How did astronomers precisely determine the length of an Astronomical Unit in the 1960s?

Use of telemetry and radar astronomy

Explanation:

An astronomical Unit (AU) is a unit of measuring distances in outer space, which is based on the approximate distance between the earth and the Sun.

After several years of trying to approximate the distance between the Sun and the Earth using several methods based on geometry and some other calculations, advancements in technology made available the presence of special motoring equipment, which can be placed in outer space to remotely monitor and measure the position of the sun.

The use of direct radar measurements to the sun (radar astronomy) have also made the determination of the AU more accurate.

A standard radar pulse of known speed is sent to the Sun, and the time with which it takes to return is measured,  once this is recorded, the distance between the Earth and the Sun can be calculated using

distance = speed X time.

However, most of these means have to be corrected for parallax errors

force required to detach it is given by

charges are taken to be fundamental charge

now we have

so above is the minimum distance at which it will detach