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I am currently getting a bit lost in reference frames and coordinate systems. From my point of view, I am currently in the ICRS frame. I know that the ECI from my point is only moving but not rotating, therefore its x,y,z-axis are always pointing in the same direction. I know that at the vernal equinox say 2021.3.20 at 9:43 the sun is directly above the equator. And at that date and time, the z-axis of the ECI is rotated -23.43653 deg around the line between earth and sun.

Position of the Earth at 2021.3.20 9:43 in ICRS

x -1.003059603498145 * AU

y = 0.0104425511782964 * AU

z = 0.0001246952057660868 AU

Still, the vernal equinox does not define the direction of the x-axis of the ECI, I just know that it is 90 deg to the z-axis.

Now I need the J2000 frame:

x -0.17566 * AU

y = 0.96599 * AU

z = 0.00020 * AU

here the x-axis of the ECI frame is defined as "The x-axis is aligned with the mean equinox" wiki and I assume, that at J2000 the x-axis of the ECEF and ECI frame are identical. Therefore, the ECI and ECEF x-axis at J2000 is aligned with the earth CG and a point on the earth surface a little bit south of Nigeria (0 deg lat/lon)

Now my question, what is exactly meant by the "mean equinox"? I can't find any definition of that. Does it mean the point where the earth is on average in the ICRS at the vernal equinox? Is it parallel to the x-axis at the vernal equinox? But if so, then where is the x-axis at the vernal equinox?

I have checked out this orbit simulation and here it seems, that the x-axis at J2000 of the ECI and ECEF point in the direction of the sun, but some degrees above. Does it mean, that the track of the x-axis in the orbital plane point at the sun? Could I use this as the additional definition of the ECI coord system to the 2021 vernal equinox?

EDIT Comment Organic Marble:

"In astronomy, an equinox is either of two places on the celestial sphere at which the ecliptic intersects the celestial equator". I have already read this article. But it still not 100% sure As I understand from this I made two drawings indicating the equinox "direction" at J2000 and the vernal equinox Vernal equinox

J2000

If these drawings are correct, then the ECI x-axsis is going to the "right" (relative to the view in the images). Therefore, the x-axsis must always align with the equinox direction (besides its precession and nutation motion). And my assumption, that at J2000 ECI and ECEF x-axis are colinear is incorrect.

Is it correct, that under the simplification, that the z axis of the ECEF and the ECI are colinear, the rotation of ECEFs relative to the sun only depends on the time and earth rotational speed? If yes where could I find the rotation around the z-axis between ECI and ECED at J2000?

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enter image description here The ECI x-axis is defined as pointing towards the intersection between Earth's equatorial plane and the ecliptic plane, which is shown in the screenshot here. The red line across the sky is the ecliptic and the blue grid is the celestial sphere (also sometimes called equatorial grid). So you can see here the red arrow labeled "J2000: X" is pointing at that intersection on 2000 January 1 12:00:00. This frame is inertially fixed, which is why it needs to be defined as a specific time since the Earth's spin axis is not inertially fixed (also meaning that Earth's equatorial plane is not inertially fixed). ECEF X-axis is not aligned with the ECI X-axis at epoch J2000.

According to a quick google search, the vernal equinox in year 2000 was March 19 23:35, which is shown here, where the ECI X-axis and Sun vectors are aligned. somewhere around 12 hours later the ECEF X-axis would also be aligned with the other 2 vectors: enter image description here

How exact these vectors are aligned / you need them to be depend on your application. Like you said, you could make the simplifying assumption that the ECI and ECEF Z-axes are always aligned, so you could find ECEF as a function of time by a simple z-axis rotation. But since Earth's spin vector is not inertially fixed, over time your model would become more and more inaccurate. Here is what the 2 z-axes look like 100 years later: enter image description here

If you're interested in learning more about the ECI and ECEF frame definitions, JPL's SPICE documentation has good sources. Here are 3:

https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Tutorials/pdf/individual_docs/23_lunar-earth_pck-fk.pdf

https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Tutorials/pdf/individual_docs/17_frames_and_coordinate_systems.pdf

https://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/pck00010.tpc

Also note that the ICRF and ECI frames are almost identical (off by less than 0.1 arcseconds, according the the frames and coordinate systems pdf linked above), and are considered the same frame in SPICE.

I'm honestly a little unsure on what your question(s) were here, but I'm thinking that these visuals and sources will help.

Edit: mean equinox question I think the "mean" part of that is a bit misleading. My understanding is that the definition given above is what they refer to as mean equinox, but use the word mean because its oscillating, so after some time in the future it will come back to that point. The ECI frame also goes by a few different names: in SPICE, its referred to just as J2000. Its also referred to as EME2000 (Earth Mean Equator). And again, ICRF is basically coinciding with ECI.

Also, theres the eclipticJ2000 frame (in SPICE its called ECLIPJ2000), which is also sometimes called EMO2000 (Earth Mean Orbit).

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    $\begingroup$ Thank you, that was an excellent answer and you helped me a lot! I got one last additional question. Is there an exact date or frame where the ECEF and ECI x-axis are aligned? You mentioned around 12h after the vernal equinox, but is it possible to find some more accurate time? I am currently working on a little hobby project to visualise our university satellite in 3D in space and therefore it must be somewhat accurate in around 50 years. And I would like to add Earth’s accurate rotation, but I don’t know like an initial frame from which I can integrate zu rotation. $\endgroup$ – Olgidos Apr 1 at 11:12
  • $\begingroup$ I explain how to calculate the ECEF frame using SPICE in this question: space.stackexchange.com/questions/51068/… Also, the ECI and ECEF x-axis will be approximately aligned every sidereal day, so you can plot out the angle between the 2 vectors for a 24 hour period and the minimum of that plot will be where they are closest aligned $\endgroup$ – Alfonso Gonzalez Apr 1 at 11:59
  • $\begingroup$ I have just checked out cosmographia, how did enabled the visible axis? $\endgroup$ – Olgidos Apr 1 at 19:50
  • $\begingroup$ Right clicking on an object (like Earth) will show a number of options, including displaying reference frames $\endgroup$ – Alfonso Gonzalez Apr 2 at 21:54

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