Your shut down issues could be due to a few different things, heat is possible but will usually be followed by a warning on restart that the shut down was performed to protect the computer or a similar statement. Note that a CPU heats up very quickly - you can get a thermal shutdown within one minute if vents are blocked or restricted. You can check the temps on most newer processors using a tool called CoreTemp available free. You can also check to ensure you feel warm air coming out of the vents when the computer is running - if the vents are blocked by dust bunnies (ie. you feel no warm air from 2" - 3" away), disassembly and cleaning is needed (see video below).
IMO Power issues are a more likely culprit here. The appearent lack of a restart message or reason for restart makes me consider power as the problem since it seems it happens fast enough that a dump file is not created. Replacing the battery (or at least checking to ensure it is good) would probably be a good start and possibly replacing the AC power pack also. Those are not the only possible problem areas for power though, there are two other possible power issue places, both require disassembly to reach. One of those, the power jack is possible to repair if you are adept at the use of a soldering iron, you can check that by attempting to wiggle the power cord at the jack. If the plug moves at all or feels like it isn't inserting properly, it's very likely the jack. The other, your power distribution is not a DIY thing. This video gives you an idea of what is involved in tearing down your laptop
http/www.youtube.com/watch?v=Y8BnFS9rSVc to repair a power jack (your's isn't the easiest)
The last two possible issues are ram and hard drive failures which, like heat issues, are usually followed by an alert as to why the restart occurred.
If there is a message explaining the restart, that will be helpful - the message may appear to be jibberish like "Fault exception at 00F00020" or something like that
I realize it's a lot but hopefully it helps