(42S/DM42) Solar Declination, Zenith, and Sunrise

[Eddie W. Shore]

Blog Link: http://edspi31415.blogspot.com/2019/12/h...ation.html

Introduction

The program SUNS will calculate:

1. Approximation declination given month and day. A 365-year is assumed.

δ ≈ 23.45° sin(.9863 * (284 + n))

where n is the nth day of the year:

If month = 1 or month = 2, then:
n = IP(30.6 * month + 368.8) + day - 399

If month ≥ 3 then:
n = IP(30.6 * month + 1.6) + day - 34

2. Zenith angle θ:

θ ≈ acos(sin δ * sin ϕ + cos δ * cos ϕ * cos hour_angle)

where:
ϕ = latitude, north is positive
hour_angle = (12 - number of hours before noon) * 15

3. Approximate time of sunrise:

ω ≈ acos(-tan ϕ * tan δ)
Sunrise time ≈ (12 - ω/15) converted to HMS

The program was entered on a HP 42S, and should work for the Free42 and Swiss Micros DM42.

HP 42S/DM42 Program SUNS

Code:

00 {206-Byte Prgm}
01 LBL "SUNS"
02 DEG
03 "MONTH?"
04 PROMPT
05 STO 01
06 "DAY?"
07 PROMPT 
08 STO 02
09 "HRS BEFORE NOON"
10 PROMPT 
11 STO 03
12 "LAT (N+)?"
13 PROMPT
14 →HR
15 STO 04
16 RCL 02     // calculate n
17 RCL 01
18 CF 00
19 3
20 X>Y?
21 SF 00
22 R↓
23 30.6
24 *
25 1.6
26 +
27 FS? 00
28 367.2
29 FS? 00
30 +
31 IP
32 +
33 34
34 -
35 FS? 00
36 365
37 FS? 00
38 -
39 CF 00
40 STO 11
41 284     // calculate δ
42 +
43 0.9863
44 *
45 SIN
46 23.45
47 * 
48 STO 12
49 "DEC="
50 ARCL ST X
51 AVIEW 
52 STOP
53 12
54 RCL- 03
55 15
56  *
57 STO 13
58 RCL 12   // calculate θ
59 SIN
60 RCL 04
61 SIN
62 *
63 RCL 12
64 COS
65 RCL 04
66 COS
67 *
68 RCL 13
69 COS 
70 *
71 +
72 ACOS
73 STO 14
74 "ZENITH="
75 ARCL ST X
76 AVIEW
77 STOP
78 RCL 04   // calculate ω
79 TAN
80 +/-
81 RCL 12
82 TAN
83 *
84 ACOS
85 15
86 ÷
87 12
88 X<>Y
89 -
90 →HMS
91 STO 15
92 "SUNRISE="
93 ARCL ST X
94 AVIEW
95 END

Variables:
R01 = month
R02 = day
R03 = hours before moon
R04 = latitude

R11 = n, nth day of the year
R12 = δ, declination
R13 = hour_angle
R14 = θ, zenith
R15 = ω, approximate sunrise

Example

March 15, 8:00 AM (4 hours before noon), latitude is 40°

Results:
DEC = -2.8191 (-2.8191°)
ZENITH = 114.4672
SUNRISE = 6.0928 (around 6:09:28)

Remember, these results are approximate!

Source:
Hewlett Packard. "Solar-Beam Irradiation" Hewlett Packard User's Library Solution: Solar Engineering. HP 41C January 1984.