<p>For a project I needed to implement some of the code to calculate the position of the sun and moon. To do this I'm using Meeus. Everything is working but in doing so I've started to wonder about the currently accepted coefficient values for the various polynomials.</p>
<p>First a concrete example. Per Chapter 22 the Earth/Sun Mean Anomaly is found using a polynomial of the Julian Ephemeris Century (jce)
357.52772 + 35999.050340jce - 0.0001603jce^2 - (1.0/300000)jce^3</p>
<p>But then Chapter 25 uses coefficients of:
357.52911, 35999.05029, -0.00011537</p>
<p>And Chapter 47 says:
357.5291092, 35999.0502909, -0.0001536, 1.0/24490000</p>
<p>Each of these is footnoted saying where Meeus got the coefficients from.  Ch22 says Astronomical Almanac for 1984 page S26. Ch25 says the source is a disk of planetary tables. Ch47 says Chapront's lunar theory ELP-2000.</p>
<p>I could give other examples such as the Moon's Argument of Latitude, or it's Mean Longitude, but I think this gives the idea. There are quite a few of these polynomials.</p>
<p>But Meeus' 2nd edition is from 1998. I rather suspect that someone has worked up new values for these. Where would I find them? I'd think someone (NOAA?) would have a publicly available list of the &quot;best&quot; values. Possibly they're calculated from the DE-nnn tables available from JPL?</p>
<p>I do want to be clear on one thing. I don't need anything better than the values in Meeus. This is just academic interest. There are people who need highly accurate and precise values. Where do they get them?</p>

CC BY-SA 4.0
2024-01-30T14:35:11.510Z
What about NREL SPA (as an alternative)?  https://midcdmz.nrel.gov/spa/ It is very often referred to as one of the most accurate.  There are implementations in Matlab.
ayr
CC BY-SA 4.0
2024-01-30T20:45:09.880Z
According to https://en.wikipedia.org/wiki/Ephemeride_Lunaire_Parisienne "ELP was fitted initially to the DE200, but improved parameters have been published up to DE405". We're now up to DE440/441. I expect that the next JPL DE release will include improvements to the Jovian system, using data derived from the Juno mission. And every new DE has improvements to the lunar model, mostly due to the Lunar Laser Ranging data.
PM 2Ring
CC BY-SA 4.0
2024-02-01T14:08:45.260Z
@dtn: The NREL SPA is simply a stripped down reiteration of Meeus with the equations in the order you need them. It's not a source for the current 'best' coefficients.
PhotoKevin
CC BY-SA 4.0
2024-02-01T14:11:41.477Z
As I said in the post, I suspect the coefficients are generated from the DE***, but they aren't in the DE***, nor do I know how to calculate them from the DE***. I'm thinking someone should be doing that and publishing it.
PhotoKevin
CC BY-SA 4.0
2024-02-01T16:31:49.627Z
@PhotoKevin Regarding the DE-nnn tables. As I understand it, there is a lot of accumulated and structured information. Maybe it is enough to make a fairly accurate prediction of these coefficients in the short term (by astronomical standards)? (I want to emphasize - only on the condition that they cannot be obtained in any other way).
ayr
CC BY-SA 4.0
2024-02-01T16:31:58.867Z
@PhotoKevin Regarding Julian Ephemeris Century. This parameter is included in the 3rd degree polynomial i.e. is one of its 3 roots. If the other two roots are not an “artifact” of the theory, i.e. have an astronomical meaning or are indirectly related to it, then one could try to determine them using calculations/measurements and for the resulting “triple-group” obtain the coefficients of its polynomial. But this is for exceptional, extreme cases and mathematical tricks. Still, it seems to me that the coefficients you need must be somewhere.
ayr
CC BY-SA 4.0
2024-02-16T05:23:47.603Z
@PhotoKevin It's easy to get osculating elements from JPL Horizons. Converting osc elements to mean elements is called "epoch point conversion", which is an iterative refinement process (think multivariable Newton's method). You estimate the mean elements, use them to calculate trial osc elements, and adjust the parameters until your trial elements closely match the real osc elements.
PM 2Ring
CC BY-SA 4.0
2024-02-16T05:25:01.150Z
It's not easy to find recent un-paywalled articles about this, and many of the articles I found focus on artificial Earth satellites, so they only need simple short term expressions for the mean elements. Whereas for planetary ephemerides we'd prefer polynomials of at least degree 3.
PM 2Ring
CC BY-SA 4.0
2024-02-28T19:32:56.713Z
@PM2Ring - Thanks. I had given up on this. The other people commenting didn't even understand the question. But the short answer is: There isn't anyplace to get updated coefficients.
PhotoKevin
CC BY-SA 4.0
2024-02-28T21:30:20.433Z
@PhotoKevin If anyone has newer coefficients, they aren't advertising them very well. ;) I suspect that anyone who needs better ephemerides is just using the JPL DEs. The Paris Observatory also do ephemeris calculations. See https://en.wikipedia.org/wiki/VSOP_model Fred Espenak *might* know where to find updated coefficients. His early eclipse calculations used VSOP87, but his more recent work uses the JPL DE. https://eclipsewise.com/eclipse.html
PM 2Ring
CC BY-SA 4.0
2024-03-07T08:24:12.273Z
Chapront; Chapront-Touzé; Francou (2002) https://doi.org/10.1051%2F0004-6361%3A20020420 gives 4th degree polys for lunar elements. I don't know if that's better than what you're currently using. And there have been significant improvements in lunar data & in the nutation model in the last decade or so. https://en.wikipedia.org/wiki/Lunar_month explains how to use those polys to calculate various lunar months.
PM 2Ring
<coordinate><algorithm>
Current sources for astronomical equation coefficients
PhotoKevin
# Current sources for astronomical equation coefficients

(Asked by: username_0 on 2024-01-30)

For a project I needed to implement some of the code to calculate the position of the sun and moon. To do this I'm using Meeus. Everything is working but in doing so I've started to wonder about the currently accepted coefficient values for the various polynomials.

First a concrete example. Per Chapter 22 the Earth/Sun Mean Anomaly is found using a polynomial of the Julian Ephemeris Century (jce)
357.52772 + 35999.050340jce - 0.0001603jce^2 - (1.0/300000)jce^3

But then Chapter 25 uses coefficients of:
357.52911, 35999.05029, -0.00011537

And Chapter 47 says:
357.5291092, 35999.0502909, -0.0001536, 1.0/24490000

Each of these is footnoted saying where Meeus got the coefficients from. Ch22 says Astronomical Almanac for 1984 page S26. Ch25 says the source is a disk of planetary tables. Ch47 says Chapront's lunar theory ELP-2000.

I could give other examples such as the Moon's Argument of Latitude, or it's Mean Longitude, but I think this gives the idea. There are quite a few of these polynomials.

But Meeus' 2nd edition is from 1998. I rather suspect that someone has worked up new values for these. Where would I find them? I'd think someone (NOAA?) would have a publicly available list of the "best" values. Possibly they're calculated from the DE-nnn tables available from JPL?

I do want to be clear on one thing. I don't need anything better than the values in Meeus. This is just academic interest. There are people who need highly accurate and precise values. Where do they get them?

**Tags:** coordinate, algorithm

> **Comments:**
> **username_1**: What about NREL SPA (as an alternative)?  https://midcdmz.nrel.gov/spa/ It is very often referred to as one of the most accurate.  There are implementations in Matlab.  
> **username_2**: According to https://en.wikipedia.org/wiki/Ephemeride_Lunaire_Parisienne "ELP was fitted initially to the DE200, but improved parameters have been published up to DE405". We're now up to DE440/441. I expect that the next JPL DE release will include improvements to the Jovian system, using data derived from the Juno mission. And every new DE has improvements to the lunar model, mostly due to the Lunar Laser Ranging data.  
> **username_0**: @dtn: The NREL SPA is simply a stripped down reiteration of Meeus with the equations in the order you need them. It's not a source for the current 'best' coefficients.  
> **username_0**: As I said in the post, I suspect the coefficients are generated from the DE***, but they aren't in the DE***, nor do I know how to calculate them from the DE***. I'm thinking someone should be doing that and publishing it.  
> **username_1**: @username_0 Regarding the DE-nnn tables. As I understand it, there is a lot of accumulated and structured information. Maybe it is enough to make a fairly accurate prediction of these coefficients in the short term (by astronomical standards)? (I want to emphasize - only on the condition that they cannot be obtained in any other way).  
> **username_1**: @username_0 Regarding Julian Ephemeris Century. This parameter is included in the 3rd degree polynomial i.e. is one of its 3 roots. If the other two roots are not an “artifact” of the theory, i.e. have an astronomical meaning or are indirectly related to it, then one could try to determine them using calculations/measurements and for the resulting “triple-group” obtain the coefficients of its polynomial. But this is for exceptional, extreme cases and mathematical tricks. Still, it seems to me that the coefficients you need must be somewhere.  
> **username_2**: @username_0 It's easy to get osculating elements from JPL Horizons. Converting osc elements to mean elements is called "epoch point conversion", which is an iterative refinement process (think multivariable Newton's method). You estimate the mean elements, use them to calculate trial osc elements, and adjust the parameters until your trial elements closely match the real osc elements.  
> **username_2**: It's not easy to find recent un-paywalled articles about this, and many of the articles I found focus on artificial Earth satellites, so they only need simple short term expressions for the mean elements. Whereas for planetary ephemerides we'd prefer polynomials of at least degree 3.  
> **username_0**: @username_2 - Thanks. I had given up on this. The other people commenting didn't even understand the question. But the short answer is: There isn't anyplace to get updated coefficients.  
> **username_2**: @username_0 If anyone has newer coefficients, they aren't advertising them very well. ;) I suspect that anyone who needs better ephemerides is just using the JPL DEs. The Paris Observatory also do ephemeris calculations. See https://en.wikipedia.org/wiki/VSOP_model Fred Espenak *might* know where to find updated coefficients. His early eclipse calculations used VSOP87, but his more recent work uses the JPL DE. https://eclipsewise.com/eclipse.html  
> **username_2**: Chapront; Chapront-Touzé; Francou (2002) https://doi.org/10.1051%2F0004-6361%3A20020420 gives 4th degree polys for lunar elements. I don't know if that's better than what you're currently using. And there have been significant improvements in lunar data & in the nutation model in the last decade or so. https://en.wikipedia.org/wiki/Lunar_month explains how to use those polys to calculate various lunar months.  

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## 0 answers

