New Dissipative Model Preserves Angular Momentum in Point-Mass N-Body Systems

A new mathematical model of celestial motion could change how scientists think about the slow, subtle effects that reshape planetary systems over millions or billions of years. Developed by mathematicians Matheus Lazarotto and Clodoaldo Ragazzo of the University of São Paulo, the model describes a gravitational many-body system in which mechanical energy steadily decreases while total angular momentum remains exactly conserved. That combination is designed to mimic one of the defining features of tidal evolution: internal friction can drain orbital energy, but the angular momentum is transferred within the system rather than simply disappearing. The work, published in Celestial Mechanics and Dynamical Astronomy, offers a new framework for studying how gravitational systems gradually circularize, collide, escape or settle into long-lived configurations.

Most textbook versions of the point-mass N-body problem are conservative. In those models, the total mechanical energy—kinetic plus gravitational potential energy—never changes. A system can exhibit regular or chaotic motion, with bodies orbiting indefinitely, colliding or escaping, but its overall energy remains fixed. Real astronomical systems are not perfectly conservative. Tidal friction inside planets and moons, drag from gas and dust, interactions with plasma and radiative effects can all remove energy from orbital motion. Conventional drag models often represent this loss through a force proportional to velocity, such as (-\alpha \dot{\mathbf r}). Although useful for describing atmospheric or interplanetary drag, such forces generally remove angular momentum as well as energy. The new model instead applies dissipation only along the line joining each pair of bodies, preserving rotational symmetry and therefore the total angular momentum.

The proposed pairwise force is proportional to the relative radial velocity of two masses. In simplified form, it is ( -\alpha Gm_i mj |{\mathbf r}{ij}|^{-d}(\dot{\mathbf r}{ij}\cdot\hat{\mathbf r}{ij})\hat{\mathbf r}{ij}), where (\mathbf r{ij}) is the separation vector, (\hat{\mathbf r}_{ij}) is its unit direction and (d) controls how sharply the effect grows at small distances. The force opposes expansion or contraction but does not resist sideways motion. This distinction is crucial: because the force is central, its torque about the system’s centre of mass is zero. Angular momentum therefore remains unchanged. At the same time, the force does negative work whenever the separation between bodies changes, causing the total energy to obey (dE/dt=-2\mathcal D\leq0), with (\mathcal D) a non-negative Rayleigh dissipation function. Energy loss stops only when every pairwise distance is constant, as in a rigidly rotating relative equilibrium.

The researchers then examined whether this special form of dissipation preserves a particularly important family of gravitational solutions known as central configurations. In a central configuration, the gravitational force on every body points toward the system’s centre of mass in a precise mass-weighted way. In the planar case, the entire arrangement can evolve homographically: its shape remains fixed while the configuration expands or contracts by a scale factor (s(t)) and rotates through an angle (\theta(t)). This includes familiar patterns such as uniformly rotating arrangements and the geometric shapes associated with some idealized multi-body orbits. For the specific exponent (d=3), the dissipative term has the same distance dependence as Newtonian gravity. That symmetry allows the many-body equations restricted to a central configuration to collapse into two equations for (s) and (\theta), mathematically equivalent to a dissipative two-body problem.

This reduction is one of the study’s most significant results. Instead of tracking all positions and velocities of (N) bodies, the researchers can study a single effective radial coordinate and an angular coordinate. The equations have the form (\ddot{s}-s\dot{\theta}^2=\lambda(s^{-2}+\alpha\dot{s}s^{-3})) and (s\ddot{\theta}+2\dot{s}\dot{\theta}=0), where (\lambda) is determined by the original central configuration. The second equation expresses angular-momentum conservation, while the additional radial term represents energy loss during contraction or expansion. This means that conclusions obtained for the dissipative Kepler problem—the idealized two-body gravitational problem—also apply to planar central motions involving any number of point masses, provided the configuration remains on this invariant family.

The two-body analysis reveals a striking phase-space landscape. With the reduced mass scaled to one, the separation (r) and radial velocity (v=\dot r) obey (\dot r=v) and (\dot v=C^2/r^3-\gamma(1/r^2+\alpha v/r^3)), where (C=r^2\dot\theta) is the conserved angular momentum and (\gamma=G(m_1+m_2)). The effective energy is (v^2/2+C^2/(2r^2)-\gamma/r), while the dissipation is (\alpha\gamma v^2/(2r^3)). The system has a circular equilibrium at (r_c=C^2/\gamma), the radius set by angular momentum and gravity. Unlike ordinary linear drag, the radial force does not directly slow angular motion; instead, it removes energy associated with changing separation, gradually driving many initially eccentric trajectories toward a circular orbit.

To map all possible long-term outcomes, the authors used Poincaré compactification, a mathematical technique that brings infinitely distant parts of phase space onto a finite disk. This makes it possible to classify trajectories that begin or end with (r) approaching infinity, rather than treating them as inaccessible boundaries. The resulting topology contains capture orbits, escape orbits, scattering trajectories, bounded paths, ejection solutions and collisions. Weak dissipation can transform the conservative boundary between bound and unbound motion into a more complicated structure. Some bodies arriving from infinity become captured and spiral toward the circular equilibrium; others pass through the system and escape again, carrying away less radial energy than they initially possessed. For sufficiently strong dissipation, the basin of attraction of the circular orbit expands, while scattering regions shrink and eventually disappear at a critical bifurcation.

The model also predicts an unexpected change in the mathematics of close encounters. In the conservative Kepler problem, an orbit with very small angular momentum can approach collision along an almost straight path and, after regularization, rebound with formally infinite speed at the origin. Any nonzero value of the proposed dissipation changes that behaviour qualitatively. When (C=0), the collision state becomes an equilibrium of a regularized system with an attracting centre manifold tangent to (v=-r). Motion along that manifold reaches the collision point with finite speed in the physical variables, rather than undergoing the idealized infinite-velocity bounce. For small but nonzero angular momentum, close approaches are associated with particularly intense energy loss because the dissipative term scales as (r^{-3}). The result is rapid circularization into a small orbit rather than the repeated near-collisions expected from an almost eccentric conservative ellipse.

The researchers stress that the model is mathematical rather than a complete tidal theory. The exponent (d) is treated as a free parameter, although a direct simplification of a tidal prescription by Mignard would suggest (d=8). The exact preservation of central configurations depends on the special (d=3) scaling, and the same invariant structure is not guaranteed for other exponents. Nevertheless, the authors find that several qualitative features of the two-body phase portrait can survive after suitable time rescaling when (d>1). They also compare their construction with earlier dissipative models involving Stokes drag, Poynting–Robertson forces and more general radial terms, emphasizing that those alternatives usually do not conserve angular momentum in the same way.

Finally, an orbital averaging analysis indicates that the dissipation changes the semimajor axis and eccentricity while leaving the argument of periapsis unchanged at the averaged level. In other words, the model can shrink and circularize an orbit without producing a secular periapsis precession. The amount of energy lost per orbit depends on the eccentricity, semimajor axis and the distance exponent controlling the force; stronger distance dependence concentrates dissipation near periapsis, where bodies are closest and radial speeds are largest. The framework therefore offers a compact way to explore how systems evolve through long-lived transients before reaching collision, escape or synchronization. Its broader message is that energy loss alone does not determine a celestial system’s fate: the geometric structure of the dissipative force—and whether it respects angular momentum—can reorganize the entire map of possible futures.