Sidereal Time Conversions

 

 
Greenwich Mean and Apparent Sidereal Time for Julian Day Date in Degrees of Arc
Notes: Greenwich Mean Sidereal Time (GMST) is the hour angle (arc span of hours) between the prime meridian (0° longitude) at Greenwich and the vernal equinox measured westward along the celestial equator. Therefore at Greenwich the GMST corresponds to the line of right ascension that passes through the zenith (the celestial meridian) above Greenwich. The Mean Time indicates that the vernal point used is the intersection of the Earth's mean equator of date (accounts for precession but not nutation) and ecliptic of date (changes with precession).
 
Greenwich Apparent Sidereal Time (GAST) differs from mean time in that the true vernal equinox point is used. This is the intersection of the ecliptic of date and true equator, which is the mean equator corrected for nutation. This correction is referred to as the equation of the equinoxes and does not exceed 1.15 seconds of time.
 
The first grayed equation below is a general formula that can convert a given JD date/time to GMST or GAST. The next two are easy to remember wrapper functions that accomplish one or the other conversion.
 
Resource: [AA: p. 84]
 
Greenwich Mean/Apparent Sidereal Time function coefficients
 
Greenwich Mean/Apparent Sidereal Time core function
Input: Julian day date.
Output: Time in degrees of arc normalized 0 - 360°.
 
Equation of the Equinoxes for date in Julian century format
Input: Julian centuries.
Output: Seconds of time.
 
Equation Converts JD to Greenwich Mean or Apparent Sidereal Time
Input: Julian day date and IsMean flag: 1 = get Mean Time, else Apparent Time
Output: Time in degrees of arc normalized 0 - 360
 
Greenwich Mean or Apparent Sidereal Time for Julian day date wrapper functions
Input: Julian day date.
Output: Time in degrees of arc normalized 0 - 360°.
 
Greenwich Mean Sidereal Time for Julian day date
Greenwich Apparent Sidereal Time for Julian day date
 
Example: Determine GMST and GAST for date and time (UT).
 
Input date and UT time (default to User Constants)
 
Convert day, hr, min and sec into day decimal:
This yields the following date to convert:
Convert civil date to JD used by GMST and GAST equations:
 
GMST and GAST results in degrees:
 
Convert arc degrees into hour angle time units
GMST
GAST
The results reveal the slight difference in time reckoning caused by Nutation.

Local Mean and Apparent Sidereal Time for Julian Day Date in Degrees of Arc
Notes: Local Mean Sidereal Time (LMST) indicates what meridian of right ascension (RA) is directly above the observers location. LMST can be determined by subtracting the observer's longitude (positive west, negative east) from GMST. At the Greenwich meridian LMST and GMST are the same. Locations west of the Greenwich Meridian are behind in sidereal time (ST) so the subtraction of (positive) west longitudes makes sense. A common use of this calculation by amateur astronomers is to determine which sky map to use for a time (in UT) of night.
 
Local Apparent Sidereal Time (LAST) is similarly figured from GAST. LAST is useful in telescope Polar alignment and other observations requiring positions at an instantaneous equinox.
(See definition GMST and GAST above and Polar Alignment from Sidereal Time.)
 
Function determines LMST or LAST
Input: JD, Longitude and IsMean flag: 1 = Mean, else Apparent.
Output: Time in degrees of arc normalized 0 - 360°
 
LMST() and LAST Wrapper Functions
These just wrap the above function so the IsMean flag is not needed.
Input: JD and Longitude of observer.
Output: Time in degrees of arc normalized 0 - 360°
 
Examples: Given Longitude and Date/Time determine Mean and Apparent Local Sidereal Times
 
Input observer's longitude (deg, min, sec), positive west (defaults are User Constants):
Input date (yr, mth, day) and UT time (hr, min, sec) (defaults are User Constants):
 
Convert longitude to decimal format:
Convert date and UT to Julian day date:
 
LMST calculation:
Result in degrees:
Translated from degrees to sidereal time:
LMST
 
LAST calculation:
Result in degrees:
Translated from degrees to sidereal time:
LAST

UT for Local Mean and Apparent Sidereal Time for Julian Day Date in Degrees of Arc
See Also: LMST and LAST functions above.
Notes: These equations converts Local Mean Sidereal Time (LMST) and Local Apparent Sidereal Time (LAST) to Universal Time (UT). They works by subtracting the Greenwich Sidereal Time at calendar day start (0.0 UT) from the target sidereal time. This quantity of sidereal time, converted to a quantity of solar time, is the mean solar time since midnight -- the UT hour for that day. Your local watch time is usually an integer number of hours different from this, depending on time zone and daylight savings adjustments. See Sidereal vs Civil Time.
 
One complication when dealing with sidereal time is that two instances of a sidereal time can happen in one solar day. In fact the first few minutes (about 3:55.9) of sidereal time at the start of a solar day will have two occurrences that solar day. This is because a sidereal day is shorter than a solar day -- which allows sidereal time to roll-over and repeat in the course of the longer solar day. This is illustrated in the example at the bottom.
 
The first equation below is a general equation that can convert LMST or LAST. The next two equations are wrapper functions that do one or the other conversion. This is followed by a formatting matrix than can display the second instance of a sidereal time or "OK" if there is only one instance that day.
 
Resource: [TAA p. B7]
 
Local Sidereal Time to UT function handles Mean and Apparent cases
Input: Julian day date the Local Sidereal Time (LST) occurs in, LST to be converted, Longitude -- all in decimal format -- and flag to indicate Mean or Apparent Time conversion: IsMean = 1 if LST is Mean, any other for Apparent.
Output: is UT in degrees of arc.

LMST and LAST to UT Conversion wrapper functions
Input: Julian day date that ST occurs in, ST to be converted and Longitude -- all in decimal format.
Output: UT in degrees of arc normalized 0 - 360°
LMST to UT Conversion Function
LAST to UT Conversion Function
Difference between Solar and Sidereal day in Solar Time expressed in degrees
Amounts to less than 4 minutes of time. Used by Test for Repeated Sidereal Times function below.

 
Test for Repeated Sidereal Times in one Solar Day
If the UT is within about 4-minutes after midnight, the ST at that instance of UT will repeat itself in about 23 hours and 56 minutes. This function will return the UT of the second instance in terms of degrees. If there is no second instance a -1 is returned.
Input: Universal Time in decimal degrees.
Output: UT of second ST occurrence in degrees of arc normalized 0 - 360° or -1 if no occurrence.

 
Formatting Matrix for Sidereal Time
Input: Time in decimal degrees.
Output: Matrix displaying first and possible second sidereal time in hr:min:sec format.
Note that nested arrays not legal in Mathcad 8 Standard Edition.

Example: Determine UT for Given LMST or LAST
 
Input calendar date, LST (considered to be Mean or Apparent) and longitude of observing site.
Calendar Date:
Local Sidereal Time:
West longitude:
 
Convert input data to decimal format for use in equation.
Julian day date at start of calendar date:
Local Sidereal Time in degrees:
Longitude of site:
 
Conversion functions generate UT solution in degrees of arc. Assume LSTDec is Local -
- Mean ST:
- Apparent ST:
 
Convert solutions to time units your alarm clock understands.
Universal Time that LMST occurs on given date
Universal Time that LAST occurs on given date
This example was contrived so that the LAST example had two occurrences of a ST at the specified UT while the same amount of LMST had one ST occurrence.

Astro Utilities Electronic Book Copyright © 1999 Pietro Carboni. All rights reserved.