Sidereal time on any date

Builds on: Sections 1.3, 2.1, 2.3 and 2.4
Reading time: 35 min
Exercises: 4
Olympiad problems: 2

At 22:00 on 15 January in Ankara, Aldebaran crossed the meridian a quarter of an hour earlier, high in the south, and Sirius will cross it at 23:54. Both facts follow from one number, the local sidereal time, which at that moment was 4h 52m. In Section 1.3 we met sidereal time as the right ascension of whatever is on the meridian, and a star's hour angle as the sidereal time minus its right ascension. In this section we find the sidereal time for any place, date and time, which is the first step of almost every problem that links the sky to a clock.

2.5.1Sidereal time from the Sun

Before you readMake a guess first

At 12:00 apparent solar time on the day of the March equinox, what is the local sidereal time?

The Sun has a right ascension and an hour angle like any star, so (1.10) applies to it. Its hour angle is the apparent solar time minus 12 hours, by (2.3). Putting the two together gives the sidereal time from the Sun.

ResultSidereal time from the Sun(2.10)

LST = T_{app} + α_{⊙} - 12 h

$LST$	local sidereal time
$T_{app}$	apparent solar time
$α_{⊙}$	right ascension of the Sun

The Sun's right ascension grows by about 2 hours a month, so at a fixed solar time the sidereal time is about 2 hours later each month, and 24 hours later after a year.

For a quick estimate, the sidereal time is about 0h at 12:00 on 21 March and runs 3 min 56.6 s ahead of solar time each day. The estimate can be out by several minutes, because the equation of time and the uneven motion of the Sun in right ascension are ignored, but it is often enough to tell which part of the sky is up.

Figure 2.6The meridian among the starsinteractive

Date in 2026Ankara clocks

Local sidereal time
Bright star nearest the meridian
Hour angle of the Sun

The figure shows the sky between declinations −60° and +60° as a flat strip, with right ascension growing to the left, as on a star chart, and stars down to magnitude 4 from a catalogue. The dotted lines are the celestial equator and the ecliptic, and the Sun is drawn at its place on the ecliptic. The heavy vertical line is the meridian of Ankara, at a right ascension equal to the local sidereal time. The sliders set the date and the time on Ankara's clocks. Moving the time by an hour moves the meridian by an hour of right ascension, while moving the date at a fixed time moves it by about four minutes a day. At the meridian has just passed Aldebaran. At it runs close to the Sun, near right ascension 0h, and at it has moved round to Pegasus, close to Enif.

Example 2.10A quick estimatefinish it

Estimate the local sidereal time at 21:00 local mean time on 21 May.

The start

At 12:00 on 21 March the sidereal time is about 0h. Sixty-one days later, at the same time, it is later by $61 \times 3 m 56.56 s = 4 h 00.5 m$ .

Your stepNow add the nine hours from 12:00 to 21:00, remembering that sidereal time runs faster than solar time by a factor of 1.002738.

hours

2.5.2Sidereal time from the date

For accurate work we start from the sidereal time at Greenwich, the Greenwich mean sidereal time, written GMST, and add the longitude. Problems usually give GMST at 0h UT on some reference date, such as 1 January of the year. From there it grows by 3 min 56.56 s for each whole day, by (2.2), and on the day itself by 1.002738 hours for each hour of universal time.

ResultSidereal time from a reference date(2.11)

\begin{aligned} GMST & = {GMST}_{0} + n \times 3 m 56.56 s + k \times UT \\ LST & = GMST + λ / 15 ° \end{aligned}

${GMST}_{0}$	Greenwich mean sidereal time at 0h UT on the reference date
$n$	number of whole days from the reference date to the day in question
$k$	1.002738, the number of sidereal hours in an hour of universal time
$λ$	longitude, positive east

Whole multiples of 24 hours are taken away at the end.

Without a reference date, the Julian date of Section 2.4 gives GMST directly. Between noon UT on 1 January 2000 and any moment there are $D = JD - 2 451 545.0$ days, and in each of them the sidereal time grows by $24 h \times 1.002738$ .

ResultSidereal time from the Julian date(2.12)

GMST = 18.697375 h + 24.0657098 h \times D

$D$	days since J2000.0, $JD - 2 451 545.0$ , including the fraction of a day

Whole multiples of 24 hours are taken away at the end. The first number is GMST at J2000.0. The formula is good to a few seconds over this century.

Example 2.11The sky over Hanoi

Find the local sidereal time in Hanoi (105.85° E, UT+7) at 21:00 on 28 September 2026.

Solution

From Example 2.8, 0h UT on 28 September 2026 is $D = 9766.5$ days after J2000.0. By (2.12),

\begin{aligned} {GMST}_{0} & = 18.697375 + 24.0657098 \times 9766.5 \\ = 235 056.452 h, \end{aligned}

and taking away $9794 \times 24$ h leaves 0.452 h, or 0h 27.1m. At 21:00 in Hanoi the universal time is 14:00, and in 14 hours the sidereal time grows by $14 \times 1.002738 = 14 h 02.3 m$ , so GMST is 14h 29.4m. Hanoi is $105.85 / 15 = 7.057$ h, or 7h 03.4m, east of Greenwich.

AnswerThe local sidereal time is

14 h 29.4 m + 7 h 03.4 m = 21 h 32.8 m

. Enif, the nose of Pegasus, at right ascension 21h 45.5m, crosses the meridian about 13 minutes later.

Quick checkUse the result above

At 22:00 on 15 January the local sidereal time in Ankara is 4h 52m. Sirius has right ascension 6h 46m. How many hours east of the meridian is Sirius?

hours

Common errorsSection 2.5

Sidereal time runs faster than solar time. An interval of universal time must be multiplied by 1.002738 before it is added to a sidereal time, which adds about 10 seconds for every hour.
GMST is the sidereal time at Greenwich. The local sidereal time needs the longitude added, east positive, at one hour for every 15°.
Days are counted from 0h UT on the reference date to 0h UT on the day in question. In a leap year February has 29 days, and forgetting it puts the answer 3 min 56 s out.
A sidereal time larger than 24h, or negative, is fine in the middle of a calculation, but the answer should be brought back between 0h and 24h.

ExercisesSection 2.54 questions

2.5.1
Use (2.12) to find GMST at 0h UT on 1 January 2026, which is JD 2 461 041.5.
hours
2.5.2
At 0h UT on 1 January 2017 the Greenwich mean sidereal time was 6h 43m. What was it at 0h UT on 1 March 2017?
hours
2.5.3
Roughly what is the local sidereal time at midnight, apparent solar time, at the September equinox?
2.5.4
Find the local sidereal time in Ankara (32.86° E, UT+3) at 22:00 on 15 January 2026, given that GMST at 0h UT on 1 January 2026 was 6h 42.6m.
hours

Olympiad problemsPast IOAA problems, restated2 problems

2018Sidereal time zero twice in a day10 points

Adapted from IOAA 2018 (Beijing), theory problem 5. We have restated it in our own words and kept the original data. The solution was last checked on 17 October 2026.

On one particular day of each year, the Greenwich mean sidereal time is 00:00:00 twice. (a) Roughly what is the right ascension of the Sun on that day? (b) Estimate the date of this day in 2018, given that GMST was 6.706 h at 0h UT on 1 January 2018 (JD 2 458 119.5).

I solved it without the solution

Answer(a) The Sun is near the September equinox, at a right ascension of about 12h. (b) It happens on 21 September 2018. The official solution states 18h in part (a), which does not fit its own reasoning about the September equinox, where the Sun's right ascension is 12h.

2020The sky above Chiayi6 points

Adapted from the 2020 Global e-Competition on Astronomy and Astrophysics (Estonia, online), held in place of the IOAA that year, theory problem 7, part (a). We have restated it in our own words and kept the original data. The solution was last checked on 17 October 2026.

An observer in Chiayi, Chinese Taipei (23.5° N, 120.4° E, UT+8) watched meteors at 21:00 local time on 25 September 2020. Find the local sidereal time of the observation. The Greenwich sidereal time at 0h UT on 1 January 2020 was 6h 40m 30s.

I solved it without the solution

AnswerThe local sidereal time was 21h 20m 51s. The official solution rounds the 268 days of growth down to 17h 36m 36s and gets 21h 20m 50s.