GBT LO Doppler tracking

by F. Ghigo, NRAO-Green Bank, August 2002
Note revisions as of May 2010, below.

The GBT LO system (see description of LO1 FITS file) provides for adjusting the local observing frequency to track the velocity of an astronomical object.

The user needs to specify:

Note handy calculator for various reference frames: Radial Velocity Calculator

The user also specifies a frequency tolerance (Ftol), i.e., by how many Hz must the topocentric frequency change before the LO is updated.

These four parameters are described in the following paragraphs.

Velocity reference frames.

The reference frames available to the GBT user are described in Table 1. The definitions of kinematical LSR, dynamical LSR, and galactocentric are the same as implemented in the Starlink package and in the "measures" utility in aips++.

Note that aips++ does not support Heliocentric, Local Group, or CMB frames. These frames should be avoided if you are using aips++ to display and process your data.

The JPL ephemeris is used to calculate the motion of the solar system barycenter and of the center of the sun (Heliocentric). Heliocentric velocities differ by no more than 0.02 km/sec from the solor system barycenter. For many purposes they can be regarded as equivalent. In general the barycenter is preferred, since it is the best approximation to an inertial reference frame in the solar system vicinity.

For the other frames of reference, the "Definition" column in Table 1 gives the definition of this frame as given in the literature. The "J2000 Vector" column gives the velocity vector between the solar system barycenter and the frame in terms of J2000 coordinates. For convenience, this J2000 vector is what is actually used by the LO1 system to convert between the specified frame and the barycenter.

Table 1: Velocity Frames
These are the frames implemented in the GBT LO1 manager as of 2002.
Frame Description Definition J2000 Vector
from Barycenter
Local (Topocentric) The LO is fixed: no tracking GBT Location:
38° 25'59.23"N;
79° 50'23.40"W;
Height: 855.6m
--- ---
Barycentric Solar System Barycenter JPL Ephemeris DE403 --- ---
Heliocentric Center of Sun JPL Ephemeris DE403 --- ---
Kinematic LSR Conventional Local Standard of Rest based on average velocity of stars in the Solar neighborhood Solar motion = 20.0 km/sec towards (18h +30°) at epoch 1900.0 20.0 km/sec towards (18h03m50.29s, +30°00'16.8") Gordon (1975)
Dynamical LSR Solar peculiar velocity with respect to a frame in circular motion about the galactic center Solar motion vector = (-9, +12, +7) km/sec in Galactic cartesian coordinates. 16.55294 km/sec towards (17h49m58.667s, +28°07'03.96") Delhaye (1965)
Galactocentric Dynamical center of the galaxy. The Dynamical LSR moves 220 km/sec towards ( l=90°, b=0°) 232.3 km/sec towards (20h55m26.77s, +47°49'23.5") Kerr and Lynden-Bell (1986)
Local Group Mean motion of Local Group Galaxies Solar Motion =
308 km/sec towards (l=105°, b=-7°)
308 km/sec towards (22h53m14.55s, +51°42'32.2") Yahil et al. 1977
Cosmic Microwave Background COBE measurements of dipole anisotropy Solar motion =
369.5 km/sec towards (l=264.4°, b=48.4°)
369.5 km/sec towards (11h12m56.40s -06°57'50.0") Kogut et al. 1993
(Note added May 2010: the GBT LO1 has not been using the Yahil et al Local Group definition, but the deVaucouleurs 1976 version listed in Table 5.)

Table 2: References
Delhaye (1965) "Solar Motion and Velocity Distribution of Common Stars," by J. Delhaye, pages 73-74, in "Stars and Stellar Systems, Volume 5: Galactic Structure", ed. Blaauw and Schmidt, Univ. of Chicago Press (1965).
Gordon (1975) "Computer Programs for Radio Astronomy," by M.A.Gordon, page 281, in
" Methods of Experimental Physics: Volume 12: Astrophysics, Part C: Radio Observations", ed. M.L.Meeks, Academic Press 1976.
Kerr and Lynden-Bell (1986) "Review of galactic constants", by F.J.Kerr and D. Lynden-Bell.
Mon.Not.Roy.Astron.Soc. vol.221, p.1023 (1986)
Kogut et al. (1993) "Dipole Anisotropy in the COBE Differential Microwave Radiometers First-year Sky Maps," by A. Kogut, C. Lineweaver, G. F. Smoot, C. L. Bennett, A. Banday, N. W. Boggess, E.S.Cheng, G. De Amici, D.J.Fixsen, G.Hinshaw, P.D.Jackson, M.Janssen, P.Keegstra, K. Loewenstein, P.Lubin, J.C.Mather, L.Tenorio, R.Weiss, D.T.Wilkinson, and E.L.Wright.
Astrophys.J. vol.419, p.1, December 10, 1993.
Yahil et al. (1977) "The Local Group: the Solar Motion Relative to its Centroid," A.Yahil, G.A. Tammann, and Allan Sandage, Astrophys.J. vol.217, p.903, November 1, 1977.

Velocity Definitions.

The user may specify the "velocity definition" to be 'Radio', 'Optical', 'Relativistic', or 'Redshift'. These actually specify the convention to be used in converting velocity to frequency.

Table 3 summarizes these conventions. 'Redshift' means the user specifies the redshift parameter 'z'; this implies the optical convention with V = cz. f0 is the rest frequency. V is the velocity component along the line of sight to the object.

The sign convention for velocity is such that V>0 for recession, i.e., increasing velocity means decreasing frequency.

Table 3: Velocity Definitions.
Velocity Definition Velocity Formula Conversion to Frequency
Radio V = c (f0 - f)/f0 f(V) = f0 ( 1 - V/c )
Optical V = c (f0 - f)/f f(V) = f0 ( 1 + V/c )-1
Redshift z = (f0 - f)/f f(V) = f0 ( 1 + z )-1
Relativistic V = c (f02 - f 2)/(f02 + f 2) f(V) = f0 { 1 - (V/c)2}1/2/(1+V/c)

The full special relativistic conversion is given by:
f = f0 { 1 - (V/c)2 }1/2 / (1 + V· S /c)

The formula given in the table is the case in which the velocity is along the line of sight, i.e, V = V· S , because in most cases we do not know the transverse velocity of the source.

Calculation of tracking frequency

The LO system transforms the rest frequency f0 to the GBT topocentric frame, given parameters Vrad, Vref, and Vdef supplied by the user. First, the rest frequency is converted according to one of the formulas given in Table 3 for the specified Vdef:
f1 = f(Vrad)

Next, the vector velocity Vb of the solar system barycenter with respect to the local (GBT) frame is calculated using the JPL ephemeris.
The velocity Vf of the specified frame with respect to the barycenter is taken from Table 1 ("J2000 Vector from Barycenter").
The resulting Vframe (topocentric to Vref) is given by Vframe = Vb + Vf .

The conversion to topocentric frequency Ftop is then calculated with the relativistic formula:

Ftop = f1{ 1 - (Vframe/c)2}1/2/(1 + Vframe· S/c)

Where Vframe is the magnitude of the vector Vframe.

Selection of Frequency tolerance

The LO system will adjust the frequency in steps of the frequency tolerance, Ftol, which is specified by the user. The minimum possible Ftol is 1 Hz. In general one should use the largest Ftol that is consistent with the desired velocity resolution.

For low velocities, the relation between a frequency change DF and the corresponding velocity change DV is given by:

(1) DF = (f0/c) DV

In general, using the relativistic formula:

(2) DF = (f0/c) DV (1+b)-1 (1 - b2)-1/2

  • (where b = V/c)
  • One can use the following graphs to select Ftol. DF is plotted versus rest Frequency f0 for two cases of velocity resolution DV=10m/s and DV=1m/s and for b = 0.0 and 0.5.


    Updated Reference Frames and alternates

    May 2010.

    Table 5. Reference Frames and alternates
    Frame Definition Galactic (l2,b2) J2000 α,&delta
    wrt Bary
    J2000 cartesian
    wrt Bary
    used ?
    kinematic LSR Solar motion 20.0 km/sec towards (18h +30°), epoch 1900.0
    (Gordon 1975)
    56.16°, +22.76° 20.0km/s, 18:03:50.24, +30:00:16.8 (0.28998, -17.31727, 10.00141) km/sec GBT/LO1; slalib
    dynamic LSR Solar motion 16.553 km/sec; vector=(+9,+12,+7) in Galactic cartesian coordinates.
    (Delhaye 1965)
    53.13°, +25.17° 16.55km/s, 17:49:58.66, +28:07:03.92 (-0.63823, -14.58542, +7.80116) GBT/LO1; slalib
    Galactocentric Dynamical LSR moves 220km/sec towards l2,b2=90,0
    (Kerr&Lynden-Bell 1986)
    90°, 0° 232.3km/s, 20:55:26.77, +47:49:23.5 (108.06585 -112.44793 172.13725) GBT/LO1; slalib
    Galactocentric Dynamical LSR moves 254km/sec towards l2,b2=90,0
    (Reid etal 2009)
    90°, 0° 266.2km/s, 20:57:32.57, +47:53:46.0 (124.86557 -127.57214 197.53465)
    Local Group Solar Motion 300km/sec towards l2,b2=(90,0)
    (deVaucouleurs, IAU 1976)
    90°, 0° 300km/sec, 21:12:01.05, 48:19:46.71 (148.23284 -133.44888 224.09467) GBT/LO1; slalib
    Local Group Solar Motion 308km/sec towards l2,b2=(105,-7)
    (Yahil et al 1977)
    105°, -7° 308km/sec, 22:53:14.58, 51:42:32.20 (182.81476 -54.80956 241.74092)
    Local Group Solar Motion 306km/sec towards l2,b2=(99,-4)
    (Courteau&vandenBergh 1999)
    99°, -4° 306km/sec, 22:10:24.03, 51:13:54.19 (170.11341 -88.17782 238.58352)
    Cosmic microwave background (COBE) Solar motion 369.5km/sec towards l2,b2=264.4,48.4.
    COBE (Kogut et al 1993)
    264.4°, 48.4° 369.5km/s, 11:12:56.43, -06:57:50.0 (-359.06915 74.78365 -44.79956) GBT/LO1
    Cosmic microwave background (WMAP) Solar motion 368.0km/sec towards l2,b2=263.85,48.25
    WMAP (Bennett et al 2003)
    263.85°, 48.25° 368km/s, 11:11:22.92, -06:52:57.02 (-357.15833 76.92350 -44.09881)

  • WMAP (Bennett 2003) gives a dipole amplitude 3.346mK towards (l,b)=263.85,48.25) and Tcmb = 2.725K; hence velocity = c(0.003346/2.725) = 368 km/sec.
  • Galactocentric (1986): J2000 vector wrt LSRD is 108.7041 -97.8625 164.3361; add this to the LSRD vector to get the result in the table for the j2000 wrt barycentric.
  • Galactocentric (2009): J2000 vector wrt LSRD is 125.5038 -112.9867 189.7335; add this to the LSRD vector to get the result in the table for the j2000 wrt barycentric.

    Table 6. References
    FITS WCS paper 3: Greisen et al.(A&A 446: 747, 2006) pdf
    WMAP: Bennett et al 2003 (ApJSupp 148:1) pdf
    Local Group: Courteau & van den Bergh 1999 (AJ 118: 337) pdf
    Local Group: deVaucouleurs 1976 : IAU Transactions 16B, 201.
    Galactic: Reid et al 2009 (ApJ 700: 137) pdf