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:









r   =    0.2316419

b1 =    0.319381530

b2 =  0.356563782

b3 =    1.781477937

b4 =  1.821255978

b5 =    1.330274429





Symbols used in Appendix G


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



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