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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 two main sets of functions here, distance estimation from parallaxes, and astrometry propagation to different epochs.

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.

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.

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.


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