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(15C and 71B) Precession of the Equinoxes - Eddie W. Shore - 11-23-2023 06:33 PM
Given a celestial object's right ascension (α) and declination (δ), we can calculate the new positions after N years by the formulas presented by The Cambridge Handbook of Physics Formulas (see Source):
α' ≈ α0 + (3.075" + 1.336" × sin α0 × tan δ0) × N
δ' ≈ δ0 + (20.043" × cos α0) × N
Decimal degrees of seconds:
3.075" ≈ 854.1677 × 10^-6
1.336" ≈ 371.1111 × 10^-6
20.043" ≈ 5.5675 × 10^-3
According to the Handbook, the formulas are good for only several centuries, as the formulas are local approximations.
HP 15C Code: New Right Ascension (LBL A) and New Declination (LBL D)
Step: Key Code: Key
New Right Ascension, α'
Code:
001: 42, 21, 11: LBL A
002: __, 43, _7: DEG
003: __, __, _1: 1
004: __, __, _3: 3
005: __, __, _3: 3
006: __, __, 48: .
007: __, __, _6: 6
008: __, __, 26: EEX
009: __, __, _6: 6
010: __, __, 16: CHS
011: __, 43, _2: →H
012: __, 45, _1: RCL 1
013: __, 43, _2: →H
014: __, __, 23: SIN
015: __, __, 20: ×
016: __, 45, _2: RCL 2
017: __, 43, _2: →H
018: __, __, 25: TAN
019: __, __, 20: ×
020: __, __, _3: 3
021: __, __, _0: 0
022: __, __, _7: 7
023: __, __, 48: .
024: __, __, _5: 5
025: __, __, 26: EEX
026: __, __, _6: 6
027: __, __, 16: CHS
028: __, 43, _2: →H
029: __, __, 40: +
030: 45, 20, _3: RCL× 3
031: __, 45, _1: RCL 1
032: __, 43, _2: →H
033: __, __, 40: +
034: __, 42, _2: →H.MS
035: __, 44, _4: STO 4
036: __, 43, 32: RTNNew Declination: δ'
Code:
037: 42, 21, 14: LBL B
038: __, 43, _7: DEG
039: __, __, _2: 2
040: __, __, 48: .
041: __, __, _0: 0
042: __, __, _0: 0
043: __, __, _4: 4
044: __, __, _3: 3
045: __, __, 26: EEX
046: __, __, _3: 3
047: __, __, 16: CHS
048: __, 43, _2: →H
049: __, 45, _1: RCL 1
050: __, 43, _2: →H
051: __, __, 24: COS
052: __, __, 20: ×
053: 45, 20, _3: RCL× 3
054: __, 45, _2: RCL 2
055: __, 43, _2: →H
056: __, __, 40: +
057: __, 42, _2: →H.MS
058: __, 44, _5: STO 5
059: __, 43, 32: RTNVariables Used:
R1 = α0: Initial Right Ascension (enter in DD.MMSS format)
R2 = δ0: Initial Declination (enter in DD.MMSS format)
R3 = N: Number of Years from 2000.
Outputs:
R4 = α': Final Right Ascension (in DD.MMSS format)
R5 = δ': Final Declination (in DD.MMSS format)
HP 71B Code: PRECES
Note: I had battery problems with the HP 71B, so I'm writing this code from written notes.
Code:
100 DEGREES
110 PRINT "PRECESSION" @ WAIT 0.25
120 PRINT "EPOCH J2000.0" @ WAIT 0.25
130 INPUT "R.A. °,M,S ?"; H,M,S
140 GOSUB 500 @ A0 = X
150 INPUT "DEC °,M,S? "; H,M,S
160 GOSUB 500 @ D0 = X
170 INPUT "# YEARS? "; N
180 A1 = A0 + (854.1667E-6 + 371.1111E-6 * SIN(A0) * TAN(D0)) * N
190 D1 = D0 + (5.5675E-3 * COS(A0)) * N
200 X = A1 @ GOSUB 600
210 H1 = H @ M1 = M @ S1 = S @ G1 = G
220 PRINT "R.A. ADJ=" @ WAIT 0.25
230 PRINT G1*H1; "°"; M1; "m"; S1; "s" @ PAUSE
240 X = D1 @ GOSUB 600
250 H2 = H @ M2 = M @ S2 = S @ G2 = G
260 PRINT "DEC ADJ=" @ WAIT 0.25
270 PRINT G2*H2; "°"; M2; "m"; S2; "s"
280 END
500 X=SGN(H) * (ABS(H) + M/60 + S/3600)
510 RETURN
600 G = SGN(X) @ X=ABS(X) @ H=IP(X)
610 M = IP(FP(X) * 60)
620 S = FP(FP(X) * 60) * 60
630 RETURNExamples
First Point of Aries (Vernal Equinox)
α0 = 0° 00' 00"
δ0 = 0° 00' 00"
N = 100
α' ≈ 5'08" (0.0508)
δ' ≈ 33'24" (0.3324)
Regulus (Alpha Leonis (Leo))
α0 ≈ 5° 55' 10"
δ0 ≈ 7° 24' 25"
N = 100
α' ≈ 6° 00' 19"
δ' ≈ 7° 57' 39"
Sagittarius A* (Center of the Milky Way Galaxy)
α0 ≈ 17° 45' 40"
δ0 ≈ -29° 00' 28"
N = 100
α' ≈ 17° 50' 25"
δ' ≈ -28° 28' 39"
Betelgeuse (Alpha Orionis (Orion))
α0 ≈ 10° 28' 22"
δ0 ≈ 11° 58' 02"
N = 100
α' ≈ 10° 13' 34"
δ' ≈ 12° 30' 55"
Source
Woan, Gaham. The Cambridge Handbook of Physics Formulas 2003 Edition. Cambridge University Press. 2000. ISBN 978-0-511-07589-6