Boletín de la Sociedad Geológica Mexicana

Volumen 67, núm. 3, 2015, p. 377-385

Methane in the Solar System

Andrés Guzmán-Marmolejo1, Antígona Segura2,*

1 Posgrado en Ciencias de la Tierra, Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacán. C.P. 04510, D.F., México.
2 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito exterior C.U. Apartado Postal 70-543. Coyoacán, C. P. 04510, D.F., México.

* This email address is being protected from spambots. You need JavaScript enabled to view it.

 

Abstract

This paper reviews the distribution of methane (CH4) in our Solar System, as well as its sources and sinks in the atmospheres of the main Solar System bodies. Methane is widely distributed in the Solar System. In general, the inner planets are methane-poor, being Earth a unique exception, whereas the outer planets have CH4-rich atmospheres. In general, the atmospheric chemistry of this compound is dominated by the solar radiation although in O2-rich atmospheres this compound participates in a reaction system that removes atmospheric CH4. In our planet most of the atmospheric CH4 is produced by lifeforms, reason why scientists have proposed that the simultaneous detection of methane signal along with oxygen (O2) or ozone (O3) signals in the atmospheric spectra of planets may be good evidence of life. Therefore, the study of this gas at planetary level is important for understanding the chemical reactions that control its abundance on the exoplanetary atmospheres and to classify possible inhabited planets.

Keywords: methane, biosignatures, Solar System.

 

Resumen

El objetivo del este trabajo es hacer una revisión sobre la distribución del metano (CH4) dentro del Sistema Solar, así como sus fuentes y sumideros en las atmósferas de sus principales cuerpos. El CH4 está ampliamente distribuido en el Sistema Solar; en general los planetas internos son pobres en este gas, con excepción de la Tierra, mientras que los planetas externos son ricos en él. La química atmosférica de este compuesto generalmente está dominada por la radiación solar, aunque en atmósferas ricas en O2, este compuesto forma parte de un sistema de reacciones que eliminan al metano atmosférico. Dado que la mayor parte de CH4 atmosférico es debido a la vida, los científicos han propuesto que su detección simultánea con oxígeno (O2) u ozono (O3) en el espectro de la atmósfera de los planetas podría ser una buena evidencia de vida. El estudio del CH4 a nivel planetario es importante para entender las reacciones que controlan su abundancia en las atmósferas de los exoplanetas y clasificar los posibles planetas habitados.

Palabras clave: metano, bioseñales, sistema solar.

 

1. Distribution of methane in the Solar System

The Solar System was formed by the gravitational collapse of a primordial gas nebula. The center of this nebula collapsed faster than its outer edge, forming the Sun at the center and a protoplanetary disc around, latter processes formed planets from the dust (Cloutier, 2007). The temperature in the inner protoplanetary disc near the Sun was high enough to evaporate volatiles like methane, which is decomposed by photolysis and it is dragged later by the solar wind. In the outer regions of the protoplanetary disc, the low temperatures allowed that ices and volatiles could be preserved. The result is CH4-poor terrestrial planets in the inner Solar System and CH4-rich big planets in the outer (Cloutier, 2007).

Methane is preserved in ices called clathrates, these solids present structures that can capture methane in their interior. They play an important role in the stabilization and dispersion of molecules in the Solar System because they are present in many kinds of environments with a wide range of pressures and temperatures (Miller, 1961; Thompson et al., 1987). Here, we review the CH4abundances in planets and small bodies of the Solar System.

 

2. Inner planets

Inner planets are the four closest planets to the Sun: Mercury, Venus, Earth, and Mars. They are small planets composed of silicates and iron. Volatiles in inner planet atmospheres as Mercury, Venus, Earth and Mars, are mainly the result of degassing from their interiors (Cloutier, 2007).

 

2.1. Mercury and Venus

Mercury has a tiny atmosphere mainly formed by He, H2, O2, Na, Ca, K and water vapor (Broadfoot et al., 1974; Potter and Morgan, 1985, 1986). Measurements with the Mercury Laser Altimeter ─MErcury Surface, Space ENvironment, GEoche-mistry, and Ranging (MESSENGER), confirmed the long held idea that Mercury contains impact-derived deposits of volatiles than may include organics (Neumann et al., 2013; Paige et al., 2013). These deposits are located in permanently shadowed zones of the north polar region where the regolith has temperatures similar to those of the icy Galilean satellites, allowing the cold-trapping of materials from comets and rich-volatile meteorites (Neumann et al., 2013). Methane is present in comets but thermal stability models do not predict its presence in cold-traps due to its higher volatility compared to water (Zhang and Paige, 2009). Gibson (1977) proposed a volatile cycle for Mercury, starting with the production of simple molecules (H2, H2O, CH4, NH3, etc.) by solar-wind ions implanted into the planet’s silicate surface. These chemical species would be outgassed and then cold-trapped in colder regions of the planet. Until now, no detection of methane has been reported for this planet.

Venus has a thick atmosphere mainly formed by CO2 (96 %) and N2 (3 %) (Niemman et al., 1980). The Pioneer spacecraft instruments detected CH4 in Venus atmosphere (1000 – 6000 ppm) and many other gases (Oyama et al., 1980). Nevertheless, measurements of CH3D/CH4 ratio of 5×10-3 caused controversy because atmospheric evolution models predicted a CH3D/CH4 ratio of 9×10-2. A plausible explanation is that the CH4 was the result of the reaction between some highly deuterated molecules in Venus atmosphere and terrestrial CH4that contaminated the instruments (Donahue and Hodges, 1993).

Based on the detection of NH3, HCl, and H2O in the Venus atmosphere, along with the fact that there is a strong possibility of electrical discharge in the atmosphere as a result of thermal convective turbulence, Otroshchenko and Surkov (1974) proposed that organic compounds could be formed in the atmosphere. Their hypothesis was experimentally tested, finding CH4and other low-mass molecules. The studies of Otroshchenko and Surkov (1974) show that presence of organic compounds in the Venus atmosphere is a strong possibility.

 

2.2. Earth

CH4 levels in the atmosphere are currently around 1.6 – 1.8 ppmv, the enhanced greenhouse effect caused by a molecule of methane is about 8 times that of a molecule of CO2 (Houghton, 2005). CH4 is homogeneously mixed in the troposphere while in the upper atmosphere the highest concentrations are at the Ecuador (http://earthobservatory.nasa.gov). CH4 levels have changed over the history of the Earth, before the emergence of life, CH4sources were geological.

The emergence of life increased the levels of CH4 in an atmosphere without free oxygen, where CH4 could have lifetimes of 5000 – 10000 years and reach concentrations of 1000 ppmv (Kasting and Siefert, 2002). Then, when oxygenic photosynthesis increased O2 levels in the atmosphere, CH4 decreased because of a set of reactions that will be described at the end of this section. Numerical models show that, today, the thermodynamic equilibrium value for CH4 is > 10-35, in volume fraction, however its abundance is approximately 1.7×10-6 (Sagan et al., 1993). CH4 is almost totally produced by biological sources and the abiotic sources represent less than 10 % (Levine et al., 2010). The pristine ice cores store a record of CH4 concentrations of thousands of years. Analysis of these cores show that CH4 abundances ranged from 0.35 ppmv to 0.8 ppmv corresponding to glacial and interglacial periods (e.g., Legrand et al., 1988; Chappellaz et al., 1990; Raynaud et al., 1993; Brook et al., 1996; Petit et al., 1999; Spahni et al., 2005; Loulergue et al., 2008). CH4 has increased its atmospheric concentration since pre-industrial time to be relatively constant around 1.7 ppmv (Dlugokencky et al., 2003). Recently, CH4 is calling the attention of scientists studying climate change due to its capability as greenhouse gas. Today near to 50 % of CH4 global emissions are produced by human activity (mining, industry, farming, and ranching) causing an imbalance between their sources and sinks of 30 Tg year-1, approximately, contributing from 4 % to 9 % of greenhouse effect (Lelieveld et al., 1998; Wuebbles and Hayhoe, 2002; Houghton, 2005).

It is estimated that all CH4 sources produce 600 Tg yr-1, approximately. There are no chemical reactions forming CH4 in atmospheres such as Earth (Levine et al., 1985). Here, almost all CH4 is produced by methanogen microorganisms. Methanogens can form CH4 by two ways: 1) using CO2 in the reaction CO2 + 4H2 → CH4 + 2H2O (Thauer, 1998) or, 2) using organic molecules as electron terminal acceptors, for example acetic acid, methanol, or methylamine, in the reaction CH3COOH → CH4 + CO2 (Fukuzaki et al., 1990). Table 1 summarizes the CH4sources. The major biological sources of methane are wetlands, followed by digestion of animals such as ruminants and decomposition of biomass. An important source, linked to human activity, is the production of energy. Other minor sources are the animal activity such as arthropods and decomposition of sediments and bacterial activity in marine environments. While the only known abiotic source is the serpentinization process and contributes with 3 % of methane emissions.

Serpentinization takes place in hydrothermal systems similar to the Lost City, located in the middle of Atlantic Ocean. In these sites, it is commonly said that CH4 is formed by serpentinization but in fact, CH4 is byproduct of a Fischer–Tropsch type reaction after to serpentinization process. In the Fischer–Tropsch reaction, CO2 is reduced by H2 forming CH4: CO2 + 4H2→ CH4 + 2H2O. This reaction needs metal catalysts such as Fe, Co, and Ni, and temperatures and pressures in the range of 200 ºC to 350 ºC and 20 bars to 30 bars (Schulz, 1999). The H2 used in the Fischer–Tropsch reaction is a product of serpentinization. In serpentinization, hydrolysis of olivine minerals ((Mg,Fe)2SiO4) form serpentine (Mg3Si2O5(OH)4), brucite (Mg(OH)2), magnetite (Fe3O4), and H2:

3Fe2SiO4 + 2H2O → 3SiO2 + 2Fe3O4 + 2H2(ac)

3Mg2SiO4 + SiO2 + 4H2O → 2Mg3Si2O5(OH)4

2Mg2SiO4 + 3H2O → Mg3Si2O5(OH)4 + Mg(OH)2

Serpentinization reactions are possible from 1 bar to 5 kbars, temperatures from 0 ºC to 500 ºC, and Fe2+ abundances from 1% to 50% (Oze and Sharma, 2005).

In contrast to the numerous CH4 sources, there are only three sinks. Table 2 summarizes the sinks of CH4. The major of those occur in the troposphere where the reaction of oxidation of CH4 by hydroxyl radical (OH) leads mainly formaldehyde (CH2O); such reaction is responsible for removing almost 90 % of atmospheric CH4. OH radical is byproduct in photolysis of O3 by UV-B radiation (Rohrer y Berresheim, 2006). OH radical rapidly reacts with CH4removing it from the atmosphere:

O2 + hv(180 - 240 nm) → O + O

O2 + O → O3

O3 + hv (200 - 300 nm) → O(1D) + O2

O(1D) + H2O → 2OH

CH4 + OH → CH3 + H2O

Where hv is the energy of a photon with frequency v and h is the Panck constant. The remaining CH4 is removed trough soil oxidation, and transport to the stratosphere (Wuebbless and Hayhoe, 2002; Houweling et al., 2006; Anderson et al., 2010). After being produced, either by biological activity or serpentinization, methane may be stored in clathrates. Gas hydrates belong to a general class of inclusion compounds commonly known as clathrates. Clathrates owe their existence to the ability of H2O molecules to assemble via hydrogen bonding and form polyhedral cavities. Molecules like methane or carbon dioxide are of an appropriate size such that they fit within cavities formed by the host material (e.g., Kvenvolden, 1993). Methane hydrates are particularly important (Mahajan et al., 2007). Within clathrates there are no chemical bond involved between the water molecules and the gas molecules other than Van der Waals forces, but the presence of guest molecules inside the ice crystals makes the structure more stable. In fact, the guest molecules stabilize the structure enough for raising the melting point of the ice to several degrees above 0 °C (Miller, 1961). There are two different reservoirs for clathrates. They can be found both within and under permafrost in arctic regions and also within a few hundred meters of the seafloor on continental slopes and in deep seas and lakes (Hester and Brewer, 2008).The permafrost is soil, sediment, or rock that is continuously frozen (temperature < 0 °C) for at least two consecutive years (Anderson et al., 2010). Permafrost is the largest CH4 reservoir in Earth. Estimates of the global inventory of methane clathrate may be 3×1018 g of carbon (Buffett and Archer, 2004). Permafrost acts as an impermeable lid, preventing CH4 escape through the seabed. Moreover, sub-sea permafrost is potentially more vulnerable to thawing than terrestrial permafrost. A consequence of climate warming is the partial thawing and failure of sub-sea permafrost and thus an increased permeability for gases. Shakhova et al. (2010a) estimate the total amount of carbon preserved within permafrost, only in the East Siberian Arctic Shelf (ESAS), to be ~1.4×1015 g. Shakhova et al. (2010b) estimated the annual outgassing from the shallow ESAS of 7.98 Tg CH4. This amount is of the same magnitude as existing estimates of total methane emissions from the entire world ocean (e.g., Anderson et al., 2010).

Because methane is also a greenhouse gas, release of even a small percentage of total deposits could have a serious effect on Earth’s atmosphere. A conservative estimate by Boswell and Collett (2011) for the global gas hydrate inventory is ~1.8×1015 g C, corresponding to a CH4 volume of ~3.0×1015 m3 if CH4 density is considered to be 0.717 kg m-3. In the unlikely event that 0.1 % (1.8 Tg C) of this CH4 were instantaneously released to the atmosphere, CH4concentrations would increase to ~2900 ppb from the 2005 value of ~1774 ppb (IPCC, 2007).

Table 1. CH4 sources on Earth.

 

Table 2. CH4 sinks on Earth.

Sinks reported by (a) Houweling et al., 2006, (b) Anderson et al., 2010, (c) Wuebbless and Hayhoe, 2002.
Sources reported by (a) Houweling et al., 2006, (b) Anderson et al., 2010.

 

2.3. Mars

Mars is an especial case. Thermodynamic calculations predict CH4 should not exist in its atmosphere (Levine et al., 2010), however a CH4 signal was discovered by Krasnopolsky et al. (1997) using the Fourier Transform Spectrometer of the Kitt Peak National Observatory (Arizona, USA). The authors estimated 0.07 ppm of atmospheric CH4. Later, in 2004, two groups (Krasnopolsky et al., 2004; Formisano et al., 2004) reported abundances of 0.01 ppm using the instruments on board of the Mars Express. Zahnle et al. (2011) doubt the detection of CH4 in Mars, arguing that CH4 abundances estimated by Krasnopolsky et al. (2004) and Formisano et al. (2004) were supported on tenuous signals slightly distinguishable from the noise, however Mumma et al. (2009) reported a clear signal of CH4 and his calculations confirm CH4abundances of 0.01 ppm.

In 2010, Fonti and Marzo made distribution map of methane on the Martian surface. They identify three localized sources on the Martian surface, related to probable underground water reservoirs. Their analyses suggest that CH4abundances vary throughout seasonal cycles.

There are some hypotheses about the sources and sinks of CH4 in Mars. For example, Krasnopolsky et al. (2004) considered that degassing from the interior of the planet is unlikely due to the lack of geologic activity. Bar-Nun and Dimitrov (2007) proposed that photolysis of H2O in the presence of CO can generate CH4, however Krasnopolsky (2007) argues that it is not possible due to the kinetic chemistry of Mars. Serpentinization has also been proposed (e.g. Oze and Sharma, 2005; Lyons et al., 2005; Szponar et al., 2013; Etiope et al., 2013), this hypothesis is supported by the spatial correlation of underground water reservoirs and volcanoes where serpentinization may be possible. The origin of CH4 on Mars is still not clear, some authors have proposed biogenic sources such as methanogenesis via metabolic pathways (e.g. Weiss et al., 2000; Chapelle et al., 2002; Jakosky et al., 2003; Varnes et al., 2003; Buford, 2010). CH4 lifetime is 340 years and methane should be uniformly mixed in the atmosphere. Heterogeneous loss of atmospheric methane is probably negligible, while the sink of CH4 during its diffusion through the regolith may be significant. There are no processes of CH4 formation in the atmosphere, so the photochemical loss must therefore be balanced by its sources (Krasnopolsky et al., 2004). It was thought that the main sink of CH4 was its direct photolysis around 80 km from the surface. Other sink is the reaction between CH4 and Martian soil, but theoretical studies calculate the collision probability between CH4 y O of 2×10-11. Therefore, this reaction is negligible versus its direct photolysis (Krasnopolsky et al., 2004).

 

3. Outer planets (Jupiter, Saturn, Uranus and Neptune)

3.1. Jupiter and Saturn

They are giant planets with atmospheres mainly constituted by H2 (> 80 %) and He as the second more important constituent. Their composition and chemistry are relatively similar in those planets. Jupiter is the largest planet in the Solar System with 318 M. Methane is the most abundant species in the upper Jovian troposphere after hydrogen and helium, accounting for approximately 0.2 % of the molecular abundance (Taylor et al., 2005). Different calculations estimate that CH4/H2 ratio is from 1.9×10-3 to 2.3×10-3 (Hanel et al., 1979; Gautier et al., 1982; Wong et al., 2004). Methane does not condense at the temperatures found on Jupiter, and is chemically stable except in the upper atmosphere (P < 1 mbar), where it is dissociated by solar ultraviolet radiation. Higher hydrocarbons are produced from methane by photochemical processes in the upper atmosphere of Jupiter (Taylor et al., 2005). Photolysis of CH4 is the only sink (Moses et al., 2000), but it is not an effective way to destroy it in the Jupiter’s atmosphere because the large excess of H2 that suggests that radicals like CH3, byproducts of the CH4 photolysis, react with the H radical reforming CH4 (McNesby, 1969). Saturn is the second largest planet in our solar system. Observations from the Cassini spacecraft suggest mole fractions of CH4 of 4.7×10-3 (Fletcher et al., 2009). The chemistry of CH4in Saturn is similar to Jupiter.

 

3.2. Uranus and Neptune

The only known photochemically active volatile in the atmosphere of Uranus is methane. From observations of the Ultraviolet Spectrometer in the Voyager spacecraft, the calculated abundance for CH4 is 10-4 near 0.1 mbar. Other species normally present in the atmospheres of Jupiter and Saturn are not likely to be gaseous in the photolytic regime of the upper troposphere and stratosphere of Uranus due to the low tropopause temperature (Atreya et al., 1991). The stratospheric CH4 is photolyzed forming acetylene, methyl-acetylene, ethane, and ethylene (Orton et al., 1987; Bézard et al., 1991; Schulz et al., 1999; Meadows et al., 2008). However, CH4 photolysis is relatively inefficient on Uranus. Only 10 to 15 % of CH4 molecules, which absorb ultraviolet photons, produce higher hydrocarbons resulting in a loss rate of 6×106 CH4 molecules cm-2 s-1 at the equator. For comparison, the loss rate on Jupiter is 30 % (Atreya et al., 1991).

In Neptune, the mixing ratios of methane suggested by photochemical models is ~2 % at pressures > 0.1 bars (e.g., Baines et al., 1995), but there is evidence from remote observations that its abundance may be up to 4 % at P > 3.3 bars (Karkoschka and Tomasko, 2011). At lower pressures, methane is not homogenously distributed at all latitudes. The expected mixing ratio at the mean temperature of Neptune’s tropopause (~52 K) is ~5×10-5 but values of (1.5 ± 0.2)×10-3 have been derived from Herschel-PACS observations (Lellouch et al., 2010). This is consistent with the hypothesis that CH4 leaking through the warm south polar tropopause (62 – 66 K) is globally redistributed by stratospheric motion (Fletcher et al., 2010). Voyager 2 observed Neptune’s atmosphere. Their images show that Neptune contains clouds of methane ice (Smith et al., 1989). Similar to Uranus, CH4 is photolyzed in the stratosphere, producing hydrocarbons like acetylene and ethane (Romani and Atreya, 1989; Romani et al., 1993).

 

4. Methane in small bodies

4.1. Pluto

Pluto’s atmosphere is the result of the sublimation of superficial ices, in consequence, it is expected that the atmosphere is in vapor-pressure equilibrium with the surface (e.g., Young et al., 1997). Owen et al. (1993) estimated that the surface contains 1.5 % of solid CH4. Later, Young et al. (1997) detected gaseous methane in Pluto for the first time, calculating a partial pressure of 0.072 µbar. In 2008 and 2012 this body was observed using the CRIRES instrument in the Very Large Telescope (VLT) to constrain the spatial and vertical distribution of methane in Pluto’s atmosphere (Lellouch et al., 2015). From these observations, the calculated methane-mixing ratio is 0.44 % with negligible longitudinal variations. Because Pluto has not yet been observed with any spacecraft, all its parameters have been inferred using instruments on the ground. In 2015, the mission New Horizons will be able to characterize the surface and atmosphere of Pluto and its satellite, Charon.

 

4.2. Triton

The Voyager 2 spacecraft observed Triton (Neptune's largest moon) in 1989 and it has been later studied using instruments on Earth’s surface and the Hubble Space Telescope (Buratti et al., 2011 and references therein). Similar to Pluto, its atmosphere is the result of the sublimation of the more volatile ices on its surface. The surface of Triton contains approximately 0.05 % CH4 in ices (Tyler et al., 1989; Cruikshank et al., 1993) and its atmospheric mixing ratio was calculated to be 10-4 from the Voyager observations (Tyler et al., 1989) and confirmed by the VLT/CRIRES instrument (Lellouch et al., 2011).

 

4.3. Titan

Titan is the largest moon of Saturn. Its bulk composition has nearly equal mass fractions of silicates and ices (Grasset et al., 2000). Titan’s atmosphere is mainly composed by N2, with 5 % of CH4 near to surface (Tobie et al., 2006). Methane was likely to be present in the materials that built Titan and is possible that cometary impacts were a significant source in the far past (e.g. Tobie et al., 2006; Mousis et al., 2009). Present abundances of CH4 have not been possible to explain, because it is photochemically active in the atmosphere and requires a constant replenishment over geologic time scales (Davies et al., 2013). Liquid filled basins in the polar regions of Titan (Stofan et al., 2007; Turtle et al., 2009) composed by methane mixed with ethane (Brown et al., 2008) and a number of other organic species (Cordier et al., 2010) were identified using Cassini spacecraft observations.

In 2005, the Huygens spacecraft descended to the surface of Titan measuring in situ the CH4 mole fraction when it descended. Huygens found that the CH4 mole fraction is relatively constant in the stratosphere; it increases between 32 and 8 km, and remains constant near the surface (Atreya et al., 2006). Titan has pressures and temperatures near to the methane triple-point, for this reason the CH4 can evaporate from the surface to atmosphere, where it can condense and rain, forming a CH4cycle similar to water on Earth (Roe, 2009; Lunine, 2012).

Mathematical models based on the Voyager’s measurements suggest that the lifetime of CH4 is from 10 to 100 millions of years (e.g. Yung et al., 1984; Lara et al., 1996; Lebonnois et al., 2001; Wilson and Atreya, 2004). In the stratosphere CH4 is photolized to CH3, CH2 or CH, forming ethane, propane, and benzene (Strobel, 1974). There are not reactions to generate CH4 in the Titan's atmosphere, so it is proposed that CH4 may come from clathrates formed in the subnebula that originated the satellite (Mousis et al., 2002, Davies et al., 2013). Other authors proposed the activity of bacteria as a likely CH4 source (e.g. McKay and Smith, 2005; Schulze-Makuch and Grinspoon, 2005), nevertheless there is not evidence about it (Atreya et al., 2006). Another possibility is the serpentinization (Niemann et al., 2005) but according to Mousis et al. (2009) this source of methane is not able to reproduce the deuterium over hydrogen (D/H) ratio observed at present in methane in its atmosphere.

 

4.4. Comets

These icy bodies have been studied with flyby missions and ground infrared and radio observations. Methane is a primary volatile in comets, this means that it is stored as ice in the cometary nucleus and released as gas into the coma. This compound has been detected in eleven comets and its abundance relative to water ranges from ~0.4 % to 2 % (Allen et al., 1988; Drapatz et al., 1987; Mumma et al., 1996; Bockelée-Morvan et al., 2000; Gibb et al., 2003; Mumma and Charnley, 2011).

 

5. Final comments

The study of methane is relevant to understand the process of synthesis and distribution of organic molecules during the formation of the Solar System. On potentially habitable planets around other stars its presence maybe the result of geological or biological activity. The bodies of our Solar System, especially Earth, serve as benchmarks for understanding the origin, sources and reservoirs of this compound to identify possible inhabitable worlds around other stars.

 

Acknowledgements

We acknowledge the support of the project PAPIIT IN119709-3.

 

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Manuscript received: April 30, 2014.

Corrected manuscript received: February 9, 2015.

Manuscript accepted: February 12, 2015.