Astrophysics

Telescopes, stellar classification, cosmology, and the expanding Universe.

Spec Points Covered
  • Describe the optical principles of converging and diverging lenses and draw ray diagrams for astronomical telescopes.
  • Explain the operation of astronomical refracting telescopes in normal adjustment and calculate angular magnificationThe ratio of the angle subtended by the image to the angle subtended by the object. For a telescope in normal adjustment, $M = f_o / f_e$. $M = f_o / f_e$.
  • Describe Cassegrain and Newtonian reflecting telescopes and explain the advantages of concave mirrors over lenses.
  • Compare reflecting and refracting telescopes in terms of chromatic aberrationThe dispersion of white light by a lens, causing different colours to focus at different points. Mirrors do not suffer from this., weight, support, and cost.
  • Apply the Rayleigh criterion $\theta \approx \lambda / D$ to calculate the minimum angular resolutionThe smallest angle between two point sources that allows them to be distinguished as separate objects. of a telescope.
  • Explain how collecting powerThe ability of a telescope to gather light. It is proportional to the area of the objective, so $\propto D^2$. depends on objective diameter and compare telescopes of different sizes.
  • Describe the principles of radio, infrared, ultraviolet, and X-ray telescopes and explain why some must be placed in space.
  • Explain the operation of charge-coupled devicesSilicon chip detectors that convert incoming photons into electrical charge. They have high quantum efficiency (over 80%) compared with photographic film (~1%). (CCDs) and their advantages over photographic film.
  • Define apparent magnitudeA logarithmic scale measuring how bright a star appears from Earth. Lower numbers mean brighter stars. A difference of 5 magnitudes corresponds to a factor of 100 in brightness. and use the brightness ratio equation $I_1/I_2 = 100^{(m_2 - m_1)/5}$.
  • Apply the inverse square lawThe intensity of radiation from a point source falls off as the inverse square of the distance: $I = L / 4\pi d^2$. of radiation $I = L / 4\pi d^2$ to relate luminosity, intensity, and distance.
  • Define and convert between astronomical distances: AU, light-year, and parsecThe distance at which a star has a parallax angle of 1 arcsecond. 1 pc = 3.09 $\times 10^{16}$ m = 3.26 ly., and use stellar parallax.
  • Define absolute magnitudeThe apparent magnitude a star would have if placed at a standard distance of 10 parsecs from Earth. and apply $m - M = 5\log(d/10)$.
  • Apply Wien's displacement law $\lambda_{\max} T = 2.898 \times 10^{-3}$ m K to determine stellar surface temperature from peak wavelength.
  • Apply Stefan's law $L = 4\pi r^2 \sigma T^4$ to calculate stellar luminosity from radius and temperature.
  • Describe how emission and absorption spectra are produced and explain what they reveal about stellar composition and temperature.
  • List the stellar spectral classes (O B A F G K M) in order of decreasing temperature and describe key features of each.
  • Describe the Jeans mass criterion, protostar collapse, and the conditions for main-sequence ignition.
  • Describe the lifecycle of a low-mass star from main sequence through red giant to white dwarfThe dense, hot remnant of a low-mass star after it sheds its outer layers. Supported by electron degeneracy pressure..
  • Describe the lifecycle of a high-mass star including the supergiant, supernova, and neutron star or black hole stages.
  • Distinguish Type Ia and Type II supernovae and describe gamma-ray bursts.
  • Explain how Cepheid variableA pulsating star whose luminosity varies periodically. The period-luminosity relationship allows its absolute magnitude to be determined, making it a standard candle. stars and Type Ia supernovae are used as standard candles to measure distances.
  • State the Chandrasekhar limitThe maximum mass of a white dwarf, approximately 1.4 solar masses. Above this, electron degeneracy pressure cannot support the star. and Oppenheimer-Volkoff limit and describe neutron stars and black holes.
  • Explain how the Hertzsprung-Russell diagramA scatter plot of stellar luminosity against surface temperature. Stars cluster into distinct groups: main sequence, red giants, supergiants, and white dwarfs. classifies stars and describe evolutionary tracks across it.
  • Apply the Doppler equation $\Delta\lambda / \lambda \approx v/c$ to calculate the radial velocity of astronomical objects.
  • Describe the light curves of eclipsing binary star systems and extract orbital period and relative stellar sizes.
  • Explain cosmological redshiftThe stretching of light wavelengths caused by the expansion of space itself, distinct from the Doppler effect. and distinguish it from Doppler redshift.
  • Apply Hubble's law $v = Hd$ to calculate recession velocity and estimate the age of the Universe from $T \approx 1/H$.
  • Describe three pieces of evidence for the Big Bang: galactic redshift, the CMBR, and hydrogen-helium abundance.
  • Explain how Type Ia supernovae observations revealed the accelerating expansion of the Universe and dark energyA mysterious form of energy that makes up approximately 68% of the Universe and is responsible for its accelerating expansion..
  • Describe the properties of quasarsQuasi-stellar objects: extremely luminous active galactic nuclei powered by supermassive black holes accreting matter. and explain how they provide evidence for supermassive black holes.
  • Describe the transit and radial velocity methods for detecting exoplanetsPlanets orbiting stars other than the Sun. Over 5000 have been confirmed, mostly using the transit and radial velocity methods..
Notes
9.1: Telescopes
01 Lenses & ray diagrams for telescopes Converging lens, Diverging lens 3.9.1.1 → 02 Refracting telescopes $M = f_o / f_e$ 3.9.1.2 → 03 Reflecting telescopes Cassegrain, Newtonian 3.9.1.3 → 04 Reflecting vs refracting telescopes Chromatic aberration 3.9.1.4 → 05 Resolving power of telescopes $\theta \approx \lambda / D$ 3.9.1.5 → 06 Collecting power of telescopes Collecting power $\propto D^2$ 3.9.1.6 → 07 Non-optical telescopes Radio, IR, UV, X-ray 3.9.1.7 → 08 Charge-coupled devices (CCDs) Quantum efficiency 3.9.1.8 →
9.2: Classification of Stars
09 Brightness & apparent magnitude Apparent magnitude 3.9.2.1 → 10 Inverse square law of radiation $I = L / 4\pi d^2$ 3.9.2.2 → 11 Astronomical distances Parsec, Light-year 3.9.2.3 → 12 Absolute magnitude $m - M = 5\log(d/10)$ 3.9.2.4 → 13 Wien's displacement law $\lambda_{\max} T = \text{const}$ 3.9.2.5 → 14 Stefan's law $L = 4\pi r^2 \sigma T^4$ 3.9.2.6 → 15 Emission & absorption spectra Line spectra 3.9.2.7 → 16 Stellar spectral classes O B A F G K M 3.9.2.8 → 17 Star formation Protostar, Jeans mass 3.9.2.9 → 18 Low-mass star lifecycle Red giant, White dwarf 3.9.2.10 → 19 High-mass star lifecycle Supergiant, Supernova 3.9.2.11 → 20 Supernovae & gamma-ray bursts Type Ia, Type II 3.9.2.12 → 21 Standard candles Cepheid variables 3.9.2.13 → 22 Neutron stars & black holes Chandrasekhar limit 3.9.2.14 → 23 Hertzsprung-Russell diagram H-R diagram 3.9.2.15 →
9.3: Cosmology
24 Doppler effect of light $\Delta\lambda / \lambda \approx v/c$ 3.9.3.1 → 25 Binary star systems Eclipsing binary 3.9.3.2 → 26 Galactic redshift Cosmological redshift 3.9.3.3 → 27 Hubble's law $v = Hd$ 3.9.3.4 → 28 Big Bang evidence CMBR, Redshift, H/He 3.9.3.5 → 29 Dark energy Accelerating expansion 3.9.3.6 → 30 Quasars Active galactic nuclei 3.9.3.7 → 31 Exoplanets Transit, Radial velocity 3.9.3.8 →
Σ Key Equations
On Data Sheet
Not on Data Sheet
Wien's displacement law
$$\lambda_{\max} T = 2.898 \times 10^{-3} \text{ m K}$$
Peak wavelength is inversely proportional to surface temperature.
Inverse square law
$$I = \frac{L}{4\pi d^2}$$
$I$ = intensity (W m$^{-2}$), $L$ = luminosity (W), $d$ = distance (m).
Stefan's law
$$L = 4\pi r^2 \sigma T^4$$
$L$ = luminosity (W), $r$ = radius (m), $T$ = surface temperature (K), $\sigma$ = Stefan-Boltzmann constant.
Brightness ratio
$$\frac{I_1}{I_2} = 100^{(m_2 - m_1)/5}$$
Relates the intensity ratio of two stars to their apparent magnitude difference.
Angular magnification
$$M = \frac{f_o}{f_e}$$
For a telescope in normal adjustment. $f_o$ = objective focal length, $f_e$ = eyepiece focal length.
Rayleigh criterion
$$\theta \approx \frac{\lambda}{D}$$
Minimum angular resolution (rad). $D$ = diameter of objective.
Magnitude-distance relation
$$m - M = 5\log\!\left(\frac{d}{10}\right)$$
$m$ = apparent magnitude, $M$ = absolute magnitude, $d$ = distance in parsecs.
Doppler shift
$$\frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$$
For $v \ll c$. Positive $\Delta\lambda$ = redshift (recession), negative = blueshift (approach).
Hubble's law
$$v = Hd$$
$v$ = recession velocity (km s$^{-1}$), $H$ = Hubble constant, $d$ = distance. Age $\approx 1/H$.
Q Retrieval Practice
Q1. State the Rayleigh criterion and explain what it tells us about a telescope's resolving power.
  • $\theta \approx \lambda / D$, where $\theta$ is the minimum angular separation (in radians) that can be resolved.
  • A larger objective diameter $D$ gives a smaller $\theta$, so the telescope can distinguish finer detail.
  • Shorter wavelengths also improve resolution, which is why radio telescopes need very large dishes.
Q2. A star has an apparent magnitude of +3 and is 50 parsecs away. Calculate its absolute magnitude.
  • $m - M = 5\log(d/10)$
  • $3 - M = 5\log(50/10) = 5\log 5 = 5 \times 0.699 = 3.495$
  • $M = 3 - 3.5 = -0.5$
  • The star is intrinsically brighter than it appears, since $M < m$.
Q3. Describe the lifecycle of a star with mass similar to the Sun.
  • Nebula collapses under gravity to form a protostar.
  • When the core is hot enough for hydrogen fusion, it joins the main sequence and stays there for billions of years.
  • When hydrogen runs out in the core, the outer layers expand and cool to form a red giant.
  • Helium fusion occurs in the core. Eventually the outer layers are shed as a planetary nebula.
  • The remaining core becomes a white dwarf, which slowly cools over time.
Q4. State Hubble's law and explain how it can be used to estimate the age of the Universe.
  • $v = Hd$: the recession velocity of a galaxy is proportional to its distance.
  • Rearranging: $d/v = 1/H$, which gives an estimate of the time since the galaxies were all at the same point.
  • So $T \approx 1/H$ gives an approximate age of the Universe (about 13.8 billion years with current values of $H$).
Q5. Describe two methods used to detect exoplanets.
  • Transit method: When a planet passes in front of its star, the observed brightness dips slightly. The size and period of the dip reveal the planet's radius and orbital period.
  • Radial velocity method: The planet's gravity causes the star to wobble. This produces a periodic Doppler shift in the star's spectral lines. The shift reveals the planet's mass and orbital period.