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 asteroid’s 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 couldn’t 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. (Kaasalainen 2001) (Pravec et al. 2005) The rotational motion can be described with the time evolution of the Euler angles (e.g., Samarasinha and A’Hearn 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: P φ, P ψ

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 tumbler’s lightcurve, we observe the frequencies f φ = P φ -1, f ψ = P ψ -1, and their linear combinations. The highest signal is often observed in the second harmonic of (f φ ± f ψ ); 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) JD Mag JD Mag JD Mag

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, Ondřejov Additional unlinked data available (check of solution’s 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 asteroid’s viewing and illumination geometry during the apparition: Geocentric (R.A., Decl.) changed from (10.7 h, -27.4°) to (7.7 h, +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 –> 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 p V = 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: f 1 = 1/30.56 h, f 2 = 1/29.04 h (uncertainties < 0.1 h). Prominent signal in 2f 1, f 1, f 2, (2f 2 – f 1 ), (2f 1 – f 2 ). More signal in other frequencies.

10. (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 p V = 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: f 1 = 1/30.56 h, f 2 = 1/29.04 h (uncertainties < 0.1 h). Prominent signal in 2f 1, f 1, f 2, (2f 2 – f 1 ), (2f 1 – f 2 ). More signal in other frequencies. Apophis’ tumbling (Henych and Pravec 2013) 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.

11. Apophis’ tumbling NPA rotation, apparent frequencies: f 1 = 1/30.56 h, f 2 = 1/29.04 h (uncertainties < 0.1 h). Prominent signal in 2f 1, f 1, f 2, (2f 2 – f 1 ), (2f 1 – f 2 ). More signal in other frequencies. For body in SAM, the main apparent frequency is f 1 = (1/P φ – 1/P ψ ). If f 2 = 1/P φ, then P ψ = 583 h (= 24.3 d) and I 2 to I 3 is very close to 1 (within < 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 f 2 = 1/34.4 h. With this, our current working hypothesis is that P ψ ~ 34.4 h and P φ ~ 16.2 h; thus (1/P φ – 1/P ψ ) = 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 ψ and P φ that would fit the observed lightcurve.

13. Slow tumblers population In the `rubble pile’ size range, tumblers predominate at spin periods: P > 60 h at D = 10 km, P > 35 h at D = 2 km, P > 14 h at D = 0.3 km. This is a very shallow dependence f lim (D) ~ D α, with α ≈ -0.42. Other potentially relevant time scales: For T damp ~ P 3 D -2 (Harris 1994), we get T damp = t life –> α = -5/6 for t life ~ D 1/2 or α ≈ -1 for t life from Bottke et al. T damp = t YORP –> α = -4/3 In the `rubble pile’ size range, tumblers predominate at spin periods: P > 60 h at D = 10 km, P > 35 h at D = 2 km, P > 14 h at D = 0.3 km. This is a very shallow dependence f lim (D) ~ D α, with α ≈ -0.42. Other potentially relevant time scales: For T damp ~ P 3 D -2 (Harris 1994), we get T damp = t life –> α = -5/6 for t life ~ D 1/2 or α ≈ -1 for t life from Bottke et al. In the `rubble pile’ size range, tumblers predominate at spin periods: P > 60 h at D = 10 km, P > 35 h at D = 2 km, P > 14 h at D = 0.3 km. This is a very shallow dependence f lim (D) ~ D α, with α ≈ -0.42.

14. Slow tumblers population The shallow slope of f lim (D) If the tumbling was original --asteroid excited in the formation in a catastrophic collision--, the slope of f lim (D) shallower than the slope for T damp = t life could be due to a dependence of μ Q on asteroid size: a decrease of μ Q with decreasing D would lead to shorter T damp 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; t YORP several times shorter than t life. (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 don’t know how YORP works for slow rotations where the assumption of the basic YORP theory that T damp << t YORP is not valid. Is PA rotation preserved? Tumblers are in the leftmost bin f < 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. Slow+fast tumblers population