Astronomy C10 Lecture Notes: Doppler Effect, Thermal Radiation, and Spectral Patterns

Bonus material: Auroras (Northern Lights) – a quick physical intuition

Solar wind from the Sun can be intensified by solar storms (flares, coronal mass ejections).
Energetic charged particles travel toward Earth and are guided by Earth's magnetic field toward the poles.
When these particles reach the upper atmosphere, they collide with oxygen and nitrogen.
Collisions excite the atmospheric atoms; as electrons return to lower energy states, photons are emitted, producing colors observed in auroras.
Colors depend on which electronic transitions occur in O and N and the altitude where these collisions happen.
The same basic mechanism underlies emission spectra: charged particles excite atoms, which then emit photons as they de-excite.
The aurora illustrates the general idea that particle or photon energy input leads to emission lines/regions in spectra, modulated by the local atmospheric composition and conditions.

Foundational topics for today

1) Doppler effect (applied to light) – radial velocities

The Doppler effect is a change in observed wavelength (or frequency) due to relative motion between source (emitter) and observer.
For light, only relative motion matters; light does not require a medium to propagate.
Rest wavelength λ0 (or rest frequency f0) is the wavelength (or frequency) measured in the emitter’s rest frame.
If the source moves away from the observer, observed wavelength λ is longer (red shift); if it moves toward the observer, λ is shorter (blue shift).
In the non-relativistic regime (v ≪ c), the approximate relation is:
- $\frac{\Delta \lambda}{\lambda0} \approx \frac{v}{c}$ where $\Delta \lambda = \lambda{\text{obs}} - \lambda_0$ .
This is an excellent approximation for speeds up to ~20% of the speed of light; relativistic corrections are negligible in most astronomy problems here.
Therefore, for small v, the radial velocity is:
- $v \approx c \ \frac{\Delta \lambda}{\lambda0} \equiv c \ \frac{\lambda{\text{obs}} - \lambda0}{\lambda0}$
Sign convention:
- Red shift (emitter moving away) gives a positive velocity (toward longer λ).
- Blue shift (emitter moving toward) gives a negative velocity (toward shorter λ).
- If the observer and emitter move perpendicularly to the line of sight, there is effectively no shift.
Important conceptual points:
- The product of wavelength and frequency is constant for a given photon: $c = \lambda f$ , so a shift in wavelength corresponds to a compensating shift in frequency.
- The Doppler shift for light depends only on relative motion, not on a medium.
Practical note on interpretation:
- In astronomy, the Doppler shift is used to measure radial velocities of stars, galaxies, gas clouds, etc.
- It helps infer dynamics, binarity, outflows, and cosmological recession (Hubble flow) in certain regimes.
Example calculation (Hydrogen-α line):
- Rest wavelength: $\lambda_0 = 656.3\ \text{nm}$ .
- Observed wavelength: $\lambda_{\text{obs}} = 656.5\ \text{nm}$ .
- $\Delta \lambda = 0.2\ \text{nm}$ .
- $v = c \frac{\Delta \lambda}{\lambda_0} = (3.0\times 10^{5}\ \text{km/s})\times\frac{0.2}{656.3} \approx 9.0\times 10^{1}\ \text{km/s}$
- Since λobs > λ0, this indicates a red shift (receding). If λobs < λ0, it would be blue shifted (approaching).
Practical use in the course: you will often compute velocities from observed wavelengths and rest wavelengths for various spectral lines.

2) Thermal radiation and blackbody concepts

A star (and many astronomical objects) can be approximated as a thermal emitter (blackbody-like) – an opaque body that absorbs all incident light.
Absorption vs emission in spectra:
- Absorption: If a hot, bright continuum source is viewed through a cooler, intervening gas, the gas absorbs photons at wavelengths corresponding to its allowed transitions, producing dark absorption lines.
- Emission: If you view a cloud of gas that is excited (e.g., by nearby hot stars) but not directly emitting a continuum source along the line of sight, you see bright emission lines at characteristic wavelengths.
The spectrum of a blackbody is determined solely by its temperature (to a good approximation): the shape and peak depend on temperature, not chemical composition or distance.
Wien’s displacement law (peak wavelength shifts with temperature):
- $\lambda_{\text{max}} T = b,\quad b \approx 2.90\times 10^{6}\ \text{nm K}$
- Hotter objects peak at shorter wavelengths; cooler objects peak at longer wavelengths.
Planck spectrum and color: hotter objects emit more of their radiation at shorter wavelengths; cooler objects emit more at longer wavelengths; in the optical, hotter objects appear blue and cooler objects appear red.
Real-world caveat: the color of an object in everyday life often reflects reflected light, not its own thermal emission (e.g., ice looks blue because of reflection, not because it emits blue thermal radiation).
Sun vs. other stars:
- The Sun (approx. 5800–6000 K) has a peak near the green-yellow region and appears white to us due to a mix of wavelengths.
- Very hot stars peak in the blue; cooler stars peak in the red/orange.
Blackbody vs real spectra:
- Real stars have absorption lines superposed on the continuum due to elements in their atmospheres.
- The continuum approximation is often used to estimate effective temperatures from peak wavelengths.
Thermally emitted energy and total output:
- The total energy emitted per unit area per unit time by a blackbody is proportional to the fourth power of its temperature:
- $\frac{dE}{dA \; dt} = \sigma T^{4}$
- Here, (\sigma) is the Stefan–Boltzmann constant, (\sigma \approx 5.67\times 10^{-8}\ \text{W m}^{-2} \text{K}^{-4}).
Practical implication:
- The hotter the star, the more radiant power per unit area it emits across all wavelengths, and the spectrum’s peak shifts to shorter wavelengths as temperature increases.
Applications to human-visible observations:
- Humans at ~300 K emit mostly in the infrared; their optical visibility arises from reflected light, not their thermal emission.
- Infrared cameras can visualize thermal emission, often recolored to visualize with our eyes.

Spectral patterns and the role of energy levels

Electronic transitions and spectral fingerprints

Each element (neutral or ionized) has a unique set of energy levels; allowed transitions between levels produce photons of specific energies.
Emission vs absorption depending on context:
- Emission line: an atom transitions from a higher energy level to a lower one, emitting a photon with energy $\Delta E = E{\text{upper}} - E{\text{lower}}$ .
- Absorption line: an atom absorbs a photon with energy $\Delta E = E{\text{upper}} - E{\text{lower}}$ as it moves to a higher energy level.
The observed spectrum is a fingerprint of the transitions present in the gas (or stellar atmosphere) along the line of sight.
A useful relation between energy and photon properties:
- Energy of a photon is $\Delta E = h f = \frac{h c}{\lambda}$ where (h) is Planck's constant, (f) is frequency, (\lambda) is wavelength, and (c) is the speed of light.
- Rearranging gives the wavelength (or frequency) associated with a given transition:
- $\lambda = \frac{h c}{\Delta E},\quad f = \frac{\Delta E}{h}$
Example focus: Hydrogen energy levels and common series (Lyman, Balmer, Paschen) illustrate how a set of lines arises for a single element; you do not need to memorize exact wavelengths for the course, but you should understand the principle.
Practical diagnostic strategy:
- A single line often isn’t enough to identify an element unambiguously.
- The full pattern of multiple lines (the spectral fingerprint) allows you to determine which elements are present.
Concrete hydrogen example mentioned in class:
- The Balmer series involves transitions to the n=2 level; the Lyman series to n=1; the Paschen series to n=3, etc. (you don’t need to memorize wavelengths).
Concrete calculation example that may appear on quizzes:
- For a hydrogen line, convert the energy difference to a wavelength and check for a corresponding dip in the observed spectrum using the relation above.
Hydrodynamic and geometric reasoning about spectra:
- When off-LOS gas (not directly between source and observer) emits, you can see emission lines without a continuum.
- If gas lies in front of a hot continuum source, you see absorption lines at the wavelengths of allowed transitions.

Concrete think-pair-share example and interpretation

Scenario: optical spectrum of a thin cool cloud of gas near a hot bright star (but not directly along the LOS to the star).
Observed spectrum result: A bright continuum and absorption lines is not the right scenario because there is no strong continuum along LOS; the cloud near the star produces emission lines due to excitation by the star’s radiation.
Correct answer: D) only emission lines when the cloud is seen without a direct line of sight to the star as a source of continuum.
Rationale:
- If the cloud is the only source, lines are from transitions in the gas; there is no continuum because the star’s light is not passing through the cloud toward the observer.
- If the cloud is along the LOS to the star, you would see absorption lines imprinted on the star’s continuum spectrum.

Orion Nebula as an illustrative case

Orion Nebula is a star-forming region (stellar nursery) with gas and dust collapsing under gravity.
Four hot, ultraviolet-emitting stars ionize surrounding gas.
Ionized gas recombines and emits photons (recombination lines):
- Hydrogen emission lines (e.g., red Hα) and doubly ionized oxygen emission (green lines from [O III]).
Emission spectrum is seen when looking at the cloud off the direct LOS to a bright background star.
Direct LOS to a star yields absorption features superposed on the star’s continuum: the star’s own photospheric absorption lines plus any absorption by intervening gas.
The Doppler concept helps distinguish whether lines arise from the cloud or from the star (broadened lines from the star vs narrower lines from the cloud).

Practical aspects of spectroscopy and interpretation

Problem of separating stellar vs interstellar absorption lines:
- Stars have broad lines due to high pressures and rapid motions in their atmospheres (Doppler broadening).
- Interstellar/intervening gas typically produces narrower lines.
- The width helps attribute lines to either the star or the cloud.
Summary of spectral fingerprints and information you can get:
- Chemical composition (which elements are present).
- Kinematics (via Doppler shift and line widths).
- Physical state (ionization, temperature, density via line ratios and continuum shape).

Star colors, temperature, and the blackbody picture

Temperature and color intuition

Observational fact: star color correlates with surface temperature.
- Hotter stars appear blue; cooler stars appear red; Sun-like stars appear white, roughly in the middle.
The color of a star is tied to its thermal spectrum (a good approximation for many stars): the distribution of emitted light shapes the observed color.
Important nuance: color in everyday life often comes from reflected light as well as the star’s own emission; the reflection/absorption properties of materials can affect the apparent color.

Blackbody radiation and the “blackbody” concept

Blackbody: an idealized opaque thermal emitter that absorbs all incident light and emits a spectrum determined only by its temperature.
The spectrum of a blackbody is the Planck distribution, with peak and shape set by temperature.
The peak wavelength is given by Wien’s law: $\lambda_{\text{max}} T = b,\quad b \approx 2.90\times 10^{6}\ \text{nm K}$ .
Temperature units and scales:
- Kelvin (K) is the standard in astronomy; 0 K is absolute zero (no thermal motion).
- Celsius (°C) and Fahrenheit (°F) are common everyday scales; conversions:
- $T{\text{K}} = T{\text{C}} + 273.15$
- $T{\text{C}} = T{\text{K}} - 273.15$
Practical example: the Sun’s effective temperature is ~5800–6000 K; its spectrum peaks around the green-yellow region, giving a white appearance overall due to a mix of wavelengths.
Hotter stars peak at shorter wavelengths (blue), cooler stars peak at longer wavelengths (red/orange).

Realized spectra and interpretation

The visible color of stars is a rough indicator of temperature; actual observed color is influenced by distance, extinction, and spectral lines.
The total power emitted per unit area increases rapidly with temperature (Stefan–Boltzmann law):
- $\frac{dE}{dA \; dt} = \sigma T^{4},\quad \sigma \approx 5.67\times 10^{-8}\ \text{W m}^{-2} \text{K}^{-4}$
The constant-through-time lesson:
- hotter objects glow brighter overall and peak at shorter wavelengths.
- cooler objects glow dimmer overall and peak at longer wavelengths.
The human eye and night vision connect to the visible portion of the spectrum; most human thermal emission is in the infrared (e.g., 300 K peaks around 10 μm), hence infrared visibility requires detectors beyond the eye.

Making sense of color versus temperature with everyday examples

Ice appears blue not because it emits blue light, but because blue light is scattered/reflected more efficiently and the emission is in the infrared (not visible).
Coals glow red-hot due to their thermal emission in the visible; the color is an imprint of the temperature on the Planck spectrum, not a pigment.
Pigments on objects (e.g., balls, shirts) reflect specific wavelengths; the observed color can be a property of the pigment rather than of thermal emission.

Connecting the physics to practical astronomy problems

Spectra as fingerprints: the entire pattern of lines (not a single line) is used to identify elements in stars, nebulae, and planetary atmospheres.
Absorption lines in a star’s spectrum reveal the star’s chemical composition and atmospheric conditions; emission lines reveal excited gas in nebulae and the presence of ionized species.
Doppler shifts in lines yield radial velocities; line widths provide clues about turbulence, pressure broadening, and source environments.
The concept of a blackbody and Wien’s law helps interpret the color and temperature of stars and the total energy output.
The Orion Nebula example illustrates both ionization by hot stars and subsequent recombination emission lines (e.g., H, O III), mapping to common spectral features.
In all cases, the combination of a continuum spectrum with superposed lines is the key observable that allows astrophysicists to infer physical properties of distant objects.

Quick reference formulas (summary)

Photon energy and wavelength:
- $\Delta E = h f = \frac{h c}{\lambda}$
Frequency and wavelength relations:
- $f = \frac{\Delta E}{h}, \quad \lambda = \frac{h c}{\Delta E}$
Doppler shift (non-relativistic):
- $\frac{\Delta \lambda}{\lambda0} \approx \frac{v}{c}, \quad v \approx c \frac{\Delta \lambda}{\lambda0}$
- Sign convention: red shift (v > 0), blue shift (v < 0).
Wien’s displacement law: peak wavelength and temperature
- $\lambda_{\text{max}} T = b, \quad b \approx 2.90\times 10^{6} \ \text{nm K}$
Stefan–Boltzmann law (total power per unit area):
- $\frac{dE}{dA \; dt} = \sigma T^{4}, \quad \sigma \approx 5.67\times 10^{-8} \ \text{W m}^{-2} \text{K}^{-4}$
Temperature scale conversions:
- $T{\text{K}} = T{\text{C}} + 273.15 \quad (\text{and})\quad T{\text{C}} = T{\text{K}} - 273.15$

Quick worked example snapshots (to study later)

Hydrogen alpha line velocity example:
- Rest λ0 = 656.3 nm; observed λ = 656.5 nm; Δλ = 0.2 nm.
- v = c (Δλ/λ0) ≈ (3.00×10^5 km/s)(0.2/656.3) ≈ 90 km/s (receding).
Wien’s law intuition:
- If a star’s spectrum peaks near 500 nm, estimate T ≈ b/λmax ≈ (2.90×10^6 nm K) / (500 nm) ≈ 5800 K, consistent with a Sun-like star.