# Multiverse Predictions for Habitability: Planetary Characteristics

^{1}

^{2}

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^{5}

^{*}

## Abstract

**:**

## 1. Introduction

**yellow**condition, that a relatively narrow energetic band is required for photosynthesis [11], which restricts the mass range of stars to be counted in ${N}_{\star}$. The

**entropy**condition, that the habitability of a planet is proportional to the entropy it receives from starlight [12], so that ${f}_{\mathrm{life}}\propto \Delta S$. The

**C/O**condition, that a carbon to oxygen ratio of close to 1 is required [13]. Furthermore, in this paper we employ the

**temp**and

**terr**conditions, which restrict our analysis to planets that are temperate (capable of retaining liquid surface water) and terrestrial (having small secondary, high molecular mean atmospheres), respectively, which further restrict our definition of ${n}_{e}$ [7]. Though these last two conditions were not found to improve agreement with observations, they significantly simplify our analysis, and will be relaxed in a future publication. Collectively, we refer to the set of these habitability factors as the

**yellow + entropy + terr + temp + C/O**condition. We parameterize stellar mass in terms of Chandrasekhar mass as $\lambda ={M}_{\star}/{M}_{\mathrm{Ch}}$. In addition, we employ the astronomical notation for characteristics of solar system bodies, or their analogues in other stellar systems: The sun (star) ⊙, Earth (temperate, terrestrial planet under consideration) ⊕, moon ☾, Jupiter (innermost gas giant planet) ♃, and Saturn (second gas giant) ♄.

## 2. Eccentricity

**yellow + entropy + terr + temp + C/O**condition:

## 3. Obliquity

#### 3.1. Why Is Obliquity So Unstable?

#### 3.2. What Sets the Size of and Distance to the Moon?

#### 3.3. Are Gentle Collisions Generic?

#### 3.4. What Sets the Moon Retention Timescale?

#### 3.5. Does a Giant Impact Phase Always Occur?

#### 3.6. Is a Large Moon Necessary for Complex Life?

## 4. Water Delivery

#### 4.1. Asteroid Injection

#### 4.2. Grand Tack

**yellow + entropy + temp + terr + C/O + grand tack**condition, we find

#### 4.3. Comets

#### 4.4. Magma Ocean

#### 4.5. How Does Habitability Depend on Water Content?

**yellow + entropy + terr + temp + C/O**condition and Table 2 for the

**nitrogen + area + Mg/Si**condition, the two conditions that give the highest probabilities found in [13].

**nitrogen + area + Mg/Si**condition, the intermediate water hypotheses of the first two rows are disfavored with an asteroid origin of Earth’s water, and the grand tack and cometary scenarios generically do best across the board. The magma ocean scenario fares best with habitability hypotheses that favor wetter planets. Additionally, we should note that some of these habitability hypotheses are taken to be proportional to powers of ${f}_{w}$ and $(1-{f}_{w})$, whereas in the original motivations these were accompanied by powers of ${A}_{\mathrm{planet}}$. In this regard, particularly the second hypothesis resembles a hard step process, which is extremely disfavored by our multiverse calculations; because they exhibit such a strong preference for large area, it is not uncommon for some of the probability values to be ∼${10}^{-15}$, similar to the results found in [12].

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**List of how various quantities depend on fundamental constants. The quantities in the left column are discussed in [11] except for ${h}_{\mathrm{mountain}}$, which is discussed in [12]. The quantities in the right column were derived in [7]. Details of the derivations, as well as original sources where applicable, can be found within these references. All prefactors have been chosen to reproduce values observed in our universe.

Quantity | Expression | Quantity | Expression |
---|---|---|---|

${\rho}_{\mathrm{rock}}$ | $0.13\phantom{\rule{0.166667em}{0ex}}{\alpha}^{3}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3}\phantom{\rule{0.166667em}{0ex}}{m}_{p}$ | ${a}_{\mathrm{ice}}$ | $297.9\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{43/30}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{2/3}}{{\alpha}^{4}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{4/3}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{1/3}}$ |

${T}_{\mathrm{mol}}$ | $0.037\phantom{\rule{0.166667em}{0ex}}\frac{{\alpha}^{2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3/2}}{{m}_{p}^{1/2}}$ | $\mathsf{\Sigma}\left(a\right)$ | $1.7\times {10}^{5}\phantom{\rule{0.166667em}{0ex}}\frac{\kappa \phantom{\rule{0.166667em}{0ex}}{\lambda}^{2/3}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{2}}{a}$ |

${M}_{\mathrm{terr}}$ | $92\phantom{\rule{0.166667em}{0ex}}\frac{{\alpha}^{3/2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3/4}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{3}}{{m}_{p}^{11/4}}$ | ${H}_{\mathrm{disk}}$ | $0.2\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{3/80}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{1/4}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{1/8}\phantom{\rule{0.166667em}{0ex}}{a}^{9/8}}{{\alpha}^{1/2}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{1/4}}$ |

${R}_{\mathrm{terr}}$ | $3.6\phantom{\rule{0.166667em}{0ex}}\frac{{M}_{pl}}{{\alpha}^{1/2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3/4}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{5/4}}$ | ${r}_{\mathrm{disk}}$ | $3.6\times {10}^{-6}\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{1/3}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}}{\kappa \phantom{\rule{0.166667em}{0ex}}{m}_{p}^{2}}$ |

${v}_{\mathrm{esc}}$ | $1.37\phantom{\rule{0.166667em}{0ex}}\frac{\alpha \phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3/4}}{{m}_{p}^{3/4}}$ | ${t}_{\mathrm{disk}}$ | $4.1\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}\frac{\lambda \phantom{\rule{0.166667em}{0ex}}{M}_{pl}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{1/4}}{{\alpha}^{3}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{9/4}}$ |

${M}_{\star}$ | $122.4\phantom{\rule{0.166667em}{0ex}}\frac{\lambda \phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{3}}{{m}_{p}^{2}}$ | ${\omega}_{\mathrm{rot}}$ | $0.02\phantom{\rule{0.166667em}{0ex}}\frac{{\alpha}^{3/2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3/2}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{1/2}}{{M}_{pl}}$ |

${R}_{\star}$ | $108.6\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{4/5}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}}{{\alpha}^{2}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{2}}$ | ${M}_{\mathrm{iso}}$ | $1.3\times {10}^{8}\phantom{\rule{0.166667em}{0ex}}\frac{{\kappa}^{3/2}\phantom{\rule{0.166667em}{0ex}}{\lambda}^{25/8}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{7/4}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{9/4}}{{\alpha}^{15/2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3}}$ |

${L}_{\star}$ | $9.7\times {10}^{-4}\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{7/2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{2}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}}{{\alpha}^{2}\phantom{\rule{0.166667em}{0ex}}{m}_{p}}$ | ${\rho}_{\mathrm{halo}}$ | $2.7\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{\kappa}^{3}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{4}$ |

${a}_{\mathrm{temp}}$ | $7.6\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{7/4}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{1/2}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{1/2}}{{\alpha}^{5}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{2}}$ | ${t}_{\mathrm{migration}}$ | ${10}^{7}\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{73/60}{m}_{p}^{1/3}{M}_{pl}^{5/6}}{{\alpha}^{4}{m}_{e}^{13/6}}$ |

${h}_{\mathrm{mountain}}$ | $0.0056\phantom{\rule{0.166667em}{0ex}}\frac{{M}_{pl}}{{\alpha}^{1/2}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{3/4}\phantom{\rule{0.166667em}{0ex}}{m}_{p}^{5/4}}$ | ${T}_{\mathrm{irr}}$ | $0.19\phantom{\rule{0.166667em}{0ex}}\frac{{\lambda}^{43/40}\phantom{\rule{0.166667em}{0ex}}{m}_{e}^{1/2}\phantom{\rule{0.166667em}{0ex}}{M}_{pl}^{1/2}}{\alpha \phantom{\rule{0.166667em}{0ex}}{m}_{p}^{3/4}\phantom{\rule{0.166667em}{0ex}}{a}^{3/4}}$ |

## Notes

1 | This frequency also depends nontrivially on the separation between Jupiter and Saturn, an effect which we ignore here. For an interesting proposal to search for observational signatures of a selection effect on this front, see [43]. |

2 | |

3 | In passing, we remark on one last scenario, that of water delivered by interstellar grains. Precluding a scenario where the feeding zone of early Earth is greatly enhanced, as in [105], this is usually not considered a viable option, because grains, and subsequently rocks, situated at 1 AU are extremely dry. However, it was suggested in [106] that these typical arguments are based off the assumption that grains are spherical, and that if instead their fractal geometry is taken into account, a much greater amount of water may adsorb onto the surface, enhancing the amount delivered. Additionally, ref. [107] find that proton irradiation on interstellar grains can form water by breaking silicate bonds. The final ratio depends on the grain size distribution, but more so on the adsorption rate, which scales as ${e}^{{E}_{\mathrm{mol}}/T}$. This factor is identical to the exponential dependence for the magma ocean scenario, where both the size and location of the Earth are dictated by ${E}_{\mathrm{mol}}$. As such, the probabilities in the grain adsorption scenario will be close, though not strictly identical to, the magma ocean scenario. |

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**Figure 1.**The effect of varying ${e}_{\mathrm{max}}$ on the probabilities of observing our constants. Above ${e}_{\mathrm{max}}=0.1$, probabilities are changed by at most $4\%$.

**Figure 2.**Ratio of frequencies for a sample of ${10}^{5}$ points. The color denotes the number of observers for a given parameter choice (relative to our values). The dashed line denotes the region of stable obliquities. The cross marks the values of the Earth-Moon system.

**Figure 3.**Dependence of the probabilities of observing our constants as a function of the fraction of disrupted planetesimals that enter the inner solar system, as opposed to becoming emplaced in the asteroid belt.

**Table 1.**Bayes factors for various combinations of water delivery mechanisms and habitability conditions. All values are the product of the probabilities of observing the five values of the constants we vary, relative to the baseline case where habitability is independent of water content. Values above $0.1$ are displayed in bold. All values are computed assuming the

**yellow + entropy + terr + temp + C/O**condition.

$\mathbb{H}$ | Asteroids | Grand Tack | Comets | Magma Ocean |
---|---|---|---|---|

${f}_{w}(1-{f}_{w})$ | 0.36 | 4.6 | 4.2 | 1.4 |

${f}_{w}^{6}{(1-{f}_{w})}^{3}$ | 0.52 | 11.0 | 3.3 | 1.4 |

${(1-{f}_{w})}^{2}$ | 0.44 | 0.067 | 1.2 | 0.12 |

${f}_{w}^{2}$ | 0.51 | 5.1 | 0.56 | 0.26 |

$\theta (0.92-{f}_{w})$ | 0.94 | 0.18 | 2.5 | 0.7 |

$\theta ({f}_{w}-0.05)$ | 0.56 | 5.1 | 1.0 | 0.97 |

**Table 2.**Same as above, but with the

**nitrogen + area + Mg/Si**condition. Values above $0.1$ are displayed in bold.

$\mathbb{H}$ | Asteroids | Grand Tack | Comets | Magma Ocean |
---|---|---|---|---|

${f}_{w}(1-{f}_{w})$ | 0.044 | 0.46 | 1.9 | 0.14 |

${f}_{w}^{6}{(1-{f}_{w})}^{3}$ | 0.055 | 0.42 | 1.6 | 0.63 |

${(1-{f}_{w})}^{2}$ | 0.82 | 0.58 | 0.41 | 0.0013 |

${f}_{w}^{2}$ | 0.24 | 1.6 | 0.77 | 0.51 |

$\theta (0.92-{f}_{w})$ | 0.93 | 0.65 | 2.1 | 0.1 |

$\theta ({f}_{w}-0.05)$ | 0.068 | 0.78 | 1.0 | 0.99 |

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Sandora, M.; Airapetian, V.; Barnes, L.; Lewis, G.F.
Multiverse Predictions for Habitability: Planetary Characteristics. *Universe* **2023**, *9*, 2.
https://doi.org/10.3390/universe9010002

**AMA Style**

Sandora M, Airapetian V, Barnes L, Lewis GF.
Multiverse Predictions for Habitability: Planetary Characteristics. *Universe*. 2023; 9(1):2.
https://doi.org/10.3390/universe9010002

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Sandora, McCullen, Vladimir Airapetian, Luke Barnes, and Geraint F. Lewis.
2023. "Multiverse Predictions for Habitability: Planetary Characteristics" *Universe* 9, no. 1: 2.
https://doi.org/10.3390/universe9010002