Fundamentals of Orbital Mechanics
This chapter applies the principles of physics for describing motion of artificial satellites and planets which are moving under the influence of gravitation force. It also presents an accurate gravity model for flight dynamic applications (for use in subsequent chapters).
2.1 Kepler’s Laws of Planetary Motion
In 1609 Kepler has presented his three laws for describing motion of planets around the Sun. These laws are empirical because Kepler has found them by analyzing the observations of the astronomer Tycho Brahe. Kepler’s Laws of Planetary Motion can be stated as follows:
Kepler’s first law:
The orbit of each planet around the Sun is an ellipse with the sun at one focus.
Kepler’s second law:
Each …show more content…
Since h is a constant vector, it is fixed in direction. Thus the motion of the planet around the Sun or the motion of an artificial satellite around a planet is two dimensional.
2.4 Equations of Motion in Polar Coordinates
The last section demonstrates that all celestial bodies (i.e. planets / satellites) under the central force move in a fixed plane. Therefore, this section uses polar coordinates to determine the equation of motion of the celestial body. The position vector r of the body is defined in the cartesian coordinates as follows:
r = x i + y j. (2.4)
The cartesian coordinates x and y can be expressed in polar coordinates as follows: x = r cos , y = r sin (2.5)
Plugging Eq. (2.5) into Eq. (2.4) we obtain r = r cos i + r sin j (r )/r = cos i + sin j r ̂ = cos i + sin j. (2.6)
Here r ̂ is unit vector along the radial direction. The orthonornal vector ̂ to is r ̂ is defined as follows
̂ = sin i + cos j. (2.7)
The position vector r of the planet can be expressed in polar coordinate as follows: r = r r
This chapter applies the principles of physics for describing motion of artificial satellites and planets which are moving under the influence of gravitation force. It also presents an accurate gravity model for flight dynamic applications (for use in subsequent chapters).
2.1 Kepler’s Laws of Planetary Motion
In 1609 Kepler has presented his three laws for describing motion of planets around the Sun. These laws are empirical because Kepler has found them by analyzing the observations of the astronomer Tycho Brahe. Kepler’s Laws of Planetary Motion can be stated as follows:
Kepler’s first law:
The orbit of each planet around the Sun is an ellipse with the sun at one focus.
Kepler’s second law:
Each …show more content…
Since h is a constant vector, it is fixed in direction. Thus the motion of the planet around the Sun or the motion of an artificial satellite around a planet is two dimensional.
2.4 Equations of Motion in Polar Coordinates
The last section demonstrates that all celestial bodies (i.e. planets / satellites) under the central force move in a fixed plane. Therefore, this section uses polar coordinates to determine the equation of motion of the celestial body. The position vector r of the body is defined in the cartesian coordinates as follows:
r = x i + y j. (2.4)
The cartesian coordinates x and y can be expressed in polar coordinates as follows: x = r cos , y = r sin (2.5)
Plugging Eq. (2.5) into Eq. (2.4) we obtain r = r cos i + r sin j (r )/r = cos i + sin j r ̂ = cos i + sin j. (2.6)
Here r ̂ is unit vector along the radial direction. The orthonornal vector ̂ to is r ̂ is defined as follows
̂ = sin i + cos j. (2.7)
The position vector r of the planet can be expressed in polar coordinate as follows: r = r r