PUBLICATIONSJournal of Geophysical Research: Planets
RESEARCH ARTICLE
10.1002/2015JE004943
Key Points:• Tidal elongation from orbital decay
produces surface stresses• These stresses align with the observed
grooves on the surface, implying aconnection to tides• Our model implies that Phobos’interior is weak with a stronger elastic
outer layer

Tidal disruption of Phobos as the causeof surface fractures
T. A. Hurford1, E. Asphaug2, J. N. Spitale3, D. Hemingway4, A. R. Rhoden2,5, W. G. Henning1,6, B. G. Bills7,
S. A. Kattenhorn8, and M. Walker9
1

NASA Goddard Space Flight Center, Greenbelt, Maryland, USA, 2School of Earth and Space Exploration, Arizona State
University, Tempe, Arizona, USA, 3Planetary Science Institute, Tucson, Arizona, USA, 4Department of Earth and Planetary
Sciences, University of California, Santa Cruz, Santa Cruz, California, USA, 5Johns Hopkins Applied Physics Laboratory, Laurel,
Maryland, USA, 6Center for Research and Exploration in Space Science and Technology, University of Maryland, College
Park, Maryland, USA, 7Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA, 8Departmentof Geological Sciences, University of Alaska Anchorage, Anchorage, Alaska, USA, 9Earth, Planetary, and Space Sciences,
University of California, Los Angeles, Los Angeles, California, USA

Correspondence to:T. A. Hurford,
terry.a.hurford@nasa.gov

Abstract Phobos, the innermost satellite of Mars, displays an extensive system of grooves that are mostly

Citation:
Hurford, T. A., E. Asphaug, J. N. Spitale,
D. Hemingway, A. R. Rhoden, W. G.
Henning, B. G. Bills, S. A. Kattenhorn, andM. Walker (2016), Tidal disruption of
Phobos as the cause of surface fractures,
J. Geophys. Res. Planets, 121, 1054–1065,
doi:10.1002/2015JE004943.

symmetric about its sub-Mars point. Phobos is steadily spiraling inward due to the tides it raises on Mars
lagging behind Phobos’ orbital position and will suffer tidal disruption before colliding with Mars in a few
tens of millions of years. We calculate the surface stress ﬁeld of the deorbiting satellite and show that the ﬁrst
signs of tidal disruption are already present on its surface. Most of Phobos’ prominent grooves have an
excellent correlation with computed stress orientations. The model requires a weak interior that has very low
rigidity on the tidal evolution time scale, overlain by an ~10–100 m exterior shell that has elastic properties
similar to lunar regolith as described by Horvath et al. (1980).

Received 21 SEP 2015
Accepted 20 MAY 2016
Accepted article online 27 MAY 2016
Published online 22 JUN 2016

1. Introduction
Phobos’ global network of ~10–100 m wide, kilometers-long linear features (“grooves”) can be seen in Figure 1.
Some resemble pit chains or graben while others resemble secondary crater chains (catenae). Some appear toradiate from the largest crater, ~9 km Stickney [Thomas et al., 1979]; however, most grooves have no geometric
relationship to Stickney, indicating that they did not form because of that impact event. Differing morphologies
suggest that either a range of evolutionary progression or more than one mechanism is responsible for groove
formation. Parallel grooves overprint all of Phobos’ major craters, including Stickney, crossing their ejecta
deposits, implying that groove formation postdates major crater formation and takes place in regolith.
Phobos’ bulk density ρav = 1880 kg/m3 and semimajor axis a = 9376 km place it signiﬁcantly inside the Roche
limit (3.19 RMars) of Mars for a ﬂuid body [Witasse et al., 2014; Dobrovolskis, 1982]. The limited available spectroscopy reveals a generally asteroid-like surface composition with little, if any, evidence of hydration [Pieters et al.,2014; Rivkin et al., 2002]. Details of the global surface geomorphology at the scale of groove formation are
unknown; the best images from Mars orbiters are a few meters/pixel locally but do not provide global coverage[Basilevsky et al., 2014]. Thermal inertia indicates that the surface is mostly powder to ~0.1–1 m depth and may
extend much deeper [Lunine et al., 1982]. On the Moon, the powdery regolith is hypothesized to behave
seismically as a uniform elastic layer to depths of ~100 m [Horvath et al., 1980], perhaps analogous to Phobos.
Many of Phobos’ grooves feature pit chains. Several mechanisms have been suggested for their formation,such as thin ejecta sheets from Mars [Murray and Heggie, 2014], cohesionless regolith draining into dilated
fractures [Horstman and Melosh, 1989], gas jets, or boulders bouncing downhill [Wilson and Head, 1989].
While some pitted grooves resemble secondary ejecta chains, or catenae, there is no suitable source crater
on Mars or on Phobos. Moreover, a relationship to major Martian craters would imply that the grooves arebillions of years old, contradicting observations of their relative geologic youthfulness.

©2016. American Geophysical Union.
All Rights Reserved.

HURFORD ET AL.

Phobos’ orbit decays due to the tides it raises on Mars lagging behind Phobos’ orbital position [Kaula, 1964;
Mignard, 1981; Dobrovolskis, 1982; Burns, 1978; Bills et al., 2005] and shortly after the Viking spacecraftobtained the ﬁrst geomorphic images of Phobos, it was proposed that stresses from orbital decay causegrooves [Soter and Harris, 1977]. But, assuming an internally homogeneous Phobos, it proved impossible
to account for the buildup of failure stress in the exterior regardless of the value assumed for Phobos’ rigidity

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10.1002/2015JE004943(see Appendix A). Also, stress from orbital
decay follows a 1/a3 dependency, where a isthe semimajor axis (currently 2.77 RMars,
decaying at ~2 m per century). Hence, thegreatest tidal stress would have built up mostrecently, whereas the geological record showsthat the most prominent grooves range inages and are not always the youngest features[Weidenschilling, 1979]. Therefore, the tidal
model languished. Here we revisit the tidal origin of surface fractures with a more detailed
treatment that shows the production of significant stress in a surface layer, with a verystrong correlation to the geometry of grooves.
Figure 1. Phobos’ surface is covered with parallel linear featuresand pit/crater chains, collectively called grooves. Stickney crateris shown in the lower left of the image. Phobos has a triaxial shapeof 13.0 km × 11.4 km × 9.1 km in radius. Image Credit: Viking
Project, Jet Propulsion Laboratory, NASA.

2. Tidal Stress andGroove Orientations

Tidal theory is built on the assumption that
undeformed planets are spherical bodies andthat their tidal deformation is small compared to their radii [Sabadini and Vermeersen, 2004]. In Phobos’ case(13.0 km × 11.4 km × 9.1 km in radius), the long and short axes deviate from a sphere by ~20% [Willner et al.,2014]. The tide-raising potential acting on Phobos can be written as an expansion of Legendre polynomials[Sabadini and Vermeersen, 2004]. The lowest-order term is a second-order Legendre polynomial and is the tideraising potential of a spherical body with average radius. We assume that perturbations on the tide-raisingpotential due to Phobos’ irregular shape are accounted for by higher-order Legendre polynomial terms. For thisstudy, the resulting tidal deformation from gravitational terms higher than second order are assumed to besmall. We also assume that stresses due to the formation of Phobos’ irregular shape have relaxed away prior
to any successive tidal deformation and that the eccentricity of Phobos has remained historically small.
We treat Phobos as a two-layered body made up of a homogeneous interior and a discrete incompressible
outer layer. Surface stresses are computed using a thin-shell spherical approximation [Sabadini andVermeersen, 2004; Kattenhorn and Hurford, 2009] (see Appendix B–Appendix D). The horizontal strain of thisshell, as global shape evolves due to orbital decay, produces stresses on the surface given by
9Mμo h2 1
1σ θθ ¼
ð5 þ 3cos2θÞ(1)
88πρav a3f a3i

9Mμo h2 1
1
σ φφ ¼ 
ð1  9cos2θÞ(2)
88πρav a3f a3i
where θ is the colatitude measured with respect to the axis through the center of the tidal bulge (i.e., θ = 0°
corresponds to the sub-Mars point on Phobos’ surface) [Melosh, 1977; Leith and McKinnon, 1996]. We deﬁnetension as positive and compression as negative. The surface stress σ θθ is along the surface in the θ-direction,
while σ φφ is along the surface orthogonal to σ θθ. Here M is the mass of Mars, and ai and af are the starting andﬁnal semimajor axes during any span of orbital decay.
The magnitude of the tidal stress on the surface is proportional to the displacement Love number h2, which
describes Phobos’ tidal response. The value of h2 depends on internal structure and material properties of ourincompressible two-layer model (see Appendix B and Appendix D). Both layers have ρav = 1880 kg/m3 and the total
radius, R, is 11.1 km. The outer, stiffer layer is modeled as 100 m thick based on the thickness implied by the drainage of regolith into underlying cracks [Horstman and Melosh, 1989] or possibly into pore space in the interior.
The rigidities of the two layers are not well constrained. However, we assume that the inner layer is strong
enough to support the overburden pressure, ρav gR, where g is the surface gravity, resulting in an overburdenof ~105 Pa. The inner layer is conservatively modeled with a rigidity μi = 106 Pa (see Discussion section). The
outer layer is modeled with μo = 108 Pa, based on the assumed regolith density and knowledge of S wave
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velocities that are found to be uniformly ~200 m/s (μ = ρv2) in Lunar
regolith [Horvath et al., 1980]. For
these model parameters, h2 = 0.021,
which corresponds to a tidal bulge
height of ~24 m at the sub-Mars point.

Figure 2. Stresses in a thin elastic shell computed for the last 10% of orbital
decay for a spherical Phobos with a weak interior. The sub-/anti-Marsregions (centered at 0 and 180° longitude, respectively) experience tension
in both principal stress directions. Over the rest of Phobos’ surface, stresses
radial to the tidal axis are tensile, while stresses concentric about the tidal
axis are compressional. Tension fractures would tend to be concentric
about the sub-/anti-Mars regions. Within the sub-/anti-Mars regions, thetensile stress is nearly isotropic near the tidal axis, and no preferredorientation is predicted.

Figure 2 shows the stress ﬁeld predicted
on Phobos from orbital migration starting at 3 RMars to the current orbital
radius (2.77 RMars). Tensile tidal stresses
range from ~50 kPa near the sub-/antiMars regions to >10 kPa at points in
between. In the sub-/anti-Mars regions,both principal stresses are tensile,whereas in the interbulge regions, stresses radial to the tidal axis are tensile andstresses concentric about the tidal axis
are compressive. This pattern qualitatively ﬁts the orientation of grooves on
Phobos’ surface.

Extension of work done on Mercury[Melosh, 1977] would predict strikeslip failure on Phobos due to the tidal
stress if the outer layer tensile strengthis sufﬁciently high. However, if tidal
tension is comparable to or greaterthan the tensile strength of the surface layer, tensile failure that would produce fractures perpendicular to the tensile stress is a possible modeof failure [Helfenstein and Parmentier, 1983]. This type of surface failure has been extensively researched on icy
satellites [McEwen, 1986; Greenberg et al., 1998]. Moreover, experiments with material designed to simulate
Lunar regolith suggest that tensile failure occurs at stresses as low as 1 kPa [Arslan et al., 2008]. Therefore,
we propose that the orbital decay tidal stress can drive tensile failure of the surface in a similar manner.
This hypothesis means that the principal tensile stress should be normal to the grooves. To test this, we
mapped ~200 of the most prominent linear surface features (Figure 3). Using the latitude, longitude, andstrike for multiple points along these fractures, we calculated the principal tidal stress tensor experienced
along each fracture in response to orbital decay. The principal stress tensor was then decomposed relative
to the local strike of the fracture, yielding the normal stress across each fracture. The magnitudes of thecomputed stresses, as well as the correlation between principal stress orientations and the azimuths ofobserved fractures, provide the critical tests of the tidal fracturing model.
There is a strong correlation between the surface stress ﬁeld and the geometry of Phobos’ grooves (Figure 3).
The majority of grooves (warmer colors in Figure 3) experience a tensile stress normal to their strikes, indicating that they could have formed (or are still forming) by tensile failure of the outer shell. Note that we do notmodel failure explicitly nor the stress perturbation associated with the opening of a fracture.
The orientations of most grooves on Phobos are closely aligned with the modeled tidal stress ﬁeld. In regionsof anisotropic horizontal stress, we expect the surface to fail perpendicular to the direction of maximum
tension. Indeed, many of the grooves experiencing tension in these regions are aligned almost perfectly perpendicular to the direction of maximum tidal tensile stress (redder colors in Figure 3). In the sub-/anti-Mars
equatorial regions, both principal stresses are tensile; however, failure will still occur perpendicular to the
maximum tension as soon as the tensile strength of the surface layer is reached. Where the principal stresses
have nearly equal magnitudes (i.e., approaching an isotropic stress ﬁeld), no preferred formation orientation
is predicted. Preferred fracture orientations in these regions are therefore less reliably compared to the tidal
stress ﬁeld (dark gray lines in Figure 3) and may be inﬂuenced by local sources of stress. For example, stresses
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Figure 3. Locations and orientations of grooves mapped for this project. The orientation of each groove can be comparedto the tidal principal stress ﬁeld to assess goodness of ﬁt to the principal tension direction. A majority of the observed
fractures experience tensile stress normal to their strike (warmer colors), while a cluster of fractures in the leading
hemisphere are oriented such that the normal stress would be compressional across them (cooler colors). Most fractures intension, and in regions with clear large asymmetry in principal stresses, are aligned nearly perfectly with the principal
stresses; i.e., the principal tensile stress is perpendicular to the fracture (redder colors). The orientation ﬁts of fractures in the
sub-/anti-Mars regions (gray) are difﬁcult to assess since the tidal stress is almost isotropically tensile. Any preferred formation orientations in these areas may be inﬂuenced by local stress anisotropy (e.g., Stickney crater, outlined for clarity).

from the impact that produced Stickney could have combined with tidal stresses to preferentially produce
east-west grooves in the sub-Mars region near Stickney crater or if the grooves postdate Stickney, topography of the crater may have inﬂuenced the surface stress concentration.

3. Discussion
There is a relative paucity of tidally aligned grooves S and SE of the sub-Mars point. This might be attributed
to the absence of a cohesive material in this location (e.g., coarser regolith [Scheeres et al., 2010]). Limited
spectroscopy data show regional compositional differences that might correspond to varying regolith elastic
properties [Witasse et al., 2014; Rivkin et al., 2002]. Nonaligned grooves are not random (cooler colors in
Figure 3) but cluster in the equatorial zones between the sub-/anti-Mars tidal bulges; they require an alternative or additional explanation for their formation. Considering that the predominant formation orientations
are at high angles to the maximum compression direction, such grooves could represent contractional failureof surface materials. Alternatively, they may have formed under different stress conditions, such as the stresses produced when Phobos experienced nonsynchronous rotation before becoming tidally locked[Weidenschilling, 1979]. They are found predominantly in the orbital leading hemisphere, which might also
allow for their formation by sweep up of coorbiting material derived from Phobos, Deimos, or even Mars
[Murray and Heggie, 2014].
Morphologically, grooves resemble extensional tectonic features. The majority of grooves form remarkably
linear, consistent topographic depressions in distinct sets. Groove widths and spacings are relatively homogeneous, as is typical of extensional fracture sets in a uniform thickness brittle layer. Scalloped edges along
many grooves are consistent with pit chain morphologic evolution, whereby initially isolated pits merge
along strike to create a continuous feature, as is described across the solar system [Horstman and Melosh,
1989; Ferrill et al., 2004; Wyrick et al., 2004; Nahm and Kattenhorn, 2015]. Initially isolated pits in loose regolithalign above an underlying dilational crack, into which regolith progressively drains. The circular edges of theoriginal pits create the scalloped margin effect, although linear margins may also develop if the underlying
structure is a dilational fault or graben [e.g., Ferrill et al., 2004; Grant and Kattenhorn, 2004]. Bright margins
along shadowed edges of some grooves imply upturned ﬂanks, which can be produced by footwall uplift
along normal faults. Inferred extensional deformation thus justiﬁes comparisons to the principal extension
direction to determine the viability of an orbital decay driving mechanism.

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Where two or more crosscutting sets withdisparate orientations exist (Figure 3),
multiple stages of tectonic extension
and direction are implied. In such cases,
relative ages of groove sets can be
determined either by direct evidence ofcrosscutting relationships or by their ages
relative to craters that formed intermediate to the two groove sets. For example,
some of the ENE oriented groove sets
west of Stickney (cooler colors in
Figure 3) have pit chain morphologies
that imply that they formed in extension,
despite currently experiencing compression in the contemporary orbital decay
stress ﬁeld. Moreover, they are older thanthe N-S grooves in the same region thatshow a strong correlation to the orbital
decay stress ﬁeld. These relationships
are observed elsewhere across Phobos.
Hence, orbital decay is likely to be the
Figure 4. The maximum tensile stress experienced on Phobos for various key driving force behind the most recent
amounts of orbital decay toward an orbital distance af and for a change in Phobos deformation. Older extensional
orbital distance Δa. Earlier in Phobos’ orbital history, signiﬁcant stresses
features imply that ancient stress ﬁelds
could be generated, allowing multiple generations of fractures to evolve on
differed from that produced spatially by
its surface with ongoing orbital decay over time. The black dot represents
orbital decay today, perhaps explaining
the case illustrated in the text for which migration starts (ai) at an orbital
radius of 3.0 RMars and ends at the current orbital radius (af = 2.77 RMars;mismatches (purple grooves) in Figure 3.
Δa = 0.23), resulting in a maximum tensile stress of ~50 kPa. The gray dot
In rare cases (e.g., the NW oriented
represents the maximum stress that could be generated if Phobos migrated
groove projecting NW away fromfrom the synchronous orbital radius, which results in >100 kPa of stress.
Stickney crater), linear features show lessresemblance to pit chains and may be catenae (aligned impact craters). In such cases, the pits (craters) vary insize and spacing and are typically larger than pits caused by the draining of regolith. The catena mentioned
above almost encircles Phobos, creating a sinusoidal trace across the surface in the fracture map (Figure 3)
and predates the youngest tectonic grooves.
Orbital decay from 3.0 to 2.77 RMars can produce ~50 kPa of tensile stress at the sub-/anti-Mars point (Figure 4;
the 0° longitude and 0° latitude point in Figure 2). While these stresses may seem low, experiments with Lunar
simulant suggest that tensile failure occurs at stresses as low as 1 kPa [Arslan et al., 2008]. We ﬁnd that stresses
from orbital decay are up to 50 times larger, making deep groove formation plausible by our proposed mechanism and high enough to drive fractures to depths of up to 5 km, thus providing a mechanism to create dilational
cracks below the surface layer into which regolith may have drained to form surface pit chains.
These stress levels were theoretically achievable early in the history of Phobos’ ongoing orbital decay(Figure 4), so in principle, there could be generations of fractures and a range of evolutionary morphologies.
Hence, grooves do not all have to be young according to our model, consistent with the apparent protracted
history of their formation [Weidenschilling, 1979]. As successive generations of grooves form, their pattern oforientations should remain relatively consistent over time as long as Phobos’ tidal alignment with Mars
remains constant. Impact events might break the tidal alignment occasionally, but Phobos’ irregular shapepresumably helps to quickly reestablish a tidally spin-locked system.
While the bulk rigidity (μB) of Phobos is not constrained, the bulk rigidity for the nominal model presented here for
Phobos is ~106 Pa. This value is lower than the lower bound estimate for Phobos’ rigidity of 5 × 108 Pa [Yoder, 1982].
Surveys of 1 km class asteroid binaries have observed asteroids with rigidities ranging from 105 Pa to 108 Pa with a
median value of ~5 × 106 Pa [Taylor and Margot, 2011], assuming the tidal quality factor for dissipation (Q) is similar
to rocky bodies, Q~100. If Phobos behaves like these smaller asteroids, it is plausible that its bulk rigidity can be~106 Pa. However, larger 100 km class binaries were found to be more rigid, ranging from 107 Pa to 1013 Pa with

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a median value of ~109 Pa (again assuming Q = 100) [Taylor and Margot, 2011],
but these bodies are gravity dominatedand Phobos’ size places it at transition
point between strength and gravitydominated bodies [Asphaug et al., 2015];
therefore, we posit that its rigidity could
be similar to the 1 km class asteroids.

Figure 5. The nominal rigidity model for Phobos, adopted in this paper,provides stresses above 100 kPa for migration from ~6 RMars to thecurrent orbital radius (gray dot in Figure 4). This model is not uniqueand a variety of combinations of models with inner layer rigidity (μi)
and outer layer rigidity (μo) can produce the stress in excess of 100 kPa,
when two-layer modes with the indicated rigidity values are evaluated(see Appendix B and Appendix C). Our nominal model produces a bulk
6
rigidity (μB) of ~10 Pa.

A ﬁnding that Phobos’ interior is weak is
consistent with hypothesized formation
scenarios. If Phobos was formed by
the accretion of debris in Martian orbit,
resulting from a large impact [Citron
et al., 2015], then the process of moonbuilding may have resulted in Phobos
being inherently weak, with a coresimilar to a rubble pile. Predictions of
the rigidity of a 10 km rubble pile yieldvalues ~107 Pa [Goldreich and Sari,
2009], which are only an order of magnitude stronger than the model in thisstudy. Regardless, the tidal model for
fracture formation can still produce
large stresses on Phobos for otherassumptions of layer rigidities (Figure 5).

Our assumption that the outer layer is 100 m thick is a conservative upper limit, as the estimated thickness of
the regolith varies from 5 to 100 m [Basilevsky et al., 2014]. The effect of thinner shells is explored in Appendix
D and in general results in higher levels of stress. Therefore, we ﬁnd that our results are robust and are valid
even if future investigations of Phobos can place different constraints on some of the values we assume for
Phobos’ interior properties. Once fractures have been formed on Phobos’ surface, they can remain active
through diurnal tidally driven motions. Phobos’ orbital eccentricity produces a dynamic diurnal stress ﬁeld,
which changes throughout Phobos’ 7.6 h orbit. While these stresses are smaller by an order of magnitude,
they can still work to dilate the grooves similar to the tidal dilation of fractures on Enceladus [Hurford et al.,2007]. The reworking of fractures by these stresses may be natural seismic sources [Lee et al., 2003], which
could be exploited by future missions to probe Phobos’ interior. However, Phobos quakes are only possible
if the interior is weak, allowing an enhancement of tidal diurnal stress. Thus, a seismometer could quickly
determine the state of Phobos’ interior; it would be rigid if there are no quakes on the diurnal time scale,and weak if there are quakes. Furthermore, Phobos quakes would allow a natural way to probe the propertiesof Phobos’ internal structure.
The presence of tidally driven fracturing on Phobos does not imply its imminent catastrophic disruption,
particularly given that a friction angle of only ~3° is sufﬁcient to prevent downslope movement today, evenin the complete absence of cohesion [Holsapple, 2001]. Tidal disaggregation of Phobos will occur only when it
is much closer to Mars and is suggested to occur in 20–40 Ma [Black and Mittal, 2015]. But meanwhile, Phobos
is weak enough internally, on the time scale of orbital migration, to permit global deformation that results instresses to build up within an elastic lunar-like outer layer. According to our model, its failure is opening up
granular ﬁssures, an interpretation that is consistent with the hypothesis that the pitted grooves are formedby regolith draining into fractures [Horstman and Melosh, 1989].

4. Conclusions
Our model results applied to surface observations imply that Phobos is a rubble pile overlain by a lunar-likecohesive regolith layer. This layer is developing extensional ﬁssures as the global body deforms in response toincreasing tides related to orbital decay. More detailed study by an orbiter or lander will provide better

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constraints on the satellite’s geologic evolution, and its suitability for human exploration given what may bean active and evolving surface.

5. Appendices
The following material serves as a reference for the reader. In general, it provides more detailed informationthat is supplemental to the main text. In particular the sections detail the following:
Reviews the problem of assuming that Phobos is a homogeneous sphere when determining tidal stresses on
its surface. This ﬁnding has not been well preserved in the literature.
Details how the tidal deformation of a two-layer model is determined. The standard theory here can provide
the Love number h2.
Details the extension of the two-layer model in Appendix B to determine the stresses on the surface of thesphere. This method is not well documented in the literature.
Discusses the evaluation of stress on the surface given by the two-layer model. It also shows how robust andsensitive the results are to model parameters. Finally, it demonstrates that a thin-shell model can approximate stresses on the surface for our nominal model parameters. Because the thin-shell equations are straightforward, we adopt them for the main text but this material serves as justiﬁcation for that simpliﬁcation.

Appendix A: Tidal Stress on a Homogeneous Sphere
Surface stresses on a homogeneous sphere due to orbital migration are given by
9Mμh2 11σ θθ ¼20πρav a3f a3i

9Mμh2 1
1
σ φφ ¼ 
ð1  3cos2θÞ
40πρav a3f a3i

(A1)
(A2)

where θ is the colatitude measured with respect to the axis through the center of the tidal bulge [Kattenhorn
and Hurford, 2009]. Tension is positive and compression negative. The surface stress σ θθ is along the surface inthe θ direction, while σ φφ is along the surface orthogonal to σ θθ. Surface radial and shear stresses are required
to be zero. Here M is the mass of Mars, μ is the rigidity, and ai and af are the initial and ﬁnal semimajor axes
during orbital decay. The average density of Phobos is given by ρav, while its second-order tidal surface displacement Love number h2 is given by the classical solution [Love, 1911],
h2 ¼

5=2

;
19μ
2ρav gR þ 1(A3)

where g is the surface gravity of Phobos and R is the average radius.
In the homogeneous solution, surface
stress σ θθ is constant and always tensile,
while surface stress σ φφ is tensile in the
sub-/anti-Mars regions, extending toangular distances of 35.3° and compressive elsewhere. Tidal stresses are maximized as ai →∼ 11
5 af , representing thetidal deformation from a sphere to thecurrent tidal shape. In the case that
Phobos is dominated by its material
strength (i.e., 2ρ19μgR ≫1 ), the maximum
av

tensile stresses go to a limiting value
2

Figure A1. Maximum tensile stress on the surface of a homogenous
Phobos depends on the assumed valueof its rigidity. For a rigid Phobos,
2
av R
the stress increases to a limit of 7GMρ
50a3 , which for the parameters adopted
in this study would yield about 1700 Pa.

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TIDAL DISRUPTION OF PHOBOS

av R
of ∼ 7GMρ
50a3 , which for the parameters
adopted in this study would yield about
1700 Pa (1.7 kPa). These stresses,
although of the correct orientation forgrooves, are not likely of a high enough

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magnitude for groove formation. Decreasing the rigidity of the homogenous Phobos allows greater tidal
deformation but will not result in the production of greater stresses (Figure A1).

Appendix B: Two-Layer Model of Tidal Deformation
In order to ﬁnd a solution for the tidal deformation of an incompressible spherical two-layer body, we follow
the derivation of Sabadini and Vermeersen [2004]. In this method, six coupled differential equations are solved
simultaneously to ﬁnd the equilibrium tidal deformation. The six differential equations solve for two displacements, ur and uθ, two stresses, σ rr and σ rθ, the tide-raising potential, V, and the continuity of the tidal potential, q (sometimes called the potential stress), at the top of each layer. These six equations can be written, tosecond order, in the following forms as
ur ¼ ηy 1 R2 P2 ðcos θÞ;(B1)uθ ¼ ηy 2 R2 S2 ðcos θÞ;(B2)

σ rr ¼ ηy 3 R2 P2 ðcos θÞ;(B3)

σ rθ ¼ ηy 4 R2 S2 ðcos θÞ;(B4)

V ¼ ηy 5 R2 P2 ðcos θÞ; and(B5)

q ¼ ηy 6 R2 P2 ðcos θÞ:(B6)

In these equations, η ¼ GM
a3 with a the orbital radius and G the gravitational constant. R is the average radius of
Phobos and S2(cos θ) is the derivative of P2(cos θ), the second-order Legendre polynomial with respect to θ.
The six functions (y1, y2, y3, y4, y5, and y6) shown here all have the same form in that each of the terms is linear
in the constants of integration such that→→y i ¼ Y ðu; ρ; g; r Þ A(B7)

where Y(μ, ρ, g, r) is the matrix of functions that describe the yi equations in terms of the constants of integra→tion, A . As noted in Sabadini and Vermeersen [2004], the Y(μ, ρ, g, r) matrix for spherical harmonic degree-2 isgiven by
3
2
r3
1
1
0
r
0
7
6
2r 2
r4
7
7
6
7
6
5r 3
r
1
7
6
0
0

0
4
7
6
2
3r
42
7
61 2
6μ
þ
ρgr
8μ
þ
ρgr
ρ
2
6 r ðμ þ ρgr Þ
2μ þ ρgr
 ρr
 3 7
7
67
3
2
2r
r
r
Y ðμ; ρ; g; r Þ ¼ 6
7
8μr 2
μ
8μ
7
6
7
6
0
μ
0
3
5
7
6
2r
3r
21
7
6
1
2
7
6
0
0

0
0

r
6
3
r 7
5
4
3 3
3γρ
3γρ
γρr
3γρr
 5r
0
2
4
7
2r
r
(B8)
where γ ¼ 43 πG.
→

The value of y i at any radius can be found, if the constants of integration are known, by evaluating Y at that→→radius. In this case, we do not know A . So we next describe an alternative way of ﬁnding y i based on knowing→

their values at a different radius. The propagator matrix P is deﬁned such that it gives y i at r = R2 based onvalues at r = R1:→→

y i ðr 2 Þ ¼ Pðμ; ρ; g1 ; g2 ; R1 ; R2 Þ y i ðr 1 Þ(B9)

P is known as the “propagator” matrix. This matrix translates the solution radially within a layer in which thematerial properties are constant. This will allow the propagation of boundary conditions from one layerboundary to the next. The propagator matrix can be expressed in terms of Y(μ, ρ, g, r):Pðμ; ρ; g1 ; g2 ; R1 ; R2 Þ ¼ Y ðμ; ρ; g2 ; R2 Þ Y1 ðμ; ρ; g1 ; R1 Þ

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(B10)

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Journal of Geophysical Research: Planets

10.1002/2015JE004943

At r = 0, the solution must be ﬁnite, yet there are three irregular solutions which approach inﬁnity at this limit,
allowing us to reduce the solution to three constants of integration just as Love did for a uniform sphere→[Arslan et al., 2008]. Therefore, the values of y i in the inner layer depend on only three constants of integration,and the matrix equation is reduced to→→y i ðμ1 ; ρ1 ; g1 ; r Þ ¼ Y ðμ1 ; ρ1 ; g1 ; r ÞI1 A1

(B11)

→
for r < R1, where A1 is a vector of the constants of integration for the inner layer, and

2

1
60
6
I1 ≡6 0
40
0
0

0
1
0
0
0
0

3
0
07
1 7:
07
5
0
0

(B12)

→

In the model, the continuity of the material requires that the displacement and potential (and thus the y i ) must
be continuous across the layer boundaries. This continuity means that at the inner layer/outer layer boundary→→y i ðμ1 ; ρ1 ; g1 ; R1 Þ ¼y i ðμ2 ; ρ2 ; g2 ; R1 Þ:

(B13)
→

Next consider the outer layer. The solution anywhere in the layer (R1 < r < R2) can also be given by y i→

ðμ2 ; ρ2 ; g2 ; r Þ ¼ Pðμ2 ; ρ2 ; g1 ; g2 ; R1 ; r Þ y i ðμ2 ; ρ2 ; g2 ; R1 Þ if the solution at r = R1 is known. However, using thecontinuity of the solution across the boundary, the solution anywhere in this layer can also be described→→using the solution for the inner layer, y i ðμ2 ; ρ2 ; g2 ; r Þ ¼ Pðμ2 ; ρ2 ; g1 ; g2 ; R1 ; r Þ y i ðμ1 ; ρ1 ; g1 ; R1 Þ which now
depends only on the three constants of integration from the inner layer. By letting r = R2, the solution at
the surface is described in terms of three constants of integration. The solution from the inner layer/outerlayer boundary at r = R1 is said to have been “propagated” to the surface.
Now the boundary conditions at the surface can be applied in order to ﬁnd the three constants of integration.
These boundary conditions are the same as for a uniform sphere. The outer surface is stress free (y3 = 0 and
y4 = 0). Also, there is continuity in the potential across the outer surface (y6 = 5/R2). Setting up the matrix
equation to match the boundary conditions gives→→

I2 Pðμ2 ; ρ2 ; g1 ; g2 ; R1 ; R2 ÞY ðμ1 ; ρ1 ; g1 ; R1 ÞI1 A1 ¼b

(B14)

→

where b is the vector of the boundary conditions on y3, y4, and y6, respectively,
"
#
0
→
0
b¼
5=R2
and I2 is a matrix that selects only the y3, y4, and y6 solutions,"
0 0 1 0 0
I2 ≡ 0 0 0 1 0
0 0 0 0 0

#
0
0 :
1

(B15)

(B16)

→

Solving equation (B14) yields A1 , the set of three constants of integration→→

A1 ¼ ½I2 Pðμ2 ; ρ2 ; g1 ; g2 ; R1 ; R2 ÞY ðμ1 ; ρ1 ; g1 ; R1 ÞI1 1 b :

(B17)

→

Once the constants of integration are known, the y i at the surface can be fully described. The tidal response of
the two-layer body as given by h2 is related to y1 as follows:h2 ¼ y 1 g2

(B18)

where g2 is the acceleration of gravity at the surface (the top of the second layer).

Appendix C: Tidal Stress on the Surface of a Two-Layer Model
Even though a full solution has been found based on this propagation technique, sometimes it is useful toknow the values of the six constants of integration at the surface of the outer layer. These six constants
can be found by solving for them using the solution at the surface,

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Journal of Geophysical Research: Planets

10.1002/2015JE004943→→

A2 ¼ Y1 ðμ2 ; ρ2 ; g2 ; R2 ÞPðμ2 ; ρ2 ; g1 ; g2 ; R1 ; R2 ÞY ðμ1 ; ρ1 ; g1 ; R1 ÞI1 A1(C1)This will be especially useful for calculating the surface stresses, which are given by the equationsσ θθ ¼ ηR2 ½y 7 P2 ðcosθÞ þ y 8 S2 ðcosθÞand(C2)

σ φφ ¼ ηR ½y 7 P2 ðcosθÞ þ y 8 T 2 ðcosθÞ

(C3)

2

where T2(cos θ) is the derivative of P2(cos θ) with respect to θ multiplied by cot θ. The new functions y7 and y8
are similar to the yi functions in that they depend on the same constants of integration. These functions canbe written in terms similar to the other functions and are
2312
ρg 2μ þ ρgr
ρ
2
ð
Þ
r
5μ
þ
ρgr
2μ
þ
ρgr
ρr

6
2r 2
r5
r3 7
(C4)
Y ′ ðμ; ρ; g; r Þ ¼ 4 7
5:
5μr 2
2μ
0
μ
0
0
 5
3r
21
Thus, the y ’ i functions can be evaluated by→→y’i ðμ2 ; ρ2 ; g2 ; R2 Þ ¼ Y ′ ðμ2 ; ρ2 ; g2 ; R2 Þ A2 :(C5)The stress from orbital migration can be found as σ θθ and σ φφ change due to the change in orbital distance,i.e., evaluating η with a speciﬁc value of a,Δσ θθ ¼ σ θθ ðaf Þ  σ θθ ðai Þand(C6)

Δσ φφ ¼ σ φφ ðaf Þ  σ φφ ðai Þ:(C7)

The tidal tensile stresses are maximized under the conditions that ai →e 11
5 af at θ = 0, the sub/anti-Marspoint.

Appendix D: Evaluation of a Two-Layer Phobos Model
We use the incompressible two-layer model (Appendix B) to evaluate the tidal response for Phobos.
The model is built of an inner layer with material properties of density ρ1 and rigidity μ1 with a radialextent of R1. The outer layer has its own material properties of density ρ2 and rigidity μ2, but its outerradius is constrained to the average radius of Phobos, R2 = 11,200 m. The model can have up to ﬁvefree parameters. However, to simplify things, we assume that the density within the two layers is constant and set it to Phobos’ average density, i.e., ρ1 = ρ2 = 1880 kg/m3. The model has now been reduced
to three free parameters, and we further assume the rigidity of the outer layer to be 108 Pa, i.e.,
μ2 = 1 × 108 Pa, based on an analog to the Lunar regolith layer [Horvath et al., 1980]. We then cansee how the stresses on the surface vary based on the rigidity of the inner layer μ1 and the thicknessof the outer shell, H (see Table D1 and Figure D1).
We ﬁnd that for interiors with rigidities < 106 Pa, the stresses on the surface are well characterized by the thinshell approximation for thinner shell thicknesses (H ≤ 100 m). In the case of an outer shell with a thickness of
100 m, the thin-shell approximation conservatively estimates the magnitude of stress but would produce
grooves with similar orientation as long as the stresses are high enough to fracture the shell.
In the main text, we adopt a nominal model with μ1 = 106 Pa and an outer shell that is 100 m in thickness. The magnitude of the tidal stress is roughly inversely proportional to the shell thickness. If theouter shell thickness is 10 m, tidal stresses as shown in Figure 3 would be increased by about a factorof 10. Thus, we ﬁnd that our nominal model is a conservative estimate for computing stresses on
Phobos’ surface.
Even though Phobos is not quite spherical (it is a triaxial ellipsoid with radii of 13.0 km × 11.4 km × 9.1 km), we
ﬁnd that the stresses produced by a spherical shell model have orientations that match the orientations ofgrooves on the surface (Figure 3 in the main text). Therefore, to ﬁrst order, we maintain that this model
can characterize stresses on Phobos due to orbital decay. Further reﬁnements to the stress model should
provide even better ﬁts between tidal stress and observed groove orientations. Future work should focus
in tidal stresses on a triaxial ellipsoid.

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Table D1. Table of Two-Layer Model Results Compared to a Thin-Shell Approximation Based on the Two-Layer Tidal Deformation, h2
Internal Rigidity μ1 (Pa)
10
100
1,000
10,000100,000
1.00E + 06
1.00E + 07
1.00E + 08
1.00E + 09
1.00E + 10
1.00E + 11
10
100
1,000
10,000100,000
1.00E + 06
1.00E + 07
1.00E + 08
1.00E + 09
1.00E + 10
1.00E + 11
10
100
1,000
10,000100,000
1.00E + 06
1.00E + 07
1.00E + 08
1.00E + 09
1.00E + 10
1.00E + 11aaShell Thickness H (m)Love Number h2

Max Tension Two-Layer
Model (Pa)Max Tension Thin-Shell
Model (Pa)% Difference Between
the Models

10
10
10
10
10
10
10
10
10
10
10
50
50
50
50
50
50
50
50
50
50
50
100
100
100
100
100
100
100
100
100
100
100

0.51
0.51
0.50
0.44
0.20
0.03
0.003
0.00030.000030.000003
0.0000003
0.12
0.12
0.12
0.12
0.09
0.03
0.003
0.00030.000030.000003
0.0000003
0.062
0.062
0.062
0.061
0.052
0.021
0.003
0.00030.000030.000003
0.0000003

4,955,560
4,948,085
4,874,481
4,238,124
1,756,866
201,703
17,847
1,746
174
17
2
1,199,983
1,199,555
1,195,296
1,154,230
854,785
214,342
19,197
1,746
171
15
0
625,595
625,481624,344
613,185
519,391
194,620
20,448
1,746
163
9
6

4,936,376
4,929,442
4,861,157
4,269,774
1,927,555
298,042
31,563
3,175
318
32
3
1,177,008
1,176,614
1,172,688
1,134,824
857,908
249,751
30,970
3,175
318
32
3
602,042
601,939
600,911590,829505,952
207,827
30,260
3,175
319
32
3

0.4
0.4
0.3
0.7
9.7
47.8
76.8
81.8
82.4
83.2
91.1
1.9
1.9
1.9
1.7
0.4
16.5
61.3
81.8
86.5
108.81299.5
3.8
3.8
3.8
3.6
2.6
6.8
48.0
81.8
95.9
252.6151.2

3

The model evaluated consists of two layers, each with a density of 1880 kg/m . The rigidity of the inner layer is given in the table, while the outer layer has a rigidity8
of 10 Pa. The thickness of the outer shell (H) is provided in the table and the total radius is 11.2 km. The nominal model used in the text is italicized. Table results are forthe case of that starts (ai) at an orbital radius of ~6.0 RMars to the current orbital radius (af = 2.77 RMars).

Figure D1. The maximum tensile stress on the surface of Phobos for various interior rigidities are shown for 2-layer models
with three different outer shell thicknesses (10 m, 50 m and 100 m). The dashed lines show the thin-shell approximation for6
these stresses and are in good agreement with the two-layer result (solid line) as long as the interior’s rigidity is ≤10 Pa.

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Journal of Geophysical Research: Planets
Acknowledgments
In general, this study is theoretical innature, and numerical evaluations of
the equations detailed herein weredone for the parameters listed at locations on Phobos’ surface determined byour mapping of surface features. Whilethe ﬁgures provide a summary of our
ﬁndings, both the data on our mapped
fractures and numerical model results
are available from the lead author uponrequest (terry.a.hurford@nasa.gov).

HURFORD ET AL.

10.1002/2015JE004943

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