Question: Here is my code for simulating planetary orbits. When I have the bodies list set up with only the Earth, Sun and Jupiter, my code works well and prints out a reasonably accurate time for Jupiter and Earth's orbit. However, when I add Saturn into the bodies list, I get a value of 43200 seconds for both Jupiter and Saturn's orbit. What's strange though is that Earth's orbit is printing out accurately even with Saturn in the list of bodies. In addition to this, the plot I'm getting is correct, as if I run the simulation for, say, 28 years, Saturn shows as not having a completed orbit. If I run the simulation for 29.5 years, Saturn just about completes its orbit, which is accurate. So the gravitational calculation is still working. Perhaps the index is failing somehow when another body is added? But I'm not sure why. It seems the line simulation.run(30*u.year,6*u.hr)  is causing this. Whenever I change the time-step to around 24*u.hr, it prints out accurate adjacent times for only Jupiter, but not accurate enough. When I change the time-step to 10 days, it prints out a value of 10600 for Saturn's orbital period. This isn't accurate enough however. import numpy as np import matplotlib.pyplot as plt import astropy.units as u import astropy.constants as c import sys import time from mpl_toolkits.mplot3d import Axes3D #making a class for Celestial Objects class CelestialObjects(): def __init__(self,mass,pos_vec,vel_vec,name=None, has_units=True): self.name=name self.has_units=has_units if self.has_units: self.mass=mass.cgs self.pos=pos_vec.cgs.value self.vel=vel_vec.cgs.value else: self.mass=mass #3d array for position of body in 3d space in AU self.pos=pos_vec #3d array for velocity of body in 3d space in km/s self.vel=vel_vec def return_vec(self): return np.concatenate((self.pos,self.vel)) def return_name(self): return self.name def return_mass(self): if self.has_units: return self.mass.cgs.value else: return self.mass v_earth=(((c.G*1.98892E30)/1.495978707E11)**0.5)/1000 v_jupiter=(((c.G*1.98892E30)/7.779089276E11)**0.5)/1000 v_saturn=(((c.G*1.98892E30)/1.421179772E12)**0.5)/1000 #set up first instance of a celestial object, Earth Earth=CelestialObjects(name='Earth', pos_vec=np.array([0,1,0])*u.AU, vel_vec=np.array([v_earth.value,0,0])*u.km/u.s, mass=1.0*c.M_earth) #set up second instance of a celestial object, the Sun Sun=CelestialObjects(name='Sun', pos_vec=np.array([0,0,0])*u.AU, vel_vec=np.array([0,0,0])*u.km/u.s, mass=1*u.Msun) Jupiter=CelestialObjects(name='Jupiter', pos_vec=np.array([0,5.2,0])*u.AU, vel_vec=np.array([v_jupiter.value,0,0])*u.km/u.s, mass=317.8*c.M_earth) Saturn=CelestialObjects(name='Saturn', pos_vec=np.array([0,9.5,0])*u.AU, vel_vec=np.array([v_saturn.value,0,0])*u.km/u.s, mass=95.16*c.M_earth) bodies=[Earth,Sun,Jupiter,Saturn] #making a class for system class Simulation(): def __init__(self,bodies,has_units=True): self.has_units=has_units self.bodies=bodies self.Nbodies=len(self.bodies) self.Ndim=6 self.quant_vec=np.concatenate(np.array([i.return_vec() for i in self.bodies])) self.mass_vec=np.array([i.return_mass() for i in self.bodies]) self.name_vec=[i.return_name() for i in self.bodies] def set_diff_eqs(self,calc_diff_eqs,**kwargs): self.diff_eqs_kwargs=kwargs self.calc_diff_eqs=calc_diff_eqs def rk4(self,t,dt): k1= dt* self.calc_diff_eqs(t,self.quant_vec,self.mass_vec,**self.diff_eqs_kwargs) k2=dt*self.calc_diff_eqs(t+dt*0.5,self.quant_vec+0.5*k1,self.mass_vec,**self.diff_eqs_kwargs) k3=dt*self.calc_diff_eqs(t+dt*0.5,self.quant_vec+0.5*k2,self.mass_vec,**self.diff_eqs_kwargs) k4=dt*self.calc_diff_eqs(t+dt,self.quant_vec+k3,self.mass_vec,**self.diff_eqs_kwargs) y_new=self.quant_vec+((k1+2*k2+2*k3+k4)/6) return y_new def run(self,T,dt,t0=0): if not hasattr(self,'calc_diff_eqs'): raise AttributeError('You must set a diff eq solver first.') if self.has_units: try: _=t0.unit except: t0=(t0*T.unit).cgs.value T=T.cgs.value dt=dt.cgs.value self.history=[self.quant_vec] clock_time=t0 nsteps=int((T-t0)/dt) start_time=time.time() orbit_completed=False orbit_start_time=clock_time initial_position=self.quant_vec[:3] min_distance=float('inf') min_distance_time=0 count=0 for step in range(nsteps): sys.stdout.flush() sys.stdout.write('Integrating: step = {}/{}| Simulation Time = {}'.format(step,nsteps,round(clock_time,3))+'\r') y_new=self.rk4(0,dt) self.history.append(y_new) self.quant_vec=y_new clock_time+=dt #checking if planet has completed an orbit current_position=self.quant_vec[:3] #**THIS IS WHERE ORBIT TIME CALCULATED** To explain, this is earths position, the first three elements of the vector, the next three elements are its velocity, and then the next three are the suns position vectors and so on. Saturns index would be self.quant_vec[18:21]. distance_to_initial=np.linalg.norm(current_position-initial_position) if distance_to_initial<min_distance and orbit_completed is False: min_distance=distance_to_initial min_distance_time=clock_time if count==1: orbit_completed=True count+=1 runtime=time.time()-start_time print(clock_time) print('\n') print('Simulation completed in {} seconds'.format(runtime)) print(f'Minimum distance reached at time: {min_distance_time:.3f} seconds. Minimum distance: {min_distance:.3e}') self.history=np.array(self.history) def nbody_solver(t,y,masses): N_bodies=int(len(y)/6) solved_vector=np.zeros(y.size) distance=[] for i in range(N_bodies): ioffset=i * 6 for j in range(N_bodies): joffset=j * 6 solved_vector[ioffset]=y[ioffset+3] solved_vector[ioffset+1]=y[ioffset+4] solved_vector[ioffset+2]=y[ioffset+5] if i != j: dx= y[ioffset]-y[joffset] dy=y[ioffset+1]-y[joffset+1] dz=y[ioffset+2]-y[joffset+2] r=(dx**2+dy**2+dz**2)**0.5 ax=(-c.G.cgs*masses[j]/r**3)*dx ay=(-c.G.cgs*masses[j]/r**3)*dy az=(-c.G.cgs*masses[j]/r**3)*dz ax=ax.value ay=ay.value az=az.value solved_vector[ioffset+3]+=ax solved_vector[ioffset+4]+=ay solved_vector[ioffset+5]+=az return solved_vector simulation=Simulation(bodies) simulation.set_diff_eqs(nbody_solver) simulation.run(30*u.year,12*u.hr) earth_position = simulation.history[:, :3] # Extracting position for Earth sun_position = simulation.history[:, 6:9] # Extracting position for Sun jupiter_position=simulation.history[:, 12:15] #Jupiter position saturn_position=simulation.history[:, 18:21] #Saturn position fig = plt.figure() ax = fig.add_subplot(111, projection='3d') # Plot the trajectories ax.plot(earth_position[:, 0], earth_position[:, 1], earth_position[:, 2], label='Earth') ax.plot(sun_position[:, 0], sun_position[:, 1], sun_position[:, 2], label='Sun') ax.plot(jupiter_position[:, 0], jupiter_position[:, 1], jupiter_position[:, 2], label='Jupiter') ax.plot(saturn_position[:, 0], saturn_position[:, 1], saturn_position[:, 2], label='Saturn') # Add labels and title ax.set_xlabel('X (AU)') ax.set_ylabel('Y (AU)') ax.set_zlabel('Z (AU)') ax.set_title('Trajectories of Earth and Jupiter Around the Sun') ax.scatter([0], [0], [0], marker='o', color='yellow', s=200, label='Sun') # Marking the Sun at the origin ax.scatter(earth_position[0, 0], earth_position[0, 1], earth_position[0, 2], marker='o', color='blue', s=50, label='Earth') # Marking the initial position of Earth ax.scatter(jupiter_position[0, 0], jupiter_position[0, 1], jupiter_position[0, 2], marker='o', color='green', s=100, label='Jupiter') # Marking the initial position of Earth ax.scatter(saturn_position[0, 0], saturn_position[0, 1], saturn_position[0, 2], marker='o', color='red', s=80, label='Saturn') # Marking the initial position of Earth # Add a legend ax.legend() # Show the plot plt.show() 

Answer: The reason for this behaviour is that the if distance_to_initial<min_distance condition for the time logging is not triggered correctly in this case. The specific cause of this is the "min_distance", which might get "hopped over" in certain circumstances (e.g. due to increasing step distance), and then only the first min_distance_time value is used, without ever "completing the orbit". To fix this, instead of using a min_distance, i would instead check if the initial position is in between the previous and the current position. Here's a run method which implements this approach:  def run(self, T, dt, t0=0): if not hasattr(self, "calc_diff_eqs"): raise AttributeError("You must set a diff eq solver first.") if self.has_units: try: _ = t0.unit except: t0 = (t0 * T.unit).cgs.value T = T.cgs.value dt = dt.cgs.value self.history = [self.quant_vec] clock_time = t0 nsteps = int((T - t0) / dt) start_time = time.time() initial_position = self.quant_vec[18:21] min_distance = None min_distance_time = None for step in range(nsteps): sys.stdout.flush() sys.stdout.write( "Integrating: step = {}/{}| Simulation Time = {}".format( step, nsteps, round(clock_time, 3) ) + "\r" ) y_new = self.rk4(0, dt) y_old = self.quant_vec self.history.append(y_new) self.quant_vec = y_new clock_time += dt current_position = self.quant_vec[18:21] last_position = y_old[18:21] if step == 0 or min_distance_time is not None: # do not check orbit criteria for initial step # or if orbit already completed continue # checking if planet has completed an orbit, i.e. if origin is between current and last position last_to_initial = np.linalg.norm(last_position - initial_position) current_to_initial = np.linalg.norm(current_position - initial_position) current_to_last = np.linalg.norm(current_position - last_position) has_passed_initial_position = ( last_to_initial <= current_to_last and current_to_initial <= current_to_last ) if has_passed_initial_position: min_distance = current_to_initial min_distance_time = clock_time runtime = time.time() - start_time print("\n") print("Simulation completed in {} seconds".format(runtime)) print( f"Minimum distance reached at time: {min_distance_time:.3f} seconds. Minimum distance: {min_distance:.3e}" ) self.history = np.array(self.history)