🔧Intro to Mechanics Unit 6 – Gravity and Orbital Mechanics

Key Concepts and Definitions

• Gravity is the force of attraction between two objects with mass
• Orbital mechanics studies the motion of objects in orbit around other objects due to gravity
• Kepler's laws describe the motion of planets around the sun and satellites around planets
  • First law: Orbits are elliptical with the central body at one focus
  • Second law: A line segment joining a planet and the sun sweeps out equal areas in equal intervals of time
  • Third law: The square of the orbital period is proportional to the cube of the semi-major axis of the orbit
• Newton's law of universal gravitation states that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them
• Escape velocity is the minimum speed needed for an object to escape a gravitational field
• Gravitational potential energy is the energy an object possesses due to its position in a gravitational field

Historical Context and Development

• Ancient Greeks observed and studied celestial motion, laying the foundation for orbital mechanics
• Ptolemy developed a geocentric model of the universe with circular orbits and epicycles to explain planetary motion
• Copernicus proposed a heliocentric model with the sun at the center and planets orbiting in circular paths
• Tycho Brahe made precise observations of planetary positions, which Kepler used to develop his laws of planetary motion
• Galileo's observations of Jupiter's moons provided evidence supporting the Copernican heliocentric model
• Newton's law of universal gravitation and laws of motion provided a mathematical framework for understanding orbital mechanics
• Einstein's theory of general relativity refined our understanding of gravity as a curvature of spacetime

Fundamental Laws and Equations

• Newton's law of universal gravitation: $F = G \frac{m_1 m_2}{r^2}$, where $F$ is the gravitational force, $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses of the objects, and $r$ is the distance between their centers
• Kepler's third law: $\frac{T^2}{a^3} = \frac{4\pi^2}{GM}$, where $T$ is the orbital period, $a$ is the semi-major axis of the orbit, $G$ is the gravitational constant, and $M$ is the mass of the central body
• Vis-viva equation: $v^2 = GM(\frac{2}{r} - \frac{1}{a})$, where $v$ is the orbital speed, $G$ is the gravitational constant, $M$ is the mass of the central body, $r$ is the distance from the center of the central body, and $a$ is the semi-major axis of the orbit
• Gravitational potential energy: $U = -\frac{GMm}{r}$, where $U$ is the gravitational potential energy, $G$ is the gravitational constant, $M$ and $m$ are the masses of the objects, and $r$ is the distance between their centers
• Escape velocity: $v_e = \sqrt{\frac{2GM}{r}}$, where $v_e$ is the escape velocity, $G$ is the gravitational constant, $M$ is the mass of the body being escaped from, and $r$ is the distance from the center of the body to the point of escape

Types of Orbits and Trajectories

• Circular orbits have a constant radius and a constant speed
  • Satellites in geostationary orbits appear stationary relative to the Earth's surface
• Elliptical orbits have a varying radius and speed, with the central body at one focus
  • Most planets and satellites follow elliptical orbits
  • Eccentricity measures the deviation of an orbit from a perfect circle
• Parabolic trajectories have an eccentricity equal to 1 and a specific energy of zero
  • Objects on parabolic trajectories have just enough energy to escape the gravitational field
• Hyperbolic trajectories have an eccentricity greater than 1 and a positive specific energy
  • Objects on hyperbolic trajectories have more than enough energy to escape the gravitational field
• Hohmann transfer orbits are elliptical orbits used to transfer a spacecraft between two circular orbits with minimal energy
• Interplanetary trajectories, such as those used by spacecraft to travel between planets, often involve a combination of different orbit types

Gravitational Fields and Potential Energy

• A gravitational field is a region of space where a mass experiences a force due to the presence of another mass
• The strength of a gravitational field decreases with the square of the distance from the center of the mass
• Gravitational potential energy is the energy stored in an object due to its position within a gravitational field
  • Objects farther from the center of a mass have higher gravitational potential energy
• The gravitational potential at a point is the work done per unit mass to move an object from infinity to that point
• Equipotential surfaces are surfaces on which the gravitational potential is constant
  • The gravitational force is always perpendicular to the equipotential surface
• The gradient of the gravitational potential gives the gravitational field strength and direction
• The Laplace equation, $\nabla^2 \phi = 0$, describes the gravitational potential in regions without mass

Orbital Mechanics Applications

• Satellite orbits are designed based on their intended purpose (communication, navigation, Earth observation, etc.)
  • Low Earth orbits (LEO) are used for Earth observation and some communication satellites
  • Medium Earth orbits (MEO) are used for navigation satellites like GPS
  • Geostationary orbits (GEO) are used for communication and weather satellites
• Spacecraft trajectories are planned using orbital mechanics principles to minimize fuel consumption and travel time
  • Gravity assist maneuvers use the gravitational field of a planet to change a spacecraft's velocity and trajectory
• Orbital perturbations, such as atmospheric drag and gravitational influences from other bodies, must be accounted for in long-term orbit predictions
• Orbital debris mitigation and removal strategies rely on understanding the orbital mechanics of space objects
• Astrodynamics applies orbital mechanics principles to the motion of natural celestial bodies and artificial satellites

Calculations and Problem-Solving Techniques

• Determine the orbital period using Kepler's third law: $T = 2\pi \sqrt{\frac{a^3}{GM}}$
• Calculate the orbital speed at a given point using the vis-viva equation: $v = \sqrt{GM(\frac{2}{r} - \frac{1}{a})}$
• Find the escape velocity at a given distance from a body: $v_e = \sqrt{\frac{2GM}{r}}$
• Determine the gravitational potential energy between two objects: $U = -\frac{GMm}{r}$
• Use conservation of energy to solve problems involving gravitational potential energy and kinetic energy
• Apply Newton's laws of motion and law of universal gravitation to analyze forces acting on orbiting objects
• Employ numerical integration techniques, such as the Runge-Kutta method, to propagate orbits over time
• Utilize software tools like STK (Systems Tool Kit) or GMAT (General Mission Analysis Tool) for complex orbital simulations and mission planning

Real-World Examples and Case Studies

• International Space Station (ISS) orbits the Earth in a nearly circular LEO at an altitude of about 400 km
• GPS satellites orbit the Earth in MEO at an altitude of approximately 20,200 km
• Geostationary satellites, such as those used for satellite TV and some weather satellites, orbit at an altitude of 35,786 km above the Earth's equator
• The Voyager 1 and 2 spacecraft used gravity assist maneuvers to explore the outer planets of the solar system
• The Hubble Space Telescope orbits the Earth in LEO, allowing it to capture high-resolution images of distant astronomical objects
• The GRACE (Gravity Recovery and Climate Experiment) mission used two satellites in a low polar orbit to map the Earth's gravitational field and study changes in water and ice distribution
• The Apollo missions used orbital mechanics principles to plan trajectories for traveling to the Moon, orbiting it, and returning to Earth
• SpaceX's Falcon 9 rockets use a series of elliptical transfer orbits to deploy payloads, such as Starlink satellites, into their target orbits