# Effect of Mars Atmospheric Loss on Snow Melt Potential in a 3.5-Gyr Mars
Climate Evolution Model

Megan Mansfield<EMAIL_ADDRESS>University of Chicago Department
of Geophysical Sciences  Edwin S. Kite  University of Chicago Department of
Geophysical Sciences  Michael A. Mischna  Jet Propulsion Laboratory,
California Institute of Technology

###### Abstract

Post-Noachian Martian paleochannels indicate the existence of liquid water on
the surface of Mars after about 3.5 Gya (Irwin et al., 2015; Palucis et al.,
2016). In order to explore the effects of variations in CO2 partial pressure
and obliquity on the possibility of surface water, we created a zero-
dimensional surface energy balance model. We combine this model with
physically consistent orbital histories to track conditions over the last 3.5
Gyr of Martian history. We find that melting is allowed for atmospheric
pressures corresponding to exponential loss rates of $dP/dt\propto t^{-3.73}$
or faster, but this rate is within $0.5\sigma$ of the rate calculated from
initial measurements made by the Mars Atmosphere and Volatile EvolutioN
(MAVEN) mission, if we assume all the escaping oxygen measured by MAVEN comes
from atmospheric CO2 (Lillis et al., 2017; Tu et al., 2015). Melting at this
loss rate matches selected key geologic constraints on the formation of
Hesperian river networks, assuming optimal melt conditions during the warmest
part of each Mars year (Irwin et al., 2015; Stopar et al., 2006; Kite et al.,
2017a, b). The atmospheric pressure has a larger effect on the surface energy
than changes in Mars’s mean obliquity. These results show that initial
measurements of atmosphere loss by MAVEN are consistent with atmospheric loss
being the dominant process that switched Mars from a melt-permitting to a
melt-absent climate (Jakosky et al., 2017), but non-CO2 warming will be
required if $<2$ Gya paleochannels are confirmed, or if most of the escaping
oxygen measured by MAVEN comes from H2O.

##  I Introduction

Large ($>10\text{ km}^{2}$), late-stage Martian alluvial fans and river deltas
provide evidence for surface liquid water on post-Noachian Mars after about
3.5 Gya (Grant and Wilson, 2012; Williams and Weitz, 2014; Irwin et al., 2015;
Palucis et al., 2016). However, by about 3.5 Gya most of the conditions
favorable to the existence of surface liquid water no longer existed: much of
Mars’s atmosphere was lost before the end of the Noachian and the Martian
dynamo shut down around the mid-Noachian (Pepin, 1994; Lillis et al., 2013).
Additionally, the amount of greenhouse gases released by volcanism on Mars is
less than on Earth because the mantle is more reducing (Stanley et al., 2011),
and by 3.5 Gya the rate of volcanic degassing had slowed down significantly
(Kite et al., 2009). Because orbiters detect only minor post-Noachian
carbonate, which may be due to SO2 or acidity (Bullock and Moore, 2007; Halevy
and Schrag, 2009), it is difficult to justify post-Noachian carbonate
sequestration of more CO2 than exists in the present, thin atmosphere (Edwards
and Ehlmann, 2015).

Atmospheric pressure has a strong effect on permitting surface liquid water
because lower atmospheric pressure can preclude melting by decreasing the
strength of greenhouse warming and increasing the amount of evaporative
cooling. Clow (1987) showed these effects in a model which determined the
minimum atmospheric pressure necessary to melt dusty snow on the surface of
Mars. They found that melting could occur at relatively low pressures if the
snowpack was assumed to be thick enough that melt accumulated at the base of
the thick snow layer while the upper layers of snow were at colder
temperatures. Hecht (2002) used a similar model to determine the necessary
surface properties, such as albedo and conductivity, to allow melting on Mars.
Additionally, Hecht emphasized the importance of evaporative cooling at low
atmospheric pressures, as first pointed out by Ingersoll (1970). Models which
have not included evaporative cooling produce much more melting on the surface
of Mars, which indicates its importance in setting melting conditions (Costard
et al., 2002; Williams et al., 2008). The high rate of evaporative cooling at
low atmospheric pressure precludes melting of pure water ice on present-day
Mars, which has an average atmospheric pressure of 600 Pa.

Although atmospheric pressure can have a strong effect on the surface energy
balance, changes in Mars’s orbital parameters, especially large changes in its
mean obliquity, can also affect melt conditions. Kite et al. (2013) determined
the conditions for snowmelt at a variety of atmospheric pressures,
obliquities, and eccentricities. They found that combining these factors could
produce intermittent melting that could provide small amounts of liquid water
for sediment induration at low latitudes.

While models that study moments in time can provide some insight into the past
Martian climate, models showing evolution over time give a better
understanding of the timing of melting in Mars’s past. Manning et al. (2006)
addressed the time evolution of the Martian climate by examining the relative
sizes of various CO2 reservoirs over the history of Mars. Their model
determined that different climate states are stable for different obliquities,
but they only considered a subset of the full range of obliquities that Mars
has experienced.

A model that simultaneously studies the effects of time evolution, orbital
variations, and changing energy balance with changing atmospheric pressure is
needed to more fully understand the timing and intermittency of melting on
Mars. To address this, we created a zero-dimensional energy balance model that
spans over 3.5 Gyr of Mars history and examines how changes in orbital
parameters, solar luminosity, and atmospheric pressure impact the melt
conditions of snow on the surface of Mars. We consider snowmelt as a source of
water as opposed to catastrophic flows after impacts because recent work
suggests lakes lasted for at least a few thousand years, and this relatively
long lifetime implies a climate favorable to melting and not just a sudden
catastrophic event that allowed melting for a brief period (Grant and Wilson,
2012; Williams and Weitz, 2014; Irwin et al., 2015; Palucis et al., 2016;
Turbet et al., 2017). We also ignore the effects of impacts and eruptions
because changes due to atmospheric loss, solar brightening, and orbital
variations are more well understood, and because the melt potential of impacts
themselves depends on the atmospheric heat capacity and therefore the
atmospheric pressure. Our model is zero-dimensional because this allows fast
computation of a full 3.5-Gyr energy balance, whereas previous 3D models have
been unable to include time evolution because of computational limits
(Wordsworth et al., 2013). This model is timely because the Mars Atmosphere
and Volatile EvolutioN (MAVEN) mission is currently in orbit around Mars
collecting data on rates of atmospheric loss, and is presumably providing
better constraints on the atmospheric pressure over time on Mars which we can
use in our model to constrain the conditions that explain melting on post-
Noachian Mars (Lillis et al., 2017). We describe our model in Section II. In
Section III we present our results, which indicate that atmospheric pressure
has a more dominant effect on surface melt conditions than obliquity
variations. In Section IV we discuss assumptions of our model and possible
future extensions, and we summarize our findings in Section V.

##  II Model Description

###  II.1 Chaotic Orbital Histories

To investigate the influence of orbital variability on post-Noachian surface
liquid water, we created a model that combined the effects of orbital
variations, atmospheric loss, and solar brightening, as illustrated in Figure
1. Physically consistent orbital histories were constructed using the mercury6
N-body integrator and an obliquity code (Chambers, 1999; Armstrong et al.,
2014). The orbital histories recorded the orbital parameters of the simulated
Mars at 200-year intervals over the last 3.5 Gyr and included quasi-periodic
variations in orbital parameters (Kite et al., 2015). Most importantly, they
included changes in obliquity, which for Mars has probably jumped from low
mean values around $20\degree$ to high mean values around $40\degree$ between
two and nine times in its history on $\gtrapprox$ 200-Myr timescales (Li and
Batygin, 2014). The orbital histories were sampled at 2,000-year intervals.
Eight possible orbital histories were studied, all of which have obliquity
jumps at different times in their timelines, but which all end with
obliquities within 12° of the modern-day value of $25.19\degree$. Several
obliquity tracks were considered because the chaotic nature of the Solar
System means that the exact value of Mars’s obliquity is not known beyond a
few hundred million years ago. Therefore, we studied a distribution of
possible orbital histories to statistically constrain the influence of the
orbital parameters on melting. The eight orbital histories all produced
similar energy histories, so only two will be examined in detail in this
paper. Obliquity histories for tracks 1 and 2, which will be examined in more
detail later, are shown in Figure 2.

Figure 1:  A flowchart showing how key aspects of the model were incorporated.
Boxes outlined in red indicate new contributions from our model. Atmospheric
loss, solar brightening, and obliquity histories were calculated as described
in Section II.1 and combined in an energy balance described in section II.3.

Figure 2:  Obliquity histories for orbital tracks 1 (top) and 2 (bottom). The
overplotted red lines show 200-Myr averages.

###  II.2 Loss of Atmospheric Pressure

The chaotic orbital histories were combined with a model of solar brightening
over time and loss of atmospheric pressure (Bahcall et al., 2001). Although
there are several possible loss mechanisms for CO2 in past Mars, such as
chemical fixation of CO2 in deep aquifers (Chassefiére and Leblanc, 2011) and
basal melting of a CO2 ice cap (Kurahashi-Nakamura and Tajika, 2006), we chose
to focus on escape to space and assumed all atmospheric loss was due to escape
to space. Additionally, we neglected carbonate formation because it is thought
to be relatively ineffective in the post-Noachian, although this is debated
(Edwards and Ehlmann, 2015; Hu et al., 2015).

The current atmospheric pressure on Mars is approximately 600 Pa, but at the
temperatures that would allow liquid water on the surface of Mars, the
approximately 600 Pa of additional CO2 currently trapped in Martian polar caps
would be released into the atmosphere, and so we assume a modern-day
atmospheric pressure of 1200 Pa (Bierson et al., 2016). Atmospheric loss to
space was parameterized as a power law

$\frac{dP}{dt}=-k\left(\frac{t_{0}}{t}\right)^{\alpha}$ (1)

where $P$ is the atmospheric pressure in Pa, $t$ is time in Gyr since the Sun
formed, $t_{0}=4.56\text{ Gyr}$, and $k=1270\text{ Pa Gyr}^{-1}$ is a constant
chosen to match the parameterization to the current rate of Mars atmospheric
loss due to solar UV flux, which we estimated MAVEN measurements of hot atomic
oxygen escape (Lillis et al., 2017). This estimate of $k$ assumes that the
oxygen loss which was measured by MAVEN corresponds entirely to loss of CO2
over geologic time, and not to loss of other atmospheric constituents such as
H2O. We assume that the oxygen loss is due to loss of CO2 because Mars does
not have significant carbonate deposits and a thick CO2 atmosphere was likely
necessary on early Mars. However, MAVEN has yet to identify a major loss
channel for carbon, and some authors (e.g., Hodges, 2002) argue that Mars has
lost negligible amounts of carbon to space since the Noachian, so that almost
all of the oxygen loss corresponds to net loss of H2O to space. Therefore, our
assumption that all of the oxygen loss is due to loss of CO2 is the most
optimistic situation for producing higher past atmospheric pressures. $\alpha$
represents the increase in atmospheric loss rate in the past relative to the
current loss rate, due to increased solar UV flux earlier in the Sun’s
lifetime Tu et al. (2015). In the model, $\alpha$ was varied between
$\alpha=0$ and $\alpha=4.24$. A variety of values of $\alpha$ were
investigated because, while the solar brightening over time is well understood
(Bahcall et al., 2001), the atmospheric pressure over time is poorly
understood. A constant loss rate with $\alpha=0$ is unrealistic, as the past
atmospheric loss should have been faster because of higher solar UV flux (Tu
et al., 2015), flaring, and solar wind, but we include this value as a
reference. $\alpha=4.24$ was the maximum value considered because it leads to
an atmospheric pressure of approximately 2 bars at 3.5 Gya. 2 bars is
estimated to be an approximate upper limit on the atmospheric pressure at
about 3.5 Gya from the size distribution of craters formed at this time (Kite
et al., 2014). The current best estimate of a realistic value for $\alpha$
comes from an estimate of modern-day atmospheric loss on Mars based on initial
MAVEN results, combined with an estimate of the evolution of the Sun’s UV flux
over time (Lillis et al., 2017; Tu et al., 2015). The initial results from the
MAVEN mission suggest that the average modern rate of photochemical loss of
oxygen is $4.3\times 10^{25}$ s-1, with upper and lower bounds of $9.6\times
10^{25}$ s-1 and $1.9\times 10^{25}$ s-1, respectively. Our estimate considers
only photochemical loss of oxygen, but not ion escape or sputtering. (See
Lillis et al. (2015) for a review of these fluxes and how they are measured).
Currently published estimates indicate that the present-day escape rate due to
ion escape is approximately one order of magnitude lower than photochemical
loss (Barabash et al., 2007; Ramstad et al., 2015; Dong et al., 2017), and
that the present-day escape rate due to sputtering is smaller still (Leblanc
et al., 2015). These escape rates would differ in the past, because the solar
wind evolves as the Sun ages. However, in part because of the uncertain
rotation history of the Sun and the difficulty of directly observing stellar
winds from solar-analog stars, there is still significant uncertainty about
how to extrapolate solar wind interaction into the past (Johnstone et al.,
2015). Therefore, based on publications to date, it appears that photochemical
loss of oxygen is the dominant loss channel for mass loss from today’s Mars.
If either ion escape or sputtering played a proportionately more important
role in the past, then this would have the effect of increasing $\alpha$ in
our two-parameter model given in Equation 1.

Table 1 shows the maximum atmospheric pressure in the model (the pressure at
3.5 Gya) for $\alpha=3.22$, which is the mean value estimated by Lillis et al.
(2017) using data from MAVEN and an estimate of the history of solar UV flux
from Tu et al. (2015), and values that are 0.5 and 1 standard deviations away
from this value. To calculate values of $\alpha$ corresponding to 0.5 and 1
standard deviations away from $\alpha=3.22$, we combined error estimates from
Lillis et al. (2017) and Tu et al. (2015). While Lillis et al. (2017) quote
$1\sigma$ errors on their measurement, Tu et al. (2015) only provide
information on the 10th, 50th, and 90th percentiles of their calculations, and
so we assumed a Gaussian distribution to estimate $0.5\sigma$ and $1\sigma$
errors from Tu et al. (2015). The ongoing MAVEN mission will continue to
improve estimates of $\alpha$ (Lillis et al., 2017).

Table 1:  Maximum pressure in the 3.5-Gyr energy balance for different rates of atmospheric loss, as given by Equation 1, assuming that all oxygen loss measured by MAVEN corresponds to loss over geologic time of CO2 and not H2O (Lillis et al., 2017). $\alpha=3.22$ is the mean value of $\alpha$ inferred from initial MAVEN results, while $\alpha=2.20$, 2.71, 3.73, and 4.24 represent values that are $-1$, $-0.5$, 0.5, and 1 standard deviations away from this value, respectively (Lillis et al., 2017; Tu et al., 2015). $\alpha$ | 2.20 | 2.71 | 3.22 | 3.73 | 4.24
---|---|---|---|---|---
Maximum Pressure [bar] | 0.243 | 0.390 | 0.654 | 1.133 | 2.020

###  II.3 Snow Surface Energy Balance Model

To evaluate whether the orbital histories could produce melting conditions, we
wrote a zero-dimensional energy balance model in MATLAB for a snow surface on
Mars. We primarily considered snow surfaces at low latitudes beween $0\degree$
and $20\degree$ because we expect these latitudes to be the most stable
locations for snow and ice when the obliquity is higher than $40\degree$, but
we also considered slightly higher latitudes of $40\degree$ at which ice would
be stable for intermediate obliquities between $30\degree$ and $40\degree$
(Jakosky and Carr, 1985; Mischna et al., 2003; Forget et al., 2006). Because
we only consider melting when the obliquity is high, the pole temperature
should be high enough to prevent atmospheric collapse and to swiftly reverse a
preexisting collapse (Forget et al., 2013; Soto et al., 2015). We assumed a
snow surface at the melting point ($T=273$ K) and calculated whether the net
energy delivered to the snow surface per second was positive, which would
indicate melting was possible, or negative, which would indicate that the snow
surface would not be warm enough to melt. We assumed an albedo for dusty snow
of 0.3, a thermal inertia of $\approx$275 J m-2 K-1 s-1/2, and a relative
humidity of 0.25 (Kite et al., 2013). These assumptions will be discussed more
in Section IV.1. The energy balance model incorporated the effects of
insolation, upwelling longwave radiation from the surface, the greenhouse
effect from the CO2 atmosphere, latent cooling due to evaporation, sensible
cooling due to atmosphere-surface temperature differences, and conduction. We
do not consider the effects of meridional advection. Additionally, we do not
include warming from water ice clouds, because the amplitude of warming
depends on the optical depth and altitude of the clouds, and these are unknown
for early Mars (Ramirez and Kasting, 2017). Figure 3 shows a schematic of the
terms in the energy balance.

Figure 3:  Schematic of the fluxes included in the energy balance model for a
snow surface at T=273 K.

The incoming solar flux, $F_{SW}$, was estimated for a peak melting period
measured over the warmest four hours in the warmest season of the year. We
chose to calculate the energy balance only for the most optimal melting
conditions in the year because periodic warm conditions could still result in
melting for part of the year that could create the observed large alluvial fan
features (Clow, 1987). In contrast to previous studies (e.g., Schorghofer,
2008, which looked at averaged temperatures for ground ice), we only consider
peak temperatures because we are interested in studying surface ice and snow.
The effect of Rayleigh scattering was included (Kite et al., 2013).

The upwelling longwave radiation, $F_{LW}$, was modeled as a grey body
following the equation

$F_{LW}=\epsilon\sigma T^{4}$ (2)

where $\sigma$ is the Stefan-Boltzmann constant, the emissivity $\epsilon$ was
assumed to be 0.98, and the temperature was held fixed at the melting point
($T=273\text{ K}$). Values for all constants used in the energy balance
equations are given in Table 2.

Table 2:  Values of key constants used in the energy balance equations. Symbol |  Parameter |  Value |  Source
---|---|---|---
$\sigma$ |  Stefan-Boltzmann constant |  $5.67\times 10^{-8}$ W m-2 K-4 |
$\epsilon$ |  emissivity of ice |  0.98 |
$k$ |  thermal conductivity of snow |  0.125 W/m/K |  Carr and Head (2003)
$\tau$ |  length of Martian day |  88200 s |
$d$ |  diurnal skin depth |  0.22 m |  Turcotte and Schubert (2014)
$T_{s}$ |  surface temperature |  273.15 K |
$k_{a}$ |  thermal conductivity of atmosphere |  0.0138 W/m/K |  Vesovic et al. (1990)
$C_{p}$ |  specific heat capacity of atmosphere |  770 J/kg/K |  Kite et al. (2013)
$g$ |  Mars surface gravity |  3.7 m/s2 |
$m_{c}$ |  molar mass of CO2 |  0.044 kg/mol |
$m_{w}$ |  molar mass of H2O |  0.018 kg/mol |
$r_{h}$ |  atmospheric relative humidity |  0.25 |
$A_{v}$ |  von Karman’s constant |  0.4 |
$z_{anem}$ |  anemometer height |  5.53 m |  Kite et al. (2013)
$z_{0}$ |  surface roughness |  $10^{-4}$ m |  Dundas and Byrne (2010)
$L_{e}$ |  latent heat of evaporation |  $2.83\times 10^{6}$ J/kg |
$M_{w}$ |  molecular mass of H2O |  $2.99\times 10^{-26}$ kg |
$k$ |  Boltzmann’s constant |  $1.381\times 10^{-23}$ J/K |

The greenhouse effect was estimated based on a fit to a Mars GCM (Mischna et
al., 2012, 2013). In the GCM, we used the radtran model from Mischna et al.
(2012) with no water vapor and present-day topography to estimate the
greenhouse effect for atmospheric pressures of 6, 60, 600, and 1200 mbar, and
for obliquities of $15\degree$, $25\degree$, $35\degree$, $45\degree$, and
$60\degree$. The model used present-day values for Mars’s eccentricity
(0.0935) and longitude of perihelion ($251\degree$). The greenhouse effect
from the atmospheric CO2 should scale with the longwave flux out of the
surface, which is proportional to the temperature to the fourth power.
Additionally, for saturated bands such as the 15 $\mu$m CO2 absorption band,
the greenhouse effect is proportional to the square root of the pressure
because of pressure broadening (Goody and Yung, 1989). Therefore, a linear
regression was performed to find a fit to an equation in the form
$F_{gh}=a+bT^{4}\sqrt{P}$, where $T$ is the surface temperature in Kelvin, $P$
is the atmospheric pressure in Pa, and $a$ and $b$ are constants. Before
performing the fit, points that had elevations lower than $-6$ km or higher
than $11$ km, albedo less than 0.15, or thermal inertia less than $150$ J m-2
K-1 s-1/2 were removed from the model output. Additionally, we considered only
points with latitude between $-25\degree$ and $25\degree$, because we are
primarily interested in melting near the equator and because this resulted in
a data set with less scatter (Kraal et al., 2008). The final fit to the
greenhouse forcing gave

$F_{gh}=(9.981\pm 0.145)+(9.662\pm 0.017)\times 10^{-9}T^{4}\sqrt{P}$ (3)

with $F_{gh}$ in W m-2, $T$ in K, and $P$ in Pa. The fit had R${}^{2}=0.98$.

The GCM was also used to estimate a diurnal temperature range, which was used
to calculate conductive cooling. The conductive cooling calculated over the
same 4-hour peak melting period as the incoming solar flux was calculated
using

$F_{cond}=\frac{\Delta Tk\sin(7200\omega)}{14400\omega d}$ (4)

where $\Delta T$ is the diurnal temperature range estimated from the Mars GCM
(Mischna et al., 2012, 2013), $k$ is the thermal conductivity of the snow
surface, $\omega=\frac{2\pi}{\tau}$ s-1 is the frequency of temperature
oscillations, $\tau$ is the length of a Martian day, and $d$ is the diurnal
skin depth.

The forced and free sensible and latent cooling fluxes were calculated
following the parameterizations in Dundas and Byrne (2010), Hecht (2002), and
Ingersoll (1970). Free sensible cooling is due to the buoyancy of air near the
surface and was described by

$S_{fr}=0.14(T_{s}-T_{a})k_{a}\left[\left(\frac{C_{p}\nu_{a}\rho_{a}}{k_{a}}\right)\left(\frac{g}{\nu_{a}^{2}}\right)\left(\frac{\Delta\rho}{\rho_{a}}\right)\right]^{1/3}$
(5)

where $T_{s}$ is the surface temperature, $T_{a}$ is the atmospheric
temperature, $k_{a}$ is the atmospheric thermal conductivity, $C_{p}$ is the
specific heat capacity of air, $\nu_{a}$ is the air viscosity, $\rho_{a}$ is
the density of the atmosphere, $g$ is Mars gravity, and
$\frac{\Delta\rho}{\rho_{a}}$ is given by

$\frac{\Delta\rho}{\rho_{a}}=\frac{(m_{c}-m_{w})e_{sat}(1-r_{h})}{m_{c}P}$ (6)

where $m_{c}$ is the molar mass of CO2, $m_{w}$ is the molar mass of water,
$e_{sat}$ is the saturation vapor pressure at temperature $T_{s}$, $r_{h}$ is
the relative humidity of the atmosphere, and $P$ is the atmospheric pressure
in Pa (Kite et al., 2013). The thermal conductivity $k_{a}$ was determined by
interpolating data from Vesovic et al. (1990) to a temperature of $T=257$ K,
which was the average atmospheric temperature in our model. The atmospheric
temperature $T_{a}$ was determined from a fit to data for the warmest part of
the warmest day of the year from the Mars GCM described above, based on the
model in Mischna et al. (2012). These data were used because our model
determines the surface energy balance for the warmest part of the year.
Atmospheric temperatures were determined from a linear fit of $\ln{T_{a}}$ vs.
$\ln{P}$, which had R${}^{2}=0.85$. The air viscosity was given by

$\nu_{a}=(1.48\times
10^{-5})\left(\frac{RT_{bl}}{m_{c}P}\right)\left(\frac{240+293.15}{240+T_{bl}}\right)\left(\frac{T_{bl}}{293.15}\right)^{\frac{3}{2}}$
(7)

where $R$ is the universal gas constant and $T_{bl}$ is the temperature of the
atmospheric boundary layer (Dundas and Byrne, 2010). The boundary layer
temperature was determined from a fit to the Mars GCM (Mischna et al., 2012)
in the same manner as the fit to the atmospheric temperature $T_{a}$. The
density of the atmosphere was determined using the ideal gas law.

Forced sensible cooling is due to advection by the wind and was given by

$S_{fo}=\rho_{a}C_{p}u_{s}A(T_{s}-T_{a})$ (8)

where $u_{s}$ is the wind speed near the surface, and $A$ is given by

$A=\frac{A_{v}^{2}}{\ln(z_{anem}/z_{0})^{2}}$ (9)

where $A_{v}$ is von Karman’s constant, $z_{anem}$ is the anemometer height,
and $z_{0}$ is the surface roughness (Dundas and Byrne, 2010). The wind speed
was also determined from a fit to the Ames Mars GCM wind speeds (Kahre et al.,
2006). A linear fit of $\ln{u_{s}}$ vs. $\ln{P}$ for pressures of 7, 50, and
80 mbar was extrapolated to higher pressures. The fit had R${}^{2}=0.96$.

Free latent cooling was given by

$L_{fr}=1.57\times
0.14\,L_{e}\Delta\eta\rho_{a}D_{a}\left[\left(\frac{\nu_{a}}{D_{a}}\right)\left(\frac{g}{\nu_{a}^{2}}\right)\left(\frac{\Delta\rho}{\rho_{a}}\right)\right]^{1/3}$
(10)

where $L_{e}$ is the latent heat of evaporation, $\Delta\eta$ is the
difference between the water mass fractions in the atmosphere and surface, and
$D_{a}$ is the diffusion coefficient of H2O in CO2 (Dundas and Byrne, 2010).
The factor of 1.57 was added to the beginning of the equation to make the
amount of latent cooling match experimental results for evaporative losses at
current Mars pressure (Moore and Sears, 2006). $\Delta\eta$ was given by

$\Delta\eta=\frac{\rho_{sat}(1-r_{h})}{\rho_{a}}$ (11)

where $\rho_{sat}$ is the saturation vapor density, determined from $e_{sat}$
and the ideal gas law (Kite et al., 2013). The diffusion coefficient of H2O in
CO2 was given by

$D_{a}=(1.387\times
10^{-5})\left(\frac{T_{bl}}{273.15}\right)^{\frac{3}{2}}\left(\frac{10^{5}}{P}\right)$
(12)

where $P$ is in Pa (Dundas and Byrne, 2010).

Forced latent cooling was given by

$L_{fo}=L_{e}\left(\frac{M_{w}}{kT_{bl}}\right)u_{s}A(e_{sat}(1-r_{h}))$ (13)

where $M_{w}$ is the molecular mass of water and $k$ is Boltzmann’s constant
(Dundas and Byrne, 2010). Following Dundas and Byrne (2010), Williams and
Malin (2008), and Toon et al. (1980), we sum all of the sensible and latent
cooling terms together in our energy balance model instead of considering one
dominant term.

The final energy balance is given by
$F_{net}=F_{SW}-F_{LW}+F_{gh}-F_{cond}-S_{fo}-S_{fr}-L_{fo}-L_{fr}$. A
positive value of $F_{net}$ represented that there was excess energy available
to melt snow on the surface, while a negative value of $F_{net}$ indicated
that melting was prohibited. Figure 4 shows each of the terms in the energy
balance model and the overall surface energy for a variety of atmospheric
pressures at 77% modern-day solar luminosity, which is approximately the solar
luminosity at 3.5 Gya. At very low pressures, latent cooling dominates over
most of the other terms and prevents melting. At higher pressures, the
greenhouse forcing dominates over latent cooling, and melting occurs. This is
because the greenhouse forcing scales as $\sqrt{P}$, while the free latent
cooling scales as $P^{-2/3}$. Figure 5 shows the minimum pressure to permit
melting as a function of time, assuming zero obliquity and values of
eccentricity and longitude of perihelion equal to those for modern-day Mars.
The minimum pressure to allow melting at 3.5 Ga is 0.676 bar. At earlier
times, with the orbital parameters held constant, the lower solar luminosity
means that a higher pressure is required in order to allow melting.

Figure 4:  Magnitude of each term in the energy balance versus pressure, at
77% modern solar luminosity, which is the approximate solar luminosity at 3.5
Gya. Solid/dashed lines indicate warming/cooling. The lines show incoming
shortwave solar radiation (light blue), outgoing longwave radiation (orange),
greenhouse forcing (yellow), conductive cooling (purple), free latent cooling
(green), forced latent cooling (magenta), free sensible cooling (dark blue),
forced sensible cooling (dark red), and the magnitude of the net forcing
(black). Small wiggles in the net forcing are artifacts of interpolation to
calculate incoming shortwave radiation. The net forcing is negative for
$P<56039$ Pa because latent cooling dominates over greenhouse forcing, and
positive for $P>56039$ Pa. Figure 5:  Minimum pressure at which surface
melting is permitted for a flat surface at $0\degree$ latitude as a function
of time, assuming zero obliquity and values of eccentricity and longitude of
perihelion equal to those for modern-day Mars. This curve was smoothed to
remove artifacts of the interpolation to calculate incoming shortwave
radiation. Earlier in time the minimum pressure to allow melting is higher
because the Sun was fainter.

For each possible orbital history, an energy balance was computed at latitudes
of $0\degree$, $20\degree$, and $40\degree$ in order to understand the
relative importance of obliquity and atmospheric pressure changes on a variety
of latitudes throughout the areas where young fluvial features are seen on
Mars. Additionally, the water source regions we are modeling are on sloped
surfaces with slopes of approximately $20\degree$, so energy balances were
computed for a surface at $20\degree$ with a $20\degree$ pole-facing slope and
for a surface at $40\degree$ with both a $20\degree$ pole-facing and
$20\degree$ equator-facing slope.

##  III Results

###  III.1 Relative Impact of Atmospheric Pressure and Obliquity

Figure 6 shows an energy balance history for a surface at $20\degree$ with a
$20\degree$ pole-facing slope, using the obliquity history for track 1. The
blue curve shows an energy balance at constant pressure, while the orange and
yellow curves show energy balances with atmospheric loss parameterized by
Equation 1 with $\alpha=3.22$ ($P$=0.654 bars at 3.5 Gya) and $\alpha=3.73$
($P$=1.133 bars at 3.5 Gya), respectively. Black lines show 200-Myr averages,
which average over variations in eccentricity, obliquity, and longitude of
perihelion. The upward trend that is visible in the earliest 1.5 Gyr of the
fit to the blue curve demonstrates the effect of solar brightening. The
downward trend of the fits to the orange and yellow curves show the effect of
atmospheric loss. The bump in all curves around 3.5 Gyr shows the effect of a
rapid change from a low mean obliquity around $10\degree$ to a high mean
obliquity around $40\degree$, which increases the energy available for melting
because a pole-facing surface at a latitude of $20\degree$ will receive more
direct sunlight for an obliquity around $40\degree$ than for an obliquity
around $10\degree$. For almost all times and values of $\alpha$, the energy
balance is negative, indicating that melting is prohibited. In all potential
orbital histories considered, only $\alpha\gtrapprox 3.73$ allows melting,
which indicates that atmospheric pressure has a stronger influence on whether
melt is permitted at the surface than obliquity. However, this value of
$\alpha$ is within $0.5\sigma$ of the value inferred from initial MAVEN data,
indicating that large post-Noachian alluvial fans and deltas might be
explained by a rate of atmospheric loss only slightly higher than that
estimated by Lillis et al. (2017).

Figure 6:  Energy balance over time for obliquity track 1, for a surface at a
latitude of $20\degree$ with a $20\degree$ pole-facing slope. The blue curve
shows energy balance at constant pressure. The orange and yellow curves show
energy balances for pressure loss parameterized by Equation 1 with
$\alpha=3.22$ and $\alpha=3.73$, respectively. $\alpha=3.22$ is the best
estimate of $\alpha$ from initial MAVEN results, found by combining an
estimate from MAVEN of the dependence of the loss rate on solar EUV flux with
an estimate of the rate at which solar UV flux has changed over time (Lillis
et al., 2017; Tu et al., 2015). $\alpha=3.73$ is 0.5 standard deviations away
from the best estimate of $\alpha=3.22$ inferred from MAVEN data(Lillis et
al., 2017; Tu et al., 2015). Black lines show 200-Myr averages, which average
over orbital variations and show changes due to solar brightening and
atmospheric loss. The increase in energy around 3.5 Gyr is where the obliquity
rapidly changes from a mean obliquity around $10\degree$ to a mean obliquity
around $40\degree$.

###  III.2 Comparison to Key Geologic Constraints

For the high values of $\alpha$ that allowed melting, we compared the amount
of melting and intermittency of melt periods to three key geologic
constraints. First, the amount of fluvial sediment transported by large post-
Noachian alluvial fans indicates that the longest runoff event must have had
liquid water at the surface for $>10^{4}$ years (Irwin et al., 2015). Here, we
use runoff event to mean a time interval during which seasonal meltwater
runoff sustains lakes at an approximately constant lake level (Irwin et al.,
2015). Second, the presence of olivine in the catchments of alluvial fans and
deltas strongly suggests that the most recent runoff event must have lasted
$<10^{7}$ years (Stopar et al., 2006; Kite et al., 2017a). Third, counting
interbedded craters indicates that there must have been $>10^{7}$ years
between the first and last runoff events (Kite et al., 2017b). Some estimates
of melting in young channels, like those found in Lyot crater, suggest that
the time between the first and last runoff events lasted for much longer than
$10^{7}$ years (Dickson et al., 2009).

In order to produce melt, high temperatures are required, but the snow and ice
also need to be available in the location where melting can occur. At
obliquities greater than $40\degree$, snow and water ice are most stable near
the equator, while for lower obliquities they are more stable at the poles
(Jakosky and Carr, 1985; Mischna et al., 2003; Forget et al., 2006). If the
amount of ice contained in the polar caps was spread over the equatorial
region, it would produce an ice sheet about 100 m - 1 km thick, which would
melt or sublimate in 100 - 10,000 years (Madeleine et al., 2009). This melting
time is much less than the length of warm periods produced by obliquity
changes in our model, and so the timescale for the ice to migrate to the
coldest location on the planet is less than the timescale at which changes in
obliquity shift the latitude of the coldest point on the planet. This means
that when the obliquity is high, we can make the assumption that all of the
snow and ice are found near the equator, where snow and ice are most stable.
Therefore, for latitudes of $0\degree$ and $20\degree$, we assumed that
melting could only have happened when the obliquity was greater than
$40\degree$. For a latitude of $40\degree$, we assumed that snow and ice would
only be available when the obliquity was between $30\degree$ and $40\degree$.
Additionally, we assumed that runoff would not begin to occur until some
amount of liquid water had pooled and refroze within the snow to create an
impermeable ice layer for runoff (Woo, 2012), and so we only began recording a
melt period when the energy available for melting went above 15 W/m2.

Figure 7 shows a plot of the net surface energy vs. the number of years that
energy was exceeded for obliquity track 1. This plot shows results for a
$20\degree$ pole-facing sloped surface at $20\degree$ latitude, with values of
$\alpha$ within 1 standard deviation of the mean value inferred by Lillis et
al. (2017). Only years for which the obliquity was greater than $40\degree$
are counted, because these are the only years when ice would be available for
melting. The vertical dashed red line indicates a surface energy of $15$ W/m2,
above which runoff would occur. For $\alpha=3.73$ ($P$=1.133 bars at 3.5 Gya)
and $\alpha=4.24$ ($P$=2.020 bars at 3.5 Gya), some runoff is produced when
the orbital history is at high obliquity. For $\alpha=3.73$, the longest
melting event lasts ${4.44\times 10^{7}}$ years, the most recent melting event
lasts ${2.40\times 10^{4}}$ years, and the total time between the first and
last melting events is ${2.78\times 10^{8}}$ years. For $\alpha=4.24$, these
values are ${2.55\times 10^{8}}$ years, ${1.80\times 10^{4}}$ years, and
${5.10\times 10^{8}}$ years, respectively. Therefore, both of these values of
$\alpha$ produce melt periods that match the three geologic constraints. For
$\alpha=3.73$, the rate of runoff production is approximately 0.05 mm/hr for
approximately 18 Myr, which is consistent with estimated runoff production
rates of 0.03-0.4 mm/hr (Irwin et al., 2015; Morgan et al., 2014; Dietrich et
al., 2017).

Figure 7: Energy available for melting vs. the number of years that energy was
exceeded for obliquity track 1 at $20\degree$ latitude with a $20\degree$
pole-facing slope, for $\alpha=2.20$, 2.71, 3.22, 3.73, and 4.24. For this
graph, only times with obliquity $>40\degree$ are plotted, because snow and
ice will only be available for melting at low latitudes during times of high
obliquity (Kite et al., 2013). The red dashed line indicates a net energy of
15 W/m2, above which runoff can occur. Only values of $\alpha$ above about
3.73 allowed any runoff.

Smaller values of $\alpha$, corresponding to lower past pressures, do not
allow melting under the current model, but this model only considered
greenhouse forcing from a pure CO2 atmosphere. Some amount of non-CO2
greenhouse forcing could increase the surface energy balance enough that lower
values of $\alpha$ would still produce some melting, as discussed in Section
IV.2.

The distribution of the length of melting periods shows how the melt
conditions depend on orbital cycles. Figure 8 shows a histogram of the length
of melting periods in obliquity track 1 for all latitudes and slopes
considered for $\alpha=3.73$. During all of these melting periods, melting
occurs every year during the warmest part of the year. The majority of time
periods when melting is allowed last around 100,000 years, which is about the
timescale of a single peak-to-trough oscillation of obliquity, eccentricity,
or longitude of perihelion. Therefore, most of the melting occurs in short
time periods about 100 kyr long during optimal orbital conditions. For
example, at high latitudes much of the melting occurs when the eccentricity
and longitude of perihelion line up such that Mars is near perihelion at
solstice and the obliquity is such that the amount of direct sunlight is
maximized during the warmest part of the day. There are fewer melting periods
of 1 to 10 Myr duration, which correspond to the length of larger oscillations
in obliquity and eccentricity. The couple of periods that last longer than 50
Myr corresponds to long periods of optimal conditions due to a much higher
mean obliquity. Between these intermittent melting periods were several dry
periods, which could affect the ability of organisms to survive on the surface
of Mars. All eight obliquity tracks contained at least one such dry period
longer than 10 Myr, which is long enough that the top few meters of soil would
be sterilized by galactic cosmic radiation even for radiation-resistant
microbes like D. radiodurans (Hassler et al., 2014).

Figure 8: Histogram of the length of periods in which melting is allowed for
obliquity track 1 for all latitudes and slopes considered, and for
$\alpha=3.73$. In all cases, we assume that a melt period begins when the
energy balance goes above 15 W/m2 and ends when it goes below 0 W/m2. We also
assume melting can only occur at low latitudes for obliquities $>40\degree$
and at higher latitudes for obliquities between $30\degree$ and $40\degree$.
See text for discussion.

##  IV Discussion

###  IV.1 Model Assumptions

Our model made several assumptions about physical characteristics of Mars’s
surface and atmosphere which made it more likely to produce liquid water. Here
we consider the key assumptions and how changes to these assumptions could
affect our conclusions. First, the surface albedo was assumed to be 0.3.
Assuming a higher albedo value would reduce the net energy flux into the
surface, and so less melting would be permitted. Second, we assumed a thermal
inertia of $\approx$275 J m-2 K-1 s-1/2. A higher thermal inertia value would
also reduce the time over which melting was permitted for a given value of
$\alpha$ because the surface would conductively cool faster. Third, we assumed
no refreezing or infiltration losses of meltwater, both of which would
decrease the amount of runoff produced. These assumptions combine to create
the most optimal conditions for producing runoff.

We also assumed the latitudes where ice is stable change abruptly at
obliquities of $40\degree$ and $30\degree$. In other words, we assumed that at
an obliquity just below $40\degree$, all ice would be located at middle or
high latitudes, and at an obliquity just above $40\degree$, all ice would be
in the equatorial region. In reality, this transition may be less abrupt.
However, since we found that atmospheric pressure was the primary control on
melting, the exact value of obliquity at which ice migrates to the equator
should not have a strong influence on the results. Additionally, it is
possible that, at high pressures, water ice will be located near the equator
for all obliquities (Wordsworth, 2016). If this were the case, melting at the
equator would potentially occur even at lower obliquities, because water ice
would still be in the equatorial region.

###  IV.2 Open Questions

We found that a rate of atmospheric loss consistent with that inferred by
Lillis et al. (2017) to within 0.5 standard deviations allowed melting and
produced an amount of runoff which matched three key geologic constraints
(Lillis et al., 2017; Tu et al., 2015; Irwin et al., 2015; Stopar et al.,
2006; Kite et al., 2017a, b). However, atmospheric loss by escape to space is
not the only process that may have affected the energy balance on Mars’s
surface. Non-CO2 forcing from a variety of sources could have permitted
melting at lower atmospheric pressures. Our model did not include clouds,
which can warm the surface (Ramirez and Kasting, 2017). At higher obliquities,
when water ice is most stable in the equatorial region, the warm temperatures
at those low latitudes might produce more water vapor, which could form into
cirrus clouds that would warm the planet. The water vapor would also provide
direct vapor warming on the order of a few Kelvin (Mischna et al., 2013). We
also did not consider warming from other minor atmospheric constituents. In
particular, CO2-H2 and CO2-CH4 collision-induced absorption (CIA) can have a
significant impact on the greenhouse forcing (Wordsworth et al., 2017).
Because melting only occurs before 3.3 Gya in our model, additional warming
may also be necessary to explain observations of anomalously young ($<2$ Gya)
supraglacial channels at the low-elevation Lyot crater (Dickson et al., 2009)
and in isolated cases elsewhere (Fassett et al., 2010). Some very young
alluvial fans, like those in the 5-Myr old Mojave crater, are likely related
to impact processes (Williams and Malin, 2008).

Another possible explanation for these young fluvial features is higher
atmospheric pressure than was estimated in our model. The atmospheric pressure
could have been higher in the past if CO2 was lost through other mechanisms,
such as chemical fixation in deep aquifers (Chassefiére and Leblanc, 2011) or
basal melting of a CO2 ice cap (Kurahashi-Nakamura and Tajika, 2006; Longhi,
2006). Higher atmospheric pressure would increase the amount of runoff
compared to our calculations. Additionally, volcanic degassing could increase
the amount of melt by adding more CO2 (and other greenhouse gases such as CH4,
SO2, and H2S) into the atmosphere. However, Stanley et al. (2011) calculate
that Mars magmas have less CO2 than on Earth because the Mars mantle is more
reducing. Stanley et al. (2011) estimate that the volcanic CO2 content is
approximately 50-70 ppm, which would correspond to about 30 mbar of volcanic
outgassing. If the Martian atmosphere was more oxidizing, the volcanic CO2
content could be as high as 500-700 ppm, but the pressure contribution would
still be relatively small, about 300 mbar (Stanley et al., 2011). These
Stanley et al. (2011) numbers imply a substantial downward revision from
Craddock and Greeley (2009), who estimated that Martian volcanoes contained
0.7 wt.% CO2. Carbonate formation may also have been underestimated, which
would mean higher past atmospheric pressures could exist without invoking
larger loss rates due to solar UV flux (Hu et al., 2015).

##  V Conclusions

We wrote a minimal zero-dimensional energy balance model containing only
effects of a CO2 atmosphere, solar brightening, and obliquity changes, to
investigate whether conditions on the surface of Mars could have produced
enough melting to explain observed large ($>10\text{ km}^{2}$) alluvial fan
features in the last 3.5 Gyr. We find that melting is only permitted for thick
atmospheres that require high rates of atmospheric loss, and that the effect
of high atmospheric pressure is more important than varying values of
obliquity in producing melt over the history of Mars. However, if we make the
end-member assumption that all atmospheric loss is due to escape to space and
all escaping oxygen is from CO2, rates of atmospheric loss consistent to
within 0.5 standard deviations with the best estimate of atmospheric loss
based on initial results from the MAVEN mission produce melting. Additionally,
the amount and timing of runoff from this melting matches three specific,
recently-obtained key geologic constraints on the formation of alluvial fans
(Lillis et al., 2017; Tu et al., 2015; Irwin et al., 2015; Stopar et al.,
2006; Kite et al., 2017a, b).

Although our model only produced melting for very high atmospheric pressures,
additional warming due to, for example, cirrus clouds, water vapor warming, or
collision-induced absorption could warm the surface enough to allow melting at
lower pressures (Ramirez and Kasting, 2017; Mischna et al., 2013; Wordsworth
et al., 2017). Further research into non-CO2 warming can determine whether it
produces a large enough effect to allow melting at lower past atmospheric
pressures.

###### Acknowledgements.

The authors would like to thank Rob Lillis, Itay Halevy, John Armstrong, and
the University of Chicago Research Computing Center. The authors would like to
thank the referees, whose comments greatly improved the paper. This paper
contains no new data, but the energy balance model is available at
https://github.com/meganmansfield/MarsEnergyBalance. We are grateful for
support from NASA grant NNX16AG55G.

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