Images of asteroid (101955) Bennu acquired by the OSIRIS-REx mission reveal a rocky world covered in rubble; Shape deviates from hydrostatic surface; Internal friction and/or cohesion even if no tensile strength; Understanding the deviation of the surface from idealized shape may help constrain mechanical properties of the interior; Geologic evolution of Bennu is driven by downslope migration of surface material and rubble; May be caused by YORP-induced spin-up, re-accumulation, impact-induced seismic shaking, thermal stresses, or tidal disruption by close encounters to larger bodies

Asad, M. M. Al

Barnouin, O. S.

Bottke, W. F.

Daly, M. G.

Daly, R. T.

Johnson, C. L.

Lauretta, D. S.

McMahon, J. W.

Michel, P.

Neumann, G. A.

Nolan, M. C.

Palmer, E. E.

Perry, M. E.

Roberts, J. H.

Scheeres, D. J.

Walsh, K. J.

Weirich, J. R.

NASA Technical Reports Server

OSIRIS-­‐REX	  |	  LUNAR	  AND	  PLANETARY	  SCIENCE	  CONFERENCE	  |	  	  MARCH	  2019
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INTRODUCTION
• Images	  of	  asteroid	  (101955)	  Bennu	  acquired	  by	  the	  OSIRIS-­‐REx	  mission	  [1]	  
reveal	  a	  rocky	  world	  covered	  in	  rubble.
• Shape	  deviates	  from	  hydrostaJc	  surface	  [2]
• Internal	  fricJon	  and/or	  cohesion	  even	  if	  no	  tensile	  strength	  [3,4]
• Understanding	  the	  deviaJon	  of	  the	  surface	  from	  idealized	  shape	  may	  help	  
constrain	  mechanical	  properJes	  of	  the	  interior
• Geologic	  evoluJon	  of	  Bennu	  is	  driven	  by	  downslope	  migraJon	  of	  surface	  
material	  [5]	  and	  rubble.
• May	  be	  caused	  by	  YORP-­‐induced	  spin-­‐up	  [e.g.,	  6,7],	  re-­‐accumulaJon	  [8,	  9],	  
impact-­‐induced	  seismic	  shaking,	  thermal	  stresses,	  or	  Jdal	  disrupJon	  by	  close	  
encounters	  to	  larger	  bodies.
EQUILIBRIUM	  FIGURES
• 	  
REFERENCES
• Adding	  internal	  fricJon	  helps	  it	  
hold	  together	  at	  higher	  
rotaJon	  rates	  (Fig.	  1a)
• An	  object	  with	  the	  observed	  
rotaJon	  rate	  and	  density	  (see	  
Table	  1)	  of	  Bennu	  [11,12]	  
requires	  φ >	  18°	  to	  prevent	  
further	  ﬂa:ening,	  despinning	  
and	  potenJally	  undergoing	  
binary	  ﬁssion.
Prolate	  Spheroid
• More	  complicated	  funcJon	  of	  
allowable	  Ω as	  funcJon	  of	  α	  
and	  φ,	  
• Both	  upper	  AND	  lower	  bounds	  
on	  Ω.
• Five	  asteroids	  for	  which	  we	  
have	  high-­‐resoluJon	  shape	  
models	  (Table	  1)	  have	  been	  
approximated	  as	  prolate	  
spheroids	  and	  plo`ed	  on	  Fig.	  
1b.
• All	  prolate	  bodies	  require	  
internal	  fricJon	  or	  cohesion
RESULTS	  FOR	  BENNU
• Shape	  model	  developed	  from	  SPC	  [13]	  (derived	  from	  images	  taken	  
during	  Preliminary	  Survey	  and	  Orbital	  A	  phases	  [1]),	  validated	  by	  limb	  
measurements,	  and	  further	  constrained	  by	  OLA	  [2,14].
• Figure	  2	  shows	  height	  of	  shape	  model	  above	  the	  equilibrium	  spheroid	  
consistent	  with	  Bennu’s	  parameters.	  
• Spherical	  harmonic	  decomposiJon	  shows	  strong	  degree	  4	  contribuJon	  
(Figure	  3,	  [15]).	  Zonal	  component	  is	  largely	  due	  to	  the	  equatorial	  ridge,	  but	  
there	  is	  also	  a	  strong	  sectoral	  component	  “Squarish”	  shape	  seen	  in	  the	  
polar	  views.	  Four	  N-­‐S	  trending	  ridges	  are	  outlined	  in	  Figure	  2.
• Figure	  4	  shows	  the	  Jlts,	  which	  further	  constrain	  φ
• Internal	  fricJon	  must	  be	  high	  enough	  to	  support	  material	  from	  sliding	  
downslope	  to	  meet	  the	  equilibrium	  surface	  
• Maximum	  Jlts	  are	  at	  lower	  laJtudes	  than	  those	  on	  a	  MacLaurin	  surface;	  
slopes	  of	  the	  equatorial	  ridge
RESULTS	  FOR	  OTHER	  ASTEROIDS
• Repeated	  analysis	  for	  10	  
asteroids	  in	  Table	  1.
• Oblate	  body	  Jlts	  peak	  ~10°
• Prolate	  body	  Jlts	  peak	  ~20°
• Both	  much	  higher	  than	  
idealized	  shape
• Very	  long	  tails	  at	  upper	  ends
DISCUSSION
• Equatorial	  ridges	  and	  N-­‐S	  ridges	  clearly	  visible	  in	  height	  diﬀerence	  map
• May	  point	  to	  underlying	  structure	  –	  few	  large	  fragments	  controlling	  shape?
• Does	  the	  equatorial	  ridge	  act	  as	  a	  barrier?	  
• It’s	  a	  gravitaJonal	  minimum,	  so	  rubble	  slides	  downhill	  to	  it.
• May	  have	  some	  larger	  blocks	  (buried	  in	  ﬁnes)	  there
• AddiJonal	  material	  may	  be	  lodged	  up	  against	  	  the	  ridge?
• SystemaJc	  variaJon	  in	  Jlt	  distribuJon	  for	  oblate	  vs.	  prolate	  asteroids
• Many	  asteroids	  be`er	  represented	  as	  triaxial	  ellipsoids.	  Requires	  
numerical	  modeling.	  
• Stability	  analysis	  assumes	  internal	  fricJon	  is	  the	  only	  source	  of	  strength.	  
Cohesion	  would	  reduce	  the	  required	  fricJon	  angle.	  
	  J.H.	  Roberts1,	  O.S.	  Barnouin1,	  C.L.	  Johnson2,3,	  M.G.	  Daly4,	  M.E	  Perry1,	  R.T.	  Daly1,	  M.M.	  Al	  Asad2,	  E.E.	  Palmer3,	  J.R.	  Weirich3,	  P.	  Michel5,	  W.F.	  
Bo`ke6,	  K.J.	  Walsh6,	  M.C.	  Nolan7,	  D.J.	  Scheeres8,	  J.W.	  McMahon8,	  G.A.	  Neumann9,	  D.S.	  Laure`a7,	  and	  the	  OSIRIS-­‐REx	  Team.
1The	  Johns	  Hopkins	  University	  Applied	  Physics	  Laboratory;	  2Department	  of	  Earth,	  Ocean	  and	  Atmospheric	  Sciences,	  University	  of	  BriJsh	  Columbia;	  3Planetary	  Science	  InsJtute;	  4The	  Centre	  for	  Research	  in	  Earth	  and	  Space	  Science,	  York	  University;	  
5Observatoire	  de	  la	  C’ôte	  d’Azur,	  University	  of	  Nice;	  6Southwest	  Research	  InsJtute;	  7Lunar	  Planetary	  Laboratory,	  University	  of	  Arizona;	  8Colorado	  Center	  for	  Astrodynamics	  Research,	  University	  of	  Colorado.	  9NASA	  Goddard	  Space	  Flight	  Center.
DEVIATION	  OF	  THE	  SHAPE	  OF	  BENNU	  FROM	  ROTATIONAL	  FIGURES	  OF	  STABILITY
Tilts from Maclaurin Spheroid Actual Tilts Bennu (v20)
Asteroid Spheroid α Ω (s-1) ρ (g cm-3)
4 Vesta Maclaurin 0.8242 3.27×10-4 3.456
21 Lutetia Maclaurin* 0.7182 2.14×10-4 3.4
243 Ida Prolate 0.3679 3.77×10-4 2.6
253 Mathilde Prolate 0.7121 4.18×10-6 1.3
433 Eros Prolate 0.6512 3.31×10-4 2.67
951 Gaspra Prolate 0.5330 2.48×10-4 2.7
25143 Itokawa Prolate* 0.4701 1.44×10-4 1.95
66391 1999 KW4a Maclaurin 0.8907 6.31×10-4 2
101955 Bennu Maclaurin 0.8874 4.07×10-4 1.19
162173 Ryugu Maclaurin 0.9317 2.29×10-4 1.27
KW4a
Bennu
RyuguVestaLutetia
Ida
Eros
Gaspra
MathildeItokawa
Figure 1:  Rotational stability for cohesionless, solid, oblate (top) and prolate 
(bottom) spheroids for a wide range of rotation rate, axis ratios, and internal 
friction angles: The curves of rotational stability for cohesionless, solid, oblate 
spheroids for a wide range of rotation rate, oblateness, and internal friction. 
Each curve describes the limits of the allowable dimensionless rotation rate 
as a function of the axis ratio. Each point marks the dimensionless spin rates 
and axis ratios consistent with observed asteroids (Table 1). 
Figure 2:  Deviation of Bennu’s shape model from the closest-fit Maclaurin spheroid consistent with Bennu’s 
observed density (1.19 g cm-3) and rotation period (4.3 h). Left: Polar view. Right: Equatorial view. Ellipses mark 
portions of the north-south ridges, which are clearly high-standing relative to locations to the east and west.
Figure 3. Amplitude spectrum of a spherical harmonic expansion for the shape model The large zonal 
degree 2 and 4 terms show the most distinctive characteristic of Bennu: the top shape with an 
equatorial ridge. The relatively low amplitudes of the degree 3 and 5 terms demonstrate that there is no 
substantial north-south asymmetry in Bennu’s shape. The degree 4 sectoral terms (C44 and S44), 
capture the ~90° longitudinal variations in shape associated with the major north-south ridges. 
Figure 4: Tilts (angle between the normal to the surface and the direction to the center) from closest-fit Maclaurin 
spheroid consistent with Bennu’s observed density and rotation period (left), and from the shape model (right).
Figure 5:  Histograms showing the distribution of tilts on closest-fit Maclaurin or prolate ellipsoids to the shapes of 
Bennu, Ryugu, Itokawa, Eros, and KW4a (left), and on shape models of these asteroids (right).
[1] DellaGiustina, D. et al. (2018), AGU Fall Meeting P21A-04. [2] Barnouin, O.S. et al. (2019) LPSC 50, 1744. [3] Zhang, Y. et 
al., (2017) Icarus 294, 98–123. [4] Hirabayashi, M. and Scheeres, D.J. (2015) ApJ Letters, 798, L8. [5] Richardson, J.E. and 
Bowling, T.J. (2014) Icarus 234, 53–65. [6] Rubincam, D.P. (2000) Icarus, 148, 2–11. [7] Walsh, K.J., et al., (2008) Nature, 454, 
188–191. [8] Michel, P. et al. (2001) Science, 294, 1696–1700. [9] Michel, P. et al. (2019) LPSC 50, 1659. [10] Holsapple, K.A. 
(2004) Icarus 172, 272–303. [11] Palmer, E. et al. (2019) LPSC 50, 2588. [12] Scheeres, D.J. et al. (2018) AGU Fall Meeting 
P22A-05. [13] Gaskell, R.W. et al. (2008) MAPS, 43, 1049–1061. [14] Daly, M.G. et al. (2017) SSR, 212, 899–924. [15] 
Barnouin, O.S. et al. (2019), Nature Geosci., in press.
Table 1: Physical properties relevant to rotational stability of ten asteroids, here approximated as either oblate or prolate 
spheroids. Asteroids denoted by * are better approximated as tri-axial ellipsoids, but for this comparison have been 
classified as oblate or prolate.
https://ntrs.nasa.gov/search.jsp?R=20190002404 2019-08-30T21:23:40+00:00Z


Deviation of the Shape of Bennu from Rotational Figures of Stability

Deviation of the Shape of Bennu from Rotational Figures of Stability

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