What is the maximum size for a solid celestial body such that it still can be tunneled down to the core?

Question

Old science fiction had lots of stories that involved people exploring the center of Earth, like for example Journey to the Center of the Earth, by Jules Verne. As Earth sciences advanced, this sort of story fell out of favor, and today even sounds somewhat ludicrous, as now is well understood pressure and temperature increase fast as one descends into Earth, so it's unfeasible to dig and hold open large cavities anywhere near the core, much less survive for any period of time there. So, despite the fact Earth radius is close to 6400km, the deepest mine in the planet, Mponeng Gold Mine, goes only 4km under ground level, or 0.0625% of the distance required to reach the center of Earth. I don't know how deeper could it go, and if the limiting factor precluding greater depths is heat removal due to high temperatures or tunnel collapse due to overburden pressure, but I don't believe it can go much beyond current depth.

But if our planet is too large for this scenario, making it infeasible here, surely there is a lower threshold for mass and size making it actually possible. So I'd like to know, what's the maximum size for a planet/dwarf planet/asteroid, where would be possible in principle to dig tunnels all the way down to the core, or perhaps even to build some sort of "corescrapper", an underground multi-floor building going from surface to the core? By tunnels I mean tunnels that people could actually pass through, using technology similar to what is used in current mines here on Earth. So just being drillable, like in a oil well, to the core, doesn't count.

If necessary, the putative celestial body can be assumed to have rocky composition, perhaps with a metallic core, and age similar to the current age of the solar system.

Answer 1

Such tunnels might be possible for objects up to a few hundred kilometres in size. Larger objects could be tunnelled if the tunnels are lined with high strength materials.

Assuming the centre of the object is cool enough not to melt the tunnel lining, then the tunnel will be limited by the compressive strength, $\sigma_c$, of the tunnel walls. Experience from deep mines tells us that the maximum pressure a tunnel can sustain is something like $P_{max}\sim 0.4\,\sigma_c$.

To get a crude estimate, let's assume the objects are spherical and have uniform density throughout. (Note: This will likely overestimate the maximum tunnelable size, since larger objects often have a dense core. For example, assuming a uniform density Earth underestimates its central pressure by a factor of two.)

The gravitational pressure at the centre of a sphere with uniform density $\rho$ and radius $R$ is

$$P=\frac{2}{3}\pi G\rho^2R^2$$

so a reasonable guess for the maximum radius of an object that could be tunnelled through is

$$R_{max}\sim \frac{1}{\rho}\sqrt{\frac{3P_{max}}{2 \pi G}}\sim\frac{1}{\rho}\sqrt{\frac{\sigma_c}{2 \pi G}}$$

Here are some example estimates of the maximum tunnelable radius for some different simple object compositions. Also shown is the maximum radius if the tunnels are lined with a hard ceramic such as alumina ($\sigma_c \sim 3000\,\mathrm{MPa}$).

Object	Density	$\mathbf{R_{max}}$ (unlined)	$\mathbf{R_{max}}$ (lined)	Notes
	($\mathrm{g/cm^3}$)	($\mathrm{km}$)	($\mathrm{km}$)
Pure ice object	1	350	2700	The compressive strength of high density ice at a temperature of 173K (typical of the outer asteroid belt) is $\sim$ 50 MPa.
Realistic icy object (e.g. similar to Ceres)	2.2	160	1200	Ceres actual radius is $\sim$ 470 km, and it appears to be geologically active, which might make even lined tunnelling difficult.
Iron-Nickel asteroid (e.g. Vesta core)	8	120	330	Assumed similar to iron-nickel meteorites that have compressive strengths as high as 400 MPa
Hard rock asteroid	3.5	170	760	Assumed $\sigma_c \sim$ 150 MPa (typical of hard rock)

Notes:

• Many smaller asteroids are rubble piles. Tunnelling through them would be like digging through sand, so a lining would be essential. Even larger objects may not be completely consolidated.
• The above maximum (unlined) tunnelable sizes are comparable to the minimum sizes for celestial objects to naturally become spherical. This is unsurprising since if gravitational forces are strong enough to make an object round, they are also strong enough to collapse any central voids. The largest tunnelable object might be one that is initially hot and malleable when formed, but then cools down to higher compressive strength solid.