What is the speed of Newtonian gravity? I already read this Phys.SE question but my question is different. I'm well aware that the effect of gravity in GR gets transmitted from point to point at the speed of light. But let's ignore GR for now. What about Newtonian gravity?
Imagine a universe in which Newtonian gravity(but neither GR nor SR ) holds. Imagine also that there's only one object(say, a planet or anything with mass) in this universe. Suddenly another planet(or any object with mass) pops out into existence(yes I know, but ignore the conservation of mass for now). How long will it take for the gravity to affect those two objects. Is there any way to calculate it from Newton's law of gravitation? 
$$F= \frac{GMm}{r^2}$$ 
It seems to that this equation does not say anything about how fast gravity should get transmitted. Or maybe In this Newtonian universe it cannot be calculated from theory but rather be measured experimentally?
A lot of people say it gets transmitted instantaneously, but from where does that follow? How does one prove this only assuming Newton's laws of motion and his gravity law as our axioms? And no don't tell me that $c$ is the upper limit, I'm assuming a Newtonian universe here.
 A: In the Newtonian theory, gravity is instantaneous. This is because the force law says that whenever you have two objects a distance $r$ apart, there will be a force between them inversely proportional to $r^2$. The force at any given instant depends on the distance at that same instant, which means that if you move one of the bodies a little bit, the other one will know about it immediately.
Of course, this is not what actually happens. It's just what you can deduce from Newtonian mechanics.
A: To add to Joe's very good answer, another way of seeing this is to consider both the SR and GR theories as you take the limit of letting the speed of light ($c$) approach $\infty$. 
In that limit SR Minkowski space (with Lorentz transformations) obviously reduces down to a three dimensional space (with Galilean transformations) plus an absolute time that is the same for all observers - in other words, Newtonian mechanics.  
Also (but not as obviously) in that limit, GR reduces to Newtonian gravitational theory.  Therefore in that limit the speed of light and the speed of transmitting gravity would both be infinite ($\infty$). QED.
A: Newtonian gravity must necessarily be instantaneous, otherwise planetary orbits wouldn't obey Kepler's laws. 
In fact, Laplace computed that if propagation of gravity was simultaneously according to Newton's laws and at the speed of light Earth would fall into the Sun within a few centuries. 
This would happen because Earth would get attracted by the Sun not toward the current position of the Sun, but toward the position where Sun used to be about 8 min ago (because it takes about 8 min for light to propagate from Sun to Earth). Thus the direction of the gravitational force would be at a slight angle away from the direction toward the center of the Sun, and that misdirection would be enough to destabilize Earth orbit and make it fall into the Sun within centuries.
A: So the fact that you don't see anything about the speed of gravity in Newton's equation is a bit of a clue; Newtonian gravitational interactions propagate instantaneously (ie. with infinite speed). If there was some finite speed of propagation, we'd have to have a constant related to that speed in the force law, and we don't.
It's a little more obvious to see if you write Newtonian gravity in terms of Poisson's equation instead of Newton's force law;
$\nabla^2 \Phi(\textbf{r},t) = 4 \pi G \rho(\textbf{r},t)$
In case you've not seen Poisson's equation before, it's the field equation for gravity. That means that given a distribution of mass $\rho(\textbf{r},t)$, we can calculate the gravitational potential $\Phi(\textbf{r},t)$. From $\Phi$ we can calculate the force of gravity at each point, and if we place a point mass at the origin we recover Newton's force law.
The reason that it's obvious from Poisson's equation that Newtonian gravity is transmitted instantaneously is because it doesn't have any time derivatives ($\nabla^2$ contains only spatial derivatives). We solve the equation to calculate a spatial distribution of the gravitational potential at a given time, and then if the matter distribution in our spacetime changes, the gravitational potential changes instantaneously at all points in space.
Perhaps the prototypical field equation that does include time derivatives, and along with those a finite propagation speed, is the wave equation, where $\Theta$ is some general field in space and time;
$-\frac{1}{v^2}\frac{\partial^2}{\partial t^2}\Theta(\textbf{r},t) + \nabla^2 \Theta(\textbf{r},t) = 0$
Where we can now see that, rather than a set of functions $\Theta$ which we solve for spatially at each time time, we have to solve for the time and space parts of the equation together. Also note the appearance of $v$, the speed of propagation of the wave, which has no equivalent in Poisson's equation.
This problem of infinite propagation of Newtonian gravity was the main motivating reason for Einstien's search for a relativistic theory of gravity.
There's a lot more information in the wikipedia article called 'speed of gravity' if you're interested.
