= Long Life Star =

The heating Sun will make the (unmodified) Earth uninhabitable in less than a billion years.  How long would a lower mass star last, and for a closer planet with the same illumination as Earth, what would the system escape velocity be?

In the mass range near the suns ( 0.5 M⊙ < M < 2.0 M⊙), the luminosity is approximately L/L⊙ = (M/M⊙)⁴ .    Assuming a 5 billion year lifetime for Earth, and an escape velocity of 30 km/s, what is the orbit radius in AU and the escape velocity for an "earthlike" planet?

'''NOTE 1''' the "lifetime" should be scaled (somehow) to the amount of photosynthesis-friendly red light compared to the amount of life-scrambling heat.  Life decays with thermal energy, and builds from red photons (blue is used by terrestrial photosynthesis, but probably is not necessary).

'''NOTE 2''' Other (artificial?) photosynthetic systems may be more optimal, using different colors of light.  Earth photosynthesis evolved for the wavelengths that penetrate water.

'''NOTE 3''' Estimating Tidal effects:  Tides from the Moon plus the Earth slow the earth's rotation by 2.3 milliseconds per century.  Solar tidal effects are 46% of lunar tidal effects, so the day slows down due to the Sun alone is 7.3 microseconds/year.  That would slow the earth to a stop in 12 billion years - though the Earth will be gone before then.  Scale tidal locking to a 17 billion year tidal lock for our G2 Sun.

||  M   ||   L  ||   r  || life || vesc || Temp || 650nm || Wien || Scaled || Orbit ||  tide ||
||  M⊙  ||  L⊙  ||   AU || GY   || km/s ||    K || Power ||   nm || GY     ||  year ||  GY   ||
|| 0.60 || 0.13 || 0.36 || 23.1 || 38.7 || 4400 ||  2.29 ||  658 ||  53.0  || 0.279 ||  1.3  ||
|| 0.65 || 0.18 || 0.42 || 18.2 || 37.2 || 4600 ||  2.06 ||  631 ||  37.4  || 0.341 ||  2.0  ||
|| 0.70 || 0.24 || 0.49 || 14.6 || 35.9 || 4780 ||  1.85 ||  606 ||  26.9  || 0.410 ||  2.9  ||
|| 0.75 || 0.32 || 0.56 || 11.9 || 34.6 || 4960 ||  1.66 ||  584 ||  19.6  || 0.487 ||  4.0  ||
|| 0.80 || 0.41 || 0.64 ||  9.8 || 33.5 || 5130 ||  1.49 ||  565 ||  14.6  || 0.572 ||  5.8  ||
|| 0.85 || 0.52 || 0.72 ||  8.1 || 32.5 || 5300 ||  1.35 ||  547 ||  11.0  || 0.666 ||  7.5  ||
||<-12> ||
|| 0.86 || 0.55 || 0.74 ||  7.8 || 32.4 || 5330 ||  1.32 ||  543 ||  10.4  || 0.686 ||  8.0  || optimum K0?G9 ||
||<-12> ||
|| 0.90 || 0.66 || 0.81 ||  6.9 || 31.6 || 5460 ||  1.22 ||  530 ||   8.3  || 0.768 ||   10  ||
|| 0.95 || 0.81 || 0.90 ||  5.8 || 30.8 || 5620 ||  1.10 ||  515 ||   6.4  || 0.880 ||   13  ||
|| 1.00 || 1.00 || 1.00 ||  5.0 || 30.0 || 5780 ||  1.00 ||  501 ||   5.0  || 1.000 ||   17  ||
|| 1.05 || 1.22 || 1.10 ||  4.3 || 29.3 || 5930 ||  0.91 ||  488 ||   3.9  || 1.130 ||   22  ||
|| 1.10 || 1.46 || 1.21 ||  3.8 || 28.6 || 6080 ||  0.83 ||  477 ||   3.1  || 1.269 ||   27  ||
|| 1.15 || 1.75 || 1.32 ||  3.3 || 28.9 || 6230 ||  0.76 ||  465 ||   2.5  || 1.418 ||   34  ||
|| 1.20 || 2.07 || 1.44 ||  2.9 || 27.4 || 6370 ||  0.69 ||  455 ||   2.0  || 1.577 ||   42  ||
|| 1.25 || 2.44 || 1.56 ||  2.6 || 26.8 || 6510 ||  0.64 ||  445 ||   1.6  || 1.747 ||   52  ||
|| 1.30 || 2.86 || 1.69 ||  2.3 || 26.3 || 6650 ||  0.59 ||  436 ||   1.3  || 1.927 ||   63  ||

Twice the escape energy for a star lasting 8 times as long.  A "perfect" star system for star-faring life might be an M = 0.7 M⊙ star (K5?), with 3 times the stellar lifetime, 0.586 the tidelocking time, 20% more escape velocity.

||Class || R/R☉ || M/M☉ || L/L☉ ||  K    || Example              ||
||   F0 || 1.3  || 1.7  || 6    || 7,240 || Gamma Virginis       ||
||   F5 || 1.2  || 1.3  || 2.5  || 6,540 || Eta Arietis          ||
||   G0 || 1.05 || 1.10 || 1.26 || 5,920 || Beta Comae Berenices ||
||   G2 || 1.00 || 1.00 || 1.00 || 5,780 || Sun                  ||
||   G5 || 0.93 || 0.93 || 0.79 || 5,610 || Alpha Mensae         ||
||   K0 || 0.85 || 0.78 || 0.40 || 5,240 || 70 Ophiuchi A        ||