The Caledonian symmetrical double binary four-body problem. I: Surfaces of zero-velocity using the energy integral (Q5945036)

The Caledonian symmetrical double-binary problem (CSDBP) is a restricted four-body problem possesing simultaneously a past-future symmetry (presence of a mirror configuration in the sense of \textit{A. E. Roy} and \textit{M. W. Ovenden} [Mon. Not. R. Astron. Soc. 115, 296--309 (1955; Zbl 0067.45802)] and a dynamical symmetry (the presence of a parallelogram configuration) that has been named after the Glasgow Caledonian University in the UK. The initial conditions are first given in terms of initial orbital parameters to derive the formula for mechanical energy \(E=-E_0\), and then the authors use new variables that benefit from the scaling features of \(E_0\): \(\rho _1=E_0r_1\), \(\rho _2=E_0r_2\), \(\rho _{12}=E_0r_{12}\), where \(r_{1,2}\) are the distances of two bodies forming an initial binary from the center of mass of the four-body system, and \(r_{12}\) represents the distance between the two bodies. They read as \(r_{12}=r_{34}=a\), \(r_{13}=r_{24}=2b+a(1-\mu )/(1+\mu )\), \(r_{14}=2(b+a/(1+\mu ))\), \(r_{23}=2(b-\mu a/(1+\mu ))\), \(r_4=r_1=b+a/(1+\mu )\), \(r_3=r_2=b-\mu a/(1+\mu )\), \(V_{34}=V_{12}=V/2\), \(v_4=v_1=v/(1+\mu )\), \(v_3=v_2=\mu v/(1+\mu )\), \(v=(M(1+\mu )((2/a)-(1/a_{12})))^{1/2}\), \(V=\) \((2M(1+\mu )((1/b)-(1/A)))^{1/2}\). Here, \(M\), \(m\) are the masses (double binary!), \(r_i\) are distances of point masses from the center of mass of the four-body system \(C\), \(r_{ij}\) are distances between different point masses, \(V_i\) are velocities with respect to \(C\), \(b\) is the initial distance of the centers of mass of the two binaries (denoted as \(C_{12}\), \(C_{34}\)) from \(C\), \(V_{ij}\) are velocities of \(C_{ij}\) with respect to \(C\), \(v_i\) are relative velocities (velocity of a point mass with respect to the center of mass of the binary), \(V\) is the velocity of \(C_{12}\) with respect to \(C_{34}\), \(v\) is the relative velocity of point masses (velocity of a point mass in a binary with respect to the other point mass), \(a_{12}\) is the initial semimajor axis of the orbit of a point mass in a binary with respect to the other point mass, \(\mu =m/M\) and \(\alpha =a/b\). The kinetic energy reads as \(T=M[(1+\mu )V_{12}^2+2V_{12}(\mu v_1-v_2)\)cos\(i+(\mu v_1^2+v_2^2)]\), where \(i\) is the initial inclination of the orbital plane of a binary to the orbital plane of \(C_{ij}\)s, while the force function is given by \(U=M^2[2\mu (1/r_{12}+1/(2(r_1^2+r_2^2)-r_{12}^2)^{1/2})+(1/2)(1/r_2+\mu ^2/r_1)]\) and has the minimum \(U_{\text{min}}=M^2[4\mu /(r_1^2+r_2^2)^{1/2}+(1/2)(1/r_2+\mu ^2/r_1)]\) for \(r_{12}=(r_1^2+r_2^2)^{1/2}\). The fundamental limitation in range (expressed in a 3-axis rectangular frame \(Or_1r_2r_{12}\)) is \(\left| r_1-r_2\right| \leq r_{12}\leq r_1+r_2\), and a further restriction of the region of possible presence of the four masses reads as \(M^2[2\mu (1/r_{12}+1/(2(r_1^2+r_2^2)-r_{12}^2)^{1/2})+(1/2)(1/r_2+\mu ^2/r_1)]-E_0\geq 0\) or, equivalently, \(M^2[2\mu (1/\rho _{12}+1/(2(\rho _1^2+\rho _2^2)-\rho _{12}^2)^{1/2})+(1/4)(1/\rho _2+\mu ^2/\rho _1)]-1/2\geq 0\) (Section 5). In the case of equal masses (\(m=M=\mu =b=1\), \(a=\alpha \), \(i=0\), \(V^2=2\), \(v^2=2/\alpha \)) the last limitation (that follows from the energy integral), namely \(2(1/r_{12}+1/(2r^2-r_{12}^2)^{1/2})+\sqrt{2}/r-E_0\geq 0\), allows for a precise and extremely relevant description of the region of possible presence. The paper focuses on the boundary of this region (the zero velocity surface), investigating via the energy integral its symmetries and topology (including double points and maximum widths); the region in \(Or_1r_2r_{12}\) consists in four tubes meeting in a transition region near the origin. Among the advantages of such an approach (carefully explained in Section 6) are the dimension of the space in which the zero velocity surface lies (previous techniques involved a five-dimensional space) as well as the lack of cumbersome computations (that occur when the angular momentum is taken into account). Besides the high interest of the topic to researchers in celestial mechanics, the paper is eloquently written, explanations abound in all the Sections, and the reader enjoys his/her trip into the subject.