Orbital Mechanics
•How earth orbit is achieved
•Laws that describe the motion of an
object orbiting another body.
•How satellites maneuver in space
•Determination of the look angle to a
satellite from the earth.
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Orbital Mechanics: Understanding Satellite Motion and Orbits - Prof. Gunnapalli, Lecture notes of Labour Law
A comprehensive overview of orbital mechanics, covering the fundamental principles and equations that govern the motion of satellites orbiting the earth. It delves into the laws of planetary motion, the forces acting on satellites, and the mathematical formulations used to describe satellite orbits. Topics such as the determination of look angles, the relationship between satellite mass and acceleration, the concept of centrifugal and centripetal forces, and the derivation of the equation of motion for a satellite in a stable orbit. It also presents detailed calculations and examples related to satellite orbital parameters, including velocity, period, and the transformation between cartesian and polar coordinate systems. The document serves as a valuable resource for understanding the complex dynamics involved in satellite operations and the factors that influence their trajectories.
Typology: Lecture notes
2023/2024
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Orbital Mechanics
• How earth orbit is achieved
• Laws that describe the motion of an
object orbiting another body.
• How satellites maneuver in space
• Determination of the look angle to a
satellite from the earth.
Developing the equations of the orbit
- Fundamental Newton‟s equations that describe the motion of a body.
- Newton‟s law of motion can be encapsulated into four equations, 1. s = ut + (1/2)at^2 2. v^2 = u^2 + 2at 3. v = u + at 4. F = ma
- Where S is the distance travelled from t=0,
- u is the initial velocity of the object at time t=0,
- v is the final velocity of the object at time t,
- a is the acceleration of the object,
- F is the force acting on the object,
- m is the mass of the object.
Note that the acceleration can be positive or negative, depending on the direction it is acting with respect to the velocity vector.
- Centrifugal force
▫ due to the kinetic energy of the satellite.
▫ attempts to fling the satellite into a
higher orbit.
- Centripetal force
▫ due to the gravitational attraction of the
planet about which the satellite is
orbiting.
▫ attempts to pull the satellite down
towards the planet.
ME
FIN=GMEM/r^2
FOUT=mv^2 /r
- The acceleration due to gravity „a‟ at a distance r from the center of the earth is, a = /r 2 km/s^2 ▫ Where the constant is the product of the universal gravitational constant and the mass of the earth ME
- The product GME is called as Kepler‟s constant and has the value, = 3.986x10^5 km^3 /s^2
- And G = 6.672x10-11^ NM^2 /kg^2 (or)
6.672x10-20^ km^3 /kg s^2
- Any Force, F = ma
- Since centripetal force, FIN = m x /r^2 FIN = m x GME/r^2
- Centrifugal acceleration is given by, a=v^2 /r FOUT=m x v^2 /r
- If the forces on the satellite are balanced, FIN = FOUT M x /r^2 = m x v^2 /r
v=(/r)1/
Which is the velocity of the satellite in a circular orbit.
Satellite Orbital height (km)
Orbital velocity (km/s)
Orbital period (hr) (min) (s)
Intelsat (GEO) 35,786.03 3.0747 23 56 4. New-ICO(MEO) 10,255 4.8954 5 55 48.
Skybridge (LEO) 1,469 7.1272 1 55 17.
Iridium (LEO) 780 7.4624 1 40 27.
Mean earth radius is 6378.137 km and GEO radius
from the center of the earth is 42,164.17 km
- Uses the Cartesian co- ordinate system with the earth at the center and the reference planes coinciding with the equator and the polar axis
- The initial coordinate system that could be used to describe the relationship between the earth and a satellite.
- Axes cx , cy , and cz are mutually orthogonal axes, with cx and cy passing through the earth‟s geographic equator.
- The vector r locates the moving satellite with respect to the center of the earth.
- The satellite mass „m‟ is located at a vector distance r, from the center of the earth,
- Then the gravitational force F on the satellite is given by, 𝐹 = −𝐺𝑀𝑟 3 𝐸 𝑚𝑟 1 Where ME is the mass of the earth G=6.672 x 10-11^ NM^2 /kg^2
- But F=ma
- Equation 1 can also be written as,
𝐹 = 𝑚 𝑑
(^2) 𝑟 𝑑𝑡^2 ^2
- To remove this dependence,
▫ a different set of coordinates can be chosen to describe the location of the satellite such that the unit vectors in the three axes are constant. ▫ The orbital plane coordinate system.
- In this coordinate system, the orbital plane of the satellite is used as the reference plane.
- The orthogonal
axes, x 0 and y 0 lie in
the orbital plane.
- The third axis, z 0 , is
perpendicular to
the orbital plane.
- This Co-ordinate system uses the plane of the satellite‟s orbit as the reference plane (Orbital plane Co-ordinate system).
- Expressing in terms of the new co-ordinate axes x 0 , y 0 , and z 0 as
𝑥 0
𝑑^2 𝑥 0
𝑑𝑡^2
𝑑^2 𝑦 0
𝑑𝑑^2
𝑥 02 + 𝑦 02 3/^
- The above equation is easier to solve if it is expressed in a polar Co-ordinate system rather than Cartesian Co-ordinate system.
- Polar coordinate system in the plane of the satellite‟s orbit.
- The plane of the orbit coincides with the plane of the paper.