<p>After quite a bit of searching <a href="https://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/tn/TechnNote13/tn13_SOLID_EARTH_TIDES.pdf?__blob=publicationFile&amp;v=1" rel="nofollow noreferrer">this old version of IERS notes</a> gives expressions for them at the end of page 53/beginning of page 54, as functions of the more frequently used Delaunay variables. Expressions for Delaunay variables can be found in IERS notes, <a href="https://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/tn/TechnNote36/tn36_043.pdf?__blob=publicationFile&amp;v=1" rel="nofollow noreferrer">for example here</a> in equations 5.43 of page 67. I have verified that these are correct, by using them to calculate ocean tide corrections to the Earth geopotential model and confirming the result indeed improves the accuracy of propagation of LEO satellites.</p>
<p>A summary of the expressions here:
<span class="math-container">$$
s=F+\Omega
$$</span>
<span class="math-container">$$
h=s-D
$$</span>
<span class="math-container">$$
p=s-\mathscr{l}
$$</span>
<span class="math-container">$$
N'=-\Omega
$$</span>
<span class="math-container">$$
p_1=s-D-\mathscr{l}'
$$</span>
<span class="math-container">$$\tau=\theta_{g}+\pi-s
$$</span>
Where:</p>
<ul>
<li><span class="math-container">$s$</span> is Moon's mean longitude</li>
<li><span class="math-container">$h$</span> is Sun's mean longitude</li>
<li><span class="math-container">$p$</span> is the longitude of Moon's mean perigee</li>
<li><span class="math-container">$N'$</span> is the negative of the longitude of Moon's mean node</li>
<li><span class="math-container">$p_1$</span>, which I have also seen referred to as <span class="math-container">$p_s$</span>, is the longitude of Sun's mean perigee</li>
<li><span class="math-container">$\tau$</span> is the time angle in lunar days from lower transit</li>
<li><span class="math-container">$\mathscr{l}$</span> is the mean anomaly of the Moon (Delaunay variable 1)</li>
<li><span class="math-container">$\mathscr{l}'$</span> is the mean anomaly of the Sun (Delaunay variable 2)</li>
<li><span class="math-container">$F$</span>, is the difference between the Moon's mean longitude and <span class="math-container">$\Omega$</span>, which is obvious from the expression for <span class="math-container">$s$</span>, but note that there are direct expressions for <span class="math-container">$F$</span> (it is Delaunay variable 3)</li>
<li><span class="math-container">$D$</span> is the mean elongation of the Moon from the sun (Delaunay variable 4)</li>
<li><span class="math-container">$\Omega$</span> is the mean longitude of the ascending node of the Moon (Delaunay variable 5)</li>
<li><span class="math-container">$\theta_g$</span> is the mean sidereal time of the conventional zero meridian, which can be calculated, for example, with SOFA's iauGmst06 function.</li>
</ul>
<p>Not sure why IERS note removed these expressions in later versions, but they are indeed quite useful.</p>

CC BY-SA 4.0
2022-03-08T02:54:17.437Z
2022-03-08T02:54:17.437Z
Rafa
<p>I am following <a href="https://iers-conventions.obspm.fr/content/chapter6/icc6.pdf" rel="nofollow noreferrer">chapter 6 of this 2010 IERS note</a> describing the calculation of a geopotential model.</p>
<p>In the notes of equation 6.8, two expressions for the parameter called <span class="math-container">$\theta_f$</span> is given. One of them involves Delaunay arguments, for which I found the recommended expressions in <a href="https://iers-conventions.obspm.fr/content/chapter5/icc5.pdf" rel="nofollow noreferrer">chapter 5 of the same IERS note</a>. This is fine for the case of calculation of solid Earth tides corrections, since multipliers for the Delaunay variables are given.</p>
<p>However, for the case of ocean tide corrections, most tide models describe the frequency of the different waves with the Doodson number, from which multipliers for the Doodson arguments can be obtained. Unfortunately, I have not been able to find in the IERS note, or in any other similar source, a recommended expression for the Doodson arguments. <a href="https://en.wikipedia.org/wiki/Theory_of_tides#Doodson_numbers" rel="nofollow noreferrer">This section of Wikipedia</a> gives an expression as a function of Delaunay's arguments, and cites <a href="https://articles.adsabs.harvard.edu/pdf/1971CeMec...4..190M" rel="nofollow noreferrer">this article</a> as the source. But after checking it, it does not seem to actually contain the expressions for Doodson arguments.</p>
<p>So my question is, is anybody aware of a good source with expressions for the Doodson arguments?</p>

CC BY-SA 4.0
2022-01-25T01:50:11.570Z
Is [The harmonic development of the tide-generating potential](https://royalsocietypublishing.org/doi/10.1098/rspa.1921.0088) of any help?
GrapefruitIsAwesome
CC BY-SA 4.0
2022-01-25T01:58:06.330Z
Nice! I see that in page 311 of the document, expressions are indeed given. However, they seem to omit terms of order higher than 2 in the expression depending on Julian centuries since 1900. Maybe a more recent source includes terms up to order 4, as is the case for Delaunay arguments?
Rafa
CC BY-SA 4.0
2022-01-25T01:59:47.923Z
My suggestion would be start with [Google Scholar](https://scholar.google.com/scholar?hl=en&as_sdt=0%2C39&q=The+harmonic+development+of+the+tide-generating+potential&btnG=) and see who referenced this paper to see if there are any further developments.  [Introduction to ocean tides](https://www.whoi.edu/cms/files/lecture01_21351.pdf) may be helpful as well.
GrapefruitIsAwesome
CC BY-SA 4.0
2022-01-25T02:39:56.177Z
[This paper](https://www.sciencedirect.com/science/article/pii/S0032063301001209#BIB5) gives some potentially useful expressions, citing a 1995 technical note by Roosbeek as the original source. However, I seem to be unable to locate this technical note. Would be great if someone has a link to it!
Rafa
CC BY-SA 4.0
2022-01-25T02:53:48.047Z
Ah, this is rather confusing... in fact, for example, for the expression for $s$ in the paper I linked, the constant term is said to be 785939.9243 arc seconds, which would be approximately equal to 218.3166 degrees. However, the original 1921 Doodson paper gives the constant value for $s$ as 277.0248 degrees
Rafa
<tidal-forces><resource-request>
Is there an official recommended expression for Doodson arguments?
Rafa
# Is there an official recommended expression for Doodson arguments?

(Asked by: username_0 on 2022-01-25)

I am following [chapter 6 of this 2010 IERS note](https://iers-conventions.obspm.fr/content/chapter6/icc6.pdf) describing the calculation of a geopotential model.

In the notes of equation 6.8, two expressions for the parameter called $\theta\_f$ is given. One of them involves Delaunay arguments, for which I found the recommended expressions in [chapter 5 of the same IERS note](https://iers-conventions.obspm.fr/content/chapter5/icc5.pdf). This is fine for the case of calculation of solid Earth tides corrections, since multipliers for the Delaunay variables are given.

However, for the case of ocean tide corrections, most tide models describe the frequency of the different waves with the Doodson number, from which multipliers for the Doodson arguments can be obtained. Unfortunately, I have not been able to find in the IERS note, or in any other similar source, a recommended expression for the Doodson arguments. [This section of Wikipedia](https://en.wikipedia.org/wiki/Theory_of_tides#Doodson_numbers) gives an expression as a function of Delaunay's arguments, and cites [this article](https://articles.adsabs.harvard.edu/pdf/1971CeMec...4..190M) as the source. But after checking it, it does not seem to actually contain the expressions for Doodson arguments.

So my question is, is anybody aware of a good source with expressions for the Doodson arguments?

**Tags:** tidal-forces, resource-request
> **Comments:**
> **username_1**: Is [The harmonic development of the tide-generating potential](https://royalsocietypublishing.org/doi/10.1098/rspa.1921.0088) of any help?  
> **username_0**: Nice! I see that in page 311 of the document, expressions are indeed given. However, they seem to omit terms of order higher than 2 in the expression depending on Julian centuries since 1900. Maybe a more recent source includes terms up to order 4, as is the case for Delaunay arguments?  
> **username_1**: My suggestion would be start with [Google Scholar](https://scholar.google.com/scholar?hl=en&as_sdt=0%2C39&q=The+harmonic+development+of+the+tide-generating+potential&btnG=) and see who referenced this paper to see if there are any further developments.  [Introduction to ocean tides](https://www.whoi.edu/cms/files/lecture01_21351.pdf) may be helpful as well.  
> **username_0**: [This paper](https://www.sciencedirect.com/science/article/pii/S0032063301001209#BIB5) gives some potentially useful expressions, citing a 1995 technical note by Roosbeek as the original source. However, I seem to be unable to locate this technical note. Would be great if someone has a link to it!  
> **username_0**: Ah, this is rather confusing... in fact, for example, for the expression for $s$ in the paper I linked, the constant term is said to be 785939.9243 arc seconds, which would be approximately equal to 218.3166 degrees. However, the original 1921 Doodson paper gives the constant value for $s$ as 277.0248 degrees  

---

## 1 answers

### Answer by username_0 on 2022-03-08. Score: 2 | Accepted answer

After quite a bit of searching [this old version of IERS notes](https://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/tn/TechnNote13/tn13_SOLID_EARTH_TIDES.pdf?__blob=publicationFile&v=1) gives expressions for them at the end of page 53/beginning of page 54, as functions of the more frequently used Delaunay variables. Expressions for Delaunay variables can be found in IERS notes, [for example here](https://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/tn/TechnNote36/tn36_043.pdf?__blob=publicationFile&v=1) in equations 5.43 of page 67. I have verified that these are correct, by using them to calculate ocean tide corrections to the Earth geopotential model and confirming the result indeed improves the accuracy of propagation of LEO satellites.

A summary of the expressions here:
$$
s=F+\Omega
$$
$$
h=s-D
$$
$$
p=s-\mathscr{l}
$$
$$
N'=-\Omega
$$
$$
p\_1=s-D-\mathscr{l}'
$$
$$\tau=\theta\_{g}+\pi-s
$$
Where:

* $s$ is Moon's mean longitude
* $h$ is Sun's mean longitude
* $p$ is the longitude of Moon's mean perigee
* $N'$ is the negative of the longitude of Moon's mean node
* $p\_1$, which I have also seen referred to as $p\_s$, is the longitude of Sun's mean perigee
* $\tau$ is the time angle in lunar days from lower transit
* $\mathscr{l}$ is the mean anomaly of the Moon (Delaunay variable 1)
* $\mathscr{l}'$ is the mean anomaly of the Sun (Delaunay variable 2)
* $F$, is the difference between the Moon's mean longitude and $\Omega$, which is obvious from the expression for $s$, but note that there are direct expressions for $F$ (it is Delaunay variable 3)
* $D$ is the mean elongation of the Moon from the sun (Delaunay variable 4)
* $\Omega$ is the mean longitude of the ascending node of the Moon (Delaunay variable 5)
* $\theta\_g$ is the mean sidereal time of the conventional zero meridian, which can be calculated, for example, with SOFA's iauGmst06 function.

Not sure why IERS note removed these expressions in later versions, but they are indeed quite useful.


---

