Class Gaia


  • public class Gaia
    extends java.lang.Object
    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 vector calculation and manipulation
    • distance estimation from parallaxes
    • astrometry propagation to different epochs

    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

    • C.A.L.Bailer-Jones, "Estimating distances from parallaxes", PASP 127, p994 (2015) 2015PASP..127..994B
    • T.L.Astraatmadja and C.A.L.Bailer-Jones, "Estimating Distances from Parallaxes. II. Performance of Bayesian Distance Estimators on a Gaia-like Catalogue", ApJ 832, a137 (2016) 2016ApJ...832..137A
    • X.Luri et al., "Gaia Data Release 2: Using Gaia Parallaxes", A&A in press (2018) arXiv:1804.09376

    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.

    Since:
    2 Mar 2018
    Author:
    Mark Taylor
    • Field Summary

      Fields 
      Modifier and Type Field Description
      static double AU_YRKMS
      This quantity is A_v, the Astronomical Unit expressed in km.yr/sec.
      static double C_KMS
      The speed of light in km/s (exact).
      static double PC_AU
      Parsec in Astronomical Units, equal to 648000/PI.
      static double PC_YRKMS
      Parsec in units of km.yr/sec.
    • Method Summary

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method Description
      static double[] astromUVW​(double[] astrom6)
      Calculates Cartesian components of velocity from quantities available in the Gaia source catalogue.
      static double[] astromUVW​(double ra, double dec, double pmra, double pmdec, double radial_velocity, double r_parsec, boolean useDoppler)
      Calculates Cartesian components of velocity from the observed position and proper motion, radial velocity and radial distance, with optional light-time correction.
      static double[] astromXYZ​(double ra, double dec, double parallax)
      Calculates Cartesian components of position from RA, Declination and parallax.
      static double[] distanceBoundsEdsd​(double plxMas, double plxErrorMas, double lPc)
      Calculates the 5th and 95th percentile confidence intervals on the distance estimate using the Exponentially Decreasing Space Density prior.
      static double distanceEstimateEdsd​(double plxMas, double plxErrorMas, double lPc)
      Best estimate of distance using the Exponentially Decreasing Space Density prior.
      static double[] distanceQuantilesEdsd​(double plxMas, double plxErrorMas, double lPc, double... qpoints)
      Calculates arbitrary quantiles for the distance estimate using the Exponentially Decreasing Space Density prior.
      static double distanceToModulus​(double distPc)
      Converts a distance in parsec to a distance modulus.
      static double[] eclToIcrs​(double[] xyz)
      Converts a 3-element vector representing ecliptic coordinates to ICRS (equatorial) coordinates.
      static double[] epochProp​(double tYr, double[] astrom6)
      Propagates the astrometry parameters, supplied as a 6-element array, to a different epoch.
      static double[] epochPropErr​(double tYr, double[] astrom22)
      Propagates the astrometry parameters and their associated errors and correlations, supplied as a 22-element array, to a different epoch.
      static double[] galToIcrs​(double[] xyz)
      Converts a 3-element vector representing galactic coordinates to ICRS (equatorial) coordinates.
      static double[] icrsToEcl​(double[] xyz)
      Converts a 3-element vector representing ICRS (equatorial) coordinates to ecliptic coordinates.
      static double[] icrsToGal​(double[] xyz)
      Converts a 3-element vector representing ICRS (equatorial) coordinates to galactic coordinates.
      static double modulusToDistance​(double distmod)
      Converts a distance modulus to a distance in parsec.
      static double[] polarXYZ​(double phi, double theta, double r)
      Converts from spherical polar to Cartesian coordinates.
      static double rvKmsToMasyr​(double rvKms, double plxMas)
      Converts from unnormalised radial velocity in km/s to normalised radial velocity in mas/year.
      static double rvMasyrToKms​(double rvMasyr, double plxMas)
      Converts from normalised radial velocity in mas/year to unnormalised radial velocity in km/s.
      • Methods inherited from class java.lang.Object

        clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
    • Field Detail

      • AU_YRKMS

        public static final double 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.
        See Also:
        Constant Field Values
      • PC_AU

        public static final double PC_AU
        Parsec in Astronomical Units, equal to 648000/PI.
        See Also:
        Constant Field Values
      • PC_YRKMS

        public static final double PC_YRKMS
        Parsec in units of km.yr/sec.
        See Also:
        Constant Field Values
      • C_KMS

        public static final double C_KMS
        The speed of light in km/s (exact).
        See Also:
        Constant Field Values
    • Method Detail

      • polarXYZ

        public static double[] polarXYZ​(double phi,
                                        double theta,
                                        double r)
        Converts from spherical polar to Cartesian coordinates.
        Parameters:
        phi - longitude in degrees
        theta - latitude in degrees
        r - radial distance
        Returns:
        3-element vector giving Cartesian coordinates
        Examples:
        polarXYZ(ra, dec, distance_estimate), polarXYZ(l, b, 3262./parallax) - calculates vector components in units of light year in the galactic system, on the assumption that distance is the inverse of parallax
      • astromXYZ

        public static double[] astromXYZ​(double ra,
                                         double dec,
                                         double 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.

        Parameters:
        ra - Right Ascension in degrees
        dec - Declination in degrees
        parallax - parallax in mas
        Returns:
        3-element vector giving equatorial space coordinates in parsec
        Examples:
        astromXYZ(ra, dec, parallax), icrsToGal(astromXYZ(ra, dec, parallax))
      • icrsToGal

        public static double[] icrsToGal​(double[] 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.

        Parameters:
        xyz - 3-element vector giving ICRS Cartesian components
        Returns:
        3-element vector giving Galactic Cartesian components
        Examples:
        icrsToGal(polarXYZ(ra, dec, distance))
      • galToIcrs

        public static double[] galToIcrs​(double[] 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.

        Parameters:
        xyz - 3-element vector giving Galactic Cartesian components
        Returns:
        3-element vector giving ICRS Cartesian components
        Examples:
        galToIcrs(polarXYZ(l, b, distance))
      • icrsToEcl

        public static double[] icrsToEcl​(double[] 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).

        Parameters:
        xyz - 3-element vector giving ICRS Cartesian components
        Returns:
        3-element vector giving ecliptic Cartesian components
        Examples:
        icrsToEcl(polarXYZ(ra, dec, distance))
      • eclToIcrs

        public static double[] eclToIcrs​(double[] 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).

        Parameters:
        xyz - 3-element vector giving ecliptic Cartesian coordinates
        Returns:
        3-element vector giving ICRS Cartesian coordinates
        Examples:
        eclToIcrs(polarXYZ(ecl_lon, ecl_lat, distance))
      • astromUVW

        public static double[] astromUVW​(double[] 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.
        Parameters:
        astrom6 - vector of 6 astrometric parameters as provided by the Gaia source catalogue
        Returns:
        3-element vector giving equatorial velocity components in km/s
        Examples:
        astromUVW(array(ra, dec, parallax, pmra, pmdec, radial_velocity)), icrsToGal(astromUVW(array(ra, dec, parallax, pmra, pmdec, radial_velocity)))
      • astromUVW

        public static double[] astromUVW​(double ra,
                                         double dec,
                                         double pmra,
                                         double pmdec,
                                         double radial_velocity,
                                         double r_parsec,
                                         boolean 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.

        Parameters:
        ra - Right Ascension in degrees
        dec - Declination in degrees
        pmra - proper motion in RA * cos(dec) in mas/yr
        pmdec - proper motion in declination in mas/yr
        radial_velocity - radial velocity in km/s
        r_parsec - radial distance in parsec
        useDoppler - whether to apply the Doppler factor to account for light-time effects
        Returns:
        3-element vector giving equatorial velocity components in km/s
        Examples:
        astromUVW(ra, dec, pmra, pmdec, radial_velocity, dist, true), icrsToGal(astromUVW(ra, dec, pmra, pmdec, radial_velocity, 1000./parallax, false))
      • epochProp

        public static double[] epochProp​(double tYr,
                                         double[] 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.
        Parameters:
        tYr - epoch difference in years
        astrom6 - astrometry at time t0, represented by a 6-element array as above (a 5-element array is also permitted where radial velocity is zero or unknown)
        Returns:
        astrometry at time t0+tYr, represented by a 6-element array as above
        Examples:
        epochProp(-15.5, array(ra,dec,parallax,pmra,pmdec,radial_velocity)) - calculates the astrometry at 2000.0 of gaia_source values that were observed at 2015.5
      • epochPropErr

        public static double[] epochPropErr​(double tYr,
                                            double[] 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.

        Parameters:
        tYr - epoch difference in years
        astrom22 - astrometry at time t0, represented by a 22-element array as above
        Returns:
        astrometry at time t0+tYr, represented by a 22-element array as above
        Examples:
        epochPropErr(-15.5, array( ra,dec,parallax,pmra,pmdec,radial_velocity, ra_error,dec_error,parallax_error,pmra_error,pmdec_error,radial_velocity_error, ra_dec_corr,ra_parallax_corr,ra_pmra_corr,ra_pmdec_corr, dec_parallax_corr,dec_pmra_corr,dec_pmdec_corr, parallax_pmra_corr,parallax_pmdec_corr, pmra_pmdec_corr)) - calculates the astrometry with all errors and correlations at 2000.0 for gaia_source values that were observed at 2015.5.
      • rvMasyrToKms

        public static double rvMasyrToKms​(double rvMasyr,
                                          double 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.

        Parameters:
        rvMasyr - normalised radial velocity, in mas/year
        plxMas - parallax in mas
        Returns:
        radial velocity in km/s
      • rvKmsToMasyr

        public static double rvKmsToMasyr​(double rvKms,
                                          double 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.

        Parameters:
        rvKms - unnormalised radial velocity, in mas/year
        plxMas - parallax in mas
        Returns:
        radial velocity in mas/year
      • distanceEstimateEdsd

        public static double distanceEstimateEdsd​(double plxMas,
                                                  double plxErrorMas,
                                                  double lPc)
        Best estimate of distance using the Exponentially Decreasing Space Density prior. This estimate is provided by the mode of the PDF.
        Parameters:
        plxMas - parallax in mas
        plxErrorMas - parallax error in mas
        lPc - length scale in parsec
        Returns:
        best distance estimate in parsec
      • distanceBoundsEdsd

        public static double[] distanceBoundsEdsd​(double plxMas,
                                                  double plxErrorMas,
                                                  double 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.

        Parameters:
        plxMas - parallax in mas
        plxErrorMas - parallax error in mas
        lPc - length scale in parsec
        Returns:
        2-element array giving the 5th and 95th percentiles in parsec of the EDSD distance PDF
      • distanceQuantilesEdsd

        public static double[] distanceQuantilesEdsd​(double plxMas,
                                                     double plxErrorMas,
                                                     double lPc,
                                                     double... 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.

        Parameters:
        plxMas - parallax in mas
        plxErrorMas - parallax error in mas
        lPc - length scale in parsec
        qpoints - one or more required quantile cut points, each in the range 0..1
        Returns:
        array with one element for each of the supplied qpoints giving the corresponding distance in parsec
        Examples:
        distanceQuantilesEdsd(parallax, parallax_error, 1350, 0.5)[0] calculates the median of the EDSD distance PDF using a length scale of 1.35kpc, distanceQuantilesEdsd(parallax, parallax_error, 3000, 0.01, 0.99) returns a 2-element array giving the 1st and 99th percentile of the distance estimate using a length scale of 3kpc
      • distanceToModulus

        public static double distanceToModulus​(double distPc)
        Converts a distance in parsec to a distance modulus. The formula is 5*log10(distPc)-5.
        Parameters:
        distPc - distance in parsec
        Returns:
        distance modulus in magnitudes
      • modulusToDistance

        public static double modulusToDistance​(double distmod)
        Converts a distance modulus to a distance in parsec. The formula is 10^(1+distmod/5).
        Parameters:
        distmod - distance modulus in magnitudes
        Returns:
        distance in parsec