Moon equatorial coordinates

Question

It took some effort, but in the end, I successfully calculated the geographical point of the Sun. I assumed that I could do the same for the Moon.

My plan is:

1. Find RA and declination of the Moon
2. Find GHA of the Moon, using the same math as for the Sun

However, after following this article almost to the letter (except for precession and perturbations), my calculated answer is much different from this site. I've examined the article and my code many times, but can't see an error.

I use Swift programming language, but here it's merely a way to represent my calculations and results.

If anyone would like to try it for yourself, here's my code in text format:

import Foundation

extension Double {
  var radians: Double {
    return self * .pi / 180
  }

  var degrees: Double {
    return self  * 180 / .pi
  }

  var rangedUpTo360: Double {
    let multiples = (self/360).rounded(.down)
    return self - 360*multiples
  }

  var rangedUpTo86400: Double {
    let multiples = (self/86400).rounded(.down)
    return self - 86400*multiples
  }
}

extension Date {
  var greenwich: Date {
    let greenwichDate = self.addingTimeInterval(TimeInterval(-TimeZone.current.secondsFromGMT()))
    return greenwichDate
  }
}

class Astronomy {

  static func julianDays(since: Date) -> Double {
    return since.timeIntervalSince1970 / 86400 + 2_440_587.5
  }

  static func daysSinceJ2000(for date: Date) -> Double {
    return julianDays(since: date) - 2_451_545
  }

  static func sunAxialTilt(for daysSinceJ200: Double) -> Double {
    return 23.4393 - 3.563e-7 * daysSinceJ200
  }

  static func greenwichHourAngle(rightAscension: Double, date: Date) -> Double {
    let greenwichTimeZone = TimeZone(secondsFromGMT: 0)!
    var calendar = Calendar.current
    calendar.timeZone = greenwichTimeZone
    let components = calendar.dateComponents([.hour, .minute, .second], from: date)
    let secondsOfDay = Double(components.hour!)*3600 + Double(components.minute!)*60 + Double(components.second!)

    let centuriesSinceJ200 = daysSinceJ2000(for: date.greenwich) / 36525
    let greenwichSiderealTime = 24110.54841 + 8640184.812866*centuriesSinceJ200 + 0.093104*pow(centuriesSinceJ200, 2) - 0.0000062*pow(centuriesSinceJ200, 3)
    let earthSiderealRotationRate = 1.00273790935 + 5.9e-11*centuriesSinceJ200
    let earthRotation = earthSiderealRotationRate*secondsOfDay
    let gmstSeconds = greenwichSiderealTime + earthRotation
    let gmstSecondsNormalized = gmstSeconds.rangedUpTo86400
    let gmst = gmstSecondsNormalized / 3600 * 15
    return gmst.rangedUpTo360 - rightAscension
  }

  static func moonPosition(on date: Date) -> (declination: Double, RA: Double, distance: Double) {
    date
    // day number
    let d = daysSinceJ2000(for: date)
    // longitude of the ascending node
    var N = 125.1228 - 0.0529538083 * d // have to be normalized
    if N < 0 {
      N += 360
    }
    // inclination to the ecliptic (plane of the Earth's orbit)
    let i = 5.1454
    // argument of perihelion
    var w = 318.0634 + 0.1643573223 * d // have to be normalized
    w = w.rangedUpTo360
    // semi-major axis, or mean distance from Sun
    let a = 60.2666 // (Earth radii)
    // eccentricity (0=circle, 0-1=ellipse, 1=parabola)
    let e = 0.054900
    // mean anomaly (0 at perihelion; increases uniformly with time)
    var M = 115.3654 + 13.0649929509 * d // have to be normalized
    M = M.rangedUpTo360
    // eccentric anomaly
    let E = M + e * sin(M.radians) * (1.0 + e * cos(M.radians))
    let xv = a * (cos(E.radians) - e)
    let yv = a * (sqrt(1.0 - e*e) * sin(E.radians))
    // true anomaly (angle between position and perihelion)
    let v = atan2(yv, xv).degrees
    let r = sqrt(pow(xv, 2) + pow(yv, 2))
    // geocentric ecliptical position
    let xG = r * (cos(N.radians) * cos((v + w).radians) - sin(N.radians) * sin((v + w).radians) * cos(i.radians))
    let yG = r * (sin(N.radians) * cos((v + w).radians) + cos(N.radians) * sin((v + w).radians) * cos(i.radians))
    let zG = r * (sin((v + w).radians) * sin(i.radians))
    // equatorial coordinates
    let ecl = sunAxialTilt(for: d)
    let xE = xG
    let yE = yG * cos(ecl.radians) - zG * sin(ecl.radians)
    let zE = yG * sin(ecl.radians) + zG * cos(ecl.radians)
    //right scension
    var rA = atan2(yE, xE)
    rA = rA.degrees
    // declinatino
    var dec = atan2(zE, sqrt(xE * xE + yE * yE) )
    dec = dec.degrees
    // geocentric distance
    let rG = sqrt(xE * xE + yE * yE + zE * zE)
    return (declination: dec, RA: rA, distance: rG)
  }
}

Astronomy.moonPosition(on: Date().greenwich)

Answer 1

It's a bit late but hopefully it will help others in the future.

The calculations can be found on this site, article from Keith Burnett (keith@xylem.demon.co.uk)

The first problem is that you use J2000 as time reference but it's not what the site takes as time reference!

"Get the days to Dec 31st 0h 2000 - note, this is NOT same as J2000"

So you have to do :

let d = daysSinceJ2000(for: date) + 1.5

The second problem is you are missing one degrees to radians conversion when you calculate E, it should be

let E = M.radians + e * sin(M.radians) * (1.0 + e * cos(M.radians))

The third problem is the calculation of the declination, it should be arctan, not arctan2 :

var dec = atan(zE / sqrt(xE * xE + yE * yE) )

On a side note, when you do :

var N = 125.1228 - 0.0529538083 * d // have to be normalized
    if N < 0 {
      N += 360
    }

N is still negative, to put a negative value within (0,360) I did

N = range(N)

func range(x: Double) -> Double{
    if x >= 0 {
        return x
    }
    else {
        return range(x: x + 360)
    }
}

For positive values, you can just use

M = M.truncatingRemainder(dividingBy: 360)

Although I don't think it's really crucial for the calculations, I tried without and it worked fine

Cheers, Lucas