Orbital Mechanics with Mathcad
Path of Ceres from 1801 January 1 to 1806 May 23. This figure shows what the path of Ceres, as a dwarf planet in the asteroid belt, looked like from the date of Ceresís discovery by Piazzi through its observation by Olbers, Harding, and Bessel in 1805-1806.
"Reconstruction of the 1801 Discovery Orbit of Ceres via Contemporary Angles-Only Algorithms" is the title of a paper that fellow astrodynamicist Dr. Gim J. Der (see
and I prepared for presentation at the Advanced Maui Optical and Space Surveillance Technologies (AMOS) Conference 2016, held in Maui, Hawaii USA, during September 20-23, 2016. To download our paper from the AMOS website, click on
The figure above is excerpted from a 48-inch wide by 36-inch high poster that accompanies our paper. The poster can be viewed by clicking here:
For this analysis we input the 19 complete discovery observations of the first known asteroid, Ceres, to our contemporary initial orbit determination (IOD) and differential correction (DC) computer algorithms in order to solve for the heliocentric ecliptic state vector and osculating orbital elements of Ceres that describe the motion of the dwarf planet in 1801.
A very brief history of the discovery and recovery of Ceres now follows, together with further information relevant to our own analyses and results with the discovery observations.
1. Brief History of the Discovery and Recovery of Ceres
Giuseppe Piazzi discovered Ceres on 1801 January 1 and observed it almost nightly until February 11. Baron Franz Xaver von Zach, editor and publisher of the German-language astronomical journal Monatliche Correspondenz ("Monthly Correspondence", abbreviated below as "MC") published Piazzi's observations in the 1801 September issue of MC on p. 280.
The astronomers Karl Friedrich Gauss, Wilhelm Olbers, Johann Burckhardt, and Piazzi each determined orbital elements and search ephemerides for Ceres as the result of independent calculations that started with Piazzi's discovery observations. Their results were discussed by von Zach in the October, November, and December issues of MC. Also discussed were the cloudy, overcast conditions in Europe during much of the year 1801, and the frustration of the European astronomers in not being able to recover and observe Ceres and update its orbital elements.
Then, finally, on the night of 1801 December 31 - 1802 January 1, von Zach recovered Ceres using only Gauss's search ephemeris. Gauss subsequently became famous as an astronomer as the result of the orbit determination procedure that he conceived and applied in order to determine the orbital elements of Ceres and compute a search ephemeris.
Gauss eventually became regarded, then as now, as one of three greatest mathematicians who have ever lived -- these three greatest mathematicians being, according to Eric Temple Bell (himself an eminent 20th century mathematician): Archimedes, Gauss, and Newton. (See Bell's Men of Mathematics, Chapter 14, for Bell's biography and assessment of Gauss.)
For an excellent mathematical and historical article that examines in greater detail how Gauss became famous as the result of his role in the discovery and recovery of Ceres, see
Donald Teets and Karen Whitehead, "The Discovery of Ceres: How Gauss Became Famous," Mathematics Magazine, Vol. 72, No. 2 (April 1999), pp. 83-93.
In the year 2000, this article won the Mathematical Association of America's prestigious Carl B. Allendoerfer Award for expository excellence.
2. Finding von Zach's Monatliche Correspondenz (MC) Changed our Research Plan
Gim and I originally thought that we would simply vet the 19 complete discovery observations of Ceres using modern computers and our contemporary angles-only algorithms. But in the course of our research we found that von Zach's MC, written in German, had recently become available as a Nabu Public Domain reprint. So we obtained the MC reprint and translated into English the key article from 1801 December, the one with Gauss's search ephemeris on p. 647. That key article, in German, and a page-by-page summary of the facts relevant to our contemporary research, can be viewed from the two links immediately below:
von Zach's Monatliche Correspondenz, Vol. 4 (1801 December), pp. 638-649
Summary of von Zach's MC, Vol. 4 (1801 December), pp. 638-649.
Given Gauss's published search ephemeris, and the obliquities of the ecliptic that we now computed for the dates of the observations, we were able to convert the geocentric ecliptic longitudes and latitudes in the search ephemeris to right ascensions and declinations using the appropriate equations from spherical trigonometry.
Now we were able to compare our contemporary solutions' predicted ephemerides directly with Gauss's own search ephemeris.
There was an unexpected finding. For more about this, download the paginated, pre-publication draft of our paper at
and see the ADDENDUM, pp. 17-19.
3. Batch UPM DC Algorithm (see our AMOS 2016 paper, Ref. 13)
Reference 13 of our AMOS 2016 paper cites two Mathcad 15 worksheets viewable here. The two worksheets, in .pdf format, are:
Batch Least Squares Differential Correction of a Heliocentric Orbit, Part 1: Test Case Specification Worksheet
Batch Least Squares Differential Correction of a Heliocentric Orbit, Part 2: Manual Correction Worksheet
These two worksheets completely specify our Batch UPM DC algorithm as applied to the 17 best observations of Piazzi during 1801 January 1 - February 11.
4. Presentation to the Denver Astronomical Society (DAS)
I have been invited to speak about my AMOS 2016 poster and paper at the DAS general meeting at 7:30 p.m. on Friday evening, April 7, 2017 at the University of Denver's Olin Hall, Room 105. Click on the link below to see my PowerPoint presentation:
DAS 2017_04_07 Presentation.pdf.
This concludes my webpage topic "Reconstruction of the 1801 Discovery Orbit of Ceres." Please click on the link below to go back to the top of this webpage.
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Trajectories of three Newtonian particles numerically integrated using Mathcad Prime 1.0.
"Nonlinear Dynamics Using Mathcad Prime 1.0"
is the title of a presentation that I recorded at PTC World Headquarters in Needham, Massachusetts on March 15, 2011 -- for viewing at the PlanetPTC Virtual Mathcad Event held on April 14, 2011.
My goal in the presentation was to show that, while Mathcad Prime 1.0 is not yet ready to take over the role of Mathcad 15, one can nevertheless use it right now to do some rather sophisticated nonlinear dynamical modeling.
To see what I mean, consider the figure above. It looks like black, red, and blue scribbles as might have been made by a child using colored pens or crayons.
But in fact the figure is the result of integrating numerically the trajectories of three massive particles obeying the three highly-nonlinear, second-order, ordinary differential equations embodied in Newton's universal law of gravitation.
For an animation of the figure, see
For the particulars of my presentation, including all worksheets in both Mathcad and PDF format, visit
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Mathematical Heart Surface plotted via Mathcad 15 and posted to PlanetPTC.
PlanetPTC is the union of new online PTC communities that includes users of the CAD programs Creo Elements/Pro (formerly Pro/ENGINEER) and Mathcad, as developed by Parametric Technology Corporation (PTC) of Needham, Massachusetts USA.
To see a clickable list of all of the PTC communities, click on
To go directly to the PTC Mathcad community, click on
In January 2011, Dan Marotta, the PlanetPTC webmaster, challenged Mathcad users to draw a heart surface such as was shown at the German website
Mathematische Basteleien (Mathematical Tinkerings). I responded with the plot shown above. It is actually a "pointillist" plot, i.e., it is a 3D scatter plot of points lying on the surface of the heart.
The mathematical heart surface depicted above has both bilateral (side-to-side) and dorsal-ventral (back-to-front) symmetry. I needed to take advantage of these symmetries in order that my plot not take too long to calculate in Mathcad. I discuss how I constructed this plot in the Mathcad worksheet posted at
If you do not have Mathcad 15, you can view my worksheet as a PDF file by clicking on Mathematical_Heart.pdf.
Anatomy enthusiasts will note that the mammalian heart has neither of the two true symmetries of the mathematical heart depicted above.
This is because, although the mammalian heart has left and right atria and ventricles, suggesting at least bilateral symmetry, the "blue" (oxygen-depleted) blood returns from the body to the right atrium and is pumped to the right ventricle, and then on to the lungs, while "red" (oxygen-enriched) blood returns from the lungs to the left atrium and is pumped to the left ventricle, and then on to the body via the aortic artery.
Why do I mention the mammalian heart in this context? Because there is actually an animation of a beating mammalian heart in the Mathcad videos at PlanetPTC. To see it, click on the link
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Tautochrone Balls animation featured in PTC Express Newsletter of August 2010.
Four balls, initially at rest, are poised to slide down a curved path, as illustrated in the figure above. Assume that the balls are acted upon by the downward pull of gravity alone, with no rolling friction nor wind resistance.
Which ball will be the first to arrive at the bottom of the curve, at the point x = pi, y = -2?
For the answer, go to the Mathcad 15 video posted at
http://communities.ptc.com/videos/1344. The answer may not be what you expect.
The Mathcad 15 worksheet is available for download from the August 2010 issue of PTC Express. If you do not have Mathcad 15, you can view the Tautochrone Balls worksheet as a PDF file by clicking on Tautochrone_Balls.pdf.
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View from Denver, Colorado at midnight on 2009 February 28. Pointer marks epsilon Aurigae.
Post of March 2011:
Mystery Solved! Thanks to the technical leadership, diligence, and perseverance of Dr. Bob Stencel and Mr. Jeff Hopkins, and with the collaboration of an international team of professional and amateur astronomers, the nature of the stellar system epsilon Aurigae has at last been elucidated.
For full details, go to Dr. Bob's University of Denver (DU) Portfolio page at
and download the PDF file with the title, "IR results on epsilon Aurigae, Jan.2011 PDF". This poster paper, presented at the AAS conference in Seattle, Washington USA in January 2011, is as visually appealing as it is informative.
Post of February 2009:
Epsilon Aurigae excitement builds as the next minimum of this mysterious stellar system nears (August 2009). The German astrophysicist Hans Ludendorff (1873-1941) brought epsilon Aurigae to the attention of the world astronomical community early in the 20th century, with the publication of two seminal papers that characterized the eclipses of 1847-48, 1874-75, and 1901-02.
But Ludendorff's papers apparently had never been translated into English. Therefore, to assist in the work of the Epsilon Aurigae Eclipse Campaign 2009-2011, I have translated from German to English Ludendorff's two main papers on epsilon Aurigae,
"Untersuchungen ueber den Lichtwechsel von epsilon Aurigae," Astronomische Nachrichten (A.N.), Vol. 164, pp. 81-114 (1904), and
"Bearbeitung der Schmidtschen Beobachtungen des Veraenderlichen epsilon Aurigae," A.N., Vol. 192, pp. 389-406 (1912).
My English translations, and the original articles in German, are now available for downloading from Jeff Hopkins's HPOSoft website, at
(Look under the heading, "Conversions of Ludendorff's 1904 and 1912 publications from German to English by Roger Mansfield of Astronomical Data Service".)
For further information about the Epsilon Aurigae Eclipse Campaign 2009-2011, click on Dr. Robert E. Stencel's Epsilon Aurigae Eclipse Campaign Homepage at
And be sure to read Dr. Stencel's feature article, "The Very Long Mystery of Epsilon Aurigae," beginning on p. 58 in the May 2009 issue of Sky & Telescope!
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Predicting Iridium Flares is a webpage that predicted the Iridium flares visible from Boulder, Colorado U.S.A.
during the period Sunday, April 27 through Saturday, May 3, 2008.
This webpage page was prepared for use by attendees at the Division on Dynamical Astronomy (DDA) 2008 Meeting of the American
Astronomical Society (AAS), April 28 - May 1, as hosted by the Southwest Research Institute (SwRI), 1050 Walnut Street,
Suite 300, Boulder, Colorado 80302.
The webpage was provided as a supplement to my DDA 2008 poster presentation, "Predicting Iridium Flares." Click on
to see the Iridium flare predictions that were made for the DDA meeting. (A concise, half-page, printed
summary flyer was also distributed to attendees at the DDA meeting, via the registration desk.)
For more information about the DDA 2008 meeting, click on
To see the followup article, "Predicting Iridium Flares," which was published in the May 2008
issue of PTC Express, monthly newsletter of Parametric Technology Corporation, click on
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Mathcad Worksheets by Astroger describes twelve Mathcad worksheets available for downloading
from Mathsoft's website. These worksheets implement key algorithms in dynamical astronomy and astrodynamics. Go to
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Topics in Astrodynamics describes my astrodynamics textbook of the same name, and provides
resources toward using the book to teach a two-semester course sequence, Astrodynamics I and
Astrodynamics II. Go to
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Orbital Mechanics with Mathcad lists the lesson plan topics for my
five-day, eight-hours-per-day course for space professionals in Colorado
Springs. For further information go to
(Other venues besides Colorado Springs are possible. Inquire at the e-mail address at the end of the astrocourse webpage.)
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Space Ornithology is a webpage that documents my contributions to artificial Earth satellite observation as regards (a) writing and publishing the Space Birds computer program in 1987, and (b) coining the term space ornithology as the study of the space bird population (satellites in low-Earth orbit visible with the naked eye), and (c) publishing sixteen quarterly issues of the Space Ornithology Newsletter during 1988-1991. Go to
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How I Got Started in Dynamical Astronomy and Astrodynamics
In the decade after the launch of Sputnik I, the U.S. Air Force took great strides in space
technology under the leadership of military space pioneers such as General Bernard A. Schriever and Dr. Louis G. Walters (for some recent links mentioning Dr. Walters and the Aeronutronic Division of Ford Motor Company, see the Wikipedia entries
Project_Space_Track (1957-1961), and
New career fields opened up for satellite controllers
and for orbital analysts. Professional space training for these new space career fields was set up at Keesler Air Force Base, Mississippi.
I attended two courses as a part of this new "Cold War" professional space training: the
three-week Space Operations Officer Course and the eight-week Orbital Analyst Course. My
fire for orbital mechanics was kindled at the first course and intensified at the
second. In my post-service career, I actively and successfully sought opportunities to do
orbital mechanics for a living, and to teach orbital mechanics (i.e., astrodynamics) as well.
Much of the orbital mechanics that I was doing applies to natural
celestial bodies and space probes as well as to artificial Earth satellites. And so
my interests began to encompass dynamical astronomy as well as astrodynamics. These
Astroger Webpages are thus devoted to both fields.
*Astroger is the union of the letters in "Astro" and the letters of my first name, "roger". Astroger is pronounced "Astro-jer" (soft "g"). It is actually a word, perhaps a contraction of "astrologer." I take "astroger" not to mean an astrologer, but rather "one who calculates trajectories using the principles of orbital mechanics."
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(c) 2011-2017 by Astronomical Data Service. Last updated 2017 April 30.