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 Ceres’s discovery by Piazzi through its observation by Olbers, Harding, and Bessel in 18051806.
"Reconstruction of the 1801 Discovery Orbit of Ceres via Contemporary AnglesOnly Algorithms" is the title of a paper that fellow astrodynamicist Dr. Gim J. Der (see
Der Astrodynamics)
and I prepared for presentation at the Advanced Maui Optical and Space Surveillance Technologies (AMOS) Conference 2016, held in Maui, Hawaii USA, during September 2023, 2016. To download our paper from the AMOS website, click on
amostech.com/TechnicalPapers/2016/Poster/Mansfield.pdf.
The figure above is excerpted from a 48inch wide by 36inch high poster that accompanies our paper. The poster can be viewed by clicking here:
ceres_1801_poster.
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 Germanlanguage 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 Carl 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. 8393.
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 anglesonly 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 pagebypage 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. 638649
Summary of von Zach's MC, Vol. 4 (1801 December), pp. 638649.
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, prepublication draft of our paper at
mansfield_der_amos_2016_09_15_preprint.pdf
and see the ADDENDUM, pp. 1719.
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 the information about my presentation to the Denver Astronomical Society.
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5. Presentation to the Rocky Mountain Section of the MAA
I have been invited to speak at the 2018 meeting of the Rocky Mountain Section (RMS) of the Mathematical Association of America (MAA), to be held April 1314 at the University of Northern Colorado, Greeley, Colorado.
The topic of my presentation is "MODERN SPACE SITUATIONAL AWARENESS  It Began with Piazzi, von Zach, and Gauss in 1801." This presentation was suggested by paragraph 7 of my AMOS 2016 paper, "7. SPACE SITUATIONAL AWARENESS: THEN AND NOW." Click on the link below to see my PowerPoint presentation:
MAA/RMS Presentation Greeley April 2018.pdf.
On p. 17 of this presentation I refer to a Mathcad worksheet that I constructed to show how to convert the geocentric ecliptic longitudes and latitudes in Gauss's search ephemeris into right ascensions and declinations for plotting on a star chart. Click on the link below to see that worksheet:
Ceres_Search_Ephemeris_by_Gauss_1801_December.pdf.
The star charts on pages 3, 19, and 20 of my presentation did not take well to being pasted into PowerPoint as .jpg images, followed by conversion to .pdf, even with highquality print selected as a conversion setting. Click on the link below to download the three star maps in a resolution suitable for printing on a laser printer:
SKY MAPS 2018.pdf.
In the "Tools" menu of Adobe Reader, be sure to specify Rotate > Counterclockwise to view the star maps in their intended landscape orientation.
Map 1 is a rectangular projection of the entire celestial sphere onto a plane. The stars are grouped into constellations by pattern lines. Pattern lines facilitate memorization of the appearance of the night sky, while the constellations themselves provide a framework for locating the brighter planets, stars, and other celestial objects.
Map 1 is "unbiased" because it is never completely upside down. For if north is up and you rotate the map 180 degrees, south is now up and you can still read the map's upper celestial hemisphere labels rightside up (try it).
Note the right ascension (RA) scales on the upper and lower borders of Map 1. These allow you to determine the local apparent sidereal time at your longitude by the following two rules.
Rule 1  Locate Celestial Meridian at Midnight on Given Date
To find the RA of the stars crossing your celestial meridian at midnight on a given date, use the dates and the RA scale for your geographical hemisphere (northern or southern based upon your latitude) as a guide to estimating the placement of a line perpendicular to the RA scales and passing through the given date on both RA scales. This perpendicular line is your celestial meridian line at midnight. Note that each "tick" on the RA scale corresponds to about four calendar days as well as to exactly 15 minutes of RA.
Rule 2  Locate Celestial Meridian at Given Hour on Rule 1 Date
To locate the celestial meridian at a given hour on the date used for Rule 1, move the celestial meridian line west (decreasing RA) one hour of RA for each hour earlier than midnight. Move the celestial meridian line east (increasing RA) one hour of RA for each hour later than midnight.
Since the celestial meridian is the extension of the great circle of your geographical longitude onto the celestial sphere, it marks the local apparent sidereal time. Its location in the sky at any point in time tells you what stars and constellations you can see as you look south (northern hemisphere observer) or north (southern hemisphere observer) along the celestial meridian.
Map 2 is a polar equidistant projection of the north celestial hemisphere onto a plane, plus another 45 degrees of southern declination. It is therefore good for northern hemisphere observers.
Map 3 is a polar equidistant projection of the south celestial hemisphere onto a plane, plus another 45 degrees of northern declination. It is therefore good for southern hemisphere observers.
The 57 celestial navigation stars are labeled on all three maps, where visible, plus other bright stars as indicated on the maps.
In Memoriam
Dr. Barbara D. Buchalter (1929 April 13  2005 November 26)
Professor of Mathematics, University of Nebraska at Omaha
"Dr. Buchalter taught me Complex Variables and Conformal Mapping"
This concludes the information about my presentation to the Rocky Mountain Section of the MAA.
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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 highlynonlinear, secondorder, ordinary differential equations embodied in Newton's universal law of gravitation.
For an animation of the figure, see
http://communities.ptc.com/videos/1867.
For the particulars of my presentation, including all worksheets in both Mathcad and PDF format, visit
http://communities.ptc.com/message/162082#162082.
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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
http://communities.ptc.com/.
To go directly to the PTC Mathcad community, click on
http://communities.ptc.com/community/mathcad.
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 (sidetoside) and dorsalventral (backtofront) 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
http://communities.ptc.com/message/158690#158690.
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" (oxygendepleted) blood returns from the body to the right atrium and is pumped to the right ventricle, and then on to the lungs, while "red" (oxygenenriched) 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
Heartbeat.
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Tautochrone Balls animation featured in PTC Express Newsletter of August 2010.
Question
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?
Answer
For the answer, click on this link:
Tautochrone_Balls.avi. The answer may not be what you expect!
(The video has been tested with Microsoft's Windows Media Player and Apple's QuickTime.)
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
https://portfolio.du.edu/pc/port?portfolio=rstencel
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 (18731941) 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 184748, 187475, and 190102.
But Ludendorff's papers apparently had never been translated into English. Therefore, to assist in the work of the Epsilon Aurigae Eclipse Campaign 20092011, 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. 81114 (1904), and
"Bearbeitung der Schmidtschen Beobachtungen des Veraenderlichen epsilon Aurigae," A.N., Vol. 192, pp. 389406 (1912).
My English translations, and the original articles in German, are now available for downloading from Jeff Hopkins's HPOSoft website, at
http://www.hposoft.com/EAur09/EAurPEPRef.html.
(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 20092011, click on Dr. Robert E. Stencel's Epsilon Aurigae Eclipse Campaign Homepage at
http://mysite.du.edu/~rstencel/epsaur.htm.
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
http://skywatcher.astroger.com/
to see the Iridium flare predictions that were made for the DDA meeting. (A concise, halfpage, 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
http://dda.harvard.edu/.
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
http://www.imakenews.com/ptcexpress/e_article001094851.cfm?x=bcG4481,b3jsqcsB,w.
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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
http://mathcadwork.astroger.com/.
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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 twosemester course sequence, Astrodynamics I and
Astrodynamics II. Go to
http://astrotopics.astroger.com/.
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Orbital Mechanics with Mathcad lists the lesson plan topics for my
fiveday, eighthoursperday course for space professionals in Colorado
Springs. For further information go to
http://astrocourse.astroger.com/.
(Other venues besides Colorado Springs are possible. Inquire at the email 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 lowEarth orbit visible with the naked eye), and (c) publishing sixteen quarterly issues of the Space Ornithology Newsletter during 19881991. Go to
http://spacebirds.astroger.com/.
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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 (19571961), and
1st_Aerospace_Surveillance_and_Control_Squadron).
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
threeweek Space Operations Officer Course and the eightweek Orbital Analyst Course. My
fire for orbital mechanics was kindled at the first course and intensified at the
second. In my postservice 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 "Astrojer" (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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Webpage Credits
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(c) 20112018 by Astronomical Data Service. Last updated 2018 October 17.
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