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Modeling the light curve of `Oumuamua: evidence for torque and disc-like shape

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We present the first attempt to fit the light curve of the interstellar visitor `Oumuamua using a physical model which includes optional torque. We consider both conventional (Lommel-Seeliger triaxial ellipsoid) and alternative ("black-and-white ball", "solar sail") brightness models. With all the brightness models, some torque is required to explain the timings of the most conspicuous features -- deep minima -- of the asteroid's light curve. Our best-fitting models are a thin disc (aspect ratio 1:6) and a thin cigar (aspect ratio 1:8) which are very close to being axially symmetric. Both models are tumbling and require some torque which has the same amplitude in relation to `Oumuamua's linear non-gravitational acceleration as in Solar System comets which dynamics is affected by outgassing. Assuming random orientation of the angular momentum vector, we compute probabilities for our best-fitting models. We show that cigar-shaped models suffer from a fine-tuning problem and have only 16 per cent probability to produce light curve minima as deep as the ones present in `Oumuamua's light curve. Disc-shaped models, on the other hand, are very likely (at 91 per cent) to produce minima of the required depth. From our analysis, the most likely model for `Oumuamua is a thin disc (slab) experiencing moderate torque from outgassing.
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MNRAS 000,121 (2019) Preprint 9 June 2019 Compiled using MNRAS L
X style file v3.0
Modeling the light curve of ‘Oumuamua: evidence for
torque and disc-like shape
Sergey Mashchenko?
Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton ON L8S 4M1, Canada
Accepted XXX. Received YYY; in original form ZZZ
We present the first attempt to fit the light curve of the interstellar visitor ‘Oumua-
mua using a physical model which includes optional torque. We consider both con-
ventional (Lommel-Seeliger triaxial ellipsoid) and alternative (”black-and-white ball”,
solar sail”) brightness models. With all the brightness models, some torque is re-
quired to explain the timings of the most conspicuous features – deep minima – of
the asteroid’s light curve. Our best-fitting models are a thin disc (aspect ratio 1:6)
and a thin cigar (aspect ratio 1:8) which are very close to being axially symmetric.
Both models are tumbling and require some torque which has the same amplitude
in relation to ‘Oumuamua’s linear non-gravitational acceleration as in Solar System
comets which dynamics is affected by outgassing. Assuming random orientation of
the angular momentum vector, we compute probabilities for our best-fitting models.
We show that cigar-shaped models suffer from a fine-tuning problem and have only
16 per cent probability to produce light curve minima as deep as the ones present in
‘Oumuamua’s light curve. Disc-shaped models, on the other hand, are very likely (at
91 per cent) to produce minima of the required depth. From our analysis, the most
likely model for ‘Oumuamua is a thin disc (slab) experiencing moderate torque from
Key words: methods: numerical – minor planets, asteroids: general – minor planets,
asteroids: individual (‘Oumuamua)
1I/2017 ‘Oumuamua is the first and only known interstellar
minor body to pass through the Solar System. It was de-
tected by Pan-STARRS1 survey on October 19, 2017, and
by October 22 was determined to have by far the largest
known hyperbolic eccentricity of 1.2 (Meech et al. 2017).
Unfortunately it was discovered when it was already on its
way out of the Solar System, after passing its perihelion
(0.25 au from the Sun) on September 9, and having a close
approach (0.16 au) to Earth on October 14. This severely
limited the number of observations which could be acquired.
Most of observations of ‘Oumuamua were done during the
5-day interval between October 25 and October 30 (Fraser
et al. 2018;Drahus et al. 2018), with a few more observa-
tions over the next two months until its final sighting on
January 2, 2018 (Micheli et al. 2018). The observations were
primarily done in visible light, though two very interesting
non-detections in other wavelengths were also reported: in
infrared by Spitzer (Trilling et al. 2018) and in radio by SETI
(Harp et al. 2019).
The second unique feature of ‘Oumuamua (in addition
to its interstellar nature) was its extreme brightness variabil-
ity. At 2.6±0.2mag, the amplitude of the brightness changes
was larger than for any Solar System minor body, suggest-
ing an extreme geometry (Drahus et al. 2018). The early
suggestion was that ‘Oumuamua has a very prolate (cigar-
like) shape (Meech et al. 2017) which was later accepted
by most of the literature on this object, though very oblate
(disc-like) shapes would work as well (Belton et al. 2018).
The idea that ‘Oumuamua’s large brightness variations are
not geometric in nature, but are primarily driven by large
albedo variations across the asteroid’s surface, was briefly
entertained and dismissed as unlikely due to the absence of
any sign of volatiles (Meech et al. 2017).
‘Oumuamua’s light curve was also unusual for another
reason. While early papers based on limited data suggested
that the asteroid is a simple rotator with the rotation period
between 7.3 and 8.1 h (Meech et al. 2017;Jewitt et al. 2017;
Bolin et al. 2018), later papers which analysed more com-
©2019 The Authors
2S. Mashchenko
plete datasets concluded (by analysing the periodograms for
the light curve) that the asteroid is in an excited (Non Prin-
cipal Axis, or tumbling) rotational state (Belton et al. 2018;
Drahus et al. 2018;Fraser et al. 2018). It is important to
note that periodograms of noisy and patchy data of a limited
size can produce fake dominant frequencies (Samarasinha &
Mueller 2015). Also, such an analysis assumes there is no
torque. Dominant frequencies found in periodograms should
be treated as suggestive only, and should be ideally verified
(confirmed or disproved) by means of physical modeling of
the light curve (Pravec et al. 2005).
The third unique feature of ‘Oumuamua is its non-
gravitational acceleration, discovered by Micheli et al.
(2018), combined with the lack of any signs of outgassing
(Drahus et al. 2018;Trilling et al. 2018;Sekanina 2019).
In Solar System comets, non-gravitational acceleration is
usually associated with active outgassing. This conundrum
spurred some non-orthodox explanations, such as the solar
sail idea of Bialy & Loeb (2018).
As evidenced by Solar System comets and expected on
theoretical grounds, the same agent (e.g. outgassing), which
drives linear non-gravitational acceleration of a minor body,
should also produce torque, which amplitude should corre-
late with the amplitude of the linear acceleration (Rafikov
2018a). Seligman et al. (2019) showed that their physical
model of ‘Oumuamua, where the outgassing point tracks
the subsolar spot on the asteroid’s surface, can reproduce
the magnitude of the non-gravitational linear acceleration
and some features of the light curve. (We have to emphasize
that the authors did not carry out computationally expen-
sive fitting of the observed light curve.) On the other hand,
recently Rafikov (2018b), based on the frequency analysis of
the light curve by Belton et al. (2018), claimed that ‘Oumua-
mua should have experienced negligible torque. We critically
assess this claim in our paper.
Despite significant research efforts, the nature of
‘Oumuamua remains a puzzle. Perhaps it is a comet, which
would be in line with the theoretical expectations predicting
200 . . . 104times more icy objects than rocky objects among
interstellar minor bodies (Meech et al. 2017), and because
it exhibited strong non-gravitational acceleration. Or per-
haps it is an asteroid, judging from the non-detection of any
signs of outgassing, but then the non-gravitation accelera-
tion remains unexplained. Or it could be something else, e.g.
a solar sail. Clearly more efforts are needed to bring some
clarity to this issue.
This paper represents the first attempt to fit a physi-
cal model to the observed light curve of ‘Oumuamua, with
all free model parameters recovered by means of multi-
dimensional optimization. The paper is organized as fol-
lows. Section 2describes the two main components of the
model: the kinematic part (spin evolution of a tumbling as-
teroid with optional constant torque), and the brightness
model part (can be either a geometric one – Lommel-Seeliger
triaxial ellipsoid, or a variable albedo one – ”black-and-
white ball”). Section 3presents our GPU-based numerical
code, describes the numerical setup, and details code vali-
dation tests. Section 4describes the observational data used
for modeling, and presents the results of fitting ‘Oumua-
mua’s light curve with our physical model (with and with-
out torque). The paper ends with Discussion (section 5) and
Conclusions and Future Work (section 6).
2.1 Overview
Our model consists of two major components: kinematic
model, and brightness model.
In terms of kinematics, ‘Oumuamua is assumed to be
a rigid body with an arbitrary shape and arbitrary den-
sity distribution, which rotates in a Non Principal Axis
(NPA) mode; in other words, it is tumbling. As the sim-
plest non-inertial extension, the model can optionally ac-
count for arbitrary torque which is fixed in time and space
(in the asteroidal comoving coordinate system). Physically,
this might correspond to semi-steady outgassing from a spe-
cific point on the asteroid’s surface. We describe the equa-
tions of motion in subsection 2.2, the initial conditions in
subsection 2.3, and the model’s coordinate transformations
in subsection 2.4.
Our main brightness model (described in subsection 2.5)
assumes that the asteroid is a triaxial ellipsoid with uniform
albedo surface with Lommel-Seeliger (LS) light scattering
properties. The ellipsoid can be either self-consistent (with
the semi-axes lengths taken from the kinematic part of the
model), or relaxed (with the semi-axes lengths not linked to
the kinematic model). Relaxing the brightness ellipsoid pa-
rameters can help to account for potential deviations of the
asteroid’s properties (e.g. shape) from the model assump-
tions (Pravec et al. 2005).
We also explore the simplest non-geometric explanation
for the large brightness variations of ‘Oumuamua: a spheri-
cal body with the two hemispheres having different albedo
values, which is oriented arbitrarily relative to the diagonal
components of the inertia tensor. This model is described in
subsubsection 2.5.2.
2.2 Equations of motion
We adopt a comoving right-handed Cartesian coordinate
system with the three principal axes – b,c, and a– co-
inciding with the three diagonal components of the aster-
oid’s inertia tensor, Ib,Ic, and Ia, respectively. The axes are
chosen in a way that the following inequalities are always
true: Ia<
=Ic. (Our axes b,c, and aare equivalent
to the axes i,s, and lof Samarasinha & A’Hearn 1991.) If
the asteroid’s shape can be described as a triaxial ellipsoid,
the corresponding semi-axes of the ellipsoid would follow the
We adopt the units where a=1and Ia=1; the time
unit is a day. In these units, the three diagonal components
of the inertia tensor of a triaxial ellipsoid are
In the comoving coordinate system, Euler’s equations
for rigid body rotation can be written as
(Landau & Lifshitz 1976, p. 115). Here b,c,aand Kb,
Kc,Kaare the components of the angular velocity vector
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 3
and the torque pseudo vector, respectively, in the comoving
(asteroidal) coordinate system. The angular velocity vector
components can be expressed in terms of the three Euler
angles (nutation angle θ, precession angle ϕ, and rotation
angle ψ) and their derivatives:
b=˙ϕsin θsin ψ+˙
θcos ψ,
c=˙ϕsin θcos ψ˙
θsin ψ,
a=˙ϕcos θ+˙
(Landau & Lifshitz 1976, p. 111).
Equations (2) and (3) can be rewritten to form a system
of six ordinary differential equations (ODEs) for the three
components and the three Euler angles:
˙ϕ=bsin ψ+ccos ψ/sin θ,
θ=bcos ψcsin ψ,
ψ=a˙ϕcos θ.
Here Tb=Kb/Ib,Tc=Kc/Ic, and Ta=Kaare the compo-
nents of the torque vector normalized by the corresponding
diagonal components of the inertia tensor.
The system (4) is the one we need to integrate numeri-
cally to describe the rotation of a rigid body in the presence
of torque. If torque is zero, there is a trick allowing one to
bypass the Euler equations, and reduce the problem to only
three ODEs, for the three Euler angles (Kaasalainen 2001).
Specifically, in the torque-free regime the angular momen-
tum vector Lof a rotating rigid body is fixed in all iner-
tial coordinate systems (angular momentum conservation),
which lets us write the following equations:
b=LI 1
bsin θsin ψ,
c=LI 1
csin θcos ψ,
a=Lcos θ
(Landau & Lifshitz 1976, p. 119). As shown by Kaasalainen
(2001), combining equations (3) and (5) results in the fol-
lowing system of three ODEs, for the three Euler angles:
˙ϕ=L(I+Icos 2ψ),
θ=LIsin θsin 2ψ,
ψ=cos θ(L˙ϕ).
Here I=1
cand I+=1
Kaasalainen (2001) set the principal axes of the co-
moving coordinate system differently for Short Axis Mode
(SAM) and Long Axis Mode (LAM) rotators, which allowed
them to simplify many model equations, with only one form
of an equation for both SAM and LAM cases. In our test-
ing, this worked well for mildly flattened objects. Unfortu-
nately, for the shortest-to-longest ellipsoid axes ratios (c/a)
smaller than 0.2we observed the ODEs integration errors
to quickly become significant (necessitating much smaller
time steps, which would make simulations much longer).
At some point (around c/a<
0.15), the ODEs integration
completely breaks down due to some numerical instability.
No such issues were observed when we used the same co-
moving axes assignment (with cand aalways correspond-
ing to the smallest and largest ellipsoid’s semi-axes, respec-
tively), for both SAM and LAM objects, as in Samarasinha
& A’Hearn (1991). Using this latter approach allowed us to
use a fairly large integration time step without noticeably
affecting the accuracy of integration. Crucially, this also al-
lowed us to fully explore the range of c/aratios needed to
explain ‘Oumuamua’s extreme brightness variations.
In our model, we solve equations (6) for torque-free
runs, and equations (4) for runs with torque (which is as-
sumed to be constant in the comoving coordinate system). It
is important to emphasize that the equations are applicable
to any rigid body (not just triaxial ellipsoids), described by
the three diagonal components of its inertia tensor – Ib,Ic,
and Ia.
2.3 Initial conditions
To solve either equations (6) or equations (4), one has to
set the initial values of the independent variables. In the
adopted comoving coordinate system, bca, precession angle
ϕinitially can have any value: ϕ0[0,2π].
Angular momentum vector modulus Lcan have any
positive value initially, L[0,[. Total allowed range for
the model parameter E02E/L2is I1
c. . . 1. (Here Eis
the rotational kinetic energy of the body.) In the short and
long axis modes (SAM and LAM), the corresponding sub-
ranges are I1
c. . . I1
band I1
b. . . 1, respectively (Samaras-
inha & A’Hearn 1991, equations A30 and A54). Parameters
Land E0are fixed in torque-free simulations, but change
with time in runs with non-zero torque. In the latter case,
only the initial values of the two parameters need to be pro-
vided; the ODEs (Equation 4) do not explicitly use them.
Rotation angle ψcan have any value (ψ0[0,2π]) in
LAM, but is constrained to the following range in SAM
(Samarasinha & A’Hearn 1991, equations A63 and A64):
ψmin =arctan sIb(Ic1/E0)
ψmax =arctan sIb(Ic1/E0)
Combining the expressions for the components of the
angular momentum vector,
Lb=Lsin θsin ψ,
Lc=Lsin θcos ψ,
La=Lcos θ
(Landau & Lifshitz 1976, p. 119) with the kinetic energy
MNRAS 000,121 (2019)
4S. Mashchenko
(Landau & Lifshitz 1976, p. 116) allows us to write the ex-
pression for the initial value of the nutation angle:
θ0=arcsin sE01
As one can see, θ0is not a free parameter (unlike ϕ0and
ψ0): it is fully determined by other model parameters (E0,
ψ0,band c). Nutation angle can vary between 0and π.
Once we set the values for the parameters L,θ0, and
ψ0, the initial values of the three components of the angular
velocity vector for simulations with non-zero torque can
be computed using equations (5).
When modeling light curves of asteroids, it is common
to use rotation period Pψand precession period Pϕ(which
can often be deduced or guessed from observations) in place
of the physical parameters E(or E0in our case) and L.
This necessitates the conversion (Pψ,Pϕ)(E0,L) for each
tested model which is computationally expensive and can
dramatically slow down the simulations. We use a compro-
mise approach in our model, which can take either Lor Pψ
as a free parameter. We keep E0(which is relatively well
constrained) as another free parameter. When Pψis a free
parameter, we derive Lfrom Pψand E0using the following
efficient computational routine.
First we compute the parameter k2(Samarasinha &
A’Hearn 1991, equations A32 and A56):
Next we compute the elliptic integral,
(Landau & Lifshitz 1976, p. 118), using very efficient AGM
(arithmetic-geometric mean) iterative method1:
for (int i=0; i<N; i++)
a1 = (a+g)/2;
g1 = sqrt(a*g);
a = a1; g = g1;
If initial values of the variables aand gare set to 1and
1k2, respectively, the above iterative loop quickly con-
verges, with π/(a+g)Keas N→ ∞. In our testing, after
only 5 iterations the error in Keis smaller than 1010, for
k2=0. . . 0.9999998.
Figure 1. Transformation from the inertial coordinate system
XY Z (Zbeing the initial orientation of the angular momentum
vector L) to the comoving coordinate system bca using Euler
angles θ,ϕ, and ψ.
Now we can derive Lfrom Pψas follows:
E0(Ib1)( IcE0−1)(LAM),
E0(IcIb)( E0−11)(SAM)
(Samarasinha & A’Hearn 1991, equations A45 and A71).
2.4 Coordinate transformations
Our model utilizes three different right-handed Cartesian
coordinate systems. The starting point is the inertial Solar
System Barycentre (SSB) coordinate system, xyz. We used
online NASA’s tool HORIZONS2to generate SSB coordi-
nates for the centres of the Sun, Earth, and ‘Oumuamua for
all the data points in ‘Oumuamua’s light curve.
The second coordinate system, X Y Z, is also inertial. The
axis Zcoincides with the angular momentum vector L. (For
runs with non-zero torque, Zcoincides with the angular
momentum vector Lat the initial moment of time.) The axis
Xis arbitrarily chosen to coincide with the vector y×Z.
The axis Ycomplements the other two axes to form a right-
handed coordinate system: Y=Z×X. The three axes of
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 5
the XY Z coordinate system can be described as unit vectors
in the SSB (xyz) coordinate system as follows:
Zx,y,z={sin θLcos ϕL,sin θLsin ϕL,cos θL},
Here free model parameters θLand ϕLare polar and az-
imuthal angles, respectively, describing the (initial) orienta-
tion of the angular momentum vector Lin the SSB coordi-
nate system.
Finally, the comoving (asteroidal) coordinate system,
bca, has its three axes coinciding with the intermediate,
largest, and smallest diagonal components of the asteroidal
inertia tensor. This coordinate system is derived by rotating
the XY Z coordinate system using three Euler angles, θ,ϕ,
and ψ(Figure 1), which are derived by solving the equa-
tions of motion (subsection 2.2). If the mass distribution of
the asteroid can be well approximated as a homogeneous tri-
axial ellipsoid, the three axes correspond to the intermediate
(b), smallest (c), and largest (a) semi-axes of the ellipsoid.
The SSB components of the three axes, b,c, and a, can
be computed via a sequence of geometric transformations as
Components of the unit node vector N, derived by ro-
tating vector Xtowards vector Yby the Euler angle ϕ
(see Figure 1), with Zbeing the rotation axis, in the SSB
coordinate system are given by
Nx,y,z=(Xxcos ϕ+Yxsin ϕ, Yysin ϕ,
Xzcos ϕ+Yzsin ϕ.(15)
Using another auxiliary unit vector, p=N×Z,
allows us to derive the axis a(a unit vector) components in
the SSB coordinate system by rotating Zby the Euler angle
θtowards p, with the node vector Nbeing the rotation
ax,y,z=(Zxcos θ+pxsin θ, Zycos θ+pysin θ,
Zzcos θ+pzsin θ.(17)
Using yet another auxiliary unit vector, w=a×N,
allows us to derive the axis b(a unit vector) components in
the SSB coordinate system by rotating the vector Nby the
Euler angle ψtowards the vector w, with the vector abeing
the rotation vector:
bx,y,z=(Nxcos ψ+wxsin ψ, Nycos ψ+wysin ψ,
Nzcos ψ+wzsin ψ.(19)
The SSB components for the third axis, c=a×b, can now
be computed as
Brightness models described in subsection 2.5 require
the knowledge of the components of the unit vectors S
and Econnecting the asteroid with the centres of the Sun
and Earth, respectively, in the comoving coordinate system
(bca). These vectors are readily obtainable in the SSB coor-
dinate system (based on HORIZONS’ data). Once the base
vectors of the comoving coordinate system, b,c, and a,
have been computed (equations 17,19,20), the components
of the vectors Sand Ein the bca coordinate system can be
calculated as
From the equations listed in this subsection, only the
equations (14) are computed once per model integration;
the rest have to be computed for each observed data point,
as they depend on time-variable Euler angles θ,ϕ, and ψ.
2.5 Brightness models
2.5.1 Lommel-Seeliger triaxial ellipsoid
We adopt the brightness model of Muinonen & Lumme
(2015) written for a triaxial ellipsoid with Lommel-Seeliger
light scattering surface. We consider the simplest case of the
isotropic single-scattering function. Disc-integrated absolute
magnitude of such an object in our comoving coordinate sys-
tem bca can be expressed as
H=V2.5 log (b0c0TT
Tcos(λ0α0)+cos λ0+sin λ0
×sin(λ0α0)ln cot 1
2λ0cot 1
MNRAS 000,121 (2019)
6S. Mashchenko
cos α0=(SbEb/b02+ScEc/c02+SaEa)/(TT),
sin α0=1cos α02,
+2TTcos α0,
cos λ0=(T+Tcos α0)/T,
sin λ0=Tsin α0/T
(Muinonen & Lumme 2015; please note that the authors
used abc coordinate system, whereas we use bca coordinate
system). Here Sb,c,aand Eb,c,aare components of the unit
vectors in the directions of the Sun and Earth, respectively,
in the asteroidal coordinate system bca (see Equations 21
and 22), b0and c0are the intermediate and smallest semi-
axes of the brightness ellipsoid expressed as a fraction of
its largest semi-axis, and the constant Vabsorbs two un-
known parameters – albedo and scale (largest semi-axis a0
in physical units) of the asteroid.
Fitting the brightness model (Equation 23) to the as-
teroid’s observed light curve, transformed to absolute mag-
nitudes, produces the value of V(offset between the model
and observed light curves). To get an estimate of the aster-
oid’s scale a0
m(the ellipsoid’s largest semi-axis in meters),
let us place the asteroid 1 au away from the Earth and Sun,
with zero phase angle. Equation 23 is then reduced to
H=V2.5 log(b0c0T).(25)
As the projected area of the ellipsoid (in square meters) is
(Muinonen & Lumme 2015)
Equation 25 can be rewritten as
The standard asteroid diameter equation (Lamy et al. 2004;
their equation 5, written for a spherical body observed at
zero phase angle) can be expressed in terms of the asteroid’s
projected area A(in square meters) as
A=π(1.49598 ×1011)2
where mis the apparent magnitude of the Sun in the same
spectral filter as the one used to observe the asteroid, and pis
the geometric albedo of the asteroid. Equating Equation 27
and Equation 28 produces the estimate of the physical scale
of the ellipsoid:
m=1.49598 ×1011
In our code, the triaxial ellipsoid brightness model can
be used in two different ways:
(i) Self-consistent case: semi-axes of the brightness ellip-
soid, b0and c0, are equal to the corresponding semi-axes of
the kinematic ellipsoid, band c. No additional free parame-
(ii) Relaxed case: semi-axes of the brightness ellipsoid are
not equal to the corresponding semi-axes of the kinematic
ellipsoid. Two additional free parameters: b0and c0.
2.5.2 Black-and-white ball
As the simplest case of a non-geometric explanation for the
large brightness variations of ‘Oumuamua, we consider a toy
brightness model consisting of a spherical body with two
hemispheres with different albedo values. As ‘Oumuamua’s
phase angle is relatively small (α=19 . . . 25in the time
interval covered by our analysis), we ignore phase effects for
Position of the darker hemisphere (with the albedo de-
scribed by a free model parameter κ[0,1[) is specified via
a unit vector h(described by two free model parameters:
polar angle θhand azimuthal angle ϕh) in the asteroidal co-
ordinate system bca. The opposite (brighter) hemisphere is
considered to have albedo =1.
Ignoring phase effects (phase angle α=0), integrated
absolute magnitude of such an object is given by
H=V2.5 log "κ1+cos ζ
2+1cos ζ
Here ζis the angle between the vector in the direction of
the observer, E(see Equation 22), and the vector h.
In total, this brightness model is specified by three free
parameters: κ,θh, and ϕh.
2.6 Free parameters
Our model can be used with different numbers of free pa-
rameters. The most basic model (tumbling self-consistent
ellipsoid with zero torque) has 8 free parameters: Lor Pψ,
θL,ϕL,ϕ0,ψ0,E0,b, and c. Here Lis the modulus of the
angular momentum vector, Pψis the rotation period, θLand
ϕLare the polar and azimuthal angles, respectively, for the
angular momentum vector, ϕ0and ψ0are the initial val-
ues of the precession and rotation Euler angles, respectively,
E0=2E/L2where Eis the rotational kinetic energy, and b
and c(used in both kinematic and brightness models) are
the intermediate and smallest semi-axes of the ellipsoid ex-
pressed in units of the largest semi-axis a.
Multiple expanded models (with larger numbers of
free parameters) are supported. In particular, relaxing the
brightness ellipsoid model adds two free parameters (bright-
ness ellipsoid’s semi-axes b0and c0), bringing the total to 10.
Non-zero torque models add three more free parameters –
normalized torque pseudo vector components in the comov-
ing coordinate system Tb,Tc, and Ta. This brings the total
to 11 and 13 free model parameters, for self-consistent and
relaxed brightness ellipsoid models, respectively.
Finally, black-and-white ball brightness model adds
three free parameters (dark-to-bright albedo ratio κand po-
lar and azimuthal angles for the dark side in the comoving
coordinate system, θhand ϕh, respectively) on top of the 8
parameters of the basic model, bringing the total to 11 free
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 7
parameters. In the presence of torque, the number grows to
One more parameter, V, is present implicitly (see
equations 23 and 30). It is the offset between the observed
and model light curves, in brightness magnitudes. It is a
byproduct of χ2fitting of the model brightness curve to the
observed one, and can be used to assign physical units to
the model (Equation 29).
Our code can also utilize other free parameters which
we did not use for the current project. In particular, our
detrending” parameter Ais designed to approximate the
gradual change of the average asteroid’s brightness with the
phase angle α. Preliminary tests showed that this additional
parameter does not improve the quality of fit for our models,
which is not surprising given that we analyse a very short
time interval, where ‘Oumuamua’s phase angle changes only
slightly (from 19.2 to 24.7).
3.1 Overview
We present our code3, which fits different models of tumbling
asteroids, described in section 2, to observed light curves.
It is written in C++ utilizing CUDA framework, and con-
sists of more than 3000 lines of code. With the exception of
the brief initialization and finalizing steps, and infrequent
checkpointing steps, the entire code runs on a GPU as tens
of thousands of independent parallel threads, each explor-
ing different optimization paths through multi-dimensional
free model parameter space. The code runs best on Pascal
P100 GPUs (CUDA capability 6.x), but can also be used on
older Tesla GPUs (CUDA capability 2.0 or larger). The mas-
sive computational power of modern GPUs makes it realis-
tic to reliably find global minima in the 8+ dimensional free
model parameter space using a Monte-Carlo style optimiza-
tion strategy. As the optimization engine we use downhill
simplex (Nelder-Mead) method4which works well for large
number of dimensions and does not require the knowledge
of the partial derivatives of the function to optimize, which
in our case is the χ2function produced by fitting the model
light curve to the observed one.
Each GPU thread repeatedly goes through the following
(i) A random initial point is generated in the scale-free
model parameter space, using CURAND library. The li-
brary allows for generation of tens of thousands of indepen-
dent quasi-random number sequences, one for each parallel
thread. In the optimization scale-free space, each model pa-
rameter is normalized to have the initial range of [0,1] (for
periodic angle parameters, the [0,1] scale-free range corre-
sponds to the [0,2π]radians range). Strongly non-linear pa-
rameters are first linearized. For example, kinematic and
brightness ellipsoid semi-axis ratios, b,c,b0, and c0, utilize
logarithmic scale, to provide a comparable sampling cover-
age for each decade of the full parameter’s range. Parameter
3The code is publicly available here:
Pψ(rotation period) is sampled uniformly in the 1/Pψ(fre-
quency) space. Some parameters (like c,c0, and Pψ) have
static limits for the allowed range of the initial random val-
ues, while other parameters (b,b0,ψ0,E0etc.) have limits
which change during optimization (they depend on the val-
ues of other parameters). The way the initial value of E0
is generated is such that SAM and LAM have equal prob-
abilities. During optimization, the values of parameters are
allowed to drift beyond the initial range (”soft limits”), as
long as they stay within the physically allowed hard limits.
(ii) The initial simplex is constructed, using a small (typi-
cally 0.001 in scale-free units) initial step in each dimension.
(iii) Every time the optimization ( χ2) function value
needs to be computed, the following substeps are performed:
(a) Free model parameters are converted from scale-
free to physical units.
(b) Initial values of the independent variables in the
equations of motions (either Equation 4 or Equation 6)
are computed (subsection 2.3).
(c) The ODEs (equations of motions; see subsec-
tion 2.2) are solved using the 4th order Runge-Kutta
method. The integration starts at the time corresponding
to the earliest observed point, and proceeds with steps not
larger than 0.01 d (which in our tests provides sufficient
accuracy for ‘Oumuamua’s modeling) from one observed
point to the next one, covering all the observed points.
This produces model values of the Euler angles θ,ϕ, and
ψ(plus the values of the angular velocity vector compo-
nents, b,c, and a, for models with non-zero torque)
for each observed point.
(d) For each observed point, the directions of the unit
vectors Sand Econnecting the asteroid with the centres
of the Sun and Earth, respectively, are computed via a
series of geometric transformations as described in sub-
section 2.4.
(e) Using one of the brightness models (subsection 2.5),
the model absolute magnitude of the asteroid is computed
for each observed point. χ2value is computed, along with
the offset Vbetween the observed and modeled light
curves. Each point uses the weight of 12
i, where σiis
the measurement error for this data point. If the observed
data were produced using more than one spectral filter,
separate values of Vare computed for each filter, inde-
(iv) The downhill simplex algorithm is used to descend to
a nearby local χ2minimum. The descent is stopped when ei-
ther the simplex has shrunk below the smallest allowed size
(a sign of being in a local minimum) or a maximum num-
ber of simplex steps (typically 5000) were taken, whichever
comes first.
3.2 Numerical runs
We tried several multi-stage optimization strategies. The one
we adopted performed the best overall in our validation tests
(see below), and consists of the following steps:
(i) Random search stage. Eight instances of the code is
run on eight P100 GPUs for 24 hours. There are 30,000
threads running in parallel on each GPU. Each GPU thread
starts at a random point in the free model parameter space.
MNRAS 000,121 (2019)
8S. Mashchenko
Once all the threads in a block of 256 threads converge to
local χ2minima or hit the simplex step limit, the best model
(lowest χ2) of the block is chosen. It is then used to seed
the second search phase (searching in the neighborhood of
the best model), where the same 256 threads would start
at points which are randomly and slightly offset from the
best model, with randomized initial simplex steps for each
dimension. The second phase ends using the same criteria:
all the block threads either converge to local minima or hit
the simplex step limit. The best model of the block in the
second phase is stored in a file. At the end, about 105best
models are written to files.
(ii) Re-optimization stage. Eight instances of the code
running on eight P100 GPUs for 24 hours is launched after
the first stage is completed. The GPU farm goes through
the sorted list of the models found in the first stage, start-
ing from the best (smallest χ2) model. Each code instance
searches in the neighborhood of a model from the first stage
in multiple (typically 20) global re-optimization steps. Each
global step consists of all parallel threads on a GPU (typ-
ically 30,000) searching for the best χ2minimum in the
neighborhood of the globally best model found by all the
parallel threads in the previous step. Around 1 per cent of
the best models from Stage One are processed in Stage Two,
resulting in 103highly optimized models.
(iii) Fine-tuning stage. Finally, a few best models from
Stage Two are subjected to additional global re-optimization
steps, using higher (double, vs. single in prior stages) floating
point precision and much smaller minimum simplex diam-
eter (1010 vs. 105, in scale-free units), until a numerical
convergence is achieved. For models where some light curve
minima still have an obvious offset from the observed ones
in the time dimension, an attempt to drive all the major
model minima towards nearest observed ones is made. This
is achieved by means of progressively decreasing the opti-
mization ( χ2) function value as the model minima converge
to the observed ones in the time dimension.
3.3 Code validation: artificial dataset test
We performed two different kinds of validation tests with our
code. The first one, described in this subsection, is designed
to test the internal self-consistency of the code, using fake
light curve data which were created to be as close to the
observed light curve data for ‘Oumuamua as possible.
To generate the fake data, we used one of our best-
fitting models for ‘Oumuamua (see subsection 4.1) which
consists of a relaxed brightness ellipsoid model and a tum-
bling rotation model subject to fixed torque (13 free model
parameters in total; see subsection 2.6). This model is for a
thin (1:6 ratio) disc-like object which is close to being self-
consistent (brightness ellipsoid semi-axes b0and c0are close
to the corresponding kinematic ellipsoid semi-axes, band
c). To generate fake light curve data based on this model,
we computed the model absolute magnitudes for the same
number of observations at exactly the same moments of time
as in ‘Oumuamua’s dataset. Next we degraded the data us-
ing Gaussian noise with the std following the same trend
as the observational uncertainties: σ100.323H8.416. (Here
His the absolute magnitude of the asteroid, and units for
σare magnitudes.) The same values of σwere later used
as fake observational uncertainties during χ2model fitting.
Figure 2. Artificial dataset test. tis the number of days since
MJD =58050, and Vis the absolute magnitude in the rspectral
filter. Artificial data points with one-sigma errorbars are shown
in green (grey in the printed version of the journal). Black line
depicts the light curve of the underlying model. Orange (grey in
the printed version of the journal) line corresponds to the best-
fitting model.
Figure 2 shows the fake data as dots with errorbars, and the
underlying model as a black curve.
We next performed a full set of model numerical runs (as
described in subsection 3.2) on the fake data, using the same
optimizations parameters and the same soft limits for free
model parameters as in ‘Oumuamua runs (subsection 4.3).
The free model parameters were the same 13 parameters
used to generate the underlying model. At the end we did
find a few very good model matches; the best one is shown as
an orange (grey in the printed version of the journal) curve
in Figure 2. Despite the significant noise present in the data
and its patchiness, the recovered model is very close to the
underlying model.
Full details of the underlying and best-fitting models are
listed in Table 1 in Fake ini and Fake fit columns, respec-
tively. The only obvious difference between the two models
is the opposite direction of the angular momentum vector,
which affected the angles θL,ϕL, and ϕ0. The rest of the
recovered parameters are close to their original values, in-
cluding the three normalized torque vector components, Tb,
Tc, and Ta. The χ2value for the best-fitting model (1.011)
is close to one, as expected.
The artificial dataset test demonstrated the following:
(i) The code is internally self-consistent (it can recover its
own models from noisy data).
(ii) The underlying model can be recovered for models as
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 9
Table 1. Models.
Parameter Fake ini Fake fit TD60ATD60BTD60CINERT DISC CIGAR SAIL BALL
Fitting parameters
χ2- 1.011 6.275 2.283 4.135 18.40 9.963 10.84 11.50 7.859
rms, mag - 0.116 0.124 0.075 0.101 0.299 0.220 0.230 0.236 0.195
V, mag 22.6720 22.5066 18.8774 18.3002 18.2139 19.3378 22.5287 20.2872 22.0703 22.5523
Free model parameters
θL, rad 2.23573 1.05433 0.97459 1.91359 2.50770 0.90522 2.27661 2.02359 1.42107 1.57272
ϕL, rad 1.62370 4.96208 5.82597 2.68432 2.68163 3.00312 1.61185 1.96810 2.36917 4.67914
ϕ0, rad 6.25839 3.94604 2.91895 0.08683 0.17613 5.12912 6.24557 2.96016 3.23746 4.58948
Tb4.02658 4.08395 - - - - 3.97579 2.55650 47.8356 15.4057
Tc1.11909 1.04747 - - - - 1.12470 7.43845 0.40040 2.23678
Ta5.85003 5.22063 - - - - 5.82852 1.01262 21.4239 5.75293
c0.16435 0.18766 0.56295 0.52170 0.53268 0.00999 0.16293 0.12972 0.00001 0.52318
b0.96483 0.96332 0.64739 0.78211 0.80860 0.06014 0.96427 0.13144 0.80683 0.93090
00.98270 0.97923 0.65202 0.58023 0.60140 0.00820 0.98204 0.03353 0.42320 0.86537
L017.0921 16.9094 97.9840 93.5938 90.4275 293.884 17.0497 525.649 51.4673 42.4590
ψ0, rad 1.97276 1.91811 1.70236 0.48206 0.28808 6.14166 1.99745 0.08955 0.07324 0.52839
c00.17488 0.14139 0.31844 0.20757 0.19353 0.01375 - - - -
b00.99941 0.87302 0.67547 0.78147 0.76291 0.06346 - - - -
θh- - - - - - - - - 2.93279
ϕh- - - - - - - - - 1.96926
κ- - - - - - - - - 0.03083
Derived parameters
Pψ,0, h 52.01 52.70 6.784 6.783 6.788 7.670 51.81 80.84 7.750 23.92
Pϕ,0, h 10.75 10.89 2.850 2.852 2.851 138.3 10.79 8.557 7.163 4.509
Pψ,1, h 32.37 33.98 - - - - 32.28 29.45 8.194 20.12
Pϕ,1, h 10.83 10.52 - - - - 10.81 8.895 7.685 3.930
Note. Different columns correspond to different models. The units for free model parameters are set by a=1and Ia=1; the time
unit is a day. For models with torque, E0
0and L0values are the initial values, for zero torque models they are fixed in time. For the
derived parameters Pψ,Pϕ, and MODE, both the initial (subscript 0) and final (subscript 1; only for models with torque) values are
provided. Models TD60Aand TD60Bare for the second half of the full dataset for 2002 TD60 (993 points); model TD60Cis for the full
dataset (1914 points). For the three 2002 TD60 models, only one value of V(corresponding to the observations calibrated to Rfilter) is
provided. In χ2computations, we arbitrarily assumed the 2002 TD60 brightness measurements to have the std of 0.05 mag. The initial
MJD moments of time in the asteroidal coordinate system are 58053.31317 (all Fake and ‘Oumuamua models), 52609.73185 (TD60Aand
TD60Bmodels), and 52585.01840 (TD60Cmodel).
complex as the one used for testing (13 free model param-
eters, including three torque parameters), and for data as
patchy and noisy as ‘Oumuamua’s light curve dataset.
3.4 Code validation: tumbling asteroid 2002 TD60
The artificial dataset test described in the previous subsec-
tion validated many important aspects of our code and nu-
merical procedure, but it lacks physical validation of the
code and model. (Being internally self-consistent does not
mean the model is correct and physical.) We addressed this
shortcoming by using our code to recover the parameters of
a well studied tumbling asteroid, 2002 TD60 (Pravec et al.
The Amor asteroid 2002 TD60 is one of the best studied
NPA rotators. It was observed on multiple telescopes during
the observational campaign in November–December 2002,
producing a significant number (1914) of high accuracy mea-
surements of the asteroid’s brightness (Pravec et al. 2005).
Around 30 per cent of the measurements (544 points) are
calibrated (Rspectral filter); the rest consist of eight uncali-
brated subsets. This necessitates the use of nine independent
fitting parameters V(one for each internally self-consistent
subset of the data) when fitting a model to the full dataset.
Pravec et al. (2005) obtained good model fits to the light
curve of 2002 TD60. In many respects their approach is sim-
ilar to ours; in particular, their brightness model is a triaxial
ellipsoid with LS reflectance law, and the optimization en-
gine is the simplex downhill method. But there are some
non-trivial differences. Importantly, their brightness model
is a numerical one (the triaxial ellipsoid is represented by
2292 flat triangles), whereas we use a more accurate and
reliable (and much faster to compute) analytical formula-
tion of Muinonen & Lumme (2015). (One has to note that
the numerical approach is more flexible as one can easily
modify the reflectance law.) Another significant difference
is the fact that Pravec et al. (2005) had to estimate many
of the model parameters (main frequencies, brightness el-
lipsoid axes ratio) before performing the simplex downhill
optimization, whereas we employ a brute force optimization
approach in which no model parameter estimates are used.
The brute force approach is more advantageous as it ex-
MNRAS 000,121 (2019)
10 S. Mashchenko
plores the whole free model parameter space, including the
regions which may be overlooked in a constrained approach.
Our approach was made possible by the dramatically faster
computing hardware available today, and also thanks to the
brightness model being analytical.
We carried out a full suite of numerical runs, as de-
scribed in subsection 3.2, to fit our model to the light curve
data for 2002 TD60 (generously provided by the author; P.
Pravec 2018, private communication). Most of our analy-
sis was restricted to the second half of the full dataset (993
out of 1914 points; MJD =52609.752617.1; five independent
Vparameters). As TD60 observations span a significantly
longer time interval (11–35 d) than ‘Oumuamua’s dataset
(5 d), we had to use a shorter ODE integration time step
(0.005 d vs. 0.01 d) to achieve a good numerical convergence.
As Figure 3 shows, our best-fitting model fits very well the
observed light curve for the asteroid. Full details for this
model are listed in the TD60Bcolumn of Table 1. The qual-
ity of the fit is substantially better than for the best-fitting
model of Pravec et al. (2005): the rms is 0.075 and 0.18 mag,
respectively. We should caution that these rms values are for
different subsets of the dataset. To make a more meaningful
comparison, we re-optimized our model TD60Bfor the full
dataset (1914 points), which produced a slightly worse fit,
with the rms of 0.10 mag (model TD60Cin Table 1) – still
almost factor of two better than the best-fitting model of
Pravec et al. (2005).
Interestingly, our best-fitting model is substantially dif-
ferent from the one derived by Pravec et al. (2005). The
rotation and precession periods are essentially identical (our
model: Pψ=6.783 h and Pϕ=2.852 h; model of Pravec
et al. 2005:Pψ=6.787 h and Pϕ=2.851 h), but the models
are rather different otherwise. Importantly, our best-fitting
model is in a SAM tumbling motion, whereas the model of
Pravec et al. (2005) is a LAM rotator.
To get more clarity, we used our code to search for a
best-fitting model in the neighbourhood of the best-fitting
model of Pravec et al. (2005), using the same dataset as
for our model TD60B(993 points). We did find a local χ2
minimum which corresponds to a model (column TD60Ain
Table 1) which is much closer to the best-fitting model of
Pravec et al. (2005). Importantly, both models are LAM
rotators. The periods are again almost identical, but now
other model parameters are close as well: c={0.56,0.54},
b={0.65,0.70},c0={0.32,0.36},b0={0.68,0.64}(for our
model and that of Pravec et al. 2005, respectively; using
our notation). The rms for TD60Ais 0.124 mag, which is 1.7
times larger than for our globally best-fitting model, TD60B.
Our light curve model fitting for the tumbling asteroid
2002 TD60 appears to be largely consistent with the previ-
ously published results (Pravec et al. 2005). The small dif-
ferences in the best-fitting model parameters can likely be
attributed to numerical (Pravec et al. 2005) versus analytical
(present paper) implementations of the LS triaxial ellipsoid
integrated brightness model. The biggest factor appears to
be our brute force optimization approach which allowed us
to thoroughly explore the free model parameter space and
as a consequence to discover a substantially better solution.
At the end, the second (physical) code validation test
proved to be not as clear-cut as we hoped. Nevertheless, we
did gain extra confidence in the code and model correctness,
and again confirmed the code’s power to discover good model
Figure 3. Our best-fitting tumbling asteroid model (TD60B, see
Table 1) for the asteroid 2002 TD60 .tis the number of days
since MJD =52500, and Vis the absolute magnitude in the R
spectral filter. Dots are the 993 observational data points from
Pravec et al. (2005). The top and bottom panels correspond to
the panels (c) and (d) of figure 4 of Pravec et al. (2005).
fits for realistic (noisy and patchy) light curve datasets, for
asteroids which shape is not exactly a triaxial ellipsoid.
4.1 Observational data
We used two sources of ‘Oumuamua’s light curve data,
together covering five days of observations, from MJD =
58051.0(October 25, 2017) to MJD =58056.3(October 30,
2017). This is the only time interval when the asteroid’s
brightness measurements were very frequent (but still rather
patchy, with large gaps between individual observational
runs; see Figure 4), resulting in reliable detection of mul-
tiple features (minima and maxima). The few other existing
observations excluded from our analysis (Belton et al. 2018)
are very sparse, lack any obvious features (significantly re-
ducing their value for model fitting), and span a much longer
time interval (one month), which would make model compu-
tations a factor of five slower. Over the course of these five
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 11
Figure 4. Observed light curve for ‘Oumuamua (770 points)
based on the datasets of Fraser et al. (2018, black points) and
Drahus et al. (2018, red points; grey points in the printed version
of the journal). tis the number of days since MJD =58050, and V
is the absolute magnitude in the r/r0spectral filter. The errorbars
are one-sigma uncertainties.
days, the asteroid–Sun distance ranged from 1.36 to 1.49 au,
the asteroid–Earth distance ranged from 0.40 to 0.58 au, and
the phase (Earth – asteroid – Sun) angle ranged from 19.2
to 24.7. The small range of the phase angle change is par-
ticularly helpful, as it minimizes the need for sophisticated
light scattering formulations in our brightness models.
The first source of data (Fraser et al. 2018) is itself a
compilation of optical photometry of ‘Oumuamua from mul-
tiple publications (Meech et al. 2017;Bannister et al. 2017;
Jewitt et al. 2017;Bolin et al. 2018;Knight et al. 2017). It
consists of 339 ‘Oumuamua brightness measurements, con-
verted to the same spectral filter (r0) using known spec-
tral properties of the asteroid. Some of these observations
have very large one-sigma uncertainties; the full range is
0.02 . . . 2.4mag, with the geometric mean of 0.19 mag. The
dataset of Fraser et al. (2018) is corrected for light travel
(times correspond to the asteroidal coordinate system), and
converted to absolute magnitudes (corresponding to both
asteroid–Earth and asteroid–Sun distances of 1 au).
The second source of data is a homogeneous set of 431
high quality ‘Oumuamua’s brightness measurements carried
out in the rspectral band using Gemini Multi-Object Spec-
trograph (GMOS-N) over the course of two nights, Octo-
ber 27–28, 2018 (Drahus et al. 2017,2018). The times are
light travel corrected to the standard epoch of MJD =58054;
we had to subtract 0.0028662 d from all times to convert
them to the asteroidal coordinate system. The brightness
values are geometry corrected to MJD =58054; to convert
them to absolute magnitudes we had to add 0.74 mag to the
published values. One-sigma brightness measurements un-
certainties range from 0.02 to 0.5 mag, with the geometric
mean of 0.08 mag. The full dataset is not publicly available,
but was generously provided by the authors (Michal Drahus
2018, private communication). The full dataset only became
available recently, in the final stages of this research project.
Most of our numerical runs used a shorter version of the
dataset, consisting of 51 points manually scanned from fig-
ure 4 of Drahus et al. (2017). Once the full dataset became
available, we made sure that our reduced dataset is fully
consistent with the actual data. The final re-optimization
steps in our analysis presented here are based on the full
dataset (431 points).
Merging the two datasets together produced a list of
either 390 (when using the scanned version of Drahus et al.
2017 dataset) or 770 (when using the full dataset of Drahus
et al. 2018) ‘Oumuamua’s absolute magnitude values in the
r/r0spectral filter with the corresponding one-sigma uncer-
tainties, with the times corrected for light travel. These are
the data we used in all our light curve modeling efforts for
this asteroid. The data are plotted in Figure 4 as black and
red (grey in the printed version of the journal) points with
one-sigma errorbars.
As noted by many authors before, the most striking
feature of ‘Oumuamua’s light curve is the presence of mul-
tiple very deep minima (see Figure 4). Some of the minima
are defined by few points with large measurement errors so
may not appear very significant individually (for example,
minima A, B, I, and L in Figure 5), but taken together
they present a very convincing case for an object undergo-
ing extreme brightness variations (with the amplitude up to
2.5–2.6 mag – larger than any known Solar System asteroid,
Jewitt et al. 2017) on a quasi-regular basis. The conventional
interpretation of these brightness variations is that they are
caused by extremely elongated (if it is cigar-like) or flattened
(if it is disc-like) shape of ‘Oumuamua (Meech et al. 2017),
though non-geometric interpretations (e.g. extreme albedo
variations across the object’s surface) cannot be ruled out.
Assuming the geometric interpretation and ignoring phase
effects, the light curve amplitude of 2.5 mag would corre-
spond to the cigar or disc largest-to-smallest axes ratio of
10:1 (Meech et al. 2017). When taking into account phase ef-
fects, the shape constraints are not as extreme (>5:1, Fraser
et al. 2018), though still quite remarkable.
4.2 Inertial ellipsoid models
The first class of models we used to simulate the light curve
of ‘Oumuamua is a 10-parameter inertial (zero torque) re-
laxed LS triaxial ellipsoid model (see subsection 2.6). The
10 free model parameters had the following soft (hard) lim-
its: θL[0, π[(same), ϕL[0,2π](any), ϕ0[0,2π](any),
c[0.01,1[ (]0,1[), b[c,1[ (same), E0[0,1] (same),
Pψ[2,4800] h (Pψ>0), ψ0[ψmin, ψmax]for SAM (same;
see Equation 7) and ψ0[0,2π]for LAM (any), c0=c
(]0,1[), b0=b([c0,1[). (Soft limits are used when generat-
ing initial random values of the parameter; hard limits are
enforced during optimization; see subsection 2.6 for the ex-
planation of the parameters.)
Early attempts of model fitting produced completely
MNRAS 000,121 (2019)
12 S. Mashchenko
Figure 5. Our best-fitting inertial model (column INERT in Ta-
ble 1) for ‘Oumuamua. tis the number of days since MJD =58050,
and Vis the absolute magnitude in the r/r0spectral filter. The
dots are the observational data (errorbars were omitted to make
trends more obvious). Blue dots (open circles in the printed ver-
sion of the journal) are the anchor points with the std reduced
to 0.05 mag, which purpose is to define major features (primarily
deep minima) in the observational data. Capital letters A–L mark
the positions of major features in the observational data.
unsatisfactory results, with hundreds of lowest χ2models
failing to reproduce the quantity and locations of the major
features (minima and maxima) of the observed light curve.
This did not occur in our artificial dataset tests (subsec-
tion 3.3), despite the fact that the fake data were as noisy
and patchy as ‘Oumuamua’s data, and the ”fake” model be-
ing more complex (three additional model parameters – the
three normalized torque vector components). This was an
early indication that the inertial ellipsoid model was not
the right one for this asteroid. To try to match at least the
quantity and locations of the obvious observed minima, we
started to reduce the std for some of the data points, which
define the major light curve features, to a small value of
0.05 mag (comparable to the best real std in the data). At
the end of this rather lengthy iterative process we ended up
with 28 ”anchor” points (see Figure 5). (It is important to
note that we only used the ”anchor” points with fake std
values during the stages one and two of our optimization
procedure; during the final – fine-tuning – stage we used
correct std values for all the data points.)
We also employed another trick during the fine-tuning
stage (subsection 3.2), when the model minima located close
to the seven most obvious observed minima (features A, B,
C, D, E, I, and L in Figure 5) would be gradually ”nudged”
towards the corresponding observed minima in the time di-
mension. This is accomplished by multiplying the χ2values
by a parameter βwhich is equal to one when the minima are
far apart, and becomes significantly smaller than one when
all seven model minima converge onto the corresponding ob-
served minima.
The above tricks allowed us to produce somewhat better
model fits. But even the best of them (our model INERT; see
Figure 5 and Table 1) was completely unsatisfactory: some
of the model minima (especially the feature B, also A and
L) were offset in the time dimension from the corresponding
observed minima by a non-trivial amount. The fact that re-
laxing of the brightness ellipsoid model (by means of adding
two more free model parameters) failed to produce models
which would be at least in a qualitative agreement with the
observed light curve is highly suggestive of significant issues
with the current model. The possible explanations for the
model failures are:
(i) The shape of ‘Oumuamua is dramatically different
from the assumed triaxial ellipsoid shape. This explanation
cannot be ruled out based on our analysis, but we consider
it to be very unlikely. As our 2002 TD60 test case shows, the
triaxial ellipsoid brightness model has no issues in fitting the
observed minima of a real (that is, not with a perfect triax-
ial ellipsoid shape) asteroid in the time dimension. It does
not do as well in terms of explaining the detailed shape of
minima and maxima, but relaxing the brightness model (by
means of adding two more free parameters, c0and b0) im-
proves the quality of the fit substantially, to a large degree
taking care of the shape mismatch. (This is fully consistent
with much more extensive results of Cellino et al. 2009 and
Muinonen et al. 2015.) A significantly different reflection
law would also unlikely fix the significant offsets between
the model and observed light curve minima in the time di-
mension (Samarasinha & Mueller 2015).
(ii) In light of the discovery of ‘Oumuamua’s non-
gravitational acceleration (Micheli et al. 2018), a natural
expansion of our model would be to assume a presence
of some torque. We consider the simplest prescription for
torque (fixed in time and spatially, in the asteroidal coordi-
nate system) in our ellipsoid models with torque simulations
(see subsection 4.3).
(iii) The large brightness variations of ‘Oumuamua are
not geometric in nature (caused by extreme shape of the
object). An alternative explanation would be an object with
a more conventional shape (say, roughly spherical), but with
extreme albedo variations across the surface. We explore this
alternative explanation via our ”black-and-white ball”model
(see subsection 4.5).
4.3 Ellipsoid models with torque
The most obvious (in light of the detected non-gravitational
acceleration of ‘Oumuamua, Micheli et al. 2018) and sim-
plest extension to our inertial tumbling rotation model is
to add constant torque (fixed in the asteroidal coordinate
system). This adds three more free parameters (normalized
torque vector components Tb,Tc, and Ta), and doubles the
number of ODEs (from 3 to 6) in the equations of motion
(Equation 4).
We carried out the standard suite of numerical runs
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 13
Figure 6. Our best-fitting disc model with torque (column DISC
in Table 1) for ‘Oumuamua. See the caption of Figure 5 for details.
Figure 7. Our best-fitting cigar model with torque (column
CIGAR in Table 1) for ‘Oumuamua. See the caption of Figure 5
for details.
(subsection 3.2) to find best-fitting ‘Oumuamua models for
both self-consistent and relaxed LS ellipsoid brightness mod-
els (11 and 13 free model parameters, respectively). We used
the same soft and hard limits for the basic model parameters
as in our inertial model runs (subsection 4.2); for the three
components of the normalized torque vector we ended up
using the soft limits [10,10] (no hard limits) in the model
physical units (where a=1and Ia=1; the time unit is
a day). Preliminary tests showed that with a factor of 10
larger soft limits the vast majority of best-fitting models end
up spinning unphysically fast (periods less than one hour) at
the end of the 5-day simulated time interval, producing light
curves which looked totally wrong. A factor of 10 smaller soft
limits, [1,1], produced best-fitting models similar to our
best-fitting inertial models, suggesting that in these models
torque was too weak to make an obvious impact.
In what we consider to be the main result of this pa-
per, we found that adding the simplest (constant) torque
prescription to the inertial tumbling asteroid model signif-
icantly improves the quality of model fits to ‘Oumuamua’s
light curve. Importantly, the timings of the well defined and
sharp observed brightness minima can now be matched very
well by the models (Figures 6and 7). In both relaxed and
self-consistent brightness ellipsoid runs we identified two
classes of models which were consistently in the top 5–10
best models in terms of the lowest χ2values, had model
minima matching well the timings of the observed minima,
and also reproduced well other major features of the ob-
served light curve (for example features J and K).
The first class of models has the best overall χ2val-
ues, and is comprised of thin discs which are almost self-
consistent and close to being axially symmetric. An inter-
esting point is that making the brightness model relaxed
(by adding two shape parameters – b0and c0) does not im-
prove the quality of fit (in terms of χ2, rms, and matching
the timings of the observed minima) for this class of models.
As a result, here we present only the self-consistent version
of this model (Figure 6; column DISC in Table 1). As one
can see, it is still far from being perfect. In particular, the
large observed depth of the minimum D is not correctly re-
produced, the shape of the minimum G is not well matched,
and the model feature L is systematically raised relative to
the observed one (Figure 6). Also, the χ2and rms values
(10.0 and 0.22 mag, respectively) are still fairly large, albeit
much smaller than for our best-fitting inertial model INERT
(18.4 and 0.30 mag, respectively; see Table 1). Despite all
of this, the ability of our torque models to match all the
main features of the observed light curve of ‘Oumuamua is
quite remarkable. The remaining deviations of the model
light curve from the observed one can be plausibly ascribed
to non-ellipsoidal shape, non-homogeneous albedo, and/or
more complex light scattering properties of the asteroid.
Our best-fitting model DISC (Table 1) is a thin (1:6.1)
disc which is very close to being axially symmetric (b/a=
0.96) and which is in a LAM rotation initially (with the ro-
tation and precession periods 51.8 and 10.8 h, respectively).
It is interesting that by the end of the simulated time inter-
val of 5 days, constant torque turns the asteroid into a SAM
rotator, reducing the rotation period by a factor of 1.6, but
keeping the precession period essentially unchanged. Assum-
ing geometric albedo p=0.1and adopting the Sun’s visual
magnitude in rfilter m=27.04 mag from Willmer (2018),
MNRAS 000,121 (2019)
14 S. Mashchenko
Figure 8. Pro jection of two models – DISC (left panel) and
CIGAR (right panel) – onto bOa (top) and bOc (bottom) planes.
Thin black lines show the extent of the asteroid. Thick red (grey
in the printed version of the journal) lines correspond to possible
locations of the outgassing point. (The invisible – behind the as-
teroid’s body – parts are shown as dashed lines.) The dots show
the centre of asteroid. We assumed geometric albedo p=0.1.
we estimate the physical dimensions (full diameters) of the
model as 115 ×111 ×19 meters (from Equation 29).
The second class of best-fitting models has slightly
larger (which is likely statistically insignificant) values of
χ2and rms, and is comprised of narrow (1:7.7) cigar-
shaped objects which are close to being axially symmetric
(c/b=0.99). The best representative of this class – model
CIGAR (Figure 7,Table 1) – has comparable χ2and rms
values for both self-consistent and relaxed brightness ellip-
soid models (same as with our DISC model), so again we
only report here the self-consistent version of the model.
The rotational state evolution here is the opposite to that
of the DISC model: it starts as a SAM rotator (rotation and
precession periods 80.8 and 8.56 h, respectively), and spins
up to become a LAM rotator, with 2.7 times shorter rota-
tion period and almost unchanged precession period at the
end of simulations. Assuming p=0.1albedo, the physical
dimensions are estimated as 324 ×42 ×42 meters.
Assuming that the torque present in our models is gen-
erated by outgassing from one point on the asteroid’s sur-
face, we can identify the locus of the plausible locations of
this point as follows.
Applying steady force per unit mass fto a point on the
asteroid’s surface will produce in a general case both con-
stant linear acceleration for the whole body, fr, and constant
tangential (torque) acceleration, ft. The linear component
is derived by projecting the vector fonto the asteroidal
radius-vector at this point, r; the torque component ftis
derived by projecting fonto the plane perpendicular to the
radius-vector. Torque pseudo vector is a cross product of the
radius-vector and the tangential component of the force vec-
tor, K=r×ft, and as such is perpendicular to both. As
a consequence, the only points on the surface of the ellip-
soid where a given torque vector can be reproduced are the
ones where the radius-vector ris perpendicular to K. These
points lay along the intersection of the plane which is pass-
ing through the center of the ellipsoid and is perpendicular
to K. The intersection line is an ellipse on this plane. (The
full force vector fhas to lay in the same plane, as its both
components – frand ft– lay in that plane.) Points which
are closer to the centre of the ellipsoid would require larger
tangential acceleration, points further from the centre would
need smaller tangential acceleration: ft=K/r. (Here Kis the
modulus of the torque vector.) Assuming that the source of
the torque is the rocket force from outgassing (which is di-
rected inwards), only the points where the angle between
the torque vector Kand the normal to the surface is larger
than 90would be physically plausible.
In Figure 8 we show the locus of physically plausible
outgassing points on the surface of the asteroid for both
models (DISC and CIGAR). As one can see, outgassing can
be happening over a wide range of the distances from the
asteroid’s centre – from the central area to the very edge of
the object.
As one can see from Figures 6and 7, torque-driven spin-
up of the asteroid is not very obvious in the model light
curves. To get a better idea whether the model torque val-
ues are physically plausible, it is instructive to compare our
best-fitting models with the data on Solar System comets
which had their both linear non-gravitational acceleration
and change of the spin (both caused by the same mecha-
nism – outgassing) measured. Rafikov (2018a) showed that
these comets (there are seven in total) show a clear correla-
tion between torque K(deduced from the rate of change of
the spin) and linear non-gravitational acceleration fr:
K=ζD fr.(31)
(Here Dis the characteristic size of the object – for a sphere
it would be its radius, and ζis a small proportionality co-
efficient which the authors call a ”lever arm” parameter.)
The log-average value for ζis 0.006, with the full range
0.0007 . . . 0.03.Rafikov (2018a) deduced torque values for
Solar System comets based on some simplifying assump-
tions, but in our case we can get Kvalues directly from
the model. Micheli et al. (2018) showed that the linear non-
gravitational acceleration of ‘Oumuamua can be described
as fr5×106m s2/R2
(here Ris the distance from
the Sun in au). Our analysis covers a narrow range of
R=1.36 . . . 1.49 au. Using the average value for R, 1.43 au,
we estimate the linear non-gravitational acceleration to be
fr=2.45 ×106m s2within the time interval of inter-
est. Using our model’s semi-major axis ain place of D, the
model’s torque value K, and ‘Oumuamua’s value of frde-
rived above, we can satisfy Equation 31 if we set ζ=0.0046
and ζ=0.014 for DISC and CIGAR models, respectively.
This is well within the range of the ζvalues deduced for So-
lar System comets, with the DISC value of 0.0046 being close
to the log-average ζvalue of 0.006. Based on our analysis,
our model torque values are consistent with being produced
by the same outgassing which presumably drives the linear
non-gravitational acceleration of ‘Oumuamua.
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 15
Figure 9. Amplitude of the brightness fluctuations for a 1:10
ratio disc (dashed line) and cigar (solid line), as a function of the
cosine of the polar angle θLfor the angular momentum vector.
There is an important caveat in the above analysis: tak-
ing our model torque assumptions (torque being fixed in
time and space, in the asteroidal coordinate system) liter-
ally, one cannot produce cumulative linear acceleration for
the asteroid: as the asteroid spins (with the outgassing point
attached to its surface), the contributions to frat different
rotation phases would all cancel out. But there are ways to
relax our model assumptions somewhat to circumvent this
difficulty. For example, if we make a reasonable assumption
that outgassing is the most active when the outgassing point
is facing the Sun, the kinematic model will not change sig-
nificantly (with our Kparameter now representing a time-
averaged value of the torque vector), but the linear acceler-
ation can now gradually accumulate, with the acceleration
vector pointing in the right (anti-Sun) direction.
4.4 Cigar or disc?
The analysis presented in subsection 4.3 demonstrated that
the two most promising candidates for a model of ‘Oumua-
mua are either a thin disc-shaped or a thin cigar-shaped
object subject to some torque. Unfortunately, it is not pos-
sible to differentiate between these two very different cases
based solely on the quality of fit of the model light curve to
the observed one.
This model degeneracy can be broken by performing a
statistical analysis of a different kind, based on the follow-
ing simple geometric considerations, under the assumption
that the initial angular momentum orientation is random
(which is a sensible assumption for an interstellar visitor).
Specifically, to produce large brightness fluctuations (com-
parable to the asteroid’s largest-to-shortest axes ratio), a
cigar-shaped object spinning around its shortest axis would
need to have its longest axis repeatedly pointing at the ob-
server with a high accuracy. As a consequence, such an ob-
ject would require a high degree of fine-tuning for its angular
momentum vector orientation to produce the desired effect.
The opposite is true for a disc-like object (at least for the
case when it is a LAM rotator): there is a fairly narrow
range of the angular momentum vector orientations (when
Table 2. Ranked observed brightness minima.
Rank Depth, mag Feature
1 25.715 D
2 25.254 E
3 25.234 C
4 25.212 A
5 24.940 B
6 24.846 F
7 24.834 L
Note. Higher rank corresponds to deeper minimum. ”Depth” de-
scribes the largest absolute magnitude of the brightness mini-
mum. Observed light curve features A–L are marked on Figure 5.
the vector is pointing towards the observer) when the ob-
server would not see large brightness fluctuations.
This effect can be easily quantified for the idealistic sit-
uation when the object (either a LAM disc or a SAM cigar)
is not tumbling, is not subject to torque, and when we ig-
nore phase effects (by assuming the phase angle is equal to
zero). Let us assume that the cosine of the angular momen-
tum polar angle θLis equal to zero when the vector is in
the plane of the sky (this will result in largest brightness
fluctuations for both disc and cigar). Figure 9 shows how
the amplitude of the brightness fluctuations changes as a
function of cos θLfor 1:10 ratio disc and cigar. (We used the
LS ellipsoid brightness model to compute the brightness;
see subsubsection 2.5.1.) For a randomly oriented angular
momentum vector L, described by its polar angle θLand
azimuthal angle ϕL, equal intervals in cos θLcorrespond to
equal probabilities. From Figure 9 one can see that the disc
model is much more likely to produce brightness fluctua-
tions larger than a given amplitude than the cigar model.
For example, amplitudes equal to or larger than nine (hori-
zontal dotted line) will occur in 44 per cent of all disc model
cases (the length of the interval AA0divided by two – the
full range of cos θL), whereas for an equally thin cigar this
will be the case in less than 5 per cent of random angular
momentum orientations (the length of the interval BB0
divided by two).
This effect should manifest itself to a similar degree in
more realistic models, which are tumbling, subject to torque,
and have phase effects, though it is more difficult to quan-
tify. We designed the following statistical analysis pipeline
which computes probabilities for our models, by answering
the following question: ”Given that the initial angular mo-
mentum vector orientation and initial precession angle are
random, how likely is it that a given model can produce light
curve minima as deep as the observed ones?”.
(i) As a starting point, we use one of our best-fitting mod-
els – either DISC or CIGAR (see Table 1).
(ii) We relax the LS ellipsoid brightness model, by as-
signing a given ”thickness” to either c0parameter (for disc
models) or both c0and b0(for cigar-shaped models).
(iii) In our numerical code ran on a GPU, we concurrently
generate 2563models which have the same model parame-
ters (except for the parameters θL,ϕL, and ϕ0) and the same
physical scale (parameter V) as our initial model. The three
variable parameters – initial angular momentum vector ori-
entation angles (θL,ϕL) and initial precession angle ϕ0
are sampled with 256 different values each. The sampling is
MNRAS 000,121 (2019)
16 S. Mashchenko
Table 3. Probabilities for different models.
Model Thickness hNminiProbability
DISC 0.139 5.89 0.50
0.10 6.84 0.91
0.05 6.86 0.92
0.01 6.87 0.92
CIGAR 0.10 1.37 0.16
0.05 1.55 0.20
0.01 1.60 0.21
Note. Thickness parameter describes c0in the disc model, and
both c0and b0in the cigar model. hNmin iis the average number
of ranked model minima which are deeper than the seven observed
ranked minima (the allowed range is 0. . . 7).
equidistant for azimuthal angles (ϕLand ϕ0), and equidis-
tant for the cosine of the polar angle θL. As a result, each
of the 2563models have equal probability.
(iv) For each of the 2563models we perform the following
(a) We compute the model light curve in absolute mag-
nitudes, and measure the depth (largest magnitude) of
the model brightness minima located within the follow-
ing time intervals: t=1.045 . . . 1.118,t=1.978 . . . 2.185,
t=3.079 . . . 3.529,t=4.093 . . . 4.514,t=5.234 . . . 5.355,
and t=6.181 . . . 6.278. (Here tis the number of days since
MJD =58050.) These time intervals correspond to the ob-
served ‘Oumuamua light curve intervals which are both
wide enough and have enough of observed points to make
it possible to resolve a minimum if it happens to be there.
(b) We rank the model minima starting with the deep-
est (largest absolute magnitude) one.
(c) We set the counter of ”good” model minima Nmin to
(d) We compare the depth of the deepest model mini-
mum (model rank #1) with the depth of the deepest ob-
served minimum (observed rank #1; see Table 2). If the
model minimum is deeper (that is, if the absolute magni-
tude is larger), we increment Nmin by one.
(e) We compare the model rank #2 minimum to the
observed #2 minimum (Table 2), and increment Nmin by
one if the model minimum is deeper.
(f) We repeat the previous step for ranks #3. . . 7.
(g) A the end, each of the 2563equally likely models
will have a value of Nmin [0,7]. If Nmin =0then none
of the model minima were as deep as the corresponding
rank observed minima. If Nmin =7then all of the model
minima were deeper than the corresponding rank observed
(v) By counting the number of models where Nmin =7and
dividing the number by the total number of models (2563)
we can estimate how likely the initial model is.
We performed the above analysis for our two best-fitting
torque models – DISC and CIGAR – for a few different
thickness values. The results are summarized in Table 3.
The most striking result here is that regardless of how thin
the model is (from the plausible value of 0.10 down to the
extreme value of 0.01), disc models are very likely (in fact
almost guaranteed, with 91 per cent probability) to pro-
duce brightness minima as deep as observed ‘Oumuamua’s
minima. Cigar models, on the other hand, are very unlikely
to reproduce the deep observed minima, with the probabil-
ity of only 16 per cent for the plausible thickness of 0.10
(which grows only slightly – to 21 per cent – for the implau-
sible thickness of 0.01). Based on this statistical analysis
(which is independent from the χ2goodness-of-fit analysis
we performed in the previous section), the expected thick-
ness of disc models is 0.14 (when the probability of the
DISC model is around 50 per cent; see Table 3), which is
slightly smaller than the value derived by means of χ2fit-
ting (c=0.16, from Table 1).
In addition to producing a single probability value for
each model, it is instructive to analyse detailed probability
maps for the initial angular momentum vector orientation
(Figure 10). As expected, the disc model is very likely for
almost any orientation of the angular momentum vector,
whereas the cigar model requires a high degree of fine-tuning
in terms of the angular momentum vector orientation.
Based on the statistical analysis presented in this sec-
tion, ‘Oumuamua is most likely a disc-shaped object, though
the cigar shape cannot be ruled out.
4.5 Alternative models
In this section we consider two additional auxiliary models
for ‘Oumuamua.
The first auxiliary model is identical to our fiducial
model (LS ellipsoid brightness model + constant torque;
subsection 4.3) in all aspects except for numerical values of
some parameters – the thickness (parameter c), which now
has the initial range (soft limits) between 104and 102(the
range was [0.01,1] in the fiducial model), and the length
of the intermediate semi-axis b(new soft limits: [0.3,1]).
These numerical runs were an attempt to use our model’s
framework to explore the idea of Bialy & Loeb (2018) that
‘Oumuamua is a solar sail – an extremely thin object, with
very low surface density (0.1g cm2).
As we already demonstrated that there is no need to use
relaxed brightness ellipsoid with the models with constant
torque to reproduce well ‘Oumuamua’s light curve (subsec-
tion 4.3), we used the same self-consistent brightness ellip-
soid in the ”solar sail” simulations. That means the total
number of free parameters was 11 (8 basic tumbling model
parameters plus 3 torque component parameters). We car-
ried out our standard set of numerical runs (subsection 3.2)
for the new model.
Our main finding here is that the quality of the light
curve fit for the ”solar sail” models is noticeably worse than
for our best-fitting fiducial models (DISC and CIGAR). As
one can see from Figure 11, our best-fitting model (SAIL)
struggles to reproduce the brightness maximum features J
and the one around t=2.0, the depth of the minimum H,
and has timing issues with the minima C and L. The model
does require some torque (Table 1). It is interesting that
the model is degenerate in the sense that the light curve is
essentially unchanged starting from c0.001 down to the
smallest value we tested (105; that is the value we use in
Figure 11 and Table 1). As this is a self-consistent brightness
ellipsoid model, the degeneracy is present for both kinematic
and brightness ellipsoids. As a result, our SAIL model is con-
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 17
0 50 100 150 200 250 300 350
cos L
L, deg
0 50 100 150 200 250 300 350
cos L
L, deg
Figure 10. Probability maps for our best-fitting models (DISC on the left and CIGAR on the right). We set the thickness parameter to
0.1 for both models. Polar (ϕL) and azimuthal (θL) angles describe the initial orientation of the angular momentum vector. Assuming
that this vector is oriented randomly, each pixel in these maps is equally likely. Each pixel of the map presents the model probability
averaged over all values of the initial precession angle ϕ0. Black corresponds to 100 per cent probability, white corresponds to 0 per cent
Figure 11. Our best-fitting alternative models: SAIL (orange
line; dotted line in the printed version of the journal) and BALL
(green line; solid line in the printed version of the journal). See
the caption of Figure 5 for details.
sistent with the extremely low surface density requirement
of the solar sail hypothesis of Bialy & Loeb (2018).
It is easy to see why the model becomes degenerate in
the c0and c00limits. The kinematic part of the
model becomes degenerate because as cis approaching zero,
the three diagonal components of the inertia tensor converge
to constant values (see Equation 1): Ia=1,Ib=b2,Ic=
(1+b2)b2. This in turn ensures that the kinematic ODEs
(Equation 4) no longer depend on c(become degenerate).
Even stronger effect is observed in the brightness model part.
In the limit of c00our triaxial ellipsoid model degenerates
into an extremely thin flat object. One can easily show that
neither different shapes nor variable albedo can affect the
light curve from such an object, as each individual element of
the object’s surface would see exactly the same phase angle
as other elements on the same side of the sail. Changing the
shape of the object by moving the elements around (in the
sail’s plane) will not affect the integrated brightness at any
given orientation of the sail. Variable albedo is also incapable
of changing the light curve shape, as at each orientation of
the sail all elements (with different albedo) would contribute
to the integrated brightness in the same proportion. The
only factors which still can affect the shape of the light curve
for a solar sail (and hence can be potentially used to further
improve the quality of fit between the model and observed
light curves) are (a) a different torque model, and (b) a
different light scattering model. A more fundamental change
would be to assume that the thin sail is not perfectly flat (it
has some curvature or ripples). Modeling these effects would
go beyond the scope of this paper.
Assuming a fairly high albedo of 0.5 (which would be
appropriate for a solar sail), the size of the SAIL disc is
64×51 m (full diameters). The torque ”lever arm”parameter
ζis 0.017, implying that less than 2 per cent of the linear
non-gravitational force experienced by ‘Oumuamua needs to
be converted to torque.
How can radiation pressure produce the required
torque? We speculate that making the shape and/or mass
distribution asymmetric may not do the trick, as once the
sail makes half a full rotation, the opposite direction torque
would be exerted, canceling out the original torque. Vari-
able albedo seems to be a more promising agent. Let us
assume ‘Oumuamua is a flat disc-shaped sail of radius R
with one of the sides consisting of a darker (albedo p1) and
brighter (albedo p2) halves. Radiation pressure can be com-
puted as P=(1+p)C, where pis the albedo and Cis a
constant when the distance from the Sun is fixed (Bialy &
Loeb 2018). An element with the surface area dSwill expe-
MNRAS 000,121 (2019)
18 S. Mashchenko
rience force df=PdS. Integrating the product rdfover the
darker half of the disc (rbeing the distance of the element
from the disc centre) gives us the torque applied to that side,
K1=(1+p1)CπR3/3; doing the same for the brighter side
gives us the second torque component, K2=(1+p2)CπR3/3.
The global torque is K=K2K1=(p2p1)CπR3/3, while the
global linear force due to solar radiation is fr=(1+hpi)CπR2.
(Here hpi=(p1+p2)/2is the average albedo of the side.) Sub-
stituting the above expressions for Kand frinto Equation 31
(and setting D=R), we derive the following expression for
the difference between the two albedos which will generate
the required torque: p2p1=3(1+hpi)ζ. Assuming hpi=0.5,
we can produce the required torque (ζ=0.017) if the albedos
differ by a fairly small amount: p2p10.08.
Our second auxiliary model uses a completely differ-
ent brightness model. Specifically, it explores an alternative
(non-geometric) explanation for the large brightness varia-
tions of ‘Oumuamua, where the asteroid is assumed to be
roughly spherical in shape but with large albedo variations
across its surface. We use the simplest possible implementa-
tion of this idea – ”black-and-white ball” brightness model
(subsubsection 2.5.2), with only three free parameters – po-
lar coordinates θh,ϕhof the dark spot on the surface of the
asteroid (in the comoving coordinate system bca), and the
dark/bright sides albedo ratio κ. Both dark and bright sides
are equal in size (both are hemispheres) for simplicity. Also,
we ignore phase effects (phase angle is assumed to be zero).
Our primary motivation to explore non-geometric ex-
planations for the large brightness variations of ‘Oumuamua
was the fact that this approach presents a completely differ-
ent coupling between the spinning/tumbling motion and the
brightness variations. Specifically, in our ”black-and-white
ball” model there is one maximum and one minimum in the
light curve per full rotation of the body. This is in contrast
with the geometric (ellipsoid) picture, where one has two
maxima and two minima per rotation. Our hope was that
this very different coupling might remove the need for torque
when trying to fit the observed light curve of ‘Oumuamua.
Our analysis showed this not to be the case. Running
our full suite of simulations for an inertial tumbling ”black-
and-white ball” failed to produce light curves which were a
noticeably better fit to ‘Oumuamua’s observed light curve
than with our inertial LS brightness ellipsoid runs (subsec-
tion 4.2). (For the model parameter κwe used logarithmic
scaling and soft limits [0.01,0.1]; the kinematic ellipsoid’s
thickness chad soft limits [0.3,1].) Crucially, similarly to the
inertial brightness ellipsoid case, the best-fitting zero torque
black-and-white ball” models had serious issues matching
the timings of the observed light curve minima.
Adding constant torque to the above model rectified
the situation (the same way it helped in the LS brightness
ellipsoid simulations). We ran a full suite of numerical sim-
ulations for a ”black-and-white ball” with torque (14 free
model parameters in total: 8 basic tumbling model parame-
ters plus 3 ”black-and-white ball” brightness model param-
eters plus 3 torque components), and show our best-fitting
model BALL in Table 1 and Figure 11. Now the timings of
the model minima match well those of the observed minima,
and overall quality of fit is decent (in fact better than for
our models DISC and CIGAR). The model BALL starts as
a SAM rotator with the rotation period of 23.9 h and the
precession period of 4.5 h, and ends up as a LAM rotator
with Pψ=20.1h and Pϕ=3.9h after 5 days. The parameter
κ(dark/bright sides albedo ratio) is 0.03. The kinematic el-
lipsoid is not exactly a ”ball”, but with the shape parameters
c=0.52 and b=0.93 its geometry is much less extreme than
in our models DISC and CIGAR. Polar angle θhis equal to
162, meaning that the dark hemisphere is fairly close to the
southern polar region” of the ”ball”.
Prior attempts to interpret ‘Oumuamua’s light curve (Fraser
et al. 2018;Belton et al. 2018;Drahus et al. 2018) were based
on searching dominant frequencies and interpreting them as
a linear combination of two frequencies – precessional and
rotational. This approach usually works very well for Solar
System asteroids and comets, but its fundamental assump-
tion is that torque is zero. If that is not the case, the domi-
nant frequencies found in light curves can no longer be used
to find the rotational state of the asteroid: at best they might
correspond to real frequencies present in some segments of
the data which are particularly well sampled, at worst they
are purely fake, reflecting the patchiness of the data. Sama-
rasinha & Mueller (2015) provide one such example, when
adding noise to the perfect model data and making it patchy
produced fake dominant frequencies.
Our research represents the first attempt to fit ‘Oumua-
mua’s light curve using a physical model. (Recently pub-
lished research by Seligman et al. 2019 did use a physical
model with torque to explain ‘Oumuamua’s light curve, but
they did not carry out multi-dimensional model fitting, so
their results are only suggestive; the computational tasks are
completely incomparable: where we had to compute hun-
dreds of millions of physical models, they only computed
a few.) The fundamental advantage of such an approach is
that torque can be modeled directly. In addition, other as-
pects affecting the light curve (variable phase angle, different
shapes, spatially variable albedo etc.) can also be directly
modeled, which removes a lot of guesswork from interpret-
ing light curves.
We started this project fully expecting that given how
limited, noisy, and patchy ‘Oumuamua’s light curve is, we
would be finding a large number of very different inertial
models which would all provide a comparable quality and
reasonable description of the data. The first big surprise was
when we realized that no inertial model we tried (LS ellip-
soid, ”black-and-white ball”, ”solar sail”) could match the
timings of the most conspicuous features of the observed
light curve – the multiple deep and narrow minima. The
simplest non-inertial extension of the model we tried (steady
torque fixed in the comoving coordinate system) was suffi-
cient to rectify this situation for all of our brightness mod-
els. In all likelihood our torque model, with only three free
parameters, is an oversimplification, as any realistic mecha-
nism producing torque would be significantly more compli-
cated (time variable, not firmly attached to the surface of
the object etc.). The important point here is that any torque
prescription, even as simple as the one we used, should be
able to fix the minima matching issues which plagued all our
inertial models.
We consider the finding that some torque is needed to
model well the light curve of ‘Oumuamua to be our main
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 19
result. It is quite remarkable that the torque required is
in line with the results for the Solar System comets for
which both linear non-gravitation acceleration and change
of the spin (both effects driven by outgassing) were mea-
sured (Rafikov 2018a). This could be viewed as an impor-
tant evidence supporting the comet hypothesis for ‘Oumua-
mua. We should caution though that this does not prove the
non-gravitational acceleration of ‘Oumuamua and its torque
are driven by outgassing. Other mechanisms where a force
is applied to the asteroid’s surface can have a comparable
relation between the torque and the linear acceleration. For
example, solar radiation can drive primarily the linear accel-
eration of a thin object (”solar sail”), but can also generate
some torque (for example, if albedo varies across one or both
sides of the sail; see subsection 4.5).
As a side note, we suggest here one mechanism which
by design will only produce linear non-gravitational acceler-
ation (or rather an appearance of such), with zero torque:
if ‘Oumuamua happens to be made of some sort of ex-
otic matter for which the gravity law deviates slightly from
the canonical form. Indeed, if the gravity constant Gfor
‘Oumuamua were only 0.0008 of fractional units smaller than
the standard value, this would completely reproduce the ef-
fect discovered by Micheli et al. (2018): the appearance of
an additional force which is radially directed away from the
Sun, scales as r2and has the right magnitude. As this is
not a real force, there would be zero torque by design.
Both the discovery of the non-gravitational acceleration
by Micheli et al. (2018) and our current findings strongly
suggest that ‘Oumuamua should have experienced a fairly
strong torque. But recently Rafikov (2018b) claimed that
‘Oumuamua experienced negligible torque, and hence cannot
be a comet. We would like to point out an internal inconsis-
tency in the argument of Rafikov (2018b). On one hand it is
true, as the author claimed, that given that the periodogram
analysis of ‘Oumuamua’s light curve (spanning 30 days) car-
ried out by Belton et al. (2018) revealed the presence of a
dominant period of (8.67 ±0.34) h, one possible explana-
tion can be the hypothesis that the frequency is physical
(corresponding to either precessional, rotational periods, or
some combination of the two) and that the torque is so weak
that the dominant frequency did not change by more than
the quoted uncertainty of 0.34 h over the 30 days. On the
other hand, once one assumes the torque is strong enough
to affect the spin of ‘Oumuamua, one can no longer inter-
pret dominant frequencies recovered from the light curve as
physical. Non-negligible torque would smear out the physi-
cal frequencies in the periodogram, leaving instead artefacts
of the patchiness of the data, or perhaps a dominant fre-
quency present in the most sampled segment(s) of the light
curve. The latter may very well be the case here, as the
DISC model minima D and H are separated by three times
8.54 h, and minima E and I are separated by three times
8.78 h (see Figure 6; all four minima are among the best
sampled in the light curve). The average of these two peri-
ods is 8.66 h (almost exactly the dominant period of 8.67 h
detected by Belton et al. 2018), and the deviations from the
average are 0.12 h – well within the uncertainty of 0.34 h
of the detected period. This period could conceivably show
up in a periodogram for the model’s light curve, despite the
fact that the model lacks a well defined period due to the
effects of steady torque.
Staying within the realm of conventional explanations
for ‘Oumuamua (asteroid versus comet), both the presence
and magnitude of torque evidenced by the current research
would appear to tilt the scales towards the cometary nature
of the object. One has to emphasize though that the lack
of any direct signs of outgassing for ‘Oumuamua is highly
troubling. Trying to reconcile the cometary hypothesis with
the lack of outgassing detections, Micheli et al. (2018) had
to assume a rather extreme composition of the object in
terms of the CN to H2O ratio and the dust properties, leav-
ing the H2O and CO as the most likely drivers of the non-
gravitational acceleration of the asteroid. The non-detection
of CO outgassing using Spitzer Space Telescope (Trilling
et al. 2018) and the argument of Sekanina (2019) that H2O
has much lower abundance than what is needed to drive
the non-gravitational acceleration of ‘Oumuamua make the
cometary explanation even more problematic. If ‘Oumua-
mua is a comet in some sense, it must be a very exotic one,
with its properties (chemical composition and geometry) be-
ing nothing like properties of Solar System comets.
This makes other (”exotic”) explanations for ‘Oumua-
mua’s nature quite competitive. Even though our model
SAIL, designed to mimic the solar sail hypothesis of Bialy
& Loeb 2018, does not provide as good fit to ‘Oumua-
mua’s light curve as our more conventional models, DISC
and CIGAR, relaxing some of our model assumptions (e.g.
changing the light scattering law, or assuming that the thin
sail has a curvature or ripples) could potentially make it
a viable option. Importantly, our model is degenerate, al-
lowing the thickness of the object to be arbitrarily small –
even as small as the solar sail requirement, 0.5mm. The
model does require some torque to match the timings of the
asteroid’s brightness minima reasonably well, but as we ar-
gued earlier solar radiation can generate torque if the albedo
varies across the surface of the sail.
Another (semi)-exotic explanation for ‘Oumuamua we
considered – a ”black-and-white ball” – was a failure in the
sense that it did not remove the need for torque. The model
has a rather extreme bright-do-dark sides albedo ratio of 32.
Given that ‘Oumuamua did not exhibit obvious color varia-
tions (with the possible exception of a ”red spot”, as noted by
Fraser et al. 2018), and that for Solar System minor bodies
shape is the main driver of large brightness variations, this
hypothesis should be treated as an interesting but unlikely
alternative explanation for the asteroid’s nature.
As our second main result, in this paper we presented
the evidence that by far the most likely shape for ‘Oumua-
mua is a disc (or slab, or pancake). Making a reasonable
assumption that ‘Oumuamua’s angular momentum vector
had no preferred direction, the requirement for the model
to produce light curve minima as deep as the observed ones
sets the likelihood of the cigar shape, popular in the liter-
ature, at only 16 per cent. A thin disc, on the other hand,
is very likely to produce brightness minima of the required
depth. Disc-shaped and cigar-shaped objects produce very
similar-looking light curves (compare Figure 6 and Figure 7).
It takes a statistical analysis of a different kind (presented
in subsection 4.4) to break this model degeneracy. This find-
ing may have interesting implications for future discussions
about the nature of the asteroid. In particular, recent re-
search providing explanations for ‘Oumuamua’s cigar shape
MNRAS 000,121 (2019)
20 S. Mashchenko
(e.g. Katz 2018;Vavilov & Medvedev 2019;Sugiura et al.
2019) may need to be revisited.
Combining our physical model fitting of ‘Oumuamua’s
light curve with our statistical analysis of the model proba-
bility based on the depth of the light curve minima points to
a tumbling thin disc experiencing some torque as the most
likely model for the asteroid. The disc diameter is 110 m
(assuming geometric albedo p=0.1), and it is very close to
being axially symmetric. The model is self-consistent (the
same ellipsoidal shape explains both the kinematics and the
brightness variations). The disc thickness is estimated at
19 m (from the light curve fitting) or 16 m (from the prob-
abilistic minima depth analysis; see Table 3). It requires a
moderate amount of torque over the 5 days covered by this
analysis, consistent with the amount of torque experienced
by Solar System comets. The remaining deviations of the
model light curve from the observed one suggest that the
shape of the object is not exactly ellipsoidal and/or there
are some albedo variations across its surface.
Our analysis only covered a very short time interval (5
days). One important question is: what is the longer term
impact of torque in our models? Will the asteroid spin up
in a fairly short time to the point that it breaks apart? To
start with, our model assumption of steady torque fixed in
the asteroidal coordinate system is a significant oversimpli-
fication. At the very least, it should go down as r2as the
object moves away from the Sun. Also, as we discussed ear-
lier (subsection 4.3), real torque has to vary with the rotation
phase (e.g., by becoming stronger when the outgassing point
is heated by the Sun), otherwise the model will not produce
the linear acceleration term. As a worst case scenario, we
ran our models DISC and CIGAR for 25 more days (bring-
ing the total evolution time to 30 days), maintaining the
same fixed values of the torque pseudo vector. During this
time, the light curves for both models remained fairly regu-
lar, extending the trend from the first 5 days (see Figure 6
and Figure 7). The effective rotation period (interval be-
tween alternate minima) for the DISC model changes from
9.6 h to 2.3 h after 30 d. For the CIGAR model the change
is much less steep (from 9.4 h to 8.4 h), which suggests that
for this model the torque primarily impacts the direction of
the angular momentum vector. Even for the most affected
model (DISC), the rotation period after 30 days (2.3 h) is
still short of what is needed to break up the asteroid (<1h;
Rafikov 2018b). Once one takes into account the r2depen-
dence of the torque on the distance from the Sun, the spin
up due to torque will be even more moderate.
Our model cannot tell us what was happening before
the 5-day interval we simulated. The asteroid was closer to
the Sun, so presumably the torque was stronger. It is very
likely that as we move backward in time, if we simply as-
sume the torque direction in the asteroidal coordinate sys-
tem is fixed, and its magnitude grows as r2, our model
would quickly become unphysical. One way out of this is to
assume that over longer time intervals our assumptions that
torque is constant and the outgassing point is fixed in the
comoving coordinate system can no longer be valid even in
an approximate sense. A more realistic picture would have
multiple outgassing points happening primarily in the Sun-
lit parts of the asteroid. The outgassing model of Seligman
et al. (2019), where the outgassing point is not fixed in the
asteroidal coordinate system but instead tracks the subso-
lar point, may be more appropriate for longer time interval
We presented the first attempt to fit the light curve of the
interstellar asteroid ‘Oumuamua using a physical model,
which consists of the kinematic part (tumbling asteroid sub-
ject to constant torque) and the brightness model part (ei-
ther Lommel-Seeliger triaxial ellipsoid or ”black-and-white
ball”). We performed exhaustive, Monte-Carlo style, multi-
dimensional optimization of the models using our numerical,
GPU-based code, developed specifically for this project. We
spent approximately one GPU-year for this project, using
Here are our main findings.
(i) Some torque is needed to explain the exact timings of
the deep light curve minima of ‘Oumuamua. This is true for
all brightness models we tried (LS ellipsoid – including the
special case of a ”solar sail” – and ”black-and-white ball”).
(ii) The amplitude of the torque required by our best-
fitting models is consistent with the torque measured for
Solar System comets which spin and radial acceleration was
affected by outgassing.
(iii) Our analysis produced two different best-fitting el-
lipsoidal models for ‘Oumuamua: either a thin disc or a thin
cigar. Both models are very close to being axially symmet-
ric, and are self-consistent (brightness ellipsoid is identical
to the kinematic ellipsoid).
(iv) Assuming random orientation of the asteroid’s angu-
lar momentum vector, we computed the probability that our
best-fitting models can produce light curve minima as deep
as the observed ones. This analysis demonstrated that the
disc shape (probability 91 per cent) is much more likely than
the cigar shape (probability 16 per cent).
(v) Our best overall model for ‘Oumuamua is a thin disc
(115 ×111 ×19 m assuming geometric albedo p=0.1) which
is initially a LAM rotator with the rotation and precession
periods of 51.8 h and 10.8 h, respectively. After five days it
evolves into a SAM rotator with the rotation period of 32.3 h
(the precession period remains essentially unchanged). The
lever arm” parameter ζ(the measure of the torque strength
in relation to the non-gravitational linear acceleration of the
asteroid) for this model is 0.0046, which is close to the log-
average value of 0.006 for Solar System comets.
(vi) Though we consider the two alternative models we
tried (”solar sail” and ”black-and-white ball”; both needed
some torque) less likely, we believe they are viable.
Our current research has definitely not exhausted the field
of physical modeling of ‘Oumuamua. The asteroid’s light
curve appears to be rich enough (with multiple sharp fea-
tures) to sustain even more advanced physical modeling. In
particular, attempts can be made to carry out a full light
curve inversion (like in Kaasalainen & Torppa 2001), to try
to recover the true shape (with no assumptions of symme-
try and convexity) of the asteroid. One could also try to
model both the variable shape (e.g. as a triaxial ellipsoid)
and albedo variations across the surface, or try different
torque prescriptions (e.g. the one used by Seligman et al.
2019). Finally, more advanced solar sail models (with some
curvature and variable albedo) could be developed, with the
MNRAS 000,121 (2019)
Modeling the light curve of ‘Oumuamua 21
hope that they can both explain the observed light curve
and have self-consistent torque and linear non-gravitational
acceleration (both driven by solar radiation).
This research was enabled in part by support provided
by SHARCNET ( and Compute Canada
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This paper has been typeset from a T
X file prepared by
the author.
MNRAS 000,121 (2019)
... As it tumbled every eight hours (see Figure 4), the brightness of sunlight reflected from it changed by a factor of ten. This meant that it has an extreme shape, which at the ~90% confidence level was disk-like 3 . The Spitzer Space Telescope did not detect any carbon-based molecules or dust around `Oumuamua, setting a tight limit on ordinary cometary activity 4 . ...
... Kornmesser), or a flattened, pancake-shaped object -as shown in the lower image (Credit: Mark Garlick). The pancake shape provides the best fit to `Oumuamua's light curve 3 . Even a razor-thin object, like a flat sheet of paper, would appear to possess some width when projected at a random orientation on the sky, so the intrinsic aspect ratio of `Oumuamua can be much smaller than the value of 1:10 inferred from its light curve. ...
Full-text available
Science offers the privilege of following evidence, not prejudice. The first interstellar object discovered near Earth, Oumuamua, showed half a dozen anomalies relative to comets or asteroids in the Solar system. All natural-origin interpretations of the Oumuamua anomalies contemplated objects of a type never-seen-before, such as: a porous cloud of dust particles, a tidal disruption fragment or exotic icebergs made of pure hydrogen or pure nitrogen. Each of these natural-origin models has major quantitative shortcomings, and so the possibility of an artificial origin for Oumuamua must be considered. The Galileo Project aims to collect new data that will identify the nature of Oumuamua-like objects in the coming years.