ASTR - lec 12

ASTR 1P02 - lec 12

Lecture 12: Starlight

Key Learning Objectives

• How star brightness is measured.

• Light and the electromagnetic spectrum.

• The meaning of the colors of stars.

• Information derived from analyzing starlight.

Introduction to Star Brightness

• Measurement Units:

  • Standard unit of energy: Joule (J)

  • Definition: 1 J is the energy required to accelerate a 1 kg mass at 1 m/s² over a distance of 1 m.

  • Power: Energy per unit time (measured in watts (W))

  • Definition: 1 W = 1 J/s.

  • Example: A 100 W lightbulb consumes 100 J every second.

Luminosity of Stars

• Luminosity: Total energy emitted per second in light.

  • Measured in watts (W).

  • Sun's luminosity: $L_\odot = 3.828 \times 10^{26} \text{ W}$

  • Symbol $\odot$ represents the Sun.

  • Sun's mass $M_\odot ≈ 2.0 \times 10^{30} \text{ kg}$.

  • Sun's radius $R_\odot ≈ 7.0 \times 10^{8} \text{ m}$.

  • Example star: BAT99-98 in the Large Magellanic Cloud has a luminosity of approximately 5,000,000 $L_\odot$.

  • Faint star 2MASS J0523-1403 has a luminosity of approximately 0.0001 $L_\odot$ .

Apparent Magnitude of Stars

• Definition: Measures how bright a star appears from Earth.

  • Depends on three factors:

  1. Luminosity of the star.

  2. Distance from Earth (light dims with distance).

  3. Interstellar dust on the line of sight (blocks light).

  • Distinction made:

  • Luminosity is an objective property.

  • Apparent magnitude is subjective, varying with the observer's location.

Understanding Light Emission

• Stars emit light in all directions; we only see a portion directed toward Earth.

• As distance increases, light spreads over a larger area, decomposing intensity.

Inverse Square Law of Brightness

• Brightness diminishes with distance:$(A = \frac{L}{4\pi r^2})$

  • Light is spread out as a sphere, increasing surface area with radius $r$.

  • Surface area of a sphere: $A = 4\pi r^2$

  • Brightness is inversely proportional to the square of distance:

  • $ext{Brightness} \propto \frac{1}{r^2}$.

• Example Calculation:

  • Distance from Sun to Earth: 1 AU $≈ 150,000,000 ext{ km}$.

  • Distance from Sun to Neptune: ~30 AU; light is 900 times less bright on Neptune than on Earth: $30^2 = 900$.

  • Distance from Sun to Mercury: ~0.4 AU; light is 6.25 times brighter on Mercury than Earth: $2.5^2 = 6.25$.

Apparent Magnitude Scale

• Apparent magnitude indicates visibility:

  • Under -25: Painfully bright.

  • Under -4: Visible during the day.

  • Under +6.5: Visible to the naked eye under ideal conditions.

  • Under +27.7: Visible through the Subaru Telescope.

  • Under +31.5: Visible to the Hubble Space Telescope.

• Examples of objects and their magnitudes:

  • Sun: -26.8, Full Moon: -12.7, Sirius: -1.5, Proxima Centauri: +11, Charon: +15.6.

Absolute Magnitude

• Definition: Apparent magnitude if viewed from a distance of 32.6 light-years without obstructions.

  • Example:

  • Betelgeuse: Apparent Magnitude +0.5; Absolute Magnitude -5.85.

  • Vega: Apparent Magnitude +0.03; Absolute Magnitude +0.58.

Properties of Light as a Wave

• Light characterized as both:

  1. Electromagnetic wave: Made of electric and magnetic fields.

  2. Particles (photons): Massless elementary particles.

• This phenomenon is known as wave-particle duality in quantum mechanics.

• Wave}: Regular disturbance in a medium; characterized by:

  • Amplitude (A): Maximum displacement from equilibrium (e.g., loudness in sound).

  • Wavelength (λ): Distance between two crests.

  • Frequency (f): Number of wavelengths per second (measured in Hz).

  • Speed (c): Speed of wave propagation.

Wave Relationships

• Relationship: $c = \lambda f$

  • Wavelength $\lambda$ and frequency $f$ are inversely proportional:

  • High frequency = Short wavelength.

• Speed of light in vacuum: $c ≈ 3 \times 10^8 ext{ m/s}$.

Color and Temperature of Stars

• Photon energy relation: $E = hf = \frac{hc}{\lambda}$ (Planck’s Equation)

  • $h ≈ 4.1 \times 10^{-15} ext{ eV/Hz}$.

• Color spectrum ranges visible to the human eye:

  • Unique wavelength ~380-750 nm; frequency ~400-790 THz.

  • Monochromatic light: Single wavelength.

  • Non-spectral colors: Combinations of wavelengths (e.g., magenta).

Electromagnetic Spectrum Overview

• Spectrum from lowest to highest frequency:

  • Radio waves, Microwaves, Infrared, Visible light, Ultraviolet, X-rays, Gamma rays.

• Wavelength examples:

  • Visible light ~0.5 × 10⁻⁶ m, Ultraviolet ~1 × 10⁻⁸ m.

Black Body Radiation

• Black body: Ideal body absorbing all incident electromagnetic radiation.

• Emission characteristic: According to Wien’s Displacement Law:

  • Peak wavelength: $\lambda_{peak} = \frac{b}{T}$ where $b ≈ 2.9 \times 10^{-3} ext{ m⋅K}$.

Stellar Temperatures and Colors

• Temperature affects star color and therefore its emitted spectrum.

  • Red stars: lower temperatures (~3,000 K).

  • Blue stars: higher temperatures (~25,000 K).

  • Planckian locus illustrates black-body spectrum transitions (red to blue).

Light Interaction within Atmosphere

• Rayleigh scattering effects on sunlight through the atmosphere.

  • Shorter wavelengths (blue) scatter more, leading to a blue sky and yellow/red appearance of the Sun.

Spectroscopy

• Absorption of specific light frequencies based on gas composition yields dark lines in the spectrum.

  • Example: Fraunhofer lines in the Sun’s spectrum indicate chemical composition.

• Heating a gas causes emission lines reflecting the same frequencies as absorption lines.

Bohr Model of the Atom

• Electrons in discrete orbits defined by quantum numbers.

• Energy emitted upon transition: $E = R<em>hc \left( \frac{1}{n</em>f^2} - \frac{1}{n_i^2} \right)$ with $R$ as Rydberg constant.