How serious is the problem of interstellar dust for interstellar space travel?

Question

Proxima Centauri b seems to be the closest exoplanet in its habitable zone. At a distance of $4.2$ lightyears, travel to this planet within a human lifetime is not impossible prima facie, however doing so requires a mean velocity that is an appreciable fraction of the speed of light, i.e. around $0.1 c \approx 3 \times 10^4 \text{km/s}$.

My questions concern the problem to travelling at this speed posed by the presence of an interstellar medium due to (1) the damage caused by collisions with microscopic particles and (2) the loss of momentum to the interstellar medium.

Firstly the problem of collisions. At this speed it is plausible that collisions with even microscopic particles pose a problem. A crude equation of the incident kinetic energy with energy of required to raise steel to its boiling point from absolute zero, suggests that a collision of a dust particle $1\text{mg}$ ($\approx$ a grain of sand) travelling at $0.1c$ would be enough to vaporize about $0.6\text{kg}$ of steel from the surface of a spacecraft, $$ \frac{(1\text{mg})\cdot(0.1 c)^2}{2 \cdot (420 \, \text{kJ} \cdot\text{kg}^{-1} \cdot\text{K}^{-1})\cdot (1643 \text{K})} = 0.65\ldots \text{kg} $$ and transfer about the same momentum as a bowling ball thrown from a moving car. Collisions with objects that are hard enough to start to penetrate steel before vaporising are potentially even more serious problems with the penetration depth of a hard hypervelocity projectile growing proportional to the incident kinetic energy (see e.g. Walker 2001).

Interstellar medium is of course very empty. However to be empty enough that such collisions could be neglected requires that dust particles large enough to do damage have a number density $n \ll 10^{-19} \text{m}^{-3}$, (roughly the inverse of the volume traced out by a small ($10\text{m}$ by $10\text{m}$) spacecraft travelling $4.2\text{ly}$. Is the density of potentially damaging particles in the interstellar medium low enough to permit interstellar travel at speeds $0.1c$?

The second problem is the problem of viscosity. Neglecting the potentially damaging nature of collisions with larger particles, the fact space is not empty implies that work must be done against it to maintain a constant velocity. Does presence of a viscous interstellar medium place any practical restrictions on the speed of interstellar travel?

Answer 1

The density of the interstellar medium varies hugely, so the specific problem of Sol-to-Proxima-Centauri travel is different from the general problem of interstellar travel.

According to WP:

In cool, dense regions of the ISM, matter is primarily in molecular form, and reaches number densities of $10^6$ molecules per $cm^3$ (1 million molecules per $cm^3$). In hot, diffuse regions of the ISM, matter is primarily ionized, and the density may be as low as $10^{−4}$ ions per $cm^3$.

The Local Interstellar Cloud, within which is our solar system, is somewhere in the middle of this enormous density range:

...not very dense, with 0.3 atoms per cubic centimetre. This is less dense than the average for the interstellar medium in the Milky Way (0.5 per $cm^3$, though six times denser than the gas in the hot, low-density Local Bubble (0.05 per $cm^3$) which surrounds the local cloud.

The Alpha/Proxima Centauri system is inside the neighboring G-Cloud; I didn't find any density estimates for the G-Cloud in my extensive five-minute survey, but David Hammen asserts in a comment on this answer that its density is similar to the LIC.

Wikipedia also tells us that:

By mass, 99% of the ISM is gas in any form, and 1% is dust. Of the gas in the ISM, by number 91% of atoms are hydrogen and 8.9% are helium, with 0.1% being atoms of elements heavier than hydrogen or helium... By mass this amounts to 70% hydrogen, 28% helium, and 1.5% heavier elements.

With the given mass breakdown, the overall mass density is straightforward to calculate; if I haven't screwed up it works out to something like $ {6 \times 10^{-25} g}$ per ${cm^3}$ in the Local Interstellar Cloud.

Without knowing more about the distribution of dust particle sizes, it's difficult to calculate the odds of a single catastrophic impact, but the mass density should give some idea of the continuous "wearing away" rate as well as the "drag" resistance of the ISM.