(19) Video lecture space 2

Introduction

Importance of Knowledge in Engineering

• Engineering relies on facts, analysis, and providing reliable answers.

• Need for sound knowledge in designing space missions: vehicles, orbits, instruments, etc.

Topic of the Lecture: Satellite Orbits

• Focus on orbits from physics and mathematics perspectives.

• Derivations will be presented, which are not included in the exam but essential for understanding.

• Importance of reproducing derivations for better comprehension.

Categories of Orbits

• Discussion will initially cover elliptical orbits, with more categories explored in subsequent lectures.

Reasons for Interest in Orbital Mechanics

1. Direct Relation to Mission Purpose

   • Example: Satellite wants to be stationary over Moscow requires specific orbit.

   • Communication satellites need to cover specific geographical areas.

2. Necessity of Position and Velocity Knowledge

   • Collision Prevention: Knowledge to avoid proximity of spacecraft (space debris concerns).

   • Communication Scheduling: Knowing satellite visibility times for ground stations.

   • Orbit Changes: Manoeuvres require precise timing to ensure correct trajectory.

3. Determining Trajectory

   • The trajectory of a satellite is determined by its initial position and velocity after launch.

   • Forces acting on the spacecraft are critical for understanding its path.

Questions Addressed through Orbital Mechanics

• Key parameters: inclination, semi-major axis sizes crucial to describe orbits.

• Low Earth Orbit (LEO) parameters vs. higher altitude specifics.

Experiments and Observations

• Simple experiment with a piece of chalk to illustrate:

  • Gravity: Chalk exhibits parabolic motion as it falls due to gravity.

  • Energy Conservation: Different velocities observed during the chalk's motion.

Fundamentals

Johannes Kepler and Laws of Planetary Motion

1. First Law: Planets move in elliptical orbits with the Sun at one focus.

2. Second Law: A line between a planet and the Sun sweeps equal areas in equal times. This says something about the velocity of the planet in relation to how close it is to the sun. If it is far away it will sweep around slower then if it is close.

3. Third Law: The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.

Isaac Newton's Contributions

• Developed laws of motion and the universal law of gravitation.

• Key points include:

  1. Objects remain at rest or uniform motion unless acted upon. in absence of a force, a body either is at rest or moves in a straight line with constant speed.

  2. Force equals mass times acceleration (F = ma). The force is proportional to the time derivative of the momentum.

  3. For every action, there is an equal and opposite reaction. whenever a first body exerts a force F, the second body exerts a force -F on the first body. F and-F are equal in magnitude and opposite in direction.

  4. Law of gravitation: Every two objects attract each other based on mass and distance. every point mass attracts other point mass by a force pointing along the line connecting both points. the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between then point masses.

     

Equation of Motion for Orbital Mechanics

• Gravitational attraction between two point Masses or between two homogeneous spheres (with masses M1 and M2:

  

• Therefore the relative acceleration between object 1 and object 2: is:

  

• we typically see that one of the masses if much much smaller than the other (e.g. planet and sun, or earth and satellite). so then we can say that when M₂«M₁:

• here μ is the gravitational parameter of the central body

  

• We can use these equations to observer the mass of other planets. Since we know that the motion of the smaller object is driven bij the mass of the larger object we can use the observations of e.g. mars’s moons to calculate mars’s weight

Conservation of angular momentum

• The vectorial product of equation of motion. with r:

  

• when the derivative of an equation is= to 0 then the integral is a constant therefore r x r(dot) is a constant. This means that the cross product of the vector of the radius r and the vector of the velocity V is constant = H.

• The vector H is perpendicular to the orbital plane and therefore can be used

• Therefore motion is in one plane. so unless and extra force is applied the orbit will remain in one direction.

  

• Area law indicated that being far away from the central body means a lower velocity and being close to the central body indicates a higher velocity.

Conservation of energy:

• The conservation of energy can be shown by taking the scalar product of the equation of motion with dr/dt.

  

Deriving Equations for Orbits:

• This can be done by taking the scalar product of the equation of motion with r

  

• from this we can derive the first Law of Kepler (the proof is in a master course)

  

Conclusion

• Importance of mastering these fundamental equations as an aerospace engineer.

• Encouragement for students to revisit these concepts in their study time.

Recap and Next Steps

• Brief overview of the results obtained from the equations presented.

• Feedback and anticipation for the next lecture.