THE SIZE, SHAPE, ALBEDO, DENSITY, AND ATMOSPHERIC LIMIT OF TRANSNEPTUNIAN OBJECT (50000) QUAOAR FROM MULTI-CHORD STELLAR OCCULTATIONS

All light curves were normalized to the unocculted star flux by applying a linear or second-degree polynomial fit to the observed flux before and after the event. The resulting light curves are shown in Figure 2. The mid-exposure time of each frame was extracted from the image headers. Only four stations (out of the six that detected the event) have an internal accuracy of a few hundredths of a second: Harlingten and ASH2 (at San Pedro de Atacama), Armazones, and UEPG. On the other two sites, only the integer part of the second is available due to the acquisition software used. In order to retrieve the fraction of a second, a linear fit was performed to the set of points (i, t_i), where i is the image number and t_i is the corresponding header time. This allowed us to retrieve the time of each image with relative accuracies of about 0.05 s. All sites except one set up their clocks using robust internet servers for time synchronization, and absolute errors larger than 1 s are not expected (Deeths & Brunette 2001). The exception is the Rivera station, where the computer clock was manually set up 1 hr before the event by looking at a Global Positioning System (GPS) display due to the lack of internet access. This point is discussed further in Section 4.3.

The start and end times of the occultation were obtained for each light curve by fitting a sharp edge occultation model. This model is convolved by Fresnel diffraction, the CCD bandwidth, the stellar diameter in kilometers, and the finite integration time; see Widemann et al. (2009).

The Fresnel scale ($F= \sqrt{\lambda D/ 2}$) for the present Quaoar's geocentric distance D = 42.4 AU =6.34 × 10⁹ km is 1.4 km for a typical wavelength of λ = 0.65 μm. The star diameter is estimated using the formulae of van Belle (1999). Its B, V, and K apparent magnitudes are 16.5, 15.8, and 12.8, respectively, in the NOMAD catalog (Zacharias et al. 2004). This yields a diameter of about 0.5 km projected at Quaoar's distance. The smallest integration time used in the positive observations was 3 s, which translates to almost 55 km in the celestial plane. Therefore, the occultation light curves are largely dominated by the integration times, not by Fresnel diffraction or star diameter.

The occultation fits consist in minimizing a classical χ² function for each light curve, as described in Sicardy et al. (2011; Section S3). The free parameter to adjust is the ingress (disappearance) or egress (re-appearance) time t_occ, which provides the minimum value of χ², denoted as $\chi ^2_{\rm min}$.

The best fits to the occultation light curves are shown in Figure 2, and the derived occultation times are listed in Table 1. The error bars are not necessarily symmetric due to the presence of gaps (caused by readout time intervals) between images. No information is gathered during those gaps, causing local plateaux in the χ² function, and hence asymmetric error bars.

The occultation times provide the chords shown in Figure 2. The most general shape for Quaoar's limb that we considered here is an ellipse characterized by five adjustable parameters: the coordinates of the body center, relative to the star in the plane of the sky (f_c, g_c); the apparent semimajor axis a'; the apparent oblateness ' = (a' − b')/a' (where b' is the apparent semiminor axis); and the position angle P of the semiminor axis b'. The quantities f_c and g_c, expressed in kilometers, are positive toward the local celestial east and north, respectively. They are calculated using the JPL#21 Quaoar's ephemeris (Giorgini et al. 1997) at each site, and the star position is given in Equation (1). The position angle P is counted positively from the direction of local celestial north to celestial east.

The six positive observations provide N = 12 chord extremities, whose positions are denoted as f_{i, obs}, g_{i, obs}. Among the six chords, two were obtained at San Pedro de Atacama with the Harlingten and ASH2 telescopes. Although consistent with the Harlingten results, the ASH2 timing has significantly higher error bars due to longer exposure time (Table 1), so that we kept only N = 10 independent points along the limb and M = 5 free parameters to be adjusted. The elliptical fit to the N = 10 chord extremities is found by minimizing the relevant χ² function (see, for instance, Sicardy et al. 2011). In that case, the value of χ² per degree of freedom (or unbiased χ²) is given by $\chi ^2_{\rm pdf}=\chi ^2/(N - M)$.

As discussed later, in the hypothesis of hydrostatic equilibrium and in the small angular momentum regime, the apparent oblateness ' is the result of the projection of an oblate Maclaurin spheroid with semi-axes a = b > c, where a and c are the true equatorial and polar radii, respectively.²⁹ The true oblateness = 1 − (c/a) is then related to the apparent oblateness through

$$\begin{equation} \epsilon ^{\prime } = 1 - \sqrt{\cos ^2 (\zeta) + (1-\epsilon)^2 \sin ^2 (\zeta)}, \end{equation} \tag{ 2 }$$

where ζ is the angle between the polar c-axis and the line of sight. We will call this quantity the "polar aspect angle," with ζ = 0 (resp. ζ = 90°) corresponding to the pole-on (resp. equator-on) geometry.

To explore the parameter space, we have fitted the chord extremities with ellipses, imposing prescribed values of the position angle P and oblateness '. We have then varied P between 0° and 180° by steps of 3°, and ' from 0 to 0.5 by steps of 0.004. For each pair of prescribed values (P, '), we have adjusted and stored the parameters a', f_c, g_c, and the resulting χ². This allowed us to construct two-dimensional colored maps (see Figures 4 and 5), where the χ² value is shown as a function of oblateness ' and apparent equivalent radius (instead of a'). This quantity is defined by $R_{\rm equiv}= \sqrt{a^{\prime }b^{\prime }}= a^{\prime } \sqrt{1-\epsilon ^{\prime }}$, and is the radius of the disk that has the same area as that enclosed by the apparent limb. The 1σ error bars on R_equiv and are obtained by varying χ² from its minimum value $\chi ^2_{\rm min}$ to $\chi ^2_{\rm min} +1$.

We note that the colored points are confined inside a V-shaped boundary. This is because the longest occultation chords impose that a' be larger than some value L, which is close to the lengths of those longest chords. Also, those chords are roughly aligned with the longer axis of the fitted ellipse (Figure 4), which imposes that b' be smaller than ∼L. Thus, we must have $L \sqrt{1-\epsilon ^{\prime }} < R_{\rm equiv} < L/\sqrt{1-\epsilon ^{\prime }}$, corresponding to the V-shaped domain seen in Figures 4 and 5. The value of L depends on the particular model considered to fit the chords, and varies in a range from about 500 to 680 km; see the next section.

The nominal solution uses the ingress and egress occultation times at face value, and fits an elliptical shape to the chord extremities, accounting for the 1σ error bar of each of them; see Figure 3. The best fit returns a best value $\chi ^2_{\rm pdf}$ = 9.1, clearly indicating that an overall elliptical limb shape is not satisfactory. This is obvious from Figure 3, and at this point it is meaningless to interpret the physical parameters derived from this fit and to discuss their error bars. We thus examine possible extreme solutions in which Quaoar has large topographic features, i.e., a deep crater or a high mountain.

The first solution we examine is the crater solution. It consists of an overall oblate object from which a part has been excavated; see Figure 4. An elliptical limb shape is again fitted to the chord extremities, except for the San Pedro de Atacama egress point, so that we now have N = 9 data points to fit. This solution provides the most elongated shape of all those obtained in this work. It has a position angle P = 21°, an apparent oblateness ' = 0.29, an equivalent radius of R_equiv = 557 km, and $\chi ^2_{\rm min}= 15.1$ (and $\chi ^2_{\rm pdf}=3.78$), better than the nominal solution, but still indicating significant departures from the model. This can be seen by the large radial residual of about −200 km at the San Pedro de Atacama station; see Figure 4.

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The limit D_c to the diameter of a possible crater on an icy (resp. rocky) body, without shattering it, is D_{c, icy} ∼ 1.2R (resp. D_{c, rocky} ∼ 1.6R), where R is the radius of the body (Leliwa-Kopystyński et al. 2008). So, considering the derived equivalent radius, the largest possible crater has a diameter of about D_{c, icy} ∼ 670 km (D_{c, rocky} ∼ 890 km). In the right panel of Figure 4, the depression related to the San Pedro de Atacama chord has a diameter of about 430 km. However, Leliwa-Kopystyński et al. (2008) give a maximum value of 0.25 for the ratio depth/diameter of a crater in case of an H₂O–ice body, and 0.3 for a CO₂–ice body (and 0.26 for a rocky body). Thus, the 430 km crater yields a maximum depth of about 110–130 km, depending on composition. But, the depression in Figure 4 has a depth of roughly 200 km, about 45% of the crater diameter. Consequently, such a crater appears to be too deep, unless Quaoar has unexpected cohesion properties.

Another solution consists of a less oblate object, but with an important prominence on its eastern limb (Figure 5). To obtain this solution, which will be referred to as the mountain solution, we have discarded the egress extremity of Rivera, the southernmost positive chord. So, again we have N = 9 data points to fit. The elliptical fit has a position angle of 7°, an oblateness of 0.138, an equivalent radius of 525 km, and a $\chi ^2_{\rm min}=15.7$ ($\chi ^2_{\rm pdf}=3.92$), indicating a fit quality comparable to the crater solution.

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The Rivera egress chord extremity has a residual of more than 140 km, or 24% of the equatorial radius (587 km) obtained here. This can be compared to the much smaller topographic features, ∼20 km at most, expected on an icy body (Johnson & McGetchin 1973). A rocky body could support higher features, about ∼150 km, comparable to the value we obtain here. However, the density of this solution would be around 2 g cm⁻³ (see Table 3), implying a significant fraction of ice. Thus, the mountain solution seems physically implausible, as the mountain is too big to be sustained at Quaoar surface.

Although we do not expect large absolute timing errors at the various stations (Section 3.1), it is difficult to assess timing errors after the fact, as some of the observations are not repeatable. In view of the physically implausible Quaoar shape obtained in the previous section, we now allow for time shifts at each station, except San Pedro de Atacama. For this site, we have two observations with independent and consistent ingress and egress times (Table 1) so that a systematic timing error is not expected for the two corresponding occultation chords.

It is known that, in an ellipse, the midpoint of a set of parallel chords should be aligned, i.e., a straight line must go through all the midpoints of the chords. So, as a first exercise, we slide the various chords shown in Figure 3 along themselves, so as to align the middle of all chords along a line perpendicular to the chords. Thus, a circle is adjusted to the chord extremities with a radius 555 ± 2.5 km and a value $\chi ^2_{\rm min}= 1.33$.

To estimate the quality of the fit, however, we have to consider that we implicitly added four adjustable parameters to our model (the time shifts of all the chords except San Pedro de Atacama). Besides, we also adjust the radius of the circle R_circ and the coordinates of its center (f_c,g_c). So, we have M = 7 parameters to be adjusted to the N = 10 chord extremities, implying a $\chi ^2_{\rm pdf}=0.44$. The fact that the $\chi ^2_{\rm pdf}$ is satisfactory is thus an encouraging argument to proceed in our analysis.

Actually, we may now align the middle of the chords, but along a line which makes an angle α relative to the line perpendicular to the chords. Doing so, we transform the original circle (corresponding to α = 0) into an infinity of possible ellipses parameterized by α, which can vary in principle between −90° and +90°, with the convention that α is increasing from celestial north to celestial east.³⁰ We will denote by β the angle between the long axis of the ellipse and the direction of the chords (with the same sign convention as for α).

Note that since we slide the chords along their own direction, all the ellipses share the same area. Consequently, the radius of the circle R_circ described before actually gives Quaoar's equivalent radius, R_equiv = 555 ± 2.5 km, common to all the ellipses. The next step is to deduce Quaoar's actual shape in space from its apparent elliptical limb.

Quaoar has a small-amplitude rotational light curve with Δm = 0.133 ± 0.028. Its rotation period is 8.8394 ± 0.0002 hr, if it has a single-peaked rotational light curve (Ortiz et al. 2003). The period can be 17.6788 ± 0.0004 hr in case of a double-peaked light curve and can be caused by albedo features, shape effects, or a combination of the two. Quaoar's large size and density suggest a body in hydrostatic equilibrium, i.e., a Maclaurin spheroid. We note that a Jacobi-type equilibrium (Lacerda & Jewitt 2007) is not expected here because a 17.7 hr rotation period would require a very low density smaller than 0.6 g cm⁻³ and an oblateness larger than 0.42.

Consequently, we will assume from now on that Quaoar is a Maclaurin spheroid of equatorial radius a, polar radius c, and true oblateness , observed with a polar aspect angle ζ and with a rotation period of 8.8394 hr (so that the amplitude of its rotational light curve is entirely due to albedo features). Then, it can be shown from Equation (2) that Quaoar's apparent surface area is $S=\pi a^2 \sqrt{\cos ^2(\zeta) + (c/a)^2 \sin ^2(\zeta)}$, where $S= \pi R^2_{\rm equiv}$. Moreover, Quaoar's density is given by ρ = 3M/(4πa²c). Elementary calculations show that

$$\begin{equation} \left\lbrace \begin{array}{@{}l}\displaystyle \epsilon = 1 - \frac{\sqrt{(R_{\rm equiv}/a)^4 - \cos ^2(\zeta)}}{\sin (\zeta)} \\ \\ \displaystyle \rho = \frac{3M}{4\pi a^3} \cdot \frac{1}{1-\epsilon } \\ \\ \displaystyle \tan ^2(\beta) = 1 - \epsilon ^{\prime }, {\rm with\ sgn(\beta)=-sgn(\alpha),} \\ \end{array} \right. \end{equation} \tag{ 3 }$$

where the apparent oblateness ' is given as a function of the true oblateness by Equation (2). The condition 0 < (1 − )² < 1 imposes $R_{\rm equiv} < a < R_{\rm equiv}/\sqrt{|\cos (\zeta)|}$. By varying a inside this allowed interval, the first two equations above yield an infinity of solutions (ρ, ) that fit equally well the occulting chord extremities (after applying the necessary time shift to each chord). These (ρ, ) solutions are shown as blue lines in Figure 6. They define a V-shape domain inside which the actual solution must reside, with the limit corresponding to the equator-on (ζ = 90°) and pole-on (ζ = 0°) geometries.

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The blue lines are parameterized by R_equiv = 555 ± 2.5 km (obtained from our observations) and M = (1.40 ± 0.21) × 10²¹ kg, derived from Weywot's orbit (Vachier et al. 2012; Fraser et al. 2013). We note that the errors on the position of the V-shaped domain (the dashed blue lines) are dominated by the uncertainties in M, not in R_equiv.

The final condition that we impose on the solutions is that they satisfy the Maclaurin equilibrium condition

$$\begin{equation} \frac{\pi G \rho }{\omega ^2} = \frac{\sin ^2(\psi) \cdot \tan (\psi)}{2\psi \left[2+\cos (2\psi)\right] - 3\sin (2\psi)}, \end{equation} \tag{ 4 }$$

where ω is the spin frequency, G is the constant of gravitation, and cos (ψ) = 1 − (Plummer 1919; Sicardy et al. 2011). Considering the single-peak case (i.e., a rotation period of 8.8394 hr as discussed above), the equation above provides the red curve plotted in Figure 6.

The intersection between the Maclaurin equilibrium curve and the V-shaped domain finally provides Quaoar's true oblateness = 0.0897 ± 0.006 and density ρ = 1.99 ± 0.14 g cm⁻³, where the error bars come from the indetermination of ζ. Also considering the uncertainty in Quaoar's mass determination (see above), we obtain larger error bars $\epsilon = 0.0897^{+0.0268}_{-0.0175}$ and ρ = 1.99 ± 0.46 g cm⁻³.

From the oblateness, density, and equivalent radius given above, the second of Equations (3) provides Quaoar's equatorial radius $a = 569^{+24}_{-17}$ km. The third equation indicates that β must be negative, since α is positive, so β = −441 ± 09 (see the position angle of the centers of the chords in Figure 7). Since the chords are themselves inclined by 125 with respect to the east–west direction, the position angle of Quaoar's north pole projected in the plane of the sky is P = −441 + 125 = −318 ± 09 (or P = 1484 ± 09, as there is an indetermination of 180° on the pole position); see Figure 7. This position angle can be compared with that of Weywot's orbital pole projected in the plane of the sky. The various solutions derived by Vachier et al. (2012) and Fraser et al. (2013) provide a position angle between 5° and 15°. Thus, our best Maclaurin solution rules out an equatorial orbit for Weywot.

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Once R_equiv is adjusted, Equations (3) and (4) impose the values of a, , and P, leaving only the center position (f_c, g_c) to be adjusted. Thus, the Maclaurin model requires the adjustment of M = 7 parameters (four time shifts, R_equiv, and (f_c, g_c)). Our preferred fit shown in Figure 7 has $\chi ^2_{\rm min}=1.33$. Consequently, we have $\chi ^2_{\rm pdf}=0.44$, indicating that we may be slightly overestimating the error bars on the chord extremities.

This satisfactory limb fitting comes at a price, that of applying time shifts of several seconds to the chords; see the values given in Figure 7. The problem is particularly acute for the Rivera station, where we have to apply a time shift of −9.1 s. However, we note that this is the only site where the computer clock was not systematically corrected by an internet server due to the lack of internet access at that station. It was set up with respect to a GPS display, half an hour before the event. Such a procedure usually provides an accuracy of a fraction of a second. It is impossible to assess the errors after the fact, but, although improbable, we cannot exclude a time drift of the clock during the observation or even that the GPS display may not reflect the actual UTC time due to unknown acquisition problem.

The clocks at the other stations were all connected to an internet time server via Network Time Protocol. We note that the time shifts applied to the Florianópolis and Ponta Grossa chords are within the 1σ level obtained for the fitted occultation times (see Section 3.2), so the shift could be explained by the errors in the occultation timing. Concerning Armazones, the shift is at about the 2.5σ level of the occultation times, a discrepancy that is more difficult to explain. Finally, we assumed that the San Pedro de Atacama timing is the most reliable because it is confirmed by two experiments. But, although unlikely, an identical systematic error for the two experiments cannot be discarded.

It is important to note that we cannot discard intermediate solutions with topographic features still compatible with a Maclaurin body, which would demand smaller time shifts but still bigger than the 1σ level.

Considering its flux ratio to Quaoar and assuming equal albedo, Weywot has a size ratio of about 1/12 with respect to the primary (Fraser & Brown 2010). From the Quaoar size determined here, we can calculate Weywot equivalent diameter. This yields a diameter of about 90 km to the secondary. According to the ephemeris³¹ calculated by Vachier et al. (2012), the satellite shadow passed at about 800 km north of Quaoar's shadow, and about 11 minutes later. The occultation would have lasted for 5 s at most.

No secondary occultation was seen at Belo Horizonte (CEAMIG) or Pico do Dias (OPD), the two nearest sites to the predicted path. However, the observation strategy was not optimized for such a detection, since the integration time at CEAMIG was 6 s, with a readout time of 3 s (Table 1). So, if an occultation by Weywot had occurred during one exposure, then a conspicuous 2.8 mag drop would have been observed. If it had happened during a readout time interval, then a magnitude drop of 1.1 would be expected. No such dimming was detected. At OPD, the exposure time was 0.95 s, with about 1.05 s of readout time between frames. If a 5 s occultation had happened, then it would have lasted for at least two complete cycles with a drop of about 3 mag; any event like that is seen in the light curve. However, the absence of detection does not bring further constraints on the satellite orbit.

On 2012 February 17, another multi-chord stellar occultation by Quaoar was observed from Europe. Also identified in the ESO/WFI program (Assafin et al. 2012), the occulted star has magnitudes R = 15.2 and K = 11.6. Astrometric updates could only be made several months in advance, before its conjunction with the Sun. The small solar elongation just prior to the event prevented last-minute predictions. The final ICRF/J2000 star position is given in Table 4.

Table 4. Astrometric Constraints Derived from the 2012 February 17 and 2012 October 15 Occultations

Date	2012 Feb 17a	2012 Oct 15b
Time of closest approach (UT)	04:30:43.76 ± 0.11	00:41:19 ± 5
Distance of closest approach	161 ± 12 mas, N	44 ± 25 mas, S
from geocenterc
Offset in right ascensiond	−101 ± 25 mas	−110 ± 26 mas
Offset in declinationd	−92 ± 24 mas	−63 ± 26 mas
Coordinates of star	17h34m21.s8453 ± 0022	17h28m10.s1257 ± 00013
(ICRF/J2000)	−15° 42' 105860 ± 0008	−15° 36' 23324 ± 00013

Notes. ^aObtained from the circular fit. ^bAssuming a Quaoar radius of R = 535 km, the middle value of the global range of R_equiv in Table 3. ^cThe label "N" (resp. "S") means that the shadow center passed north (resp. south) of geocenter. ^dQuaoar's offset relative to the JPL#21/DE405 ephemeris.

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Fifteen sites reported attempts, but the majority was unable to observe due to poor weather or technical issues. The event was detected from four different sites (Table 5), but three sites were separated by 3 km only cross-track. This means that we have only two effective chords to fit Quaoar's limb.

Table 5. Circumstances of Observation for the Quaoar Occultation of 2012 February 17

Site	Latitude	Aperture	Exposure	Ingress (UT)
City	Longitude	Instrument	S/N	Egress (UT)
Observer	Altitude	Pixel Scale ('')
Gnosca/CHE	46° 13' 532 N	0.4 m	10.24 s	4:28:37.6 ± 4.5
S. Sposetti	09° 01' 265 E	Watec 120N+	4.4	4:29:09.4 ± 5.5
	260 m	19
TAROTa	43° 45' 073 N	0.25 m	3 × 180b s	4:28:08.1 ± 3.0
OCAc/Calern/FRA	06° 55' 251 E	CCD Andor	4.6	4:29:06.5 ± 3.5
A. Klotz and	1270 m	33
E. Frappa
	43° 51' 524 N	0.21 m	5.12d s	4:28:09.2 ± 2.1
Valensole/FRA	06° 00' 230 E	Watec 120N+	4.0	4:29:03.2 ± 2.2
J. Lecacheux	622 m	084
Tourrette-Levens/FRA	43° 47' 222 N	0.356 m	10 s	4:28:17.6 ± 2.2
	07° 15' 472 E	Apogee Alta U1	30	4:29:04.3 ± 0.5
P. Tanga	385 m	27

Notes. ^aTélescopes à Action Rapide pour les Objets Transitoires. ^bThree images obtained with drift scan method; the telescope tracking was set to 96.4% so the time resolution per pixel was 6 s pixel⁻¹. ^cObservatoire de la Côte d'Azur. ^dThe observation started with 10.24 s of integration time, but about 1 minute before the beginning of the event it was set to 5.12 s.

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The observations were performed with video and CCD cameras. Drift scan mode was used with the TAROT instrument, with a 6 s pixel⁻¹ resolution, resulting in a low accuracy chord length. At Tourrette-Levens, a sequence of CCD images was acquired, but during ingress a tracking problem occurred. Moreover, the measured chord is too short compared to the other two along the same track, possibly pointing toward a timing problem. As the stations TAROT, Tourrette-Levens, and Valensole observed the same chord, we use the occultation times given by the Valensole observation, as it presents the best accuracy among them (Table 5).

With two chords at hand, we can perform a circular fit, with radius R and center (f_c, g_c) as free parameters. This provides $R= 685^{+445}_{-155}$ km (1σ), and thus a lower limit of R_min = 530 km. The observed chords are compatible with the Maclaurin solution obtained from the 2011 May 4 event, but do not bring further constraints (Figure 10).

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This event can be used, however, to improve Quaoar's ephemeris by giving its time of closest approach to the star, its distance to the star at that moment, and its offset relative to the JPL#21/DE405 ephemeris; see Table 4.

A third stellar occultation by Quaoar was observed on 2012 October 15. The occulted star has magnitudes R = 17.3 and K = 14.4, and was also taken from the Assafin et al. (2012) catalog. Astrometric updates suggested an occultation path crossing the south of Chile. This unfavorable prediction explains why only four telescopes in Chile were scheduled for that event. The PROMPT³² instrument was the only one that could eventually monitor the star and detect the event. Two 0.4 m telescopes with identical setups were used with 8 s integration times and 1 s readout times. The acquisitions were offset by 4 s so as to overlap the effective observing intervals and thus avoid information loss caused by readout times.

The event lasted from 00:45:30 ± 2.5 to 00:45:50 ± 2.5 UT, which translates into a chord of 400 km in length. Consequently, this observation does not provide new constraints on Quaoar's size (considering the two positive detections presented above), but we may use this result for astrometric purposes. Even not knowing if the chord probed the north or the south part of Quaoar (i.e., we have two possible solutions), this uncertainty is only about 26 mas in the plane of the sky. Table 4 provides the time of Quaoar's closest approach to the star, and its angular distance to the star at that moment.