How can I simulate the gravity of a sphere instead of a plane?

Question

How can I simulate the gravity of a sphere instead of a plane?

For a naval warfare game, I wish to implement a surface that simulates a curved plane with horizon. The plane is rendered with water physics, which needs, along with all physical objects, avatars and projectiles, to respond to gravity as a focal point rather than a distributed surface.

Answer 1

It depends on the effect you are looking for.

If you are looking for a constant gravity the answer is:

object.AddForce(((Vector) sphere.center - (Vector) object.pos).normalized * gravityForce);

If you are looking for a gravity based on the distance between the object and the sphere then you replace gravityForce with this formula:

gravityForce =  6.673×10^(−11) * sphere.mass * object.mass / Distance(sphere.center, object.pos); 

Answer 2

You simply need a reworked Newton's law of universal gravitation.

So F=G(M1*M2)/R2 translates as code into.

// Create force scalar based off of distance.

distanceSquared = (gravityCenter.y - object.y)^2 + (gravityCenter.x - object.x)^2

force = gravitationalConstant*distanceSquared

// Get direction from object to gravity epicenter.

direction = arctangent2(gravityCenter.y - object.y, gravityCenter.x - object.x)

// Transform vector into into coordinate based impulse.

object.xVelocity += force*cosine(direction)
object.yVelocity += force*sine(direction)

Then adjust gravitationalConstant to suit.

NOTE: Unless you want to send ships into space, implementing gravity based on distance is completely a waste of cycles as a constant "downwards" force is more processor friendly.