[Curious Cat (OP)]

.

[Eternal Student]

Hi.

This is unusual formatting with everything written down one side.

A body
under the surface experiences greater g towards the centre than the
surface, and that's the case all the way down, to the centre

   Let's take that point first.  It might seem reasonable to think this but the force of gravity doesn't increase at you approach the centre of the earth.  This a consequence of the Shell theorem:

A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of mass.

[Taken from Wikipedia,  https://en.wikipedia.org/wiki/Shell_theorem]

The density isn't exactly uniform throughout the earth and it's not straightforward to measure the density at different depths.  Our best estimates tend to produce an over-all result like this:


See  Wikipedia for more info:    "Shell Theorem" (linked above)    and   also   https://en.wikipedia.org/wiki/Gravity_of_Earth#Depth  which is where the diagram came from.

Best Wishes.

Late editing:   Sorry, it looks like someone else has already replied.

[evan_au]

What is the ... temperature and pressure at the centre of the Earth?

This part of the question is somewhat independent of the gravitational aspects
- Although temperature and pressure do affect density a bit

It is thought that the center of the Earth is a nickel/iron alloy similar to that seen in metallic meteorites.
- It is still cooling down from the temperature of Earth's formation
- A process which is slowed by the decay of radioactive elements

It is thought that the center of the Earth has a temperature of around 5,400C, which is similar to the surface of the Sun.
- The pressure is about 3 million times atmospheric pressure at the surface.

See: https://en.wikipedia.org/wiki/Earth%27s_inner_core

An upcoming space probe will visit a metal-rich asteroid Psyche, which may give some ideas about the center of the Earth. Planned launch date is July 2022.
https://en.wikipedia.org/wiki/Psyche_(spacecraft)

[Eternal Student]

Hi again,

    This is odd, I'll guess you (curious cat) sent mail to both Halc and me and then posted the replies on the thread anyway.  It would have been easier just to put the question on the thread where everyone could answer directly.
    Anyway, the good news is that the answers are similar.   Halc has identified the fuel rods as the main component that would actually melt.  Meanwhile, I took your question at face value and allowed "the nuclear reactor" (including the reaction vessel) to start melting through the earth.

Best Wishes.

[evan_au]

So, if a nuclear reactor could actually melt its way into/thru the Earth, where would it end up?

The premise in the title of the 1979 movie "The China Syndrome" was that a nuclear reactor would melt its way all the way through the Earth and come out at China (supposedly a distant and exotic place on the other side of the Earth).
https://www.imdb.com/title/tt0078966/

At the start of the outer core, where the max/imum g is, or the centre?

Silicate materials like concrete are not dense enough to sink into the core - they would stay in the mantle.
- The mantle is pretty viscous stuff - how far the reactor components would fall into the mantle depends on convection currents - if there are upwards convection currents in the mantle, then it won't sink at all, and will stay near the crust. But if there are descending convection currents in the mantle, it will descend faster.
- How far into the core it travels depends on solubility - iron and nickel components will dissolve into the nickel/iron core, but silicates will not.
- How far the radioactive iron would travel into the core depends on whether the core is solid or liquid. Movement through liquid outer core is assisted by convection, but it will be blocked at the inner solid solid core.
- Uranium forms stable oxides, which means it will tend to stay in Earth's crust (a "lithophile"), rather than descend to the core and dissolve in the iron core.
See the orange lithophile group of elements here: https://en.wikipedia.org/wiki/Goldschmidt_classification

In reality, at a real reactor meltdown like Fukushima, it is believed that a majority of the radioactive omelet of fuel/cladding/reactor shell is still in the basement of the reactor building.
- This is because it takes a careful mix of enriched uranium and moderator to sustain an ongoing nuclear chain reaction.
- Once you mix the fuel with concrete and rock, and dilute the moderator with normal water, the chain reaction slows down enormously, and the heat output is significantly reduced.
- It is still an environmental disaster with a large exclusion zone, but it won't get through enough of the Earth's crust  to join the subducting basaltic ocean plate descending into the mantle below Fukushima (which is what caused the massive earthquake and tsunami in the first place).