USER-SPECIFIED TWO-LINE ELEMENTS (TLEs)

Geocentric SGP4/SDP4 Two-Line Element (TLE) format data can be specified by users to define an Earth-orbiting artificial satellite. Such data is primarily available from Space-Track.org and CelesTrak.com.

Horizons maintains trajectories for only a small subset of Earth-orbiting satellites, so this user-input option allows specification of other objects not in Horizons. These add-ins can then be used as targets (to observe) and coordinate origins (observing sites from which to observe).

The standard TLE format can be cut-and-pasted into Horizons here. For example:

SC-1
1 87820U 11053A   11273.79990913  .00099611  00000-0  64461-3 0  9991
2 87820 042.7843 189.7738 0014383 039.8647 002.5266 15.74868665   196
1 87820U 11053A   11273.86983630 -.00085102 +00000-0 -55758-3 0  9998
2 87820 042.7804 189.3478 0014258 040.7498 038.5752 15.74826749000204

 *  PRESS <return> ON A BLANK LINE WHEN DONE.

 *  Type '-' alone on a line to abandon TLE input (discard)

 *  The first line can be the object's name ("SC-1"), as can every 3rd line.
     Names are a maximum of 24 characters in length.

 *  Name is optional, but if no name is provided, you will be prompted for
     an optional name after TLE data input is complete.

 *  Multiple TLE sets can be input, up to some system limit which may change
     but is nominally 600 sets (1200 data lines ... name lines are not counted)

 *  Input TLEs can be extrapolated over a limited time-span before and after
     the first and last TLE epochs. This limit may change, but is nominally
     +/- 14 days. Extrapolation errors outside the TLE epoch bounds grow at
     perhaps 1-3 km per day, but this is highly variable and could be much
     worse if there is thruster activity. If there are large gaps in the data,
     ephemeris results could be heavily degraded (i.e., worthless ... passing
     through the Earth, etc.).

 *  The data does not have to be in chronological order, though that
     would be a normal thing to do.

 *  Once a TLE object has been defined, you can go off and do other things
     in Horizons (when using it interactively via telnet), then come back
     later and select the previously input dataset as a target by typing
     "TLE" at the main Horizons prompt.

 *  An input TLE object can be also used as a coordinate origin (an observing
     point) at any time, as explained below.

There are programs that use the max slew rate of 1800x for pulse movement of the mount which doesn't track in a smooth manner but which keeps the object in the field in a back-forth manner. This is not controlled tracking but it does work to keep a fast moving satellite in the field, just not perfectly centered.

I use SkyX. In it you can download satellite elements and then use it to track the satellite across the sky.