Uhura: Mr. Scott, Computer Central reports that we're coming up on the coordinates of the last-established position of the shuttlecraft.
Scott: Thank you, Lieutenant.
Sulu: Mr. Scott, bearing 310, mark 35 just cleared. No antimatter residue.

(Antimatter residue indicates the use of warp drive engines. I believe this is another clue as to the nature of the Galileo II's drive capabilities.)

Scott: All scanners, spherical sweep. Range-maximum. They'll have to pick it up.

(Question-What is the maximum range of a starship's scanners)

Uhura: If the shuttlecraft powered away,but if it were just towed?
Scott: There would still be traces of residal matter floating around, Lieutenant.

(Question-What is the nature of this residual matter, and from what location of the Galileo II would the matter originate from?)

Sulu: Bearing 210, mark 40. Strong particle concentration. We're on it, Mr. Scott.
Scott: Lay on that course. Maintain scanning.
Sulu: Course laid in, sir. Particle density decreasing. Gone, sir. No readings.
Scott: Steady as she goes, Mr. Sulu.
Uhura: What do you think it means, Mr. Scott?
Scott: The shuttlecraft was on schedule until it was shy 5 hours rendezvous. Then something happened.
Uhura: Well I'd feel a whole lot better if you were a little more definite.
Scott: It didn't wreck. There was no debris. There's no trace of expelled internal atmosphere, no residual radioactivity. Ah, it's...Something took over, tractor beam maybe...They dragged it away on the heading we're now on.

(Questions:
1. How does a starship scan for the 'expelled internal atmosphere' of a destroyed ship in space?
2. The residual radioactivity. Does matter and antimatter have radioactivity? Or does the radioactivity originate from another source found on the Galileo II?)

Uhura: If there are no further traces, how are we going to follow them?
Scott: We stay on this course, see what comes up.
Uhura: It's a big galaxy, Mr. Scott.
Scott: Aye.

A personal note: I enjoy the performance of Glen Corbett as Zefram Cochrane. His portrayal of this character gives, for me, the sense of knowing and feeling for this individual.

1) The scanning hardware
2) The environment
3) The target

In this case, the target was rather small (even in TNG, wreckage was always scanned at "close visual range"). The environment included these nasty asteroids. This might have reduced the range to a few lightseconds even if the hardware was top-notch subspace-scanning stuff.

The fourth factor, theoretical limits deriving from the scanning principle, is probably what the 17-ly range for the E-D (in the TNG Tech Manual) refers to.

The "residual matter" from an intact towed-away ship would have to be some sort of regular exhaust. Perhaps vented drive plasma (which could come from impulse engines, too)? Perhaps "bilge" like waste gases and liquids from a not-completely-closed-cycle life support system?

Scanning for internal atmosphere would probably include spectroscopy for oxygen's radiation-absorption or emission traces.

And matter and antimatter do often create exotic if rather short-lived particles instead of pure radiation quanta when annihilating. The possible dust and small-scale debris from the exploded shuttle would probably become radioactive from a short exposure to these particles or to the original gamma rays of the annihilation.

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