Sidereal time and the Lagna formula
Local sidereal time is the hour angle of the vernal equinox, and it is the quantity that links the Lagna to the local place and time.
What it is
The Lagna is found from the local sidereal time (in degrees, as RAMC) plus the longitude, the obliquity and the latitude. The site uses the standard ascendant formula, and then subtracts the ayanamsa to give the sidereal Lagna.
How it is worked out or read
The greenwich apparent sidereal time comes from the library, and the longitude of the place is added. The formula is in the how the Lagna is calculated article.
Where it appears on this site
The Lagna is shown in the birth chart. The method page gives the formula.
What sidereal time is
Sidereal time is time measured by the rotation of the Earth relative to the stars rather than the Sun. A sidereal day, one full turn relative to the stars, lasts about 23 hours 56 minutes 4 seconds, nearly four minutes shorter than the solar day. Local sidereal time at a place tells which part of the sky is crossing the meridian there: in technical terms, it equals the right ascension of the meridian. Because the stars return to the same position about four minutes earlier each day, sidereal time runs slightly faster than clock time.
From clock time to sidereal time
To find local sidereal time, a calculator first converts the local clock time to Universal Time. It then computes Greenwich sidereal time for that moment from a standard formula based on the number of days since a reference epoch. Finally it adds the place's longitude, converted to time at four minutes per degree, east positive. A place at 77 degrees east, for example, adds 5 hours 8 minutes to Greenwich sidereal time.
This step is where the birth place enters the Lagna calculation: the same UT moment gives different sidereal times at different longitudes.
From sidereal time to the Lagna
With local sidereal time known, the rising degree of the ecliptic follows from a formula that uses three quantities: the right ascension of the meridian, the obliquity of the ecliptic and the latitude of the place. The latitude enters because the horizon tilts differently at different latitudes. The result is the tropical ascendant, from which the ayanamsa is subtracted to give the sidereal Lagna used in Jyotish. The tenth-house point, the midheaven, comes from the same sidereal time without the latitude.
Why the Lagna changes so fast
Because the Earth turns once a day, the full zodiac rises over the horizon in about one sidereal day, so on average a sign rises every two hours and a degree every four minutes. That is why the birth time matters so much for the Lagna and the houses, and much less for the planets, which hardly move in a few minutes. An error of eight minutes in birth time typically moves the Lagna by about two degrees, which matters when the Lagna is near the edge of a sign.
Older methods
Traditional astrologers computed the Lagna from the time elapsed since sunrise, in ghatis and palas, together with the Sun's position and tables of rising times for the latitude. This method reaches the same result by a different route, since sunrise fixes the Sun on the horizon and the rising-time tables account for the obliquity and latitude. Modern calculators use sidereal time because it is direct and precise, but the older method is worth knowing when reading traditional charts that record the birth time as so many ghatis after sunrise.