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In this paper, a reliable computational tool will be developed for the determination of the parameters of the stellar density function in a region of the sky with complete error controlled estimates. Of these error estimates are, the variance of the fit, the variance of the least squares solutions vector, the average square distance between the exact and the least-squares solutions, finally the variance of the density stellar function due to the variance of the least squares solutions vector. Moreover, all these estimates are given in closed analytical forms.

Modern observational astronomy has been characterized by an enormous growth of data stimulated by the advent of new technologies in telescopes, detectors and computations. The new astronomical data give rise to innumerable statistical problems [

On the other hand, one of the most crucial pieces of information needed in astronomy is the stellar density function in a region of the sky, due to the wealth of information on galactic structure gained directly from a study of the variations in the stellar density (e.g. [

Although the least-squares method is the most powerful technique that has been devised for the problems of astrostatistics in general [

At this stage we should point out that 1) the accuracy of the estimators and the accuracy of the fitted curve are two distinct problems; and 2) an accurate estimator will always produce small variance, but small variance does not guarantee an accurate estimator. This could be seen from Equation (2) by noting that the lower bounds for the average square distance between the exact and the least-squares values is

The importance of the stellar density function as mentioned very briefly as in the above and the existing practical difficulties due to the deficiency of the error estimation and controlling had motivated our work: to develop a reliable computational tool for the determination of the parameters of the stellar density with complete error estimates. Of these error estimates are, the variance of the fit, the variance of the least squares solutions vector, the average square distance between the exact and the least-squares solutions, finally the variance of the density stellar function due to the variance of the least squares solutions vector.

By this we aim at giving an idea on what may called an “accepted solution set” for the parameters of the stellar density functions and the associated variances by the selected tolerances for the error estimates.

Before starting the analysis, it is profitable, to give brief notes on the structure of the paper as follows.

1-Using Fourier transform to obtain analytical solution of the density function;

2-Using the least squares method to find second order polynomial for each of the apparent and absolute magnitudes distributions;

3-Using steps 1 & 2, we established analytical expressions of the density function with coefficients directly obtained from observations.

Let

where

where

According to the least squares criterion, it could be shown that [

1-The estimators

2-The estimators

3-The variance-covariance matrix

where

4-The average squared distance between

Also it should be noted that, if the precision is measured by probable error

Finally, if

then [

where

The absolute magnitude, M of a star is given in terms of the apparent magnitude

where M is thus defined in terms of the standard distance of 10 parsecs. We write, for convenience,

so that

In the above formulae the base of the logarithm is 10.

We shall refer to

Let

Let

then

Let

then

Consequently

Hence

or

where

Equation (4) is the basic integral equation to be solved for the density function

Let the distributions of the apparent and absolute magnitudes are Maxwellian in form. We assume that

As regards Equation (7.1), this is the form found to satisfy the star counts for a given galactic latitude in the exhaustive investigation by many authors. The parameters

Equation (7.2) must be regard as applicable only to a particular spectral type or subdivision of spectral type. In many studies of the distribution of absolute magnitudes, the separation of stars into the giant and dwarf classes is recognized, that in dealing with a given spectral type we represent the function

The condition (7.3) implies that the dispersion about the mean is less for absolute magnitudes of a given spectral type than for the apparent magnitudes. This is in accordance with observations, for the giants or for the dwarfs.

Taking the natural logarithm of Equations (7.1) and (7.2) we get,

where

and

Since,

where

In the following two sections, the solutions of the normal equations for

The solutions of the normal Equations (11) for

where

According to Section 2, we deduce for

1-The variance of the fit is:

2-The variance of the solutions are:

3-The average squared distance between the least square solutions and the exact solutions is

Recalling the Fourier transform

while its inverse is

Multiply Equation (4) by

where

Let

also

Then Equation (20) reduces to

also

The inverse of Fourier transform of

then

where

Using Equation (7.1) in Equation (22) the later becomes:

or, on setting

evaluating the integral on the right hand side we get

Similarly, as in deriving Equation (23) we can get for

Now, substituting Equations (23) and (24) into Equation (21) we get,

Using Equation (6) and remembering that

In what follows empirical expression of the density function

Substituting Equations (9) and (10) into Equation (25), we get for the density function

where

Since

Define

therefore we have

From Equations (29) we get

Multiply Equations (28.1) and (28.2) and then summing, we get

similarly

Since

then summing we have

similarly

Multiply Equation (33) by

similarly

Since

consequently,

According to Section 2, we have

where

Multiply Equations (38) and (37) by

similarly

let us take the error,

then assuming that the errors

where

where

Now, in Equation (40), e is linear function of the errors

Using Equations (31), (32) and (17) we finally get

where

Since each of the constants

where

Due to the above mentioned practical difficulties encountered in most applications of the least squares method we should at this stage reformalize the concept of an “acceptably small” variance. We may define an acceptable solution set to the determination of

where Tol and

In conclusion, a reliable computational tool was developed in the present paper for the determination of the parameters of the stellar density function in a region of the sky with complete error controlled estimates. Of these error estimates are, the variance of the fit, the variance of the least squares solutions vector, the average square distance between the exact and the least-squares solutions, finally the variance of the density stellar function due to the variance of the least squares solutions vector. Moreover, all these estimates are given in closed analytical forms.