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 |
|
|
|||||||||||
0.86 |
0.55 |
0.74 |
7.8 |
32.4 |
5330 |
1.32 |
543 |
10.4 |
0.686 |
8.0 |
optimum K0?G9 |
|
|||||||||||
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 |