In June 2014 while on assignment for the Applied Technology Institute in Riva, Maryland,
Logsdon and his professional colleague, Dr. Moha El-Ayachi, a professor at Rabat, Morocco,
taught a group of international students who were flown into the United Nations Humanitarian
Services Center in Brindisi, Italy. The students came in from such far-flung locales as Haiti,
Liberia, Georgia, Western Sahara, the South Sudan, Germany, and Senegal to learn how to
better survey land parcels in their various countries. Studies have shown that if clear,
unequivocal boundaries defining property ownership can be assured to the citizens of a Third-
World Country, financial prosperity inevitably follows. By mastering modern space-age
surveying techniques using Trimble Navigation’s highly precise equipment modules, the
international students were able to achieve quarter-inch (1 centimeter) accuracy levels for
precise benchmarks situated all over the globe.
This was Logsdon’s second year of teaching the course in Brindisi and the Applied Technology
Institute has already been invited to submit bids for another, similar course with the same two
instructors for the spring of 2015. The students who converged on Brindisi were all fluent in
English and well-versed in American culture. Their special skills were especially helpful to their
instructors, Tom and Moha, who trained them to use the precisely timed navigation signals
streaming down from the 31 GPS satellites circling the Earth 12,500 miles high.
The DOD’s Request for Proposal for the GPS navigation system was released in 1973.
Rockwell International won that contract to build 12 satellites with the total contract value of
$330 million. Over the next dozen years, the company was awarded a total of $3 billion in
contracts to build more than 40 GPS navigation satellites. Today 1 billion GPS navigation
receivers are serving satisfied users all around the globe. The course taught by Tom and Moha
covered a variety of topics of interest to specialized GPS users: What is the GPS? How does it
work? What is the best way to build or select a GPS receiver? How is the GPS serving its user
base? And how can specialize users find clever new ways accentuate its performance?
The GPS constellation currently consists of 31 satellites. That specialized constellation provides
at least six-fold coverage to users everywhere in the world. Each of the GPS satellites transmits
precisely timed electromagnetic pulses down to the ground, that require about one 11th of a
second to make that quick journey. The electronic circuits inside the GPS receiver measure the
signal travel time and multiply it by the speed of light to obtain the line-of-sight range to that
particular satellite. When it has made at least four ranging measurements to a comparable
number of satellites, the receiver employees a four-dimensional analogy of the Pythagorean
theorem to determine its exact position and the exact time. This solution utilizes four equations
in four unknowns: the receiver’s three position coordinates and the current time. The GPS
system must keep track of time intervals to an astonishing level of precision. A radio wave
moving through a vacuum travels a foot in a billionth of a second. So an accurate and effective
GPS system must be able to keep track of time to within a few billionths of a second. This is
accomplished by designing and building satellite clocks that are so accurate and reliable they
would lose or gain only one second every 300,000 years. These amazingly accurate clocks are
based on esoteric, but well-understood principles, from quantum mechanics. Despite their
amazing accuracy, the clocks on board the GPS satellites must be re-synchronized using
hardware modules situated on the ground three times each and every day.
The timing measurements for the GPS system are so accurate and precise Einstein’s two
famous Theories of Relativity come into play. The GPS receivers located on or near the ground
are in a one-g environment and they are essentially stationary compared the satellites whizzing
overhead. A GPS satellite travels around its orbit at a speed of 8600 miles per hour and the
gravity at its 12,500-mile altitude above the earth is only six percent as strong as the gravity
being experienced by a GPS receiver situated on or near the ground. The difference in speed
creates a systematic distortion in time due to Einstein’s Special Theory of Relativity. And the
difference in gravitational attraction creates a systematic (and predictable) time distortion due to
Einstein’s General Theory Of Relativity. If the designers of the GPS navigation system did not
understand and compensate for these relativistic time-dilation effects, the GPS radionavigation
system would, on average, be in error by about 7 miles. Fortunately, today’s scientists and
engineers have gradually developed a firm grasp of the mathematics associated with relativity
so they are able to make extremely accurate compensations to all of the GPS navigation
solutions. The positions provided by the GPS, for rapidly moving users such as race cars and
military airplanes, are typically accurate to within 15 or 20 feet. For the stationary benchmarks of
interest to professional surveyors, the positioning solutions can be accurate to within one
quarter of an inch, or about one centimeter.
Tom Logsdon has been teaching short courses for the Applied Technology Institute
(www.ATIcourses.com) for more than 20 years. During that interval, he has taught nearly 300
short courses, most of which have spanned 3 to 5 days. His specialties include “Orbital and
Launch Mechanics”, “GPS Technology”, “Team-Based Problem Solving”, and “Strapped-
Down Inertial Navigation Systems”.
Logsdon has written and sold 1.8 million words including 33 nonfiction books. These have
included The Robot Revolution (Simon and Schuster), Striking It Rich in Space (Random
House), The Navstar Global Positioning System (Van Nostrand Reinhold), Mobile
Communications Satellites (McGraw-Hill), and Orbital Mechanics (John Wiley & Sons). All of
his books have sold well, but his best-selling work has been Programming in Basic, a college
textbook that, over nine printings, has sold 130,000 copies. Logsdon also, on occasion, writes
magazine articles and newspaper stories and, over the years, he has written 18,000 words for
Encyclopaedia Britannica. In addition, he has applied for a patent, help design an exhibit for
the Smithsonian Institution, and helped write the text and design the illustrations for four full-
color ads that appeared in the Reader’s Digest.
In 1973 Tom Logsdon received his first assignment on the GPS when he was asked to figure
out how many GPS satellites would be required to provide at least fourfold coverage at all times
to any receiver located anywhere on planet Earth. What a wonderful assignment for a budding
young mathematician! Working in Technicolor— with colored pencils and colored marking pens
on oversize quad-pad sheets four times as big as a standard sheet of paper— Logsdon used
his hard-won knowledge of three-dimensional geometry, graphical techniques, and integral
calculus to puzzle out the salient characteristics of the smallest constellation that would provide
the necessary fourfold coverage. He accomplish this in three days— without using any
computers! And the constellation he devised was the one that appeared in the winning
proposal that brought in $330 million in revenues for Rockwell International.
Even as a young boy growing up wild and free in the Bluegrass Region of Kentucky, Tom
Logsdon always seemed to have an intuitive understanding of and subtle mathematical
relationships of the type that proved to be so useful in the early days of the American space
program. His family had always been “gravel-driveway poor.” At age 18 he had never eaten in a
restaurant; he had never stayed in a hotel; he had never visited a museum. But, somehow, he
managed to work his way through Eastern Kentucky University as a math-physics major while
serving as the office assistant to Dr. Smith Park, head of the mathematics department. He also
worked as the editor of the campus newspaper, at a noisy Del Monte Cannery in Markesan,
Wisconsin, and as a student trainee at the Naval Ordnance Laboratory in Silver Spring,
Maryland.
Later he earned a Master’s Degree in Mathematics from the University of Kentucky where he
wrote a regular column for the campus newspaper, played ping-pong with the number 9
competitor in the America, and specialized in a highly abstract branch of mathematics called
combinatorial topology. In his 92-page thesis, jam-packed with highly abstract mathematical
symbols, he evaluated the connectivity and orientation properties of simplicial and cell
complexes and various multidimensional analogies of Veblin’s Theorem.
Soon after he finished his thesis, Logsdon accepted a position as a trajectory and orbital
mechanics expert at Douglas Aircraft in Santa Monica, California. His most famous projects
there included the giant 135 foot-in-diameter Echo Balloon, the six Transit Navigation Satellites,
the Thor-Delta booster, and the third stage of the Saturn V moon rocket. A few years later, he
moved on to Rockwell International in Downey, California, where he worked his mathematical
magic on the second stage of the Saturn V, the four manned Skylab missions, the 24-satellite
constellation of GPS radionavigation satellites, the manned Mars mission of 2016, various
unmanned asteroid and comet probes, and the solar-power satellite project which, if it had
reached fruition, would have incorporated at least 100 geosynchronous satellites each with a
surface area equal to that of Manhattan Island (about 20 square miles).
Among his proudest accomplishments at Rockwell International was the clever utilization of nine
different branches of advanced mathematics, in partnership with his friend, Bob Africano, to
increase the performance capabilities of the Saturn V moon rocket by 4700 extra pounds of
payload bound for the moon — each pound of which was worth five times its weight in 24 karat
gold! These important performance gains were accomplished without changing any of the
hardware elements on the rocket. Logsdon and Africano, instead, employed their highly
specialized knowledge of mathematics and physics to work out ways to operate the mighty
Saturn V more efficiently. This involved shaping the trajectories of the rocket for maximum
propulsive efficiency, shifting the burning mixture ratio in mid flight in an optimal manner, and
analyzing their six-degree-of-freedom post-flight trajectory simulations to minimize the heavy
reserve propellants necessary to assure completion of the mission. These powerful
breakthroughs in math and physics led to a saving of $3.5 billion for NASA – an amount equal to
the lifetime earnings of 2000 average American workers!
Currently, Logsdon and his wife, Cyndy, live in Seal Beach, California. Logsdon is now retired
from Rockwell International, but he is still writing books, acting as an expert witness in a variety
of aerospace-related legal cases, lecturing professionally at big conventions, and teaching
short courses on rocket science, orbital mechanics, and GPS technology at major universities,
NASA bases, military installations, and at a variety of international locations. Prior to his recent
trips to Italy, Logsdon delivered two lectures at Hong Kong University in southern China and
taught two short courses at Stellenbach University near Cape Town, South Africa. Over the past
30 years or so he has taught and lectured at 31 different countries scattered across six
continents. At the International Platform Association meetings in Washington, DC, two of his
presentations in successive years placed in the top 10 among the 45 professional platform
lecturers making presentations there. Colleges and Universities that have sponsored his
presentations have included Johns Hopkins, Berkeley, USC, Oxford, North Texas University,
the International Space University in Strasbourg, France, Saddleback.
