During a power outage, which occurs frequently in Florida, I would like to
use a battery to power some communication gear. Although the battery is
12v
and not 13.8v, I think that the gear will still work OK (hopefully).

Question 1. I believe that car batteries have a low internal resistance
because their primary function is to provide a lot of current for a short
time. There are other 12v large battery types that will last as long as a
typical car battery. They are designed not to provide peak current, but to
supply steady state current for a long time.
For the life of me, I can't remember the name of those type batteries so I
can't do a google to find a supplier. I hope this is making some sense.

Question 2. Where can I find articles & schematics for a solar cell to
provide a trickle charge for the aforementioned battery?
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1) If radio still works at 12 V you can use a 12 V battery

2)If a lead acid battery is charged to 13.8 V and subsequebtly discharged it
still has 70 -80 % of its full charge when the voltage has dropped to 12.6
V.

3) The construction of car - versus leisure (deep cycle) batteries is that
the plates of a car battery have a larger area but are thinner . This
enables the chemical reaction to be stronger resulting in a high current
capability (engine start) ,be it for a relatively short time. A leisure
battery has thicker plates but with a smaller area ,hence these batteries
can provide a more modest current ,but for a longer time .
Marine batteries have similar characteristics.
Leisure batteries are available as 'wet' units which can be topped up , with
distilled or de-ionised water ,or are of the sealed type (as used in UPS
=Uninterruptible Power Supply units and for example golf carts )
It is recommended by some manufacturers to charge SLAs = sealed lead acid
batteries to 13.5 -13.8 V ,not higher ,when used for stand by service .
( I restrict the SLAs I use to 13.5 V)

4) If only a solar panel is used for charging a lead acid battery , a simple
comparator circuit with any opamp ,sensing (a fraction of ) the battery
voltage against a zener reference can operate a relay disconnecting the
battery from the solar panel and shorting the solar panel . Alternatively
,when battery is fully charged the generated current is dumped via a
transistor - resistor or transistor- car bulb arrangement .
The current dumping system is necessary when other renewable energy sources
are used as well ,like a wind generator. The latter system is the one I use
..

Frank GM0CSZ / KN6WH