GBT LO Doppler tracking

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:

• the radial velocity (Vrad) of the object,
• the velocity definition (Vdef), and
• the reference frame (Vref) to which the radial velocity refers.

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.

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	Reference
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.

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)

• f0 is the frequency of radiation emitted by a source moving at a velocity V with respect to the observer.
• V is the velocity vector, V = (Vx, Vy, Vz).
• S is the unit vector in the direction of the source.
• V is the magnitude of the vector V.

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

• Vrad = radial velocity of source in frame Vref.
• Vdef = one of the velocity definitions given in Table 3.
• Vref = one of the frames listed in Table 1.

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

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

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.