C:\Users\Fred\Desktop>perl fall2.pl
cos tau = - tan phi * tan delta
onset of fall ==> delta equal zero
delta is 0
delta equals zero will zero out rhs for entire globe
phi is latitude: 45 in portland (close enough)
phi is 0.785398163397448
rhs is 0
tau is 1.5707963267949
degrees is 90
estimate is 12
agrees with theory given mathematical objects with primitive assumptions

C:\Users\Fred\Desktop>type fall2.pl
#!/usr/bin/perl -w
use strict;
use 5.010;
use Math::Trig;
use Math::Trig ':pi';

say "cos tau = - tan phi * tan delta";
say "onset of fall ==> delta equal zero";
my $delta = tan( 0);
say "delta is $delta";
say "delta equals zero will zero out rhs for entire globe";
say "phi is latitude: 45 in portland (close enough)";
my $phi = deg2rad(45);
say "phi is $phi";
my $rhs = - tan ($phi) * tan ($delta);
say "rhs is $rhs";
my $tau = acos($rhs);
say "tau is $tau";
my $degrees = rad2deg($tau);
say "degrees is $degrees";
my $estimate = 2 * $degrees/15;
say "estimate is $estimate";
say "agrees with theory given mathematical