1988 XJS. Need to replace the alternator. Bought a Bosch AL192X. Pulled the old one and there is a difference in the size of the "arbor" where the pulley goes. The old one is smaller that this one. So my question is what can I do? Is there a replacement I can use?

 

With all the trouble replacement alternators have caused I would recommend getting your OEM one rebuilt. Hopefully with a higher Amp output rating as well.
It sounds like you do still have the original unit?
.
.
.