{"openapi":"3.1.0","info":{"title":"Tidal Forces API","version":"1.0.0","description":"Tidal-physics and gravitational-dominance astrophysics as an API, computed locally and deterministically. The tidal-force endpoint computes the tidal (differential) acceleration that stretches a body, a = 2·G·M·r/d³, from the primary mass, the radius (half-size) of the affected body and the centre-to-centre distance — and the force if a body mass is given; tidal effects fall off as the inverse cube of distance, far faster than gravity's inverse square, which is why they matter only close in. The roche-limit endpoint computes the Roche limit, the distance inside which tidal forces tear a satellite apart, for both rigid bodies, d = R·(2·ρM/ρm)^(1/3), and fluid bodies, d = 2.44·R·(ρM/ρm)^(1/3), from the primary radius and the two densities — Saturn's rings sit inside its Roche limit. The hill-sphere endpoint computes the Hill-sphere radius, r_H ≈ a·(1−e)·(m/3M)^(1/3), the region where a body's own gravity dominates so it can keep moons, from the orbital distance, eccentricity and the two masses. Masses are in kilograms, distances and radii in metres and densities in kg/m³, with G = 6.674×10⁻¹¹. Everything is computed locally and deterministically, so it is instant and private. Ideal for astronomy, astrophysics, planetary-science, simulation and education app developers, ring-system and moon-stability tools, and physics education. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 3 endpoints. This is tidal and gravitational-dominance physics; for Newtonian gravity use a gravitation API and for orbital periods an orbital-mechanics API.","contact":{"name":"PremiumApi","url":"https://www.oanor.com/by/premiumapi"}},"servers":[{"url":"https://api.oanor.com/tidal-api","description":"oanor gateway"}],"tags":[{"name":"Tidal"},{"name":"Meta"}],"components":{"securitySchemes":{"oanorKey":{"type":"apiKey","in":"header","name":"x-oanor-key","description":"Get your key at https://www.oanor.com/developer/keys"}}},"security":[{"oanorKey":[]}],"paths":{"/v1/hill-sphere":{"get":{"operationId":"get_v1_hill_sphere","tags":["Tidal"],"summary":"Hill sphere","description":"","parameters":[{"name":"orbital_distance","in":"query","required":true,"description":"Orbital distance a (m)","schema":{"type":"string"},"example":"1.496e11"},{"name":"body_mass","in":"query","required":true,"description":"Body mass m (kg)","schema":{"type":"string"},"example":"5.972e24"},{"name":"primary_mass","in":"query","required":true,"description":"Primary mass M (kg)","schema":{"type":"string"},"example":"1.989e30"},{"name":"eccentricity","in":"query","required":false,"description":"Eccentricity","schema":{"type":"string"},"example":"0"}],"security":[{"oanorKey":[]}],"responses":{"200":{"description":"OK","content":{"application/json":{"example":{"data":{"note":"Hill sphere r_H ≈ a·(1−e)·(m/3M)^(1/3) — the region where a body's gravity dominates over the primary's, so it can hold moons. Earth's Hill sphere is about 1.5 million km.","inputs":{"body_mass":5.972e+24,"eccentricity":0,"primary_mass":1.989e+30,"orbital_distance":149600000000},"hill_radius_m":1496417737.0931,"hill_radius_km":1496417.7371},"meta":{"timestamp":"2026-06-05T19:50:33.973Z","request_id":"ecca9d55-eb41-4ba9-a474-fa0607b9f1d4"},"status":"ok","message":"Hill sphere","success":true}}}},"401":{"description":"Missing or invalid x-oanor-key header"},"402":{"description":"Active subscription required"},"429":{"description":"Rate-limit or monthly quota reached"},"502":{"description":"Upstream did not respond"}}}},"/v1/roche-limit":{"get":{"operationId":"get_v1_roche_limit","tags":["Tidal"],"summary":"Roche limit","description":"","parameters":[{"name":"primary_radius","in":"query","required":true,"description":"Primary radius R (m)","schema":{"type":"string"},"example":"6.371e6"},{"name":"primary_density","in":"query","required":true,"description":"Primary density (kg/m³)","schema":{"type":"string"},"example":"5514"},{"name":"satellite_density","in":"query","required":true,"description":"Satellite density (kg/m³)","schema":{"type":"string"},"example":"3344"}],"security":[{"oanorKey":[]}],"responses":{"200":{"description":"OK","content":{"application/json":{"example":{"data":{"note":"Roche limit — inside it tidal forces tear a satellite apart. Rigid bodies: d = R·(2·ρM/ρm)^(1/3); fluid bodies: d = 2.44·R·(ρM/ρm)^(1/3). Saturn's rings lie inside its Roche limit.","inputs":{"primary_radius":6371000,"primary_density":5514,"satellite_density":3344},"roche_limit_fluid_m":18365281.7634,"roche_limit_rigid_m":9483116.8365,"roche_limit_fluid_km":18365.2818,"roche_limit_rigid_km":9483.1168},"meta":{"timestamp":"2026-06-05T19:50:34.062Z","request_id":"9ca5cb91-121f-4171-b3ae-6f6608a0745d"},"status":"ok","message":"Roche limit","success":true}}}},"401":{"description":"Missing or invalid x-oanor-key header"},"402":{"description":"Active subscription required"},"429":{"description":"Rate-limit or monthly quota reached"},"502":{"description":"Upstream did not respond"}}}},"/v1/tidal-force":{"get":{"operationId":"get_v1_tidal_force","tags":["Tidal"],"summary":"Tidal acceleration","description":"","parameters":[{"name":"primary_mass","in":"query","required":true,"description":"Primary mass M (kg)","schema":{"type":"string"},"example":"7.342e22"},{"name":"body_radius","in":"query","required":true,"description":"Affected-body radius r (m)","schema":{"type":"string"},"example":"6.371e6"},{"name":"distance","in":"query","required":true,"description":"Centre-to-centre distance d (m)","schema":{"type":"string"},"example":"3.844e8"},{"name":"body_mass","in":"query","required":false,"description":"Affected-body mass (kg) for force","schema":{"type":"string"}}],"security":[{"oanorKey":[]}],"responses":{"200":{"description":"OK","content":{"application/json":{"example":{"data":{"note":"Tidal (differential) acceleration across a body a = 2·G·M·r/d³ — the stretch between the near and far sides. It falls off as 1/d³, far faster than gravity's 1/d².","inputs":{"distance":384400000,"body_radius":6371000,"primary_mass":7.342e+22},"tidal_acceleration_ms2":1.0992780787710049e-6},"meta":{"timestamp":"2026-06-05T19:50:34.170Z","request_id":"c95912f8-6c24-4692-bdc9-abc830e05ff9"},"status":"ok","message":"Tidal acceleration","success":true}}}},"401":{"description":"Missing or invalid x-oanor-key header"},"402":{"description":"Active subscription required"},"429":{"description":"Rate-limit or monthly quota reached"},"502":{"description":"Upstream did not respond"}}}},"/v1/meta":{"get":{"operationId":"get_v1_meta","tags":["Meta"],"summary":"Spec","description":"","parameters":[],"security":[{"oanorKey":[]}],"responses":{"200":{"description":"OK","content":{"application/json":{"example":{"data":{"notes":"Masses in kg, distances and radii in m, densities in kg/m³. G = 6.674×10⁻¹¹. Tidal effects scale as 1/d³.","service":"tidal-api","formulae":{"hill":"r_H = a·(1−e)·(m/3M)^(1/3)","tidal":"a = 2·G·M·r/d³","roche_fluid":"d = 2.44·R·(ρM/ρm)^(1/3)","roche_rigid":"d = R·(2·ρM/ρm)^(1/3)"},"constants":{"gravitational_constant":6.6743e-11},"endpoints":{"GET /v1/meta":"This document.","GET /v1/hill-sphere":"Hill-sphere radius from the orbital distance and the two masses.","GET /v1/roche-limit":"Roche limit (rigid and fluid) from the primary radius and the two densities.","GET /v1/tidal-force":"Tidal (differential) acceleration a = 2·G·M·r/d³, and force if a body mass is given."},"description":"Tidal-physics and gravitational-dominance calculator: tidal acceleration across a body, the Roche limit (rigid and fluid) and the Hill-sphere radius."},"meta":{"timestamp":"2026-06-05T19:50:34.266Z","request_id":"fdc8a0a7-cfb3-46ae-b269-883e05bdacd2"},"status":"ok","message":"Meta","success":true}}}},"401":{"description":"Missing or invalid x-oanor-key header"},"402":{"description":"Active subscription required"},"429":{"description":"Rate-limit or monthly quota reached"},"502":{"description":"Upstream did not respond"}}}}},"x-oanor-pricing":[{"slug":"free","name":"Free","price_cents_month":0,"monthly_call_quota":3700,"rps_limit":2,"hard_limit":true},{"slug":"starter","name":"Starter","price_cents_month":500,"monthly_call_quota":35000,"rps_limit":6,"hard_limit":true},{"slug":"pro","name":"Pro","price_cents_month":1400,"monthly_call_quota":214000,"rps_limit":15,"hard_limit":true},{"slug":"mega","name":"Mega","price_cents_month":4400,"monthly_call_quota":1295000,"rps_limit":40,"hard_limit":true}],"x-oanor-marketplace-url":"https://www.oanor.com/api/tidal-api"}