Goldplating Planets with Kilonovas

How much gold are we talking about?

Well, let's use the approximation $$\Delta V\approx4\pi r^2\Delta r$$ If $r$ is about the radius of Earth and $\Delta r$ is $1\text{ meter}$, then $\Delta V\approx5\times10^{14}\text{ m}^3$. The density of gold is $\rho\approx19320\text{ kg/m}^3$, and so $\Delta m=\rho\Delta V\approx10^{19}\text{ kg}$.

Let's say that the cloud of material expands isotropically outward from the source with a total mass $M$, radius $R$ (at a given time) and a thickness $\Delta R$. It therefore has a volume $$V\approx4\pi R^2\Delta R$$ and an area mass density of $$\sigma=\frac{M}{4\pi R^2}$$ The planet has a cross-section of $A=\pi r^2$,¹ meaning that the total amount of mass that hits the planet is $$\Delta m=\sigma A=\sigma\pi r^2=\frac{1}{4}M\frac{r^2}{R^2}$$ If we set this equal to our earlier value of $\Delta m$, then we find $$\frac{M}{R^2}\approx10^6\text{ kg/m}^2$$

The typical amount of gold produced in a kilonova via the r-process is . . . not well-known. We've got a limited dataset. That said, if we assume (conservatively!) that $M\approx100M_{\oplus}$, then we need $R\approx2\times10^{9}\text{ m}$, or roughly $0.01\text{ AU}$. The planet would almost certainly be in orbit around the binary system, which could be dangerously close. I certainly wouldn't want to be anywhere nearby!

Things you could do to solve the problem

You do have some options here:

• Lower the thickness of the gold coating you want!
• Lower the density of the coating. I honestly don't know what a good value would be; my $\rho$ assumed that the gold would be totally solid (which is probably not an excellent approximation).

Both of these let you place the planet further from the melee.

It's worth noting, by the way, that only the half of the planet facing the binary system would be significantly coated, unless an accretion disk of sorts was the form around it. Additionally, the depth of the coating would be non-uniform.

What would be the effects on the planet from the merger?

Radiation from such a merger should be emitted isotropically, i.e. most of should spread out equally in all directions.² It's possible that some emissions could be constrained to thin beams, but this certainly won't be the case for all of the electromagnetic emission.

Let's look at the energies and luminosities involved:

• Gamma rays: $\sim10^{39}\text{ J}$, $\sim10^{40}\text{ W}$^[1]
• X-rays: $\sim10^{32}\text{ W}$^[2]

By comparison, the luminosity of the sun is $\sim10^{26}\text{ W}$. This sort of event is brief but very violent. I don't know what the long-term effects of this radiation will be for the planet, but short-term, it won't be pleasant. The deposition of radioactive isotopes might also be a problem, but I don't know just how bad it would be.

¹ Okay, it's a bit more than that because gravity's involved, so there's going to be some accretion. In particular, gravitational focusing will give an effective cross-section (p. 22) (see also here) of $$A_{eff}=A\left[1+\frac{v_{esc}^2}{v_0^2}\right]$$ where $v_{esc}$ is the scape velocity at the surface of the planet and $v_0$ is the initial relative velocity of the gas and the planet. Let's assume that the planet's escape velocity is the same as that of Earth, $\sim11\text{ km/s}$. The speed of the ejected gas is expected to be $0.1-0.2c$. This means that $v_0\gg v_{esc}$, and so $A_{eff}\approx A$.
² For what it's worth, there will be delays before the radiation is emitted, even without the effects of interstellar dispersion. It probably won't help you that much, though - the planet will still be orbiting the remnant.

posted about 7 years ago

HDE 226868‭

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