Doppler Tracking with Multiple Spectral Windows

Doppler Tracking with Multiple Spectral Windows on the GBT

The GBT allows multiple spectral windows to be observed simultaneously, a very useful feature for spectral line studies. But with the present LO/IF system, all windows are not equal. These inequalities can be ignored for Galactic objects, with velocities less than a few hundred km/sec, but for much larger velocities, one must understand the complications.

The Basics

The tracking LO
The RF from the sky is mixed with a first LO reference frequency, known as "LO1". The LO1 synthesizer can be tuned from 0 to 20 GHz in steps of 1 Hz. For the 26-40 GHz receiver, LO1 is tripled, and for the 40-50 GHz receiver LO1 is quadrupled.
The software that drives LO1 adjusts the frequency in real time in order to track a specified velocity. It understands 3 velocity definitions (Radio, Optical, and Relativistic), and several reference frames (e.g., Local, Barycentric, LSR, etc). Refer to the GBT Doppler Tracking Page for descriptions of the various frames and definitions.
The Restfreq configuration list
In the GBT Configuration tool, one may specify a list of rest frequencies as follows:
restfreq = freq1, freq2, freq3, ...
The first frequency in the list is special. It is the rest frequency given to the LO1 software. Thus LO1 will correctly track the first spectral line. The Configuration tool will set up the frequency bands in the hardware such that the specified frequencies will be centered in the bands.
The second LOs
After the first LO mix, which occurs in the front end receiver box, the IF band comes to the GBT equipment room on fibers. Here it is split into up to 8 dual-polarization sub-bands. Secondary LOs, called "LO2s" allow tuning of these sub-bands independently of each other.
Configuration vs Scans
The Configuration tool sets up all the parameters for the GBT systems including all the LOs and connections. It attempts to put each rest frequency specified by the "restfreq" keyword to approximately the center of each spectral window, using the specified velocity frame and definition, and using the velocity equal to the average of "vhigh" and "vlow". After the configuration, the frequency differences between the spectral windows remain fixed with respect to each other in the local frame.
Typically, a user does a configuration very seldom and does many scans without changing the configuration. During a scan, the LO manager changes LO1 in real time in order to track the velocity of the source. This velocity is usually given in the source catalog.
Thus, while LO1 is tracking, the center frequencies of all the spectral windows are changing. But the LO2s stay fixed, so that the frequency differences between the windows remain fixed, in the local frame.

So how about some specific examples !

Let's assume the following Configuration keywords:
nwin = 4 restfreq = 1420.4057, 1665.4018, 1667.3590, 1720.5300 vlow = 60000 vhigh = 90000 vdef = 'Radio' vframe = 'lsrk'
and define the following:
FR1 = rest frequency number 1 (=1420.4057 MHz) FRn = rest frequency number n (n = 2,3,4) Vav = 0.5*(vlow + vhigh) = 75,000 km/sec Vs = the velocity of a source being observed, which may be different from Vav. local frequency f(V) = f0(1 - V/c) using the Radio definition. floc1 = local center frequency for spectral window number 1 flocn = local center frequency for spectral window number n
the Configuration tool will set LO1 and LO2 such that:
floc1 = FR1 * ( 1 - (Vav /c)) flocn = FRn * ( 1 - (Vav /c))
Note that the Config tool has not used the velocity frame! When the tracking scan starts, the LO1 manager changes the local center frequency to the correct value for the specified frame. The local frequencies of windows 2,3,4,... shift accordingly.
Now What?
If the user now wants to track a velocity Vs, then local frequency n should be : flocn' = FRn * (1 - (Vs /c)) but it actually is : flocn'' = FR1(1 - (Vs /c)) + (flocn-floc1) so the error in the band center is : Efn = flocn' - flocn'' and that simplifies to: Efn = (FRn - FR1) (Vav - Vs)/c** (and I have assumed the part due to the velocity frame has canceled out -- not strictly true!) So for example, if FR1=1420.4057; FRn=1720.53; Vav=75,000; Vs=85,000 then Efn = 10 MHz So if your bandwidth is 12.5 MHz, then the line is shifted out of the window.
So the lesson is:
If you want your line to be positioned within 0.1 of the center of the band, then you should re-Configure whenever *Efn > 0.1bandwidth** (set Config keywords: vlow=vhigh=Vs) Conversely, if you want the frequency error Efn to be < 1 MHz, then the change in velocity (Vav - Vs) should be <1000 km/sec
What about GBTIDL??
The local frequencies are stored with the FITS data files, so if you give GBTIDL the rest frequency, it will display the correct velocity. It will take account of the specified velocity frame and definition. None of the foregoing remarks affect the accuracy of the velocity information provided by GBTIDL; they only relate to the centering of a line in a spectral window, or lack of it.
What if we had used the "Optical" definition?
In this case, f(V) = f0/(1+V/c) so *Efn = (FRn - FR1) (Vav - Vs) / {c * (1+(Vs/c))(1+(Vav/c))}** and for the example: if FR1=1420.4057; FRn=1720.53; Vav=75,000; Vs=85,000 then Efn = 6.2 MHz
That's all there is to it, folks !