How Do They Do That?

Originally, a two-stage system was planned, that features two piloted fly-back vehicles, a Launch Booster and a made-from-scratch Time Shuttle, both using liquid propellant for conventional propulsion (the Time Shuttle employs a Tachyon Drive to cross time). The booster carries the Time Shuttle to a set altitude, then detaches to return to base.

The Time Shuttle's actual launch system employs a repurposed and retrofitted STS Program Orbiter, using supplementary liquid propellant tank and twin solid-fuel booster rockets, just as the Space Shuttles were originally launched. The boosters and auxiliary fuel tank are jettisoned at altitude, separating from the vehicle and are later recovered for re-use. This method allows the Time Shuttle to retain most of its onboard liquid fuel supply for return.

At apogee, the Time Shuttle engines ignite for a hypersonic descent, and at about 765 knots, the Tachyon Drive is engaged manually by the Chrononaut, and propels the vehicle across the time barrier.

Once in the past, the Time Shuttle lands conventionally while undetectable to the naked eye and all sensing technology developed prior to the 2060's.

Advances in propulsion allow the Time Shuttle to re-launch without a booster, but limited fuel supply allows just one re-launch attempt.
To return, the Time Shuttle uses nearly all remaining onboard liquid fuel to launch horizontally like a conventional aircraft, climb to necessary altitude, then reach hypersonic speed on a descent.

At 765 knots -or 880 MPH -the pilot again engages the Tachyon drive, time-warping back to the future.