This video covers the two body assumptions, Newton's universal law of gravitation, Newton's 1st law, and Kepler's first law, to explain the two body problem of orbital motion.

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The two body problem assumes:
1. All that exists in the universe is one large body (like the Earth) and one small body (like a spacecraft)
2. The large body is spherical and its mass is evenly distributed, which means that its gravitational pull can be modeled as coming from its geometric center
3. The small body does not influence the large body gravitationally
4. We use an inertial frame centered at the geometric center of the large body
Using these 4 assumptions, we arrive at Newton's Universal Law of Gravitation, which states that the force due to gravity on the small body is equal to the mass of the large body times the mass of the small body times the gravitational constant in our universe divided by the distance between them squared

However, we are interested in the acceleration of the small body (not the gravitational force), because we use acceleration to simulate the motion of orbiting bodies. We can solve for the acceleration using Newton's first law (F=ma). This acceleration equation is a differential equation, because we want to calculate the position of the small body over time, and the acceleration is the second derivative of position.

In order to create 3D simulations of orbits, we need to extend that equation into 3D using vectors. So we turn the radial distance into a vector, which is the vector pointing from the center of the large body to the small body. We also need to add a negative sign, because the acceleration due to gravity is pointing in the exact opposite direction from the position of the small body. And this 3D equation is exactly how we plug it into software to create these simulations.

Links to the Space Engineering Podcast (YouTube, Spotify, Google Podcasts, SimpleCast):
   • Space Engineering Podcast Full Episodes  
https://open.spotify.com/show/01Gcgly...
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Link to Mecánica Orbital con Python (videos en Español):
   • Mecánica Orbital con Python  

Link to old video on the Two Body Problem and Ordinary Differential Equation (ODE) Solvers:
   • The Two Body Problem / ODE Solvers | ...  

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