---
title: "Where the photons go when ionized bubbles meet"
description: "Overlapping regions contain one copy of the gas. Follow excess supply into neutral volume, then distinguish volume conservation from bubble shape."
canonical_url: "https://sandboxphysics.com/stories/reionization/when-ionized-bubbles-overlap/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/reionization/when-ionized-bubbles-overlap/"
last_modified: "2026-11-12"
---

[Starlight and intergalactic hydrogen](https://sandboxphysics.com/stories/reionization/)

# Where the photons go when ionized bubbles meet

Sandbox Physics · 2026-11-12

When two ionized regions meet, their overlap contains one copy of the gas. Taking the union of two independently grown regions can leave some of the supplied light unaccounted for. A spatial calculation must decide where that supply can ionize gas next.

Start with [Supplying photons to an expanding volume](https://sandboxphysics.com/stories/reionization/photons-in-an-expanding-box/). That chapter determines how much ionized volume can remain. Here we distribute its increase through space, using fractions, sums and distances.

## An overlap does not need a second copy of the gas

Consider uniform gas that starts neutral. For this short example, neglect recombination and let each absorbed photon cause one primary hydrogen ionization. Measure volume in equal cells containing equal numbers of nuclei; a fractional cell volume is allowed.

Two sources can each ionize forty cell volumes. Suppose their independently assigned regions overlap by twenty. Their union contains

$$
40+40-20=60
$$

cell volumes, yet the sources have supplied enough photons for eighty. If neutral gas remains available to absorb them, the extra supply can ionize twenty more cell volumes. Simply keeping the union loses that part of the supply from the calculation. This gives a direct example of the photon-conservation problem. [Choudhury and Paranjape, 2018, §4.1](https://arxiv.org/abs/1807.00836v2)

Recombination and propagation matter in real gas. The preceding chapter already accounts for escaped photons and recombination when calculating the retained ionized volume. Our spatial prescription assigns only its **net increase**. Assigning the cumulative emitted photons again would spend the photons needed to replace recombinations a second time. [Madau, Haardt and Rees, 1999, §3.1](https://arxiv.org/abs/astro-ph/9809058v1)

## Keep the excess before moving it

Let $f_j$ be the fraction of cell $j$ occupied by the ionized phase before an increment. The fraction of its volume that remains neutral is

$$
c_j=1-f_j.
$$

For example, $f_j=0.7$ leaves three tenths of the cell volume. This is a two-phase volume fraction, as defined earlier; it is different from assigning a local particle ionized fraction of 0.7 to every parcel.

In the first round, sources distribute their individual shares to nearby regions that still contain neutral gas. Each starts from the same old field, so several sources can add to one cell. Call their temporary combined result $g_j$. If it reaches 1.25, one cell volume can remain there and the extra 0.25 must be tracked.

Define the retained base and excess by

$$
h_j=\min(g_j,1),\qquad \epsilon_j=\max(g_j-1,0).
$$

Their sum is $g_j$. Capping a cell at one therefore has two outputs: retained volume and an excess for the next round. Discarding the excess would remove exactly $\sum_j\epsilon_j$ from the total.

The second round starts at excess cells and fills nearby neutral volume. Each new allocation uses the fraction still neutral at that moment. When enough neutral volume remains, all the excess can be accepted and the total matches the first-round sum.

We need a definite rule for equally distant recipients. This teaching prescription fills the same fraction of each recipient's remaining neutral volume. Suppose only two cells in a distance shell still contain neutral gas. Their neutral volumes are 0.2 and 0.3 cell volumes, with 0.25 to assign. The common fraction is

$$
\lambda=\frac{0.25}{0.2+0.3}=0.5.
$$

The recipients gain 0.1 and 0.15, respectively. They accept the same fraction of what they can still hold, and together receive the full 0.25.

## Divide the net growth among the sources

A box with $N^3$ cells and filling-factor increase $\Delta Q$ gains $N^3\Delta Q$ cell volumes. Write the ionizing photon rate of source $s$ as $\dot N_s$. All sources share one escape fraction, so their relative weights are the same before and after escape. Give that source the share

$$
b_s=N^3\Delta Q\frac{\dot N_s}{\sum_r\dot N_r}.
$$

The rate ratio is dimensionless; $b_s$ counts cell volumes. In a thousand-cell box with $\Delta Q=0.02$, two sources with rates in a three-to-one ratio receive fifteen and five cell volumes. Recombination has already entered the calculation of $\Delta Q$.

The first round adds $\sum_s b_s$. Redistributing all its excess gives

$$
\sum_j f_j^{\mathrm{new}}
=\sum_j f_j^{\mathrm{old}}+\sum_s b_s
=\sum_j f_j^{\mathrm{old}}+N^3\Delta Q.
$$

This equation checks the total by following every added volume. It leaves a spatial question: which arrangement of that volume best approximates the gas?

One step of a preliminary synthetic calculation in uniform gas adds about 67.2 cell volumes. Capping the temporary first-round field would remove 0.561 cell volumes; the second round delivers all of that excess into the remaining neutral volume. The excerpt illustrates volume conservation, with the detailed boundaries still approximate. [Saved step values and provenance](https://sandboxphysics.com/stories/reionization/materials/saved-example-inputs.json)

## Distance gives one arrangement

![Two synthetic records with uniform density and a common recombination budget have identical global counters at this native epoch and different patterns. Colors show original cell volume fractions in 32 display levels; white points project source positions. The allocation is noncausal.](https://sandboxphysics.com/figures/reionization/same-budget-slices-en-desktop.svg)

Two synthetic records with uniform density and a common recombination budget have identical global counters at this native epoch and different patterns. Colors show original cell volume fractions in 32 display levels; white points project source positions. The allocation is noncausal.

This incremental redistribution is inspired by a photon-conserving allocation method. Opposite box faces are joined, and distance is periodic. A source near one edge can affect gas near the opposite edge, with each cell counted once.

The companion calculation first splits a source's photon rate among the cell centres that bracket its position in each coordinate, at most eight. The linear weights sum to one, and coincident contributions are combined before allocation. These contributions belong to the same source. Refining the grid narrows the source smoothing in comoving length.

The order of redistribution can affect the boundaries. If two excess cells compete for the same nearby neutral volume, the first fills part of it and the second must look farther away. The same total volume can occupy different cells. Smaller increments and finer cells can also change that arrangement. Bubble sizes, connections and spatial fluctuations therefore call for comparisons of these numerical choices. [Choudhury and Paranjape, 2018, §4.1–4.2](https://arxiv.org/abs/1807.00836v2)

The uniform model gives every region the same effective recombination cost. Changing the fixed source arrangement then changes the shape, with the total still set by the global equation. In gas with resolved density structure, changing which regions are ionized can also change recombination losses and hence the total.

The prescription assigns each increment instantaneously, a noncausal approximation to the spatial distribution. A radiation-transport calculation is needed for front speeds, absorption and spectral evolution. This prescription produces an incrementally growing ionized-phase distribution whose volume can be checked. Next we use that same distribution to calculate an [ideal neutral-hydrogen 21 cm signal](https://sandboxphysics.com/stories/reionization/reading-a-neutral-hydrogen-map/).

## Problems with explained answers

1. Two supplies can each ionize thirty-five cell volumes. Their independent regions overlap by ten. Neutral gas is sufficient, with recombination and other losses neglected. Find the union and the volume still to assign.
2. A cell starts at $f=0.6$ and receives first-round additions of 0.25 and 0.3. Find its temporary value, retained base and excess. If the nearest cells have neutral volumes of 0.1 and 0.2, how much does each receive under the common-fraction rule?
3. Two source arrangements have the same total rate and supply history. Why can this model give the same $Q$ but different maps? What additional quantity must be recalculated in nonuniform gas?

For the first, the union is $35+35-10=60$ cell volumes. The sources can ionize seventy, so ten remain to be assigned. That supply was counted twice in the overlap.

For the second, $g=0.6+0.25+0.3=1.15$. One is retained and the excess is 0.15. The next shell has 0.3 neutral cell volumes, so it fills halfway: the recipients gain 0.05 and 0.1.

For the third, the global equation uses the fixed total supply and common recombination cost; the spatial rule decides where the ionized phase goes. Nonuniform gas can expose different densities to ionization. Electron–proton encounters then cost different numbers of recombinations, which must enter the updated total.

## Sources

1. [Choudhury and Paranjape, 2018, §4.1](https://arxiv.org/abs/1807.00836v2)
2. [Madau, Haardt and Rees, 1999, §3.1](https://arxiv.org/abs/astro-ph/9809058v1)

## Keep reading

Starlight and intergalactic hydrogen · Reading path

### Photons, ionized volume and a radio signal

1. [Article **How starlight changes intergalactic hydrogen**](https://sandboxphysics.com/stories/reionization/starlight-and-hydrogen/)
2. [Article **What does half ionized mean?**](https://sandboxphysics.com/stories/reionization/ionized-volume-and-local-fraction/)
3. [Article **Supplying photons to an expanding volume**](https://sandboxphysics.com/stories/reionization/photons-in-an-expanding-box/)
4. [You are here **Where the photons go when ionized bubbles meet**](https://sandboxphysics.com/stories/reionization/when-ionized-bubbles-overlap/)
5. [Article **Reading ionized bubbles through neutral hydrogen**](https://sandboxphysics.com/stories/reionization/reading-a-neutral-hydrogen-map/)
6. [Article **Recompute a fixed-source ionization history**](https://sandboxphysics.com/stories/reionization/recompute-a-fixed-source-history/)
7. [Article **Reading HERA's 21 cm upper limits**](https://sandboxphysics.com/stories/reionization/reading-hera-upper-limits/)