.24 in the CGS system, the above function accounts for the B-H law. Another approach of the problem is made by considering fluctuations of the potentialU(r, t) and of the density of matter ρ(r, t). For very small fluctuations, it may be written down the Poisson equation $$\Delta \tilde U(r,t_0 ) + 4\pi G\tilde \rho (r,t_0 ) = 0$$ , withU(r, t)=U ₀(r)+?(r, t ₀ ) and \(\tilde \rho (r,t_0 )\) . It suffices to postulate \(\tilde \rho (r,t_0 ) = k[\tilde U(r,t_0 )/r^2 ](k = cte)\) for finding the solution $$\tilde U(r,t_0 ) = \frac{{cte}}{{r^{1/2} }}\cos [a\log r - \zeta (t_0 )]$$ . Fora=14.94 and ζ(t ₀)=434.48 in CGS system, the successive maxima of ?(r,t ₀) account again for the B-H law. In the last approach we try to write Ψ(r, t) under a wave function form $$\Psi ^2 (r,t) = \frac{{A^2 }}{r}\sin ^2 \left[ {\omega \log \left( {\frac{r}{v} - t} \right)} \right].$$ It is emphasized that all calculations are made under mathematical considerations.     

A relation between Giorgilli-Galgani's and Deprit's algorithms

If \(T = \sum\nolimits_{i = 1}^\infty {\varepsilon ^i } T_i\) and \(W = \sum\nolimits_{n = 1}^\infty {n\varepsilon ^{n - 1} } W^{\left( n \right)}\) are respectively the generators of Giorgilli-Galgani's and Deprit's transformations, we show that the change of variables generated byT is the inverse of the one generated byW, ifT _i =W ⁽ⁱ⁾ for anyi. The method used is to show that the recurrence which defines the first algorithm can also be obtained with the second one.     

Molecular and low excitation emission in high mass post-main-sequence nebulae

We have undertaken mapping and spectroscopy of a broad range of type I post-Main-Sequence nebulae in COJ=1→0,J=2→1, andJ=3→2, using the 12 m antenna at Kitt Peak, and the 45 m facility of the Nobeyama Radio Observatory. As a consequence, we find COJ=2→1 emission associated with NGC 3132 and NGC 6445, determine the location of COJ=1→0 emission in the nucleus of NGC 6302, and obtain (for the first time) COJ=3→2 spectroscopy for a substantial cross-section of type I sources. LVG analysis of the results suggests densitiesn(H₂) ～ 10⁴ cm^?3, and velocity gradients dv/dr ～ 2×10² in both NGC 7027 and CRL 618, commensurate with uniform expansion of a constant velocity outflow, whilst for the case of NGC 2346 these values probably exceedn(H₂) ～ 4.0×10⁵ cm^?3. dv/dr ～ 2.6×10³ km s^?1 andT _k ～10² K, implying appreciable compression (and shock heating?) of the CO excitation zone. Hi masses extend over a typical range 0.01<M(Hi)/M _⊙<1, whilst corresponding estimates of the progenitor mass imply 0.7<M _prog/M _⊙<2.3; values significantly in excess of those pertinent for normal PN, although somewhat at the lower end of the type I mass range. COJ=3→2 profiles for CRL 2688 confirm the presence of an extended plateau with width Δv～85 km s^?1, whilst modestJ=3→2 enhancement is also observed for the high-velocity components in NGC 7027. TheJ=3→2 spectrum for NGC 2346 appears to mimic lower-frequency results reasonably closely, confirming the presence of a double-peaked structure towards the core, and predominantly unitary profiles to the north and south, whilst there is also evidence to suggest appreciableJ=3→2 asymmetry in CRL 618 compared to lower-frequency measures. The status of an extended cloud near HB 5 remains uncertain, although this clearly represents a remarkably complex region with velocity span ΔV～50 km s^?1. Our presentJ=3→2 results appear to track lower frequency measures extremely closely, implying local densitiesn(H₂)>3×10³ cm^?3—although temperatures close to theV _lsr of HB 5 are relatively weak, and of orderT _MB (J=3→2)≤0.9 K. Finally, as a result of both this, and previous investigations we find that of type I sources so far observed in CO, some ～42% appear to possess detectable levels of emissionT _r ^* >0.1 K. Similarly, in cross-correlating this data with other results, we note a closely linear relation betweenJ=2→1 antenna temperaturesT _MB, and the surface brightness of H₂ S(1) quadrupole emissionS(H₂)—a trend which appears also to be reflected betweenS(H₂) and corresponding parameters for [Oi], [Oii], [Ni], [Nii], and [Sii]. Such relations almost certainly arise from comparable secular variations in line intensities, although the CO, H₂, and optical emission components are likely to derive from disparate line excitation zones. As a consequence, it is clear that whilst H₂ S(1) emission is probably enhanced as a result of local shock activity, the evidence for post-shock excitation of the CO and optical forbidden lines is at best marginal. Similarly, although it seems likely that CO emission derives from circum-nebular Hi shells with kinetic temperatureT _k ～ 30 K or greater, the predominant fraction of low-excitation emission arises from a mix of charge exchange reactions, nebular stratification and, probably most importantly, the influence of UV shadow zones and associated neutral inclusions.     

Magnetized string cosmological model in cylindrically symmetric inhomogeneous universe-revisited

Cylindrically symmetric inhomogeneous magnetized string cosmological model is investigated. The source of the magnetic field is due to an electric current produced along x-axis. F ₂₃ is the only non-vanishing component of electromagnetic field tensor. To get the deterministic solution, it has been assumed that the expansion (θ) in the model is proportional to the eigen value σ ₁ ¹ of the shear tensor σ _j ⁱ . The physical and geometric properties of the model are also discussed in presence and absence of magnetic field.     

Some identities for spheroidal harmonics

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.     

Comparative analysis of photometric variability of TT ARI in the years 1994–1995 and 2001, 2004

We present the results of photometric observations of a bright cataclysmic variable TT Ari with an orbital period of 0.13755 days. CCD observations were carried out with the Russian-Turkish RTT 150 telescope in 2001 and 2004 (13 nights). Multi-color photoelectric observations of the system were obtained with the Zeiss 600 telescope of SAO RAS in 1994–1995 (6 nights). In 1994–1995, the photometric period of the system was smaller than the orbital one (0 _. ^d 132 and 0 _. ^d 134), whereas it exceeded the latter (0 _. ^d 150 and 0 _. ^d 148) in 2001, 2004. An additional period exceeding the orbital one (0 _. ^d 144) is detected in 1995 modulations. We interpret it as indicating the elliptic disc precession in the direction of the orbital motion. In 1994, the variability in colors shows periods close to the orbital one (0 _. ^d 136, b-v), as well as to the period indicating the elliptic disk precession (0 _. ^d 146, w-b). We confirm that during the epochs characterized by photometric periods shorter than the orbital one, the quasi-periodic variability of TT Ari at time scales about 20 min is stronger than during epochs with long photometric periods. In general, the variability of the system can be described as a “red” noise with increased amplitudes of modulations at characteristic time scales of 10–40 min.     

Recent progress in the theory of Trojan asteroids

In previous publications the author has constructed a long-periodic solution of the problem of the motion of the Trojan asteroids, treated as the case of 1:1 resonance in the restricted problem of three bodies. The recent progress reported here is summarized under three headings:

1. The nature on the long-periodic family of orbits is re-examined in the light of the results of the numerical integrations carried out by Deprit and Henrard (1970). In the vicinity of the critical divisor $$D_k \equiv \omega _1 - k\omega _2 ,$$ not accessible to our solution, the family is interrupted by bifurcations and shortperiodic bridges. Parametrized by the normalized Jacobi constant α², our family may, accordingly, be defined as the intersection of admissible intervals, in the form $$L = \mathop \cap \limits_j \left\{ {\left| {\alpha - \alpha _j } \right| > \varepsilon _j } \right\};j = k,k + 1, \ldots \infty .$$ Here, {α_j(m)} is the sequence of the critical α_j corresponding to the exactj: 1 commensurability between the characteristic frequencies ω₁ and ω₂ for a given value of the mass parameterm. Inasmuch as the ‘critical’ intervals |α?α_j|<ε_j can be shown to be disjoint, it follows that, despite the clustering of the sequence {α_j} at α=1, asj→∞, the family extends into the vicinity of the separatrix α=1, which terminates the ‘tadpole’ branch of the family.
2. Our analysis of the epicyclic terms of the solution, carrying the critical divisorD _k , supports the Deprit and Henrard refutation of the E. W. Brown conjecture (1911) regarding the termination of the tadpole branch at the Lagrangian pointL ₃. However, the conjecture may be revived in a refined form. “The separatrix α=1 of the tadpole branch spirals asymptotically toward a limit cycle centered onL ₃.”
3. The periodT(α,m) of the libration in the mean synodic longitude λ in the range $$\lambda _1 \leqslant \lambda \leqslant \lambda _2$$ is given by a hyperelliptic integral. This integral is formally expanded in a power series inm and α² or \(\beta \equiv \sqrt {1 - \alpha ^2 }\) .

The large amplitude of the libration, peculiar to our solution, is made possible by the mode of the expansion of the disturbing functionR. Rather than expanding about Lagrangian pointL ₄, with the coordinatesr=1, θ=π/3, we have expandedR about the circler=1. This procedure is equivalent to analytic continuation, for it replaces the circle of convergence centered atL ₄ by an annulus |r?1|<ε with 0≤θ<2π.     

Revisiting Noether gauge symmetry for F(R) theory of gravity

Noether gauge symmetry for F(R) theory of gravity has been explored recently. The fallacy is that, even after setting gauge to vanish, the form of F(R)∝R ⁿ (where n≠1 is arbitrary) obtained in the process, has been claimed to be an outcome of gauge Noether symmetry. On the contrary, earlier works proved that any nonlinear form other than $F(R) \propto R^{\frac{3}{2}}$ is obscure. Here, we show that, setting gauge term zero, Noether equations are satisfied only for n=2, which again does not satisfy the field equations. Thus, as noticed earlier, the only form that Noether symmetry admits is $F(R) \propto R^{\frac{3}{2}}$ . Noether symmetry with non-zero gauge has also been studied explicitly here, to show that it does not produce anything new.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, elliptic expansions in terms of the sectorial variables θ _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularise highly oscillating perturbation force of some orbital systems will be explored for the first four categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, numerical results are included to provide test examples for constructing computational algorithms.     

Reconstruction of f(R), f(T) and f(\mathcal{G}) models inspired by variable deceleration parameter

We study an special law for the deceleration parameter, recently proposed by Akarsu and Dereli, in the context of f(R), f(T) and $f(\mathcal{G})$ theories of modified gravity. This law covers the law of Berman for obtaining exact cosmological models to account for the current acceleration of the universe, and also gives the opportunity to generalize many of the dark energy models having better consistency with the cosmological observations. Our aim is to reconstruct the f(R), f(T) and $f(\mathcal{G})$ models inspired by this law of variable deceleration parameter. Such models may then exhibit better consistency with the cosmological observations.     

Spectra of radiation from a strongly magnetized plasma

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.     

Physical properties of X-ray gas in galaxy cluster CL0024+17

We analyzed the X-ray data obtained by the Chandra telescope for the galaxy cluster CL0024+17 (z = 0.39). The mean temperature of the cluster is estimated (kT = 4.35 _?0.44 ^+0.51 keV) and the surface brightness profile is derived. We generated the mass and density profiles for dark matter and gas using numerical simulations and the Navarro-Frenk-White dark matter density profile (Navarro et al., 1995) for a spherically symmetric cluster in which gas is in hydrostatic equilibrium with the cluster field. The total mass of the cluster is estimated to be M ₂₀₀ = 3.51 _?0.47 ^+0.38 × 10 _Sun ¹⁴ within a radius of R ₂₀₀ = 1.24 _?0.17 ^+0.12 Mpc of the cluster center. The contribution of dark matter to the total mass of the cluster is estimated as ${{M_{200_{DM} } } \mathord{\left/ {\vphantom {{M_{200_{DM} } } {M_{tot} }}} \right. \kern-0em} {M_{tot} }} = 0.89$ .     

A statistical analysis of NaI D1 profile fluctuations at the center of the solar disk

Three radial-velocity fluctuation arrays V(Δλ, Y) and line-formation fluctuation arrays L(Δλ, Y),where Δλ is wavelength displacement from the center of Nai D₁ and Y is displacement on the Sun's surface along the spectrograph slit, were obtained from Sacramento Peak Observatory spectrograms. The variations of these line profile fluctuations are qualitatively described. The RMS_υ's, coherences, and power spectra shapes for V(Δλ, Y) fluctuations are examined at different Δλ with the corresponding effective heights of formation calculated with Mein weighting functions. Results include: (a) possible anticorrelation between continuum fluctuations and those near line center; (b) RMS _υ ^(cr) 's, which are root-mean-square values of the radial velocity corrected for instrumental and atmospheric blurring, are large (1.5 to 4.0 km s^?1) primarily due to large corrections for atmospheric blurring; (c) RMS _υ ^(cr) minima at effective heights of formation above 350 km suggest penetration of granulation velocities into the upper photosphere; (d) very rough determinations of RMS _υ ^(cr) 's, which are additionally corrected for line-of-sight averaging, range from around 5 km s^?1 in the low chromosphere to a sharp minimum ≤ 0.5 km s^?1 located in the upper photosphere; (e) power spectra shapes reflect decreasing average fluctuation scales above the temperature minimum (possibly high-frequency oscillations) and in the low and middle photosphere (possibly penetration of granulation); and (f) RMS _υ ^(cr) 's and average fluctuation scales suggest changes in the resolvable velocity field occurring near the temperature minimum.     

On The Ideal Resonance Problem

The Ideal Resonance Problem in its normal form is defined by the Hamiltonian (1) $$F = A (y) + 2B (y) sin^2 x$$ with (2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?^1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by (3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance, (4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration ₁/n₂ of the natural frequencies of the motion generates a sequence. (5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form.     

Sur un modele explicitant les differents types morphologiques des galaxies

In the now classical Lindblad-Lin density-wave theory, the linearization of the collisionless Boltzmann equation is made by assuming the potential functionU expressed in the formU=U ₀ + \(\tilde U\) +... WhereU ₀ is the background axisymmetric potential and \(\tilde U<< U_0 \) . Then the corresponding density distribution is \(\rho = \rho _0 + \tilde \rho (\tilde \rho<< \rho _0 )\) and the linearized equation connecting \(\tilde U\) and the component \(\tilde f\) of the distribution function is given by $$\frac{{\partial \tilde f}}{{\partial t}} + \upsilon \frac{{\partial \tilde f}}{{\partial x}} - \frac{{\partial U_0 }}{{\partial x}} \cdot \frac{{\partial \tilde f}}{{\partial \upsilon }} = \frac{{\partial \tilde U}}{{\partial x}}\frac{{\partial f_0 }}{{\partial \upsilon }}.$$ One looks for spiral self-consistent solutions which also satisfy Poisson's equation $$\nabla ^2 \tilde U = 4\pi G\tilde \rho = 4\pi G\int {\tilde f d\upsilon .} $$ Lin and Shu (1964) have shown that such solutions exist in special cases. In the present work, we adopt anopposite proceeding. Poisson's equation contains two unknown quantities \(\tilde U\) and \(\tilde \rho \) . It could be completelysolved if a second independent equation connecting \(\tilde U\) and \(\tilde \rho \) was known. Such an equation is hopelesslyobtained by direct observational means; the only way is to postulate it in a mathematical form. In a previouswork, Louise (1981) has shown that Poisson's equation accounted for distances of planets in the solar system(following to the Titius-Bode's law revised by Balsano and Hughes (1979)) if the following relation wasassumed $$\rho ^2 = k\frac{{\tilde U}}{{r^2 }} (k = cte).$$ We now postulate again this relation in order to solve Poisson's equation. Then, $$\nabla ^2 \tilde U - \frac{{\alpha ^2 }}{{r^2 }}\tilde U = 0, (\alpha ^2 = 4\pi Gk).$$ The solution is found in a classical way to be of the form $$\tilde U = cte J_v (pr)e^{ - pz} e^{jn\theta } $$ wheren = integer,p =cte andJ _v (pr) = Bessel function with indexv (v ² =n ² + α²). By use of the Hankel function instead ofJ _v (pr) for large values ofr, the spiral structure is found to be given by $$\tilde U = cte e^{ - pz} e^{j[\Phi _v (r) + n\theta ]} , \Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).$$ For small values ofr, \(\tilde U\) = 0: the center of a galaxy is not affected by the density wave which is onlyresponsible of the spiral structure. For various values ofp,n andv, other forms of galaxies can be taken into account: Ring, barred and spiral-barred shapes etc. In order to generalize previous calculations, we further postulateρ ₀ =kU ₀/r ², leading to Poisson'sequation which accounts for the disc population $$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$ AsU ₀ is assumed axisymmetrical, the obvious solution is of the form $$U_0 = \frac{{cte}}{{r^v }}e^{ - pz} , \rho _0 = \frac{{cte}}{{r^{2 + v} }}e^{ - pz} .$$ Finally, Poisson's equation is completely solvable under the assumptionρ =k(U/r ². The general solution,valid for both disc and spiral arm populations, becomes $$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$ The density distribution along the O _z axis is supported by Burstein's (1979) observations.     

A review of recent observations of dust in Hii regions

The upper limit for the absorption cross section σ _H ^ext , of dust in Hii regions in the wave-length range 912–504 Å derived by Mezgeret al. (1974), is compatible with that expected for large dust grains, and a gas-to-dust ratio equal to that in the general interstellar medium. The albedo of the small grains must be high for λ>504 Å. This restriction is lifted if the visual extinction cross section of the grains in Hii regions is less than that for grains in the general interstellar medium. New observations of the Orion Nebula indicate that the visual extinction cross section is within a factor 2 of the value in the general interstellar medium.     

Ultraviolet solar identifications based on extended absorption series observed in the laboratory spectrum of Sii

The absorption spectrum of Sii in the wavelength region 1500–1900 Å has been photographed at high resolution. The silicon vapour was produced in a 122 cm long King furnace at 1800–2300°C. Forty-two Rydberg series have been observed from the ground state terms 3p ^{2 3} P and¹ D to terms associated with the 3pns and 3pnd configurations. All of the series from these configurations withJ<4 have been extended with the 3pnd ³ D ₃ ^o levels reachingn=56. Numerous perturbations have been observed. This laboratory work has provided the basis for extending the identification of silicon lines in the solar spectrum. Nearly all lines found in the laboratory spectrum are also found in rocket spectrograms of the solar chromosphere. More than 300 lines have been attributed to Sii. The excellent correlation between laboratory and solar Sii lines will be illustrated.