Analysis of SN1006 GBT observations – combination with VLA

Tim Cornwell, Kristy Dyer, Ron Maddalena

 

SN1006 is about 30 arcmin in diameter – the same size as the FWHM of the VLA primary beam at 20cm. To image it correctly at arc-second resolution at 20cm requires VLA observations in A, B, C, and D configurations, supplemented by GBT continuum observations to get the large-scale structure.

 

Calibration and background removal is described here

 

There are many ways to combine single dish and interferometer data. The main possibilities are:

 

• Merge: Add the images after deconvolution of the respective point spread functions (GBT primary beam and VLA synthesized beam).
• Fake Synthesis: Make fake synthesis data from the deconvolved single dish image, add it to the real synthesis data and deconvolve together.
• Joint deconvolution: Perform a joint deconvolution of the single dish and synthesis data.
• Single dish as starting model: Use the deconvolved single dish image as the starting point in a deconvolution of the synthesis data. For Clean, we would use it as the initial image, and for MEM, we would use it as the default image.

 

Since the deconvolution process is non-linear, the merge technique does not take optimum advantage of the entire data set. The fake synthesis approach is possible, and joint deconvolution is known to produce very good results on simulated data, our software does not easily support either. For this reason, we adopt the single dish as starting model approach.

 

First we need to deconvolve the GBT primary beam. Although the GBT primary beam is known to be very good, there are surprisingly few measurements. We have therefore used an analytical beam corresponding to Gaussian taper of the unblocked aperture with 15dB taper at the edge of the aperture. We have to be concerned about the OTF effects. In these measurements, the GBT was set to scan at 600 arcmin / minute, and the dump time was 100msec. Hence the telescope moves an arc-minute during one integration. This will smear the image in Right Ascension by an arc-minute, or 1/3 grid sampling. Since the primary beam of the GBT is about 8.5 arcmin, this is a very small effect, but we include it anyway. We calculate the effective beam via an AIPS++ script. This also calculates an image of the corresponding voltage pattern that can used during the gridding and degridding process instead of the canned primary beam in AIPS++. In the current implementation of single dish gridding, the convolution function is constrained to be a function of radius only. However, the smearing effect is more important in the deconvolution process.

 

To make the single dish image, we have a number of possibilities for the gridding function used to put the samples onto a regular grid.

 

• BOX: nearest neighbor gridding – a sample is moved to the nearest grid point. This works well if the data are actually sampled on a regular grid in RA, Dec. Synthesis types like to think of this approach as an analog to uniform weighting – the noise is highest and the resolution finest. The disadvantage is that the image must have the same sampling as the data.
• PB: primary beam gridding – the samples are convolved with a model of the primary beam and then resampled on the desired grid. This is an analog to synthesis natural weighting - it produces the image with the least variance but low resolution. The image and data sampling need not be the same in this case.
• SF: spheroidal function gridding – the samples are convolved with a spheriodal function, fixed in pixel space, and then resampled on the desired grid. We choose a spheroidal function since it is limited in extent but also reasonably smooth (superior to, for example, a truncated Gaussian).

 

The GBT can observe an RA, Dec grid so BOX gridding is an easy choice when no deconvolution is required. Below we show our GBT image of SN1006 made using BOX gridding:

 

If the data and sample grids are the same, then the relationship between the true sky I and the BOX image is the classic convolution equation:

 

where P is the GBT primary beam and N is the additive noise. For the PB and SF images:

 

 

where G is the spheroidal function.

 

The AIPS++ deconvolver tool can solve these types of equation using Maximum Entropy (script). The resulting images (shown below – first BOX and then PB) have finer resolution, the primary beam having been removed, but there is a trough around the remnant. The trough is not too troublesome since it corresponds to the spacings beyond the maximum sampling by the GBT for which the VLA D configuration will help.

 

 

 

 

Now that we have a model for the short spacings (from a few meters up to 100m), we can check the relative calibration between GBT and VLA. For this we must predict model visibilities from the GBT image after tapering with the VLA primary beam (script). In the next figure, we plot the ratio of the observed VLA visibilities to those predicted from the deconvolved GBT image.

 

 

 

At less than about 30m and more than 45m, the visibility function tends close to zero and so the errors in the model are magnified. In the well-determined range, 35 – 40m, the relative calibration can be seen to be very good, within a few percent of unity.

 

The final deconvolved image is obtained by Multi-Scale CLEAN, starting from the deconvolved GBT image (script).