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B.1.10 Gaia

Functions related to astrometry suitable for use with data from the Gaia astrometry mission.

The methods here are not specific to the Gaia mission, but the parameters of the functions and their units are specified in a form that is convenient for use with Gaia data, in particular the gaia_source catalogue available from http://gea.esac.esa.int/archive/ and copies or mirrors.

There are currently three main sets of functions here:

Position and velocity vectors

Functions are provided for converting the astrometric parameters contained in the Gaia catalogue to ICRS Cartesian position (XYZ) and velocity (UVW) vectors. Functions are also provided to convert these vectors between ICRS and Galactic or Ecliptic coordinates. The calculations are fairly straightforward, and follow the equations laid out in section 1.5.6 of The Hipparcos and Tycho Catalogues, ESA SP-1200 (1997) and also section 3.1.7 of the Gaia DR2 documentation (2018).

These functions will often be combined; for instance to calculate the position and velocity in galactic coordinates from Gaia catalogue values, the following expressions may be useful:

    xyz_gal = icrsToGal(astromXYZ(ra,dec,parallax))
    uvw_gal = icrsToGal(astromUVW(array(ra,dec,parallax,pmra,pmdec,radial_velocity)))
 
though note that these particular examples simply invert parallax to provide distance estimates, which is not generally valid. Note also that these functions do not attempt to correct for solar motion. Such adjustments should be carried out by hand on the results of these functions if they are required.

Functions for calculating errors on the Cartesian components based on the error and correlation quantities from the Gaia catalogue are not currently provided. They would require fairly complicated invocations. If there is demand they may be implemented in the future.

Distance estimation

Gaia measures parallaxes, but some scientific use cases require the radial distance instead. While distance in parsec is in principle the reciprocal of parallax in arcsec, in the presence of non-negligable errors on measured parallax, this inversion does not give a good estimate of distance. A thorough discussion of this topic and approaches to estimating distances for Gaia-like data can be found in the papers

The functions provided here correspond to calculations from Astraatmadja & Bailer-Jones, "Estimating Distances from Parallaxes. III. Distances of Two Million Stars in the Gaia DR1 Catalogue", ApJ 833, a119 (2016) 2016ApJ...833..119A based on the Exponentially Decreasing Space Density prior defined therein. This implementation was written with reference to the Java implementation by Enrique Utrilla (DPAC).

These functions are parameterised by a length scale L that defines the exponential decay (the mode of the prior PDF is at r=2L). Some value for this length scale, specified in parsec, must be supplied to the functions as the lpc parameter.

Note that the values provided by these functions do not match those from the paper Bailer-Jones et al. "Estimating Distances from Parallaxes IV: Distances to 1.33 Billion stars in Gaia Data Release 2", accepted for AJ (2018) arXiv:1804.10121. The calculations of that paper differ from the ones presented here in several ways: it uses a galactic model for the direction-dependent length scale not currently available here, it pre-applies a parallax correction of -0.029mas, and it uses different uncertainty measures and in some cases (bimodal PDF) a different best distance estimator.

Epoch Propagation

The Gaia source catalogue provides, for at least some sources, the six-parameter astrometric solution (Right Ascension, Declination, Parallax, Proper motion in RA and Dec, and Radial Velocity), along with errors on these values and correlations between these errors. While a crude estimate of the position at an earlier or later epoch than that of the measurement can be made by multiplying the proper motion components by epoch difference and adding to the measured position, a more careful treatment is required for accurate propagation between epochs of the astrometric parameters, and if required their errors and correlations. The expressions for this are set out in section 1.5.5 (Volume 1) of The Hipparcos and Tycho Catalogues, ESA SP-1200 (1997) (but see below), and the code is based on an implementation by Alexey Butkevich and Daniel Michalik (DPAC). A correction is applied to the SP-1200 treatment of radial velocity uncertainty following Michalik et al. 2014 2014A&A...571A..85M because of their better handling of small radial velocities or parallaxes.

The calculations give the same results, though not exactly in the same form, as the epoch propagation functions available in the Gaia archive service.

polarXYZ( phi, theta, r )
Converts from spherical polar to Cartesian coordinates.

astromXYZ( ra, dec, parallax )
Calculates Cartesian components of position from RA, Declination and parallax. This is a convenience function, equivalent to:
    polarXYZ(ra, dec, 1000./parallax)
 

Note that this performs distance scaling using a simple inversion of parallax, which is not in general reliable for parallaxes with non-negligable errors. Use at your own risk.

icrsToGal( xyz )
Converts a 3-element vector representing ICRS (equatorial) coordinates to galactic coordinates. This can be used with position or velocity vectors.

The input vector is multiplied by the matrix AG', given in Eq. 3.61 of the Gaia DR2 documentation, following Eq. 1.5.13 of the Hipparcos catalogue.

The output coordinate system is right-handed, with the three components positive in the directions of the Galactic center, Galactic rotation, and the North Galactic Pole respectively.

galToIcrs( xyz )
Converts a 3-element vector representing galactic coordinates to ICRS (equatorial) coordinates. This can be used with position or velocity vectors.

The input vector is multiplied by the matrix AG, given in Eq. 3.61 of the Gaia DR2 documentation, following Eq. 1.5.13 of the Hipparcos catalogue.

The input coordinate system is right-handed, with the three components positive in the directions of the Galactic center, Galactic rotation, and the North Galactic Pole respectively.

icrsToEcl( xyz )
Converts a 3-element vector representing ICRS (equatorial) coordinates to ecliptic coordinates. This can be used with position or velocity vectors.

The transformation corresponds to that between the coordinates (ra,dec) and (ecl_lon,ecl_lat) in the Gaia source catalogue (DR2).

eclToIcrs( xyz )
Converts a 3-element vector representing ecliptic coordinates to ICRS (equatorial) coordinates. This can be used with position or velocity vectors.

The transformation corresponds to that between the coordinates (ecl_lon,ecl_lat) and (ra,dec) in the Gaia source catalogue (DR2).

astromUVW( astrom6 )
Calculates Cartesian components of velocity from quantities available in the Gaia source catalogue. The output is in the same coordinate system as the inputs, that is ICRS for the correspondingly-named Gaia quantities.

The input astrometry parameters are represented by a 6-element array, with the following elements:

 index  gaia_source name  unit    description
 -----  ----------------  ----    -----------
   0:   ra                deg     right ascension
   1:   dec               deg     declination
   2:   parallax          mas     parallax
   3:   pmra              mas/yr  proper motion in ra * cos(dec)
   4:   pmdec             mas/yr  proper motion in dec
   5:   radial_velocity   km/s    barycentric radial velocity
 
The units used by this function are the units used in the gaia_source table.

This convenience function just invokes the 7-argument astromUVW function using the inverted parallax for the radial distance, and without invoking the Doppler correction. It is exactly equivalent to:

    astromUVW(a[0], a[1], a[3], a[4], a[5], 1000./a[2], false)
 
Note this naive inversion of parallax to estimate distance is not in general reliable for parallaxes with non-negligable errors.

astromUVW( ra, dec, pmra, pmdec, radial_velocity, r_parsec, useDoppler )
Calculates Cartesian components of velocity from the observed position and proper motion, radial velocity and radial distance, with optional light-time correction. The output is in the same coordinate system as the inputs, that is ICRS for the correspondingly-named Gaia quantities.

The radial distance must be supplied using the r_parsec parameter. A naive estimate from quantities in the Gaia source catalogue may be made with the expression 1000./parallax, though note that this simple inversion of parallax is not in general reliable for parallaxes with non-negligable errors.

The calculations are fairly straightforward, following Eq. 1.5.74 from the Hipparcos catalogue. A (usually small) Doppler factor accounting for light-time effects can also optionally be applied. The effect of this is to multiply the returned vector by a factor of 1/(1-radial_velocity/c), as discussed in Eq. 1.2.21 of the Hipparcos catalogue.

Note that no attempt is made to adjust for solar motion.

epochProp( tYr, astrom6 )
Propagates the astrometry parameters, supplied as a 6-element array, to a different epoch.

The input and output astrometry parameters are each represented by a 6-element array, with the following elements:

 index  gaia_source name  unit    description
 -----  ----------------  ----    -----------
   0:   ra                deg     right ascension
   1:   dec               deg     declination
   2:   parallax          mas     parallax
   3:   pmra              mas/yr  proper motion in ra * cos(dec)
   4:   pmdec             mas/yr  proper motion in dec
   5:   radial_velocity   km/s    barycentric radial velocity
 
The units used by this function are the units used in the gaia_source table.

epochPropErr( tYr, astrom22 )
Propagates the astrometry parameters and their associated errors and correlations, supplied as a 22-element array, to a different epoch.

The input and output astrometry parameters with associated error and correlation information are each represented by a 22-element array, with the following elements:

 index  gaia_source name      unit    description
 -----  ----------------      ----    -----------
   0:   ra                    deg     right ascension
   1:   dec                   deg     declination
   2:   parallax              mas     parallax
   3:   pmra                  mas/yr  proper motion in RA * cos(dec)
   4:   pmdec                 mas/yr  proper motion in Declination
   5:   radial_velocity       km/s    barycentric radial velocity
   6:   ra_error              mas     error in right ascension
   7:   dec_error             mas     error in declination
   8:   parallax_error        mas     error in parallax
   9:   pmra_error            mas/yr  error in RA proper motion * cos(dec)
  10:   pmdec_error           mas/yr  error in Declination proper motion
  11:   radial_velocity_error km/s    error in barycentric radial velocity
  12:   ra_dec_corr                   correlation between ra and dec
  13:   ra_parallax_corr              correlation between ra and parallax
  14:   ra_pmra_corr                  correlation between ra and pmra
  15:   ra_pmdec_corr                 correlation between ra and pmdec
  16:   dec_parallax_corr             correlation between dec and parallax
  17:   dec_pmra_corr                 correlation between dec and pmra
  18:   dec_pmdec_corr                correlation between dec and pmdec
  19:   parallax_pmra_corr            correlation between parallax and pmra
  20:   parallax_pmdec_corr           correlation between parallax and pmdec
  21:   pmra_pmdec_corr               correlation between pmra and pmdec
 
Note the correlation coefficients, always in the range -1..1, are dimensionless.

This is clearly an unwieldy function to invoke, but if you are using it with the gaia_source catalogue itself, or other similar catalogues with the same column names and units, you can invoke it by just copying and pasting the example shown in this documentation.

This transformation is only applicable for radial velocities determined independently of the astrometry, such as those obtained with a spectrometer. It is not applicable for the back-transformation of data already propagated to another epoch.

rvMasyrToKms( rvMasyr, plxMas )
Converts from normalised radial velocity in mas/year to unnormalised radial velocity in km/s.

The output is calculated as AU_YRKMS * rvMasyr / plxMas, where AU_YRKMS=4.740470446 is one Astronomical Unit in km.yr/sec.

rvKmsToMasyr( rvKms, plxMas )
Converts from unnormalised radial velocity in km/s to normalised radial velocity in mas/year.

The output is calculated as rvKms * plxMas / AU_YRKMS, where AU_YRKMS=4.740470446 is one Astronomical Unit in km.yr/sec.

distanceEstimateEdsd( plxMas, plxErrorMas, lPc )
Best estimate of distance using the Exponentially Decreasing Space Density prior. This estimate is provided by the mode of the PDF.

distanceBoundsEdsd( plxMas, plxErrorMas, lPc )
Calculates the 5th and 95th percentile confidence intervals on the distance estimate using the Exponentially Decreasing Space Density prior.

Note this function has to numerically integrate the PDF to determine quantile values, so it is relatively slow.

distanceQuantilesEdsd( plxMas, plxErrorMas, lPc, qpoints, ... )
Calculates arbitrary quantiles for the distance estimate using the Exponentially Decreasing Space Density prior.

Note this function has to numerically integrate the PDF to determine quantile values, so it is relatively slow.

distanceToModulus( distPc )
Converts a distance in parsec to a distance modulus. The formula is 5*log10(distPc)-5.

modulusToDistance( distmod )
Converts a distance modulus to a distance in parsec. The formula is 10^(1+distmod/5).

AU_YRKMS
This quantity is A_v, the Astronomical Unit expressed in km.yr/sec. See the Hipparcos catalogue (ESA SP-1200) table 1.2.2 and Eq. 1.5.24.

PC_AU
Parsec in Astronomical Units, equal to 648000/PI.

PC_YRKMS
Parsec in units of km.yr/sec.

C_KMS
The speed of light in km/s (exact).


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