# A Mathematical Model of Circumstellar and Circumplanetary Habitable Zones Accounting for Multiple Heat Sources

## Abstract

The habitable zone is the range of orbital radii at which celestial body’s temperature is conducive to liquid water. The ability to predict exoplanetary habitability allows for understanding of the range of orbital parametersin which life as we know it may arise and survive. In this paper, elements of mathematical models in the literature developed to determine energy contributions by stellar radiation and tidal heating were combined, producing a Maple 18 model accounting for both circumstellar and circumplanetary habitability. Using input star, planet, and moon properties, the model outputs calculated temperatures for both the planet and moon, as well as coloured maps representing the habitable zone. Using this model, the effect on the habitable zone when tidal heating is considered in addition to stellar radiation was assessed, giving a better understanding of how these factors influence habitability. Specifically, tidal heating pushes both edges of the habitable zone outward, but overall, tends to decrease the size of the habitable zone. However, accounting for the circumplanetary habitable zonein addition to the circumstellar habitable zone expands the overall habitable zone. By synthesizing a single model from existing literature models, a more accurate prediction of exoplanetary system habitability may be obtained, allowing scientists to better determine which exoplanets are more likely to have life and target these for further study. Therefore, the model was applied to existing exoplanet data to demonstrate this application of the model in predicting exoplanet and exomoon temperatures.

### Keywords

## Introduction

The relatively new field of astrobiology is concerned with the origins, evolution, distribution, and future of life in the universe (National Aeronautics and Space Administration, 2014a). An important aspect of this field is understanding the range of conditions under which life as we know it can develop and exist (Des Marais, et al., 2008). One condition often considered is the presence of liquid water: for life as we know it, water is the universal solvent, and its polar nature grants it many properties that are useful to life (Mottl, et al., 2007). Thus, the “habitable zone” around a star is often defined as the set of orbital radii around a star that allows liquid water to exist on a celestial body (Doyle, Billingham and DeVincenzi, 1998). Traditionally, most calculations of habitable zones have only considered the amount of energy received by a planet due to its star’s illumination: while this is a major component for assessing habitability, other factors are also involved (Kaltenegger and Segura, 2011). One of the most intriguing of these additional factors is tidal heating. Tidal heating is the heat generated by the flexing of a celestial body’s surface due to the gravitational forces exerted by an orbiting body. While the heat generated by the moon’s tidal heating of the Earth’s oceans is negligible compared to the incoming stellar radiation, tidal heating is considered a major source of energy for celestial bodies like Jupiter’s moon Europa (Patiño Douce, 2011). Tidal heating by Jupiter’s gravity delivers 6TW of power to Europa, which is enough to generate what is believed to be a subsurface liquid water ocean on Europa (Greenberg, 2007).

In this paper, I produce a model that takes into account both solar illumination and tidal heating as a sources of heat for a celestial body. This model considers both planets and moons together, whereas previous studies have generally considered one or the other in isolation. Thus, this paper works to combines previous work including Heller and Barnes (2013), who consider exomoons alone, and Jackson, Barnes and Greenberg (2008), who examine only tidal heating of planets.

### Factors Affecting Habitability

In this work, the habitable zone refers to the set of regions in space around a star in which a celestial body receives enough heat energy for liquid water to exist. Thus, factors affecting habitability are the various sources of heat that a celestial body may have.

As noted above, stellar illumination is one of the greatest factors and is most commonly used to calculate habitable zones. Another factor affecting the amount of heat energy a celestial body receives is its atmosphere. The atmosphere may reflect some of the incoming stellar radiation, for example, due to clouds. This, combined with the light reflected by the body’s surface, is quantified as the albedo. However, the atmosphere can also help the planet retain the heat it absorbs through the greenhouse effect (Kaltenegger and Segura, 2011).

Besides the factors involving stellar radiation (stellar illumination, albedo, and the greenhouse effect), tidal heating is another factor that can affect habitability. Tidal heating occurs due to the flexing of a celestial body’s surface caused by gravitational attraction by an orbiting body. This heating can occur through multiple mechanisms. When the orbiting bodies are not tidally locked (they do not rotate at the same speed and so do not always present the same face to one another), the tidal bulge raised on one body by the other rotates away as the first body rotates. This causes the bulge to “travel” across the surface in order to continue facing the other body; the required deformation of material causes the release of heat. However, tidal heating can also occur for tidally locked systems. When one body is tidally locked to another, the same face is always facing the other body. Thus, the bulge does not travel across the surface and no tidal heat is generated through the first mechanism. However, if the tidally locked body’s orbit is eccentric, then at different points in its orbit, it will be at different distances to the second body. This means it will experience varying levels of the gravitational force over its orbit, so the tidal bulge will change in height at different points in the orbit. Again, this change in height requires the deformation of material, so heat is released in the process (Patiño Douce, 2011). Another tidally-‐ induced form of heating involves Rossby waves, which are oceanic waves with long wavelength and low amplitude that can transfer energy over distances (Tyler, 2008).

In addition, a celestial body can also gain heat from other sources. For example, heat can be released by the radioactive decay of radionuclides in a body’s interior (Frank, Meyer and Mojzsis, 2014). As well, there are various types of heat associated with planetary formation including heat from accretion, gravitational compression, and core formation (Horedt, 1980).

## Materials and Methods

### Factors Considered in the Model

While it would be ideal to consider all the factors described previously, this is not feasible due to the dependence of many of these factors on parameters for which data is not widely available. Thus, this model considers only stellar radiation and tidal heating. While the greenhouse effect plays a major role on planets with substantial atmospheres, it is difficult to model atmospheric effects, so this factor is neglected. Furthermore, calculations of radiogenic and primordial heat require knowledge of the physical composition of celestial bodies, and exact quantitative data on this is not easily accessible, especially for exoplanets and exomoons.

Tidal heating, as noted above, consists of various different components. Attempts were made to include all components discussed, but the mathematics behind Rossby waves were too complex to incorporate in the available timeframe. Additionally, they require the assumption of a fluid covering the surface of the celestial body, an assumption that would be in contrast to other assumptions made while calculating the other components of tidal heating, namely, the assumption of a solid, Earth-‐like surface (see the Assumptions section below).

### Mathematical Equations for the Factors

An equation quantifying the amount of heat contributed by a particular factor is necessary in order to be able to incorporate that factor into the mathematical model. In order to describe the amount of energy received by a celestial body per unit time (power), we use the equation

where is the (unitless) albedo of the planet, is the flux at the given orbital radius in , is the radius of the planet (or moon), is the Stefan-‐ Boltzmann constant, and is the temperature of the celestial body in Kelvin — the value being solved for (Seager, 2010). The equation can be understood in physical terms as describing the celestial body in a state of thermal equilibrium where the incoming heat (left side) is balanced by the heat being lost (right side). On the left side, the incoming flux is . This is multiplied by the area of the celestial body facing the star, which appears as a disc and is therefore represented by . This value can then be multiplied by the term to account for incoming energy that is not absorbed due to albedo. On the left side, the entire surface of the celestial body is assumed to radiate heat equally, so the full surface area of a sphere, , is used. This is multiplied by , which is equal to the flux leaving the planet (as stated by the Stefan-‐Boltzmann law).

To calculate the stellar flux at a given distance, a second formula is required:

where is the radius of the star, is the temperature of the star, is the semi‐major axis length in meters, and is the flux as a function of the semi-major axis length, in (Seager, 2010). This formula can be interpreted as an expression of the conservation of energy: a star radiates of power (surface area multiplied by flux ), which is conserved as it moves to greater radii, so the same amount of energy is spread out over a larger surface area, . The cancels out, and the above expression is obtained. This equation also works for stellar radiation falling on a moon if and are used instead.

There are many different formulae used to calculate the tidal heat. Here, four formulae are compared, from Heller and Barnes (2013), Henning, O’Connell and Sasselov (2009), Jackson, Barnes and Greenberg (2008), and Greenberg (2007), with additional information from Carone and Pätzold (2007) and the National Aeronautics and Space Administration (2014b, 2014c), along with the authors’ estimates of the remaining parameters (see Appendix A). It was found that the calculated tidal energy was all of the same order of magnitude, indicating that all models are fairly good. However, the model by Heller and Barnes (2013) gave the value closest to the 6TW figure calculated by Greenberg (2007) by scaling Europa’s physical and orbital parameters to those of Io, for which the tidal heating has been estimated through measurement. Thus, the model by Heller and Barnes (2013) was selected:

where G is the gravitational constant, k2,s is the second-order potential Love number (a dimensionless measure of the rigidity of a celestial body), Mp is the mass of the parent body (the planet in a moon-planet interaction or the star in a star-planet interaction), Ms is the mass of the satellite, aps is the orbital radius (semi-major axis) of the satellite about the parent body, Rs is the radius of the satellite, τs is the constant time lag factor (the duration of time lag for the bulge to reach its equilibrium position (Hut, 1981), eps is the eccentricity of the satellite’s orbit around the parent body, and E is the energy released due to tidal heating per unit time, in watts.

### Parameters Required for the Model

Based on the above equations, the parameters that the model would require are listed in the table below:

Star | Planet | Moon |
---|---|---|

Radius | Radius | Radius |

Temperature | ||

Mass | Mass | Mass |

Circumstellar semi-major axis | Circumplanetary semi-major axis | |

Eccentricity | Eccentricity | |

Second-order Love number | Second-order Love number | |

Time lag constant | Time lag constant |

Most of these parameters can easily be found for any given planet/moon in the solar system, and for several exoplanetary systems too. However, the Love number and time lag constant are based on physical properties of the bodies and are empirically determined. Thus, in this paper, they are approximated using known values from the Earth, meaning that the planets and moons are assumed to have an Earth-like composition. Similarly, the albedo of the planet is assumed to be equal to that of the Earth for the purposes of calculation, since data for this only exists for planets that can be directly observed, i.e., planets within this solar system, and not for exoplanets.

### Assumptions

As with most mathematical models, assumptions must be made in order for the system to be describable in mathematical terms. In this model, various factors are neglected due to feasibility. First of all, the greenhouse effect is neglected: this normally has a fairly large effect on the habitability of a planet. For example, Venus’ greenhouse effect makes it hundreds of Kelvin warmer than predicted by solar radiation alone (Lang, 2011). As well, in tidal heating interactions, both bodies exert tidal forces on one another (Patiño Douce, 2011), but the tidal force of the satellite on the parent body is neglected since it will be much smaller than the effect the parent body has on the satellite, unless the satellite is close in size to the parent body. As well, it is neglected that in these interactions, the eccentricity and semi-major axis length change over time because the bodies slow each other down, and conservation of angular momentum leads the bodies to increase in distance from one another over time, which would change the magnitude of the tidal force over time (Heller and Barnes, 2013). Another limitation of the model used is that it does not account for Rossby wave tidal heating, which has been calculated to contribute a large portion of heat which is orders of magnitude greater than radiogenic heating and the types of tidal heating included in this model (Tyler, 2008). Finally, it is assumed that an exomoon receives the same stellar flux as its parent body, neglecting the fact that it is likely for the parent body to block stellar radiation from reaching the satellite for some portion of the satellite’s orbit (Heller and Barnes, 2013).

### Implementation of Model and Visualization

The combined equation from above was input into the computer algebra software Maple 18 (Maplesoft, 2014). A code was then developed to associate different temperatures with different colours. In this model, the freezing and melting points of water were used as the thresholds for habitability. Temperatures greater than the boiling point of water were mapped to red colours. Temperatures from the boiling point of water to the stellar temperature were mapped linearly to the intensity of the red colour, with the most intense red colour corresponding to 1010K, and the boiling point having the least intense red colour. Likewise, temperatures from 0K to the melting point of water ice were mapped linearly to the colour blue, with the most intense blue corresponding to absolute zero. Temperatures considered habitable were green, with the most intense green corresponding to the temperature in the middle of the habitable range, with the intensity of colour decreasing linearly towards both edges.

Initially, the model was intended to superimpose a circumstellar habitable zone (accounting for stellar radiation and stellar tidal heating on planets) with the circumplanetary habitable zone (accounting for stellar radiation and planetary tidal heating of moons) for the given planetary and moon data. However, it turned out that the circumplanetary habitable zone was extremely small relative to the circumstellar habitable zone, such that except at extreme orbital eccentricities, the circumplanetary habitable zone was almost invisible (Figure 1).

Therefore, the model was redesigned to superimpose a circumstellar habitable zone for planets, based on stellar radiation and tidal heating of the planet by the star, with a circumstellar habitable zone for moons, based on stellar radiation and tidal heating of the moon by the planet, given a certain circumplanetary orbital semi-major axis of the moon.

Superimposition was accomplished by the following precedence rules, applied in the given order:

- Habitable zones
- “Hot” zones
- Planetary temperature
- Moon temperature The results of applying these rules are summarized in the table below, where the subscript letters refer to the planet or the moon being the source of the temperature used to calculate the colour.

Planet | ||||

Cold | Habitable | Hot | ||

Cold | ||||

Moon | Habitable | |||

Hot |

The complete mathematical model can be found in Appendix B.

## Results and Discussion

Although stellar illumination is the dominant factor in determining the habitable zone for planets, there are other factors involved. In this model, tidal heating is also considered: tidal heating provides additional heat to the planet, pushing the outer boundary of the habitable zone away. At the same time, the inner edge also moves outward due to the additional heat energy. What is not immediately evident is whether the orbital range of the habitable zone changes. It has been suggested that tidal heating shrinks the habitable zone (Barnes, et al., 2012). This theory is supported by the equations used in this model, as stellar radiation energy decreases with the square of distance to the star (aplanet), while tidal heating decreases with the ninth power of distance to the star (aplanet). This would suggest that the inner edge of the habitable zone should be pushed out a greater amount than the outer edge, shrinking the habitable zone. To ascertain that this was indeed the case, the equations used in the model were solved to find the orbital semi-major axis lengths where the planet temperature was equal to the freezing and melting points of water. As the results in Appendix C show, increased eccentricity, which corresponds to increased tidal heating, shrinks the habitable zone range.

As can be expected, the superimposition method used in this paper, where habitability of either the planet or moon was sufficient for an orbital radius to be considered habitable, increased the overall habitable zone. However, this effect is not very noticeable at low moon eccentricities like those that are found for most moon-planet systems in this solar system (Figure 2).

By synthesizing a model from multiple sources in the existing literature, we are able to consider a wider range of factors. This allows us to more accurately predict the habitability of an exoplanet (and potential exomoon) using data available through today’s technologies. To demonstrate this, the model is applied to the Sun/Jupiter/Europa system, as well as the Kepler-22/Kepler-22b system (no exomoons have yet been detected due to limitations of technology). Figure 3 shows that the Jupiter-Europa system falls outside of the habitable zone: a temperature of 110.65K is calculated for Jupiter, and a temperature of 111.26K is calculated for Europa. These temperatures are far below the freezing point of water at atmospheric pressure, indicating that the presence of liquid water is unlikely.

For Kepler-22b (Figure 4), the calculated temperature is 295.84K, indicating it could be habitable.

The calculated temperature for Europa is much colder than the freezing point of water, despite evidence suggesting that Europa has a liquid water ocean. Likewise, when the model is run on the Earth, it predicts a temperature on Earth lower than the freezing point of water (see Appendix B). These results are indicative of the simplifications made in this model. For instance, Rossby waves are expected to contribute greatly to Europa’s heat, while the Earth is kept warmer than predicted due to the greenhouse effect.

## Conclusion

In this paper, a model of circumstellar planetary and moon habitable zones was produced by combining aspects of previous work in the literature. The resulting model accounts for both stellar radiation and tidal heating when calculating the habitable zone around a star for both a planet and for a moon orbiting the planet at a given circumplanetary radius. Through the calculations used in the model, it was shown that the addition of tidal heating shrinks habitable zones compared to those calculated by models that account only for stellar radiation. However, because moons may be tidally heated to temperatures allowing liquid water at circumstellar radii where water would normally be frozen, the model also demonstrates that the habitable zone may extend further than previously thought. However, the model bears some weaknesses due to several simplifying assumptions made to avoid factors that are difficult to calculate using available astronomical data. Nevertheless, this model is a good starting place for gaining a more accurate understanding of the habitable zone around a star. Future work could build on this by adding more factors into the calculations.

## Acknowledgements

The author would like to acknowledge the supervision of Dr. George Dragomir (McMaster University), and the assistance provided throughout this project.

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