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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?
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 planeso Equatorial orbit above earth’s equatoro Polar orbit passes over both poleso Other orbits referred to as inclined orbits
Altitude of satelliteso Low earth orbit (LEO)o Medium earth orbit (MEO)o Geostationary orbit (GEO)o High elliptical orbit (HEO)
• 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)
LEO Categories
Little LEOs • Frequencies below 1 GHz
Big LEOs• Frequencies above 1 GHz
Mega (Super) LEOs• 20-30 GHz range
• 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)
•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)
•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)
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
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
• Laws of Planetary MotionLaw 1 - Law of EllipsesLaw 2 - Law of Equal AreasLaw 3 - Harmonic Law
(P2=ka3)
• Kepler’s laws provide a concise and simple description of the motions of the planets
Kepler’s Laws
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
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 ( ) pp
s p
R HT R H
g R
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 23
( ) ˆ 0G m mr rr
relative position vectorrelative acceleration vector
G=universal gravitational constant
rr
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 numericalvalue for μ is
8 3 2µ =GMe = 3986004.418 x 10 m /s
Keplerian motion
Keplerian motionPerigee 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
Keplerian orbital parameters
Keplerian motion defined by six orbital parameters
Mean MotionMean 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
AnomaliesThe 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
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
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
• 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
•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.
•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.
Disturbing accelerationsIn 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
Non sphericity of the earthThe earth’s potential V can be represented by a spherical harmonic expansion
212mv eV
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-
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
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
