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!