Theoretical Prediction of a Shepherd Planet at the Kuiper Cliff Based on the Space Flow Hydrodynamic Model (v_L)

Title: Theoretical Prediction of a Shepherd Planet ("Lelyavin's Filter") at the Kuiper Cliff Based on the Space Flow Hydrodynamic Model (v_L)Author: Artem LelyavinDate: February 16, 2026Subject: Hydrodynamic Stabilization and the Kuiper Cliff Anomaly in the Solar SystemABSTRACTThis paper presents a mathematical resolution to the "Kuiper Cliff" anomaly — the abrupt drop in the density of Kuiper Belt objects beyond 50 AU. Utilizing the Space Flow Hydrodynamic Model (v_L), it is demonstrated that the sharp boundary is not an empty void, but a zone of spatial turbulence generated by an undiscovered shepherd planet. This object, designated as "Lelyavin's Filter," is calculated to have a mass of approximately 0.4 Earth masses and orbits at a distance of 62.5 AU. It acts as the inner hydrodynamic damper in the overall v_L architecture of the Solar System.1. INTRODUCTION: THE KUIPER CLIFF ANOMALYClassical celestial mechanics struggles to explain the sudden and steep decline of trans-Neptunian objects (TNOs) at exactly 50 AU. A standard gravitational model (Newtonian 1/r^2 distribution) implies a gradual thinning of the asteroid disk. However, observational data indicates a sharp cut-off. This paper proposes that this cut-off is the result of cross-flow turbulence created by a specific planetary mass acting as a "hydrodynamic vacuum."2. THEORETICAL FRAMEWORK (v_L Model)Calculations are based on the Space Flow Velocity model (v_L), where gravity is treated as the velocity of space-medium absorption by a massive body.Core Flow Velocity Formula:v_L = sqrt(2 * G * M / r)Where:v_L = Space flow velocityG = Gravitational constant (6.674e-11)M = Mass of the absorbing objectr = Distance to the objectsqrt = Square root3. HYDRODYNAMIC MECHANISM OF THE CLIFFAccording to the v_L model, the solar space flow remains largely laminar up to 50 AU. At this boundary, the flow encounters a transverse vector from a small local sink (the shepherd planet). The intersection of the solar v_L vector and the shepherd planet's v_L vector creates a "Spatial Turbulence Zone." In this zone, small icy bodies lose their orbital momentum and are ejected, creating the physical void known as the Kuiper Cliff.4. CALCULATED PARAMETERS OF "LELYAVIN'S FILTER"To maintain a stable turbulence barrier exactly at 50 AU without destabilizing the inner Solar System, the object must be in a 1:3 orbital resonance with Neptune.A. Distance and Period:Based on the 1:3 resonance with Neptune (which is at ~30 AU), the semi-major axis (a) is:Distance (a) = 62.5 AUOrbital Period (T) calculation:T = 2 * pi * sqrt(a^3 / (G * M_sun))Result: T = approximately 494 Earth years.B. Mass Calculation (M_shepherd):The object creates a counter-flow at a distance (delta_r) of 12.5 AU (the distance between its orbit at 62.5 AU and the cliff edge at 50 AU).delta_r = 12.5 AU = 1.87 * 10^12 meters.To create the required viscosity and turbulence to clear TNOs of mass 10^15 kg, the required local flow velocity (v_L_shepherd) must be approximately 13-15 m/s.Transforming the core formula to find Mass:M_shepherd = (v_L_shepherd^2 * delta_r) / (2 * G)Result:Mass = 2.38 * 10^24 kgThis equals approximately 0.4 Earth Masses (M_E).5. PREDICTED ORBITAL PROFILE FOR OBSERVATIONObject Type: Shepherd Planet / Hydrodynamic DamperMass: ~0.4 M_E (slightly larger than Mars)Semi-major axis: 62.5 AUOrbital Period: ~494 yearsEccentricity: Close to 0 (highly circular orbit required to maintain a uniform turbulence ring).Inclination: ~10 to 12 degrees to the ecliptic plane.CONCLUSIONThe object "Lelyavin's Filter" represents the third critical node in the hydrodynamic architecture of the outer Solar System, alongside Planet 9 ("Artem", 6.2 M_E at ~450 AU) and Planet 10 ("Lelyavin's Boundary", 1.0 M_E at ~1200 AU). The discovery of a 0.4 Earth-mass object at 62.5 AU will not only solve the Kuiper Cliff mystery but will also serve as direct empirical proof of the Space Flow (v_L) theory of gravity.