View allAll Photos Tagged polynomials

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

"Limits of Optimal Polynomial Approximants"

polynomials orthogonal over the unit disk

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 1920 / length of data axis 1

NAXIS2 = 1080 / length of data axis 2

NAXIS3 = 3 / length of data axis 3

EXTEND = T / FITS dataset may contain extensions

COMMENT FITS (Flexible Image Transport System) format is defined in 'Astronomy

COMMENT and Astrophysics', volume 376, page 359; bibcode: 2001A&A...376..359H

BZERO = 0. / Offset data range to that of unsigned short

BSCALE = 1. / Default scaling factor

PROGRAM = 'Siril 1.4.0-beta2' / Software that created this HDU

DATE = '2025-05-25T04:36:06' / UTC date that FITS file was created

DATE-OBS= '2025-05-24T06:42:54.562902' / YYYY-MM-DDThh🇲🇲ss observation start,

IMAGETYP= 'Light ' / Type of image

EXPTIME = 10. / [s] Exposure time duration

TELESCOP= 'S50_f1c10930' / Telescope used to acquire this image

FILTER = 'LP_Starless' / Active filter name

APERTURE= 5. / Aperture of the instrument

FOCALLEN= 250. / [mm] Focal length

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 2.90000009536743 / [um] Pixel X axis size

YPIXSZ = 2.90000009536743 / [um] Pixel Y axis size

INSTRUME= 'Seestar S50' / Instrument name

CCD-TEMP= 16.5 / [degC] CCD temperature

GAIN = 80 / Sensor gain

BAYERPAT= 'GRBG ' / Bayer color pattern

FOCPOS = 1587 / [step] Focuser position

STACKCNT= 193 / Stack frames

OBJECT = 'M 57 ' / Name of the object of interest

SITELAT = 41.7559 / [deg] Observation site latitude

SITELONG= -88.1817 / [deg] Observation site longitude

OBJCTRA = '18 53 41.708' / [H M S] Image center Right Ascension

OBJCTDEC= '+32 59 57.534' / [D M S] Image center Declination

RA = 283.423783955069 / [deg] Image center Right Ascension

DEC = 32.999314905147 / [deg] Image center Declination

CTYPE1 = 'RA---TAN-SIP' / TAN (gnomic) projection + SIP distortions

CTYPE2 = 'DEC--TAN-SIP' / TAN (gnomic) projection + SIP distortions

CUNIT1 = 'deg ' / Unit of coordinates

CUNIT2 = 'deg ' / Unit of coordinates

EQUINOX = 9.87654321E+107 / Equatorial equinox

CRPIX1 = 1027.74934387 / Axis1 reference pixel

CRPIX2 = 638.474120617 / Axis2 reference pixel

CRVAL1 = 283.5005666 / [deg] Axis1 reference value

CRVAL2 = 32.9546001127 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000659799075280385 / [deg] X pixel size

CDELT2 = 0.000659608978753408 / [deg] Y pixel size

PC1_1 = 0.00518729418360521 / Linear transformation matrix (1, 1)

PC1_2 = -0.99998654589902 / Linear transformation matrix (1, 2)

PC2_1 = -0.999981686632843 / Linear transformation matrix (2, 1)

PC2_2 = -0.00605197479621091 / Linear transformation matrix (2, 2)

A_ORDER = 2 / SIP polynomial degree, axis 1, pixel-to-sky

A_0_0 = 0.

A_1_0 = 0.

A_0_1 = 0.

A_2_0 = 2.86109521904E-07

A_1_1 = -1.29394838394E-06

A_0_2 = 2.3587804816E-06

B_ORDER = 2 / SIP polynomial degree, axis 2, pixel-to-sky

B_0_0 = 0.

B_1_0 = 0.

B_0_1 = 0.

B_2_0 = 9.35141297562001E-08

B_1_1 = -9.84136886845E-07

B_0_2 = 3.21558028496E-06

AP_ORDER= 2 / SIP polynomial degree, axis 1, sky-to-pixel

AP_0_0 = 0.000192584989749

AP_1_0 = -5.29841367311E-07

AP_0_1 = 2.19440235099E-06

AP_2_0 = -2.86178110142E-07

AP_1_1 = 1.29499382405E-06

AP_0_2 = -2.36192723025E-06

BP_ORDER= 2 / SIP polynomial degree, axis 2, sky-to-pixel

BP_0_0 = 0.000298064397267

BP_1_0 = -7.71663053267E-07

BP_0_1 = 3.35419644196E-06

BP_2_0 = -9.36096853414001E-08

BP_1_1 = 9.85512218419001E-07

BP_0_2 = -3.22041417503E-06

HISTORY Saturation enhancement (amount=0.31)

HISTORY Saturation enhancement (amount=0.43)

HISTORY Rotation (90 deg)

CREATOR = 'ZWO Seestar S50' / Capture software

PRODUCER= 'ZWO ' / Powered by ZWO

XORGSUBF= 0 / Subframe X position in binned pixels

YORGSUBF= 0 / Subframe Y position in binned pixels

EQMODE = 1 / Equatorial mode

CCDXBIN = 1 / Camera X Bin

CCDYBIN = 1 / Camera Y Bin

TOTALEXP= 1930. / Total Exposure Time in seconds

IMAGEW = 1080 / Image width, in pixels.

IMAGEH = 1920 / Image height, in pixels.

BG-EXTR = 'GraXpert'

CBG-1 = 0.011884219

CBG-2 = 0.011884219

CBG-3 = 0.011884219

INTP-OPT= 'RBF '

HIERARCH SMOOTHING = 0.9

HIERARCH CORR-TYPE = 'Subtraction'

HIERARCH SAMPLE-SIZE = 15

HIERARCH RBF-KERNEL = 'thin_plate'

HIERARCH SPLINE-ORDER = 3

BG-PTS = '[[555, 9, 1], [627, 9, 1], [699, 9, 1], [741, 9, 1], [813, 9, 1], &'

CONTINUE '[411, 111, 1], [555, 111, 1], [627, 111, 1], [699, 111, 1], [771, &'

CONTINUE '111, 1], [813, 111, 1], [885, 111, 1], [339, 183, 1], [381, 183, &'

CONTINUE '1], [453, 183, 1], [525, 183, 1], [597, 183, 1], [699, 183, 1], &'

CONTINUE '[771, 183, 1], [843, 153, 1], [885, 183, 1], [987, 183, 1], [1029, &'

CONTINUE '183, 1], [267, 255, 1], [339, 255, 1], [396, 240, 1], [453, 255, &'

CONTINUE '1], [525, 255, 1], [597, 255, 1], [699, 255, 1], [771, 225, 1], &'

CONTINUE '[828, 240, 1], [885, 255, 1], [987, 255, 1], [1044, 240, 1], [195, &'

CONTINUE '297, 1], [267, 327, 1], [324, 312, 1], [396, 312, 1], [453, 327, &'

CONTINUE '1], [540, 312, 1], [597, 297, 1], [699, 327, 1], [771, 297, 1], &'

CONTINUE '[843, 297, 1], [915, 327, 1], [972, 312, 1], [1029, 327, 1], [195, &'

CONTINUE '399, 1], [237, 399, 1], [309, 369, 1], [381, 399, 1], [453, 399, &'

CONTINUE '1], [555, 369, 1], [597, 399, 1], [699, 369, 1], [756, 384, 1], &'

CONTINUE '[843, 399, 1], [915, 399, 1], [987, 369, 1], [1059, 399, 1], [21, &'

CONTINUE '471, 1], [108, 456, 1], [195, 471, 1], [267, 471, 1], [339, 471, &'

CONTINUE '1], [381, 471, 1], [483, 441, 1], [525, 471, 1], [627, 441, 1], &'

CONTINUE '[699, 471, 1], [771, 471, 1], [813, 441, 1], [915, 471, 1], [987, &'

CONTINUE '441, 1], [1029, 471, 1], [21, 543, 1], [93, 543, 1], [195, 543, &'

CONTINUE '1], [237, 543, 1], [339, 543, 1], [381, 543, 1], [453, 543, 1], &'

CONTINUE '[555, 543, 1], [627, 543, 1], [699, 543, 1], [741, 543, 1], [843, &'

CONTINUE '513, 1], [915, 513, 1], [957, 543, 1], [1059, 543, 1], [51, 615, &'

CONTINUE '1], [123, 615, 1], [165, 585, 1], [237, 585, 1], [324, 600, 1], &'

CONTINUE '[381, 615, 1], [453, 615, 1], [555, 615, 1], [597, 615, 1], [699, &'

CONTINUE '615, 1], [771, 615, 1], [843, 615, 1], [915, 585, 1], [987, 615, &'

CONTINUE '1], [1059, 615, 1], [51, 687, 1], [93, 687, 1], [195, 687, 1], &'

CONTINUE '[267, 687, 1], [309, 657, 1], [396, 672, 1], [453, 657, 1], [525, &'

CONTINUE '687, 1], [597, 657, 1], [699, 687, 1], [771, 657, 1], [843, 687, &'

CONTINUE '1], [779.7394136807818, 704.6254071661238, 1.0], [987, 657, 1], &'

CONTINUE '[1029, 687, 1], [21, 729, 1], [123, 759, 1], [180, 744, 1], [267, &'

CONTINUE '759, 1], [324, 744, 1], [396, 744, 1], [453, 729, 1], [525, 759, &'

CONTINUE '1], [627, 729, 1], [699, 729, 1], [771, 759, 1], [843, 729, 1], &'

CONTINUE '[855.830618892508, 766.1237785016286, 1.0], [972, 744, 1], [1029, &'

CONTINUE '759, 1], [51, 831, 1], [123, 831, 1], [165, 831, 1], [267, 831, &'

CONTINUE '1], [309, 801, 1], [396, 816, 1], [453, 801, 1], [555, 801, 1], &'

CONTINUE '[627, 801, 1], [684, 816, 1], [741, 801, 1], [813, 831, 1], [915, &'

CONTINUE '801, 1], [987, 831, 1], [1059, 801, 1], [51, 873, 1], [123, 873, &'

CONTINUE '1], [195, 873, 1], [267, 873, 1], [309, 873, 1], [381, 873, 1], &'

CONTINUE '[483, 873, 1], [525, 873, 1], [597, 873, 1], [699, 903, 1], [771, &'

CONTINUE '903, 1], [813, 903, 1], [915, 873, 1], [972, 888, 1], [1059, 903, &'

CONTINUE '1], [51, 945, 1], [123, 945, 1], [165, 945, 1], [252, 960, 1], &'

CONTINUE '[309, 945, 1], [381, 945, 1], [453, 945, 1], [699, 975, 1], [771, &'

CONTINUE '945, 1], [843, 975, 1], [885, 975, 1], [972, 960, 1], [1059, 975, &'

CONTINUE '1], [51, 1047, 1], [123, 1017, 1], [165, 1017, 1], [237, 1047, 1], &'

CONTINUE '[309, 1017, 1], [381, 1047, 1], [483, 1047, 1], [627, 1047, 1], &'

CONTINUE '[699, 1017, 1], [771, 1047, 1], [813, 1017, 1], [885, 1017, 1], &'

CONTINUE '[972, 1032, 1], [1029, 1017, 1], [51, 1119, 1], [123, 1089, 1], &'

CONTINUE '[165, 1119, 1], [252, 1104, 1], [309, 1089, 1], [396, 1104, 1], &'

CONTINUE '[453, 1119, 1], [525, 1089, 1], [627, 1119, 1], [699, 1119, 1], &'

CONTINUE '[741, 1089, 1], [843, 1089, 1], [885, 1089, 1], [957, 1119, 1], &'

CONTINUE '[1029, 1119, 1], [51, 1161, 1], [123, 1191, 1], [165, 1191, 1], &'

CONTINUE '[267, 1191, 1], [309, 1161, 1], [381, 1191, 1], [453, 1161, 1], &'

CONTINUE '[525, 1161, 1], [627, 1191, 1], [699, 1161, 1], [741, 1161, 1], &'

CONTINUE '[813, 1161, 1], [885, 1161, 1], [972, 1176, 1], [1029, 1191, 1], &'

CONTINUE '[51, 1233, 1], [108, 1248, 1], [195, 1233, 1], [267, 1233, 1], &'

CONTINUE '[339, 1233, 1], [396, 1248, 1], [453, 1233, 1], [525, 1233, 1], &'

CONTINUE '[627, 1233, 1], [699, 1233, 1], [771, 1263, 1], [813, 1233, 1], &'

CONTINUE '[915, 1233, 1], [957, 1233, 1], [1029, 1263, 1], [51, 1335, 1], &'

CONTINUE '[108, 1320, 1], [195, 1305, 1], [237, 1305, 1], [324, 1320, 1], &'

CONTINUE '[381, 1335, 1], [483, 1335, 1], [540, 1320, 1], [627, 1305, 1], &'

CONTINUE '[699, 1305, 1], [771, 1335, 1], [813, 1335, 1], [915, 1335, 1], &'

CONTINUE '[957, 1335, 1], [1059, 1335, 1], [36, 1392, 1], [93, 1377, 1], &'

CONTINUE '[165, 1377, 1], [267, 1377, 1], [309, 1377, 1], [381, 1407, 1], &'

CONTINUE '[483, 1407, 1], [555, 1407, 1], [627, 1407, 1], [699, 1377, 1], &'

CONTINUE '[741, 1407, 1], [813, 1407, 1], [885, 1407, 1], [987, 1407, 1], &'

CONTINUE '[1029, 1407, 1], [51, 1449, 1], [93, 1449, 1], [267, 1449, 1], &'

CONTINUE '[309, 1449, 1], [411, 1479, 1], [483, 1449, 1], [555, 1479, 1], &'

CONTINUE '[627, 1479, 1], [699, 1479, 1], [771, 1479, 1], [843, 1479, 1], &'

CONTINUE '[915, 1479, 1], [987, 1479, 1], [1059, 1449, 1], [36, 1536, 1], &'

CONTINUE '[93, 1521, 1], [165, 1551, 1], [237, 1551, 1], [339, 1521, 1], &'

CONTINUE '[381, 1551, 1], [483, 1521, 1], [555, 1521, 1], [627, 1521, 1], &'

CONTINUE '[684, 1536, 1], [771, 1551, 1], [843, 1551, 1], [885, 1521, 1], &'

CONTINUE '[987, 1551, 1], [1029, 1521, 1], [51, 1623, 1], [93, 1593, 1], &'

CONTINUE '[195, 1593, 1], [237, 1623, 1], [309, 1623, 1], [411, 1623, 1], &'

CONTINUE '[453, 1623, 1], [525, 1623, 1], [627, 1623, 1], [699, 1623, 1], &'

CONTINUE '[771, 1623, 1], [843, 1593, 1], [915, 1623, 1], [987, 1623, 1], &'

CONTINUE '[1029, 1593, 1], [51, 1695, 1], [123, 1665, 1], [195, 1665, 1], &'

CONTINUE '[267, 1665, 1], [339, 1695, 1], [411, 1695, 1], [483, 1695, 1], &'

CONTINUE '[555, 1695, 1], [627, 1695, 1], [699, 1665, 1], [771, 1695, 1], &'

CONTINUE '[843, 1695, 1], [915, 1695, 1], [987, 1695, 1], [1029, 1695, 1], &'

CONTINUE '[21, 1767, 1], [108, 1752, 1], [165, 1767, 1], [267, 1767, 1], &'

CONTINUE '[309, 1767, 1], [396, 1752, 1], [483, 1737, 1], [555, 1767, 1], &'

CONTINUE '[627, 1767, 1], [699, 1767, 1], [771, 1767, 1], [843, 1767, 1], &'

CONTINUE '[915, 1767, 1], [987, 1767, 1], [1029, 1767, 1], [21, 1839, 1], &'

CONTINUE '[93, 1839, 1], [165, 1839, 1], [252, 1824, 1], [339, 1809, 1], &'

CONTINUE '[411, 1839, 1], [483, 1839, 1], [555, 1839, 1], [627, 1809, 1], &'

CONTINUE '[699, 1809, 1], [771, 1839, 1], [843, 1809, 1], [915, 1839, 1], &'

CONTINUE '[987, 1839, 1], [1059, 1839, 1], [21, 1911, 1], [93, 1881, 1], &'

CONTINUE '[165, 1911, 1], [267, 1881, 1], [309, 1881, 1], [411, 1881, 1], &'

CONTINUE '[483, 1881, 1], [555, 1881, 1], [627, 1881, 1], [699, 1911, 1], &'

CONTINUE '[771, 1911, 1], [843, 1911, 1], [915, 1911, 1], [987, 1911, 1], &'

CONTINUE '[1059, 1911, 1], [32, 322, 1], [51, 339, 1], [123, 309, 1], [36, &'

CONTINUE '396, 1], [123, 411, 1], [123, 267, 1], [53, 93, 1], [93, 51, 1], &'

CONTINUE '[180, 36, 1], [51, 93, 1], [108, 108, 1], [195, 123, 1], [21, 165, &'

CONTINUE '1], [123, 165, 1], [74, 290, 1], [51, 267, 1], [214, 202, 1], &'

CONTINUE '[267, 195, 1], [321, 48, 1], [309, 51, 1], [411, 51, 1], [483, 51, &'

CONTINUE '1], [267, 93, 1], [917, 48, 1], [885, 51, 1], [987, 21, 1], [1044, &'

CONTINUE '36, 1], [1029, 123, 1], [926, 106, 1], [927, 213, 1], [741, 339, &'

CONTINUE '1], [456, 1570, 1], [339, 1563, 1], [195, 1491, 1], [237, 1491, &'

CONTINUE '1], [495, 1570, 1], [568, 1559, 1]]'

END

 

polynomials orthogonal over the unit disk

Draw a similar shape, where the radius R, the distance α and the angle ω with ω>0° have random values.

 

The consequences are

 

1. x1 + x2 + x3 + x4 = 0

 

Of course, the values ​​of the points x are the roots of a reduced quartic equation. Then

 

2. The radius R is calculated by the coefficients of this equation, while it is the only radius for which the rectangularity of the lines of the figure applies.

 

This is part of the geometric interpretation of the quartic equation that leads to multiple analytical solutions.

 

Can this work for higher degree equations?

 

The following result shows whether it would be possible for such an equation (with degree n) to have a similar model with parameters R, α, ω (where the angles formed by the lines at point 0,0 are equal to 360°/n each).

 

If R,α and n have constant values with R>|α|>=0 and n>2, then for every angle ω will be valid

 

x1+x2+x3+...+xn=δcos(nω)

 

where δ is a constant. Therefore δ will be equal to the sum of the roots for ω=0°. When n is an odd number it will be δ=0 only if α=0, while when n is an even number it will always be δ=0 due to the symetry that these systems have with respect to the y-axis for ω=0°.

 

So the indications are encouraging for equations that have an even degree, but the problem remains open.

 

"The single biggest problem we face is that of visualisation", Richard Neiman.

more on viXra, The Secret Geometry of the Quartic Equation

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

THese were gonna be a MOVIE! Polynomial Lost on Algebra Island wiht her sister Leucine and bro Tryptophan. I let Luci got o family and Luci gave a litter of hte best pups (Rubisco and Kinase) and Tryp was Tryp! (In middle)

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

Isaac Newton -I have a fundamental belief in the Bible as the Word of God, written by those who were inspired. I study the Bible daily. -Mathematician, Astronomer and Physicist

polynomials orthogonal over the unit disk

SIMPLE = T / file does conform to FITS standard

BITPIX = -32 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 5444 / length of data axis 1

NAXIS2 = 3803 / length of data axis 2

NAXIS3 = 3 / length of data axis 3

EXTEND = T / FITS dataset may contain extensions

COMMENT FITS (Flexible Image Transport System) format is defined in 'Astronomy

COMMENT and Astrophysics', volume 376, page 359; bibcode: 2001A&A...376..359H

BZERO = 0. / Offset data range to that of unsigned short

BSCALE = 1. / Default scaling factor

PROGRAM = 'Siril 1.4.0-beta4' / Software that created this HDU

DATE = '2025-10-06T12:04:35' / UTC date that FITS file was created

DATE-OBS= '2025-10-06T10:09:06.886626' / YYYY-MM-DDThh🇲🇲ss observation start,

ROWORDER= 'TOP-DOWN' / Order of the rows in image array

EXPTIME = 836.25 / [s] Exposure time duration

OBSERVER= 'DaveMartin' / Observer name

FILTER = 'Dual ' / Active filter name

FOCALLEN= 956.82 / [mm] Focal length

CENTALT = 43.875 / [deg] Altitude of telescope

CENTAZ = 197.023888888889 / [deg] Azimuth of telescope

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 3.76 / [um] Pixel X axis size

YPIXSZ = 3.76 / [um] Pixel Y axis size

INSTRUME= 'ZWO ASI2600MC Pro' / Instrument name

CCD-TEMP= 0. / [degC] CCD temperature

SET-TEMP= 0. / [degC] CCD temperature setpoint

GAIN = 100 / Sensor gain

OFFSET = 12 / Sensor gain offset

CVF = 0.2636 / [e-/ADU] Electrons per A/D unit

FOCPOS = 19865 / [step] Focuser position

FOCTEMP = 8.30000019073486 / [degC] Focuser temperature

OBJECT = 'NGC1976 (M42, Great Orion Nebula, Orion Nebula)' / Name of the object

AIRMASS = 1.44068164488539 / Airmass at frame center (Gueymard 1993)

SITELAT = 39.3488888888889 / [deg] Observation site latitude

SITELONG= -78.8116666666667 / [deg] Observation site longitude

OBJCTRA = '05 35 15.799' / [H M S] Image center Right Ascension

OBJCTDEC= '-05 22 47.677' / [D M S] Image center Declination

RA = 83.8158281898562 / [deg] Image center Right Ascension

DEC = -5.37991021764815 / [deg] Image center Declination

CTYPE1 = 'RA---TAN-SIP' / TAN (gnomic) projection + SIP distortions

CTYPE2 = 'DEC--TAN-SIP' / TAN (gnomic) projection + SIP distortions

CUNIT1 = 'deg ' / Unit of coordinates

CUNIT2 = 'deg ' / Unit of coordinates

EQUINOX = 2000. / Equatorial equinox

CRPIX1 = 2693.5 / Axis1 reference pixel

CRPIX2 = 1938.5 / Axis2 reference pixel

CRVAL1 = 83.8212837066781 / [deg] Axis1 reference value

CRVAL2 = -5.38889204929805 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000225139323345704 / [deg] X pixel size

CDELT2 = 0.000225169400022428 / [deg] Y pixel size

PC1_1 = -0.348038991134063 / Linear transformation matrix (1, 1)

PC1_2 = -0.937480058801457 / Linear transformation matrix (1, 2)

PC2_1 = 0.937494027453252 / Linear transformation matrix (2, 1)

PC2_2 = -0.348001362769574 / Linear transformation matrix (2, 2)

A_ORDER = 3 / SIP polynomial degree, axis 1, pixel-to-sky

A_0_0 = 0.

A_1_0 = 0.

A_0_1 = 0.

A_2_0 = -1.64277020685452E-07

A_1_1 = 9.87549346279314E-09

A_0_2 = -8.69160424952199E-08

A_3_0 = -3.02073434291275E-10

A_2_1 = 5.06449749491843E-12

A_1_2 = -3.06521889392484E-10

A_0_3 = -7.11631122483371E-12

B_ORDER = 3 / SIP polynomial degree, axis 2, pixel-to-sky

B_0_0 = 0.

B_1_0 = 0.

B_0_1 = 0.

B_2_0 = -1.28288782073535E-08

B_1_1 = -5.68511798589425E-08

B_0_2 = -5.88039813473346E-08

B_3_0 = 2.51466443610045E-12

B_2_1 = -3.03411808507867E-10

B_1_2 = 1.88074302834745E-11

B_0_3 = -3.02158975629337E-10

AP_ORDER= 3 / SIP polynomial degree, axis 1, sky-to-pixel

AP_0_0 = -0.00282500133597869

AP_1_0 = -1.43205078008135E-05

AP_0_1 = 6.69821284630872E-07

AP_2_0 = 1.66963688254827E-07

AP_1_1 = -1.00792430236529E-08

AP_0_2 = 8.76805314355769E-08

AP_3_0 = 3.06432679354613E-10

AP_2_1 = -5.24743960913225E-12

AP_1_2 = 3.1135956544913E-10

AP_0_3 = 6.95417774079078E-12

BP_ORDER= 3 / SIP polynomial degree, axis 2, sky-to-pixel

BP_0_0 = -0.000587530962498615

BP_1_0 = 6.73660948648711E-07

BP_0_1 = -1.0370590796116E-05

BP_2_0 = 1.30208895607198E-08

BP_1_1 = 5.76814378001768E-08

BP_0_2 = 5.93230277432801E-08

BP_3_0 = -2.60709033909064E-12

BP_2_1 = 3.07758434362221E-10

BP_1_2 = -1.92588333919524E-11

BP_0_3 = 3.05579270177439E-10

PLTSOLVD= T / Siril internal solver

HISTORY Plate Solve

HISTORY Crop (x=431, y=223, w=5444, h=3803)

HISTORY Background extraction (Correction: Subtraction)

HISTORY StatStretch: m=0.20 l=True n=True c=True b=0.00

HISTORY GraXpert AI BGE: subtraction

HISTORY GraXpert AI denoise: strength 0.50

HISTORY GraXpert AI deconvolve: strength 0.50

HISTORY GraXpert AI deconvolve: strength 0.50

HISTORY GHS LINEAR BP: 0.08

HISTORY GHS pivot: 0.138, amount: 1.25, local: 0.00 [0.00 1.00]

HISTORY GHS LINEAR BP: 0.06

HISTORY GHS INV pivot: 0.116, amount: 0.82, local: 0.00 [0.00 1.00]

HISTORY Apply Signature

HISTORY GHS pivot: 0.179, amount: 2.38, local: 0.00 [0.00 1.00]

HISTORY GHS LINEAR BP: 0.03

HISTORY HDR Multiscale: 7 scales

HISTORY HDR Multiscale: 7 scales

ADCBITS = 16 / Bit depth of camera sensor ADC in current mode

BIASADU = 121.16593933105469 / ADU for bias level (no photons) at current sett

CAMID = '1E1E560D0B020900' /

COLORTYP= 'RGB' /

DATE-AVG= '2025-10-06T10:24:13.1694766' / System Clock:Est. Frame Mid Point

DATE-END= '2025-10-06T10:39:19.4523264' / System Clock:Est. Frame End

EGAINSAV= 0.26355 / Electrons per ADU at saved bit depth

JD_UTC = 2460954.933485758 / Julian Date at mid exposure

PIERSIDE= 'EAST' /

RDNOISE = 1.77 / Read noise in electrons

RELGAIN = 2.964 / Multiplicative gain relative to minumum

SUBEXP = 3.75 /

END

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

SIMPLE = T / file does conform to FITS standard

BITPIX = 16 / number of bits per data pixel

NAXIS = 3 / number of data axes

NAXIS1 = 5007 / length of data axis 1

NAXIS2 = 3313 / length of data axis 2

NAXIS3 = 3 / length of data axis 3

EXTEND = T / FITS dataset may contain extensions

COMMENT FITS (Flexible Image Transport System) format is defined in 'Astronomy

COMMENT and Astrophysics', volume 376, page 359; bibcode: 2001A&A...376..359H

BZERO = 0. / Offset data range to that of unsigned short

BSCALE = 1. / Default scaling factor

MIPS-HI = 65535 / Lower visualization cutoff

PROGRAM = 'Siril 1.4.0-beta3' / Software that created this HDU

DATE = '2025-09-05T17:06:14' / UTC date that FITS file was created

DATE-OBS= '2025-09-03T07:47:09.267542' / YYYY-MM-DDThh🇲🇲ss observation start,

ROWORDER= 'TOP-DOWN' / Order of the rows in image array

EXPTIME = 1830. / [s] Exposure time duration

OBSERVER= 'DaveMartin' / Observer name

FILTER = 'mixed ' / Active filter name

FOCALLEN= 350.23 / [mm] Focal length

CENTALT = 24.0991666666667 / [deg] Altitude of telescope

CENTAZ = 115.277222222222 / [deg] Azimuth of telescope

XBINNING= 1 / Camera binning mode

YBINNING= 1 / Camera binning mode

XPIXSZ = 3.76 / [um] Pixel X axis size

YPIXSZ = 3.76 / [um] Pixel Y axis size

INSTRUME= 'ZWO ASI2600MC Pro' / Instrument name

CCD-TEMP= 0. / [degC] CCD temperature

SET-TEMP= 0. / [degC] CCD temperature setpoint

GAIN = 100 / Sensor gain

OFFSET = 12 / Sensor gain offset

CVF = 0.2636 / [e-/ADU] Electrons per A/D unit

FOCPOS = 33201 / [step] Focuser position

FOCTEMP = 9.84000015258789 / [degC] Focuser temperature

STACKCNT= 2 / Stack frames

LIVETIME= 3660. / [s] Exposure time after deadtime correction

OBJECT = 'IC0434 (Horsehead Nebula, Orion B)' / Name of the object of interest

AIRMASS = 2.43479916985599 / Airmass at frame center (Gueymard 1993)

SITELAT = 39.3488888888889 / [deg] Observation site latitude

SITELONG= -78.8116666666667 / [deg] Observation site longitude

OBJCTRA = '05 40 31.794' / [H M S] Image center Right Ascension

OBJCTDEC= '-02 24 35.066' / [D M S] Image center Declination

RA = 85.1324766100387 / [deg] Image center Right Ascension

DEC = -2.40974050855275 / [deg] Image center Declination

CTYPE1 = 'RA---TAN-SIP' / TAN (gnomic) projection + SIP distortions

CTYPE2 = 'DEC--TAN-SIP' / TAN (gnomic) projection + SIP distortions

CUNIT1 = 'deg ' / Unit of coordinates

CUNIT2 = 'deg ' / Unit of coordinates

EQUINOX = 2000. / Equatorial equinox

CRPIX1 = 2565.5 / Axis1 reference pixel

CRPIX2 = 1486.5 / Axis2 reference pixel

CRVAL1 = 85.2341341042268 / [deg] Axis1 reference value

CRVAL2 = -2.4558836476365 / [deg] Axis2 reference value

LONPOLE = 180. / Native longitude of celestial pole

CDELT1 = -0.000615265260750159 / [deg] X pixel size

CDELT2 = 0.000614968375501954 / [deg] Y pixel size

PC1_1 = 0.0794083577694203 / Linear transformation matrix (1, 1)

PC1_2 = 0.996842170414336 / Linear transformation matrix (1, 2)

PC2_1 = -0.996753807467596 / Linear transformation matrix (2, 1)

PC2_2 = 0.0805099204995885 / Linear transformation matrix (2, 2)

A_ORDER = 3 / SIP polynomial degree, axis 1, pixel-to-sky

A_0_0 = 0.

A_1_0 = 0.

A_0_1 = 0.

A_2_0 = -3.79677154119047E-08

A_1_1 = 7.38342030330063E-08

A_0_2 = 3.79720296820246E-08

A_3_0 = -4.0324918048076E-11

A_2_1 = -3.29565885448712E-12

A_1_2 = -3.15341842919888E-11

A_0_3 = -5.64825366267551E-12

B_ORDER = 3 / SIP polynomial degree, axis 2, pixel-to-sky

B_0_0 = 0.

B_1_0 = 0.

B_0_1 = 0.

B_2_0 = -2.95818634164737E-08

B_1_1 = -7.01783412603652E-09

B_0_2 = 6.37999469584622E-08

B_3_0 = -1.1830595845234E-12

B_2_1 = -3.6090721607276E-11

B_1_2 = -3.56608103304676E-12

B_0_3 = -3.98995352329986E-11

AP_ORDER= 3 / SIP polynomial degree, axis 1, sky-to-pixel

AP_0_0 = -6.78978013124636E-05

AP_1_0 = -1.98812997398434E-07

AP_0_1 = -1.76803289727692E-08

AP_2_0 = 3.80426138529127E-08

AP_1_1 = -7.39458324637942E-08

AP_0_2 = -3.80036649006719E-08

AP_3_0 = 4.03953162484519E-11

AP_2_1 = 3.29680404894911E-12

AP_1_2 = 3.15972819765301E-11

AP_0_3 = 5.66102140439871E-12

BP_ORDER= 3 / SIP polynomial degree, axis 2, sky-to-pixel

BP_0_0 = -5.9377198142655E-06

BP_1_0 = -2.57587180113559E-08

BP_0_1 = -1.40977118756602E-07

BP_2_0 = 2.96142451666252E-08

BP_1_1 = 7.03465727719834E-09

BP_0_2 = -6.38693631944821E-08

BP_3_0 = 1.18996527009289E-12

BP_2_1 = 3.61456285132576E-11

BP_1_2 = 3.57731662253042E-12

BP_0_3 = 3.99567245871633E-11

SUBEXP = 30 /

CAMID = '1E1E560D0B020900' /

PIERSIDE= 'WEST' /

ADCBITS = 16 / Bit depth of camera sensor ADC in current mode

BIASADU = 121.16593933105469 / ADU for bias level (no photons) at current sett

EGAINSAV= 0.26355 / Electrons per ADU at saved bit depth

RELGAIN = 2.964 / Multiplicative gain relative to minumum

RDNOISE = 1.77 / Read noise in electrons

COLORTYP= 'RGB' /

DATE-END= '2025-09-03T08:23:51.4900595' / System Clock:Est. Frame End

DATE-AVG= '2025-09-03T08:05:30.3788009' / System Clock:Est. Frame Mid Point

JD_UTC = 2460921.837157162 / Julian Date at mid exposure

END

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

polynomials orthogonal over the unit disk

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polynomials orthogonal over the unit disk

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