The Unexpected Finger plowed through interstellar gas (99% protons) at semi-relativistic speed for eighty years, yet felt negligible drag. But negligible drag was not the only thing she felt.

Proton impacts deposited kinetic energy, which depended on their mass and impact velocity.

What was their impact velocity?

Impact velocity is the ship's velocity plus the proton's drift. Proton drift was ignored when we computed drag, because drift added zero momentum, on average. (Some drifted toward the Finger, others drifted away.) But we cannot take that shortcut when computing kinetic energy, since energy increases as velocity squared. This means drift toward the collision is more important than drift away.

But do protons drift fast enough to worry about?

The typical speed of drifting photons (I.e., their RMS velocity toward the impact) is called Thermal velocity. It is given by the formula V_rms  = (kT/M_p)^½, where T is temperature, M_p is the proton mass (1.7×10^‑27kg), and k is the Boltzmann constant, 1.4×10^‑23kg m²s^‑2K^‑1.

The temperature T is the temperature of our region of space, the local interstellar cloud. Current thinking puts that temperature at 6000K, which is about the same as the surface of the sun, so it's a good thing space has low humidity.

Plugging in the numbers gives us the typical proton drift velocity, V_rms  = (kT/M_p)^½ = (1.4×10^‑23kg m²s^‑2K^‑1 × 6000K / 1.7×10^‑27kg)^½ = (4.9×10⁷ m²s^‑2)^½ = 7×10³m/s.

In other words, in our neighborhood of space, the typical proton is drifting along at seven kilometers per second. This sounds fast, and it is. But it was only one ten-thousandth of the Finger's velocity, so we will continue to ignore proton drift.

The kinetic energy of impacting protons depended on their mass and the Finger's velocity.

Click to Embiggen

Previously we showed the total mass of impacting protons was M  = 1.5×10^‑2 kg, and the Finger's velocity was v  = 6×10⁷m/s.

So the total kinetic energy of impacting protons was E  = ½Mv² = ½(1.5×10^‑2 kg) × (6×10⁷m/s)² = 2.7×10¹³ kg m²/s² = 2.7×10¹³ J.

Wow. Floating protons whacked the pillow with 27 terajoules, almost half the Hiroshima nuke. Good thing they didn't all hit at once.

But those impacts slowed the Finger by only four parts per billion. How could interstellar gas collisions be so destructive while exerting so little drag?

At the Finger's velocity our intuition fails, because in daily life we see only slow-moving objects, whose momentum and kinetic energy are comparable. Momentum grows linearly with speed, while energy grows as speed squared. This is why the American M-16 rifle (small, fast bullet) kicks noticably less than an AK-47 (heavy, slow bullet) even though the two weapons deliver similar muzzle energy. At relativistic speed, this effect is exaggerated to a ridiculous extreme.

For more stuff about guns and recoil, see this.

Yes I know, an AK47 offers 13% more muzzle energy. This counts as "similar."

You might be thinking about the constant patter of Brownian dings we all put up with because we live in atmosphere. Good thought, but room-temperature air molecules bounce off us elastically, and hardly ever knock little chips off our skull.

Knocking off chips. Four million billion protons per second, knocking off little chips.

Uh oh.