Refraction2

Astronomical Refraction(2) for the HP-41

Overview

1°) Program#1-One Polynomial of degree 7
2°) Program#2-One Polynomial of degree 13
3°) Program#3-Three Polynomials of degree 6
4°) Program#4-With XM-Data File
5°) Auer & Standish Model of the Atmosphere
6°) Faster Programs

    a) REFR
    b) RADAU
    c) PH0-H & PH-H0

-The programs listed in paragraphs 1 to 4 employs the data given in https://ccmc.gsfc.nasa.gov/modelweb/models/msis_vitmo.php
-After fitting these data to functions of the type:

n - 1 = ( n₀ - 1 ) exp [ p(x)] where n is the refractive index of the atmosphere, p(x) a polynomial and x = the altitude

the refraction is calculated by a numerical integration, as described by Auer & Standish ( cf reference [2] for the details )

R = § ( r dn/dr ) / ( n + r dn/dr ) . d(psi)

-So, you can use these programs for h₀ < 0

-Gauss-Legendre 3-point formula is used to evaluate this integral.
-The interval of integration is divided in 3 parts, corresponding to the altitudes [0,11] [11,28] & [28,87] ( in kilometers )

-In paragraph 5, the atmospheric model given by Auer & Standish is used - with small modifications.

-The value of ( n₀ - 1 ) is computed by Owens' simplified formula
-The azimuth is also taken into account to calculate the radius of curvature Rho: Fermat's formula yields:

Rho = 1 / [ ( Cos² Az ) / r₁ + ( Sin² Az ) / r₂ ]

where r₁ = a ( 1 - e² ) ( 1 - e² Sin² b )^-3/2 and r₂ = a ( 1 - e² Sin² b )^-1/2

( a = semi-major axis of the Earth , e = eccentricity , b = latitude of the observer )

-The refractivity at an altitude 0 ( or at the altitude of the observer ) is given by Owens' formula:

( n₀ - 1 ) 10⁸ = [ 2371.34 + 683939.7 ( 130 - 1/L² )^-1 + 4547.3 ( 38.9 - 1/L² )^-1 ] D_S
+ [ 6487.31 + 58.058 / L² - 0.71150 / L⁴ + 0.08851 / L⁶ ] D_W

where D_S = ( P_S / T ) [ 1 + P_S ( 57.90 10^-8 - 9.325 10^-4 / T + 0.25844 / T² ) ]
and D_W = ( F / T ) [ 1 + F ( 1 + 0.00037 F ) ( -0.00237321 + 2.23366 / T - 710.792 / T² + 77514.1 / T³ ) ]

with P_S = pressure of dry air in mbar P = P_S + F L = wavelength in µm
F = pressure of water vapor in mbar T = temperature in Kelvin

-In paragraph 6, simplified formulas are used to get faster - but less accurate - results.

1°) Program#1-One Polynomial of degree 7

-Here, we assume that p(x) is a polynomial of degree 7 for x between 0 and 87 km

p(x) = c₁ x + c₂ x² + ............... + c₇ x⁷

-So, n - 1 = ( n₀ - 1 ) exp [ c₁ x + c₂ x² + ............... + c₇ x⁷ ] ( approximately ) with:

    c₁ = t / 1250 - 0.109142               c₄ = 8.89464 E-6               c₇ = -3.63388 E-12
    c₂ = -1/97162 - t x 9 E-6              c₅ = -1.53611 E-7
    c₃ = -204894 E-9                         c₆ = 1.21088 E-9

-You have to initialize R01 thru R06:

t = temperature
P = total pressure
F = pressure of water-vapor

Data Registers: R00 = dn/dx ( Registers R01 thru R06 are to be initialized before executing "REF7" )

                                      • R01 = t ( °C )           • R04 = wave-length ( µm )
                                      • R02 = P ( mbar )       • R05 = Latitude ( ° ' " )
                                      • R03 = F ( mbar )       • R06 = Altitude of the observer ( in meters ) < 11000

R07 = c₁ .................... R13 = c₇ R14 to R40: temp

Flag: F01 CF 01 = t , P , F are measured for an altitude = 0
SF 01 = t , P , F are measured at the observer's station
Subroutines: /

-Lines 08 & 218 are three-byte GTOs

01 LBL "REF7"
02 DEG
03 HR
04 STO 22
05 COS
06 STO 20
07 X=0?
08 GTO 12
09 STO 32
10 X<>Y
11 HR
12 1
13 P-R
14 X^2
15 0.99330562
16 /
17 X<>Y
18 X^2
19 RCL 05
20 HR
21 SIN
22 X^2
23 669438 E-8
24 *
25 1
26 X<>Y
27 -
28 SQRT
29 ST* Z
30 ST* Z
31 ST* Z
32 *
33 +
34 6378.137
35 X<>Y
36 /
37 STO 23
38 RCL 06
39 E3
40 /
41 STO 20
42 +
43 STO 36
44 RCL 01
45 273.15
46 +
47 1/X
48 ENTER
49 ENTER
50 STO 00
51 77514.1
52 *

53 710.792
54 -
55 *
56 2.23366
57 +
58 *
59 237321 E-8
60 -
61 RCL 03
62 ST* Y
63 37 E-5
64 *
65 1
66 +
67 *
68 1
69 +
70 RCL 03
71 *
72 *
73 STO 07
74 RCL 04
75 X^2
76 1/X
77 ENTER
78 ENTER
79 STO 08
80 .08851
81 *
82 .7115
83 -
84 *
85 58.058
86 +
87 *
88 6487.31
89 +
90 ST* 07
91 RCL 00
92 .25844
93 *
94 9325 E-7
95 -
96 RCL 00
97 *
98 579 E-9
99 +
100 RCL 02
101 RCL 03
102 -
103 ST* 00
104 *

105 1
106 +
107 RCL 00
108 *
109 4547.3
110 38.9
111 RCL 08
112 -
113 /
114 130
115 RCL 08
116 -
117 683939.7
118 X<>Y
119 /
120 +
121 2371.34
122 +
123 *
124 RCL 07
125 +
126 E8
127 /
128 STO 39
129 1.6
130 STO 31
131 FRC
132 SQRT
133 STO 30
134 RCL 01
135 1250
136 /
137 .109142
138 -
139 STO 07
140 97162
141 1/X
142 CHS
143 RCL 01
144 9 E-6
145 *
146 -
147 STO 08
148 ST+ X
149 STO 14
150 204894 E-9
151 CHS
152 STO 09
153 3
154 *
155 STO 15
156 8.89464 E-6

157 STO 10
158 4
159 *
160 STO 16
161 1.53611 E-7
162 CHS
163 STO 11
164 5
165 *
166 STO 17
167 1.21088 E-9
168 STO 12
169 6
170 *
171 STO 18
172 3.63388 E-12
173 CHS
174 STO 13
175 7
176 *
177 STO 19
178 CLX
179 X<> 20
180 XEQ 04
181 LASTX
182 STO 00
183 FS? 01
184 RCL 39
185 RCL 36
186 ST* Y
187 +
188 RCL 32
189 *
190 STO 35
191 FC? 01
192 GTO 00
193 RCL 00
194 RCL 39
195 /
196 ST/ 39
197 LBL 00
198 3 E-6
199 STO 38
200 11
201 XEQ 02
202 STO 21
203 6
204 STO 40
205 XEQ 13
206 28
207 XEQ 02
208 X<> 21

209 STO 22
210 4
211 STO 40
212 XEQ 13
213 87
214 XEQ 02
215 X<> 21
216 STO 22
217 XEQ 13
218 GTO 12
219 LBL 13
220 RCL 22
221 RCL 21
222 STO 24
223 -
224 RCL 40
225 STO 28
226 /
227 STO 27
228 2
229 /
230 ST+ 24
231 RCL 30
232 *
233 STO 25
234 CLX
235 STO 26
236 LBL 14
237 RCL 24
238 RCL 25
239 -
240 XEQ 01
241 ST+ 26
242 RCL 24
243 XEQ 01
244 RCL 31
245 *
246 ST+ 26
247 RCL 24
248 RCL 25
249 +
250 XEQ 01
251 ST+ 26
252 RCL 27
253 ST+ 24
254 DSE 28
255 GTO 14
256 RCL 26
257 *
258 ST+ 20
259 RTN
260 LBL 02
261 STO 34

262 XEQ 04
263 X<>Y
264 RCL 23
265 +
266 *
267 RCL 35
268 X<>Y
269 /
270 ACOS
271 RTN
272 LBL 01
273 COS
274 STO 29
275 LBL 03
276 RCL 34
277 XEQ 04
278 STO 37
279 X<>Y
280 RCL 23
281 +
282 STO 33
283 *
284 RCL 35
285 RCL 29
286 /
287 -
288 RCL 00
289 RCL 33
290 *
291 RCL 37
292 +
293 /
294 ST- 34
295 ABS
296 RCL 38
297 X<Y?
298 GTO 03
299 RCL 37
300 RCL 33
301 RCL 00
302 *
303 STO Z
304 +
305 /
306 RTN
307 LBL 04
308 ENTER
309 STO Z
310 RCL 19
311 *
312 RCL 18
313 +
314 *

315 RCL 17
316 +
317 *
318 RCL 16
319 +
320 *
321 RCL 15
322 +
323 *
324 RCL 14
325 +
326 *
327 RCL 07
328 +
329 STO 00
330 CLX
331 RCL 13
332 *
333 RCL 12
334 +
335 *
336 RCL 11
337 +
338 *
339 RCL 10
340 +
341 *
342 RCL 09
343 +
344 *
345 RCL 08
346 +
347 *
348 RCL 07
349 +
350 *
351 E^X
352 RCL 39
353 *
354 ST* 00
355 SIGN
356 LASTX
357 +
358 RTN
359 LBL 12
360 RCL 20
361 E3
362 *
363 STO Y
364 3600
365 /
366 HMS
367 END

( 689 bytes / SIZE 041 )

STACK	INPUTS	OUTPUTS
Y	Az ( ° ' " )	R ( " )
X	h₀ ( ° ' " )	R ( ° ' " )

Where h₀ = apparent height

Example: t = 10°C P = 1010 mbar F = 6 mbar Lambda = 0.577 µm Latitude = 33°21'22" Altitude = 1706 m

    10    STO 01        0.577    STO 04
1010 STO 02      33.2122 STO 05
     6     STO 03        1706     STO 06

1°) CF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345   XEQ "REF7" >>>> R = 0°18'08"747                    ---Execution time = 7m24s---
                                          RDN   R = 1088"747

2°) SF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345       R/S     >>>> R = 0°21'45"293
                                   RDN   R = 1305"293

Notes:

-With Az = 84° , we get 1090"366 & 1307"268 in the examples above.

-You can of course use your own constants:
-Change lines 134 to 174.

2°) Program#2-One Polynomial of degree 13

-Here, we assume that p(x) is a polynomial of degree 13 for x between 0 and 87 km

p(x) = c₁ x + c₂ x² + ............... + c₁₃ x¹³

-So, n - 1 = ( n₀ - 1 ) exp [ c₁ x + c₂ x² + ............... + c₁₃ x¹³ ] ( approximately ) with:

    c₁ = t / 1250 - 0.109671                  c₄ = -1.553002 E-4           c₇ = 1.012373 E-8            c₁₀ = 2529916 E-20            c₁₃ = -41167039 E-28
    c₂ = -0.0026952 - t x 95 E-7           c₅ = 1.137826 E-5            c₈ = -1054348 E-16           c₁₁ = -31539538 E-23
    c₃ = 958131 E-9                             c₆ = -4.532222 E-7           c₉ = -3737867 E-19           c₁₂ = 1805402 E-24

( lines 136 to 176 )

Data Registers: R00 = dn/dx ( Registers R01 thru R06 are to be initialized before executing "REF13" )

R07 = c₁ .................... R19 = c₁₃ R20 to R52: temp

Flags: F00 & F01 CF 00 = The constants in R07 to R19 are those listed below
SF 00 = Store your own constants in R07 to R19 before executing "REF13"

CF 01 = t , P , F are measured for an altitude = 0
SF 01 = t , P , F are measured at the observer's station
Subroutines: /

-Lines 08-135-233 are three-byte GTOs

01 LBL "REF13"
02 DEG
03 HR
04 STO 33
05 COS
06 STO 51
07 X=0?
08 GTO 12
09 STO 43
10 X<>Y
11 HR
12 1
13 P-R
14 X^2
15 .99330562
16 /
17 X<>Y
18 X^2
19 RCL 05
20 HR
21 SIN
22 X^2
23 .00669438
24 *
25 1
26 X<>Y
27 -
28 SQRT
29 ST* Z
30 ST* Z
31 ST* Z
32 *
33 +
34 6378.137
35 X<>Y
36 /
37 STO 34
38 RCL 06
39 E3
40 /
41 STO 51
42 +
43 STO 47
44 RCL 01
45 273.15
46 +
47 1/X
48 ENTER
49 ENTER
50 STO 00
51 77514.1
52 *
53 710.792
54 -
55 *
56 2.23366
57 +
58 *
59 .00237321
60 -

61 RCL 03
62 ST* Y
63 37 E-5
64 *
65 1
66 +
67 *
68 1
69 +
70 RCL 03
71 *
72 *
73 STO 21
74 RCL 04
75 X^2
76 1/X
77 ENTER
78 ENTER
79 STO 22
80 .08851
81 *
82 .7115
83 -
84 *
85 58.058
86 +
87 *
88 6487.31
89 +
90 ST* 21
91 RCL 00
92 .25844
93 *
94 9325 E-7
95 -
96 RCL 00
97 *
98 579 E-9
99 +
100 RCL 02
101 RCL 03
102 -
103 ST* 00
104 *
105 1
106 +
107 RCL 00
108 *
109 4547.3
110 38.9
111 RCL 22
112 -
113 /
114 130
115 RCL 22
116 -
117 683939.7
118 X<>Y
119 /
120 +

121 2371.34
122 +
123 *
124 RCL 21
125 +
126 E8
127 /
128 STO 50
129 1.6
130 STO 42
131 FRC
132 SQRT
133 STO 41
134 FS? 00
135 GTO 00
136 RCL 01
137 1250
138 /
139 .109671
140 -
141 STO 07
142 .0026952
143 CHS
144 RCL 01
145 95 E-7
146 *
147 -
148 STO 08
149 958131 E-9
150 STO 09
151 1.553002 E-4
152 CHS
153 STO 10
154 1.137826 E-5
155 STO 11
156 4.532222 E-7
157 CHS
158 STO 12
159 1.012373 E-8
160 STO 13
161 1054348 E-16
162 CHS
163 STO 14
164 3737867 E-19
165 CHS
166 STO 15
167 2529916 E-20
168 STO 16
169 31539538 E-23
170 CHS
171 STO 17
172 1805402 E-24
173 STO 18
174 41167039 E-28
175 CHS
176 STO 19
177 LBL 00
178 31.019
179 STO 48
180 19

181 13
182 LBL 07
183 RCL IND Y
184 RCL Y
185 *
186 STO IND 48
187 CLX
188 SIGN
189 ST- Z
190 -
191 DSE 48
192 GTO 07
193 CLX
194 X<> 51
195 XEQ 04
196 LASTX
197 STO 00
198 FS? 01
199 RCL 50
200 RCL 47
201 ST* Y
202 +
203 RCL 43
204 *
205 STO 46
206 FC? 01
207 GTO 00
208 RCL 00
209 RCL 50
210 /
211 ST/ 50
212 LBL 00
213 3 E-6
214 STO 49
215 11
216 XEQ 02
217 STO 32
218 6
219 STO 52
220 XEQ 13
221 28
222 XEQ 02
223 X<> 32
224 STO 33
225 4
226 STO 52
227 XEQ 13
228 87
229 XEQ 02
230 X<> 32
231 STO 33
232 XEQ 13
233 GTO 12
234 LBL 13
235 RCL 33
236 RCL 32
237 STO 35
238 -
239 RCL 52
240 STO 39

241 /
242 STO 38
243 2
244 /
245 ST+ 35
246 RCL 41
247 *
248 STO 36
249 CLX
250 STO 37
251 LBL 14
252 RCL 35
253 RCL 36
254 -
255 XEQ 01
256 ST+ 37
257 RCL 35
258 XEQ 01
259 RCL 42
260 *
261 ST+ 37
262 RCL 35
263 RCL 36
264 +
265 XEQ 01
266 ST+ 37
267 RCL 38
268 ST+ 35
269 DSE 39
270 GTO 14
271 RCL 37
272 *
273 ST+ 51
274 RTN
275 LBL 02
276 STO 45
277 XEQ 04
278 X<>Y
279 RCL 34
280 +
281 *
282 RCL 46
283 X<>Y
284 /
285 ACOS
286 RTN
287 LBL 01
288 COS
289 STO 40
290 LBL 03
291 RCL 45
292 XEQ 04
293 STO 48
294 X<>Y
295 RCL 34
296 +
297 STO 44
298 *
299 RCL 46
300 RCL 40

301 /
302 -
303 RCL 00
304 RCL 44
305 *
306 RCL 48
307 +
308 /
309 ST- 45
310 ABS
311 RCL 49
312 X<Y?
313 GTO 03
314 RCL 48
315 RCL 44
316 RCL 00
317 *
318 STO Z
319 +
320 /
321 RTN
322 LBL 04
323 ENTER
324 STO Z
325 RCL 31
326 *
327 RCL 30
328 +
329 *
330 RCL 29
331 +
332 *
333 RCL 28
334 +
335 *
336 RCL 27
337 +
338 *
339 RCL 26
340 +
341 *
342 RCL 25
343 +
344 *
345 RCL 24
346 +
347 *
348 RCL 23
349 +
350 *
351 RCL 22
352 +
353 *
354 RCL 21
355 +
356 *
357 RCL 20
358 +
359 *

360 RCL 07
361 +
362 STO 00
363 CLX
364 RCL 19
365 *
366 RCL 18
367 +
368 *
369 RCL 17
370 +
371 *
372 RCL 16
373 +
374 *
375 RCL 15
376 +
377 *
378 RCL 14
379 +
380 *
381 RCL 13
382 +
383 *
384 RCL 12
385 +
386 *
387 RCL 11
388 +
389 *
390 RCL 10
391 +
392 *
393 RCL 09
394 +
395 *
396 RCL 08
397 +
398 *
399 RCL 07
400 +
401 *
402 E^X
403 RCL 50
404 *
405 ST* 00
406 SIGN
407 LASTX
408 +
409 RTN
410 LBL 12
411 RCL 51
412 E3
413 *
414 STO Y
415 3600
416 /
417 HMS
418 END

( 843 bytes / SIZE 053 )

STACK	INPUTS	OUTPUTS
Y	Az ( ° ' " )	R ( " )
X	h₀ ( ° ' " )	R ( ° ' " )

Example: t = 10°C P = 1010 mbar F = 6 mbar Lambda = 0.577 µm Latitude = 33°21'22" Altitude = 1706 m

    10    STO 01        0.577    STO 04
1010 STO 02      33.2122 STO 05
     6     STO 03        1706     STO 06

1°) CF 00 CF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345   XEQ "REF13" >>>> R = 0°17'56"321                    ---Execution time = 10m39s---
                                             RDN   R = 1076"321

2°) CF 00 SF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345      R/S       >>>> R = 0°21'35"834
                                    RDN   R = 1295"834

Note:

-With Az = 84° , we get 1077"921 & 1297"595 in the examples above.

3°) Program#3-Three Polynomials of degree 6

-Here, we assume that the logarithm of the densities d(x) is a polynomial of degree 6 for 0 < x < 11 km
-Another polynomial of degree 6 for 11 < x < 28 km
-And a 3rd polynomial of degree 6 for 28 < x < 87 km

    d₁(x) = exp [ c₀ + c₁ x + c₂ x² + ............... + c₆ x⁶ ]     ( 0 < x < 11 )
    d₂(x) = exp [ c'₀ + c'₁ x + c'₂ x² + ............... + c'₆ x⁶ ]     ( 11 < x < 28 )
    d₃(x) = exp [ c"₀ + c"₁ x + c"₂ x² + ............... + c"₆ x⁶ ]     ( 28 < x < 87 )     with:

    c₀ = -6.704085          c₃ = -5.19398 E-4          c₆ = 2.811385 E-8            ( lines 136 to 153 )
    c₁ = -0.111511          c₄ = 2.309197 E-5
    c₂ = 3.835206 E-3     c₅ = -9.619965 E-7

    c'₀ = -6.05731            c'₃ = -0.00341631           c'₆ = 2.638265 E-8            ( lines 198 to 215 )
    c'₁ = -0.3472742        c'₄ = 1.461133 E-4
    c'₂ = 0.03978828       c'₅ = -3.109464 E-6

    c"₀ = -27.374327        c"₃ = 0.003395521           c"₆ = -9.3999402 E-10        ( lines 231 to 248 )
    c"₁ = 2.5006517          c"₄ = -4.655209 E-5
    c"₂ = -0.1331043        c"₅ = 3.284335 E-7

-You can use the constants listed below ( in this case, clear flag F00 )
-Or your own coefficients ( store these 21 numbers in a DATA file and set flag F00 )

Data Registers: R00 = dn/dx ( Registers R01 thru R06 are to be initialized before executing "REF6+" )

R07 = c₀ .................... R13 = c₆ R14 to R40: temp

Flags: F00 & F01 CF 00 = The constants in R07 to R13 are those listed below
SF 00 = Store the 3x7 coefficients in a DATA file before executing "REF6+"

CF 01 = t , P , F are measured for an altitude = 0
SF 01 = t , P , F are measured at the observer's station
Subroutines: /

-Lines 08 & 260 are three-byte GTOs

01 LBL "REF6+"
02 DEG
03 HR
04 STO 22
05 COS
06 STO 19
07 X=0?
08 GTO 12
09 STO 32
10 X<>Y
11 HR
12 1
13 P-R
14 X^2
15 0.99330562
16 /
17 X<>Y
18 X^2
19 RCL 05
20 HR
21 SIN
22 X^2
23 669438 E-8
24 *
25 1
26 X<>Y
27 -
28 SQRT
29 ST* Z
30 ST* Z
31 ST* Z
32 *
33 +
34 6378.137
35 X<>Y
36 /
37 STO 23
38 RCL 06
39 E3
40 /
41 STO 19
42 +
43 STO 36
44 RCL 01
45 273.15
46 +
47 1/X
48 ENTER
49 ENTER
50 STO 00
51 77514.1
52 *
53 710.792
54 -
55 *
56 2.23366
57 +
58 *
59 237321 E-8
60 -

61 RCL 03
62 ST* Y
63 37 E-5
64 *
65 1
66 +
67 *
68 1
69 +
70 RCL 03
71 *
72 *
73 STO 07
74 RCL 04
75 X^2
76 1/X
77 ENTER
78 ENTER
79 STO 08
80 .08851
81 *
82 .7115
83 -
84 *
85 58.058
86 +
87 *
88 6487.31
89 +
90 ST* 07
91 RCL 00
92 .25844
93 *
94 9325 E-7
95 -
96 RCL 00
97 *
98 579 E-9
99 +
100 RCL 02
101 RCL 03
102 -
103 ST* 00
104 *
105 1
106 +
107 RCL 00
108 *
109 4547.3
110 38.9
111 RCL 08
112 -
113 /
114 130
115 RCL 08
116 -
117 683939.7
118 X<>Y
119 /
120 +

121 2371.34
122 +
123 *
124 RCL 07
125 +
126 E8
127 /
128 STO 39
129 1.6
130 STO 31
131 FRC
132 SQRT
133 STO 30
134 FS? 00
135 GTO 00
136 6.704085
137 CHS
138 STO 07
139 .111511
140 CHS
141 STO 08
142 3.835206 E-3
143 STO 09
144 5.19398 E-4
145 CHS
146 STO 10
147 2.309197 E-5
148 STO 11
149 9.619965 E-7
150 CHS
151 STO 12
152 2.811385 E-8
153 STO 13
154 GTO 06
155 LBL 00
156 CLX
157 SEEKPT
158 7.013
159 STO 42
160 GETRX
161 LBL 06
162 RCL 39
163 RCL 07
164 E^X
165 /
166 STO 40
167 XEQ 07
168 CLX
169 X<> 19
170 XEQ 04
171 LASTX
172 STO 00
173 FS? 01
174 RCL 39
175 RCL 36
176 ST* Y
177 +
178 RCL 32
179 *
180 STO 35

181 FC? 01
182 GTO 00
183 RCL 00
184 RCL 39
185 /
186 ST/ 40
187 LBL 00
188 3 E-6
189 STO 38
190 11
191 XEQ 02
192 STO 21
193 6
194 STO 20
195 XEQ 13
196 FS? 00
197 GTO 00
198 6.05731
199 CHS
200 STO 07
201 .3472742
202 CHS
203 STO 08
204 .03978828
205 STO 09
206 .00341631
207 CHS
208 STO 10
209 1.461133 E-4
210 STO 11
211 3.109464 E-6
212 CHS
213 STO 12
214 2.638265 E-8
215 STO 13
216 GTO 06
217 LBL 00
218 RCL 42
219 GETRX
220 LBL 06
221 XEQ 07
222 28
223 XEQ 02
224 X<> 21
225 STO 22
226 4
227 STO 20
228 XEQ 13
229 FS? 00
230 GTO 00
231 27.374327
232 CHS
233 STO 07
234 2.5006517
235 STO 08
236 .1331043
237 CHS
238 STO 09
239 .003395521
240 STO 10

241 4.655209 E-5
242 CHS
243 STO 11
244 3.284335 E-7
245 STO 12
246 9.3999402 E-10
247 CHS
248 STO 13
249 GTO 06
250 LBL 00
251 RCL 42
252 GETRX
253 LBL 06
254 XEQ 07
255 87
256 XEQ 02
257 X<> 21
258 STO 22
259 XEQ 13
260 GTO 12
261 LBL 07
262 18.013
263 STO 41
264 13
265 6
266 LBL 08
267 RCL IND Y
268 RCL Y
269 *
270 STO IND 41
271 CLX
272 SIGN
273 ST- Z
274 -
275 DSE 41
276 GTO 08
277 RTN
278 LBL 13
279 RCL 22
280 RCL 21
281 STO 24
282 -
283 RCL 20
284 STO 28
285 /
286 STO 27
287 2
288 /
289 ST+ 24
290 RCL 30
291 *
292 STO 25
293 CLX
294 STO 26
295 LBL 14
296 RCL 24
297 RCL 25
298 -
299 XEQ 01
300 ST+ 26

301 RCL 24
302 XEQ 01
303 RCL 31
304 *
305 ST+ 26
306 RCL 24
307 RCL 25
308 +
309 XEQ 01
310 ST+ 26
311 RCL 27
312 ST+ 24
313 DSE 28
314 GTO 14
315 RCL 26
316 *
317 ST+ 19
318 RTN
319 LBL 02
320 STO 34
321 XEQ 04
322 X<>Y
323 RCL 23
324 +
325 *
326 RCL 35
327 X<>Y
328 /
329 ACOS
330 RTN
331 LBL 01
332 COS
333 STO 29
334 LBL 03
335 RCL 34
336 XEQ 04
337 STO 37
338 X<>Y
339 RCL 23
340 +
341 STO 33
342 *
343 RCL 35
344 RCL 29
345 /
346 -
347 RCL 00
348 RCL 33
349 *
350 RCL 37
351 +
352 /
353 ST- 34
354 ABS
355 RCL 38
356 X<Y?
357 GTO 03
358 RCL 37
359 RCL 33
360 RCL 00
361 *

362 STO Z
363 +
364 /
365 RTN
366 LBL 04
367 ENTER
368 STO Z
369 RCL 18
370 *
371 RCL 17
372 +
373 *
374 RCL 16
375 +
376 *
377 RCL 15
378 +
379 *
380 RCL 14
381 +
382 *
383 RCL 08
384 +
385 STO 00
386 CLX
387 RCL 13
388 *
389 RCL 12
390 +
391 *
392 RCL 11
393 +
394 *
395 RCL 10
396 +
397 *
398 RCL 09
399 +
400 *
401 RCL 08
402 +
403 *
404 RCL 07
405 +
406 E^X
407 RCL 40
408 *
409 ST* 00
410 SIGN
411 LASTX
412 +
413 RTN
414 LBL 12
415 RCL 19
416 E3
417 *
418 STO Y
419 3600
420 /
421 HMS
422 END

( 905 bytes / SIZE 043 )

STACK	INPUTS	OUTPUTS
Y	Az ( ° ' " )	R ( " )
X	h₀ ( ° ' " )	R ( ° ' " )

Example: t = 10°C P = 1010 mbar F = 6 mbar Lambda = 0.577 µm Latitude = 33°21'22" Altitude = 1706 m

    10    STO 01        0.577    STO 04
1010 STO 02      33.2122 STO 05
     6     STO 03        1706     STO 06

1°) CF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345   XEQ "REF6+" >>>> R = 0°17'40"473                    ---Execution time = 7m12s---
                                             RDN   R = 1060"473

2°) SF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345      R/S     >>>> R = 0°21'21"145
                                  RDN   R = 1281"145

Notes:

-With Az = 84° , we get 1062"060 & 1283"096 in the examples above.
-Register R42 is unused if CF 00.

4°) Program#4-With XM-Data File

-Instead of fitting Ln d to a polynomial, the following program uses directly the densities at altitude 0km, 1km, ..... , 89km
-Lagrange interpolation & differentiation formulas are employed to evaluate n & dn/dr ( 5 point-formulas, cf reference [6] )

-One of the numerous atmospheric models given in reference [4] gives the following values for the density as a function of the altitude ( in kilometers ):

    0.0 1.226E-03
    1.0 1.101E-03
    2.0 9.932E-04
    3.0 8.987E-04
    4.0 8.136E-04
    5.0 7.354E-04
    6.0 6.627E-04
    7.0 5.944E-04
    8.0 5.303E-04
    9.0 4.701E-04
   10.0 4.138E-04
   11.0 3.618E-04
   12.0 3.144E-04
   13.0 2.718E-04
   14.0 2.340E-04
   15.0 2.008E-04
   16.0 1.718E-04
   17.0 1.467E-04
   18.0 1.251E-04
   19.0 1.065E-04
   20.0 9.071E-05
   21.0 7.723E-05
   22.0 6.576E-05
   23.0 5.601E-05
   24.0 4.771E-05
   25.0 4.066E-05
   26.0 3.466E-05
   27.0 2.957E-05
   28.0 2.523E-05
   29.0 2.155E-05
   30.0 1.842E-05
   31.0 1.576E-05
   32.0 1.349E-05
   33.0 1.156E-05
   34.0 9.909E-06
   35.0 8.503E-06
   36.0 7.306E-06
   37.0 6.288E-06
   38.0 5.423E-06
   39.0 4.687E-06
   40.0 4.062E-06
   41.0 3.530E-06
   42.0 3.077E-06
   43.0 2.690E-06
   44.0 2.360E-06
   45.0 2.078E-06
   46.0 1.836E-06
   47.0 1.626E-06
   48.0 1.444E-06
   49.0 1.284E-06
   50.0 1.143E-06
   51.0 1.018E-06
   52.0 9.066E-07
   53.0 8.070E-07
   54.0 7.177E-07
   55.0 6.375E-07
   56.0 5.654E-07
   57.0 5.007E-07
   58.0 4.427E-07
   59.0 3.909E-07
   60.0 3.447E-07
   61.0 3.035E-07
   62.0 2.669E-07
   63.0 2.344E-07
   64.0 2.056E-07
   65.0 1.800E-07
   66.0 1.575E-07
   67.0 1.375E-07
   68.0 1.200E-07
   69.0 1.045E-07
   70.0 9.088E-08
   71.0 7.892E-08
   72.0 6.843E-08
   73.0 5.914E-08
   74.0 5.119E-08
   75.0 4.426E-08
   76.0 3.821E-08
   77.0 3.291E-08
   78.0 2.827E-08
   79.0 2.420E-08
   80.0 2.065E-08
   81.0 1.754E-08
   82.0 1.485E-08
   83.0 1.251E-08
   84.0 1.050E-08
   85.0 8.779E-09
   86.0 7.308E-09
   87.0 6.061E-09
   88.0 5.009E-09
   89.0 4.125E-09

-Store the logarithms of these densities in a DATA file of 90 numbers before executing "REFXM"

>>> This file must be the current file.

Data Registers: R00 = dn/dx ( Registers R01 thru R06 are to be initialized before executing "REFXM" )

R07 to R46: temp

Flag: F01 CF 01 = t , P , F are measured for an altitude = 0
SF 01 = t , P , F are measured at the observer's station
Subroutines: /

-Lines 08 & 199 are three-byte GTOs

01 LBL "REFXM"
02 DEG
03 HR
04 STO 17
05 COS
06 STO 36
07 X=0?
08 GTO 12
09 STO 27
10 X<>Y
11 HR
12 1
13 P-R
14 X^2
15 .99330562
16 /
17 X<>Y
18 X^2
19 RCL 05
20 HR
21 SIN
22 X^2
23 .00669438
24 *
25 1
26 X<>Y
27 -
28 SQRT
29 ST* Z
30 ST* Z
31 ST* Z
32 *
33 +
34 6378.137
35 X<>Y
36 /
37 STO 18
38 RCL 06
39 E3
40 /
41 STO 36
42 +
43 STO 31
44 RCL 01
45 273.15
46 +
47 1/X
48 ENTER
49 ENTER
50 STO 00
51 77514.1
52 *
53 710.792
54 -
55 *
56 2.23366
57 +
58 *
59 .00237321
60 -
61 RCL 03
62 ST* Y
63 37 E-5
64 *
65 1

66 +
67 *
68 1
69 +
70 RCL 03
71 *
72 *
73 STO 07
74 RCL 04
75 X^2
76 1/X
77 ENTER
78 ENTER
79 STO 08
80 .08851
81 *
82 .7115
83 -
84 *
85 58.058
86 +
87 *
88 6487.31
89 +
90 ST* 07
91 RCL 00
92 .25844
93 *
94 9325 E-7
95 -
96 RCL 00
97 *
98 579 E-9
99 +
100 RCL 02
101 RCL 03
102 -
103 ST* 00
104 *
105 1
106 +
107 RCL 00
108 *
109 4547.3
110 38.9
111 RCL 08
112 -
113 /
114 130
115 RCL 08
116 -
117 683939.7
118 X<>Y
119 /
120 +
121 2371.34
122 +
123 *
124 RCL 07
125 +
126 E8
127 /
128 STO 34
129 0
130 SEEKPT

131 CLX
132 GETX
133 E^X
134 /
135 STO 46
136 1.6
137 STO 26
138 FRC
139 SQRT
140 STO 25
141 SIGN
142 STO 08
143 2
144 STO 09
145 +
146 STO 10
147 4
148 STO 11
149 *
150 STO 45
151 5
152 STO 12
153 6
154 STO 13
155 8
156 STO 14
157 37.041
158 STO 07
159 CLX
160 X<> 36
161 XEQ 04
162 LASTX
163 STO 00
164 FS? 01
165 RCL 34
166 RCL 31
167 ST* Y
168 +
169 RCL 27
170 *
171 STO 30
172 FC? 01
173 GTO 00
174 RCL 00
175 RCL 34
176 /
177 ST/ 46
178 LBL 00
179 3 E-6
180 STO 33
181 11
182 XEQ 02
183 STO 16
184 6
185 STO 35
186 XEQ 13
187 28
188 XEQ 02
189 X<> 16
190 STO 17
191 4
192 STO 35
193 XEQ 13
194 87
195 XEQ 02

196 X<> 16
197 STO 17
198 XEQ 13
199 GTO 12
200 LBL 13
201 RCL 17
202 RCL 16
203 STO 19
204 -
205 RCL 35
206 STO 23
207 /
208 STO 22
209 2
210 /
211 ST+ 19
212 RCL 25
213 *
214 STO 20
215 CLX
216 STO 21
217 LBL 14
218 RCL 19
219 RCL 20
220 -
221 XEQ 01
222 ST+ 21
223 RCL 19
224 XEQ 01
225 RCL 26
226 *
227 ST+ 21
228 RCL 19
229 RCL 20
230 +
231 XEQ 01
232 ST+ 21
233 RCL 22
234 ST+ 19
235 DSE 23
236 GTO 14
237 RCL 21
238 *
239 ST+ 36
240 RTN
241 LBL 02
242 STO 29
243 XEQ 04
244 RCL 15
245 RCL 18
246 +
247 *
248 RCL 30
249 X<>Y
250 /
251 ACOS
252 RTN
253 LBL 01
254 COS
255 STO 24
256 LBL 03
257 RCL 29
258 XEQ 04
259 STO 32
260 RCL 15

261 RCL 18
262 +
263 STO 28
264 *
265 RCL 30
266 RCL 24
267 /
268 -
269 RCL 00
270 RCL 28
271 *
272 RCL 32
273 +
274 /
275 ST- 29
276 ABS
277 RCL 33
278 X<Y?
279 GTO 03
280 RCL 32
281 RCL 28
282 RCL 00
283 *
284 STO Z
285 +
286 /
287 RTN
288 LBL 04
289 ENTER
290 STO 15
291 INT
292 RCL 09
293 -
294 X<0?
295 CLX
296 SEEKPT
297 RCL 07
298 GETRX
299 CLX
300 RCL 09
301 +
302 -
303 ENTER
304 STO Z
305 RCL 09
306 *
307 RCL 10
308 -
309 *
310 RCL 08
311 -
312 *
313 RCL 08
314 +
315 RCL 37
316 *
317 STO 00
318 CLX
319 RCL 11
320 *
321 RCL 10
322 -
323 *
324 RCL 14

325 -
326 *
327 RCL 11
328 +
329 ST+ X
330 RCL 38
331 *
332 ST- 00
333 RDN
334 X^2
335 ST+ X
336 RCL 12
337 -
338 *
339 RCL 13
340 *
341 RCL 39
342 *
343 ST+ 00
344 CLX
345 RCL 11
346 *
347 RCL 10
348 +
349 *
350 RCL 14
351 -
352 *
353 RCL 11
354 -
355 ST+ X
356 RCL 40
357 *
358 ST- 00
359 CLX
360 RCL 09
361 *
362 RCL 10
363 +
364 *
365 RCL 08
366 -
367 *
368 RCL 08
369 -
370 RCL 41
371 *
372 ST+ 00
373 RDN
374 X^2
375 RCL 08
376 -
377 STO 43
378 X<>Y
379 RCL 09
380 -
381 *
382 *
383 RCL 37
384 *
385 STO 42
386 RDN
387 X^2
388 RCL 11

389 -
390 STO 44
391 *
392 X<>Y
393 RCL 08
394 -
395 *
396 RCL 11
397 *
398 RCL 38
399 *
400 ST- 42
401 CLX
402 RCL 43
403 RCL 44
404 *
405 RCL 13
406 *
407 RCL 39
408 *
409 ST+ 42
410 CLX
411 RCL 44
412 *
413 X<>Y
414 RCL 08
415 +
416 *
417 RCL 11
418 *
419 RCL 40
420 *
421 ST- 42
422 CLX
423 RCL 09
424 +
425 *
426 RCL 43
427 *
428 RCL 41
429 *
430 RCL 42
431 +
432 RCL 45
433 ST/ 00
434 ST+ X
435 /
436 E^X
437 RCL 46
438 *
439 ST* 00
440 SIGN
441 LASTX
442 +
443 RTN
444 LBL 12
445 RCL 36
446 E3
447 *
448 STO Y
449 3600
450 /
451 HMS
452 END

( 746 bytes / SIZE 047 )

STACK	INPUTS	OUTPUTS
Y	Az ( ° ' " )	R ( " )
X	h₀ ( ° ' " )	R ( ° ' " )

Example: t = 10°C P = 1010 mbar F = 6 mbar Lambda = 0.577 µm Latitude = 33°21'22" Altitude = 1706 m

    10    STO 01        0.577    STO 04
1010 STO 02      33.2122 STO 05
     6     STO 03        1706     STO 06

1°) CF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345   XEQ "REFXM" >>>> R = 0°17'40"358                    ---Execution time = 15m21s---
                                               RDN   R = 1060"358

2°) SF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345       R/S       >>>>    R = 0°21'19"933
                                     RDN    R = 1279"933

Notes:

-The number of subintervals ( lines 184 & 191 ) is perhaps not quite enough.
-Try another number if need be.

-If you want to use the densities at altitudes 0 0.5 1 1.5 ... km, create a data file of 177 numbers and store

Ln d(0) Ln d(0.5) Ln d(1) Ln d(1.5) .................. Ln d(88)

replace the first instructions in LBL 04 ( lines 288 to 304 ) by

LBL 04          RCL 09        RCL 07        /                               after replacing   lines 432 to 435 by
ENTER          -                   GETRX       -
STO 15          X<0?            CLX           ST+ X                       RCL 45      ST/ 00     /
ST+ X            CLX             RCL 09      ENTER                      RCL 09     RCL 11
INT                SEEKPT       ST+ Y       STO Z    ........             /                 *

-Here are the densities for the same atmospheric model:

    0.0 1.226E-03
    0.5 1.161E-03
    1.0 1.101E-03
    1.5 1.045E-03
    2.0 9.932E-04
    2.5 9.446E-04
    3.0 8.987E-04
    3.5 8.552E-04
    4.0 8.136E-04
    4.5 7.738E-04
    5.0 7.354E-04
    5.5 6.985E-04
    6.0 6.627E-04
    6.5 6.280E-04
    7.0 5.944E-04
    7.5 5.619E-04
    8.0 5.303E-04
    8.5 4.997E-04
    9.0 4.701E-04
    9.5 4.414E-04
   10.0 4.138E-04
   10.5 3.873E-04
   11.0 3.618E-04
   11.5 3.375E-04
   12.0 3.144E-04
   12.5 2.925E-04
   13.0 2.718E-04
   13.5 2.523E-04
   14.0 2.340E-04
   14.5 2.168E-04
   15.0 2.008E-04
   15.5 1.858E-04
   16.0 1.718E-04
   16.5 1.588E-04
   17.0 1.467E-04
   17.5 1.355E-04
   18.0 1.251E-04
   18.5 1.154E-04
   19.0 1.065E-04
   19.5 9.830E-05
   20.0 9.071E-05
   20.5 8.370E-05
   21.0 7.723E-05
   21.5 7.126E-05
   22.0 6.576E-05
   22.5 6.069E-05
   23.0 5.601E-05
   23.5 5.169E-05
   24.0 4.771E-05
   24.5 4.404E-05
   25.0 4.066E-05
   25.5 3.754E-05
   26.0 3.466E-05
   26.5 3.201E-05
   27.0 2.957E-05
   27.5 2.731E-05
   28.0 2.523E-05
   28.5 2.332E-05
   29.0 2.155E-05
   29.5 1.992E-05
   30.0 1.842E-05
   30.5 1.703E-05
   31.0 1.576E-05
   31.5 1.458E-05
   32.0 1.349E-05
   32.5 1.248E-05
   33.0 1.156E-05
   33.5 1.070E-05
   34.0 9.909E-06
   34.5 9.178E-06
   35.0 8.503E-06
   35.5 7.881E-06
   36.0 7.306E-06
   36.5 6.777E-06
   37.0 6.288E-06
   37.5 5.838E-06
   38.0 5.423E-06
   38.5 5.040E-06
   39.0 4.687E-06
   39.5 4.362E-06
   40.0 4.062E-06
   40.5 3.785E-06
   41.0 3.530E-06
   41.5 3.294E-06
   42.0 3.077E-06
   42.5 2.876E-06
   43.0 2.690E-06
   43.5 2.519E-06
   44.0 2.360E-06
   44.5 2.214E-06
   45.0 2.078E-06
   45.5 1.952E-06
   46.0 1.836E-06
   46.5 1.727E-06
   47.0 1.626E-06
   47.5 1.532E-06
   48.0 1.444E-06
   48.5 1.361E-06
   49.0 1.284E-06
   49.5 1.211E-06
   50.0 1.143E-06
   50.5 1.079E-06
   51.0 1.018E-06
   51.5 9.607E-07
   52.0 9.066E-07
   52.5 8.554E-07
   53.0 8.070E-07
   53.5 7.611E-07
   54.0 7.177E-07
   54.5 6.765E-07
   55.0 6.375E-07
   55.5 6.005E-07
   56.0 5.654E-07
   56.5 5.321E-07
   57.0 5.007E-07
   57.5 4.709E-07
   58.0 4.427E-07
   58.5 4.161E-07
   59.0 3.909E-07
   59.5 3.672E-07
   60.0 3.447E-07
   60.5 3.235E-07
   61.0 3.035E-07
   61.5 2.847E-07
   62.0 2.669E-07
   62.5 2.502E-07
   63.0 2.344E-07
   63.5 2.196E-07
   64.0 2.056E-07
   64.5 1.924E-07
   65.0 1.800E-07
   65.5 1.684E-07
   66.0 1.575E-07
   66.5 1.472E-07
   67.0 1.375E-07
   67.5 1.285E-07
   68.0 1.200E-07
   68.5 1.120E-07
   69.0 1.045E-07
   69.5 9.747E-08
   70.0 9.088E-08
   70.5 8.470E-08
   71.0 7.892E-08
   71.5 7.350E-08
   72.0 6.843E-08
   72.5 6.368E-08
   73.0 5.914E-08
   73.5 5.502E-08
   74.0 5.119E-08
   74.5 4.761E-08
   75.0 4.426E-08
   75.5 4.114E-08
   76.0 3.821E-08
   76.5 3.548E-08
   77.0 3.291E-08
   77.5 3.052E-08
   78.0 2.827E-08
   78.5 2.617E-08
   79.0 2.420E-08
   79.5 2.236E-08
   80.0 2.065E-08
   80.5 1.904E-08
   81.0 1.754E-08
   81.5 1.615E-08
   82.0 1.485E-08
   82.5 1.364E-08
   83.0 1.251E-08
   83.5 1.147E-08
   84.0 1.050E-08
   84.5 9.607E-09
   85.0 8.779E-09
   85.5 8.014E-09
   86.0 7.308E-09
   86.5 6.659E-09
   87.0 6.061E-09
   87.5 5.512E-09
   88.0 5.009E-09

5°) Auer & Standish Model of the Atmosphere

-In this model, the refractivity of air is given by formulas of the form:

n - 1 = ( n₀ - 1 ) [ 1 + C ( r_T / ( r_T + x ) - 1 ) ]⁵ if the altitude x < 11016 m
n - 1 = ( n₀ - 1 ) Exp [ C' ( r_T / ( r_T + x ) - 1 ) ] if the altitude x > 11016 m

-Cf reference [2] for the details

-Here, there are a few modifications to take the latitude & the azimuth into account.

Data Registers: R00 = dn/dx ( Registers R01 thru R06 are to be initialized before executing "AUSTD" )

                                      • R01 = t ( °C )           • R04 = wave-length ( µm )
                                      • R02 = P ( mbar )       • R05 = Latitude ( ° ' " )
                                      • R03 = F ( mbar )       • R06 = Altitude of the observer ( in meters ) < 11019

R07 to R38: temp

Flag: F01 CF 01 = t , P , F are measured for an altitude = 0
SF 01 = t , P , F are measured at the observer's station
Subroutines: /

-Lines 08 & 218 are three-byte GTOs

01 LBL "AUSTD"
02 DEG
03 HR
04 STO 22
05 COS
06 STO 20
07 X=0?
08 GTO 12
09 STO 32
10 X<>Y
11 HR
12 1
13 P-R
14 X^2
15 .99330562
16 /
17 X<>Y
18 X^2
19 RCL 05
20 HR
21 SIN
22 X^2
23 669438 E-8
24 *
25 1
26 X<>Y
27 -
28 SQRT
29 ST* Z
30 ST* Z
31 ST* Z
32 *
33 +
34 6378.137
35 X<>Y
36 /
37 STO 23
38 RCL 06
39 E3
40 /
41 STO 20
42 STO 38
43 +
44 STO 36
45 RCL 01
46 273.15
47 +
48 STO 15
49 1/X
50 ENTER
51 ENTER
52 STO 00
53 77514.1
54 *

55 710.792
56 -
57 *
58 2.23366
59 +
60 *
61 237321 E-8
62 -
63 RCL 03
64 ST* Y
65 37 E-5
66 *
67 1
68 +
69 *
70 1
71 +
72 RCL 03
73 *
74 *
75 STO 07
76 RCL 04
77 X^2
78 1/X
79 ENTER
80 ENTER
81 STO 08
82 .08851
83 *
84 .7115
85 -
86 *
87 58.058
88 +
89 *
90 6487.31
91 +
92 ST* 07
93 RCL 00
94 .25844
95 *
96 9325 E-7
97 -
98 RCL 00
99 *
100 579 E-9
101 +
102 RCL 02
103 RCL 03
104 -
105 ST* 00
106 *
107 1
108 +

109 RCL 00
110 *
111 4547.3
112 38.9
113 RCL 08
114 -
115 /
116 130
117 RCL 08
118 -
119 683939.7
120 X<>Y
121 /
122 +
123 2371.34
124 +
125 *
126 RCL 07
127 +
128 E8
129 /
130 STO 19
131 1.6
132 STO 31
133 FRC
134 SQRT
135 STO 30
136 5
137 CHS
138 STO 07
139 287053 E-9
140 STO 09
141 RCL 15
142 *
143 RCL 05
144 HR
145 SIN
146 X^2
147 STO 00
148 228
149 *
150 51631
151 +
152 RCL 00
153 *
154 9780325
155 +
156 E9
157 /
158 RCL 23
159 X^2
160 *
161 X<>Y
162 /

163 STO 11
164 6
165 /
166 STO 10
167 SF 10
168 CLX
169 X<> 20
170 XEQ 04
171 RCL 14
172 FS? 01
173 RCL 19
174 RCL 36
175 ST* Y
176 +
177 RCL 32
178 *
179 STO 35
180 FC? 01
181 GTO 00
182 RCL 14
183 RCL 19
184 /
185 ST/ 19
186 LBL 00
187 3 E-6
188 STO 18
189 11.019
190 XEQ 02
191 STO 21
192 RCL 14
193 STO 15
194 RCL 12
195 CF 10
196 XEQ 04
197 RCL 14
198 RCL 08
199 /
200 ST* 15
201 6
202 STO 17
203 SF 10
204 XEQ 13
205 CF 10
206 28
207 XEQ 02
208 X<> 21
209 STO 22
210 4
211 STO 17
212 XEQ 13
213 E2
214 XEQ 02
215 X<> 21
216 STO 22

217 XEQ 13
218 GTO 12
219 LBL 13
220 RCL 22
221 RCL 21
222 STO 24
223 -
224 RCL 17
225 STO 28
226 /
227 STO 27
228 2
229 /
230 ST+ 24
231 RCL 30
232 *
233 STO 25
234 CLX
235 STO 26
236 LBL 14
237 RCL 24
238 RCL 25
239 -
240 XEQ 01
241 ST+ 26
242 RCL 24
243 XEQ 01
244 RCL 31
245 *
246 ST+ 26
247 RCL 24
248 RCL 25
249 +
250 XEQ 01
251 ST+ 26
252 RCL 27
253 ST+ 24
254 DSE 28
255 GTO 14
256 RCL 26
257 *
258 ST+ 20
259 RTN
260 LBL 02
261 STO 34
262 XEQ 04
263 STO 16
264 RCL 12
265 RCL 23
266 +
267 *
268 RCL 35
269 X<>Y

270 /
271 ACOS
272 RTN
273 LBL 01
274 COS
275 STO 29
276 LBL 03
277 RCL 34
278 XEQ 04
279 STO 37
280 RCL 12
281 RCL 23
282 +
283 STO 33
284 *
285 RCL 35
286 RCL 29
287 /
288 -
289 RCL 00
290 RCL 33
291 *
292 RCL 37
293 +
294 /
295 ST- 34
296 ABS
297 RCL 18
298 X<Y?
299 GTO 03
300 RCL 37
301 RCL 33
302 RCL 00
303 *
304 STO Z
305 +
306 /
307 RTN
308 LBL 04
309 STO 12
310 RCL 23
311 RCL X
312 RCL Z
313 +
314 STO 13
315 *
316 /
317 CHS
318 FC? 10
319 GTO 05
320 RCL 10
321 *
322 PI

323 SIGN
324 +
325 ENTER
326 X^2
327 X^2
328 STO 00
329 *
330 RCL 19
331 ST* 00
332 *
333 STO 14
334 SIGN
335 LASTX
336 +
337 X<> 00
338 RCL 10
339 *
340 RCL 07
341 *
342 RCL 13
343 X^2
344 /
345 X<> 00
346 RTN
347 LBL 05
348 RCL 11
349 *
350 E^X
351 RCL 15
352 *
353 SIGN
354 LASTX
355 STO 08
356 +
357 STO 00
358 LASTX
359 CHS
360 RCL 11
361 *
362 RCL 13
363 X^2
364 /
365 X<> 00
366 RTN
367 LBL 12
368 RCL 20
369 E3
370 *
371 STO Y
372 3600
373 /
374 HMS
375 END

( 669 bytes / SIZE 039 )

STACK	INPUTS	OUTPUTS
Y	Az ( ° ' " )	R ( " )
X	h₀ ( ° ' " )	R ( ° ' " )

Example: t = 10°C P = 1010 mbar F = 6 mbar Lambda = 0.577 µm Latitude = 33°21'22" Altitude = 1706 m

    10    STO 01        0.577    STO 04
1010 STO 02      33.2122 STO 05
     6     STO 03        1706     STO 06

1°) CF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345   XEQ "AUSTD" >>>> R = 0°17'54"337                    ---Execution time = 5m28s---
                                              RDN   R = 1074"337

2°) SF 01 Az = 12°41' h₀ = 1°23'45"

       12.41    ENTER^
      1.2345       R/S       >>>>    R = 0°21'28"454
                                     RDN   R = 1288"454

6°) Faster Programs

a) REFR

-"REFR" approximate the results given by "REF13" for

t = 10°C P = 1010 mbar F = 0 mbar Lambda = 0.59 µm Latitude = 45° Altitude = 0

-The error ( compared to "REF13" ) are:

    err < 0"003 over [5°,90°]
    err < 0"004 over [1°,5°]
    err < 0"005 over [0°,1°]
    err < 0"011 over [-1°,0°]

-Such an accuracy is of course illusory near the horizon...
-And the precision is much smaller for other atmospheric conditions.

Formula for h₀ > 5°

R ~ 57"91214 / Tan h₀ - 0"06675061 / Tan³ h₀ + 1"97745 E-4 / Tan⁵ h₀ - 6"652813 E-7 Tan⁷ h₀ + 1"306196 E-9 / Tan⁹ h₀

Formula for h₀ < 5°

R ~ Exp ( A + B h₀ + C h₀² + D h₀³ + E h₀⁴ + F h₀⁵ + G h₀⁶ + H h₀⁷ + I h₀⁸ + J h₀⁹ + K h₀¹⁰ + L h₀¹¹ + M h₀¹² ) with

   A = +7.631589            E = - 0.010024199        I = - 4.8712695 E-4      M = - 3.7223778 E-8
   B = - 0.3890402          F = +0.007237414        J = +1.0159107 E-4
   C = +0.03649829        G = - 0.0039216984     K = - 1.3748284 E-5
   D = +0.006352585      H = +0.0016179943     L = +1.0796128 E-6

Data Registers: R00 = h₀ ( deg ) ( Registers R01 thru R06 are to be initialized before executing "REFR" )

                                      • R01 = t ( °C )           • R04 = wave-length ( µm )
                                      • R02 = P ( mbar )       • R05 = Latitude ( ° ' " )
                                      • R03 = F ( mbar )       • R06 = Altitude of the observer ( in meters )

Flags: /
Subroutines: /

01 LBL "REFR"
02 DEG
03 HR
04 STO 00
05 5
06 X<>Y
07 X>Y?
08 GTO 01
09 1.0796128 E-6
10 RCL Y
11 3.7223778 E-8
12 *
13 -
14 *
15 1.3748284 E-5
16 -
17 *
18 1.0159107 E-4
19 +
20 *
21 4.8712695 E-4
22 -
23 *
24 .0016179943
25 +
26 *

27 .0039216984
28 -
29 *
30 .007237414
31 +
32 *
33 .010024199
34 -
35 *
36 .006352585
37 +
38 *
39 .03649829
40 +
41 *
42 .3890402
43 -
44 *
45 7.631589
46 +
47 E^X
48 GTO 02
49 LBL 01
50 TAN
51 1/X
52 X^2

53 6.652813 E-7
54 CHS
55 RCL Y
56 1.306196 E-9
57 *
58 +
59 *
60 1.97745 E-4
61 +
62 *
63 .06675061
64 -
65 *
66 57.91214
67 +
68 X<>Y
69 SQRT
70 *
71 LBL 02
72 RCL 02
73 *
74 3.56701
75 /
76 RCL 01
77 273.15
78 +

79 /
80 1
81 RCL 03
82 18 E4
83 /
84 6579
85 1/X
86 +
87 RCL 03
88 *
89 -
90 *
91 5981
92 RCL 04
93 X^2
94 /
95 982818
96 +
97 E6
98 /
99 *
100 RCL 05
101 HR
102 ST+ X
103 COS

104 RCL 00
105 49
106 *
107 197
108 +
109 RCL 00
110 *
111 500
112 +
113 /
114 CHS
115 1
116 +
117 *
118 RCL 06
119 11000
120 /
121 CHS
122 E^X
123 *
124 STO Y
125 3600
126 /
127 HMS
128 END

( 350 bytes / SIZE 007 )

STACK	INPUTS	OUTPUTS
Y	/	R ( " )
X	h₀ ( ° ' " )	R ( ° ' " )

Example: t = 10°C P = 1010 mbar F = 6 mbar Lambda = 0.577 µm Latitude = 33°21'22" Altitude = 1706 m

    10    STO 01        0.577    STO 04
1010 STO 02      33.2122 STO 05
     6     STO 03        1706     STO 06

1°) h₀ = 1°23'45"

1.2345 XEQ "REFR" >>>> R = 0°18'20"742 ---Execution time = 10s---
RDN R = 1100"742

2°) h₀ = 12°34'56"

12.3456 R/S >>>> R = 0°03'37"253 ---Execution time = 7s---
RDN R = 217"253

Note:

-The results rapidly become meaningless for h₀ < -1°

b) RADAU

-"RADAU" approximates the table of refraction given in reference [5] - pages A42 & A43 - over [-1°,+90°]
for a temperature = 0°C , atmospheric pressure = 760 mmHg , water-vapor pressure = 6 mmHg , latitude = 45° , altitude = 0

Formula:

R ~ A / Tan ( h₀ + B / ( h₀ + C / ( h₀ + D / ( h₀ + E / ( h₀ + F ) ) ) ) )

with A = 1°/59.79268 B = 3.68278 C = 7.37814 D = 15.08593 E = 64.96944 F = 13.55049

h = h₀ - R

precision ~ 0"06

01 LBL "RADAU"
02 DEG
03 HR
04 64.96944
05 RCL Y
06 13.55049
07 +
08 /
09 +
10 15.08593
11 X<>Y
12 /
13 +
14 7.37814
15 X<>Y
16 /
17 +
18 3.68278
19 X<>Y
20 /
21 +
22 TAN
23 1/X
24 59.79268
25 /
26 ST- Y
27 HMS
28 X<>Y
29 HMS
30 END

( 82 bytes / SIZE 000 )

STACK	INPUTS	OUTPUTS
Y	/	R ( ° ' " )
X	h₀ ( ° ' " )	h ( ° ' " )

Example: h₀ = -0°12'34"

0.1234 CHS XEQ "RADAU" >>>> h = -0°52'22"71 ---Execution time = 3s---
RDN R = 0°39'48"71

Formula:

R ~ A / Tan ( h + B / ( h + C / ( h + D / ( h + E / ( h + F ) ) ) ) )

with A = 1°/59.76866 B = 4.67605 C = 7.93897 D = 16.24011 E = 73.68457 F = 14.61994

h₀ = h + R

precision ~ 0"10

01 LBL "RADAU"
02 DEG
03 HR
04 73.68457
05 RCL Y
06 14.61994
07 +
08 /
09 +
10 16.24011
11 X<>Y
12 /
13 +
14 7.93897
15 X<>Y
16 /
17 +
18 4.67605
19 X<>Y
20 /
21 +
22 TAN
23 1/X
24 59.76866
25 /
26 ST+ Y
27 HMS
28 X<>Y
29 HMS
30 END

( 82 bytes / SIZE 000 )

STACK	INPUTS	OUTPUTS
Y	/	R ( ° ' " )
X	h ( ° ' " )	h₀ ( ° ' " )

Example: h = -1°

1 CHS XEQ "RADAU" >>>> h₀ = -0°18'31"14 ---Execution time = 3s---
RDN R = 0°41'28"86

c) PH0-H & PH-H0

-"PH0-H" uses the refraction tables of Pulkovo for standard atmospheric conditions:

t = 15°C , P= 1013.25 mbar , F = 0 , lambda = 0.59 µ , latitude = 45° , altitude = 0

-It takes the apparent height of a star and returns the true height.

-PH-H0" performs the reverse transformation.

-With "PH0-H", the precision is ~ 0"26 over [0°,20°]

-Better than 0"02 over [20°,23°]
-Better than 0"01 over [23°,30°]
-Better than 0"002 over [30°,90°]

-With "PH-H0" the precision is ~ 0"11 over [0°,20°]

Formulae:

1°) "PH0-H"

R ~ A / Tan ( h₀ + B / ( h₀ + C / ( h₀ + D / ( h₀ + E / ( h₀ + F ) ) ) ) ) if h₀ < 20°

with A = 1°/63.05561 B = 3.81451 C = 6.04529 D = 8.42681 E = 23.82074 F = 7.40780

R ~ 57"085 / Tan h₀ - 0"0666 / Tan³ h₀ if h₀ > 20°

2°) "PH-H0"

R ~ A / Tan ( h + B / ( h + C / ( h + D / ( h + E / ( h + F ) ) ) ) ) if h < 20°

with A = 1°/62.93951 B = 4.80017 C = 6.90263 D = 10.06891 E = 31.76812 F = 8.87360

R ~ 57"0684 / Tan h - 0"081674 / Tan³ h if h > 20°

Data Registers: /
Flags: /
Subroutines: /

01 LBL "PH0-H"
02 DEG
03 HR
04 STO M
05 20
06 X<>Y
07 X<Y?
08 GTO 01
09 TAN
10 1/X
11 ENTER
12 X^2
13 .0666
14 CHS
15 *
16 57.085
17 +
18 *
19 3600
20 /

21 GTO 02
22 LBL 01
23 23.82074
24 RCL Y
25 7.4078
26 +
27 /
28 +
29 8.42681
30 X<>Y
31 /
32 +
33 6.04529
34 X<>Y
35 /
36 +
37 3.81451
38 X<>Y
39 /
40 +

41 TAN
42 1/X
43 63.05561
44 /
45 LBL 02
46 0
47 X<> M
48 X<>Y
49 -
50 HMS
51 RTN
52 LBL "PH-H0"
53 DEG
54 HR
55 STO M
56 20
57 X<>Y
58 X<Y?
59 GTO 01
60 TAN

61 1/X
62 ENTER
63 X^2
64 .081674
65 *
66 CHS
67 57.0684
68 +
69 *
70 3600
71 /
72 GTO 02
73 LBL 01
74 31.76812
75 RCL Y
76 8.8736
77 +
78 /
79 +
80 10.06891

81 X<>Y
82 /
83 +
84 6.90263
85 X<>Y
86 /
87 +
88 4.80017
89 X<>Y
90 /
91 +
92 TAN
93 1/X
94 62.93951
95 /
96 LBL 02
97 0
98 X<> M
99 +
100 HMS
101 END

( 233 bytes / SIZE 000 )

STACK	INPUT	OUTPUT
X	h₀ or h ( ° ' " )	h or h₀ ( ° ' " )

Example1: h₀ = 1°23'45"

1.2345 XEQ "PH0-H" >>>> h = 1°02'51"39 R/S >>>> h₀ = 1°23'45"02 ---Execution time = 2 or 3s---

Example2: h = 24°12'57"

24.1257 XEQ "PH-H0" >>>> h₀ = 24°15'02"99 R/S >>>> h = 24°12'57"00 ---Execution time = 2 or 3s---

Note:

-Synthetic register M is cleared.

References:

[1] Jean Kovalevsky et al. - "Introduction aux Ephemerides Astronomiques" - EDP Sciences - ISBN 2-86883-298-9 ( in French )
[2] Lawrence H. Auer & E. Myles Standish - "Astronomical Refraction, Computational method for all zenith angles" 2000 AJ, 119 , 2472
[3] Andrew T. Young - "Understanding Astronomical Refraction" - 2006, The Observatory, Vol. 126, N° 1191, pp. 82-115
[4] https://ccmc.gsfc.nasa.gov/modelweb/models/msis_vitmo.php
[5] "Connaissance des Temps 1977" - Gauthier-Villars - Paris
[6] Abramowitz and Stegun - "Handbook of Mathematical Functions" - Dover Publications - ISBN 0-486-61272-4

hp41programs

Astronomical Refraction(2) for the HP-41