If a spacecraft travels at 10% c will it be destroyed by interstellar dust and particles?

Question

If a spacecraft travels at 10% the speed of light will it be destroyed by collisions with interstellar dust and particles?

The spacecraft will be traveling to nearby stars, not going through a nebula.

Answer 1

Based on this answer:

$$E \approx \frac{1}{2} m v^2 = \frac{1}{2} m c^2 \left(\frac{v}{c}\right)^2$$

The mass of a proton $m_P c^2$ is about 938 MeV, so if an innocent atom of hydrogen or bare proton in space were hit by a spacecraft at 0.1 c, in the spacecraft's frame it would look like a speedy proton. Of what energy?

$$E = \frac{938}{2} 0.1^2 = 4.7 \text{ MeV}$$

From here I clicked proton projected range and found out it will be about $1 \times 10^{-2}$ g/cm$^3$, or about 5-10 microns in some low-density material.

Cold interstellar medium is about $1 \times 10^{12}$ protons/m$^3$ or $1 \times 10^{6}$ protons/cm$^3$, but comments below indicate that typical values are way lower, only 0.05 to 0.3 protons/cm$^3$. So I will use 0.1 /cm$^3$ ($1 \times 10^{5}$ /m$^3$) going forward.

At 0.1 c each square meter of spacecraft intercepts $3 \times 10^{7}$ m$^3$/s or $1 \times 10^{12}$ protons/sec.

This is not a lot of heat

The power deposited in the 10 micron thick layer can be gotten from multiplying 4.7 million volts by $1.5 \times 10^{-7}$ amps (coulombs/second) per square meter, or 0.7 watts. In order to calculate what temperature can re-radiate that power use the Stefan–Boltzmann law with $\sigma = 5.67 \times 10^{-8}$:

$$P = \sigma T^4$$

which gives us a skin temperature of about 60 Kelvin, not even enough to keep standard electronics working.

Sputtering is not a problem

Our 10 micron thick square meter has of order 0.1 mole or about $10^{23}$ of atoms.

From random links 1, 2, and Figure 1 in 3 I will ballpark the sputter rate at $10^{-4}$ atom/atom which is conservative.

So at $10^{12}$ protons per second we'll sputter $10^{8}$ per second, meaning we'll loose 10 microns every 30 million years.

Note that the sputter rate may be a few orders of magnitude faster because my estimate was conservative and at 3300 K the atoms are already pretty energetic and predisposed to leave.

But what about things bigger than protons, like cosmic dust?

Damage by things bigger than a proton is going to be a big problem, but I don't know how big. I will leave that aspect to another answer

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