Does gravity increase the closer to the core you get?

This is a companion to the other answer. The other answer gives values for what the acceleration actually is based on knowledge about the composition of the planet. In this answer I want to introduce some of the mathematical tools that allow you to reason about how gravity varies in and around planets. David Hammen's answer, which I think he wrote overlapping with the second part of this one, also has versions of some of the maths in the second part of this one, and is very worth reading.

An idealised (and incorrect) Earth

First of all consider an idealised Earth: in this model Earth let's assume that the planet is:

• spherical;
• not spinning;
• has constant density throughout.

The first two of these are reasonably good approximations. the third is not a good approximation: the density of the Earth in fact varies significantly with depth.

For this simplified model we can use a famous result, proved by Newton, known as the shell theorem. This says two things:

1. a spherically-symmetric mass distribution can be treated gravitationally outside it as if it was the same mass concentrated at a point at the centre of the sphere;
2. a spherically-symmetric shell of mass (so a hollow sphere) has no gravitational influence on any body inside the shell.

From this it immediately follows that, at any point inside our model Earth, you can ignore all the mass outside that point and then just consider the gravitational influence of the mass inside the point, and we can also consider that mass to be concentrated at the centre of the model Earth.

So, well, let the density of the planet be $rho$. Then the mass within a radius $r$ is $m(r) = 4pi r^3 rho/3$, for $rle R_E$, where $R_E$ is the radius of the Earth. And using Newton's law of gravitation, we get the acceleration due to gravity, $g(r)$ to be

$$begin{aligned} g(r) &= frac{Gm(r)}{r^2} &=frac{4pi Grho r}{3}quad r le R_E end{aligned}$$

Which shows you that $g$ reduces linearly as you approach the centre of the Earth in this model.

A more realistic model Earth

A nice thing is that you can use most of the above even if $rho$ is not constant: so long as $rho = rho(r)$, only, then you can still use the shell theorem to compute $g(r)$, but you now have to do some integral over $r$ to compute $m(r)$. So you can relax the third requirement above, at the cost of a little more work. I believe that, in practice, $rho$ does depend only on $r$ to a fairly good approximation, so this model will let us compute how gravity actually does vary.

In the case where $rho = rho(r)$ we can use the shell theorem to write $g(R)$ at some radius $R$ (I'm using $R$ not because I want to integrate over $r$).

$$begin{aligned} g(R) &= frac{G m(R)}{R^2} &= frac{G}{R^2}int_0^R 4pi r^2 rho(r),dr end{aligned}$$

Where the integral is the mass of the planet to radius $R$. It's clear from this that $g$ is constant if $rho(r) = rho_0/r$: a planet with a density which goes down as $1/r$ has gravitational acceleration which is constant throughout its volume. Note that this planet has infinite density at its centre: since that's never true for planets – we're not dealing with black holes here, and if we were we could not use Newtonian gravitation – this means that for all real planets, $g to 0$ as $r to 0$.

More generally we can differentiate the above to get an expression for $dg/dR$:

$$frac{dg}{dR} = 4pi Grho(R) - frac{2G}{R^2}int_0^R 4pi R^2rho(r),dr $$

Defining $bar{rho}(R)$ as the mean density of the planet to radius $R$, then the integral in the second term is just $4pi R^3bar{rho}(R)/3$, so we get

$$frac{dg}{dR} = 4pi Gleft(rho(R) - frac{2}{3}bar{rho}(R)right)$$

This tells you how $g$ behaves with depth, and in particular whether it goes up or down with depth for a general spherically-symmetric mass distribution, if you know the radius, the density at that radius, and the mean density inside that radius (or equivalently the radius, the density, and the mass inside that radius).

So, if we can measure how the density of Earth varies with depth, and that it varies only with depth to a good approximation, then this model can be used to compute how $g$ varies with depth, to a good approximation.

This is a companion answer to both Erik's and tfb's answers.

tfb's answer assumes a uniform density throughout the Earth. This is highly inaccurate. Not to disparage tfb's answer; I've run across physics PhDs who thought that this was the correct model of gravitation inside the Earth.

The problem with this model is that the uniform density model is invalid for any object large enough to pull itself into a more or less spherical shape. In the case of our Earth, the mantle is significantly denser than is the crust, and the mantle's density increases with increasing depth due to increasing pressure. The core is much denser than is the mantle, by over a factor of two. The core's density increases with depth, and then jumps even a bit higher at the outer core / inner core boundary.

Erik's answer (which is correct and should be the accepted answer) shows the flaws in this assumption. The uniform density assumption would suggest that gravitational acceleration at the core/mantle boundary should be a bit more than half of the gravitational acceleration at the surface. Erik's answer shows that it is in fact a bit over the gravitational acceleration at the surface. (No halfsies!) Rather than being roughly half the surface acceleration, gravitational acceleration reaches its maximum at the core/mantle boundary.

The graph in Erik's answer shows that in some places (e.g., the Earth's core) gravitational acceleration increases with increasing radial distance from the center of the Earth, but it decreases with increasing radial distance in other places. This answer addresses the conditions that make gravitational acceleration increase or decrease with increasing radial distance from the center of the Earth.

I'll make the same assumption made in the Preliminary Reference Earth Model, which is that density inside the Earth depends only on radial distance from the center of the Earth. This means that Newton's shell theorem applies: The gravitational acceleration at some point inside the Earth depends only on the material closer to the center of the Earth than the point in question. Denoting $M(r)$ as the mass of all of the material at a radial distance less than $r$, then the gravitational acceleration at some point inside the Earth is $$g(r) = frac{G M(r)}{r^2}tag{1}$$ Differentiating with respect to $r$ results in $$frac{dg(r)}{dr} = frac G{r^3}left(rfrac{dM(r)}{dr} - 2M(r)right)tag{2}$$ One way to calculate $M(r)$ is as a volume integral:$$M(r) = int_0^r 4 pi x^2 rho(x) dx$$ where $rho(r)$ is the local density at a radial distance $r$ from the center of the Earth. Differentiating with respect to $r$ results in $$frac{dM(r)}{dr} = 4 pi r^2 rho(r)tag{3}$$ Another way to express $M(r)$ is via the average density $barrho(r)$ of all of the material at a radial distance less than $r$:$$bar rho(r) equiv frac{M(r)}{V(r)} = frac {M(r)}{frac43 pi r^3}$$ or $$M(r) = frac43 pi r^3 bar rho(r)tag{4}$$ Applying equations (3) and (4) to (2) yields $$frac{dg(r)}{dr} = 4 pi Gleft(rho(r) - frac23barrho(r)right)tag{5}$$ In other words, gravitational acceleration increases with increasing distance from the center of a planet in regions where the local density is more than two thirds of the average density of all of the stuff at a lesser distance, but decreases with increasing distance in regions where the local density is less than this limit. In the case of a hypothetical uniform density planet, the local density is equal to the average density and thus gravitational acceleration increases from the center up to the surface.

But in the case of the Earth, there's a very large drop in density at the core-mantle boundary, enough of a drop to make the local density less than the critical two thirds of the average density of all the stuff below. The core mantle boundary is where gravitational acceleration reaches a global maximum. Another marked drop in density occurs at the upper mantle-lower mantle boundary, making gravitational acceleration decrease from there to the surface. The upper mantle-lower mantle boundary is where gravitational acceleration achieves a local (but not global) maximum.