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 ↓


Kepler's laws of planetary motion


Kepler's laws of planetary motion are three scientific laws that describe the motion of planets around the Sun. These laws, formulated in the early 17th century, revolutionized our understanding of the solar system. German mathematician and astronomer Johannes Kepler proposed that planets move in elliptical orbits rather than circular ones, challenging the long-held belief in circular orbits. These laws are fundamental to the study of celestial mechanics and have laid the groundwork for classical mechanics and astronomy.

First law: The law of ellipses

The first of Kepler's laws is the law of ellipses. It states that the orbit of a planet is elliptical, with the Sun at one of two foci.

Ellipse: A regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (foci) is constant.

In simple terms, this law tells us that a planet does not revolve around the Sun in a perfect circle. Instead, it follows an elliptical path. Imagine an elongated circle; it is an ellipse.

Focus 1 Focus 2 Oval Sun

In the ellipse above, the two focus points, labeled as focus 1 and focus 2, represent important points in the ellipse. The Sun is located at one of these focus points, not at the center.

In the actual solar system structure, each planet orbits around the Sun in a unique ellipse. The shape of each planet's orbit is characterized by its eccentricity, which is a measure of how stretched the ellipse is. Circles are a special case of ellipses with eccentricity of zero. Extremely elongated ellipses have large eccentricities.

Second law: Law of equal areas

Kepler's second law, also known as the law of equal areas, states that the line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time.

If a planet moves such that the time to travel from point A to B is equal to the time from C to D, then the area swept out by line AB is equal to the area swept out by line CD.

This means that the speed of the planet's orbit around the sun is not uniform. When a planet is closer to the sun, such as point B in the diagram, it moves faster, and covers a greater distance in the same time. Conversely, when it is farther from the sun, it moves slower.

The consequence of this law is that planets have variable speeds in their orbit; they speed up as they approach the Sun and slow down as they move away. This concept helps us understand why some planets take more or less time to complete their orbit, depending on their position in the ellipse.

Third law: The law of harmony

Kepler's third law, the Law of Harmony, states that the square of the orbital period of any planet is proportional to the cube of the semi-major axis of its orbit.

(T^2) ∝ (a^3)

Expressed mathematically, if T represents the time taken by a planet to complete one revolution around the Sun (the period of revolution), and a represents the semi-major axis of its elliptical orbit, then:

T^2 = k * a^3

Here, k is a constant that is the same for all planets orbiting the same star. This law reveals a harmonious relationship between the distance of the planets from the Sun and their orbital periods. This can be seen by comparing different planets:

Sun Planet 1 Planet 2 Planet 3

In the diagram above, different circular paths represent different planets. Planet 1, which is closer to the Sun, will complete its orbit more quickly than Planet 3, which is farther away. This is why planets like Mercury, which are closer to the Sun, have shorter years than distant planets like Jupiter.

Applications and significance

Kepler's laws of planetary motion laid the foundation for Isaac Newton's theory of universal gravitation. Newton was able to provide a physical explanation and mathematical basis for Kepler's descriptions. Additionally, these laws are important for understanding celestial navigation and space exploration, such as determining the trajectories of spacecraft.

On a broader scale, Kepler's laws help deepen our understanding of how gravitational forces operate in space. They solidify the notion that natural forces guide the motions of our solar system's celestial inhabitants and the planets we observe.

Conclusion

Kepler's laws of planetary motion provide detailed descriptions of planetary orbits and revolutionize the way humans interpret the universe. These laws support the idea that observations must be matched with accurate mathematical models to predict new discoveries and understanding.

The study of these laws marks an important period in scientific history when rational thinking began to take precedence over earlier beliefs. Despite their age, Kepler's laws remain relevant and continue to influence modern astronomical studies and celestial mechanics.


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Kepler's laws of planetary motion

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