1 Abstract
TABThis investigation attempts to examine the habitability of Mars should scientists consider landing a settlement on Mars in 2033.
As living conditions on a planet are contingent on the geology and
resources available, many factors play an integral role in determin-ing whether Mars can be a “second home” to the human race. In the scope of this investigation, Mars’ habitability is examined against two criteria: the surface temperature of Mars and the rate of hydrogen escape. To accomplish this goal, this paper is structured into four sections: first, a descriptive synthesis of existing research into planetary habitability is performed to discover all criteria which make a planet habitable. Second, the equation of Mars’ orbit is computed to locate Mars’ apoastron and periastron. Third, the surface temperature of Mars at each apsis is calculated to provide a range in which this temperature varies per orbit. The maximum rate of hydrogen escape is also calculated to predict the number of years left before the depletion of all atmospheric hydrogen. At last, these results are assessed against the habitability criteria to conclude that Mars offers a habitable environment in the short-term, but it may not be a suitable candidate for a long-term human settlement such as the one planned in 2033 due to the exposure to HZE-charged particles and predicted depletion of atmospheric hydrogen after 383.72 years.
2 Introduction
TABIn the last century, Mars has been the target of many unmanned spacecrafts to unravel its potential for extraterrestrial life. The missions to Mars interspersed throughout the years have led to an enormously deeper understanding of the biosignatures of the planet.
TABDue to similar lengths of days and patterns in seasonal changes comparable to that of the Earth and the resembling elliptic orbit which brings about a similar rotational period and tilts in the rotational axis, Mars creates an environment that is comparatively suitable for the upbringing and survival of primary species. Although liquid water is in lack due to Mars’ low atmospheric pressure, water can be found in polar ice caps and underground repositories – which, if melted, surmount to a volume sufficient to support life. If these facts prove to be true, Mars could offer the human race a possible environment to survive should issues such as overpopulation and climate change exacerbate.
TABHowever, other facets of Mars may diminish this twinkling hope. As the opponents of Martian inhabitation highlight, Mars is a terrestrial planet with a thin atmosphere and a surface marred by impact craters. These facts alone may expose the lifeforms existent to the impact of celestial bodies, not to mention the possibility of suffering from hypoxia should oxygen continue to be entrained with hydrogen escape. Nevertheless, the excitement toward establishing a second colony in the universe surmounts the skepticism directed toward Mars’ possible inhabitable nature. In year 2033, for example, NASA is planning its first manned mission to establish a human settlement on Mars.
TABMany studies have hypothesized the possible ramifications of this mission. However, the possibility of inhabitation has never been closely examined in consideration of Mars’ orbit in the solar system. As Mars is located at a closer orbit to the sun, the average apparent brightness of the sun supersedes that of the Earth, which might render inhabitants of Mars under exposure of higher spectral intensities of ultraviolet (UV) radiation. The length and shape of each orbit (i.e., orbital eccentricity e) also changes as a function of angle from the polar axis (Figure 6), suggesting the different locations of the perihelion and aphelion may also give rise to different fluctuation patterns of temperature and flux density at Mars’ surface.
3 An Overview of Planetary Habitability
TABThe study of planetary habitability first commenced in the late 1900s by the work of Williams and Kasting, who introduced models and specific parameters against which habitability can be quantified (Kasting et al., 1993). The radiative-convective model, for example, explicitly correlated habitability with profiles of atmospheric
temperature. Atmospheric temperature, then, can be interpreted for its net energy by taking account of all solar radiation, emission of radiation, and the planet’s energy absorption and potential greenhouse effects. This model, along with the energy balance model (Williams et al., 1997; a model which designates the planetary atmosphere as a site of thermal equilibrium, such that the received and radiated stellar flux are equal, resulting in spatially uniform temperature), served as a jumping-off point for the study of habitability, which led to the advent of new theories that resulted in major advances in the field.
3.1 Mean Flux Approximation
TABOne prominent theory that emerged from this definition is the mean flux approximation, which states that a planet can be called habitable if it receives on average a flux compatible with the presence of surface liquid water.
TABPlanets which fulfill this criterion of habitability share the characteristic of lying within the “insolation habitable zone” (HZ) (Schwieterman, E., 2021). The HZ, primordially posited by Kasting et al. in 1993, denotes a zone in which solar radiation facilitates the sustenance of surface liquid water. Most planets lying in the HZ are gaseous and massive; however, there exists a vital few which are possibly rocky, whose radii are smaller than 2R⊕ and eccentricity measure ~ 0.1 (Méndez, A., 2021). Unlike the gaseous planets with large orbital eccentricities whose surface liquid water predisposes to evaporation and condensation at the periastron and apoastron, these smaller planets sustain surface liquid water on the dayside with an ice cap on the nightside (Kopparapu et al., 2019).
TABIn addition to the approximation, a direct correlation was observed between the mass of stars and temperature of its orbiting planets. As stellar mass is directly correlated with its luminosity, and luminosity is directly correlated with its surface temperature by the Stefan-Boltzmann law*, the stars with lower masses have lower surface temperatures, which translate to a “redder” spectrum by Wien’s displacement law. As the wavelengths emitted are larger, the albedo of the ice and snow decreases (Forget et al., 2012; von Paris et al., 2012) such that radiation of the sun would not be able to drive the planets into a glaciation state. Because of this,
the planets reflect less of the radiation; thus, the atmosphere absorbs more incoming flux, contributing to a higher surface temperature.
TABIn the planets that fulfill all the aforementioned criteria of habitability — lying with the HZ, low eccentricity, revolving around a sun with low luminosity — habitability is also strongly contingent on the abundance of minerals in the planet’s silicate mantle (Claudi and Alei, 2019). As life only emerges when water comes in direct contact with elements such as phosphorus, sulfur, iron, magnesium and nitrogen (Pinotti et al., 2003.), the planets with a high-pressure ice layer between the liquid water ocean and the silicate mantle (i.e., the planets characterized by larger orbital eccentricities) do not offer optimal environments for the emergence* of life, as the layer impedes such a contact (unless it is by solid convection). On the other hand, planets with a small mass fraction of water (1-2%) is more probable to be a harbor of life (Ronco et al., 2015).
TABThe mass of surface liquid water, thus, needs to be large in order for a planet to be habitable. As water only occupies the liquid state within a set range of temperatures, the flux received at the planetary surface also needs to reside within a narrow range to ensure that water neither evaporates or condenses at the periastron and apoastron. In reference to Kasting et al’s 1993 study, we can thereby also conclude that the desirable radii of planets must be relatively small (≤2R⊕) to maximize habitability (Kasting et al., 1993).
3.2 Atmospheric Composition as Determined by Jeans Escape
TABIsotope fractionation in the Martian atmosphere serve as an important evidence of hydrogen escape (Hunten et al., 1974). It is predicted that there might have existed an episode of hydrodynamic escape in early Mars that have resulted in the depletion of noble gases in the atmosphere (Stone et al., 2020). Recent evidence also highlights the continuous thinning of the Martian atmosphere (ESA Science & Technology - The Martian atmosphere, 2021), suggesting that such escapes are not a mere event of the past, but are events of the present and the future as well.
TABHydrogen escape poses significant risks to the inhabitation of any species, as hydrogen outflow also causes the entrainment of heavier elements in the atmosphere (oxygen, nitrogen, argon, water vapor). Since oxygen is indispensable for respiration (hence the survival of a species), its atmospheric depletion would indubitably render a planet inhabitable. Although the Earth is also subject to hydrogen escape, its rate is very limited as compared to that on Mars by energy in the exobase. As atmospheric escape (in the case of Jean escape) only occurs when a molecule reaches a “critical velocity” such that its kinetic energy overcomes the planet’s gravitational pull, the rate of escape is limited by the strong gravitational pull exerted by the Earth (Hunten et al., 1974). Another factor that limits the rate of atmospheric escape on Earth is the number of molecules diffused toward the upper atmosphere. As the atmosphere below the homopause is characterized by collisions of hydrogen atoms with other molecules, and the concentration of these molecules is inversely proportional to the planet’s gravitational pull, the Earth’s atmosphere right below the homopause is populated with less air molecules, which results in lower frequencies of collisions, hence a lower rate of diffusion toward the exobase (Hunten et al., 1974).
TABMars, on the other hand, has a thinner atmosphere, a smaller mass, hence a smaller gravitational pull on its air particles. This causes the ascension of hydrogen atoms into the homopause, which escape upon reaching a velocity equivalent to or greater than the critical velocity (Stone et al., 2020).
TABAccording to Sengupta, 2016, the maximum rate of hydrogen escape (ξ) is provided by the equation (Sengupta, 2016),
(3.1)
TABWhere g is the surface gravity of the planet, b is the binary diffusion coefficient for the escape of hydrogen as well as heavier species (=4.8 × 1017(T/K)0.75 cm−1s−1), Mp is the mass of the planet, mH is the mass of hydrogen atom, k0 is the thermal conductivity of air, k is the Boltzmann’s constant, mo is the mass of oxygen atom (=16mH), XO and XH are the molar mixing ratios at the exobase for oxygen and hydrogen.
TABAs evident by the complexity of the equation, a full appreciation of its derivation process cannot be achieved in this present investigation. For an in-depth overview of the derivation of this formula, refer to Sujan Sengupta’s 2016 work in the reference section.
TABIn the scope of this investigation, we are only concerned with the maximum rate of hydrogen escape at Mars’ exobase. Hence, we will only need to utilize this formula and insert appropriate values to calculate such a result.
3.3 Galactic Cosmic Rays and HZE-Charged Particles
TABA vital criterion for a planet to be considered habitable is the good health of its inhabitants. As Mars has a thin atmosphere, its surface is invariably exposed to higher-than-Earth doses of Galactic Cosmic Rays (GCR). As GCR is the result of supernovae, it is emitted in the form of highly energetic heavy-ions also known as HZE charged particles. These HZE-charged particles are usually harmless on Earth’s surface but can constitute a health hazard when encountered in the upper atmosphere. This is because of the attenuation effects of the atmosphere, which reduces the received HZE flux at the planetary surface.
TABThe Martian atmosphere, however, does not possess this inherent thickness that provide the attenuation required to reduce HZE flux to an extent that does not cause harm. Therefore, its surface is invariably exposed to higher quantities of HZE-charged particles, which constitute a major hazard to any inhabitants at its surface.
TABUsually, manned mars missions can avoid such exposure by allocating the mission to a limited time window, in which solar activity is a minimum. However, as the 2033 mission involves establishing a human settlement which is planned to last forever (in an unspecified time window), the population on Mars will undeniably come in contact with harmful fluxes of HZE-charged particles during high solar activity periods.
TABIn a 2019 study, Kamsali et al. generated predictions up to the end of 2031 of the HZE fluxes perceived at Mars’ surface in different lengths of solar cycles (solar cycle 24* and 25*). By utilizing the Fourier and Wavelet transform analyses of the long duration annual sunspot data, Kamsali et al. determined explicit time windows in which HZE flux is a minimum and maximum. Their results provided insight to when missions to Mars may be best arranged, but in the case of human settlements, their study did not provide a conclusive statement as to whether exposure to HZE fluxes in such long-term voyages will promise the health of its inhabitants.
TABNonetheless, it was indicated in their research that protection measures, such as spacecraft shielding, spacesuit, and related safeguards should constitute a major concern should any mission be prolonged.
4 Calculating the Orbit of Mars
4.1 The Heliocentric Orbit in a Simple Model
TABThe planetary system governing the physical relationships between mars and its heliocentric orbit is similar to that of an individual particle, with point mass m, subject to a centripetal force F whose magnitude |F| is inversely proportional to the radius of m from a fix point O. To simplify the problem and to reduce the redundant factors usually entailed in computations for planetary motion (e.g., mass of Mars, orbital eccentricity, etc.), this section will commence by ascribing an orbital function to a simple particle whose laws can be extended to a celestial body such as Mars. For such a particle, three properties can be assigned to its orbital motion:
TABProperty A – the particle moves in a plane.
TABProperty B – the particle satisfies the law of conservation of momentum.
TABProperty C – the particle’s orbit sweeps out equal areas in equal times.
TABThe force governing the particle’s motion, instead of having to consider the multiple vectors acting on it at the same time (in that case, we would need to compute for the respective forces and a sum that expresses the system’s net force), we may consider the dynamics of the system to be re represented by a single vector-point function F, which provides the magnitude and direction of gravitational force acting on the particle at any single point in the gravitational field. Hence, due to the uniformity of field strength and the predictability of the model by which this field can be constructed (i.e., all other factors of influence are controlled, with only independent variables considered), particles present at all points within this field follow the conversation of energy, which enables its orbit to be determined with relative ease.
TABIn effort to determine the orbit of particles in a gravitational field, there are a sequence of necessary computations. First, radial (i.e., the vector that points in the centrifugal direction from P away from O) and radial-perpendicular vector (i.e., the vector perpendicular to the radial vector, this may be envisioned as the tangent of the particle’s motion) components of a particle’s motion through a gravitational field need to be determined in order to set the foundation for the determination of the orbital path of the particle. There are two methods by which this first step can be approached. The first involves the use of a rectangular coordinate (x, y) system, whereas the second involves a polar coordinate (r, θ) system. This section will be devoted to the derivation of the orbital equation by each of these coordinate systems.
4.1.1 Applying a Rectangular Coordinate System
TABIn rectangular coordinates, the position of a particle can be written in the form P(x, y). F is composed of two components respective to the x- and y- directions, denoted Fx and Fy.
(4.1.1)
TABBy Newton’s law of universal gravitation, the gravitational force, F, is defined as:
(4.1.2)
TABAs the primary body, M, in both cases considered (the fix point O in this simple model or the sun in the actual orbit of Mars), is uniform, it may be treated as a control variable with gravitational constant G, by assigning it an appropriate substitution μ, the gravitational parameter.
(4.1.3)
(4.1.4)
TABIn which μ essentially expresses the product of G and M.
TABIn this scenario, the next step requires the substitution of x- and y- values into the equation to find an expression in terms of each vector component:
(4.1.5)
(4.1.6)
TABThus, in combination to the x and y components given in (4.1.1), the horizontal and vertical components of the gravitational force yield the following expressions:
(4.1.7)
(4.1.8)
TABDue to the complexity of variables in this expression, it would be effortful to compute for individual vector components (observe, each component of force possesses three variables: θ, x and y; unlike we able to simplify these variables into only two variables in polar coordinates by a useful substitution of x and y by r, we are unable to perform such a substitution here). Hence, the rectangular coordinate system does not offer a convenient approach to this problem.
TAB(Note that these respective force components is not necessary for this section’s computations. I have only chosen this as an example to illustrate the complexity of using Cartesian coordinates to compute the components of a vector.)
4.1.2 Applying the Polar Coordinate System
TABPolar coordinates, on the other hand, offers a simple approach. Based on the x-y system of Cartesian coordinates, a polar coordinate system introduces the notion of a vector r, which subtends from a fix point O – the pole – at an angle (θ) from the x-axis – the polar axis.
TABr measures the distance from the pole to point P, whose coordinates are defined by two variables: r and θ, written as the coordinate (r, θ). The convention by which θ is measured is in the direction counterclockwise from the polar axis.
TABSimilar to how the base of a right angle is
related to its hypotenuse, the vector r is
related to x and y by Pythagoras’ theorem:
(4.1.9)
TABApplying a simple trigonometric ratio, we
can express x and y as the cosine and sine
function of r, such that,
(4.1.10)
Figure 2. The polar coordinate system.
TABWith the inclusion of vector r, the polar coordinate system enables x and y to be expressed in similar trigonometric ratios of the angle θ. Different from how we need to compute for separate x and y components of F using Cartesian coordinates, polar coordinates allow such computations to be achieved fairly easily. The application of a polar coordinate system relinquishes the need to compute for the x and y components (of gravitational force; also, of the position, velocity and acceleration of the particle in later computations) by the inclusion of three variables—θ, x and y. Instead of requiring three, it imposes variations (trigonometric ratios) within two— r, θ—which simplifies the computations by manifolds (you may consider the following derivations of r and θ components of velocity and acceleration as paragons of this simplicity).
TABConsider the following velocity components:
(4.1.11)
(4.1.12)
TABNote that, in such a polar coordinate-scenario, instead of “substituting” the notion of the x-coordinate as r and th y-coordinate as θ, we need to venture a bit further to elaborate on these concepts of r and θ. In any case involving the r and θ components of a vector, the r component is referred to as the “radial” component (i.e., it is parallel to the radius, or centrifugally, away the fix point O), and the θ component is referred to as the “radial-perpendicular” component (i.e., the component is runs perpendicularly to the radial component).
Figure 3. Radial and radial-perpendicular (tangential) components of velocity in polar coordinates, and x- and y- components of velocity in Cartesian coordinates.
TABIf we apply the same components to acceleration,
(4.1.13)
(4.1.14)
TAB*Note: the velocity and acceleration components are differentiated in respect to dt (i.e., both parameters are functions of time, such that velocity is the first-order derivative and acceleration is the second-order derivative of displacement P(r, θ) in relation to time). This is because the system we are dealing with concerns that of orbital motion, such that we cannot aim to determine the velocity and acceleration in a duration of time on average, but an instantaneous value that pertains only to one point in time. Thus, to fulfill such a goal, we differentiate in respect to time.
TABThese formulas (4.1.11) (4.1.12) for velocity and (4.1.13) (4.1.14) acceleration hold for every value of θ. However, these formulas will not be easy to compute for real values of velocity and acceleration at P(r, θ) as the orbit described by these relations is not continuous. A continuous function would be necessary as the orbit of a
particle is a smooth conic curve, not a polygon. Hence, a limit is applied to minimize θ at individual points on the orbit. Therefore, (4.1.11), (4.1.12), (4.1.13), (4.1.14) for a smooth orbit become:
(4.1.17)
(4.1.18)
(4.1.19)
(4.1.20)
TABWhere,
(4.1.21)
4.2 Elaborating the Model by Incorporating r, θ Components of Gravitational Force
TABUntil this step, we have applied the notion of a particle in orbital motion around a fix point O. The use of a particle with point mass offers a simplistic approach to determining these respective vector components, however, if such a model is applied to further computations for the equation of its free orbit around O, parameters such as the particle’s mass and force exerted by the gravitational field will need to be incorporated into the calculation.
TABTo begin this computation, consider Newton’s second law,
(4.2.1)
TABIn a polar coordinate system, the vector-point function can be defined as the sum of the vector components and , the components of force pointing in the radial and radial-perpendicular directions at P(r, θ), which can be individually expressed as:
(4.2.2)
TABAs the particle orbits around O, 𝐹% points in the direction of the tangential velocity (derivative of the displacement function at P(r, θ)). As the particle remains in orbit, its radial acceleration 𝑎θ equals zero. Hence,
(4.2.3)
TABMultiplying both sides of the equation by r,
(4.2.4)
TABAs the derivative of equals zero, should equal an arbitrary constant, h.
(4.2.5)
TABThe constant, h, specifies the specific angular momentum of the particle travelling in the orbit (this conclusion is concurrent with Property A stated earlier, which postulates that a particle in orbit about a fix point O satisfies the law of conservation of angular moentum.) Therefore, an expression for may be written by dividing both sides by :
(4.2.6)
TAB points radially from P(r, θ) to O. Because the only component of force perpendicular to this radial direction is the gravitational force, in which the radial acceleration ( ) is the centripetal acceleration ( ), will be equivalent to the gravitational force exerted on the particle by O.
(4.2.7)
TABThe point mass m of the particle can be eliminated from both sides of the equation. Substituting GM by 𝜇:
(4.2.8)
TABSubstituting (4.1.19) for and (4.2.6) for gives:
(4.2.9)
(4.2.10)
4.3 Solving the Differential Equation for r
TABUp until this point, we have formulated two equations which express and 𝑟̈ in terms of r and constants h and μ. In order to derive an equation for the particle’s free orbit around fix point O, we need to find an expression for r. (4.2.10) is a second order nonlinear differential equation of r and its second-order derivative 𝑟̈. To solve this equation, we will first introduce a substitution to simplify (1.2.6) and (1.2.10) by eliminating the denominator:
(4.3.1)
TABHence,
(4.3.2)
(4.3.3)
TABApplying substitutions of the first part of (4.1.21) and (4.3.1):
(4.3.4)
TABSubstituting (4.3.3) into (4.3.4),
(4.3.5)
TABDifferentiating for 𝑟̈,
(4.3.6)
TABSubstituting (4.3.2) into (4.3.6),
(4.3.7)
TABAs (4.3.3) and (4.3.7) both express 𝑟̈ in terms of u, the expressions can be combined and simplified:
(4.3.8)
TABUsing Lagrange’s notations:
(4.3.9)
TABTo generalize the solution to this differential equation, we will need to solve two components of the equation, its complementary solution and homogeneous solution. First, observe that the 𝑢 + is equivalent to a constant. As the second derivative of any variable will be reduced by a power of 2, if , is equivalent to a constant,
(4.3.9) can be presented as the following:
(4.3.10)
TABThis equation indicates that the sum of the derivative and derivative is equivalent to a constant, TABTB. Hence, deducing from the decrease in the power of u as the n increases, (4.3.10) suggests:
(4.3.11)
TABHence,
(4.3.12)
TABThis solves the homogeneous part of the differential equation. To solve for the complementary part, we will need to identify an appropriate substitution for 𝑢 + = 0:
(4.3.13)
TABBoth solutions satisfy the differential equation. Therefore:
(4.3.14)
TABWhere C is an arbitrary constant. Hence, if we combine (4.3.12) and (4.3.14), the solution to the differential equation (1.1.2.29) is:
(4.3.15)
4.4 Adjusting the General Solution of the Differential Equation for the Formula of the Free Orbit
TAB(4.3.15) may be simplified by the following sequence of computations:
(4.4.1)
(4.4.2)
(4.4.3)
(4.4.4)
TAB*Note, the incorporation of both C1 and C2 isn’t absolutely necessary as all C, C1 and C2 represent arbitrary constants, i.e., C = C1 + C2. It is only for the coherency of computations that C1 is converted to C2 at the factorization of .
TABWhere C1, C2 and 𝜃0 are arbitrary constants. Applying a trigonometric substitution for 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜃0 + 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜃0:
(4.4.5)
TABSubstituting u=1/r:
(4.4.6)
TABLet 𝜃0 = 0 and substitute C2 for orbital eccentricity e, we obtain:
(4.4.7)
TABThis gives the equation of the free orbit of a point mass in a gravitational field.
4.5 Substituting h, μ and e Values for Mars’ Orbit
4.5.1 Properties of an Orbit
TABIt is fascinating to observe how the universe, despite its complexity, possesses an underlying uniformity that can be unraveled through observation and the subsequent identification of a pattern. The calculations preceding this section have been toilsome in an effort to identify such a pattern (1.4.7) for all planetary orbits, but as this has been completed, the rest toward finding Mars’ orbit would be simple – since having found the pattern, we only need to adjust the equation for its constants pertaining to Mars’ specific parameters.
TABIn section 1.1 and 1.4, we have briefly introduced the following constants: (a) specific angular momentum h, (b) gravitational parameter μ, and (c) orbital eccentricity e. In regard to specific angular momentum, we have mentioned the property such that all particles exhibiting orbital motion adhere to the law of conversation of
Figure 4. An elliptical orbit (e < 0) of a secondary body m around a central body M. The major axis extends from one apsis (periastron or perigee) to the other (apoastron or apogee). The major axis is the perpendicular bisector of the major axis and extends to meet the orbit at its foci, which intersection points are the widest points of the perimeter. The semi-latus rectum is a chord through the focus to the orbit which runs parallel to the semimajor axis and perpendicular to the major axis (Mathworld.wolfram.com, 2021.), which magnitude is also equivalent to orbital eccentricity e.
angular momentum (Property A, section 1.1), and subsequently generated a brief proof for this property in (1.2.5). For the gravitational parameter, we have only also gone as far as to defining it as a product of the gravitational constant G and mass of the central body M; the same applies to orbital eccentricity e, not much has been explained in respect to its particularity and the role that it plays in the equation.
TABAs these three constants not only comprise an integral part of (1.4.7) but also provide a unique description of a secondary body’s orbit, they are pivotal components in the computation of Mars’ orbit, since, taken literally, they allow for the differentiation between the planetary orbits of each secondary body by its uni-que parameters.
4.5.2 Orbital Eccentricity, Specific Angular Momentum, and Gravitational Parameter
Orbital eccentricity e: the orbital eccentricity is defined as the dimensionless quantity which value corresponds to the length of the semi-latus rectum. As shown in Diagram 1.5.2, variations in its value are straightly reflected in the degree by which the orbit deviates from a perfect circle.
Specific angular momentum, h: the specific angular momentum measures the angular momentum of a body per unit mass (Zipcon.net. 2021). Hence, suppose angular momentum may be denoted by 𝛚, then h may be defined accordingly as 𝛚/m, where m is the mass of the secondary body.
Gravitational parameter, μ: the gravitational parameter is a product of gravitational constant G, which is equivalent to 6.67408 × 10-11 m3 kg-1 s-2, and M, the mass of the primary body. As our calculation concerns only the orbit of Mars which revolves around the sun (Marspedia. 2021), the primary body will be the sun, whose mass is given by 1.989 × 1030 kg
TABThese three parameters are treated as constants due to the relative non-variance in their numerical values. For example, the absolute values* of spin angular momentum and orbital angular momentum – two important invariants in the orbital motion of a celestial body – change with a period that range to as long as 106 years. As most problems involving these parameters are not likely to stretch over long periods as such, slight fluctuations in their values will not effectuate significant influence over the data collected to an extent such that the effects are nonnegligible.
Figure 5. Orbit eccentricity (0≤e<1) of five orbits with identical major axes.
TABErgo, the values of these parameters can be obtained directly from any reliable source. Below are the values collected for each parameter from NASA’s database (Mars Fact Sheet, 2020):
TAB• Orbital eccentricity of mars, e♂ = 0.0934.
TAB• Specific angular momentum of mars, h♂ = 5.48 × .
TAB• Gravitational parameter (for a heliocentric orbit*), μ⊙=(1.32712440042 ± ) ×
TAB*The uncertainty inherent in the gravitational parameter—“± 10-1”—corresponds to the uncertainty in the measurement of sun’s mass. However, because of its small magnitude, the value is negligible.
(4.5.2.1)
(4.5.2.2)
(4.5.2.3)
4.5.3 Optimization to Locate the Apoastron and Periastron
TABThese values are of particular significance to the calculation of the distance between the sun and the periastron and apoastron in Mars’ orbit. However, as this calculation requires knowledge of the positions at which the periastron and apoastron reside on the orbit, we need to first perform an optimization of equation (1.4.7) to determine the angles (θ) at which r is respectively a maximum or minimum, which may be computed by differentiating (1.4.7):
(4.5.3.1)
TABThe local minimum and maximum points of a function occurs at values of θ at which the derivative is zero, hence, by equating (4.5.3.1) to 0, we find the critical points of the distance function:
(4.5.3.2)
(4.5.3.3)
TABNotice that the orbit only consists of one revolution around the fix point O. Hence, if the domain of θ is restricted to 2𝜋,
(3.5.3.4)
TABThere are three critical points on the function. However, note that θ = 0, θ = 2𝜋 occupies the same point on one orbit. Hence, the two distinct critical points on one orbit is θ1 = 0, θ2 = 𝜋.
TABBy 1.5.1, these critical points may be determined directly by observing that the periastron and apoastron occupy both ends of the major axis. Consequently, because the major axis is parallel to the polar axis (refer to Diagram 1.2.1) where we designate θ =0, it will be reasonable to conclude from the diagrams that 0 and 𝜋 are the only critical points. In such a case, some may even take it further as to make an assumption for which value of θ constitutes a maximum and which constitutes a minimum. However, this further assumption must be predicated on another assumption of where we intend to designate the focus (or fix point O) of the orbit to be, but then again, as the focus can take two positions and neither is directly implied through any known and established principle, we will need to find the respective r values for each critical point to consolidate any assumption on the
minimum/maximum nature of the two points.
TABTo compute the maximum and minimum distances, we will need the values to the constants, e♂, h♂, and μ⊙ in (4.5.2.1), (4.5.2.2), (4.5.2.3). Inserting these values into equation (4.4.7),
(4.5.3.5)
TABThis will specify the formula for the free orbit of Mars around the sun. And because,
(4.5.3.6)
TABFurther computations of the distance at each critical point (θ1=0, θ2= 𝜋) yield:
(4.5.3.7)
(4.5.3.8)
TABNote that 2.06952374 × m < 2.49593786 × m. Therefore, having established that apoastron measures the farthest distance of Mars from the sun, and vice versa, we may conclude the following:
Apoastron =𝜃1= 𝜃a = 2.06952374 × m
Periastron = 𝜃2= 𝜃p = 2.49593786 × m
TABIf the distances are to be presented using the astronomical unit (AU):
1AU=1.495978707 × m
Apoastron = 1.668431405AU ≈ 1.67AU
Periastron = 1.383391176AU ≈ 1.38AU
TABTherefore, in Cartesian coordinates, we find that Mars resides at (𝜋, 1.67AU) at the apoastron and (0, 1.38AU) at the periastron.
(4.5.3.9)
(4.5.3.10)
(4.5.3.11)
(4.5.3.12)
(4.5.3.13)
5 Assessing the Habitability of Mars
5.1 Habitability Criteria of Mars
TABIn section 2, we defined habitability as a planet’s ability to hold standing water within one orbit around its sun, including the extrema—the apoastron and periastron, at which water must neither condense nor evaporate, that is, by remaining in a liquid state. We elaborated on the basis of this definition by introducing two criteria that a star must meet in order to harbor life, namely:
TAB1. The stellar flux which it receives at the planetary surface must remain within a narrow range, with a mean
flux similar to that of the surface of the Earth.
TAB2. (Corollary to 1) The surface temperature must also fluctuate within a narrow range, such that that the
extrema in temperature neither exceed nor go below the freezing and evaporation points of water.
TAB*Provided that the freezing point of water is 0°C and the evaporation point of water is 100°C (note that water evaporates at any temperature above its freezing point; hence, the “evaporation point” refers to the boiling point), we may specify this appropriate temperature range to be (0°C, 100°C).
TABTo fulfill these criteria, we thereby concluded a set of conditions that must be fulfilled should a planet be considered habitable:
TAB1. The planet must be located within the insolation habitable zone of its star.
TAB2. With a radius ≤2R⊕.
TAB3. With an orbital eccentricity approximately equal to 0.1.
TAB4. The planet must revolve around a star whose stellar luminosity measure within the range [ L⊙, 1L⊙],
with the lower end being more desirable.
TAB5. (Corollary to 4) the star must possess a “redder” spectrum.
TABIn addition to these conditions, we also considered hydrogen escape, in which the rate of hydrogen escape must not exceed a certain critical value above which heavier elements (e.g., oxygen) are entrained with the outflow of hydrogen. This comprises another condition necessary for the existence of life (condition 6) on a
planet.
TABHowever, it is important to note that an environment facilitative for the emergence of life – although it will also be important for the survival of existent life – will not be strictly required to support the lifeforms that are already existent (see Figure 6).
Figure 6. Let B represent the set of conditions
necessary for the emergence of new life, and A
represent the set of conditions that support the
survival of existent life. Then, A⊆B, i.e., the conditions that make the apparition of new life
possible exceed those required to merely sustain the survival of present species. As the aim of our investigation is to determine whether Mars provide an appropriate environment for human survival, we only need to satisfy the list of conditions that support the survival of the human species, i.e, set A.
TABTherefore, although it may be fundamental for a planet to contain liquid water, the presence of water in liquid form is not strictly required for the survival of the human species, as with the advance of technology, a planet is habitable as long as water (regardless of which form) is present. Hence, the set of criteria to support
human life may be modified as the following:
TAB1. The stellar flux which it receives at the planetary surface must remain within an appropriate range, such
that with sufficient artificial insulation, an average body temperature of 37°C may be sustained.
TAB2. (Corollary to 1) The surface temperature must fluctuate within a narrow range, such that that the
maximum in temperature will not exceed the evaporation point of water and the minimum is not less than
a “critical temperature” necessary to sustain a body temperature of 37°C with proper artificial insulation.
TABAlso, adjusting the five conditions to the case of Mars, we see that Mars’ radius of 0.53R⊕ ≤2R⊕ (satisfies condition 2), and its orbital eccentricity of 0.093 ≈ 0.1 (satisfies condition 3). The star around which Mars revolves, the sun, measures 1L⊙ (condition 4). Hence, we can already prove its fulfillment of condition 2, 3 and 4
without performing any calculations.
TABTherefore, this section will be devoted to the assessment of Mars’ habitability by calculations corresponding to conditions 1, 5 and 6.
5.2 Surface Temperature
TABThis section will be devoted to the calculation of Mars’ surface temperature by taking into account of the flux received and emitted by its atmosphere. Firstly, there are a few terms worthy of definition:
5.2.1 Luminosity
TABLuminosity measures the rate at which an object radiates away energy from its center. In this scenario, the “object” is the sun, whose luminosity is given by the constant,
(5.2.1.1)
TABWhere luminosity is expressed in Watts (W). When calculating luminosity, we construct a hypothetical shell around the star called the Dyson sphere. A Dyson
sphere contains all the energy radiated from the sun to an object located at distance , i.e., the radius of the sphere,
where it receives a luminosity equivalent to a fraction of stellar luminosity (𝐿⨀),
expressed as a ratio of the area of the object with the surface area of the Dyson sphere.
(5.2.1.2)
Figure 7. (a) A Dyson sphere – a hypothetical “shell” extending from the surface of the Sun to Mars. (b) The Sun radiating energy to Mars. (c) As the distance between the Sun and Mars is very large, Mars’ half-spherical surface that faces the sun may be simplified to a 2-dimensional plane, whose surface area is equivalent to that of a circle.
TABTherefore, the luminosity received at the object’s surface is given by multiplying the stellar luminosity 𝐿⨀ with the fraction X:
(5.2.1.3)
5.2.2 Flux
TABAnother concept crucial for the computation of Mars’ surface temperature is the flux it receives at its surface. Flux is the quantity of luminosity received per unit area (the energy radiated from a blackbody per unit time per unit area). Hence, if expressed in relation to luminosity, flux may be written as:
𝑑𝐿 = 𝑓𝑑𝐴 (5.2.2.1)
TABWritten in the form of an integral:
(5.2.2.2)
TABThis expression indicates that the luminosity received at an object’s surface is equivalent to the flux received at its surface multiplied by the total surface area of a sphere. This makes sense, as all astronomical objects may be viewed as spherical objects, therefore, if we assume light is emitted isotropically from a blackbody (i.e., emitted equally in all directions), flux will be equally distributed per unit area of the Dyson sphere.
TABHence, the expression of flux at Mars’ surface can be written as:
(5.2.2.3)
5.2.3 Calculating the Surface Temperature
TABHaving established the notion of the Dyson sphere, we may compute the relative positions of the apoastron and periastron of Mars’ orbit on two separate Dyson spheres, denoted (at apoastron) and (periastron).
TABIn section 4, the position of Mars at the two apsides are denoted and , which, for convenience, we will call and . As both and are measured from a fix point O to Mars, they encompass the radius of the Sun. Therefore, the individual radii of two Dyson spheres at the apoastron and periastron are:
(5.2.3.1)
(5.2.3.2)
TABThese Dyson-radii are very important, because if we are to calculate the actual distance of light travelled from the sun’s surface to Mars, we will be considering the Dyson radius, not the actual radius of Mars in its orbit around the sun. For example, to compute the flux received at the apoastron and periastron, and ,
(5.2.3.3)
(5.2.3.4)
Figure 8. Dyson radii at the apoastron and periastron.
TABHowever, note that the flux calculated above is specific to per unit area. Therefore, to compute for the total flux of radiation intercepted by Mars, we will need to multiply the individual fluxes received at the apoastron and periastron with the area of the hypothetical “circle” we constructed in diagram 5.2.1.1 (as observed from afar, the curvature of Mars may be ignored, hence allowing the surface exposed to radiation to be simplified to a straight plane, whose surface area is given by , where is the radius of Mars.)
(5.2.3.5)
TABWhere denotes the total flux intercepted, is given by Mars’ volumetric mean radius, which equals 3389.5km. Therefore, to compute the total fluxes received at the apoastron and periastron, we need to plug in the values of respective fluxes received at the two focal points:
(5.2.3.6)
(5.2.3.7)
TABIn which is the total flux received at the apoastron, and is the total flux received at the periastron.
Figure 9. Conservation of
energy of the total stellar flux
received and emitted by Mars
TABAs solar radiation hits Mars, it is worth noticing that despite the total flux that reaches the atmosphere, the flux is not entirely absorbed into the atmosphere so that it reaches the planetary surface. A portion of this flux is also emitted to outer space as radiation.
TABIf we consider other components of flux to be negligible and designate there only to be two flux components from which the total flux separates to, we may conclude, by the conservation of energy, that the sum of these two components of flux (the flux emitted and the flux absorbed) is equivalent to the total flux that reaches Mars’ atmosphere, .
TABHence, if is the total flux absorbed and is the total flux reflected:
(5.2.3.8)
TABLet us direct our attention to derive an expression for . When considering the amount of flux absorbed into an atmosphere, a portion of flux is reflected ( ) while the rest is absorbed ( ). This portion can be
denoted by albedo, a constant that determines how much of solar radiation is reflected. Hence, if albedo varies within a range of 0 to 1, where 1 denotes complete reflection (no absorption) and 0 no reflection (full absorption), then (1 – albedo) will denote how much of total flux is absorbed into Mars’ atmosphere. To compute this flux, we simply multiply the total flux with the quantity (1 − 𝛼♂), where 𝛼♂ is Mars’ albedo (=0.250).
(5.2.3.9)
TABSimilarly, the flux absorbed into Mars’ atmosphere at each apsis will be:
(5.2.3.10)
(5.2.3.11)
TABThis flux absorbed will account for the surface temperature on Mars, and also, account for the amount of radiation emitted by Mars. If we consider that the energy generated by Mars is negligible, then, the energy radiated by the sun should equate that received on Mars. Hence, if the energy radiated by the sun is given by Stefan-Boltzmann’s law:
(5.2.3.12)
TABWhere 𝜎 is the Stefan-Boltzmann’s constant (=5.6704 × ) and 𝑇⨀ is the surface temperature of the sun. Then, this luminosity should correspond to the radiant flux absorbed by Mars, which put simply, is the overall flux received on Mars. Hence, the flux radiated from Mars can be written as:
(5.2.3.13)
TABHaving established that energy is conserved, such that the energy absorbed and emitted equate, then, we can determine the surface temperature on Mars by the following sequence of calculations:
(5.2.3.14)
(5.2.3.15)
(5.2.3.16)
TABWhere, if we calculate the temperature separately when Mars is located at each apsis, then, the respective surface temperatures at the apoastron and periastron are:
(5.2.3.17)
(5.2.3.18)
TABThe surface temperature at the apoastron is -72.1084°C, the surface temperature at the periastron is -52.6740°C.
5.3 Hydrogen Escape
5.3.1 Calculating the Maximum Rate Using Sengupta’s Equation
TABIn section 2, we briefly mentioned the maximum rate of hydrogen escape per unit volume of atmospheric air, which, according to (Sengupta, 2016), is given by:
(5.3.1.1)
TABGiven the following constants: RP is the radius of Mars (=3389.5km), b is the binary diffusion coefficient of a gas ( ), g is the surface gravity on Mars (=3.721 ), mH is the mass of a hydrogen atom (=1.6735575× ), mO is the mass of an oxygen atom (=16×1.6735575× ), XH is the molar mixing ratio of diatomic hydrogen (=500× ), XO is the molar mixing ratio of oxygen(=0.21mol/mol), k refers to the Boltzmann’s constant (=1.38064852× ), κ0 is the thermal conductivity of hydrogen (=0.172W/mK), G is the gravitational constant (=6.67430× ), MP is the mass of Mars (=6.39× ). Then, the maximum rate of hydrogen escape on Mars is:
(5.3.1.2)
(5.3.1.3)
The maximum rate of hydrogen escape in the Martian atmosphere is approximately
5.3.2 Calculating the Volume of Mars’ Atmosphere
According to NASA’s Mars Atmospheric Model, Mars has a radius of 3376km at its polar regions and a radius of 3396km at the equator. The atmosphere has a scale height of 10.8km, and, if we suppose that there are no external gravitational forces that could influence the height of the atmosphere at specific regions on the planet, Mars’ atmosphere can be modelled by an ellipsoid shell (Figure 10). To calculate its volume, it is necessary to obtain respective volumes of two ellipsoids: Mars without its atmosphere and Mars with its atmosphere. Then, a subtraction of the two volumes will result in a volume of its atmosphere.
Figure 10. Mars and Mars with its atmosphere as two concentric ellipses (Mars without atmosphere:
dark grey; atmosphere: light grey)
An ellipse is modelled by the relation:
(5.3.2.1)
Where a is the major axis and b is the minor axis.
Using this formula, Mars without its atmosphere can be modelled by:
(5.3.2.2)
And Mars with its atmosphere can be modelled by:
(5.3.2.3)
Employing the technique of volume by revolution, we will need to adjust the equations (5.3.2.2) (5.3.2.3) to the following format:
(5.3.2.4)
To begin with, isolate at one side of the equation:
(5.3.2.5)
(5.3.2.6)
The lower and upper limits in the ellipse are respectively -3396 and 3396 for the bare volume of Mars (denoted 𝑉1) and -3406.8 and 3406.8 for the volume of Mars with its atmosphere (denoted 𝑉2). However, as the ellipses are symmetrical about the y-axis, we can simplify the integral by halving each domain and doubling the
integral. Hence, substituting and these limits into V gives:
(5.3.2.7)
(5.3.2.8)
Solving for the respective integrals:
(5.3.2.9)
(5.3.2.10)
Subtraction of 𝑉1 from 𝑉2 gives the volume of Mars’ atmosphere:
(5.3.2.11)
Hydrogen is present as water vapor in the Martian atmosphere, which constitutes 0.03% of the total atmospheric mass (Mars Fact Sheet, 2020). This deficiency is due to hydrogen escape in the past, which will continue through Mars’ life. The other known sources of water surmount to a volume of .
Suppose the density of Mars’ atmosphere is (Mars Fact Sheet, 2020), then, the total atmospheric mass can be computed by:
(5.3.2.12)
Where M is the mass, V is the volume, and ρ is the density of Mars’ atmosphere. Then:
(5.3.2.13)
Therefore, total mass of water vapor in the atmosphere is:
(5.3.2.14)
Water vapor is composed of two hydrogen atoms and one oxygen atom. As the relative molecular mass of the water molecule is 18u and the relative atomic mass of the hydrogen atom is approximately 1u, hydrogen atoms constitute of the mass of a water molecule:
(5.3.2.15)
This indicates that there is a total of of hydrogen atoms in Mars’ atmosphere. n the other hand, the known reservoir of water on Mars measures a volume of approximately . As this water exists as ice, its density measures , thus, its mass can be obtained by the same volume-density relation used in (5.3.2.12).
(5.3.2.16)
The mass of hydrogen atoms is then:
(5.3.2.17)
Therefore, the total mass of hydrogen atoms on Mars is approximately equal to 1.04×1012kg + 5.09×1023kg≈5.09×1023k . Through this sum, we can see that the mass of hydrogen atoms in the atmosphere is too small to be accounted for in the calculation. Thus, the mass of hydrogen atoms in the atmosphere and at surface level will be calculated separately for the rest of this paper.
5.3.3 Hydrogen Escape in Mars’ Atmosphere
TABHaving calculated that the maximum rate of hydrogen escape in Mars’ atmosphere is 1.3791657 × 10,'A𝑘𝑔!𝑐= 1.3791657×105kg2km-3s-1 , the rate of hydrogen escape in the total volume of Mars’ atmosphere will be:
(5.3.3.1)
TABIf there is a total of 1.04 × 10'!𝑘𝑔 of hydrogen atoms in Mars’ atmosphere, then, the time before hydrogen is depleted in the atmosphere is:
(5.3.3.2)
TABAs there are 3.1536×107 seconds in a year, this result is equivalent to 383.72 years, or 383 years 8 months and 20 days.
5.3.4 Hydrogen Escape at the Surface of Mars
TABSection 5.2.3’s calculations indicated that the surface temperature of Mars at its periastron and apoastron is respectively -52.6740°C and -72.1084°C. This means that the range at which the surface temperature fluctuates is [-72.1084°C, - 52.6740°C], where the maximum temperature is -52.6740°C. As hydrogen exists as H2O at the surface of Mars, the maximum temperature of -52.6740°C is insufficient to cause ice to melt and hydrogen atoms to escape into the atmosphere. Therefore, there will be no significant hydrogen escape at the surface of Mars.
6 Discussion
TABIn this investigation, I found that Mars’ surface temperature varies within a range of [-72.1084°C, -52.6740°C], where the minimum temperature occurs at the apoastron and the maximum temperature occurs at the periastron. Furthermore, the maximum rate of hydrogen escape in Mars’ atmosphere is found to be
1.3791657 × 10,'A𝑘𝑔!𝑐𝑚,K𝑠,', , which corresponds to a duration of 383.72 years before the depletion of atmospheric hydrogen. Together with existing research of the HZE-flux anticipated on Mars’ surface during high solar activity periods and the continuous discoveries of ice reservoirs on Mars, my findings indicate that Mars
does not satisfy all criteria of planetary habitability, but it does provide ample opportunities for a future human settlement.
6.1 Insolation Habitable Zone and Surface
Temperature
TABAn insolation habitable zone is the region around a star in which all planets may hold standing liquid water. For water to remain in the liquid state, temperature fluctuations on the planetary surface must remain within the range (0°C, 100°C).
TABHowever, as evident from the calculations in section 5.2.3, the surface temperature on Mars inarguably falls below the minimum temperature for liquid water. Thus, Mars does not reside in the insolation habitable zone of the Sun. However, as noted in Figure 6, being in an insolation habitable zone is not entirely necessary to harbor life. Despite water being a necessity for all lifeforms, liquid water is only indispensable when it comes to the apparition of new life. Existent life, like the human species, has developed proper technology and solutions to endure harsh environments.
TABOn Earth, the coldest survivable temperature reported is -93.2°C (Los Angeles Times. 2021). With adequate insulation, this is the temperature the human body can endure without experiencing hypothermia (livescience.com. 2021). As the minimum temperature on Mars is higher than -93.2°C, survival is possible if proper insulation is in place. However, it is worth noting that the orbital period of Mars is approximately 687 Earth days, which means that, unlike a winter on Earth which will subside over 3 months, the extreme cold temperatures on Mars will sustain for another 3 months. This “stretched” winter will implicate significant challenges for adaptation to the sustained cold temperature.
TABNonetheless, research indicates that the cold temperature may not be permanent. A 2003 post from NASA indicates that Mars is discovered to harbor “water-carved gullies, glacier-like flows, regional buried ice and possible snow packs” that provide evidence of a potential “icy past.” (NASA - Mars May Be Emerging from an Ice Age, 2003; Vincendon, M., 2015). Furthermore, recent research brings into light that Mars’ surface which consists of a cover of water ice mixed with dust is slowly retreating and degrading (Christensen, 2006). The number of impact craters in these features and the patterns of changes in Mars’ orbit and tilt also provide insight into the nature of Mars as a planet, which indicates that Mars may be even more responsive to climate changes than the Earth (Vincendon et al., 2011). These evidence indicate that Mars may evolve to harbor a much friendlier environment for a human settlement should it emerge from its glaciation state. However, further research into whether Mars is situated in such a post-glaciation period will be necessary to confirm this hypothesis.
6.2 Hydrogen Escape
TABComputations using Sengupta’s formula (Sengupta, 2006) indicate that hydrogen effuses out of Mars’ atmosphere at a maximum rate of 1.3791657×105kg2km-3s-1. As hydrogen is present as water vapor in the atmosphere, oxygen will also be entrained along hydrogen outflow. This indicates that with the depletion of atmospheric hydrogen, depletion of atmospheric oxygen present in water vapor as well as diatomic oxygen is likely to ensue.
TABAs we have determined in section 5.3.3 that it will take approximately 383.72 years before all atmospheric hydrogen is depleted, it is reasonable to make the prediction that the concentration of oxygen will severely be reduced along hydrogen efflux, which will eventually lead to its depletion in Mars’ atmosphere.
TABTherefore, as humans suffer from anoxia at oxygen concentrations lower than 12%, Mars is unlikely to be a habitable environment after 380 years from the present due to severe reductions in the concentration of atmospheric oxygen. Therefore, either oxygenating equipment is developed and supplied to the population, or humans will need to venture the universe for a new home.
7 Limitations and Future Directions
TABDespite the novelty of this investigation, there are plenty limitations to be acknowledged. First, the hydrogen escape in Mars’ atmosphere is computed using a formula in an unfamiliar domain, which could have impeded the acquisition of a full understanding of the formula and the accuracy of the result obtained. Second, the result obtained only provides an upper limit approximation of the rate of hydrogen escape, and thus, the findings may not be generalized to provide a mean rate of hydrogen escape. Third, as the concentration of hydrogen does not play a direct role in determining planetary habitability, calculations targeted to hydrogen escape, not oxygen escape, only suffice to provide a rough prediction as to when oxygen will be depleted. Fourth, the volume of Mars’ atmosphere was calculated only under the assumption that it is evenly distributed around the planetary surface. This, however, is not true, as tidal effects and climate changes impede the formation of an atmosphere of uniform scale height (Chapter 2. Atmospheric Pressure, 1999).
TABMoreover, a full assessment of habitability of Mars cannot be achieved as there are certain limitations in my mathematical ability. An individual assessment of the flux of HZE-charged particles on Mar’s surface will constitute a fuller analysis of habitability on Mars, but advanced computational strategies such as Fourier and Wavelet Transform 1-D Analysis will be required to complete this calculation. Thus, I will not be able to make any quantitative conclusions regarding the HZE-flux received at Mars’ surface, but rather just a statement that sums up existing research about the anticipated HZE-fluxes during high and low solar activity periods on Mars. Then again, as this is my first attempt at calculating large volumes and computing with large numbers, it is possible that some of my computations lack accuracy.
TABTherefore, the current results do not allow an exact conclusion to be drawn about Mars’ habitability or a statement to be made about the precise duration before the depletion of atmospheric oxygen. Because of the narrow scope of this investigation, it is also unknown of how Mars will evolve in the next three decades in consideration of polar warming effects and the emergence from its previous glaciation state. It is also noteworthy that the present investigation only assessed Mars’ habitability in respect to its surface temperature and the rate of hydrogen escape. There are far more criteria to examine, and without them, it is impossible to make an accurate and overall judgement of Mars’ habitability. Predicated on the basis of this investigation, these other criteria will be fruitful directions for future research. Longitudinal studies that examine the evolution of Mars’ climate will also be very helpful in disentangling how living conditions on Mars evolve and in answering whether Mars is habitable for a human settlement such as the one planned for 2033.
8 Conclusion
TABTo conclude, in the current investigation I found that Mars is a short-term habitable environment for a human settlement. Despite the minimum temperature of -72.1°C and the maximum temperature of -52.7°C at Mars’ surface, both temperatures are higher than the minimum survivable temperature on Earth, which is -93.2°C. This indicates that with adequate insulation and possible technological advancement in the next 12 years, survival on Mars is more than a possibility. On the other hand, destitute hydrogen in Mars’ atmosphere and the rapid rate of hydrogen outflow suggest that Mars may only be a habitable environment for the next 383.72 years. Existing research from HZE-charged particles also indicates that inhabitants of Mars may be subject to health hazards during high solar activity periods (Kamsali et al., 2019). These results shed light on the possible opportunities and risks of establishing a settlement on Mars, and demonstrate the possibility of applying an approach of analyzing the current rate of hydrogen escape to predict the possible depletion of oxygen in the future for investigating the habitability of Mars.
Acknowledgements
TABThis work is supported by Mr. Hughes for his support and encouragement during my pursuit of this project, Mr. Cely for his recommendation of Orbital Mechanics for Engineering Students (Fourth Edition), and the Department of Mathematics at K. International School Tokyo for the provision of such a wonderful opportunity to dive into this issue of my interest.
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Investigation of the Habitability of Mars in Specific to
Surface Temperature and Hydrogen Escape
Shuonan Yu
May 2021
