Evolution of Planetary Nebulae Envelopes: An Empirical Approach

We carried out the empirical investigation of the evolution of gas density distribution in the envelopes of planetary nebulae (PN). For this purpose we analysed the isophotal maps of 10 PN in the lines H alpha, H beta or in the optically thin radio continuum. To obtain the spatial radial distribution of gas density n(r) (where n = n(H)+n(He)) we used Abell's integral equation in the simplest, spherically-symmetric case. We found that n(r) for all PN envelopes in our sample can be described by an approximative expression:

On the Normal Distribution in the Orientation Space

The properties of model distributions used in texture analysis up to now are discussed. The normal distribution in the G-space (recently investigated by T. I. Savjolova) is analysed. Its connection with the central limit theorem of probability theory is demonstrated in a mathematically simplified manner. An analytically closed approximative expression (with very high precision for halfwidths of practical interest) for the normal distribution is derived. Possible correlations between forms of texture components and mechanisms of texture development are mentioned.