Centripetal acceleration with vector components

Question

I'm working on solar system simulation and I need to implement centripetal acceleration. The problem is I don't know how it looks like in vector form. I mean I have these formulas:

1. Velocity $$ velocity.x = initialVelocity.x + (acceleration.x * motionTime.x)\\ velocity.y = initialVelocity.y + (acceleration.y * motionTime.y)\\ velocity.z = initialVelocity.z + (acceleration.z * motionTime.z) $$

2. Position $$ position.x = initialPosition.x + ((initialVelocity.x * motionTime.x) + (acceleration.x * (motionTime.x * motionTime.x) * 0.5f))\\(...) $$

I'm looking for similiar form for centripetal acceleration. Now I have: $$ acceleration.x = ((velocity.x * velocity.x) / distance) * direction.x\\(...) $$ But it doesn't work. Could you help me?

Answer 1

In your situation, the only force acting on your planets will be the gravitational attraction from the sun. (This is not perfectly true, because all the other planes will cause acceleration too, but usually we can ignore this.

Anyway, you will be able to calculate the force of gravity by Newton's law of universal gravitation.

$$ F_{gravity} = \frac{G M_{sun}M_{planet}}{r^2}$$

where r is the separation of the planet and the sun. Once you have the force acting on the planet, divide by $M_{planet}$ to get its acceleration.

It seems like your method is of simulation is numerical, so from my experience you will probably want a time step on the order of a few seconds.