Sidereal day and solar day
A sidereal day is the time Earth takes to turn once against the distant stars, about 23 hours 56 minutes. A solar day, the usual day, is about 24 hours because Earth also moves along its orbit.
What it is
The difference of about four minutes a day adds up to a full day over a year, which is why the stars rise about four minutes earlier each night.
How it is worked out or read
The sidereal time used in the Lagna formula is the local sidereal time, measured from the equinox point rather than the Sun. The conversion from the solar time uses this difference.
Where it appears on this site
The sidereal time is part of the Lagna calculation.
Why the sidereal day is shorter
Earth turns once relative to the stars in about 23 hours 56 minutes. In that time it has also moved a little along its orbit around the Sun, so the Sun has not returned to the same place in the sky. Earth must turn a little further, about four minutes more, before the Sun is at the meridian again. That extra four minutes is the difference between the sidereal and solar days, and it accumulates to a full day in a year.
What this means for a chart
The sidereal day is the natural clock of the stars, and it is the one that sets the local sidereal time used in the Lagna. For this reason, the Lagna changes at a steady rate against the clock: about one sign every two hours, with the exact timing depending on the latitude. A reader who wants the Lagna at a given clock time must convert the solar time to sidereal time first, which is what the tool does with the Greenwich sidereal time for the birth moment.
A simple observation
You can see the difference yourself. Note a bright star that rises just before the Sun on one evening, and check when it rises the next evening. It will rise about four minutes earlier each night. Over a month the difference is about two hours, which is why the same star rises at very different clock times through the seasons. The sky keeps the sidereal time, and the clock keeps the solar time.
Why the difference matters for timing
The difference between the sidereal and solar day matters whenever a reader converts a clock time into a sidereal position. A reader who forgets it may assign the Lagna to the wrong time, because the sidereal clock runs about four minutes faster each day.
Sidereal time and the stars' rising
Because the sidereal day is set by the stars, a reader can observe its effect by noting when particular stars rise on successive nights. The shift of about four minutes each night is the visible sign of the difference between the two days.
Where the four minutes go
Over a full year the four-minute daily gap adds up to exactly one extra sidereal day. That is why a year contains about 366 sidereal days but 365 solar days.