GPS Satellite Orbit

1. GNSS Surveying, GE 205
Kutubuddin ANSARI
kutubuddin.ansari@ikc.edu.tr
Lecture 6, April 13, 2015
Satellite Orbit

2. Satellite Orbits
• At what location is the satellite looking?
• When is the satellite looking at a given location?
• How often is the satellite looking at a given
location?
• At what angle is the satellite viewing a given
location?

3. Classification of Satellite Orbits
 Circular or elliptical orbit
o Circular with center at earth’s center
o Elliptical with one foci at earth’s center
 Orbit around earth in different planes
o Equatorial orbit above earth’s equator
o Polar orbit passes over both poles
o Other orbits referred to as inclined orbits
 Altitude of satellites
o Low earth orbit (LEO)
o Medium earth orbit (MEO)
o Geostationary orbit (GEO)
o High elliptical orbit (HEO)

4. • Circular/slightly elliptical orbit under 2000 km
• Orbit period ranges from 1.5 to 2 hours
• Diameter of coverage is about 8000 km
• Round-trip signal propagation delay less than 20 ms
• Maximum satellite visible time up to 20 min
• System must cope with large Doppler shifts
• Atmospheric drag results in orbital deterioration
Low Earth Orbit (LEO)

5. LEO Categories
Little LEOs
• Frequencies below 1 GHz
Big LEOs
• Frequencies above 1 GHz
Mega (Super) LEOs
• 20-30 GHz range

6. • Circular orbit at an altitude in the range of 5000 to
12,000 km
• Orbit period of 6 hours
• Diameter of coverage is 10,000 to 15,000 km
• Round trip signal propagation delay less than 50 ms
• Maximum satellite visible time is a few hours
Medıum Earth Orbit (MEO)

7. •Geostationary satellites orbit the Earth's axis as fast as
the Earth spins.
• They hover over a single point above the Earth at an
altitude of about 36,000 kilometers (22,300 miles).
• This orbit allows these satellites to continuously look
at the same spot on the earth .It is important for
locating the position of hurricanes and monitoring
developing severe storms.
Geostationary Earth Orbit (GEO)

8. •National Oceanic and Atmospheric Administration
(NOAA) typically operates two geostationary satellites
called Geostationary Operational Environment
Satellite (GOES).
•One has a good view of the East Coast (GOES-East)
while the other focuses on the West Coast (GOES-
West).
Geostationary Earth Orbit (GEO)

9. Highly elliptical orbit (HEO)
• Eccentricity = 0.737
• Semi-major axis = 26,553 km
• Altitude 3,960 km higher than GEO)
• Inclination = 63.4°
• Period = 717.7 min (≈12 hr)
• Used as communications satellites

10. Applications
• Weather forecasting
• Radio and TV broadcast satellites
• Military satellites
• Satellites for navigation
• Global telephone backbones
• Connections for remote or developing areas
• Global mobile communications

11. • Laws of Planetary Motion
Law 1 - Law of Ellipses
Law 2 - Law of Equal Areas
Law 3 - Harmonic Law (P2
=ka3
)
• Kepler’s laws provide a concise and simple description
of the motions of the planets
Kepler’s Laws

12. Kepler's First Law:
Each planet’s orbit around the Sun is
an ellipse, with the Sun at one focus.
Kepler's Second Law: Line
joining planet and the Sun sweeps out
equal areas in equal times
Kepler's Third Law: The squares of
the periods of the planets are proportional to
the cubes of their semi-major axes:
Kepler’s Laws

13. The orbital period of a satellite around a planet
is given by
where
T0 = orbital period (sec)
Rp =planet radius (6380 km for Earth)
H= orbit altitude above planet’s surface (km)
gs =acceleration due to gravity (0.00981 km s-2
for
Earth)
Orbital Period of a Satellite
0 2
2 ( )
p
p
s p
R H
T R H
g R
π
+
= +

14. Keplerian motion
Assume two point masses m1 and m2 separated by the
distance r. Considering for the moment only the
attractive force between the masses and applying
Newtonian mechanics, the movement of mass m2
relative to m1 is defined by the homogeneous
differential equation of second order
1 2
3
( )
ˆ 0
G m m
r r
r
+
+ =&&
relative position vector
relative acceleration vector
G=universal gravitational constant
r
r
=
=&&

15. In the case of an artificial satellite orbiting the earth, the
mass of the satellite can be neglected. The product of G
and the earth’s mass Me is denoted as the geocentric
gravitational constant μ. According to the current IERS
conventions, the numerical
value for μ is
8 3 2
µ =GMe = 3986004.418 x 10 m /s
Keplerian motion

16. Keplerian motion
Perigee The point of closest position of the satellite with respect
to the earth’s center of mass is called perigee
Apogee The point of most distant position of the satellite with
respect to the earth’s center of mass is called the apogee

17. Keplerian orbital parameters
Keplerian motion defined by six orbital parameters

18. Mean Motion
Mean motion is a measure of how fast a satellite
progresses around its elliptical orbit. Unless the orbit is
circular, the mean motion is only an average value, and
does not represent the instantaneous angular rate.
The mean angular satellite velocity n also known as the
mean motion with revolution period P and it follows
from Kepler’s third law given by

19. Anomalies
The instantaneous position of the satellite within its
orbit is described by angular quantities known as
anomalies. The mean anomaly M(t) is a mathematical
abstraction relating to mean angular motion, while
both the eccentric anomaly E(t) and the true anomaly
v(t) are geometrically producible .
Where e is eccentricity of ellipse

20. In celestial mechanics, the mean anomaly is a
parameter relating position and time for a body
moving in a Kepler orbit. The mean anomaly increases
uniformly from 0 to 2π radians during each orbit.
The mean anomaly can be used instead of T0 as a
defining parameter for the orbit
Mean Anomaly

21. where n is the mean motion, a is the length of the
orbit's semi-major axis, M* and m are the orbiting
masses, and G is the gravitational constant.
The mean anomaly is usually denoted by the letter M,
and is given by the formula
Mean Anomaly

22. • The eccentric anomaly is
an angular parameter that
defines the position of a
body that is moving along
an elliptic Kepler orbit.
•For the point P orbiting
around an ellipse, the
eccentric anomaly is the
angle E in the figure.
Eccentric Anomaly

23. •It is determined by drawing a vertical line from the
major axis of the ellipse through the point P and
locating its intercept P′ with the auxiliary circle, a
circle of radius a that describes the entire ellipse.
Eccentric Anomaly
•This intersection P′ is called the corresponding point
to P. The radius of the auxiliary circle passing
through the corresponding point makes an angle E
with the major axis.

24. •It is the angle between the
direction of periapsis and the
current position of the body,
as seen from the main focus of
the ellipse (the point around
which the object orbits).
•The true anomaly is usually
denoted by the Greek letters ν
or θ, or the Roman letter f.
True Anomaly
•The true anomaly is an angular parameter that
defines the position of a body moving along a
Keplerian orbit.

25. Disturbing accelerations
In reality, many disturbing accelerations act on a
satellite and are responsible for the temporal
variations of the Keplerian elements. Roughly
speaking, they can be divided into two groups, namely
those of gravitational and those of non gravitational
origin

26. Non sphericity of the earth
The earth’s potential V can be represented by a
spherical harmonic expansion
21
2
mv eV=

27. Tidal effects
A differential gravitation force along an extended body
as a result of the varying distance from a gravitational
source to the different parts of the body such as the
force on the moon on the earth's oceans closed to and
farthest from the moon.
The disturbing acceleration is
given by-

28. Solar radiation pressure
The perturbing acceleration due to the direct solar
radiation pressure has two components. The principal
component is directed away from the sun and the
smaller component acts along the satellite’s y-axis

29. Relativistic effect
The relativistic effect on the satellite orbit is caused by
the gravity field of the earth and gives rise to a
perturbing acceleration which is given by