Time of the Seasons

This post is a Micromite eXtreme program that calculates the calendar date and UTC (Universal Coordinated Time) time of the seasons. The only user input required is the calendar year of interest.

Here’s a typical user interaction with the software and the computed results.

time of the seasons
===================

please input the calendar year
? 2017

spring equinox
--------------

calendar date March 20 2017

UTC time 10 hours 28 minutes 31.97 seconds

summer solstice
---------------

calendar date June 21 2017

UTC time 4 hours 23 minutes 51.66 seconds

fall equinox
------------

calendar date September 22 2017

UTC time 20 hours 1 minutes 26.39 seconds

winter solstice
---------------

calendar date December 21 2017

UTC time 16 hours 27 minutes 54.68 seconds

Here’s a copy of the MATLAB PDF document for this program.

2017-03-06_114618_seasons.pdf

Here’s the MMBASIC source code for this program;

' seasons.bas         March 6, 2017
  
  ' calendar date and UTC time of the seasons
  
  ' Micromite eXtreme version
  
  '''''''''''''''''''''''''''
  
  ' dimension global arrays
  
  dim sl(50) as float, sr(50) as float, sa(50) as float, sb(50) as float
  
  dim jdleap(27) as float, leapsec(27) as float, month$(12) as string
  
  ' global constants
  
  const pi2 = 2.0 * pi, pidiv2 = 0.5 * pi, dtr = pi / 180.0
  
  ' read solar ephemeris data
  
  for i% = 1 to 50
    
    read sl(i%), sr(i%), sa(i%), sb(i%)
    
  next i%
  
  data 403406,      0, 4.721964,      1.621043
  data 195207, -97597, 5.937458,  62830.348067
  data 119433, -59715, 1.115589,  62830.821524
  data 112392, -56188, 5.781616,  62829.634302
  data   3891,  -1556, 5.5474  , 125660.5691
  data   2819,  -1126, 1.5120  , 125660.9845
  data   1721,   -861, 4.1897  ,  62832.4766
  data      0,    941, 1.163   ,      0.813
  data    660,   -264, 5.415   , 125659.310
  data    350,   -163, 4.315   ,  57533.850
  data    334,      0, 4.553   ,    -33.931
  data    314,    309, 5.198   , 777137.715
  data    268,   -158, 5.989   ,  78604.191
  data    242,      0, 2.911   ,      5.412
  data    234,    -54, 1.423   ,  39302.098
  data    158,      0, 0.061   ,    -34.861
  data    132,    -93, 2.317   , 115067.698
  data    129,    -20, 3.193   ,  15774.337
  data    114,      0, 2.828   ,   5296.670
  data     99,    -47, 0.52    ,  58849.27
  data     93,      0, 4.65    ,   5296.11
  data     86,      0, 4.35    ,  -3980.70
  data     78,    -33, 2.75    ,  52237.69
  data     72,    -32, 4.50    ,  55076.47
  data     68,      0, 3.23    ,    261.08
  data     64,    -10, 1.22    ,  15773.85
  data     46,    -16, 0.14    , 188491.03
  data     38,      0, 3.44    ,  -7756.55
  data     37,      0, 4.37    ,    264.89
  data     32,    -24, 1.14    , 117906.27
  data     29,    -13, 2.84    ,  55075.75
  data     28,      0, 5.96    ,  -7961.39
  data     27,     -9, 5.09    , 188489.81
  data     27,      0, 1.72    ,   2132.19
  data     25,    -17, 2.56    , 109771.03
  data     24,    -11, 1.92    ,  54868.56
  data     21,      0, 0.09    ,  25443.93
  data     21,     31, 5.98    , -55731.43
  data     20,    -10, 4.03    ,  60697.74
  data     18,      0, 4.27    ,   2132.79
  data     17,    -12, 0.79    , 109771.63
  data     14,      0, 4.24    ,  -7752.82
  data     13,     -5, 2.01    , 188491.91
  data     13,      0, 2.65    ,    207.81
  data     13,      0, 4.98    ,  29424.63
  data     12,      0, 0.93    ,     -7.99
  data     10,      0, 2.21    ,  46941.14
  data     10,      0, 3.59    ,    -68.29
  data     10,      0, 1.50    ,  21463.25
  data     10,     -9, 2.55    , 157208.40
  
  ' read leap second data
  
  for i% = 1 to 27
    
    read jdleap(i%), leapsec(i%)
    
  next i%
  
  data 2441317.5,  10.0
  data 2441499.5,  11.0
  data 2441683.5,  12.0
  data 2442048.5,  13.0
  data 2442413.5,  14.0
  data 2442778.5,  15.0
  data 2443144.5,  16.0
  data 2443509.5,  17.0
  data 2443874.5,  18.0
  data 2444239.5,  19.0
  data 2444786.5,  20.0
  data 2445151.5,  21.0
  data 2445516.5,  22.0
  data 2446247.5,  23.0
  data 2447161.5,  24.0
  data 2447892.5,  25.0
  data 2448257.5,  26.0
  data 2448804.5,  27.0
  data 2449169.5,  28.0
  data 2449534.5,  29.0
  data 2450083.5,  30.0
  data 2450630.5,  31.0
  data 2451179.5,  32.0
  data 2453736.5,  33.0
  data 2454832.5,  34.0
  DATA 2456109.5,  35.0
  data 2457204.5,  36.0
  
  ' calendar months
  
  month$(1) = "January"
  month$(2) = "February"
  month$(3) = "March"
  month$(4) = "April"
  month$(5) = "May"
  month$(6) = "June"
  month$(7) = "July"
  month$(8) = "August"
  month$(9) = "September"
  month$(10) = "October"
  month$(11) = "November"
  month$(12) = "December"
  
  ''''''''''''''''''
  ' begin simulation
  ''''''''''''''''''
  
  print " "
  print "time of the seasons"
  print "==================="
  print " "
  
  ' request calendar year
  
  print "please input the calendar year"
  
  input year
  
  ' process each season
  
  for iequsol% = 1 to 4
    
    select case iequsol%
        
      case (1)
        
        cmonth = 3
        
        day = 15
        
        julian(cmonth, day, year, jdayi)
        
      case (2)
        
        cmonth = 6
        
        day = 15
        
        julian(cmonth, day, year, jdayi)
        
        along2 = 0.5 * pi
        
      case (3)
        
        cmonth = 9
        
        day = 15
        
        julian(cmonth, day, year, jdayi)
        
      case (4)
        
        cmonth = 12
        
        day = 15
        
        julian(cmonth, day, year, jdayi)
        
        along2 = 1.5 * pi
        
    end select
    
    ' find event
    
    x1 = 0.0
    
    x2 = 10.0
    
    realroot1(x1, x2, 1.0e-8, xroot, froot)
    
    ' TDB julian day of event
    
    jdtdb = jdayi + xroot
    
    ' compute UTC julian day
    
    tdb2utc(jdtdb, jdutc)
    
    ' print results for this event
    
    print " "
    
    select case iequsol%
        
      case (1)
        
        print "spring equinox"
        print "--------------"
        
      case (2)
        
        print "summer solstice"
        print "---------------"
        
      case (3)
        
        print "fall equinox"
        print "------------"
        
      case (4)
        
        print "winter solstice"
        print "---------------"
        
    end select
    
    print " "
    
    gdate (jdutc, cmonth, day, year)
    
    print "calendar date  ", month$(cmonth), " ", STR$(int(day)), " ", str$(year)
    
    print " "
    
    thr0 = 24.0 * (day - int(day))
    
    thr = int(thr0)
    
    tmin0 = 60.0 * (thr0 - thr)
    
    tmin = int(tmin0)
    
    tsec = 60.0 * (tmin0 - tmin)
    
    print "UTC time       ", str$(thr) + " hours " + str$(tmin) + " minutes " + str$(tsec, 0, 2) + " seconds"
    
  next iequsol%
  
  print " "
  
end
  
  '''''''''''''''
  '''''''''''''''
  
sub esfunc(x, fx)
  
  ' equinox/solstice objective function
  
  '''''''''''''''''''''''''''''''''''''
  
  local jday, rlsun, rasc, decl
  
  jday = jdayi + x
  
  solar(jday, rlsun, rasc, decl)
  
  if (iequsol% = 1 or iequsol% = 3) then
    
    fx = decl
    
  else
    
    fx = along2 - rlsun
    
  end if
  
end sub
  
  ''''''''''''''''''''''''
  ''''''''''''''''''''''''
  
sub tdb2utc (jdtdb, jdutc)
  
  ' convert TDB julian day to UTC julian day subroutine
  
  ' input
  
  '  jdtdb = TDB julian day
  
  ' output
  
  '  jdutc = UTC julian day
  
  '''''''''''''''''''''''''
  
  local x1, x2, xroot, froot
  
  jdsaved = jdtdb
  
  ' set lower and upper bounds
  
  x1 = jdsaved - 0.1
  
  x2 = jdsaved + 0.1
  
  ' solve for UTC julian day using Brent's method
  
  realroot2(x1, x2, 1.0e-8, xroot, froot)
  
  jdutc = xroot
  
end sub
  
  '''''''''''''''''''
  '''''''''''''''''''
  
sub jdfunc (jdin, fx)
  
  ' objective function for tdb2utc
  
  ' input
  
  '  jdin = current value for UTC julian day
  
  ' output
  
  '  fx = delta julian day
  
  ''''''''''''''''''''''''
  
  local jdwrk, tai_utc
  
  findleap(jdin, tai_utc)
  
  utc2tdb(jdin, tai_utc, jdwrk)
  
  fx = jdwrk - jdsaved
  
end sub
  
  '''''''''''''''''''''''''''''''
  '''''''''''''''''''''''''''''''
  
sub solar (jd, rlsun, rasc, decl)
  
  ' precision ephemeris of the Sun
  
  ' input
  
  '  jd = julian ephemeris day
  
  ' output
  
  '  rlsun = ecliptic longitude of the sun
  '          (0 <= rlsun <= 2 pi)
  '  rasc  = right ascension of the Sun (radians)
  '          (0 <= rasc <= 2 pi)
  '  decl  = declination of the Sun (radians)
  '          (-pi/2 <= decl <= pi/2)
  
  ''''''''''''''''''''''''''''''''''
  
  local u, a1, a2, psi, deps, meps, eps, seps, ceps
  
  local dl, dr, w, srl, crl, srb, crb, sra, cra
  
  u = (jd - 2451545.0) / 3652500.0
  
  ' compute nutation in longitude
  
  a1 = 2.18 + u * (-3375.7 + u * 0.36)
  
  a2 = 3.51 + u * (125666.39 + u * 0.1)
  
  psi = 0.0000001 * (-834.0 * sin(a1) - 64.0 * sin(a2))
  
  ' compute nutation in obliquity
  
  deps = 0.0000001 * u * (-226938 + u * (-75 + u * (96926 + u * (-2491 - u * 12104))))
  
  meps = 0.0000001 * (4090928.0 + 446.0 * cos(a1) + 28.0 * cos(a2))
  
  eps = meps + deps
  
  seps = sin(eps)
  
  ceps = cos(eps)
  
  dl = 0.0
  
  dr = 0.0
  
  for i% = 1 to 50
    
    w = sa(i%) + sb(i%) * u
    
    dl = dl + sl(i%) * sin(w)
    
    if (sr(i%) <> 0.0) then
      
      dr = dr + sr(i%) * cos(w)
      
    end if
    
  next i%
  
  dl = modulo(dl * 0.0000001 + 4.9353929 + 62833.196168 * u)
  
  dr = 149597870.691 * (dr * 0.0000001 + 1.0001026)
  
  rlsun = modulo(dl + 0.0000001 * (-993.0 + 17.0 * cos(3.1 + 62830.14 * u)) + psi)
  
  rb = 0.0
  
  ' compute geocentric declination and right ascension
  
  crl = cos(rlsun)
  srl = sin(rlsun)
  crb = cos(rb)
  srb = sin(rb)
  
  decl = asin(ceps * srb + seps * crb * srl)
  
  sra = -seps * srb + ceps * crb * srl
  
  cra = crb * crl
  
  rasc = atan3(sra, cra)
  
end sub
  
  ''''''''''''''''''''''''''''''''''''''
  ''''''''''''''''''''''''''''''''''''''
  
sub realroot1(x1, x2, tol, xroot, froot)
  
  ' real root of a single non-linear function subroutine
  
  ' input
  
  '  x1  = lower bound of search interval
  '  x2  = upper bound of search interval
  '  tol = convergence criter%ia
  
  ' output
  
  '  xroot = real root of f(x) = 0
  '  froot = function value
  
  ' note: requires sub esfunc
  
  '''''''''''''''''''''''''''
  
  local eps, a, b, c, d, e, fa, fb, fcc, tol1
  
  local xm, p, q, r, s, xmin, tmp
  
  eps = 2.23e-16
  
  e = 0.0
  
  a = x1
  
  b = x2
  
  esfunc(a, fa)
  
  esfunc(b, fb)
  
  fcc = fb
  
  for iter% = 1 to 50
    
    if (fb * fcc > 0.0) then
      
      c = a
      
      fcc = fa
      
      d = b - a
      
      e = d
      
    end if
    
    if (abs(fcc) < abs(fb)) then
      
      a = b
      
      b = c
      
      c = a
      
      fa = fb
      
      fb = fcc
      
      fcc = fa
      
    end if
    
    tol1 = 2.0 * eps * abs(b) + 0.5 * tol
    
    xm = 0.5 * (c - b)
    
    if (abs(xm) <= tol1 or fb = 0.0) then exit for
    
    if (abs(e) >= tol1 and abs(fa) > abs(fb)) then
      
      s = fb / fa
      
      if (a = c) then
        
        p = 2.0 * xm * s
        
        q = 1.0 - s
        
      else
        
        q = fa / fcc
        
        r = fb / fcc
        
        p = s * (2.0 * xm * q * (q - r) - (b - a) * (r - 1.0))
        
        q = (q - 1.0) * (r - 1.0) * (s - 1.0)
        
      end if
      
      if (p > 0.0) then q = -q
      
      p = abs(p)
      
      min = abs(e * q)
      
      tmp = 3.0 * xm * q - abs(tol1 * q)
      
      if (min < tmp) then min = tmp
      
      if (2.0 * p < min) then
        
        e = d
        
        d = p / q
        
      else
        
        d = xm
        
        e = d
        
      end if
      
    else
      
      d = xm
      
      e = d
      
    end if
    
    a = b
    
    fa = fb
    
    if (abs(d) > tol1) then
      
      b = b + d
      
    else
      
      b = b + sgn(xm) * tol1
      
    end if
    
    esfunc(b, fb)
    
  next iter%
  
  froot = fb
  
  xroot = b
  
end sub
  
  ''''''''''''''''''''''''''''''''''''''
  ''''''''''''''''''''''''''''''''''''''
  
sub realroot2(x1, x2, tol, xroot, froot)
  
  ' real root of a single non-linear function subroutine
  
  ' input
  
  '  x1  = lower bound of search interval
  '  x2  = upper bound of search interval
  '  tol = convergence criter%ia
  
  ' output
  
  '  xroot = real root of f(x) = 0
  '  froot = function value
  
  ' note: requires sub jdfunc
  
  '''''''''''''''''''''''''''
  
  local eps, a, b, c, d, e, fa, fb, fcc, tol1
  
  local xm, p, q, r, s, xmin, tmp
  
  eps = 2.23e-16
  
  e = 0.0
  
  a = x1
  
  b = x2
  
  jdfunc(a, fa)
  
  jdfunc(b, fb)
  
  fcc = fb
  
  for iter% = 1 to 50
    
    if (fb * fcc > 0.0) then
      
      c = a
      
      fcc = fa
      
      d = b - a
      
      e = d
      
    end if
    
    if (abs(fcc) < abs(fb)) then
      
      a = b
      
      b = c
      
      c = a
      
      fa = fb
      
      fb = fcc
      
      fcc = fa
      
    end if
    
    tol1 = 2.0 * eps * abs(b) + 0.5 * tol
    
    xm = 0.5 * (c - b)
    
    if (abs(xm) <= tol1 or fb = 0) then exit for
    
    if (abs(e) >= tol1 and abs(fa) > abs(fb)) then
      
      s = fb / fa
      
      if (a = c) then
        
        p = 2.0 * xm * s
        
        q = 1.0 - s
        
      else
        
        q = fa / fcc
        
        r = fb / fcc
        
        p = s * (2.0 * xm * q * (q - r) - (b - a) * (r - 1.0))
        
        q = (q - 1.0) * (r - 1.0) * (s - 1.0)
        
      end if
      
      if (p > 0) then q = -q
      
      p = abs(p)
      
      xmin = abs(e * q)
      
      tmp = 3.0 * xm * q - abs(tol1 * q)
      
      if (xmin < tmp) then xmin = tmp
      
      if (2.0 * p < xmin) then
        
        e = d
        
        d = p / q
        
      else
        
        d = xm
        
        e = d
        
      end if
      
    else
      
      d = xm
      
      e = d
      
    end if
    
    a = b
    
    fa = fb
    
    if (abs(d) > tol1) then
      
      b = b + d
      
    else
      
      b = b + sgn(xm) * tol1
      
    end if
    
    jdfunc(b, fb)
    
  next iter%
  
  froot = fb
  
  xroot = b
  
end sub
  
  ''''''''''''''''''''''''''''''''
  ''''''''''''''''''''''''''''''''
  
sub julian(month, day, year, jday)
  
  ' Gregorian date to julian day subroutine
  
  ' input
  
  '  month = calendar month
  '  day   = calendar day
  '  year  = calendar year (all four digits)
  
  ' output
  
  '  jday = julian day
  
  ' special notes
  
  '  (1) calendar year must include all digits
  
  '  (2) will report October 5, 1582 to October 14, 1582
  '      as invalid calendar dates and exit
  
  '''''''''''''''''''''''''''''''''''''''''
  
  local a, b, c, m, y
  
  y = year
  
  m = month
  
  b = 0.0
  
  c = 0.0
  
  if (m <= 2.0) then
    
    y = y - 1.0
    
    m = m + 12.0
    
  end if
  
  if (y < 0.0) then c = -0.75
  
  if (year < 1582.0) then
    
    ' null
    
  elseif (year > 1582.0) then
    
    a = fix(y / 100.0)
    
    b = 2.0 - a + fix(a / 4.0)
    
  elseif (month < 10.0) then
    
    ' null
    
  elseif (month > 10.0) then
    
    a = fix(y / 100.0)
    
    b = 2.0 - a + fix(a / 4.0)
    
  elseif (day <= 4.0) then
    
    ' null
    
  elseif (day > 14.0) then
    
    a = fix(y / 100.0)
    
    b = 2.0 - a + fix(a / 4.0)
    
  else
    
    print "this date does not exist!!"
    
    exit
    
  end if
  
  jday = fix(365.25 * y + c) + fix(30.6001 * (m + 1.0)) + day + b + 1720994.5
  
end sub
  
  ''''''''''''''''''''''''''''''''
  ''''''''''''''''''''''''''''''''
  
sub gdate (jday, month, day, year)
  
  ' Julian day to Gregorian date subroutine
  
  ' input
  
  '  jday = julian day
  
  ' output
  
  '  month = calendar month
  '  day   = calendar day
  '  year  = calendar year
  
  ''''''''''''''''''''''''
  
  local a, b, c, d, e, f, z, alpha
  
  z = fix(jday + 0.5)
  
  f = jday + 0.5 - z
  
  if (z < 2299161) then
    
    a = z
    
  else
    
    alpha = fix((z - 1867216.25) / 36524.25)
    
    a = z + 1.0 + alpha - fix(alpha / 4.0)
    
  end if
  
  b = a + 1524.0
  
  c = fix((b - 122.1) / 365.25)
  
  d = fix(365.25 * c)
  
  e = fix((b - d) / 30.6001)
  
  day = b - d - fix(30.6001 * e) + f
  
  if (e < 13.5) then
    
    month = e - 1.0
    
  else
    
    month = e - 13.0
    
  end if
  
  if (month > 2.5) then
    
    year = c - 4716.0
    
  else
    
    year = c - 4715.0
    
  end if
  
end sub
  
  '''''''''''''''''''''''''''''''''
  '''''''''''''''''''''''''''''''''
  
sub utc2tdb (jdutc, tai_utc, jdtdb)
  
  ' convert UTC julian date to TDB julian date
  
  ' input
  
  '  jdutc   = UTC julian day
  '  tai_utc = TAI-UTC (seconds)
  
  ' output
  
  '  jdtdb = TDB julian day
  
  ' Reference Frames in Astronomy and Geophysics
  ' J. Kovalevsky et al., 1989, pp. 439-442
  
  '''''''''''''''''''''''''''''''''''''''''
  
  local corr, jdtdt, t
  
  ' TDT julian date
  
  corr = (tai_utc + 32.184) / 86400.0
  
  jdtdt = jdutc + corr
  
  ' time argument for correction
  
  t = (jdtdt - 2451545.0) / 36525.0
  
  ' compute correction in microseconds
  
  corr = 1656.675 * sin(dtr * (35999.3729 * t + 357.5287))
  corr = corr + 22.418     * sin(dtr * (32964.467  * t + 246.199))
  corr = corr + 13.84      * sin(dtr * (71998.746  * t + 355.057))
  corr = corr +  4.77      * sin(dtr * ( 3034.906  * t +  25.463))
  corr = corr +  4.677     * sin(dtr * (34777.259  * t + 230.394))
  corr = corr + 10.216 * t * sin(dtr * (35999.373  * t + 243.451))
  corr = corr +  0.171 * t * sin(dtr * (71998.746  * t + 240.98 ))
  corr = corr +  0.027 * t * sin(dtr * ( 1222.114  * t + 194.661))
  corr = corr +  0.027 * t * sin(dtr * ( 3034.906  * t + 336.061))
  corr = corr +  0.026 * t * sin(dtr * (  -20.186  * t +   9.382))
  corr = corr +  0.007 * t * sin(dtr * (29929.562  * t + 264.911))
  corr = corr +  0.006 * t * sin(dtr * (  150.678  * t +  59.775))
  corr = corr +  0.005 * t * sin(dtr * ( 9037.513  * t + 256.025))
  corr = corr +  0.043 * t * sin(dtr * (35999.373  * t + 151.121))
  
  ' convert corrections to days
  
  corr = 0.000001 * corr / 86400.0
  
  ' TDB julian date
  
  jdtdb = jdtdt + corr
  
end sub
  
  ''''''''''''''''''''''''''''
  ''''''''''''''''''''''''''''
  
sub findleap(jday, leapsecond)
  
  ' find number of leap seconds for utc julian day
  
  ' input
  
  '  jday = utc julian day
  
  ' input via global
  
  '  jdleap  = array of utc julian dates
  '  leapsec = array of leap seconds
  
  ' output
  
  '  leapsecond = number of leap seconds
  
  ''''''''''''''''''''''''''''''''''''''
  
  if (jday <= jdleap(1)) then
    
    ' date is <= 1972; set to first data element
    
    leapsecond = leapsec(1)
    
    exit sub
    
  end if
  
  if (jday >= jdleap(27)) then
    
    ' date is >= end of current data
    ' set to last data element
    
    leapsecond = leapsec(27)
    
    exit sub
    
  end if
  
  ' find data within table
  
  for i% = 1 to 26
    
    if (jday >= jdleap(i%) and jday < jdleap(i% + 1)) then
      
      leapsecond = leapsec(i%)
      
      exit sub
      
    end if
    
  next i%
  
end sub
  
  '''''''''''''''''''''''''
  '''''''''''''''''''''''''
  
function modulo(x) as float
  
  ' modulo 2 pi function
  
  ''''''''''''''''''''''
  
  local a
  
  a = x - pi2 * fix(x / pi2)
  
  if (a < 0.0) then
    
    a = a + pi2
    
  end if
  
  modulo = a
  
end function
  
  '''''''''''''''''''''''''''
  '''''''''''''''''''''''''''
  
function atan3(a, b) as float
  
  ' four quadrant inverse tangent function
  
  ' input
  
  '  a = sine of angle
  '  b = cosine of angle
  
  ' output
  
  '  atan3 = angle (0 =< atan3 <= 2 * pi; radians)
  
  ''''''''''''''''''''''''''''''''''''''''''''''''
  
  local c
  
  if (abs(a) < 1.0e-10) then
    
    atan3 = (1.0 - sgn(b)) * pidiv2
    
    exit function
    
  else
    
    c = (2.0 - sgn(a)) * pidiv2
    
  endif
  
  if (abs(b) < 1.0e-10) then
    
    atan3 = c
    
    exit function
    
  else
    
    atan3 = c + sgn(a) * sgn(b) * (abs(atn(a / b)) - pidiv2)
    
  endif
  
end function

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' seasons.bas March 6, 2017

' calendar date and UTC time of the seasons

' Micromite eXtreme version

'''''''''''''''''''''''''''

' dimension global arrays

dim sl(50) as float, sr(50) as float, sa(50) as float, sb(50) as float

dim jdleap(27) as float, leapsec(27) as float, month$(12) as string

' global constants

const pi2 = 2.0 * pi, pidiv2 = 0.5 * pi, dtr = pi / 180.0

' read solar ephemeris data

for i% = 1 to 50

read sl(i%), sr(i%), sa(i%), sb(i%)

next i%

data 403406, 0, 4.721964, 1.621043

data 195207, -97597, 5.937458, 62830.348067

data 119433, -59715, 1.115589, 62830.821524

data 112392, -56188, 5.781616, 62829.634302

data 3891, -1556, 5.5474 , 125660.5691

data 2819, -1126, 1.5120 , 125660.9845

data 1721, -861, 4.1897 , 62832.4766

data 0, 941, 1.163 , 0.813

data 660, -264, 5.415 , 125659.310

data 350, -163, 4.315 , 57533.850

data 334, 0, 4.553 , -33.931

data 314, 309, 5.198 , 777137.715

data 268, -158, 5.989 , 78604.191

data 242, 0, 2.911 , 5.412

data 234, -54, 1.423 , 39302.098

data 158, 0, 0.061 , -34.861

data 132, -93, 2.317 , 115067.698

data 129, -20, 3.193 , 15774.337

data 114, 0, 2.828 , 5296.670

data 99, -47, 0.52 , 58849.27

data 93, 0, 4.65 , 5296.11

data 86, 0, 4.35 , -3980.70

data 78, -33, 2.75 , 52237.69

data 72, -32, 4.50 , 55076.47

data 68, 0, 3.23 , 261.08

data 64, -10, 1.22 , 15773.85

data 46, -16, 0.14 , 188491.03

data 38, 0, 3.44 , -7756.55

data 37, 0, 4.37 , 264.89

data 32, -24, 1.14 , 117906.27

data 29, -13, 2.84 , 55075.75

data 28, 0, 5.96 , -7961.39

data 27, -9, 5.09 , 188489.81

data 27, 0, 1.72 , 2132.19

data 25, -17, 2.56 , 109771.03

data 24, -11, 1.92 , 54868.56

data 21, 0, 0.09 , 25443.93

data 21, 31, 5.98 , -55731.43

data 20, -10, 4.03 , 60697.74

data 18, 0, 4.27 , 2132.79

data 17, -12, 0.79 , 109771.63

data 14, 0, 4.24 , -7752.82

data 13, -5, 2.01 , 188491.91

data 13, 0, 2.65 , 207.81

data 13, 0, 4.98 , 29424.63

data 12, 0, 0.93 , -7.99

data 10, 0, 2.21 , 46941.14

data 10, 0, 3.59 , -68.29

data 10, 0, 1.50 , 21463.25

data 10, -9, 2.55 , 157208.40

' read leap second data

for i% = 1 to 27

read jdleap(i%), leapsec(i%)

next i%

data 2441317.5, 10.0

data 2441499.5, 11.0

data 2441683.5, 12.0

data 2442048.5, 13.0

data 2442413.5, 14.0

data 2442778.5, 15.0

data 2443144.5, 16.0

data 2443509.5, 17.0

data 2443874.5, 18.0

data 2444239.5, 19.0

data 2444786.5, 20.0

data 2445151.5, 21.0

data 2445516.5, 22.0

data 2446247.5, 23.0

data 2447161.5, 24.0

data 2447892.5, 25.0

data 2448257.5, 26.0

data 2448804.5, 27.0

data 2449169.5, 28.0

data 2449534.5, 29.0

data 2450083.5, 30.0

data 2450630.5, 31.0

data 2451179.5, 32.0

data 2453736.5, 33.0

data 2454832.5, 34.0

DATA 2456109.5, 35.0

data 2457204.5, 36.0

' calendar months

month$(1) = "January"

month$(2) = "February"

month$(3) = "March"

month$(4) = "April"

month$(5) = "May"

month$(6) = "June"

month$(7) = "July"

month$(8) = "August"

month$(9) = "September"

month$(10) = "October"

month$(11) = "November"

month$(12) = "December"

''''''''''''''''''

' begin simulation

''''''''''''''''''

print " "

print "time of the seasons"

print "==================="

print " "

' request calendar year

print "please input the calendar year"

input year

' process each season

for iequsol% = 1 to 4

select case iequsol%

case (1)

cmonth = 3

day = 15

julian(cmonth, day, year, jdayi)

case (2)

cmonth = 6

day = 15

julian(cmonth, day, year, jdayi)

along2 = 0.5 * pi

case (3)

cmonth = 9

day = 15

julian(cmonth, day, year, jdayi)

case (4)

cmonth = 12

day = 15

julian(cmonth, day, year, jdayi)

along2 = 1.5 * pi

end select

' find event

x1 = 0.0

x2 = 10.0

realroot1(x1, x2, 1.0e-8, xroot, froot)

' TDB julian day of event

jdtdb = jdayi + xroot

' compute UTC julian day

tdb2utc(jdtdb, jdutc)

' print results for this event

print " "

select case iequsol%

case (1)

print "spring equinox"

print "--------------"

case (2)

print "summer solstice"

print "---------------"

case (3)

print "fall equinox"

print "------------"

case (4)

print "winter solstice"

print "---------------"

end select

print " "

gdate (jdutc, cmonth, day, year)

print "calendar date ", month$(cmonth), " ", STR$(int(day)), " ", str$(year)

print " "

thr0 = 24.0 * (day - int(day))

thr = int(thr0)

tmin0 = 60.0 * (thr0 - thr)

tmin = int(tmin0)

tsec = 60.0 * (tmin0 - tmin)

print "UTC time ", str$(thr) + " hours " + str$(tmin) + " minutes " + str$(tsec, 0, 2) + " seconds"

next iequsol%

print " "

end

'''''''''''''''

sub esfunc(x, fx)

' equinox/solstice objective function

'''''''''''''''''''''''''''''''''''''

local jday, rlsun, rasc, decl

jday = jdayi + x

solar(jday, rlsun, rasc, decl)

if (iequsol% = 1 or iequsol% = 3) then

fx = decl

else

fx = along2 - rlsun

end if

end sub

''''''''''''''''''''''''

sub tdb2utc (jdtdb, jdutc)

' convert TDB julian day to UTC julian day subroutine

' input

' jdtdb = TDB julian day

' output

' jdutc = UTC julian day

'''''''''''''''''''''''''

local x1, x2, xroot, froot

jdsaved = jdtdb

' set lower and upper bounds

x1 = jdsaved - 0.1

x2 = jdsaved + 0.1

' solve for UTC julian day using Brent's method

realroot2(x1, x2, 1.0e-8, xroot, froot)

jdutc = xroot

end sub

'''''''''''''''''''

sub jdfunc (jdin, fx)

' objective function for tdb2utc

' input

' jdin = current value for UTC julian day

' output

' fx = delta julian day

''''''''''''''''''''''''

local jdwrk, tai_utc

findleap(jdin, tai_utc)

utc2tdb(jdin, tai_utc, jdwrk)

fx = jdwrk - jdsaved

end sub

'''''''''''''''''''''''''''''''

sub solar (jd, rlsun, rasc, decl)

' precision ephemeris of the Sun

' input

' jd = julian ephemeris day

' output

' rlsun = ecliptic longitude of the sun

' (0 <= rlsun <= 2 pi)

' rasc = right ascension of the Sun (radians)

' (0 <= rasc <= 2 pi)

' decl = declination of the Sun (radians)

' (-pi/2 <= decl <= pi/2)

''''''''''''''''''''''''''''''''''

local u, a1, a2, psi, deps, meps, eps, seps, ceps

local dl, dr, w, srl, crl, srb, crb, sra, cra

u = (jd - 2451545.0) / 3652500.0

' compute nutation in longitude

a1 = 2.18 + u * (-3375.7 + u * 0.36)

a2 = 3.51 + u * (125666.39 + u * 0.1)

psi = 0.0000001 * (-834.0 * sin(a1) - 64.0 * sin(a2))

' compute nutation in obliquity

deps = 0.0000001 * u * (-226938 + u * (-75 + u * (96926 + u * (-2491 - u * 12104))))

meps = 0.0000001 * (4090928.0 + 446.0 * cos(a1) + 28.0 * cos(a2))

eps = meps + deps

seps = sin(eps)

ceps = cos(eps)

dl = 0.0

dr = 0.0

for i% = 1 to 50

w = sa(i%) + sb(i%) * u

dl = dl + sl(i%) * sin(w)

if (sr(i%) <> 0.0) then

dr = dr + sr(i%) * cos(w)

end if

next i%

dl = modulo(dl * 0.0000001 + 4.9353929 + 62833.196168 * u)

dr = 149597870.691 * (dr * 0.0000001 + 1.0001026)

rlsun = modulo(dl + 0.0000001 * (-993.0 + 17.0 * cos(3.1 + 62830.14 * u)) + psi)

rb = 0.0

' compute geocentric declination and right ascension

crl = cos(rlsun)

srl = sin(rlsun)

crb = cos(rb)

srb = sin(rb)

decl = asin(ceps * srb + seps * crb * srl)

sra = -seps * srb + ceps * crb * srl

cra = crb * crl

rasc = atan3(sra, cra)

end sub

''''''''''''''''''''''''''''''''''''''

sub realroot1(x1, x2, tol, xroot, froot)

' real root of a single non-linear function subroutine

' input

' x1 = lower bound of search interval

' x2 = upper bound of search interval

' tol = convergence criter%ia

' output

' xroot = real root of f(x) = 0

' froot = function value

' note: requires sub esfunc

'''''''''''''''''''''''''''

local eps, a, b, c, d, e, fa, fb, fcc, tol1

local xm, p, q, r, s, xmin, tmp

eps = 2.23e-16

e = 0.0

a = x1

b = x2

esfunc(a, fa)

esfunc(b, fb)

fcc = fb

for iter% = 1 to 50

if (fb * fcc > 0.0) then

c = a

fcc = fa

d = b - a

e = d

end if

if (abs(fcc) < abs(fb)) then

a = b

b = c

c = a

fa = fb

fb = fcc

fcc = fa

end if

tol1 = 2.0 * eps * abs(b) + 0.5 * tol

xm = 0.5 * (c - b)

if (abs(xm) <= tol1 or fb = 0.0) then exit for

if (abs(e) >= tol1 and abs(fa) > abs(fb)) then

s = fb / fa

if (a = c) then

p = 2.0 * xm * s

q = 1.0 - s

else

q = fa / fcc

r = fb / fcc

p = s * (2.0 * xm * q * (q - r) - (b - a) * (r - 1.0))

q = (q - 1.0) * (r - 1.0) * (s - 1.0)

end if

if (p > 0.0) then q = -q

p = abs(p)

min = abs(e * q)

tmp = 3.0 * xm * q - abs(tol1 * q)

if (min < tmp) then min = tmp

if (2.0 * p < min) then

e = d

d = p / q

else

d = xm

e = d

end if

else

d = xm

e = d

end if

a = b

fa = fb

if (abs(d) > tol1) then

b = b + d

else

b = b + sgn(xm) * tol1

end if

esfunc(b, fb)

next iter%

froot = fb

xroot = b

end sub

''''''''''''''''''''''''''''''''''''''

sub realroot2(x1, x2, tol, xroot, froot)

' real root of a single non-linear function subroutine

' input

' x1 = lower bound of search interval

' x2 = upper bound of search interval

' tol = convergence criter%ia

' output

' xroot = real root of f(x) = 0

' froot = function value

' note: requires sub jdfunc

'''''''''''''''''''''''''''

local eps, a, b, c, d, e, fa, fb, fcc, tol1

local xm, p, q, r, s, xmin, tmp

eps = 2.23e-16

e = 0.0

a = x1

b = x2

jdfunc(a, fa)

jdfunc(b, fb)

fcc = fb

for iter% = 1 to 50

if (fb * fcc > 0.0) then

c = a

fcc = fa

d = b - a

e = d

end if

if (abs(fcc) < abs(fb)) then

a = b

b = c

c = a

fa = fb

fb = fcc

fcc = fa

end if

tol1 = 2.0 * eps * abs(b) + 0.5 * tol

xm = 0.5 * (c - b)

if (abs(xm) <= tol1 or fb = 0) then exit for

if (abs(e) >= tol1 and abs(fa) > abs(fb)) then

s = fb / fa

if (a = c) then

p = 2.0 * xm * s

q = 1.0 - s

else

q = fa / fcc

r = fb / fcc

p = s * (2.0 * xm * q * (q - r) - (b - a) * (r - 1.0))

q = (q - 1.0) * (r - 1.0) * (s - 1.0)

end if

if (p > 0) then q = -q

p = abs(p)

xmin = abs(e * q)

tmp = 3.0 * xm * q - abs(tol1 * q)

if (xmin < tmp) then xmin = tmp

if (2.0 * p < xmin) then

e = d

d = p / q

else

d = xm

e = d

end if

else

d = xm

e = d

end if

a = b

fa = fb

if (abs(d) > tol1) then

b = b + d

else

b = b + sgn(xm) * tol1

end if

jdfunc(b, fb)

next iter%

froot = fb

xroot = b

end sub

''''''''''''''''''''''''''''''''

sub julian(month, day, year, jday)

' Gregorian date to julian day subroutine

' input

' month = calendar month

' day = calendar day

' year = calendar year (all four digits)

' output

' jday = julian day

' special notes

' (1) calendar year must include all digits

' (2) will report October 5, 1582 to October 14, 1582

' as invalid calendar dates and exit

'''''''''''''''''''''''''''''''''''''''''

local a, b, c, m, y

y = year

m = month

b = 0.0

c = 0.0

if (m <= 2.0) then

y = y - 1.0

m = m + 12.0

end if

if (y < 0.0) then c = -0.75

if (year < 1582.0) then

' null

elseif (year > 1582.0) then

a = fix(y / 100.0)

b = 2.0 - a + fix(a / 4.0)

elseif (month < 10.0) then

' null

elseif (month > 10.0) then

a = fix(y / 100.0)

b = 2.0 - a + fix(a / 4.0)

elseif (day <= 4.0) then

' null

elseif (day > 14.0) then

a = fix(y / 100.0)

b = 2.0 - a + fix(a / 4.0)

else

print "this date does not exist!!"

exit

end if

jday = fix(365.25 * y + c) + fix(30.6001 * (m + 1.0)) + day + b + 1720994.5

end sub

''''''''''''''''''''''''''''''''

sub gdate (jday, month, day, year)

' Julian day to Gregorian date subroutine

' input

' jday = julian day

' output

' month = calendar month

' day = calendar day

' year = calendar year

''''''''''''''''''''''''

local a, b, c, d, e, f, z, alpha

z = fix(jday + 0.5)

f = jday + 0.5 - z

if (z < 2299161) then

a = z

else

alpha = fix((z - 1867216.25) / 36524.25)

a = z + 1.0 + alpha - fix(alpha / 4.0)

end if

b = a + 1524.0

c = fix((b - 122.1) / 365.25)

d = fix(365.25 * c)

e = fix((b - d) / 30.6001)

day = b - d - fix(30.6001 * e) + f

if (e < 13.5) then

month = e - 1.0

else

month = e - 13.0

end if

if (month > 2.5) then

year = c - 4716.0

else

year = c - 4715.0

end if

end sub

'''''''''''''''''''''''''''''''''

sub utc2tdb (jdutc, tai_utc, jdtdb)

' convert UTC julian date to TDB julian date

' input

' jdutc = UTC julian day

' tai_utc = TAI-UTC (seconds)

' output

' jdtdb = TDB julian day

' Reference Frames in Astronomy and Geophysics

' J. Kovalevsky et al., 1989, pp. 439-442

'''''''''''''''''''''''''''''''''''''''''

local corr, jdtdt, t

' TDT julian date

corr = (tai_utc + 32.184) / 86400.0

jdtdt = jdutc + corr

' time argument for correction

t = (jdtdt - 2451545.0) / 36525.0

' compute correction in microseconds

corr = 1656.675 * sin(dtr * (35999.3729 * t + 357.5287))

corr = corr + 22.418 * sin(dtr * (32964.467 * t + 246.199))

corr = corr + 13.84 * sin(dtr * (71998.746 * t + 355.057))

corr = corr + 4.77 * sin(dtr * ( 3034.906 * t + 25.463))

corr = corr + 4.677 * sin(dtr * (34777.259 * t + 230.394))

corr = corr + 10.216 * t * sin(dtr * (35999.373 * t + 243.451))

corr = corr + 0.171 * t * sin(dtr * (71998.746 * t + 240.98 ))

corr = corr + 0.027 * t * sin(dtr * ( 1222.114 * t + 194.661))

corr = corr + 0.027 * t * sin(dtr * ( 3034.906 * t + 336.061))

corr = corr + 0.026 * t * sin(dtr * ( -20.186 * t + 9.382))

corr = corr + 0.007 * t * sin(dtr * (29929.562 * t + 264.911))

corr = corr + 0.006 * t * sin(dtr * ( 150.678 * t + 59.775))

corr = corr + 0.005 * t * sin(dtr * ( 9037.513 * t + 256.025))

corr = corr + 0.043 * t * sin(dtr * (35999.373 * t + 151.121))

' convert corrections to days

corr = 0.000001 * corr / 86400.0

' TDB julian date

jdtdb = jdtdt + corr

end sub

''''''''''''''''''''''''''''

sub findleap(jday, leapsecond)

' find number of leap seconds for utc julian day

' input

' jday = utc julian day

' input via global

' jdleap = array of utc julian dates

' leapsec = array of leap seconds

' output

' leapsecond = number of leap seconds

''''''''''''''''''''''''''''''''''''''

if (jday <= jdleap(1)) then

' date is <= 1972; set to first data element

leapsecond = leapsec(1)

exit sub

end if

if (jday >= jdleap(27)) then

' date is >= end of current data

' set to last data element

leapsecond = leapsec(27)

exit sub

end if

' find data within table

for i% = 1 to 26

if (jday >= jdleap(i%) and jday < jdleap(i% + 1)) then

leapsecond = leapsec(i%)

exit sub

end if

next i%

end sub

'''''''''''''''''''''''''

function modulo(x) as float

' modulo 2 pi function

''''''''''''''''''''''

local a

a = x - pi2 * fix(x / pi2)

if (a < 0.0) then

a = a + pi2

end if

modulo = a

end function

'''''''''''''''''''''''''''

function atan3(a, b) as float

' four quadrant inverse tangent function

' input

' a = sine of angle

' b = cosine of angle

' output

' atan3 = angle (0 =< atan3 <= 2 * pi; radians)

''''''''''''''''''''''''''''''''''''''''''''''''

local c

if (abs(a) < 1.0e-10) then

atan3 = (1.0 - sgn(b)) * pidiv2

exit function

else

c = (2.0 - sgn(a)) * pidiv2

endif

if (abs(b) < 1.0e-10) then

atan3 = c

exit function

else

atan3 = c + sgn(a) * sgn(b) * (abs(atn(a / b)) - pidiv2)

endif

end function

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