Xbox runs for a while before turning off

Hello! I have an original Xbox, and it runs very well, but only for maybe an hour and a half, before it suddenly turns off. It can be turned back on, but it will then run for only a few minutes before shutting off. If you turn it on again after that, it is only on for seconds. If I leave it alone for a few days I get my hour and a half again, only for it to repeat.

I have tried a new power cable and that did not seem to solve the problem. Not sure what I should be looking for. I was planning on cracking it open soon, but wanted to pose this question first. Any help is appreciated.