Another hobbyist, John Locker of Britain, posted photos of the satellite on a Web site, galaxypix.com.

John E. Pike, director of GlobalSecurity.org, a private group in Alexandria, Va., that tracks military and space activities, said the hobbyists exemplified fundamental principles of openness and of the power of technology to change the game.

“It has been an important demystification of these things,” Mr. Pike said, “because I think there is a tendency on the part of these agencies just to try to pretend that they don’t exist, and that nothing can be known about them.”

But the spotters are also pursuing a thoroughly unusual pastime, one that calls for long hours outside, freezing in the winter and sweating in the summer, straining to see a moving light in the sky and hoping that a slip of the finger on the stopwatch does not delete an entire night’s work. And for the adept, there is math. Lots of math.

“It’s somewhat time consuming and tedious,” Mr. Molczan said, acknowledging that the precise and methodical activities might seem, to the uninitiated, “a close approximation to work.”

When a new spy satellite is launched, the hobbyists will collaborate on sightings around the world to determine its orbit, and even guess at its function, sharing their information through the e-mail network SeeSat-L, which can be found via the Web site satobs.org.

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From his 23rd-floor balcony, or the roof of his 32-floor building, Mr. Molzcan will peer through his binoculars at a point in the sky he expects the satellite to cross, which he locates with star charts. When the moving dot appears, he determines its direction and the distance it travels across the patch of sky over time, which he can use to calculate its speed.

Mr. Molzcan declined a request to visit him in Toronto and to be photographed for this article, saying: “No offense intended, but this is beginning to sound like more of a human interest story than one about the substance of the hobby. My preference is for the latter. Also, I prefer not to have photos of myself published.”

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Mr. Locker, who favors a telescope for his camerawork, said that people like him and Mr. Molczan were not, as he put it, “nerdy buffs who lie on our backs and look into the sky and try to undermine governments.” Spotting, he said, is simply a hobby.

“There are people who look at train timetables and go watch trains,” he said. People are drawn to what interests them, he said, and “it’s what draws people to any hobby.”

While recent news coverage has focused on the current satellite’s threat to people when it falls from above, that threat is, statistically, very small. Even when the space shuttle Columbia broke up over Texas five years ago and rained debris over two states, no one on the ground was injured.

Gordon Johndroe, a spokesman for the National Security Council, noted that 328 satellites had come down in the past five years without injury to anyone. While Mr. Johndroe declined to divulge much about the current satellite aside from the fact that it carries no nuclear material, he said that the government would take responsibility in the remote chance of damage or injury.

The government’s relationship with the hobbyists is not a comfortable one. Spokesmen for the National Reconnaissance Office have stated that they would prefer the hobbyists not publish their information, and suggest that foreign countries try to hide their activities when they know an eye in the sky will be passing overhead.

The satellite spotters acknowledge that this may be so, though they doubt that such tactics are effective. Mr. Molczan said he believed that the hobbyists hurt no one but that “you can’t say with absolute certainty what effect you’re having.”

Mr. Pike said the officials who complained about the hobbyists “don’t like it, but they’ve got to lump it.” Despite the many clever ways that the spy agencies try to minimize the likelihood that their satellites will be spotted, he said, they will be. And that, he said, is a valuable warning: a world with so many eyes on the skies renders deep secrets shallow.

“If Ted can track all these satellites,” Mr. Pike said, “so can the Chinese.”