How long was Sputnik 1's second stage rocket body in orbit? Are there TLEs for it?

Question

Early artificial satellites were tracked by radio and visually by amateurs, scientists and military personnel for various reasons including fun and improving the understanding of orbital dynamics including Earth's oblateness or J2 and atmospheric drag.

The image below is of Sputnik 1's second stage rocket body photographed from Earth from a Operation Moon Watch station in Schenectady New York. It will be fun to try to reconstruct when this was from the position of the Moon and some TLEs for this object if they exist, (or other orbital information about it). The first steps are to bracket the time and to find out if TLEs are available, so here I'd like to ask:

Question: How long was Sputnik 1's third stage rocket body in orbit? Are there TLEs for it?

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This question was inspired by the segment in the vintage video "SCIENCE IN SPACE" EARLY 1960s SPACE EXPLORATION FILM SPUTNIK & EXPLORER VANGUARD ROCKET 12494 linked below, from the Periscope Film collection, from 12:06 to 14:16.

Some of them can be seen by the naked eye or with binoculars, but only around twilight. However scientists have developed systems far more powerful and precise than the eye to follow space laboratories and to communicate with them.

Special radio devices track a satellite by measuring the direction from which its radio signals come. From successive angles of direction the satellite’s orbit can be computed. Amateur radio operators also pick up and report the satellite signals. Radio tracking networks stretch around the world.

Another international network consists of optical observation stations, manned by volunteer teams engaged in Operation Moon Watch. Using Telescopes, these observers are especially helpful in finding a newly-launched satellite, and in following a dying satellite in its final variable course. Each observer is assigned his own sector of sky. Upon sighting the satellite, the observer such as this member of a Philippine team calls out; “In. Time. Out.”

A series of Moon watch reports coordinated with a recorded time signal helps in plotting the path of the satellite. Photographed by a Moon watch observer in Schenectady New York, this actual fast motion shot shows the tumbling third stage rocket of Sputnik 1 crossing the sky in front of the Moon.

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Answer 1

Wikipedia indicates it lasted about two months:

The core stage of the R-7 remained in orbit for two months until 2 December 1957, while Sputnik 1 orbited for three months, until 4 January 1958, having completed 1,440 orbits of the Earth.2

The cited source is RussianSpaceWeb:

The core stage of the R-7 rocket made 882 orbits and reentered on December 2, 1957. The satellite made 1,440 orbits and then burned up on the reentry into the Earth atmosphere on January 4, 1958, after 92 days in space. (84)

The booster was indexed as 1957-001A (the satellite was B) - that entry also gives an estimated time of deorbit, about 0846 UTC on the 2nd. There is a bit of trajectory information given by NSSDC but it looks essentially the same as that for the satellite so may not be very accurate.

However, a contemporary study noted that it was "observed in advance of the satellite proper in approximately the same orbital plane", and has some notes on the exact period of orbit and decay rate that might help produce a more detailed trajectory.