Why does the Earth have a z-component at the start of the J2000 epoch?

Question

When I go to the NASA HORIZONS webpage

https://ssd.jpl.nasa.gov/horizons.cgi

and ask for the VECTOR coordinates of the Earth on 1 January 2000, it tells me that Earth has the following coordinates in the J2000 epoch:

X =-1.756637922977122E-01 Y = 9.659912850526895E-01 Z = 2.020629118443605E-04

These are measured in astronomical units. Quoting from the webpage:

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Reference epoch: J2000.0

XY-plane: plane of the Earth's orbit at the reference epoch

X-axis : out along ascending node of instantaneous plane of the Earth's orbit and the Earth's mean equator at the reference epoch

Z-axis : perpendicular to the xy-plane in the directional (+ or -) sense of Earth's north pole at the reference epoch.

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But these figures don't make sense! The z value should be EXACTLY zero, since we are measuring at 1 January 2000. Also the x should be exactly 1, and the y should be zero.

What is going on?

Answer 1

You made the same fundamental mistake that Anton Gromov made in his question on the sister Space Exploration StackExchange network site: You used the solar system barycenter rather than the Sun as the frame origin. Had you used the Sun, the apparent discrepancy would have dropped by almost two orders of magnitude.

A much lesser flaw is that you apparently used midnight rather than noon. The J2000.0 epoch is 12 noon Terrestrial Time on 1 January 2000. This doesn't change things by much, but it is nonetheless important. That half day offset has bitten me more than once.

The z coordinate of the Earth's position would not have dropped to zero had you made the above corrections (Sun-centered rather than solar system barycenter, and 12 noon rather than midnight). It would not have dropped to zero even if you had used the Earth-Moon barycenter rather than the center of the Earth.

One reason it would not have dropped to zero is that the Earth's orbit about the Sun is not an ellipse. The Earth's orbit doesn't even lie on a plane! The key cause of this non-planar motion is that the Moon's orbit about the Earth is inclined by about 5.15° with respect to the ecliptic, making the Earth bob up and down with respect to the ecliptic¹. Because the Earth's orbit is not truly elliptical or even planar, the ecliptic plane at some epoch is a time-averaged plane centered about the epoch time that makes the Earth's position with respect to the Sun (or the Sun's position with respect to the Earth) average to zero.

A lesser reason is that the J2000.0 mean ecliptic and mean equinox frame was defined over 36 years ago. HORIZONS uses DE431, which was released 6 years ago. JPL's ability to model the solar system improved significantly in the intervening three decades.

_{¹ Strictly speaking, it makes the Sun appear to bob up and down with respect to the ecliptic. The concept of the ecliptic plane dates back to the ancient Greeks, and it retains a vestige of this definition to this day.}

Answer 2

In addition to the small z-coordinate discrepancy explained in the previous answer and links, the assertion that the x-coordinate should be 1 AU at JD 2451545, and the y-axis zero, is incorrect. In the question you included the following:

X-axis : out along ascending node of instantaneous plane of the Earth's orbit and the Earth's mean equator at the reference epoch

In other words, the x-axis points outward from the Vernal equinox at the J2000.0 epoch along the line of nodes, which is the intersection of the ecliptic and equatorial planes. The Sun is about 2 2/3 months away from the ascending node on Jan 1, so the Earth-Sun radius is not directly on the x-axis.