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I am not an astronomer, but am trying to calculate the trajectory of an interstellar space probe, for example to Proxima Centauri. The HelioCoordinates seem appropriate, as typically used within the solar system. (I am assuming HelioCoordinates is an ecliptic Cartesian coordinate system centered on the Sun. Unfortunately StarData "Definition" is missing for "HelioCoordinates" and all the other properties I have checked.) But I am having difficulty with characterizing the proper motion of Proxima in these coordinates. As a check, I used a nearby star Hadar for comparison -- since it is much farther away, its proper motion is small. This is confirmed:

In[168]:= {StarData["ProximaCentauri", "ProperMotion"], 
 StarData["Hadar", "ProperMotion"]}
Out[168]= {Quantity[1895.42, ("MilliarcSeconds")/("Years")], 
 Quantity[30.16, ("MilliarcSeconds")/("Years")]}

Now I try to calculate the same motion in heliocoordinates. The results don't agree with the reported proper motion, and further are almost identical for the two stars:

In[169]:= motionInOneYear[star_] := Module[{p1, p2},
  p1 = StarData[star, 
     Dated["HelioCoordinates", DateObject[{2020, 1, 1}]]]/
    Quantity["ly"];
  p1 = p1/Norm[p1]; (* normalize so the point is on a unit sphere *);
  (* one year later *)
  p2 = StarData[star, 
     Dated["HelioCoordinates", DateObject[{2021, 1, 1}]]]/
    Quantity["ly"];
  p2 = p2/Norm[p2];
  (* distance between points on unit sphere is angular change in \
radians *)
  Norm[p1 - p2] Quantity["radians"]/Quantity["mas"] 
  ]
{motionInOneYear["ProximaCentauri"], motionInOneYear["Hadar"]}

Out[170]= {435.456, 424.801}

I conclude that there are two things going on: (a) StarData in HelioCoordates does not take into account the proper motion of the star (it just uses data from an epoch, even though it seems to accept a date) and (b) what I am seeing is a change in star position due to the rotational motion of the solar system. Or in other words, in HelioCoordinates the far star field is moving slowly due to a rotational motion of the solar system ecliptic plane. Is my interpretation correct? If so, what is the best way to characterize the position of Proxima Centauri in HelioCoordinates that does take the star's proper motion into account? Or perhaps I should be using a different coordinate system from the get-go?

Addition: After further investigation it seems clear that the variation in star positions I am seeing is not due to proper motion but rather due to the precession/nutation of the heliocentric coordinate system in question. I have therefore decided to do all my space probe trajectory calculations in the heliocentric coordinates, so at least they are consistent. Then I will have to characterize the star proper motion separately from that. I would criticize Wolfram for not properly documenting what database they are using or its limitations. No wonder nobody has answered my question.

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  • 5
    $\begingroup$ Let me welcome you to mathematica.stackexchange and thank you for asking such a well-written and detailed question as your first post. $\endgroup$ – halirutan♦ Apr 18 '19 at 18:57
  • $\begingroup$ Thanks for the compliment from @halirutan. I am a long-time user of Mathematica, but new to the astronomy aspects. I tend to avoid the more obscure features to avoid getting into deep issues. $\endgroup$ – Dave M Apr 20 '19 at 22:51
  • $\begingroup$ I would love to see you answer your own question, if you have the time and a suitable solution or workaround. It seems an extremely interesting area and I'd be very curious to see how to solve your problem. Sorry I am unable to help, but this stuff is way out of my knowledgebase :) $\endgroup$ – Carl Lange Apr 21 '19 at 8:34
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I concluded that the StarData[] "HelioCoordinates" database (a) must not not take account the proper motion of a star and (b) relative to the galactic background precesses slowly. The former is disappointing, and the latter makes sense since there is no reason to expect that the ecliptic plane of the solar system is at an absolutely fixed orientation relative to the galactic background.

To address these issues, I divide the calculation of a space probe into several pieces. I calculate a space probe trajectory to a fixed star position (at one point of time, so no precession), but over one year's orbit of the earth about the sun. That separates out the parallax effect due to the earth orbit. Then I will have to independently make a calculation of the star proper motion, inferring it from the "ProperMotion" property of the star. (Unfortunately that is specified with respect to a different geocentric equatorial coordinate system.) The precession of the solar system I will neglect since I am interested in the trajectory in relation to earth rather than the galactic background. Comments/objections are helpful!

Again I would appeal to Mathematica to document the origin and limitations of their astronomical databases.

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