We hosted a special session with R. Asa Gordon, a retired astrodynamicist who worked at NASA’s Goddard Space Flight Center for 25 years. Asa shared about his career, his leap from punch‑card FORTRAN on mainframes to a Timex/Sinclair 2068, and how constrained hardware inspired a practical, elegant approach to predicting satellite orbits.
From Jim Crow schooling to Goddard: the formative years
Asa began by grounding his path in history. He grew up and received his early education under Jim Crow laws, attended Hampton Institute on out‑of‑state aid (because the state of Georgia would not admit him to the University of Georgia), and majored in physics with a minor in mathematics. After college and years of civil rights activism, he found his way to Howard University’s computer lab and learned FORTRAN on punch cards and tape — the start of his programming journey.
Goddard Space Flight Center recruited him as a programmer/mathematician during the Apollo era, and he landed in the Orbital Analysis/Prediction branch under African‑American leadership — a dramatic early contrast to the broader workplace. Asa credits this support for enabling him to transition from punching cards and running prewritten routines to learning the underlying theory and then inventing his own methods.
Problem solving during Apollo: penumbra, umbra, and radio blackout
One of Asa’s first assignments at Goddard was to analyze radio blackout during the Apollo missions when the spacecraft passed behind the Moon. The branch’s trajectory determination system modeled shadow as a simple cylindrical region, and it produced coarse predictions for acquisition and loss of signal (AOS/LOS).
Asa visualized the geometry differently. He described making models with styrofoam spheres and a flashlight to see penumbra and umbra effects, then bringing geometry and physics to bear. Instead of treating eclipse shadow as a cylinder, he modeled partial illumination (penumbra) and total shadow (umbra). He implemented a routine to compute the fraction of solar flux available to a spacecraft as discs intersect and used that to predict AOS/LOS more accurately for the Apollo 8 mission.
“I went to an art store, got some styrofoam balls and a flashlight… I imagined myself being a little particle out there, the satellite in space… it gave me an idea.”
On Christmas Eve the calculation ran. When Asa’s code predicted the exact second of signal acquisition and loss, the prediction matched reality. The result became a milestone in his career and in how his colleagues perceived his capabilities.
Thinking outside the box: analytical over brute force
Asa repeatedly emphasized that his training in physics — taught to visualize and seek closed‑form or analytical insight — helped him avoid always reaching for brute‑force numerical integration. He preferred deriving compact, interpretable formulas that could be evaluated efficiently.
This mindset led to two major themes in his work:
- Semi‑analytical orbit theory: A hybrid approach that mixes analytic averaging and perturbation theory with lightweight numerical iteration, capturing both short‑period oscillations and long‑term secular trends efficiently.
- Pseudo‑physical drag modeling: A phenomenological secular model for drag that averaged energy loss per orbit, avoiding the need for large, detailed atmospheric lookup tables on tiny computers.
Those choices made it possible to get mainframe‑level accuracy on far smaller machines.
Why the Timex/Sinclair 2068 mattered
Asa’s real breakthrough in democratizing orbital mechanics came when he ported his semi‑analytical propagator to the Timex/Sinclair 2068. The reasons this specific home computer worked for his purposes included:
- Improved floating point and precision: The TS2068’s BASIC ROM and floating‑point implementation reduced cumulative rounding drift that plagued earlier micros.
- Expanded RAM: With 48 KB of RAM, the 2068 could hold larger lookup tables and intermediate orbit elements, making in‑machine ephemeris generation feasible.
- Higher resolution graphics: The 2068 offered improved plotting that let students and hobbyists visualize ground tracks and perturbation effects immediately.
In Asa’s words, the timing was perfect: the 2068 arrived just as his ideas had matured and as the hardware was finally capable enough to realize them on a microcomputer.
Technical innovations: making drag and perturbations practical on an 8‑bit machine
Asa emphasized two technical aspects that were key to why his approach worked.
Pseudo‑physical secular drag
Rather than using large atmospheric density tables and expensive exponential modeling, Asa implemented a secular drag relationship directly into the semi‑major axis update routine. The concept was simple and powerful: average the net energy loss (or Δa per orbit) over each revolution, fit that average as a secular (slowly varying) term, and apply a recursive update to the semi‑major axis.
This avoids complex per‑step density lookups and keeps runtime compact. The tradeoff is conceptual simplicity for slightly coarser modeling detail — but in practice that averaging captured the real net effect of drag and other retarding influences (including aspects of solar‑driven atmosphere changes), and it produced surprising accuracy.
Hybrid analytic/numeric event prediction
Asa modified Kepler’s equation with Brown perturbation theory terms and devised a closed‑form recursive scheme to predict special events — perigee/apogee passages, nodal crossings (ascending/descending nodes), and maximum/minimum sub‑latitudes — without brute‑force stepwise integration.
By combining Kepler constraints with perturbation series and a simple contraction‑mapping proof of convergence for his iterative solver, he achieved efficient event detection suitable for onboard use or quick ground predictions. This explicit link between physical insight and computation gave operators more intuition about how drag, oblateness (J2), and other perturbations affect timing.
Validation: simulated tests and real tracking data
Asa tested his methods two ways:
- Simulated model validation: He used full numerical integration including multiple zonal harmonics and drag models to generate ephemerides, then fit his analytic model to that simulated arc and projected forward to compare errors.
- Real tracking data: He used NORAD tracking observations (real satellite tracking) and differential correction techniques based on a truncated J2 secular state matrix to fit ephemerides and evaluate out‑of‑arc prediction errors.
The results were striking: multi‑day ephemeris projections from the Timex/Sinclair implementation matched or outperformed full numerical integration in several test cases, with sub‑kilometer to few‑kilometer in‑track error over days — comparable to what mainframes produced for those scenarios.
“It turned out to be as good as results other folks were getting on their IBM mainframe.” — Asa Gordon
Asa attributes much of this success to averaging the messy, rapidly varying influences into a stable secular term: “I’m averaging all that out… I don’t care if you’re facing full on or sideways — I just average it every period.”
Impact: democratizing astrodynamics and anticipating on‑board autonomy
By proving that professional‑grade orbit propagation could run on an 8‑bit microcomputer, Asa’s work lowered the barrier between institutional astrodynamics and private/educational experimentation. The practical outcomes included:
- Educators and hobbyists could experiment with semi‑analytical theories on a sub‑$200 machine and visualize ground tracks, perturbations, and orbit events.
- His approach anticipated today’s on‑board orbit determination concept — the idea that satellites can predict and schedule their own passes or even operate autonomously without constant ground contact.
- His work informed later efforts to manage tracking gaps (e.g., TDRS / tracking data relay satellite systems) and to design satellite systems resilient to limited ground coverage.
Asa also reflected on a broader lesson: constraints often spur ingenuity. Limited memory, limited numeric precision, and constrained CPU cycles forced elegant, physically meaningful algorithms rather than brute‑force numerics that grow up around abundant computing power.