2005 Ford Five Hundred. Turn ignition key, clicking as if starter solenoid is bad, if released and turned again and held for a couple seconds (while clicking) eventually will turn the engine over. Is this caused by a solenoid on the starter or a relay in the battery junction box? this car only has 48,000 miles on it.

You are in luck! You can program a 2006 Ford Five Hundred key fob yourself and don't need to go to a dealer or professional. The steps are pretty easy, it's the timing of the steps that you have to pay attention to. You most likely will not get the programming correct on the first try, keep at it, you will get it.

Here are the key fob programming instructions for a 2006 Ford Five Hundred:
*** THERE ARE 2 POSSIBLE PROGRAMMING PROCEDURES. ***Procedure #1Note: Obtain all remote transmitters for this vehicle.
Note: Do not depress the brake pedal during procedure.

1. Turn ignition from OFF to RUN 8 times within 10 seconds, with the 8th time ending in RUN. Door locks will cycle to confirm programming mode. (Off is the position where you can remove your key, RUN is the position just shy of starting your car.)
2. Within 20 seconds press any button on the first keyless remote to be programmed. Door locks will cycle to confirm programming. If you only have one remote then skip to step 4.
3. Within 20 seconds, press any button on the second keyless remote. Door locks will cycle again to confirm programming. Repeat step 3. For all other remotes.
4. Turn ignition to OFF. Locks will again cycle to indicate end of programming mode.
   
   Procedure #2Note: Obtain all remote transmitters for this vehicle.1Fasten seat belts and close all doors.
5. Turn ignition from OFF to RUN 4 times within 6 seconds, with the 4th time ending in RUN. (Off is the position where you can remove your key, RUN is the position just shy of starting your car.)
6. Turn ignition to OFF. Acoustic signal will sound and system warning lamp will illuminate to confirm programming mode.
7. Within 10 seconds, press and hold any button on the first keyless remote. Wait for acoustic sound and warning lamp to flash (confirming programming) then release button on remote. Wait 2 seconds but within 10 seconds repeat step 4. For all other remotes. If you only have one remote then continue to step 5.
8. Turn ignition to Crank or wait 10 seconds to end programming mode.

Make sure you have a spare key fob too. You can save a bunch of money ordering a spare key fob online at:
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Sounds like the solenoid for the starter has a burnt spot on it. I don't think leaving the key on is heating up the starter so I do not know what is happening there. Are you getting a lot of water up in the starter so it is freezing over night. I would test the solenoid or replace it.

Check connections and make sure starter has good ground

Click here to see,Click here to see how to remove cylinder.

You are in luck, a 2005 Ford Five Hundred Key Fob Remote is self programmable. Now these procedures can be a bit frustrating at times, so be patient, it's all about timing. If you get it programmed on the first try you are among a very few talented programming individuals! Also, if you need an affordable key fob replacement for your Ford Five Hundred, buy it here:
https://www.keylessentryremotefob.com/Ford-Five-Hundred-Remotes-and-Keys-s/5167.htm

Here are the programming instructions:

FORD REMOTE - THIS PROCEDURE RESETS ALL REMOTES FOR THE VEHICLE. ALL FUNCTIONING REMOTES WILL HAVE TO BE REPROGRAMMED DURING THE PROCEDURE There are a few different procedures that will program your remotes depending on what vehicle that you have. You can find your exact procedure in your owners manual. If you do not have your owners manual you can try any of the procedures below until you find the one that works for your vehicles
Make sure that you don't press the brake pedal at any time during or before the procedure.
Procedure A:
1.Close all doors, then open drivers door and press the Unlock button on driver's door while it is open, insert key into ignition.
2.Turn ignition from OFF to RUN 8 times within 8 seconds, with the 8th time ending in RUN. Door locks will cycle to confirm programming mode. (if the door locks do not cycle, then it was done incorrectly. You must start over by removing key and beginning from step 1)
(Off is the position where you can remove your key, RUN is the position just shy of starting your car.)
3.Within 10 seconds press any button on the first keyless remote to be programmed. Door locks will cycle to confirm programming. If you only have one remote then skip to step 5.
4.Within 10 seconds, press any button on the second keyless remote. Door locks will cycle again to confirm programming. Repeat step 4. for all other remotes.
5.Turn ignition to OFF. Locks will again cycle to indicate end of programming mode.
*If this does not work try the procedure with the door closed.
Procedure B:
1. Fasten seatbelt and close all doors
2. Turn the ignition from OFF to RUN 4 times within 3 seconds.
3. Turn the ignition OFF. The system will chime to confirm programming mode.
4. Within 10 seconds, press and hold any button on first keyless remote. The system will chime once to confirm programming.
5. Within 10 seconds, press and hold any button on second keyless remote. System will chime once to confirm programming.
6. Turn ignition to RUN to exit programming mode.
Please note: All steps should be performed in fairly rapid succession. If you never get the initial chime or door lock cycle, that means the vehicle has never entered programming mode. You should remove key from ignition, open and close the door and start over.

Sounds like your on the right path.. I would make sure the car is in park... or you can try to start it in neutral . But it sounds to me like you may have a pats issue.. That means the chip in the key isn't Bing read corrctly.. see if there is a flashing light on the dash when you turn the key on.

Most likely you need a new Ignition switch. The small "tickler" wires on your solenoid are staying on. You can disconnect the small wire from the Solenoid and then put a test light on it. Turn the Ignition to Start and see if the wire stays hot when you release the key.

Another problem you could have is the Starter has such a bad intermittent short in it, that it keeps the points welded together after the key is released from the Start position. You can get a starter Draw test done at some parts stores like Autozone or Oreillys to name a few, they do it free and on the truck.

Let me know the result please,

this sound slike you have a bad computer ignition module. if it just turns over repeatedly and doesn't start. need to replace it.

Is the security light flashing? If so it thinks you are trying to steal it. You first need to determine if power is getting to the solenoid wire, its the small wire at the starter solenoid, if so, the starter is bad, if not, find out why.

no this is either caused by a defective starter solenoid (on starter) or a defective positive battery (red) cable, both are common on this year of explorer.