al, please find appended my analysis of mark's simulations. these are *not* the surface error ones, because as far as i know he hasn't gotten those to work yet. i also haven't gotten to the IRAM sims yet for comparison, but will attempt to over the weekend. -bryan 'science' analysis of the debris disk model. bjb. 2001aug21-2001sep27 I. checking out the model. the observing and disk geometry parameters: distance = 12 pc radius = 100 AU inclination angle = 45 deg frequency = 345 GHz the total flux density in the image is 10.3 mJy. the flux density in the central pixel is 240.2 uJy. this is almost exactly what i get for the sun at 12 pc: F* ~ 6 Teff Rs^2 / Dpc^2 uJy where Teff is the effective temperature, Rs is the radius in solar radii, and Dpc is the distance in parsecs. let Teff = 6000 K; Rs = 1; Dpc = 12, then F* ~ 250 uJy so this seems right... now, assume that temperature, emissivity, and opacity don't vary in the debris disk. then, the total emission is: Fd = t e B O where t = dust opacity, e = dust emissivity, B = dust brightness O = solid angle of disk. given a disk of radius Rd at inclination angle p, the solid angle is of course O ~ pi cos(p) Rd^2 / D^2. the brightness is just the planck equation. the effective opacity is t = t' / cos(p), where t' is the opacity that would be observed if it were face-on (note that this causes the inclination angle factors [cos(p)] to cancel out - as expected...). dust emissivity is roughly .015 at 345 GHz (very rough, but good enough for this kind of order-of-magnitude analysis). opacities vary from something like 1E-3 (beta-pic) to 1E-7 (our own solar system's zodiacal dust). so, given lee's disk geometry, and a pretty dusty disk (t' = 1E-3), for a temperature of 50 K (the bulk of the emission comes from the cold dust, because it occupies the bulk of the solid angle), you get: F ~ 11.9 mJy so, 10 mJy in the model is not so far from reality. but it *is* still a pretty dusty system. a more 'typical' debris disk dust opacity might be something like 1E-5, and slightly colder temperatures (20 K), giving a total flux density more like 50 uJy (factor of 100 less emission). so, i'd say that this is a good model for a very dusty system. if you want more typical systems, reduce the flux density numbers by a factor of 10-100. II. the simulations/reconstructions A. holdaway's simulations 1. restrict myself to 345 GHz, since that is where the model was really calculated anyway (see lee's model description). 2. retrieve the necessary files from mark's website: DEBRIS.1.345.MODEL2 - the model DEBRIS.1.345.1.CVM2 - image1 - 12 m INT; 12 m TP DEBRIS.2.345.1.CVM2 - image2 - 12 m INT; 12 m TP; 7 m INT DEBRIS.3.345.1.CVM2 - image3 - 12 m INT; 7 m INT; 7 m TP FLUXSCALE - the 'sault weighting' correction map note that you don't need all 3 different '*.MODEL2' files, because they are identical. 3. convolve the model image with the appropriate gaussian beam (task CONVL in AIPS, with BMAJ=1;BMIN=.91;BPA=83.39). 4. make the 'sault weighting' correction to the convolved model image (task COMB in AIPS, with OPCOD='MULT'; APARM=1,0; input map # 1 is the result from step 3 above; input map # 2 is the FLUXSCALE image). 5. fix the 'flux density scale'. because mark fudged the flux density in the model, i need to make a correction for that. i thought that at one point mark had said (during one of the imcal telecons) that the multiplication factor was 10000, but it seems to me that that's not right. for example, consider the central pixel value (the max in the images, due to the star emission). in the original input model, the value of this pixel is 240.2 uJy/pixel. in mark's input model, the value of this pixel is 9.30 mJy/pixel. the ratio of these two is 38.7, not 10000. but, the problem here is in the resampling, since the original input model had pixel size of 16.67 masec/pixel, and mark's has 143.2 masec/pixel, or, a factor of 100 larger pixels. what happens if you convolve both the original model and mark's model with the beam of the reconstructed images and compare? since this beam size (1.0 X 0.91 asec) is much larger than either pixel size, the differing pixel size in the two inputs to the convolution shouldn't matter. in the original model (convolved), the central pixel value goes to 0.5447 uJy/pixel. in mark's model (convolved), the central pixel value goes to 214.4 uJy/pixel. now, multiply these by pixels/beam ({ pi*bmaj*bmin/[4*ln2] / pixsiz }^2), and the convolved original has a max of 2.085 mJy/bm, and mark's convolved has a max of 11.12 mJy/bm. so, it looks to me like you only have to divide the reconstructed images (and mark's convolved model) by a factor of 11.12/2.085 = 5.33. use COMB in AIPS to do this. this is a *serious* confusion, and is a problem with the way that mark did his simulations. why not just take the flux density as given by the model? why does he have to scale it at all? harumph... 6. make difference images difference image = reconstruction - model 7. make fractional error images i don't know what is the 'standard' for definition of fidelity any more, so i used fractional error instead (i like it better anyway): difference image fractional error image = ------------------ model this is like 1/fidelity, i think, as it is being defined by most folks. i used a cutoff of 27 uJy/bm in the model image, which is 0.25% of the peak, roughly. i found this by trial and error, just messing around with the value until it gave somewhat 'reasonable' resultant images. 8. qualititative assessment of the reconstructed images. the first thing to say is that the reconstructed images all look pretty good, to first order. in all 3 of them, you can see the inner gap, inner rings, middle gap, outer ring, and both 'condensations' (and associated debris pileups). of the 3 simulations, the 2nd is the best, however. you can see that it gets the stuff in the inner portion of the disk much better than the other 2. also, the 'junk' at the upper left is of smaller amplitude. image 1, in the fractional error image, displays a large scale error where to the upper left, it's positive-biased, and to the lower right, it's negative-biased. the other 2 seem to do a much better job of spreading the errors around more evenly. 9. quantitative assessment of the reconstructed images. a) statistical measures first, some raw statistics - use FLXRM, with 'equatorial radius' = 12", 'polar radius' = 9" (eyeballed from model): model: peak flux density = 10.8 mJy total flux density = 19.5 mJy image 1: peak flux density = 10.8 mJy/bm total on-source flux density = 18.9 mJy off-source rms = 1.97 uJy/bm off-source total flux density = -0.147 mJy off-source min, max = -13.1, 17.4 uJy/bm dynamic range = 5480 range in difference image = -21.8 to 24.3 uJy/bm range in fractional error image = -.190 to .065 median in fractional error image = -0.072 -> median fidelity ~ 14 image 2: peak flux density = 10.8 mJy/bm total on-source flux density = 19.1 mJy off-source rms = 1.72 uJy/bm off-source total flux density = -0.646 mJy off-source min, max = -14.4, 13.4 uJy/bm dynamic range = 6280 range in difference image = -17.8 to 27.5 uJy/bm range in fractional error image = -.184 to .104 median in fractional error image = -0.032 -> median fidelity ~ 31 image 3: peak flux density = 10.8 mJy/bm total on-source flux density = 18.8 mJy off-source rms = 1.71 uJy/bm off-source total flux density = -0.266 mJy off-source min, max = -12.2, 14.0 uJy/bm dynamic range = 6320 range in difference image = -16.8 to 27.8 uJy/bm range in fractional error image = -.205 to .020 median in fractional error image = -0.059 -> median fidelity ~ 17 so, by these measures, image 2 wins. and by a substantial margin, at least in the fractional error/fidelity. the median fractional error is less than half as large in that image than in image 1, and similar for image 3. b) star flux density & hence effective temperature this is impossible to do, because of the poor resolution. too much disk dust is included in that central pixel, and so any derived flux density is *much* too high. the star flux density is of order 240 uJy, whereas we've got roughly 2 mJy (factor of 10 too high) in that central beam. c) disk geometry factors (1) disk diameter & inclination i took cuts in the vertical and horizontal directions to estimate these. i took the the 'size' in either dimension as the number of pixels between points where the flux density went to 1 uJy/bm. this is cheating (and not strictly the right way to do it in any case), but i didn't want to go to the full trouble of ellipse fitting, and it will be close enough, i think. image 1 - 'y-size' = 109 pixels = 15.61 asec 'x-size' = 156 pixels = 22.34 asec image 2 - 'y-size' = 109 pixels = 15.61 asec 'x-size' = 152 pixels = 21.77 asec image 3 - 'y-size' = 107 pixels = 15.32 asec 'x-size' = 177 pixels = 25.35 asec but, remember that these are convolved effective sizes, so the roughly 1" beam has to be 'deconvolved' out of them. to first order, this beam convolution increases the size on either side of the center by 2" or so, so subtract 4" from the above sizes, giving: image 1 - 'y-size' = 11.61 asec = 140 AU 'x-size' = 18.34 asec = 220 AU -> inclination ~ 40 deg. image 2 - 'y-size' = 11.61 asec = 140 AU 'x-size' = 17.77 asec = 210 AU -> inclination ~ 42 deg. image 3 - 'y-size' = 11.32 asec = 130 AU 'x-size' = 21.35 asec = 260 AU -> inclination ~ 30 deg. the sizes should be 140 AU and 200 AU. so, simulations 1 & 2 did pretty well here (though #2 is again better), but simulation 3 is pretty far off. (2) total disk dust mass this must come from the total flux density in the reconstructed images. from the above statistical section, all 3 of the reconstructions do nearly equally well. they find 18.9, 19.1, and 18.8 mJy respectively, out of 19.5 mJy in the model - all about 97% of the total flux density. so, image 2 finds about 1% more total flux density, but at that level, there is no believable/realistic difference. (3) disk surface density falloff with radius from IRING - using APARM=0,0,90,45,0; CPARM=0,15,.5,0 this makes a list of the average flux density as you go out in rings. plotted this up for the 3 images. no practical difference between the 3 of them. (4) size of gaps & holes, etc... i can see no appreciable difference in the 3 images. if i take the 3 unique differences between the 3 images (i.e., image 1 - image 2, image 2 - image 3, and image 1 - image 3), there is no structure necessarily associated with the disk gaps or holes, which means that they reproduce these structures equally well. III. summary i would say that qualitatively, there is no appreciable difference between any of the 3 images. anybody studying the formation of planetary systems would be knocked out to see an image like any of these 3. there is certainly a statistical difference between them, with image 2 being the clear winner here, but i see no practical effect on the science that would come out of this kind of image. i would comment, however, that i think a better test would be to not use the most compact configuration to do the mosaicing here - use instead a configuration that gives roughly 10 X better resolution (roughly 0.1"). i suspect that alot of folks are going to want to image large structures with much better resolution than just that afforded by the most compact configuration (note that BIMA and OVRO *already* get to roughly 1"). i do not know that the answer will be any different, but it would be a better test, i think.