From be4ff2cfcc405a431633e2c1dc2739427dc4eea0 Mon Sep 17 00:00:00 2001
From: xfy <i@rua.plus>
Date: Thu, 16 Jul 2026 17:43:58 +0800
Subject: [PATCH] =?UTF-8?q?feat(physics):=20Kerr=20equatorial=20orbital=20?=
 =?UTF-8?q?frequency=20=CE=A9=5F=CF=86=20(Bardeen=201972)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 src/physics.rs | 31 +++++++++++++++++++++++++++++++
 1 file changed, 31 insertions(+)

diff --git a/src/physics.rs b/src/physics.rs
index 1f2bb86..c0155d2 100644
--- a/src/physics.rs
+++ b/src/physics.rs
@@ -79,6 +79,14 @@ pub fn kerr_horizon(chi: f32) -> f32 {
     m + (m * m - a * a).max(0.0).sqrt()
 }
 
+/// Kerr 赤道 prograde 圆轨角速度 (Rs=1, M=0.5). Bardeen 1972 eqn 2.16.
+/// `Ω_φ = 1 / (r^1.5 + a)`, a = χM = 0.5χ. χ=0 退化为牛顿 1/r^1.5.
+pub fn kerr_orbital_frequency(r: f32, chi: f32) -> f32 {
+    let m = 0.5;
+    let a = chi * m;
+    1.0 / (r.powf(1.5) + a)
+}
+
 /// Kerr bending acceleration (CPU mirror of the shader `deriv` accel).
 /// `chi = a/M ∈ [0,1]`. At chi=0 this equals `bending_accel`.
 pub fn kerr_bending_accel(pos: Vec3, dir: Vec3, chi: f32) -> Vec3 {
@@ -279,4 +287,27 @@ mod tests {
         assert!(b > BCRIT);
         assert!(!is_captured(eye, dir, 2000, 0.1), "ray above bcrit should escape");
     }
+
+    #[test]
+    fn orbital_frequency_reduces_to_newton_at_zero_spin() {
+        // χ=0: Ω = 1/r^1.5 (牛顿开普勒, Rs=1)
+        for r in [4.0_f32, 6.0, 10.0, 20.0] {
+            let newton = 1.0 / r.powf(1.5);
+            let kerr = kerr_orbital_frequency(r, 0.0);
+            assert!(
+                (kerr - newton).abs() < 1e-6,
+                "χ=0 at r={}: expected {} (newton), got {}",
+                r, newton, kerr
+            );
+        }
+    }
+
+    #[test]
+    fn orbital_frequency_decreases_with_spin_at_fixed_r() {
+        // prograde 轨道 (a>0): Ω_φ 随 χ 减小 (分母 r^1.5+a 增大)
+        let r = 8.0;
+        let omega_0 = kerr_orbital_frequency(r, 0.0);
+        let omega_1 = kerr_orbital_frequency(r, 1.0);
+        assert!(omega_1 < omega_0, "prograde Ω should decrease with spin");
+    }
 }