Starlight

An ionised sphere that is a count of photons

A hot star sits in a cloud of hydrogen and ionises it, out to a radius, and there the ionisation stops within a tiny fraction of the distance. That edge was predicted in 1939 from a single count: every recombination uses up one photon, so the star can hold ionised only as many atoms as its light can pay for. By the equation that describes a stellar atmosphere at the same temperature, the same gas would hold ten orders of magnitude fewer neutral atoms than it does. In a nebula the temperature is not what decides the ionisation. A budget is.

Assumes Ionisation, Interstellar medium and Stellar colour.

16 min read 5 figures Light is the only messenger Measuring the unreachable

In a star’s atmosphere the ionisation of an atom is a matter of temperature and pressure. Saha’s equation says how many atoms are ionised in a gas in thermal equilibrium, and it is what makes the spectral sequence a thermometer. Radiation and matter are in close contact, each photon absorbed is matched by a photon emitted, and the gas has one temperature that decides everything. It is natural to carry that reasoning into a nebula, where the gas shines at about ten thousand kelvin and is plainly ionised. Saha’s equation at that temperature gives a nearly fully ionised gas. So the answer looks right.

It is right for quite the wrong reason, and the size of the error is ten orders of magnitude. A nebula is not in thermal equilibrium with anything. The radiation in it comes from a star, some parsecs away, whose light has been diluted by a factor of about $10^{16}$ . An atom sits there unionised for months at a time, and is ionised by a photon far hotter than the gas. The state of the gas is set by a balance between two rare processes. Once this is written down, an unexpected thing follows. The ionised region has a size, it can be predicted with no reference to the temperature, and it has an edge.

Bengt Strömgren worked this out in 1939, in a paper of startling economy. He had the sharp edges of what are now called H II regions in mind, and the relevant measurements came from the spectra Otto Struve and his collaborators had just been taking of hydrogen across the sky. His sphere is one of the very few objects in astrophysics that is both simple enough to compute by hand and accurate enough to use.

A budget of photons

Take a star emitting $Q$ ionising photons a second, in a uniform gas of $n$ hydrogen atoms per cubic centimetre. In a steady state, each ionisation is balanced by a recombination somewhere in the volume. A recombination takes place at a rate $n_e n_p\alpha$ per unit volume, with $\alpha$ a coefficient that depends only slightly on temperature. If the gas in the region is almost completely ionised, this gives a total rate for a sphere of radius $R$ . Setting that equal to $Q$ :

Q = \frac{4\pi}{3}R^3 n^2\alpha_B .

The subscript on $\alpha$ is the only subtle point: a recombination straight to the ground state emits a photon that can ionise another atom, so it pays nothing back; only those to the excited levels count, and that coefficient is denoted by B. It has a value of about $2.6\times10^{-13}$ cubic centimetres a second at $10^4$ K.

Fig. 1 The radius within which a star’s ionising photons are used up by recombinations, against photons per second, for gas densities from 1 to 10,000 per cubic centimetre. An O7 star,

10^{49}

photons a second, makes a sphere of 14.6 parsecs at

n = 10

. A star a hundred times fainter in this count makes one only 4.6 times smaller. A density a hundred times higher shrinks it by 22.

The behaviour of that radius with its inputs is what makes it useful. It goes as $Q^{1/3}$ , so a hundredfold difference in the star’s output makes under a fivefold difference in the size. It goes as $n^{-2/3}$ , so in practice the size of a real region is a measure of its density more than of its star. That is a surprising fact, given how much effort goes into the star. For a given measured $R$ and emission brightness, an observer has two equations and can find both $n$ and $Q$ , and so can estimate the star’s ionising output from the nebula without ever having to measure the star’s ultraviolet directly. For stars hidden by dust this is often the only way to do it.

The count is a sort of energy budget, but it has no energy in it. It does not ask what any photon’s energy is, how hot the gas is, or what the temperature of the star might be. It is a tally of events, of one photon per ionisation and one recombination per ionisation, and that kind of tally recurs in cosmology where reionisation is the same arithmetic on a larger scale.

Why it has an edge

The count says how much gas is ionised but not how sharply that stops. A smoothly fading boundary would be the first guess, because a star’s flux falls as the inverse square of distance and nothing seems to mark a special radius. But the actual equilibrium gives an ionised fraction $x$ at radius $r$ from

(1-x)\,\Gamma = x^2 n\alpha_B,\qquad \Gamma=\frac{Q\,e^{-\tau}\sigma}{4\pi r^2},\qquad \frac{d\tau}{dr}=n(1-x)\sigma,

where $\sigma\approx3\times10^{-18}$ square centimetres is a photoionisation cross-section averaged over the spectrum.

$Almost everything is ionised, and then almost nothing is. The ionised fraction of hydrogen against distance from an O7 star in gas of 10 per cubic centimetre, from integrating photoionisation balance outward, in units of the counted radius of 14.62 parsecs. The fraction is near one until about the edge and then falls from 90% to 10% within 0.192 parsecs, 1.3% of the radius. That width is about 18 mean free paths of an ionising photon in neutral gas (0.011 pc). Nothing about the star sets it. A region with no gradual boundary is a surprising result of a smooth input, and it comes from the fact that a photon that gets as far as the neutral gas is absorbed at once.$

Fig. 2 The ionised fraction of hydrogen against distance from an O7 star in gas of 10 per cubic centimetre, from integrating the balance outward, in units of the counted radius of 14.62 parsecs. The fraction falls from 90 per cent to 10 per cent within 0.19 parsecs, 1.3 per cent of the radius. That width is about 18 mean free paths of an ionising photon in neutral gas, and the integration lands on the counted radius.

The reason is the size of the cross-section. It is enormous on the scale of the region. In completely neutral gas at a density of 10, a typical ionising photon goes only a hundredth of a parsec. Inside the ionised region it goes much further, because the neutral fraction is only about a thousandth. A photon that arrives near the boundary is met by a rapidly rising number of neutral atoms, each of which increases the absorption, which reduces the photons reaching further out, which leaves still more atoms neutral, and so on. It is a runaway in optical depth. The transition takes place over a small number of mean free paths in the neutral gas. For gas with this density, it is a tiny fraction of the radius.

The sharpness is one of those things that surprise in proportion to how smooth the setup was. A uniform gas and an inverse-square law yield a region with a nearly discontinuous boundary. In photographs of real regions, of the Rosette Nebula for instance, the bright gas does have a clear edge where the dark cloud begins, and bright rims mark the ionisation fronts, in ways that cannot be explained by clouds alone. Real edges have some further width, as the dust around them also shows, because an actual stellar spectrum has more energetic photons with smaller cross-sections, which penetrate further and spread the boundary by a factor of a few. The figure uses a single grey cross-section, and hence underestimates that.

The same gas, by the wrong equation

The balance equation shows what a nebula is, and comparing it with thermal equilibrium shows how far from equilibrium this is.

$The same gas, by the wrong equation. The neutral fraction of hydrogen inside the sphere, logarithmic, from photoionisation balance, against distance, with Saha's equation at the same 10,000 K and density as the dashed line, at 10^-13.5. The balance gives about 6e-4 at half the radius, so the Saha answer is too small by 10 orders of magnitude. Saha is exact in thermal equilibrium, where radiation and matter share one temperature and its radiation field is intense. Here the gas is at 10⁴ K and the star's radiation, at 40,000 K, is diluted by 2e-16.$

Fig. 3 The neutral fraction of hydrogen inside the sphere, logarithmic, from photoionisation balance, with Saha’s equation at the same 10,000 K and density as the dashed line. The balance gives about

6\times10^{-4}

at half the radius, and the Saha answer is too small by ten orders of magnitude. The star’s radiation, at 40,000 K, is diluted here by a factor of about

2\times10^{-16}

A comparison of ten orders of magnitude means these are not two estimates of the same quantity. Saha’s equation is the statement that ionisation and recombination balance detail by detail in a gas where radiation is thermal at the gas’s own temperature. In a stellar atmosphere, a hydrogen atom sits in a bath of Planck radiation which ionises it from the excited states as often as electrons recombine to them. In a nebula, a neutral atom meets an ionising photon every few months, and the photon has a colour temperature of about forty thousand kelvin, four times hotter than the gas, and all the intervening space is empty.

This has practical importance for the reader of a spectrum. Saha is a formula with temperature and density in it, and it can be used only if local thermodynamic equilibrium is a reasonable approximation. That is not something the formula can say about itself. It gave the qualitatively right answer here: “mostly ionised”. Anyone using it to find the amount of neutral hydrogen inside the region, for computing its absorption of ionising photons, or its emission in the Lyman lines, would make an enormous error, and no sign of the problem would appear in the arithmetic. It is the same kind of danger that a temperature assigned to a gas that has none carries. A number comes out, and it means nothing.

The photoionisation balance has its own surprising form. Since the neutral fraction $1-x$ depends on $n/\Gamma$ , the only quantity in the equation with information about the physics is the ratio of gas density to photon density, called the ionisation parameter. That single dimensionless parameter sets the nebula’s whole ionisation structure, and it is what observers actually fit. The temperature does not enter at all, apart from a small dependence in $\alpha_B$ . So a nebula at ten thousand kelvin and one at twenty thousand kelvin, at the same ionisation parameter, have essentially the same ionisation.

The front, and then the pressure

A star does not switch on in a finished sphere. It starts with neutral gas around it, and the front has to run out. Early on there are few recombinations, since little has been ionised, so the photons reach the edge and ionise a further shell. The front moves very fast at first. Then, as the ionised volume grows, more of the star’s output goes to balancing recombinations inside, until after several recombination times the whole supply is used and the front comes to a halt at $R_S$ .

Fig. 4 The radius of the ionised region against time, for the same star and gas. The ionisation front runs out through still gas, reaching 14.6 parsecs on the recombination time of about 12,000 years. The ionised gas is at

10^4

K, about a hundred times the pressure of the cold gas around it, so it then expands at the sound speed. After a few million years the radius has grown to about 69 parsecs.

The stage after that is a separate story and it was worked out by Lyman Spitzer in 1978 and others. Ionisation heats the gas to about $10^4$ K. Neutral gas just outside it, in a cloud, can be at less than a hundred kelvin. So there is a pressure jump of a factor of a hundred or more at the boundary, and the hot gas begins pushing out at about the sound speed, about eleven kilometres a second, into the cold gas in front of it, sweeping a dense shell ahead. As the region expands its density falls, and since $R\propto n^{-2/3}$ the count allows it to be larger.

The two parts have rather different characters. The first is a fast, simple process of counting, and it is over in ten thousand years. The second lasts for the star’s whole life, and it is responsible for dispersing the parent cloud. Most of what makes a star-forming cloud stop forming stars comes from this, since an expanding H II region can take hundreds of solar masses of cloud with it. So a single count of photons determines what fraction of a molecular cloud survives to form stars, in part, and this is one of the main processes through which massive stars control star formation in galaxies.

Zones inside zones

Hydrogen is not the only element here. Helium has a higher ionisation threshold at 24.6 electronvolts, and doubly ionised oxygen needs 35.1. Each ion has its own sphere, determined by the number of photons above its own threshold, and they are nested.

$Spheres inside spheres, set by the hardness of the star. The radius of the helium-ionised and doubly-ionised oxygen zones, as a fraction of the hydrogen sphere, against a blackbody star's temperature in thousands of kelvin. Each zone is set by the number of photons above its own threshold compared with the number of atoms, helium being about a tenth of hydrogen and competing for photons. Above about 42 thousand kelvin helium is ionised throughout. The doubly ionised oxygen sits inside a shell whose size says how hot the star is. Nebular images in the lines of different ions are therefore pictures of the star's ultraviolet spectrum, which cannot itself be seen. The oxygen weighting is a stand-in for its ionisation balance and only roughly placed.$

Fig. 5 The radius of the helium-ionised and doubly-ionised oxygen zones as a fraction of the hydrogen sphere, against a blackbody star’s temperature. Above about 42 thousand kelvin helium is ionised throughout. The doubly ionised oxygen sits inside a shell whose size says how hot the star is. The oxygen weighting is a stand-in for its ionisation balance and is only roughly placed.

This is a way of seeing what cannot otherwise be seen. A hot star’s flux below 912 ångströms is absorbed in the interstellar medium before reaching the Earth, and it has been directly measured for very few stars. But the gas around the star is an instrument for it: the relative sizes of the different zones, and the ratios of emission lines from different ions, are a direct measurement of the shape of the star’s ultraviolet spectrum. Hermann Zanstra in 1927 had already used that kind of reasoning to measure the temperatures of the central stars of planetary nebulae, the first such method. He used the number of hydrogen recombination photons from the nebula, compared with the star’s visible light.

The image from a narrow-band filter therefore becomes a kind of spectrum. Pictures of nebulae, with the oxygen shown in blue-green at the centre and the hydrogen and nitrogen in red further out, are colour-coded maps of the energy distribution of light that no telescope could detect directly. That the colour-coded nebula looks so structured, given a uniform gas, comes down to these thresholds, and to the same runaway at each ion’s boundary.

It is also how the same kind of analysis gets done at great distance. The line ratios in distant star-forming galaxies come from this process, and they are used to estimate the hardness of the ionising starlight in galaxies seen when the universe was young. Whether those show harder radiation than local galaxies, and what that means for massive stars, is a current question, and it depends on nothing more complicated than a count of photons, repeated at each threshold.

What the sphere leaves out

Real nebulae are not uniform. They are clumped, with most of the emission coming from a small fraction of the volume. A filling factor enters as a modifier, with the emission going as the density squared and the radius going up accordingly. Gas in filaments can leave low-density channels through which ionising photons escape.

Not all the photons are used on the gas. Dust within the nebula, whose temperature follows the light it catches, absorbs a fraction of the ionising photons directly, and the fraction can reach a half or more in dense regions. This makes the estimate of $Q$ from a region’s recombination lines into a lower limit. Stars embedded in dense gas may have their real output underestimated by a factor of two or more.

Case B is an approximation. It assumes that a Lyman-continuum photon emitted after a recombination to the ground state is immediately reabsorbed nearby. This is good at high density and a very poor approximation at low density and near edges, where such photons can escape. In real calculations, a whole radiative transfer is performed, and it gives somewhat different answers near the boundary.

The figures take a single grey opacity and a blackbody. Actual stellar spectra of hot O stars differ from a blackbody considerably at these wavelengths, including at the helium edge, with the result that the size of the helium zone in the last figure is an order of magnitude estimate only.

A connection with the end of the dark ages

The same balance applies in a very different setting. When the first stars and galaxies formed, every source was surrounded by its own growing ionised bubble, and those grew until they overlapped: the process that ended the cosmic neutral era. The physics differs mainly in that the expansion of the universe thins the gas between source and edge, so recombination becomes less effective, and there is no static sphere: at early enough times the recombination time is comparable with the age of the universe, so a bubble just goes on growing.

This also brings up a useful version of the original count, one that needs no sphere at all. Over the whole universe the number of ionising photons emitted per hydrogen atom has to exceed one, plus the number of recombinations per atom, to complete the job. Reionisation is thus a Strömgren count on a cosmological scale, and its timing is set by the production of photons by galaxies and by the fraction of them that escape. That last fraction, the escape fraction, is the one number that cannot be measured at those early epochs directly, and it depends on how leaky a local H II region is. So the structure of a local nebula has come to have direct bearing on the first billion years.

A nebula as a calorimeter

There is an unusual way to read the brightness of such a nebula. For each ionising photon the star emits, one recombination occurs in the nebula, and a fixed fraction of recombinations produce an $\mathrm{H}\alpha$ photon, about 45 per cent. So if the region absorbs all of the star’s ionising light, the star’s $\mathrm{H}\alpha$ luminosity counts the star’s ultraviolet photons directly. Indeed this has become the standard method to measure the star formation rate of a galaxy: count $\mathrm{H}\alpha$ photons, and from that obtain the number of massive stars alive at that moment, because only those stars emit such photons and they live only a few million years.

The calibration of that method rests on the Strömgren count and it depends on no geometry. It is an unusually direct form of the argument, because a measurement of a bright visible line has told how many ultraviolet photons a star cluster emitted, in the same way that the visible glow of a fluorescent tube tells how many ultraviolet photons are being produced inside it. Zanstra’s insight was exactly that. Only a thin theoretical thread is involved, and it has been stretched from a planetary nebula’s central star to all the galaxies in the universe.

Still open: what lets ionising photons out

The count assumes the star’s whole output is absorbed. In a real region some photons escape, through low-density channels or because the cloud is smaller than the star’s sphere, and they go on to ionise the diffuse gas of the galaxy. About a fifth of a galaxy’s hydrogen recombination light actually comes from outside any visible H II region, from diffuse ionised gas, and this has to have received its photons somehow. How far those photons travel, and what paths they take, is still being worked out.

It matters on the largest scale too. Whether galaxies in the early universe allowed ten per cent of their photons to escape or one per cent makes the difference between a universe reionised by small galaxies and one that needs more exotic sources. In nearby analogue galaxies the fraction has been measured directly in a few tens of objects, with large scatter and with a dependence on properties such as the compactness of the star formation and the strength of the oxygen lines that is only partly understood. For a single nebula with a star inside it, the question is simple to state, and it comes down to the exact structure of a gas that the sphere assumes uniform. The edge in the figure is sharp, but it cannot be known from the sphere how many holes it has.