Apophis

1
Apophis tumbling

• P. Pravec, P. Scheirich, J. Pollock, P. Kušnirák,
  K. Hornoch,
• A. Galád, E. Jehin, J. Manfroid, C. Opitom, M.
  Gillon, J. Oey,
• J. Vraštil, D.E. Reichart, J.B. Haislip, K.M.
  Ivarsen, and A.P. LaCluyze
• The 8th Workshop on Catastrophic Disruption in
  the Solar System
• Hawaii, the Big Island, 2013 June 24 - 27

2
(99942) Apophis
In December 2012, impact on 2036 April 13 was not
ruled out yet (Giorgini et al. 2008, Farnocchia
et al. 2013). The most significant uncertainty
in the prediction - an unknown magnitude and sign
of the Yarkovsky drift of the Apophis orbit.
The Yarkovsky drift depends on asteroids
rotation state, angular momentum vector, and
size. First lightcurve observations by the group
of Raoul Behrend from 2005 Jan. 5 to Feb.
1. Suggested a rotation period of 30.4 h. Was
only relative photometry a possible tumbling
couldnt be constrained. We find asteroids with
spin rates and sizes similar to Apophis being
mostly in Non-Principal Axis (NPA) rotation
states .
3
Force-free precession
Non-principal axis rotation (free precession,
tumbling) is a spin state with higher than
minimal rotational kinetic energy for given
angular momentum L.
The rotational motion can be described with the
time evolution of the Euler angles (e.g.,
Samarasinha and AHearn 1991, Kaasalainen
2001). It is a rotation around one of the
extreme principal inertia axes and a precession
of the axis around the L vector. Two periods
Pf, P?
(Pravec et al. 2005)
(Kaasalainen 2001)
4
Tumbler lightcurve
Lightcurve of a tumbling asteroid can be expanded
with the Fourier series in the two angular
variables (Kaasalainen 2001, Pravec et al.
2005). In a tumblers lightcurve, we observe the
frequencies ff Pf-1, f? P?-1, and their
linear combinations. The highest signal is often
observed in the second harmonic of (ff f?) it
is the actual mean frequency of rotation of the
body around the instantaneous spin axis.
(Pravec et al. 2005)
5
Lightcurve of a fast spinning tumbler (2002 TD60)
(Pravec et al. 2005)
6
Lightcurve of a fast spinning tumbler (2008 TC3)
Mag
JD
Mag
JD
Mag
JD
7
2008 TC3 numerical model

• Best-fit shape (convex model)

Dimensions ratio z/x 2.4 y/x 1.3
(Scheirich et al. 2010)
8
Photometry of Apophis

• Apparition 2012 Dec. 23 2013 April 15
• Data from
• 31 nights with the 1.54-m Danish telescope, La
  Silla
• 30 nights with the 0.41-m PROMPT 1 telescope,
  Cerro Tololo
• 4 nights with the 0.6-m TRAPPIST telescope, La
  Silla
• 3 nights with the 0.35-m, Leura, Australia
• 1 night with the 0.65-m, Ondrejov
• Additional unlinked data available (check of
  solutions consistency).
• All observations transformed or linked to the
  Cousins R system, absolute errors 0.03 mag for
  all subsets. Additional points in Johnson V
  taken with the 1.54-m, the VR measurements
  calibrated with absolute errors below 0.01 mag.
• Substantial change of asteroids viewing and
  illumination geometry during the apparition
• Geocentric (R.A., Decl.) changed from (10.7h,
  -27.4) to (7.7h, 18.0), i.e.,
• 45 in both R.A. and Declination.
• Solar phase changed from 77.6 down to 32.4 (on
  2013 Jan. 24) to 73.2.
• Data covering an arc needed to get unique
  spin/shape model.
• Modeling of observations at high solar phases
  difficult (scattering sensitive to local
• topography especially in lightcurve
  minima gt amplitude-phase effect).

9
Apophis tumbling
(V R) 0.453 0.01, consistent with the SQ
classification. The mean absolute magnitude H
19.09 0.19, derived assuming G 0.24 0.11
for the SQ type (Warner et al. 2009). Assuming
pV 0.197 0.051 for S type asteroids (Pravec
et al. 2012), we estimate the mean effective
diameter D 0.46 0.08 km. NPA rotation,
apparent frequencies f1 1/30.56 h, f2
1/29.04 h (uncertainties lt 0.1 h). Prominent
signal in 2f1, f1, f2, (2f2 f1), (2f1 f2).
More signal in other frequencies.
10
Apophis tumbling
(V R) 0.453 0.01, consistent with the SQ
classification. The mean absolute magnitude H
19.09 0.19, derived assuming G 0.24 0.11
for the SQ type (Warner et al. 2009). Assuming
pV 0.197 0.051 for S type asteroids (Pravec
et al. 2012), we estimate the mean effective
diameter D 0.46 0.08 km. NPA rotation,
apparent frequencies f1 1/30.56 h, f2
1/29.04 h (uncertainties lt 0.1 h). Prominent
signal in 2f1, f1, f2, (2f2 f1), (2f1 f2).
More signal in other frequencies.
Apophis lightcurve resembles simulated curves
for tumblers in Short-Axis Mode with the wobbling
angle 20 to 25 Apophis spin may be only
moderately excited.
(Henych and Pravec 2013)
11
Apophis tumbling
NPA rotation, apparent frequencies f1 1/30.56
h, f2 1/29.04 h (uncertainties lt 0.1 h).
Prominent signal in 2f1, f1, f2, (2f2 f1), (2f1
f2). More signal in other frequencies. For
body in SAM, the main apparent frequency is f1
(1/Pf 1/P?). If f2 1/Pf, then P? 583 h (
24.3 d) and I2 to I3 is very close to 1 (within lt
0.01) such symmetric body is physically
unlikely. We suspect that the signal (full
amplitude of 20 of mean light) in the apparent
frequency of 1/29.04 h is an artifact due to
presence of the secular Amplitude-Phase effect
causing higher lightcurve amplitude both at the
beginning and at the end of the fitted interval
(42-day long) where the solar phase was higher.
12
Apophis tumbling
A check of the other frequencies gave another
good candidate f2 1/34.4 h. With this, our
current working hypothesis is that P? 34.4 h
and Pf 16.2 h thus (1/Pf 1/P?) 1/30.56 h
the main observed period. We work on a physical
model to test the working hypothesis, or to
derive other combinations of P? and Pf that would
fit the observed lightcurve.
13
Slow tumblers population
In the rubble pile size range, tumblers
predominate at spin periods P gt 60 h at D 10
km, P gt 35 h at D 2 km, P gt 14 h at D 0.3
km. This is a very shallow dependence flim(D)
Da, with a -0.42. Other potentially relevant
time scales For Tdamp P3 D-2 (Harris 1994), we
get Tdamp tlife gt a -5/6 for tlife D1/2
or a -1 for tlife from
Bottke
et al. Tdamp tYORP gt a -4/3
In the rubble pile size range, tumblers
predominate at spin periods P gt 60 h at D 10
km, P gt 35 h at D 2 km, P gt 14 h at D 0.3
km. This is a very shallow dependence flim(D)
Da, with a -0.42. Other potentially relevant
time scales For Tdamp P3 D-2 (Harris 1994), we
get Tdamp tlife gt a -5/6 for tlife D1/2
or a -1 for tlife from
Bottke
et al.
In the rubble pile size range, tumblers
predominate at spin periods P gt 60 h at D 10
km, P gt 35 h at D 2 km, P gt 14 h at D 0.3
km. This is a very shallow dependence flim(D)
Da, with a -0.42.
14
Slow tumblers population

• The shallow slope of flim(D)
• If the tumbling was original --asteroid excited
  in the formation in a catastrophic collision--,
  the slope of flim(D) shallower than the slope for
  Tdamp tlife could be due to a dependence of µQ
  on asteroid size a decrease of µQ with
  decreasing D would lead to shorter Tdamp for
  smaller asteroids than expected (Harris 1994),
  hence it would result in PA rotations
  predominating to lower spin frequencies at
  smaller sizes, as we observe.
• But YORP must be taken into account!

15
Slow tumblers population
Spin rate distribution of asteroids with D 3-15
km (median 6.5 km) flattened by YORP effect
tYORP several times shorter than tlife.
(Pravec et al. 2008)
NPA rotations likely not original asteroids
moved to the slow-rotators bin by YORP slow down
from faster rotations where they were in PA
rotation states. But we dont know how YORP
works for slow rotations where the assumption of
the basic YORP theory that Tdamp ltlt tYORP is not
valid. Is PA rotation preserved? Tumblers are
in the leftmost bin f lt 1 d-1 where there is
observed the excess of slow rotators. The
excess may be due to asteroids staying in the
slow rotation for a prolonged time does
tumbling inhibit YORP?
16
Origin of the slow tumbling

• Candidate excitation mechanisms
• YORP effect on slow spins (Breiter and
  Vokrouhlický, in prep.)
• Sub-catastrophic impacts (Henych and Pravec 2013)
• Planetary flybys (for near-Earth asteroids
  Scheeres et al. 2005)

17
Conclusions

• Apophis is in a non-principal axis rotation
  state. It is a member of the population of
  small, slowly tumbling asteroids.
• Actual excitation and damping mechanisms for slow
  tumblers are unknown yet.

18
Slowfast tumblers population