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Calculating Sidereal Time

Sidereal time is the angular distance between a terrestrial meridian and the vernal equinox. The sidereal time at a given location on earth is equal to the right ascension of a star when it appears to pass over the meridian at said location. Because both right ascension and sidereal time are based on the vernal equinox, they cyclically change concurrently overtime due to changes in the earths’s axial tilt. Sidereal time is used to locate stars and other celestial objects relative to the earth because of its equivalence to right ascension.

How is sidereal time calculated?

Most of the following formulas came from the Astronomical Almanac published in 2012. The methods for calculating Greenwich mean sidereal time changes as new information about the earth’s motion is discovered. As far as I know, these formulas are current as I am writing this. The formulas for determining the Julian data since the epoch J2000 came from Wolfram. Links to the Wolfram page and a scanned version of the 2012 Astronomic Almanac will be linked at the bottom of this post.

A lot of unexplained constants follow. Most of them were derived from Simon Newcomb’s work which he tabulated in his papers “Tables of the Motion of the Earth on its Axis and Around the Sun” and “The elements of the four inner planets and the fundamental constants of astronomy“. These constants are a result of observations and pattern finding over a long period of time (most of his adult life).

Because these constants are probability based, they need updating once in a while. The Jet Propulsion Laboratory’s DE series gives these updated values.

LST = GMST + east longitude

local sidereal time (LST) is the Greenwich mean sidereal time (GMST) corrected for the location by adding the east longitude in terms of hours. Greenwich mean sidereal time is, in essence, a close approximation of the sidereal time at Greenwich and is based on the average position of the vernal equinox.

GMST ignores the earths nutation but accounts for precession. Because precession makes up the majority of the vernal equinox’s movement and nutation is very small, GMST is sufficient for most purposes.

GMST(Du, T) = θ(Du) + GMSTp(T)

Where,

θ(Du) = 2pi(0.7790572732640 + 1.00273781191135448 * Du )

And,

GMSTp(T) = 0.014506 + 4612.156534T +  1.3915817T2 – 0.00000044T3 – 0.000029956T4 – 0.0000000368T5

Where,

Du = JDUTC – 2451545.0

And,

T = (JDTT – 2451545.0)/36525

θ(Du) is the function for the earth’s rotation angle (ERA) where Du is the number of days since the epoch J2000. 1.00273781191135448 is the earths angular velocity and 0.7790572732640 was the ERA at noon on January 1st 2000 (the J2000 epoch)

GMSTp(T) is the polynomial part (similar to Newcomb’s model for the earth’s obliquity) where T is the number of centuries since the epoch. It is separated because it is used in other calculations. I kept it separate in an attempt to keep things neat.

The ERA is the angular distance from a mostly fixed point on the equator called the celestial intermediate origin (CIO) to Greenwich. The polynomial part is the angular distance from the CIO to the vernal equinox.

θ(Du) is measured in radians while GMSTp(T) is measured in arc seconds. Before adding them together, they will need to be converted to the same unit of measurement.

To calculate JDTT, we need to convert UTC to TT like so,

TT = UTC + leap seconds added to UTC to date + 32.184

At the bottom of this post is a link to the Wiki page for leap seconds. The Wiki article has a table of all the leap seconds that have been added to UTC.

JDTT = 367Y – INT(7(Y + INT((M + 9)/12))/4) – INT(3(INT((Y + (M – 9)/7)/100) + 1)/4) + INT(275M/9) + D + 1721028.5 + TIMETT/24

JDUTC = 367Y – INT(7(Y + INT((M + 9)/12))/4) – INT(3(INT((Y + (M – 9)/7)/100) + 1)/4) + INT(275M/9) + D + 1721028.5 + TIMEUTC/24

Y is the year, M is the month, D is the day, and TIME is the time. INT means remove the decimal part. TIME is converted to decimal hours.

An example

I got the information for this example from the Astro databank.

Marilyn Manson

Born on January 5th 1969 at 20:05 EST in Canton, Ohio 40n48, 81w23

Y =1969

M = 1

D = 5

UTC TIME = 25.083333333333333

JDUTC = 367Y – INT(7(Y + INT((M + 9)/12))/4) – INT(3(INT((Y + (M – 9)/7)/100) + 1)/4) + INT(275M/9) + D + 1721028.5 + TIME/24

JDUTC = 367*1969 – INT(7*(1969 + INT((1 + 9)/12))/4) – INT(3*(INT((1969 + (1 – 9)/7)/100) + 1)/4) + INT(275*1/9) + 5 + 1721028.5 + 25.083333333333333/24

JDUTC = 2440227.54513888889

TT = UTC + leap seconds added to UTC to date + 32.184

TT = 25.083333333333333 + ((leap seconds added to UTC to date + 32.184)/3600)

TT = 25.0922733333

JDTT = 367Y – INT(7(Y + INT((M + 9)/12))/4) – INT(3(INT((Y + (M – 9)/7)/100) + 1)/4) + INT(275M/9) + D + 1721028.5 + TIME/24

JDTT = 367*1969 – INT(7*(1969 + INT((1 + 9)/12))/4) – INT(3*(INT((1969 + (1 – 9)/7)/100) + 1)/4) + INT(275*1/9) + 5 + 1721028.5 + 25.0922733333/24

JDTT = 2440227.54551138889

Du = JDUTC – 2451545.0

Du = 2440227.54513888889 – 2451545.0

Du = -11317.4548611

T = (JDTT – 2451545.0)/36525

T = (2440227.54551138889- 2451545.0)/36525

T = -0.30985501679

θ(Du) = 2pi(0.7790572732640 + 1.00273781191135448 * Du ) modulo 2pi

θ(Du) = 2pi(0.7790572732640 + 1.00273781191135448 * -11317.4548611) modulo 2pi

θ(Du) = -71299.4560272 modulo 2pi

θ(Du) = 2.13083867395

θ(Du) in degrees = θ(Du) * 180 / pi

θ(Du) in degrees = 2.13083867395 * 180 / pi

θ(Du) in degrees = 122.088062841

θ(Du) in hours = θ(Du) in degrees / 15

θ(Du) in hours = 122.088062841 / 15

θ(Du) in hours = 8.1392041894

GMSTp(T) = 0.014506 + 4612.156534T +  1.3915817T2 – 0.00000044T3 – 0.000029956T4 – 0.0000000368T5

GMSTp(T) = 0.014506 + 4612.156534* -0.30985501679 +  1.3915817* -0.309855016792 – 0.00000044* -0.309855016793 – 0.000029956* -0.309855016794 – 0.0000000368* -0.309855016795

GMSTp(T) = -1428.9517286

GMSTp(T) in degrees = (GMSTp(T) / 3600) Modulo 360

GMSTp(T) in degrees = (-1428.9517286 / 3600) Modulo 360

GMSTp(T) in degrees = (-0.39693103572) Modulo 360

GMSTp(T) in degrees = 359.603068964

GMSTp(T) in hours = GMSTp(T) in degrees / 15

GMSTp(T) in hours = 359.603068964 / 15

GMSTp(T) in hours = 23.9735379309

GMST(Du, T) = θ(Du) + GMSTp(T) modulo 24
GMST(Du, T) = 8.1392041894 + 23.9735379309 modulo 24

GMST(Du, T) = 32.1127421203 modulo 24

GMST(Du, T) = 8.1127421203

east longitude = 81w23 = -81.38333333333 = -5.42555555556

LST = GMST + east longitude modulo 24

LST = 8.1127421203 + -5.42555555556 modulo 24

LST = 2.68718656474 modulo 24

LST = 2.68718656474 = 2:41:13.871633064


The Astro databank, linked below, has birth dates, times, and locations for many public figures. Try calculating the sidereal time of birth for some of them. Leave your results and any questions that come up in the comments.

Links

2012 Astronomical Almanac

https://archive.org/details/astronomicalalma00naut/page/n132

Wolfram article about Julian dates

http://scienceworld.wolfram.com/astronomy/JulianDate.html

Wiki page about leap seconds

https://en.m.wikipedia.org/wiki/Leap_second

Astro databank

https://www.astro.com/astro-databank/Main_Page

Sidereal time calculator

http://neoprogrammics.com/sidereal_time_calculator/index.php

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