# ELP2000

by Phill Edwards

## ELP2000 Data

ELP2000 data is stored in 36 data files. The data files each contain components of the lunar longitude, latitude and distance relative to the Earth.

### Arguments

The Delaunay arguments describe aspects of lunar motion:

• l = W1 - W2 is the Moon's mean anomaly, the distance of the mean longitude of the Moon from the mean longitude of its perigee Γ
• l' = T - Π is the Sun's mean anomaly, the distance of the mean longitude of the Sun from the mean longitude of its perigee Γ', where T is the Earth-Moon barycentre mean longitude, Π is the mean longitude of perihelion
• F = W1 - W3 is the Moon's mean argument of latitude, the distance of the mean longitude of the Moon from the mean longitude of its ascending (northward-bound) node Ω
• D = W1 - T + π the Moon's mean (solar) elongation, the distance of the mean longitude of the Moon from the mean longitude of the Sun
Where:
• W1 is the mean longitude of the Moon
• W2 is the mean longitude of the lunar perigee
• W3 is the mean longitude of the lunar ascending node

There are power series for the arguments which fit them to the JPL DE200/LE200 numerical integration data:
W1 = 218°18'59".95571 + 1732559343".73604t - 5".8883t2 + 0".006604t3 - 0".00003169t4
W2 = 83°21'11".67475 + 14643420".2632t - 38".2776t2 - 0".045047t3 + 0".00021301t4
W3 = 125°02'40".39816 - 6967919".3622t + 6".3622t2 + 0".007625t3 - 0".00003586t4
T = 100°27'59".22059 + 129597742".2758t - 0".0202t2 + 0".000009t3 + 0".00000015t4
D = 297°51'00".73512 + 1602961601".4603t - 5".8681t2 + 0".006595t3 - 0".00003184t4
l' = 357°31'44".79306 + 129596581".0474t - 0".5529t2 + 0".000147t3
l = 134°57'48".28096 + 1717915923".4728t + 32".3893t2 + 0".051651t3 - 0".00024470t4
F = 93°16'19".55755 + 1739527263".0983t - 12".2505t2 - 0".001021t3 + 0".00000417t4

Where t is the time in Julian centuries after the J2000 epoch:
t = ( JDE - 2451545.0 ) / 36525.0

### Constants

There are a number of constants and terms in the theory:
ν = 1732559343".18/cy is the sidereal mean motion of the Moon
n' = 129597742".34/cy is the sidereal mean motion of the Sun
α = 0.002571881335 is the ratio of the semi-major axis of the Moon to the sem-major axis of the Earth-Moon barycenter

There are corrective constants which are needed to fit to the JPL DE200/LE200 numerical integration data:
δν = 0".55604
δE = 0".01789
δΓ = -0".08066
δn' = -0".0642
δe' = -0".12879

### Main Problem

The main problem data in files ELP1, ELP2 and ELP3 give the main components of the lunar longitude, latitude and distance respectively. Each record contains the values i1, i2, i3, i4, A, B1, B2, B3, B4, B5, B6

The longitude λ and latitude Β where A is in arc seconds are calculated as: $\Sigma A\mathrm{sin}\left({i}_{1}D+{i}_{2}\mathrm{l\text{'}}+{i}_{3}l+{i}_{4}F\right)$ The distance Δ where A is in kilometers is calculated as: $\Sigma A\mathrm{cos}\left({i}_{1}D+{i}_{2}\mathrm{l\text{'}}+{i}_{3}l+{i}_{4}F\right)$

The A coefficients need to be modified to fit to the JPL DE200/LE200 data.
For longitude and latitude A = A + δA where:

$\mathrm{\delta A}=-m\left({B}_{1}+\frac{\mathrm{2\alpha }}{\mathrm{3m}}{B}_{5}\right)\frac{\mathrm{\delta \nu }}{\nu }+\left({B}_{1}+\frac{\mathrm{2\alpha }}{\mathrm{3m}}{B}_{5}\right)\frac{\mathrm{\delta n\text{'}}}{\nu }+\left({B}_{2}\mathrm{\delta \Gamma }+{B}_{3}\mathrm{\delta E}+{B}_{4}\mathrm{\delta e\text{'}}\right)/206264.81$
For distance:

$A=A+\mathrm{\delta A}-\frac{\mathrm{2A}}{3}\frac{\mathrm{\delta \nu }}{\nu }$
Where:
m = n'/ν
The Bi are derivatives of A and in the same units as A.
The longitude term needs to havethe Moon's mean mean longitude W1 added to it.

### Earth Figure Perturbations

Earth figure perturbations are effects of the Earth's shape on the lunar orbit. These are corrections to the values calculated by the main problem.

The Earth figure data in files ELP4, ELP5 and ELP6 give the perturbations in the lunar longitude, latitude and distance respectively. Files ELP7, ELP8 and ELP9 give the derivative the perturbations in the lunar longitude, latitude and distance respectively which must be multiplied by the time offset from J2000 in Julian centuries. Each record contains the values i1, i2, i3, i4, i5, φ, A, P

Each offset is calculated as: $\Sigma A\mathrm{sin}\left({i}_{1}\zeta +{i}_{2}D+{i}_{3}\mathrm{l\text{'}}+{i}_{4}l+{i}_{5}F+\phi \right)$
Where:
The Delaunay and W1 terms are only calculated to the first power of t.
ζ = W1 + pt
p = 5029".0966/cy the precession constant in J2000

### Planetary Perturbations

The planetary perturbations are the effects of the gravitational attraction of the other planets of the solar system on the Moon.

The planetary perturbation data in files ELP10, ELP11 and ELP12 give the planatary perturbations in the lunar longitude, latitude and distance respectively. The files ELP13, ELP14 and ELP15 give the derivative the perturbations in the lunar longitude, latitude and distance respectively which must be multiplied by the time offset from J2000 in Julian centuries. Each record contains the values i1, i2,i3, i4, i5, i6, i7, i8, i9, i10, i11, φ, A, P

Each offset is calculated as: $\Sigma A\mathrm{sin}\left({i}_{1}{M}_{e}+{i}_{2}V+{i}_{3}T+{i}_{4}{M}_{a}+{i}_{5}J+{i}_{6}S+{i}_{7}U+{i}_{8}N+{i}_{9}D+{i}_{10}l+{i}_{11}F+\phi \right)$ Where:
The Delaunay, T and W1 terms are reduced to the first power of t.
The planetary longitudes, for ELP10, ELP11 and ELP12 and their mean motions ("/cy) for ELP13, ELP14 and ELP15 are derived from the contant and first order times terms from VSOP86.

Similarly, the planetary perturbation data in files ELP116, ELP17 and ELP18 give the planatary perturbations in the lunar longitude, latitude and distance respectively. The files ELP19, ELP20 and ELP21 give the derivative the perturbations in the lunar longitude, latitude and distance respectively which must be multiplied by the time offset from J2000 in Julian centuries. Neptune's longitude is replaced by that of the Sun. Each record contains the values i1, i2,i3, i4, i5, i6, i7, i8, i9, i10, i11, φ, A, P

Each offset is calculated as: $\Sigma A\mathrm{sin}\left({i}_{1}{M}_{e}+{i}_{2}V+{i}_{3}T+{i}_{4}{M}_{a}+{i}_{5}J+{i}_{6}S+{i}_{7}U+{i}_{8}D+{i}_{9}\mathrm{l\text{'}}+{i}_{10}l+{i}_{11}F+\phi \right)$

### Remaining files

The remaining files are:

• ELP22. ELP23, ELP24 Tidal effects longitude, latitude and distance
• ELP25. ELP26, ELP27 Tidal effects longitude, latitude and distance mean motions ("/cy)
• ELP28. ELP29, ELP30 Moon figure perturbations longitude, latitude and distance
• ELP31. ELP32, ELP33 Relatavistic perturbations longitude, latitude and distance
• ELP25. ELP26, ELP27 Planetary perturbations (solar eccentricity) longitude, latitude and distance mean motions2 ("/cy2)

These files have the same format and are calculated in the same way as for the Earth Figure files.