What is the distance two spacecraft will spark/arc at in SSO? Eg if one has just launched and one has been there a long time (eg a 10-20kV worst case voltage difference). I think Paschen's Law is the right equation, but I'm not sure how to deal with this in the space environment...
I'm unable to give you precise numbers, but for orders of 10kV this will be a sub-millimeter distance.
The mechanism of arcing in vacuum is significantly different than in air. While in air the air particles, excited, become the plasma and start conducting electricity, there is no such medium in vacuum. There's Field Electron Emission, especially from any sharp points; either into empty space, or towards a positively charged target (the other spaceship). Now this emission must achieve such intensity as to heat up the electrode until its own material evaporating is emitted into space and carried by the electrons reaches the target - creates a plasma bridge. Only then arcing can occur, as it excites more of the electrode material producing more arc-sustaining plasma.
This largely depends on ability to emit enough electrons in an area small enough so that the current vaporizes the electrode surface. And here the actual calculation problems begin: what is the evaporation temperature of the material? What is its resistance - corresponding to wattage of the current being emitted. What is its shape? Sharp blades produce much higher field emission than flat surfaces. How fast and how far will the emitted material fly before it cools down enough to become resistive? It ceases to be an electrical problem, and becomes a problem of calculating behavior of mechanics of production and travel of evaporated metal plasma in vacuum.
edit: Wikipedia gives:
High vacuum (field emission limited) 20 - 40MV/m (depends on electrode shape)
It is going to depend upon the orbital region and so this is a partial answer relating specifically to the ISS.
The space-flight environment: the International Space Station and beyond provides this description:
Spacecraft in low orbit around the earth have a complex interaction with ionospheric plasma. The solar arrays on the International Space Station operate at 160V, and the distribution system is at 120V DC. The negative side of the power system is grounded to the structure of the space station, resulting in a large amount of energy stored in the structure at –140V. High voltage solar arrays, coupled with the design and material properties of the International Space Station, can lead to detrimental interactions with the ionospheric plasma.
Two plasma contactor units have been placed on the International Space Station to provide a “ground wire” to prevent arc discharging. These devices emit a low-energy stream of electrons during spacewalks that reduces the buildup of electrical charge. As long as the plasma contactor units are functional, an astronaut floating freely during a spacewalk has no risk of exposure to arcing. However, the steel tethers used by astronauts to attach themselves to the structure of the space station and the exposed metallic surfaces of the spacesuit or tools used during the spacewalk are potential sources for arcing if both of the plasma contactor units were to fail during a spacewalk.
For full credit please note that the reference "" in the quote refers to this book:
Tribble AC. The space environment and its impact on spacecraft design. 31st American Institute of Aeronautics and Astronautics (AIAA), Aerospace Sciences Meeting and Exhibit; 1993 Jan 11–14; Reno (NV). Webster (TX): American Institute of Aeronautics and Astronautics; 1993. p. 491.
To address the question: its not clear what distance can be inferred here. The risk of metal hand tools appears to be that if left "floating" in electrical terms then at some point in their own charge build-up a discharge event can bridge insulating materials such as in a space suit. It seems reasonable that an arc could also occur when such a tool is returned to close proximity with the space station.
Its not clear what the role of the steel tether would be. Superficially it would appear to be a good opportunity to reduce the risks if each loose item could be connected back to the space station 0 volt reference, but it appears that a different design philosophy applies from this context.
To get more on the distance answer I'd suggest, as a next stop, reading more into the plasma contactor unit purpose and design.