Mathcad Worksheets by Astroger
My career, starting in 1967, has been mainly about developing astronomy & astrodynamics algorithms and applications in FORTRAN, BASIC, Pascal, and C. And leading/teaching groups of others to do the same.
Why I Adopted Mathcad. Along the way, in 1997, I became enamored with Mathsoft Mathcad PLUS 6 for Macintosh. This was when Mathsoft introduced the capability to create user-defined functions in Mathcad worksheets. Now, with the capability to construct these user-defined functions, Mathcad worksheets became computer programs. Programs analogous to FORTRAN main programs and subroutines, or C/C++ main functions and subordinate functions.
Mathcad Worksheet Advantages. I realized then that Mathcad worksheets also provide an effective way to document highly technical algorithms, right in the notations of mathematics. And furthermore, that these functions are live, in the sense that the user-defined Mathcad functions can be modified right in the Mathcad worksheet itself, and executed there as well.
PTC Took over Mathcad from Mathsoft. But much has happened since the late 1990s. Mathcad PLUS 6 for Macintosh was the last version of Mathcad offered for Macintosh computers. Parametric Technology Corporation of Needham, Massachusetts (now just PTC, and now with headquarters in Boston rather than in Needham) purchased Mathcad from Mathsoft.
Mathsoft's Mathcad Evolved to PTC's Mathcad Prime. PTC then started a new incarnation of Mathcad called Mathcad Prime, but still supported the original version of Mathcad for Windows up to Mathcad 15. Mathcad Prime was very primitive at first, yet I supported it as a beta tester and also even once as an alpha tester. I went to Needham in March of 2011 and recorded a presentation about the usefulness of Mathcad Prime 1.0 in my own work.
Mathcad 15 and Mathcad Prime 10. Now Mathcad 15 is no longer available. PTC introduced Mathcad Prime Version 10 earlier this year (2024). Although even now, in Mathcad Prime 10, some of the original features of Mathsoft Mathcad 6 have so far not been included. Yet I continue my support of Mathcad Prime.
Mathcad 15 to Mathcad Prime 10. Indicative of my continuing support of Mathcad Prime is that I recently updated all twelve of my Mathcad Worksheets by Astroger to Mathcad 15. Then, using the Mathcad Prime 10 converter, I converted them all to Mathcad Prime 10.
How to Obtain My Worksheets
The .pdf (portable document format) versions of all twelve of my Mathcad Prime 10 worksheets are treated in the links below. Open them with the Adobe Reader and save them to your Windows computer.
1. Ephemeris of a Comet Via Uniform Path Mechanics (UPM)
2. Herget's Method for an Asteroid
3. Orbit Propagation via State Space Analysis
4. Effect of a Radial Impulse on a Circular Orbit
5. Herget's Method with Cassini's Earth Flyby
6. Sun Altitudes for Sextant Practice
7. Sun-Sight Solutions Without Tables
8. Rectilinear Two-Body Motion ("Earth Falls Into the Sun")
9. Gauss's Angles-Only Method with "Killer Asteroid"
10. Tracking Data Reduction for Galileo's Earth 1 Flyby
Finally, in Author's Background, at the very end of this webpage, I give a brief outline of my space career and my related interests for the reader who wishes to know more about my professional background and credentials.
1. Ephemeris of a Comet Via Uniform Path Mechanics (UPM)
This worksheet uses the orbit of Comet Hale-Bopp to illustrate UPM, a mathematical procedure for calculating the orbital path of a space object. This procedure has the same form (i.e., is "uniform") for all possible path eccentricities.
The two original Mathcad worksheets created in 1998, hm1.mcd (the "initiator") and hmc.mcd (the "iterator") used astrometric observations of the orbit of the critical-list minor planet 1035 Amata to illustrate Herget's Method of computing a preliminary orbit. The observations were taken by USAF 2Lt Dan Burtz.
Regarding Herget's method,
Project Pluto provides a Windows application called FIND_ORB that lets you "find" an orbit from the astrometric observations of a minor planet or comet. FIND_ORB uses Herget's method to solve for the preliminary orbit, but its capabilities do not stop there. It can fit a more precise orbit to the observations using the method of batch least squares differential correction. You can specify which planets are presumed to be "perturbers" of the comet's or asteroid's orbit as fitted to the observations. To download FIND_ORB, go to the Project Pluto website at
http://www.projectpluto.com.
This worksheet uses the Earth escape trajectory of the Near-Earth Asteroid Rendezvous (NEAR) spacecraft, following its launch on 1996 February 17, to illustrate how state space analysis can be applied to the problem of propagating a trajectory through space and time.
This worksheet considers the effect of a 1.0 km/sec radial impulse on
the orbit of an artificial Earth satellite at geostationary altitude (35786 km),
both outward and inward.
This set of worksheets applied Herget's method to determine the flyby path of the Cassini/Huygens space probe, launched 1997 October 15, following its return to Earth for a gravity assist on 1999 August 18.
This worksheet provided an algorithm for predicting the sun's altitude and azimuth with respect to an Earth-fixed observer. Given your latitude and longitude, and an input file of local times of interest on a date of interest, sunalts.mcd would calculate for you the sun's altitude and azimuth at those times.
Thus, if you took a sequence of sun altitude measurements with a sextant, the worksheet would compute for you what results you should get, as a check on your skill in making "sun shots".
You needed not to have an interest in celestial navigation to have found this worksheet useful. Suppose you wanted to know the sun's altitude and its azimuth direction from your house, at the beginning of each daylight hour on some date of interest. The sunalts.mcd worksheet would have allowed you to calculate these data quickly and accurately. All you had to do was to (1) use, say, Windows Notepad to set up the local times of interest in the file times.prn, (2) change the year, month, and day in the worksheet itself, and (3) also change the latitude, longitude, height, and time zone offset in the worksheet. Then clicked on "Calculate Worksheet" in Mathcad's Math menu and scrolled down to the page containing the worksheet's results.
This worksheet presented a novel two-sight fix method and applied it to the angles.prn data in Richard R. Shiffman's "Sextant Noon-Day Sun Sightings" (Shiffman's worksheet could also be found in the Mathcad Library). It referenced the sunalts.mcd worksheet*, so you needed to have also sunalts.mcd and its two input files, times.prn and angles.prn, in order for this worksheet, sunsols.mcd to have worked.
(*If the reference in the very first page of the sunsols.mcd worksheet wasn't working, you went to Mathcad's Insert menu and clicked on "Reference", typed in the name "sunalts.mcd", clicked on the "use relative reference" box, and then clicked "O.K". Then went to Mathcad's Math menu and clicked on "Calculate Worksheet". The reference to sunalts.mcd in the sunsols.mcd worksheet should then have worked.)
Because sunsols.mcd referenced sunalts.mcd, it was able to take advantage of all of the procedural functions defined there: functions for calculating solar position in the ECI reference frame, referred to the mean equator and equinox of J2000.0; aberration (light-time correction); precession; nutation; ECI-to-topocentric transformation; refraction, etc. Thus the navigation solution was effected without reference to the Nautical Almanac, as well as without reference to sextant sight reduction tables.
If Earth were to stop in its orbit and fall into the sun, for how many days would it fall? Greg Neill, editor and publisher of The Orrery, posed and solved this problem in issue #43 (December 2001). Upon reading Greg's solution, I was reminded of a paper I presented at a conference of the American Institute of Aeronautics and Astronautics (AIAA) in 1986. This worksheet solved the problem of "how long would it take for Earth to fall into the sun" by means of the theory that I presented in that 1986 AIAA paper.
Asteroid 1997 XF11 received much attention in March 1998 when astronomers determined that it might hit Earth in October 2028. After the alarm was sounded, it was a matter of just one day before new information came in and it could be determined that a direct hit was not really very likely after all. In this worksheet I determined an orbital solution, via Gauss's angles-only method of orbit determination, using three observations taken in December 1997. In the worksheet I compared my solution with the definitive solution published by the Minor Planet Center in Cambridge, Massachusetts.
The Galileo mission to Jupiter was one of the most exciting and successful endeavors in
the history of robotic planetary exploration. Launched on 1989 October 18, the
Galileo spacecraft orbited the Sun for more than six years, receiving one gravity
assist from Venus and two from Earth, before achieving insertion into jovicentric
orbit on 1995 December 7. After almost eight years of orbiting Jupiter and
exploring the planet, its rings, and its moons, the Galileo spacecraft flew into
Jupiter on command, and burned up in Jupiter's atmosphere, on 2003 September 21. Thus
ended the Galileo mission.
I was privileged to be able to assist in the near-realtime reduction of tracking data from Galileo's Earth 1 flyby on 1990 December 8. I published a related article in 1993 ["Algorithms for Reducing Radar Observations of a Hyperbolic Near-Earth Flyby," Journal of the Astronautical Sciences (April-June 1993), pp. 249-259].
The two worksheets in this Mathcad application, Gd1.mcdx and Gdc.mcdx, rigorously solved for the flyby trajectory of the Galileo spacecraft. They agreed perfectly with the results I presented in my 1993 paper. (I did the calculations in the paper in 1992 by means of a Turbo Pascal 1.1 program that I developed and ran on an accelerated Apple Macintosh SE.)
The tracking data were reduced by opening worksheet Gd1.mcdx and clicking on "Calculate Worksheet". Then opened Gdc.mcdx and clicked on "Calculate Worksheet". Scrolled down to page 14 of the Gdc.mcdx worksheet to see the RMS error matrix. The RMS error for the first iteration of the differential correction (DC) was 5.043 km. Clicked a second time and watched the RMS error go to 4.584 km on the second iteration. Noted that the DC had converged, on the second iteration, to the orbital solution that could be seen by scrolling down to pages 17 and 18. The results on page 18 showed that the Galileo spacecraft flew to within about 961 km of Earth's surface at its Earth 1 flyby.
What does calculating the photoperiod have to
do with astronomy? The answer is this: the
photoperiod is the duration of daylight each
day, i.e., the time of sunset minus the time of
sunrise. It is a function of both the geographic
latitude (or the astronomical declination) and
the season, expressed as a count of days since
the beginning of the year.
This contribution to the Mathcad Library
consisted of two worksheets, as follows.
1. Calculating Photoperiod - II.mcd derived
simple formulas for the photoperiod, in
minutes, and its diurnal rate, in minutes per
day, as functions of declination and days
since the beginning of the year.
See also the chapter on photoperiodism in
Plant Physiology, a textbook by
Frank B. Salisbury and Cleon W. Ross
(Wadsworth, 1992). The chapter on photoperiodism
contains graphs prepared by Michael J. Salisbury
circa 1983, using equations and data provided by
Astroger back then. The "Calculating Photoperiod"
worksheet generated these same graphs using new,
analytical (vs. numerical) photoperiod and
photoperiod rate formulas as derived in the
worksheet.
2. Photoperiod 2003.mcd performed rigorous
astronomical calculations to provide and plot
the photoperiod, as a function of latitude and
day count, for various latitudes. This worksheet
was prepared in December 2003, in response to an
e-mail request from Dr. Lilian B. P. Zaidan,
plant physiologist at the Instituto de Botanica,
Sao Paulo, Brazil.
2005 was the "World Year of Physics" and it marked the 100th
anniversary of Albert Einstein's publication of papers on the
photoelectric effect, Brownian motion, and the special theory
of relativity. All four papers on these three topics appeared
in the year 1905.
But modern physics, i.e., modern quantum physics really began
in 1901, when Max Planck propounded the notion that material
bodies, but especially blackbodies*, emit and absorb
thermal radiation in discrete quanta of energy, rather than
continuously.
Planck's hypothesis of quantized absorption and emission of
radiation made it possible for him to derive a radiation law
that applies to blackbody emission at all wavelengths and all
frequencies, a universal law that succeeds in spectral regions
where the prior radiation laws of Rayleigh, Jeans and Wien had
failed. Planck received the Nobel Prize in physics in 1918 for
his quantum theory of radiation.
The Mathcad 12 worksheet, "Modeling Blackbody Radiation,"
revisited how Max Planck integrated the blackbody radiation
curve for an arbitrary Kelvin temperature, T, over all
possible wavelengths of thermal emission, to arrive at the
Stefan-Boltzmann law. The Maple symbolic processing
capability of Mathcad was invoked at key points of the
derivation and Bernoulli numbers were used to evaluate the
infinite series that is crucial to the derivation. Finally,
Mathcad's X-Y Plot capability was used to plot the blackbody
radiation curve for 2.725 degrees Kelvin.
*A blackbody is an ideal body in thermal equilibrium that absorbs all
incident radiation and re-emits it as light energy distributed over
the entire electromagnetic spectrum.
(Note: the worksheet macupm.mcd is no longer available for download from the Mathcad Library. If you are a Power Mac user and have Mathcad PLUS 6 for Power Macintosh, e-mail me at the address at the very end of this web document and I'll send macupm.mcd back to you as an e-mail attachment.)
The worksheets use Comet Hale-Bopp (C/1995 O1) for their example, and they generate ephemeris points at 5-day intervals during the period 1997 March 17 - 1997 May 16. If you have Mathcad, PLUS 6 or some later Professional version (e.g., 7, 8 or 2000), you can use the live worksheet to generate an ephemeris for any comet that you have orbital elements for, for any time period of interest. (Actually, Mathcad Explorer will even let you modify the worksheets to run for your own test case, but you cannot save your modifications unless you have Mathcad PLUS 6 or later.)
The UPM theory is the result of the author's research into universal variables methods of orbit propagation, for application to gravity-assist flybys (swingbys) of Earth by manmade spacecraft such as Galileo, NEAR, and Cassini. It is intended to be used with orbits having eccentricity 0.95 or greater. Such orbits are typically Earth escape and flyby or swingby trajectories.
The full UPM theory is implemented in the tracking data reduction software of U.S. Space Command's Space Defense Operations Center (SPADOC) near Colorado Springs. It was used to reduce tracking data on the Galileo spacecraft's Earth 2 flyby trajectory (1992 December 8) and on the Near-Earth Asteroid Rendezvous (NEAR) spacecraft's Earth escape trajectory (1996 February 17). Other recent or upcoming applications of the UPM theory are the NEAR spacecraft's swingby of Earth on 1998 January 23 and the Cassini/Huygens spacecraft's gravity-assist flyby of Earth expected on 1999 August 18.
For more on the full UPM theory, see "Algorithms for Reducing Radar Observations of a Hyperbolic Near-Earth Flyby," Journal of
the Astronautical Sciences (April-June 1993).
In Herget's time, images of comets and asteroids were obtained via photographic astrometry. Photographic astrometry employs a telescope with a camera that positions a glass photographic plate in the focal plane to record an image of a comet or asteroid against the star background. This method, which still finds use today for high-resolution work, is a tedious, time consuming, and expensive process. CCD astrometry facilitates the taking of multiple images of the same object (e.g., comet or asteroid) on multiple nights via CCD camera, telescope, and computer. The CCD camera replaces the photographic plate in the focal plane with a CCD array that does not have to be replaced after an image is recorded, nor does it require post-collection chemical development of the image, as does a photographic plate.
How the CCD imaging process works is that the CCD camera uses the charge-coupled device array in the focal plane to record images that are stored on the computer as digital image files. Desktop computer programs such as Astrometrica and Charon permit the reduction of multiple CCD image files, collected over successive nights, to topocentric right ascension and declination measurements. Herget devised his method expressly to work with these topocentric right ascension and declination measurements, and as many of them as are available.
Herget's method works by assuming that the direction lines -- from observer to asteroid -- for the first and last observations (obs) are exact, and by improving initial estimates of the observer-to-asteroid distances for the first and last obs. The two estimated distances are fitted to the remaining observations by treating their (the remaining observations') residuals as functions of the two estimated distances. The partials of the residuals with respect to the two estimated distances are calculated by numerical differencing and the fitting process is non-linear least squares.
J.M.A. Danby gave Herget's method a particularly lucid exposition in the second edition of his textbook, Fundamentals of Celestial Mechanics (Willmann-Bell, 1988, pp. 235-238). This book is also highly recommended for its seminal exposition of Karl J. Stumpff's universal variables theory as based upon Stumpff's c-functions.
Project Pluto
has implemented Herget's method in the freely downloadable FIND_ORB (16-bit Windows, introduced in March 1997) and FIND_O32 (32-bit Windows, introduced in July 1998). FIND_ORB solves for a preliminary orbit by Herget's method (by taking "Herget steps") and optionally improves the preliminary orbit by a full, six-element differential correction, taking into account gravitational perturbations by the major planets and Earth's moon (each iteration of the differential correction is a "Full step").
Don't limit your study of Herget's method to my worksheets. Get a copy of Danby's book from Willmann-Bell at
http://www.willbell.com
and download FIND_ORB or FIND_O32 from the Project Pluto website.
I should mention that I found out about FIND_ORB when it was described by Stuart J. Goldman in the June 1998 "Software Showcase" department of Sky and Telescope magazine. Note that while FIND_ORB is an easy-to-use program with a simple Windows interface, its "engine" is comprised of computer codes of great astronomical and mathematical sophistication.
Project Pluto publishes the highly successful Guide and Charon programs for CCD astrometry, as advertised in Sky and Telescope (see the January 1999 issue, p. 139). Project Pluto is to be commended for making the FIND_ORB program available to the astronomical community at no charge via the Internet.
Mathcad worksheets for "Herget's Method for an Asteroid", hm1.mcd and hmc.mcd, should be available for downloading from MathSoft's website sometime in January 1999. These two worksheets
(The UPM-extended Gaussian iteration assumes that Gauss's hypergeometric X-function is a "quotient of c-functions", while the original Gaussian iteration assumes that Gauss's hypergeometric X-function is a "hypergeometric series".)
The first worksheet, hm1.mcd, is an "initiator", in that it sets up the test case input for hmc.mcd. Thus hm1.mcd is run before hmc.mcd, and its window should be kept open in the background when the hmc.mcd window is opened.
Hmc.mcd is the "iterator" in that each time you click on "Calculate Worksheet" in the Mathcad Math menu, hmc performs an iteration of Herget's method. To see this happen, scroll down to page 11, where the RMS matrix is accumulated, and watch the behavior of the RMS (root mean square of the residuals) as you iterate. For the test case supplied via hm1, hmc should converge on the eighth iteration to an RMS of 0.209 kilometers.
I note in closing that although "Herget's Method for an Asteroid" takes up 23 printed pages, total, for the two worksheets, it illustrates only a portion of the total body of orbit computational code that Project Pluto has built into FIND_ORB.
(a) to help with the tracking and cataloging of asteroids and comets, which it is now known have impacted Earth in the past, and may do so again at some future date, with possibly disastrous consequences for life on Earth;
(b) for the sheer intellectual challenge: you want to apply your knowledge of orbital mechanics, telescopes, computers, and CCDs, or you want to learn more about these areas of high technology in an "active" manner.
In what follows I assume the reason for your interest in comet and asteroid orbit determination is (a) above; if it is (b), then you are at least interested in (a)!
In CCD astrometry, as opposed to CCD astronomy, you (a) image comets and asteroids -- this much is in common with CCD astronomy; (b) reduce the images to topocentric right ascension (R.A.) and declination (DEC) measurements; (c) determine their orbits using the topocentric R.A. and DEC measurements, and/or (d) report your observations to the IAU's
Minor Planet Center in Cambridge, Massachusetts.
Thus in CCD astrometry your "make your own observations". You need a program such as the Guide program of
Project Pluto
to help you determine where to look, and you need a program such as Astrometrica or Project Pluto's Charon to reduce the CCD observations that you make. You can report your observations to the IAU's Minor Planet Center in Cambridge, Massachusetts, which uses them to determine and publish an improved orbit for the object. Recently now, thanks to Project Pluto, you can use their "observations to orbits" program, called FIND_ORB, to check the quality of your topocentric R.A. and DEC observations before you send them to the Minor Planet Center.
The Minor Planet Center is invaluable if you wish to do any serious work with comet and minor planet observations and orbits. The full web address, again, is
Minor Planet Center.
I should note here that the mathematics of comet and asteroid orbits is pretty much the same as that of interplanetary spacecraft when in coast phase. So if you are interested in the dynamics of interplanetary spacefaring, work with comet and minor planet orbits is not wasted.
You can find articles about CCD astronomy and the advertisements of vendors of telescopes and CCD cameras in the pages of Sky & Telescope. Richard Berry's book, Introduction to Astronomical Image Processing, published by
Willmann-Bell is a good reference here.
If you are already doing CCD astrometry, then you are already at least imaging comets and asteroids and reducing their images to topocentric right ascension and declination measurements. You may have read about the FIND_ORB program and how it can be used to determine an orbit from your observations (see Sky & Telescope, June 1998 Software Showcase, "Orbits from Observations").
Since downloading FIND_ORB myself, I have been carrying out an ongoing e-mail dialogue with Project Pluto, the FIND_ORB program publisher. I have checked out and verified all aspects of the program, and have used the program to check my own work.
FIND_ORB can be relied upon as an independent means of verifying the quality of your own CCD astrometric observations before sending them to the Minor Planet Center. But it is good for much more than that. If you generate simulated observations of an interplanetary trajectory, FIND_ORB can determine and propagate that trajectory. Given the high quality of implementation, FIND_ORB can be used to generate and/or check your own interplanetary orbital data. It is sure to become the standard for CCD observing experts who want to extend their expertise to include orbit determination from CCD observations.
My own area of mathematical expertise encompasses the mathematics that is contained in FIND_ORB. I am doing research in this area, and I wish to help interested folks to understand how to use (and how not to misuse) the capabilities that this program provides. One project I have just completed in this area is my Mathcad worksheet on Herget's method. Herget's method is implemented as a preliminary orbit determination method in FIND_ORB; my Mathcad worksheets provide a detailed exposition of the method.
Why did I choose to work with Mathcad? After many years of programming mathematical algorithms in Fortran, BASIC, Pascal, and C, and many years of supervising others doing the same, I have learned well that the codes themselves are not the best way to teach the math. Yet documenting the math via word processors was a task just as difficult as doing the original programming until Mathcad PLUS 6 came along (previous versions of Mathcad did not have the power of procedural function programming).
While I do not claim that developing and/or reading Mathcad worksheets is a "piece of cake", I do say that Mathcad is the best way that I have found so far to specify, communicate, and validate orbital mechanics algorithms. So this is why I have adopted MathSoft's Mathcad as the medium by which I communicate my astronomical algorithms for orbit determination and orbit improvement in CCD astrometry.
Of course I encourage all who are interested in orbit determination and orbit improvement to follow my example: get Mathcad and work with it. But I recognize that not all who are interested in CCD astrometry will want to make this investment of time. If you cannot justify getting and working with Mathcad, then at least download FIND_ORB from Project Pluto's website.
The article shows how state space analysis is actually used to integrate the orbit of an artificial Earth satellite. It provides a QuickBASIC 4.5 program (Figure 2 in the article) and its output (Figure 1) for the NEAR Earth escape test case, as reference material for validation of the "orbit propagation via state space analysis" mathematics. The text of the Mathcad worksheet orbpro.mcd is also provided in the body in the article. The article shows that, by including calculations of the 6-by-1 vector of perturbative accelerations, P(X), where X is the state (cartesian position and velocity), the two-body state equations, Xdot = S(X) X, can be transformed into the perturbed state equations, Xdot = S(X) X + P(X).
Subscriptions to The Orrery are presently $12.00 per year in the United States, and I believe that back issues are available. For further information write directly to Greg Neill at the address given, or e-mail him at
gneill@allstream.net.
(Note: Greg Neill ceased publication of The Orrery some time ago, and I have lost touch with him. But I leave in the references to him out of respect for his work. As a side note, Caxton C. Foster actually started The Orrery. There is a book by Caxton Foster, with that title, now available from
Willmann-Bell.
I was a contributor to Caxton Foster's book -- see Section 14.7, "Tracking the NEAR Launch.")
Anyone who has studied the topic of orbital maneuvers knows that if you fire a pulse along your instantaneous velocity vector, in the direction of motion, then you will fall downward along an elliptical orbit whose apogee point is where you fired the pulse, and whose perigee point is 180 degrees away, and at a lower altitude than the fixed altitude of your original circular orbit. And if you fire a pulse of just the right magnitude again when you get to perigee, once more in the same direction as your instantaneous velocity vector, you will enter into a new circular orbit of lower altitude than your original orbit. The two-impulse sequence for maneuvering into the lower orbit is called a Hohmann transfer.
Similarly, if you fire a pulse along your instantaneous velocity vector, but opposite to your direction of motion, you will climb upward along an elliptical orbit whose perigee point is where you fired the pulse, and whose apogee point is 180 degrees away, and at a higher altitude than the fixed altitude of your original circular orbit. If you now fire a pulse of just the right magnitude again when you get to apogee, once more in the direction opposite to your direction of travel, you will enter into a new circular orbit of higher altitude than your original orbit.
The second of the two types of Hohmann transfer just described is quite familiar to orbital analysts, because it is used to maneuver Earth satellites into high-altitude orbits. The original orbit is called the parking orbit, the elliptical half-orbital segment is called the transfer orbit, and the final orbit is called the final orbit (of course!).
Historically, impulses along the instantaneous velocity vector have proved to be the
most useful kind of orbital maneuver. But what do you think would happen if you fire
a pulse purely along the radius vector, and perpendicular to the instantaneous
velocity vector (assuming again a circular orbit), either "straight up" or "straight
down"? Will you then go straight up or straight down, or something else? The Mathcad
worksheet impulse.mcd answers this question.
But in this pair of worksheets gh1.mcd and ghc.mcd we apply Herget's method to geocentric orbital motion, rather than to heliocentric orbital motion. The CCD astrometric observations were taken in this case by Gordon J. Garradd at Loomberah, Australia, and by Rob McNaught at Siding Spring Observatory, Australia. Bill J. Gray made the data available at the Project Pluto website, along with his own solution using his FIND_ORB program (click on
http://www.projectpluto.com/cassini.htm
to see Bill's analysis).
What is most remarkable about the results is that the spacecraft Cassini was on the outbound hyperbolic arm of its Earth flyby trajectory, and at a distance that ranged from 50 - 350 Earth radii from the geocenter, when the 11 observations actually used in the Mathcad worksheet solution were taken. (The Herget's method solution uses 11 observations taken over a span of three days, and uses the two-body UPM theory as the orbit propagator.)
Despite its use with only a relatively small number of CCD (angles-only) observations, and using only two-body mechanics (UPM), Herget's method converged to a solution yielding a time of perigee passage within about four minutes of JPL's definitive result, and a closest approach distance (1155 km) within about 16 km of JPL's final result (1171 km). Bill Gray's FIND_ORB solution, which accounted for perturbations by the sun and moon, did even better. Using all 15 CCD astrometric observations, Bill got to within better than a minute of JPL's time of perigee passage, and to within about 5 km of JPL's final closest approach distance estimate (1176 km vs. 1171 km).
This pair of worksheets is the basis for a paper that I presented at the American Astronomical Society's Division on Dynamical Astronomy (AAS/DDA) conference at Yosemite National Park, April 9-12, 2000. The title of the paper is "Astrometry-Based Analysis of Cassini's Earth Flyby", and I presented it at 9:00 a.m. on Wednesday, April 12. To request a copy of the paper, e-mail me with your postal mailing address.
There is a non-trivial modicum of dynamical astronomy in the calculations. Here is an outline of the steps for each time of interest:
a. Calculate the sun's true Earth-centered, inertial (ECI) cartesian coordinates, referred to the mean equator and equinox of the J2000.0 epoch.
b. Convert the sun's coordinates from true to astrometric by correcting for aberration (the light-time correction).
c. Apply a precession matrix to refer the coordinates to the mean equator and equinox of date.
d. Apply a nutation matrix to refer the coordinates to the true equator and equinox of date.
e. Calculate the observer's ECI cartesian coordinates and subtract these from the sun's apparent coordinates in order to obtain the sun's cartesian coordinates in the topocentric, horizon-referenced reference frame.
f. Transform the sun's topocentric, horizon-referenced cartesian coordinates to altitude, azimuth, and topocentric distance in the local horizon reference frame.
g. Correct the sun's apparent altitude for atmospheric refraction.
Harrison was trying to win the Longitude Prize, created by the British Parliament in 1714 in order to stimulate a solution to one of the British Navy's (and every other contemporary navy's) most enduring problems: the lack of a way to compute a ship's longitude accurately when it is out of sight of land. In order to win the prize, Harrison invented a timepiece that could keep time accurately on a ship in storm-tossed seas, and through extremes of barometric pressure, temperature, and humidity.
When one knows the longitude of the home port and the mean solar time difference between the home port and the location at sea, one can immediately calculate the local longitude. This is because Earth rotates 15 degrees of longitude per mean solar hour. One simply multiplies the time difference, in hours, by the conversion factor of 15 degrees per hour to arrive at the longitude difference. (If the "home port" were Greenwich, England, and one's marine chronometer kept Greenwich mean time, the longitude difference at each point of a voyage to the West Indies would be precisely the west longitude, since the longitude at Greenwich is zero degrees.)
Thus Harrison was fully entitled to the prize no later than March 1762, following successful demonstration of his timekeeper's requisite accuracy during a sea trial consisting of a voyage from England to Jamaica and back. (John Harrison's son, William, had accompanied the Harrisons' fourth marine chronometer, H-4, on the voyage. H-4 lost only five seconds of time after 81 days at sea.)
Yet Nevil Maskelyne, as a member of the Board of Longitude (the Board of Longitude constituted Parliament's duly appointed, independent panel of judges for the Longitude Prize), managed to keep the Harrisons from being awarded the Longitude Prize for another eight years, while he continued to work on his own astronomical solution to the longitude problem. We should note that Maskelyne's astronomical solution was the forerunner of the modern celestial approach: sight reduction tables, used together with the annual Nautical Almanac to reduce sextant sightings. (Today's Nautical Almanac, as issued jointly and annually by Great Britain and the United States, descends directly from Maskelyne's Nautical Almanac and Astronomical Ephemeris of 1767. So, while Dava Sobel's book turns most of us into partisans for Harrison and against Maskelyne, we should acknowledge that we are beneficiaries of both mens' genius.)
Celestial navigation is, quite simply, the use of sextant sightings of celestial bodies to calculate one's latitude and longitude. The bodies useful for celestial navigation are the sun and moon, the brighter planets Venus, Mars, Jupiter, and Saturn, and the 57 navigation stars (the celestial positions of 57 bright stars, well distributed over the whole celestial sphere, are tabulated in the Nautical Almanac for use in celestial navigation).
The marine navigator of today still uses a sextant to measure the apparent altitudes of the celestial bodies visible from his or her geographical location, and then uses sight reduction tables, together with the annual Nautical Almanac, to work up a position solution. Indeed, no finer instrument has emerged for this purpose than the modern sextant, virtually unchanged since World War II, which was independently invented in 1730 by an Englishman, John Hadley, and an American, Thomas Godfrey.
In my worksheet sunsols.mcd, I apply a two-star fix method which I devised in 1982 (it was published in Navigation, Journal of the Institute of Navigation, at that time -- see sunsols.mcd for the reference). I test the method with "perfect" sun altitude measurements from the sunalts.mcd worksheet, then move on to the realworld, i.e., "imperfect" sun altitude measurements made by Richard R. Shiffman for his "Sextant Noon-Day Sun Sightings" worksheet. Using Mathcad's "regress" and "interp" functions, I smooth Shiffman's actual measurements and then show that pairwise solutions are possible by applying my two-star fix method to adjacent pairs of smoothed sun altitude measurements.
Inspection of a plot of the smoothed measurements vs. the actual measurements suggests that Shiffman's measurement taking skill improved noticeably as he took more and more measurements. From this arises my conclusion that the last pair of adjacent, smoothed measurements yields the best solution available from the totality of data, i.e., from the 30 local time and sun altitude measurement pairs.
Planck's radiation law is not just of historical interest.
In 1964 Arno Penzias and Robert Wilson discovered radio noise
emanating from all directions of the sky that is consistent
with thermal emission from a blackbody at an equilibrium
temperature of just a few degrees Kelvin. They deduced in 1965
that this radio noise is the cosmic microwave background
(CMB). For this they were awarded a Nobel Prize in 1978 [1].
By the 1960s there were two competing theories of the origin
of the cosmos, the "steady state" theory and the "Big Bang"
theory. Existence of the CMB was predicted by the Big Bang
theory, but not by the steady state theory. So when the CMB was
found by Penzias and Wilson, most physicists and astronomers
came to accept the Big Bang theory and to reject the steady
state theory.
More recently, the Cosmic Background Explorer (COBE) spacecraft
measured the CMB in all directions of space, from space (i.e.,
from Earth orbit). Its measurements of energy density vs.
frequency fit almost perfectly on Planck's radiation curve
for a blackbody at 2.725 degrees Kelvin. But small "ripples"
in energy density were in fact found; these are believed to
be evidence of variations in the early universe's energy
density. Since these variations are thought to have seeded
star and galaxy formation, it would have been a setback for
the Big Bang theory had they not been found.
REFERENCE
[1] Mather, John C. and Boslough, John, The Very First Light,
Basic Books, New York, 1996; pp. 49-50 and 64. John Cromwell
Mather was the original proposer and project scientist for
the COBE mission. The COBE satellite was launched on November
18, 1989.
Astroger has pursued the following activities from 1967 to the present.
Air Force Officer. Astroger began his space career in 1967 as a weather satellite
orbital analyst for the Defense Meteorological Satellite Program. He taught
mathematics at the U.S. Air Force Academy and, after serving a total of seven
years in the Air Force, worked for 21 more years on Air Force space systems
developmental projects. He earned a Bachelor of Science degree with high honors in
chemistry from the University of Cincinnati and a Master of Arts degree in
mathematics from the University of Nebraska at Omaha.
Space Engineer. Astroger has held positions as (1) ballistic & orbital systems
engineer, (2) 427M Space Computational Center (SCC) team leader for space
applications software maintenance, (3) Space Defense Operations Center (SPADOC)
Block 4B section supervisor for astrodynamic software development, and (4)
SPADOC principal engineer for space surveillance applications. He held all of
these positions while working for a single company that went by the names
Philco-Ford, Aeronutronic Ford, Ford Aerospace & Communications Corporation,
and Loral Aerospace Corporation during the years that he was employed
there (1974-1995).
Astroger was present in the NORAD Cheyenne Mountain Complex to assist with
tracking data reduction for the Earth 1 and Earth 2 flybys of the Galileo
spacecraft (December 1990 and December 1992, respectively), for the Mars Observer
launch and Earth escape (September 1992), and for the NEAR launch and Earth
escape (February 1996).
Educator. In the course of his space career, Astroger has taught classes to cadets, to graduate space engineers, and to active duty Air Force operations personnel. He has presented papers at AAS/AIAA astronautics conferences and his work has been published in the Journal of the Astronautical Sciences.
Educational Publisher. Astroger founded Astronomical Data Service (ADS) in 1976 to provide custom-prepared educational publications to science teachers. ADS distributed a mail order sales catalog annually in the fall. ADS accepted and filled orders by U.S. postal mail. Science teachers could write to P.O. Box 885, Palmer Lake, CO 80133, call (719) 597-4068 and leave a message, or click on the e-mail link at the bottom of this web page to request a catalog. ADS ceased mail-order operations at the end of 2018, but the telephone number, postal mail address, and email address below are still active.
Assistant Professor. Astroger held an appointment as Assistant Professor at
CU-Colorado Springs during 1996-98. He taught astrodynamics and numerical methods
to engineers working at the Lockheed Martin Astronautics Waterton Canyon facility,
and then continued to advise his former students as they proceeded through the Master of
Engineering in Space Operations program. Lockheed Martin engineers working at Waterton
Canyon have been intimately involved with space missions such as Mars Pathfinder, Mars
Global Surveyor, and Cassini; they build and operate the Titan and Atlas launchers.
Book Author. Astroger's 378-page book,
Topics in Astrodynamics,
commenced publication on October 6, 2003.
Amateur Radio Hobbyist. Astroger is a licensed amateur radio technician (KB0VJS) and operated his own ground equipment for direct image readout from U.S. and Russian polar-orbiting weather satellites. He received his best images using a quadrifilar helix antenna built by his son, Jase (AB9BI), from plans in QST magazine.
What is an "astroger"? See
The Astroger Webpages.
(c) 1997-2024 by Astroger. Last updated 2024 August 08.
The two "Ephemeris of a Comet" worksheets, winupm.mcd (for Windows users) and macupm.mcd (for Power Mac users) are Mathcad PLUS 6 worksheets that let you generate the Earth-relative sky coordinates of any celestial object, provided that you have orbital elements for the object. The worksheets first became available for downloading from MathSoft's website in September 1997.
Paul Herget, director of the Cincinnati Observatory from 1943 to 1978, and first director of the IAU's Minor Planet Center (1947-1978), published in 1965 an ingenious method for determining a preliminary orbit for a comet or asteroid, from all available angles-only observations -- not from just three angles-only observations, as is typical of other preliminary orbit determination methods. Herget's method lends itself perfectly to the task of determining comet and asteroid orbits from observations taken via modern CCD ("charge-coupled device") astrometry.
Advances in CCD astronomy and in desktop computing technology have made it possible for amateurs to image comets and asteroids, and to determine their orbits. Why do so? I offer two principal reasons:
Where do I get observations of comets and asteroids?
I'm Interested in CCD astronomy, how do I get started?
I'm already doing CCD astrometry -- what do you have to say to me?
State space analysis is a way of modeling dynamical systems. It employs systems of first-order, ordinary differential equations in matrix form. It has been adopted with great success in the field of control systems engineering. I have written an article about this worksheet, orbpro.mcd, for publication in The Orrery, a bimonthly publication about "Models of Astronomical Systems" edited and published by Greg Neill / 4541 Anderson / Pierrefonds, Quebec / Canada H9A 2W6.
Suppose that you are in a spacecraft in a perfectly circular Earth orbit, and that you can maneuver by means of a powerful rocket motor that you can point in any direction you wish. Suppose also, for the sake of simplicity, that you can only operate the motor by firing it in one or more brief, but powerful pulses in any direction. This seems contrived, but in actual fact, orbital maneuvers are modeled and analyzed in this way. Realworld motor firings are controlled in such a way as to result in idealized impulsive velocity changes.
In the pair of worksheets associated with "Herget's Method for an Asteroid" we solved for the orbital elements of the asteroid (1035) Amata using Dan Burtz's CCD astrometric data, as collected by Dan at the U.S. Air Force Academy's Observatory while he was a graduate student at the University of Colorado, Colorado Springs.
Recall that the sun's altitude is the angle, in degrees (0 degrees = on the horizon, 90 degrees = directly overhead), that it makes with the plane of the local horizon, while its azimuth is the angle that it makes with the direction "true north", as measured in the horizon plane clockwise from true north (0 degrees = due north, 90 degrees = due east, 180 degrees = due south, and 270 degrees = due west). Given that you know your latitude and longitude, how can you accurately calculate the sun's altitude and azimuth, i.e., its position in your local horizon reference frame, at times of interest during a day of interest? This is the problem that sunalts.mcd solves for you.
My most recent spell of interest in celestial navigation, which has culminated in the worksheets sunalts.mcd and sunsols.mcd, was inspired by Dava Sobel's best-selling Longitude (Walker Publishing Company, 1995). Her book is about John Harrison, inventor the first handheld marine chronometer ("H-4", completed in 1759), and his struggles with Nevil Maskelyne, creator of The Nautical Almanac and Astronomical Ephemeris (the annual astronomical tables for navigation, first published in 1766 for use in the year 1767).
E-mail: astroger@att.net
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