Sunday, December 17, 2023

HP 15C and HP 71B: Precession of the Equinoxes

HP 15C and HP 71B:   Precession of the Equinoxes



How Far Does the Celestial Object Move?  


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'


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:  RTN


New Declination:  δ'


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:  RTN



Variables 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.  


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  RETURN



Examples



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



Eddie


All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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