x_1 and y_1 convert from orbital plane coordinates to observed coordinates.
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Expression 22: "M" left parenthesis, "t" , right parenthesis equals theta Subscript, "m" , Baseline plus StartFraction, 2 pi Over "p" , EndFraction left parenthesis, "t" minus 2000 , right parenthesisMt=θm+2πpt−2000
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M(t) gives mean anomaly in terms of time
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Expression 24: "E" left parenthesis, "m" , right parenthesis equals "m" plus sine left parenthesis, "m" , right parenthesis minus StartFraction, sin left parenthesis, "m" , right parenthesis minus "e" Subscript, 0 , Baseline sin left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis Over 1 minus "e" Subscript, 0 , Baseline cos left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis , EndFraction minus StartFraction, sin left parenthesis, "m" , right parenthesis minus StartNestedFraction, sin left parenthesis, "m" , right parenthesis minus "e" Subscript, 0 , Baseline sin left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis NestedOver 1 minus "e" Subscript, 0 , Baseline cos left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis , EndNestedFraction minus "e" Subscript, 0 , Baseline sin left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis minus StartNestedFraction, sin left parenthesis, "m" , right parenthesis minus "e" Subscript, 0 , Baseline sin left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis NestedOver 1 minus "e" Subscript, 0 , Baseline cos left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis , EndNestedFraction , right parenthesis Over 1 minus "e" Subscript, 0 , Baseline cos left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis minus StartNestedFraction, sin left parenthesis, "m" , right parenthesis minus "e" Subscript, 0 , Baseline sin left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis NestedOver 1 minus "e" Subscript, 0 , Baseline cos left parenthesis, "m" plus sin left parenthesis, "m" , right parenthesis , right parenthesis , EndNestedFraction , right parenthesis , EndFractionEm=m+sinm−sinm−e0sinm+sinm1−e0cosm+sinm−sinm−sinm−e0sinm+sinm1−e0cosm+sinm−e0sinm+sinm−sinm−e0sinm+sinm1−e0cosm+sinm1−e0cosm+sinm−sinm−e0sinm+sinm1−e0cosm+sinm
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E(m) is a second-order Newton's method approx. of E in terms of M. Good for e<0.9ish
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Expression 26: "a" Subscript, 2 , Baseline left parenthesis, "x" , "y" , right parenthesis equals mod left parenthesis, arctan left parenthesis, StartFraction, "y" Over "x" , EndFraction , right parenthesis plus StartFraction, StartNestedFraction, pi NestedOver 2 , EndNestedFraction StartAbsoluteValue, "x" , EndAbsoluteValue Over "x" , EndFraction plus StartFraction, 3 pi Over 2 , EndFraction , 2 pi , right parenthesisa2x,y=modarctanyx+π2xx+3π2,2π
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atan2() function
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Expression 28: "T" left parenthesis, "E" Subscript, 0 , Baseline , right parenthesis equals pi minus 2 times "a" Subscript, 2 , Baseline left parenthesis, StartRoot, 1 plus "e" Subscript, 0 , Baseline , EndRoot sin left parenthesis, StartFraction, "E" Subscript, 0 , Baseline Over 2 , EndFraction , right parenthesis , StartRoot, 1 minus "e" Subscript, 0 , Baseline , EndRoot cos left parenthesis, StartFraction, "E" Subscript, 0 , Baseline Over 2 , EndFraction , right parenthesis , right parenthesisTE0=π−2·a21+e0sinE02,1−e0cosE02
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T(E_0) gives true anomaly in terms of eccentric anomaly
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R^2 calculation
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Expression 31: "d" Subscript, 0 , Baseline equals StartRoot, left parenthesis, "x" Subscript, 1 , Baseline left parenthesis, "r" Subscript, 0 , Baseline left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis cos left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis , "r" Subscript, 0 , Baseline left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis sin left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis , right parenthesis minus "x" Subscript, 0 , Baseline , right parenthesis squared plus left parenthesis, "y" Subscript, 1 , Baseline left parenthesis, "r" Subscript, 0 , Baseline left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis cos left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis , "r" Subscript, 0 , Baseline left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis sin left parenthesis, "T" left parenthesis, "E" left parenthesis, "M" left parenthesis, "t" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis plus omega , right parenthesis , right parenthesis minus "y" Subscript, 0 , Baseline , right parenthesis squared , EndRootd0=x1r0TEMt0+ωcosTEMt0+ω,r0TEMt0+ωsinTEMt0+ω−x02+y1r0TEMt0+ωcosTEMt0+ω,r0TEMt0+ωsinTEMt0+ω−y02
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Expression 32: "d" Subscript, 1 , Baseline equals StartRoot, left parenthesis, "x" Subscript, "a" "v" "g" , Baseline minus "x" Subscript, 0 , Baseline , right parenthesis squared plus left parenthesis, "y" Subscript, "a" "v" "g" , Baseline minus "y" Subscript, 0 , Baseline , right parenthesis squared , EndRootd1=xavg−x02+yavg−y02
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Expression 33: "x" Subscript, "a" "v" "g" , Baseline equals Start sum from "m" equals 1 to length left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , end sum, StartFraction, "x" Subscript, 0 , Baseline left bracket, "m" , right bracket Over length left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , EndFractionxavg=lengthx0∑m=1x0mlengthx0
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Expression 34: "y" Subscript, "a" "v" "g" , Baseline equals Start sum from "m" equals 1 to length left parenthesis, "y" Subscript, 0 , Baseline , right parenthesis , end sum, StartFraction, "y" Subscript, 0 , Baseline left bracket, "m" , right bracket Over length left parenthesis, "y" Subscript, 0 , Baseline , right parenthesis , EndFractionyavg=lengthy0∑m=1y0mlengthy0
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R^2 value:
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Expression 36: 1 minus StartFraction, left parenthesis, Start sum from "q" equals 1 to length left parenthesis, "d" Subscript, 0 , Baseline , right parenthesis , end sum, "d" Subscript, 0 , Baseline left bracket, "q" , right bracket squared , right parenthesis Over Start sum from "r" equals 1 to length left parenthesis, "d" Subscript, 1 , Baseline , right parenthesis , end sum, "d" Subscript, 1 , Baseline left bracket, "r" , right bracket squared , EndFraction1−lengthd0∑q=1d0q2lengthd1∑r=1d1r2