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.     

Photometric elements and apsidal motion of the eclipsing binary NSV 18773

The photometric elements of the eclipsing binary NSV 18773 (HD 99898) have been determined for the first time by analyzing its V-and I-band light curves from the ASAS-2 and ASAS-3 catalogs. Based on these elements and using other published spectroscopic and photometric data, we constructed a consistent system of geometrical and physical parameters for the system that consists of two stars (M ₁ = 20M _⊙, Sp₁=B0V, R ₁ = 5.0R _⊙ and M ₂ = 14M _⊙, Sp₂ = B1V, R ₂ = 6.5R _⊙) in elliptical orbits (P = 5 _. ^d 049, e = 0.365, a = 40.1R _⊙). The distance to the system is d = 3.3 kpc, the interstellar extinction is A _V = 2 _. ^m 0, and the age is t = 2.8 × 10⁶ yr. NSV 18773 is a visual binary with components V _A = 9 _. ^m 9 and V _B = 10 _. ^m 3 separated by 0 _. ^" 8. The third light (L ₃ = 0.61) that we found by analyzing the light curves shows that the eclipsing binary is the system’s fainter component B. We confirmed the rapid apsidal motion of the star detected by Otero and Wils (2006) and refined its observed period: U _obs = 150 ± 6 yr. Our photometric elements and physical parameters allowed the apsidal parameter $\bar k_2^{obs} = 0.0135(14)$ , which reflects the density distribution along the radii of the component stars, to be determined. Within the error limits, the derived parameter agrees with its theoretically expected value, $\bar k_2^{th} = 0.0119(8)$ , from current evolutionary models of stars of the corresponding masses and ages.     

New activity cycle of the classical symbiotic star V1413 Aql

We present the results of our photometric UBV JHKLM observations in 2008?C2011 for the classical symbiotic star V1413 Aql. At the end of 2008, the hot component of V1413 Aql experienced the next strong outburst (??V > 2 _· ^m 5). According to the photometric criterion (B-V ?? 0 _· ^m 9 ± 0 _· ^m 2), the star was in an active state even in the period preceding the strong 2008 outburst. Two eclipsing minima of the same amplitude were observed for V1413 Aql in 2010 and 2011. Our analysis of the 2011 eclipse has allowed us to estimate the sizes of the components with respect to the orbital semimajor axis if the system is seen edge-on: the radius of the cool component is R _g/a = 0.28 and the radius of the hot component is R _h/a = 0.17. However, judging by the B-V color index, the eclipse may be noncentral.     

Variability and possible rapid evolution of the hot post-AGB stars Hen 3-1347, Hen 3-1428, and LSS 4634

We present the results of spectroscopic and photometric observations for three hot southern-hemisphere post-AGB objects, Hen 3-1347 = IRAS 17074-1845, Hen 3-1428 = IRAS 17311-4924, and LSS 4634 = IRAS 18023-3409. In the spectrograms taken with the 1.9-m telescope of the South African Astronomical Observatory (SAAO) in 2012, we have measured the equivalent widths of the most prominent spectral lines. Comparison of the new data with those published previously points to a change in the spectra of Hen 3-1428 and LSS 4634 in the last 20 years. Based on ASAS data, we have detected rapid photometric variability in all three stars with an amplitude up to 0 _· ^m 3-0 _· ^m 4 in the V band. A similarity between the patterns of variability for the sample stars and other hot protoplanetary nebulae is pointed out. We present the results of UBV observations for Hen 3-1347, according to which the star undergoes rapid irregular brightness variations with maximum amplitudes ΔV = 0 _· ^m 25, ΔB = 0 _· ^m 25, and ΔU = 0 _· ^m 30 and shows color-magnitude correlations. Based on archival data, we have traced the photometric history of the stars over more than 100 years. Hen 3-1347 and LSS 4634 have exhibited a significant fading on a long time scale. The revealed brightness and spectrum variations in the stars, along with evidence for their enhanced mass, may be indicative of their rapid post-AGB evolution.     

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.     

Optical spectral decomposition of NGC 4051

Considering the host galaxy contribution, a spectral decomposition method is used to reanalyzed the archive data of optical spectra for a narrow line Seyfert 1 galaxy, NGC 4051. The light curves of the continuum f _λ (5100 Å), and Hβ, He ii, Fe ii emission lines are given. We find strong flux correlations between line emissions of Hβ, He ii, Fe ii and the continuum f _λ (5100 Å). These low-ionization lines (Hβ, Fe ii, He ii) have “inverse” intrinsic Baldwin effects. Using the methods of the cross-correlation function and the Monte Carlo simulation, we find the time delays, with respect to the continuum, are $3.45^{+12.0}_{-0.5}~\mbox{days}$ with the probability of 34 % for the intermediate component of Hβ, $6.45^{+13.0}_{-1.0}~\mbox{days}$ with the probability of 65 % for the intermediate component of He ii. From these intermediate components of Hβ and He ii, the calculated central black hole masses are $0.86^{+4.35}_{-0.33}\times 10^{6}$ and $0.82^{+3.12}_{-0.45}\times 10^{6}~M_{\odot }$ . We also find that the time delays for Fe ii are $9.7^{+3.0}_{-5.0}~\mbox{days}$ with the probability of 36 %, $8.45^{+1.0}_{-2.0}~\mbox{days}$ with the probability of 18 % for the total epochs and “subset 1” data, respectively. It seems that the Fe ii emission region is outside of the Hβ emission region.     

A photometric study of faint galaxies in the field of GRB 000926

We present our B, V, R_c, and I_c observations of a \(3'.6 \times 3'\) field centered on the host galaxy of GRB 000926 (α_2000.0=17^h04^m11^s, \(\delta _{2000.0} = + 51^ \circ 47'9\mathop .\limits^{''} 8\)). The observations were carried out on the 6-m Special Astrophysical Observatory telescope using the SCORPIO instrument. The catalog of galaxies detected in this field includes 264 objects for which the signal-to-noise ratio is larger than 5 in each photometric band. The following limiting magnitudes in the catalog correspond to this limitation: 26.6 (B), 25.7 (V), 25.8 (R), and 24.5 (I). The differential galaxy counts are in good agreement with previously published CCD observations of deep fields. We estimated the photometric redshifts for all of the cataloged objects and studied the color variations of the galaxies with z. For luminous spiral galaxies with M(B)z～1.     

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 ${{M_{200_{DM} } } \mathord{\left/ {\vphantom {{M_{200_{DM} } } {M_{tot} }}} \right. \kern-0em} {M_{tot} }} = 0.89$ .     

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.     

New open-source approaches to the modeling of stellar collapse and the formation of black holes

We present new approaches to the simulation of stellar collapse, the formation of black holes, and explosive core-collapse supernova nucleosynthesis that build upon open-source codes and microphysics. We discuss the new spherically-symmetric general-relativistic (GR) collapse code GR1D that is endowed with an approximate 1.5D treatment of rotation, comes with multiple nuclear equations of state, and handles neutrinos with a multi-species leakage scheme. Results from a first set of spinning black hole formation simulations are presented. We go on to discuss the derivative code GR1D+N which is tuned for calculations of explosive nucleosynthesis and includes a NSE/non-NSE equation of state treatment, and a nuclear reaction network. We present sample results showing GR1D+N??s performance in reproducing previous results with thermal-bomb-driven explosions. Finally, we introduce the 3?+?1 GR Zelmani core collapse simulation package and present first results obtained in its application to the 3D modeling of failing core-collapse supernovae.     

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.     

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.     

Nova Monocerotis 2002 (V838 Mon) at early outburst stages

We obtained U BV R photometric and spectroscopic observations during the outburst of V838 Mon. Before its outburst, the B brightness of the star had been stable ( $\tilde15.^m 85$ ) for 45 years. This was a blue star with the color index $(B - V)_0 = - 0\mathop .\limits^m 03 \pm 0\mathop .\limits^m 1$ and may have been a cataclysmic variable. In the middle of March 2002, the outburst amplitude reached $8\mathop .\limits^m 1$ in B. The star has the counterpart V 1006/7 in M 31 in luminosity at maximum and in spectrum. The unusual spectrum at the premaximum stage originated in the expanding photosphere of a cool K-type giant. The expansion velocity of the photosphere was 150 km s^?1; the maximum velocity in the expanding stellar envelope reached 500 km s^?1. The absorption components of neutral metal lines were enhanced by a factor of 3 or 4 compared to a normal K-type star. No overabundance of s-process elements was found. One day before the brightness peak, an intense Hα emission line with broad wings, FWZI=3100 km s^?1, and numerous lines of ionized metals appeared in V838 Mon, which is characteristic of normal classical novae. We show light, color, and spectral variations of the object.     

Exotic ion H 3 ++ in strong magnetic fields

1. The exotic system H ₃ ⁺⁺ (which does not exist without magnetic field) exists in strong magnetic fields:
   1. In triangular configuration for B≈10⁸–10¹¹?G (under specific external conditions)
   2. In linear configuration for B>10¹⁰?G
2. In the linear configuration the positive z-parity states 1σ _g , 1π _u , 1δ _g are bound states
3. In the linear configuration the negative z-parity states 1σ _u , 1π _g , 1δ _u are repulsive states
4. The H ₃ ⁺⁺ molecular ion is the most bound one-electron system made from protons at B>3×10¹³?G

Possible application: The H ₃ ⁺⁺ molecular ion may appear as a component of a neutron star atmosphere under a strong surface magnetic field B=10¹²–10¹³?G.     

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 few observations of and remarks on V1500 Cygni

The development of the post-nova light curve of V1500 Cyg inUBV andHβ, for 15 nights in September and October 1975 are presented. We confirm previous reports that superimposed on the steady decline of the light curve are small amplitude cyclic variations. The times of maxima and minima are determined. These together with other published values yield the following ephemerides from JD 2 442 661 to JD 2 442 674: $$\begin{gathered} {\text{From}} 17 {\text{points:}} {\text{JD}}_{ \odot \min } = 2 442 661.4881 + 0_{^. }^{\text{d}} 140 91{\text{n}} \hfill \\ \pm 0.0027 \pm 0.000 05 \hfill \\ {\text{From}} 15 {\text{points:}} {\text{JD}}_{ \odot \max } = 2 442 661.5480 + 0_{^. }^{\text{d}} 140 89{\text{n}} \hfill \\ \pm 0.0046 \pm 0.0001 \hfill \\ \end{gathered} $$ with standard errors of the fits of ±0 _. ^d 0052 for the minima and ±0 _. ^d 0091 for the maxima. Assuming V1500 Cyg is similar to novae in M31, we foundr=750 pc and a pre-nova absolute photographic magnitude greater than 9.68.     

Pulsations and apsidal motion in the eclipsing variable system DI Herculis

In 2003–2008, highly accurate photoelectric and CCD observations of the close binary system DI Her were performed in the V band. The light curves of three primary and three secondary eclipses were constructed. These observations, along with the highly accurate photoelectric observations of other authors obtained in different years from 1963 to 1986, have confirmed the difference between the observed (1 _. ^o 3 ± 0 _. ^o 1/100 yr) and theoretical (4 _. ^o 3/100 yr) rates of apsidal motion. Our photometric data are indicative of a possible variability in the system with period P′ = 1.175 days and amplitude A′ = 0 _. ^m 011, which is probably related to the pulsations of one of the components. There may be a third body in the system that produces in-phase variations in the times of primary and secondary minima with a period of 10.5 yr and an amplitude of 1 _. ^m 5.     

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.     

A photometric study of the star V1016 Ori

New photoelectric UBVRI observations of the eclipsing variable V 1016 Ori have been obtained with the AZT-11 telescope at Crimean Astrophysical Observatory and with the Zeiss-600 telescope at Mount Maidanak Observatory. Light curves are constructed from the new observations and from published and archival data. We use a total of 340, 348, 386, 185, and 62 magnitude estimates in the bands from U to I, respectively. An analysis of these data has yielded the following results. The photometric elements were refined; their new values are $Min I = JDH 2441966.820 + 65\mathop .\limits^d 4331E$ . The UBVRI magnitudes outside eclipse were found to be $5\mathop .\limits^m 95$ , $6\mathop .\limits^m 77$ , $6\mathop .\limits^m 75$ , $6\mathop .\limits^m 68$ , and $6\mathop .\limits^m 16$ , respectively. No phase effect was detected. We obtained two light-curve solutions: (1) assuming that the giant star was in front of the small one during eclipse, we determined the stellar radii, r _s=0.0141 and r _g=0.0228 (in fractions of the semimajor axis of the orbit); and (2) assuming that the small star was in front of the giant one, we derived r _g=0.0186 and r _s=0.0180 for the V band. The brightness of the primary star in the bands from U to I is L ₁=0.96, 0.92, 0.90, 0.89, and 0.88, the orbital inclination is $i = 87^\circ .1$ , and the maximum eclipse phase is α₀= 0.66. In both cases, we accepted the U hypothesis, assumed the orbit to be elliptical, and took into account the flux from the star Θ¹ Ori E that fell within the photometer aperture. The first solution leads to a discrepancy between the primary radius determined by solving the light curve and the radial-velocity curve and its value estimated from the luminosity and temperature. This discrepancy is eliminated in the second solution, and it turns out that, by all parameters, the primary corresponds to a normal zero-age main-sequence star.     

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.