Study of the Plutino Object (208996) 2003 AZ₈₄ from Stellar Occultations: Size, Shape, and Topographic Features

For the events described as follows, the flux from the star (plus the faint, background contribution due to the occulting body) was obtained through differential aperture photometry using the PRAIA package (Assafin et al. 2011). The resulting light curve (flux versus time) was then normalized to its unocculted value, using polynomial fits—first or third degree depending on the data quality—to the flux before and after the event.

3.1.1. Single-chord Events: 2011 and 2013

The single-chord occultation of 2011 January 8 had a typical shadow velocity of 26 km s⁻¹. It was observed at San Pedro de Atacama Celestial Explorations Observatory (longitude 68° 10' 4870 W, latitude 22° 57' 0980 S, altitude 2400 m) in Chile by Alain Maury using the C. Harlinten 0.5 m Planewave telescope (Figure 2) and by Nicolas Morales with the remotely operated 0.4 m ASH2 telescope. Alternatively, the single-chord event of 2013 December 2 had a shadow velocity of about 17 km s⁻¹ and was observed by Steve Kerr near Rockhampton (longitude 150° 30' 0080 E, latitude 23° 16' 0960 S, altitude 50 m) in Australia. The event was recorded with a Watec 120N+ video camera attached to a 0.30 m telescope attached, with cycle 2.56 s exposure time.

The occultation durations (see Figure 2 and Table 4) correspond to chord lengths of 573 ± 21 km and 657 ± 35 km for the 2011 and 2013 events, respectively, and provide lower limits for the major-axis of the object (Braga-Ribas et al. 2011, 2014b).

Table 4.  Ingress and Egress Times

2011 January 8
Site	Ingress (UT)	Error (s)	Egress (UT)	Error (s)
San Pedro Atacama (SPACE)	06:29:48.30	0.80	06:30:10.00	0.8
2012 February 3
Mt. Abu	19:45:13.89	0.70	19:45:35.95	2.5
Kraar	19:47:39.20	0.70	19:48:06.15	0.68
IUCAA Girawali	19:45:17.16a	7.3	19:45:25.43	0.10
2013 December 2
Rockhampton	14:52:48.44	1.40	14:53:28.17	1.0
2014 November 15
Thai National Telescope	17:58:09.57	0.02	17:59:13.80	${}_{-0.03}^{+0.04}$
Yunnan	17:57:21.65	0.13	17:57:36.95	${}_{-0.14}^{+0.13}$
Moriyama (CCD)	17:52:15.32	0.18	17:52:59.90	0.20
Moriyama (Video)	17:52:15.25	${}_{-3.99}^{+3.24}$	17:52:55.69	${}_{-4.01}^{+4.40}$

Note.

^aWhereas the ingress must happen between 19:45:09.88 and 19:45:24.43 UTC, it can be represented by the mid-time between them, with the error being the difference between the two extremes and the midpoint.

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3.1.2. 2012 February 3

Three occulting sites recorded this event: Mount Abu Observatory and IUCAA (Inter-University Centre for Astronomy and Astrophysics) Girawali Observatory in India, plus the Kraar Observatory in Israel. The event had a typical shadow velocity of 25 km s⁻¹; see details in Table 2.

Although all the sites had robust clock synchronizations, the acquisition software used for all of them recorded only the integer part of the second in each image header. In order to retrieve the fractional part of the second for the mid-exposure time t_i of each image i, we performed linear fits to the set $(i,{t}_{i})$, as used by Sicardy et al. (2011).

The residuals of this linear fit show a saw pattern with dispersion between −0.5 and +0.5 s (Figure 3 for example), showing that there is truncation and the period in which the acquisition cycle is regular. From this fit we obtain the acquisition cycle t_c, which is used to obtain the time of each image t_fit, now with the fraction of the second, and an initial time t₀:

$$\begin{eqnarray}&&{t}_{{fit}}={t}_{0}+({t}_{c}\cdot i)+0.5.\end{eqnarray} \tag{ 3 }$$

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Half a second is added to the times to correct the offset imposed by the truncation. With this procedure, the times of each image have an internal precision that depends on the square root of the number of images used and is less than 1 second. In the present case, we could retrieve the individual times t_fit to within an accuracy of 0.06 s.

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Note that the IUCAA light curve was obtained with an exposure time of 2 s, and a long read-out time of 14.5 s, which provides a single occultation point with partial flux drop, and a large uncertainty on the ingress time at that station (see discussion that follows, Table 4 and Figure 6).

3.1.3. 2014 November 15

This event was the slowest of all described here, with a shadow velocity of about 9.4 km s⁻¹. Four light curves were obtained at three different sites: the Yunnan station in China, the Thai National Telescope (TNT) in Thailand, and the Moriyama station in Japan (see Table 3 and Figure 4). For the three CCD recordings, the mid-exposure times t_i were extracted from the image headers. The headers from the Thai National Telescope images provide t_i down to the millisecond, while for both the Yunnan and Moriyama images, only the integer part of the second was recorded, thus requiring the same procedure as described in the previous subsection.

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The Moriyama site also recorded the event with a video camera attached to a 25.4 cm telescope. The data were collected by integrating 256 video frames to provide a complete exposure of approximately 8 s, with GPS time inserted in each frame. Using the Audela software,³⁹ we extracted individual frames in FITS format from the video. This conversion process requires special care because of possible dropped frames, duplicated fields or any software incompatibility (Buie & Keller 2016). Because of that, detailed checks were performed in all sets of images to ensure that the extracted time corresponds to the time printed at each frame.

Also, AUDELA conversion software create a FITS image for each video frame, instead of for each exposure. So it is important to check exactly how many of the exported FITS frames belong to a single acquisition, in order to combine them properly into a single image. Using photometry and observing the step-like variations of the target brightness, we can identify in each step the 256 frames that represent a single acquisition and that must be averaged out.

More precisely, in order to avoid taking into account any field from the previous or next acquisition, we excluded the first and last frame from a sequence, and then averaged out the remaining 254 frames. To ensure a correct time-stamping of each of the final images, we used the mid-exposure time of the mid-frame of each sequence.

A noteworthy feature observed during this occultation is the gradual decrease of the stellar flux observed over three acquisition intervals during the star disappearance at the Yunnan site (Figure 5). This feature is analyzed in more detail as follows and was interpreted as a topographic feature. Note that this effect cannot be the result of photometric noise, as the typical photometric error of each data point is about 0.0236 mag, less than 6% of the observed flux variation, thus ruling out photometric problems. Moreover, a careful visual analysis of the three corresponding images clearly shows a gradual disappearance of the star, without any similar behavior for the other reference objects acquired in the field of view.

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To determine the start and end times of the occultation, we fitted for each light curve a sharp edge occultation model convolved by Fresnel diffraction, CCD bandwidth, and stellar diameter projected at the body and finite integration time (see Widemann et al. 2009 and Braga-Ribas et al. 2013).

With 2003 AZ₈₄'s geocentric distances of D = 44.3 au in 2012 and D = 44.5 au in 2014, the Fresnel scale ($F=\sqrt{\lambda D/2}$) for a typical wavelength of λ = 0.65 μm (and λ = 1.65 μm for Mt. Abu) is derived for each event, F = 1.47 km (2.34 km for Mt. Abu) in 2012, and F = 1.46 km in 2014.

The star diameter projected at 2003 AZ₈₄'s distance was estimated using the B, V, and K apparent magnitudes provided by the NOMAD catalog (Zacharias et al. 2004) and the formulae of van Belle (1999). The 2012 star (B = 15.8 mag, V = 15.35 mag, K = 14.1 mag) then has an estimated projected diameter of 0.24 km, while it is 0.20 km for the 2014 star (B = 15.5 mag, V = 15.5 mag, and K = 14.0 mag).

Using the apparent TNO motion relative to the star, it is possible to translate the integration times into actual distances traveled in the sky plane. The smallest integration times were 2.0 s in 2012 and 0.786 s in 2014, corresponding to 49.7 km and 7.50 km in the celestial plane, respectively. Therefore our light curves are dominated by the integration times, and not by Fresnel diffraction or stellar diameter. The same is true for the 2011 and 2013 events.

The free parameters to adjust in the fits are the times ${t}_{\mathrm{occ}}$ of ingress (star disappearance) or egress (star re-appearance) for each station. The value of ${t}_{\mathrm{occ}}$ is obtained by minimizing a classical ${\chi }^{2}$ function, as described in Sicardy et al. (2011). The resulting best fits are shown in Figure 6, and the occultation times are listed in Table 4.

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The IUCAA light curve is special, however, because it has only one data point with a partial drop of the stellar flux (Figure 6), meaning that the star was occulted during a fraction of this 2-s acquisition interval. Calibration data taken with the same instrument on 2012 February 10 provide the separate fluxes of 2003 AZ₈₄ and the star, allowing us to estimate the fraction of time when the star was occulted during that interval. Note that from the light curve, we cannot discriminate between an ingress or egress point. However, considering the location of the point in the sky plane and comparing it to the other chords, it seems more likely to be an egress point, with an ingress that happened during 14.5 s gap between the end of the previous exposure, and before the start of the current exposure. The alternative solution (with egress happening during the gap after the single drop) would result in a very elongated body, which is unlikely (see Figure 6).

The ingress and egress times ${t}_{\mathrm{occ}}$ provide the positions of the star in the sky plane relative to 2003 AZ₈₄'s center, once the star position is specified (Equations (1) and (2)), and for a given ephemeris (NIMA V1 in our case). More precisely, each timing gives the position (f, g) of the star with respect to the body center, with f and g being measured positively toward local celestial east and celestial north, respectively.

For each event, the general shape for the body's limb was assumed to be an ellipse characterized by M = 5 adjustable parameters: the coordinates of the ellipse center, $({f}_{c},{g}_{c});$ the apparent semimajor axis, $a^{\prime} ;$ the apparent oblateness, $\epsilon ^{\prime} =(a^{\prime} -b^{\prime} )/a^{\prime}$ (where $b^{\prime}$ is the apparent semiminor axis); and the position angle of the pole P_p of $b^{\prime}$, which is the geocentric position angle of the pole measured eastward from the north. Note that the center $({f}_{c},{g}_{c})$ actually measures the offsets in R.A. and decl. to be applied to the ephemeris in use, assuming that the star position is correct. Note also that the quantities $a^{\prime}$, $b^{\prime}$, f_c, and g_c are all expressed in kilometers.

The ingress and egress times from each light curve provide the extremities of the corresponding occultation chord. As both ingress and egress times from Moriyama's light curves agree at the 1-σ level, we consider only one chord for that station (the one obtained with CCD) for the limb fitting. Thus for each event there are three positive observations, providing N = 6 chord extremities (ingress and egress), with positions ${f}_{i,\mathrm{obs}}$, ${g}_{i,\mathrm{obs}}$ (Figures 6 and 7). The best elliptical fit to the chord extremities is found by minimizing the relevant ${\chi }^{2}$ function (see, e.g., Sicardy et al. 2011). In that case, the statistical significance of the fit is evaluated from the ${\chi }^{2}$ per degree of freedom (pdf) defined as ${\chi }_{\mathrm{pdf}}^{2}={\chi }^{2}/(N-M)$, which should be close to unity. The individual 1-σ error bar of each parameter is obtained by varying manually the given parameter from its nominal solution value (keeping the other ones fixed), so that ${\chi }^{2}$ varies from its minimum value ${\chi }_{\min }^{2}$ to ${\chi }_{\min }^{2}+1$.

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The parameters of the best elliptical fits for the 2012 and 2014 events are given in Table 5, which provides the shapes of the limb and the offsets to be applied to the used ephemeris. Those fits clearly show a change in the limb shape between those two dates (see Figures 6 and 7). With this said, the fits provide the best values of ${\chi }_{\mathrm{pdf}}^{2}=0.73$ (2012) and ${\chi }_{\mathrm{pdf}}^{2}=0.98$ (2014), which indicates satisfactory fits.

Table 5.  Physical Parameters for 2003 AZ₈₄ from the 2012 and 2014 Multi-chord Events

Solution	2012	2014
Apparent semimajor axis $a^{\prime}$ (km)	${426}_{-16}^{+26}$	393 ± 1
Equivalent radius Re (km)	346 ± 6	382 ± 3
Projected oblateness $\epsilon ^{\prime}$	${0.340}_{-0.086}^{+0.097}$	0.054 ± 0.003
fc (km)a	${6800}_{-17}^{+23}$	${2862}_{-1.8}^{+2.1}$
gc (km)a	${2850}_{-11}^{+14}$	${336}_{-2.7}^{+4.0}$
Position angle of the limb PL (deg)	${41}_{-5}^{+8}$	36 ± 4
${\chi }_{\mathrm{pdf}}^{2}$	0.73	0.98
Jacobi solution
Semimajor axis a (km)	470 ± 20
Axis ratios	$b/a=0.82\pm 0.05$, $c/a=0.52\pm 0.02$
Density ρ (g cm−3)	0.87 ± 0.01
Visible geometric albedo pV	0.097 ± 0.009

Note.

^aThe quantities $({f}_{c},{g}_{c})$ are the offsets, expressed in km, in R.A. (${\rm{\Delta }}\alpha \cos (\delta )$) and decl. (${\rm{\Delta }}\delta$) to be applied to 2003 AZ₈₄'s ephemeris (NIMA V1), as deduced from the elliptical fits to the chord extremities shown in Figures 6 and 7, using the star positions of Equations (1) and (2). The NIMA ephemeris provides the following reference J2000 geocentric positions for the body: $({07}^{{\rm{h}}}{45}^{{\rm{m}}}54\buildrel{\rm{s}}\over{.} 79070,+11^\circ 12^{\prime} 43\buildrel{\prime\prime}\over{.} 0422)$, and heliocentric distance ${\rm{\Delta }}=6.6209\times {10}^{9}$ km on 2012 February 3, at 19:45 UTC, and $({08}^{{\rm{h}}}{03}^{{\rm{m}}}51\buildrel{\rm{s}}\over{.} 28512,+09^\circ 57^{\prime} 18\buildrel{\prime\prime}\over{.} 7956)$, ${\rm{\Delta }}=6.6634\times {10}^{9}$ km on 2014 November 15 at 17:55 UTC. Applying the $({f}_{c},{g}_{c})$ offsets tabulated here, we obtain the following corrected geocentric positions for 2003 AZ₈₄: $({07}^{{\rm{h}}}{45}^{{\rm{m}}}54\buildrel{\rm{s}}\over{.} 80510,+11^\circ 12^{\prime} 43\buildrel{\prime\prime}\over{.} 1310)$ and $({08}^{{\rm{h}}}{03}^{{\rm{m}}}51\buildrel{\rm{s}}\over{.} 29112,+09^\circ 57^{\prime} 18\buildrel{\prime\prime}\over{.} 8060)$ at each date, respectively. Those positions are independent of the ephemeris, but depend on the assumed star positions (Equations (1) and (2)).

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The main change concerns the apparent oblateness of the body, which varies from $\epsilon ^{\prime} ={0.340}_{-0.086}^{+0.097}$ in 2012 to $\epsilon ^{\prime} \,=0.054\pm 0.003$ in 2014, with a difference of more than 3-σ. Conversely, the apparent semimajor axes are $a=426\pm 20$ km (2012) and $a=393\pm 1$ km (2014), with a barely significant difference at 2-σ level. Note in passing that these results agree with the lower limits on the major-axis 573 ± 21 km and 657 ± 35 km quoted earlier, derived from the single-chord events of 2011 and 2013 events. Similarly, the two fits provide consistent (1-σ level) position angles of the limb: ${P}_{L}={41^\circ }_{-5}^{+8}$ (2012) and ${P}_{L}=36^\circ \pm 4^\circ$ (2014). The P_L is geocentric position angle of the northern semiminor axis of the projected limb, measured eastward from the north (not to be confounded with the position angle of the pole P_p).

We may attribute the change of the retrieved $\epsilon ^{\prime}$ to a very irregular shape, but reconciling the limbs of Figures 6 and 7 at face value would require topographic features of up to 40 km above an average elliptical limb. Hydrostatic equilibrium is expected for objects with diameter on the order of 1000 km. More precisely, the critical diameter necessary to reach equilibrium for bodies with slow rotation may vary between 500 and 1200 km for rocky bodies, and between 200 and 900 km for icy bodies (Tancredi & Favre 2008). In this context, the body assumes either the shape of a Jacobi ellipsoid with principal axes $a\gt b\gt c$ or that of a Maclaurin spheroid $a=b\gt c$, depending on their angular momentum (Chandrasekhar 1987).

Assuming first a Maclaurin solution, the variation of $\epsilon ^{\prime}$ between 2012 and 2014 could in principle be caused by a variation with time of the opening angle⁴⁰ B due to a mere change of viewing geometry. The true oblateness $\epsilon =1-(c/a)$ is related to the apparent oblateness through $\epsilon ^{\prime} =1-\sqrt{{\sin }^{2}{(B)+(1-\epsilon )}^{2}{\cos }^{2}(B)}$, and it is easy to see that the larger , the smaller the change of B necessary to change $\epsilon ^{\prime}$ by a given amount. As the Maclaurin solutions have $\epsilon \lt 0.417$ (Tancredi & Favre 2008), it results that changing $\epsilon ^{\prime}$ from ${0.340}_{-0.086}^{+0.097}$ to 0.054 ± 0.003 requires the changing of B by at least ${45}_{-15}^{+20}$ degrees. This cannot be caused just by a variable viewing geometry, because of the large geocentric distance of 2003 AZ₈₄ (more than 43 au) and of its slow heliocentric motion. In fact, testing a set of random pole positions, the maximum difference in B between 2012 and 2014 is found to be less than ∼5 degrees.

An alternative explanation for a changing B is a precession caused by the solar torque on 2003 AZ₈₄. Classical calculations (e.g., Goldstein 1950) provide a precession rate of $-3{{GM}}_{\odot }/(2{r}^{3}\omega )\epsilon \cos (\theta )$ for a spheroid, where G is the constant of gravitation, ${M}_{\odot }$ and r are the Sun mass and distance, respectively, ω is the spin frequency, and θ is the obliquity. At $r\gt 43\,\mathrm{au}$ and with a rotation period of 6.75 ± 0.04 hr (Thirouin et al. 2010), we have $3{{GM}}_{\odot }/(2{r}^{3}\omega )\sim {10}^{-15}$ rad s⁻¹, which is far too small to explain any significant change of pole orientation in only 2 years.

The same conclusion follows if the precession is caused by the satellite reported by Brown & Suer (2007). From the numbers quoted in the Introduction, and assuming same albedos, the satellite should have a typical radius of 40 km, corresponding to a mass M of a few times 10¹⁷ kg for an icy composition. This implies $3{GM}/(2{r}^{3}\omega )\sim {10}^{-10}$ rad s⁻¹ (taking $r\,\sim$ 10,000 km), which is still too small for explaining any significant changes of the pole orientation in 2 years.

Finally, the object aspect angle might change due to a misalignment between 2003 AZ₈₄'s angular momentum and its minor axis, causing a wobbling—for instance as a consequence of a collision. Burns & Safronov (1973) estimated alignment times for typical small bodies from 4 to 200 km in radius, and found typical times from $\sim 6\times {10}^{7}$ to $\sim {10}^{5}\,\mathrm{years}$, respectively. Therefore, for 2003 AZ₈₄, the alignment time should be very short ($\lt {10}^{5}$ years), making the wobbling hypothesis highly unlikely. At this point, we are left with the Jacobi shape alternative, which is explored in more detail in the next section.

Here we look for a unique Jacobi body with fixed pole position that can account for both 2012 and 2014 occultations (Figures 6 and 7). A second condition that we impose is that this solution is also compatible with the (single peaked) amplitude of 2003 AZ₈₄'s rotational light curve, ${\rm{\Delta }}m=0.07\,\pm 0.01$ mag, reported by Thirouin et al. (2010) and Ortiz et al. (2006).

For a stable Jacobi ellipsoid, the adimensional quantity ${\rm{\Omega }}={\omega }^{2}/(G\pi \rho )$ is bounded between 0.284 and 0.374 (Chandrasekhar 1987). If we assume that the body is a triaxial ellipsoid, the full rotation period should actually be $6.75\times 2=13.5\,\mathrm{hr}$. This would imply, from the definition of Ω, that the density lies in the interval $0.21\lt \rho \lt 0.28$ g cm⁻³, unrealistic low values for ρ. If we adopt a rotational period of 6.75 hr (thus assuming that the rotational light curve is caused by albedo features, not by shape effects; see discussion that follows), then the density is restricted to the interval $0.85\lt \rho \lt 1.12$ g cm⁻³.

Now, for each value of ρ, there is a unique Jacobi ellipsoid solution, and more precisely unique ratios b/a and c/b, where $a\gt b\gt c$ are the principal axes of the body. We simplify the problem further by assuming that the object does not change its pole orientation relative to the observer between the 2012 and 2014 occultations. In this way, the changes of its limb shape are solely due to the rotation around the pole axis. As commented before, exploring random pole orientations, the maximum difference in B between 2012 and 2014 is less than ∼5 degrees, while the typical change in the position angle of the pole is ∼7 degrees, making our assumption reasonable.

Then, we determine numerically a set of combinations of B and ρ for which there is a pair of rotation angle⁴¹  Q and common value of a that is compatible (to within error bars) with the projected sizes and oblatenesses $\epsilon ^{\prime}$ of both 2012 and 2014 events. Note that given the current accuracy on 2003 AZ₈₄'s rotation period (6.75 ± 0.04 hr) and the interval of more than 2 years between the two observations, the corresponding values of Q are considered as uncorrelated between 2012 and 2014.

Next, for a set of densities ρ between 0.85 and 1.12 g cm⁻³, we consider the corresponding Jacobi solutions, with their associated ratios b/a and c/a. This constrains $(\rho ,B)$ to lie on a banded-shape domain that reflects the 1-σ uncertainties on the limb parameters (see the hatched region in Figure 8). We note that near equator-on solutions ($B\sim 0^\circ$) are ruled out, as they generate very elongated projected shapes, especially for low density ellipsoids, that are incompatible with the moderate apparent oblatenesses of Table 5 (and with the very small amplitude derived from the rotational light curve). On the other hand, near pole-on solutions ($B\sim 90^\circ$) are also ruled out, as they generate little changes in projected oblateness (whatever the value of Q is), while the oblatenesses between 2012 and 2014 are significantly different.

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We now use the amplitude of the rotational light curve (${\rm{\Delta }}m=0.07\pm 0.01$ mag) as an independent constraint. Since the 6.75 hr rotational light curve is single peaked, while we assume here a Jacobi shape, this means that the observed variability is caused by albedo features, not by the shape. In this context, we can only state that the Jacobi solution must be such that it causes variations of apparent cross section that have an amplitude smaller than 0.07 magnitude (Binzel et al. 1989). This defines another possible domain for $(\rho ,B)$ (see the gray band in Figure 8). The intersection of the two domains puts a stringent constraint $\rho =0.87\pm 0.01$ g cm⁻³ for the density of the Jacobi body we are looking for, with axis ratios $b/a=0.82\pm 0.05$ and $c/a=0.52\pm 0.02$. Exploring the intersection domain shown in Figure 8, we find that Jacobi ellipsoids with a fixed pole orientation can fit both 2012 and 2014 limbs with semiaxes ranges of $a=470\pm 20$ km, $b=383\pm 10$ km, and $c=245\pm 8$ km (1-σ level).

An example of a possible solution is given in Figure 9, corresponding to a = 456 km, $\rho =0.86$ g cm⁻³, and rotation angles $Q=24^\circ$ (2012) and $Q=10^\circ$ (2014). A full treatment that provides a global best fit and finer error bars would require a Bayesian approach. This remains out of the scope of this paper, but our example shows that a unique Jacobi solution meeting all the constraints (our occultation results plus the rotational light curve amplitude) do exist in narrow intervals of ρ and a.

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Finally, the intersection region in Figure 8 provides a range for the equivalent radius of the body, ${R}_{\mathrm{eq}}=386\pm 6$ km (as observed in 2012 and 2014)—that is, the radius of the disk that has the same apparent area as the observed body. Using this value, we obtain the geometric albedo in visible band through

$$\begin{eqnarray}&&{p}_{V}={({{au}}_{\mathrm{km}}/{R}_{\mathrm{eq}})}^{2}\times {10}^{0.4({H}_{\odot ,V}-{H}_{V})},\end{eqnarray} \tag{ 4 }$$

where ${{au}}_{\mathrm{km}}\,=\,1.49597870700\,\times \,{10}^{8}$ km, and ${H}_{\odot ,V}=\,-26.74$mag is the Sun magnitude at 1 au in the visible. From 2003 AZ₈₄'s visible absolute magnitude, ${H}_{V}=3.74\pm 0.09$ mag (Mommert et al. 2012), and for the value ${R}_{\mathrm{eq}}$ obtained previously, we derive ${p}_{V}=0.097\pm 0.009$.

Figure 5 shows the gradual disappearance of the star during ingress at the Yunnan station. It took more than 8 s for the stellar flux to go from its unocculted level to complete disappearance. This corresponds to a displacement of the stellar image of more than 80 km along the local limb. Since this is the only instance of gradual dimming observed during this event, it must come from a localized feature on the limb and cannot stem from a star duplicity. Various causes might explain this behavior, the most plausible one being a topographic feature revealed by the grazing geometry of the chord (Figure 7).

Beyond a topographic origin, only exotic explanations may be envisaged, like the presence of a local dust cloud lingering over the surface, or a local atmosphere that refracts the stellar rays. The drops of signal observed during intervals A and B in Figure 5 would imply average tangential optical depths of ${\tau }_{A}=0.53$ and ${\tau }_{B}=0.83$, respectively, for dusty material (assuming that point C is then be the usual star disappearance behind the solid limb). This would imply in a very dense local dusty cloud with length ∼80 km and height ∼10 km. An intense cometary-like activity would be required to create such cloud, an unlikely process so far away from the Sun (∼45 au), and such an activity has not been reported before for 2003 AZ₈₄.

Alternatively, Ortiz et al. (2012) analyzed the effects of a localized atmosphere during an occultation. Typical surface pressures of a few microbars are required to create the feature of Figure 5. A stationary local atmosphere could result from a delicate balance between local sublimation and recondensation farther away on the surface. To analyze this scenario further, it is necessary to know the pole position of the object in order to determine the sub-solar latitude along the limb (where sublimation would be favored). Moreover, an albedo map of 2003 AZ₈₄ would be useful to identify warmer/cooler terrains where sublimation/condensation could be favored. These pieces of information are missing right now, preventing a more detailed discussion of the local atmosphere hypothesis.

Returning to the topographic feature explanation, we note that each integration interval lasts for 3 s (Table 3), corresponding to about 28 km traveled by the star parallel to the limb, while the complete duty cycle (4.3 s) corresponds to about 40 km. In that context, there is an infinity of solutions for the limb structure that fit equally well the observed light curve. For instance, the star may disappear and re-appear several times behind local reliefs during a single acquisition interval.

We examine here two extreme, simple solutions that provide possible ranges for the width and depth of the limb feature. One solution (Solution 1) assumes that the observed light curve is caused by a chasm with vertical, abrupt walls (i.e., perpendicular to the global fitted limb displayed in Figure 7), while Solution 2 assumes that the topographic features are smooth with very shallow slopes, corresponding to a local limb that is always parallel to the global limb. In both solutions, we have to convolve the sharp shadow edge profile by the Fresnel diffraction pattern, the stellar diameter, and the integration time. The Fresnel pattern corresponding to the 2014 November 15 event is displayed in Figure 10. As the projected stellar diameter is estimated to be 0.2 km, it is negligible compared to the Fresnel scale, ∼1.5 km (see Section 3.2). Thus the occultation light curve is dominated by Fresnel diffraction, and of course, by the large integration time (3 s) at the Yunnan station.

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In the Solution 1 scenario, the star first disappears behind the global limb; then it re-appears in the chasm during interval B, and disappears again during interval C. During intervals D and E, the star is completely hidden behind the body, and it re-appears from behind the global limb near the start of interval F. The locations of the walls are then adjusted so as to reproduce the observed fluxes (Figure 5).

Solution 2 is a bit more complex to implement, as the star velocity relative to the body center varies significantly during each acquisition interval. To generate a synthetic light curve, we calculate the position of the star relative to the body center at regular time steps of 0.1 s inside each integration interval (black dots in Figure 11). Using the Fresnel pattern of Figure 10, we can generate the synthetic flux by averaging the theoretical fluxes calculated at each 0.1 s step inside a given interval. The free parameter in this calculation is the radial offset of the local limb (corresponding to the blue lines in Figure 10), adjusted so as to fit the observed profile. This offset in turn provides the difference between the positions of the local limb and that of the global limb (red lines in Figure 11), and eventually the average elevation of the terrain in each interval (Figure 12).

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Figure 13 summarizes our results. The first panel sketches our inferred chasm structure (Solution 1) with a width of 22.6 ± 0.4 km, while only a lower limit of about 8 km can be given for its depth. The second panel (Solution 2) reveals a general depression that reaches more than 13 km in depth and extends over more than 80 km along the limb.

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These structures can be compared with topographic features recently revealed on Pluto's and Charon's surfaces during the New Horizons flyby (Stern et al. 2015; Nimmo et al. 2017). On both bodies, features reaching depths of up to 10 km are observed, some of them extending over tens of kilometers on the surface. This supports our finding of a chasm or depression on 2003 AZ₈₄'s surface, as this body is smaller (semimajor axis radius $a\lt 500$ km; see previous section) than Pluto and Charon (radii 1188 km and 606 km respectively; see Nimmo et al. 2017), and thus can in principle sustain comparable or deeper topographic features.