Instead of just starting it to keep the battery charged you're better off keeping a battery maintainer on the battery all the time.
I use my LS430 in the exact same way as you do your Explorer. Long trips once month or so.
I don't know how to explain it but I have lived through many car problems in my life that were traced back to a weak battery. Idling will not charge your battery. You will need some 30 minute drives to get any significant charging done.
When you start seeing all these weird codes and your Explorer starts acting funny, you will become a believer in using a battery maintainer. I use a CTEK. I got a couple of pics for your entertainment. Notice how I have connected the battery. All I gotta do is plug and play. See the connector?
This is one of best investments.