Step 3: Your first N-body program
In this step, we will write our first N-body program. We have set up the initial conditions in step 1 and the acceleration function in step 2. Now, we will need to solve the equations of motion for the particles. For Newtonian mechanics, we have the following coupled equations:
\[
\frac{\mathrm{d}\textbf{r}}{\mathrm{d}t}
= \textbf{v}, \quad
\frac{\mathrm{d}\textbf{v}}{\mathrm{d}t}
= \textbf{a}(\textbf{r}) \left( = \frac{\textbf{F}(\textbf{r})}{m} \right).
\]
They are ordinary differential equations (ODEs). To solve them, we will need a ODE solver.
Euler method
The simplest ODE solver is the Euler method, where the update is approximated as
\[
\Delta \mathbf{r} = \mathbf{v} \Delta t, \quad
\Delta \mathbf{v} = \mathbf{a}(\mathbf{r}) \Delta t,
\]
where \(\Delta t\) is the time step. By Taylor expansion, we can see that this is a first-order approximation,
\[
x(t + \Delta t) = x(t) + x'(t) \Delta t + \mathcal{O}(\Delta t^2),
\]
where \(\mathcal{O}(\Delta t^2)\) is the local truncation error if \(\Delta t \to 0\). Because the total number of steps scales with \(1/\Delta t\), the global error is
\[
\text{global error} =
\text{number of steps} \times \text{error per step}
\propto \frac{1}{\Delta t} \mathcal{O}(\Delta t^2) = \mathcal{O}(\Delta t).
\]
If you are not familiar with the big-O notation, you can think of it as the error is being bounded by a polynomial in the order of \(\Delta t\).
Implementing the Euler integrator is very easy. With
\[
\mathbf{r}_{n + 1} = \mathbf{r}_n + \mathbf{v}_n \Delta t, \quad
\mathbf{v}_{n + 1} = \mathbf{v}_n + \mathbf{a}(\mathbf{r}_{n}) \Delta t,
\]
we have the following code:
common.py
def euler(a: np.ndarray, system: System, dt: float) -> None:
"""
Advance one step with the Euler's method.
Parameters
----------
a : np.ndarray
Gravitational accelerations array with shape (N, 3).
system : System
System object.
dt : float
Time step.
"""
acceleration(a, system)
system.x += system.v * dt
system.v += a * dt
Now, we will build all the components we need for our N-body program.
Solution output
We will need a way to store the solution. A naive way is to store the system at every time step or every few time steps. However, this is a terrible idea because the output size will depends on your choice of time step. A better way is to store the solution at regular intervals. In our simulation, we will use a output interval of 0.1 years. For a simulation of 200 years, we will have 2000 time steps.
Tip
For the Solar system, we only have 9 particles and it takes very little memory to store. So, you don't need to worry too much about the solution size. Just be careful and don't set the output interval too small.
Before the simulation, we will need to set up the output array and store the initial conditions.
step3.py
OUTPUT_INTERVAL = 0.1 * 365.24 # years to days
def main() -> None:
...
# Solution array
sol_size = int(TF // OUTPUT_INTERVAL + 2) # +2 for initial and final time
sol_x = np.zeros((sol_size, system.num_particles, 3))
sol_v = np.zeros((sol_size, system.num_particles, 3))
sol_t = np.zeros(sol_size)
sol_x[0] = system.x
sol_v[0] = system.v
sol_t[0] = 0.0
output_count = 1
Also, we need to calculate the output time.
step3.py
def main() -> None:
...
for i in range(NUM_STEPS):
...
if current_time >= next_output_time:
sol_x[output_count] = system.x
sol_v[output_count] = system.v
sol_t[output_count] = current_time
output_count += 1
next_output_time = output_count * OUTPUT_INTERVAL
Finally, we resize the arrays to the actual size.
step3.py
def main() -> None:
...
sol_x = sol_x[:output_count]
sol_v = sol_v[:output_count]
sol_t = sol_t[:output_count]
Putting it all together
Let's put everything together. We first need to setup the simulation parameters.
step3.py
# Default units is AU, days, and M_sun
TF = 200.0 * 365.24 # years to days
DT = 1.0
OUTPUT_INTERVAL = 0.1 * 365.24 # years to days
NUM_STEPS = int(TF / DT)
Before running the simulation, it is a good idea to print the simulation information.
common.py
def print_simulation_info_fixed_step_size(
system: System,
tf: float,
dt: float,
num_steps: int,
output_interval: float,
sol_size: int,
) -> None:
print("----------------------------------------------------------")
print("Simulation Info:")
print(f"num_particles: {system.num_particles}")
print(f"G: {system.G}")
print(f"tf: {tf} days (Actual tf = dt * num_steps = {dt * num_steps} days)")
print(f"dt: {dt} days")
print(f"Num_steps: {num_steps}")
print()
print(f"Output interval: {output_interval} days")
print(f"Estimated solution size: {sol_size}")
print("----------------------------------------------------------")
Finally, we have the main function that runs the main simulation loop.
I have added a timer to measure the runtime, and a print statement
to show the simulation progress every time we save a solution. The \r at the end
of the print statement should overwrite the previous line, but you can remove
the print statement if it is spamming your terminal.
step3.py
def main() -> None:
# Get initial conditions
system, labels, colors, legend = common.get_initial_conditions(INITIAL_CONDITION)
# Initialize memory
a = np.zeros((system.num_particles, 3))
# Solution array
sol_size = int(TF // OUTPUT_INTERVAL + 2) # +2 for initial and final time
sol_x = np.zeros((sol_size, system.num_particles, 3))
sol_v = np.zeros((sol_size, system.num_particles, 3))
sol_t = np.zeros(sol_size)
sol_x[0] = system.x
sol_v[0] = system.v
sol_t[0] = 0.0
output_count = 1
# Launch simulation
common.print_simulation_info_fixed_step_size(
system, TF, DT, NUM_STEPS, OUTPUT_INTERVAL, sol_size
)
next_output_time = output_count * OUTPUT_INTERVAL
start = timeit.default_timer()
for i in range(NUM_STEPS):
common.euler(a, system, DT)
current_time = i * DT
if current_time >= next_output_time:
sol_x[output_count] = system.x
sol_v[output_count] = system.v
sol_t[output_count] = current_time
output_count += 1
next_output_time = output_count * OUTPUT_INTERVAL
print(f"Current time: {current_time:.2f} days", end="\r")
sol_x = sol_x[:output_count]
sol_v = sol_v[:output_count]
sol_t = sol_t[:output_count]
end = timeit.default_timer()
print()
print(f"Done! Runtime: {end - start:.3g} seconds, Solution size: {output_count}")
You should see the following output:
----------------------------------------------------------
Simulation Info:
num_particles: 9
G: 0.00029591220828411956
tf: 73048.0 days (Actual tf = dt * num_steps = 73048.0 days)
dt: 1.0 days
Num_steps: 73048
Output interval: 36.524 days
Estimated solution size: 2001
----------------------------------------------------------
Current time: 73012.00 days
Done! Runtime: 1.1 seconds, Solution size: 2000
Plotting the trajectory
To visualize the trajectory, we have the following function. Notice that we
have one ax.plot and one ax.scatter calls. The first one is to plot the
trajectory, and the second one is to plot the final position with a circle marker.
common.py
def plot_trajectory(
sol_x: np.ndarray,
labels: list,
colors: list,
legend: bool,
) -> None:
"""
Plot the 2D trajectory.
Parameters
----------
sol_x : np.ndarray
Solution position array with shape (N_steps, num_particles, 3).
labels : list
List of labels for the particles.
colors : list
List of colors for the particles.
legend : bool
Whether to show the legend.
"""
fig = plt.figure()
ax = fig.add_subplot(111, aspect="equal")
ax.set_xlabel("$x$ (AU)")
ax.set_ylabel("$y$ (AU)")
for i in range(sol_x.shape[1]):
traj = ax.plot(
sol_x[:, i, 0],
sol_x[:, i, 1],
color=colors[i],
)
# Plot the last position with marker
ax.scatter(
sol_x[-1, i, 0],
sol_x[-1, i, 1],
marker="o",
color=traj[0].get_color(),
label=labels[i],
)
if legend:
fig.legend(loc="center right", borderaxespad=0.2)
fig.tight_layout()
plt.show()
Then, we add the function call to the end of the main function.
You should see the following plot:
Congrats! 
You have just written your first N-body simulation program!
However, we can see that the results are not very accurate, especially for those inner planets.
(Mercury has drifted to a orbit beyond Saturn within 200 years!)
In the next step, we will implement some higher-order integrators to improve the accuracy.
Full scripts
The full scripts are available at 5_steps_to_n_body_simulation/python/,
or https://github.com/alvinng4/grav_sim/blob/main/5_steps_to_n_body_simulation/python/
step3.py (Click to expand)
common.py (Click to expand)
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