Abstract: Any interstellar ship carrying precious cargo must attempt to avoid all fatal impacts. It turns out this exercise is shockingly expensive. We show that the power required to detect inbound fatal pebbles at velocity v scales as v16/3. For any ship of reasonable size this extends well into the petawatt range even before reaching relativistic speeds. This suggests the actual speed limit for non-expendable craft is around 0.003c. About 2 million mph.

 

Any interstellar ship carrying precious cargo must attempt to avoid all fatal impacts. It turns out this exercise is shockingly expensive. We make three assumptions:

• The ship has shielding that can survive impacts up to energy E.
• The ship can avoid impacts if it has warning time at least T.
• The ship can sense inbound objects only if the signal exceeds some threshold S.

Suppose an approaching impactor at distance d has mass m and relative velocity v.

To be fatal, its impact energy must exceed the shield threshold: ½mv² > E

So the smallest fatal mass scales as: m ∝ 1 / v²

To get out of the way in time, the ship needs warning distance at least: d > Tv

So the required detection distance scales as: d ∝ v

For a roughly spherical impactor, cross section scales as: A ∝ m^2/3

Interstellar space is cold and dark. The ship must detect inbound threats using electromagnetic radiation it sends out and receives back. That puts the problem into the active-sensing regime.

If the ship transmits sensing power P, outbound intensity at distance d falls as: P / d²

The reflected return is then proportional to target cross section and falls by another inverse-square factor on the way back, so the received signal scales as:

Now substitute the fatality and reaction-time scalings:

To preserve the same detection margin as speed rises, the required sensing power must scale as:

This is the key result.

To safely increase ship speed by a factor of 10, one must increase sensing power by:

For any fixed sensing power budget, some velocity will push the return from the smallest fatal impactor below the noise floor. This is made even worse by the reality that the ship is very likely headed toward the glare of a destination star.

For calibration, consider a 2cm iron impactor at 775 km/s. This is 10GJ, comparable to the explosive energy budget of a modern GBU-57 MOP bunker-buster warhead [1]. Assuming we have just enough shield to endure this, and just enough power budget to detect all 2.01cm objects from 50,000km away, then 775km/s is also our maximum speed. This is 0.0026c. Go any faster and smaller objects become fatal, required detection distance becomes larger, and the power expense grows faster than the 5th power of velocity.

The uncomfortable conclusion is that propulsion is not the ultimate barrier to interstellar travel. While unmanned Daedalus [2] drone ships might still be pushed to relativistic speeds near 0.3c, generation ships carrying colonists may have a maximum safe speed around 0.003c, or 2 million mph, about 5 times the peak speed of the Parker Solar Probe [3].

References:

• [1] GBU-57A/B MOP. The GBU-57 is a 30,000 lb-class bunker-buster with an explosive weight listed as 5,342 lb.
• [2] Project Daedalus. Daedalus famously considered a thin beryllium erosion shield and an artificial particle cloud generated by "dust bugs" about 200 km ahead of the vehicle to disperse larger obstacles.
• [3] Parker Solar Probe. NASA's Parker Solar Probe reached a peak speed of 430,000mph on Dec 24, 2024.