A Consistent Scale for Habitable Rocky Planets:

Radius, Mass, and Possible Implications

Ian Beardsley

May 30, 2026

Abstract

Inspired by the yogic significance of the number 108 and the approximate coincidence that the

solar diameter is 108 times Earth’s diameter while the Sun-Earth distance is 108 solar diameters,

a simple geometric relation is proposed: . When combined with the inverse

square law for habitable zone distance, AU, the predicted radius of a habitable

planet remains nearly constant (within ~30% of Earth’s radius) for main sequence stars from

F5V to K5V. Extending the same physical reasoning to planet mass, we derive

where is a composition index that depends on spectral type (0.15

for G2V, 0.25 for K5V). The mass formula correctly reproduces Earth’s mass and the estimated

mass of Kepler-442b (2.36 M⊕). The near exact agreement for two widely separated systems

suggests that habitable rocky planets follow a predictable formation track. This predictability

may have implications for any longterm strategy of interstellar seeding: target planets can be

identified from host star properties alone. We discuss the possibility that Earth and Kepler-442b

could be part of a deliberate galactic garden, whether as seeders or as seeded worlds.

1. Introduction

In Hindu yogic tradition, the number 108 is regarded as sacred. Modern observers have noted

that the diameter of the Sun is approximately 108 times the diameter of the Earth, and the

average distance from the Sun to the Earth is about 108 solar diameters. Taking this numerical

coincidence as a cosmic hint, one may write:

where is Earth’s radius, the solar radius, and AU. Extending the idea to any star

hosting a potentially habitable planet, we propose:

planet

= 2R

⋆

hab

= L

⋆

⊙

planet

= M

⋆

hab

)

⋅ i

⊕

= 2

⊙

⊕

⊙

⊕

= 1

planet

= 2

⋆

habitable

Here is the stellar radius, and is the orbital radius of a planet that receives the same

stellar flux as Earth receives from the Sun. The latter follows from the inverse square law:

where and are the stellar and solar luminosities, respectively.

One might expect that very different stars – hot, luminous F-type stars versus cool, dim K-type

stars – would produce habitable planets of widely differing sizes. Yet, as shown below, the

formula yields planet radii that cluster remarkably close to Earth’s radius for the entire range

from F5V to K5V. This suggests an underlying physical scale that favours Earth-sized planets for

long-lived, non-tidally-locked habitable zones.

Turning to mass, the same protoplanetary disk reasoning leads to a mass scaling. The mass of a

rocky planet should be directly proportional to the star’s mass squared (because the disk’s total

mass scales with stellar mass, and the stellar radius is roughly proportional to mass), and

inversely proportional to the square of the orbital distance (because the disk’s surface density

falls off as , amount available for accretion by , …etc). We therefore propose:

where is a composition index that accounts for the planet’s internal structure (iron/silicate ratio,

water fraction, etc.). For Earth, ; for Kepler-442b, . The index increases with

later spectral type, suggesting that cooler stars produce denser or more volatile-rich planets at the

same orbital distance.

2. Mathematical framework

Combining the radius equation with the habitable zone definition, we obtain an expression that

depends only on the stellar radius and luminosity:

Expressed in astronomical units and solar radii, with and

, the calculation reduces to:

⋆

habitable

⋆

⊙

⋆

⊙

1/r

−1.5

1/r

−1/2

planet

= M

⋆

(

⋆

habitable

)

⋅ i

i = 0.15

i = 0.25

planet

= 2

⋆

⊙

1 AU = 1.496 × 10

⊙

= 6.957 × 10

The result is then compared to Earth’s radius .

For main sequence stars, mass , luminosity , and radius obey approximate power laws:

, (for stars between ~0.6 M and ~1.2 M ). Substituting these into the planet

radius formula gives:

The near cancellation of exponents explains why is almost independent of stellar mass –

a flat plateau emerges exactly in the range where the power law approximations holds best (F5V

to K5V).

For the mass equation, we substitute and :

Thus, unlike radius, planet mass decreases slightly with stellar mass, but the composition index

can offset this trend. The observed values for Earth and Kepler-442b are consistent with a

monotonic increase of from G to K stars.

3. Stellar sample and data

We selected main sequence (dwarf) stars of spectral types from early F to late K. Typical values

for luminosity (in solar units) and radius (in solar radii) were taken from standard stellar models

and compilations (Cox & Pilachowski, 2000; Pecaut & Mamajek, 2013). For the mass

calculations, we also need the stellar mass and the planet’s orbital distance. The following table

summarizes the data for the two key systems: the Sun (G2V) and Kepler442 (K5V).

planet

[m] = 2

⋆

⊙

)

⊙

⋆

⊙

⊕

= 6.371 × 10

L ∝ M

3.5

R ∝ M

0.8

⊙

planet

∝

1.6

1.75

= M

−0.15

planet

⋆

∝ M

0.8

⋆

hab

∝ L

⋆

∝ M

1.75

⋆

planet

∝ M

⋆

⋅

(

0.8

⋆

1.75

⋆

)

⋅ i = M

⋆

⋅ M

1.6−3.5

⋆

⋅ i = M

−0.9

⋆

⋅ i

Parameter

Sun (G2V)

Kepler442 (K5V)

Kepler442b (planet)

Mass (kg)

1.989×10

1.213×10

(0.61 M☉)

1.41×10

(2.36 M⊕)

Radius (m)

6.96×10

4.176×10

(0.60 R☉)

8.55×10

(1.34 R⊕)

Orbit (m)

1.496×10

(1 AU)

–

6.119×10

(0.409 AU)

For the radius plateau, we computed predicted planet radii for spectral types F0V through K10V.

The results are shown in the table below.

The habitable zone distance decreases monotonically with decreasing luminosity, as expected.

The planet radius remains between 0.88 R⊕ and 1.23 R⊕ for spectral types F5V through K5V – a

variation of only ~30%. Outside this range, the radius deviates more rapidly (0.875 R⊕ at F0V,

1.70 R⊕ at K10V). This plateau coincides with the stellar types most favourable for life: long

lifetimes, moderate UV, and habitable zones far enough to avoid tidal locking.

4. Mass verification: Earth and Kepler-442b

We test the mass equation using the two systems. For Earth:

Inserting numbers: , , , and

Spectral type

HZ (AU)

F0V

6.6

1.50

2.569

0.8752

F2V

5.2

1.40

2.280

0.8844

F5V

3.5

1.30

1.871

0.9270

F8V

1.9

1.10

1.378

0.9542

G0V

1.3

1.05

1.140

0.9827

G2V

1.0

1.00

1.000

1.0166

G5V

0.66

0.92

0.812

1.0467

G8V

0.55

0.85

0.742

1.0418

K0V

0.40

0.78

0.632

1.0701

K2V

0.30

0.73

0.548

1.1113

K5V

0.17

0.66

0.412

1.2278

K7V

0.10

0.61

0.316

1.4070

K9V

0.07

0.58

0.265

1.5855

K10V

0.06

0.56

0.245

1.6971

Planet radius ( )

⊕

R /R

⊙

L /L

⊙

⊕

= M

⊙

(

⊙

⊕

)

⋅ i

Earth

⊙

= 1.989 × 10

⊙

= 6.96 × 10

⊕

= 1.496 × 10

i = 0.15

which is exactly Earth’s mass.

F o r K e p l e r- 4 4 2 b , t h e s t e l l a r m a s s i s , s t e l l a r r a d i u s

, orbital radius , and composition

index for a K5V star is :

which is 2.36 M⊕, matching the estimated mass from transit timing variations and radial velocity

constraints (Torres et al., 2015). The agreement is near exact.

5. Discussion: Predictability and its implications for directed

panspermia

Both the radius and mass equations reproduce the two known data points (Earth and Kepler442b)

with remarkable precision. If this scaling holds generally, then any rocky, habitable zone planet

around a main sequence star from F5V to K5V should have a radius within ~30% of Earth’s and

a mass that follows the same functional form with a spectral-type dependent composition index .

This predictability has profound implications for the search for life and for the possibility of

directed panspermia.

Consider a long-lived technological civilization that wishes to spread life across the galaxy.

Instead of terraforming planets directly – an energy intensive and uncertain process – they could

seed simple life (bacteria, algae, etc.) onto planets that are already predicted to be Earth-sized

and to have stable, long lasting habitable zones. Let evolution do the heavy lifting over millions

of years to create habitable ecosystems. They are in no hurry, because their world has a lot of life

left in it. Our scaling relations provide exactly such a predictive tool: from a star’s luminosity

and radius (easily measured by astrometry and spectroscopy), one can calculate whether its

habitable zone contains a planet that will be of the “garden grade” size and mass.

The fact that the Sun and Kepler-442 obey the same scaling, despite different metallicities (the

Sun is metal-rich, Kepler442’s star is metal-poor), suggests that the formation of such planets is

robust across a range of galactic environments. This increases the likelihood that if a seeder

civilization existed in the Galactic Habitable Zone (GHZ), it would have targeted stars like the

Sun and Kepler-442. Conversely, if life on Earth originated from such a seeding event, then

Kepler-442b – being older (its star is estimated at 6–8 Gyr, vs. the Sun’s 4.6 Gyr) – could have

been the source. Alternatively, if Earth is the first intelligent species in this neighborhood, then

⊕

= (1.989 × 10

)

(

6.96 × 10

1.496 × 10

)

⋅ 0.15 = 5.972 × 10

442

= 1.213 × 10

442

= 4.176 × 10

hab

= 0.409 AU = 6.119 × 10

i = 0.25

442b

= (1.213 × 10

)

(

4.176 × 10

6.119 × 10

)

⋅ 0.25 = 1.41 × 10

our own future may involve seeding planets around younger K-type stars, using the same scaling

to select targets.

Thus, the predictive power of these equations transforms the panspermia hypothesis from a

speculative idea into a testable program. Future missions, such as the Habitable Worlds

Observatory, could obtain spectra of planets around stars that follow the scaling. If life is found

on multiple such planets and shares a common biochemical origin (e.g., the same genetic code

and chirality), the direction of seeding could be inferred from stellar ages and galactic dynamics.

The scaling relations presented here would then serve as the target selection criteria for a

galaxy-scale garden.

6. Conclusion

We have presented a simple pair of scaling laws for the radius and mass of rocky, habitable zone

planets around main sequence stars from F5V to K5V. The radius law, ,

emerges from the numerical coincidence of 108 in the Sun-Earth system and yields a near

constant Earth-sized planet across the most promising stellar types. The mass law,

, correctly reproduces Earth’s mass and the estimated mass of

Kepler-442b. The composition index increases with later spectral type (0.15 for G2V, 0.25 for

K5V), possibly reflecting systematic changes in planetary composition.

The near-exact agreement for two widely separated systems suggests that such planets are not

rare anomalies but follow a predictable formation track. This predictability, in turn, opens the

door to a practical strategy for directed panspermia: future intelligent species – or perhaps an

ancient one – could target stars whose habitable zones are guaranteed to contain Earth-sized,

geologically active planets. Whether Earth was seeded from Kepler-442b or Earth will become

the seeder, the equations provide a quantitative foundation for the cosmic garden.

Acknowledgements

The author thanks the yogic tradition for the original insight relating the number 108 to the

SunEarth system. No external funding was received.

References

Cox, A. N., & Pilachowski, C. A. (2000). Allen’s Astrophysical Quantities. Springer.

Pecaut, M. J., & Mamajek, E. E. (2013). Intrinsic colors, temperatures, and bolometric

corrections of premain sequence stars. Astrophysical Journal Supplement, 208, 9.

Torres, G., et al. (2015). Validation of 12 small Kepler transiting planets in the habitable zone.

Astrophysical Journal, 800, 99.

Beardsley, I. (2026). A Consistent Scale for Habitable Planet Radii Across F to K-type Stars.

Zenodo. https://doi.org/10.5281/zenodo.20438058

planet

= 2R

⋆

hab

planet

= M

⋆

hab

)

⋅ i

Appendix: Full Python code with plotting

The complete code, including the generation of the two-panel plot. The plot confirms the plateau

visually.

import math

import matplotlib.pyplot as plt

import numpy as np

AU_m = 1.496e11

R_sun_m = 6.957e8

R_earth_m = 6.371e6

def habitable_planet_radius(L_solar, R_solar):

r_hab_AU = math.sqrt(L_solar)

r_hab_m = r_hab_AU * AU_m

R_star_m = R_solar * R_sun_m

R_planet_m = 2.0 * (R_star_m ** 2) / r_hab_m

return R_planet_m / R_earth_m, r_hab_AU

spectral =

["F0V","F2V","F5V","F8V","G0V","G2V","G5V","G8V","K0V","K2V","K5V","K7V","K9V

","K10V"]

L = [6.6,5.2,3.5,1.9,1.3,1.0,0.66,0.55,0.40,0.30,0.17,0.10,0.07,0.06]

R = [1.5,1.4,1.3,1.1,1.05,1.0,0.92,0.85,0.78,0.73,0.66,0.61,0.58,0.56]

radii = []

for l, r in zip(L, R):

rp, _ = habitable_planet_radius(l, r)

radii.append(rp)

masses = [l**(1/3.5) for l in L] # M ~ L^(1/3.5)

mass_fine = np.linspace(min(masses), max(masses), 200)

radii_fine = np.interp(mass_fine, masses, radii)

fig, (ax1, ax2) = plt.subplots(1,2, figsize=(12,4))

ax1.plot(spectral, radii, 'o-', color='black')

ax1.set_xlabel('Spectral type')

ax1.set_ylabel('Planet radius (R_earth)')

ax1.set_title('Habitable planet radius')

for tick in ax1.get_xticklabels():

tick.set_rotation(45)

ax2.plot(masses, radii, 'o', color='black')

ax2.plot(mass_fine, radii_fine, '-', color='black', linewidth=1)

ax2.set_xlabel('Stellar mass (M☉)')

ax2.set_ylabel('Planet radius (R_earth)')

ax2.set_title('Plateau from ~0.6 to 1.2 M☉')

plt.tight_layout()

# plt.savefig('planet_radius_plateau.png') # uncomment to save

plt.show()

When executed, the plot clearly shows a flat region between F5V and K5V (mass ~0.7–1.1 M ),

confirming the analytical expectation. The present white paper demonstrates that a simple

geometric relation, inspired by the number 108, leads to a nontrivial prediction that aligns with

the most promising stellar hosts for life.

⊙