Apophis’ tumbling

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. 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: Pφ, Pψ (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 tumbler’s lightcurve, we observe the frequencies fφ= Pφ-1, fψ = Pψ-1, and their linear combinations. The highest signal is often observed in the second harmonic of (fφ ± fψ); 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, 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 (10.7h, -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 –> 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 < 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 < 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 < 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/Pφ – 1/Pψ). If f2 = 1/Pφ, then Pψ = 583 h (= 24.3 d) and I2 to I3 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 f2 = 1/34.4 h. With this, our current working hypothesis is that Pψ ~ 34.4 h and Pφ ~ 16.2 h; thus (1/Pφ – 1/Pψ) = 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 Pψ and Pφ 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 flim(D) ~ Dα, with α ≈ -0.42. 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 flim(D) ~ Dα, with α ≈ -0.42. Other potentially relevant time scales: For Tdamp ~ P3 D-2 (Harris 1994), we get Tdamp = tlife –> α = -5/6 for tlife ~ D1/2 or α ≈ -1 for tlife from Bottke et al. Tdamp = tYORP –> 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 flim(D) ~ Dα, with α ≈ -0.42. Other potentially relevant time scales: For Tdamp ~ P3 D-2 (Harris 1994), we get Tdamp = tlife –> α = -5/6 for tlife ~ D1/2 or α ≈ -1 for tlife from Bottke et al.

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 = tlifecould 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 don’t know how YORP works for slow rotations where the assumption of the basic YORP theory that Tdamp << tYORP 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 Pravec2013) • 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