First stars and galaxies as source populations

Gas must reach conditions suitable for luminous sources

The formation of a gravitationally bound matter structure does not specify a unique stellar population. Gas must accumulate and evolve, with cooling, chemistry and fragmentation affecting whether and how luminous sources form.

In primordial gas, the available chemical and radiative processes influence cooling. A source-population calculation therefore needs a gas model as well as a description of the underlying matter structures.

Radiation and mechanical effects from sources can alter later gas conditions. Such feedback introduces dependencies between a forming population and its environment. It should not be replaced by a universal sequence or a fixed stellar mass distribution without an explicit model.

The photon ledger below begins after source groups have been supplied. It does not derive their formation, identify observed first stars or determine an initial stellar mass function. Its purpose is to separate three factors in the resulting radiation budget.

An escaping budget weights each source by its radiation

At one common epoch, place three invented groups in the same chosen comoving catalogue volume V_0. For group i, supply a source count N_i, an intrinsic ionizing-photon rate per source P_i and an escape fraction f_esc,i.

The escaped rate from that group is N_iP_i f_esc,i, with units of photons per second. Summing the groups gives the total escaped rate. Dividing by the same V_0 gives a comoving emissivity, not a local per-atom photoionization rate.

A count share divides N_i by the total source count. An escaped-photon share instead divides N_iP_i f_esc,i by the total escaped rate. The denominators answer different questions and need not give the same ranking.

Escape fraction here means the supplied fraction of the group’s intrinsic ionizing photons that enters the external budget. No geometry, absorption distribution or empirical escape-fraction measurement has been computed to establish those input values.

The most numerous group need not dominate the escaped light

Choose P_0 = 10⁵⁰ photons per second as a rate unit. Group A contains 20 sources, each producing 4P_0 intrinsically with escape fraction 0.1. Its intrinsic total is 80P_0 and its escaped total is 8P_0.

Group B contains only two sources, each producing 30P_0 with escape fraction 0.2. Its intrinsic total is 60P_0, but it supplies 12P_0 to the escaped budget. Group C has 80 sources, each producing P_0 with escape fraction 0.05, and supplies only 4P_0.

There are 102 sources and 24P_0 of escaped rate in total. Group B therefore supplies half the escaping photons despite containing only about 1.96 percent of the sources. Group C is the most numerous but supplies one sixth of the escaped photons.

These groups are deliberately invented. They are not luminosity-function bins, identified primordial stellar classes or an inference that a particular real population dominates reionization.

The effective escape fraction is photon weighted

For the supplied ledger, total escaped rate divided by total intrinsic rate is 24/220 ≈ 0.109091. This effective escape fraction weights each group by its intrinsic photon production, N_iP_i.

A source-count-weighted average instead gives [20(0.1) + 2(0.2) + 80(0.05)]/102 ≈ 0.0627451. It is a different valid average of the inputs, but it cannot replace the photon-weighted fraction in the total escaped budget.

An escaped rate also leaves a product degeneracy. For one group-B source, 30P_0 times escape fraction 0.2 gives the same escaped rate as 60P_0 times 0.1. That single output does not independently determine intrinsic production and escape.

Additional information about source spectra, stellar populations, absorption and selection is needed to separate those factors in a real inference. Increasing source counts alone does not remove a shared uncertainty in a conversion or escape model.

A source census needs a matched conversion and propagation account

Observed non-ionizing light or an inferred star-formation rate does not automatically equal an escaped ionizing-photon rate. A model must specify how its measured or inferred population property converts into intrinsic ionizing production and then into escaping radiation.

Conversion conventions must remain explicit. For example, an efficiency defined per unit star-formation rate multiplies a star-formation-rate density. It cannot be relabelled as the same efficiency per ultraviolet luminosity density without a corresponding conversion.

Unobserved faint populations, dust, selection, population models and escape assumptions can affect that inference. Their roles are distinct from the arithmetic once a complete ledger has already been supplied.

The next lesson follows the radiation’s local interaction with gas. Converting an emitted comoving photon budget into a photoionization rate requires propagation, spectrum and cross-section weighting; the total 24P_0 alone does not perform that step.

Worked reasoning

Try the reasoning

1. Must the group with the most sources dominate the escaped-photon budget?

   Show the answer and reasoning

   No.

   Group C contains 80 of 102 sources but contributes only 4 of 24P_0 to the escaped rate.

2. Can 30P_0 × 0.2 and 60P_0 × 0.1 describe the same escaped rate per source?

   Show the answer and reasoning

   Yes.

   Both products equal 6P_0, so escaped output alone does not separate intrinsic production and escape fraction.

3. Does the total escaped rate divided by V_0 directly give a local per-atom ionization rate?

   Show the answer and reasoning

   No.

   It gives a comoving emissivity. Propagation, spectral intensity and atomic cross sections are still needed for a local rate.

What to carry forward

• Gas evolution and source-population formation require more than a matter-collapse description.
• Escaping photons depend jointly on source counts, intrinsic output and escape fraction.
• The relevant weights and conversion conventions must remain explicit when connecting a source census to ionization.