Zero-Velocity Surfaces

Consider the surface

$$V(x,y,z) = C,$$ (1089)

where

$$V = - 2\,U = \frac{2\,\mu_1}{\rho_1} + \frac{2\,\mu_2}{\rho_2} +x^2+y^2.$$ (1090)

Note that $V\geq 0$. It follows, from Equation (1064), that if the mass $m_3$ has the Jacobi integral $C$, and lies on the surface specified in Equation (1089), then it must have zero velocity. Hence, such a surface is termed a zero-velocity surface. The zero-velocity surfaces are important because they form the boundary of regions from which the mass $m_3$ is dynamically excluded: i.e., regions in which $V< C$. Generally speaking, the regions from which $m_3$ is excluded grow in area as $C$ increases, and vice versa.

Let $C_i$ be the value of $V$ at the $L_i$ Lagrange point, for $i=1,5$. When $\mu_2\ll 1$, it is easily demonstrated that

$$C_1 \simeq 3 + 3^{4/3}\,\mu_2^{\,2/3}-10\,\mu_2/3,$$ (1091)

$$C_2 \simeq 3 + 3^{4/3}\,\mu_2^{\,2/3}-14\,\mu_2/3,$$ (1092)

$$C_3 \simeq 3 + \mu_2,$$ (1093)

$$C_4 \simeq 3 - \mu_2,$$ (1094)

$$C_4 \simeq 3 - \mu_2.$$ (1095)

Note that $C_1>C_2>C_3>C_4=C_5$.

Figure 50: The zero-velocity surface $V=C$, where $C> C_1$, calculated for $\mu _2=0.1$. The mass $m_3$ is excluded from the region lying between the two inner curves and the outer curve.

Figure 51: The zero-velocity surface $V=C$, where $C= C_1$, calculated for $\mu _2=0.1$. The mass $m_3$ is excluded from the region lying between the inner and outer curves.

Figure 52: The zero-velocity surface $V=C$, where $C= C_2$, calculated for $\mu _2=0.1$. The mass $m_3$ is excluded from the region lying between the inner and outer curve.

Figure 53: The zero-velocity surface $V=C$, where $C =C_3$, calculated for $\mu _2=0.1$. The mass $m_3$ is excluded from the regions lying inside the curve.

Figure 54: The zero-velocity surface $V=C$, where $C_4< C < C_3$, calculated for $\mu _2=0.1$. The mass $m_3$ is excluded from the regions lying inside the two curves.

Figures 50-54 show the intersection of the zero-velocity surface $V=C$ with the $x$-$y$ plane for various different values of $C$, and illustrate how the region from which $m_3$ is dynamically excluded--which we shall term the excluded region--evolves as the value of $C$ is varied. Of course, any point not in the excluded region is in the so-called allowed region. For $C> C_1$, the allowed region consists of two separate oval regions centered on $m_1$ and $m_2$, respectively, plus an outer region which lies beyond a large circle centered on the origin. All three allowed regions are separated from one another by an excluded region--see Figure 50. When $C= C_1$, the two inner allowed regions merge at the $L_1$ point--see Figure 51. When $C= C_2$, the inner and outer allowed regions merge at the $L_2$ point, forming a horseshoe-like excluded region--see Figure 52. When $C =C_3$, the excluded region splits in two at the $L_3$ point--see Figure 53. For $C_4< C < C_3$, the two excluded regions are localized about the $L_4$ and $L_5$ points--see Figure 54. Finally, for $C < C_4$, there is no excluded region.

Figure 55: The zero-velocity surfaces and Lagrange points calculated for $\mu _2=0.01$.

Figure 55 shows the zero-velocity surfaces and Lagrange points calculated for the case $\mu _2=0.01$. It can be seen that, at very small values of $\mu _2$, the $L_1$ and $L_2$ Lagrange points are almost equidistant from mass $m_2$. Furthermore, mass $m_2$, and the $L_3$, $L_4$, and $L_5$ Lagrange points all lie approximately on a unit circle, centered on mass $m_1$. It follows that, when $\mu _2$ is small, the Lagrange points $L_3$, $L_4$ and $L_5$ all share the orbit of mass $m_2$ about $m_1$ (in the inertial frame) with $C_3$ being directly opposite $m_2$, $L_4$ (by convention) $60^\circ$ ahead of $m_2$, and $L_5$ $60^\circ$ behind.