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

NOTE 2 Other photosynthetic systems may be more optimal, using different colors of light.

not so good table

Class

Radius

Mass

Lum

Temp

Example

B

V

R

H

Planet

vesc

R/R☉

M/M☉

L/L☉

K

GY

445

551

658

1630

AU

km/s

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

not so good table

M

L

r

life

vesc

Wp

Temp

B

V

G

R

H

M⊙

L⊙

AU

GY

km/s

mag

K

445

551

658

1630

0.50

0.0625

0.2500

40.0000

42.4264

M1V

0.55

0.0915

0.3025

30.0526

40.4520

0.60

0.1296

0.3600

23.1481

38.7298

M0V

0.65

0.1785

0.4225

18.2066

37.2104

0.70

0.2401

0.4900

14.5773

35.8569

K5V

0.75

0.3164

0.5625

11.8519

34.6410

K3V

0.80

0.4096

0.6400

9.7656

33.5410

K2V

0.85

0.5220

0.7225

8.1417

32.5396

G9V

0.90

0.6561

0.8100

6.8587

31.6228

G7V

0.95

0.8145

0.9025

5.8318

30.7794

G6V

1.00

1.0000

1.0000

5.0000

30.0000

G2V

1.05

1.2155

1.1025

4.3192

29.2770

G3V

1.10

1.4641

1.2100

3.7566

28.6039

G1V

1.15

1.7490

1.3225

3.2876

27.9751

F9V

6130

1.20

2.0736

1.4400

2.8935

27.3861

F7V

1.25

2.4414

1.5625

2.5600

26.8328

F6V

6520

25 Cancri

1.30

2.8561

1.6900

2.2758

26.3117

F4V

1.35

3.3215

1.8225

2.0322

25.8199

F3V

6740

14.2

13.3

13.1

11.7

COROT-3

1.40

3.8416

1.9600

1.8222

25.3546

F1V

Twice the escape energy for a star lasting 8 times as long. A "perfect" star system for star-faring life might be a K5, with 3 times the lifetime and 20% more escape velocity.