Formulae and Data for the Stars and Planets

Julian Date,J

Algorithm

if Y<0, Y=Y+1 if M‰2, Y=Y-1 M=M+12

b=0

if D/M/YŠ15/10/1582, a=[Y/100],b=2-a+[a/4]

J=[1461Y/4]+b+D+UT/24 + [306001(M+1)/10000] + 1720994.5

Examples

Inversion of J

Z = [J+0.5], z=[J+0.5-Z],
a = [(Z-1867216.25)/36524.25]
A = (Z<2299161) ? Z : (Z+1+a-[a/4])
B = A + 1524
C = [(B-122.1)/365.25]
D = [365.25C]
E = [(B-D)/30.6001]
DDdddddd = B - D - [30.6001E] + z DD = [DDdddddd]
dddddd = DDdddddd - DD
Hour = [24dddddd]
Min = 60(24dddddd - Hour)
Month = (E<14) ? (E-1) : (E-13)
Year = (Month>2) ? (C-4716):(C-4715)
if Year‰0, Year--

Siderial time

In degrees GAST = GHA‘

To compute GHA‘

• obtain JD for 12:00 on same day - J°
  J°=[J+0.5] - 0.5
• compute T°
  T°=(J°-2451545)/36525
• GMST° = -0.0000062T°³ + 0.093104T°² + 8640184.812866T° + 24110.54841
  
• GMST° = GMST°/3600
  
• GMST = GMST° + 1.00273791UT
• GMST = 15GMST
• d = J - 2449352.5,
  ›έ = -0.0048.sin (241.1-0.053d) - 0.0004.sin (198.9+1.971d),
  GHA‘ = GMST + ›έ.cos “
  

Obliquity, “

Geographical position

• ’ = Declination
  
• GHA = Greenwhich Hour Angle
  = GHA‘-Œ
• Œ = Right Ascension

Altitude & Azimuth

G.P.  H,Z

• x = [cos ’. cos -GHA, cos ’. sin -GHA, sin ’]
  
• Rotate about z-axis thru -long
  x'=Rz(-long).x
  
  Rz(•)=
  [[cos• -sin• 0]
  [sin• cos• 0]
  [0 0 1]]
  
  
• Rotate about y-axis thru 90-lat
  x"=Ry(90-lat).x'
  
  Ry(•)=
  [[cos• 0 -sin• ]
  [0 1 0 ]
  [sin• 0 cos• ]]
  
  
• perform the tsfm:
  
  x'"=-x"
  y'"=y"
  z'"=z"
  
  
• Then
  
  cos H. cos Z = x'"
  cos H. sin Z = y'"
  sin H = z'"
  

Heliocentric coords for the planets

orbital params

R*SD - Semidiameters at unit distance

Computing [x y z]

1. Compute a,e,i,L,š, from a,e,i,L,š,
   
2. M=L-š
   w=š-
   
   
3. Solve Keplers equation for eccentric anomaly, Ζ from:
   
   Ζ - e.sin Ζ = M rads
   
   
4. Coords in plane of orbit are then:
   
   x = a*[cos Ζ - e, ƒ(1-e²)sin Ζ, 0]
   
   v = .0172021ƒa/r*[-sin Ζ, ƒ(1-e²)cos Ζ, 0]
   
   Where r=|x|
   
   
5. Use ,i,w to transform to a fixed Heliocentric coord system.
   1. rotate thru w about z-axis.
      Rz(w) =
      [[cos w -sin w 0]
      [sin w cos w 0]
      [0 0 1]]
      
      
   2. rotate thru i about x-axis
      Rx(i) =
      [[1 0 0 ]
      [0 cos i -sin i]
      [0 sin i cos i]]
      
      
   3. rotate thru  about z-axis
      Rz() =
      [[cos  -sin  0]
      [sin  cos  0]
      [0 0 1]]
      
   
   X=Rz().Rx(i).Rz(w).x
   V=Rz().Rx(i).Rz(w).v
   

Geocentric coords of the Planets

1. Calculate X & V for the Earth for the current time. - Remember to correct for barycentre.
   let E=X, E'=V
   
   
2. Set ™=0.
   
   
3. Calculate X(J-™) for a chosen Planet. In the case of the Sun this will be a zero vector.
   let Q(J-™)=X(J-™)
   J-™ = p-™/2280
   
   
4. Form P=Q(J-™)-E(J)
   
   
5. Compute ™ from:
   c™=P+2΅/c².ln[(E+P+Q)/(E-P+Q)]
   
   Where,
   E=|E(J)|,
   P=|P(J)|,
   Q=|Q(J-™)|
   ΅/c²=0.00000000987 au
   c=173.1446 au/day
   
   
6. Follow steps 3. to 5. using the new ™ to calc. a more accurate ™. If ™ is has converged sufficiently stop iterating.
   
   
7. keeping the most recent values for E=E(J), Q=Q(J-™), & P=Q-E compute the unit vectors:
   e=E/E
   q=Q/Q
   p=P/P
   
   
8. Correct p for gravitational deflection
   
   p=p + 2΅/c²E((p.q)e-(e.p)q)/(1+q.e)
   
   
9. Correct for aberration using:
   
   p=(p/ί + (1+p.V/(1+1/ί))V)/(1+p.V)
   
   Where V=E'/c = .0057755E',
   ί=1/ƒ(1-V²)
   
   
10. Convert p to Geocentric coords by matrix mult. with Rx(“).
    
    
11. Correct for precession by mult. with:
    
    Rz(zA).Ry(•A).Rz(”A)
    
    Where,
    ”A = .6406161T + .0000839T² + .0000050T³,
    •A = .5567530T + -.0001185T² - .0000116T³,
    zA = .6406161T +.0003041T² + .0000051T³.
    
    
12. Correct for nutation.
    
    d = J - 2449352.5,
    ›έ = -.0048 sin (241.1 - .053d) - .0004 sin (198.9 + 1.971d),
    ›“ = .0026 cos (241.1 - .053d) + .0002 cos (198.9 + 1.971d).
    
    rotation matrix,N =
    [[1 -›έcos“ -›έsin“]
    [›έcos“ 1 -›“ ]
    [›έsin“ ›“ 1 ]]
    
    
13. Convert to spherical polar coords, [r ’ Œ]
    
    where
    x = r cos ’ cos Œ,
    y = r cos ’ sin Œ,
    z = r sin ’
    

Geocentric coords of the Sun.

1. Obtain E(J)
2. Q = 0
3. P = -E
4. p = P/P
5. Correct for aberration with the same formula used for the Planets.
6. mult. by Rx(“)
7. correct for precession & nutation.
8. convert to sperical polar coords.

Example
Date: 1/1/1999 @12:00
Planet: Mars
J = 2451180 T = -.009993155373
“ = 23.439420953
p = .517324561404
UT = 12
Earth:
a = 1.00000171565
e = .0167098757517
i = -.000190894597857
L = 1180.71648033
š = 102.985170166
 = 181.403122398
M = 1077.73131016
w = -78.417952232
Kepler:
E = 1077.692767
E = [.982481130768 -.0402523784359 0]
E' = [.0174941737308 .000703610028595 0]
w Rotation:
E = [.157820936952 -.970557560449 0]
E' = [.0042016057623 -.0169966964618 0]
i Rotation:
E = [.157820936952 -.970557560443 .0000032336447252]
E' = [.0042016057623 -.0169966964617 .0000000566285608389]
 Rotation:
E = [-.18153932645 -.966402038091 .0000032336447252]
E' = [-.00461653823249 .0168887168353 .0000000566285608389]
Barycentre correction:
L = -4591.38590007
›E' = [.000410983412134 -.000009943019451 0]
›E = [-.000000754608212 -.0000311908731273 0]
E = [-.181540081058 -.966370847218 .0000032336447252]
E' = [-.00420555482036 .0168787738158 .0000000566285608389]
GHA‘ = 280.69688732
E = 0.983274842246
Mars:
™ = 0:
a = 1.52367319121
e = .0933684157644
i = 1.84988214909
L = 884.179109559
š = 336.056040842
 = 49.5623262866
M = L-š = 548.123068717
w = š- = 286.493714555
Q = [-1.58989550847 .491078700057 .0493707475428]
P = Q-E = [-1.40835542741 -.475292147161 .0493675138981]
P = 1.48721376624
Q = 1.66474126765
 ™ = .00858943213545
™ = .00858943213545:
a = 1.52367319122
e = .0933684155136
i = 1.84988215084
L = 884.1746084
š = 336.05604108
 = 49.5623263336
Q = [-1.58986454062 .491183287029 .0493721773397]
P = [-1.40832445956 -.475187560189 .049368943695]
P = 1.48715106608
Q = 1.66474258997
 ™ = .00858907000946
™ = .00858907000946:
a = 1.52367319122
e = .0933684155136
i = 1.84988215084
L = 884.17460859
š = 336.05604108
 = 49.5623263336
Q = [-1.58986454191 .491183282608 .04937217729]
P = [-1.40832446085 -.47518756461 .0493689436343]
P = 1.48715106872
Q = 1.6647425899
 ™ = .00858907002471
Grav. defl.
e = [-.184628013713 .982808473988 .00000328864787979]
q = [-.955021245661 .295050589556 .0296575444093]
p = [-.946994888732 -.319528778619 .0331969930108]
p = [-.946994892474 -.319528767523 .0331969930655]
aberration
p = [-.947026901759 -.319433888601 .033197264018]
Geocentric coords
p = [-.947026901759 -.306279757148 -.0966063035518]
Precession
p = [-.947104696981 -.306068122538 -.0965143383236]
Nutation
p = [-.947119528467 -.306031616354 -.096484564604]
Polar coords
’ = -5.5367715964
Π= 197.906592189
R = 1.4871510938

The Stars

1. Compute – = –₀ + ΅_–0T; ί = ί₀ + ΅_ί0T
   
   
2. Form p = [cos ί cos –, cos ί sin –, sin ί]
   
   
3. correct for grav. deflection
   
   p = p+2΅/c²(e-(p.e)p)/(1+p.e)
   
   
4. correct for aberration using same formula as for the planets.
   
5. mult. by matrix, Rx(“)
   
6. correct for precession & nutation.
   
7. transform to spherical polar coords.

–, ί for bright stars