TN 0070: Mark IV CD #5, measures of coma & saturation

Herb Johnson, with notes from Ted Newhouse and Tom Droege
First draft: Dec 9, 1999
Updated Dec 24, 1999: measures of spherical field distortion
Updated Jan 1, 2000: Revisions to field distortions, updates, re-edits for clarity. Jan 1 2000 Updated Nov 4, 2000.
Keywords: image, coma, saturation

Introduction
Reference values and methods of measurement
Conclusions, considerations, consequences
Coma and saturation in one image file
Saturated stars
Saturated stars from several image files
Comments: sky curvature, spherical coordinates. References.

Introduction:

For all the chat and software about the Mark IV disk #5 data images during Nov-Dec 1999, I did not think enough was said quantatively about the amount of coma in these images; or the measurement of saturation. Coma is when an optic introduces a distortion in imaging that produces a "stretch" radially from the center of the image. Stars go from point images at the center of the image, to teardrops near the edge of the image and expecially at the corners. Saturation also came up in discussion: that is a physical phenomena in the CCD device when a pixel "well" is so full of electrons, they overflow into adjacent pixels - typically along the column toward either end, then into adjacent columns as the volume of electrons increases.

The work in this note represents my efforts to characterize coma and saturation in the Mark IV disk #5 images obtained in November 1999. Related effects and considerations will also be reported on or at least mentioned. Subsequent to these findings, it was found that a lens was reversed in repairs to these cameras: with the lens re-oriented correctly, most of the coma went away. But my findings have some reference value so they will be retained as written. I intend to write a subsequent Tech Note on saturation features of later Mark IV camera images, so I will also retain the findings and comments in this note as written in 1999 and refer to them in subsequent TN's.

Reference values and methods of measurement

It is necessary to discuss how my measurements were made and the technical features of the CCD device and optics to interpret my findings and results.

According to Tom Droege's TASS correspondence (2 Sept 1999) the CCD saturates at 80K to 90K electrons. The electron to ADU gain and offset will set the saturation value in ADU's. (The gain is presumably the same for images in Tom's CD#5 set as was Tom's subsequent set of 3 CD's as observed on 29 Nov 1999 and sent to Glenn Gombert. It will NOT be the same as that set for Tom's CD #3 set.)

On 24 March 1998, Tom said "The 442A [chip] has 15 micron pixels and there are 2048 of them. This is 30.72mm or 1.209 inches square. That puts the diagonal at 1.71 inches or 43.4 mm." The optic's focal length is about 400mm (it has been measured but was modified afterward). Tom says the camera readout/data conversion time is about 50 seconds. Michael Richmond in TN 59 says image "[H3R1458.895] is roughly 4.3 degrees on a side. North is to the left, and East is down. The plate scale is about 7.5 arcseconds per pixel." I've checked, all these numbers are modestly self-consistent, as the tangent of the angle of the field of view should be the ratio of focal length to the width of the CCD. (See Kitchin for details.)

For much of my work in this note, the image columns and rows pixel values were examined manually, and obtained via the Astrometria program, a very straightforward and useful program for examining CCD star images which runs under MS-DOS. (Check the tass_survey Web site for a reference.) The first datapoint in an image in Astrometria is (1,1), expressed as (row, column). The data points were manually transferred from Astrometria's display, entered into an Excel spreadsheet and plotted. Data points outside the spreadsheet range are approximately at image background level. Excel plots were converted to .jpg files. Astrometria uses the BZERO value in the TASS Mark IV FITS header of 32768 and subtracts this from each pixel value upon examination. Thus typical background pixels with ADU (absolute) counts of 45056 are reported by Astrometria as 12288. All pixel values in this note are as reported by Astrometria.

Conclusions to date

As regards saturation values: because of coma and tracking errors, both of which act to spread out the individual star images across the CCD, it is not a simple matter to determine a saturation value. Tom Droege has also suggested that there is a loss in CCD transfer efficiency either by column transfers (the last row would measure lower values than the first) or in the readout row (the last column pixels would measure lower values than the first). But from the measurements done in this note, it would appear that ADU values below 40K ADU's are below the saturation of the CCD. It also appears that values up to about 50K are also below saturation. It would also seem that when values of 50K to 60K occur, they are generally associated with saturation "spikes" in a bright star image, a typical sign of CCD saturation.

As for coma: while it is apparent and clearly it increases with radial distance from the center of the image file, I simply lack the tools, and knowledge of accepted methods of measure, to quantify it. From a sampling of near-saturated stars it would appear that an aperture of about 8 rows and 11 columns, used within a central square of about 1/3 of the image width of center, would encompass all but the brightest (and saturated) individual star images. However coma effects beyond the center would require stretching this aperture further. I also note that coma shifts the otherwise central maximum bright pixel in a star point-image to far from that center in a radial direction. I do not know the relationship between the maximal pixel value and the astrometric location of the star: as I will not do astrometry in this note I cannot make that determination here.

Related considerations

I mention the following for completeness, and to suggest further study. Spherical distortion refers to the errors from projecting a spherical sky view onto a flat CCD. I will differ further discussion to comments at the end of this note. There is an astrometric consideration: the widening or narrowing of RA with declination was discussed in the TASS maillist in Dec 1999. I include some numbers for this effect in my comments section. Tracking error is probably a factor in images in the CD #5 data set: I reference it later in this note. Also, there has been reported some pincushion distortion in the images, an optical phenomena where the corners image areas outside the edges. I will not attempt to measure or confirm this as it would require astrometry of the star images.

Consequences

Coma, tracking error, spherical distortion and pincushioning act to "stretch" the star image across the CCD and so will clearly impact the kind and size of aperture used in our photometric measurements; or the astrometrical results obtained from even an optimal aperture. Spherical distortion and the variation of RA across the image are strictly astrometric considerations. To correct coma, I can imagine some kind of measure of coma based on relative position from the center, and possibly some kind of acceptance criteria for coma which would imply a maximum circle of acceptance around the center. Possibly, an algorithm for aperture photometry would adapt to the degree of coma based on radius and angle from the center. A "kernal" of such apertures, precomputed, would save computer time during photometric analysis. Alternative considerations for aperture photometry include rejecting the corners and the edge areas of the image, and/or rejecting the brighter star images to support a smaller fixed aperture. Other TASS members have suggested a variable aperture, such as methods to collect all the signifigant pixels values around the star image. They have reported consistent results with such a method.

Evaluations of coma and saturation in one image file

The first efforts to evaluate coma and saturation were based on data taken of several star images from the Mark IV CD-ROM #5 file H3R1438.903, uncorrected with dark subtraction or flat-fielding. I chose bright but unsaturated star images to consider coma, and several saturated stars otherwise. Of the comaed stars, I chose one as a star image central in the field of view to show minimal coma; and one halfway out from the center by columns but central by rows, to show modest but not maximal coma..

I selected and examined one clearly saturated star in depth - it exhibited classic streaks due to saturation, which are discussed. Subsequently, I looked for several other saturated stars. These stars were selected ad-hoc, presuming a criterion of 40,000 ADUs (BZERO of 32768 not subtracted from this value) as near to saturation. Seven such observations were located. Of them, four had these "saturation" values in only one column; three had these or larger values in multiple columns. The differences were striking enough by inspection that I segregated these as described and plotted the results seperately and draw seperate conclusions.

Two star images and coma

The first star image I selected in H3R1438.903 has a maximum at (1121,988); a star image roughly centered in the overall field of view. The central star images have minimal coma, but apparently have some stretch along the columns due (says Tom) to tracking errors. Table 1 below shows column and row values in the vicinity of that star:

Table 1: center of image, star pixels

	col 1	col 2	col 3	col 4	col 5
row 1	13309	13834	13134	12499	12195
row 2	14408	20203	19184	13336	12323
row 3	15849	24903	29987	15919	12704
row 4	16280	27408	37601	19466	12809
row 5	16070	23283	35504	20027	13002
row 6	14941	18360	27498	18897	12906
row 7	13753	15360	18136	15005	12527
row 8	13016	13635	14459	13174	12344
row 9	12424	12677	12849	12470	
row 10	12209	12271	12294

Chart 1 below plots the distribution of each row above. Rows 4 and 5, roughly central of the 10 rows, has the maximum values.

[chart1.jpg]

Chart 2 below plots the distribution of each column above. Column 3 of 5, the central column, has the maximum values.

[chart2.jpg]

The next star image in H3R1438.903 has a maximum at (464, 1045), central by rows but left of center by columns. Table 2 below shows pixel values for that star.

Table 2: left of center, star pixels

	col 1	col 2	col 3	col 4	col 5	col 6	col 7	col 8
row 1	12236	12155	12201	12330	12374	12382	12292	12416
row 2	12118	12289	12608	12748	12653	12683	12544	12426
row 3	12265	13279	13774	13381	13293	13054	12723	12553
row 4	12860	19664	17928	15037	13708	13365	12946	12761
row 5	12986	22900	23949	16646	14093	13407	13268	12851
row 6	12815	25928	28002	17841	14223	13399	13074	12874
row 7	12393	19816	25897	17742	14122	13380	12940	12772
row 8	12451	15163	19872	15936	13793	13136	12842	12605
row 9	12301	13152	14422	13840	13155	12772	12589	12418
row 10	12056	12513	13041	12792	12546	12565	12548	12390
row 11	12030	12309	12463	12263	12349	12274	12353	12240

Chart 3 below plots the distribution of each column. Note that columns 2 and 3 (rather than central column 6) have the maximum values. The "tail" of the coma'ed star image thus points to the right toward the rightmost edge of the image. This star image is about halfway between the center column and the leftmost (lowest numbered) column.

[chart3.jpg]

Chart 4 below plots the distribution of each row. Central rows 5 and 6 have the maximum values. As this star image is near the central row, this star image has minimum coma along the rows.

[chart4.jpg]

Saturated stars

For the first saturated star in H3R1438.903, I found one roughly located at (1120,1022). The central area of the star image is about 10-12 column pixels wide, but the effects of saturation reveal themselves as two central columns of about 26 and 20 pixels above the background level.

Table 3 below shows pixel values for this saturated star by relevant columns and rows. My determination of saturation is based on this spike, almost certainly caused by overflow of charge down the columns, and the large number of relatively constant and large ADU counts in the center of the star image. At this point it's simpler to list them out. The background count is approximately 12100; Astrometria reports a background of 12105 +/- 386. I stopped tallying values when they approached that level. The third column of data below shows *'s to represent the location and size of the star image adjacent to these two columns, and two *'s to represent the approximate central row. In the central row, the image extended about five pixels to either side of the two rows. Coma effects were likely minimal due to the central location of this observation in the field of view.

An ad-hoc value of saturation would seem to be below 50000; but the CCD may not respond linearly near saturation, and some cells may have a lower saturation value.

Table 3: Saturated star pixels for one star, brightest columns


col 1   col2    other cols

12146
38221   12107
47072   12486
49760   46262
49968   48970
50097   49692
50208   49944
50315   50082	*
50392   50159	*
50435   50179	*
50448   50217	*
50464   50240	*
50465   50240	**
50483   50208	*
50475   50200	*
50488   50080	*
50416   49984	*
50344   49769	*
50336   49304
50218   43667
50184   12408
50104
49889
49730
49141
46432
12195

Additional saturated stars

After some consideration, I looked for other saturated or near-saturated stars in H3R1438.903 to see if the effects of saturation could be empirically observed, despite coma. I noted that large ADU values seemed to cluster around 40K and around 60K, and the latter often occurred in multiple adjacent rows. While these methods are ad-hoc and the sample size is small (and for one image), the results are very suggestive.

Table 4 below shows column runs in H3R1438.903 from 4 star images with nearly saturated or saturated pixels (i.e >40K ADU's) but with adjacent columns with pixels less than 40K ADU. They are identified by column number, which is relatively unique given their intensity. The row value of the maximum pixel is also given.

Table 4: other saturated star pixels, central columns only


row	1934	1751	1055	1741 (maximum pixel value only) 
col 	1673	577	234	1912 (column values shown below around maximum)

pixel 12991 12529	12744	12877
vals	13358	12899	16134	13788
	13746	13353	25964	15366
	14424	13937	37768	18471
	15248	14714	41120	24355
	16443	15589	37345	40456
	19695	16968	23814	43392
	27525	19478	15547	44721
	34305	26972	13314	44544
	35332	40523	12398	33097
	28448	43048		17942
	18754	44232		14193
	13937	43682		12506
	12730	34369		
	17768		
	14338		
	12664		
	12282

Chart 6 below is a simple plot of these values for each column (image). It would appear there is a "knee" in the curve somewhere 40,000 ADU's, at which the response flattens out. My initial engineering guess was that 40K ADU's corresponds to where the CCD cells in the column start to "bleed" over to adjacent cells in the same column. Verifying and determining this threshold would require more sampled stars and more images from each CCD. (However, after discussion with other TASS members and in consideration of more star images, this guess did not take into account the effects of coma which would also act to "flatten" the shape of the star image.)

[chart6.jpg]

Table 5 has column runs from three star images in H3R1438.903 with "very" saturated or saturated pixels (i.e >40K or >60K ADU's) with adjacent columns with pixels above 40K ADU. They are identified by column number, which is relatively unique given their intensity. The row value of the maximum pixel is also given.

Table 5: very saturated star pixels, brightest column only


row	1892	415	217  (maximum pixel value only)
col 	273	449	1730 (column values shown below around maximum)

pixel	19043	12157	12149
vals	22468	20984	15555
	29105	55656	58688
	41920	58673	61377
	43776	59481	62004
	44401	59840	62305
	44585	60067	62400
	44586	60168	62481
	44512	60208	62441
	60232	62120
	60208	61169
	60192	50820
	60104	25443
	59920	21849
	59699	18817
	59200	16688
	57935	
	34424	
	15360	
	13589	
	12853	
	12232

Chart 5 below is a simple plot of these values for each column (image). It would appear there is another "knee" or flat spot in the curve at about 60,000 ADU's. My engineering guess was that 60K ADU's corresponds to where the CCD cells in the column start to "bleed" over to adjacent cells in the adjacent columns. Clearly this is well beyond the useful range of the CCD. However, note that in the first saturated star image I examined at (1120,1022) has a flattened maximum around 50,000 ADU's. More sampled star images would hopefully clarify this situation.

[chart5.jpg]

Bright stars vs. saturated stars from several image files

The previous analysis suggested that coma as well as CCD pixel saturation may act to flatten the maximum values of imaged stars. So I decided on a strategy to look at the least-coma'ed star images across several image files in CD-ROM #5. Coma is least in the center of the image.

Table 6 are a series of data points taken from unmodified Mark IV CD-ROM #5 files as named, from the central 1/9th of each image file to minimize the effects of coma. These files were, on visual inspection, relatively free of background haze (i.e. local variations in background count). These data points are column values through the maximum pixel value, for stars that have a maximum well above 20000 ADU's (above BZERO of 32767) but which (with two exceptions) do not have obvious saturation characteristics such as a long column spike, or a series of almost constant values in the column. Column values from "below" the maximum (later rows) to "above" (earlier rows) were taken 3 pixels away. Also, a manual estimate was made of the row and column diameters of the imaged star, presuming a minimum value of 12,500.

table 6: representative saturated stars, pixels by ???

star	image    	row & col max	i 7	i 6	i 5	i 4	i 3	i 2	i 1	row and col width	
1	h3r1438.878	932	1095	13718	19043	26020	28385	28584	20199	15737	6	10	
2	h3r1438.878	836	728	25356	37084	41664	42304	29504	22659	17605	6	10	
3	h3r1438.878	809	1189	13605	15384	18221	22449	21907	19873	16099	6	10	
4	h3r1438.882	1044	1066	16216	22897	35210	42568	39328	28504	15651	7	10	
5	h3r1438.882	976	889	51032	51049	51072	51089	51081	50947	50944			
6	h3r1438.882	1206	918	15064	19640	30267	40320	38029	31731	16921			
7	h3r1438.882	1325	835	17753	26504	43988	44258	40850	28618	13904	8	10	
8	h3r1438.882	1280	798	13302	14681	17573	20544	18892	16788	12989	7	9	
9	h3r1438.886	752	952	15101	19083	26371	29784	26555	20021	13309	6	9	
10	h3r1438.886	1072	983	14743	18289	24262	25500	24896	19201	13983	6	9	
11	h3r1438.886	752	952	15101	19083	26371	29784	26555	20021	13309	6	9	
12	h3r1438.890	1251	1163	17856	24989	33672	36692	32763	25199	15785	7	11	
13	h3r1438.890	1254	1153	18370	25568	35762	41715	38350	28459	16883	7	11	
14	h3r1438.890	1255	1153	20683	29743	35247	33742	25804	17640	14042	7	11	
15	h3r1438.890	1431	922	16192	19565	24068	28399	24361	19551	13629	11	11	
16	h3r1438.890	971	804	21800	34344	45539	47552	44744	36596	20001	8	11	
17	h3r1438.890	970	805	19666	25585	34104	44770	46976	41520	29512	8	11	
18	h3r1438.895	881	970	18476	27864	41571	45048	41953	33763	15438	7	9	
19	h3r1438.895	882	970	18302	30779	39874	41112	31641	19929	14347	7	9	
20	h3r1438.895	1349	843	14223	17577	23304	23951	21045	16056	12521	8	9	
21	h3r1438.895	734	929	15193	21453	29024	31512	27315	20139	13579	6	9	
22	h3r1438.895	863	1164	15419	21552	34354	36928	35497	22256	14283	6	9	
23	h3r1438.895	864	1164	17985	30144	34321	34584	25683	17528	14166	6	9	
24	h3r1438.895	698	1333	48400	48400	48394	48448	48416	48368	48316

Stars #5 and #24 were clearly saturated, to the extent that several column values in each were large (>>20K) and very similar. Star #5 had a vertical spike characteristic of saturation effects. Most of the star images chosen had adjacent columns that had no pixels near the maximum value. However, stars #13 and #14, #18 and #19, and #22 and #23, are in fact data from the same star as taken from adjacent columns.

A plot (not shown) of the above "i-values", centered about the maximum pixel value, strongly suggests that the flat-top effect of saturation does not occur for maximal pixel values up to the maximum for star #16 of 47.5K ADU's. The saturated stars as noted above are flattened at 51K and 48.4K ADU's. Further inspection of these and other images for more sample star images may resolve this apparent boundary further, but it would appear that a lower bound for saturation would be 40K ADUs, and an upper boundary would be 50K ADUs.

Charge transfer, aperture size, tracking error

I attempted to find some correlation between pixel maximum and the row or column width of the star image, as measured by pixels above 12.5K for the longest row or column through the image. A consistent change would suggest a charge transfer inefficiency. The saturated stars were not included in this attempt. There is probably some correlation, but it would seem more realistic to say that a box which is 8 pixel rows by 11 pixel columns would be sufficient to encompass all the pixels at or above 12.5K ADU's. A stronger correlation might be seen if star images with max pixels below 20K were included, or if these samples were not chosen manually and ad-hoc.

The 8 by 11 box does suggest a measure of the elongation of star images in the center of the image, again presumably due to tracking error and some degree of coma; or possibly some tilt of the image plane relative to the CCD plane.

Comments: Distortions due to sky curvature and spherical coordinates

The projection of the curved sky onto the flat CCD means that the field of view of the Mark IV will be distorted. I believe this is called "spherical distortion". The Mark III camera minimized this effect with its narrower field of view, but Mark IV has a four-degree field of view. I'm not sure how to represent this at this time. In addition, the coordinates for RA become compressed on the end of the image that is closest to the celestial pole. The Mark III camera minimized this effect by operating near the celestial equator. This effect does not matter for my measurements as I'm not determining location. However, Ted Newhouse, in private email on Dec 21 1999, responded to my request for specific measures of this, based on a 7.33 arc sec/pixel measure.

"It depends on the declination. The further north or south of the equator you go, the more the plane coordinates of the image differ from the spherical coordinates of the sky. Here are some figures. They're tentative, as I only wrote the program yesterday, and it's subject to further testing.

At the equator, correction to a point midway along any side is about 0.4 pixel, and to a corner 1 pixel. At 5 degrees N (where disk 5 images were taken), it's about 0.4 pixels at north or south side, 3 pixels at east or west side, 7 pixels in a corner away from the equator, and 1 pixel in a corner toward the equator.

At 85 degrees north, it's still only about 0.4 pixels north or south, but 9380 east or west, 14985 in the corner away from the equator, and 6608 in a corner toward the equator.

That's as far as I went when I was testing. Above 88 degrees of course, you'll get all possible RAs from 0 to 360 in the field of view."

References

According to Kitchin, if the angular change relative to the center is r, and the corresponding flat coordinate is x, then the relationship is r = arctan(x/F), where F is the focal length. From Astrophysical Techniques, second edition by C R Kitchin, Institute of Physics Publishing (IOP Publishing), 1991