Gyrocopter VHF antenna length

I'm setting up an antenna to fit under the keel of an open-frame
two-place gyro. The antenna will extend down underneath the keel from
a point about a foot forward of the main gear (made of spring
aluminum).

The only thing that will approach being a ground plane will be the 2x2"
keel forward and aft of the antenna mount and whatever effect the
spring aluminum gear has.

I know aircraft band VHF antennas need about 22", but where can I find
out the exact length that would best suit the freqs I use the most --
123.05 and 122.8?

I have also read that the thickness of the antenna has an effect on the
width of the band covered, but I don't know where to dig that info up
either. I'm using a stainless steel whip that is about 5/32" at the
base tapering to 1/8" at the tip.

Two other questions: Does having a small metal ball at the tip of the
antenna have any effect? Does bending back the last 6" of the antenna
about 45 degrees have any effect?

This antenna will be driven by a Sporty's handheld navcom with a BNC
connector at the radio.

Thanks for any info.