The simple version of the question is: what was the GPS time expressed in week number (pure number) and time of week (SI seconds) at the J2000 epoch? I am asking because my hand calculations differ from what I get from the https://gnsscalc.com/ website.
Long version (no need to read it all, if you are not interested in the details).
We know that J2000.0 is defined as Julian Date 2451545.0 TT, which is equal to 2000-01-01 12:00:00 TT, which is equal to 2000-01-01 11:59:27.816 TAI. The GPS time is calculated as weeks (pure number) and time of week (SI seconds) elapsed since 1980-01-06 00:00:00 UTC. Moreover GPS time, TAI and TT are not affected by leap seconds and they all share the same definition of a SI second, that is the Cesium standard.
So it should be straightforward to obtain the GPS time at J2000 epoch by the following calculation:
GPS time origin -> Sunday 06 Jan 1980 00:00:00 GPST - 00:00:19 TAI - 00:00:51.184 TT + 1042 weeks Sunday 26 Dec 1999 00:00:00 GPST - 00:00:19 TAI - 00:00:51.184 TT + 6 days + 11 hours + 59 minutes + 08 seconds + 816 milliseconds = 561548.816 seconds J2000 epoch -> Saturday 1 Jan 2000 11:59:08.816 GPST - 11:59:27.816 TAI - 12:00:00 TT
In total, I get that the GPS time at J2000.0 epoch (
GPS time at J2000 epoch - GPS time origin) is 1042 weeks and 561548.816 seconds that is different from the value of 1042 weeks and 561535.816 seconds I get from the website. In the website notes, it is written that the leap second is taken into account for TAI and TT calculations but I do not understand how.