---
title: 1:1 Distant Prograde Orbit in Earth–Moon PCR3BP
url: https://www.emergentmind.com/topics/1-1-distant-prograde-orbit
type: topic
---

# 1:1 Distant Prograde Orbit in Earth–Moon PCR3BP

In the Earth–Moon setting, a **1:1 distant prograde orbit** (DPO) is a periodic orbit around the Moon in the planar circular restricted three-body problem (PCR3BP) belonging to the classical \(g\) and \(g'\) families discovered by Hénon in the Hill problem and confirmed in the PCR3BP. The specific 1:1 DPO selected in recent transfer-design work is defined by the period \(T=2\pi\ \text{TU}\) and the symmetry-section initial state \(\bm{X}_0=[x_0,0,0,v_0]^{\text{T}}\), with \(x_0=1.007819412874657\) and \(v_0=1.082615000979063\) [2508.01769]. In this context, “prograde” means that the spacecraft moves around the Moon in the same rotational sense as the Moon’s motion in the rotating frame. These orbits are unstable over a range of Jacobi energies, and that instability is operationally useful because it supports cislunar surveillance, relay / communication architectures, low-energy lunar transfers, and chaining with \(L_1/L_2\) libration-point dynamics [2508.01769].

## 1. Definition and dynamical identity

A DPO is a lunar-centered periodic orbit in the Earth–Moon three-body problem rather than a two-body Keplerian ellipse about the Moon. The 1:1 designation used for the selected orbit refers to the chosen periodic solution with \(T=2\pi\ \text{TU}\), not to a generic descriptive label. The orbit is represented on the symmetry section by
\[
\bm{X}_0=[x_0,0,0,v_0]^{\text{T}},
\]
with the numerical values stated above, and it sits on the \(g\) and \(g'\) branch structure identified in the Earth–Moon PCR3BP [2508.01769].

The orbit is dynamically notable because the relevant DPO family is unstable over part of the Jacobi-energy range. In ordinary mission design language instability is often treated as an impediment, but here the instability is exploited: it enables natural homoclinic/heteroclinic chaining and low-energy access pathways. A plausible implication is that the 1:1 DPO is best understood as a phase-space transport object as much as a parking orbit, since its value lies in how it connects regions of cislunar phase space rather than only in long-term passive residence [2508.01769].

## 2. Earth–Moon PCR3BP formulation

The standard model used for the 1:1 DPO transfer problem is the Earth–Moon PCR3BP. In the rotating frame, the state is
\[
\bm{X}=[x,y,u,v]^{\text{T}},
\]
and the equations of motion are
\[
\left[ \begin{array}{c} \dot x\\ \dot y\\ \dot u\\ \dot v \end{array} \right]
=
\left[ \begin{array}{c}
u\\
v\\
2v + \frac{\partial \Omega_3}{\partial x}\\
-2u + \frac{\partial \Omega_3}{\partial y}
\end{array} \right].
\]
The effective potential is
\[
\Omega_3 = \frac{1}{2}\left(x^2+y^2+\mu(1-\mu)\right) +\frac{1-\mu}{r_1}+\frac{\mu}{r_2},
\]
with
\[
r_1 = \sqrt{(x+\mu)^2+y^2},\qquad r_2 = \sqrt{(x+\mu-1)^2+y^2},
\]
and
\[
\mu = \frac{m_M}{m_E+m_M}.
\]
The Jacobi constant is
\[
C = -(u^2+v^2) + (x^2+y^2) + \frac{2(1-\mu)}{r_1} + \frac{2\mu}{r_2} + \mu(1-\mu)
\]
[2508.01769].

A periodic orbit must satisfy the closure condition
\[
\bm{c}(\bm{X}_0)=\phi_{t_0}^{t_0+T}(\bm{X}_0)-\bm{X}_0=0.
\]
For the selected 1:1 DPO, the initial condition is restricted to the symmetry section \([x_0,0,0,v_0]^{\text{T}}\). The family distribution in \((x_0,C)\) and \((x_0,v_0)\) space shows two branches, \(g\) and \(g'\), and the reported branch structure is unlike the classic Hill-problem bifurcation picture [2508.01769].

## 3. Transfer from circular low Earth orbit

The transfer problem studied for the 1:1 DPO is a **bi-impulsive transfer** from a **167 km circular low Earth orbit** to the selected DPO. The scenario consists of an Earth injection impulse \(\Delta v_i\), a coast arc, and a Moon insertion impulse \(\Delta v_f\), with tangential conditions imposed at both endpoints [2508.01769].

The construction is parameterized at the DPO arrival point by
\[
\bm{y}=[\tau_f,\ \beta_f,\ \text{TOF}]^{\text{T}},
\]
where \(\tau_f\) is the phase along the DPO, \(\beta_f\) is the insertion-point velocity ratio, and TOF is the time of flight. The DPO state at insertion is obtained by propagation along the orbit,
\[
\bm{X}_{\text{DPO}}=\phi_{t_0}^{t_0+\tau_f}(\bm{X}_0),
\]
after which the final transfer state is matched to the DPO position and to a scaled DPO velocity [2508.01769].

The total impulsive cost is
\[
\Delta v=\Delta v_i+\Delta v_f.
\]
A key structural result is that the insertion cost \(\Delta v_f\) decreases substantially as the number of Earth revolutions increases, while \(\Delta v_i\) stays relatively clustered around \(3.12{-}3.15\) km/s. This produces the characteristic time–fuel trade across the transfer families: short transfers are relatively expensive, whereas long multi-revolution transfers reduce total cost primarily through a reduced lunar-end insertion requirement [2508.01769].

## 4. Construction methodology

The computational strategy combines **grid search**, **backward propagation**, **trajectory correction**, and **predictor-corrector continuation**. This is not merely a search for isolated trajectories; it is an explicit attempt to map the transfer solution space for the 1:1 DPO [2508.01769].

The grid search scans
\[
\tau_f \in [0,2\pi],\quad \Delta\tau_f=\pi/5000,
\]
and
\[
\beta_f \in [1,2],\quad \Delta\beta_f=0.0001.
\]
For each candidate insertion state, the trajectory is propagated backward in time for up to
\[
12\pi,
\]
and an initial guess is recorded when the departure constraint is approximately satisfied with
\[
\|\bm{\psi}_i\|<10^{-4}.
\]
Those guesses are then corrected using MATLAB `fsolve` with Levenberg–Marquardt, requiring
\[
\|\bm{\psi}_i\|<5\times 10^{-8},
\]
which is reported to keep the altitude error below \(1\) km [2508.01769].

Continuation is built from a linear predictor in \((\tau_f,\beta_f,\text{TOF})\)-space. Linearization of the departure constraint gives
\[
\left.\frac{\partial \bm{\psi}_i}{\partial \bm{y}}\right|_{\bm{y}=\bm{y}^0}\delta\bm{y}
=\bm{A}\delta\bm{y}
=\bm{0},
\]
with \(\bm{A}\in\mathbb{R}^{2\times 3}\). The feasible direction is obtained by SVD,
\[
\bm{A}=\bm{U}_{2\times 2}\bm{\Sigma}_{2\times 3}\bm{V}_{3\times 3}^{\text{T}}, 
\qquad \delta\bm{y}=\bm{V}_3,
\]
and the predictor step is
\[
\tilde{\bm{y}}^{1}=\bm{y}^0+\delta\bm{y}\Delta s, 
\qquad \Delta s=\pm 10^{-5}.
\]
The continuation process stops when either the corrected solution violates
\[
\|\bm{\psi}_i\|>5\times 10^{-8}
\]
or the step count exceeds \(100000\) [2508.01769].

## 5. Transfer families and mission classes

The reported search identifies **5,663,373 solutions** grouped into **12 transfer families**, labeled **F1–F12**. Most are described as new or previously underexplored. All belong to **interior transfers** in Topputo’s sense, meaning that they do not require the spacecraft to go on very large exterior arcs beyond about \(3\)–\(4\) Earth–Moon distances [2508.01769].

| Family | Geometry | \(\Delta v\), TOF |
|---|---|---|
| F1 | Direct; no complete Earth revolution | \(3.464{-}3.758\) km/s; \(4{-}11\) d |
| F2 | One Earth revolution | \(3.457{-}3.563\) km/s; \(14{-}16\) d |
| F3 | About three Earth revolutions | \(3.404{-}3.509\) km/s; \(38{-}43\) d |
| F4 | About six Earth revolutions | \(3.370{-}3.457\) km/s; \(67{-}69\) d |
| F5 | About five Earth revolutions | \(3.476{-}3.570\) km/s; \(67\) d |
| F6 | About eight Earth revolutions | \(3.427{-}3.445\) km/s; \(86\) d |
| F7 | About eight Earth revolutions; high-altitude lunar flyby | \(3.319{-}3.452\) km/s; \(84{-}87\) d |
| F8 | About nine Earth revolutions | \(3.383{-}3.391\) km/s; \(94{-}95\) d |
| F9 | About eight Earth revolutions | \(3.420{-}3.464\) km/s; \(95\) d |
| F10 | About ten Earth revolutions | \(3.337{-}3.354\) km/s; \(100{-}101\) d |
| F11 | About eleven Earth revolutions | \(3.335{-}3.350\) km/s; \(108{-}112\) d |
| F12 | About eleven Earth revolutions | \(3.386{-}3.435\) km/s; \(109{-}111\) d |

The family structure exposes a clear operational taxonomy. **F1** is a direct transfer with short TOF and high \(\Delta v\), and **F2** remains relatively short while adding one full Earth revolution. These two families are explicitly suggested for **fast transfers / crewed missions**. **F3** and **F5** occupy intermediate regimes: F3 offers lower \(\Delta v\) and lower \(\Delta v_f\) than F1–F2 at the expense of longer flight time, whereas F5 is an alternative medium-energy medium-time solution set [2508.01769].

The longer multi-revolution families dominate the low-energy end of the atlas. **F4** and **F6–F12** are explicitly suggested for **lower fuel-consumption transport missions**. Among them, **F7** is singled out because it contains the **minimum total \(\Delta v\)** found in the study and includes a **high-altitude lunar flyby**, which helps reduce \(\Delta v\). A plausible implication is that the 1:1 DPO is not associated with a single canonical transfer but with a structured design space spanning rapid-access, logistics, and long-horizon cislunar support architectures [2508.01769].

## 6. Model dependence and comparison with earlier constructions

A central methodological issue is whether the transfer geometry is primarily intrinsic to the DPO or strongly dependent on the dynamical model used to construct it. The comparison with **Mingotti et al. (2012)** shows that the answer is the latter: model choice materially affects transfer construction [2508.01769].

The earlier approach computed invariant manifolds of the 1:1 DPO, patched them with Sun–Earth PCR3BP manifolds, and refined the result in the Sun–Earth/Moon PBCR4BP. That construction yielded **single-impulse** transfers with
\[
\Delta v_f=0.
\]
By contrast, the Earth–Moon PCR3BP study finds slightly lower \(\Delta v_i\) but higher \(\Delta v_f\), and therefore slightly worse total \(\Delta v\) than the PBCR4BP solution. The stated reason is that, in the PCR3BP, solar perturbation is absent, so the DPO manifold does not reach close enough to the Earth/LEO region for a single-impulse-like match [2508.01769].

This comparison addresses a common oversimplification in reduced-order cislunar design: a family map in the PCR3BP is not equivalent to a family map in a model that includes solar forcing. The PCR3BP remains useful because it supports global family exploration with reduced complexity, whereas the PBCR4BP can lower insertion cost and enable better capture-like transfer structure. The papers therefore imply complementary rather than interchangeable roles for the two models [2508.01769].

## 7. Terminological scope and related dynamical uses

The expression **“1:1 distant prograde orbit”** sits at the intersection of several established celestial-mechanics vocabularies, and adjacent literatures use the component terms differently. In the cislunar DPO context, the term denotes a periodic orbit around the Moon in the Earth–Moon PCR3BP. In coorbital-resonance work, by contrast, **1:1** refers to mean-motion resonance with a planet, with prograde coorbital modes such as tadpoles and horseshoes and retrograde modes such as R1–R4. That literature finds that retrograde capture is intrinsically more efficient than prograde capture, and that about half the objects cross the coorbital region for any eccentricity and any inclination below \(120^\circ\) [1704.00550]. The shared notation therefore masks different dynamical objects.

A second nearby usage appears in extrasolar three-body dynamics. There, the **1/1 resonance** supports a stable family of symmetric periodic orbits with a **planetary branch** and a **satellite branch**. Under drag, a system starting on the stable planetary branch can migrate along the family and end as a close planet–satellite binary orbiting the star [1105.2713]. This suggests that “1:1” can denote either a coorbital mean-motion resonance or a specific periodic-orbit family, depending on the model and configuration space.

The qualifier **prograde** is likewise context-dependent. In tidal-stripping theory, the main result for a **1:1 prograde coplanar orbit** is that the tidal radius is smaller than for a retrograde orbit, and in the synchronous prograde limit \(\Omega_s/\Omega=1\) the tidal radius reduces to the classical Jacobi form
\[
r_t=x_h\left(\frac{m}{3M}\right)^{1/3}
\]
[1508.03149]. In irregular-satellite dynamics, **S/2016 J2 (Valetudo)** is identified as the most distant prograde satellite around any planet at \(0.36\) Hill radii and is dynamically remarkable because its orbit overlaps Jupiter’s distant retrograde satellites [1809.00700]. In exoplanet spin-orbit work, **CoRoT-18b** is called prograde because the Rossiter–McLaughlin anomaly shows that the planet orbits in the same sense as the stellar rotation, with \(\lambda=-10^\circ\pm20^\circ\) and \(\psi=20^\circ\pm20^\circ\) [1107.2032]. For that reason, the cislunar 1:1 DPO should be read as a specific three-body periodic-orbit term, not as a generic synonym for any distant orbit moving in the same sense as a primary’s rotation or revolution.

Source: https://www.emergentmind.com/topics/1-1-distant-prograde-orbit