Copernicus, Kepler, and Galileo's Discoveries

Discoveries of Copernicus, Kepler, and Galileo

• Nicolaus Copernicus proposed the heliocentric model, positioning the Sun at the center of the solar system.
• The heliocentric model offered a more natural explanation for retrograde motion.
• Johannes Kepler defined three laws of planetary motion.
  • Planets orbit the Sun in ellipses, with the Sun at one focus.
  • A line connecting a planet to the Sun sweeps equal areas during equal time intervals.
  • The square of a planet's orbital period (P) is proportional to the cube of its semi-major axis (a), expressed as P^2 = a^3.
• Galileo Galilei made key observations using a telescope.
  • Discovered Jupiter's moons and demonstrated that not all celestial bodies orbit Earth.
  • Observed phases of Venus, proving it orbits the Sun.
  • Noticed sunspots, indicating the Sun isn't perfect.
  • Resolved the Milky Way into countless stars.

Newton's Laws of Motion and Gravity

• Newton's first law of motion, inertia, states that an object remains in its current state of motion unless acted upon by a force.
• Newton's second law is F = ma, which defines force as mass times acceleration.
• Newton's third law states that for every action, there is an equal and opposite reaction.
• Newton's Law of Universal Gravitation is F = Gm1m2/r^2.
  • F is the gravitational force.
  • G is the gravitational constant (6.674 × 10^-11 m³/kg/s²).
  • m1 and m2 are the masses of the objects.
  • r is the distance between the objects.
• The Law of Universal Gravitation explained Kepler's laws and elliptical planetary orbits.

Escape Velocity

• Escape velocity is the minimum speed required to escape a planet’s gravity.
• The formula is v_esc = √(2GM/R).
  • G = gravitational constant.
  • M = the planet’s mass.
  • R = the planet’s radius.
• Earth's escape velocity is approximately 11.2 km/s.

Finite Speed of Light

• Ole Rømer first measured the speed of light in 1676 using observations of Io's eclipses.
• The speed of light (c) is 299,792,458 m/s.
• Light takes about 8.3 minutes to travel from the Sun to Earth.

Newtonian "Black Holes"

• John Michell (1783) and Pierre-Simon Laplace speculated that if an object's escape velocity exceeded the speed of light, nothing, including light, could escape.
• This concept was a precursor to general relativistic black holes.

Absolute Space and Absolute Time

• Newton believed in absolute space as a motion reference and absolute time as a constant flow.
• These concepts were later disproven by Special and General Relativity.

Perihelion Shift of Mercury

• Mercury’s orbit precesses slightly more than predicted by Newtonian mechanics.
• General Relativity explains this discrepancy through spacetime curvature near the Sun.

Michelson-Morley Experiment (1887)

• The experiment aimed to detect Earth’s movement through the "luminiferous aether."
• The experiment found no variance in the speed of light, regardless of direction.
• The result supported Einstein's Special Relativity.

Fitzgerald Contraction

• The hypothesis of Fitzgerald contraction suggested objects moving through the aether contract in length.
• Special Relativity later explained this contraction in terms of length contraction.

Problem of "Open Field Lines" in Electromagnetism

• In theory, electric and magnetic field lines should always close.
• Some astrophysical observations suggest that they might not close.
• Open field lines remain an issue in magnetohydrodynamics and plasma physics.

Special Theory of Relativity

• The Special Theory of Relativity is built on two postulates:
  • The laws of physics are the same in all inertial frames of reference.
  • The speed of light is constant in all reference frames.

Length Contraction and Time Dilation

• Length contraction: L = L0√(1 - v²/c²).
  • Moving objects contract in the direction of motion.
• Time dilation: t = t0/√(1 - v²/c²).
  • Moving clocks tick slower.

Basic Tests of Special Relativity

• Atomic clock experiments confirm that moving clocks run slower.
• GPS satellites account for relativistic effects.
• Atmospheric muons survive longer due to time dilation.

Principle of Equivalence

• Einstein's principle of equivalence states that gravity and acceleration are indistinguishable.

Gravitational Deflection of Light

• Light bends when passing near massive objects.
• This was confirmed during a 1919 solar eclipse expedition led by Eddington.

Gravitational Lensing

• Light from background galaxies is distorted due to the bending of light by foreground mass.

Gravity as the Curvature of Spacetime

• Free-falling objects follow curved paths, known as geodesics, in curved spacetime.
• Einstein’s field equations describe how matter and energy dictate the curvature of spacetime.

Schwarzschild Spacetime Geometry

• Describes a non-rotating black hole.
• Defines the Schwarzschild radius: rs = 2GM/c².
• Nothing, not even light, can escape beyond the Schwarzschild radius.

Gravitational Redshift

• Light escaping from a gravitational field loses energy.
• This energy loss causes the wavelength to increase, resulting in a redshift.

Difference Between Newtonian & GR Black Holes

• Newtonian black holes require an escape velocity greater than c.
• GR black holes feature event horizons, singularities, and extreme spacetime curvature.

Normal Stars as Gravitationally Bound Fusion Reactors

• Stars produce energy via nuclear fusion of hydrogen into helium in their cores.
• Stars maintain hydrostatic equilibrium where gravity pulling inward balances radiation pressure from fusion pushing outward.
• More massive stars experience higher core temperatures, faster fusion rates, and shorter lifespans.

Possible Fates for Stars

• Low-mass stars (< 8 M⊙) transform into white dwarfs.
• Intermediate-mass stars end up as neutron stars.
• High-mass stars (> 20 M⊙) may become black holes.

Eta Carinae as a Black Hole Progenitor

• Eta Carinae is a highly massive and unstable star.
• This star is expected to explode as a supernova or hypernova.
• The explosion will likely leave behind a black hole.

Bosons and Fermions

• Bosons (e.g., photons) can occupy the same quantum state and are force carriers.
• Fermions (e.g., electrons, neutrons) obey the Pauli exclusion principle and cannot occupy the same quantum state.

Pauli Exclusion Principle & Degeneracy Pressure

• The Pauli Exclusion Principle dictates that no two fermions can occupy the same quantum state, resulting in degeneracy pressure.
• Electron degeneracy pressure supports white dwarfs.
• Neutron degeneracy pressure supports neutron stars.

White Dwarfs

• White dwarfs are supported by electron degeneracy pressure.
• Sirius B is a well-known white dwarf.
• The Chandrasekhar limit is about 1.4 M⊙.
• White dwarfs exceeding this limit collapse into neutron stars or black holes.

Neutron Stars

• Neutron stars are supported by neutron degeneracy pressure.
• Neutron stars have a maximum mass of about 2.1-2.3 M⊙.
• Exceeding this mass leads to collapse into a black hole.
• Pulsars are rapidly rotating neutron stars that emit radio waves.

Black Hole Formation

• Oppenheimer & Snyder (1939) determined that stars exceeding a certain mass inevitably undergo gravitational collapse.
• John Wheeler coined the term "black hole."

Finding Black Holes

• Gravitational lensing identifies black holes by observing how they bend light from background objects.
• X-ray binaries indicate black holes when a star orbits an unseen massive object, emitting X-rays from accretion.

Accretion Basics

• Wind accretion occurs as gas flows from a companion star.
• Stream accretion involves matter flowing through the Roche lobe into an accretion disk.

Accretion Disks

• Gas spirals into a black hole, forming a hot, rotating disk.
• Accretion is highly efficient, releasing about 10% of the mass-energy, compared to 0.7% for fusion.
• Cygnus X-1 is a key observational example as the first identified black hole candidate.

Discovery & Properties of Gamma-Ray Bursts (GRBs)

• GRBs were first detected in the 1960s by military satellites searching for nuclear tests.
• GRBs are brief but intensely energetic events.

Challenges in Understanding GRBs

• GRBs appear to come from all directions, showing an isotropic distribution.
• It was initially unclear if GRBs originated in our galaxy or were extragalactic.

BeppoSAX & Afterglows

• The BeppoSAX satellite identified GRB afterglows in X-rays and visible light.
• It was determined that GRBs occur in distant galaxies.

GRB Models

• GRBs are thought to be caused by relativistic blast waves.
• Material jets move at near the speed of light.
• The hypernova model suggests GRBs result from the collapse of massive stars.
• Neutron star mergers lead to short GRBs when two neutron stars collide.

Swift Gamma-Ray Burst Explorer

• Swift, launched in 2004, revolutionized GRB studies.
• It confirmed that long GRBs result from collapsing massive stars.
• Swift also found that short GRBs originate from neutron star mergers.

Number of Galaxies

• There are an estimated 200 billion galaxies in the observable universe.
• Each galaxy contains millions to trillions of stars.

Galactic Nuclei & Supermassive Black Holes (SMBHs)

• Most galaxies host a supermassive black hole (SMBH) at their center.
• SMBHs range from millions to billions of solar masses (M⊙).

Sgr A* (Milky Way’s SMBH)

• Sagittarius A* (Sgr A*) is located at the center of the Milky Way.
• Stellar orbits around Sgr A* confirm its mass is about 4 million M⊙.
• Direct evidence was obtained through observations from Keck and VLT telescopes.

Formation of SMBHs

• SMBHs grow through mergers and accretion.
• Active Galactic Nuclei (AGN) are SMBHs actively accreting material, emitting enormous amounts of energy.

Explore the groundbreaking discoveries of Copernicus, Kepler, and Galileo. Copernicus proposed the heliocentric model. Kepler defined the laws of planetary motion. Galileo's telescopic observations supported the heliocentric view and revealed new celestial phenomena.