On pulsar-driven isothermal blast-wave models of supernova remanants

We investigate the ‘equilibrium’ and stability of spherically-symmetric self-similar isothermal blast waves with a continuous post-shock flow velocity expanding into medium whose density varies asr ^?ω ahead of the blast wave, and which are powered by a central source (a pulsar) whose power output varies with time ast ^ω?3. We show that:

1. for ω<0, no physically acceptable self-similar solution exists;
2. for ω>3, no solution exists since the mass swept up by the blast wave is infinite;
3. ? must exceed zero in order that the blast wave expand with time, but ?<2 in order that the central source injects a finite total energy into the blast wave;
4. for 3>ω_min(?)>ω>ω_max(?)>0, where $$\begin{gathered} \omega _{\min } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} + {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} + {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \omega _{\max } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} - {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} - {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \end{gathered} $$ two critical points exist in the flow velocity versus position plane. The physically acceptable solution must pass through the origin with zero flow speed and through the blast wave. It must also pass throughboth critical points if \(\varphi > \tfrac{5}{3}\) , while if \(\varphi< \tfrac{5}{3}\) it must by-pass both critical points. It is shown that such a solution exists but a proper connection at the lower critical point (for ?>5/3) (through whichall solutions pass with thesame slope) has not been established;
5. for 3>ω>ω_min(?) it is shown that the two critical points of (iv) disappear. However a new pair of critical points form. The physically acceptable solution passing with zero flow velocity through the origin and also passing through the blast wave mustby-pass both of the new critical points. It is shown that the solution does indeed do so;
6. for 3>ω_min(?)>ω_max(?)>ω it is shown that the dependence of the self-similar solution on either ω or ? is non-analytic and therefore, inferences drawn from any solutions obtained in ω>ω_max(?) (where the dependence of the solutionis analytic on ω and ?) are not valid when carried over into the domain 3>ω_min(?)>ω_max(?)>ω;
7. all of the physically acceptable self-similar solutions obtained in 3>ω>0 are unstable to short wavelength, small amplitude but nonself-similar radial velocity perturbations near the origin, with a growth which is a power law in time;
8. the physical self-similar solutions are globally unstable in a fully nonlinear sense to radial time-dependent flow patterns. In the limit of long times, the nonlinear growth is a power law in time for 5<ω+2?, logarithmic in time for 5>ω+2?, and the square of the logarithm in time for 5=ω+2?.

The results of (vii) and (viii) imply that the memory of the system to initial and boundary values does not decay as time progresses and so the system does not tend to a self-similar form. These results strongly suggest that the evolution of supernova remnants is not according to the self-similar form.     

A polarimetric investigation of the eclipsing binary XY ursae majoris

On three nights in February 1976 we carried out polarimetric measurements, in V, of the short periodic eclipsing binary XY UMa, covering a complete cycle. The results are as follows:

1. Within all phase intervals the linear polarization does not exceed 0.1%.
2. In the phase range 0 ^p .95–1 ^p .35 the scatter of the Stokes parametersQ andU is about twice that within the phase interval 0 ^p .35–0 ^p .95.
3. A periodogram analysis of these data revealed a period of 21000 s, which is equal to half the orbital periodP _o=0^d.47899 within 1.5%.

From these we derive the conclusions that no circumstellar envelope can be made responsible for the observed long-term changes of the light curve and system brightness, supporting the earlier spectroscopic finding. The different scatter of the Stokes parameters at different phase intervals and theP _o/2 periodicity are in favor of the star spot model for XY UMa proposed by one of the authors (E. G.).     

Study of physical properties of spectroscopic binary stars

The main results of a study of a catalogue of physical parameters of 1041 spectroscopic binaries are presented. The distribution of spectroscopic binaries over all main parametersM ₁, a, e, M₁/M₂, P, and certain dependencies between some of them have been found.

1. It appears that among bright (m _v?3 ^m –5 ^m ) stars withM?1M _⊙, about 40% are apparently spectroscopic binaries with comparable masses of components.
2. The majority of spectroscopic binaries with the ratio of the large semiaxis of the orbit to the radius of the primarya/R ₁?20, have eccentricities close to zero. This is probably a consequence of the tidal circularization of orbits of close binaries by viscous friction.
3. The discovery of duplicity of double-line spectroscopic binaries is possible only if the semiamplitude of radial velocityK ₁ is almost 10 times higher than the semiamplitude of the radial velocity of a single-line spectroscopic binary of the same mass.
4. Double-line spectroscopic binaries witha/R _⊙?6(M ₁/M _⊙)^1/3,M ₁≈M ₂?1.5M _⊙ are almost almost absent, and the number of stars witha/R _⊙?6(M ₁/M _⊙)^1/3,M ₁≈1.5M _⊙ is relatively low.
5. The distribution of unevolved SB stars over the large semiaxis may be described by the expression d(N _d/N_t)≈0.2 d loga for 6(M ₁/M _⊙)^1/3?a/R _⊙?100.
6. The intial mass-function for primaries of spectroscopic binaries is the same Salpeter function dN _d≈M ₁ ^?2.35 dM ₁ for 1?M ₁/M _⊙?30.
7. It is possible to explain the observed ratio of the number of single-line spectroscopic binaries to the number of double-line binaries if one assumes that the average initial mass ratio is close to 1 and that the mass of the postmass-exchange remnant of the primary exceeds the theoretical one and/or that half of the angular momentum of the system is lost during mass-exchange.
8. The above-mentioned distributions ofM ₁ anda and assumptions on the mass of remnant and/or momentum loss also allow us to explain the observed shapes of dN/dM, dN/dq, and dN/da distributions after some selection effects are taken into account.

     

On the spin distributions of ΛCDM haloes

We used merger trees realizations, predicted by the extended Press-Schechter theory, in order to study the growth of angular momentum of dark matter haloes. Our results showed that:

1. The spin parameter λ′ resulting from the above method, is an increasing function of the present day mass of the halo. The mean value of λ′ varies from 0.0343 to 0.0484 for haloes with present day masses in the range of 10⁹h^?1 M _⊙ to 10¹⁴h^?1 M _⊙.
2. The distribution of λ′ is close to a log-normal, but, as it is already found in the results of N-body simulations, the match is not satisfactory at the tails of the distribution. A new analytical formula that approximates the results much more satisfactorily is presented.
3. The distribution of the values of λ′ depends only weakly on the redshift.
4. The spin parameter of an halo depends on the number of recent major mergers. Specifically the spin parameter is an increasing function of this number.

     

Distribution of quasars and formation of large-scale structures

According to the proposal developed by Fanget al. (1984, referred to as Paper I) on formation of large-scale structure in the Universe, we should expect that:

1. The distributions of quasars should be different from that of galaxies by no strong inhomogeneity on the scale of 10–100 Mpc.
2. The distributions of quasars withZ>2 andZ<2 should be different from each other with no large-scale structure in the former, and conversely in the latter.

Our various statistical results obtained from quasar distributions are consistent with these predictions. Particularly, the nearest neighbour test for the complete quasar sample given by Savage and Bolton (1979) clearly shows that the distribution ofZ>2 quasars is rather homogeneous while theZ<2 quasars have a tendency to cluster.     

Energy changes of cosmic rays in the interplanetary region

A clarification and discussion of the energy changes experienced by cosmic rays in the interplanetary region is presented. It is shown that the mean time rate of change of momentum of cosmic rays reckoned for a fixed volume in a reference frame fixed in the solar system is 〈p〉 =p V·G/3 (p=momentum,V is the solar wind velocity andG=cosmic-ray density gradient). This result is obtained in three ways:

1. by a rearrangement and reinterpretation of the cosmic-ray continuity equation;
2. by using a scattering analysis based on that of Gleeson and Axford (1967);
3. by using a special scattering model in which cosmic-rays are trapped in ‘magnetic boxes’ moving with the solar wind.

The third method also gives the rate of change of momentum of particles within a moving ‘magnetic box’ as 〈p〉_ad = ?p ?·V/3, which is the adiabatic deceleration rate of Parker (1965). We conclude that ‘turnaround’ energy change effects previously considered separately are already included in the equation of transport for cosmic rays.     

The mode of creation of stars,stellar systems and the hubble system of galaxies

It is confirmed that the creation of stars in spiral (and perhaps also Irri) galaxies requires a physical parameter (X factor) additional to gas density. Consequently theX factor is an essential feature of stellar patterns and perhaps of stellar systems (spiral and other disk and spheroidal systems, globular clusters) and may be the key to the origin of the few, yet remarkably varied Hubble system of galaxies.

1. It is shown that theX factor is organized over the whole galaxy and is a function of azimuth φ as well as radiusr. Only a galaxy-wide force field seems capable of explaining such anX(r, φ) factor either magnetic or gravitational in origin.
2. If gravitational in origin, theX factor must be a shock wave, but a survey of observations in eight galaxies, including our own, shows no large-scale shocks associated with star creation. This provides further strong evidence against the density-shock theory of twin spiral arms.
3. It is confirmed that galaxies of different Hubble types did not evolve from one another, so that each protogalaxy must possess a genetic factor which predetermines its evolution, and in particular its stellar systems. Thus the protogalactic genetic factor may be identical with theX factor.
4. The case for a primordial magnetic field is strengthened, and it is shown that in our Galaxy and some others the field must be generally oblique to the disk. Such a field can explain theX(r, φ) factor in terms of a magneto-gravitational mechanism of gas clumping.
5. An earlier, hydromagnetic theory of the Hubble types and of radio galaxies is extended by including theX factor to explain the various stellar systems observed in spiral, elliptical, lenticular and irregular galaxies.

     

Large scale circulation in the convection zone and solar differential rotation

In this paper we study the dependence on depth and latitude of the solar angular velocity produced by a meridian circulation in the convection zone, assuming that the main mechanism responsible for setting up and driving the circulation is the interaction of rotation with convection. We solve the first order equations (perturbation of the spherically symmetric state) in the Boussinesq approximation and in the steady state for the axissymmetric case. The interaction of convection with rotation is modelled by a convective transport coefficient k _c = k _co + ?k _c2 P ₂(cos θ) where ? is the expansion parameter, P ₂ is the 2nd Legendre polynomial and k _c2 is taken proportional to the local Taylor number and the ratio of the convective to the total fluxes. We obtain the following results for a Rayleigh number 10³ and for a Prandtl number 1:

1. A single cell circulation extending from poles to the equator and with circulation directed toward the equator at the surface. Radial velocities are of the order of 10 cm s^?1 and meridional ones of the order of 150 cm s^?1.
2. A flux difference between pole and equator at the surface of about 5 percent, the poles being hotter.
3. An angular velocity increasing inwards.
4. Angular velocity constant surfaces of spheroidal shape. The model is consistent with the fact that the interaction of convection with rotation sets up a circulation (driven by the temperature gradient) which carries angular momentum toward the equator against the viscous friction. Unfortunately also a large flux variation at the surface is obtained. Nevertheless it seems that the model has the basic requisites for correct dynamo action.

     

Effects of temperature and velocity gradients on doppler widths

We have investigated how the gradients of temperature and expansion velocities will change the emergent profiles from an extended medium in spherical symmetry. Variation of the source function and expansion velocities are assumed. The following variations of temperature are employed:

1. T(r) ; T₀ (isothermal case)
2. T(r) ; T₀(r/r₀)^1/2
3. T(r) ; T₀(r/r₀)^-1
4. T(r) ; T₀(r/r₀)^-2
5. T(r) ; T₀(r/r₀)^-3

The profiles calculated present an interesting feature of broadening.     

Magnetic field versus gas density,in different physical conditions

A search is made here for possible variations in the behavior of magnetic field valuesB at various gas density valuesn, when comparing low density gas versus high density gas, and when comparing compressed gas versus quiescent gas.

1. For thequiescent microturbulent interstellar gas (e.g., clouds, interclumps — see TableI), the statistical relationB ~n ^k yieldsk = 0.46 ± 0.07 forhigh gas densityn > 100 cm^?3, andk = 0.17 ± 0.03 forlow gas densityn < 100 cm^?3 (see Figure 1).
2. For thecompressed macroturbulent interstellar gas (e.g., masers, expanding shells — see Table II), the statistical relationB ~n ^K yieldsK = 0.61 ± 0.09 forhigh gas densityn > 100 cm^?3, andK = 0.37 ± 0.2 forlow gas densityn < 100 cm^?3 (see Figure 2).
3. The separation betweenlow density gas andhigh density gas is statistically significant. The 2 different physical behaviors (below and above the break at 100 cm^?3) are confirmed statistically (about 2 to 4 sigma away for the quiescent gas alone; about 3 to 6 sigma away for the combined quiescent plus compressed gas).
4. The separation betweencompressed gas andquiescent gas is not statistically significant now (see Figure 3). Atn > 100 cm^?3, a comparison of quiescent gas versus compressed gas shows no statistically significant differences in behavior (they are only about 1 sigma away). Atn < 100 cm^?3, a comparison of quiescent versus compressed gas also shows no statistically significant differences in behavior (less than 1 sigma away).
5. A relation between the densityn and the galactic-wide Star Formation Rate (SFR) can be made for galactic magnetic fields, i.e.: (SFR) ~n ⁿ . For galactic-wide parameters using quiescent, low densityn < 100 cm^?3, and the known relationshipsB ~n ^k/j withk = 0.17,B ~ (SFR) ^j withj = 0.13, then one gets here a lawSFR ~n ^k/j with an exponentk/j = 1.3. This is in rough accord with known data for the Milky Way and for NGC6946.

     

Orbites periodiques de satellites

In a previous paper (Stellmacher, 1981, hereafter mentioned as Paper I), we have given an algorithm for the construction of periodic orbits in a rotating frame, for satellites around an oblate planet. In the present paper, we apply this theory to the Mimas-Tethys case; we obtain the following results:

1. Without resonance, it is possible to find a rotating system in which the solution is a periodic one. The angular velocity of this rotating frame is calculated as function of the masses of the two satellites.
2. Including the resonant terms and assuming an exact commensurability of the implied frequencies, we demonstrate that the condition for periodic solutions in the rotating system as defined in (a) is: the initial position of the satellites at conjunction lies on an axis defined by (Ω₁+Ω₂)/2 or (Ω₁+Ω₂)/2 + π/2;Ω₁ and Ω₂ are the longitudes of the ascending nodes of the satellite's orbits. The solution still is a periodic one, thus all the conjunction occur in either axis.
3. In the Mimas Tethys case there is only approximately commensurability between these frequencies. The two satellites are considered as oscillators whose amplitudes and phases are functions of time. The equation of the libration can be established; we find the usual form, but for each satellite the generating solution is a periodic solution (as defined in Paper I), but not a Keplerian one. It follows a determination of the masses which slightly differs from that given by Kozai (1957), when the same values of the observed quantities are used for calculations.
4. The equation of the libration is: $$\ddot z + n_1^2 h^2 \sin z + n_1 q\dot z\sin z = 0$$

     

Systematics of Dwarf Novae

From about 30 Dwarf Novae with the best determined distances the following relationships are found.

1. a tight correlation between absolute magnitude at maximum light, M_v(max), and orbital period, P.
2. a correlation between M_v(min) and P showing wide scatter.
3. a correlation between M_v(mean), the mean absolute magnitude averaged over normal outbursts, and P, again with wide scatter. The scatter is shown to correlate strongly with ouburst timescale T_n.
4. a strong correlation between range, M_v(min)-M_v(max), and T_n (the Kukarkin-Parenago relationship).
5. a strong correlation between range M_v(mean)-M_v(max), and both T_n and P.

This final correlation is interpreted in terms of the disc instability model of dwarf novae and successfully predicts the observed width of outburst versus P relationship.     

Drift theory of charged particles in electric and magnetic fields

In this paper we review the drift theory of charged particles in electric and magnetic fields. No new physical interpretations are added to this classical topic, but through an alternative, simplified derivation of the guiding centre velocity, several complexities are eliminated and possible misconceptions of the theory are clarified. It is shown that:

1. The curvature/gradient drift velocity in the magnetic field, averaged over a particle distribution function is to lowest order in the direction of?×B/B ², while the average particle velocity is in the direction ofB×? P withP the scalar particle pressure.
2. These drift directions are correct for first-order expansions of the particle distribution function, and only second-order or higher expansions change these directions.
3. The?×B/B ² drift, which is the standard gradient plus curvature drift, and which is usually considered as a ‘single particle’ drift, need not be ‘reconciled’ with theB×? P, or ‘macroscopic, collective’ drift, as is often asserted in the literature. They are in fact related per definition and we show how.
4. When viewed in fixed momentum intervals (p,p+dp), the so-called Compton-Getting factor enters into the electric field (E×B)/B ² drift term.
5. The results are independent of the scale length of variation ofE andB, in contrast to existing drift theory. We discuss the implications of this result for three important cases.

     

A statistical study of the spiral spots on the solar disc

After adding the data observed in the years from 1979 to 1982 to those obtained earlier (Ding et al., 1981), we re-examine the previous results and conclude:

1. The longitudinal distribution of spiral spots on the solar disc is generally the same as that of sunspot groups with areas of S _p ≥ 400, but their active longitudes seem to be more concentrated.
2. The distribution of spiral patterns in the southern and northern hemispheres shows that the differential rotation may be a fundamental solar dynamo for the formation of the spiral spots.
3. The statistical directions of the emerging twisted magnetic vectors in the active regions in the southern and northern hemispheres are synchronously inverse with a period of about two years. This period seems to be detected in other solar observations.

     

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.     

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.     

Sur les solutions Lagrangiennes du problème des trois corps solides avec la loi de Weber

In this work we consider the problem of translational-rotational motion of three solid bodies, for which the elementary particles attract each other according to different Weber's laws for each pair of bodies. This problem represents a special case of the generalized problem of three solids considered in a previous work, (Dubochin, 1974) and it gives an example of the verification of the existence conditions for the Lagrangian solutions. In these solutions, the centers of mass always for m an equilateral triangle. Each body has axial symmetry with the plane of symmetry perpendicular to the axis of symmetry rotates uniformly around this axis, which at any instant stays perpendicular to the plane of the triangle formed by the centers of mass. According to Weber's law (Tisserand, 1896) the elementary particles of two bodiesT _i andT _j (i, j=0, 1, 2) are attracted by forces which are proportional to the function $$F_{ij} (W) = \frac{{f_{ij} }}{{\Delta _{ij^2 } }}\left[ {1 - a_{ij} \dot \Delta _{ij^2 } + 2a_{ij} \Delta _{ij} \ddot \Delta _{ij} } \right]$$ wheref _ij anda _ij (in generalf _ji ≠f _ij anda _ji ≠a _ij ) are functions of the timet, and where the real quantities Δ_ij are the mutual distances between the particles of the bodiesT _i andT _j , and where \(\dot \Delta _{ij} \) and \(\ddot \Delta _{ij} \) are their derivatives with respect to the time. The analysis of the general conditions for the Lagrangian solutions gives the following results for the case of Weber's laws.

1. Only the invariant Lagrangian solutions, (the traingle of the centres of mass does not change in time) are possible in this problem.
2. Besides the conditions (NL) obtained in the case of the Newton-Coulomb law, (all thea _ij are zero), the complementary conditions (WL) must be satisfied.

In particular, if all the bodies are spheres or homogeneous ellipsoids, they must necessarily have the same dimensions, but they can have different masses.     

Some properties of finite energy constant-α force-free magnetic fields in a half-space

Some useful properties of a finite energy, constant-α, force-free magnetic field B _α occupying a half-space D are presented. In particular:

1. Fourier and Green representations of B _α are obtained and used to derive conditions for the existence and uniqueness of a B _α having a given normal component B _z on the boundary ?D.
2. The asymptotic behaviour of B _α at infinity as well as stability results against changes in the boundary condition on ?D and in the value of α are established.
3. The energy of B _α is shown to be smaller than the energy of the open field having the same B _z on ?D, thus confirming an earlier conjecture (Aly, 1984).
4. B _α is proved to not be a Taylor-Heyvaerts-Priest state, in spite of the fact that its relative helicity H is finite and that it is the only solution of the Lagrange-Euler equation associated with the problem of minimizing the energy among all the fields having the same value of H and the same B _z on ?D.

     

Hydrodynamic and turbulent motions in the galactic disk,II

From a comparative study between stellar and gas data it is seen that turbulent and hydrodynamic motions in the Galaxy are common to both types of materials:

1. Galactic clusters have sizes and intrinsic dispersions compatible with the modified form of the Kolmogorov law seen in molecular clouds: undimensional velocities σ(km s^?1)=0.54d ^0.38 (pc). This indicates that ‘typic’ clusters were born from ‘typic’ dark clouds as these of the Lynds's catalogue (diametersd<10 pc, dispersions σ<1.5 km s^?1 hydrogen densitiesn _H>200 atom cm^?3). These clouds have mass enough to form galactic clusters (1000–3000M _⊙).
2. The cluster formation is related to the supersonic range of the Kolmogorov relationship σ(d>1 pc) while the AFGKM stars are related to the subsonic range of the same relationship σ(d<0.3 pc), the intermediate transition zone is probably related to OB stars and/or trapezia.
3. The effects of the magnetic fields in the clouds are also discussed. It seems to be that in the clouds the magnetic energy does not exceed the kinetic energy (proportional toσ ²(d)) and that this determinates the freezing criteria. The hypotheses introduced here can be checked with 21 cm Zeeman splitting.
4. Low-density globular clusters are also coherent with the Kolmogorov relationship. Some hypotheses about their origin and the type of clouds where they were born are discussed. This last part of the study lets open the possibility of further studies about evolution of globular clusters.

     

Radio spectrum of two active regions

The results of the total solar eclipse of November 12, 1966, observed at 8 different wave-lengths between 3 and 21 cm, are studied and the spectrum of two active regions present on the disk is deduced. It is shown that the observed increase of the flux of the most intense source in the range 3–10 cm is due to geometrical effects. Neglecting the influence of the magnetic field, the following quantities are deduced.

1. the mean and central temperature of the coronal condensation.
2. the ∫ _corona N ²dh (N = electron density).

Both these quantities are in good agreement with optical observations.