SciFi Saturday – Black Holes

For my Star Frontiers: Knight Hawks game, I'm currently in the process of modeling some planets as ping-pong balls (they're almost exactly the right size according to the spatial scale of action). I was trying to come up with some clever designs (the pack I got has a whole bunch of ping-pong balls), and then it occurred to me: probably the easiest thing to do is spray paint one all-black and call it a black hole. But what would the effects of such a body be on the tactical game?

There's one official adventure for SFKH that featured a black hole: Dragon #88 (one of my favorite issues of the magazine -- hello, Marvel Super Heroes Thor!) contained a scenario called "The Battle at Ebony Eyes" by William Tracy, and it features not one but actually two black holes closely orbiting each other. In this game, ships can orbit the black holes just like any planet (from one hex away), but the primary wrinkle is that they cause illusory duplicates of everything in the surrounding area -- so basically, all the ships in the battle have something like a mirror image effect going on. Clearly that's a bit gamey-fantastic (although high-gravity lensing is a real thing, as illustrated in the picture above).

So I was wondering: What would the actual gravitational effect of a black hole be? One issue with my ping-pong model (for both planets and black holes) is that the SFKH ship models are at a radically different scale than the surrounding space scale (something like 1" = 50 meters for the former, 1" = 10,000 km for the latter). So if we set down a planet at the space scale, then it's a whole lot smaller than the ship models orbiting it, and it looks a little ridiculous. With the black hole maybe I have the option of declaring it to be at the same scale as the ships -- but which is better for gameplay?

Fortunately, I've previously worked on alternate orbital possibilities for smaller or larger planets, and worked up a spreadsheet to quickly summarize the possibilities. The key unknown in that spreadsheet is, what's the mass of my black hole? Wikipedia comes to the rescue with a formula relating black hole mass to size. As a simplifying assumption, we'll assume that this is a "Schwarzschild black hole": basically symmetric, with no angular momentum or electric charge, and so it acts like any other gravitational mass at a distance. Then the radius of the event horizon in kilometers is related to mass by about: r ~ 3 M/M(sun), where M(sun) is about 2×10^30 kg. Turning this around algebraically, we get M ~ r×M(sun)/3. So for my two candidate black hole sizes (a ping-pong ball being 40mm or about 1.5 inches in diameter):

Black Hole Type I -- Ship scale, about 50 meters radius. M ~ r×M(sun)/3 = 0.05×2×10^30/3 ~ 3×10^28 kg.

Black Hole Type II -- Planet scale, about 8,000 km radius. M ~ r×M(sun)/3 = 8000×2×10^30/3 ~ 5×10^33 kg.


Then I can fill out the orbital spreadsheet and see the results below.

What we see is this: At the smaller ship scale, the black hole is at least conceivably usable in the tactical game. At a range of 50 inches (maybe the very edge of your gaming table), a ship can orbit at a rate of 4 inches/turn using the black hole's gravity. At a middle range 10 inches, the orbital speed is 8 inches/turn (i.e., making a cycle about every 8 turns or so, kind of like standard orbiting behavior in the core game). At a short range of 2 inches (like standard orbit expectation), an orbiting vessel would flash around at a speed of almost 20 inches, that is, almost making two complete orbits every turn!

If we consider using the planet-scale black hole, then things get quasi-comical (remember that the black hole is on the order of a billion times more dense than a like-sized planet). Even on the furthest edge of your standard playing space the orbital velocity is 1,500 inches (i.e., about 5 full orbits around the perimeter of your table in a single turn). At a mid-range the speed is over 3,000 inches (almost 60 orbits in a turn), and at standard close orbit the speed approaches 8,000 inches (a whiplash-inducing 600+ cycles in a single game turn).

So clearly if I use this ping-pong modeled black hole in my Star Frontiers: Knight Hawks game, then I'll declare it to be at ship-scale of about 50 meters radius, with effects that are reasonable near the edge of the board, and challenging to deal with (but not utterly insane) near the center. Also this has the advantage of looking nicer, next to the same-scale ship miniatures. You could consider using different-sized black holes in your game but the preceding is about the range of possibilities. See Wikipedia: Micro-black holes with 0.1mm size and Moon-mass would have no effect on ship movement; Stellar-size and above would already be several hundred times more powerful than my "Type I" above, and thus likely unusable for game purposes.

Oh, one final thing: Run into the black hole and you're dead. (Or at least playing a different game system.)

[Revised ODS spreadsheet here if you want it.]