Comprehensive Study Guide on Celestial Motion and Orbital Mechanics

Mass and Weight

• Mass:

  • Mass is the amount of matter an object has. It only depends on the object itself.

  • How many masses can an object have? An object has strictly 1 mass.

• Weight:

  • Weight is the force an object exerts due to gravity. It depends on the object and the gravitational field it's in.

  • How many weights can an object have? An object can have infinitely many weights, depending on the gravitational field.

Kinematics and Motion Variables

• Definition of Kinematics: Kinematics is the study of motion.

• Kinematic Variables:

  • Position ($x$): Position is the location of an object.

  • Speed ($v$): Speed is the change in an object's position over time ($v = \frac{\delta x}{\delta t}$).

  • Acceleration ($a$): Acceleration is the change in an object's speed over time ($a = \frac{\delta v}{\delta t} = \frac{\delta^2 x}{\delta t^2}$). A change in direction is also acceleration.

• Describing Kinematic Variables:

  • Qualitative: Large, small, increasing, decreasing.

  • Quantitative: A specific number (including zero).

• Application Examples:

  • An asteroid coasts through empty outer space: Involves position and speed, with constant speed and zero acceleration ($a = 0$).

  • Dust particles gather due to gravity: Involves position, speed, and acceleration.

  • A black hole's location is stationary: Stationary position, zero speed ($v = 0$), and zero acceleration ($a = 0$).

  • A planet orbits a star at constant speed: Changing position, constant speed, and non-zero acceleration (change in direction is also acceleration).

Centripetal Acceleration and Force

- Centripetal Acceleration:

- Objects that travel in circles accelerate, even if their linear speed ($v$) remains constant.

- The direction of the speed changes, requiring acceleration (directed toward the center of the path).

• Force:

  • An interaction that causes the motion (speed) of an object to change.

  • Forces cause acceleration. Newton's laws and gravity refer to forces.

### Energy in Motion and Gravitational Systems

• Definition of Energy: The capacity of an object to move or cause motion.

• Kinetic Energy ($KE$): Objects that move on their own have kinetic energy.

• Potential Energy ($PE$): Objects that could move on their own have potential energy.

- Energy System Examples:

- **An asteroid traveling through outer space**: Kinetic energy.

- **A rock sitting at the top of a cliff**: Potential energy.

- **A meteor passing into a planet's atmosphere**: Both kinetic and potential energy.

- **A star's surface matter being pulled into a nearby black hole**: Both kinetic and potential energy.

Gravity on Earth

- Definition of Gravity: A mutual attraction between matter based on mass and separation.

• Freefall Acceleration at Earth's Surface:

  • Earth's gravity causes objects at surface to accelerate at $9.8\,m/s^2$ (meters per second per second).

  • With each second, an object in freefall travels $9.8\,m/s$ faster.

• Freefall Speed Progression:

  • Fall Time $0\,s$: Speed = $0\,m/s$

  • Fall Time $0.5\,s$: Speed = $4.9\,m/s$

  • Fall Time $1\,s$: Speed = $9.8\,m/s$

  • Fall Time $3\,s$: Speed = $29.4\,m/s$ (Calculated as $3\,s \times 9.8\,m/s^2 = 29.4\,m/s$)

• Geographic and Altitudinal Variations of Acceleration ($g$):

- Gravity is constant at the surface of Earth.

• Gravity decreases as you travel further from sea level (climbing a mountain or flying will cause a change).

Gravity in Outer Space

• Universal Gravitational Attraction:

  • All celestial objects exert gravity on each other.

• Formula for Gravitational Acceleration: $a = \frac{G \times M}{r^2}$

  • $a$: Gravitational acceleration.

  • $G$: Constant ($6.67 \times 10^{-11}\,N\cdot m^2/kg^2$).

  • $M$: Mass of attracting object.

  • $r$: Separation distance.

• Proportionality Principles:

  • Acceleration depends on mass and separation distance.

  • An object experiences greater gravity when the other object is more massive or closer to it.

• Gravity Increase or Decrease Scenarios:

  • The Earth (A) drifts closer to the Sun (B): Gravity increases.

  • An asteroid (A) approaches a growing star (B): Gravity increases.

  • A rocket (A) launches from the Earth (B): Gravity decreases.

  • A spaceship (A) ejects cargo while orbiting Mars (B): Gravity on spaceship remains unchanged.

Newton's Laws of Motion

- Historical Background:

- Isaac Newton wrote the *Principia* (*Philosophiæ Naturalis Principia Mathematica*) in 1687, describing physical laws for the motion of objects.

- Main idea: forces change speed or direction of motion, but constant motion does not require force to continue. Previous theories argued that even constant motion requires force.

• Newton's First Law:

  • "An object in motion stays in motion unless acted upon by a net force."

  • Forces change motion (speed/direction), but do not sustain motion.

- Examples:

- A hockey puck sliding across frictionless ice.

- A spacecraft coasting in outer space.

• Newton's Second Law:

  • "The acceleration an object has is proportional to the net force upon it."

  • Formula: $F = m \times a$ (as force increases, acceleration increases).

• Newton's Third Law:

  • "For every action, there is an equal and opposite reaction."

- Examples:

- If we lean against a wall, the wall pushes back and the net force is $0$.

- A rocket expels fuel, and the fuel pushes against the rocket.

Conservation of Energy in Orbits

- Principle: The total energy of an object is conserved unless a force adds or subtracts energy. Celestial orbits conserve energy.

• Orbital Energy Dynamics:

  • Closer: More $KE$, less $PE$.

  • Further: Less $KE$, more $PE$.

Orbital Mechanics and Center of Mass

- Mutual Orbital Motion:

- Celestial objects orbit each other as gravity provides centripetal force to both. Neither object is stationary.

• Center of Mass ($CM$):

  • Two objects in an orbit pair orbit the center of mass ($CM$).

  • The center of mass is the point where matter is equally distributed.

- Mass Disparity Effects:

- It will shift towards the more massive object in the orbit pair.

- In cases where one object is much more massive than the other, it can appear as though that object is stationary.

- **$3M$ vs $M$ System Configuration**: Center of mass is positioned along the connecting axis, shifted towards the $3M$ mass ($1/4$ distance from $3M$, $3/4$ distance from $M$).

Orbital Shapes and Energy States

- Energy Ratio and Geometry:

- The shape of an orbit is determined by the balance between kinetic and potential energy.

- **Closed Orbits ($KE \approx PE$)**: Only orbits where $KE$ is similar to $PE$ are closed.

- **Open Trajectories ($KE > PE$ or $KE \gg PE$)**: Too much $KE$ or $PE$ results in an open orbit (spiral or parabola).

- **Inward Spiral Trajectories ($KE < PE$)**: Insufficient kinetic energy leads to spiral paths.

Planetary Orbits in the Solar System

• Elliptical Nature: The orbit of every planet in the solar system is elliptical, although many are nearly circular. Measure with circularity parameter.

• Circularity Parameter:

  • Earth: $98\%$ ($98.3\%$)

  • Venus: $99\%$ ($99.3\%$)

  • Mercury: $77\%$ ($79.5\%$)

  • Halley's Comet: $3\%$

  • Hale-Bopp Comet: $0.5\%$

Space Flight Mechanics

- Mission Trajectory Design: Orbital mechanics are used in designing space flight paths. Minimizing fuel use is essential for successful missions.

• Apollo 11 Flight Path Overview (1969):

  1. Earth Launch.

  2. Insertion into Earth Parking Orbit.

  3. Translunar Injection burn.

  4. Spacecraft / rocket stage separation.

  5. First midcourse correction.

  6. Second midcourse correction.

  7. Lunar orbit insertion by spacecraft.

  8. Lunar orbit descent and landing by Lunar Module (LM).

  9. Command and Service Module (CSM) / Lunar Module ascent stage rejoin.

  10. Transearth injection burn.

  11. Third midcourse correction.

  12. Command Module / Service Module separation.

  13. Earth entry and landing.

Kepler's Laws of Planetary Motion

• Historical Background:

  • Kepler published three laws of planetary motion starting in 1609.

  • They describe the motion of planets around the Sun in the solar system.

  • They were refinements to the laws proposed by Copernicus in about 1514.

• Kepler's First Law:

  • "Planets orbit the Sun in ellipses, with the Sun at one focus."

  • The foci define the shape of the ellipse.

• Kepler's Second Law:

  • "A line connecting the Sun to a planet sweeps out equal areas for equal time intervals."

  • A planet travels faster when closer to the Sun, and slower when further away (explained by conservation of energy and momentum: Less $KE$ / More $PE$ vs More $KE$ / Less $PE$).

• Kepler's Third Law:

  • "The square of the orbital period is proportional to the cube of the semi-major axis."

  • Formula: $T^2 = \frac{4 \pi^2}{G (M + m)} a^3$

  • A planet further from the Sun has a longer orbital period.

  • A more massive planet will have a smaller orbital period.

Questions & Review Exercises

• Freefall Speed Calculation:

  • Question: How fast would an object in freefall travel after $3$ seconds?

  • Answer: $29.4\,m/s$ (With each second, an object in freefall travels $9.8\,m/s$ faster: $3 \times 9.8\,m/s = 29.4\,m/s$).

• Energy Conversion Dynamics:

  • Question: As Earth moves towards the Sun, does its potential energy increase or decrease?

  • Answer: Decrease (Closer: Less $PE$, More $KE$).

  • Question: As Earth moves away from the Sun, does its kinetic energy increase or decrease?

  • Answer: Decrease (Further: Less $KE$, More $PE$).

• Kepler's Laws Summary:

  • 1st Law: "Planets orbit the Sun in ellipses, with the Sun at one focus."

  • 2nd Law: "A line connecting the Sun to a planet sweeps out equal areas for equal time intervals."

  • 3rd Law: "The square of the orbital period is proportional to the cube of the semi-major axis."