How Cecilia Payne-Gaposchkin Inferred What Stars Contain

Imagine looking at a star’s spectrum and finding that a particular hydrogen absorption line is weaker than a line belonging to a heavier element. Would you conclude that the star contains less hydrogen?

That seems reasonable if you assume stronger absorption means more atoms. But it can give you the completely wrong answer. An atom can only absorb a particular wavelength if it is in the correct energy state and ionisation state. Most atoms of an element might not be able to produce the line you’re measuring.

This was the problem Cecilia Payne-Gaposchkin tackled in 1925. By combining stellar spectra with the developing theories of atomic excitation and thermal ionisation, she found evidence that hydrogen and helium were vastly more abundant in stars than astronomers had previously believed.

01The problem: a spectrum is not a chemical inventory

Astronomers already knew that stars produced absorption spectra. These spectra contained dark lines at particular wavelengths, and each chemical element produced its own characteristic pattern.

The basic physics was understood. A star’s hot, dense photosphere produces a nearly continuous spectrum of light. As this light travels through the cooler gas in the star’s outer atmosphere, atoms and ions absorb photons at specific wavelengths.

The result is an absorption spectrum.

The wavelength of an absorption line identifies the atomic transition responsible for it. Since different elements have different electron energy levels, their spectral lines form recognisable patterns.

Think of the pattern as a barcode. It can tell you which element is present, even when the light has travelled enormous distances.

But identifying an element is different from measuring how much of it is present.

Astronomers had often treated the intensity of an absorption line as a rough measure of the abundance of its element.

The reasoning was straightforward:

• More atoms should mean more opportunities to absorb photons.
• More absorption should produce a stronger spectral line.
• Therefore, stronger lines should indicate greater abundance.

There is some truth in this reasoning. If everything else stays the same, increasing the number of suitable absorbing atoms generally increases absorption.

The problem is that everything else does not stay the same.

Temperature changes the proportion of atoms in different energy states. It also changes how many atoms are ionised. Consequently, two stars containing exactly the same amount of an element can produce very different absorption spectra.

Payne realised that astronomers needed to calculate how many atoms were actually capable of producing each spectral line before using its strength to estimate abundance.

02How Payne examined the starlight

Payne’s work was based on the extensive collection of stellar spectra at Harvard College Observatory.

The observational equipment was relatively simple compared with modern astronomical instruments, but it provided an enormous amount of information.

The process followed several stages.

Step 1: Collect the starlight.

A telescope collected light from a distant star and directed it towards a prism.

Step 2: Separate the wavelengths.

The prism spread the incoming light into its component wavelengths. Different wavelengths appeared at different positions along the resulting spectrum.

Step 3: Record the spectrum.

The dispersed light was focused onto a photographic glass plate. The plate recorded the spectrum, which could then be examined under magnification.

Step 4: Identify and compare the lines.

Astronomers identified absorption lines by comparing their wavelengths with known atomic spectra. They then examined how the lines changed in intensity across different stellar spectra.

The Harvard photographic records were particularly useful because they allowed Payne to compare many stars using consistent observational material.

She was not simply counting the number of dark lines. She was investigating how particular lines appeared, strengthened, reached maximum intensity, and disappeared as stellar temperature changed.

This brought her to a much more interesting question:

If two stars contain similar amounts of an element, why might that element produce a strong line in one star and a weak line in another?

The answer involves two processes: excitation and ionisation.

03First complication: excitation changes the strength of absorption lines

An atom must start in the right energy state

Electrons in atoms cannot have arbitrary energies. They occupy particular allowed energy levels.

The lowest available energy level is called the ground state. When an electron gains energy and moves to a higher level, the atom is described as excited.

For an absorption line to occur, an electron must absorb a photon whose energy matches the difference between its initial and final energy levels.

Importantly, the atom must already have an electron in the correct lower energy state.

This is easy to overlook.

Suppose an absorption line requires an electron to move from energy level 2 to energy level 3. An atom with its electron in the ground state cannot produce that particular absorption line through this transition.

It first needs to be excited into level 2.

The fraction of atoms occupying each energy level depends on temperature.

At higher temperatures, collisions between particles transfer energy more frequently. A greater proportion of atoms can therefore occupy excited states.

This distribution is described by the Boltzmann distribution.

For two energy levels, a simplified form is:

\[ \frac{N_2}{N_1} = \frac{g_2}{g_1} e^{-\Delta E/(kT)} \]

Where:

• \(N_1\) is the number of atoms in the lower energy level.
• \(N_2\) is the number of atoms in the higher energy level.
• \(g_1\) and \(g_2\) are the statistical weights of the two levels.
• \(\Delta E\) is the energy difference between the levels, measured in joules or electronvolts.
• \(k\) is Boltzmann’s constant.
• \(T\) is the absolute temperature in kelvin.

The important feature is the exponential term.

A large energy difference compared with the available thermal energy makes the excited state much less populated.

Why this matters for hydrogen

The visible absorption lines of hydrogen belong to the Balmer series.

These transitions involve electrons initially occupying the second energy level, \(n=2\).

However, the ground state of hydrogen is much lower in energy. An electron needs approximately \(10.2\text{ eV}\) to move from \(n=1\) to \(n=2\).

At the Sun’s photospheric temperature, approximately \(5700\text{ K}\), this is a substantial energy requirement.

Only a tiny fraction of neutral hydrogen atoms occupy the second energy level.

The American Physical Society’s historical account of Payne’s work gives a useful estimate: approximately one in 200 million hydrogen atoms occupies the first excited state under the relevant solar-atmosphere assumptions.

Think about what this means.

Even if hydrogen is overwhelmingly abundant, only a very small fraction of its atoms can absorb photons through the Balmer transitions at any particular moment.

A weak Balmer line therefore does not imply that hydrogen is scarce.

It may simply mean that very few hydrogen atoms occupy the required energy level.

04Second complication: ionisation changes which lines an element produces

Excitation is only part of the problem.

At sufficiently high temperatures, collisions can transfer enough energy to remove electrons from atoms altogether.

This process is called ionisation.

For example:

\[ \mathrm{Mg}\rightarrow\mathrm{Mg}^{+}+e^- \]

A neutral magnesium atom has become a singly ionised magnesium ion.

With further energy:

\[ \mathrm{Mg}^{+}\rightarrow\mathrm{Mg}^{2+}+e^- \]

The element is still magnesium, but its ionisation state has changed.

This matters because neutral magnesium and ionised magnesium have different electron energy levels. They therefore produce different spectral lines.

A neutral magnesium line cannot be produced by an ionised magnesium atom. The same applies to other elements.

The Saha ionisation equation

Meghnad Saha developed the thermal ionisation theory that made this problem much easier to analyse.

His work, published from 1920 onwards, connected the relative populations of different ionisation states to temperature and electron pressure.

A simplified relationship is:

\[ \frac{N_{\mathrm{ionised}}} {N_{\mathrm{neutral}}} \propto \frac{T^{3/2}}{n_e} e^{-\chi/(kT)} \]

Where:

• \(N_{\mathrm{ionised}}\) is the number of atoms in the ionised state.
• \(N_{\mathrm{neutral}}\) is the number of neutral atoms.
• \(T\) is absolute temperature.
• \(n_e\) is the electron number density.
• \(\chi\) is the ionisation energy.
• \(k\) is Boltzmann’s constant.

The complete Saha equation includes additional constants and statistical-weight factors. This simplified expression shows the important relationships.

At a fixed electron density, increasing temperature generally increases ionisation.

Electron density also matters. At the same temperature, a lower electron density generally favours the ionised state.

This means that temperature alone does not completely determine a star’s spectrum. Pressure and electron density must also be considered.

Why a spectral line can become weaker as a star gets hotter

Consider a spectral line produced by singly ionised magnesium, \(\mathrm{Mg}^{+}\).

For this line to appear, two conditions must be satisfied:

1. Magnesium must be singly ionised.
2. The ion must occupy the correct lower energy state for the absorption transition.

As temperature increases, these conditions compete.

At relatively low temperatures, there may be very few magnesium ions in the required excited state. The line is weak.

As temperature increases, more magnesium becomes ionised, and more of the Mg+ ions reach the required excited state. The line strengthens.

But keep increasing the temperature and another process becomes important: Mg+ begins losing another electron to form Mg2+.

The number of Mg+ ions available to produce the line decreases.

Eventually, the line becomes weaker again.

This produces a maximum in line strength at an intermediate temperature.

Payne examined this behaviour using magnesium’s absorption lines, including the Mg+ line at 4481 Å.

Her analysis showed that the fraction of atoms able to produce a particular line could increase and then decrease as temperature rose, even when the total magnesium abundance remained unchanged.

This was a major step forward. The changing appearance of spectral lines could be explained through atomic physics rather than requiring different chemical compositions for different stars.

Helium provides another important example

Helium also demonstrates why the absence of a particular line does not establish the absence of an element.

Many of helium’s visible absorption lines require atoms to occupy excited energy states.

At relatively low temperatures, very few neutral helium atoms occupy the required states. Its visible absorption lines may therefore be weak or absent.

As temperature increases, helium lines become more prominent.

At still higher temperatures, helium becomes increasingly ionised, producing a different set of spectral lines.

Payne’s 1925 analysis identified a temperature-dependent sequence:

• Neutral helium lines reached their strongest representation around spectral class B3.
• Singly ionised helium lines became prominent in the much hotter O-type stars.

The helium was not suddenly appearing in these stars. The conditions were changing which helium atoms could produce observable lines.

05Putting excitation and ionisation together

We can now construct a more useful model of absorption-line strength.

An element may be extremely abundant, but only some of its atoms can produce a particular spectral line.

The number of suitable absorbing atoms depends on:

• The total number of atoms of that element.
• The fraction in the required ionisation state.
• The fraction in the required lower energy state.

A simplified relationship is:

\[ N_{\mathrm{active}} = N_{\mathrm{total}} f_{\mathrm{ion}} f_{\mathrm{exc}} \]

Where:

• \(N_{\mathrm{active}}\) is the number of atoms capable of absorbing the selected wavelength.
• \(N_{\mathrm{total}}\) is the total number of atoms of the element.
• \(f_{\mathrm{ion}}\) is the fraction in the required ionisation state.
• \(f_{\mathrm{exc}}\) is the fraction of those atoms in the required lower energy state.

For a weak, unsaturated absorption line, its strength is approximately proportional to the number of active absorbing atoms, provided the transition probability and other atmospheric conditions are accounted for.

This is the key distinction:

A spectral line measures the population of atoms capable of producing that line, not the total abundance of the element.

06How Payne used this understanding to estimate stellar composition

By the early 1920s, Saha’s thermal ionisation theory had been extended by physicists Ralph Fowler and Edward Milne.

Their work provided a way to calculate the fraction of an element’s atoms occupying the states responsible for particular spectral lines.

Payne applied these ideas to the stellar spectra collected at Harvard.

Her approach involved comparing absorption lines across different spectral classes and examining two particularly useful features:

• The temperature at which a line reached maximum strength.
• The spectral class at which a line first appeared or became too weak to detect.

The second feature was especially important.

Marginal appearance: when a line is just visible

Payne used the concept of marginal appearance.

This is the point in a sequence of stellar spectra where an absorption line is just about to disappear or become detectable.

Why is this useful?

Imagine two elements, X and Y.

Suppose a certain number of active atoms is needed to produce a just-visible absorption line for each element.

However, only 1% of element X’s atoms occupy the required state, while only 0.01% of element Y’s atoms do.

Element Y has a much smaller active fraction.

To produce the same number of absorbing atoms, the star must therefore contain substantially more Y than X.

Mathematically, under the assumptions of Payne’s method:

\[ N_{\mathrm{total}}\propto \frac{1}{f_{\mathrm{active}}} \]

This gives a way to estimate relative abundance without assuming that line strength directly represents the total number of atoms.

Payne’s 1925 paper, Astrophysical Data Bearing on the Relative Abundance of the Elements, explains this method in detail.

She also took care to use comparable photographic spectra. Differences in photographic quality, spectral dispersion, and line visibility could otherwise distort the comparison.

Her calculations relied on several assumptions, including:

• Stellar atmospheres had broadly uniform chemical compositions.
• Lines at marginal appearance were unsaturated.
• The absorbing efficiencies of different atoms were sufficiently comparable.
• Conditions such as electron pressure were adequately represented by the adopted model.

The assumption about comparable absorbing efficiencies was particularly important. Different atoms do not necessarily absorb equally effectively, so the method could only provide a first approximation.

Payne acknowledged these limitations in her original publication.

07The hydrogen and helium result

Now we can return to the original puzzle.

Why did Payne conclude that hydrogen and helium were extraordinarily abundant in stars, even though their absorption lines did not always appear as strong as their abundance would suggest?

The answer came from correcting for the tiny fractions of atoms capable of producing the observed lines.

Hydrogen

The Balmer lines require hydrogen atoms to occupy the \(n=2\) energy level.

At solar temperatures, this is an extremely small fraction of the neutral hydrogen population.

Once Payne accounted for this, the observed hydrogen absorption could no longer be interpreted as evidence of a small hydrogen abundance.

Instead, her calculations indicated an enormous total hydrogen population.

There was another important clue.

Hydrogen’s Balmer lines appeared across the normal stellar sequence, although their strength changed substantially with temperature.

The lines reached their greatest strength around spectral class A0, corresponding to approximately 10,000 K in the temperature scale used in her analysis.

This behaviour was consistent with the changing balance between excitation and ionisation.

Helium

Helium presented a similar problem.

Its visible absorption lines required atoms in particular excited states. The absence of these lines in cooler stars therefore did not establish that helium was absent.

Payne considered both neutral helium and ionised helium when analysing the stellar spectra.

Her calculations suggested that helium was also present in extraordinary abundance.

The evidence was particularly interesting because helium lines became prominent in hot B-type stars, while ionised helium appeared in hotter O-type stars.

The observed pattern was consistent with changes in the atomic populations rather than helium appearing and disappearing between different stars.

What did Payne actually publish?

There is an important historical distinction here.

Payne’s 1925 research produced extraordinarily high calculated abundances for hydrogen and helium.

Her paper included a table of logarithmic relative-abundance estimates. It listed hydrogen with a lower limit above 11, helium at 8.3, and singly ionised helium with a lower limit above 12. Many of the heavier elements had values around 3 to 6.

These were not percentages. They were values on the logarithmic relative-abundance scale used in her calculations.

Hydrogen’s value was only a lower limit because its Balmer lines persisted across the stellar sequence, making its marginal disappearance difficult to establish.

However, Payne did not present these hydrogen and helium estimates as established quantitative results.

In the comparison table that followed, she omitted both elements and described their calculated abundances as improbably high and almost certainly not real.

Why did she retreat from such a remarkable conclusion?

The result conflicted with the prevailing belief that the Sun and Earth had broadly similar chemical compositions.

Henry Norris Russell, one of the leading astronomers of the period, also challenged the result. In a letter dated 14 January 1925, he argued that hydrogen could not possibly be a million times more abundant than the metals and questioned the reliability of the theory used to calculate its excited-state population.

Russell suggested that hydrogen’s unusual spectral behaviour might require a different explanation.

Payne took these criticisms seriously. Her published work retained the extraordinary calculations but treated the hydrogen and helium abundances cautiously.

The distinction matters: she had identified the implication of her calculations, but the scientific community had not yet accepted it as a reliable conclusion.

How the result was eventually confirmed

Further work strengthened the case for the high abundance of hydrogen and helium.

In 1928, Albrecht Unsöld developed calculations using the observed contours of solar absorption lines to estimate the number of atoms responsible for their production. E. A. Milne reported that Unsöld’s relative abundance estimates agreed with Payne’s results for the elements examined.

In 1929, Russell published On the Composition of the Sun’s Atmosphere.

Using a different, more quantitative approach to the solar spectrum, he reached the conclusion that hydrogen and helium were the dominant constituents of the Sun’s atmosphere.

This was a major development because Russell had previously rejected the extraordinary hydrogen abundance suggested by Payne.

The result eventually became part of the accepted picture of stellar composition.

Today, the Sun is known to consist of approximately 75% hydrogen and 25% helium by mass, with only a small proportion of heavier elements.

Payne’s central insight was correct.

08What modern astronomy adds to Payne’s interpretation

Payne’s work established an important way of thinking about stellar spectra, but modern abundance calculations are more sophisticated.

Astronomers now use detailed models of stellar atmospheres to account for:

• Temperature variations through the atmosphere.
• Electron density and pressure.
• The populations of different excitation and ionisation states.
• The probability of absorption for individual atomic transitions.
• Line broadening, saturation, and blending.
• The transfer of radiation through the atmosphere.

Modern calculations also distinguish between situations where the atomic populations follow the Boltzmann and Saha distributions and situations where the assumption of local thermodynamic equilibrium breaks down.

The strength of an absorption line is often measured using its equivalent width, which represents the integrated absorption across the line rather than simply its central depth.

These improvements allow astronomers to estimate chemical abundances much more accurately than was possible in 1925.

There is also an important limitation to remember: stellar spectra directly reveal the composition of the layers from which the observed radiation emerges. They do not provide a direct measurement of the star’s core.

Nor should we assume that every star has exactly the same composition. Hydrogen and helium dominate most normal stars, but the relative abundance of heavier elements varies between stars and can change during stellar evolution.

The enduring lesson from Payne’s work is that an observation must be interpreted using a physical model of the system producing it.

A weak absorption line might indicate low abundance. But it might also indicate that very few atoms occupy the required energy state, that most atoms are in a different ionisation state, or that the relevant transition is difficult to observe.

You cannot distinguish these explanations by looking at line strength alone.

Historical sources and further reading

The following sources provide access to the original research and its historical development.

• Cecilia H. Payne, Stellar Atmospheres: A Contribution to the Observational Study of High Temperature in the Reversing Layers of Stars (1925). Her original doctoral thesis, including the discussion of ionisation, spectral line intensities, and relative abundances.

• Cecilia H. Payne, Astrophysical Data Bearing on the Relative Abundance of the Elements (1925). Her original paper explaining the marginal-appearance method and presenting her abundance estimates.

• Harvard & Smithsonian Wolbach Library, Stellar Spectra and Quantum Mechanics. Historical explanation of the photographic spectra and atomic physics used in Payne’s work.

• American Physical Society, Cecilia Payne-Gaposchkin and the day the universe changed. Historical discussion of her hydrogen and helium calculations and their significance.

• E. A. Milne, Pressure of Calcium in the Sun’s Atmosphere (1928). A contemporary discussion of Unsöld’s independent calculations.

• Henry Norris Russell, On the Composition of the Sun’s Atmosphere (1929). Russell’s later analysis of solar composition.

• Floe Foxon, How much iron is in the Sun? (2024). A modern historical overview of the development of stellar abundance measurements.

09Questions and solutions

Question 1

A stellar atmosphere contains \(5.0\times10^{11}\) atoms of element X in the region responsible for an absorption line.

Of these atoms:

• 4.0% are in the required ionisation state.
• 10% of those ions occupy the lower energy state needed to produce the absorption line.

Assuming the line is unsaturated, calculate the number of atoms capable of absorbing at the selected wavelength.

Explain why this number is more useful than the total number of X atoms when interpreting the line.

Solution 1

There are \(2.0\times10^9\) active absorbing atoms.

The relevant quantity is the total population multiplied by the two fractions required for absorption.

\[ \begin{aligned} N_{\mathrm{active}} &=N_{\mathrm{total}}f_{\mathrm{ion}}f_{\mathrm{exc}}\\ &=5.0\times10^{11}\times0.040\times0.10\\ &=2.0\times10^9 \end{aligned} \]

Therefore:

\[ \boxed{N_{\mathrm{active}}=2.0\times10^9} \]

Although the atmosphere contains \(5.0\times10^{11}\) atoms of X, only \(2.0\times10^9\) can absorb the selected wavelength.

The distinction is important because the remaining atoms may be in a different ionisation state or energy level. They cannot contribute to this particular absorption transition.

Question 2

Two stars have identical total magnesium abundances but different surface temperatures.

For a neutral magnesium absorption line, the fraction of magnesium atoms capable of absorbing is:

• Cool star: \(f_{\mathrm{active}}=0.30\).
• Hot star: \(f_{\mathrm{active}}=0.015\).

For a singly ionised magnesium line, the corresponding fractions are:

• Cool star: \(f_{\mathrm{active}}=0.002\).
• Hot star: \(f_{\mathrm{active}}=0.10\).

Assume both lines are unsaturated and that their absorption probabilities and other relevant conditions are comparable between the stars.

(a) Calculate the ratio of neutral magnesium line strengths, cool star to hot star.

(b) Calculate the ratio of singly ionised magnesium line strengths, hot star to cool star.

(c) Explain why the two lines show opposite trends despite the stars having identical magnesium abundances.

Solution 2

The neutral magnesium line is 20 times stronger in the cool star, while the singly ionised magnesium line is 50 times stronger in the hot star.

(a) Neutral magnesium

For comparable unsaturated lines, strength is proportional to the active population.

\[ \begin{aligned} \frac{W_{\mathrm{cool}}}{W_{\mathrm{hot}}} &=\frac{0.30}{0.015}\\ &=20 \end{aligned} \]

The neutral magnesium line is therefore 20 times stronger in the cool star.

(b) Singly ionised magnesium

\[ \begin{aligned} \frac{W_{\mathrm{hot}}}{W_{\mathrm{cool}}} &=\frac{0.10}{0.002}\\ &=50 \end{aligned} \]

The Mg+ line is 50 times stronger in the hot star.

(c) Physical explanation

The temperature changes the distribution of magnesium between neutral and ionised states.

In the cooler star, a greater fraction of magnesium remains neutral, allowing the neutral magnesium line to be stronger.

In the hotter star, more magnesium is ionised. The population capable of producing the Mg+ line increases, so this line becomes stronger instead.

The total magnesium abundance is identical in both stars.

The apparent difference comes from the changing populations of the atomic states, not a change in chemical composition.

Question 3

Payne used the marginal appearance of spectral lines to estimate relative elemental abundances.

Consider two elements, X and Y, whose absorption lines are just detectable in comparable stellar spectra.

The calculated fractions of atoms capable of producing the respective lines are:

\[ f_X=2.0\times10^{-3} \]

\[ f_Y=2.0\times10^{-5} \]

Assume that the same number of active absorbing atoms is required to produce a just-detectable line for both elements.

(a) Determine the ratio of the total abundance of Y to X.

(b) Explain why the element with the smaller active fraction must have the greater total abundance.

(c) Identify one important assumption in this method that could introduce error into the abundance estimates.

Solution 3

Element Y is estimated to be 100 times more abundant than element X.

(a) Calculate the abundance ratio.

At marginal appearance, assume the number of active atoms is the same for both elements.

Therefore:

\[ N_Xf_X=N_Yf_Y \]

Rearranging:

\[ \frac{N_Y}{N_X}=\frac{f_X}{f_Y} \]

Substitute:

\[ \begin{aligned} \frac{N_Y}{N_X} &=\frac{2.0\times10^{-3}}{2.0\times10^{-5}}\\ &=10^2\\ &=100 \end{aligned} \]

Thus:

\[ \boxed{\frac{N_Y}{N_X}=100} \]

(b) Explain the result.

Only a very small fraction of element Y’s atoms can produce its absorption line.

To achieve the same number of active absorbing atoms as element X, the star must contain more total Y atoms.

This is why comparing the fractions in the required atomic states is essential.

(c) Identify a limitation.

The method assumes that approximately the same number of active atoms is required to produce a just-detectable line for different elements.

However, different atoms can have different absorption probabilities.

If one element absorbs much more effectively than another, fewer atoms may be needed to produce a detectable line. This would affect the estimated abundance ratio.

Payne recognised this limitation and treated her method as a first approximation rather than an exact determination.

Question 4

A student writes the following statement in an examination:

“Cecilia Payne-Gaposchkin proved in 1925 that hydrogen and helium made up nearly all stars. Her results were immediately accepted, and Henry Norris Russell independently confirmed them in 1926.”

Evaluate the accuracy of this statement.

Your response should discuss:

• The conclusions suggested by Payne’s calculations.
• How she presented her hydrogen and helium estimates in her published work.
• Russell’s response.
• The later historical evidence supporting her conclusion.

Solution 4

The statement is historically inaccurate because it overstates the certainty of Payne’s published conclusion and gives the wrong date for Russell’s later confirmation.

Payne’s 1925 calculations indicated that hydrogen and helium were extraordinarily abundant in stellar atmospheres.

Her calculations accounted for excitation and ionisation, allowing her to estimate how many atoms were actually capable of producing the observed absorption lines.

The logarithmic abundance estimates in her 1925 PNAS paper placed hydrogen and helium far above many heavier elements.

However, Payne did not present these estimates as established quantitative results. She omitted hydrogen and helium from her subsequent comparison table, describing their calculated abundances as improbably high and almost certainly not real.

This caution reflected both the limitations of the theory and the prevailing belief that the Sun and Earth had broadly similar chemical compositions.

Russell challenged Payne’s results in a letter dated 14 January 1925. He considered the calculated hydrogen abundance implausible and questioned the treatment of hydrogen’s excited states.

The high hydrogen abundance was not independently supported by Russell until his 1929 paper, On the Composition of the Sun’s Atmosphere. His later analysis reached a conclusion consistent with Payne’s original calculations.

Modern measurements establish that the Sun is approximately 75% hydrogen and 25% helium by mass.

The historically accurate conclusion is that Payne identified the extraordinary abundance of hydrogen and helium through her 1925 analysis, but the result was initially treated with caution and required further supporting evidence before becoming widely accepted.

Question 5

Two stars, P and Q, have been observed using two absorption lines belonging to the same element, iron.

One line belongs to neutral iron, Fe I. The other belongs to singly ionised iron, Fe II.

An atmospheric model gives the following fractions of iron atoms capable of producing each line:

The observed equivalent-width ratios are:

\[ \frac{W_{\mathrm{Fe\,I,P}}} {W_{\mathrm{Fe\,I,Q}}}=4 \]

\[ \frac{W_{\mathrm{Fe\,II,P}}} {W_{\mathrm{Fe\,II,Q}}}=0.10 \]

Assume that:

• Both lines are unblended and unsaturated.
• The oscillator strength of each transition is identical between the two stars.
• Differences in continuum intensity and absorbing column geometry have already been corrected.
• For each transition, the simplified relationship \(W\propto A_{\mathrm{Fe}}f_{\mathrm{active}}\) applies.

Here, \(W\) is equivalent width, \(A_{\mathrm{Fe}}\) is the total iron abundance, and \(f_{\mathrm{active}}\) is the fraction capable of producing the selected line.

(a) Use the Fe I line to calculate the abundance ratio \(A_{\mathrm{Fe,P}}/A_{\mathrm{Fe,Q}}\).

(b) Repeat the calculation using the Fe II line.

(c) Can a single iron abundance ratio explain both observations? Explain the physical significance of your result and identify what an astronomer should investigate next.

Solution 5

No single iron abundance ratio satisfies both observations. The two lines produce conflicting abundance estimates, indicating that the simplified atmospheric model is inadequate.

The important constraint is that both lines belong to the same element.

Iron has one total abundance in each star, regardless of whether its atoms are neutral or ionised. Therefore, two reliable abundance measurements using different ionisation stages should produce consistent results.

(a) Calculate the abundance ratio using Fe I.

Starting with:

\[ W\propto A_{\mathrm{Fe}}f_{\mathrm{active}} \]

The ratio between the stars is:

\[ \frac{A_P}{A_Q} = \frac{W_P/W_Q} {f_P/f_Q} \]

For the Fe I line:

\[ \begin{aligned} \frac{A_P}{A_Q} &=\frac{4}{0.20/0.02}\\ &=\frac{4}{10}\\ &=0.40 \end{aligned} \]

The neutral iron line suggests that star P contains 40% as much iron as star Q.

(b) Calculate the abundance ratio using Fe II.

For the singly ionised iron line:

\[ \begin{aligned} \frac{A_P}{A_Q} &=\frac{0.10}{0.01/0.20}\\ &=\frac{0.10}{0.05}\\ &=2.0 \end{aligned} \]

The Fe II line instead suggests that star P contains twice as much iron as star Q.

(c) Evaluate the conflicting results.

The two estimates are:

These estimates disagree by a factor of five.

It would be tempting to select whichever line appears more convincing or simply average the two abundance estimates.

Neither approach is justified.

The total iron abundance must be consistent across its different ionisation states. Changing the total abundance cannot independently fix the disagreement between the Fe I and Fe II measurements.

The discrepancy indicates that something in the simplified physical model or its inputs needs to be investigated.

An astronomer could examine:

• Whether the assumed temperatures are accurate.
• Whether the electron pressure has been modelled correctly.
• Whether the two lines form at different atmospheric depths.
• Whether the atomic populations depart from local thermodynamic equilibrium.
• Whether the calculated active fractions accurately describe the actual line-forming regions.

This is a more important conclusion than simply choosing one of the two abundance ratios.

A spectrum is not a direct chemical inventory. Its interpretation depends on whether the physical model can consistently explain multiple independent observations.

The next useful step is to combine spectral analysis with the Hertzsprung-Russell diagram. Spectral lines help determine a star’s surface temperature and chemical composition, while the H-R diagram combines temperature and luminosity to reveal information about its radius and evolutionary stage. Together, these tools allow astronomers to investigate not just what stars contain, but how their physical properties change throughout their lives.