Introduction to
Global Navigation Satellite System (GNSS)
Satellite Orbits
Dinesh ManandharCenter for Spatial Information Science
The University of Tokyo
Contact Information: dinesh@iis.u-Tokyo.ac.jp

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Orbital Mechanics• Orbital mechanics or astrodynamics is the application of celestialmechanics to the practical problem concerning the motion ofspacecraft.
• A core discipline within space mission design, control, and operation.

• Celestial mechanics treats the orbital dynamics of natural
astronomical bodies such as star systems, planets, and moons.

Sputnik-1
The first artificial Earth satellite
launched by the Soviet Union in1957.
Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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History• There is little distinction between orbital and celestial mechanics. The
fundamental techniques are the same.
• Johannes Kepler was the first to successfully model planetary orbits to a highdegree of accuracy, publishing his laws of planetary motion in 1605.

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Kepler’s Laws of Planet Motion• The orbit of every planet is an ellipse with the Sun at one of the twofoci (plural of focus).
• A line joining a planet and the sun sweeps out equal area during
equal intervals of time.
• The square of the orbital period of planet is proportional to the cubeof the semi-major axis of its orbit.
B Planet
A-aSunaC

D

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Kepler Orbit• Kepler orbit can be uniquely defined by six parameters know
as Keplerian elements.
• Semi-major axis (a)• Eccentricity (e)• Inclination (i)• Right ascension of the ascending node (RAAN) (Ω)• Argument of perigee (ω)• True anomaly (v: Greek letters nu)

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Orbital Plane
The shape of an elliptic orbit can be defined by
the semi-major axis and eccentricity.

yThe satellite positionin the orbital plane
can be defined bytrue anomaly.

vaxFocus(Earth)
Perigee

ApogeeDinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Equatorial Coordinate System• The most common coordinate frame for describing satellite orbits is
the geocentric equatorial coordinate system, which is also called an
Earth-Centered Inertial (ECI) coordinate system.
z

North pole

Equatorial plane
The fundamental
plane is consisting of
the projection of theEarth’s equator

Vernal equinox
The direction of the Sun
as seen from Earth at thebeginning of spring time.

Oyx

The Origin at thecenter of the Earth

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Orientation of the Orbital PlanezNorthΩ ωxOrbital Plane to ECI𝒓𝐸𝐶𝐼 = 𝑅𝑧 −Ω 𝑅𝑥 −𝑖 𝑅𝑧 −𝜔 𝒓

Perigeeiy

VernalEquinoxDinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Rotation Matrices
zz’

Around the x-axis

Rotationaroundthe z-axis
y’
θ

xyx’

 cos Rz     sin  0sin cos 00
0
1

0
1Rx    0 cos 0  sin 0 sin  cos  

Around the y-axiscos R y     0 sin 

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

0  sin  
10 0 cos  
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Typical GPS Orbit• 26,560 km semi-major axis (20,200 km altitude)• The orbital period is approximately 12 hours• Less than 0.01 eccentricity (near circular)• 55 degree inclination• 6 orbital planes with at least 4 satellites in each plane• The ascending nodes of the orbital planes are separated by 60 degree

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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GPS YUMA ALMANAC FILEID

PRN ID of SV

Health000 = usable

Eccentricity

This shows the amount of the orbit deviation from circular (orbit). It is the distance between the foci divided by the length of
the semi-major axis (our orbits are very circular).

Time of applicability

The number of seconds in the orbit when the almanac was generated. Kind of a time tag.

Orbital Inclination

The angle to which the SV orbit meets the equator (GPS is at approximately 55 degrees). Roughly, the SV's orbit will not rise
above approximately 55 degrees latitude. The number is part of an equation: # = π/180 = the true inclination.

Rate of RightAscension

Rate of change in the measurement of the angle of right ascension as defined in the Right Ascension mnemonic.

SQRT(A) Square Rootof Semi-Major Axis

This is defined as the measurement from the center of the orbit to either the point of apogee or the point of perigee.

Right Ascension atTime of Almanac (TOA)

Right Ascension is an angular measurement from the vernal equinox ((OMEGA)0).

Argument of Perigee

An angular measurement along the orbital path measured from the ascending node to the point of perigee, measured in thedirection of the SV's motion.

Mean Anomaly

Angle (arc) traveled past the longitude of ascending node (value = 0±180 degrees). If the value exceeds 180 degrees,
subtract 360 degrees to find the mean anomaly. When the SV has passed perigee and heading towards apogee, the mean
anomaly is positive. After the point of apogee, the mean anomaly value will be negative to the point of perigee.

Af0

SV clock bias in seconds.

Af1

SV clock drift in seconds per seconds.

Af2

GPS week (0000–1023), every 7 days since 1999 August 22.

GPS Yuma Almanac FileGPS Week

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Example of Yuma Almanac File for GPS• ******** Week 887 almanac for PRN-01 ********• ID: 01
• Health: 000• Eccentricity: 0.5854606628E-002• Time of Applicability(s): 589824.0000• Orbital Inclination(rad): 0.9652777840• Rate of Right Ascen(r/s) : -0.7714607059E-008• SQRT(A) (m 1/2)
: 5153.593750• Right Ascen at Week(rad) : 0.2492756606E+001• Argument of Perigee(rad) : 0.531310874• Mean Anom(rad): 0.3110215331E+001• Af0(s): 0.3147125244E-004• Af1(s/s): 0.0000000000E+000• Week: 887
http://qz-vision.jaxa.jp/USE/en/almanac

https://celestrak.com/GPS/almanac/Yuma/definition.asp

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Perturbation Forces• Satellite orbit will be an ellipse only if treating each of satellite
and Earth as a point mass.
• In reality, Earth’s gravitational field is not a point mass.
• Main force acting on GNSS satellites is Earth’s central
gravitational force, but there are many other significant
perturbations.
• Non sphericity of the Earth’s gravitational potential• Third body effect• Direct attraction of Moon and Sun• Solar radiation pressure• Impact on the satellite surfaces of photons emitted by the Sun

Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Accelerations Acting on GNSS Satellites
TermAcceleration [m/s2]

Earth’s central gravity

0.56

Flatness of the Earth (J2)

5×10-5

Other gravity

3×10-7Moon and Sun

5×10-6

Solar Radiation Pressure

10-7Effects of SRP on GNSS satellite position: 5~10 m
Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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Satellite orbit in Navigation Message• Broadcast ephemeris• Kepler orbit parameters and satellite clock corrections• 9 orbit perturbation corrections parameters• 2 m satellite position accuracy for 2 hours• Each GNSS satellite broadcasts only its own ephemeris data

• Almanac• Kepler orbit parameters and satellite clock corrections• Less accurate but valid for up to several months• Each GNSS satellite broadcasts almanac data for all satellites in theconstellation
Dinesh Manandhar, CSIS, The University of Tokyo, dinesh@iis.u-tokyo.ac.jp

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