Sidereal Time: From the Sidereal Day to LAST #

Sidereal time is the bridge between the time on the clock and the orientation of the sky. It tells you which part of the celestial sphere is currently on the meridian, and therefore determines the Midheaven and — through the Ascendant formula — the entire framework of houses. No quantity in chart calculation is more important to get right. This article defines sidereal time from first principles, gives the IAU 2006 computation, and traces the path from a civil clock reading to the local apparent sidereal time that enters the Ascendant formula.

The Sidereal Day #

The sidereal day is the time it takes the Earth to rotate once relative to the stars (more precisely, relative to the vernal equinox). It is approximately 23 hours, 56 minutes, 4.091 seconds of mean solar time — about 3 minutes 56 seconds shorter than a solar day.

The difference arises because, in one solar day, the Earth both rotates on its axis and advances about 1° in its orbit around the Sun. The extra ~1° of rotation needed to bring the Sun back to the meridian takes about 3 minutes 56 seconds. Relative to the stars (which are effectively at infinity), the Earth completes a full 360° rotation in one sidereal day.

In one tropical year, there are approximately 366.2422 sidereal days but only 365.2422 solar days. The extra sidereal day is “used up” by the Sun’s apparent annual circuit around the ecliptic.

Greenwich Mean Sidereal Time (GMST) #

GMST is the mean sidereal time at the Greenwich meridian (longitude 0°). It equals the hour angle of the mean vernal equinox at Greenwich. “Mean” here indicates that the short-period oscillation of the equinox due to nutation has been averaged out.

GMST is computed from UT1 (not UTC, not TT — this distinction matters; see Time Scales). The IAU 2006 expression, as a function of the Julian UT1 date, is:

$$ \text{GMST} = \text{ERA} + 0.014506’’ + 4612.156534’’ , T_u + 1.3915817’’ , T_u^2 - 0.00000044’’ , T_u^3 - 0.000029956’’ , T_u^4 - 0.0000000368’’ , T_u^5 $$

where $T_u$ is the number of Julian centuries of UT1 from J2000.0:

$$ T_u = \frac{JD_{\text{UT1}} - 2451545.0}{36525} $$

and ERA is the Earth Rotation Angle (defined below). The polynomial terms after ERA represent the accumulated precession from J2000.0.

The Classical Polynomial (Direct Computation) #

An equivalent and commonly used formula computes GMST directly in seconds of time from the Julian Date. The IAU 1982 expression (Aoki et al.), which remains adequate for chart calculation at the level of a few hundredths of a second:

$$ \text{GMST}_{0h} = 24110.54841 + 8640184.812866 , T_0 + 0.093104 , T_0^2 - 6.2 \times 10^{-6} , T_0^3 $$

This gives the GMST at 0h UT1 in seconds of sidereal time, where:

$$ T_0 = \frac{JD_{0h,\text{UT1}} - 2451545.0}{36525} $$

is the Julian centuries from J2000.0 to 0h UT1 on the date of interest. To get the GMST at a non-zero UT1 hour, add the UT1 time of day multiplied by the ratio of sidereal to solar time:

$$ \text{GMST} = \text{GMST}{0h} + \text{UT1}{\text{hours}} \times 1.00273790935 $$

The factor $1.00273790935$ is the ratio of sidereal seconds to mean solar seconds (one sidereal day has $86164.0905$ solar seconds; $86400/86164.0905 = 1.00273790935$).

The result is in seconds of sidereal time. Divide by 3600 for hours, then normalize to $[0, 24)$.

Worked Example #

Date: January 10, 2024, at 14:30:00 UT1.

Step 1 — Julian Date at 0h UT1: $JD_{0h} = 2460319.5$.

Step 2 — $T_0 = (2460319.5 - 2451545.0)/36525 = 8774.5/36525 = 0.240173$

Step 3 — GMST at 0h UT1 (seconds):

$$ \text{GMST}_{0h} = 24110.54841 + 8640184.812866 \times 0.240173 + 0.093104 \times 0.240173^2 - 6.2\times 10^{-6} \times 0.240173^3 $$

$$ = 24110.54841 + 2075917.268 + 0.005372 - 0.000000086 $$

$$ = 2100027.822 \text{ seconds} $$

Step 4 — Normalize to $[0, 86400)$: $2100027.822 \mod 86400 = 2100027.822 - 24 \times 86400 = 2100027.822 - 2073600 = 26427.822$ seconds $= 7^h 20^m 27.82^s$.

Step 5 — Add the UT1 time of day: $14.5 \times 3600 \times 1.00273790935 = 52342.9$ seconds.

$$ \text{GMST} = 26427.822 + 52342.9 = 78770.7 \text{ s} = 21^h 52^m 50.7^s $$

At 14:30 UT1 on January 10, 2024, the GMST is approximately $21^h 52^m 51^s$. This means the point on the celestial equator at right ascension $21^h 53^m$ is on the Greenwich meridian.

The Equation of the Equinoxes #

The vernal equinox is not fixed in space — nutation causes it to oscillate with a dominant period of 18.6 years (and many shorter periods). The position of the true equinox at any moment differs from the mean equinox by a small angle.

The equation of the equinoxes ($Ee$) is the difference in right ascension between the true and mean equinoxes:

$$ Ee = \Delta\psi \cos\varepsilon + \text{small terms} $$

where $\Delta\psi$ is the nutation in longitude and $\varepsilon$ is the true obliquity. The “small terms” are complementary terms of order $0.001’'$, negligible for chart calculation. See Precession and Nutation for the full development of $\Delta\psi$.

The dominant term has an amplitude of about $\pm 1.1$ seconds of time (corresponding to $\pm 17’'$ of arc), with a period of 18.6 years. Smaller terms contribute oscillations at various periods.

Greenwich Apparent Sidereal Time (GAST) #

GAST is the hour angle of the true vernal equinox at Greenwich. It differs from GMST by the equation of the equinoxes:

$$ \text{GAST} = \text{GMST} + Ee $$

GAST is the sidereal time that corresponds to the actual direction of $\gamma$ at the moment — the direction that defines the tropical zodiac. For chart calculation, GAST (or its local equivalent, LAST) is the quantity that enters the Ascendant and house-cusp formulas, because these formulas reference the true equinox.

In practice, the difference between GMST and GAST is at most about $\pm 1.1$ seconds of time ($\pm 0.28’$ in ecliptic longitude). For a chart where the Ascendant moves at roughly $1°$ per 4 minutes of time, $1.1$ seconds of sidereal time corresponds to about $0.005°$ of Ascendant shift — negligible. But including the correction is straightforward and costs nothing.

Local Sidereal Time #

The local sidereal time at any geographic longitude $L$ (positive east) is simply the Greenwich sidereal time plus the longitude expressed in time:

$$ \text{LMST} = \text{GMST} + L / 15° $$

$$ \text{LAST} = \text{GAST} + L / 15° $$

Normalize the result to $[0^h, 24^h)$.

LAST is the quantity that directly equals the right ascension of the meridian (RAMC). This is the connection to chart angles: the MC has the same right ascension as the LAST, and the Ascendant is computed from the LAST and the obliquity, as developed in Chart Angles.

Continuing the Worked Example #

Location: 12°30′ E longitude.

$$ \text{LAST} = 21^h 52^m 51^s + \frac{12.5°}{15°/\text{h}} = 21^h 52^m 51^s + 0^h 50^m 00^s = 22^h 42^m 51^s $$

The right ascension of the meridian at this location and time is $22^h 42^m 51^s \approx 340.71°$. The MC’s ecliptic longitude is found by converting this right ascension to ecliptic longitude (a single application of the equatorial-to-ecliptic transformation for a point on the equator; see Chart Angles).

Earth Rotation Angle (ERA) and the CIO Paradigm #

The IAU 2000/2006 resolutions introduced a new paradigm that separates Earth rotation from precession-nutation more cleanly than the classical sidereal-time approach.

Earth Rotation Angle #

The Earth Rotation Angle (ERA, denoted $\theta$) is the angle between the Celestial Intermediate Origin (CIO) and the Terrestrial Intermediate Origin (TIO), measured along the equator of the Celestial Intermediate Pole (CIP). It is a linear function of UT1:

$$ \theta = 2\pi (0.7790572732640 + 1.00273781191135448 \times D_u) $$

where $D_u = JD_{\text{UT1}} - 2451545.0$ is the number of days from J2000.0 (UT1). This formula is exact — ERA has no polynomial terms beyond linear — because, by design, all non-linear effects (precession and nutation) are absorbed into the CIO and CIP definitions.

ERA is the modern replacement for sidereal time as the measure of Earth rotation. It increases uniformly with UT1, at a rate of one full revolution per sidereal day.

CIO and the Separation from Precession #

In the classical paradigm, sidereal time mixes two things: (1) the Earth’s rotation and (2) the precession of the equinox. GMST includes accumulated precession (the polynomial terms), and the equation of the equinoxes adds nutation. In the CIO paradigm, ERA is purely rotational, and precession-nutation is handled separately through the celestial-to-intermediate (GCRS-to-CIRS) transformation.

For chart calculation, the practical difference is nil. Both approaches give the same final answer — the right ascension of the meridian — by different computational routes. The classical GMST+Ee=GAST approach is simpler to implement and is used by most astrological software. The CIO-based approach is used internally by modern astronomical software (SOFA, ERFA) but produces the same LAST.

Why ERA Matters #

ERA is useful for understanding what sidereal time is: a measure of the Earth’s rotation, contaminated (in the classical formulation) by precession. ERA strips the precession out, revealing the pure rotation. For a developer implementing high-precision calculations using SOFA or a similar library, ERA is the starting point; the library then adds precession-nutation to produce the equivalent of LAST.

The LAST → RAMC Identity #

The single most important relationship for chart calculation is:

$$ \text{LAST} = \text{RAMC} $$

The local apparent sidereal time is the right ascension of the Midheaven. This is not an approximation; it is a definition. The meridian is the great circle through the poles and the zenith, and the sidereal time at any moment is the right ascension of the point where the equator crosses the upper meridian. That point is the MC (in the equatorial system).

From RAMC, the MC’s ecliptic longitude, the Ascendant, and all house cusps follow. The details of those computations are in Chart Angles and House Systems.

Summary of the Computation Chain #

The complete path from civil time to LAST, with article references:

1. Civil time → UTC (Time Zones)
2. UTC → UT1 (subtract DUT1; Time Scales)
3. UT1 → Julian Date (Julian Date)
4. JD UT1 → GMST (polynomial above, or via ERA)
5. GMST → GAST (add equation of the equinoxes; Precession and Nutation)
6. GAST → LAST (add longitude)

Steps 4–6 are the core of this article. Steps 1–3 are covered in the referenced articles.

Implementation Notes #

1. Use UT1, not UTC or TT, for sidereal time computation. UT1 tracks the actual rotation of the Earth. UTC can differ from UT1 by up to 0.9 seconds (the DUT1 correction); TT differs by $\Delta T \approx 69$ seconds as of 2024. Using TT instead of UT1 would shift the Ascendant by about $0.3’$ — small, but unnecessary.

2. Normalize aggressively. The GMST polynomial can produce very large numbers (millions of seconds). Normalize to $[0, 86400)$ seconds (or $[0°, 360°)$) after every computation.

3. The sidereal-to-solar ratio $1.00273790935$ is the ratio of the mean sidereal day to the mean solar day. It is a constant to the precision needed here (it changes by about $10^{-10}$ per century due to the slowing of Earth’s rotation, but this is completely negligible).

4. Accuracy: the classical GMST polynomial (Aoki et al. 1982) is accurate to better than 0.1 second over the range 1900–2100. The IAU 2006 expression via ERA is accurate to better than 0.001 seconds over the same range. For astrological chart calculation, either is more than sufficient.

5. Testing: at 0h UT1 on J2000.0 (January 1.5, 2000), the GMST should be $18^h 41^m 50.5^s$ (approximately $280.46°$). This is a standard check value.

References #

• Capitaine, N. et al. (2003). “Expressions for IAU 2000 precession quantities.” Astronomy & Astrophysics, 412, 567–586.
• Capitaine, N., & Gontier, A.-M. (2007). “Expressions for the celestial intermediate pole and celestial ephemeris origin consistent with the IAU 2000A precession-nutation model.” Astronomy & Astrophysics, 275, 645–650.
• Aoki, S. et al. (1982). “The new definition of universal time.” Astronomy & Astrophysics, 105, 359–361.
• Meeus, J. (1998). Astronomical Algorithms, 2nd ed. Willmann-Bell. Chapter 12.
• Explanatory Supplement to the Astronomical Almanac, 3rd ed. (2013). University Science Books. Chapters 2, 6.
• IERS Conventions (2010). IERS Technical Note 36. Chapter 5.