# Reflected light and angular scales

Run `python reproduce.py` with numpy and matplotlib. Standalone analytic teaching study with runtime versions and generator SHA256. There are818 CSVrows:181phase,130brightness-degeneracy,500separation/aperture and7independent quadrature checks.

Uniform Lambert phase: Phi(alpha)=[sin(alpha)+(pi-alpha)cos(alpha)]/pi, angle0..pi. alpha is star–planet–observer angle: full phase0, zero reflected illumination atpi. Endpoints are explicitly handled. Assigned planet/star flux ratio=Ag(Rp/a)²Phi, with geometric albedoAg=.3, radius6,371km and star–planet separation1AU=149,597,870.7km. No thermal emission, clouds, glint or atmospheric model is included.

Brightness degeneracy uses radius ratio0.50..2.00 in.01 steps and Ag=.3/ratio², retaining130 values with Ag<=0.6 (ratios.71..2). All give the same full-phase flux. This is a formal amplitude degeneracy, not a family of established materials. For a strictly passive Lambert sphere, phase integral is3/2 and energy conservation additionally requires Ag<=2/3. All retained rows satisfy this limit; the figure marks the bound for context. Geometric albedo in general is not identical to Bond albedo, and the chosen cap0.6 is not a universal geometric-albedo theorem.

Angular separation theta=sAU/dpc arcseconds is the small-angle relation for a specified projected sky-plane separation, not generally the orbital radius. The default1AU at10pc gives0.1arcsec. CSV covers1..50pc and apertures1..10m. Ideal Rayleigh scale is1.22lambda/D radians converted by180*3600/pi, lambda550nm. Crossing these curves is not a detection threshold: stellar contrast, coronagraph response, inner working angle, detector sampling, noise and exposure remain unmodeled. The projected separation and phase are separate inputs here and are not a self-consistent simulated orbit.

Independent phase verification integrates illumination times visibility over a sphere: visible hemisphere parameter mu=n dot observer from0to1 and azimuth0to2pi; integrand mu*max(n dot star,0), normalized by full-phase2pi/3. Midpoint grids400x800 and800x1600 are compared for seven angles. The finer-grid max absolute phase error is3.90e-6; this is numerical quadrature error, not observational uncertainty. Analytic endpoints, nonnegativity, monotonicity, quadrature and equal-brightness scaling pass. Invalid angles reject. No detection, life or habitability score is produced.

Primary formula source verified: [NASA-hosted research paper on reflected-light flux and Lambert phase](https://ntrs.nasa.gov/api/citations/20240007032/downloads/Nemati%20Stahl%20HWO%20EB%20Roman%20Lessons%20Final%20SPIE.pdf). [Cowan, Fuentes and Haggard, reflected-light kernel](https://academic.oup.com/mnras/article/434/3/2465/1041737) supports independent surface integration. Parameter choices and figures here are synthetic, not that paper's measurements.

The three horizontal diffraction lines descend from 1 m to 3 m to 10 m aperture and use distinct dash patterns plus direct labels.