[VA] SRC #011 - April 1st, 2022 Bizarro Special

[Albert Chan]

This version improve accuracy by removing square roots. Let x=u^2, v=1-u

∫(f(1-√x), x=0..1) = ∫(f(1-u)*(2u), u=0..1) = 2*∫(f(v)*(1-v), v=0..1)

Transformed integral (with discontinuity at x=0, evaluate in 2 pieces)

\(\displaystyle I = 4 \int_{-1}^{1} \int_0^1 \int_0^1
\frac{(d+x)\,(1-|x|)\,(1-y)\,(1-z)}
{[(d+x)^2 + y^2 + z^2]^{3/2}}\;dz\;dy\;dx \)

10 P=.0001 @ D=1 @ T=TIME
20 DEF FNZ(X,Y)=INTEGRAL(0,1,P,X/(X*X+Y*Y+IVAR*IVAR)^1.5*(1-IVAR))
30 DEF FNY(X)=INTEGRAL(0,1,P,FNZ(X,IVAR)*(1-IVAR))
40 I1=INTEGRAL(0,1,P,FNY(D+IVAR)*(1-IVAR))*4 @ DISP I1,IBOUND*4,TIME-T
50 I2=INTEGRAL(0,1,P,FNY(D-IVAR)*(1-IVAR))*4 @ DISP I2,IBOUND*4,TIME-T
60 DISP I1+I2

>RUN
 .236198653356      2.35758565639E-5      128.35
 .689787720788      6.90121464808E-5      763.87
 .925986374144

1st integral is sum of force for mass separated by D or more.
2nd integral is sum of force for all mass separated by D or less.

2nd integral (for D=1) contributed about 3/4 of full force.
Its singularity (where planets are touching) make it harder to calculate.