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 )
phi (floating point): longitude in degreestheta (floating point): latitude in degreesr (floating point): radial distancepolarXYZ(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 parallaxastromXYZ( ra, dec, parallax )
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.
ra (floating point): Right Ascension in degreesdec (floating point): Declination in degreesparallax (floating point): parallax in masastromXYZ(ra, dec, parallax)
icrsToGal(astromXYZ(ra, dec, parallax))
icrsToGal( xyz )
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.
xyz (array of floating point): 3-element vector giving ICRS Cartesian componentsicrsToGal(polarXYZ(ra, dec, distance))
galToIcrs( xyz )
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.
xyz (array of floating point): 3-element vector giving Galactic Cartesian componentsgalToIcrs(polarXYZ(l, b, distance))
icrsToEcl( xyz )
The transformation corresponds to that between the coordinates
(ra,dec) and (ecl_lon,ecl_lat) in the
Gaia source catalogue (DR2).
xyz (array of floating point): 3-element vector giving ICRS Cartesian componentsicrsToEcl(polarXYZ(ra, dec, distance))
eclToIcrs( xyz )
The transformation corresponds to that between the coordinates
(ecl_lon,ecl_lat) and (ra,dec) in the
Gaia source catalogue (DR2).
xyz (array of floating point): 3-element vector giving ecliptic Cartesian coordinateseclToIcrs(polarXYZ(ecl_lon, ecl_lat, distance))
astromUVW( astrom6 )
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 velocityThe 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.
astrom6 (array of floating point): vector of 6 astrometric parameters
as provided by the Gaia source catalogueastromUVW(array(ra, dec, parallax, pmra, pmdec,
radial_velocity))
icrsToGal(astromUVW(array(ra, dec, parallax, pmra, pmdec,
radial_velocity)))
astromUVW( ra, dec, pmra, pmdec, radial_velocity, r_parsec, useDoppler )
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.
ra (floating point): Right Ascension in degreesdec (floating point): Declination in degreespmra (floating point): proper motion in RA * cos(dec) in mas/yrpmdec (floating point): proper motion in declination in mas/yrradial_velocity (floating point): radial velocity in km/sr_parsec (floating point): radial distance in parsecuseDoppler (boolean): whether to apply the Doppler factor to account
for light-time effectsastromUVW(ra, dec, pmra, pmdec,
radial_velocity, dist, true)
icrsToGal(astromUVW(ra, dec, pmra, pmdec,
radial_velocity, 1000./parallax, false))
epochProp( tYr, astrom6 )
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 velocityThe units used by this function are the units used in the
gaia_source table.
Null values for parallax, pmra,
pmdec and radial_velocity
are treated as if zero for the purposes of propagation.
The documentation of the equivalent function in the Gaia archive
comments "This is a reasonable choice for most stars because
those quantities would be either small (parallax and proper motion)
or irrelevant (radial velocity).
However, this is not true for stars very close to the Sun,
where appropriate values need to be taken from the literature
(e.g. average velocity field in the solar neighbourhood)."
The effect is that the output represents the best estimates available for propagated astrometry; proper motions, parallax and RV are applied if present, but if not the output values are calculated or simply copied across as if those quantities were zero.
tYr (floating point): epoch difference in yearsastrom6 (array of floating point): 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)t0+tYr,
represented by a 6-element array as aboveepochProp(-15.5,
array(ra,dec,parallax,pmra,pmdec,radial_velocity))
- calculates the astrometry at 2000.0 of gaia_source values
that were observed at 2015.5epochPropErr( tYr, astrom22 )
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 pmdecNote the correlation coefficients, always in the range -1..1, are dimensionless.
Null values for parallax, pmra,
pmdec and radial_velocity
are treated as if zero for the purposes of propagation.
The documentation of the equivalent function in the Gaia archive
comments "This is a reasonable choice for most stars because
those quantities would be either small (parallax and proper motion)
or irrelevant (radial velocity).
However, this is not true for stars very close to the Sun,
where appropriate values need to be taken from the literature
(e.g. average velocity field in the solar neighbourhood)."
The effect is that the output represents the best estimates available for propagated astrometry; proper motions, parallax and RV are applied if present, but if not the output values are calculated or simply copied across as if those quantities were zero.
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.
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.
tYr (floating point): epoch difference in yearsastrom22 (array of floating point): astrometry at time t0,
represented by a 22-element array as aboveepochPropErr(-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( rvMasyr, plxMas )
The output is calculated as
AU_YRKMS * rvMasyr / plxMas,
where AU_YRKMS=4.740470446
is one Astronomical Unit in km.yr/sec.
rvMasyr (floating point): normalised radial velocity, in mas/yearplxMas (floating point): parallax in masrvKmsToMasyr( rvKms, plxMas )
The output is calculated as
rvKms * plxMas / AU_YRKMS,
where AU_YRKMS=4.740470446
is one Astronomical Unit in km.yr/sec.
rvKms (floating point): unnormalised radial velocity, in mas/yearplxMas (floating point): parallax in masdistanceEstimateEdsd( plxMas, plxErrorMas, lPc )
plxMas (floating point): parallax in masplxErrorMas (floating point): parallax error in maslPc (floating point): length scale in parsecdistanceBoundsEdsd( plxMas, plxErrorMas, lPc )
Note this function has to numerically integrate the PDF to determine quantile values, so it is relatively slow.
plxMas (floating point): parallax in masplxErrorMas (floating point): parallax error in maslPc (floating point): length scale in parsecdistanceQuantilesEdsd( plxMas, plxErrorMas, lPc, qpoints, ... )
Note this function has to numerically integrate the PDF to determine quantile values, so it is relatively slow.
plxMas (floating point): parallax in masplxErrorMas (floating point): parallax error in maslPc (floating point): length scale in parsecqpoints (floating point, one or more): one or more required quantile cut points,
each in the range 0..1qpoints
giving the corresponding distance in parsecdistanceQuantilesEdsd(parallax, parallax_error,
1350, 0.5)[0]
calculates the median of the EDSD distance PDF
using a length scale of 1.35kpcdistanceQuantilesEdsd(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 3kpcdistanceToModulus( distPc )
5*log10(distPc)-5.
distPc (floating point): distance in parsecmodulusToDistance( distmod )
10^(1+distmod/5).
distmod (floating point): distance modulus in magnitudesAU_YRKMS
PC_AU
PC_YRKMS
C_KMS