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How to use this tutorial?

The source code for the python scripts and the markdown files are available on

but I strongly recommend reading this tutorial on our website:

Snippets of the code will be shown as we progress, but you don't have to copy them one by one as the full code is always available at the end of each step. However, I do recommend you to translate it into your own code in order to have a better understanding.

Step 1: Initial setup

Welcome to the first step of 5 steps to N-body simulation! This is a series of tutorials with the goal of teaching beginners how to write fast and clean N-body gravity simulations code in Python. In this step, we will set up the required python environment and implement some basic functions to be used in the following steps.

First of all, make sure you have Python 3 installed (to be safe, I recommend Python 3.9 or later). You can check the installation by running the following command in your terminal: 

python --version

Two packages are required for this tutorial: numpy and matplotlib. They are popular packages in Python for scientific computing and data visualization. You can install them using pip: 

pip install numpy matplotlib

Now we are ready to start coding. First, let us create two files:

  • common.py: This file will contain the common functions and classes used in all steps.
  • step1.py: This file will contain the code for this step.

We first focus on common.py. Import the required packages:

common.py
import numpy as np
import matplotlib.pyplot as plt

System

To keep the code clean and organized, we will create a class called System to represent the N-body system. It has the following attributes:

  • num_particles (float): Number of particles N in the system.
  • x (np.ndarray): Positions of the particles in 3D space (shape: (N, 3)).
  • v (np.ndarray): Velocities of the particles in 3D space (shape: (N, 3)).
  • m (np.ndarray): Masses of the particles (shape: (N,)).
  • G (float): Gravitational constant.
common.py
class System:
    def __init__(
        self, num_particles: int, x: np.ndarray, v: np.ndarray, m: np.ndarray, G: float
    ) -> None:
        self.num_particles = num_particles
        self.x = x
        self.v = v
        self.m = m
        self.G = G

We will also implement a method to set the center of mass position and velocity to zero. This is to prevent drifting of the system during the simulation, and to set the center of mass of the system to the origin. To do this, we subtract the center of mass position and velocity from the particles,

\[ \mathbf{r}_{\mathrm{com}} = \frac{1}{M} \sum_{i=1}^{N} m_i \mathbf{r}_i, \quad \mathbf{v}_{\mathrm{com}} = \frac{1}{M} \sum_{i=1}^{N} m_i \mathbf{v}_i. \]

where \(M\) is the total mass of the system, \(m_i\), \(\mathbf{r}_i\), and \(\mathbf{v}_i\) are the mass, position, and velocity of the i-th particle respectively. If you don't care about the performance, you may just use a for loop to iterate over all particles. By the way, x_cm is a 3D vector, m[i] is a scalar, and self.x[i] is a 3D vector.

common.py
class System:
    ...
    def center_of_mass_correction(self) -> None:
        """Set center of mass of position and velocity to zero"""
        x_cm = np.zeros(3)
        v_cm = np.zeros(3)
        M = 0.0
        for i in range(self.num_particles):
            x_cm += self.m[i] * self.x[i]
            v_cm += self.m[i] * self.v[i]
            M += self.m[i]

        x_cm /= M
        v_cm /= M
        self.x -= x_cm
        self.v -= v_cm

Optimization

Python loops are very slow. Here, we introduce a more efficient approach using numpy's broadcasting feature. First, we compute \(m_i \mathbf{r}_i\) as 

self.m[:, np.newaxis] * self.x
where np.newaxis is used to "broadcast" the mass array along axis 1 (column).

\[ \begin{bmatrix} m_{1} \\ m_{2} \\ \vdots \\ m_{N} \end{bmatrix} \to \begin{bmatrix} m_{1} \dots \\ m_{2} \dots \\ \vdots \\ m_{N} \dots \end{bmatrix} \]

The shape of self.m[:, np.newaxis] is now (N, 1) and the shape of self.x is (N, 3). The multiplication is then done element-wise as

\[ \begin{bmatrix} m_{1} r_{1,1} & m_{1} r_{1,2} & m_{1} r_{1,3} \\ m_{2} r_{2,1} & m_{2} r_{2,2} & m_{2} r_{2,3} \\ \vdots & \vdots & \vdots \\ m_{N} r_{N,1} & m_{N} r_{N,2} & m_{N} r_{N,3} \end{bmatrix} \]

Then, to get \(\sum_{i = 1}^N m_i \mathbf{r}_i\) we perform the summation along the axis 0 (row) with length N by 

np.sum(self.m[:, np.newaxis] * self.x, axis=0)
A even faster way is to use np.einsum. 
np.einsum("i,ij->j", self.m, self.x)
Why i,ij->j? Denote the axis 0 and 1 as \(i\) and \(j\) respectively.

  • m is a 1D vector broadcasted along axis 1: \(i\)
  • x is a 2D vector: \(ij\)
  • Final sum is done along axis 0: \(ij \to j\)

Finally, the total mass \(M\) can be simply computed as 

M = np.sum(self.m)
Putting it all together, we have the following code:
common.py
class System:
    ...
    def center_of_mass_correction(self) -> None:
        """Set center of mass of position and velocity to zero"""
        M = np.sum(self.m)
        x_cm = np.einsum("i,ij->j", self.m, self.x) / M
        v_cm = np.einsum("i,ij->j", self.m, self.v) / M

        self.x -= x_cm
        self.v -= v_cm

Initial conditions (Solar System)

With the System class ready, we can now implement a function to get the initial conditions. Since it is tedious to prepare the initial conditions, I have prepared the data for you. Simply input the name of the system and it will return the initial conditions and information for plotting.

Data Sources

The Solar System initial conditions at 1/Jan/2024 are generated using the JPL Horizons System1. Gravitational constant and masses of the solar system objects are calculated using the data from R.S. Park, et. al.2.

Tip

In our tutorial, we will stick with the following units:

  • Length: AU (Astronomical Unit), i.e. the distance from the Earth to the Sun.
  • Mass: \(M_\odot\) (Solar mass)
  • Time: days

They are convenient units for solar system simulations. If you want to use different units, make sure to convert all data to the same units and be consistent.

Code (Click to expand)
common.py
def get_initial_conditions(
    initial_condition: str,
) -> Tuple[System, List[Optional[str]], List[Optional[str]], bool]:
    """
    Returns the initial conditions for solar system,
    with units AU, days, and M_sun.

    Parameters
    ----------
    initial_condition : str
        Name for the initial condition.

    Returns
    -------
    system: System
        System object with initial conditions.
    labels: list
        Labels for the particles.
    colors: list
        Colors for the particles.
    legend: bool
        Whether to show the legend.
    """
    # Conversion factor from km^3 s^-2 to AU^3 d^-2
    CONVERSION_FACTOR = (86400**2) / (149597870.7**3)

    # GM values (km^3 s^-2)
    # ref: https://ssd.jpl.nasa.gov/doc/Park.2021.AJ.DE440.pdf
    GM_KM_S = {
        "Sun": 132712440041.279419,
        "Mercury": 22031.868551,
        "Venus": 324858.592000,
        "Earth": 398600.435507,
        "Mars": 42828.375816,
        "Jupiter": 126712764.100000,
        "Saturn": 37940584.841800,
        "Uranus": 5794556.400000,
        "Neptune": 6836527.100580,
        "Moon": 4902.800118,
        "Pluto": 975.500000,
        "Ceres": 62.62890,
        "Vesta": 17.288245,
    }

    # GM values (AU^3 d^-2)
    GM_AU_DAY = {
        "Sun": 132712440041.279419 * CONVERSION_FACTOR,
        "Mercury": 22031.868551 * CONVERSION_FACTOR,
        "Venus": 324858.592000 * CONVERSION_FACTOR,
        "Earth": 398600.435507 * CONVERSION_FACTOR,
        "Mars": 42828.375816 * CONVERSION_FACTOR,
        "Jupiter": 126712764.100000 * CONVERSION_FACTOR,
        "Saturn": 37940584.841800 * CONVERSION_FACTOR,
        "Uranus": 5794556.400000 * CONVERSION_FACTOR,
        "Neptune": 6836527.100580 * CONVERSION_FACTOR,
        "Moon": 4902.800118 * CONVERSION_FACTOR,
        "Pluto": 975.500000 * CONVERSION_FACTOR,
        "Ceres": 62.62890 * CONVERSION_FACTOR,
        "Vesta": 17.288245 * CONVERSION_FACTOR,
    }

    # Solar system masses (M_sun^-1)
    SOLAR_SYSTEM_MASSES = {
        "Sun": 1.0,
        "Mercury": GM_KM_S["Mercury"] / GM_KM_S["Sun"],
        "Venus": GM_KM_S["Venus"] / GM_KM_S["Sun"],
        "Earth": GM_KM_S["Earth"] / GM_KM_S["Sun"],
        "Mars": GM_KM_S["Mars"] / GM_KM_S["Sun"],
        "Jupiter": GM_KM_S["Jupiter"] / GM_KM_S["Sun"],
        "Saturn": GM_KM_S["Saturn"] / GM_KM_S["Sun"],
        "Uranus": GM_KM_S["Uranus"] / GM_KM_S["Sun"],
        "Neptune": GM_KM_S["Neptune"] / GM_KM_S["Sun"],
        "Moon": GM_KM_S["Moon"] / GM_KM_S["Sun"],
        "Pluto": GM_KM_S["Pluto"] / GM_KM_S["Sun"],
        "Ceres": GM_KM_S["Ceres"] / GM_KM_S["Sun"],
        "Vesta": GM_KM_S["Vesta"] / GM_KM_S["Sun"],
    }

    G = GM_AU_DAY["Sun"]

    # Solar system position and velocities data
    # Units: AU-D
    # Coordinate center: Solar System Barycenter
    # Data dated on A.D. 2024-Jan-01 00:00:00.0000 TDB
    # Computational data generated by NASA JPL Horizons System https://ssd.jpl.nasa.gov/horizons/
    SOLAR_SYSTEM_POS = {
        "Sun": [-7.967955691533730e-03, -2.906227441573178e-03, 2.103054301547123e-04],
        "Mercury": [
            -2.825983269538632e-01,
            1.974559795958082e-01,
            4.177433558063677e-02,
        ],
        "Venus": [
            -7.232103701666379e-01,
            -7.948302026312400e-02,
            4.042871428174315e-02,
        ],
        "Earth": [-1.738192017257054e-01, 9.663245550235138e-01, 1.553901854897183e-04],
        "Mars": [-3.013262392582653e-01, -1.454029331393295e00, -2.300531433991428e-02],
        "Jupiter": [3.485202469657674e00, 3.552136904413157e00, -9.271035442798399e-02],
        "Saturn": [8.988104223143450e00, -3.719064854634689e00, -2.931937777323593e-01],
        "Uranus": [1.226302417897505e01, 1.529738792480545e01, -1.020549026883563e-01],
        "Neptune": [
            2.983501460984741e01,
            -1.793812957956852e00,
            -6.506401132254588e-01,
        ],
        "Moon": [-1.762788124769829e-01, 9.674377513177153e-01, 3.236901585768862e-04],
        "Pluto": [1.720200478843485e01, -3.034155683573043e01, -1.729127607100611e00],
        "Ceres": [-1.103880510367569e00, -2.533340440444230e00, 1.220283937721780e-01],
        "Vesta": [-8.092549658731499e-02, 2.558381434460076e00, -6.695836142398572e-02],
    }
    SOLAR_SYSTEM_VEL = {
        "Sun": [4.875094764261564e-06, -7.057133213976680e-06, -4.573453713094512e-08],
        "Mercury": [
            -2.232165900189702e-02,
            -2.157207103176252e-02,
            2.855193410495743e-04,
        ],
        "Venus": [
            2.034068201002341e-03,
            -2.020828626592994e-02,
            -3.945639843855159e-04,
        ],
        "Earth": [
            -1.723001232538228e-02,
            -2.967721342618870e-03,
            6.382125383116755e-07,
        ],
        "Mars": [1.424832259345280e-02, -1.579236181580905e-03, -3.823722796161561e-04],
        "Jupiter": [
            -5.470970658852281e-03,
            5.642487338479145e-03,
            9.896190602066252e-05,
        ],
        "Saturn": [
            1.822013845554067e-03,
            5.143470425888054e-03,
            -1.617235904887937e-04,
        ],
        "Uranus": [
            -3.097615358317413e-03,
            2.276781932345769e-03,
            4.860433222241686e-05,
        ],
        "Neptune": [
            1.676536611817232e-04,
            3.152098732861913e-03,
            -6.877501095688201e-05,
        ],
        "Moon": [
            -1.746667306153906e-02,
            -3.473438277358121e-03,
            -3.359028758606074e-05,
        ],
        "Pluto": [2.802810313667557e-03, 8.492056438614633e-04, -9.060790113327894e-04],
        "Ceres": [
            8.978653480111301e-03,
            -4.873256528198994e-03,
            -1.807162046049230e-03,
        ],
        "Vesta": [
            -1.017876585480054e-02,
            -5.452367109338154e-04,
            1.255870551153315e-03,
        ],
    }

    SOLAR_SYSTEM_COLORS = {
        "Sun": "orange",
        "Mercury": "slategrey",
        "Venus": "wheat",
        "Earth": "skyblue",
        "Mars": "red",
        "Jupiter": "darkgoldenrod",
        "Saturn": "gold",
        "Uranus": "paleturquoise",
        "Neptune": "blue",
    }

    SOLAR_SYSTEM_PLUS_COLORS = {
        "Sun": "orange",
        "Mercury": "slategrey",
        "Venus": "wheat",
        "Earth": "skyblue",
        "Mars": "red",
        "Jupiter": "darkgoldenrod",
        "Saturn": "gold",
        "Uranus": "paleturquoise",
        "Neptune": "blue",
        "Pluto": None,
        "Ceres": None,
        "Vesta": None,
    }

    if initial_condition == "pyth-3-body":
        # Pythagorean 3-body problem
        R1 = np.array([1.0, 3.0, 0.0])
        R2 = np.array([-2.0, -1.0, 0.0])
        R3 = np.array([1.0, -1.0, 0.0])
        V1 = np.array([0.0, 0.0, 0.0])
        V2 = np.array([0.0, 0.0, 0.0])
        V3 = np.array([0.0, 0.0, 0.0])

        x = np.array([R1, R2, R3])
        v = np.array([V1, V2, V3])
        m = np.array([3.0 / G, 4.0 / G, 5.0 / G])

        system = System(
            num_particles=len(m),
            x=x,
            v=v,
            m=m,
            G=G,
        )
        system.center_of_mass_correction()

        labels: List[Optional[str]] = [None, None, None]
        colors: List[Optional[str]] = [None, None, None]
        legend = False

        return system, labels, colors, legend

    elif initial_condition == "solar_system":
        m = np.array(
            [
                SOLAR_SYSTEM_MASSES["Sun"],
                SOLAR_SYSTEM_MASSES["Mercury"],
                SOLAR_SYSTEM_MASSES["Venus"],
                SOLAR_SYSTEM_MASSES["Earth"],
                SOLAR_SYSTEM_MASSES["Mars"],
                SOLAR_SYSTEM_MASSES["Jupiter"],
                SOLAR_SYSTEM_MASSES["Saturn"],
                SOLAR_SYSTEM_MASSES["Uranus"],
                SOLAR_SYSTEM_MASSES["Neptune"],
            ]
        )

        R1 = np.array(SOLAR_SYSTEM_POS["Sun"])
        R2 = np.array(SOLAR_SYSTEM_POS["Mercury"])
        R3 = np.array(SOLAR_SYSTEM_POS["Venus"])
        R4 = np.array(SOLAR_SYSTEM_POS["Earth"])
        R5 = np.array(SOLAR_SYSTEM_POS["Mars"])
        R6 = np.array(SOLAR_SYSTEM_POS["Jupiter"])
        R7 = np.array(SOLAR_SYSTEM_POS["Saturn"])
        R8 = np.array(SOLAR_SYSTEM_POS["Uranus"])
        R9 = np.array(SOLAR_SYSTEM_POS["Neptune"])

        V1 = np.array(SOLAR_SYSTEM_VEL["Sun"])
        V2 = np.array(SOLAR_SYSTEM_VEL["Mercury"])
        V3 = np.array(SOLAR_SYSTEM_VEL["Venus"])
        V4 = np.array(SOLAR_SYSTEM_VEL["Earth"])
        V5 = np.array(SOLAR_SYSTEM_VEL["Mars"])
        V6 = np.array(SOLAR_SYSTEM_VEL["Jupiter"])
        V7 = np.array(SOLAR_SYSTEM_VEL["Saturn"])
        V8 = np.array(SOLAR_SYSTEM_VEL["Uranus"])
        V9 = np.array(SOLAR_SYSTEM_VEL["Neptune"])

        x = np.array(
            [
                R1,
                R2,
                R3,
                R4,
                R5,
                R6,
                R7,
                R8,
                R9,
            ]
        )
        v = np.array(
            [
                V1,
                V2,
                V3,
                V4,
                V5,
                V6,
                V7,
                V8,
                V9,
            ]
        )

        system = System(
            num_particles=len(m),
            x=x,
            v=v,
            m=m,
            G=G,
        )
        system.center_of_mass_correction()

        labels = list(SOLAR_SYSTEM_POS.keys())
        colors = list(SOLAR_SYSTEM_COLORS.values())
        legend = True

        return system, labels, colors, legend

    elif initial_condition == "solar_system_plus":
        m = np.array(
            [
                SOLAR_SYSTEM_MASSES["Sun"],
                SOLAR_SYSTEM_MASSES["Mercury"],
                SOLAR_SYSTEM_MASSES["Venus"],
                SOLAR_SYSTEM_MASSES["Earth"],
                SOLAR_SYSTEM_MASSES["Mars"],
                SOLAR_SYSTEM_MASSES["Jupiter"],
                SOLAR_SYSTEM_MASSES["Saturn"],
                SOLAR_SYSTEM_MASSES["Uranus"],
                SOLAR_SYSTEM_MASSES["Neptune"],
                SOLAR_SYSTEM_MASSES["Pluto"],
                SOLAR_SYSTEM_MASSES["Ceres"],
                SOLAR_SYSTEM_MASSES["Vesta"],
            ]
        )

        R1 = np.array(SOLAR_SYSTEM_POS["Sun"])
        R2 = np.array(SOLAR_SYSTEM_POS["Mercury"])
        R3 = np.array(SOLAR_SYSTEM_POS["Venus"])
        R4 = np.array(SOLAR_SYSTEM_POS["Earth"])
        R5 = np.array(SOLAR_SYSTEM_POS["Mars"])
        R6 = np.array(SOLAR_SYSTEM_POS["Jupiter"])
        R7 = np.array(SOLAR_SYSTEM_POS["Saturn"])
        R8 = np.array(SOLAR_SYSTEM_POS["Uranus"])
        R9 = np.array(SOLAR_SYSTEM_POS["Neptune"])
        R10 = np.array(SOLAR_SYSTEM_POS["Pluto"])
        R11 = np.array(SOLAR_SYSTEM_POS["Ceres"])
        R12 = np.array(SOLAR_SYSTEM_POS["Vesta"])

        V1 = np.array(SOLAR_SYSTEM_VEL["Sun"])
        V2 = np.array(SOLAR_SYSTEM_VEL["Mercury"])
        V3 = np.array(SOLAR_SYSTEM_VEL["Venus"])
        V4 = np.array(SOLAR_SYSTEM_VEL["Earth"])
        V5 = np.array(SOLAR_SYSTEM_VEL["Mars"])
        V6 = np.array(SOLAR_SYSTEM_VEL["Jupiter"])
        V7 = np.array(SOLAR_SYSTEM_VEL["Saturn"])
        V8 = np.array(SOLAR_SYSTEM_VEL["Uranus"])
        V9 = np.array(SOLAR_SYSTEM_VEL["Neptune"])
        V10 = np.array(SOLAR_SYSTEM_VEL["Pluto"])
        V11 = np.array(SOLAR_SYSTEM_VEL["Ceres"])
        V12 = np.array(SOLAR_SYSTEM_VEL["Vesta"])

        x = np.array(
            [
                R1,
                R2,
                R3,
                R4,
                R5,
                R6,
                R7,
                R8,
                R9,
                R10,
                R11,
                R12,
            ]
        )
        v = np.array(
            [
                V1,
                V2,
                V3,
                V4,
                V5,
                V6,
                V7,
                V8,
                V9,
                V10,
                V11,
                V12,
            ]
        )

        system = System(
            num_particles=len(m),
            x=x,
            v=v,
            m=m,
            G=G,
        )
        system.center_of_mass_correction()

        labels = list(SOLAR_SYSTEM_POS.keys())
        colors = list(SOLAR_SYSTEM_PLUS_COLORS.values())
        legend = True

        return system, labels, colors, legend

    else:
        raise ValueError(f"Initial condition not recognized: {initial_condition}.")

Plotting initial conditions

Finally, we will implement a function to plot the initial conditions of the solar system. We will use the matplotlib package to plot the positions of the particles in 2D. Colors and labels are optional, but they make the plot look nicer. If plt.show() does not work in your environment, you may need to change it to plt.savefig(file_name) to save the plot.

common.py
def plot_initial_conditions(
    system: System,
    labels: list,
    colors: list,
    legend: bool,
) -> None:
    """
    Plot the initial positions.

    Parameters
    ----------
    system : System
        System object.
    labels : list
        Labels for the particles.
    colors : list
        Colors for the particles.
    legend : bool
        Whether to show the legend.
    """
    fig, ax = plt.subplots()
    ax.set_xlabel("$x$ (AU)")
    ax.set_ylabel("$y$ (AU)")

    for i in range(system.num_particles):
        ax.scatter(
            system.x[i, 0], system.x[i, 1], marker="o", color=colors[i], label=labels[i]
        )

    if legend:
        ax.legend()

    plt.show() # Here, you may need to change to plt.savefig(file_name) if 
               # plt.show() does not work in your environment.

Test the code

Now we could try to run the code with step1.py.

step1.py
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import common

INITIAL_CONDITION = "solar_system"


def main():
    # Get initial conditions
    system, labels, colors, legend = common.get_initial_conditions(INITIAL_CONDITION)
    print("Number of particles:", system.num_particles)
    print("Initial positions (AU):\n", system.x)
    print("Initial velocities (AU/day):\n", system.v)
    print("Masses (M_sun):\n", system.m)
    print("Gravitational constant (AU^3 / day^2 / M_sun):", system.G)

    # Plot the initial conditions
    common.plot_initial_conditions(
        system=system,
        labels=labels,
        colors=colors,
        legend=legend,
    )


if __name__ == "__main__":
    main()

As you run the code, you should see the following output: 

Number of particles: 9
Initial positions (AU):
 [[-7.96712825e-03 -2.90611166e-03  2.10213120e-04]
 [-2.82597500e-01  1.97456095e-01  4.17742433e-02]
 [-7.23209543e-01 -7.94829045e-02  4.04286220e-02]
 [-1.73818374e-01  9.66324671e-01  1.55297876e-04]
 [-3.01325412e-01 -1.45402922e+00 -2.30054066e-02]
 [ 3.48520330e+00  3.55213702e+00 -9.27104467e-02]
 [ 8.98810505e+00 -3.71906474e+00 -2.93193870e-01]
 [ 1.22630250e+01  1.52973880e+01 -1.02054995e-01]
 [ 2.98350154e+01 -1.79381284e+00 -6.50640206e-01]]
Initial velocities (AU/day):
 [[ 4.87524241e-06 -7.05716139e-06 -4.57929038e-08]
 [-2.23216589e-02 -2.15720711e-02  2.85519283e-04]
 [ 2.03406835e-03 -2.02082863e-02 -3.94564043e-04]
 [-1.72300122e-02 -2.96772137e-03  6.38154172e-07]
 [ 1.42483227e-02 -1.57923621e-03 -3.82372338e-04]
 [-5.47097051e-03  5.64248731e-03  9.89618477e-05]
 [ 1.82201399e-03  5.14347040e-03 -1.61723649e-04]
 [-3.09761521e-03  2.27678190e-03  4.86042739e-05]
 [ 1.67653809e-04  3.15209870e-03 -6.87750693e-05]]
Masses (M_sun):
 [1.00000000e+00 1.66012083e-07 2.44783829e-06 3.00348962e-06
 3.22715608e-07 9.54791910e-04 2.85885670e-04 4.36624961e-05
 5.15138377e-05]
Gravitational constant (AU^3 / day^2 / M_sun): 0.00029591220828411956
You should also see the following plot:

Initial Condition

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/

step1.py (Click to expand)
5_steps_to_n_body_simulation/python/step1.py
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import common

INITIAL_CONDITION = "solar_system"


def main():
    # Get initial conditions
    system, labels, colors, legend = common.get_initial_conditions(INITIAL_CONDITION)
    print("Number of particles:", system.num_particles)
    print("Initial positions (AU):\n", system.x)
    print("Initial velocities (AU/day):\n", system.v)
    print("Masses (M_sun):\n", system.m)
    print("Gravitational constant (AU^3 / day^2 / M_sun):", system.G)

    # Plot the initial conditions
    common.plot_initial_conditions(
        system=system,
        labels=labels,
        colors=colors,
        legend=legend,
    )


if __name__ == "__main__":
    main()
common.py (Click to expand)
5_steps_to_n_body_simulation/python/common.py
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from typing import Tuple, List, Optional

import numpy as np
import matplotlib.pyplot as plt


##### Step 1 #####
class System:
    def __init__(
        self, num_particles: int, x: np.ndarray, v: np.ndarray, m: np.ndarray, G: float
    ) -> None:
        self.num_particles = num_particles
        self.x = x
        self.v = v
        self.m = m
        self.G = G

    def center_of_mass_correction(self) -> None:
        """Set center of mass of position and velocity to zero"""
        M = np.sum(self.m)
        x_cm = np.einsum("i,ij->j", self.m, self.x) / M
        v_cm = np.einsum("i,ij->j", self.m, self.v) / M

        self.x -= x_cm
        self.v -= v_cm


def get_initial_conditions(
    initial_condition: str,
) -> Tuple[System, List[Optional[str]], List[Optional[str]], bool]:
    """
    Returns the initial conditions for solar system,
    with units AU, days, and M_sun.

    Parameters
    ----------
    initial_condition : str
        Name for the initial condition.

    Returns
    -------
    system: System
        System object with initial conditions.
    labels: list
        Labels for the particles.
    colors: list
        Colors for the particles.
    legend: bool
        Whether to show the legend.
    """
    # Conversion factor from km^3 s^-2 to AU^3 d^-2
    CONVERSION_FACTOR = (86400**2) / (149597870.7**3)

    # GM values (km^3 s^-2)
    # ref: https://ssd.jpl.nasa.gov/doc/Park.2021.AJ.DE440.pdf
    GM_KM_S = {
        "Sun": 132712440041.279419,
        "Mercury": 22031.868551,
        "Venus": 324858.592000,
        "Earth": 398600.435507,
        "Mars": 42828.375816,
        "Jupiter": 126712764.100000,
        "Saturn": 37940584.841800,
        "Uranus": 5794556.400000,
        "Neptune": 6836527.100580,
        "Moon": 4902.800118,
        "Pluto": 975.500000,
        "Ceres": 62.62890,
        "Vesta": 17.288245,
    }

    # GM values (AU^3 d^-2)
    GM_AU_DAY = {
        "Sun": 132712440041.279419 * CONVERSION_FACTOR,
        "Mercury": 22031.868551 * CONVERSION_FACTOR,
        "Venus": 324858.592000 * CONVERSION_FACTOR,
        "Earth": 398600.435507 * CONVERSION_FACTOR,
        "Mars": 42828.375816 * CONVERSION_FACTOR,
        "Jupiter": 126712764.100000 * CONVERSION_FACTOR,
        "Saturn": 37940584.841800 * CONVERSION_FACTOR,
        "Uranus": 5794556.400000 * CONVERSION_FACTOR,
        "Neptune": 6836527.100580 * CONVERSION_FACTOR,
        "Moon": 4902.800118 * CONVERSION_FACTOR,
        "Pluto": 975.500000 * CONVERSION_FACTOR,
        "Ceres": 62.62890 * CONVERSION_FACTOR,
        "Vesta": 17.288245 * CONVERSION_FACTOR,
    }

    # Solar system masses (M_sun^-1)
    SOLAR_SYSTEM_MASSES = {
        "Sun": 1.0,
        "Mercury": GM_KM_S["Mercury"] / GM_KM_S["Sun"],
        "Venus": GM_KM_S["Venus"] / GM_KM_S["Sun"],
        "Earth": GM_KM_S["Earth"] / GM_KM_S["Sun"],
        "Mars": GM_KM_S["Mars"] / GM_KM_S["Sun"],
        "Jupiter": GM_KM_S["Jupiter"] / GM_KM_S["Sun"],
        "Saturn": GM_KM_S["Saturn"] / GM_KM_S["Sun"],
        "Uranus": GM_KM_S["Uranus"] / GM_KM_S["Sun"],
        "Neptune": GM_KM_S["Neptune"] / GM_KM_S["Sun"],
        "Moon": GM_KM_S["Moon"] / GM_KM_S["Sun"],
        "Pluto": GM_KM_S["Pluto"] / GM_KM_S["Sun"],
        "Ceres": GM_KM_S["Ceres"] / GM_KM_S["Sun"],
        "Vesta": GM_KM_S["Vesta"] / GM_KM_S["Sun"],
    }

    G = GM_AU_DAY["Sun"]

    # Solar system position and velocities data
    # Units: AU-D
    # Coordinate center: Solar System Barycenter
    # Data dated on A.D. 2024-Jan-01 00:00:00.0000 TDB
    # Computational data generated by NASA JPL Horizons System https://ssd.jpl.nasa.gov/horizons/
    SOLAR_SYSTEM_POS = {
        "Sun": [-7.967955691533730e-03, -2.906227441573178e-03, 2.103054301547123e-04],
        "Mercury": [
            -2.825983269538632e-01,
            1.974559795958082e-01,
            4.177433558063677e-02,
        ],
        "Venus": [
            -7.232103701666379e-01,
            -7.948302026312400e-02,
            4.042871428174315e-02,
        ],
        "Earth": [-1.738192017257054e-01, 9.663245550235138e-01, 1.553901854897183e-04],
        "Mars": [-3.013262392582653e-01, -1.454029331393295e00, -2.300531433991428e-02],
        "Jupiter": [3.485202469657674e00, 3.552136904413157e00, -9.271035442798399e-02],
        "Saturn": [8.988104223143450e00, -3.719064854634689e00, -2.931937777323593e-01],
        "Uranus": [1.226302417897505e01, 1.529738792480545e01, -1.020549026883563e-01],
        "Neptune": [
            2.983501460984741e01,
            -1.793812957956852e00,
            -6.506401132254588e-01,
        ],
        "Moon": [-1.762788124769829e-01, 9.674377513177153e-01, 3.236901585768862e-04],
        "Pluto": [1.720200478843485e01, -3.034155683573043e01, -1.729127607100611e00],
        "Ceres": [-1.103880510367569e00, -2.533340440444230e00, 1.220283937721780e-01],
        "Vesta": [-8.092549658731499e-02, 2.558381434460076e00, -6.695836142398572e-02],
    }
    SOLAR_SYSTEM_VEL = {
        "Sun": [4.875094764261564e-06, -7.057133213976680e-06, -4.573453713094512e-08],
        "Mercury": [
            -2.232165900189702e-02,
            -2.157207103176252e-02,
            2.855193410495743e-04,
        ],
        "Venus": [
            2.034068201002341e-03,
            -2.020828626592994e-02,
            -3.945639843855159e-04,
        ],
        "Earth": [
            -1.723001232538228e-02,
            -2.967721342618870e-03,
            6.382125383116755e-07,
        ],
        "Mars": [1.424832259345280e-02, -1.579236181580905e-03, -3.823722796161561e-04],
        "Jupiter": [
            -5.470970658852281e-03,
            5.642487338479145e-03,
            9.896190602066252e-05,
        ],
        "Saturn": [
            1.822013845554067e-03,
            5.143470425888054e-03,
            -1.617235904887937e-04,
        ],
        "Uranus": [
            -3.097615358317413e-03,
            2.276781932345769e-03,
            4.860433222241686e-05,
        ],
        "Neptune": [
            1.676536611817232e-04,
            3.152098732861913e-03,
            -6.877501095688201e-05,
        ],
        "Moon": [
            -1.746667306153906e-02,
            -3.473438277358121e-03,
            -3.359028758606074e-05,
        ],
        "Pluto": [2.802810313667557e-03, 8.492056438614633e-04, -9.060790113327894e-04],
        "Ceres": [
            8.978653480111301e-03,
            -4.873256528198994e-03,
            -1.807162046049230e-03,
        ],
        "Vesta": [
            -1.017876585480054e-02,
            -5.452367109338154e-04,
            1.255870551153315e-03,
        ],
    }

    SOLAR_SYSTEM_COLORS = {
        "Sun": "orange",
        "Mercury": "slategrey",
        "Venus": "wheat",
        "Earth": "skyblue",
        "Mars": "red",
        "Jupiter": "darkgoldenrod",
        "Saturn": "gold",
        "Uranus": "paleturquoise",
        "Neptune": "blue",
    }

    SOLAR_SYSTEM_PLUS_COLORS = {
        "Sun": "orange",
        "Mercury": "slategrey",
        "Venus": "wheat",
        "Earth": "skyblue",
        "Mars": "red",
        "Jupiter": "darkgoldenrod",
        "Saturn": "gold",
        "Uranus": "paleturquoise",
        "Neptune": "blue",
        "Pluto": None,
        "Ceres": None,
        "Vesta": None,
    }

    if initial_condition == "pyth-3-body":
        # Pythagorean 3-body problem
        R1 = np.array([1.0, 3.0, 0.0])
        R2 = np.array([-2.0, -1.0, 0.0])
        R3 = np.array([1.0, -1.0, 0.0])
        V1 = np.array([0.0, 0.0, 0.0])
        V2 = np.array([0.0, 0.0, 0.0])
        V3 = np.array([0.0, 0.0, 0.0])

        x = np.array([R1, R2, R3])
        v = np.array([V1, V2, V3])
        m = np.array([3.0 / G, 4.0 / G, 5.0 / G])

        system = System(
            num_particles=len(m),
            x=x,
            v=v,
            m=m,
            G=G,
        )
        system.center_of_mass_correction()

        labels: List[Optional[str]] = [None, None, None]
        colors: List[Optional[str]] = [None, None, None]
        legend = False

        return system, labels, colors, legend

    elif initial_condition == "solar_system":
        m = np.array(
            [
                SOLAR_SYSTEM_MASSES["Sun"],
                SOLAR_SYSTEM_MASSES["Mercury"],
                SOLAR_SYSTEM_MASSES["Venus"],
                SOLAR_SYSTEM_MASSES["Earth"],
                SOLAR_SYSTEM_MASSES["Mars"],
                SOLAR_SYSTEM_MASSES["Jupiter"],
                SOLAR_SYSTEM_MASSES["Saturn"],
                SOLAR_SYSTEM_MASSES["Uranus"],
                SOLAR_SYSTEM_MASSES["Neptune"],
            ]
        )

        R1 = np.array(SOLAR_SYSTEM_POS["Sun"])
        R2 = np.array(SOLAR_SYSTEM_POS["Mercury"])
        R3 = np.array(SOLAR_SYSTEM_POS["Venus"])
        R4 = np.array(SOLAR_SYSTEM_POS["Earth"])
        R5 = np.array(SOLAR_SYSTEM_POS["Mars"])
        R6 = np.array(SOLAR_SYSTEM_POS["Jupiter"])
        R7 = np.array(SOLAR_SYSTEM_POS["Saturn"])
        R8 = np.array(SOLAR_SYSTEM_POS["Uranus"])
        R9 = np.array(SOLAR_SYSTEM_POS["Neptune"])

        V1 = np.array(SOLAR_SYSTEM_VEL["Sun"])
        V2 = np.array(SOLAR_SYSTEM_VEL["Mercury"])
        V3 = np.array(SOLAR_SYSTEM_VEL["Venus"])
        V4 = np.array(SOLAR_SYSTEM_VEL["Earth"])
        V5 = np.array(SOLAR_SYSTEM_VEL["Mars"])
        V6 = np.array(SOLAR_SYSTEM_VEL["Jupiter"])
        V7 = np.array(SOLAR_SYSTEM_VEL["Saturn"])
        V8 = np.array(SOLAR_SYSTEM_VEL["Uranus"])
        V9 = np.array(SOLAR_SYSTEM_VEL["Neptune"])

        x = np.array(
            [
                R1,
                R2,
                R3,
                R4,
                R5,
                R6,
                R7,
                R8,
                R9,
            ]
        )
        v = np.array(
            [
                V1,
                V2,
                V3,
                V4,
                V5,
                V6,
                V7,
                V8,
                V9,
            ]
        )

        system = System(
            num_particles=len(m),
            x=x,
            v=v,
            m=m,
            G=G,
        )
        system.center_of_mass_correction()

        labels = list(SOLAR_SYSTEM_COLORS.keys())
        colors = list(SOLAR_SYSTEM_COLORS.values())
        legend = True

        return system, labels, colors, legend

    elif initial_condition == "solar_system_plus":
        m = np.array(
            [
                SOLAR_SYSTEM_MASSES["Sun"],
                SOLAR_SYSTEM_MASSES["Mercury"],
                SOLAR_SYSTEM_MASSES["Venus"],
                SOLAR_SYSTEM_MASSES["Earth"],
                SOLAR_SYSTEM_MASSES["Mars"],
                SOLAR_SYSTEM_MASSES["Jupiter"],
                SOLAR_SYSTEM_MASSES["Saturn"],
                SOLAR_SYSTEM_MASSES["Uranus"],
                SOLAR_SYSTEM_MASSES["Neptune"],
                SOLAR_SYSTEM_MASSES["Pluto"],
                SOLAR_SYSTEM_MASSES["Ceres"],
                SOLAR_SYSTEM_MASSES["Vesta"],
            ]
        )

        R1 = np.array(SOLAR_SYSTEM_POS["Sun"])
        R2 = np.array(SOLAR_SYSTEM_POS["Mercury"])
        R3 = np.array(SOLAR_SYSTEM_POS["Venus"])
        R4 = np.array(SOLAR_SYSTEM_POS["Earth"])
        R5 = np.array(SOLAR_SYSTEM_POS["Mars"])
        R6 = np.array(SOLAR_SYSTEM_POS["Jupiter"])
        R7 = np.array(SOLAR_SYSTEM_POS["Saturn"])
        R8 = np.array(SOLAR_SYSTEM_POS["Uranus"])
        R9 = np.array(SOLAR_SYSTEM_POS["Neptune"])
        R10 = np.array(SOLAR_SYSTEM_POS["Pluto"])
        R11 = np.array(SOLAR_SYSTEM_POS["Ceres"])
        R12 = np.array(SOLAR_SYSTEM_POS["Vesta"])

        V1 = np.array(SOLAR_SYSTEM_VEL["Sun"])
        V2 = np.array(SOLAR_SYSTEM_VEL["Mercury"])
        V3 = np.array(SOLAR_SYSTEM_VEL["Venus"])
        V4 = np.array(SOLAR_SYSTEM_VEL["Earth"])
        V5 = np.array(SOLAR_SYSTEM_VEL["Mars"])
        V6 = np.array(SOLAR_SYSTEM_VEL["Jupiter"])
        V7 = np.array(SOLAR_SYSTEM_VEL["Saturn"])
        V8 = np.array(SOLAR_SYSTEM_VEL["Uranus"])
        V9 = np.array(SOLAR_SYSTEM_VEL["Neptune"])
        V10 = np.array(SOLAR_SYSTEM_VEL["Pluto"])
        V11 = np.array(SOLAR_SYSTEM_VEL["Ceres"])
        V12 = np.array(SOLAR_SYSTEM_VEL["Vesta"])

        x = np.array(
            [
                R1,
                R2,
                R3,
                R4,
                R5,
                R6,
                R7,
                R8,
                R9,
                R10,
                R11,
                R12,
            ]
        )
        v = np.array(
            [
                V1,
                V2,
                V3,
                V4,
                V5,
                V6,
                V7,
                V8,
                V9,
                V10,
                V11,
                V12,
            ]
        )

        system = System(
            num_particles=len(m),
            x=x,
            v=v,
            m=m,
            G=G,
        )
        system.center_of_mass_correction()

        labels = list(SOLAR_SYSTEM_PLUS_COLORS.keys())
        colors = list(SOLAR_SYSTEM_PLUS_COLORS.values())
        legend = True

        return system, labels, colors, legend

    else:
        raise ValueError(f"Initial condition not recognized: {initial_condition}.")


def plot_initial_conditions(
    system: System,
    labels: list,
    colors: list,
    legend: bool,
) -> None:
    """
    Plot the initial positions.

    Parameters
    ----------
    system : System
        System object.
    labels : list
        Labels for the particles.
    colors : list
        Colors for the particles.
    legend : bool
        Whether to show the legend.
    """
    fig, ax = plt.subplots()
    ax.set_xlabel("$x$ (AU)")
    ax.set_ylabel("$y$ (AU)")

    for i in range(system.num_particles):
        ax.scatter(
            system.x[i, 0], system.x[i, 1], marker="o", color=colors[i], label=labels[i]
        )

    if legend:
        ax.legend()

    plt.show()


##### Step 2 #####
def acceleration(
    a: np.ndarray,
    system: System,
) -> None:
    """
    Compute the gravitational acceleration

    Parameters
    ----------
    a : np.ndarray
        Gravitational accelerations array to be modified in-place,
        with shape (N, 3)
    system : System
        System object.
    """
    # Empty acceleration array
    a.fill(0.0)

    # Declare variables
    x = system.x
    m = system.m
    G = system.G

    # Compute the displacement vector
    r_ij = x[:, np.newaxis, :] - x[np.newaxis, :, :]

    # Compute the distance
    r_norm = np.linalg.norm(r_ij, axis=2)

    # Compute 1 / r^3
    with np.errstate(divide="ignore", invalid="ignore"):
        inv_r_cubed = 1.0 / (r_norm * r_norm * r_norm)

    # Set diagonal elements to 0 to avoid self-interaction
    np.fill_diagonal(inv_r_cubed, 0.0)

    # Compute the acceleration
    a[:] = G * np.einsum("ijk,ij,i->jk", r_ij, inv_r_cubed, m)


##### Step 3 #####
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


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("----------------------------------------------------------")


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()


##### Step 4 #####
def compute_rel_energy_error(
    sol_x: np.ndarray, sol_v: np.ndarray, system: System
) -> np.ndarray:
    """
    Compute the relative energy error of the simulation.

    Parameters
    ----------
    sol_x : np.ndarray
        Solution position array with shape (N_steps, num_particles, 3).
    sol_v : np.ndarray
        Solution velocity array with shape (N_steps, num_particles, 3).
    system : System
        System object.

    Returns
    -------
    energy_error : np.ndarray
        Relative energy error of the simulation, with shape (N_steps,).
    """
    # Allocate memory and initialize arrays
    n_steps = sol_x.shape[0]
    num_particles = system.num_particles
    m = system.m
    G = system.G
    rel_energy_error = np.zeros(n_steps)

    # Compute the total energy (KE + PE)
    for count in range(n_steps):
        x = sol_x[count]
        v = sol_v[count]
        for i in range(num_particles):
            # KE
            rel_energy_error[count] += 0.5 * m[i] * np.linalg.norm(v[i]) ** 2
            # PE
            for j in range(i + 1, num_particles):
                rel_energy_error[count] -= G * m[i] * m[j] / np.linalg.norm(x[i] - x[j])

    # Compute the relative energy error
    initial_energy = rel_energy_error[0]
    rel_energy_error = (rel_energy_error - initial_energy) / initial_energy
    rel_energy_error = np.abs(rel_energy_error)

    return rel_energy_error


def plot_rel_energy_error(rel_energy_error: np.ndarray, sol_t: np.ndarray) -> None:
    """
    Plot the relative energy error.

    Parameters
    ----------
    rel_energy_error : np.ndarray
        Relative energy error of the simulation, with shape (N_steps,).
    sol_t : np.ndarray
        Solution time array with shape (N_steps,).
    """
    plt.figure()
    plt.plot(sol_t, rel_energy_error)
    plt.yscale("log")
    plt.xlabel("Time step")
    plt.ylabel("Relative Energy Error")
    plt.title("Relative Energy Error vs Time Step")
    plt.show()


def euler_cromer(a: np.ndarray, system: System, dt: float) -> None:
    """
    Advance one step with the Euler-Cromer method.

    Parameters
    ----------
    a : np.ndarray
        Gravitational accelerations array with shape (N, 3).
    system : System
        System object.
    dt : float
        Time step.
    """
    acceleration(a, system)
    system.v += a * dt
    system.x += system.v * dt


def rk4(a: np.ndarray, system: System, dt: float) -> None:
    """
    Advance one step with the RK4 method.

    Parameters
    ----------
    a : np.ndarray
        Gravitational accelerations array with shape (N, 3).
    system : System
        System object.
    dt : float
        Time step.
    """
    num_stages = 4
    coeff = np.array([0.5, 0.5, 1.0])
    weights = np.array([1.0, 2.0, 2.0, 1.0]) / 6.0

    # Allocate memory and initialize arrays
    x0 = system.x.copy()
    v0 = system.v.copy()
    xk = np.zeros((num_stages, system.num_particles, 3))
    vk = np.zeros((num_stages, system.num_particles, 3))

    # Initial stage
    acceleration(a, system)
    xk[0] = v0
    vk[0] = a

    # Compute the stages
    for stage in range(1, num_stages):
        # Compute acceleration
        system.x = x0 + dt * coeff[stage - 1] * xk[stage - 1]
        acceleration(a, system)

        # Compute xk and vk
        xk[stage] = v0 + dt * coeff[stage - 1] * vk[stage - 1]
        vk[stage] = a

    # Advance step
    # dx = 0.0
    # dv = 0.0
    # for stage in range(num_stages):
    #     dx += weights[stage] * xk[stage]
    #     dv += weights[stage] * vk[stage]

    dx = np.einsum("i,ijk->jk", weights, xk)
    dv = np.einsum("i,ijk->jk", weights, vk)

    system.x = x0 + dt * dx
    system.v = v0 + dt * dv


def leapfrog(a: np.ndarray, system: System, dt: float) -> None:
    """
    Advance one step with the LeapFrog method.

    Parameters
    ----------
    a : np.ndarray
        Gravitational accelerations array with shape (N, 3).
    system : System
        System object.
    dt : float
        Time step.
    """
    # Velocity kick (v_1/2)
    acceleration(a, system)
    system.v += a * 0.5 * dt

    # Position drift (x_1)
    system.x += system.v * dt

    # Velocity kick (v_1)
    acceleration(a, system)
    system.v += a * 0.5 * dt


##### Step 5 #####
def print_simulation_info_adaptive_step_size(
    system: System,
    tf: float,
    tolerance: float,
    initial_dt: float,
    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")
    print(f"tolerance: {tolerance}")
    print(f"Initial dt: {initial_dt} days")
    print()
    print(f"Output interval: {output_interval} days")
    print(f"Estimated solution size: {sol_size}")
    print("----------------------------------------------------------")


def plot_dt(sol_dt: np.ndarray, sol_t: np.ndarray) -> None:
    """
    Plot the time step.

    Parameters
    ----------
    sol_dt : np.ndarray
        Time step array with shape (N_steps,).
    sol_t : np.ndarray
        Solution time array with shape (N_steps,).
    """
    plt.figure()
    plt.semilogy(sol_t, sol_dt)
    plt.xlabel("Time")
    plt.ylabel("dt")
    plt.show()


##### Extra #####
def set_3d_axes_equal(ax: plt.Axes) -> None:
    """
    Make axes of 3D plot have equal scale

    Parameters
    ----------
    ax : matplotlib axis
        The axis to set equal scale

    Reference
    ---------
    karlo, https://stackoverflow.com/questions/13685386/how-to-set-the-equal-aspect-ratio-for-all-axes-x-y-z
    """

    x_limits = ax.get_xlim3d()  # type: ignore
    y_limits = ax.get_ylim3d()  # type: ignore
    z_limits = ax.get_zlim3d()  # type: ignore

    x_range = abs(x_limits[1] - x_limits[0])
    x_middle = np.mean(x_limits)
    y_range = abs(y_limits[1] - y_limits[0])
    y_middle = np.mean(y_limits)
    z_range = abs(z_limits[1] - z_limits[0])
    z_middle = np.mean(z_limits)

    # The plot bounding box is a sphere in the sense of the infinity
    # norm, hence I call half the max range the plot radius.
    plot_radius = 0.5 * max([x_range, y_range, z_range])

    ax.set_xlim3d([x_middle - plot_radius, x_middle + plot_radius])  # type: ignore
    ax.set_ylim3d([y_middle - plot_radius, y_middle + plot_radius])  # type: ignore
    ax.set_zlim3d([z_middle - plot_radius, z_middle + plot_radius])  # type: ignore


def plot_3d_trajectory(
    sol_x: np.ndarray,
    labels: list,
    colors: list,
    legend: bool,
) -> None:
    """
    Plot the 3D 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, projection="3d")
    ax.set_xlabel("$x$ (AU)")
    ax.set_ylabel("$y$ (AU)")
    ax.set_zlabel("$z$ (AU)")  # type: ignore

    for i in range(sol_x.shape[1]):
        traj = ax.plot(
            sol_x[:, i, 0],
            sol_x[:, i, 1],
            sol_x[:, i, 2],
            color=colors[i],
        )
        # Plot the last position with marker
        ax.scatter(
            sol_x[-1, i, 0],
            sol_x[-1, i, 1],
            sol_x[-1, i, 2],
            marker="o",
            color=traj[0].get_color(),
            label=labels[i],
        )

    set_3d_axes_equal(ax)

    if legend:
        ax.legend(loc="center right", bbox_to_anchor=(1.325, 0.5))
        fig.subplots_adjust(right=0.7)

    plt.show()

  1. Jet Propulsion Laboratory. Horizons system. 2024. Accessed: April 2024. URL: https://ssd.jpl.nasa.gov/horizons/. ↩

  2. Ryan S. Park, William M. Folkner, James G. Williams, and Dale H. Boggs. The jpl planetary and lunar ephemerides de440 and de441. The Astronomical Journal, 161(3):105, feb 2021. doi:10.3847/1538-3881/abd414. ↩

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