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I ran VSOP2013 in Fortran, and I now have the following position $x, y, z$ as well as velocity $v_x, v_y, v_z$ data.

How do I convert the position to heliocentric ecliptic spherical coordinates $l, b, r$? 

PLANETARY EPHEMERIS VSOP2013  EMB     
Ecliptic   Heliocentric Coordinates:  X,Y,Z (au)  X',Y',Z' (au/d)  - Dynamical Frame J2000

Julian Day 2415021.0   
-0.2054467990    0.9615310525    0.0002141233   -0.0171063024   -0.0036576954   -0.0000011815
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    $\begingroup$ You're asking how to convert rectangular coordinates to spherical? $\endgroup$
    – user21
    Commented Jul 23, 2017 at 15:23
  • $\begingroup$ whats L, B, R? I have never heard of those coordinates. At least none by that name or TLA. $\endgroup$
    – Natsfan
    Commented Jul 23, 2017 at 19:21
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    $\begingroup$ l, b,r are heliocentric solar coordinates. l is longitude (measured from vernal equinox) b is ecliptic latiude (north positive) r is heliocentric distance. en.wikipedia.org/wiki/Ecliptic_coordinate_system $\endgroup$
    – James K
    Commented Jul 23, 2017 at 21:19
  • $\begingroup$ thanks, that sure sounds like spherical coordinates to me. lol $\endgroup$
    – Natsfan
    Commented Jul 23, 2017 at 23:35
  • $\begingroup$ x,y,z - do you know if they are earth rotating coordinates or just earth centered non-rotating? $\endgroup$
    – Natsfan
    Commented Jul 23, 2017 at 23:42

1 Answer 1

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Looking at the numbers and comparing to JPL Horizons, the best match is to the position of the Earth-Moon Barycenter in heliocentric coordinates. That's of course consistent with the data being labeled as: EMB, Ecliptic Heliocentric Coordinates.

If you can live with some small uncertainties due to the details of how coordinate transforms may evolve over time, I believe you can just apply some simple trigonometry. You can confirm here.

$$r = \sqrt{x^2 + y^2 + z^2}$$

$$l = tan^{-1}(y/x)$$

$$b = sin^{-1}(z/r)$$

enter image description here

Data in Question: for JD 2415021.0
position (AU):     [-0.2054467990,  0.9615310525,  0.0002141233]
velocity (AU/day): [-0.0171063024, -0.0036576954, -0.0000011815]

JPL Horizions Earth-Moon barycenter J2000 Heliocentric for JD 2415021.0
position (AU):     [-0.2054465857,  0.9615310983,  0.00021393097] 
velocity (AU/day): [-0.0171063032, -0.0036576916, -0.00000118253]

JPL Horizions Earth J2000 Heliocentric for JD 2415021.0
position (AU):     [-0.2054522688,  0.9615603123,  0.00021303115] 
velocity (AU/day): [-0.0171136285, -0.0036594066, -0.00000182603]

JPL Horizions Earth J2000 Solar System barycenter for JD 2415021.0
position (AU):     [-0.2022720210,  0.9679285296,  0.00010939456]
velocity (AU/day): [-0.0171209826, -0.0036556277, -0.00000165111]
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    $\begingroup$ I see you've been asking python questions about VSOP2013 in SO. You might want to look at the two Python packages PyEphem based on VSOP87 according to this answer and also definitely check out Skyfield. Skyfield uses the same Development Ephemerides as Horizons but is very new, modern, and easy to use. $\endgroup$
    – uhoh
    Commented Jul 24, 2017 at 19:01

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