A Mathematical Guide for Doppler Tracking of Satellites in Low Earth Orbit
, 2
Abstract
For a low-cost, low-power, consistent, and accurate method of satellite tracking, which may be
an optimal option for those involved in the private space industry or academia, Doppler tracking
is a proven and versatile choice. By using foundational concepts from various fields of
mathematics with little known background information, a satellite’s complete orbital profile can
be determined from a single overhead (horizon-to-horizon) pass worth of Doppler data and
further updated with consecutive passes. Despite not being a focal point, this study’s algorithm
could be further improved through the incorporation of software, allowing for quick and efficient
conversions and calculations.
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Introduction
Significantly increasing space accessibility for those in academia and private industry,
CubeSats (Cube satellites) have become a popular method of sending small-scale payloads to
space for a variety of missions such as for scientific research and demonstrations of technology.
Launching smaller payloads provides several advantages including cost and development time,
but also presents unique challenges. Confining a payload within a smaller, standardized structure
limits the overall weight, size, and power of the hardware required, often resulting in trade-offs
during the spacecraft’s design. One such trade designers may choose is not equipping their
satellite with a GPS receiver fit for spaceflight. Further influencing this decision could be limits
on communication bit rates or power budgets (Dykstra, 2015). Another reason mission designers
may exclude a highly precise navigation instrument may be because it is not necessary to know
with a certain degree of accuracy the satellite’s position to satisfy the requirements of the
mission.
No matter the case, some sufficient knowledge of a satellite’s orbital characteristics and
position over time is necessary for the scheduling and operations of a mission. For instance, to
send commands, receive scientific data or telemetry, or to diagnose a problem during its flight,
ground station operators need to know where the satellite is so they can properly direct antennas
for communication (Dykstra, 2015). For low-cost, low-power (further compounding some of the
benefits provided by CubeSats), accurate, and consistent satellite tracking, Doppler tracking is an
effective method proven on spacecraft in low-Earth orbit (LEO) as well as deep space.
Background
When the Soviet Union ushered in the space age with Sputnik in 1957, the need for a
satellite tracking system became apparent, and with an omnidirectional antenna receiving radio
, 4
frequencies from the satellite, researchers at the Applied Physics Laboratory (APL) at Johns
Hopkins University in Maryland addressed that need (Guier and Weiffenbach, 1998). Although
tracking methods for guided missiles were already developed, determining a complete set of
orbital elements for a satellite was yet to be demonstrated. Despite other organizations using
angle measurements from radio interferometers, APL was the only organization at the time
conducting satellite tracking research using Doppler phenomenon.
Using data obtained from Sputnik and Sputnik II, it was determined that all six elements
of a satellite can be “inferred” with a single horizon-to-horizon pass worth of Doppler frequency
information and further updated with future passes (Guier and Weiffenbach, 1998). This method
influenced the Navy’s creation of TRANSIT, the first constellation of satellites purposed for
navigation (Dykstra, 2015). Although surpassed today by modern satellite systems, the utility of
Doppler Tracking remains significant and could potentially serve as a foundation for a global
navigation satellite system (GNSS) beyond Earth.
Statement of Purpose
This independent study reviews mathematical equations and theory relevant to satellite
Doppler tracking with potential application in satellite ground stations at the University of North
Dakota (UND) John D. Odegard School of Aerospace Sciences. By defining and detailing an
algorithm for executing this method of satellite tracking, this paper supports the concept of
determining all orbital elements (and therefore position) from a single overhead satellite pass and
updating as necessary with consecutive passes.
Scope and Limitations
Due to the scope of this study paired with time constraints, no data collection was done
for this paper. With this study primarily focused on the mathematical theory and equations
, 2
Abstract
For a low-cost, low-power, consistent, and accurate method of satellite tracking, which may be
an optimal option for those involved in the private space industry or academia, Doppler tracking
is a proven and versatile choice. By using foundational concepts from various fields of
mathematics with little known background information, a satellite’s complete orbital profile can
be determined from a single overhead (horizon-to-horizon) pass worth of Doppler data and
further updated with consecutive passes. Despite not being a focal point, this study’s algorithm
could be further improved through the incorporation of software, allowing for quick and efficient
conversions and calculations.
, 3
Introduction
Significantly increasing space accessibility for those in academia and private industry,
CubeSats (Cube satellites) have become a popular method of sending small-scale payloads to
space for a variety of missions such as for scientific research and demonstrations of technology.
Launching smaller payloads provides several advantages including cost and development time,
but also presents unique challenges. Confining a payload within a smaller, standardized structure
limits the overall weight, size, and power of the hardware required, often resulting in trade-offs
during the spacecraft’s design. One such trade designers may choose is not equipping their
satellite with a GPS receiver fit for spaceflight. Further influencing this decision could be limits
on communication bit rates or power budgets (Dykstra, 2015). Another reason mission designers
may exclude a highly precise navigation instrument may be because it is not necessary to know
with a certain degree of accuracy the satellite’s position to satisfy the requirements of the
mission.
No matter the case, some sufficient knowledge of a satellite’s orbital characteristics and
position over time is necessary for the scheduling and operations of a mission. For instance, to
send commands, receive scientific data or telemetry, or to diagnose a problem during its flight,
ground station operators need to know where the satellite is so they can properly direct antennas
for communication (Dykstra, 2015). For low-cost, low-power (further compounding some of the
benefits provided by CubeSats), accurate, and consistent satellite tracking, Doppler tracking is an
effective method proven on spacecraft in low-Earth orbit (LEO) as well as deep space.
Background
When the Soviet Union ushered in the space age with Sputnik in 1957, the need for a
satellite tracking system became apparent, and with an omnidirectional antenna receiving radio
, 4
frequencies from the satellite, researchers at the Applied Physics Laboratory (APL) at Johns
Hopkins University in Maryland addressed that need (Guier and Weiffenbach, 1998). Although
tracking methods for guided missiles were already developed, determining a complete set of
orbital elements for a satellite was yet to be demonstrated. Despite other organizations using
angle measurements from radio interferometers, APL was the only organization at the time
conducting satellite tracking research using Doppler phenomenon.
Using data obtained from Sputnik and Sputnik II, it was determined that all six elements
of a satellite can be “inferred” with a single horizon-to-horizon pass worth of Doppler frequency
information and further updated with future passes (Guier and Weiffenbach, 1998). This method
influenced the Navy’s creation of TRANSIT, the first constellation of satellites purposed for
navigation (Dykstra, 2015). Although surpassed today by modern satellite systems, the utility of
Doppler Tracking remains significant and could potentially serve as a foundation for a global
navigation satellite system (GNSS) beyond Earth.
Statement of Purpose
This independent study reviews mathematical equations and theory relevant to satellite
Doppler tracking with potential application in satellite ground stations at the University of North
Dakota (UND) John D. Odegard School of Aerospace Sciences. By defining and detailing an
algorithm for executing this method of satellite tracking, this paper supports the concept of
determining all orbital elements (and therefore position) from a single overhead satellite pass and
updating as necessary with consecutive passes.
Scope and Limitations
Due to the scope of this study paired with time constraints, no data collection was done
for this paper. With this study primarily focused on the mathematical theory and equations
