1. Introduction

A planetary nebula is created when a star blows off its outer layers after it has run out of fuel to burn. These outer layers of gas expand into space, forming a nebula, which is often the shape of a ring, bubble or other types (^{[4, 5, 11]}). Only about 20% of planetary nebulae are spherically symmetric (for example, see Abell 39) ^[2] and likely produced by the old stars similar to the Sun ^[8]. Planetary nebulae play a very important role in galactic evolution. The early universe consisted almost entirely of hydrogen and helium, but stars create heavier elements via nuclear fusion. In recent studies (^{[3, 9]}), out of 200 billion stars, about 3500 planetary nebulae are now known to exist in our galaxy, more than double what it was a decade ago. They are found mostly near the plane of the Milky Way, with the greatest concentration near the galactic center ^[7].

The present paper, is devoted for the statistical simulations of galactic planetary nebulae, for such studies we used two catalogues and of VizieR database.

The first catalogue (its original reference is ^[14]) was utilized for establishing: (1) The correlation coefficients between the entireties of the data, (2) the position of the maximum (minimum) of each entry in the data together with the values of the other entries at this position, (3) the histograms of the entireties, (4) descriptive location, dispersion and shape statistics, (5) finally the best and its error analysis was developed between LH &LR where LH is the logarithm of HI Zanstrs luminosity and LR is logarithm of nebular radius.

The second catalogue (the original reference of the catalogue is ^[1]) was utilized for establishing the two dimensional distribution of the its nebulae and for determining the equatorial coordinates of the north galactic pole (NGP). The results of this application are in good agreements with those of IAU(1959), ^[15] and ^[13]. A good review to the history of the Galactic coordinate system is given in ^[6].

2. Statistical Analysis of Planetary Nebulae Properties

For the present analysis, we used catalogue: of VizieR database (the original reference of the catalogue is ^[14]). What concerning us among the entries of the catalogue are the rows {6, 7, 8, 9, 10, 12, 16, 17}, the explanation of each row together with its used abbreviation are listed in Table 1.

A correlation coefficient is a statistical measure of the degree to which two variables are related to each other. The linear correlation coefficient between is

(1)

This coefficient is a dimensionless quantity; that is, it does not depend on the units employed. Its value is always between, such that, the closest the value of to one is the more correlation between the two variables will be. The sign of r only fells us whether y is increasing or decreasing when x increases.

We find for the correlation coefficients between the entries of Table 1 (N=49), the values listed in Table 2.

Table 1. Explanations and the used abbreviations of the catalogue entries

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Table 2. Correlation coefficients between the entries of Table 1

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Table 3 (Table 4) gives the position of the maximum (minimum) of each entry in the data together with the values of the other entries at this position.

Table 3. The position of the maximum of each entry V and the values of the other entries at this position

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Table 4. The position of the minimum of each entry V and the values of the other entries at this position

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The following figures show the histograms of the entries

Figure 1. Histograms of the entries

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The following are some descriptive statistics of the entries

Table 5. Basic Descriptive statistics of entries

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Table 6. Location statistics of entries

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Table 7. Dispersion statistics of entries

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where, variance of mean, standard error of sample mean, coefficient of variation, mean deviation,median deviation and sample range.

Table 8. Shape statistics of entries

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1. We established for the best fit between the analytical representations:

(2)

where the coefficients and their probable errors are:

2. The estimated variance.

3. Residuals at some points

Table 9. Residuals at some points of the entries

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4. The mean prediction confidence interval for some values of x are shown in the following table:

Table 10. The mean prediction confidence interval

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3. Galactic Pole Using Planetary Nebulae

For the present analysis, we used catalogue: of VizieR database (the original reference of the catalogue is ^[1]), hereafter will be referred to as Paper I). What concerning us among the entries of the catalogue, are the galactic coordinates and the equatorial coordinates ( right ascension & declination).

The sample used in Paper I is 117 planetary nebulae, the two dimensional distribution of the sample is shown in Figure 2.

Figure 2. Two-dimensional distribution of the sample

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Figure 2 confirms the result of Paper I in that, the majority of nebulae of the sample are located close to the galactic plane.

For the determination of the equatorial coordinates of the galactic pole, we obviously shall except to get the best results choosing objects, which exhibit strong concentration towards the galactic plane. So the data of the sample of Paper I is suitable for such determination

It was shown that ^[12], the determination of is a typical constrained minimization problem with:

1. the objective function is the sum of the N (say) squared distances of the selected objects from the plane of galactic plane.

(3)

where, on the assumption that, all objects having the equatorial coordinates are all at unit desistance from the Sun, so

(4)

and are direction cosines of the perpendicular to the plane drawn from the origin.

2. The constrained of the problem is:

(5)

Now, the solution of the constrained minimization problem is found by Lagrange's method so, the objective function becomes:

(6)

where is Lagrange’s multiplier to be determined.

The necessary conditions for the minimum are:

(7)

(8)

where, the elements of the symmetric matrix A are given by:

(9)

(10)

It is well known that ^[10], the symmetric least square matrix A has all its eigen values real and distinct.

Now after solving the eigen value problem of Equation (8) for select from this set, the values that gives the global minimum of F, then the required values of could be determined from:

(11)

Applying the above analytic developments for the data of Paper I, result as the average of five values corresponding to galactic latitudes of the members yields.

In concluding the present paper, two catalogues and of VizieR database were used for the statistical simulations of galactic planetary nebulae. Each catalogue was utilized for certain purposes, the first catalogue for, the correlation coefficients between the entireties of the data, the determination of the position of the maximum (minimum) of each entry in the data together with the values of the other entries at this position. In addition, the histograms and descriptive, location, dispersion and shape statistics for the entries of the data are also developed. Finally, the best fit and its error analysis was established between LH &LR where LH is the logarithm of HI Zanstrs luminosity and LR is the logarithm of nebular radius.

The second catalogue was used to display two-dimensional distributions of the interacting planetary nebulae, moreover IAU and other references used the equatorial coordinates of these nebulae to determine the equatorial coordinates of the north galactic pole (NGP). The results of this application are in good agreement with those given.

References

[1]	Ali A., Sabin L., Snaid S. and Basurah H. M., “Interacting planetary nebulae. I. Classification and orientation”, A & A 541, p. 1-10, 2012.
	In article
[2]	Jacoby, George. H., Ferland, Gary. J. and Korista, Kirk T., “The Planetary Nebula A39: An Observational Benchmark for Numerical Modeling of Photoionized Plasmas”, Ap. J. 560(1): 272-86, 2001.
	In article		CrossRef
[3]	Kwitter, K. B. et al., “The Present and Future of Planetary Nebulae Research”, A White Paper by the IAU. Planetary Nebula Working Group, Revista Mexicana de Astronom´ıa y Astrof´ısica, 50, 203-223, 2014.
	In article
[4]	Kwok, Sun, and Su, Kate Y. L., “Discovery of Multiple Coaxial Rings in the Quadrupolar Planetary Nebula NGC 6881”, Ap. J. 635 (1): L49-52, 2005.
	In article		CrossRef
[5]	Kwok, Sun, “The origin and evolution of planetary nebulae”, Cambridge University Press, 2000.
	In article		CrossRef
[6]	Liu, J.-C., Zhu, Z. and Zhang, H., “Reconsidering the Galactic coordinate system”, A & A, 526 A16, p. 1-7, 2011.
	In article
[7]	Majaess, D. J., Turner, D. and Lane, D., “In Search of Possible Associations between Planetary Nebulae and Open Clusters”, PASP 119 (862): 1349-6, 2007.
	In article		CrossRef
[8]	Osterbrock, Donald E. and Ferland, G. J., Ferland, G. J., ed., Astrophysics of gaseous nebulae and active galactic nuclei, University Science Books, 2005.
	In article
[9]	Parker, Quentin, A. et al., “The Macquarie/AAO/Strasbourg Hα Planetary Nebula Catalogue: MASH”, MNRAS, 373 (1): 79-94, 2006.
	In article		CrossRef
[10]	Ralston, A. and Rabinowitz, P., “A First Course in Numerical Analysis”, McGraw-Hill Kogakusha, Ltd., Tokyo, 1978.
	In article
[11]	Rees, B. and Zijlstra, A. A., “Alignment of the Angular Momentum Vectors of Planetary Nebulae in the Galactic Bulge”, MNRAS, 435(2), 975-991, 2013.
	In article		CrossRef
[12]	Sharaf, M. A., “Determination of a Celestial Plane Orientation: Application to Milky Way”, ROAJ, 18, no. 1, p. 61-67, 2008.
	In article
[13]	Schneider, P., “Extragalactic Astronomy and Cosmology: An Introduction”, Berlin, Springer, 2006.
	In article
[14]	Stasinska, G., Gorny, S. K, and Tylenda, R., “On the mass distribution of planetary nebulae central stars”, A & A, 327, p.736-742, 1997.
	In article
[15]	Martinez, Vicent J., and Saar, Enn, “Statistics of the Galaxy Distribution”, CRC Press, 2001.
	In article		CrossRef