diff --git a/README.md b/README.md index 1bbba71..6350e6c 100644 --- a/README.md +++ b/README.md @@ -18,11 +18,13 @@ The simulator consists of several key components: The simulator uses Newton's law of universal gravitation to calculate the forces between celestial bodies. The gravitational force between two bodies is given by: -$$ F = G \frac{m_1 m_2}{r^2} $$ +$$ +F = G \frac{m_1 m_2}{r^2} +$$ Where: - $F$ is the gravitational force between the two bodies -- $G$ is the gravitational constant ($$6.67430 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2$$) +- $G$ is the gravitational constant ($6.67430 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2$) - $m_1$ and $m_2$ are the masses of the two bodies - $r$ is the distance between the centers of the two bodies @@ -31,9 +33,18 @@ Where: The motion of each celestial body is updated using numerical integration. We use a simple Euler method for updating positions and velocities: 1. Calculate the net force on each body -2. Calculate acceleration: $$ \vec{a} = \frac{\vec{F}}{m} $$ -3. Update velocity: $$ \vec{v}_{new} = \vec{v}_{old} + \vec{a} \Delta t $$ -4. Update position: $$ \vec{x}_{new} = \vec{x}_{old} + \vec{v}_{new} \Delta t $$ +2. Calculate acceleration: +```math +$$ \vec{a} = \frac{\vec{F}}{m} $$ +``` +3. Update velocity: +```math +$$ \vec{v}_{new} = \vec{v}_{old} + \vec{a} \Delta t $$ +``` +4. Update position: +```math +$$ \vec{x}_{new} = \vec{x}_{old} + \vec{v}_{new} \Delta t $$ +``` Where $\Delta t$ is the time step of the simulation.