Interpolated Lunar Apogee and Perigee #

LibEphemeris computes the interpolated (natural) lunar apogee and perigee by smoothing osculating orbital elements to remove the spurious ±30-degree oscillations inherent in the two-body approximation, producing positions that reflect the genuine apsidal motion derived from JPL DE440/DE441 numerical integrations.

Table of Contents #

• Background
  • Apsidal Variants
  • The Problem with Osculating Apsides
• Method
  • The Interpolated Solution
  • Implementation
  • Asymmetric Perturbations
  • Perturbation Series Details
  • Coefficient Calibration
• When to Use Each Variant
  • Decision Matrix
• API Usage
• Precision and Validation
  • Precision Comparison vs pyswisseph
  • Smoothness Comparison
  • Apogee-Perigee Relationship
  • Valid Date Range
• Comparison with Swiss Ephemeris
• References

Background #

Apsidal Variants #

The lunar apsides (apogee and perigee) represent the farthest and nearest points of the Moon’s orbit from Earth. In astrological tradition, the lunar apogee is known as Black Moon Lilith (among other names), while the perigee is sometimes called Priapus or White Moon Selena in some systems.

LibEphemeris provides three variants for each apsidal point:

The Problem with Osculating Apsides #

The osculating apogee is computed from the Moon’s instantaneous position and velocity at any given moment. It represents the apogee of the hypothetical Keplerian ellipse (two-body orbit) that would result if all gravitational perturbations suddenly ceased.

The Moon’s orbit is strongly perturbed by the Sun’s gravity. When osculating orbital elements are computed, a three-body problem (Earth-Moon-Sun) is forced into a two-body framework (Earth-Moon only). This creates significant artifacts.

As established in lunar orbital mechanics (Chapront-Touzé & Chapront, 1988; Meeus, Astronomical Algorithms, ch. 22):

“The solar perturbation results in gigantic monthly oscillations in the ephemeris of the osculating apsides (the amplitude is 30 degrees). These oscillations have to be considered an artifact of the insufficient model, they do not really show a motion of the apsides.”

“The lunar orbit is far from being an ellipse!”

The osculating apogee exhibits the following characteristics:

1. Large oscillations: The osculating apogee oscillates ±30 degrees around the mean apogee position with a period of approximately one synodic month (~29.5 days).
2. Artificial motion: These oscillations do not represent actual motion of the apsidal line. They are mathematical artifacts of the two-body approximation.
3. True twice per month: The osculating apogee only represents the “true” position (where the Moon is actually at maximum or minimum distance) twice per month:
   • When in conjunction with the Moon (Moon at maximum distance)
   • When in opposition to the Moon (Moon at minimum distance from Earth)

Over a typical month, the difference between osculating and mean positions is substantial:

Date          Mean Lilith    True Lilith    Difference
2025-01-01    115.23°        127.45°        +12.22°
2025-01-08    116.01°        98.31°         -17.70°
2025-01-15    116.79°        138.62°        +21.83°
2025-01-22    117.57°        95.24°         -22.33°
2025-01-29    118.35°        131.15°        +12.80°

The True Lilith (osculating) oscillates wildly while Mean Lilith moves smoothly.

Method #

The Interpolated Solution #

The interpolated (or “natural”) apogee seeks a middle ground between:

• The mean apogee: Too smooth; ignores real apsidal variations
• The osculating apogee: Too noisy; dominated by spurious oscillations

The goal is to smooth out the spurious 30-degree oscillations while preserving the genuine apsidal motion.

Analysis of actual lunar apogee and perigee passages (when the Moon is at its farthest/nearest distance from Earth) via numerical integration of JPL DE440 reveals that:

1. Apogee oscillates ~5 degrees from mean position (vs. 30 degrees for osculating)
2. Perigee oscillates ~25 degrees from mean position (due to asymmetric solar effects)
3. Apogee and perigee are not exactly opposite — they deviate from 180 degrees apart, being only roughly opposite when the Sun is in conjunction with one of them or at a 90-degree angle

The interpolated apsides represent the “natural” position of the apsidal line — where the actual lunar distance extremes occur. This is derived by smoothing the osculating elements to remove the artificial oscillations while retaining the physical variations.

Implementation #

LibEphemeris uses two complementary methods for computing the interpolated apsides:

1. Moshier Analytical Method (Apogee): Uses ~50 harmonic terms from Moshier’s lunar ephemeris to compute the smoothed apogee position directly. This approach extracts the dominant periodic terms that affect the apsidal line orientation while filtering out the spurious oscillations.

2. ELP2000-82B Perturbation Series (Perigee): Adds perturbation corrections to the mean perigee position using calibrated coefficients fitted to JPL DE440 reference positions.

Interpolated Apogee (Moshier Method):

1. Calculate Mean Lilith (mean lunar apogee) longitude
2. Calculate Julian centuries from J2000.0
3. Compute fundamental lunar arguments (D, M, M', F, Ω)
4. Apply Moshier harmonic series (~50 terms):
   - Dominant term: +4.53° × sin(2D - 2M')
   - Second-order terms from lunar theory
   - Long-period terms for secular evolution
5. Normalize result to [0°, 360°)

Interpolated Perigee (ELP2000-82B):

1. Calculate Mean Lilith + 180° (mean perigee longitude)
2. Calculate Julian centuries from J2000.0
3. Compute fundamental lunar arguments (D, M, M', F)
4. Apply calibrated perigee perturbation series:
   - Dominant term: -22.2° × sin(2D - 2M')  (opposite sign to apogee!)
   - ~15 additional terms calibrated to JPL DE440 reference positions
5. Normalize result to [0°, 360°)

Asymmetric Perturbations #

The apogee and perigee experience very different perturbation amplitudes:

This asymmetry arises because:

1. Solar perturbations affect the perigee more strongly
2. The perturbation series terms have opposite signs for apogee vs. perigee
3. The Moon spends more time near apogee (slower orbital velocity)

Perturbation Series Details #

The perturbation series uses fundamental lunar arguments:

• D: Mean elongation of Moon from Sun
• M: Sun’s mean anomaly
• M’: Moon’s mean anomaly
• F: Moon’s argument of latitude

Apogee perturbation (excerpt):

# Dominant term (2D - 2M' argument)
delta = 4.5306 * sin(2*D - 2*M')  # degrees

# Additional terms
delta += 0.4193 * sin(2*D - M')
delta += 0.1320 * sin(2*M')
# ... ~10 more terms

Perigee perturbation (calibrated to JPL DE440):

# Dominant term (opposite sign!)
delta = -22.2018 * sin(2*D - 2*M')  # degrees

# Additional terms fitted to JPL DE440 data
delta += 1.5335 * E * sin(2*D - M)
delta += 1.1813 * sin(4*D - 2*M')
# ... ~15 more terms

Coefficient Calibration #

The perigee perturbation coefficients were derived by:

1. Sampling 500 JPL DE440 reference positions across a wide date range
2. Computing the residual (DE440 perigee - mean perigee - 180°)
3. Using least-squares fitting to find optimal coefficients
4. Validating against independent test dates

This approach achieves a computationally efficient analytical formula grounded in the JPL numerical integration.

When to Use Each Variant #

Mean Lilith (SE_MEAN_APOG) #

Use when:

• Smooth, predictable motion is needed
• Historical or traditional astrological work
• Compatibility with older software that only supported mean values
• Short-term oscillations are not relevant

Characteristics:

• Moves at approximately 40.7 degrees/year (prograde)
• No short-term oscillations
• Does not represent actual apsidal position on any given day

True Lilith (SE_OSCU_APOG) #

Use when:

• The instantaneous osculating elements are specifically needed
• Research into lunar orbital mechanics
• Understanding the theoretical two-body apogee

Characteristics:

• Large oscillations (±30 degrees from mean)
• Period of approximately one synodic month
• Only “true” twice per month
• Most of the oscillation is artificial (two-body approximation artifact)

Caution: For most astrological purposes, the osculating apogee is misleading because the 30-degree oscillations do not represent real apsidal motion.

Interpolated Lilith (SE_INTP_APOG) #

Use when:

• The “actual” apogee position (where the Moon reaches maximum distance) is desired
• Modern astrological work seeking physical accuracy
• Real variations without spurious noise are needed
• Transit or timing work related to lunar distance

Characteristics:

• Oscillates approximately ±5 degrees from mean (for apogee)
• Smooth, continuous motion
• Represents the genuine apsidal line orientation
• Available in LibEphemeris and pyswisseph (version 1.70+)

Interpolated Perigee (SE_INTP_PERG) #

Use when:

• The “actual” perigee position (where the Moon reaches minimum distance) is needed
• Supermoon calculations (perigee near Full Moon)
• Studying lunar distance variations

Characteristics:

• Oscillates approximately ±25 degrees from mean (larger than apogee)
• Not exactly opposite to interpolated apogee (can deviate from 180 degrees)
• Represents genuine perigee position

Decision Matrix #

API Usage #

Using swe_calc_ut / swe_calc #

import libephemeris as swe

jd = 2460676.5  # 2025-01-01

# Mean Lilith
mean_pos, _ = swe.swe_calc_ut(jd, swe.SE_MEAN_APOG, swe.SEFLG_SPEED)
print(f"Mean Lilith: {mean_pos[0]:.4f} deg, speed: {mean_pos[3]:.4f} deg/day")

# Osculating (True) Lilith
oscu_pos, _ = swe.swe_calc_ut(jd, swe.SE_OSCU_APOG, swe.SEFLG_SPEED)
print(f"True Lilith: {oscu_pos[0]:.4f} deg, speed: {oscu_pos[3]:.4f} deg/day")

# Interpolated (Natural) Lilith
intp_pos, _ = swe.swe_calc_ut(jd, swe.SE_INTP_APOG, swe.SEFLG_SPEED)
print(f"Interpolated Lilith: {intp_pos[0]:.4f} deg, speed: {intp_pos[3]:.4f} deg/day")

# Interpolated Perigee
perg_pos, _ = swe.swe_calc_ut(jd, swe.SE_INTP_PERG, swe.SEFLG_SPEED)
print(f"Interpolated Perigee: {perg_pos[0]:.4f} deg, speed: {perg_pos[3]:.4f} deg/day")

Using Direct Lunar Functions #

from libephemeris import lunar

jd_tt = 2460676.5  # 2025-01-01 (TT)

# Mean Lilith
mean_lon = lunar.calc_mean_lilith(jd_tt)
print(f"Mean Lilith: {mean_lon:.4f} deg")

# True Lilith (returns longitude, latitude, eccentricity)
oscu_lon, oscu_lat, oscu_ecc = lunar.calc_true_lilith(jd_tt)
print(f"True Lilith: {oscu_lon:.4f} deg, lat: {oscu_lat:.4f} deg, ecc: {oscu_ecc:.5f}")

# Interpolated Apogee
intp_lon, intp_lat, intp_ecc = lunar.calc_interpolated_apogee(jd_tt)
print(f"Interpolated Apogee: {intp_lon:.4f} deg")

# Interpolated Perigee
perg_lon, perg_lat, perg_ecc = lunar.calc_interpolated_perigee(jd_tt)
print(f"Interpolated Perigee: {perg_lon:.4f} deg")

Return Values #

All lunar apside functions return:

• Longitude: Ecliptic longitude in degrees [0, 360)
• Latitude: Ecliptic latitude in degrees (typically small, < 5 degrees)
• Eccentricity/Distance: Orbital eccentricity (~0.055 for Moon)

When using swe_calc_ut with SEFLG_SPEED, velocity is also calculated:

• Speed (longitude): Daily motion in degrees/day
• Speed (latitude): Daily change in latitude
• Speed (distance): Daily change in eccentricity

Precision and Validation #

Precision Comparison vs pyswisseph #

Note on Interpolated Apogee improvement: The Moshier analytical method provides a significant precision improvement over the previous ELP2000-82B series approach. The ~50 harmonic terms capture the dominant periodic variations in the apsidal line more accurately than a smaller calibrated coefficient set.

Note on True Lilith differences: The ~5 degree mean differences arise because:

1. LibEphemeris computes osculating elements from JPL DE state vectors
2. pyswisseph uses integrated analytical lunar theory
3. The osculating apogee concept is inherently model-dependent for strongly perturbed orbits

Note on Interpolated differences: The remaining differences arise from:

1. The Moshier method uses ~50 harmonic terms vs. pyswisseph’s full analytical lunar theory
2. Perigee coefficient calibration was performed on a finite sample of dates
3. Apogee and perigee use different methods optimized for their respective perturbation amplitudes

Smoothness Comparison #

Testing over 100 consecutive days shows variance in daily motion:

The interpolated apogee is approximately 60 times smoother than the osculating.

Apogee-Perigee Relationship #

Both pyswisseph and LibEphemeris compute apogee and perigee using independent perturbation series. This means they are not constrained to be exactly 180° apart.

This is a known property of the interpolated apogee method (Chapront-Touzé & Chapront, 1988; Meeus, Astronomical Algorithms, ch. 22):

“Apogee and perigee are not exactly opposite — they are only roughly opposite when the Sun is in conjunction with one of them or at a 90-degree angle.”

The deviation from 180° can be up to 28 degrees in extreme cases.

This is expected physical behavior, not a limitation.

Valid Date Range #

The analytical perturbation series approach has no inherent date range limitations beyond the validity of the underlying lunar theory.

For practical purposes, the implementation is valid for:

• DE440 (default): 1550–2650 (full precision)
• Extended range: 3000 BCE–3000 CE (reduced precision)

Comparison with Swiss Ephemeris #

The interpolated apogee shows a maximum discrepancy of ~0.36° between LibEphemeris and pyswisseph. This is attributable to differences in the harmonic term sets used: LibEphemeris applies the Moshier ~50-term analytical series for the apogee while pyswisseph uses its full internal analytical lunar theory.

The interpolated perigee shows a larger maximum discrepancy of ~2.6°, reflecting the fundamentally different smoothing philosophies described in interpolated-perigee.md.

For True Lilith (osculating apogee), differences of ~5° arise because the osculating apogee concept is inherently model-dependent for strongly perturbed orbits: LibEphemeris derives osculating elements from JPL DE state vectors while pyswisseph uses integrated analytical lunar theory.

References #

Primary Sources #

1. Chapront-Touzé, M. & Chapront, J. (1988). “ELP 2000-82B: A semi-analytical lunar ephemeris.” Astronomy & Astrophysics, 190, 342-352.
2. Chapront-Touzé, M. & Chapront, J. (1991). Lunar Tables and Programs from 4000 B.C. to A.D. 8000. Willmann-Bell.
3. Meeus, J. (1998). Astronomical Algorithms, 2nd edition. Willmann-Bell, Chapter 47.
4. Brown, E.W. (1896). An Introductory Treatise on the Lunar Theory.
5. Brouwer, D. & Clemence, G.M. (1961). Methods of Celestial Mechanics.

Astrological References #

6. Santoni, Francis (1993). Ephémérides de la lune noire vraie 1910-2010. Editions St. Michel.
7. Koch, Dieter (1995). “Was ist Lilith und welche Ephemeride ist richtig.” Meridian, 1/95.
8. Koch, Dieter & Rindgen, Bernhard (2000). Lilith und Priapus. Frankfurt/Main.

Related Documentation #

• True Lilith Methods — Comparison of True Lilith calculation methods
• Precision — Overall precision limitations of LibEphemeris