Jupiter isn’t close.
The smallest Brown Dwarf (we know of) to achieve stellar ignition is 50 Jupiter masses.
Made for a nice Science Fiction story tho.
For the OP, if you really want to blow their ‘minds’ tell them our star even has a name, and they used it before system.
50 Jupiter masses is still a brown dwarf. It is massive enough to achieve deuterium ignition (13 Jupiter masses minimum), which will not last long. Not quite enough mass to start burning what little lithium is there.
Still not enough mass to initiate a proton-proton chain reaction.