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 slow down due to the Sun alone is 7.3 microseconds/year .

M

L

r

life

vesc

Temp

650nm

Wien

Scaled

Orbit

tide

M⊙

L⊙

AU

GY

km/s

K

Power

nm

GY

year

ratio

0.35

0.0150

0.1225

116.62

50.7093

3303

3.69

877

430.81

0.072

0.005

0.40

0.0256

0.1600

78.125

47.4342

3547

3.43

817

268.29

0.101

0.010

0.45

0.0410

0.2025

54.870

44.7214

3777

3.14

767

172.13

0.136

0.018

0.50

0.0625

0.2500

40.000

42.4264

3995

2.84

725

113.51

0.177

0.031

0.55

0.0915

0.3025

30.053

40.4520

4203

2.55

689

76.74

0.224

0.050

0.60

0.1296

0.3600

23.148

38.7298

4402

2.29

658

53.06

0.279

0.078

0.65

0.1785

0.4225

18.207

37.2104

4594

2.06

631

37.44

0.341

0.116

0.70

0.2401

0.4900

14.577

35.8569

4779

1.85

606

26.90

0.410

0.168

0.75

0.3164

0.5625

11.852

34.6410

4958

1.66

584

19.65

0.487

0.237

0.80

0.4096

0.6400

9.7656

33.5410

5132

1.49

565

14.57

0.572

0.328

0.85

0.5220

0.7225

8.1417

32.5396

5300

1.35

547

10.96

0.666

0.444

0.90

0.6561

0.8100

6.8587

31.6228

5464

1.22

530

8.34

0.768

0.590

0.95

0.8145

0.9025

5.8318

30.7794

5624

1.10

515

6.42

0.880

0.773

1.00

1.0000

1.0000

5.0000

30.0000

5780

1.00

501

5.00

1.000

1.000

1.05

1.2155

1.1025

4.3192

29.2770

5932

0.91

488

3.93

1.130

1.276

1.10

1.4641

1.2100

3.7566

28.6039

6081

0.83

477

3.12

1.269

1.611

1.15

1.7490

1.3225

3.2876

27.9751

6227

0.76

465

2.49

1.418

1.011

1.20

2.0736

1.4400

2.8935

27.3861

6370

0.69

455

2.01

1.577

2.488

1.25

2.4414

1.5625

2.5600

26.8328

6510

0.64

445

1.63

1.747

3.052

1.30

2.8561

1.6900

2.2758

26.3117

6648

0.59

436

1.33

1.927

3.713

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