// Functions for calculating the positions of the planets to high accuracy // using the full VSOP87 theory. // // use sun.frink // A data structure used to hold planet information. class Planet { // The name of the planet var name // A function to calculate the heliocentric coordinates var coordFunc // A function to calculate the magnitude as seen from earth. var magFunc // Constructor that takes a name, a function to calculate heliocentric // coordinates, and a function to calculate magnitude new[n, cfunc, mfunc] := { name = n coordFunc = cfunc magFunc = mfunc } // Returns the heliocentric coordinates at the given time getCoordinates[date] := { return coordFunc[date] } // Returns the magnitude at the given time. getMagnitude[date] := { return magFunc[date] } // Returns the name of this planet. getName[] := name class var Mercury = new Planet["Mercury", MercuryHeliocentricCoordinates, MercuryMagnitude] class var Venus = new Planet["Venus", VenusHeliocentricCoordinates, VenusMagnitude] class var Earth = new Planet["Earth", EarthHeliocentricCoordinates, EarthMagnitude] class var Mars = new Planet["Mars", MarsHeliocentricCoordinates, MarsMagnitude] class var Jupiter = new Planet["Jupiter", JupiterHeliocentricCoordinates, JupiterMagnitude] class var Saturn = new Planet["Saturn", SaturnHeliocentricCoordinates, SaturnMagnitude] class var Uranus = new Planet["Uranus", UranusHeliocentricCoordinates, UranusMagnitude] class var Neptune = new Planet["Neptune", NeptuneHeliocentricCoordinates, NeptuneMagnitude] class var planets = [Planet.Mercury, Planet.Venus, Planet.Earth, Planet.Mars, Planet.Jupiter, Planet.Saturn, Planet.Uranus, Planet.Neptune] // Calculates geocentric coordinates of a planet (other than earth) at the // specified time. // // returns: // [ra, decl] geocentricCoordinates[date, timeConverge=.1 s, FK5=false] := { // println["geocentricCoordinates: this planet is " + this]; [x,y,z,d,sunLong] = Planet.Earth.apparentPosition[date, this, timeConverge] lambda = arctan[y,x] beta = z / sqrt[x^2+y^2] // eq. 33.2 // println["lambda is " + ((lambda mod circle) -> "degrees")] // println["beta is " + (beta -> "degrees")] // This is from Meeus ch. 23, before eq. 23.2 e = earthEccentricity[date] T = meeusT[date] longPi = (102.93735 + 1.71946 T + .00046 T^2) degrees // println["e=$e"] // println["longPi = " + (longPi->"degrees")] // println["sunLong = " + (sunLong->"degrees")] // Eq. 23.2 k = 20.49552 arcsec deltaLambda = (-k cos[sunLong - lambda] + e k cos[longPi-lambda])/cos[beta] deltaBeta = -k sin[beta] ( sin[sunLong - lambda] - e sin[longPi - lambda]) // println["deltaLambda is " + ((deltaLambda) -> "degrees")] // println["deltaBeta is " + (deltaBeta -> "degrees")] lambda = lambda + deltaLambda beta = beta + deltaBeta // println["New lambda is " + ((lambda mod circle) -> "degrees")] // println["New beta is " + (beta -> "degrees")] if (FK5) { [deltaLambda, deltaBeta] = correctionDynamicalToFK5[date, lambda, beta] lambda = lambda + deltaLambda beta = beta + deltaBeta } // println["deltaLambda is " + ((deltaLambda) -> "degrees")] // println["deltaBeta is " + (deltaBeta -> "degrees")] // println["New lambda is " + ((lambda mod circle) -> "degrees")] // println["New beta is " + (beta -> "degrees")] // Correct for nutation [deltapsi, deltaepsilon] = highAccuracyNutation[date] // println["deltaPsi is " + ((deltapsi) -> "arcsec")] // println["deltaEpsilon is " + (deltaepsilon -> "arcsec")] lambda = lambda + deltapsi // println["New lambda is " + ((lambda mod circle) -> "degrees")] epsilon = trueObliquityOfEcliptic[date] // println["Epsilon is " + (epsilon -> "degrees")] [alpha, delta] = eclipticalToRADecl[lambda, beta, epsilon] return [alpha, delta] } // Find the apparent position of one planet from this planet at the // specified time. See chapter 33 in Meeus. apparentPosition[date, toPlanet is Planet, timeConverge=.1 s] := { // println["FromPlanet is " + this]; // println["toPlanet is " + toPlanet]; [L0, B0, R0] = getCoordinates[date] oldLightTime = 0 s var x var y var z var d do { [L,B,R] = toPlanet.getCoordinates[date - oldLightTime] [x,y,z,d] = distanceBetweenSphericalCoordinates[L,B,R, L0,B0,R0] lightTime = d/c err = lightTime - oldLightTime oldLightTime = lightTime // println["$x $y $z $d $err"] } while err > timeConverge return[x, y, z, d, L0+180 deg, L, B, R] } // Calculate the azimuth and altitude of a planet as seen from earth. // Arguments: // [date, planet, lat, long] // // date: time of observation // planet: An object of type Planet // lat: Latitude of observer (North is positive) // long: Longitude of observer (West is positive) // // Returns: // [azimuth, altitude] as angles. Azimuth is calculated as angle west from // south. To get compass bearing, use // (azimuth + 180 degrees) mod circle // // Implements equations 13.5 and 13.6 from Meeus. // NOTE: This is not corrected for refraction nor parallax! These are // geocentric coordinates, that is, relative to the center of the earth. // Due to parallax, the altitude figure may be significantly in error for // nearby bodies like the moon. To get the corrected altitudes, use the // refractedAzimuthAltitude method below. // // returns: // [azimuth, altitude] getAirlessAzimuthAltitude[date, lat, long, timeConverge =.1 s] := { [ra,decl] = geocentricCoordinates[date, timeConverge] return airlessAzimuthAltitude[date, ra, decl, lat, long] } // Calculate the apparent azimuth and altitude of an object with atmospheric // refraction and parallax corrections. // Arguments: // [date, planet, lat, long, distance, temp, pressure] // // date: time of observation // planet: An object of type Planet // lat: Latitude of observer (North is positive) // long: Longitude of observer (West is positive) // temp: Temperature of atmosphere // pressure: Atmospheric pressure // // Returns: // [azimuth, altitude] as angles. Azimuth is calculated as angle west from // south. To get compass bearing, use // (azimuth + 180 degrees) mod circle // // Implements equations 13.5 and 13.6 from Meeus. // refractedAzimuthAltitude[date, lat, long, temp=283K, pressure=1010 millibars, timeConverge = .1 s] := { // println["In refractedAzimuthAltitude"] [azimuth, altitude] = getAirlessAzimuthAltitude[date, lat, long, timeConverge] distance = trueDistance[date, Planet.Earth, this] // println["Distance is " + (distance-> "au")] altitude = altitude - parallaxAngleAlt[distance, altitude] // println["Parallax angle is " + (parallaxAngleAlt[distance, altitude] -> "degrees")] corrAltitude = altitude + refractionAngle[altitude, temp, pressure] return[azimuth, corrAltitude] } // Secant method solver to find when a planet is at a given azimuth to // an observer at the specified latitude and longitude on the Earth. // It takes a first guess at date1 and adds an hour to that to find // its two starting points. // // This function corrects for refraction. // // This currently doesn't handle the case where the planet doesn't pass // through the desired azimuth on that date and may give unpredictable // results in this case or the case where the initial guess is far off. // // desiredAzimuth should be specified in the coordinate system specified by // Meeus, which is angle west of south. To convert from normal true compass // bearings, use (trueAzimuth + 180 degrees) mod circle secantAzimuth[date1, lat, long, desiredAzimuth, temperature = 283 K, pressure=1010 millibars ] := { date2 = date1 + 5 minutes [azimuth1, altitude1] = refractedAzimuthAltitude[date1, lat, long, temperature, pressure] [azimuth2, altitude2] = refractedAzimuthAltitude[date2, lat, long, temperature, pressure] azimuth1 = azimuth1 - desiredAzimuth azimuth2 = azimuth2 - desiredAzimuth // println["Azimuth 1: $azimuth1"] // println["Azimuth 2: $azimuth2"] while (true) { correction = (azimuth1 * (date1 - date2)) / (azimuth1-azimuth2) date = date1 - correction // println[date] if abs[correction] < 1 second return date date2 = date1 date1 = date azimuth2 = azimuth1 [azimuth1, altitude1] = refractedAzimuthAltitude[date, lat, long, temperature, pressure] azimuth1 = azimuth1 - desiredAzimuth } } // Calculates the phase angle of a planet for the specified date. // Returns i, the phase angle of the planet. // See chapter 41 in Meeus. phaseAngle[date] := { arccos[cosPhaseAngle[date]] } // Calculates the *cosine* of the phase angle of a planet for a given // date. This is eq. 41.3 in Meeus. The cosine of the angle is returned // as it's easier to use it in the illuminateFraction[date] function. // For calculating the phase angle directly, use the phaseAngle function // above. cosPhaseAngle[date] := { [L, B, R] = getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] return (R - R0 cos[B] cos[L-L0])/dist // Eq. 41.3 (returns cos[i]) } // Returns the illuminated fraction of a planet for the given date, as // seen from Earth. // This is calculated by equations 41.1 in Meeus. illuminatedFraction[date] := { (1 + cosPhaseAngle[date])/2 // Eq. 41.1 } } // End of Planet class. // Magnitude functions that will be passed to the constructors. // These are the Astronomical Almanac formulas as given in Meeus, // p. 286. MercuryMagnitude = {|date| i = Planet.Mercury.phaseAngle[date] [L,B,R] = Planet.Mercury.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] id = i / degree return -0.42 + 5 log[R/au * dist/au] + 0.0380 id - 0.000273 id^2 + 0.000002 id^3 } VenusMagnitude = {|date| i = Planet.Venus.phaseAngle[date] [L,B,R] = Planet.Venus.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] id = i / degree return -4.40 + 5 log[R/au * dist/au] + 0.0009 id + 0.000239 id^2 - 0.00000065 id^3 } EarthMagnitude = {|date| return undef } MarsMagnitude = {|date| i = Planet.Mars.phaseAngle[date] [L,B,R] = Planet.Mars.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] id = i / degree return -1.52 + 5 log[R/au * dist/au] + 0.016 id } JupiterMagnitude = {|date| i = Planet.Jupiter.phaseAngle[date] [L,B,R] = Planet.Jupiter.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] id = i / degree return -9.40 + 5 log[R/au * dist/au] + 0.005 i } // The calculations for Saturn include terms for the ring, which make it // non-negligibly brighter. SaturnMagnitude = {|date| i = Planet.Saturn.phaseAngle[date] [L,B,R] = Planet.Saturn.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] // Calculate the position of the rings [deltaU, sB] = SaturnRingPosition[date] return -8.88 + 5 log[R/au * dist/au] + 0.044 (deltaU/degree) - 2.60 sin[abs[sB]] + 1.25 sin[sB]^2 } UranusMagnitude = {|date| i = Planet.Uranus.phaseAngle[date] [L,B,R] = Planet.Uranus.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] id = i / degree return -7.19 + 5 log[R/au * dist/au] } NeptuneMagnitude = {|date| i = Planet.Neptune.phaseAngle[date] [L,B,R] = Planet.Neptune.getCoordinates[date] [L0, B0, R0] = Planet.Earth.getCoordinates[date] [x,y,z,dist] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] id = i / degree return -6.87 + 5 log[R/au * dist/au] } // PlutoMagnitude = -1.00 + 5 log[R/au * dist/au] // Saturn rings calculations. See chapter 45 of Meeus. SaturnRingPosition[date] := { T = meeusT[date] //println["T=$T"] // Step 1. These are equations 45.1 i = (28.075216 - 0.012998 T + 0.000004 T^2) degrees omega = (169.508470 + 1.394681 T + 0.000412 T^2) degrees //println["i=" + (i->"degrees")] //println["omega=" + (omega->"degrees")] // Step 2. [l0, b0, r0] = Planet.Earth.getCoordinates[date] //println["l0=" + (l0->"degrees")] //println["b0=" + (b0->"degrees")] //println["r0=" + (r0->"au")] // Steps 3 and 4 [x, y, z, d, L0prime, l, b, r] = Planet.Earth.apparentPosition[date, Planet.Saturn] //println["l=" + (l->"degrees")] //println["b=" + (b->"degrees")] //println["r=" + (r->"au")] //println["x=" + (x->"au")] //println["y=" + (y->"au")] //println["z=" + (z->"au")] //println["d=" + (d->"au")] // Step 5 lambda = arctan[y,x] beta = arctan[z, sqrt[x^2+y^2]] //println["lambda=" + (lambda->"degrees")] //println["beta=" + (beta->"degrees")] // Step 6 B = arcsin[sin[i] cos[beta] sin[lambda-omega] - cos[i] sin[beta]] aAxis = 375.35 arcsec / (d/au) bAxis = aAxis sin[abs[B]] //println["B=" + (B->"degrees")] //println["a=" + (aAxis->"arcsec")] //println["b=" + (bAxis->"arcsec")] // Step 7 N = (113.6655 + 0.8771 T) degrees //println["N=" + (N->"degrees")] rau = r/au lprime = l - (0.01759 degrees/rau) bprime = b - 0.000764 degrees cos[l-N]/rau //println["lprime=" + (lprime->"degrees")] //println["bprime=" + (bprime->"degrees")] // Step 8 Bprime = arcsin[sin[i] cos[bprime] sin[lprime-omega] - cos[i] sin[bprime]] //println["Bprime=" + (Bprime->"degrees")] // Step 9 U1 = arctan[sin[i] sin[bprime] + cos[i] cos[bprime] sin[lprime-omega], cos[bprime] cos[lprime-omega]] U2 = arctan[sin[i] sin[beta] + cos[i] cos[beta] sin[lambda-omega], cos[beta] cos[lambda-omega]] deltaU = abs[U1-U2] //println["U1=" + (U1->"degrees")] //println["U2=" + (U2->"degrees")] //println["deltaU=" + (deltaU->"degrees")] return [deltaU, B] } // Calculate the distance between planets given their // spherical coordinates. If using this for calculating geocentric // coordinates, the coordinates of earth should be the second set of // coordinates (L0, B0, R0). // arguments: // L, L0: heliocentric longitude // B, B0: heliocentric latitude // R, R0: distance from sun // // // returns: // [x,y,z,d] // Where x,y,z are the componentwise distances, and d is the total distance. distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] := { // eq. 33.1 in Meeus x = R cos[B] cos[L] - R0 cos[B0] cos[L0] y = R cos[B] sin[L] - R0 cos[B0] sin[L0] z = R sin[B] - R0 sin[B0] d = sqrt[x^2 + y^2 + z^2] return [x,y,z,d] } // Returns the true distance between two planets at the specified date. trueDistance[date, fromPlanet is Planet, toPlanet is Planet] := { [L0, B0, R0] = fromPlanet.getCoordinates[date] // println["L0=$L0, B0=$B0, R0=$R0"] [L, B, R] = toPlanet.getCoordinates[date] // println["L=$L, B=$B, R=$R"] [x,y,z,d] = distanceBetweenSphericalCoordinates[L, B, R, L0, B0, R0] return d } // Returns the corrections to lambda and beta when converting between // heliocentric coordinates referred to the dynamical ecliptic to the FK5 // system. // returns: // [deltaL, deltaB] // Meeus eq. 32.3 correctionDynamicalToFK5[date, L, B] := { T = meeusT[date] Lprime = L - 1.397 degrees T - 0.00031 degrees T^2 deltaL = (0.09033 + 0.03916 cos[Lprime] + sin[Lprime]) arcsec deltaB = 0.03916 arcsec (cos[Lprime] - sin[Lprime]) return [deltaL, deltaB] } // --------------------------------------------------------------------- // Data below this point is the output of vsop87.frink and should not be // modified by hand! // ---------------------------------------------------------------------- // // Mercury // // This function calculates the heliocentric coordinates of Mercury // referred to the mean equinox *of the date*. You may want to convert this // to another coordinate system, such as FK5. // // arguments: // d: the date/time to be calculated for // // returns: // [L, B, R] // // Where // L is the heliocentric longitude, // B is the heliocentric latitude // R is the distance from the sun. MercuryHeliocentricCoordinates = {|d| tau = meeusT[d] / 10 L0 = 5e-11 * cos[1.14175711692 + 188898.39784101078 * tau] + 4e-11 * cos[2.66996288682 + 25572.43927048119 * tau] + 4e-11 * cos[4.17772291197 + 53132.09543911899 * tau] + 4e-11 * cos[5.34965278511 + 24787.0763233692 * tau] + 4e-11 * cos[5.1333479743 + 26387.0295358434 * tau] + 5e-11 * cos[4.46570663623 + 104237.17327761157 * tau] + 5e-11 * cos[2.2014651331 + 82865.65954368559 * tau] + 4e-11 * cos[2.91542426579 + 130215.17091216908 * tau] + 4e-11 * cos[2.9409789397 + 129902.71119577558 * tau] + 4e-11 * cos[6.14515791357 + 34102.10134449819 * tau] + 4e-11 * cos[1.10868110225 + 52072.0240945678 * tau] + 4e-11 * cos[6.24881318031 + 18073.7049386502 * tau] + 5e-11 * cos[2.50519016528 + 79845.66877300559 * tau] + 5e-11 * cos[2.12658067178 + 130542.60848208958 * tau] + 4e-11 * cos[4.56426377076 + 46131.577701773 * tau] + 4e-11 * cos[4.90170477242 + 104888.41707839219 * tau] + 4e-11 * cos[6.27884818175 + 170068.84005193636 * tau] + 4e-11 * cos[2.90835831307 + 61299.38303816579 * tau] + 4e-11 * cos[3.59370682931 + 46564.4708559004 * tau] + 4e-11 * cos[2.33941435272 + 23389.451694197 * tau] + 5e-11 * cos[0.93391333656 + 52102.50915728939 * tau] + 4e-11 * cos[2.18529936793 + 23384.2869868986 * tau] + 4e-11 * cos[4.40507427126 + 129483.22655190038 * tau] + 4e-11 * cos[0.00216060134 + 213255.17862965238 * tau] + 5e-11 * cos[3.64294402198 + 27331.3907432526 * tau] + 4e-11 * cos[1.10749672173 + 51086.09749195479 * tau] + 4e-11 * cos[1.28935961583 + 234364.53008329177 * tau] + 6e-11 * cos[2.35439456378 + 139543.42270198859 * tau] + 5e-11 * cos[1.4054582464 + 233202.05537888398 * tau] + 5e-11 * cos[0.36682731238 + 146505.58156934578 * tau] + 5e-11 * cos[3.5709323278 + 53654.67285721279 * tau] + 5e-11 * cos[4.61789756581 + 20894.5360827764 * tau] + 6e-11 * cos[4.68979746906 + 26149.62911631439 * tau] + 6e-11 * cos[0.00392034365 + 73891.83305794839 * tau] + 5e-11 * cos[0.08842684725 + 27993.3679065146 * tau] + 5e-11 * cos[3.59670550074 + 131072.29944718418 * tau] + 6e-11 * cos[4.21216601811 + 182595.65223011958 * tau] + 6e-11 * cos[0.5782905701 + 50600.9604818662 * tau] + 5e-11 * cos[4.04820376371 + 129277.04100346318 * tau] + 5e-11 * cos[2.69389382491 + 180496.55813064099 * tau] + 6e-11 * cos[0.84754795471 + 104466.051854982 * tau] + 5e-11 * cos[2.71005052987 + 104427.87863757239 * tau] + 5e-11 * cos[3.77056538505 + 164407.50800278416 * tau] + 6e-11 * cos[0.20069428083 + 50490.75416064679 * tau] + 5e-11 * cos[0.80991017339 + 25661.9943650602 * tau] + 6e-11 * cos[3.62607025643 + 27883.16158529519 * tau] + 5e-11 * cos[5.35670104895 + 26513.8119180882 * tau] + 7e-11 * cos[5.00361119157 + 209762.60706278277 * tau] + 5e-11 * cos[2.46087822928 + 156681.19765992998 * tau] + 7e-11 * cos[2.86077141334 + 179172.74704605396 * tau] + 5e-11 * cos[3.98361684334 + 54862.5283776896 * tau] + 5e-11 * cos[3.20880924837 + 259769.65286364855 * tau] + 5e-11 * cos[4.47086994598 + 27360.58416720139 * tau] + 6e-11 * cos[2.47972848111 + 266540.36346489414 * tau] + 5e-11 * cos[5.69376389474 + 52400.1510788503 * tau] + 6e-11 * cos[4.66124811472 + 65717.22748563958 * tau] + 7e-11 * cos[2.57540094308 + 77218.55458853499 * tau] + 6e-11 * cos[2.17116807817 + 156373.64003896038 * tau] + 5e-11 * cos[0.08842088266 + 79859.89586700719 * tau] + 6e-11 * cos[1.00031747475 + 51538.81001112419 * tau] + 6e-11 * cos[1.48244229587 + 260349.34045064733 * tau] + 6e-11 * cos[1.44535366888 + 20043.6745601988 * tau] + 6e-11 * cos[1.86021237763 + 96471.52341295779 * tau] + 5e-11 * cos[4.19687137282 + 111032.83741969639 * tau] + 8e-11 * cos[0.55724966367 + 129806.73196855778 * tau] + 7e-11 * cos[1.76338847974 + 55484.27217852879 * tau] + 6e-11 * cos[5.29985761538 + 175376.64639941938 * tau] + 7e-11 * cos[1.99199973963 + 88285.54670482199 * tau] + 7e-11 * cos[1.17954845926 + 209232.91609768818 * tau] + 7e-11 * cos[5.89987227798 + 53757.76563143139 * tau] + 6e-11 * cos[3.70620257597 + 182608.20844401857 * tau] + 6e-11 * cos[2.59291082745 + 52186.8519834123 * tau] + 6e-11 * cos[5.14998820697 + 130364.73410930368 * tau] + 7e-11 * cos[3.67760971466 + 229129.79622501558 * tau] + 6e-11 * cos[5.55141392261 + 207747.62539098498 * tau] + 6e-11 * cos[5.50261387027 + 26238.95081141709 * tau] + 6e-11 * cos[3.04358438526 + 83591.18553310538 * tau] + 6e-11 * cos[2.17221286567 + 27573.88326263939 * tau] + 7e-11 * cos[5.19369769843 + 91919.56991589899 * tau] + 6e-11 * cos[2.25676540184 + 118007.47305747319 * tau] + 8e-11 * cos[4.05701479596 + 60170.3347251726 * tau] + 6e-11 * cos[2.30839350284 + 27353.47062020059 * tau] + 7e-11 * cos[0.19039890992 + 117077.06600107178 * tau] + 8e-11 * cos[1.46359871781 + 74935.57376844239 * tau] + 6e-11 * cos[2.24614788518 + 131079.41299418498 * tau] + 6e-11 * cos[4.63624566094 + 57369.17334194619 * tau] + 6e-11 * cos[1.42098831764 + 26026.177166834 * tau] + 6e-11 * cos[1.4831694091 + 26191.68533015479 * tau] + 6e-11 * cos[3.27000773736 + 104557.79811473397 * tau] + 6e-11 * cos[0.04385251847 + 182634.99175191917 * tau] + 6e-11 * cos[4.54553590954 + 26235.9818660005 * tau] + 6e-11 * cos[3.92166942434 + 182402.02289558138 * tau] + 6e-11 * cos[6.2231897778 + 261988.40996783535 * tau] + 7e-11 * cos[1.85784871452 + 28096.46068073319 * tau] + 6e-11 * cos[2.25546242417 + 79955.87509422498 * tau] + 7e-11 * cos[0.89727780818 + 28213.78054895339 * tau] + 7e-11 * cos[5.46082675395 + 76352.50994150538 * tau] + 7e-11 * cos[4.68484570125 + 129971.55071751658 * tau] + 8e-11 * cos[3.9435380026 + 77929.85348395319 * tau] + 7e-11 * cos[5.55018242223 + 148532.89040742096 * tau] + 7e-11 * cos[0.78772735446 + 104044.0549453272 * tau] + 7e-11 * cos[4.38077281912 + 87648.55043279778 * tau] + 7e-11 * cos[1.2898265788 + 127791.06088239799 * tau] + 7e-11 * cos[5.83295581556 + 78339.97549599818 * tau] + 7e-11 * cos[1.524098908 + 224577.84272795678 * tau] + 7e-11 * cos[3.70718359263 + 61165.27398858079 * tau] + 7e-11 * cos[0.63731291763 + 132350.71519108818 * tau] + 1.0e-10 * 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cos[5.65335125838 + 77734.01845962799 * tau] + 2.7504e-7 * cos[0.98010127839 + 45892.73043315699 * tau] + 2.9538e-7 * cos[0.69771244088 + 213.299095438 * tau] + 2.6253e-7 * cos[0.64807043102 + 1589.0728952838 * tau] + 2.6215e-7 * cos[5.24158618719 + 22645.32819660879 * tau] + 2.8496e-7 * cos[0.65048992658 + 426.598190876 * tau] + 3.0800e-7 * cos[5.77017310191 + 103292.23063610759 * tau] + 3.4044e-7 * cos[0.47470299167 + 65697.55772473979 * tau] + 3.5964e-7 * cos[1.4238083863 + 2218.7571041868 * tau] + 3.0560e-7 * cos[5.84045074182 + 43071.8992890308 * tau] + 3.3951e-7 * cos[2.78618091049 + 14765.2390432698 * tau] + 3.5392e-7 * cos[3.36964859355 + 36301.18868778519 * tau] + 4.8007e-7 * cos[5.49260554912 + 51749.20809227239 * tau] + 3.5627e-7 * cos[0.81390126585 + 32858.61374281979 * tau] + 3.8045e-7 * cos[2.43117327523 + 12566.1516999828 * tau] + 4.1882e-7 * cos[5.19309298528 + 6283.0758499914 * tau] + 4.4744e-7 * cos[1.22366857463 + 77154.33087262919 * tau] + 5.1458e-7 * cos[5.47786463494 + 50586.73338786459 * tau] + 4.1764e-7 * cos[5.64185159566 + 53131.406024757 * tau] + 5.6531e-7 * cos[5.11920557675 + 73711.75592766379 * tau] + 4.9567e-7 * cos[3.98985863874 + 6770.7106012456 * tau] + 4.7560e-7 * cos[5.49722099211 + 3.881335358 * tau] + 5.4442e-7 * cos[3.14331542453 + 27147.28507176339 * tau] + 5.5376e-7 * cos[4.05312663019 + 30639.856638633 * tau] + 6.4830e-7 * cos[0.0476292581 + 33326.5787331742 * tau] + 5.9481e-7 * cos[2.74692752 + 16983.9961474566 * tau] + 6.9729e-7 * cos[3.57201709671 + 25132.3033999656 * tau] + 6.8493e-7 * cos[0.63424819267 + 83925.04147387479 * tau] + 6.3463e-7 * cos[3.14700877722 + 7238.6755916 * tau] + 6.9247e-7 * cos[4.19446437496 + 19.66976089979 * tau] + 8.4971e-7 * cos[4.33100364958 + 79373.08797681599 * tau] + 8.9987e-7 * cos[5.85243631094 + 41962.5207369374 * tau] + 0.00000106422 * cos[4.20572116254 + 19804.8272915828 * tau] + 8.8329e-7 * cos[5.41338795963 + 26617.5941066688 * tau] + 8.6723e-7 * cos[1.9595304265 + 46514.4742339962 * tau] + 8.6819e-7 * cos[2.64219349385 + 51646.11531805379 * tau] + 0.00000125219 * cos[3.72079804425 + 39609.6545831656 * tau] + 9.6860e-7 * cos[6.2039820274 + 234791.12827416777 * tau] + 0.00000118233 * cos[2.78149786369 + 77204.32749453338 * tau] + 0.00000137942 * cos[0.29098447849 + 10213.285546211 * tau] + 0.00000142316 * cos[3.36003948842 + 37410.5672398786 * tau] + 0.00000172643 * cos[2.45200164173 + 24498.8302462904 * tau] + 0.00000208995 * cos[2.09178234008 + 47623.8527860896 * tau] + 0.00000181629 * cos[2.43413502466 + 25661.3049506982 * tau] + 0.00000175965 * cos[4.53636829858 + 51066.427731055 * tau] + 0.00000183359 * cos[2.62878670784 + 27043.5028831828 * tau] + 0.00000216645 * cos[0.65987207348 + 13521.7514415914 * tau] + 0.00000264336 * cos[3.91705094013 + 57837.1383323006 * tau] + 0.00000238793 * cos[0.11343953378 + 1059.3819301892 * tau] + 0.0000023483 * cos[0.266721189 + 11322.6640983044 * tau] + 0.00000272947 * cos[2.49451163975 + 529.6909650946 * tau] + 0.00000345212 * cos[2.79211901539 + 15874.6175953632 * tau] + 0.00000259587 * cos[0.98732428184 + 4551.9534970588 * tau] + 0.00000325335 * cos[1.3367433478 + 53285.1848352418 * tau] + 0.00000451137 * cos[6.04989275289 + 51116.4243529592 * tau] + 0.00000339214 * cos[5.86327765 + 25558.2121764796 * tau] + 0.00000343313 * cos[5.76531885335 + 955.5997416086 * tau] + 0.00000352441 * cos[5.24156297101 + 20426.571092422 * tau] + 0.000004042 * cos[3.28228847025 + 208703.22513259358 * tau] + 0.00000643759 * cos[5.30266110787 + 21535.9496445154 * tau] + 0.00000714182 * cos[1.54144865265 + 24978.5245894808 * tau] + 0.00001017332 * cos[0.8803143904 + 31749.2351907264 * tau] + 0.00001589923 * cos[2.99510417815 + 25028.521211385 * tau] + 0.00001364682 * cos[4.59918318745 + 27197.2816936676 * tau] + 0.00001803463 * cos[4.1033317841 + 5661.3320491522 * tau] + 0.00001726012 * cos[0.35832239908 + 182615.32199101939 * tau] + 0.0000355974 * cos[1.51202669419 + 1109.3785520934 * tau] + 0.00007583476 * cos[3.7134840051 + 156527.41884944518 * tau] + 0.00034561897 * cos[0.77930765817 + 130439.51570787099 * tau] + 0.00165590362 * cos[4.11969163181 + 104351.61256629678 * tau] + 0.00855346843 * cos[1.16520322351 + 78263.70942472259 * tau] + 0.05046294199 * cos[4.4778548954 + 52175.8062831484 * tau] + 0.40989414976 * cos[1.48302034194 + 26087.9031415742 * tau] + 4.40250710144 * cos[0.0 + 0.0 * tau]; L1 = 2e-11 * cos[3.49751807681 + 259819.64948555276 * tau] + 2e-11 * cos[5.10682926241 + 159969.99379441059 * tau] + 2e-11 * cos[5.66467419812 + 25466.159340735 * tau] + 2e-11 * cos[5.57750243319 + 25928.601406782 * tau] + 2e-11 * cos[4.49873870855 + 50007.0458008658 * tau] + 2e-11 * cos[0.63373893542 + 1265.5674786264 * tau] + 2e-11 * cos[0.54898566845 + 153084.84390447979 * tau] + 2e-11 * cos[0.23150025826 + 24448.8336243862 * tau] + 3e-11 * cos[6.19860078008 + 25004.8229092806 * tau] + 2e-11 * cos[2.93412437896 + 63786.3582415226 * tau] + 2e-11 * cos[5.22579437397 + 6885.14988993081 * tau] + 2e-11 * cos[0.11511311275 + 27170.98337386779 * tau] + 3e-11 * cos[5.43138498093 + 106684.80895916879 * tau] + 2e-11 * cos[2.15515419441 + 51951.46148744649 * tau] + 2e-11 * cos[2.11477268878 + 25773.71696170459 * tau] + 3e-11 * cos[1.10832319418 + 25600.26839032 * tau] + 2e-11 * cos[5.1750816963 + 163298.12945069079 * tau] + 2e-11 * cos[5.91297170155 + 87367.61641884019 * tau] + 2e-11 * cos[4.41481991835 + 65538.25598994759 * tau] + 2e-11 * cos[5.97797555746 + 50167.24874398939 * tau] + 2e-11 * cos[3.56126103854 + 26189.8656598398 * tau] + 3e-11 * cos[5.60392923976 + 104466.051854982 * tau] + 2e-11 * cos[3.56733241522 + 26073.67604757259 * tau] + 2e-11 * cos[6.08909078783 + 116783.65521669458 * tau] + 2e-11 * cos[2.51564214512 + 26667.590728573 * tau] + 2e-11 * cos[0.63548334553 + 23384.2869868986 * tau] + 2e-11 * cos[4.74203919229 + 76887.93562487679 * tau] + 2e-11 * cos[1.90218644384 + 24356.7807886416 * tau] + 3e-11 * cos[1.11245383788 + 192828.60753723037 * tau] + 3e-11 * cos[2.06087055668 + 52065.59996192899 * tau] + 3e-11 * cos[3.05479754348 + 162810.49469943657 * tau] + 2e-11 * cos[3.75076268715 + 130005.80396999417 * tau] + 3e-11 * cos[1.3503644855 + 23919.1426592916 * tau] + 3e-11 * cos[5.29581547615 + 57503.2823915312 * tau] + 3e-11 * cos[6.1660165005 + 20043.6745601988 * tau] + 3e-11 * cos[5.02808401414 + 130432.40216087017 * tau] + 3e-11 * cos[2.80509981839 + 104358.2411251968 * tau] + 3e-11 * cos[3.28684847821 + 208276.62694171758 * tau] + 4e-11 * cos[3.60184879954 + 207114.15223730978 * tau] + 3e-11 * cos[5.95559848753 + 104344.98400739678 * tau] + 3e-11 * cos[1.86490629185 + 51707.84129279399 * tau] + 3e-11 * cos[1.06499703474 + 26734.913974889 * tau] + 3e-11 * cos[4.29158350083 + 154408.65498906678 * tau] + 3e-11 * cos[2.12651302584 + 78469.89497315978 * tau] + 3e-11 * cos[3.6290381593 + 130435.63437251298 * tau] + 3e-11 * cos[0.98764974659 + 128843.32926558638 * tau] + 3e-11 * cos[2.44134109777 + 156520.30530244438 * tau] + 3e-11 * cos[3.80375047171 + 26402.0893214438 * tau] + 3e-11 * cos[0.77610404005 + 142871.55835826878 * tau] + 4e-11 * cos[5.67939124426 + 106470.37642667518 * tau] + 3e-11 * cos[1.00008179165 + 50264.6067999312 * tau] + 3e-11 * cos[0.53381434741 + 26709.6469424134 * tau] + 3e-11 * cos[0.40695406256 + 52278.89905736699 * tau] + 3e-11 * cos[4.09573256493 + 155460.92337225519 * tau] + 3e-11 * cos[2.88091404327 + 78149.27013603736 * tau] + 3e-11 * cos[5.89324219451 + 50483.640613646 * tau] + 3e-11 * cos[4.05932083484 + 52822.81711646319 * tau] + 3e-11 * cos[4.27304181959 + 130593.29451835579 * tau] + 3e-11 * cos[3.46632047879 + 45494.58142974879 * tau] + 3e-11 * cos[3.99050522839 + 71378.55953479178 * tau] + 4e-11 * cos[6.01723954923 + 104237.17327761157 * tau] + 4e-11 * cos[3.24764679291 + 28286.9904848612 * tau] + 4e-11 * cos[1.65814724695 + 50800.03248330259 * tau] + 3e-11 * cos[2.67114319601 + 24998.19435038059 * tau] + 3e-11 * cos[5.22780381742 + 78338.49102328988 * tau] + 4e-11 * cos[3.42019897755 + 61279.713277266 * tau] + 3e-11 * cos[0.98973904852 + 105418.10804348679 * tau] + 4e-11 * cos[0.20412928769 + 169093.57054942798 * tau] + 4e-11 * cos[0.68519128486 + 104276.83096772949 * tau] + 4e-11 * cos[1.34568004641 + 182828.62108645738 * tau] + 4e-11 * cos[4.20598588282 + 218916.51067880458 * tau] + 4e-11 * cos[0.30056239753 + 233202.05537888398 * tau] + 5e-11 * cos[3.75130732 + 62197.64356324779 * tau] + 4e-11 * cos[1.21404057369 + 139543.42270198859 * tau] + 4e-11 * cos[3.38866894732 + 104888.41707839219 * tau] + 4e-11 * cos[3.08522628552 + 175844.61138977378 * tau] + 4e-11 * cos[4.26267840541 + 132658.27281205778 * tau] + 4e-11 * cos[0.51045647492 + 107692.22499299659 * tau] + 4e-11 * cos[3.24471199066 + 130020.03106399579 * tau] + 4e-11 * cos[4.36228014188 + 207747.62539098498 * tau] + 4e-11 * cos[5.12386500502 + 182622.43553802017 * tau] + 4e-11 * cos[1.88512485652 + 54879.422437824 * tau] + 4e-11 * cos[2.53135799952 + 79487.52726550119 * tau] + 4e-11 * cos[1.51752118872 + 103814.80805420138 * tau] + 4e-11 * cos[5.55015227967 + 149846.19399604558 * tau] + 4e-11 * cos[5.65734972357 + 51868.2486621788 * tau] + 4e-11 * cos[0.30384598125 + 285907.55262712695 * tau] + 4e-11 * cos[5.40862384292 + 104984.39630560997 * tau] + 4e-11 * cos[0.67741502331 + 131079.41299418498 * tau] + 4e-11 * cos[3.43760746038 + 234261.43730907318 * tau] + 4e-11 * cos[3.86524951456 + 209762.60706278277 * tau] + 5e-11 * cos[5.23264312237 + 126996.94076290558 * tau] + 4e-11 * cos[3.25642623421 + 195181.47369100217 * tau] + 4e-11 * cos[0.80000960641 + 1485.2907067032 * tau] + 4e-11 * cos[5.22571179686 + 261988.40996783535 * tau] + 4e-11 * cos[0.51369916437 + 181563.05360783098 * tau] + 4e-11 * cos[4.43048849829 + 118828.96374949679 * tau] + 5e-11 * cos[3.31232177694 + 48847.6706268682 * tau] + 4e-11 * cos[2.74756161638 + 144916.86689107098 * tau] + 4e-11 * cos[5.24580055005 + 52309.9153327334 * tau] + 5e-11 * cos[5.90579064813 + 1478.8665740644 * tau] + 5e-11 * cos[5.65870377214 + 1300.826818205 * tau] + 5e-11 * cos[4.42201446083 + 78057.52387628538 * tau] + 4e-11 * cos[0.10839681347 + 38.1330356378 * tau] + 4e-11 * cos[1.8932118711 + 2111.6503133776 * tau] + 5e-11 * cos[5.13889133658 + 130907.48069822539 * tau] + 4e-11 * cos[1.46002058771 + 935.92998070881 * tau] + 6e-11 * cos[4.78710650697 + 102755.42612401219 * tau] + 5e-11 * cos[2.1751453684 + 51329.7234483972 * tau] + 4e-11 * cos[1.6456663118 + 78697.42116259939 * tau] + 4e-11 * cos[4.47492094862 + 137678.19129947099 * tau] + 4e-11 * cos[5.77579934605 + 129799.61842155698 * tau] + 4e-11 * cos[0.85257120719 + 118007.47305747319 * tau] + 4e-11 * cos[6.20316213523 + 79116.90580647459 * tau] + 5e-11 * cos[4.29231489707 + 220025.88923089796 * tau] + 4e-11 * cos[2.00085806148 + 189386.03259226495 * tau] + 5e-11 * cos[1.42813778372 + 266540.36346489414 * tau] + 5e-11 * cos[2.55069425936 + 154308.66174525839 * tau] + 4e-11 * cos[1.42876132655 + 52252.07235442399 * tau] + 4e-11 * cos[5.71176695959 + 81591.84508100279 * tau] + 5e-11 * cos[4.68929437146 + 79315.97780791098 * tau] + 4e-11 * cos[2.71436739189 + 182402.02289558138 * tau] + 4e-11 * cos[0.64833718455 + 156377.85565231058 * tau] + 4e-11 * cos[4.0062366874 + 175376.64639941938 * tau] + 4e-11 * cos[0.588581477 + 323.5054166574 * tau] + 4e-11 * cos[4.65165864815 + 195047.36464141717 * tau] + 5e-11 * cos[2.02580304988 + 259769.65286364855 * tau] + 5e-11 * cos[1.82284966431 + 1884.9011634174 * tau] + 6e-11 * cos[4.70286246928 + 91805.13062721379 * tau] + 5e-11 * cos[2.8002233534 + 4437.51420837359 * tau] + 4e-11 * cos[4.33627478046 + 89.485747846 * tau] + 4e-11 * cos[3.93130462476 + 130289.95251073639 * tau] + 4e-11 * cos[1.1697417665 + 193937.98608932378 * tau] + 5e-11 * cos[2.90596877294 + 154194.22245657316 * tau] + 5e-11 * cos[0.26610630357 + 235746.72801577636 * tau] + 6e-11 * cos[5.57902902188 + 87253.17713015496 * tau] + 5e-11 * cos[3.3535670841 + 205260.65018762815 * tau] + 5e-11 * cos[1.76549372525 + 53771.99272543299 * tau] + 5e-11 * cos[3.95096384085 + 78188.92782615528 * tau] + 5e-11 * cos[0.55904094194 + 2383.1930147762 * tau] + 5e-11 * cos[0.76195152302 + 417406.45026518717 * tau] + 7e-11 * cos[3.69898590349 + 81604.32185142238 * tau] + 5e-11 * cos[4.87459278742 + 78413.27262185719 * tau] + 5e-11 * cos[5.10195955921 + 112231.70171963578 * tau] + 5e-11 * cos[0.02805624095 + 149756.70824819959 * tau] + 5e-11 * cos[1.46111595893 + 101011.00013959699 * tau] + 6e-11 * cos[3.03452226064 + 77741.13200662879 * tau] + 5e-11 * cos[0.51603916276 + 171292.65789271498 * tau] + 6e-11 * cos[3.58276567647 + 198489.93958638258 * tau] + 5e-11 * cos[4.54130791881 + 156740.71794488319 * tau] + 5e-11 * cos[1.05826639118 + 27780.06881107659 * tau] + 5e-11 * cos[1.13701088228 + 26312.2479372761 * tau] + 5e-11 * cos[1.95647355963 + 76571.54375522019 * tau] + 5e-11 * cos[0.76610110351 + 24864.08530079559 * tau] + 5e-11 * cos[0.96987247797 + 149288.74325784518 * tau] + 5e-11 * cos[2.36158299954 + 78896.49316403578 * tau] + 7e-11 * cos[1.34355308244 + 1731.1223529326 * tau] + 5e-11 * cos[4.06443360712 + 130285.73689738619 * tau] + 6e-11 * cos[1.20953605164 + 130459.18546877075 * tau] + 5e-11 * cos[2.52891655111 + 123758.29085447139 * tau] + 6e-11 * cos[2.6662090451 + 229129.79622501558 * tau] + 5e-11 * cos[5.97271372078 + 156314.11975400719 * tau] + 5e-11 * cos[2.15640465605 + 235900.50682626116 * tau] + 6e-11 * cos[1.88665916947 + 80174.90890793978 * tau] + 7e-11 * cos[5.94496795198 + 256327.07791868312 * tau] + 5e-11 * cos[1.35980682609 + 94329.77528619739 * tau] + 5e-11 * cos[0.0534317532 + 27044.1922975448 * tau] + 5e-11 * cos[1.7231082929 + 103718.82882698359 * tau] + 6e-11 * cos[5.45690475475 + 1243.4876016784 * tau] + 7e-11 * cos[0.15300257128 + 2014.9816717978 * tau] + 5e-11 * cos[2.79143401592 + 51315.49635439559 * tau] + 7e-11 * cos[2.08385155566 + 104819.57755665119 * tau] + 5e-11 * cos[6.03800330793 + 84944.93427812219 * tau] + 7e-11 * cos[6.25185953525 + 74935.57376844239 * tau] + 6e-11 * cos[2.7051772072 + 52492.19815280499 * tau] + 5e-11 * cos[6.28085632767 + 31415.379249957 * tau] + 7e-11 * cos[6.02219529486 + 156507.74908854539 * tau] + 6e-11 * cos[4.29792825666 + 52812.8025551726 * tau] + 6e-11 * cos[3.74091556543 + 155475.15046625677 * tau] + 8e-11 * cos[4.96432797037 + 52396.2189255872 * tau] + 6e-11 * cos[5.0693978209 + 51841.950342379 * tau] + 6e-11 * cos[2.61401898803 + 80382.47328510099 * tau] + 6e-11 * cos[5.03417122369 + 78786.28684281638 * tau] + 6e-11 * cos[0.36162525319 + 26823.77965510599 * tau] + 6e-11 * cos[0.15202166498 + 51859.41441349179 * tau] + 6e-11 * cos[6.23319937932 + 182619.20332637738 * tau] + 6e-11 * cos[3.65663654813 + 70383.6202713836 * tau] + 6e-11 * cos[3.99809593088 + 50380.54783942739 * tau] + 7e-11 * cos[2.89745929648 + 230239.17477710896 * tau] + 6e-11 * cos[3.49486002612 + 66653.15746634839 * tau] + 6e-11 * cos[4.09613031145 + 53867.97195265079 * tau] + 6e-11 * cos[3.6498851175 + 151199.94274106238 * tau] + 6e-11 * cos[4.08906102427 + 91919.56991589899 * tau] + 6e-11 * cos[2.04066112896 + 156534.53239644598 * tau] + 7e-11 * cos[3.15542325331 + 53029.0026649004 * tau] + 6e-11 * cos[3.48561717935 + 61560.64729122359 * tau] + 7e-11 * cos[0.762437199 + 183674.70392120858 * tau] + 7e-11 * cos[2.36435599398 + 58220.0348645238 * tau] + 6e-11 * cos[4.82915128356 + 79859.89586700719 * tau] + 7e-11 * cos[0.37969316028 + 102872.74599223239 * tau] + 6e-11 * cos[4.84865195534 + 213255.17862965238 * tau] + 7e-11 * cos[1.50054882304 + 78313.70604662679 * tau] + 6e-11 * cos[0.27837413876 + 76137.83201734339 * tau] + 7e-11 * cos[5.53062936347 + 917.9302859818 * tau] + 6e-11 * cos[4.22528261713 + 79330.20490191258 * tau] + 6e-11 * cos[4.9361756676 + 215473.93573383916 * tau] + 6e-11 * cos[0.45279066685 + 225687.22128005016 * tau] + 7e-11 * cos[2.39268679107 + 80596.9058175946 * tau] + 6e-11 * cos[0.30098706976 + 54824.2611086214 * tau] + 7e-11 * cos[4.32694078051 + 104248.51979207818 * tau] + 7e-11 * cos[4.68299502093 + 838.9692877504 * tau] + 7e-11 * cos[2.01982955946 + 176332.24614102798 * tau] + 7e-11 * cos[6.15588549143 + 75930.51303185058 * tau] + 7e-11 * cos[6.04804574225 + 103189.13786188899 * tau] + 6e-11 * cos[1.04156861985 + 104197.83375581198 * tau] + 8e-11 * cos[3.27628873745 + 183145.01295611398 * tau] + 7e-11 * cos[6.19019459405 + 37698.4550999484 * tau] + 7e-11 * cos[0.4481172726 + 104301.61594439259 * tau] + 6e-11 * cos[3.88021248158 + 104991.50985261079 * tau] + 8e-11 * cos[5.07207947299 + 202420.14928260217 * tau] + 7e-11 * cos[1.41908776548 + 39450.3528483734 * tau] + 6e-11 * cos[1.24806705522 + 181659.72224941078 * tau] + 8e-11 * cos[1.24101149187 + 128857.55635958798 * tau] + 7e-11 * cos[4.99225219771 + 102659.44689679438 * tau] + 8e-11 * cos[3.72793390212 + 77795.74443436819 * tau] + 7e-11 * cos[4.53777069646 + 78339.97549599818 * tau] + 7e-11 * cos[2.81152781596 + 92741.06060792258 * tau] + 8e-11 * cos[4.90310056069 + 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