Astrophysics and cosmology
Module 5: Newtonian world and astrophysics5.5 Astrophysics and cosmology |
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| 5.5.1 | Stars
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| 5.5.2 | Electromagnetic radiation from stars
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| 5.5.3 | Cosmology
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1. Stars:
- a) Stars and Stellar Evolution
- Planets:
- – Celestial bodies orbiting a star, massive enough to be rounded by their own gravity but not massive enough to initiate fusion.
- Planetary Satellites:
- – Objects orbiting planets, such as moons.
- Comets:
- – Small icy bodies with elliptical orbits around a star, developing tails when close to the star due to sublimation of ice.
- Solar Systems:
- – A system of a star and all objects gravitationally bound to it (e.g., planets, comets, asteroids).
- Galaxies:
- – Massive collections of stars, gas, dust, and dark matter bound by gravity (e.g., Milky Way).
- The Universe:
- – The totality of space, time, matter, and energy encompassing all galaxies and cosmic structures.
- b) Formation of a Star
- Interstellar Dust and Gas:
- – Stars form from vast clouds of gas and dust, primarily hydrogen.
- Gravitational Collapse:
- – Gravitational attraction pulls the gas and dust together.
- – As the density increases, temperature rises, forming a protostar.
- Nuclear Fusion:
- – When the core temperature reaches [math]\sim 10^7 \, \text{K}[/math] hydrogen nuclei fuse into helium, releasing energy:
- [math]4H \rightarrow \text{He} + \text{energy}[/math]
- Radiation and Gas Pressure:
- – Energy from fusion generates radiation and gas pressure, balancing gravitational collapse and stabilizing the star.
- c) Evolution of a Low-Mass Star
- Main Sequence:
- – Hydrogen fusion occurs in the core; the star remains stable for billions of years (e.g., the Sun).
- Red Giant:
- – Hydrogen in the core depletes, and helium fusion begins in a shell around the core.
- – The outer layers expand and cool, forming a red giant.
- Planetary Nebula and White Dwarf:
- – The outer layers are ejected, forming a planetary nebula.
- – The remaining core becomes a white dwarf, which slowly cools over time.
- d) Characteristics of a White Dwarf
- Electron Degeneracy Pressure:
- – Prevents further gravitational collapse by the Pauli exclusion principle, where no two electrons can occupy the same quantum state.
- Chandrasekhar Limit:
- – Maximum mass for a stable white dwarf: [math]1.4 M_{\odot}[/math].
- – Beyond this, the core collapses into a neutron star or black hole.
- e) Evolution of a Massive Star
- Red Supergiant:
- – After exhausting hydrogen, heavier elements fuse (e.g., helium, carbon, oxygen) in successive layers.
- Supernova:
- – Core collapse triggers a violent explosion, dispersing elements into space.
- Neutron Star or Black Hole:
- – If the core mass:
- – Is [math]1.4 M_{\odot} < M < 3 M_{\odot}[/math] it becomes a neutron star.
- – Exceeds [math]3 M_{\odot}[/math], it forms a black hole.
- f) Characteristics of Neutron Stars and Black Holes
- Neutron Star:
- – Extremely dense core composed of neutrons.
- – Diameter: [math]\sim 10 – 20 \, \text{km}, \, \text{Mass}: 1.4 M_{\odot} < M < 3 M_{\odot}[/math]
- – Supported by neutron degeneracy pressure.
- Black Hole:
- – Region of spacetime with gravitational pull so strong that not even light can escape.
- – Defined by the event horizon, with radius:
- [math]r_s = \frac{2GM}{c^2}[/math]
- – Where [math]r_s[/math] is the Schwarzschild radius.
- g) Hertzsprung–Russell (HR) Diagram
- Definition:
- – A plot of stellar luminosity (L) against surface temperature (T).
- Key Features:
- – Main Sequence: Stable stars burning hydrogen in their cores.
- – Red Giants/ Supergiant: Luminous, cool stars with expanded outer layers.
- – White Dwarfs: Hot but faint stars due to their small size.
- Significance:
- – Demonstrates stellar evolution stages based on mass and temperature.
2. Electromagnetic Radiation from Stars
- a) Energy Levels of Electrons in Isolated Gas Atoms
- Discrete Energy Levels:
- – Electrons in atoms occupy quantized energy levels.
- – Transitions between these levels involve absorbing or emitting discrete amounts of energy as photons.
- b) Negative Energy Values:
- – Energy levels are negative because they are measured relative to the zero-energy state of a free electron (at infinity).
- – The more negative the value, the closer the electron is to the nucleus (e.g.[math]E_1 < E_2 < 0[/math]).
- c) Emission Spectral Lines
- ⇒ Emission of Photons:
- – When an electron transitions from a higher energy level ([math]E_2[/math]) to a lower one ([math]E_1[/math]), it emits a photon with energy:
- [math]\Delta E = E_2 – E_1[/math]
- d) Equations:
- – Relating photon energy ([math]E[/math]) to frequency ([math]f[/math]) and wavelength ([math]{\lambda}[/math]):
- [math]hf = \Delta E \quad \text{and} \quad \Delta E = \frac{hc}{\lambda}[/math]
- Where:
- – [math]h: \text{Planck’s constant} \, (6.63 \times 10^{-34} \, \text{Js}),[/math]
- – [math]c: \text{speed of light} \, (3 \times 10^8 \, \text{m/s})[/math]
- e) Spectral Lines:
- – Different atoms have unique energy levels, producing unique emission spectra, which serve as “fingerprints” for identifying elements in stars.
- f) Spectra Types
- ⇒ Continuous Spectrum:
- – Produced by hot, dense objects.
- – Contains all wavelengths without gaps.
- ⇒ Emission Line Spectrum:
- – Produced by hot, low-density gases.
- – Bright lines at specific wavelengths corresponding to emitted photons.
- ⇒ Absorption Line Spectrum:
- – Occurs when a continuous spectrum passes through a cooler gas.
- – Dark lines appear at wavelengths absorbed by the gas, corresponding to energy transitions.
- g) Transmission Diffraction Grating and Wavelength Determination
- A transmission diffraction grating is an optical device consisting of many closely spaced parallel slits that diffract light passing through it. This diffraction separates light into its component wavelengths (spectrum), allowing the determination of the light’s wavelength.
- ⇒ Principles of Diffraction and Interference
- When light passes through the slits of a diffraction grating, diffraction and interference occur:
- – Diffraction: The bending of light as it passes through the slits.
- – Interference: The constructive and destructive superposition of light waves emerging from the slits.
- The constructive interference forms bright fringes (maxima), and the condition for this is given by:
- [math]d \sin \theta = n \lambda[/math]
- ⇒ Diffraction Grating Equation
- The relationship between the grating parameters and the wavelength is summarized as:
- [math]\lambda = \frac{d \sin \theta}{n}[/math]
- Parameters:
- Grating Spacing (d):
- – It is related to the number of lines per unit length (N) on the grating:
- [math]d = \frac{1}{N}[/math]
- – If the grating has N=5000lines per millimeter, then:
- [math]d = \frac{1}{N} \\
d = \frac{1}{5000 \, \text{mm}} \\
d = 2 \times 10^{-7} \, \text{m}[/math] - h) Condition for Maxima:
- – The path difference between adjacent slits must equal an integer multiple of the wavelength ([math][/math]):
- [math]d \sin \theta = n \lambda[/math]
- Where:
- – d: Grating spacing (inverse of lines per meter),
- – [math]\theta[/math]: Angle of diffraction,
- – n: Order of the maximum.
- ⇒ Applications:
- – Used to measure precise wavelengths of light emitted by stars.
- i) Wien’s Displacement Law
- ⇒ Definition:
- – The wavelength at which a star’s intensity is maximum ([math]\lambda_{\text{max}}[/math]) is inversely proportional to its surface temperature (T):
- [math]\lambda_{\text{max}} T = b[/math]
- Where:
- – b: Wien’s constant [math](2.9 \times 10^{-3} \, \text{m} \cdot \text{K}^{-1})[/math]
- ⇒ Applications:
- – By measuring[math]{\lambda_{\text{max}}}[/math], the surface temperature of a star can be estimated:
- [math]T = \frac{b}{\lambda_{\text{max}}}[/math]
- j) Luminosity and Stefan’s Law
- ⇒ Luminosity:
- – The total energy radiated per second by a star (L) depends on its radius (r) and surface temperature (T):
- [math]L = 4 \pi r^2 \sigma T^4[/math]
- Where:
- – [math]\sigma: \text{Stefan-Boltzmann constant} \, (5.67 \times 10^{-8} \, \text{W/m}^2 \, \text{K}^4)[/math]
- – r: Radius of the star,
- – T: Surface temperature.
- k) Estimating Radius:
- – Combining Wien’s law and Stefan’s law:
- [math]r = \sqrt{\frac{L}{4 \pi \sigma T^4}}[/math]
3. Cosmology:
- a) Distance in Astronomy:
- ⇒ Astronomical unit (AU):
- – Astronomical Unit (AU): The average distance between the Earth and the Sun, approximately 93 million miles or 149.6 million kilometers.
- – Light-Year (ly): The distance light travels in one year, approximately 5.88 trillion miles or 9.46 trillion kilometers.
- – Parsec (pc): A unit of distance equal to 3.26 light-years or [math]3.085 \times 10^{16}[/math] kilometers.
- b) Stellar Parallax:
- – Parallax is the apparent shift of a nearby star’s position against the background of more distant stars due to the Earth’s motion around the Sun.
- – The parallax angle (p) is measured in seconds of arc (“).
- – The distance (d) in parsecs is related to the parallax angle by the equation:[math]p = \frac{1}{d}[/math]
- c) The equation [math]p = \frac{1}{d}[/math] where p is the parallax in seconds of arc and d is the distance in parsec:
- The equation [math]p = \frac{1}{d}[/math] relates the parallax angle (p) of a star to its distance (d) from Earth. This relationship is fundamental in astronomy and provides a direct method to measure distances to stars.
- ⇒ Explanation:
- Parallax (p):
- – Definition: Parallax is the apparent shift in the position of a nearby star against the background of distant stars, observed from two different positions of Earth in its orbit around the Sun (typically six months apart).
- – Unit: Parallax (p) is measured in seconds of arc (arcseconds).
- ⇒ Distance (d):
- Definition: The distance to the star, measured in parsecs (pc), where 1 parsec is defined as the distance at which a star would have a parallax of 1 arcsecond.
- 1 parsec (pc): Equal to approximately [math]3.26 \, \text{light-years} \, \text{or} \, 3.086 \times 10^{16} \, \text{m}[/math] .
- ⇒ The Equation:
- [math]p = \frac{1}{d}[/math]
- Where:
- – p = parallax in arcseconds (arcsec),
- – d = distance in parsecs (pc).
- d) Cosmological Principle:
- – The universe is homogeneous (uniform in composition and structure) on large scales.
- – The universe is isotropic (looks the same in all directions) on large scales.
- – The laws of physics are universal and apply everywhere in the universe.
- The cosmological principle is a fundamental concept in cosmology, suggesting that the universe is consistent and predictable on large scales. This principle has been supported by numerous observations and experiments, including:
- – Cosmic Microwave Background Radiation (CMB)
- – Large-scale structure of the universe
- – Supernovae observations
- – Baryon acoustic oscillations (BAO)
- e) The Doppler effect:
- – The Doppler effect is the change in frequency or wavelength of electromagnetic radiation due to the relative motion between a source and an observer. This phenomenon applies to all types of waves, including light, radio waves, and sound waves.
- ⇒ Doppler Shift:
- – When the source is moving towards the observer, the frequency increases (blueshift).
- – When the source is moving away from the observer, the frequency decreases (redshift).
- f) Doppler Equation:
- e) The Doppler equation relates the change in wavelength (Δλ) or frequency (Δf) to the relative velocity (v) between the source and observer, and the speed of light (c):
- [math]\frac{\Delta \lambda}{\lambda} \approx \frac{\Delta f}{f} \approx \frac{v}{c}[/math]
- f) where:
- – Δλ is the change in wavelength
- – λ is the original wavelength
- – Δf is the change in frequency
- – f is the original frequency
- – v is the relative velocity between the source and observer
- – c is the speed of light
- g) This equation shows that the Doppler shift is proportional to the relative velocity and inversely proportional to the speed of light.
- h) Applications:
- – Astronomical observations: measuring velocity of stars, galaxies, and cosmological expansion
- – Radar technology: measuring velocity of targets
- – Medical imaging: Doppler ultrasound measures blood flow velocity
- – Particle physics: measuring particle velocities in high-energy collisions
- i) The Doppler effect has far-reaching implications in various fields, allowing us to study motion and velocity in diverse contexts.
- g) Hubble’s Law:
- [math]v \approx H_0 d[/math]
- This equation relates the recession velocity (v) of a galaxy to its distance (d) from us, with H0 being the Hubble constant.
- Hubble Constant ([math]H_0[/math]):
- – Units: [math]\text{km/s/Mpc (kilometers per second per megaparsec)} \, \text{or} \, \text{s}^{-1} \, \text{(inverse seconds)}[/math]
- – Value: [math]\text{approximately} \, 67 \, \text{km/s/Mpc} \, \text{or} \, 2.2 \times 10^{-18} \, \text{s}^{-1}[/math]
- Hubble’s Law indicates that:
- – Galaxies are moving away from us
- – Velocity increases with distance
- – Universe is expanding
- Galactic Redshift:
- – Light from receding galaxies is shifted towards the red end of the spectrum
- – Redshift increases with distance
- – Evidence for expansion
- j) Big Bang Theory:
- The universe began as a singularity, expanding rapidly around 13.8 billion years ago
- Expansion continues, with galaxies moving away from each other
- Supported by multiple lines of evidence, including:
- – Cosmic Microwave Background Radiation
- – Abundance of light elements
- – Large-scale structure of the universe
- Hubble’s Law and the Expanding Universe:
- – Hubble’s Law describes the velocity-distance relationship for galaxies
- – Expanding universe model explains the redshift of light from receding galaxies
- – Big Bang Theory provides the framework for understanding the universe’s evolution
- The Hubble constant ([math]H_0[/math]) is a fundamental parameter in cosmology, relating the expansion rate to the distance. Its value has been refined over time through various observations and experiments.
- k) Experimental Evidence:
- Cosmic Microwave Background Radiation (CMB):
- – Discovered in 1964 by Arno Penzias and Robert Wilson
- – Radiation is thought to be the residual heat from the initial explosion
- – Temperature: approximately 2.7 Kelvin (°K)
- CMB is key evidence for the Big Bang Theory, as it:
- – Confirms the universe’s thermal history
- – Supports the idea of a hot, dense universe in the past
- l) Expansion of Space-Time:
- – The Big Bang led to the expansion of space-time itself
- – This expansion is still ongoing, with galaxies moving away from each other
- – Hubble’s Law describes this expansion
- m) Age of the Universe:
- – Estimated using Hubble’s constant ([math]H_0[/math])
- [math]t \approx H_0^{-1}[/math]
- Where t is the age of the universe and H0 is the Hubble constant
- Current estimate: approximately 13.8 billion years
- ⇒ Additional Evidence:
- – Abundance of light elements (e.g., hydrogen, helium, lithium)
- – Large-scale structure of the universe (galaxy distributions, clusters, superclusters)
- – Gravitational lensing and baryon acoustic oscillations
- ⇒ The Big Bang Theory is widely accepted by scientists as the most accurate explanation for the origins and evolution of our universe. The evidence from multiple fields of astronomy and physics converges to support this theory.
- n) The Evolution of the Universe:
- ⇒ Timeline:
- – Big Bang (13.8 billion years ago)
- – Expansion and Cooling (first fraction of a second)
- – Proton-Neutron Era (first few minutes)
- – Nucleosynthesis (first 20 minutes)
- – Photon Era (first 380,000 years)
- – Cosmic Dark Ages (380,000 – 1 billion years)
- – Star and Galaxy Formation (1 billion years – present)
- – Large-scale Structure Formation (1 billion years – present)
- – Present Day Universe (13.8 billion years after Big Bang)
- ⇒ Current Ideas:
- Universe Composition:
- – Dark Energy (~68%): mysterious energy driving acceleration
- – Dark Matter (~27%): unknown particles holding galaxies together
- – Ordinary Matter (~5%): regular matter we can see and interact with
- O) Universe Evolution:
- Expansion continues, with galaxies moving away from each other
- Dark energy drives acceleration, while dark matter slows it down
- Ordinary matter clumps together, forming stars, galaxies, and structures
- The universe has evolved significantly since the Big Bang, transforming from a hot, dense plasma to the vast, complex cosmos we observe today. Ongoing research aims to better understand dark energy and dark matter, refining our understanding of the universe’s evolution and fate.