Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Gravitational force ↓


Orbital mechanics


Orbital mechanics, also known as celestial mechanics, is a branch of classical mechanics that deals with the motion of objects in space under the influence of gravitational forces. It deals primarily with the orbits of planets, moons, and artificial satellites. In this talk, we will explore the fundamental principles and rules that govern orbital mechanics, focusing on the role of gravity as described by Isaac Newton's law of universal gravitation, as well as on some of the key orbital parameters and types of orbits. This foundational knowledge helps us understand how objects move through the universe, from Jupiter's moons to our adventurous spacecraft.

Newton's law of universal gravitation

At the core of orbital mechanics is the force of gravity, a universal force that attracts two bodies toward each other. Newton's law of universal gravitation is expressed by the formula:

F = G * (m1 * m2) / r^2

In this equation:

  • F is the gravitational force between the two masses.
  • G is the gravitational constant, approximately 6.674 × 10^-11 N(m/kg)^2.
  • m1 and m2 are the masses of the two objects.
  • r is the distance between the centers of the two masses.

This formula highlights that the gravitational force is directly proportional to the product of two masses and inversely proportional to the square of the distance between them. This forms the basis for understanding how celestial bodies interact with each other.

Kepler's laws of planetary motion

Before Newton's insights, Johannes Kepler had formulated three empirical laws describing the motion of planets. These laws were derived from careful observations of the sky:

Kepler's first law: the law of ellipses

Kepler's first law states that a planet's orbit around the Sun is an ellipse with the Sun at one of two foci. Unlike a perfect circle, an ellipse is an elongated circle. This means that the distance between the planet and the Sun changes as the planet moves along its orbit.

r = a(1 - e^2) / (1 + e * cos(θ))

Where:

  • r is the orbital radius at angle θ.
  • a is the semi-major axis of the ellipse.
  • e is the eccentricity of the orbit, which shows how much it deviates from the circle.

Kepler's second law: the law of equal areas

Kepler's second law, the law of equal areas, states that the line segment joining a planet and the Sun clears equal areas in equal time intervals. This means that planets move faster when they are closer to the Sun and slower when they are farther from the Sun.

In the visual example above, the shaded area represents the area of the orbit the planet completes in a given time. The orange area is the same over different time periods, reflecting Kepler's second law.

Kepler's third law: harmonic law

Kepler's third law, the harmonic law, provides a relationship between the period of a planet's orbit and the semi-major axis of its ellipse. Mathematically, it is expressed as:

T^2 ∝ a^3

This law means that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (a) of its orbit. This relation helps in calculating the time taken by a planet to orbit the Sun based on its distance from the Sun.

Conic sections in orbital mechanics

The orbits of celestial objects can be described using conic sections, which include circles, ellipses, parabolas, and hyperbolas. In orbital mechanics, the type of conic section depends on the energy and eccentricity of the orbiting object.

Circular and elliptical orbits

Both circular and elliptical orbits are closed paths around a central object. The difference lies in the eccentricity:

  • A circular orbit has an eccentricity of 0, which represents a perfect circle.
  • The eccentricity of an elliptical orbit is between 0 and 1, indicating its elliptical shape.
Circular orbit Elliptical orbit

Parabolic and hyperbolic trajectories

Parabolic and hyperbolic trajectories describe open paths where a celestial body is not bound by gravity to its central body:

  • A parabolic trajectory has eccentricity 1 and represents the escape path with escape velocity.
  • A hyperbolic trajectory has an eccentricity greater than 1 and indicates that the object is moving greater than the escape velocity.
Parabolic path Hyperbolic path

Orbital velocity and energy

Orbital velocity is the speed at which an object must travel to maintain a stable orbit around a celestial body. It depends on the mass of the central body and the distance to the orbiting object. Orbital velocity is given by:

v = sqrt(G * M / r)

Here:

  • v is the orbital velocity.
  • G is the gravitational constant.
  • M is the mass of the central body.
  • r is the distance from the center of the central body.

The concept of escape velocity is also important in orbital mechanics. It is the minimum speed required for an object to "break free" from the gravitational attraction of the central body without any additional propulsion. Escape velocity is calculated as:

v_escape = sqrt(2 * G * M / r)

Orbital parameters

Several parameters help describe the shape and direction of an orbit. These include:

  • Semi-major axis (denoted a): Half the longest diameter of the ellipse, it determines the shape of the orbit.
  • Eccentricity (denoted by e): Describes the shape of the orbit. A value of 0 is a circle, while one close to 1 is a more elongated ellipse.
  • Inclination: The inclination of the plane of the orbit compared to a reference plane, such as the equatorial plane of the central body.
  • Longitude of ascending node: The angle from the reference direction to the direction of the ascending node of the orbit.
  • Argument of periapsis: The angle from the ascending node to the periapsis (the closest point in the orbit to the central body).
  • True anomaly: The angle between the direction of periapsis and the body's current position on the orbit.

Key concepts in orbital transfer

Spacecraft often need to change orbit, known as orbital transfers, which include:

  • Hohmann transfer orbit: An efficient path between two circular orbits using two engine burns. This is the most fuel-efficient way to transfer between orbits when time is not a constraint.
  • Bi-elliptical transfer: a transfer involving two elliptical orbits and two burns, used when the shapes of the orbits differ significantly.
  • Gravity assist: A technique that uses the gravity of a celestial body to alter the path and speed of a spacecraft, thereby saving fuel.

Conclusion

Orbital mechanics is crucial for understanding how objects move in space under the influence of gravity. Gravitational forces and orbital parameters define the complex dance of our solar system, guiding planets, moons, and man-made satellites. Through Newton's law of gravity, Kepler's laws of motion, and a deep dive into conic sections and orbital parameters, we gain insight into the cosmic ballet of celestial bodies. Furthermore, orbital mechanics allows us to plan space missions, ensure satellite functionality, and explore the planets and beyond.


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Orbital mechanics

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