## 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.

### TI 30Xa Algorithms: Greatest Common Divisor

TI 30Xa Algorithms: Greatest Common Divisor To find the greatest common divisor between two positive integers U and V: Let U ≥ V. ...