This is a simple N-body simulation of Celestials.

1. 2D vs 3D: For now we choose to use 3D, but we can also use 2D.

2. System invariance and consistency of state
   Regardless of how and where the bodies are initialized, the states of these initialized points should also be consistent.

• potential solution:
  • center of mass frame
  • fixing the initial positions of the bodies
  • use invariant encodings of the state

3. Collision detection

1. Properties

   • mass (m)
   • position (x, y, z)
   • velocity (vx, vy, vz)

2. Default Units

   • ( G = 1 )
   • ( M_{\text{planet}} = 1 )
   • ( M_{\text{star}} = 1 )

   A one star and one planet system is used as the default system.

3. Solar-sys like Units

   • ( Yr = 365.25 ) days (Julian)
   • ( AU = 1.495978707 \times 10^{11} ) m
   • ( M_{\text{sun}} = 1.98855 \times 10^{30} ) kg
   • ( G = 4 \pi^2 \left( \frac{AU^3}{M_{\text{sun}} \times Yr^2} \right) = 39.47841760435743 )

   Details explaination about the units and gravitational constant.

1. Properies (State)

   • mass (m)
   • position (x, y)
   • velocity (vx, vy)

2. Energy

   • kinetic energy
   • potential energy

3. Default Units

   • ( G = 1 )
   • ( M_{\text{celestial}} = 1 )

4. Return Values

   • Position and Velocity of the bodies in time steps (x, y, vx, vy)
   • Derivatives of the position and velocity of the bodies in time steps (dx/dt, dy/dt, dvx/dt, dvy/dt)
   • Energy of the system in time steps (e = potential + kinetic)

A more advanced implementation: Using the Rebound library to calculate the gravitational interactions between the bodies. The simulation is done in 3D, but the output is a 2D projection of the orbits on the XY plane.

1. Properties
   • mass (m)
   • position (x, y, z)
   • velocity (vx, vy, vz)