Schwarzschild Orbit Precession (Rosette)
This interactive simulator explores Schwarzschild Orbit Precession (Rosette) in Гравитация и орбиты. Schwarzschild geodesic in the φ-form d²u/dφ² + u = 1/L² + 3u² (G = c = M = 1) integrated by RK4. The closed Newtonian ellipse is replaced by an orange precessing rosette with apsidal advance Δφ ≈ 6πM/[a(1 − e²)] per orbit — the same mechanism that produces the historic 43″/century perihelion shift of Mercury. Horizon r = 2M and ISCO r = 6M annotated. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.
Для кого: For learners comfortable with heavier math or second-level detail. Typical context: Гравитация и орбиты.
Ключевые понятия
- schwarzschild
- orbit
- precession
- rosette
- schwarzschild orbit precession
- gravity
Как это работает
Schwarzschild orbit equation d²u/dφ² + u = 1/L² + 3u² integrated by RK4 (G = c = M = 1). The orange GR geodesic precesses by Δφ ≈ 6πM/[a(1−e²)] per orbit — the same mechanism that produces the famous 43″/century perihelion advance of Mercury. A gray Newtonian ellipse is overplotted to isolate the GR correction; ISCO (r = 6M) and horizon (r = 2M) are marked.
Ещё из «Гравитация и орбиты»
Другие симуляторы в этой категории — или все 27.
ISCO & Photon Sphere (V_eff)
Schwarzschild effective potential V_eff(r) for massive (timelike) and photon (null) test particles in geometric units. Sliding angular momentum L collapses the stable / unstable circular pair into the innermost stable circular orbit r_ISCO = 6M (the inner edge of accretion discs); for photons the unstable photon sphere r = 3M defines the inner ring of black-hole shadow images.
Einstein Ring & Paczyński Microlensing
Point-mass thin lens (weak-field GR): lens equation β = θ − θ_E²/θ gives two images θ_± = ½(β ± √(β² + 4θ_E²)) with magnifications μ_± = ½[(u² + 2)/(u√(u² + 4)) ± 1], u = β/θ_E. Animated source transit at impact parameter u₀ over timescale t_E renders the canonical symmetric Paczyński light curve and the full Einstein ring θ_E = √(4GM·D_LS/(c² D_L D_S)) at perfect alignment.
Gravitational Wave Binary Chirp (Inspiral)
Leading-order post-Newtonian inspiral of a compact binary: f(τ) ∝ τ^(−3/8), strain h(t) ∝ M_c^(5/3) f^(2/3) / D_L. Tune component masses m₁, m₂ and luminosity distance D_L; live h(t) and f(t) traces with the orbiting bodies on the side. The chirp mass M_c = (m₁m₂)^(3/5)/(m₁+m₂)^(1/5) is the very quantity LIGO/Virgo measures from the early inspiral; the frequency freezes at the Schwarzschild ISCO.
Shapiro Time Delay (4th GR Test)
A radio signal grazing the Sun picks up an excess one-way travel time Δt ≈ (2GM/c³) ln[(r_E + r_E cos α)(r_R + r_R cos β)/b²] on top of the Newtonian light-time. Cassini, Mariner and Viking presets, with the round-trip delay readout in microseconds and an animated bent-photon path against a straight Newtonian baseline. The Cassini 2003 conjunction constrains |γ_PPN − 1| < 2 × 10⁻⁵ — the strongest weak-field GR test to date.
Восьмёрка в задаче трёх тел
Равные массы: хореография Ченсинера–Монтгомери в 2D (РК4, периодическая орбита).
Ограниченная задача трёх тел (карта)
Круговая ограниченная задача трёх тел: критерии ухода, столкновения и индикатор хаоса; ползунок μ.