APPENDIX G                                         

         Analytical Approximation of the Normal Integral

 

 

The flux capture fraction , is introduced in Chapter 9.  It is simply the fraction of reflected flux from a parabola surface, which falls within a beam having the width of n standard deviations of the total angular error. If the reflected flux is assumed to be normally distributed, the flux capture fraction is simply the area under the normal distribution curve when integrated from   n / 2  to + n / 2.

 

For the flux capture fraction , where the number of standard deviations of interest n is known, a polynomial approximation to the area under the normal curve from Abramowitz and Stegun (1970) can be calculated as:

 

                                                                                                                (G.1)

where                                                   

                                

                and                                   

                and                            

                and                               

                with

r   =    0.2316419

b₁ =    0.319381530

b₂ =  0.356563782

b₃ =    1.781477937

b₄ =  1.821255978

b₅ =    1.330274429

 

 

 

 

Symbols used in Appendix G

Variables

f(x)               defined function

n                   number of standard deviations

Q(x)              area in one ‘tail’ of normal curve

r                   constant

t(x)               defined parameter

x                    error limit

Γ                 area under normal curve

                  = 3.1416

Reference

 

Abramowitz and Stegun (1970), Handbook of Mathematical Functions, National Bureau of Standards, Washington D.C.