Alevel Astironomu and Cosmology

Luminosity \(L\) 光度

The total power of radiation emitted by a star, measured in \(watts (W)\). \[L = 4\pi \sigma r^{2}T^{4}\]

  • \(r\) = Radius of the star
  • \(T\) = Surface Temperature of the star

Radiant Flux Intensity \(F\) 辐射通量强度

The amount of radiation energy received per second per unit surface area from a star, measured at normal incidence on a surface.

Star light Figure: Star light

\[F = \frac{L}{4\pi d^2}, \quad F \propto \frac{1}{d^2}\] Inverse square law

  • F: Radiant Flux intensity at a point P
  • L: Luminosity of the star
  • d: Distance from the star to the point P

Standard Candle 标准烛光

  1. An object with a known luminosity.

    • Type Ia Supernova: A supernova event where a star explodes, significantly increasing in brightness before fading away. The peak luminosity is determined by the time taken to reach this peak.
    • Cepheid Variable Stars 造父变星: Stars whose brightness varies periodically (with periods ranging from 1 to 100 days). The luminosity is determined by the period of variation in brightness.

Procedure for Measuring the Distance to Galaxies

  1. Identify a "Standard Candle" within the galaxy.
  2. Record the variation of its radiant flux intensity over time.
  3. Determine its luminosity.
  4. Apply the formula \[F = \frac{L}{4πd^2}\] to calculate the distance d.

Blackbody Radiation 黑体辐射

  1. Thermal electromagnetic radiation that occurs within or around a body in thermodynamic equilibrium with its surroundings, emitted by a black body (an idealized object that absorbs all incident radiation).
  2. Black Body: An object that absorbs all incident radiation without reflecting any.
    Examples include hot filaments and stars.

Wavelength Figure: Wavelength

Laws of Thermal Radiation

  1. Wien's Displacement Law: \[\lambda_{max}T=constant\ b(=0.0029mK)\]

    • \(λ_{max}\propto \frac{1}{T}\)
    • \(λ_{max}\): Wavelength at peak intensity
    • \(T\): Temperature at the surface of the star

  2. Stefan-Boltzmann Law: \[L= \sigma AT^{4}=4\pi \sigma R^{2}T^{4}\]

    • Stefan-Boltzmann constant: \[\sigma = 5.67\times 10^{-8}\ Wm^{-2}K^{-4}\]
    • A: Surface area of the star
    • R: Radius of the star

  3. The radiant flux intensity at the surface of a star is directly related to the temperature at the surface.
    \[F=\frac{L}{4\pi d^{2}}\]
    Generally, the thermal radiation power at the surface of an object is proportional to the fourth power of the local temperature.

Procedure for Measuring the Stellar Radius

  1. Identify a standard candle to determine the distance to the star's galaxy.
  2. Measure the radiant flux intensity of the star as observed from Earth.
  3. Determine the star's surface temperature using Wien's Law.
  4. Calculate the star's luminosity, given the known distance.
  5. Use Stefan-Boltzmann Law to determine the star's radius.

The luminosity of a star is influenced by its distance from us, surface temperature, and size.
The color of a star is determined by its surface temperature.

Color Temperature

Color temperature Figure: Color temperature

Doppler Effect of Light

  1. Redshift: The observed wavelength is greater than the expected value due to the star moving away from the observer.

\[\frac{\Delta f}{f} = \frac{\Delta\lambda}{\lambda} = \frac{v}{c}\]

  • c: Speed of light
  • v: Velocity of the moving star
  • Δλ: Change in wavelength
  • λ: Observed wavelength

Doppler effect Figure: Doppler effect

Expanding Universe

  1. Plotting the Receding Speed Against Distance

    • Measure the redshift to calculate the recession speed of a galaxy (i.e., the speed at which it moves away).

    • Determine the distance from Earth by observing individual stars of known luminosity, using them as standard candles.

    • All galaxies are observed to be receding from us.

    • This indicates that all parts of the universe are moving away from each other.

Expanding universe

Hubble's Law and the Big Bang Theory

Hubble's Law

The speed of recession of a galaxy is directly proportional to its distance from the observer, described by the Hubble constant H_0.
\[v=H_{0}d\]
\[H_{0}=2.2\times 10^{18} s^{-1}\]
(Note: The exact value of H_0 is still under refinement.)

Hubble
  • The observation that more distant objects are receding faster supports the idea that the universe is expanding, with all parts moving away from each other.
  • This suggests that matter was once much closer together.

Big Bang Theory

The universe originated from a highly dense and hot state, undergoing a massive expansion, and has continued expanding ever since.

Age of the Universe

\[ vt = d\\ v = H_{0}d\\ t = \frac{1}{H_{0}}\approx \frac{1}{2.2\times 10^{18}}=4.55\times 10^{-17} s = 14 \text{ billion years} \]

Evidence Supporting the Big Bang Theory

  • Hubble's Law \(v=H_{0}d\).

  • Cosmic Microwave Background Radiation \((CMBR)\).

  • Which is similar to blackbody radiation at approximately \(2.7 K\), with peak intensity at a wavelength of \(1 mm\) in the microwave range.