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Astrophysics and Space Science

, Volume 81, Issue 1–2, pp 387–395 | Cite as

Sur un modele explicitant les differents types morphologiques des galaxies

  • R. Louise
Article
  • 23 Downloads

Sommaire

En 1964, Lin et Shu avaient montré que l'équation de Poisson admettait des solutions harmoniques permettant ainsi de comprendre la structure spirale des galaxies. Leurs calculs étaient faits sur les composantes\(\tilde \rho \) de la densité et\(\tilde U\) du potential, définies par:
$$\rho = \rho _0 + \tilde \rho + \ldots , \tilde \rho<< \rho _0 $$
$$U = U_0 + \tilde U + \ldots , \tilde U<< U_0 .$$
U0 et ρ0 sont respectivement les composantes du potentiel et de la densité liées à la population stellaire du disque; elles sont supposées axisymétriques. Dans le présent article, nous adopterons une démarche différente. Tout en gardant les hypothèses de base de la théorie d'onde de densité, nous nous donnerons comme postulat une relation nouvelle
$$\rho = \frac{{kU}}{{r^2 }} (k = cte).$$
Cette relation nous a été suggérée par un travail précédent (Louise, 1981) lorsque nous étudions la loi de Titius Bode (révisée par Balsano et Hughes, 1979) donnant la répartition des distances planétaires du système solaire. La solution de la nouvelle équation de Poisson
$$\Delta U - \frac{{\alpha ^2 }}{{r^2 }}U = 0$$
est
$$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$
n est un nombre entier,p=cte positive,\(v^2 = \alpha ^2 + n^2 ,\Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).\)

Le deuxième terme à l'intérieur des crochets est responsable de la structure spirale, annulaire, barrée ou spirale-barrée des galaxies suivant les valeurs den, ν etp. Le termer−νepz représente la population du disque.

On a model accounting for various morphological structures of galaxies

Abstract

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 +\(\tilde U\)+... WhereU0 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 (v2 =n2 + 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 postulateρ0 =kU0/r2, leading to Poisson'sequation which accounts for the disc population
$$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$
AsU0 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/r2. 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.

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Copyright information

© D. Reidel Publishing Co 1982

Authors and Affiliations

  • R. Louise
    • 1
  1. 1.Observatoire de MarseilleMarseilleFrance

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