In the previous article, we explored various time scales. In this article, I will explain how these time scales are applied in astrological calculations.
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Orbital Motion
In reality, all orbital calculations are first performed using a uniform, theoretically ideal timescale. To predict the exact positions of planets in their orbits, we require a highly precise, repeatable process that defines a "unit of time." Knowing this unit and the initial positions of planets relative to the fixed stars, we can accurately predict where a planet will be after a given number of time units (on the backdrop of the fixed stars).
Initially, the time unit was defined as the "mean solar day." However, it was later discovered that solar days vary slightly in length due to factors such as Earth’s axial tilt, orbital eccentricity, and tidal interactions. Later, for a more stable reference, atomic standards (TAI) were adopted, while Earth's rotation began to be measured independently using very-long-baseline interferometry (VLBI). This led to the development of UT1 (Universal Time 1), which accounts for Earth’s rotational irregularities.
Crucially, all orbital calculations are performed in a uniform theoretical timescale. The goal is to define a future moment as "X time units" from the current global reference.
However, UT1 is still imperfect because Earth’s rotation is gradually slowing due to tidal friction. If we relied solely on UT1, the Moon’s motion would appear to accelerate over time, not because it is moving faster, but because Earth’s rotation is decelerating.
To resolve this, we needed a more stable definition of time. This arrived in the 1950s with the adoption of International Atomic Time (TAI), where the second is defined by the hyperfine transition frequency of cesium-133 (9,192,631,770 oscillations per second).
Now, planetary positions are predicted in atomic time. For example, we might say, "The Moon will reach 0° Aries after X cesium-133 oscillations." However, for practical use, we convert these atomic intervals into UT1 (reflecting the portion of the solar day).
Here's how it happens:
Practical Calculations
1) First, planetary motion is calculated in the barycentric reference frame (relative to the Solar System's center of mass) using Barycentric Dynamical Time (TDB). This timescale accounts for effects of general relativity—for example, clocks closer to the Sun run slightly slower due to gravitational time dilation and kinematic corrections from Earth's motion. TDB is necessary to accurately model gravitational interactions between all celestial bodies, especially for Mercury and GPS satellites.
2) Next, the coordinates are converted to the geocentric system (relative to Earth's center) using Terrestrial Time (TT)—the uniform timescale kept by atomic clocks on Earth (TT = TAI + 32.184 seconds, a fixed offset from historic Ephemeris Time). The difference between TDB and TT is small (up to 0.0017 seconds) but crucial for precision.
3) Finally, for an Earth-based observer, TT time is converted to UT1 and then
- to sidereal time (to determine where planets will be visible locally) as described here.
- to UTC (to express the same moment in a familiar civil time)
This approach allows calculating planetary motion in perfectly uniform time (TDB/TT) first, then "mapping" the results onto Earth's actual rotation (UT1) and then to sidereal and civil time scales.
The modern equivalents of the old "Ephemeris Time" are TDB (for barycentric calculations) and TT (for geocentric ones).
N.B.: Astrologers' claims like "we calculate orbital motion in UT1" or "Ephemeris Time is longitude-dependent" are scientifically incorrect. TDB and TT are global scales; local corrections (like sidereal time) apply only when converting to an observer's coordinates.
Local Observer's Perspective
In the final stage, we determine how the celestial sphere appears to an observer at a specific location. This is a purely geometric transformation—we do not alter the actual positions of celestial bodies but instead rotate the celestial sphere based on the observer’s longitude and latitude.
This is where sidereal time becomes relevant. It measures how much the celestial sphere has rotated relative to a given meridian. By convention, we first calculate Greenwich Sidereal Time (GST) and then adjust it for the observer’s longitude to obtain Local Sidereal Time (LST).
The process involves:
- Spherical trigonometry to account for the observer’s position.
- Rotation matrices to transform equatorial coordinates (right ascension and declination) into horizontal coordinates (altitude and azimuth).
- House system calculations.
Summary
The main takeaways from this article are:
- Time, in theoretical calculations, is uniform and continuous. In practice, it is measured in atomic seconds (TAI), with geocentric (TT) and barycentric (TDB) variants for precision accounting for relativistic effects. Zodiacal positions are calculated in the geocentric frame (TT).
- Ephemerides are computed in these uniform dynamical timescales (TDB/TT), not in Earth-rotation dependent timescales like UT1.
- For practical applications, terrestrial Time (TT) is converted to UT1 using ΔT (= TT - UT1) to align with Earth's actual rotation. UT1 serves dual purposes:
- As a basis for UTC to serve a reference for civil time
- To calculate Greenwich Sidereal Time (GST), which encodes Earth's rotation angle relative to the fixed stars in the 0-meridian.
- From a mathematical perspective, GST and LST represent rotation angles of the celestial sphere. GST specifies the rotation angle of the celestial sphere relative to the Greenwich meridian. This angle is then adjusted for the observer's longitude, yielding the local rotation angle of the celestial sphere (LST). This LST angle is implemented in the rotation matrix to:
- Transform equatorial coordinates to horizontal coordinates, and
- Determine planetary positions relative to astrological houses.
