Barycentric corrections at 1 mm/s!

OK, the newborn-induced blogging hiatus is over.  Back to business!

Measuring precise radial velocities of stars to 1 m/s means “subtracting off” the ~ 30 km/s motion of the Earth around the Sun (or, more precisely, the Solar System barycenter).  That is, we measure Doppler shifts of about a part in 104 for all of our stars because of this “barycentric motion”, and we have to measure that shift to another part in 104 to get even 3 m/s precision on our way to detecting planets.

This is actually a completely solved problem; the JPL Ephemeris gives precise positions and velocities for the Earth in the appropriate frame, and the International Earth Rotation Service provides precise orientations and angular velocities of the Earth so we can add in the telescope’s motion.  Finally, we know the figure of the Earth well enough that a GPS receiver can tell where the telescope is on the Earth to complete the problem.

John Johnson, chalkboard mathy guy, probably doing special relativistic corrections to Doppler shifts.

But that doesn’t mean it’s easy.  Back in grad school, John Johnson plotted up the measured velocities of one of our “standard” stars as a function of the “barycentric correction” we were subtracting from them, and found quadratic dependence.  We had apparently missed some sort of special relativistic effect in the problem. (Then-Miller-Fellow) Eric Ford and I independently solved the SR problem and introduced a kludgy correction to the Berkeley Doppler code to take care of it — we had been doing a Galilean transformation (just subtracting velocities) instead of a proper relativistic subtraction.

But annual signals remain in the data!  Eventually, I identified an error in the proper motion correction

Ryan Chornock, nature lover, now professor at Ohio University, where he watches stars explode and die for his edification and amusement.

algorithm, as well (actually, I had a feeling I had found it but couldn’t put my finger on it until I turned around and asked my officemate to recite the formula — Ryan Chornock said “cosine dec” and the problem was solved!).

But annual signals remain in the data!  Some of it is telluric lines and some of it might be imperfect iodine cell spectra (Sharon Wang is working on this), but we’ve always had a nagging feeling that there might be more bugs in “the bary code”.

Since those grad school days I’ve been noodling around the problem, refining those original notes from the John Johnson correction, and eventually I decided to check things against TEMPO.  TEMPO is the timing package used by the pulsar community, and its effective precision is much better than we need (case in point: it was used to find planets much much smaller than anything we find around our stars).  The problem is tricking TEMPO into reporting barycentric corrections as redshifts, which is not something it’s actually designed to do.

I had been doing this in my spare time, when Debra Fischer asked me to join on the CHIRON project to perfect barycentric corrections for it.  This gave me the motivation (and funding!) to put the real time in do the problem right.

Eric Mamajek, another chalkboard mathy guy, formerly of CTIO and purveyor of inordinate efforts to get answers to interesting astronomy-related questions.

First, we had to get the proper coordinates for the CTIO 1.5m.  The Web was no help — there were many different coordinates listed for it.  Fortunately, Eric Mamajek was now working there, so I simply asked him.

Mamajek is my hero.  The thing about Eric is, if you give him a problem that interests him, he’ll work on it until he gets the best possible answer (just ask Kevin Luhman).  Indeed, the answer he returned to us after inordinate effort was essentially perfect, and while he was at it he refined some other coordinates as well.

Paul Demorest, NRAO pulsar astronomer and patient explainer to optical astronomers of the obvious and arcane.

TEMPO, it turns out, is not trivial to install.  It’s mostly written in FORTRAN, and if you don’t regularly install things from source it can take some expertise to get going.  Fortunately, I know Andrea Lommen from Berkeley, who pointed me to Nataniel Garver-Daniels who helped me get things going on my Mac.  Also, if you’re not a pulsar astronomer, some of the documentation can be… opaque, let’s say.  Fortunately, I know Paul Demorest at NRAO (also from Berkeley), so I managed to invite myself down to give a talk and park myself in a spare office down there and pester him over the course of two days (and a longer time over email) about the details.  In the end, I got it working.

Jason Eastman, LCOGT astronomer and happy happy writer of barycentric correction code in IDL.

I had my manuscript all written up, but what I really wanted to do was release public code that did the right thing (to save others the work of installing TEMPO for this purpose).  Coincidentally, Jason Eastman was writing IDL code to perform the barycentric correction, and after lots of back and forths on the topic we joined forces:  he implemented my algorithm cleanly (“from scratch”) and released the public version, we added lots of corrections and descriptions of the code to my manuscript, and it became Wright & Eastman.

So what did we find?  Well, we can reproduce TEMPO’s results to within 1mm/s — this retires the barycentric correction algorithm  from the Doppler error budget for anyone using our code.

For work at 10 cm/s (we wish!) you need:

  • Observatory positions to better than 1km (!)
  • Stellar positions to 1”
  • Proper motions to 10 mas/yr
  • Systemic radial velocities of the star to 1,000 km/s (!)
  • Parallax to better than 10 mas (but getting it wrong only introduces a linear trend at this level)
  • Timing of the observation to 1s

The effects you have to worry about are:

  • Nutation and precession of the Earth (prospective formulae are OK)
  • Special relativistic addition of velocities (a 3 m/s effect)
  • Secular acceleration (the radial vector moves with the star’s proper motion, slowly mixing tangential space motion into the radial direction — the “pitch” of all stars is dropping with time)
  • Parallax-proper motion: like the secular acceleration, but from the parallactic motion of the star, instead of the proper motion, so it has an annual signal at ~ 10 cm/s for nearby stars.
  • At 3 cm/s, you need to worry about the variable time dilation of the Earth and variable gravitational blueshift from the Sun because of the eccentricity of the Earth’s orbit.
  • Below 1 cm/s you need to worry about the (time variable) Shapiro delay past the Sun, and potentially the astrometric effects of stellar companions on your target star.

It’s on the arXiv today.  We went to PASP instead of ApJ because the paper is intentionally didactic and we were worried an ApJ referee would complain that there is no new physics here.*  You can find Jason Eastman’s code here on his EXOFAST page.

*Altered to emphasize that it’s not that PASP papers don’t have new physics, but that in my experience ApJ referees want to see significantly new results and little elementary explanation, while PASP is much more open to didactic, review, and “documentation” papers.

2 thoughts on “Barycentric corrections at 1 mm/s!

  1. jtw13 Post author

    Yup. At worst,
    Orbital acceleration gives 30 km/s * 2pi *(1s/1 yr) ~ 6 mm/s
    Rotational acceleration gives 300 m/s * 2pi *(1s/1 day) ~ 30 mm/s

  2. michael bottom

    fantastic–something bound to be useful from here to eternity! the term that really surprised me was the timing accuracy of 1 second, that seemed too lenient for 1 mm/s…is the main reason because the angular frequencies of the earth’s rotation and orbit are small enough when multiplied twice (omega^2) in the acceleration term?

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