Phoenix in July. 47°C at noon. What if we parked a sunshade in geostationary orbit and dimmed the city by 10%? Not the whole planet — just one stubborn dot in the Sonoran Desert.

The geometry problem (and why GEO is the only option). Geostationary orbit sits at 35,786 km altitude. From there, an object appears fixed over one point on Earth's surface — essential, because Phoenix doesn't move and we want continuous shade. Lower orbits race across the sky in minutes; only GEO holds station.

Phoenix metro covers roughly 1,340 km² — call it a 41 km diameter circle. To cast a useful penumbra over that footprint from 35,786 km away, we need to account for the Sun's angular diameter (0.53°, or 9.3 milliradians). The Sun's disk, projected from GEO distance, has a "fuzz radius" of:

r_fuzz = 35,786 km × tan(0.265°) ≈ 165 km

That's the catch. Any shade smaller than ~330 km across casts only a partial penumbra — no full umbra ever touches Earth. A 1-km shade dims Phoenix by a vanishingly small fraction. Let's compute it honestly.

Back-of-envelope dimming. A disk of area A_shade blocks sunlight reaching a ground patch in proportion to its fraction of the Sun's apparent disk. The Sun's apparent area at GEO distance:

A_sun_apparent = π × (165 km)² ≈ 85,500 km²

A 1-km diameter shade has area 0.785 km². Dimming fraction:

0.785 / 85,500 ≈ 0.0009%

Phoenix receives ~1,000 W/m² at noon. Our kilometer-wide shade blocks 9 mW/m². A single sheet of printer paper held overhead does more.

Scaling up to actually matter. To dim Phoenix by 10% (a meaningful 5°C drop in peak heat), we need to block 10% of the solar disk as seen from the city. That requires a shade area of 8,550 km² — roughly 104 km in diameter. Now we're building a structure larger than Rhode Island, floating in GEO.

Mass budget. Use mylar-like film at 5 g/m² (the JWST sunshield is ~25 g/m², but solar sails have hit 3 g/m²). Mass:

8.55 × 10⁹ m² × 5 g/m² = 42,750 tonnes

SpaceX Starship promises ~100 tonnes to GEO. That's 428 launches, ignoring the tugs needed to actually deploy and tension a film the size of Connecticut. At an optimistic $50M/launch, lift alone runs $21 billion — before structure, attitude control, or station-keeping against solar radiation pressure.

The cruel torque. Solar radiation pressure at 1 AU is ~9 μN/m². On 8.55 × 10⁹ m², that's 77 kN of constant thrust pushing the shade away from the Sun. Any off-axis tilt creates torque that will spin the structure unless continuously corrected. The shade is, structurally, a colossal solar sail trying to escape.

And it misses half the time. GEO satellites drift north and south of the equator seasonally, and the Sun-Earth-shade geometry shifts with Earth's axial tilt. Without active repositioning of ~10° of inclination twice a year, the shadow wanders off Phoenix entirely in spring and fall. You need ion thrusters firing constantly.

Meanwhile, painting every Phoenix roof white achieves the same albedo effect for the price of a single launch.