Spectroscopy and Astronomical Applications
In 1925 Cecilia Payne at Harvard College Observatory analysed the spectra of more than 300 stars and concluded that stars are 73% hydrogen and 25% helium by mass — far more hydrogen-rich than anyone had believed. Her supervisor Henry Russell dismissed the finding; four years later he confirmed it independently and acknowledged her priority. Payne's result, enabled entirely by spectroscopy, is the most important finding in the history of stellar physics.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
When white light from the Sun passes through a prism, it produces a continuous rainbow spectrum. But when passed through cool gas before reaching the prism, dark lines appear at specific colours.
- Why are the dark lines dark? Where did that specific colour of light go?
- Why would a hot gas produce bright lines at those same colours?
- How might an astronomer use these lines to determine what a distant star is made of?
Write your predictions before reading on — you will revisit them at the end.
Warm-up — which type of spectrum is produced by a hot thin gas?
Know — Spectra Types
- Continuous, emission and absorption spectra
- Kirchhoff's laws of spectroscopy
- Spectral lines as atomic fingerprints
Understand — Doppler Effect for Light
- Redshift: source moving away ($\lambda$ increases)
- Blueshift: source moving toward ($\lambda$ decreases)
- $\Delta\lambda/\lambda_0 = v/c$ for $v \ll c$
Can Do — Analyse Spectral Data
- Identify elements from spectral lines
- Calculate velocities from Doppler shifts
- Interpret astronomical spectra
Core Content
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:
- A hot solid, liquid or dense gas produces a continuous spectrum — a complete rainbow without gaps.
- A hot thin gas produces an emission spectrum — bright lines at specific wavelengths, unique to each element.
- 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.
Figure 1 — Kirchhoff's three spectra: continuous (hot dense source), emission (bright lines from hot thin gas), and absorption (dark lines on continuous background from cool gas in front of hot source)
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.
Astronomers observe a star and find dark lines in its spectrum at the exact wavelengths where hydrogen emits light. Explain what this tells us about the star's atmosphere. Why don't we see hydrogen emission lines from the star itself?
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.
A cool cloud of sodium gas sits between an observer and a hot star. The observer sees a continuous spectrum with dark lines. Those dark lines appear at exactly the same wavelengths as the bright lines in sodium's emission spectrum because:
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:
$$\dfrac{\Delta\lambda}{\lambda_0} = \dfrac{v}{c}$$
where $\Delta\lambda = \lambda_{obs} - \lambda_0$; positive $v$ = receding (redshift); negative $v$ = approaching (blueshift)
$$z = \dfrac{\Delta\lambda}{\lambda_0} = \dfrac{v}{c} \qquad \Rightarrow \qquad \lambda_{obs} = \lambda_0(1 + z)$$
Figure 2 — The H-alpha line (656 nm at rest) shifts to longer wavelengths (red) when the source recedes, and to shorter wavelengths (blue) when it approaches
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): Edwin Hubble discovered that distant galaxies are redshifted, with recession velocity proportional to distance: $v = H_0 d$. This was the first evidence for an expanding universe.
- Galactic rotation: One side of a rotating galaxy is blueshifted, the other redshifted, revealing rotation speeds and evidence for dark matter.
The H$\alpha$ line of hydrogen has a rest wavelength of 656.3 nm. In the spectrum of a distant galaxy, this line is observed at 675.0 nm. Calculate the galaxy's recession velocity as a fraction of $c$. Is this galaxy moving toward or away from us?
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.
A spectral line with rest wavelength 500 nm is observed at 505 nm. The source's recession velocity is:
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.
A distant star has a spectral line of ionised calcium (Ca II) with a rest wavelength of 393.3 nm. In the star's observed spectrum, this line appears at 394.2 nm.
- Calculate the Doppler shift $\Delta\lambda$.
- Calculate the star's radial velocity. Is it approaching or receding?
- Another star shows the same Ca II line at 392.5 nm. Calculate its radial velocity.
- A binary star system shows periodic variation in the Ca II line from 393.0 nm to 393.6 nm with a period of 14 days. Explain what this reveals about the system.
$\Delta\lambda = \lambda_{obs} - \lambda_0 = 394.2 - 393.3 = $ 0.9 nm
$v = c \times \dfrac{\Delta\lambda}{\lambda_0} = 3.00\times10^8 \times \dfrac{0.9}{393.3} = 6.87\times10^5\ \text{m/s} \approx \textbf{687 km/s}$
Positive $\Delta\lambda$ → receding (redshift).
$\Delta\lambda = 392.5 - 393.3 = -0.8\ \text{nm}$
$v = 3.00\times10^8 \times \dfrac{-0.8}{393.3} = -6.10\times10^5\ \text{m/s} \approx \textbf{-610 km/s}$
Negative velocity → approaching (blueshift).
The periodic variation reveals the star is in orbit around an unseen companion. When moving toward us the line is blueshifted (393.0 nm); when moving away it is redshifted (393.6 nm). The 14-day period is the orbital period. From the velocity amplitude and period, astronomers can determine the companion's minimum mass — a key method for detecting exoplanets.
The most common error: using $\lambda_{obs}/\lambda_0 = v/c$ instead of $\Delta\lambda/\lambda_0 = v/c$. The correct form uses the shift, not the observed wavelength. Sign rule: positive $\Delta\lambda$ = receding (redshift); negative $\Delta\lambda$ = approaching (blueshift). Also note: this non-relativistic formula is only valid for $v \ll c$. For distant galaxies with large $z$, the relativistic formula is required.
The sodium D-lines have rest wavelengths of 589.0 nm and 589.6 nm. In a galaxy's spectrum, they appear at 601.0 nm and 601.6 nm. Calculate the galaxy's redshift $z$ and recession velocity.
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.
A star's spectrum shows periodic Doppler shifts with a period of 3 years, with maximum blueshift and redshift of 20 km/s each. This is best explained by:
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:
$$v = H_0 \, d$$
$v$ = recession velocity (km/s), $H_0 \approx 70$ km/s/Mpc = Hubble constant, $d$ = distance (Mpc)
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.
Figure 3 — Hubble diagram: recession velocity of galaxies is proportional to their distance. The slope of the best-fit line is the Hubble constant $H_0$
A galaxy is observed to have a redshift of $z = 0.05$. (a) Calculate its recession velocity. (b) Using $H_0 = 70$ km/s/Mpc, estimate its distance. (c) Explain why spectroscopy is described as the most important observational tool in astronomy.
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.
A galaxy has a redshift $z = 0.05$. Its recession velocity is approximately:
Activities
Observe Kirchhoff's laws and Doppler shifts interactively
- Select Emission spectrum. Identify the four hydrogen lines shown. Increase the redshift and observe how the lines move to longer wavelengths.
- Select Absorption spectrum. At z = 0, explain why the dark lines appear at the same wavelengths as the emission lines of hydrogen.
- Set z = 0.03. Calculate the recession velocity in km/s. At what observed wavelength does H$\alpha$ ($\lambda_0 = 656$ nm) now appear?
- A star shows a blueshift of 0.5% in its spectral lines (z = −0.005). Calculate its approach velocity. Would this be detectable with a modern spectrograph?
Apply Kirchhoff's laws and Doppler shift to real scenarios
- A spectrum of a distant star shows dark lines at 397 nm, 410 nm, 434 nm, 486 nm and 656 nm. Identify the element present and state which of Kirchhoff's laws applies. Explain what the star's atmospheric structure must be.
- The H$\alpha$ line ($\lambda_0 = 656.3$ nm) in a galaxy's spectrum is observed at 664.0 nm. (a) Calculate $\Delta\lambda$, (b) calculate $z$, (c) calculate the recession velocity, and (d) estimate the distance using $H_0 = 70$ km/s/Mpc.
- A planet is detected around a star using the radial velocity method. The star's Ca II line (393.3 nm) shifts between 393.29 nm and 393.31 nm with a period of 365 days. (a) Calculate the maximum Doppler velocity of the star. (b) Explain what this tells us about the planet. (c) Why does the method give only a minimum mass for the planet?
A fresh five-question set drawn from this lesson's bank — feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence — that tells the system what to drill next.
ApplyBand 4(3 marks) 1. (a) State Kirchhoff's three laws of spectroscopy and give the condition required for each type of spectrum to be produced. (b) Using the quantum model of the atom, explain why absorption lines and emission lines for the same element appear at identical wavelengths. (c) The dark lines in the Sun's spectrum (Fraunhofer lines) include lines due to hydrogen, calcium and sodium. Explain what this reveals about the structure of the Sun.
1 mark: correct statement of all three laws with conditions · 1 mark: quantum explanation linking discrete energy levels to photon energy · 1 mark: correct interpretation of solar structure (cool absorbing atmosphere over hot interior)
EvaluateBand 6(5 marks) 2. (a) The H$\alpha$ line ($\lambda_0 = 656.3$ nm) from a galaxy is observed at 664.0 nm. Calculate the galaxy's recession velocity. (b) Explain how the periodic Doppler shifts in a star's spectral lines can be used to detect an exoplanet. Why does this method give only the planet's minimum mass, not its true mass? (c) Edwin Hubble found that more distant galaxies have larger redshifts. Explain how this provides evidence that the universe is expanding, and briefly describe how this relates to the Big Bang model.
1 mark: correct $\Delta\lambda$ and $v$ calculation · 1 mark: correct Doppler velocity method for exoplanet detection · 1 mark: minimum mass explained (inclination of orbit unknown) · 1 mark: expanding universe interpretation · 1 mark: Big Bang link (running expansion back in time)
ApplyBand 5(3 marks) 3. (a) A binary star system has two stars orbiting their common centre of mass. The Ca II line ($\lambda_0 = 393.3$ nm) from one star varies between 392.7 nm and 393.9 nm with a period of 10 days. Calculate: (i) the maximum Doppler velocity when the line is at 392.7 nm, (ii) the maximum Doppler velocity when the line is at 393.9 nm, and (iii) explain what the equality (or difference) of these speeds tells us about the two stars. (b) Why are binary stars particularly useful for determining stellar masses?
1 mark each: correct velocity calculations · 1 mark: mass-ratio interpretation · 1 mark: Kepler's third law and mass determination from orbital parameters
Show all answers
Multiple choice
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.
Short Answer — Model Answers
Q1 (a): (1) A hot dense source produces a continuous spectrum. (2) A hot thin gas produces an emission spectrum (bright lines at characteristic wavelengths). (3) A cool thin gas in front of a continuous source produces an absorption spectrum (dark lines at the same wavelengths as the emission lines of that gas) (1 mark).
Q1 (b): Atoms have discrete energy levels. An emission line is produced when an electron drops from a higher to a lower level, emitting a photon of energy $E = hf = \Delta E$. An absorption line is produced when an electron absorbs a photon of exactly the same energy and jumps to a higher level. Since the energy difference $\Delta E$ is identical for both transitions, both produce photons of the same frequency and wavelength (1 mark).
Q1 (c): The presence of Fraunhofer lines indicates that the Sun has a cool, thin outer atmosphere (chromosphere) containing hydrogen, calcium and sodium. This cool gas absorbs specific wavelengths from the continuous spectrum emitted by the hot, dense solar interior (photosphere), creating the absorption lines (1 mark).
Q2 (a): $\Delta\lambda = 664.0 - 656.3 = 7.7$ nm. $v = c \times \Delta\lambda/\lambda_0 = 3.00\times10^8 \times 7.7/656.3 = 3.52\times10^6$ m/s $\approx$ 3520 km/s (receding) (1 mark).
Q2 (b): The orbiting planet causes its host star to wobble around the common centre of mass. This wobble produces tiny periodic Doppler shifts in the star's spectral lines — blueshift as the star moves toward us, redshift as it moves away. The period of the shift equals the planet's orbital period, and the amplitude gives the star's orbital velocity, from which the planet's minimum mass can be derived. It is only a minimum mass because the orbital inclination is unknown — if the orbit is tilted, only the radial (line-of-sight) component of velocity is measured, which is less than the true orbital speed (1 mark + 1 mark).
Q2 (c): If the universe were static, galaxies would have random motions with both red and blueshifts. Instead, nearly all galaxies show redshift, and the redshift increases proportionally with distance (Hubble's Law: $v = H_0 d$). This can only be explained if all parts of the universe are moving apart — i.e., the universe is expanding (1 mark). Running the expansion backward in time, all matter converges to a single point at a finite time in the past. This is the basis of the Big Bang model — the universe originated from an extremely hot, dense state approximately 13.8 billion years ago (1 mark).
Q3 (a)(i): $\Delta\lambda = 392.7 - 393.3 = -0.6$ nm. $v = 3.00\times10^8 \times (-0.6)/393.3 = -4.58\times10^5$ m/s $\approx$ -458 km/s (approaching).
Q3 (a)(ii): $\Delta\lambda = 393.9 - 393.3 = +0.6$ nm. $v = +4.58\times10^5$ m/s $\approx$ +458 km/s (receding). The equal speeds suggest the two stars have equal masses (since both orbit their common centre of mass at the same speed if their masses are equal).
Q3 (b): The orbital period $T$ and semi-major axis $a$ (from the Doppler velocity and period) give the total mass via Kepler's third law: $M_1 + M_2 = 4\pi^2 a^3/(GT^2)$. The ratio of the orbital speeds gives the mass ratio: $M_1/M_2 = v_2/v_1$. Together these give individual stellar masses — the only direct way to measure stellar masses accurately.
Five timed questions on spectroscopy and astronomical applications. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
Enter the arenaLook back at your Think First answers:
- Did you predict that dark absorption lines form because atoms in the cool gas absorb photons of exactly the right energy, exciting electrons to higher levels? The absorbed light is re-emitted in all directions, so little reaches the observer along the original path — creating a dark gap.
- Did you predict that hot gas produces bright emission lines because excited electrons drop back down, releasing photons? There is no continuous background because the gas is thin and transparent at other wavelengths.
- Did you predict that matching observed spectral lines to known element fingerprints reveals composition? Every element has unique energy levels, giving it a unique spectral "barcode".
The answer to the hook question: the specific colour of light is absorbed by atoms in the cool gas, which re-emit it in random directions — so almost none travels forward toward the observer, leaving a dark line.
The historical anchor for this lesson is Cecilia Payne's 1925 Harvard College Observatory analysis of more than 300 stellar spectra. By matching absorption line strengths to atomic physics models, she determined that stars are 73% hydrogen and 25% helium by mass — a result so unexpected that her supervisor Henry Russell initially dismissed it. Four years later Russell confirmed her findings independently, acknowledging her priority. Payne's work is the definitive demonstration that spectroscopy can reveal the composition of objects light-years away without any physical sample.