The Vulcan in Me

Readers, here is a puzzle adventure in naive physics and topology. The author respectfully denies being a Trekkie. Enjoy!

***

I graduated last in my class at Starfleet Academy. Not much of a story, so don't ask. Anyway, you probably know what came next: it was either find a civilian job or go guard some worthless corner of deep space.

But I'm not exactly sociable, so I took the assignment with no hesitation. Sure enough, it was a real plum: I was put in charge of two monitoring stations (practically bathroom-sized, one of which I controlled remotely) orbiting neighboring stars, with nothing else even close.

For two years, there was nothing but silence as I idled there, chuckling at my fate and deepening my acquaintance with some bootlegged Romulan ale. Then, finally, something happened. I got an alert:

"Reports of extremely massive unidentified vessel, may be in your sensor range. Report: are gravitational readings consistent with null hypothesis (no vessel)?"

I almost panicked. I hadn't checked the sensors in ages, and I didn't even begin to remember how to interpret the readings--how did gravity work again? If I asked I'd definitely be canned. I fumbled through my old Academy lecture notes, struggling to penetrate the chicken-scratch handwriting and the alien sex doodles everywhere. Here is what I was able to recover:

1) Newton's laws govern the universe; quantum mechanics and other such exotic junk were conclusively disproved in 2247.

2) The world contains massive point-particles; the gravitational field at a point in space is the vector sum of its gravitational attractions to all point-masses. Beyond a critical range and below a critical mass, these attractions can be assumed zero. (so I could discard the mass of myself, the two monitoring stations, and everything beyond the two stars and, maybe, this weird vessel... all of which I could model as point-masses.)

3) The gravitational attraction v to a mass m felt by a point p in space is a vector pointing from p in the direction of m, with magnitude w*f(d), where w is the mass of m, d is the distance from p to m, and f is a function given by...

...but I couldn't make out the formula for f; it was totally obscured by a lascivious Ferengi.

Still, I'm not as thick as you think. I reasoned that this much is true:

4) f, the function governing the magnitude of the attraction, must be continuous, positive, and decreasing on the positive real numbers, tending to infinity as d goes to zero, and tending to zero as d approaches infinity. That is, it looks something like this:



Now, the readings from my two stations looked something like this:



(s[1], s[2] are the two stations, and v[1], v[2] are the two gravitational field readings.)

The picture's not much to go by--the vectors didn't all lie in a plane together, and I can't swear by the magnitudes I drew. The thing to notice is that a, b were obtuse--of that I'm sure.

Recall that there were definitely two stars kicking around in the vicinity. Don't ask me how massive they are; it was on record somewhere, but I was too frazzled to even look up the numbers. Also, my windows were frosted over and I had no way of getting a bead on the stars' position. The question was, could some placement of those two masses have generated the gravitational field readings?

What can I say... I may be lazy, but I'm also one-eighth Vulcan, and my genes chose that do-or-die moment to kick in. I confidently delivered my report: "Readings consistent with null hypothesis."


Well, they never found that vessel; still, doing my duty with flair like that was a high point for me, despite my usual tendency to shirk. But two more years have passed, and all this booze has given the Vulcan in me a sore beating.

So help me remember--how the hell did I know what to say?