Roger L. Mansfield is a space professional with more than 30 years of military, industrial, and academic experience. His personal web page describes just a few of the Mathcad worksheets he has constructed since 1997 to solve problems in the mechanics of Earth orbital, escape, flyby, and interplanetary trajectories. His freely downloadable Mathcad worksheets provide live, graphical examples of many of the algorithms and procedures in his book, Topics in Astrodynamics including Mathcad worksheets, which can be purchased from his Web site.

Most recently Dr. Mansfield presented three workshops at the 17th AAS/AIAA Space Flight Mechanics Meeting, January 31, 2007, Sedona, Arizona, U.S.A. to space scientists and engineers who do orbital mechanics.

Session 1: Mathcad as an Electronic Scratchpad

Attendees constructed Mathcad worksheets that propagated the orbit of a Molniya satellite in the perifocal plane.

Session 2: Mathcad as a Programming Language

Participants used Mathcad's programming constructs to propagate the orbit of a Molniya satellite in Earth-centered, inertial (ECI) space. Mr. Mansfield also described the evolution of Mathcad as an electronic scratchpad (1984-1995) and of Mathcad as a programming language (1996-present).

Session 3: Orbital Mechanics Applications via Mathcad

Mr. Mansfield presented three Mathcad programmed applications:

1. Modeling Extrasolar Planet Orbits

Can a planet be almost as massive as its star? Using Mathcad's animation feature, participants could see that three distinct orbits arise in the case of a secondary, massive planet orbiting its primary star.

2. The Effect of a Purely Radial Impulse

What happens when a spacecraft in a circular orbit undergoes an inward or outward impulsive velocity change? Participants used the "live" features of Mathcad to investigate the possibilities and were able to find both analytical and quantitative answers.

3. Tracking Data Reduction for the Galileo Spacecraft's Earth 1 Flyby

This sequenced pair of Mathcad worksheets (an "initiator" and an "iterator") provided the full details of Mr. Mansfield's analysis of tracking data from Galileo's Earth 1 flyby. Participants used Mathcad to illustrate the tracked ephemeris points in ECI space.

How do you maneuver a spacecraft orbiting the earth? 

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 that you can only operate the motor by firing it in one or more brief, but powerful pulses in any direction. This may seem contrived, but in actual fact, orbital maneuvers are modeled and analyzed in this way. Real world motor firings are controlled in such a way as to result in idealized impulsive velocity changes.

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 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 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 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 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"? Do you go straight up or straight down, or something else? The Mathcad worksheet linked below answers this question.