<p>Ok, so I managed to figure things out.</p>
<p>Basically, the process to find the heliocentric ecliptic coordinates of the centre of the Earth is as follows:</p>
<ol>
<li><p>Determine the heliocentric ecliptic J2000 rectangular coordinates for the EMB using VSOP2013,i.e. <span class="math-container">$X_{EMB}$</span>, <span class="math-container">$Y_{EMB}$</span>, and <span class="math-container">$Z_{EMB}$</span>.</p>
</li>
<li><p>Calculate the geocentric ecliptic coordinates in kilometers for the Moon (in this case from ELP/Mpp02), i.e. <span class="math-container">$X_{moon}$</span>, <span class="math-container">$Y_{moon}$</span>, and <span class="math-container">$Z_{moon}$</span>.</p>
</li>
<li><p>Convert <span class="math-container">$X_{moon}$</span>, <span class="math-container">$Y_{moon}$</span>, and <span class="math-container">$Z_{moon}$</span> from kilometers to AU.</p>
</li>
<li><p>Using <span class="math-container">$GM_{moon}$</span> and <span class="math-container">$GM_{earth}$</span>, calculate the geocentric ecliptic coordinates for the EMB, i.e. <span class="math-container">$X_{geoEMB}$</span>, <span class="math-container">$Y_{geoEMB}$</span>, and <span class="math-container">$Z_{geoEMB}$</span>.</p>
</li>
<li><p>Finally, calculate the heliocentric coordinates for the centre of the earth:</p>
</li>
</ol>
<p><span class="math-container">\begin{align*}
X_{earth} &amp;= X_{EMB}-X_{geoEMB}\\
Y_{earth} &amp;= Y_{EMB}-Y_{geoEMB}\\
Z_{earth} &amp;= Z_{EMB}-Z_{geoEMB}\\
\end{align*}</span></p>
<p>After calculating <span class="math-container">$\lambda$</span> and <span class="math-container">$\beta$</span> one then jut simply to the necessary equations to bring the J2000 coordinates of VSOP2013 to the equinox and ecliptic of date (found in Meeus, pp. 136-137).</p>
<p>The values seem to correlate almost exactly with the JPL's.</p>

CC BY-SA 4.0
2024-09-12T14:32:55.223Z
2024-09-12T14:32:55.223Z
Coenraad van Schoor
<p>I've seen this question posed before but not answered fully at <a href="https://astronomy.stackexchange.com/questions/21827/converting-earth-moon-barycenter-coordinate-to-geocentric-coordinate">Converting Earth Moon Barycenter coordinate to Geocentric coordinate</a>. There is an insightful answer given at <a href="https://space.stackexchange.com/questions/19035/converting-orbital-state-vectors-from-one-origin-to-another">https://space.stackexchange.com/questions/19035/converting-orbital-state-vectors-from-one-origin-to-another</a>, but again no practical equations were given to easily calculate the coordinates of the centre of the earth by using the earth-moon-barycentre heliocentric coordinates calculated by VSOP2013 to enable one to calculate the positions of the planets for earth-based observation purposes by astronomers (this is of course not an issue for the VSOP87 coordinates as it includes a model that calculates the LBR coordinates of the centre of the earth).</p>
<p>According to my understanding it should be a relatively easy task to determine the centre of the earth in heliocentric coordinates if one knows the position of the Moon relative to the earth, for example by using the ELP/MPP02 model.</p>
<p>As an example of my problem I'll calculate the heliocentric coordinates for the Earth (using VSOP2013) and the geocentric coordinates of the Moon (using ELP/MPP02) for Julian date 2460564.50061489 11 September 2024 at midnight GMT) :</p>
<ol>
<li>Using VSOP2013 I calculated the Cartesian heliocentric ecliptic coordinates for the dynamic J2000 ecliptic for the EMB as</li>
</ol>
<p><span class="math-container">\begin{align*}
X&amp;= 0.98622458 \\
Y&amp;= -0.20214571 \\
Z&amp;= 0,00000567\\
\end{align*}</span></p>
<ol start="2">
<li>Using ELP/Mpp02 I calculated the inertial mean ecliptic and equinox of J2000 ecliptic coordinates for the Moon to be</li>
</ol>
<p><span class="math-container">\begin{align*}
X&amp;= -98286.95858km\\ 
Y&amp;= -378384.79064km \\
Z&amp;= -33687.61320km\\
\end{align*}</span></p>
<ol start="3">
<li>To calculate the coordinates of the Earth-Moon-Barycentre in the same coordinates as those of ELP/MPP02, I'll use the following formulae:</li>
</ol>
<p><span class="math-container">\begin{equation*}
X_{EMB}=\frac{mass_{moon} \cdot X_{moon}}{mass_{earth}+mass_{moon}}\\
Y_{EMB}=\frac{mass_{moon} \cdot Y_{moon}}{mass_{earth}+mass_{moon}}\\
Z_{EMB}=\frac{mass_{moon} \cdot Z_{moon}}{mass_{earth}+mass_{moon}}\\
\end{equation*}</span></p>
<p>The resulting coordinates of the EMB will then be:
<span class="math-container">\begin{align*}
X_{EMB}&amp;=-1363.8226km\\
Y_{EMB}&amp;=-4561.4700km\\
Z_{EMB}&amp;=-404.2623km\\
\end{align*}</span></p>
<p>or, as expressed in AU:
<span class="math-container">\begin{align*}
X_{EMB}&amp;=-7.9830279638\times10^{-6}\\
Y_{EMB}&amp;=-3.0733033237\times10^{-5}\\
Z_{EMB}&amp;=-2.7361631909\times10^{-6}\\
\end{align*}</span></p>
<p>Now, my question is: how do I calculate the centre of the earth in heliocentric coordinates given the VSOP2013 heliocentric coordinates of the EBM and the geocentric ELP/MPP02 coordinates of the EMB. Using VSOP87 the Earth's heliocentric rectangular coordinates (in AU) are:</p>
<p><span class="math-container">\begin{align*}
X_{Earth}&amp;=0.987431488\\
Y_{Earth}&amp;=-0.196173931\\
Z_{Earth}&amp;=2.08708\times10^{-6}\\
\end{align*}</span></p>
<p>I'm assuming there's a simple way to figure out how to use the geocentric rectangular coordinates of EMB calculated via ELP/MPP02 above in combination with the heliocentric EMB coordinates calculated via VSOP2013 to find the heliocentric rectangular coordinates of the centre of the Earth which should then enable one to calculate the geocentric positions of the other planets for terrestrial observation purposes. Any idea on how to go about this?</p>

CC BY-SA 4.0
2024-09-10T09:48:42.430Z
Your question is a little unclear. I presume the EMB (Earth-Moon Barycentre) coords are in au. Are both sets of coords using the same reference plane? I presume they're using the J2000 ecliptic plane, with the +X axis pointing towards the First Point of Aries (the equinox point). But another option is the equatorial plane.
PM 2Ring
CC BY-SA 4.0
2024-09-10T09:49:36.913Z
BTW, you can get high precision GM values for the Earth & Moon from https://ssd.jpl.nasa.gov/astro_par.html
PM 2Ring
CC BY-SA 4.0
2024-09-10T13:16:18.887Z
The coordinates for VSOP2013 is in Cartesian heliocentric ecliptic coordinates for the dynamic J2000 ecliptic. The ELP/MPP02 is in geocentric ecliptic as opposed to the heliocentric coords of VSOP2013. I calculated the ecliptic longitude of the Sun earlier for a random time using VSOP2013 and VSOP87, and they differ by a whopping 15'!
Coenraad van Schoor
CC BY-SA 4.0
2024-09-10T17:03:58.800Z
The ELP/MMP02 documentation says it's the "inertial mean ecliptic of date", but also contains a matrix to convert it to the "inertial mean ecliptic and equinox of J2000" ecliptic.
Greg Miller
CC BY-SA 4.0
2024-09-10T18:53:22.457Z
Thank you for the correction, Greg 
Coenraad van Schoor
CC BY-SA 4.0
2024-09-10T22:20:44.313Z
Ok. So if you have the heliocentric coords of the Moon & Earth, do you know how to compute the barycentre?
PM 2Ring
CC BY-SA 4.0
2024-09-11T05:55:58.673Z
With ELP/MPP02 the lunar coordinates are geocentric and given in ecliptic longitude, latitude and distance, which is then converted to rectangular coordinates for XYZ with a matrix calculation. The issue is that the lunar coordinates is not heliocentric, and that's exactly my question actually: how do we calculate the heliocentric co-ordinates of the Moon? If we know the heliocentric co-ordinates of the Moon and EMB, them we can use the Earth/Moon masses to calculate the offset of the EMB relative to the centre of the earth. But how do we calculate the heliocentric co-ordinates of the Moon?
Coenraad van Schoor
CC BY-SA 4.0
2024-09-11T06:00:19.890Z
The calculations to determine the EMB  coordinates in km relative to the ecliptic and the XYZ coordinates of the Moon, but in kilometers, are quite straight forward and given in the post https://space.stackexchange.com/questions/19035/converting-orbital-state-vectors-from-one-origin-to-another.
Coenraad van Schoor
CC BY-SA 4.0
2024-09-11T08:12:19.703Z
I just edited my question to make my question more clear. I hope it helps
Coenraad van Schoor
CC BY-SA 4.0
2024-09-11T10:40:40.977Z
Those geocentric EMB equations are correct, but I get different values for the coordinates. What masses did you use?
PM 2Ring
CC BY-SA 4.0
2024-09-11T10:48:15.097Z
Sorry, PM 2Ring, I meant the calculated value of the EMB's heliocentric coordinates from which one then calculates the GEOCENTRIC coordinates of the Sun.
Coenraad van Schoor
CC BY-SA 4.0
2024-09-11T11:11:52.520Z
I used mass_{earth} = 5.972 x 10^24 kg and mass_{moon} = 7.342 x 10^22 kg. I just recalculated and see I made some mistakes. I'll emend it in the original question. Thank you for pointing it out!
Coenraad van Schoor
CC BY-SA 4.0
2024-09-11T11:52:19.037Z
The new numbers look a lot better. To get more precision, you should use GM rather than M. See https://astronomy.stackexchange.com/a/48616/16685
PM 2Ring
CC BY-SA 4.0
2024-09-11T12:29:15.940Z
Thank you for the link to the other thread. Let me update the new values I calculated. Now the question still remains on how to calculate the heliocentric coordinates of the centre of earth from these values
Coenraad van Schoor
CC BY-SA 4.0
2024-09-11T13:35:37.020Z
I've changed the data in the original question to help you guys work using a real-world example of the question. Sorry for the numerous changes and emendations to the questions....I'm a bit OCD and can't help it sometimes
Coenraad van Schoor
<celestial-mechanics>
Determining the heliocentric co-ordinates of the earth's centre from the earth-moon barycentric coordinates using VSOP2013 model
Coenraad van Schoor
# Determining the heliocentric co-ordinates of the earth's centre from the earth-moon barycentric coordinates using VSOP2013 model

(Asked by: username_0 on 2024-09-10)

I've seen this question posed before but not answered fully at [Converting Earth Moon Barycenter coordinate to Geocentric coordinate](https://astronomy.stackexchange.com/questions/21827/converting-earth-moon-barycenter-coordinate-to-geocentric-coordinate). There is an insightful answer given at <https://space.stackexchange.com/questions/19035/converting-orbital-state-vectors-from-one-origin-to-another>, but again no practical equations were given to easily calculate the coordinates of the centre of the earth by using the earth-moon-barycentre heliocentric coordinates calculated by VSOP2013 to enable one to calculate the positions of the planets for earth-based observation purposes by astronomers (this is of course not an issue for the VSOP87 coordinates as it includes a model that calculates the LBR coordinates of the centre of the earth).

According to my understanding it should be a relatively easy task to determine the centre of the earth in heliocentric coordinates if one knows the position of the Moon relative to the earth, for example by using the ELP/MPP02 model.

As an example of my problem I'll calculate the heliocentric coordinates for the Earth (using VSOP2013) and the geocentric coordinates of the Moon (using ELP/MPP02) for Julian date 2460564.50061489 11 September 2024 at midnight GMT) :

1. Using VSOP2013 I calculated the Cartesian heliocentric ecliptic coordinates for the dynamic J2000 ecliptic for the EMB as

\begin{align\*}
X&= 0.98622458 \\
Y&= -0.20214571 \\
Z&= 0,00000567\\
\end{align\*}

2. Using ELP/Mpp02 I calculated the inertial mean ecliptic and equinox of J2000 ecliptic coordinates for the Moon to be

\begin{align\*}
X&= -98286.95858km\\
Y&= -378384.79064km \\
Z&= -33687.61320km\\
\end{align\*}

3. To calculate the coordinates of the Earth-Moon-Barycentre in the same coordinates as those of ELP/MPP02, I'll use the following formulae:

\begin{equation\*}
X\_{EMB}=\frac{mass\_{moon} \cdot X\_{moon}}{mass\_{earth}+mass\_{moon}}\\
Y\_{EMB}=\frac{mass\_{moon} \cdot Y\_{moon}}{mass\_{earth}+mass\_{moon}}\\
Z\_{EMB}=\frac{mass\_{moon} \cdot Z\_{moon}}{mass\_{earth}+mass\_{moon}}\\
\end{equation\*}

The resulting coordinates of the EMB will then be:
\begin{align\*}
X\_{EMB}&=-1363.8226km\\
Y\_{EMB}&=-4561.4700km\\
Z\_{EMB}&=-404.2623km\\
\end{align\*}

or, as expressed in AU:
\begin{align\*}
X\_{EMB}&=-7.9830279638\times10^{-6}\\
Y\_{EMB}&=-3.0733033237\times10^{-5}\\
Z\_{EMB}&=-2.7361631909\times10^{-6}\\
\end{align\*}

Now, my question is: how do I calculate the centre of the earth in heliocentric coordinates given the VSOP2013 heliocentric coordinates of the EBM and the geocentric ELP/MPP02 coordinates of the EMB. Using VSOP87 the Earth's heliocentric rectangular coordinates (in AU) are:

\begin{align\*}
X\_{Earth}&=0.987431488\\
Y\_{Earth}&=-0.196173931\\
Z\_{Earth}&=2.08708\times10^{-6}\\
\end{align\*}

I'm assuming there's a simple way to figure out how to use the geocentric rectangular coordinates of EMB calculated via ELP/MPP02 above in combination with the heliocentric EMB coordinates calculated via VSOP2013 to find the heliocentric rectangular coordinates of the centre of the Earth which should then enable one to calculate the geocentric positions of the other planets for terrestrial observation purposes. Any idea on how to go about this?

**Tags:** celestial-mechanics
> **Comments:**
> **username_1**: Your question is a little unclear. I presume the EMB (Earth-Moon Barycentre) coords are in au. Are both sets of coords using the same reference plane? I presume they're using the J2000 ecliptic plane, with the +X axis pointing towards the First Point of Aries (the equinox point). But another option is the equatorial plane.  
> **username_1**: BTW, you can get high precision GM values for the Earth & Moon from https://ssd.jpl.nasa.gov/astro_par.html  
> **username_0**: The coordinates for VSOP2013 is in Cartesian heliocentric ecliptic coordinates for the dynamic J2000 ecliptic. The ELP/MPP02 is in geocentric ecliptic as opposed to the heliocentric coords of VSOP2013. I calculated the ecliptic longitude of the Sun earlier for a random time using VSOP2013 and VSOP87, and they differ by a whopping 15'!  
> **username_2**: The ELP/MMP02 documentation says it's the "inertial mean ecliptic of date", but also contains a matrix to convert it to the "inertial mean ecliptic and equinox of J2000" ecliptic.  
> **username_0**: Thank you for the correction, Greg   
> **username_1**: Ok. So if you have the heliocentric coords of the Moon & Earth, do you know how to compute the barycentre?  
> **username_0**: With ELP/MPP02 the lunar coordinates are geocentric and given in ecliptic longitude, latitude and distance, which is then converted to rectangular coordinates for XYZ with a matrix calculation. The issue is that the lunar coordinates is not heliocentric, and that's exactly my question actually: how do we calculate the heliocentric co-ordinates of the Moon? If we know the heliocentric co-ordinates of the Moon and EMB, them we can use the Earth/Moon masses to calculate the offset of the EMB relative to the centre of the earth. But how do we calculate the heliocentric co-ordinates of the Moon?  
> **username_0**: The calculations to determine the EMB  coordinates in km relative to the ecliptic and the XYZ coordinates of the Moon, but in kilometers, are quite straight forward and given in the post https://space.stackexchange.com/questions/19035/converting-orbital-state-vectors-from-one-origin-to-another.  
> **username_0**: I just edited my question to make my question more clear. I hope it helps  
> **username_1**: Those geocentric EMB equations are correct, but I get different values for the coordinates. What masses did you use?  
> **username_0**: Sorry, PM 2Ring, I meant the calculated value of the EMB's heliocentric coordinates from which one then calculates the GEOCENTRIC coordinates of the Sun.  
> **username_0**: I used mass_{earth} = 5.972 x 10^24 kg and mass_{moon} = 7.342 x 10^22 kg. I just recalculated and see I made some mistakes. I'll emend it in the original question. Thank you for pointing it out!  
> **username_1**: The new numbers look a lot better. To get more precision, you should use GM rather than M. See https://astronomy.stackexchange.com/a/48616/16685  
> **username_0**: Thank you for the link to the other thread. Let me update the new values I calculated. Now the question still remains on how to calculate the heliocentric coordinates of the centre of earth from these values  
> **username_0**: I've changed the data in the original question to help you guys work using a real-world example of the question. Sorry for the numerous changes and emendations to the questions....I'm a bit OCD and can't help it sometimes  

---

## 1 answers

### Answer by username_0 on 2024-09-12. Score: 2

Ok, so I managed to figure things out.

Basically, the process to find the heliocentric ecliptic coordinates of the centre of the Earth is as follows:

1. Determine the heliocentric ecliptic J2000 rectangular coordinates for the EMB using VSOP2013,i.e. $X\_{EMB}$, $Y\_{EMB}$, and $Z\_{EMB}$.
2. Calculate the geocentric ecliptic coordinates in kilometers for the Moon (in this case from ELP/Mpp02), i.e. $X\_{moon}$, $Y\_{moon}$, and $Z\_{moon}$.
3. Convert $X\_{moon}$, $Y\_{moon}$, and $Z\_{moon}$ from kilometers to AU.
4. Using $GM\_{moon}$ and $GM\_{earth}$, calculate the geocentric ecliptic coordinates for the EMB, i.e. $X\_{geoEMB}$, $Y\_{geoEMB}$, and $Z\_{geoEMB}$.
5. Finally, calculate the heliocentric coordinates for the centre of the earth:

\begin{align\*}
X\_{earth} &= X\_{EMB}-X\_{geoEMB}\\
Y\_{earth} &= Y\_{EMB}-Y\_{geoEMB}\\
Z\_{earth} &= Z\_{EMB}-Z\_{geoEMB}\\
\end{align\*}

After calculating $\lambda$ and $\beta$ one then jut simply to the necessary equations to bring the J2000 coordinates of VSOP2013 to the equinox and ecliptic of date (found in Meeus, pp. 136-137).

The values seem to correlate almost exactly with the JPL's.


---

