Spectroscopy and Astronomical Applications

Kirchhoff's laws and the quantum explanation

In the 1850s, Gustav Kirchhoff and Robert Bunsen systematically studied the spectra of different elements heated in flames. They established three laws that form the foundation of spectroscopy:

1. A hot solid, liquid or dense gas produces a continuous spectrum a complete rainbow without gaps.
2. A hot thin gas produces an emission spectrum bright lines at specific wavelengths, unique to each element.
3. A cool thin gas in front of a continuous source produces an absorption spectrum a rainbow with dark lines at the same wavelengths as the emission lines of that gas.

The quantum mechanical explanation is elegant: atoms exist in discrete energy levels. When an electron drops from a higher level to a lower one, it emits a photon with energy equal to the difference: $E = hf = E_{upper} - E_{lower}$. Since each element has a unique set of energy levels, each produces a unique pattern of spectral lines, an atomic fingerprint.

In absorption, a photon of exactly the right energy excites an electron from a lower level to a higher one, removing that wavelength from the transmitted beam. The dark absorption lines in the Sun's spectrum (called Fraunhofer lines) reveal the presence of elements like hydrogen, sodium, iron and calcium in the cooler outer layers of the Sun.

Kirchhoff's three spectral laws: (1) hot dense source → continuous spectrum; (2) hot thin gas → emission spectrum (bright lines); (3) cool thin gas in front of hot source → absorption spectrum (dark lines at the same wavelengths as that element's emission lines). Each element's unique energy levels produce a unique spectral fingerprint.

Pause, copy the highlighted three laws into your book before moving on.

NESA syllabus (IQ1, dot point 4): spectra produced by discharge tubes, incandescent filaments and reflected sunlight

We just saw Kirchhoff's three abstract categories, continuous, emission and absorption spectra. That raises a question: what real, physical equipment in a lab or in nature actually produces each type? This card answers it → the tungsten filament, the discharge tube and reflected sunlight, three concrete real-world sources you could point to and photograph.

Kirchhoff's laws describe categories of spectra, but the NESA syllabus asks you to identify the actual apparatus that produces each one. Three real sources map directly onto the three spectrum types.

Incandescent filaments (continuous spectrum): A tungsten light bulb filament is heated by an electric current to roughly 2,700-3,000 K, hot enough to glow white. Because tungsten is a hot, dense solid, its atoms are packed closely together and constantly colliding, so instead of isolated sharp transitions, an enormous range of energy transitions overlap and smear together, producing a smooth, continuous (approximately black-body) spectrum across all visible wavelengths. This is the classic real-world source of Kirchhoff's first law.

Discharge tubes (emission spectrum): A discharge tube is a sealed glass tube containing a low-pressure gas, hydrogen or neon are common examples, with electrodes at each end. A high voltage strips electrons from the gas atoms and accelerates them, and as excited electrons fall back to lower energy levels they emit photons only at the specific wavelengths corresponding to that gas's discrete energy-level gaps. A hydrogen discharge tube shows four visible lines (656 nm red, 486 nm blue-green, 434 nm blue-violet, 410 nm violet), the same Balmer lines used to identify hydrogen in stars. A neon discharge tube glows orange-red because neon's strongest visible transitions cluster in that part of the spectrum, the physical basis of "neon" signage. This is the classic real-world source of Kirchhoff's second law.

Reflected sunlight (absorption spectrum): The Sun's photosphere is a hot, dense layer that behaves like an incandescent filament, producing a continuous spectrum. But that light must pass through the Sun's own cooler outer atmosphere (and reflect off the Moon, a planet, or a mirror, and pass through Earth's atmosphere) before it reaches your spectroscope. Cooler gas atoms along the path absorb photons at their characteristic wavelengths, removing those exact colours from the transmitted beam. Passing reflected sunlight through a prism or spectroscope reveals a continuous rainbow interrupted by thousands of dark Fraunhofer absorption lines, the classic real-world source of Kirchhoff's third law, and the same physical principle whether the sunlight reaches you directly or reflected from the Moon.

Three real-world sources, one per Kirchhoff law: incandescent filaments (tungsten light bulb, ~2900 K) produce continuous spectra; discharge tubes (hydrogen or neon gas excited by high voltage) produce emission (bright-line) spectra; reflected sunlight through a prism or spectroscope shows a continuous spectrum with dark Fraunhofer absorption lines from the Sun's cooler outer atmosphere.

Copy the three apparatus-to-spectrum matches into your book before the check below.

Motion written in wavelengths, redshift and blueshift

We just saw that spectral lines act as element fingerprints. That raises a question: what happens to those lines when a source is moving relative to us? This card answers it → the Doppler effect shifts wavelengths, giving redshift (receding) or blueshift (approaching), quantified by $\Delta\lambda/\lambda_0 = v/c$.

When a light source moves relative to an observer, the observed wavelength shifts. This is the Doppler effect for light. Unlike the Doppler effect for sound, the shift in light depends only on the relative velocity between source and observer.

• Moving away (receding): Wavelength increases → redshift ($\lambda_{obs} > \lambda_0$)
• Moving toward (approaching): Wavelength decreases → blueshift ($\lambda_{obs} < \lambda_0$)

For speeds much less than $c$ ($v \ll c$), the fractional shift is:

Astronomical applications of the Doppler effect:

• Stellar radial velocities: Measuring how fast stars move toward or away from us along the line of sight.
• Binary stars: Periodic Doppler shifts reveal stars orbiting each other, the line shifts back and forth with the orbital period.
• Exoplanet detection: A star's small wobble induced by an orbiting planet produces tiny but measurable Doppler shifts (the radial velocity method).
• Galaxy recession (Hubble's Law): Measurements showed that more distant galaxies generally have larger redshifts, approximately described at low redshift by $v = H_0 d$. This distance-redshift relation became major evidence for cosmic expansion; it was not the only evidence.
• Galactic rotation: One side of a rotating galaxy is blueshifted, the other redshifted, revealing rotation speeds and evidence for dark matter.

Doppler shift (non-relativistic): $\Delta\lambda/\lambda_0 = v/c$, so $v = c\Delta\lambda/\lambda_0$. Positive $\Delta\lambda$ → redshift (receding); negative → blueshift (approaching). Redshift parameter $z = \Delta\lambda/\lambda_0$; observed wavelength $\lambda_{obs} = \lambda_0(1+z)$.

Add the highlighted Doppler formula and sign convention to your notes before the check below.

Reading velocity from spectral lines

We just saw the Doppler formula $v = c\Delta\lambda/\lambda_0$. That raises a question: how do you apply this to find stellar velocities, distinguish approaching from receding stars, and interpret binary systems? This card answers it → multi-part worked example with sign convention and binary interpretation.

Applying the Doppler formula: $v = c \cdot \Delta\lambda/\lambda_0$ where $\Delta\lambda = \lambda_{obs} - \lambda_0$. Positive $\Delta\lambda$ → receding; negative → approaching. Periodic Doppler shifts in binary systems reveal orbital period and unseen companion mass; tiny periodic shifts indicate exoplanet "wobble."

Pause, write the highlighted calculation method into your book before the check below.

Stellar composition, temperature, and the expanding universe

We just saw how Doppler shifts reveal stellar radial velocities. That raises a question: what else can spectroscopy tell us about stars and the universe? This card answers it → composition from Fraunhofer lines, temperature from spectral class, and Hubble's Law as evidence for expansion.

Spectroscopy is the dominant observational tool in astrophysics. Without ever visiting a star, astronomers can determine its composition, temperature, luminosity, magnetic field strength, rotation, and radial velocity.

Stellar composition from Fraunhofer lines: The Sun's spectrum contains thousands of dark absorption lines. By matching their wavelengths to laboratory spectra of known elements, astronomers identified hydrogen (most abundant), helium, calcium, sodium, magnesium, iron and many other elements. The same technique works for all stars.

Stellar temperature from spectral class: Hotter stars excite higher energy transitions, producing spectral lines from more highly ionised elements. Cooler stars show molecular bands. This is the basis of the stellar classification system (O, B, A, F, G, K, M), from hottest (O, ~50,000 K) to coolest (M, ~3,000 K).

Hubble's Law and the expanding universe: Edwin Hubble (1929) discovered that almost all galaxies show redshift, and that more distant galaxies recede faster:

This was the first direct evidence that the universe is expanding. Running the expansion backward suggests all matter originated in a single event, the Big Bang.

Fraunhofer (absorption) lines reveal stellar composition; spectral class (O→M) reflects surface temperature. Hubble's Law: $v = H_0 d$ ($H_0 \approx 70$ km/s/Mpc), more distant galaxies recede faster, providing evidence for an expanding universe and the Big Bang model.

Add the highlighted laws and their significance to your notes before the check below.

NESA syllabus (IQ1, dot point 6): the spectra of stars reveal density, not just composition, temperature and velocity

We just saw that Doppler shifts reveal a star's velocity, Fraunhofer lines reveal composition, and spectral class reveals temperature. That raises a question: the syllabus also says stellar spectra reveal density, so what spectral feature could possibly encode that? This card answers it → pressure broadening, the width of a spectral line, not just its position, carries information about the density of the gas that produced it.

So far every property (composition, temperature, velocity) has come from the position of spectral lines, which wavelength they sit at. Density is different: it is read from the width of the lines, how sharp or smeared out they are.

In a low-density gas, atoms are far apart and rarely collide during the brief moment they emit or absorb a photon. Each atom's energy levels are essentially undisturbed, so every atom of a given element produces a photon at almost exactly the same wavelength, giving a narrow, sharp spectral line. In a high-density (high-pressure) gas, atoms are packed closely together and collide with each other far more frequently. Each collision briefly perturbs a nearby atom's electric field and energy levels, so the exact transition energy varies slightly from atom to atom and from instant to instant. Averaged over billions of atoms, this collisional perturbation, called pressure broadening (or collisional broadening), smears a would-be sharp line into a noticeably wider band of wavelengths.

This gives astronomers a direct observational handle on stellar atmospheric density: a compact, high-gravity star (a dwarf star, including white dwarfs) has a dense, high-pressure atmosphere with frequent particle collisions, producing broad, smeared spectral lines. A large, extended, low-gravity star (a giant star) has a diffuse, low-pressure atmosphere with far less frequent collisions, producing narrow, sharp spectral lines, even if the giant and the dwarf have the same surface temperature (the same spectral class, so the same set of lines at the same wavelengths). Line width is therefore how astronomers distinguish luminosity classes, dwarfs from giants of the same spectral type, an axis of information completely independent of the line's wavelength.

Pressure (collisional) broadening: in a dense, high-pressure stellar atmosphere, frequent particle collisions perturb atomic energy levels, smearing spectral lines into broad profiles. In a diffuse, low-pressure atmosphere, rare collisions leave lines narrow and sharp. Line width (not position) is how spectroscopy reveals stellar density, letting astronomers distinguish a dense dwarf star from a diffuse giant of the same temperature.

Add the highlighted pressure-broadening principle to your notes before the check below.