--- title: "Did the gas get hotter, or more strongly ionized?" description: "Heating and stronger ionizing light can both weaken hydrogen absorption. Follow the atoms and their motions to see what line width adds to the picture." canonical_url: "https://sandboxphysics.com/stories/lyman-alpha-forest/temperature-and-ionization/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/lyman-alpha-forest/temperature-and-ionization/" last_modified: "2026-11-05" --- [The Lyman-alpha forest](https://sandboxphysics.com/stories/lyman-alpha-forest/) The Lyman-alpha forest · L1 # Did the gas get hotter, or more strongly ionized? Sandbox Physics · 2026-11-05 Either change can make a hydrogen absorption line shallower. Stronger ionizing light leaves fewer neutral atoms to absorb. Heating can reduce their number too, and it also spreads their absorption over a wider range of colours. To separate those possibilities, we need to keep the shape of the spectrum as well as the amount of light that gets through. ## Before you begin You only need to know that light can be separated into colours and that atoms move randomly in a gas. Ionization, recombination and thermal broadening are explained as they enter the picture. ## Which hydrogen atoms can absorb the light? Imagine a bright distant object as a lamp behind a thin cloud of hydrogen. The gas absorbs some colours more strongly than others. A dip in the light reaching a telescope is an absorption line. For Lyman-alpha absorption, an electron bound to a hydrogen atom takes up a photon and moves to a higher energy level. Neutral hydrogen has an electron bound to its proton, with no net electric charge. A more energetic ultraviolet photon can remove the electron entirely. This is ionization. The remaining proton cannot produce the same Lyman-alpha absorption. A cloud can therefore become more transparent even when its total amount of hydrogen stays the same. Electrons can be captured again, a process called recombination. In the dilute, pure-hydrogen comparison used here, photoionization and radiative recombination balance to determine the neutral fraction. At fixed temperature and total density, increasing the photoionization rate removes more of the atoms that could absorb. Reading: Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1 · Independent hydrogen and sightline calculation materials ## Heating changes the atoms' motions too Temperature acts in two ways. Over the range considered here, from 5,000 to 30,000 kelvin, the radiative recombination rate falls as the gas gets hotter. With the ionizing light held fixed, fewer neutral atoms remain. That change in population already weakens the absorption. The remaining atoms also move faster. Some move towards us along the line of sight; others move away. Their motions shift the colours at which they absorb, so the collection of atoms absorbs across a wider range. This is thermal Doppler broadening. For an isolated cloud, spreading the absorption can make its centre shallower even if the beam encounters the same amount of neutral hydrogen. The thermal width grows with the square root of absolute temperature. Heating hydrogen from 10,000 to 30,000 kelvin multiplies the temperature by three, but increases this width by only the square root of three, about 1.73. Going from 5,000 to 10,000 kelvin gives a factor of about 1.41. These factors describe the thermal motion of one gas component; the width of several overlapping dips includes their different bulk velocities as well. $$ b=\sqrt{\frac{2k_{\mathrm B}T}{m_{\mathrm p}}} $$ The Doppler parameter $b$ expresses thermal width as a speed. Here $T$ is temperature in kelvin, $k_B$ is Boltzmann's constant, and the proton mass $m_p$ approximates the hydrogen atom's mass. This parameter is not the full width of an observed absorption trough. Reading: Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1 · Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1 · Independent hydrogen and sightline calculation materials ## Two ways to let four fifths of the light through A numerical example makes the ambiguity concrete. Keep one prescribed gas distribution, its large-scale velocities and the same sightline. At 10,000 kelvin and the default ionizing rate, the mean transmission across the selected spectral interval is about 70.2 percent. On average across that interval, roughly seven tenths of the background light gets through. Double only the photoionization rate and the mean transmission rises to about 80.3 percent. Now restore the original rate and heat the gas to 30,000 kelvin instead. The mean transmission is about 80.4 percent. Almost the same mean can result from two different changes. The first leaves thermal motions unchanged; the second changes both recombination and thermal broadening. Those percentages belong to this constructed sightline and finite spectral interval. They are not measurements of the universe's mean transmission. They show why reducing a spectrum to one transparency number loses useful information. The detailed shapes retain clues that the two averages have hidden. Reading: Independent hydrogen and sightline calculation materials ## A real spectrum carries a history Bulk motion can broaden or blend absorption features. An instrument can blur a narrow feature too. Higher gas density also favours recombination. Interpreting a real spectrum means considering these effects alongside the random motions that temperature controls. Earlier heating can leave another memory: pressure can spread out small density structures. Two regions with the same temperature today may have different density patterns because they were heated at different times. The numerical comparison above fixes the density and changes the present absorption physics; changing its temperature does not calculate a new expansion history. To use these spectra to study intergalactic temperature or dark matter, astronomers must also model gas structure and motion, and account for which kinds of absorption their observations can pick out. Reading: Peeples et al., Pressure Support versus Thermal Broadening in the Lyman-alpha Forest, arXiv:0910.0256v3 · Independent hydrogen and sightline calculation materials ## Exercises and explained answers 1. At fixed temperature, does doubling the photoionization rate double the thermal Doppler width? Read the explanation No. In this separated comparison it changes the number of neutral atoms, but leaves their thermal motions unchanged. The plotted interval below a chosen transmission threshold can still change as absorption weakens; that interval is not the thermal Doppler parameter. 2. Why does heating hydrogen from 10,000 to 30,000 kelvin increase its thermal width by only about 1.73? Read the explanation The width scales with the square root of temperature. A temperature ratio of three therefore gives a width ratio of the square root of three, about 1.73. Changes in the neutral population require the ionization and recombination calculation as well. ## Try the prediction ### [Compare two causes of weaker hydrogen absorption](https://sandboxphysics.com/labs/lyman-alpha-sightline/) 1. Keep the same sightline at 10,000 kelvin and double the ionizing rate. Compare transmission and spectral shape. 2. Restore the default rate, then set the temperature to 30,000 kelvin. Look for differences between the two more transparent spectra. 3. Move the sightline through another part of the gas. Explain how density adds another possibility when interpreting an absorption dip. [Previous: Why quasar light contains a forest of dark lines](https://sandboxphysics.com/stories/lyman-alpha-forest/why-quasar-light-holds-a-forest/) [Next: From a patch of hydrogen to an absorption spectrum](https://sandboxphysics.com/stories/lyman-alpha-forest/from-hydrogen-to-an-absorption-line/) ## Sources and calculation scope 1. [Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1](https://arxiv.org/abs/astro-ph/9509083v1) Targeted reading of equation 4, the hydrogen row of the fitted-parameter table, and the rate definitions. Pure-hydrogen case-A radiative recombination is used over 5,000–30,000 K. No helium, collisional ionization, self-shielding or complete ion network is solved. This is not a full reading of every ion in the paper. 2. [Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1](https://arxiv.org/abs/astro-ph/0602124v1) Targeted equations 1–8 establish the complete Lorentz/Maxwell convolution and normalization. Our calculation uses the full Voigt function, not the paper’s small-damping approximation. The atomic scalars are explicitly rounded assumptions. Hydrogen mass is approximated by proton mass. 3. [Peeples et al., Pressure Support versus Thermal Broadening in the Lyman-alpha Forest, arXiv:0910.0256v3](https://arxiv.org/abs/0910.0256v3) Targeted abstract, sections 2–3 and equations 6–7: photoionization equilibrium, uniform Gunn–Peterson opacity, and distinctions between thermal broadening and pressure history. This reading does not reproduce the hydrodynamic runs or claim a full-paper review. 4. [Independent hydrogen and sightline calculation materials](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/lyman-alpha-science-v1.zip) Original prescribed periodic gas screen and linear-parent peculiar velocity; fixed redshift three, no hydrodynamic pressure evolution. Full Voigt integration is checked against independent SciPy and finite-grid refinements. The offline Python and own synthetic examples are separate from the protected browser Engine.