Minor Planet Astrometry With CCD Images Glenn A. Snyder, Project CLEA CLEA Summer Workshop June 20, 2010 Outline I. Introduction II. Mathematics of Astrometry III. Methodology IV. Reference Catalogs V. Minor Planet Orbits I. Introduction Astrometry: “Positional Astronomy” “…the branch of astronomy concerned with the precise measurement of positions of objects on the celestial sphere.” The oldest branch of astronomy. Two kinds - absolute and relative (“differential”). Fundamental (Absolute) Astrometry Measure positions over entire sky (including Sun). Determination of Fundamental (Inertial) Reference frame. Determination of Astronomical Constants. Timekeeping. Traditionally done with Meridian Circle. Very few sites now doing this. “Differential” Astrometry Positions are measured relative to reference stars in the same field whose positions are known. Applications include parallax, proper motion, astrometric binaries, positions of comets and minor planets. Effects of precession, nutation, aberration etc. nearly constant across field and can (usually) be ignored. Equipment (Historically) Long-focus refractor, or “Astrographic Camera” Large, fragile photographic plates. Bulky, expensive blink comparators and measuring engines. Tedious error-prone measuring process and reductions via calculator, math tables, etc. Only a few dedicated astronomers & sites. Discovery of Pluto The Pluto Discovery Telescope Clyde Tombaugh Blinking Plates at Lowell Observatory, 1930 Equipment (Today) CCD chip and personal computer Taking & storing images vastly simplified. Limiting magnitude improvements allow use of smaller telescopes. Blinking and measuring via computer. Reductions are lighting fast and accurate. Meaningful astrometry now possible from small institutions, students, amateurs. Catalog Improvements “Paper” catalogs difficult to work with, bright stars only, often no charts. Proliferation of digital catalogs simplifies selection of standards. Great improvement in limiting magnitude. Small size of CCD fields places heavier requirements on catalogs. II. Mathematics of Astrometry Standard Coordinates Our problem is to relate spherical coordinates (Right Ascension and Declination) projected onto a flat image plane with linear measurements in that plane. We begin by defining Standard Coordinates. Consider an orthogonal coordinate system in the image plane of the telescope, with its origin at the intersection of the optical axis with the image plane, and axes running East - West and North - South. The Standard Coordinates ( X, Y ) of point (P) are then measured in this system in units of the focal length. Standard Coordinates II Let ( , ) sky coordinates of an object at point P ( 0 , 0 ) projected coordinates of optical axis (field center). We then obtain the following relationships : X 0 arctan cos( ) Y sin( ) 0 0 sin( 0 ) Y cos( 0 ) arcsin 2 2 1 X Y Standard Coordinates III Similarly : X Y cos( ) sin( 0 ) cos( 0 ) cos( ) cos( 0 ) sin( 0 ) sin( ) sin( 0 ) cos( ) cos( 0 ) cos( 0 ) sin( ) cos( 0 ) cos( ) cos( 0 ) sin( 0 ) sin( ) Standard Coordinates IV We can therefore compute standard coordinates for objects whose RA & Dec we know (reference stars), and... Compute the RA & Dec of an object if we know its standard coordinates. But... How do we go from image positions measured in pixels to standard coordinates? Instrumental Errors Displacement of Origin: Yields constant differences between measured & true coordinates. Error of Orientation: The x and y axes of the measurements will be rotated by some angle from true N-S, E-W. Non-Perpendicularity of Axes: The axes of measurement will not be strictly orthogonal. Scale Errors: The standard coordinates are expressed in terms of the focal length, which will not be constant, and may differ in x and y. Plate Solution All of the errors just discussed are accounted for by solutions of the form : X ax by c Y dx ey f where ( X , Y ) are the standard coordinates and ( x, y ) are the measured coordinates in any convenient units. The constants ( a, b, c, d , e, f ) are referred to as the " Plate Constants". Plate Solution II The goal of the plate solution is to determine the plate constants. We do this by measuring the positions of the reference stars, whose standard coordinates we can compute. Since we must determine 6 constants, we must measure at least 3 reference stars (each star yields a pair (x,y) of measurements). Plate Solution III In practice, more than the minimum # of reference stars should be used. The plate constants are then determined via least squares. In addition to strengthening the solution, this gives residual information that can be used to eliminate bad reference stars. More than 20-25 reference stars is probably overkill (if not impossible). Instrumental Errors II Plate Tilt: Errors due to non-perpendicularity of the image surface to the optical axis can be shown to be quadratic in nature, and are not accounted for by the 6constant solution. Others: Other potential non-linear errors include sphericity of the focal surface and coma. Plate Solution IV The non - linear error terms will generally be very small for CCDs. If desired, we can account for them by solutions of the form : X ax by cxy dx ey f 2 2 Y gx hy ixy jx ky l 2 2 Since we now have 12 plate constants to determine, this solution requires a minimum of 6 reference stars. Centering Errors Errors due to miss-centering on the star images should average out in the solution and be reflected in the residuals. But… The target is frequently much fainter than the references and thus be more likely to have a greater than average centering error. Automated centering should minimize centering errors for all but the most experienced measurers. III. Methodology General Technique Blink images, identify object. Select reference stars, get solution. “Improve” the solution. Repeat for each image. Assemble observations and report to the Minor Planet Center (MPC). Entire sequence supported by CLEA Toolkit software. Blinking Select alignment stars that are well exposed, well separated diagonally. Adjust image contrast for visibility. Blink area can be selected for magnification. Selecting Reference Stars Should cover a good range in both dimensions on the image. Ideally should surround the unknown. One or two close to the unknown in brightness is useful for error assessment. Sometimes have to live with what you get. CLEA Astrometry Toolkit Software Demo Getting Started Visit MPC Website via link in Toolkit. Follow links “How Do I Report Material to the MPC?” http://cfa-www.harvard.edu/iau/mpc.html “Observation Format” Read (as a minimum): “Guide to Minor Body Astrometry” “Format for Optical Astrometric Observations of Comets, Minor Planets and Natural Satellites” “Packed Provisional Designations” Reporting to the MPC A specific format is required - supported by CLEA Toolkit software. First time? - need site ID. Save report, insert in e-mail message to MPC. Must be in-line, MPC will not open attachments. Preserve text format, fixed font, no wrap. (May have to force e-mail software to do this.) “Junk” rating in acknowledgement! A New Observation Format Current format dates from 1940s, designed for 80 column punched cards. New format described in “The New MPC Observation Format” 132-column records, some multiple. Larger ID fields, greater precision, error estimates, etc. Records split after column 66 to avoid “butchering” by e-mail software. A New Observation Format - II CLEA Toolkit now includes the new format for output as an option. BUT… Has not been verified. MPC has yet to provide examples to check against or verification of submitted samples. June 1, 2006 was set as conversion date. (Now cancelled. “…details to be provided later.”) Not clear when MPC will begin accepting observations in new format - both formats will be accepted for “a period of time”. IV. Reference Catalogs Reference Star Errors Random: Random errors in the catalog positions are likely to be small, and are averaged out by the least squares method. Systematic: Catalog positions can have systematic errors that vary from the center to the edge of the original plates, and from field to field. Since these may be as large as 3”, they can clearly affect the final accuracy. Reference Star Errors II Proper Motions: Until recently, the most commonly used catalogs were made from plates taken decades ago, and did not include proper motions. This frequently resulted in large reference star residuals. New Catalogs now available have greatly reduced systematic errors, & include proper motion. “Obsolete” Catalogs The HST Guide Star Catalog (GSC) The USNO Precision Measuring Machine Project (PMM) Catalogs A1.0, A2.0, SA1.0, SA2.0 Distributed on CD-ROM, none currently available from original source. Biggest problem is age of plate material combined with lack of proper motion data. MPC no longer favors use of these catalogs for submitted measurements. HST Guide Star Catalog Number of Stars: ~19,000,000 Limiting Magnitude: V=~16, but many omissions Source: North - Palomar Schmidt plates (1982) South - UK Schmidt plates (1975,82) Availability of the GSC on 2 CD-ROMs made PC-based CCD astrometry possible. USNO PMM Catalogs Large, fast, highly precise measuring engine for photographic plates. Deep, dense stellar catalogs by digitization of major photographic surveys. Versions: A1.0, A2.0, B1.0 Subset versions: SA1.0, SA2.0 http://ftp.nofs.navy.mil/projects/pmm USNO A2.0 Catalog Number of Stars: 526,230,881 Limiting Magnitude: B~21, R~20 (detection in both colors required for inclusion) Source: North - Palomar Sky Survey I (1950s) South - UK SCR-J, ESO-R Surveys (1980s) Media: 11(!) CD-ROMs (Last PMM catalog available on CD-ROM.) On Intl. Earth Rotation Service (ICRS) system Earlier A1.0 on GSC system USNO A2.0 Catalog II Availability: CD-ROMs no longer available from USNO. Catalog is accessible on-line via NVO-Compliant Web Services. Advantages are completeness & limiting magnitude. Biggest drawback is age of plate material (no proper motions). MPC no longer favors use of this catalog, or its predecessors/subsets (A1.0, SA1.0, SA2.0) USNO SA2.0 Catalog Spatially sub-sampled version of A2.0. Uniform “grid” of stars in intermediate magnitude range. Number of Stars: 54,787,624 Magnitude Range: ~14.0 <= B <= 19.0 Media: 1 CD-ROM (now download only). Useful primarily as adjunct to GSC (more stars, fainter magnitude). New Catalogs USNO CCD Astrographic Catalog (UCAC) USNO B1.0* (NOMAD)* *Available via NVO-Compliant Web Services. Toolkit provides real time on-line access. USNO CCD Astrographic Catalog (UCAC) Homogeneous observations - same telescope & detector for entire sky (starting in Southern Hemisphere). Magnitude range 8-16 (passband between V & R). Includes proper motions(!) 20 mas accuracy (10<m<14), 70 mas at m=16. “Photometry is poor, with errors on the order of 0.1 to 0.3 magnitude in a single, non-standard color.” UCAC Catalog II UCAC1 - partial coverage of Southern Sky UCAC2 - declinations -90 to +40..+50. 3 CDs – no longer available from USNO. Bright Star Supplement (BSS) also available. 430,000 stars from Hipparchos & Tycho-2 UCAC3 - Full sky coverage. Released Aug. 2009 on 2-sided DVD. B, R, I photometry added from SuperCosmos project, J, H, K from 2MASS. http://www.usno.navy.mil/USNO/astrometry/o ptical-IR-prod/ucac USNO B1.0 1,000,000,000 Entries Positions, magnitudes(B,R,I) and proper motions 80 GBytes Available by download only, not circulated on CDROM/DVD. Toolkit accesses via Web Service. Catalog Summary CLEA Format GSC plus USNO SA2.0 Convenient: 1 on HD + 1 CD-ROM Useful for charts when no Web access available. USNO B1.0 Best choice for all purposes if Web access available. Only choice for faint magnitudes. UCAC3 Accuracy is excellent, probably the best. Download from CDS, but not directly as Web service. Limitation is magnitude limit (~16.5). NOMAD Naval Observatory Merged Astrometric Dataset 100 GBytes, 1.1 billion stars. Astrometry with Proper Motions plus Photometry (B,V,R,J,H,K) Source Catalogs: Hipparcos, Tycho-2, UCAC2, YellowBlue 6, USNO-B1 plus 2MASS. Toolkit (& VIREO) access via Web Service. Not a “compiled” catalog. http://www.nofs.navy.mil/nomad.html Photometry Catalog magnitudes (from photographic plates) are not very useful as standards for accurate photometry. Accuracy of photographic photometry at best is 0.1 magnitude (this isn’t best). In some cases passbands vary from plate to plate. In most cases calibrations not good. These are not photometric catalogs. Use R magnitudes with unfiltered CCD. V. Minor Planet Orbits “Keplerian” Orbital Elements 6 elements total plus associated date (epoch) 2 elements define size and shape of ellipse 3 elements define position of ellipse in space 1 element defines position of body in ellipse Orbital Elements II The " Classical" Orbital Elements : a : Semi - Major Axis e : Eccentricity i : Inclination : Longitude of Ascending Node : Argument of Perihelion T : Time of Perihelion Orbital Elements III Other Forms of the Elements Perihelion Distance : q a(1 e) Longitude of Perihelion : Mean Anomaly at Epoch : M True Longitude at Epoch : L Orbital Elements IV Osculating Elements: a set of elements defining the two-body orbit corresponding to an object’s state vector at a particular instant of time. The osculating orbit is expected to be a good approximation to the object’s true path for (some) period of time. Epoch of Osculation: the time associated with a set of osculating elements. Minor Planet Orbital Elements Two “databases” available for download. Osculating elements for over 500,000 minor planets. Both revised daily to add new discoveries and orbit updates. All osculation epochs are near the current date. CLEA Toolkit works with either or both for identification of “discoveries” and observation planning, including potential targets and finding charts. Orbit Databases Lowell Database: ASTORB.DAT ftp://ftp.lowell.edu/pub/elgb/astorb.html Best information on observing priorities Minor Planet Center: MPCORBcr.DAT ftp://cfa-ftp.harvard.edu/pub/MPCORB/ Fewer orbits. Better information on orbit “classes”. (Download usually significantly faster.) Toolkit includes links to both. Orbit Determination At least 3 nights observations required to compute orbital elements. Even a very crude orbit is (much) better for planning additional observations than mean motion. An excellent program FIND_ORB is available as shareware from Project Pluto. CLEA Toolkit interfaces easily with FIND_ORB. Project Pluto Website http://www.projectpluto.com (Link now in Toolkit.) Was supplying recalibrated GSC and/or software to do your own conversion. Also loaning copies of USNO A2.0 for copying. FIND_ORB Program. Discovery Statistics 1801: (1) Ceres Discovered. (2) Pallas, (3) Juno, (4) Vesta added by 1807. 1972: 1779 Numbered 1999: 50,000 Identified, 10,000 Numbered 2001: 125,000 Identified 2004: 251,884 Identified, 85,117 Numbered 2010: 518,834 Identified, 241,562 Numbered “The increasing number of new asteroids will eventually overwhelm observers who do the follow-up.” “What Can I Contribute?” Automated searches have SEVERELY reduced chances of new discoveries. Best chances at very faint magnitudes and/or far from the ecliptic. But… A lot of follow-up observations are required to firmly establish orbits. Object must be recovered at 3 oppositions for permanent number & name. Toolkit (& MPC) will identify potential targets for any night & location. A fertile field for student projects!