Functions: Lagrange Points

Functions for computing Lagrange point equilibrium positions in the Circular Restricted Three-Body Problem (CR3BP). Lagrange points are the five positions where a small body can maintain a stable (or quasi-stable) position relative to two larger bodies. L1, L2, and L3 are collinear (unstable), while L4 and L5 form equilateral triangles with the two primaries (stable for mass ratios below the Routh critical value). All functions in this section are IMMUTABLE STRICT PARALLEL SAFE.

Tip

point_id values: 1=L1, 2=L2, 3=L3, 4=L4, 5=L5. Use lagrange_point_name(point_id) to get the human-readable label. See Body ID Reference for planet and moon body_id values.

---

lagrange_heliocentric

Heliocentric ecliptic J2000 position of a Sun-planet Lagrange point. The CR3BP quintic solver finds the equilibrium position in the co-rotating frame, which is then rotated into the inertial ecliptic frame using VSOP87 planetary positions.

Signature

lagrange_heliocentric(body_id int4, point_id int4, t timestamptz) → heliocentric

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1=Mercury, 2=Venus, 3=Earth, 4=Mars, 5=Jupiter, 6=Saturn, 7=Uranus, 8=Neptune
point_id	int4	Lagrange point: 1=L1, 2=L2, 3=L3, 4=L4, 5=L5
t	timestamptz	Evaluation time

Returns

A heliocentric position in AU (ecliptic J2000 frame).

Example

-- Sun-Earth L1: SOHO lives here (~0.97 AU from Sun)
SELECT round(helio_distance(lagrange_heliocentric(3, 1, '2000-01-01 12:00:00+00'))::numeric, 2) AS sun_dist_au;
-- -> 0.97

-- All planets' L1 distances from the Sun
SELECT body_id,
       round(helio_distance(lagrange_heliocentric(body_id, 1, '2000-01-01 12:00:00+00'))::numeric, 4) AS l1_dist_au
FROM generate_series(1, 8) AS body_id;

---

lagrange_observe

Observe a Sun-planet Lagrange point from a ground station. Computes the heliocentric Lagrange position, subtracts Earth’s geocentric position, and transforms to topocentric azimuth/elevation.

Signature

lagrange_observe(body_id int4, point_id int4, obs observer, t timestamptz) → topocentric

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)
point_id	int4	Lagrange point: 1-5 (L1-L5)
obs	observer	Observer location on Earth
t	timestamptz	Observation time

Returns

A topocentric with azimuth, elevation (degrees), range (km), and range rate (km/s).

Example

-- Where is the Sun-Earth L2 (JWST's home) from Greenwich?
SELECT round(topo_azimuth(t)::numeric, 2) AS az,
       round(topo_elevation(t)::numeric, 2) AS el
FROM lagrange_observe(3, 2, '51.4769N 0.0005W 0m'::observer, now()) AS t;

---

lagrange_equatorial

Geocentric RA/Dec of a Sun-planet Lagrange point. Converts the heliocentric ecliptic position to geocentric equatorial coordinates, precessed to the date of observation.

Signature

lagrange_equatorial(body_id int4, point_id int4, t timestamptz) → equatorial

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

An equatorial with RA (hours), Dec (degrees), and distance (km).

Example

-- Sun-Earth L2 sky position (near the anti-solar point)
SELECT round(eq_ra(lagrange_equatorial(3, 2, now()))::numeric, 4) AS ra_hours,
       round(eq_dec(lagrange_equatorial(3, 2, now()))::numeric, 4) AS dec_deg;

---

lagrange_distance

Distance (AU) from a heliocentric position to a Sun-planet Lagrange point. Computes the Lagrange point position at the given time and returns the Euclidean distance.

Signature

lagrange_distance(body_id int4, point_id int4, pos heliocentric, t timestamptz) → float8

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)
point_id	int4	Lagrange point: 1-5 (L1-L5)
pos	heliocentric	Heliocentric position to measure from
t	timestamptz	Evaluation time (determines the Lagrange point position)

Returns

Distance in AU from the given position to the Lagrange point.

Caution

This function measures the distance from any heliocentric position to a Lagrange point. Useful for identifying Trojan asteroids near L4/L5 or objects temporarily captured near L1/L2.

Example

-- Self-test: distance from Jupiter L4 to itself
SELECT round(lagrange_distance(
    5, 4,
    lagrange_heliocentric(5, 4, '2000-01-01 12:00:00+00'),
    '2000-01-01 12:00:00+00'
)::numeric, 10) AS self_distance;
-- -> 0.0000000000

---

lagrange_distance_oe

Distance (AU) from an asteroid or comet (specified by orbital_elements) to a Sun-planet Lagrange point. Propagates the small body’s orbit to the given time via Keplerian mechanics, then measures the distance to the computed Lagrange position.

Signature

lagrange_distance_oe(body_id int4, point_id int4, oe orbital_elements, t timestamptz) → float8

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)
point_id	int4	Lagrange point: 1-5 (L1-L5)
oe	orbital_elements	Orbital elements for the asteroid or comet
t	timestamptz	Evaluation time

Returns

Distance in AU.

Example

-- Check if (588) Achilles is near Jupiter's L4 (Trojan territory)
SELECT round(lagrange_distance_oe(
    5, 4,
    oe_from_mpc('00588 14.39 0.15 K249V  41.50128  169.10254  334.19917   13.04512  0.0760428  0.22963720   5.1763803  0 MPO752723  4285  88 1992-2024 0.49 M-v 30h MPCW       0000      (588) Achilles          20240913'),
    '2024-06-21 12:00:00+00'
)::numeric, 4) AS dist_au;

---

lunar_lagrange_observe

Observe an Earth-Moon Lagrange point from a ground station. The Earth-Moon system is implied --- no body_id is needed. The Moon’s position is computed via ELP2000-82B.

Signature

lunar_lagrange_observe(point_id int4, obs observer, t timestamptz) → topocentric

Parameters

Parameter	Type	Description
point_id	int4	Lagrange point: 1-5 (L1-L5)
obs	observer	Observer location on Earth
t	timestamptz	Observation time

Returns

A topocentric with azimuth, elevation (degrees), range (km), and range rate (km/s).

Example

-- Earth-Moon L1 from Boulder (Artemis Gateway territory)
SELECT round(topo_elevation(lunar_lagrange_observe(1, '40.0N 105.3W 1655m'::observer, now()))::numeric, 2) AS el_deg;

---

lunar_lagrange_equatorial

Geocentric RA/Dec of an Earth-Moon Lagrange point. The L1 point lies between Earth and Moon at roughly 84% of the Earth-Moon distance.

Signature

lunar_lagrange_equatorial(point_id int4, t timestamptz) → equatorial

Parameters

Parameter	Type	Description
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

An equatorial with RA (hours), Dec (degrees), and distance (km).

Example

-- Earth-Moon L1 distance (~326,000 km from Earth)
SELECT round(eq_distance(lunar_lagrange_equatorial(1, '2000-01-01 12:00:00+00'))::numeric, 0) AS dist_km;

---

galilean_lagrange_observe

Observe a Jupiter-Galilean moon Lagrange point from a ground station. Uses L1.2 theory (Lieske 1998) for the Galilean moon position and VSOP87 for Jupiter’s heliocentric position.

Signature

galilean_lagrange_observe(body_id int4, point_id int4, obs observer, t timestamptz) → topocentric

Parameters

Parameter	Type	Description
body_id	int4	Galilean moon: 0=Io, 1=Europa, 2=Ganymede, 3=Callisto
point_id	int4	Lagrange point: 1-5 (L1-L5)
obs	observer	Observer location on Earth
t	timestamptz	Observation time

Returns

A topocentric with azimuth, elevation (degrees), range (km), and range rate (km/s).

Example

-- Jupiter-Ganymede L3 from Greenwich
SELECT round(topo_elevation(galilean_lagrange_observe(2, 3, '51.4769N 0.0005W 0m'::observer, now()))::numeric, 2) AS el_deg;

---

galilean_lagrange_equatorial

Geocentric RA/Dec of a Jupiter-Galilean moon Lagrange point.

Signature

galilean_lagrange_equatorial(body_id int4, point_id int4, t timestamptz) → equatorial

Parameters

Parameter	Type	Description
body_id	int4	Galilean moon: 0=Io, 1=Europa, 2=Ganymede, 3=Callisto
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

An equatorial with RA (hours), Dec (degrees), and distance (km).

Example

-- Jupiter-Io L4 sky position
SELECT round(eq_ra(galilean_lagrange_equatorial(0, 4, now()))::numeric, 4) AS ra_hours,
       round(eq_dec(galilean_lagrange_equatorial(0, 4, now()))::numeric, 4) AS dec_deg;

---

saturn_moon_lagrange_observe

Observe a Saturn moon Lagrange point from a ground station. Uses TASS17 theory (Vienne & Duriez 1995) for the moon position and VSOP87 for Saturn’s heliocentric position.

Signature

saturn_moon_lagrange_observe(body_id int4, point_id int4, obs observer, t timestamptz) → topocentric

Parameters

Parameter	Type	Description
body_id	int4	Saturn moon: 0=Mimas, 1=Enceladus, 2=Tethys, 3=Dione, 4=Rhea, 5=Titan, 6=Iapetus, 7=Hyperion
point_id	int4	Lagrange point: 1-5 (L1-L5)
obs	observer	Observer location on Earth
t	timestamptz	Observation time

Returns

A topocentric with azimuth, elevation (degrees), range (km), and range rate (km/s).

Example

-- Saturn-Titan L1 from Boulder
SELECT round(topo_azimuth(t)::numeric, 2) AS az,
       round(topo_elevation(t)::numeric, 2) AS el
FROM saturn_moon_lagrange_observe(5, 1, '40.0N 105.3W 1655m'::observer, now()) AS t;

---

saturn_moon_lagrange_equatorial

Geocentric RA/Dec of a Saturn moon Lagrange point.

Signature

saturn_moon_lagrange_equatorial(body_id int4, point_id int4, t timestamptz) → equatorial

Parameters

Parameter	Type	Description
body_id	int4	Saturn moon: 0=Mimas, 1=Enceladus, 2=Tethys, 3=Dione, 4=Rhea, 5=Titan, 6=Iapetus, 7=Hyperion
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

An equatorial with RA (hours), Dec (degrees), and distance (km).

Example

-- Saturn-Titan L1 sky position
SELECT round(eq_ra(saturn_moon_lagrange_equatorial(5, 1, now()))::numeric, 4) AS ra_hours;

---

uranus_moon_lagrange_observe

Observe a Uranus moon Lagrange point from a ground station. Uses GUST86 theory (Laskar & Jacobson 1987) for the moon position and VSOP87 for Uranus’s heliocentric position.

Signature

uranus_moon_lagrange_observe(body_id int4, point_id int4, obs observer, t timestamptz) → topocentric

Parameters

Parameter	Type	Description
body_id	int4	Uranus moon: 0=Miranda, 1=Ariel, 2=Umbriel, 3=Titania, 4=Oberon
point_id	int4	Lagrange point: 1-5 (L1-L5)
obs	observer	Observer location on Earth
t	timestamptz	Observation time

Returns

A topocentric with azimuth, elevation (degrees), range (km), and range rate (km/s).

Example

-- Uranus-Titania L2 from Greenwich
SELECT round(topo_elevation(uranus_moon_lagrange_observe(3, 2, '51.4769N 0.0005W 0m'::observer, now()))::numeric, 2) AS el_deg;

---

uranus_moon_lagrange_equatorial

Geocentric RA/Dec of a Uranus moon Lagrange point.

Signature

uranus_moon_lagrange_equatorial(body_id int4, point_id int4, t timestamptz) → equatorial

Parameters

Parameter	Type	Description
body_id	int4	Uranus moon: 0=Miranda, 1=Ariel, 2=Umbriel, 3=Titania, 4=Oberon
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

An equatorial with RA (hours), Dec (degrees), and distance (km).

Example

-- Uranus-Oberon L4 sky position
SELECT round(eq_ra(uranus_moon_lagrange_equatorial(4, 4, now()))::numeric, 4) AS ra_hours,
       round(eq_dec(uranus_moon_lagrange_equatorial(4, 4, now()))::numeric, 4) AS dec_deg;

---

mars_moon_lagrange_observe

Observe a Mars moon Lagrange point from a ground station. Uses MarsSat theory (Jacobson 2014) for the moon position and VSOP87 for Mars’s heliocentric position.

Signature

mars_moon_lagrange_observe(body_id int4, point_id int4, obs observer, t timestamptz) → topocentric

Parameters

Parameter	Type	Description
body_id	int4	Mars moon: 0=Phobos, 1=Deimos
point_id	int4	Lagrange point: 1-5 (L1-L5)
obs	observer	Observer location on Earth
t	timestamptz	Observation time

Returns

A topocentric with azimuth, elevation (degrees), range (km), and range rate (km/s).

Example

-- Mars-Phobos L1 from Boulder
SELECT round(topo_azimuth(t)::numeric, 2) AS az,
       round(topo_elevation(t)::numeric, 2) AS el
FROM mars_moon_lagrange_observe(0, 1, '40.0N 105.3W 1655m'::observer, now()) AS t;

---

mars_moon_lagrange_equatorial

Geocentric RA/Dec of a Mars moon Lagrange point.

Signature

mars_moon_lagrange_equatorial(body_id int4, point_id int4, t timestamptz) → equatorial

Parameters

Parameter	Type	Description
body_id	int4	Mars moon: 0=Phobos, 1=Deimos
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

An equatorial with RA (hours), Dec (degrees), and distance (km).

Example

-- Mars-Deimos L5 sky position
SELECT round(eq_ra(mars_moon_lagrange_equatorial(1, 5, now()))::numeric, 4) AS ra_hours,
       round(eq_dec(mars_moon_lagrange_equatorial(1, 5, now()))::numeric, 4) AS dec_deg;

---

hill_radius

Hill sphere radius (AU) for a Sun-planet system. The Hill sphere is the region where a planet’s gravity dominates over the Sun’s --- objects beyond this radius are more strongly influenced by the Sun. Computed as r_H = a * (m_p / (3 * m_sun))^(1/3), where a is the instantaneous Sun-planet distance from VSOP87.

Signature

hill_radius(body_id int4, t timestamptz) → float8

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)
t	timestamptz	Evaluation time

Returns

Hill sphere radius in AU.

Example

-- Jupiter's Hill sphere (~0.35 AU)
SELECT round(hill_radius(5, '2000-01-01 12:00:00+00')::numeric, 3) AS jupiter_hill_au;

-- All planets
SELECT body_id,
       round(hill_radius(body_id, '2000-01-01 12:00:00+00')::numeric, 4) AS hill_au
FROM generate_series(1, 8) AS body_id;

---

hill_radius_lunar

Hill sphere radius (AU) for the Earth-Moon system. Much smaller than planetary Hill spheres since the Moon is far less massive relative to Earth than planets are relative to the Sun.

Signature

hill_radius_lunar(t timestamptz) → float8

Parameters

Parameter	Type	Description
t	timestamptz	Evaluation time

Returns

Hill sphere radius in AU.

Example

SELECT hill_radius_lunar('2000-01-01 12:00:00+00') AS lunar_hill_au;

---

lagrange_zone_radius

Approximate libration zone radius (AU) around a Sun-planet Lagrange point. For L4/L5, this is related to the tadpole/horseshoe orbit domain where Trojan asteroids can remain trapped. For collinear points (L1/L2/L3), it is the linearized stability boundary.

Signature

lagrange_zone_radius(body_id int4, point_id int4, t timestamptz) → float8

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)
point_id	int4	Lagrange point: 1-5 (L1-L5)
t	timestamptz	Evaluation time

Returns

Zone radius in AU.

Example

-- Jupiter L4 libration zone (Trojan swarm extent)
SELECT round(lagrange_zone_radius(5, 4, '2000-01-01 12:00:00+00')::numeric, 4) AS zone_au;

---

lagrange_mass_ratio

CR3BP mass parameter mu = M_planet / (M_sun + M_planet). A diagnostic function useful for verifying the CR3BP solver or understanding the relative gravitational influence of a planet in its system.

Signature

lagrange_mass_ratio(body_id int4) → float8

Parameters

Parameter	Type	Description
body_id	int4	Planet identifier: 1-8 (Mercury-Neptune)

Note

No timestamp parameter --- mass ratios are compiled-in constants derived from IAU standard gravitational parameters.

Returns

Dimensionless mass ratio (small positive number; Jupiter is ~0.001, Earth is ~0.000003).

Example

SELECT lagrange_mass_ratio(5) AS jupiter_mu,
       lagrange_mass_ratio(3) AS earth_mu;

---

lagrange_point_name

Human-readable name for a Lagrange point ID. A simple lookup that converts integer IDs to their standard labels.

Signature

lagrange_point_name(point_id int4) → text

Parameters

Parameter	Type	Description
point_id	int4	Lagrange point: 1-5

Returns

Text label: 'L1', 'L2', 'L3', 'L4', or 'L5'.

Example

SELECT lagrange_point_name(1) AS name;
-- -> 'L1'

-- Use in a query for readable output
SELECT lagrange_point_name(p) AS point,
       round(helio_distance(lagrange_heliocentric(3, p, now()))::numeric, 4) AS sun_dist_au
FROM generate_series(1, 5) AS p;