# Optimal Escape from Sun-Earth and Earth-Moon L2 with Electric Propulsion

^{*}

^{†}

## Abstract

**:**

_{3}is high. As applications of such Lagrangian Point trajectories, results include considerations about escape maneuvers from different SEL2 high-fidelity Lyapunov orbits and escape for interplanetary trajectories towards near-earth asteroids.

## 1. Introduction

_{3}(${C}_{3,f}$) below 0.5 (km/s)${}^{2}$; these values are suitable, for instance, for missions to NEAs. These kinds of trajectories are the focus of the present paper, which aims at an initial evaluation of escape from SEL2 and EML2 for an electric propulsion (EP) spacecraft.

## 2. Dynamic Model

**r**, velocity

**V**, and mass, m, of the spacecraft, and are ruled by the following differential equations:

**T**. The available power and thrust are inversely proportional to the squared distance from the Sun, assuming efficiency, ${\eta}_{T}$, and effective exhaust velocity, c, as constants—$T=2({\eta}_{T}/c){P}_{1}/{R}_{s}^{2}={T}_{1}/{R}_{s}^{2}$—where ${P}_{1}$ and ${T}_{1}$ are the values at 1 AU and ${R}_{s}$ is expressed in AU.

**I**,

**J**,

**K**are unit vectors along the reference axes of EME2000. Precession and nutation are neglected. Right ascension $\vartheta $ and declination $\phi $ are introduced along with the radius magnitude r to write the position vector as $\mathit{r}=rcos\vartheta cos\phi \mathit{I}+rsin\vartheta cos\phi \mathit{J}+rsin\phi \mathit{K}$. The topocentric RF, identified by unit vectors

**𝚤**(radial),

**𝚥**(eastward), and

**k**(northward) is introduced. The unit vectors are defined by the following:

#### Lunisolar Effect

## 3. Optimization

_{3}may be added to the boundary conditions. Thrust (magnitude, constrained between 0 and the maximum available value, and direction) is the control variable. Adjoint variables $\mathbf{\lambda}$ are coupled to the state differential equations and the Hamiltonian is defined as $H={\mathbf{\lambda}}_{r}^{T}\mathit{V}+{\mathbf{\lambda}}_{V}^{T}(-\mu \mathit{r}/{r}^{3}+\mathit{T}/m+{\mathit{a}}_{J}+{\mathit{a}}_{\ell}+{\mathit{a}}_{s})-{\lambda}_{m}T/c$. Euler–Lagrange equations provide differential equations for the adjoint variables; they are presented in Appendix A, and additional details can be found in Ref. [15]. The Hamiltonian must be maximized by the optimal controls, i.e., thrust magnitude and direction, in agreement with Pontryagin’s maximum principle (PMP). The thrust

**T**must be parallel to the adjoint vector ${\mathbf{\lambda}}_{V}$, also named the primer vector. The Hamiltonian is rewritten as $H={\mathbf{\lambda}}_{r}^{T}\mathit{V}+{\mathit{\lambda}}_{V}^{T}(-\mu \mathit{r}/{r}^{3}+{\mathit{a}}_{J}+{\mathit{a}}_{\ell}+{\mathit{a}}_{s})+T({\lambda}_{V}/m-{\lambda}_{m}/c)$, and is linear with respect to T; a bang–bang control arises and the thrust must be maximized, $T={T}_{max}$, when the switching function ${S}_{F}={\lambda}_{V}/m-{\lambda}_{m}/c>0$, while the thrust shall be null $T=0$, when ${S}_{F}<0$. Singular arcs, usually associated with atmospheric flight, are here excluded.

**q**are assumed for the unknowns (switching/final times and unknown adjoint initial values) and the error on the boundary conditions is determined after integration. Each i-th component of

**q**is perturbed by a small quantity $\delta {q}_{i}$ (e.g., ${10}^{-7}$) and the variation of the error $\Delta \mathit{\chi}$ is evaluated, again after integration. The unknowns are corrected at each iteration, aiming at nullifying the errors by assuming linear behavior and evaluating numerically the error–gradient matrix $\partial \mathit{\chi}/\partial \mathit{q}=\Delta \mathit{\chi}/\partial \mathit{q}$, so that

_{3}, etc.).

_{3}, may also be specified. It follows that the same geocentric C

_{3}can produce different heliocentric energies based on the escape direction, which is here not constrained. The Results Section will present some observations about the heliocentric escape conditions, but a detailed analysis is left for future studies.

## 4. Escape Trajectories from Sun-Earth L2

^{3}/s

^{2}is Earth’s gravitational parameter. The normalized gravitational parameter of the Earth is equal to one.

_{3}equal to 0.2 and 0.5 (km/s)${}^{2}$, with no constraints on the escape time will be considered.

#### 4.1. Sun-Earth L2 Escapes with Constrained Final Time

_{3}values are lowest. The spacecraft is relatively far from the Moon, and Moon’s gravity acts mainly on the Earth. The initial geometry on the reference departure date shows that the Moon reduces the spacecraft’s geocentric energy by pulling the Earth; this perturbation modifies the Sun-Earth–spacecraft geometry so that the Sun’s pull on the spacecraft is increased, and a lower propulsive effort is needed. The opposite happens after half the lunar period. The final value of C

_{3}depends on the energy gain provided by thrusting and the overall effect of the perturbing bodies, causing oscillations in the trajectory performance.

#### 4.2. Sun-Earth L2 Escapes with Constrained Final Energy

_{3}could be sought. If the final C

_{3}value is constrained, then the escape trajectory is modified to change the overall effect of Sun’s pull. Two case studies have been analyzed, by imposing the final C

_{3}equal to 0.2 and 0.5 (km/s)${}^{2}$.

_{3}have been found one-by-one starting from a set of tentative solutions built on user experience (initial thrust aligned with velocity, thrust–coast structure). The other solutions have been found similarly or via an ad hoc continuation method. A gradual variation is imposed, starting from the free final C

_{3}solution towards the constrained one (or from the lower to the higher constrained final C

_{3}) or using the closest available solution in terms of departure date as a tentative guess. It can happen that a forward search from a certain date and a backward search from a later date do not end up to the same point; therefore, for the same departure date, solutions belonging to different “families” are found.

_{3}but with different costs and mission durations.

_{3}value, additional time constraints) may change the scenario.

## 5. Escape Trajectories from Earth-Moon L2

_{3}fixed at a lower (0.2 (km/s)${}^{2}$) and higher (0.5 (km/s)${}^{2}$) value with free time to escape.

#### 5.1. Earth-Moon L2 Escapes with Constrained Final Time

#### 5.2. Earth-Moon L2 Escapes with Constrained Final Energy

_{3}case assume the two-burn T–C–T–C structure when constrained to ${C}_{3}=0.2$ (km/s)${}^{2}$, especially those belonging to the second family. Moreover, a peculiarity to be observed, clearly shown in the corresponding Table 5 and Table 6, is that low-energy and high-energy fixed solutions, for selected studies, tend to invert the optimal thrust structure from 2 to 4 phases and vice versa.

_{3}. This increases the overall length of the trajectory with a strong reduction in the first burn phase, showing that the optimal strategy relies mainly on the solar perturbation to achieve the desired energy. Indeed, the majority of the escape, after the initial 30 days, remains in a region in which the Sun perturbation has a negative effect, retarding the energy increase.

## 6. Applications

_{3}are considered. The initial orbits are characterized by their Jacobi constant and start from the x-axis at about $200\times {10}^{3}$ km, $400\times {10}^{3}$ km, and $800\times {10}^{3}$ km from L2. The solution starting from L2 is used as a tentative solution for departures from Lyapunov orbits and convergence is readily obtained. It is worth noting that the new solutions can, in turn, be used as tentative guesses for other studies (e.g., for the optimization of departure point and/or escape duration). Table 7 and Figure 21 compare the escape trajectories from L2 and from the three Lyapunov orbits. The escape transfers show similar propulsive requirements. Escape C

_{3}tends to be smaller for the larger Lyapunov orbits; this value is however sensibly influenced by the escape duration and different control structures (e.g., a second thrust arc) could eventually become optimal for constrained C

_{3}values.

_{3}and departure on 15 October 2025, is again taken as an example. The small C

_{3}value suggests that this trajectory can be fit for transfers to NEAs and a set of 75 asteroids considered in previous works [36,37] is explored. The escape direction has a specific orientation in the Sun-Earth system (see Figure 6), which should be suited to reach only the targets that require a similar orbit correction at the selected departure.

_{3}= 0 (km/s)${}^{2}$ on 13 January 2026 (i.e., 15 October 2025, +90 days) is assumed as the reference trajectory. The length of the heliocentric transfer to rendezvous with the asteroid is 3 years. The reference solutions are compared to trajectories with departure from the actual heliocentric position and velocity corresponding to the 90-day escape trajectory departing 15 October 2025.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BVP | boundary value problem |

DSG | Deep Space Gateway |

EME2000 | Earth Mean Equator and Equinox of Epoch J2000 |

EML2 | Earth-Moon Lagrangian point 2 |

EP | electric propulsion |

ICRF | International Celestial Reference Frame |

LOP-G | Lunar Orbital Platform Gateway |

LP(s) | Lagrangian point(s) |

MOID | minimum orbit intersection distance |

NEA(s) | near-earth asteroid(s) |

NRHO(s) | near rectilinear halo orbit(s) |

PMP | Pontryagin’s maximum principle |

RF | reference frame |

SEL2 | Sun-Earth Lagrangian point 2 |

SOI | sphere of influence |

SRP | solar radiation pressure |

Nomenclature | |

a | semi-major axis, AU |

$\mathrm{a}$ | perturbing acceleration |

c | effective exhaust velocity, km/s |

${C}_{3}$ | characteristic energy, (km/s)${}^{2}$ |

H | Hamiltonian |

m | mass, kg |

p | radiation pressure, N/m^{2} |

P | Power, W |

q | unknowns tentative values |

r | radius |

R | relative distance |

${S}_{F}$ | switching function |

t | time |

T | thrust, N |

u | velocity, radial component, km/s |

$\widehat{u}$ | unit vector |

v | velocity, tangential component, km/s |

V | velocity, km/s |

w | velocity, out-of-plane component, km/s |

$\eta $ | normalized solar perturbation contribution |

${\eta}_{R}$ | reflectivity |

${\eta}_{T}$ | thruster global efficiency |

$\Delta \vartheta $ | Sun–spacecraft relative angular difference, deg |

$\vartheta $ | right ascension, deg |

$\lambda $ | adjoint variables |

$\mu $ | specific gravitational parameter, km^{3}/s^{2} |

$\xi $ | proportionality term |

$\phi $ | declination, deg |

$\chi $ | error on boundary conditions |

Subscripts | |

0 | initial |

f | final |

𝚤 | radial direction |

𝚥 | tangential direction |

J | non-sphericity perturbation |

ℓ | lunar |

$lsg$ | lunisolar gravity perturbation |

s | sun |

$srp$ | solar radiation pressure perturbation |

## Appendix A. Euler–Lagrange Equations

^{2}Assuming reflectivity ${\eta}_{R}=0.7$, the acceleration on a spherical body of mass m and surface S at a distance from the Sun ${R}_{s}$ is

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**Figure 3.**SEL2 escape trajectories over a lunar month, fixed ${t}_{f}=90$ days, free ${C}_{3}$, EME2000 RF.

**Figure 9.**SEL2 escape trajectories over a lunar month—fixed ${C}_{3}=0.2$ (km/s)${}^{2}$ (

**left**), 0.5 (km/s)${}^{2}$ (

**right**), free ${t}_{f}$, EME2000 RF.

**Figure 10.**Influence of departure date to escape from SEL2—families of solutions—fixed ${C}_{3}=0.2$ (km/s)${}^{2}$, free ${t}_{f}$.

**Figure 12.**SEL2 selected escapes—spacecraft semi-major axis in the heliocentric ecliptic RF—fixed ${C}_{3}=0.2$ (km/s)${}^{2}$ (

**left**), 0.5 (km/s)${}^{2}$ (

**right**) −$2.5\times {t}_{f}$ propagation.

**Figure 14.**Influence of departure date to escape from EML2—families of solutions—fixed ${t}_{f}=75$ days, free ${C}_{3}$.

**Figure 16.**EML2 selected escape trajectories (

**left**), departure detailed view (

**right**), Sun-Earth synodic RF.

**Figure 18.**EML2 escape trajectories over a lunar month—fixed ${C}_{3}=0.2$ (km/s)${}^{2}$ (

**left**), 0.5 (km/s)${}^{2}$ (

**right**), free ${t}_{f}$, EME2000 RF.

**Figure 19.**Time evolution of Sun–spacecraft relative angle for all EML2 solutions—fixed ${C}_{3}=0.2$ (km/s)${}^{2}$ (

**left**), 0.5 (km/s)${}^{2}$ (

**right**), free ${t}_{f}$.

**Figure 20.**Spacecraft energy over time (

**left**) and Sun–spacecraft relative angle (

**right**) for selected EML2 escape dates—comparison between free and fixed final ${C}_{3}$.

**Figure 21.**Performance comparison for escapes from L2 and Lyapunov orbits—Sun-Earth rotating RF (

**left**), energy over time (

**right**)—free C

_{3}, fixed ${t}_{f}$ 90 days.

C | f | p | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{u}}$ | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{v}}$ | $\overline{{\mathit{\eta}}_{\mathit{S}\mathit{P}}}$ | $\mathbf{\Delta}\mathit{V}$ m/s | ${\mathit{C}}_{3,\mathit{f}}$ (km/s) ^{2} |
---|---|---|---|---|---|---|---|

1 | I | 2 | 0.975 | 0.208 | 0.592 | 30.676 | 0.1833 |

2 | I | 2 | 0.979 | 0.174 | 0.576 | 36.419 | 0.1931 |

3 | I | 2 | 0.978 | 0.178 | 0.578 | 55.369 | 0.2050 |

4 | I | 2 | 0.974 | 0.217 | 0.596 | 54.982 | 0.1956 |

5 | I | 2 | 0.976 | 0.210 | 0.593 | 33.572 | 0.1828 |

**Table 2.**SEL2 escapes performance—fixed ${C}_{3}=0.2.{(\mathrm{km}/\mathrm{s})}^{2}$, free ${t}_{f}$.

C | f | p | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{u}}$ | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{v}}$ | $\overline{{\mathit{\eta}}_{\mathit{S}\mathit{P}}}$ | $\Delta \mathit{V}$ m/s | ${\mathit{t}}_{\mathit{f}}$ days |
---|---|---|---|---|---|---|---|

1 | I | 2 | 0.782 | 0.160 | 0.471 | 22.155 | 95.92 |

2 | II | 2 | 0.842 | 0.162 | 0.502 | 19.416 | 103.08 |

3 | II | 2 | 0.795 | 0.181 | 0.488 | 46.462 | 97.78 |

1 | III | 2 | 0.980 | 0.193 | 0.586 | 1.346 | 119.75 |

2 | III | 2 | 0.965 | 0.164 | 0.564 | 8.153 | 117.87 |

3 | III | 2 | 0.914 | 0.158 | 0.536 | 30.253 | 111.72 |

4 | III | 2 | 0.841 | 0.173 | 0.507 | 37.757 | 103.07 |

5 | III | 2 | 0.796 | 0.162 | 0.479 | 23.155 | 97.44 |

**Table 3.**SEL2 escapes performance—fixed ${C}_{3}=0.5.{(\mathrm{km}/\mathrm{s})}^{2}$, free ${t}_{f}$.

C | f | p | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{u}}$ | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{v}}$ | $\overline{{\mathit{\eta}}_{\mathit{S}\mathit{P}}}$ | $\Delta \mathit{V}$ m/s | ${\mathit{t}}_{\mathit{f}}$ days |
---|---|---|---|---|---|---|---|

1 | I | 3 | 0.805 | 0.199 | 0.502 | 220.205 | 98.60 |

2 | I | 3 | 0.983 | 0.173 | 0.578 | 222.191 | 119.15 |

3 | I | 3 | 0.898 | 0.237 | 0.568 | 242.771 | 109.95 |

4 | I | 3 | 0.822 | 0.253 | 0.537 | 246.049 | 101.40 |

5 | I | 3 | 0.817 | 0.205 | 0.511 | 221.774 | 99.83 |

C | f | p | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{u}}$ | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{v}}$ | $\overline{{\mathit{\eta}}_{\mathit{S}\mathit{P}}}$ | $\Delta \mathit{V}$ m/s | ${\mathit{C}}_{3,\mathit{f}}$ (km/s) ^{2} |
---|---|---|---|---|---|---|---|

1 | I | 2 | 0.833 | 0.064 | 0.448 | 48.763 | 0.3510 |

2 | I | 4 | 0.783 | −0.429 | 0.177 | 61.183 | 0.1523 |

3 | I | 2 | 0.503 | −0.712 | −0.105 | 161.279 | 0.0026 |

3 | II | 2 | 0.434 | 0.750 | 0.592 | 20.476 | 0.4789 |

3 | III | 2 | 0.805 | −0.106 | 0.349 | 17.412 | 0.3108 |

4 | III | 2 | 0.801 | −0.376 | 0.212 | 38.572 | 0.1821 |

5 | I | 2 | 0.775 | 0.302 | 0.538 | 51.373 | 0.4430 |

**Table 5.**EML2 escapes performance—fixed ${C}_{3}=0.2.{(\mathrm{km}/\mathrm{s})}^{2}$, free ${t}_{f}$.

C | f | p | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{u}}$ | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{v}}$ | $\overline{{\mathit{\eta}}_{\mathit{S}\mathit{P}}}$ | $\Delta \mathit{V}$ m/s | ${\mathit{t}}_{\mathit{f}}$ days |
---|---|---|---|---|---|---|---|

1 | I | 2 | 0.702 | −0.274 | 0.214 | 46.297 | 78.59 |

2 | I | 4 | 0.711 | −0.195 | 0.258 | 66.200 | 75.12 |

3 | II | 4 | 0.190 | 0.542 | 0.366 | 73.777 | 88.05 |

3 | III | 2 | 0.715 | −0.241 | 0.237 | 17.354 | 80.81 |

4 | III | 2 | 0.678 | −0.295 | 0.192 | 39.099 | 73.93 |

5 | I | 2 | 0.659 | −0.314 | 0.172 | 49.933 | 78.43 |

**Table 6.**EML2 escapes performance—fixed ${C}_{3}=0.5{(\mathrm{km}/\mathrm{s})}^{2}$, free ${t}_{f}$.

C | f | p | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{u}}$ | ${\mathit{\eta}}_{\mathit{S}\mathit{P},\mathit{v}}$ | $\overline{{\mathit{\eta}}_{\mathit{S}\mathit{P}}}$ | $\Delta \mathit{V}$ m/s | ${\mathit{t}}_{\mathit{f}}$ days |
---|---|---|---|---|---|---|---|

1 | I | 4 | 0.695 | 0.541 | 0.618 | 74.853 | 75.01 |

2 | I | 2 | 0.632 | 0.301 | 0.467 | 147.635 | 61.40 |

3 | I | 2 | 0.449 | 0.713 | 0.581 | 19.934 | 72.74 |

4 | I | 2 | 0.654 | 0.308 | 0.481 | 77.468 | 62.79 |

5 | I | 4 | 0.683 | 0.539 | 0.611 | 64.702 | 76.59 |

**Table 7.**Comparison of escapes performance for escape from L2 and Lyapunov orbits—free C

_{3}fixed ${t}_{f}$ 90 days.

Departure | JC (km/s) ^{2} | $\Delta \mathit{V}$ m/s | ${\mathit{C}}_{3,\mathit{f}}$ (km/s) ^{2} |
---|---|---|---|

L2 | 3.000887 | 30.676 | 0.1833 |

Lyapunov 1 | 3.000880 | 37.607 | 0.1298 |

Lyapunov 2 | 3.000858 | 34.916 | 0.0836 |

Lyapunov 3 | 3.000776 | 32.869 | 0.0263 |

Asteroid | Reference kg | 90-Day Escape kg | % Saving |
---|---|---|---|

2016 TB57 | 111.1 | 76.5 | 31.2 |

2013 XY20 | 82.4 | 59.7 | 27.5 |

2016 CF137 | 110.5 | 82.5 | 25.3 |

2017 BF29 | 88.9 | 66.8 | 24.8 |

2011 AA37 | 64.1 | 53.1 | 17.2 |

2012 BA35 | 185.7 | 152.8 | 17.7 |

2007 DD | 185.3 | 152.6 | 17.6 |

2017 HK1 | 106.8 | 88.6 | 17.1 |

2015 TJ1 | 86.0 | 73.6 | 14.4 |

2014 EK24 | 80.0 | 69.6 | 13.0 |

2017 HZ4 | 123.4 | 108.9 | 11.8 |

2006 FH36 | 143.9 | 126.8 | 11.9 |

2015 BM510 | 88.6 | 78.8 | 11.0 |

2010 HA | 76.5 | 68.3 | 10.8 |

2009 OS5 | 77.7 | 69.4 | 10.7 |

1996 XB27 | 70.3 | 63.6 | 9.6 |

2016 FY2 | 106.3 | 97.0 | 8.8 |

2015 VV | 75.2 | 70.0 | 6.9 |

2001 QJ142 | 82.7 | 79.6 | 3.8 |

2013 EM89 | 66.3 | 64.9 | 2.1 |

2013 RV9 | 103.7 | 102.3 | 1.4 |

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**MDPI and ACS Style**

Mascolo, L.; Casalino, L.
Optimal Escape from Sun-Earth and Earth-Moon L2 with Electric Propulsion. *Aerospace* **2022**, *9*, 186.
https://doi.org/10.3390/aerospace9040186

**AMA Style**

Mascolo L, Casalino L.
Optimal Escape from Sun-Earth and Earth-Moon L2 with Electric Propulsion. *Aerospace*. 2022; 9(4):186.
https://doi.org/10.3390/aerospace9040186

**Chicago/Turabian Style**

Mascolo, Luigi, and Lorenzo Casalino.
2022. "Optimal Escape from Sun-Earth and Earth-Moon L2 with Electric Propulsion" *Aerospace* 9, no. 4: 186.
https://doi.org/10.3390/aerospace9040186