Need help installing a capacitor
In order to pick a capacitor for this application, you're going to need to know two things:


How much current does the antenna mechanism draw from the control signal when it is active? Call this I.
How low can the voltage drop on the control signal before the antenna starts to retract? Call this Vmin.


You have already stated that the hold time required is 10 seconds. Call this t.

We'll call the nominal voltage of the battery (when not cranking) Vsupply.

Therefore, the charge required on the capacitor is

$$\Delta Q = I\cdot t$$

The amount of voltage drop we can tolerate is

$$\Delta V = V_{supply} - V_{min}$$

Therefore, the minimum capacitance required is

$$C = \frac{\Delta Q}{\Delta V}$$

You should probably go with 2&times; this minimum value, in order to allow for variations in Vsupply and I under different operating conditions.

Finally, you're going to want to put a diode between the radio and the capacitor in order to keep the capacitor from discharging back through the radio.