 # Approximate Position of the Sun (Altitude and Azimuth) from any Location at any Time (for low accuracy calculation)

Based on Yallop, Nautical Almanac Office,
NAO Technical Note No. 46 (1978)

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examples-

(a) Cape Town       Feb 15   10:30   1995
(b) Bloemfontein    May 20   13:35   1996
(c) Johannesburg   Sept 25   16:45   1997

(1) find Y, the year minus 1900:

(a) Y = 95
(b)     96
(c)     97

(2) find Z(J) from this table:
Jan   J= 1   Z(J)=-0.5*        Jul   J= 7   Z(J)=180.5
Feb        2            30.5*        Aug      8             211.5
Mar        3            58.5         Sep      9             242.5
Apr        4            89.5         Oct     10             272.5
May        5           119.5         Nov     11             303.5
Jun        6           150.5         Dec     12             333.5
(* reduce by one for a leap year)

(a) Z(J) =  30.5
(b)        119.5
(c)        242.5

(3) find D the number of days from this formula:
D = integer(365.25 x Y) + Z(J) + K + UT/24
where K is the day of the month and UT is the universal time

(a) D = int(365.25 x 95) +  30.5 + 15 +  8.500/24  = 34743.854
(b)        int(365.25 x 96) + 119.5 + 20 + 11.583/24  = 35203.983
(c)        int(365.25 x 97) + 242.5 + 25 + 14.750/24  = 35697.115

(4) find T the fraction of a julian century from this formula:
T = D/36525

(a) T = 0.9512349
(b)     0.9638325
(c)     0.9773337

(5) find L the mean longitude of the sun from this formula:
L = 279.697 + 36000.769 x T

(a) L = 34524.885  => 324.885   (removing multiples of 360 degrees)
(b)     34978.408  =>  58.408
(c)     35464.462  => 184.462

(6) find M the mean anomaly of the sun from this formula:
M = 358.476 + 35999.050 x T

(a) M = 34602.029 =>  42.029   (removing multiples of 360 degrees)
(b)        35055.530 => 135.530
(c)        35541.561 => 261.561

(7) find epsilon the obliquity from this formula:
epsilon = 23.452 - 0.013 x T

(a) epsilon = 23.4396
(b)           23.4395
(c)           23.4393

(8) find lambda the ecliptic longitude of the sun from this formula:
lambda = L + (1.919 - 0.005 x T) x sin(M) + 0.020 x sin(2M)

(a) lambda = 324.885 + 1.9142 x  0.6695 + 0.020 x  0.9946 = 326.186
(b)                   58.408 + 1.9142 x  0.7005 + 0.020 x -0.9998 =  59.729
(c)                   184.462 + 1.9141 x -0.9892 + 0.020 x  0.2903 = 182.574

(9) find alpha the right ascension of the sun from this formula:
alpha = arctan (tan(lambda) x cos(epsilon))      in same quadrant as
lambda
(a) alpha = 328.428
(b)          57.537
(c)         182.362

(10) find delta the declination of the sun from this formula:
delta = arcsin (sin(lambda) x sin(epsilon))

(a) delta = -12.789
(b)          20.093
(c)          -1.024

(11) to proceed you need to know LONG the east-longitude of your location:

east-longitude          latitude

Windhoek                            17.10                -22.57
Cape Town                           18.37                -33.92
P.E.                                25.67                -33.97
Bloemfontein                        26.12                -29.20
Johannesburg                        28.00                -26.25
Durban                              30.93                -29.92

(12) find HA the hour angle of the sun from this formula:
HA = L - alpha + 180 + 15 x UT + LONG

(a) HA = 324.885 - 328.428 + 180 + 15 x  8.500 + 18.37  = -37.673
(b)           58.408 -  57.537 + 180 + 15 x 11.583 + 26.12  =  20.736
(c)          184.462 - 182.362 + 180 + 15 x 14.750 + 28.00  =  71.350

(13) find the altitude of the center of the sun ALT from this formula:

ALT [degrees] =
ARCSIN [ SIN(LAT) x SIN(DEC)  +  COS(LAT) x COS(DEC) x COS(HA) ]

(a) ALT = ARCSIN ( -.5580 x -.2214  +  .8298 x .9752 x .7915 ) = 49.822

(b)            ARCSIN ( -.4879 x  .3435  +  .8729 x .9391 x .9352 ) = 36.800

(c)            ARCSIN ( -.4423 x -.0182  +  .8969 x .9998 x .3198 ) = 17.147

(14) find the azimuth of the sun AZ from this formula:

AZ [degrees] = ARCTAN [ SIN(HA) /

(COS(HA) x SIN(LAT)  -  TAN(DEC) x COS(LAT) ]

(a) AZ = ARCTAN [ -.6112/ ( .7915 x -.5580  -  -.2270 x .8298 ) ]
= ARCTAN (  2.4130 ) =  67.49   {i.e. east of true north}

(b)          ARCTAN [  .3541/ ( .9352 x -.4879  -   .3658 x .8729 ) ]
ARCTAN ( -0.45656 ) = -24.54 = 335.46  {i.e. west of true north}

(c)          ARCTAN [  .9475/ ( .3198 x -.4423  -  -.01787 x .8969 ) ]
ARCTAN ( -7.5546 ) = -82.46 = 277.54   {i.e. west of true north}

COMPARISON WITH COMPUTER ALMANAC PROGRAM

rough calculation here      computer almanac calculation    difference

(a) alt = 49.8, az =  67.5       ALT = 49.8, AZ =  67.5           none
(b) alt = 36.8, az = 335.5       ALT = 36.8, AZ = 335.5           none
(c) alt = 17.1, az = 277.5       ALT = 17.1, AZ = 277.5           none```