<p>Reda and Andreas 2008 equation 28 comes from Jean Meeus, <a href="http://www.willbell.com/math/mc1.htm" rel="nofollow noreferrer"><em>Astronomical Algorithms</em></a>, equation 12.4.
He derived it from <a href="http://adsabs.harvard.edu/abs/1982A%26A...105..359A" rel="nofollow noreferrer">Aoki et al. 1982</a> equation 13, which IAU used from 1984 through 2002.</p>

<p>Let's break it down:</p>

<ul>
<li>The constant term is GMST in degrees at 2000-01-01 12:00 UT, also known as JD 2451545.0 or J2000.0.</li>
<li>The linear term is Earth's mean sidereal rotation rate in degrees per solar day, multiplied by the number of days since JD 2451545.</li>
<li>The JC^2 and JC^3 terms approximate the long-term effects of Earth's precession and change in rotation rate. Between 1900 and 2100, they amount to less than 0.1 second of time and may be omitted unless you require high precision.</li>
</ul>

<p>The set of reference frames which IAU adopted in 2000 and implemented in 2003 was motivated by the extreme precision requirements of very long baseline radio interferometry (VLBI).
The difference between the old and new expressions for GMST is zero on 2003-01-01 and negligible within a century of that date unless you are doing VLBI yourself.</p>

<p>The results you previously got for 2003-10-17 could be due to inverting an addition or subtraction somewhere, maybe in normalizing to [0, 360].
If you try again, it may come out right as it did for 1987-04-10.</p>

CC BY-SA 4.0
2018-07-08T12:45:57.617Z
@terry-s @ Mike G Thanks for great help. 

i took complete Eq. Cap’ (2003) with tu as UT1, t as TT.

For 17.10.2003 05:30:30 UTC; x:x:29.6 (UT1); x:x:34.184 (TT). DeltaT is 64.584 s.

JD at midn’ 7.10.2003 at 00:00:00 UTC is 2452929.5 with tu 0.037905544.

t for JDTT = 0.037911841.

MST for 17.10.2003 at 05:30:30 UTC is 07:11:45.3460.
URL: 07:11:45.7484. 

tu and t is apt than tcy as suggested?

For any UT of a date, Eq. 19 Aoki et al., 1982 [1.002737909350795 + (5.9006E-11 × tu) – (5.9E-15 × tu)] is ok?

45.3460 s is acceptable over 45.7484 s, right?
Smarty
CC BY-SA 4.0
2018-07-08T13:18:26.320Z
If 0.4 s is close enough for you, it's close enough for me. :)
Mike G
CC BY-SA 4.0
2018-05-20T21:06:45.003Z
2018-05-20T22:47:10.470Z
2018-05-20T22:47:10.470Z
Mike G
<p>The formula 28 from SPA that you linked is given with units and in a form that do not match official versions. It takes a little work to 'translate' it: also, it is needlessly susceptible to roundoff error (because all significant figures for both the date and time-of-day are strung together in one very long number).  The authors of SPA do not appear to have been striving for full official accuracy, probably their application did not require that.</p>

<p>I can't quite work out from the data in the question what might have happened to the calculations, online or otherwise, but maybe the following reference data could be of help.</p>

<p>GMST for 0h UT is still tabulated in the Astronomical Almanac, and there are also two rather easy formulae derived from the official definitions that you can use (see below), they make somewhat better preservation of end-figures than a calculation in the SPA form. </p>

<p>In the meantime, just for reference, the Astronomical Almanac for 2003 gave the GMST for 0h UT on 2003 Oct 17 as 1h 40m 21.4554s (1.6726265 h).  That is for 0h UT, and during the following 24h, the GMST for x hr UT is is easily found, it is very close to  $ [ (GMST for 0h)  +  x * 1.00273791 ] $ . </p>

<p>2003 was a changeover year for the official (IAU) calculations that relate GMST and UT1. Before 2003, the formula of Aoki et al (1982), "The new definition of universal time" (<a href="http://adsabs.harvard.edu/abs/1982A&amp;A...105..359A" rel="nofollow noreferrer">http://adsabs.harvard.edu/abs/1982A&amp;A...105..359A</a>) was the one adopted. From 2003 Jan 1 the formula of N Capitaine et al. (2003), "Expressions to implement the IAU 2000 definition of UT1", Astronomy &amp; Astrophysics 406, 1135-1149 (<a href="http://adsabs.harvard.edu/abs/2003A%26A...406.1135C" rel="nofollow noreferrer">http://adsabs.harvard.edu/abs/2003A%26A...406.1135C</a>)  was (and still is) adopted. The difference between the results during 2003 was small (+0.0003 time-secs to get the result of the newer calculation), but it looks as if AA for 2003 was still using the older 1982 formula. </p>

<p>The Aoki et al (1982) formula gave :</p>

<p>GMST of 0h UT (in time-secs) = </p>

<p>$ 24110.54841 + 8640184.812866 *Tcy + 0.093104 *Tcy^2 -0.0000062*Tcy^3 $</p>

<p>plus a <em>very</em> small correction for DeltaT: this can be had from the version of the Aoki formula given in the Capitaine reference at B.3 on page 1149, (which seems to be the one that was used in official calculations) but it can practically be left out at the precision of the data in the question. Tcy means the time interval from J2000 (Julian Date 2451545.0) reduced to centuries of 36525 days. (The JD for 0h on 2003 Oct 17 0h UT was JD(UT) 2452929.5 days and DeltaT was close to +64.55 secs.) </p>

<p>The current formula from the Capitaine (2003) reference (most conveniently from formula B.2 on page 1149) gives </p>

<p>$ 24110.5493771 + 8640184.7945360 *Tcy + 0.0931118*Tcy^2 -0.0000062*Tcy^3 $ </p>

<p>plus a very small fourth-order term (given in the reference but omitted here) and again a very small correction for the slight effect of DeltaT which can be left out or included if desired as </p>

<p>$ +307.4771600 * DeltaT/(86400.0*36525.0) $ (time-secs), where DeltaT is still in seconds.</p>

CC BY-SA 4.0
2018-05-17T16:45:49.650Z
2018-05-17T16:55:45.757Z
2018-05-17T16:55:45.757Z
terry-s
<p>Using Equation 28 in Section 3.8.1 of <a href="https://www.nrel.gov/docs/fy08osti/34302.pdf" rel="nofollow noreferrer">Solar Position Algorithm</a> (Reda, I. and A. Andreas, Jan. 2008), I computed the mean sidereal time at Greenwich for 2003 October 17 for four different times (UT), using a Casio fx-991MS. The computed values were cross-schecked online using the URL <a href="http://dc.zah.uni-heidelberg.de/apfs/times/q/form#_queryForm" rel="nofollow noreferrer">Computation of GAST, GMST, and ERA</a></p>

<p>Equation 28 used to compute GMST is:</p>

<pre><code>v0=280.46061837 + 360.98564736629*(JD - 2451545) + 0.000387933*JC^2 - JC^3/38710000
</code></pre>

<p>JC is computed as:</p>

<pre><code>JC = (JD - 2451545)/36525
</code></pre>

<p>The computed values of JD and JC are as follows:</p>

<pre><code>1. 00:00:00 UT: 2452929.5   (JD) &amp; 0.037905544 (JC)
2. 05:30:00 UT: 2452929.729 (JD) &amp; 0.037911818 (JC)
3. 11:00:00 UT: 2452929.958 (JD) &amp; 0.037918083 (JC)
4. 19:21:00 UT: 2452930.306 (JD) &amp; 0.037927611 (JC)
</code></pre>

<p>However, there is a clear offset between the computed values and the values provided by the aforementioned URL. For example:</p>

<pre><code>1. 00:00:00 UT: 22.32738 hour    (Computed) &amp; 1.6726265 hour (URL)
2. 05:30:00 UT: 16.7231 hour     (Computed) &amp; 7.1960077 hour (URL)
3. 11:00:00 UT: 11.30528667 hour (Computed) &amp; 12.702743 hour (URL)
4. 19:21:00 UT: 2.93042 hour     (Computed) &amp; 21.075605 hour (URL)
</code></pre>

<p>Moreover, if I subtract 22.32738 hour from 24.0, I will get 1.67262, which is the value provided by the URL.</p>

<p>At the same time, using the same set of equation , if I compute GMST for 1987 April 10 at 00:00:00, the computed and URL values are the same.</p>

<p>April 10 at 00:00:00:</p>

<pre><code>JD=2446895.5
JC=-0.127296372
GMST = 13.18 hour (Computed) &amp; 13.17954 hour (URL)
</code></pre>

<p>Kindly help me to understand where exactly lies the problem. If the problem lies with the concept regarding the interpretation, please help for a better understanding.</p>

<p>Note: Knew gmst is now deprecated , but since i am a beginner thought of understanding gmst foremost and then will dwell into current scenario.</p>

<p>Thanks.</p>

<the-sun><time>
Computation of mean sidereal time at Greenwich using solar position algorithm
Smarty
# Computation of mean sidereal time at Greenwich using solar position algorithm

(Asked by: username_0 on 2018-05-17)

Using Equation 28 in Section 3.8.1 of Solar Position Algorithm (Reda, I. and A. Andreas, Jan. 2008), I computed the mean sidereal time at Greenwich for 2003 October 17 for four different times (UT), using a Casio fx-991MS. The computed values were cross-schecked online using the URL Computation of GAST, GMST, and ERA
Equation 28 used to compute GMST is:
v0=280.46061837 + 360.98564736629*(JD - 2451545) + 0.000387933*JC^2 - JC^3/38710000

JC is computed as:
JC = (JD - 2451545)/36525

The computed values of JD and JC are as follows:
1. 00:00:00 UT: 2452929.5   (JD) & 0.037905544 (JC)
2. 05:30:00 UT: 2452929.729 (JD) & 0.037911818 (JC)
3. 11:00:00 UT: 2452929.958 (JD) & 0.037918083 (JC)
4. 19:21:00 UT: 2452930.306 (JD) & 0.037927611 (JC)

However, there is a clear offset between the computed values and the values provided by the aforementioned URL. For example:
1. 00:00:00 UT: 22.32738 hour    (Computed) & 1.6726265 hour (URL)
2. 05:30:00 UT: 16.7231 hour     (Computed) & 7.1960077 hour (URL)
3. 11:00:00 UT: 11.30528667 hour (Computed) & 12.702743 hour (URL)
4. 19:21:00 UT: 2.93042 hour     (Computed) & 21.075605 hour (URL)

Moreover, if I subtract 22.32738 hour from 24.0, I will get 1.67262, which is the value provided by the URL.
At the same time, using the same set of equation , if I compute GMST for 1987 April 10 at 00:00:00, the computed and URL values are the same.
April 10 at 00:00:00:
JD=2446895.5
JC=-0.127296372
GMST = 13.18 hour (Computed) & 13.17954 hour (URL)

Kindly help me to understand where exactly lies the problem. If the problem lies with the concept regarding the interpretation, please help for a better understanding.
Note: Knew gmst is now deprecated , but since i am a beginner thought of understanding gmst foremost and then will dwell into current scenario.
Thanks.

**Tags:** the-sun, time

---

## 2 answers

### Answer by username_1 on 2018-05-20. Score: 1

Reda and Andreas 2008 equation 28 comes from Jean Meeus, [*Astronomical Algorithms*](http://www.willbell.com/math/mc1.htm), equation 12.4.
He derived it from [Aoki et al. 1982](http://adsabs.harvard.edu/abs/1982A%26A...105..359A) equation 13, which IAU used from 1984 through 2002.

Let's break it down:

* The constant term is GMST in degrees at 2000-01-01 12:00 UT, also known as JD 2451545.0 or J2000.0.
* The linear term is Earth's mean sidereal rotation rate in degrees per solar day, multiplied by the number of days since JD 2451545.
* The JC^2 and JC^3 terms approximate the long-term effects of Earth's precession and change in rotation rate. Between 1900 and 2100, they amount to less than 0.1 second of time and may be omitted unless you require high precision.

The set of reference frames which IAU adopted in 2000 and implemented in 2003 was motivated by the extreme precision requirements of very long baseline radio interferometry (VLBI).
The difference between the old and new expressions for GMST is zero on 2003-01-01 and negligible within a century of that date unless you are doing VLBI yourself.

The results you previously got for 2003-10-17 could be due to inverting an addition or subtraction somewhere, maybe in normalizing to [0, 360].
If you try again, it may come out right as it did for 1987-04-10.

> **Comments:**
> **username_0**: @terry-s @ Mike G Thanks for great help.   i took complete Eq. Cap’ (2003) with tu as UT1, t as TT.  For 17.10.2003 05:30:30 UTC; x:x:29.6 (UT1); x:x:34.184 (TT). DeltaT is 64.584 s.  JD at midn’ 7.10.2003 at 00:00:00 UTC is 2452929.5 with tu 0.037905544.  t for JDTT = 0.037911841.  MST for 17.10.2003 at 05:30:30 UTC is 07:11:45.3460. URL: 07:11:45.7484.   tu and t is apt than tcy as suggested?  For any UT of a date, Eq. 19 Aoki et al., 1982 [1.002737909350795 + (5.9006E-11 × tu) – (5.9E-15 × tu)] is ok?  45.3460 s is acceptable over 45.7484 s, right?  
> **username_1**: If 0.4 s is close enough for you, it's close enough for me. :)  

---

### Answer by username_2 on 2018-05-17. Score: 1

The formula 28 from SPA that you linked is given with units and in a form that do not match official versions. It takes a little work to 'translate' it: also, it is needlessly susceptible to roundoff error (because all significant figures for both the date and time-of-day are strung together in one very long number). The authors of SPA do not appear to have been striving for full official accuracy, probably their application did not require that.

I can't quite work out from the data in the question what might have happened to the calculations, online or otherwise, but maybe the following reference data could be of help.

GMST for 0h UT is still tabulated in the Astronomical Almanac, and there are also two rather easy formulae derived from the official definitions that you can use (see below), they make somewhat better preservation of end-figures than a calculation in the SPA form.

In the meantime, just for reference, the Astronomical Almanac for 2003 gave the GMST for 0h UT on 2003 Oct 17 as 1h 40m 21.4554s (1.6726265 h). That is for 0h UT, and during the following 24h, the GMST for x hr UT is is easily found, it is very close to $ [ (GMST for 0h) + x \* 1.00273791 ] $ .

2003 was a changeover year for the official (IAU) calculations that relate GMST and UT1. Before 2003, the formula of Aoki et al (1982), "The new definition of universal time" (<http://adsabs.harvard.edu/abs/1982A&A...105..359A>) was the one adopted. From 2003 Jan 1 the formula of N Capitaine et al. (2003), "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics 406, 1135-1149 (<http://adsabs.harvard.edu/abs/2003A%26A...406.1135C>) was (and still is) adopted. The difference between the results during 2003 was small (+0.0003 time-secs to get the result of the newer calculation), but it looks as if AA for 2003 was still using the older 1982 formula.

The Aoki et al (1982) formula gave :

GMST of 0h UT (in time-secs) =

$ 24110.54841 + 8640184.812866 \*Tcy + 0.093104 \*Tcy^2 -0.0000062\*Tcy^3 $

plus a *very* small correction for DeltaT: this can be had from the version of the Aoki formula given in the Capitaine reference at B.3 on page 1149, (which seems to be the one that was used in official calculations) but it can practically be left out at the precision of the data in the question. Tcy means the time interval from J2000 (Julian Date 2451545.0) reduced to centuries of 36525 days. (The JD for 0h on 2003 Oct 17 0h UT was JD(UT) 2452929.5 days and DeltaT was close to +64.55 secs.)

The current formula from the Capitaine (2003) reference (most conveniently from formula B.2 on page 1149) gives

$ 24110.5493771 + 8640184.7945360 \*Tcy + 0.0931118\*Tcy^2 -0.0000062\*Tcy^3 $

plus a very small fourth-order term (given in the reference but omitted here) and again a very small correction for the slight effect of DeltaT which can be left out or included if desired as

$ +307.4771600 \* DeltaT/(86400.0\*36525.0) $ (time-secs), where DeltaT is still in seconds.


---

