## 3 Synthesis and Analysis of Planar SALT Trajectories
### The Quadrant Rule  
Researchers numerically designing Sun-assisted low-energy transfers are familiar with the property that the apogee of a SALT trajectory is located in either the second or the fourth quadrant of the Sun-Earth rotating reference frame centered at the Earth, as shown in Figure 4. Below, we explain this feature analytically. In the patched three-body approximation, the perturbation due to the Sun's gravity can be estimated by analyzing the increment of the Jacobi integral of the Earth-Moon three-body system along the exterior leg of a trajectory. Differentiating (4) in the BR4 BP dynamics gives  
\[\frac{dI}{dt}=2\left(\frac{\partial\Omega_{3}}{\partial x}\,\dot{x}+\frac{ \partial\Omega_{3}}{\partial y}\,\dot{y}+\frac{\partial\Omega_{3}}{\partial z }\,z-\dot{x}\dot{x}-\dot{y}\dot{y}-\dot{z}\dot{z}\right)=-2\left(\frac{ \partial\Omega_{5}}{\partial x}+\frac{\partial\Omega_{5}}{\partial y}+\frac{ \partial\Omega_{5}}{\partial z}\right)\]  
or, equivalently,  
\[\frac{dJ}{dt}=2\left(\frac{\partial\Omega_{5}}{\partial\theta_{S}}\frac{d \theta_{S}}{dt}-\frac{d\Omega_{5}}{dt}\right).\]  
Hence, we have  
\[\Delta J=-2\Delta\Omega_{5}+2\int\frac{\partial\Omega_{5}}{\partial\theta_{S} }\,d\theta_{S}. \tag{9}\]  
Figure 3: Earth–Moon region of prevalence boundaries for different Sun phase angles and the elliptical boundary of the mean-square averaged region of prevalence (bold orange). It is convenient to write \(\Omega_{S}\) and its derivative \(\partial\Omega_{S}/\partial\theta_{S}\) in the Sun-barycenter frame with the \(x^{\prime}\)-axis directed along the line connecting the Sun and the Earth-Moon barycenter (Figure 5). Note that we retain the prime notation for the axes of this frame: the only difference from the Sun-Earth rotating reference frame of Figure 4 is a slight displacement of the origin from the Earth's center to the Earth-Moon barycenter. If we introduce the polar coordinates \(x^{\prime}=\rho\cos\alpha,y^{\prime}=\rho\sin\alpha\) in the \(Cx^{\prime}y^{\prime}\) plane and make use of the relations \(r^{2}-z^{2}=x^{2}+y^{2}=x^{\prime 2}+y^{\prime 2}=\rho^{2},\ x=-\rho\cos( \theta_{S}+\alpha),\ y=-\rho\sin(\theta_{S}+\alpha)\), Equation (6) is simplified to  
\[r_{3}=\sqrt{L^{2}+r^{2}+2L\rho\cos\alpha}.\]  
Substituting in Equation (5) and taking into account \(\rho/L\leq r/L\ll 1\) yields  
\[\Omega_{S}=\frac{Gm_{S}}{\sqrt{L^{2}+r^{2}+2L\rho\cos\alpha}}+\frac{Gm_{S}}{L ^{2}}\,\rho\cos\alpha\approx\frac{Gm_{S}}{L}-\frac{Gm_{S}}{2L^{3}}\,r^{2}+ \frac{3 Gm_{S}}{2L^{3}}\,\rho^{2}\cos^{2}\alpha.\]  
Finally, the partial derivative of \(\Omega_{S}\) with respect to \(\theta_{S}\) equals minus the partial derivative with respect to \(\alpha\):  
\[\frac{\partial\Omega_{S}}{\partial\theta_{S}}=-\frac{\partial\Omega_{S}}{ \partial\alpha}\approx\frac{3}{2}\frac{Gm_{S}}{L^{3}}\,\rho^{2}\sin 2\alpha. \tag{10}\]  
To reduce the spacecraft's orbital energy and ensure ballistic capture, the Sun's gravitational perturbation should lead to the positive increment of the Earth-Moon Jacobi integral. The predominant contribution to such an increment is due to the integral term in Equation (9): the spacecraft spends many weeks near the apogee of a SALT trajectory where \(\rho^{2}\simeq 10\). In order for the integral  
\[\int\frac{\partial\Omega_{S}}{\partial\theta_{S}}\,d\theta_{S}=\int\frac{ \partial\Omega_{S}}{\partial\theta_{S}}\,\omega_{S}\,dt\]  
Figure 4: Examples of planar SALT trajectories in the BR4 BP with the departing (red), arriving (green) and exterior (black) legs patched on the boundary of the Earth–Moon region of prevalence. The orbit of the Moon is marked by the dashed gray line. The apogee of every SALT trajectory in the Earth-centered Sun–Earth rotating frame is located in the 2 nd or 4 th quadrant. to be positive, the partial derivative (10) should be negative. This is equivalent to the inequality \(\sin 2\alpha<0\) valid for either the second or the fourth quadrant of the Sun-barycenter reference frame.