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 rollover 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 4minutes 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. 
