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Ex. 44.a of Meeus Astronomical Algorithm, Second Edition, provide the position of the Galileans on December 16 1992 at 0h UT (p.311/488).

Procedure 1: For Io, in terms of the Jupiter's equatorial radius (71,492 km), the solution provides, X1 = -3.44 and Y1 = +0.21. Thus, X1 = -3.44 = -2.4593248e+5 km, and Y1 = +0.21 = 1.501332e+4 km.

In terms of 1 AU = 1.49598e+8 km, the semi-major axis of Jupiter of 5.2038 AU translates to 7.784780724e+8 km.

Thus, on December 16 1992 at 0h UT, the distance between the Sun and the Io was approx. X1 = 7.784780724e+8 km - 2.4593248e+5 km = 7.7823213992e+8 km. Y1 = 7.784780724e+8 km + 1.501332e+4 km = 7.7849308572e+8

Procedure 2: Now, using JPL Horizons ephemeris app at https://ssd.jpl.nasa.gov/horizons/app.html#/ with the input parameters:

  1. Ephemeris Type; Vector Table
  2. Target Body: Io (JI)
  3. Coordinate Center: Sun (body centre) [500@10]
  4. Time Specification: Start=1992-12-16 TDB , Stop=1992-12-17, Step=1 (days)
  5. Table Settings: 1. Position components {x, y, z} only ephemeris provide: x = -8.136759055015339E+08 km and y = -2.827656223176936E+07 km.

Contents of the Batch Edit File are:

!$$SOF
MAKE_EPHEM=YES
COMMAND=501
EPHEM_TYPE=VECTORS
CENTER='500@10'
START_TIME='1992-12-16'
STOP_TIME='1992-12-17'
STEP_SIZE='1 DAYS'
VEC_TABLE='1'
REF_SYSTEM='ICRF'
REF_PLANE='ECLIPTIC'
VEC_CORR='NONE'
CAL_TYPE='M'
OUT_UNITS='KM-S'
VEC_LABELS='YES'
VEC_DELTA_T='NO'
CSV_FORMAT='YES'
OBJ_DATA='YES'

Summary of Procedure 1: X1 = 7.7823213992e+8 km. Y1 = 7.7849308572e+8 km.

Summary of Procedure 2: x = -8.136759055015339E+08 km y = -2.827656223176936E+07 km.

May I know why a difference exist between two procedures please. I hope the methodology for the procedure 1 is correct. Am I missing some points?

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    $\begingroup$ X and Y have different definitions between Meeus and Horizons. Without a lot of coordinate transformation, you cannot compare X to X and Y to Y. In Meeus, X and Y are in the plane of the sky as seen from Earth. Without knowing the third coordinate, you cannot add X and Y to the distance from Sun to Jupiter to get the distance of the satellites from the Sun. X and Y in Horizons are most likely measured from the vernal equinox, so X and Y are in entirely different directions. $\endgroup$
    – JohnHoltz
    Commented Sep 7, 2023 at 19:58
  • $\begingroup$ I havea Python script that sends Batch Data to Horizons. gist.github.com/PM2Ring/b1fec75e78cc08f6fc28c6f6c43529c3 Or just click on the link in the next comment. $\endgroup$
    – PM 2Ring
    Commented Sep 8, 2023 at 3:19
  • $\begingroup$ sagecell.sagemath.org/… $\endgroup$
    – PM 2Ring
    Commented Sep 8, 2023 at 3:19
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    $\begingroup$ As John said, it's not meaningful to add the semi-major axis of Jupiter to those Io coordinates. You need to do proper vector arithmetic. Also, Jupiter's orbit is eccentric, and the Sun-Jupiter distance varies by ~0.25 au from the mean. $\endgroup$
    – PM 2Ring
    Commented Sep 8, 2023 at 3:33
  • $\begingroup$ Thanks for adding the Horizons Batch Data. It's still not clear why you're adding Jupiter's SMA (semi-major axis) to the X & Y coordinates of Io's position vector. Do you want the position vector of Io relative to the Sun, or do you just want the distance from Io to the Sun? Also, I don't think those Io coordinates relative to Jupiter are correct. They don't match the info from Horizons, either in the J2000 ecliptic frame, or the body frame of Jupiter's equator (which are very similar), but I guess Meeus may be using a different X axis for the Jupiter satellite orbits. $\endgroup$
    – PM 2Ring
    Commented Sep 8, 2023 at 22:23

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tldr; The orientations of X and Y for Jupiter's moons are different than X and Y for the position of Jupiter from the Sun. Therefore, the values cannot be added to get the distance from the Sun to Jupiter's moon.

For the remainer of this description, I will use Xm and Ym as the coordinates of Jupiter's moons calculated by Meeus, and X and Y as the coordinates of Jupiter from the Sun.

The positions of Earth and Jupiter relative to the Sun on Dec 16 1992 are shown below (from Horizon). The important things to note are that X is towards the vernal equinox (0 h Right Ascension), and the XY plane is in the plane of the ecliptic.

Position of Earth and Jupiter projected onto the ecliptic

(Figure approximately to scale but ignores the inclination of Jupiter's orbit.)

Meeus' formulas give the position of the Jovian satellites in the plane of the sky as seen from Earth. For the position Xm = -3.44, the Moon would be located at either point 1 or 2 shown in the figure below. These are the important things to note from this figure:

  • The Xm coordinate axis points to the west as seen from Earth. Since this direction is almost at right angles to X on this date, the X coordinate from the Sun to the satellite is not X+Xm.
  • The Ym coordinate is the distance of the moon above or below the plane of Jupiter's equator. Ignoring the tilt of Jupiter's equatorial plane relative to the ecliptic plane, the Ym coordinate would be above or below the ecliptic plane. Ym cannot be added to Y to get a meaningful value.
  • The Zm coordinate of the satellite is required to determine where it is located in its orbit (position 1 or 2). Without that value, the distance to the Sun cannot be calculated.

Figure in Jupiter's equatorial plane

(Zooming in on Jupiter from the above figure. This figure is not to scale, but the orientation of Zm is approximately correct.)

Even if the inclination of Jupiter's orbit and the tile of Jupiter's rotational axis were ignored (or close to zero), the calculation of the distance from the Sun to the satellite would require estimating the Zm coordinate, and then determining the angle between the X and Zm axes so that the proper components of Xm and Zm can be added to X and Y.

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