diff --git a/docs/index.md b/docs/index.md
index 154f5656e..f0e6490c2 100644
--- a/docs/index.md
+++ b/docs/index.md
@@ -10,8 +10,8 @@ REBOUND is an N-body integrator, i.e. a software package that can integrate the
 * No dependencies on external libraries.
 * Runs natively on Linux, MacOS, and Windows. 
 * Symplectic integrators ([WHFast](integrators/#whfast), [SEI](integrators/#sei), [LEAPFROG](integrators/#leapfrog), [EOS](integrators/#embedded-operator-splitting-method-eos))
-* Hybrid symplectic integrators for planetary dynamics with close encounters ([MERCURIUS](integrators/#mercurius))
 * Hybrid reversible integrators for planetary dynamics with arbitrary close encounters ([TRACE](integrators/#trace))
+* Hybrid symplectic integrators for planetary dynamics with close encounters ([MERCURIUS](integrators/#mercurius))
 * High order symplectic integrators for integrating planetary systems ([SABA](integrators/#saba), WH Kernel methods)
 * High accuracy non-symplectic integrator with adaptive time-stepping ([IAS15](integrators/#ias15))
 * Can integrate arbitrary user-defined ODEs that are coupled to N-body dynamics for tides, spin, etc
diff --git a/docs/integrators.md b/docs/integrators.md
index bfabb25ac..8d97a5b74 100644
--- a/docs/integrators.md
+++ b/docs/integrators.md
@@ -404,7 +404,7 @@ The `reb_integrator_mercurius` structure contains the configuration and data str
 
 TRACE is a hybrid time-reversible integrator, based on the algorithm described in [Hernandez & Dehnen 2023](https://ui.adsabs.harvard.edu/abs/2023MNRAS.522.4639H/abstract). 
 It uses WHFast for long term integrations but switches time-reversibly to BS or IAS15 for all close encounters. TRACE is appropriate for systems with a dominant central mass that will occasionally have close encounters. 
-A paper describing the TRACE implementation is in preparation. 
+The TRACE implementation is described in [Lu, Hernandez & Rein](https://ui.adsabs.harvard.edu/abs/2024arXiv240503800L/abstract). 
 
 
 The following code enables TRACE and sets the critical radius to 4 Hill radii
diff --git a/rebound/integrators/trace.py b/rebound/integrators/trace.py
index 6f979be7d..e42d746db 100644
--- a/rebound/integrators/trace.py
+++ b/rebound/integrators/trace.py
@@ -67,7 +67,6 @@ def __repr__(self):
                 ("_current_C", ctypes.c_uint),
                 ("_force_accept", ctypes.c_uint),
                 ]
-    # To be honest I'm not sure what these do: do we need this? - Tiger
     @property
     def S(self):
         raise AttributeError("You can only set C function pointers from python.")
diff --git a/rebound/tests/test_trace_basic.py b/rebound/tests/test_trace_basic.py
index 604a1366c..8691e9361 100644
--- a/rebound/tests/test_trace_basic.py
+++ b/rebound/tests/test_trace_basic.py
@@ -36,6 +36,19 @@ def test_trace_encounter_condition(self):
                 for j in range(i+1,sim.N):
                     self.assertEqual(sim.ri_trace._current_Ks[i*sim.N+j],0)
 
+    def test_trace_encounter_prediction(self):
+        sim = rebound.Simulation()
+        sim.add(m=1)
+        sim.add(m=9.55e-4,x=5.2)
+        sim.add(x=5.3,y=0.36,vy=-7.2) # Non-encounter prediction misses this
+        sim.integrator = "TRACE"
+        sim.dt = 0.01
+        sim.ri_trace.r_crit_hill = 1
+        sim.step()
+        
+        self.assertEqual(sim.ri_trace._encounter_N,3)
+
+
 
 if __name__ == "__main__":
     unittest.main()
diff --git a/src/integrator_trace.c b/src/integrator_trace.c
index 2194db43e..a93f856cf 100644
--- a/src/integrator_trace.c
+++ b/src/integrator_trace.c
@@ -41,40 +41,76 @@
 
 int reb_integrator_trace_switch_default(struct reb_simulation* const r, const unsigned int i, const unsigned int j){
     struct reb_integrator_trace* const ri_trace = &(r->ri_trace);
+    const double h2 = r->dt/2.;
+    
+    const double dxi  = r->particles[i].x;
+    const double dyi  = r->particles[i].y;
+    const double dzi  = r->particles[i].z;
+
+    const double dxj  = r->particles[j].x;
+    const double dyj  = r->particles[j].y;
+    const double dzj  = r->particles[j].z;
+
+    const double dx = dxi - dxj;
+    const double dy = dyi - dyj;
+    const double dz = dzi - dzj;
+    const double rp = dx*dx + dy*dy + dz*dz;
+    
     double dcriti6 = 0.0;
     double dcritj6 = 0.0;
 
     const double m0 = r->particles[0].m;
 
     if (r->particles[i].m != 0){
-        const double dxi  = r->particles[i].x;  // in dh
-        const double dyi  = r->particles[i].y;
-        const double dzi  = r->particles[i].z;
         const double di2 = dxi*dxi + dyi*dyi + dzi*dzi;
         const double mr = r->particles[i].m/(3.*m0);
         dcriti6 = di2*di2*di2*mr*mr;
     }
 
     if (r->particles[j].m != 0){
-        const double dxj  = r->particles[j].x;  // in dh
-        const double dyj  = r->particles[j].y;
-        const double dzj  = r->particles[j].z;
         const double dj2 = dxj*dxj + dyj*dyj + dzj*dzj;
         const double mr = r->particles[j].m/(3.*m0);
         dcritj6 = dj2*dj2*dj2*mr*mr;
     }
 
-    const double dx = r->particles[i].x - r->particles[j].x;
-    const double dy = r->particles[i].y - r->particles[j].y;
-    const double dz = r->particles[i].z - r->particles[j].z;
-    const double d2 = dx*dx + dy*dy + dz*dz;
-
     double r_crit_hill2 = ri_trace->r_crit_hill*ri_trace->r_crit_hill;
     double dcritmax6 = r_crit_hill2 * r_crit_hill2 * r_crit_hill2 * MAX(dcriti6,dcritj6);
 
-    return d2*d2*d2 < dcritmax6;
+    if (rp*rp*rp < dcritmax6) return 1;
+    
+    const double dvx  = r->particles[i].vx - r->particles[j].vx;
+    const double dvy  = r->particles[i].vy - r->particles[j].vy;
+    const double dvz  = r->particles[i].vz - r->particles[j].vz;
+    const double v2 = dvx*dvx + dvy*dvy + dvz*dvz;
+    
+    const double qv = dx*dvx + dy*dvy + dz*dvz;
+    int d;
+
+    if (qv == 0.0){ // Small
+        // minimum is at present, which is already checked for
+	return 0;
+    }
+    else if (qv < 0){
+        d = 1; 
+    }
+    else{
+        d = -1;
+    }
+
+    double dmin2;
+    double tmin = -d*qv/v2;
+    if (tmin < h2){
+	// minimum is in the window
+	dmin2 = rp - qv*qv/v2;
+    }
+    else{
+	dmin2 = rp + 2*d*qv*h2 + v2*h2*h2;
+    }
+
+    return dmin2*dmin2*dmin2 < dcritmax6;
 }
 
+
 int reb_integrator_trace_switch_peri_distance(struct reb_simulation* const r, const unsigned int j){
     const struct reb_integrator_trace* const ri_trace = &(r->ri_trace);
     const double peri = ri_trace->peri_crit_distance;
@@ -519,9 +555,9 @@ void reb_integrator_trace_part1(struct reb_simulation* r){
         // These arrays are only used within one timestep.
         // Can be recreated without loosing bit-wise reproducibility.
         ri_trace->particles_backup       = realloc(ri_trace->particles_backup,sizeof(struct reb_particle)*N);
+        ri_trace->particles_backup_kepler   = realloc(ri_trace->particles_backup_kepler,sizeof(struct reb_particle)*N);
         ri_trace->current_Ks             = realloc(ri_trace->current_Ks,sizeof(int)*N*N);
         ri_trace->encounter_map          = realloc(ri_trace->encounter_map,sizeof(int)*N);
-        ri_trace->particles_backup_kepler   = realloc(ri_trace->particles_backup_kepler,sizeof(struct reb_particle)*N);
         ri_trace->N_allocated = N;
     }
 
@@ -842,7 +878,6 @@ void reb_integrator_trace_reset(struct reb_simulation* r){
     free(r->ri_trace.particles_backup_additional_forces);
     r->ri_trace.particles_backup_additional_forces = NULL;
 
-
     free(r->ri_trace.encounter_map);
     r->ri_trace.encounter_map = NULL;