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We test numerically the recently proposed linear relationship between thescale-invariant period $T_{\rm s.i.} = T |E|^{3/2}$, and the topology of anorbit, on several hundred planar Newtonian periodic three-body orbits. Here $T$is the period of an orbit, $E$ is its energy, so that $T_{\rm s.i.}$ is thescale-invariant (s.i.) period, or, equivalently, the period at unit energy $|E|= 1$. All of these orbits have vanishing angular momentum and pass through alinear, equidistant configuration at least once. Such orbits are classified inten algebraically well-defined sequences. Orbits in each sequence follow anapproximate linear dependence of $T_{\rm s.i.}$, albeit with slightly differentslopes and intercepts. The orbit with the shortest period in its sequence iscalled the "progenitor": six distinct orbits are the progenitors of these tensequences. We have studied linear stability of these orbits, with the resultthat 21 orbits are linearly stable, which includes all of the progenitors. Thisis consistent with the Birkhoff-Lewis theorem, which implies existence ofinfinitely many periodic orbits for each stable progenitor, and in this wayexplains the existence and ensures infinite extension of each sequence.

Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler's third law: a numerical test