Unfortunately not. Because there are diagonal forces imposed by objects on the theoretical plane, you cannot use Gauss's Law to model it. That isn't to say there aren't other ways of modelling it, but Gauss' Law is not it. You cannot 'merely add' the variations because they don't add up to a Gaussian model.
Again, that's not to say an infinite plane wouldn't be possible to model, merely not with the current Gaussian derivation.
You don't use the same surface for each one. You use multiple "models" to create a unified model. It wont have a simple relation like the normal ones though. Just like we can do with multiple spheres
That's what I'm saying; you can't use the same surface for each one. You have to create another function than a derived version of Gauss' Law of Gravitation. Gauss' Law doesn't work for it. I just don't know what that new function would look like.
My bad, just noticed I missed the simple model.
Yes, you can't use the same surface for each one, but you can get a function which takes the x, y and z position (potentially as a piece-wise/step function) and use that to calculate the force.
You can do that for each part, and then add it all together.
For an infinite plane, located at the origin (i.e. 0,0,0) parallel to the x,y plane, with a thickness of t, and density of p, then the acceleration due to gravity will be:
<0,0,-2*pi*G*p*t> if z>t/2;
<0,0,2*pi*G*p*t> if z<-t/2;
<0,0,-4*pi*G*p*z> otherwise
For a point mass, of mass m, located at x0,y0,z0, the acceleration due to gravity will be:
[Gm/((x-x0)^2+(y-y0)^2+(z-z0)^2)^1.5]*<x0-x,y0-y,z0-z>
You can expand that out to give you:
<(x0-x)*Gm/((x-x0)^2+(y-y0)^2+(z-z0)^2)^1.5,(y0-y)*Gm/((x-x0)^2+(y-y0)^2+(z-z0)^2)^1.5,(z0-z)*Gm/((x-x0)^2+(y-y0)^2+(z-z0)^2)^1.5>
And then you can add the 2 together.
It becomes a lot more complex because you can no longer just set the object to be at the origin and just measure the distance from it.
Instead you need to note the position of the object.