Parallax

Diurnal Parallax for the HP-41

Overview

1°) Azimuthal Coordinates
2°) Equatorial Coordinates

-The difference between the topocentric coordinates and the geocentric coordinates is called "parallax"
-The vectorial relation OP = OT + TP gives us the required formulae to compute the parallax for the Moon , the planets ...

where O = the observer , T = the center of the Earth and P = planet , Sun , Moon , .....

1°) Azimuthal Coordinates

-Let Az = geocentric azimuth , h = geocentric height and Az' & h' the corresponding topocentric values.
-Let D = distance TP , D' = distance OP
-Let b = geographic ( i-e geodetic, i-e astronomical ) latitude of the observer, b' its geocentric latitude
-Let Alt = altitude of the observer
-And a = Earth equatorial radius = 6378137 m , f = Earth flattening = 1/298.257

-We have the following rigorous formulae:

         D' cos Az' cos h' = D cos Az cos h - a r sin ( b - b' )
         D' sin Az' cos h' = D sin Az cos h
              D' sin h'         = D sin h - a r cos ( b - b' )

tan u = ( 1 - f ) tan b

r sin b' = ( 1 - f ) sin u + ( Alt / 6378137 ) sin b ( r = OT with Earth equatorial radius as unity )
r cos b' = cos u + ( Alt / 6378137 ) cos b

Data Registers: R00 & R04 thru R07: temp ( Registers R01 thru R03 are to be initialized before executing "PAR" )

                                      • R01 = D ( AU )
                                      • R02 = b ( ° ' " )
                                      • R03 = Alt ( m )
Flags: /
Subroutines:   "R-S" & "S-R" ( cf "Spherical Coordinates" or "Transformation of Coordinates" )

01 LBL "PAR"
02 DEG
03 HR
04 X<>Y
05 HR
06 X<>Y
07 RCL 01
08 XEQ "S-R"
09 STO 05
10 RDN
11 STO 06
12 X<>Y
13 STO 07

14 RCL 02
15 HR
16 STO 04
17 1
18 P-R
19 X<>Y
20 LASTX
21 298.257
22 1/X
23 -
24 STO 00
25 *
26 X<>Y

27 R-P
28 SIGN
29 P-R
30 X<>Y
31 RCL 00
32 *
33 RCL 03
34 6378137
35 /
36 RCL 04
37 X<>Y
38 P-R
39 ST+ T

40 RDN
41 +
42 X<>Y
43 R-P
44 RCL 04
45 RCL Z
46 -
47 X<>Y
48 P-R
49 23454.79
50 ST/ Z
51 /
52 ST- 07

53 X<>Y
54 ST- 05
55 RCL 07
56 RCL 06
57 RCL 05
58 XEQ "R-S"
59 X<> Z
60 HMS
61 X<>Y
62 HMS
63 END

( 104 bytes / SIZE 008 )

STACK	INPUTS	OUTPUTS
Y	h ( ° ' " )	h' ( ° ' " )
X	Az ( ° ' " )	Az' ( ° ' " )

Example: At the Palomar Obsevatory:

    D = 0.003 AU   STO 01
    b = 33°21'22"   STO 02
Alt = 1706 m       STO 03

h = 60°12' , Az = 41°16'

     60.12    ENTER^
     41.16    XEQ "PAR"   >>>>   Az' = 41°16'11"97                             ---Execution time = 8s---
                                        RDN     h' = 59°47'32"06

-With Alt = 0 STO 03 , it yields

Az' = 41°16'11"97
h' = 59°47'32"46

Notes:

-The distance D' is in Z-register. Here, D' = 0.002963056 AU
-Line 49 may be replace by 23455 without a great loss of accuracy.
( This value is 149597871 / 6378.137 = 1 Astronomical Unit / Earth's equatorial radius )

-In order to avoid using the subroutines "S-R" & "R-S", we can rewrite the formulae as follows:

Tan Az' = ( sin Az cos h ) / [ cos Az cos h - ( a r / D ) sin ( b - b' ) ]
Tan h' = [ sin h - ( a r / D ) cos ( b - b' ) ] / [ cos² h - 2 ( a r / D ) cos Az cos h sin ( b - b' ) + ( a r / D )² sin² ( b - b' ) ]^1/2

-And if Alt = 0 , we have the approximate relations: r = 1 - f sin² b and sin ( b - b' ) = f sin 2b , cos ( b - b' ) = 1
-"PAR0" uses these formulae:

Data Registers: R00-R03-R04-R05: temp ( Registers R01 & R02 are to be initialized before executing "PAR0" )

• R01 = D ( AU )
• R02 = b ( ° ' " )
Flags: /
Subroutines: /

01 LBL "PAR0"
02 DEG
03 HR
04 STO 00
05 X<>Y
06 HR
07 1
08 P-R
09 STO 03
10 CLX
11 RCL 01
12 23454.79

13 *
14 1/X
15 RCL 02
16 HR
17 SIN
18 X^2
19 298.257
20 STO 04
21 /
22 1
23 X<>Y
24 -

25 *
26 ST- Y
27 RCL 02
28 HR
29 ST+ X
30 SIN
31 RCL 04
32 /
33 *
34 STO 04
35 RCL 00
36 COS

37 RCL 03
38 *
39 ST+ X
40 -
41 RCL 04
42 *
43 RCL 03
44 X^2
45 +
46 SQRT
47 R-P
48 X<>Y

49 HMS
50 RCL 00
51 RCL 03
52 P-R
53 RCL 04
54 -
55 R-P
56 RDN
57 HMS
58 END

( 83 bytes / SIZE 005 )

STACK	INPUTS	OUTPUTS
Y	h ( ° ' " )	h' ( ° ' " )
X	Az ( ° ' " )	Az' ( ° ' " )

Example:

D = 0.003 AU STO 01
b = 33°21'22" STO 02 ( Alt = 0 )

h = 60°12' , Az = 41°16'

     60.12    ENTER^
     41.16    XEQ "PAR0"   >>>>   Az' = 41°16'11"97                             ---Execution time = 6s---
                                          RDN     h' = 59°47'32"46

Notes:

-So, we get almost the same results: error < 0"01
-In the case of the Palomar observatory, neglecting Alt does not produce an error greater than 1 arcsecond for the Moon, much less for the planets.

Other Formulae:

-If a great precision is not required, we can use:

Az' = Az
and h' = h - ( a / D ) cos h

-These last formulas are very simple and better relations may be found.
-Here are some of them:

1°) Az' = Az + f ( a r / D ) sin Az sin 2b / cos h
h' = h + ( a r / D ) ( - cos h + f cos Az sin 2b sin h ) and even better:

2°) Az' - Az = Arctan { [ ( r a / D ) sin ( b - b' ) sin Az ] / [ cos h - ( r a / D ) sin ( b - b' ) cos Az ] }
h' - h = Arctan { - [ ( r a / D ) cos ( h + ( b - b' ) cos Az ] / [ 1 - ( r a / D ) sin ( h + ( b - b' ) cos Az ] }

-The last ones above produce the same accuracy as "PAR0" ... but it seems there are some problems if h is close to 90° !
-Moreover, they lead to programs that are neither faster nor shorter than "PAR0".
-So I think we can keep "PAR0" if the altitude of the observer may be neglected and "PAR" otherwise...

2°) Equatorial Coordinates

-Similar methods are available if we know the hour angle H and the declination d ( H = Local Sidereal Time minus Right Ascension )
-Their corrected values H' and d' are related to H and d by:

Formulae:

         D' cos d' cos H' = D cos d cos H - a r cos b'
         D' cos d' sin H' = D cos d sin H
              D' sin d'        = D sin d - a r sin b'

tan u = ( 1 - f ) tan b

r sin b' = ( 1 - f ) sin u + ( Alt / 6378137 ) sin b
r cos b' = cos u + ( Alt / 6378137 ) cos b

Data Registers: R00 & R04 thru R07: temp ( Registers R01 thru R03 are to be initialized before executing "EPAR" )

01 LBL "EPAR"
02 DEG
03 X<>Y
04 HR
05 X<>Y
06 HR
07 15
08 *
09 RCL 01
10 XEQ "S-R"
11 STO 05
12 RDN
13 STO 06

14 X<>Y
15 STO 07
16 RCL 02
17 HR
18 STO 04
19 1
20 P-R
21 X<>Y
22 LASTX
23 298.257
24 1/X
25 -
26 STO 00

27 *
28 X<>Y
29 R-P
30 SIGN
31 P-R
32 X<>Y
33 RCL 00
34 *
35 RCL 03
36 6378137
37 /
38 RCL 04
39 X<>Y

40 P-R
41 ST+ T
42 RDN
43 +
44 23454.79
45 ST/ Z
46 /
47 ST- 07
48 X<>Y
49 ST- 05
50 RCL 07
51 RCL 06
52 RCL 05

53 XEQ "R-S"
54 X<> Z
55 HMS
56 X<>Y
57 15
58 /
59 24
60 MOD
61 HMS
62 END

( 106 bytes / SIZE 008 )

STACK	INPUTS	OUTPUTS
Y	d ( ° ' " )	d' ( ° ' " )
X	H ( hh.mnss )	H' ( hh.mnss )

Example: At the Palomar Obsevatory:

    D = 0.003 AU   STO 01
    b = 33°21'22"   STO 02
Alt = 1706 m       STO 03

d = -15°28' H = 19h16m

    -15.28    ENTER^
      19.16   XEQ "EPAR"   >>>>    H' = 19h13m19s02                                       ---Execution time = 7s---
                                          RDN     d' = -15°57'17"00

-With Alt = 0 STO 03 , it yields

H' = 19h13m19s06
d' = -15°57'16"53

Notes:

-Like with the azimuthal coordinates, we can also avoid using the subroutines "S-R" & "R-S" by rewriting the formulae as follows:

Tan H' = ( cos d sin H ) / [ cos d cos H - ( a r / D ) cos b' ]
Tan d' = [ sin d - ( a r / D ) sin b' ] / [ cos² d - 2 ( a r / D ) cos d cos H cos b' + ( a r / D )² cos² b' ]^1/2

-And if Alt = 0 , we have the approximate relations: r = 1 - f sin² b and b' = b - f sin 2b
-The variant hereunder uses these formulae:

Data Registers: R00-R03-R04-R05: temp ( Registers R01 & R02 are to be initialized before executing "EPAR0" )

• R01 = D ( AU )
• R02 = b ( ° ' " )
Flags: /
Subroutines: /

01 LBL "EPAR0"
02 DEG
03 HR
04 15
05 *
06 STO 00
07 X<>Y
08 HR
09 1
10 P-R
11 STO 03
12 CLX
13 23454.79
14 RCL 01
15 *

16 1/X
17 RCL 02
18 HR
19 STO 04
20 SIN
21 X^2
22 298.257
23 STO 05
24 /
25 1
26 X<>Y
27 -
28 *
29 RCL 04
30 ST+ X

31 SIN
32 RCL 05
33 /
34 R-D
35 RCL 04
36 X<>Y
37 -
38 X<>Y
39 P-R
40 RDN
41 -
42 R^
43 STO 04
44 STO 05
45 RCL 00

46 RCL 03
47 P-R
48 ST- 04
49 X<>Y
50 RDN
51 ST+ X
52 -
53 RCL 05
54 *
55 RCL 03
56 X^2
57 +
58 SQRT
59 R-P
60 X<>Y

61 HMS
62 R^
63 RCL 04
64 CHS
65 R-P
66 CLX
67 15
68 /
69 24
70 MOD
71 HMS
72 END

( 101 bytes / SIZE 006 )

STACK	INPUTS	OUTPUTS
Y	d ( ° ' " )	d' ( ° ' " )
X	H ( hh.mnss )	H' ( hh.mnss )

Example:

D = 0.003 AU STO 01
b = 33°21'22" STO 02 ( Alt = 0 )

d = -15°28' H = 19h16m

    -15.28    ENTER^
      19.16   XEQ "EPAR"   >>>>    H' = 19h13m19s06                                       ---Execution time = 6s---
                                          RDN     d' = -15°57'16"52

References:

[1] Jean Meeus - "Astronomical Algorithms" - Willmann-Bell - ISBN 0-943396-61-1
[2] F. Brunnow - "Spherical Astronomy"- New-York D. Van Nostrand - ISBN B002KW46KG ( ? )

hp41programs

Diurnal Parallax for the HP-41