Python Gravity Simulator only plots straight lines

Question

I'm trying to make a basic gravity sim using py but for some reason it plots all the lines as straight, I looked at a couple of examples when I got stuck but they all use similar equations / ways of plotting the data so not sure where I went wrong with it

class Body:

    # A class to initialise a body of mass to plot

    def __init__(self, id, mass, coordinates, velocities):
        self.id = id
        self.coordinates = np.array(coordinates, dtype=float)
        self.v = np.array(velocities, dtype=float)
        self.mass = mass
        self.a = np.zeros(3, dtype=float)
        MOTION_LOG.append({"name": self.id, "x":[coordinates[0]], "y": [coordinates[1]], "z": [coordinates[2]]})


    # Procedure for checking gravity effects on body
    def gravity(self):
        self.a = 0
        for body in bodies:
            if body != self:
                dist = body.coordinates - self.coordinates
                r = np.sqrt(np.sum(dist**2))
                self.a += (SETTINGS['G'] * body.mass * dist / r**3) ** 2


# Procedure to plot the new coordinates of the body         
    def move(self):
        self.v += self.a * SETTINGS["deltaT"]
        self.coordinates += self.v * SETTINGS['deltaT']

Then to actually simulate it I did

# For loop to run a simulation for a specific time set
for step in range(int(SETTINGS["tLimit"] / SETTINGS["deltaT"])):
    SETTINGS['elapsedT'] += SETTINGS['deltaT']
    if SETTINGS['elapsedT'] % SETTINGS["frequency"] == 0:
        prog = ((SETTINGS['elapsedT'] / SETTINGS['tLimit'])*100)//1
        for index, location in enumerate(MOTION_LOG):
            location["x"].append(bodies[index].coordinates[0])
            location["y"].append(bodies[index].coordinates[1])
            location["z"].append(bodies[index].coordinates[2])
        print(f"{prog}%")

    for body in bodies:
        body.gravity()
    for body in bodies:
        body.move()

Then finally for plotting it on the graph I did

fig = plt.figure()
ax = fig.add_subplot(1,1,1, projection='3d')

for body in MOTION_LOG:
    ax.plot(body["x"], body["y"], body["z"])

plt.show()

Not sure if I've just done the acceleration wrong or I'm simply plotting the points wrong but other examples I've seen don't do things much differently

Example data

SETTINGS = {
    'G' : 6.67e-11,
    'deltaT' : 172800,
    'elapsedT' : 0,
    'tLimit' : 315360000,
    "frequency": 1,
}

MOTION_LOG = []
bodies = []
set_bodies = {
    "Sun": {
        "id": "Sun",
        "mass": 1e20,
        "coordinates": [0, 0, 0],
        "velocities": [0, 0, 0]
    },
    "Earth": {
        "id": "Earth",
        "mass": 1e3,
        "coordinates": [1e2, -1e2, 0],
        "velocities": [0, 0, 0]
    }
}
for body, body_data in set_bodies.items():
    bodies.append(Body(**body_data))

Answer 1

Firstly, your acceleration is squared. The line you have is:

self.a += (SETTINGS['G'] * body.mass * dist / r**3) ** 2

What you want is:

self.a += (SETTINGS['G'] * body.mass * dist / r**3) 

Second, your numbers are all out of whack. Never, never just guess at nice combinations of numbers to use for simulations like this. Either use completely natural units (where the grav constant, solar mass, and AU are all equal to one), or use all the actual numbers. When you don't do that, you're just guessing at what timesteps are appropriate, and it's really difficult to get that right. For instance, with the numbers you're using for the Solar mass, grav constant, and Earth orbital radius, one unit of time is about 7.7 years. To get something that looks decent, then you'd want a timestep of about 7e-4 and run until a final time of about 1.3, and use an initial velocity of [5000, 5000, 0] for Earth (you'll also need to change how you add to MOTION_LOG since small floats don't play well with mod (%)). But don't do that - use sensible numbers. You'll be happier in the end when you don't have to spend hours upon hours converting back and forth into units you actually want.

Third, your "Earth" object doesn't have an initial velocity. It's falling into the Sun, no matter how you fix your equations or your units.

Here are some about-right "real" numbers using SI units that will work:

set_bodies = {
    "Sun": {
        "id": "Sun",
        "mass": 2e30,
        "coordinates": [0, 0, 0],
        "velocities": [0, 0, 0]
    },
    "Earth": {
        "id": "Earth",
        "mass": 6e24,
        "coordinates": [1.5e11, 0, 0],
        "velocities": [0, 3e4, 0]
    }
}