Mike,

I once had a problem with a large moth setting off my alarm, only in my case
it happened at night when the pest flew up to try and mate with the nearest
streetlight.
Could there be a little wildlife in your E39 that likes the sun?

Martin

Had a problem last year with my E38.  Similar situation with heat and thought
the same but it also happened late evening when cool.  I cleaned all the
switches and used the key to lock the car for a couple of days.

Investigated further and read that some unscrupulous persons were using radio
door bell pushes to set off car alarms and also the JAM the radio key fob
signals thus you think the car is locked and really it isn't.

The alarm can also be triggered by computer wireless networks if strong enough
and I had it trigger 3 times at work when parked in my usual bay but NEVER 40
foot away in another place in the car park.

My neighbor told me that he had installed a WiFi network at home and soon after
my alarm triggered.  I asked him it if would be possible to move it to another
room or away from the front windows - he did this and the alarm has not
triggered since.

Several routes to try and investigate here.  Good luck

I've experienced the same problem, except, the alarm has once gone off
in the middle of the night when it is cooler. My car sits in a garage.
I've reduced the frequency of the problem by making sure that my
windows and sunroof are completely closed.

I had not thought of the WiFi problem before. Probably easy enough to
check. Some WiFi run in the 2.4GHz range. If this is the case, then
perhaps cordless phones operating in this range could also potentially
effect the car alarm.