Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== I compute number of photons and times for representative photon energies and fluxes.  (I assume absorption efficiency α=η=1\alpha=\eta=1α=η=1 in the numerical examples below; realistic values will reduce the capture rate.) ===

Photon counts needed (no losses)
* visible photon, λ=500 nm\lambda=500\ \mathrm{nm}λ=500 nm (hf≈3.97×10−19 Jh f \approx 3.97\times10^{-19}\,\mathrm{J}hf≈3.97×10−19J, ~2.48 eV): Nvis≈8.187×10−143.97×10−19≈2.06×105 photons.N_{\rm vis}\approx \frac{8.187\times10^{-14}}{3.97\times10^{-19}} \approx 2.06\times10^5\ \text{photons}.Nvis≈3.97×10−198.187×10−14≈2.06×105 photons.
* UV photon, λ=100 nm\lambda=100\ \mathrm{nm}λ=100 nm (hf≈1.99×10−18 Jh f \approx 1.99\times10^{-18}\,\mathrm{J}hf≈1.99×10−18J): NUV≈4.12×104 photons.N_{\rm UV}\approx 4.12\times10^4\ \text{photons}.NUV≈4.12×104 photons.
* single 511 keV511\ \mathrm{keV}511 keV gamma photon (hf≈mec2h f \approx m_e c^2hf≈mec2): Nγ≈1(one gamma of 511 keV deposits mec2).N_{\gamma}\approx 1 \quad(\text{one gamma of 511 keV deposits }m_e c^2).Nγ≈1(one gamma of 511 keV deposits mec2).

Times to collect mec2m_e c^2mec2 (two alternative cross-section choices = projected geometric area πR2\pi R^2πR2 vs Thomson cross-section σT\sigma_TσT):

Let FFF be directional flux (W·m⁻²). For sunlight at Earth: F⊙≈1361 W/m2F_{\odot}\approx 1361\ \mathrm{W/m^2}F⊙≈1361 W/m2. For example lasers we use two sample intensities: moderate high Flaser1=1012 W/m2F_{\rm laser1}=10^{12}\ \mathrm{W/m^2}Flaser1=1012 W/m2, extreme Flaser2=1020 W/m2F_{\rm laser2}=10^{20}\ \mathrm{W/m^2}Flaser2=1020 W/m2.
* Sunlight (directional), F=1361 W/m2F=1361\ \mathrm{W/m^2}F=1361 W/m2: - using projected geometry πR2\pi R^2πR2: t≈9.65×1012 s≈3.06×105 years.t \approx 9.65\times10^{12}\ \mathrm{s} \approx 3.06\times10^5\ \text{years}.t≈9.65×1012 s≈3.06×105 years. - using Thomson cross-section σT\sigma_TσT (larger intercept): t≈9.04×1011 s≈2.87×104 years.t \approx 9.04\times10^{11}\ \mathrm{s} \approx 2.87\times10^4\ \text{years}.t≈9.04×1011 s≈2.87×104 years.
* High-intensity laser, F=1012 W/m2F=10^{12}\ \mathrm{W/m^2}F=1012 W/m2: - projected geometry: t ⁣≈ ⁣1.31×104 s≈3.65 hours.t\!\approx\!1.31\times10^4\ \mathrm{s}\approx 3.65\ \mathrm{hours}.t≈1.31×104 s≈3.65 hours. - Thomson area: t ⁣≈ ⁣1.23×103 s≈20.5 minutes.t\!\approx\!1.23\times10^3\ \mathrm{s}\approx 20.5\ \mathrm{minutes}.t≈1.23×103 s≈20.5 minutes.
* Ultra-intense laser, F=1020 W/m2F=10^{20}\ \mathrm{W/m^2}F=1020 W/m2: - projected geometry: t ⁣≈ ⁣1.31×10−4 st\!\approx\!1.31\times10^{-4}\ \mathrm{s}t≈1.31×10−4 s (0.13 ms). - Thomson area: t ⁣≈ ⁣1.23×10−5 st\!\approx\!1.23\times10^{-5}\ \mathrm{s}t≈1.23×10−5 s (12 µs).
* Cosmic microwave background (isotropic), T≈2.725 KT\approx2.725\ \mathrm{K}T≈2.725 K: - ambient isotropic absorption (blackbody formula) gives astronomically long times (∼1021\sim10^{21}∼1021 s → many orders of the Universe's age) — i.e. negligible.

Photon arrival rates (visible, projected area):
* sunlight: ~ 2.1×10−82.1\times10^{-8}2.1×10−8 visible photons/sec (one visible photon every ~170 days, at geometric projected area)
* laser 1012 W/m210^{12}\ \mathrm{W/m^2}1012 W/m2: ~ 161616 visible photons/sec (projected area)
* laser 1020 W/m210^{20}\ \mathrm{W/m^2}1020 W/m2: ~ 1.6×1091.6\times10^91.6×109 visible photons/sec (projected area)

(If you use σ_T instead of the geometric cross-section, photon arrival rates grow by factor ≈σT/(πR2)≈10.66\approx\sigma_T/(\pi R^2)\approx 10.66≈σT/(πR2)≈10.66.)