Jacobi Integral (Jacobi Constant)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The Jacobi integral is the only known analytic integral of the Circular Restricted Three-Body Problem (CR3BP) in the synodic frame, given by Jacobi in 1836. Its integration constant $C$ (the Jacobi constant, also written $C_J$ , or $J$ under a different sign convention; see below) reads, in non-dimensional units (distance unit = primary separation, $\omega=1$ ):

C \;=\; 2\Omega(x,y,z) - v^2 \;=\; (x^2+y^2) + \frac{2(1-\mu)}{r_1} + \frac{2\mu}{r_2} - (\dot x^2+\dot y^2+\dot z^2),

where $\mu=m_2/(m_1+m_2)$ is the mass parameter, $r_1$ , $r_2$ are distances to the two primaries, and $v$ is the spacecraft speed in the synodic frame (Vallado 2022, Eq. 12-15; Szebehely 1967, §1.6). Vallado calls it a pseudo-integral: not because the derivation is suspect, but because it exists only in the synodic frame and only for the restricted problem.

The sign of $C$ is opposite to that of energy: larger $C$ means lower total energy and a more confined trajectory; smaller $C$ means higher energy and broader accessibility.

Derivation: the Coriolis term does no work

The CR3BP equations of motion (non-dimensional) read

\ddot{\mathbf r} + 2\,\hat{\mathbf z}\times\dot{\mathbf r} = \nabla\Omega,

where $\nabla\Omega$ collects gravitational and centrifugal terms, while the Coriolis term $2\hat{\mathbf z}\times\dot{\mathbf r}$ is shown explicitly. Dotting both sides with $\dot{\mathbf r}$ :

\dot{\mathbf r}\cdot\ddot{\mathbf r} + \underbrace{\dot{\mathbf r}\cdot(2\hat{\mathbf z}\times\dot{\mathbf r})}_{=0} = \dot{\mathbf r}\cdot\nabla\Omega = \frac{d\Omega}{dt}.

The Coriolis force is always perpendicular to the velocity and does no instantaneous work, so it drops out. The remainder integrates at once to $v^2 = 2\Omega - C$ (Vallado 2022, §12.7.1; Szebehely 1967, §1.6). The deeper reason a conserved quantity exists at all is that the synodic-frame equations are autonomous; in any inertial frame the same derivation fails.

The effective potential $\Omega$ and its many names

$\Omega$ has two parts: the centrifugal potential $\tfrac12(x^2+y^2)$ plus the gravitational potentials $(1-\mu)/r_1 + \mu/r_2$ of the two primaries. Across this family it appears under at least five names, all denoting the same $\Omega$ :

Effective potential: the standard term, emphasizing that, having absorbed the centrifugal contribution, the equations reduce to $\ddot{\mathbf r}+2\hat{\mathbf z}\times\dot{\mathbf r}=\nabla\Omega$ .
Pseudo-potential / pseudopotential: historical, because the centrifugal contribution is not a true gravitational potential.
Effective pseudo-potential: a hybrid of the two.

A common slip is to fold the Coriolis force into $\Omega$ . It is not: the Coriolis force does no work and does not appear in $\Omega$ ; it remains only as a velocity-coupling term in the equations of motion. The level surfaces $\Omega=C/2$ are the zero-velocity surfaces; the stationary points of $\Omega$ give the five libration points.

Notation and sign conventions

At least three notational conventions coexist in the literature; trust the formula, not the symbol:

Symbol	Definition	Direction	Range (Earth-Moon)	Used by
$C$ , $C_J$	$C=2\Omega-v^2$	large = low energy	$[2.988,+\infty)$	Szebehely 1967; Vallado 2022; Parker & Anderson 2014
$J$	$J=\Omega-\tfrac12 v^2=-C/2$	large = high energy	$(-\infty,1.494]$	Mingotti et al. 2011; Sánchez & Yárnoz 2016
$E_{\text{rot}}$	$E_{\text{rot}}=\tfrac12 v^2-\Omega=J$	large = high energy	same as $J$	Scott 2010 (uses C but means $J$ )

Jacobi energy is merely a colloquial name for $C$ or $J$ , not a new concept; this glossary folds it into the present entry.

Critical values $C_i$ and accessible regions

Setting the velocity to zero at the five libration points gives five critical Jacobi constants $C_i = 2\Omega(L_i)$ . Values for the Earth-Moon ( $\mu=0.01215$ ) and Sun-Earth ( $\mu=3.04\times10^{-6}$ ) systems (Parker & Anderson 2014, Table 2-2):

Libration point	Earth-Moon $C_i$	Sun-Earth $C_i$
$L_1$	3.188341	3.000898
$L_2$	3.172161	3.000894
$L_3$	3.012147	3.000003
$L_4=L_5$	2.987997 = $3-\mu(1-\mu)$	2.999997

They satisfy $C_1>C_2>C_3>C_4=C_5$ and stratify spacecraft by energy: $C>C_1$ locks the spacecraft into one of three disconnected regions (near Earth, near Moon, or exterior); $C_2<C<C_1$ opens the $L_1$ neck for Earth-Moon transfers; $C<C_2$ opens $L_2$ for access to deep space; $C\le C_4$ opens the entire space. The topology transitions are detailed under zero-velocity surface.

Engineering applications

Hard constraint on transfer feasibility

$C$ is the first-order criterion for where a spacecraft can reach on dynamics alone (no $\Delta v$ ): to reach the Moon from low Earth orbit one must reduce $C$ below $C_1$ ; to reach beyond $L_2$ one must go below $C_2$ . The art of low-energy cislunar transfer is, in essence, finding the right instant and the smallest $\Delta v$ that takes $C$ from its LEO value (about $3.44$ , Earth-gravity-dominated) to just under $C_1 \approx 3.188$ .

The $\Delta C$ vs $\Delta v$ relation

For an impulsive maneuver at a fixed position ( $\Omega$ unchanged), $C=2\Omega-v^2$ gives at once

\Delta C = -2\,\mathbf v\cdot\Delta\mathbf v.

Corollaries (same direction as the two-body $\Delta E=\mathbf v\cdot\Delta\mathbf v$ ):

$\Delta C$ is opposite in sign to $\mathbf v\cdot\Delta\mathbf v$ : reducing $C$ (raising energy) requires $\Delta v$ along the velocity.
The change in $C$ per unit $|\Delta v|$ is largest where speed is largest, so maneuvers near periapsis are more efficient than near apoapsis.
$|\Delta C|$ is maximized when $\Delta\mathbf v$ is aligned with $\mathbf v$ .

This relation underlies the choice of maneuver points in two- and four-impulse cislunar transfers (Qiao & Yang 2024). Under continuous low-thrust the differential form is $\dot C = -2\,\mathbf a_T\cdot\mathbf v$ , with the same most-efficient-at-high-speed conclusion (Scott 2010, Eq. 5.9, under the sign convention noted above).

The Tisserand parameter: estimating $J$ from orbital elements

Before and after a planetary encounter, a comet's or asteroid's heliocentric elements $(a,e,i)$ are linked to the same $J$ approximately by the Tisserand parameter:

J \;\approx\; \frac{a_p}{a} + 2\cos i\,\sqrt{\frac{a}{a_p}(1-e^2)},

with $a_p$ the perturbing planet's semi-major axis. The relation follows from the Jacobi integral of the restricted problem in the far-encounter limit (Murray & Dermott 1999, §3.4; Sánchez & Yárnoz 2016, Eq. 2). It is used to re-identify the same comet across encounters or to screen asteroids as temporary-capture candidates.

Numerical-integration fidelity check

In CR3BP numerical integration $C$ should be exactly conserved; the drift $|C(t)-C_0|$ is one of the hardest diagnostics of integrator error and is used as a stopping criterion in differential correction (the Jacobi constant error constraint). Trajectory refinement in a high-fidelity ephemeris model (Dei Tos & Topputo 2017) and libration-point stationkeeping (target point method) both grade a solution by the magnitude of $C$ -drift.

Continuation parameter for periodic-orbit families

When tracing families of Halo, Lyapunov, Lissajous or NRHO orbits by numerical continuation, $C$ is the most natural one-parameter continuation variable: one either fixes $C$ and solves for a periodic orbit, or fixes an amplitude and traces $C$ along the family, exposing bifurcations.

References

Szebehely V. Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, 1967, §1.6, Ch. 4.
Vallado D. A. Fundamentals of Astrodynamics and Applications. 5th ed., 2022, §12.7.
Parker J. S., Anderson R. L. Low-Energy Lunar Trajectory Design. JPL, 2014, Ch. 2, Table 2-2.
Murray C. D., Dermott S. F. Solar System Dynamics. Cambridge Univ. Press, 1999, §3.4 (Tisserand criterion).
Sánchez J. P., García Yárnoz D. "Asteroid retrieval missions enabled by invariant manifold dynamics." Acta Astronautica, 2016.
Scott C. J. Transfer and Capture into Distant Retrograde Orbits. Ph.D. thesis, Purdue, 2010, §5.3.
Mingotti G., Topputo F., Bernelli-Zazzera F. "Optimal low-thrust invariant manifold trajectories via attainable sets." JGCD, 2011.
Qiao C., Yang L. Design and optimization of low-energy transfers to Earth-Moon $L_1$ . 2024.