From sign and degree to one longitude
The tropical zodiac divides the ecliptic into twelve 30° signs beginning at the equinox direction, 0° Aries. To convert a sign placement into a continuous longitude, multiply the sign index by 30 and add the degree and arc minutes. For instance, 12° 30′ Scorpio is 210° + 12.5° = 222.5°. A 0° Aries target is also 360° after a complete circuit; the calculator treats those as the same direction.
The input degree must stay below 30 because 30° of one sign is 0° of the next. Arc minutes range from 0 to under 60. These are angle minutes, each one-sixtieth of a degree; they are not clock minutes. The search interval begins at 00:00 UTC on the first date and includes every instant through 23:59:59 on the final date.
How a crossing is found
At regular steps, the calculator evaluates the selected body's apparent Earth-centred longitude using Astronomy Engine's geocentric vector and true ecliptic-of-date conversion. It compares each longitude with the target as a signed angle. When the sign of that difference changes near zero, it brackets the crossing and repeatedly halves the time bracket. Result times are displayed to the nearest second; the residual column tells you how closely the position at that rounded second matches the target.
Longitude wraps from 359.999° back to 0°. A raw numeric comparison at that seam would create a false jump. The search instead compares angles modulo 360°, so a real crossing of 0° Aries is retained while the opposite 180° seam of the signed difference is discarded. Each sampling interval also checks its midpoint, which helps reveal a short backward loop whose two crossings would otherwise sit between adjacent samples.
Retrograde dates and stations
Retrograde motion is the apparent reversal of a planet against the zodiac as seen from Earth. For a target degree inside the loop, a direct crossing can precede a retrograde crossing and a later direct one. The dates all satisfy the same longitude condition; the motion column distinguishes them. Close to an exact stationary extremum, a planet may merely touch the target without crossing it. This method searches crossings, so it may omit such a tangency or extremely close pairs that fall between samples.
Sampling steps are three hours for the Moon, six for Mercury and Venus, twelve for the Sun and Mars, and a day for Jupiter through Pluto. Broad date ranges take longer to compute. A result of zero crossings can mean the body simply never reaches the chosen degree in those dates; it does not mean that degree has a special interpretation.
Coordinate and time choices
These are apparent geocentric tropical longitudes in the true ecliptic and equinox of each date. A sidereal zodiac, heliocentric origin, or fixed J2000 frame answers a different coordinate question and yields different dates. Other ephemerides may also differ slightly because of planetary models and timescale details. The interface reports UTC rather than guessing a location's civil time or daylight-saving rules.
Use the monthly ephemeris archive to compare a calculated crossing with surrounding daily positions. A crossing at 0° Aries marks the selected body's passage through that coordinate; it does not itself specify a chart's ascendant, houses, or interpretation. These require additional choices and context.