1. News travels at the speed of light
In general relativity a change in a gravitational field spreads at c. The 2017 neutron-star merger GW170817 and its gamma-ray burst arrived within about 1.7 s of each other after travelling ~130 million light-years, so gravity's speed equals c to better than one part in 1015 (Abbott et al. 2017, ApJ Lett. 848, L13). If the Sun vanished, Earth would keep feeling its pull and keep seeing it for the light-travel time:
t = d / c Earth: d = 1 AU = 149,597,870.7 km
t = 149,597,870.7 / 299,792.458 = 499.0 s = 8 min 19 s
Mercury 3.2 min, Venus 6.0 min, Mars 12.7 min, Jupiter 43.3 min, Saturn 79.7 min, Uranus 2.67 h, Neptune 4.18 h, Pluto 5.47 h (mean orbital radii).
2. Then a straight line (Newton's first law)
Once the pull stops, a planet leaves along the tangent to its orbit at its orbital speed. Circular-orbit approximation:
v = 2*pi*a / P , P = a^(3/2) years (Kepler III, a in AU)
Earth: v = 2*pi * 149.6e6 km / 3.156e7 s = 29.78 km/s
Earth and the Moon stay bound to each other (Earth's own gravity), so the pair drifts away together. At 29.8 km/s even a direct course to the nearest star (4.2 light-years) would take ~42,000 years; the real direction is the orbit's tangent, not toward any star in particular.
3. The sky and the Moon
The Moon, planets and the daytime sky shine by sunlight. Sunlight stops reaching a body at t = r/c; its reflection stops reaching Earth one light-travel time later. For the Moon that is about 1.3 s after the Sun's last light arrives at Earth (the Earth-Moon distance is 1.28 light-seconds). Scattered sky light vanishes in a fraction of a second.
4. Tides
The Sun raises about 46% as much tide as the Moon. After the Sun's gravity stops arriving, tides remain (lunar only) but the spring/neap cycle ends.
5. The surface cools: energy balance
Before: absorbed sunlight = outgoing infrared = about 240 W/m2 (solar constant S = 1361 W/m2, albedo 0.30: S(1-0.30)/4 = 238 W/m2; Kopp & Lean 2011; Stephens et al. 2015). After: sunlight = 0, so the surface only loses heat:
C dT/dt = -[ e*sigma*T^4 - DLR ] - H + G
sigma = 5.670e-8 W/m2/K4, e = 0.97
DLR = e_a*sigma*T_air^4 (e_a falls as water vapour condenses)
H = h (T_s - T_air) (sensible + latent; weaker when cold or under ice)
G = conduction from the ground + geothermal 0.09 W/m2 (Davies & Davies 2010)
Heat stores per m2: air about 1e7 J/K (1e4 kg x 1004 J/kg/K); ground k = 2 W/m/K, rho*c = 2e6 J/m3/K (36-layer implicit conduction); ocean mixed layer 150 m of water (4e6 J/m3/K; range 50-400 m).
Sea ice (Stefan): rho_i L dh/dt = k_i (T_f - T_s)/h - F_b
k_i = 2.2 W/m/K, rho_i = 917 kg/m3, L = 3.34e5 J/kg, T_f = -1.8 C
limit: h ~ sqrt(2 k_i dT t / (rho_i L)) = about 7 m after 1 yr at dT = 100 K
6. Life
Photosynthesis stops the instant the light does (T = 499 s). Phytoplankton, the base of the ocean food web, turn over in roughly a week (Field et al. 1998, Science 281, 237), so ocean food webs begin to starve within days to weeks. Land plants live on stored sugars for longer, until it is cold enough to freeze them.
What is uncertain or simplified
- The temperature curves are an estimate from a toy energy-balance model, not a climate-model result. Real values depend on winds, ocean circulation, clouds, snow cover and the water vapour and CO2 that remain; the bands show only a spread of heat-capacity and mixing parameters.
- The ocean is one mixed layer plus ice; the deep ocean's stored heat and geothermal heat would keep water liquid under the ice for a very long time (the ice keeps thickening for thousands of years), far beyond the 11-year range shown.
- Orbits are circular and coplanar; start angles are illustrative; the Earth map is a stylised procedural world, not a real map.
- Body sizes in the system view are enlarged for visibility. Sun granulation, city lights and the atmosphere glow are dramatised.
- "City lights" assume electricity still flows; it would not for long.