World of Aerospace
What are Orbital Elements, Newton's Law of Motion and Gravitation, Kepler's Law of Planetary Motion, Two Body Problem,Trajectory Equation, Flight Path Elevation Angle, Zenith Angle and more...
Orbital Elements and Orbital Mechanics
The understanding and know-how
of orbital elements and orbital mechanics is essential to understanding
the secrets of space-flight.
The topics covered in orbital mechanics section
are,
·
Kepler's Law of Planetary Motion
·
Equation Governing Two Body Problem
Newton’s Law of Gravitation
The Newton’s Law of Gravitation
states that, Any two bodies attract each other with a force that is
directly proportional to the product of their masses and the force is
inversely proportional to the square of the distance between them
(between there centers).
1^{st} Law of
Motion: Every body continues or
remains in the state of rest or of uniform motion in a straight line
unless it is acted upon by an external force.
2^{nd} Law of
Motion: The time rate of change of
momentum is proportional to the force applied on the body and it acts in
the same direction as that of the force or When a force is applied on a
body it produces acceleration in it.
3^{rd} Law of
Motion: To every action, there is
always an equal and opposite reaction.
Kepler’s Law
of Planetary Motion
1^{st} Law (Law of
Orbit): The
orbit of the planet about the Sun is an ellipse with the Sun at one
focus.
2^{nd} Law (Law of
Area): The line joining the planet to
the Sun sweeps out equal areas in equal intervals of time.
3^{rd} Law (Law of
Period): The
square of the period of a planet is proportional to the cube of the
semi-major axis of its elliptical orbit about the Sun.
Equation
Governing Two Body Problem
Basic Assumptions:
The bodies are assumed to
be spherical and symmetric. This enables us to treat the bodies as point
masses with their masses concentrated at their centers.
There are no forces acting
on the system (absence of both external and internal forces) other than
the gravitational forces which acts along the line joining the centers
of the two bodies.
E, Conservation of
Mechanical Energy
Specific Mechanical Energy which
is the sum of the kinetic and potential energies per unit mass and
denoted by E remains constant in an orbit. This means that the value of
E doesn’t change during motion in an orbit.
h, Conservation of
Angular Momentum
The specific angular momentum of
a body remains constant along its orbit or during its motion in an
orbit.
Inferences:
Since h the specific
angular momentum is conserved in an orbit and since h is a vector
quantity. So this means that both r and V must always
remain in the same plane referred to as the orbital plane. That is why
it is the orbit of the planet about the sun lies in a plane.
Local Vertical, The local
vertical is obtained when the location of the satellite or body
coincides with the direction of the vector r.
Local Horizontal is
perpendicular to the local vertical and lines in the plane.
The angle between the local
horizontal and the local velocity vector is called as the flight path
angle or flight path elevation angle and is denoted by the symbol phi.
The angle between the local
vertical and the local velocity vector and that shows the direction of
the velocity vector is given by the symbol gamma and is called as the
Zenith angle.
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