# Supplementary Information for Attosecond Delays in X-ray Molecular
Ionization

##  1 Equivalence of the phase of the photoionization matrix element and a
delay measured in angular streaking

Below, we develop a description of the effect of the phase of the dipole
matrix element on the photoelectron distribution which is measured in an
angular streaking experiment. Our analysis is performed within the strong-
field approximation (SFA) of the laser matter interaction [1]:

$b(\vec{p},\tau)=i\int_{t_{0}}^{\infty}dt\vec{E}_{X}(t-\tau)\cdot\vec{d}\left(\vec{p}-\vec{A}(t)\right)e^{-i\Phi_{V}(t)},~{}$
(1)

where $|b(\vec{p},\tau)|^{2}$ is the probability of measuring an electron with
momentum $\vec{p}$ following the combined interaction of the target system
with the electric field of the x-ray pulse ($\vec{E}_{X}$), delayed by $\tau$,
with respect to the peak of the field of the infrared laser field,
$\vec{E}_{L}=-\partial_{t}\vec{A}$.
$\vec{d}(\vec{p})=\langle\vec{p}|r|\Phi_{0}\rangle$ is the photoionization
dipole matrix element describing the x-ray ionization of the ground state
($\Phi_{0}$) with ionization potential $I_{p}$. $\Phi_{V}$ in Eq. 1 is the so-
called Volkov phase and is given by,

$\Phi_{V}(t)=\int_{t}^{\infty}dt^{\prime}\frac{1}{2}\left[\vec{p}-\vec{A}(t^{\prime})\right]^{2}-I_{p}t^{\prime}.$
(2)

In angular streaking experiments, the streaking laser field is circularly-
polarized, and can be written as,

$\vec{A}(t)=A_{0}(t)\left[\hat{x}\cos(\omega t)+\hat{y}\sin(\omega t)\right],$
(3)

where $A_{0}(t)$ is the slowly-varying envelope of the pulse and
$T=\nicefrac{{2\pi}}{{\omega}}$ is the laser period. Expanding the expression
for the Volkov phase, Eq. 2, yields

$\displaystyle\Phi_{V}(t)$ $\displaystyle=$
$\displaystyle\int_{t}^{\infty}\left(\frac{p^{2}}{2}-I_{p}\right)dt^{\prime}$
(4) $\displaystyle-$ $\displaystyle
A_{0}(t)\left[p_{x}\int_{t}^{T_{x}}dt{{}^{\prime}}\cos(\omega
t^{\prime})+p_{y}\int_{t}^{T_{y}}dt{{}^{\prime}}\sin(\omega
t^{\prime})\right],$ (5)

where we have assumed that the streaking laser field envelope is approximately
constant over a single period of the field. In addition we assume
$|\vec{p}|\gg|\vec{A}_{0}|$ and thus neglect any terms proportional to
$|\vec{A}(t)|^{2}$. We have defined a few special times ($T_{x}$ and $T_{y}$)
where the electric field ($\vec{E}_{IR}(t)=\partial_{t}\vec{A}(t)$) is
directed along the $x-$ or $y-$axis respectively:

$\displaystyle\vec{A}(T_{x})=A_{0}(T_{x})\hat{y}$
$\displaystyle\vec{A}(T_{y})=A_{0}(T_{y})\hat{x},$ (6)

and $T_{x}=T_{y}+\frac{T}{4}$. In writing Eq 5 we have used the fact that

$\displaystyle\int_{T_{x}}^{\infty}dt{{}^{\prime}}A_{0}(t^{\prime})\cos(\omega
t^{\prime})=\int_{T_{y}}^{\infty}dt{{}^{\prime}}A_{0}(t^{\prime})\sin(\omega
t^{\prime})=0.$ (7)

The expansion in Eq. 5 demonstrates the interesting property of the SFA; the
momentum shift of the electron in a streaking experiment is acquired in the
first quarter cycle of propagation, i.e. between $t$ and $(T_{x},T_{y})$. Then
we can rewrite Eq. 1 as,

$\displaystyle b(\vec{p},\tau)$ $\displaystyle=$ $\displaystyle
i\int_{t_{0}}^{\infty}dt\vec{E}_{X}(t-\tau)\cdot\vec{d}\left(\vec{p}-\vec{A}(t)\right)e^{-i\int_{t}^{\infty}(\frac{p^{2}}{2}-I_{p})}$
(8)
$\displaystyle\times\mbox{Exp}\left[iA_{0}(t)\left(p_{x}\int_{t}^{T_{x}}dt^{\prime}\cos(\omega
t^{\prime})+p_{y}\int_{t}^{T_{y}}dt^{\prime}\sin(\omega
t^{\prime})\right)\right].$

To show that the phase of the dipole matrix element can be related to a time
delay of the x-ray electric field we consider the same exemplary case as
Schultze et al. [2], namely we define a photoionization matrix element with a
phase that depends linearly on the electron energy:

$\vec{d}(\vec{p})=\hat{d}\left|{d}(\vec{p})\right|e^{-i\alpha(\vec{p})\frac{p^{2}}{2}}=\frac{\vec{p}}{\left(\vec{p}^{2}+2I_{p}\right)^{3}}e^{-i\alpha(\vec{p})\frac{p^{2}}{2}}.$
(9)

In the second equality we have assumed that magnitude of the dipole matrix
element is given by the first-order Born approximation to ionization of a
hydrogenic $1s$ system. We can write

$\displaystyle\vec{d}\left(\vec{p}-\vec{A}(t)\right)$ $\displaystyle=$
$\displaystyle\left|{d}\left(\vec{p}-\vec{A}(t)\right)\right|e^{-i\alpha\frac{\left[\vec{p}-\vec{A}(t)\right]^{2}}{2}}$
$\displaystyle\sim$
$\displaystyle\left|{d}\left(\vec{p}-\vec{A}(t)\right)\right|\mbox{Exp}\left\\{-i\alpha\left[\frac{p^{2}}{2}-A_{0}(t)\left(p_{x}\cos(\omega
t)+p_{y}\sin(\omega t)\right)\right]\right\\}$ $\displaystyle=$
$\displaystyle\left|{d}\left(\vec{p}-\vec{A}(t)\right)\right|e^{-i\alpha\frac{p^{2}}{2}}\mbox{Exp}\left[i\alpha
A_{0}(t)\left(p_{x}\int_{t}^{T_{x}}\omega\sin(\omega
t^{\prime})dt^{\prime}-p_{y}\int_{t}^{T_{y}}\omega\cos(\omega
t^{\prime})dt^{\prime}\right)\right].$

Then plugging Eq. 1 into Eq. 8 we arrive at the expression,

$\displaystyle b(\vec{p},\tau)$ $\displaystyle=$ $\displaystyle
i\int_{t_{0}}^{\infty}dt\vec{E}_{X}(t-\tau)\left|{d}\left(\vec{p}-\vec{A}(t)\right)\right|e^{-i\left(\int_{t}^{\infty}\left(\frac{p^{2}}{2}-I_{p}\right)dt^{\prime}+\alpha\frac{p^{2}}{2}\right)}$
(11)
$\displaystyle\times\mbox{Exp}\left[i\sqrt{1+(\alpha\omega)^{2}}A_{0}(t)\left(p_{x}\int_{t}^{T_{x}}\cos(\omega[t+\tilde{\tau}_{\alpha}])+p_{y}\int_{t}^{T_{y}}\sin(\omega[t+\tilde{\tau}_{\alpha}])\right)\right]$
$\displaystyle=$ $\displaystyle
i\int_{t_{0}}^{\infty}dt\vec{E}_{X}(t-\tau)\left|{d}\left(\vec{p}-\vec{A}(t)\right)\right|e^{-i\int_{t-\tau_{\alpha}}^{\infty}\left(\frac{p^{2}}{2}-I_{p}\right)dt^{\prime}}$
$\displaystyle\times\mbox{Exp}\left[i\sqrt{1+(\alpha\omega)^{2}}A_{0}(t)\left(p_{x}\int_{t-\tilde{\tau}_{\alpha}}^{T_{x}}\cos(\omega
t)+p_{y}\int_{t-\tilde{\tau}_{\alpha}}^{T_{y}}\sin(\omega t)\right)\right]$
$\displaystyle\sim$ $\displaystyle
i\int_{t_{0}}^{\infty}dt\vec{E}_{X}(t-\tau+\tau_{\alpha})\left|{d}\left(\vec{p}-\vec{A}(t+\tau_{\alpha})\right)\right|e^{-i\int_{t}^{\infty}\left(\frac{p^{2}}{2}-I_{p}\right)dt^{\prime}}$
$\displaystyle\times\mbox{Exp}\left[i\sqrt{1+(\alpha\omega)^{2}}A_{0}(t)\left(p_{x}\int_{t}^{T_{x}}\cos(\omega
t)+p_{y}\int_{t}^{T_{y}}\sin(\omega t)\right)\right]$

where
$\tilde{\tau}_{\alpha}=\frac{1}{\omega}\arctan(\alpha\omega)\sim\alpha=\tau_{\alpha}$,
for small values of $\alpha\omega$, and we have used the identities,

$\displaystyle\alpha\omega\cos(\omega t)+\sin(\omega
t)=\sqrt{1+(\alpha\omega)^{2}}\sin(\omega t+\arctan(\alpha\omega))$
$\displaystyle\alpha\omega\sin(\omega t)+\cos(\omega
t)=\sqrt{1+(\alpha\omega)^{2}}\cos(\omega t+\arctan(\alpha\omega)).$

Moreover, if $\alpha$ is small compared to the laser period
($\frac{2\pi}{\omega}$), then
$\left|{d}\left(\vec{p}-\vec{A}(t+\tau_{\alpha})\right)\right|\sim\left|{d}\left(\vec{p}-\vec{A}(t)\right)\right|$.
Comparing Eq. 11 to Eq. 8 reveals that the effect of the linear dependence of
the dipole matrix element phase is to shift the x-ray field, $E_{X}$ along the
time-axis by $\tau_{\alpha}$ and increase the apparent strength of the
streaking field by $\sqrt{1+(\alpha\omega)^{2}}$.

Figure 1: Simulated photoelectron momentum distribution in the strong field
approximation ($|b(\vec{p},\tau)|^{2}$) for a $1.3~{}\mu$m ($T=6.7$ fs) laser
field. Each panel shows the $p_{z}=0$ slice. Panel a shows the result for
$\tau=0$ fs, $\alpha=0$ fs. In panel b the arrival time of the x-ray pulse is
shifted by 700 as ($\tau=0.7$ fs, $\alpha=0$ fs). Panel c shows the result
when a linear phase of $\alpha=0.7$ fs is applied ($\tau=0$ fs, $\alpha=0.7$
fs). Panel d shows the result when the linear phase is offset by the delay of
the attosecond pulse ($\tau=-0.7$ fs, $\alpha=0.7$ fs).

We demonstrate the equivalence numerically by integrating Eq. 1 and comparing
the final electron momentum distributions. For these simulation we consider a
$300$ as (full width at half maximum) x-ray pulse with a central frequency of
$54.4$ eV and a $1.3~{}\mu$m wavelength laser field interacting with a
hydrogen atom. We analyze the change in the measured electron momentum
distribution from adding a delay of $700$ as to the arrival time of the x-ray
pulse and adding group delay of $\alpha=700$ as to the photoelectron
wavepacket. The calculated momentum distributions are shown in Fig. 1.

##  2 Modeling of Photoemission Delay

Our measurement is performed with randonly oriented molecular ensembles,
meaning the calculations need to be summed over all orientations of the
molecular axis with respect to the x-ray polarization axis. In addition we
need to sum over all out-going electron emission directions. We evaluate the
dipole matrix element for 121 orientations of the x-ray polarization with
respect to the molecular axis, sampled uniformly on a sphere. The distribution
of polar and azimuthal angles describing these orientations is shown in Fig.
2. The same set of angles is used to define 121 emission directions $\vec{k}$
of the outgoing photoelectron in the laboratory frame, and the (one-photon)
photoemission delay $\tau$ weighted over all molecular orientations $\vec{R}$
and all outgoing electron directions $\vec{k}$ is found by weighting over
cross-section according to [3]:

$\tau=\int d\hat{k}\int
d{\hat{R}}~{}\frac{|\sum_{l,m}I_{\hat{R},l,m}Y_{l,m}(\hat{k})|^{2}}{\sum_{l,m}|I_{\hat{R},l,m}|^{2}}\times\\\
\frac{\partial}{\partial E}(\sum_{l,m}I_{\hat{R},l,m}Y_{l,m}(\hat{k})).~{}$
(12)

The ground state of the nitric oxide molecule is a ${}^{2}\Pi$ state with
electronic configuration
$1\sigma^{2}~{}2\sigma^{2}~{}3\sigma^{2}~{}4\sigma^{2}~{}1\pi^{4}~{}5\sigma^{2}~{}2\pi^{1}$.
An additional cross-section-weighted average of the photoemission delay is
performed across orthogonal channels with different azimuthal quantum numbers
of the initial ($M_{i}$) and final ($M_{f}$) molecular ion and channels for
which the total (ion + electron) final state is a $\Sigma$ state with
different (+/-) symmetry.

Figure 2: Distribution of polar and azimuthal angles used to define the
orientation of the molecular axis with respect to the x-ray polarization axis.

The MCSCI calculations are performed at three different nuclear geometries: a
nuclear separation, $r_{0}$, of 1.1 Å, 1.15 Å and 1.2 Å. Figure 3 shows the
calculated cross-section (top panels) and angle- and orientation-averaged
photoemission delay (bottom panels) for the three different nuclear
geometries. It is notable that a change in bond axis length of only 5
picometers is shown to shift the energetic position of the shape resonance
(i.e. the electron kinetic energy at which the shape resonance appears) by
several electron volts. Dashed lines represent the cross-section and delay for
ionization to the triplet $\sigma(^{3}\Pi)$ state of the molecular ion and
solid lines to the singlet $\sigma(^{1}\Pi)$ state. The cross sections roughly
follow the expected statistical behavior of
$\frac{\sigma(^{3}\Pi)}{\sigma(^{1}\Pi)}\sim 3$. The sharp features visible in
the seven-channel calculation correspond to autoionizing resonances accessed
through coupling between the different ionization channels. To account for the
finite photon energy resolution in our experiment, we convolve the simulated
photoemission delay with a gaussian kernel with a full width at half maximum
of $3$ eV).These convolved results are shown in Fig. 4.

Figure 3: Angle- and orientation-averaged cross-section (top panel) and
photoemission delay (bottom panel) for the MCSCI calculations at different
internuclear separations. Figure 4: Angle- and orientation-averaged cross-
section (top panel) and photoemission delay (bottom panel) for the MCSCI
calculations at different internuclear separations. The results have been
convolved with a gaussian kernel of $\sigma$=3 eV to account for the photon
energy resolution in our measurement

We simulate the overall cross-section and photoemission delay from our
measurement by incoherently averaging the calculations for the three different
internuclear separations and the singlet and triplet states of the molecular
ion. The contribution from each geometry is weighted by the estimated
population of that geometry in the ground vibrational state of the molecule.
The weights are calculated by integrating the probability distribution shown
in Fig. 5 across the three independent regions which are closest to each of
the three geometries. This probability distribution was calculated according
to an internuclear separation $r_{0}$ of 1.15077 Å and a fundamental
vibrational constant $\omega_{e}$ of 1904.2 cm${}^{-}1$ [4].

Figure 5: Calculated population of internuclear geometries for ground state
harmonic wavefunctino with fundamenta vibrational constant $\omega_{e}$ of
1904.2 cm${}^{-}1$.

##  3 Photoemission Delay Extraction

The delay between two photoemission events is directly proportional to the
difference in the angular deviation $\Delta\theta$ the corresponding electrons
are given by the streaking laser. In the case of the measurement described
here, the oxygen and nitrogen photoelectrons experience a change in final
momentum due to the streaking field of $\theta_{O},i$ and $\theta_{N},i$ at
each XFEL shot $i$. The difference between these two angles,
$\theta_{O},i-\theta_{N},i=\delta\theta$, maps linearly to the photoemission
delay:

$\Delta\tau=\frac{\Delta\theta}{2\pi}\times T_{L}.$ (13)

The measurement of $\Delta\theta$ is complicated by the inherent temporal
jitter between the x-ray pulse and the streaking laser, which has an error
distribution of width $\sim$500 fs [5]. As a result of this temporal jitter,
$\theta_{O},i$ and $\theta_{N},i$ change across every shot. This precludes a
standard analysis scheme where the reference is held fixed. However, we can
exploit this temporal jitter to extract the difference in streaking angle
between the two photolines through correlation.

Our analysis framework is formally analyzed in an upcoming publication and was
employed in Ref. [6]. We summarize the salient details in section 4 below. We
define two radial regions on the detector, corresponding to the high momentum
portion of each of the two photolines resulting from ionization of the N and O
$1s$ orbitals. The lower limit of each region is defined by the momentum at
which the the radial gradient of the 2D projection of the unstreaked electron
distribution is maximized. The upper limit is chosen to ensure all high energy
electrons are captured. The radial position of the maximal gradient depends on
the kinetic energy of the electrons, which in turn depends on the incident
photon energy. In this measurement, we do not have access to an independent
measurement of the incoming x-ray spectrum. However, we do perform a single-
shot measurement of the energy of the electron beam which produces the x-ray
pulse. The energy of the electron beam is related to the x-ray photon energy
through the FEL resonance equation: formula [7]:

$\lambda_{r}=\lambda_{u}\frac{1+\frac{K_{u}^{2}}{2}}{2\gamma^{2}},$ (14)

where $\gamma$ is the Lorentz factor of the electron beam, $\lambda_{u}$ and
$K_{u}$ are the undulator period and strength parameters, and $\lambda_{r}$ is
the wavelength of the XFEL radiation. The energy of the electrons in the
electron beam is not monochromatic and the energy distribution within the
electron beam is particularly broad for ESASE operation [8]. The measurement
of the electron beam energy performed at the LCLS is derived from the position
of the entire electron bunch in a dispersive section of the electron beamline
and represents a measurement of the average energy within the bunch. This
means that the calculation of the x-ray photon energy from this number has an
error distribution which we have estimated from previous measurements [9, 8]
to be $\sim$3 eV at the photon energies used in this measurement. We can
estimate the average photon energy of a number of different shots with much
greater accuracy.

To define the lower limit on a single-shot basis, we first performed a fit of
the maximal value of the radial gradient for collections of shots with the
streaking laser mistimed, at different central photon energies as determined
by the electron beam energy. On each shot, we estimate the central photon
energy using the electron beam energy and Eq. 14 and use the fit to define the
lower limit for both the nitrogen and oxygen photoelectrons. We radially
integrate the electron yield measured in these two regions to produce two
angle-resolved vectors $I_{N}(\theta)$ and $I_{O}(\theta)$, which we calculate
on a shot-to-shot basis. We calculate the partial correlation maps shown in
Fig. 2 b in the main text by calculating the partial correlation coefficient
between the radially integrated intensities $I_{N}$ and $I_{O}$ at each pair
of angles $\theta_{i},\theta_{j}$, according to:

$\displaystyle\rho(I_{N},I_{O})$
$\displaystyle=\frac{\mathrm{PCov}\left(I_{N},I_{O};P\right)}{\sqrt{\mathrm{PCov}\left(I_{N},I_{N};P\right)\mathrm{PCov}\left(I_{O},I_{O};P\right)}},$
(15)

where

$\displaystyle\mathrm{PCov}\left(X,Y;I\right)=\mathrm{Cov}\left(X,Y\right)-\frac{\mathrm{Cov}\left(X,I\right)\mathrm{Cov}\left(I,Y\right)}{\mathrm{Cov}\left(I,Y\right)}$
(16)

and

$\displaystyle\mathrm{Cov}\left(X,Y\right)=\langle XY\rangle-\langle
X\rangle\langle Y\rangle.$ (17)

Here, angular brackets denote the average across multiple XFEL shots.

The partial correlation controls for the single-shot x-ray pulse intensity,
since the single-shot intensity is known to flucuate in a SASE FEL. The pulse
intensity is monitored by recording the number of electrons produced by the
x-ray pulse as it passes through a dilute gas before the beamline [10]. From
the partial correlation maps, we identify the offset of the maximum value of
the correlation coefficient from the diagonal line described by
$\theta_{N}=\theta_{O}$, in those regions of the detector where the radial
gradient dominates. As described in section 4, this offset corresponds to the
angular difference between the momentum shift of the two photolines. We
identify the offset by transforming the axes to
($\theta_{O},\pi+\theta_{O}-\theta_{N}$) and averaging across $\theta_{N}$ to
produce a one-dimensional trace. The peak of this trace corresponds to the
difference in the angular direction of the momentum shift imparted on the two
photolines by the streaking laser. We identify the peak by finding the root of
the derivative of the polynomial which fits this trace $\pm 40~{}{{}^{\circ}}$
about the maximum. The quoted error bar is the uncertainty in this root due to
the uncertainty in the fit coefficients for the polynomial.

In addition to the formal theory developed below, we simulate our analysis
scheme and verify our method of delay extraction. Appealing to the equivalence
between phase of the photoionization matrix element and a delay measured in
angular streaking described in section 1, we simulate the independent
photoelectron distributions produced by two 200 as duration x-ray pulses
separated by a delay $\Delta\tau$ and the same infrared streaking field. For
five different delay values ($\Delta\tau$ = 0 as, 200 as, 400 as, 600 as, 800
as), we simulate the photoelectron distribution for 10,000 different shots
with random phase between the streaking field and the attosecond x-ray pulses.
We identify two regions $I_{N}(\theta)$ and $I_{O}(\theta)$ with lower bound
at the maximum of the radial gradient of the averaged streaked photoelectron
distribution and calculate the correlation maps as defined in Eq. eq:sm:pcorr.
The calculated partial correlation maps are shown in Fig. 7. We simulate for
two different sets of asymmetry parameters $\beta$:
$\beta(I_{N})=\beta(I_{O})=2$ and $\beta(I_{N})=2,\beta(I_{O})=0$. These
represent the limiting cases for the photoelectron distributions measured in
our experiment (see Fig. 6: the N 1$s$ photoline has an asymmetry parameter
$\beta=2$ across the spectral region we scan, while asymmetry parameter for
the O 1$s$ photoline is measured at $\beta=0$ just above threshold and
increases with photon energy). These values are extracted from XFEL shots
where the streaking laser was intentionally mistimed from the x-ray pulse,
using the CPBASEX algorithm [11].

Figure 6: Measured photoelectron asymmetry parameters across the oxygen NEXAFS
for nitrogen (left) and oxygen (right) 1$s$ ionization.

As for the experimental data, to extract the delay from these maps as defined
in Eq. 24b, we transform the x-axis from $\theta_{N}$ to
$\pi+\theta_{O}-\theta_{N}$ and average across $\theta_{O}$, using regions
where the radial gradient dominates. The resulting traces for the different
delays are shown in panels a and b of Fig. 8. The position of the peak of
these traces corresponds to the photoemission delay, as described by Eq. 24b.
As explained above, these simulations indicate that for the inclusion of
angles $\pm$60 ∘ from the antinodes of the distribution, the additional error
accumulated due to neglecting the angular gradient is below $\sim$ 10 as at
the $2.3$ µm streaking wavelength used in our experiment. This error grows
significantly with the inclusion of angles $\sim 70{{}^{\circ}}$ from the
antinodes. Therefore, for both simulation and experiment, we use angles which
are within 60∘ of the x-ray polarization on both photolines. The simulated
delay is compared with the extracted delay in panels b and d of Fig. 8.

Figure 7: Simulated correlation maps according to Eq. 15 for simulated
photoelectron angular distributions of a $\beta(I_{N})=\beta(I_{O})=2$ and b
$\beta(I_{N})=2,\beta(I_{O})=0$. The offset of the main diagonal feature from
the line $I_{N}=I_{O}$ increases linearly with the simulated delay $\tau$.
Figure 8: Simulation of extraction of photoemission delay from correlation
maps. Panels a and c show the average over $\theta_{O}$ of the partial
correlation maps shown in panels a and b of Fig. respectively. The average has
been calculated over the boxes indicated in black in Fig. 7 and following the
coordinate transformation $\theta_{N}\rightarrow\pi+\theta_{O}-\theta_{N}$.
Panels b and d show the delay extracted from identifying the peaks of the
traces in panels a and c, respectively.

Previous measurements have been reported for the cross-section and angular
anisotropy of x-ray ionization of nitric oxide in the oxygen NEXAFS region
[12]. We compare these measurements with our extracted delay in Fig. 9.

Figure 9: Comparison of the extracted photoemission delay with O $K$-shell
ionization $\sigma$ and ionization asymmetry parameter $\beta$ measured by
Kosugi et al. [12].

##  4 Formal Theory of Correlation Analysis

The two sets of angle-resolved electron yields, $I_{N}(\theta)$ and
$I_{O}(\theta)$, correspond to the regional integral of the distribution
density $X_{N}(\vec{p}),X_{O}(\vec{p})$ where $\vec{p}$ is the two-dimensional
VMI projection of the three-dimension photoelectron momentum distribution.
These densities depend on properties of the photoemission process as well as
the vector potential $\vec{A}(t)$. When the electron emission process is much
shorter than the IR cycle, we approximate the distribution of streaked
electrons with the displacement of the unstreaked distribution by an
instantaneous IR vector potential

$X_{\alpha}(\vec{p};\vec{A})\simeq
X_{\alpha}(\vec{p}-e\vec{A}(t_{X}+\tau_{\alpha});0)~{},\quad\alpha=N,O~{},~{}$
(18)

where $t_{X}$ is the arrival time of the x-ray pulse, and $\tau_{\alpha}$ is
the additional photoemission delay of each feature. For simplicity, we denote
$\vec{k}_{\alpha}\equiv e\vec{A}(t_{X}+\tau_{\alpha})$ and $X_{0,\alpha}\equiv
X_{\alpha}(\cdot;0)$.

Expanding Eq. (18) w.r.t. the displacement and ignoring the angular gradient,
the streaked distribution density can be further approximated as

$X_{\alpha}(\vec{p};\vec{A})\approx X_{0,\alpha}(\vec{p})-\frac{\partial
X_{0,\alpha}(\vec{p})}{\partial
r}\vec{e}_{r}\cdot\vec{k}_{\alpha}=X_{0,\alpha}(\vec{p})-\frac{\partial
X_{0,\alpha}(\vec{p})}{\partial r}k\cos(\Theta_{\alpha}-\theta)~{},$ (19)

where $\vec{e}_{r}$ is the radial unit vector, $\theta$ the direction of
$\vec{p}$, $k$ the magnitude of $\vec{k}_{N}$ and $\vec{k}_{O}$, and
$\Theta_{\alpha}=\frac{2\pi}{T_{L}}(t_{X}+\tau_{\alpha})$ is the direction of
$\vec{k}_{\alpha}$ that is uniformly random across $[0,2\pi)$ due to the
arrival time jitter. Ignoring the angular gradient requires the dominance of
the radial gradient, which for a dipole electron distribution (assymetry
parameter $\beta$=2) is the case in the vicinity of the antinodes of the
distribution, and for an isotropic distribution ($\beta$=0) is the case at
every angle. Following Eq. (19), the regional yields $I_{\alpha}$ are
approximated as $I_{\alpha}(\theta)\approx
I_{0,\alpha}(\theta)-G_{r,\alpha}(\theta)k\cos(\Theta_{\alpha}-\theta)$, with
$G_{r,\alpha}$ being the regional integral of $\partial_{r}X_{0,\alpha}$. For
unstreaked distribution that is stable from shot to shot, i.e.
$I_{0,\alpha},G_{r,\alpha}$ are stable, the covariance between electron yields
should be

$\mathrm{Cov}\left(I_{N}(\theta_{N}),I_{O}(\theta_{O})\right)=G_{r,N}(\theta_{N})G_{r,O}(\theta_{O})\frac{k^{2}}{2}\cos(\theta_{N}-\theta_{O}+\Delta\theta)~{},$
(20)

with $\Delta\theta=\frac{2\pi}{T_{L}}(\tau_{O}-\tau_{N})$ as defined in the
main text Eq. (2), and the Pearson correlation should be simply
$\cos(\theta_{N}-\theta_{O}+\Delta\theta)$ because the $G_{r,\alpha}$ cancel
out.

In the experiment the unstreaked distributions also fluctuate with the single-
shot x-ray pulse properties, so the model of total covariance decomposes [13]
as

$\mathrm{Cov}\left(I_{N},I_{O}\right)=\mathrm{Cov}\left(\mathbb{E}\left[I_{N}|\vec{k}\right],\mathbb{E}\left[I_{O}|\vec{k}\right]\right)+\mathbb{E}\left[\mathrm{Cov}(I_{N},I_{O}|\vec{k})\right]~{},$
(21)

with $I_{\alpha}$ evaluated at the respective angular bins $\theta_{\alpha}$.
The second term in Eq. (21) is approximately the covariance of unstreaked
distributions, which can be most removed by taking partial covariance [14]
with respect to the pulse energy, leaving the first term, up to a global
scaling factor $s_{P}$:

$\displaystyle\mathrm{PCov}\left(I_{N}(\theta_{N}),I_{O}(\theta_{N});P\right)$
$\displaystyle\approx\mathrm{Cov}\left(\mathbb{E}\left[I_{N}|\vec{k}\right],\mathbb{E}\left[I_{O}|\vec{k}\right]\right)s_{P}$
(22)
$\displaystyle\approx\bar{G}_{r,N}\bar{G}_{r,O}\frac{k^{2}}{2}\cos(\theta_{N}-\theta_{O}+\Delta\theta)s_{P}~{}.$
(23)

Similar to the radial gradients, this global scaling factor is cancelled in
the partial correlation calculation:

$\displaystyle\rho(\theta_{N},\theta_{O};P)$
$\displaystyle=\frac{\zeta(I_{N}(\theta_{N}),I_{O}(\theta_{O}))-\zeta(I_{N}(\theta_{N}),P)\zeta(I_{O}(\theta_{O}),P)}{\sqrt{(1-\zeta^{2}(I_{N}(\theta_{N}),P))(1-\zeta^{2}(I_{O}(\theta_{O}),P))}}$
(24a)
$\displaystyle=\frac{\mathrm{PCov}\left(I_{N},I_{O};P\right)}{\sqrt{\mathrm{PCov}\left(I_{N},I_{N};P\right)\mathrm{PCov}\left(I_{O},I_{O};P\right)}}\approx\cos(\theta_{N}-\theta_{O}+\Delta\theta)~{}.$
(24b)

The result is that, for regions of the detector where the radial gradient
dominates the 2D projection of the photoelectron distribution, the numerical
modeling indicates that the additional error accumulated due to neglecting the
angular gradient is below $\sim$ 10 as for the $2.3$ µm streaking wavelength
used in this work, for the inclusion of angles $\pm$60 ∘ from the antinodes of
the distribution.

Figure 10: Number of XFEL shots recorded at each central photon energy. In
addition to the NEXAFS region of interest examined in this work, it is also
possible to record the photoemission delay at $\sim$40 eV and $\sim$60 eV
above the oxygen $K$-edge.

##  5 Experiment

The geometry of the c-VMI spectrometer differs from a standard VMI
spectrometer because the electric field produced by the electrostatic lens
plates projects the electrons onto the position-sensitive detector along the
same direction as the propagation of the incident laser pulses, as shown in
Fig. 1 of the main paper. A 6.4 mm hole at the center of the MCP allows both
the x-ray and infrared light to pass through the detector after the
interaction point. This geometry optimizes the measurement resolution of the
photoelectron momentum distribution in the polarization plane of the
circularly polarized laser pulse [15]. The thick-lens design of the c-VMI
supports detection of electrons of kinetic energies up to a few hundred
electron volts, with energy resolution of a few percent. The c-VMI is
installed in the experimental end station inside a 10-inch vacuum chamber. The
sample was delivered through a pulse supersonic Even-Lavie gas jet, consisting
of a 2 mm diameter skimmer (Beam Dynamics, Model 2) $\sim$120 mm from the
interaction region. The skimmed molecular beam intersects with the x-ray pulse
and the streaking laser pulse in the interaction region between the repeller
and extractor electrodes in the c-VMI.

###  5.1 Streaking Laser

The IR streaking laser pulse was generated using a commercial optical
parametric amplifier (OPA, TOPAS-HE), pumped with $10$ mJ, $\sim 50$ fs, $800$
nm pulses from a Ti:sapphire laser system. The resultant $\sim 100~{}\mu$J
idler pulses were tuned to be centered at 2.3 $\mu$m, and separated from the
signal pulse with a dichroic beamsplitter. The beam polarization is set by a
broad bandwidth quarter-waveplate (Thorlabs), after which it is focused with
an $f=750$ mm lens. The focused beam is reflected from a dichroic mirror (HR:
2400 nm / T: visible) before being coupled into the vacuum chamber. The IR
laser is coupled into the interaction point by a holey mirror which the x-rays
pass through, allowing the IR beam and the x-ray pulse to co-propagate to the
interaction point. The spectrum of the streaking laser pulse, recorded
immediately before it is coupled into the vacuum chamber, is shown in Fig. 11
d.

We adjusted the intensity of the streaking laser with an iris just before the
focusing lens. We estimate the streaking laser intensity at the interaction
point to be $\sim 5.4\times 10^{12}$W/cm2, which was set so that the streaking
laser produced $\leq 1$ electrons/shot due to above-threshold ionization of
the sample.

We characterized the polarization of the streaking laser by increasing the
power at the interaction region and characterizing the resultant electron
distribution produced by above-threshold ionization. Figure 11 a shows the
photoelectron spectrum generated by above-threshold ionization (ATI) of NO
molecules using the increased IR intensity and the same quarter-wave plate
rotation as in the streaking measurements.

Thanks to the high nonlinearity of the ATI interaction, the angular anisotropy
of the resultant photoelectron distribution enables accurate determination of
the streaking field ellipticity. To characterize the polarization, we radially
integrate the photoelectron spectrum in the region defined by the red circles
in Fig. 11 a and plot this yield as a function of angle in panel d. We fit
this curve to a sinusoidal curve with varying amplitude, phase offset and DC
offset. This allows us to extract the ratio of the ionization rate along the
major and minor axes of the polarization. We compare the angular maxima and
minima in photoelectron yield with ADK (Ammosov-Delone-Krainov) predictions of
tunnel ionization rates [16] and identify an ellipticity of $\geq$0.95.

Figure 11: a Above-threshold ionization of nitric oxide by the 2.3 $\mu$m
laser field used for the streaking measurement. The region marked by the
concentric red circles is radially integrated and this angle-resolved yield is
plotted in panel b. The background (marked by the magenta circles) has been
subtracted. From this angle-resolved ATI yield, we estimate the contrast ratio
between the major and minor axes of the laser polarization in our measurement
and determine the ellipticity by simulating this ratio using ADK theory for
different laser intensities (panel c). The dashed vertical line shows the
ratio extracted from b. d Streaking laser intensity measured immediately
before coupling into the vacuum chamber.

##  6 Coulomb Laser Coupling

The Coulomb potential is long-range and its presence affects the behavior of
the electron driven by the streaking field. For electrons with a asymptotic
kinetic energy of several electron volts, it is well understood that the
additional contribution to the measured photoemission delay due to this
Coulomb-laser interaction can be independently calculated and added to the
Wigner delay calculated in section 2. This is not the case for very low energy
electrons, where the terms are no longer straightforwardly separable. This is
a significant contributing factor to the divergence of our calculations from
the measured photoemission delay at electron kinetic energies $\leq$2 eV.

This additional delay accrued through the interaction of the electron with the
streaking field in the presence of the Coulomb potential is often referred to
as the Coulomb-laser coupling (CLC), or continuum-continuum delay. This delay
can be understood as a classical effect and successfully quantified by a
classical Monte-Carlo trajectory (CTMC) simulations [17, 18, 19]. It has been
well-studied for the case of linear polarization of the streaking laser, but
exhibits a strong dependence on the angle between the electric field of the
streaking field at the time of ionization and the final position of the
electron on the detector, and thus requires careful consideration for the case
of a circularly polarized streaking field. We perform a CTMC simulation to
calculate the CLC delay for the streaking laser parameters used in our
experiment. We incorporate this additional delay by adding it to the
calculated photoemission delay described above.

We can resolve the delay for trajectories with different angles between the
laser field at the time of ionization and final position of the electron on
the detector. As shown in Fig. 12, for electron trajectories along the
polarization axis of the streaking laser, we find the CTMC calculation
converges to the analytical result of Pazourek et al. [17]. However since we
average over several emission angles in the measurement, we must calculate a
new CLC contribution, which is shown in the right panel of Fig. 12.

Figure 12: (left) CLC delay as a function of angle between final electron
position at detector and electric field at x-ray pulse arrival time. For
electrons with trajectory along the polarization axis, the values converge to
those of Pazourek et al.. (right) angle-integrated delay calculated withing
CTMC framework, compared with delay from Pazourek et al.

##  7 Post-Collisional Interaction

Figure 13: a Schematic of Auger-Meitner decay following oxygen 1$s$
ionization. b For the slow electron, the charge of the doubly charged ion
following Auger-Meitner emission is shielded by the Auger-Meitner electron
until time $t=\tau$, when the Auger-Meitner electron overtakes the slow
electron.

The oxygen $K$-shell photoelectron acquires an additional delay due to its
post-collisional interaction with a fast photoelectron produced through Auger-
Meitner decay of the core-ionized NO. This process is illustrated in panel a
of Fig. 13. X-ray ionization removes an electron from the core 1$s$ orbital of
the oxygen atom in the molecule. In our experiment, this photoionizied
electron has energies between $\sim$ 1 eV and $\sim$60 eV. We label this as
the ‘slow’ electron. The core-level vacancy in the molecular cation is highly
unstable and it subsequently decays on a short (few femtosecond) timescale.
For light elements, the dominant decay process for a core-ionized system is
non-radiative Auger-Meitner decay, where a higher-lying electron in the
valence manifold fills the core vacancy and energy is conserved by the
emission of a second valence electron. This emitted electron is the Auger-
Meitner electron, and it has kinetic energy equal to the difference between
the binding energies of the initial $1s$ orbital and the valence electron
which fills the vacancy, less the double ionization potential of the final
doubly charged final state. For oxygen $1s$ ionization, this kinetic energy is
around $450$ eV. We label this Auger-Meitner electron as the ‘fast’ electron.

Panel b depicts the post-collisional interaction between the slow and fast
electrons. The three panels show different times following the emission of the
Auger-Meitner electron. In our experiment, we measure the photoemission delay
of the slow electron. Immediately following the emission of the Auger-Meitner
electron, the slow electron is further from the molecular ion because it was
emitted before the Auger-Meitner decay took place. The Auger-Meitner electron
provides a charge screening to the slow photoelectron, which is thus emerging
from an effective Coulomb potential of charge $q_{eff}$ = 1 +. This shielding
can be approximately described by a Gauss sphere which grows in radius with
the fast electron, and is depicted in panel b as a dashed circle. At time
$t=\tau$, the photoelectron and Auger-Meitner electron cross because the Auger
electron is moving faster than the photoelectron. At times $t>\tau$, the
Auger-Meitner electron has overtaken the slow photoelectron, which no longer
enjoys the screening and thus emerges from an effective Coulomb potential of
charge $q_{eff}$ = 2 +.

In some systems, the effect of this change in the potential the electron is
emerging from can result in the recapturing of the slow photoelectron by the
molecular ion [20, 21]. The interaction also affects the lineshape (i.e.
kinetic energy distribution) of Auger-Meitner electrons. This change in
lineshape was successfully calculated for the case of inner shell ionization
of argon and xenon by the classical model of Russek and Mehlhorn [22]. We use
this model to estimate the difference in propagation time between two
electrons of the same asymptotic kinetic energy emerging from an ionic
potential: both are now traveling in a 2+ Coulomb potential, but one of the
electrons began its propagation in a 1+ potential prior to the post-
collisional interaction. To calculate this time difference, we find the time
$\tau$ at which the slow electron and the fast Auger-Meitner electron cross,
illustrated by the central panel in Fig. 13 b. This takes place when the two
electrons are at the same radius, $\rho$, from the molecular ion. If the
Auger-Meitner electron is emitted at time $t$, the time at which it overtakes
the slow electron is

$\displaystyle\tau(t;E_{P},E_{A})$
$\displaystyle=\int_{R_{i}}^{\rho}\frac{dr}{\sqrt{2(E_{P}+1/r)}}$ (25)
$\displaystyle=t+\int_{R_{i}}^{\rho}\frac{dr}{\sqrt{2(E_{A}+2/r)}}$ (26)

Once the electrons have crossed, the final eKE of the photoelectron drops by
$\Delta\simeq 1/\rho$, $E_{Pf}=E_{P}-\Delta$. Therefore, for a given final
position $R_{f}$ and final photoelectron energy $E_{Pf}$, the elapsed time is

$\displaystyle
T(E_{Pf},\rho)=\int_{R_{i}}^{\rho}\frac{dr}{\sqrt{2(E_{Pf}+\Delta+1/r)}}+\int_{\rho}^{R_{f}}\frac{dr}{\sqrt{2(E_{Pf}+2/r)}}$
(27)

and for the reference photoelectron that always propagates through a 2+
potential, this time is

$T(E_{Pf},\rho=0)=\int_{R_{i}}^{R_{f}}\frac{dr}{\sqrt{2(E_{Pf}+2/r)}}$

Therefore, the additional delay accrued by the slow photoelectron due to the
post-collisional interaction is

$\displaystyle\Delta T(E_{Pf},\rho)$
$\displaystyle=\int_{R_{i}}^{\rho}dr\left(\frac{1}{\sqrt{2(E_{Pf}+1/\rho+1/r)}}-\frac{1}{\sqrt{2(E_{Pf}+2/r)}}\right)$
(28)
$\displaystyle=\left(I(E_{Pf}+\Delta,1,1/r)-I(E_{Pf},2,1/r)\right)\Big{|}_{r=R_{i}}^{\rho}$
(29)

Solving for $\rho(t,E_{Pf},E_{A})$ according to

$\displaystyle\int_{R_{i}}^{\rho}\frac{dr}{\sqrt{2(E_{Pf}+1/\rho+1/r)}}=t+\int_{R_{i}}^{\rho}\frac{dr}{\sqrt{2(E_{A}+2/r)}}$
(30)

and using the following indefinite integral from Eq. (18$b$) in Ref. [22]:

$\displaystyle
I({E},Z,S)=\frac{({E}+ZS)^{1/2}}{2^{1/2}{E}S}+\frac{Z}{(2{E})^{3/2}}\ln\left(\frac{(1+ZS/{E})^{1/2}-1}{(1+ZS/{E})^{1/2}+1}\right)$
(31)

yields the final additional photoemission delay for a slow photoelectron of
asymptotic kinetic energy $E_{Pf}$, a fast Auger-Meitner electron of
asymptotic kinetic energy $E_{A}$ and emission time $t$. For our measurement,
the additional photoemission delay is averaged over different Auger-Meitner
emission times corresponding to the decay profile shown in dotted black in
Fig. 14, with a decay constant $\Gamma$ of 3.9 fs. We use an Auger-Meitner
electron kinetic energy of 450 eV. The dependence of the additional
photoemission delay on Auger-Meitner emission time for several different slow
photoelectron kinetic energies is by the solid colored lines.

Figure 14: Decay profile for Auger-Meitner decay with decay constant
$\Gamma$=3.7 fs (black dotted line). Dependence of additional photoemission
delay due to post-collisional interaction on Auger-Meitner emission time, for
different asymptotic kinetic energy of the slow photoelectron (solid colored
lines).

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11institutetext: Federal University of Technology - Paraná (PPGFA/UTFPR-
Curitiba), Rua Sete de Setembro, 3165, Curitiba, PR, Brazil, 22institutetext:
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ.
Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon,
France, 33institutetext: Laboratório Interinstitucional de e-Astronomia -
LIneA & INCT do e-Universo, Brazil, 44institutetext: Observatório
Nacional/MCTIC, R. General José Cristino 77, Bairro Imperial de São Cristóvão,
Rio de Janeiro, RJ, Brazil, 55institutetext: Instituto de Astrofísica de
Andalucía, IAA-CSIC, Glorieta de la Astronomía s/n, 18008 Granada, Spain,
66institutetext: Institut Polytechnique des Sciences Avancées IPSA, 63
boulevard de Brandebourg, F-94200 Ivry-sur-Seine, France, 77institutetext:
Institut de Mécanique Céleste et de Calcul des Éphémérides, IMCCE,
Observatoire de Paris, PSL Research University, CNRS,Sorbonne Universités,
UPMC Univ Paris 06, Univ. Lille, 77 Av. Denfert-Rochereau, F-75014 Paris,
France, 88institutetext: Planetary Science Institute, 1700 East Fort Lowell
Road, Suite 106, Tucson, AZ 87519 99institutetext: Universidade Estadual de
Ponta Grossa, O.A. - DEGEO, Av. Carlos Cavalcanti, 4748, Ponta Grossa
84030-900, PR, Brazil, 1010institutetext: Observatório do Valongo/UFRJ,
Ladeira Pedro Antônio 43, Rio de Janeiro, RJ, Brazil, 1111institutetext:
Institute of Physics, Federal University of Uberlândia, Uberlândia, MG,
Brazil, 1212institutetext: European Asteroidal Occultation Network (EAON)
1313institutetext: International Occultation Timing Association - European
Section (IOTA-ES), 1414institutetext: Observatoire de Nandrin, Société
Astronomique de Liège, Avenue des Platanes 17, B 4000 Liège (Cointe), Belgium,
1515institutetext: Ciel et Espace, Paris, France, 1616institutetext:
University of the Free State, 205, Nelson Mandela Dr, Park West, Bloemfontein,
9301, South Africa, 1717institutetext: Observatorio astronómico de Forcarei
OAF 36556 Forcarei Pontevedra, Spain, 1818institutetext: Planetarium Ludiver,
1700, rue de la Libértion, Tonneville, 50460 La Hauge, France,
1919institutetext: Landessternwarte, Universität Heidelberg, Königstuhl,
D-69117 Heidelberg, Germany.

# Constraints on (2060) Chiron’s size, shape, and surrounding material from
the November 2018 and September 2019 stellar occultations

F. Braga-Ribas, 112233<EMAIL_ADDRESS>C. L. Pereira 443311 B. Sicardy 22
J. L. Ortiz 55 J. Desmars 6677 A. Sickafoose 88 M. Emilio 99 B. Morgado 1010 3
3 G. Margoti 11 3 3 F. L. Rommel 1133 J. I. B. Camargo 4433 M. Assafin 101033
R. Vieira-Martins 4433 A. R. Gomes-Júnior 111133 P. Santos-Sanz 55 N. Morales
55 M. Kretlow 551212 J. Lecacheux 22 F. Colas 77 R. Boninsegna 1212 O.
Schreurs 13131414 J. L. Dauvergne 1515 E. Fernandez 13131414 H. J. van Heerden
1616 H. González 1717 D. Bihel 1818 F. Jankowsky 1919

(Received 02/05/2023; accepted 15/06/2023)

###### Abstract

Context.  After the discovery of rings around the largest known Centaur
object, (10199) Chariklo, we carried out observation campaigns of stellar
occultations produced by the second-largest known Centaur object, (2060)
Chiron, to better characterize its physical properties and presence of
material on its surroundings.

Aims.  We aim to provide constraints on (2060) Chiron’s shape for the first
time using stellar occultations. We investigate the detectability of material
previously observed in its vicinity using the 2018 occultation data obtained
from South Africa Astronomical Observatory (SAAO).

Methods.  We predicted and successfully observed two stellar occultations by
Chiron. These observations were used to constrain its size and shape by
fitting elliptical limbs with equivalent surface radii in agreement with
radiometric measurements. We also obtained the properties of the material
observed in 2011 with the same technique used to derive Chariklo’s ring
properties in our previous works, used to obtain limits on the detection of
secondary events in our 2018 observation.

Results. Constraints on the (2060) Chiron shape are reported for the first
time. Assuming an equivalent radius of $R_{\rm equiv}=105^{+6}_{-7}\,\rm{km,}$
we obtained a semi-major axis of $a=126\pm 22\,\rm{km}$. Considering Chiron’s
true rotational light curve amplitude and assuming it has a Jacobi equilibrium
shape, we were able to derive a 3D shape with a semi-axis of $a=126\pm 22$ km,
$b=109\pm 19$ km, and $c=68\pm 13$ km, implying in a volume-equivalent radius
of Rvol= 98 $\pm$ 17 km. We determined the physical properties of the 2011
secondary events around Chiron, which may then be directly compared with those
of Chariklo rings, as the same method was used. Data obtained from SAAO in
2018 do not show unambiguous evidence of the proposed rings, mainly due to the
large sampling time. Meanwhile, we discarded the possible presence of a
permanent ring similar to (10199) Chariklo’s C1R in optical depth and
extension.

Conclusions. Using the first multi-chord stellar occultation by (2060) Chiron
and considering it to have a Jacobi equilibrium shape, we derived its 3D
shape, implying a density of $1,119\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 4\leavevmode\nobreak\ \rm{kg\leavevmode\nobreak\
m}^{-3}$. New observations of a stellar occultation by (2060) Chiron are
needed to further investigate the material’s properties around Chiron, such as
the occultation predicted for September 10, 2023.

###### Key Words.:

comets: individual: (2060) Chiron; minor planets, asteroids: general; planets
and satellites: rings; Kuiper belt: general

##  1 Introduction

After discovering rings around the Centaur object (10199) Chariklo, it was
proposed that this feature may be common among distant small Solar System
objects (Braga-Ribas et al., 2014). The centaur object (2060) Chiron is the
second largest object of its kind. It has an equivalent diameter of
210${}^{+11}_{-14}$ km (model dependent), measured using data from the space
telescopes Spitzer and Herschel, as well as from the Atacama Large Millimeter
Array (ALMA) (Lellouch et al., 2017).

Fundamental properties such as size and shape for objects of this kind can be
obtained from Earth-based observations using stellar occultations (Ortiz et
al., 2020) 111An updated list of the detected stellar occultations by outer
Solar System objects that we are aware of can be found at
http://occultations.ct.utfpr.edu.br/results (Braga-Ribas et al., 2019). High-
cadence observations in terms of time and high signal-to-noise ratio (S/N)
allow for detecting or setting upper limits to the presence of material around
the occulting body, for instance, on dust shells, jets, or rings (Elliot et
al., 1995; Ortiz et al., 2020).

Secondary events on previous stellar occultation by (2060) Chiron have been
reported for events in 1993, 1994, and 2011 (Bus et al., 1996; Elliot et al.,
1995; Ruprecht et al., 2015). They were all interpreted as dust or jets coming
from Chiron’s surface, as it has presented periods of cometary activity
(Tholen et al., 1988; Belskaya et al., 2010). In 1989, Meech & Belton (1989)
were the first to detect a comma around Chiron. Later, Luu & Jewitt (1990)
analyzed 1988 to 1990 data, further investigating Chiron’s activity. As far as
we know, no non-gravitational acceleration has been detected so far.

The November 29, 2011 occultation was observed using the Infrared Telescope
Facility (IRTF), which detected the main body event, and from Faulkes
Telescope North (FTN), which had a much higher cadence and S/N, but with no
detection of the main body (Ruprecht et al., 2015). The latter data presented
two pairs of sharp secondary events that resembled those observed around
Chariklo in 2013. These data were further analyzed by Sickafoose et al.
(2020).

Ortiz et al. (2015) analyzed the reported absolute magnitude variation and the
decrease of the amplitude of its rotation light curve, together with the
secondary events relative to Chiron positions detected thus far, to argue that
Chiron may also have a ring system. The final word on the presence of rings
around Chiron depends on new observations of a stellar occultation with high
cadence and S/N or on a visit of a spacecraft such as $Centaurus$ (Singer et
al., 2019).

In this work, we presented the first multi-chord stellar occultation by
Chiron, observed on September 8, 2019, from four different sites. It gives
further constraints on Chiron’s size and shape. We also present a single chord
detection observed from South Africa Astronomical Observatory (SAAO) on
November 28, 2018, setting an upper limit on the detection of secondary
events.

##  2 Stellar occultation of September 08, 2019

This stellar occultation event was predicted by the Lucky-Star
project222https://lesia.obspm.fr/lucky-star/ using the Numerical Integration
of the Motion of an Asteroid (NIMA), as described in Desmars et al. (2015),
and the second Gaia Data Release (DR2) catalog (Gaia Collaboration et al.,
2018). The prediction path was updated using the astrometric position obtained
at Pico dos Dias and Calar Alto Observatories and the astrometric position
derived from the November 2018 occultation event (see Sect. 3).

The occulted star has the number 2741744913738549760 on the third Gaia Data
Release (DR3) with a G magnitude of 16.52. The event had a shadow velocity of
22.8 $\rm{km\,s}^{-1}$, relative to the geocenter. The position of the star
from the Gaia DR3 catalog was propagated using parallax and proper motion to
the event date, resulting in the following geocentric position at the epoch:

$\alpha=00h\leavevmode\nobreak\ 10m\leavevmode\nobreak\ 12s.74303\pm
0.1600\leavevmode\nobreak\ {\rm mas,}$ $\delta=+04^{\circ}\leavevmode\nobreak\
37^{\prime}\leavevmode\nobreak\ 04^{\prime\prime}.9141\pm
0.2824\leavevmode\nobreak\ {\rm mas},$

where mas stands for milliarcsec.

An observation campaign was organized and observations were eventually made
from five sites. The observation setups can be found in Table 1. Two sites in
France (Ludiver and Paris) and two in Belgium (Nandrin and Dourbes) positively
detected the event (Fig. 1). Unfortunately, the Belgian sites were in the same
line with respect to Chiron’s shadow path. Therefore, even if both were used,
only three effective chords were available to derive the object’s visible limb
at the event time. Observations from Forcarei/ES were negative, that is, it
did not detect the occultation by Chiron since it was south of the actual
shadow path.

Table 1: Observational circumstances of the September 8, 2019 Chiron
occultation.

Site | Longitude E | Latitude N | Elevation | Telescope | Camera | Cycle† | $\sigma_{flux}$ | Observers
---|---|---|---|---|---|---|---|---
(m) | aperture (mm) | time (s)
Nandrin/BE | 05 26 29.5 | 50 31 24.8 | 261 | 406 | Watec-910HX | 2.56 | 0.16 | O. Schreurs,
E. Fernandez
Dourbes/BE | 04 34 56.0 | 50 05 25.9 | 195 | 400 | Watec-910HX | 1.28 | 0.23 | R. Boninsegna
Ludiver/FR | -01 43 42.1 | 49 37 46.9 | 179 | 400 | RAPTOR Kite | 0.50 | 0.42 | F. Colas, D. Bihel,
J. Lecacheux
Paris/FR | 02 20 26.5 | 48 49 26.9 | 79 | 250 | ASI174MM | 1.50 | 0.33 | J. L. Dauvergne
Forcarei/ES | -08 22 15.2 | 42 36 38.4 | 670 | 507 | Watec-910HX | 2.56 | 0.11 | H. González

333†Read-out time is negligible for these cameras, so the exposure time equals
the cycle time.

Figure 1: Post-diction map of the event observed on September 8, 2019. The
arrow at the bottom right indicates the motion of the shadow. The black lines
represent Chiron’s shadow path, the blue dots are the sites that detected the
occultation, and the red one shows where data were acquired but no event was
detected.

The occultation light curves (see Appendix 15) were obtained using
differential aperture photometry using the Package for the Reduction of
Astronomical Images Automatically (PRAIA) (Assafin et al., 2011). The ingress
and egress times were calculated by modeling the light curve considering a
sharp-edge occultation model convolved with Fresnel diffraction, the apparent
stellar diameter at the object’s distance, and the finite exposure time of
each data set (more details can be found in Braga-Ribas et al., 2013; Souami
et al., 2020). The shortest integration time was 0.5 seconds at the Ludiver
site, translating into 11.4 km per data point along the track. Considering
that the stellar apparent diameter equals 0.08 km, estimated using its B, V,
and K magnitudes and van Belle (1999) equations for a main sequence star, and
the Fresnel scale at $D=17.85$ au (astronomical units) is $L_{f}=\sqrt{\lambda
D/2}=0.932\leavevmode\nobreak\ $km, the light curves are smoothed mainly by
the exposure time.

The timings of the event are presented in Table 2. Time-stamping was ensured
to be referenced to the Universal Time Coordinate (UTC) on the order of a
millisecond by Global Positioning System (GPS) devices for all stations,
except at Paris/FR, which had no reliable time source. The observation at
Planetarium Ludiver/FR was made using a Raptor CMOS Kite camera and TimeBox
system (Gallardo et al., 2021) and registered the images in ser444Simple
uncompressed video format for astronomical capturing: http://www.grischa-
hahn.homepage.t-online.de/astro/ser/ format. Nandrin/BE site used a
GPSBoxSprite-VTI, and Dourbes/BE site used an International Occultation Timing
Association Video Time Inserter (IOTA-VTI) system555https://occultations.org/,
both with Watec 910HX video cameras, and the avi video format was used for
registering the images. Paris/FR used a ZWO ASI174MM camera and registered the
event using ser file format. All the data sets were converted to individual
fits frames using Tangra software version
3.7.3666http://www.hristopavlov.net/Tangra3/. The video GPS times are inserted
at the end of the video camera exposures, so the time of individual frames
were subtracted by half an exposure time to give the time of the middle of the
integration correctly (details in Benedetti-Rossi et al., 2016; Barry et al.,
2015).

Table 2: Event times and chord length of the four positive sites on September
8, 2019 occultation.

Site | Ingress (UTC) | Egress (UTC) | Chord length (km)
---|---|---|---
Dourbes/BE | 23:04:12.10 $\pm$ 0.42 | 23:04:17.79 $\pm$ 0.70 | 138 $\pm$ 13
Nandrin/BE | 23:04:08.00 $\pm$ 0.43 | 23:04:13.40 $\pm$ 0.92 | 146 $\pm$ 28
Ludiver/FR | 23:04:27.96 $\pm$ 0.22 | 23:04:33.94 $\pm$ 0.42 | 135 $\pm$ 11
Paris/FR† | 23:04:20.39 $\pm$ 0.56 | 23:04:25.47 $\pm$ 0.79 | 116 $\pm$ 22
777†Times not shifted, see text.

Figure 2: Occultation chords in blue, with error bars in red, observed on
September 08, 2019. From north to south, we have the chords from Ludiver/FR,
Nandrin, and Dourbes/BE (same line), and Paris/FR. Gaia DR3 propagated star
position and NIMAv10 provide the f0 and g0 during the geocenter’s closest
approach.

###  2.1 Fit of the elliptical limb

The corrected times were projected in the sky plane and used to determine
Chiron’s limb shape at the time of the occultation. It is possible to see in
Fig. 2 that the Paris chord is clearly shifted with respect to the other
sites. As it did not benefit from a reliable time source, we shifted it
forward by 0.9 seconds by minimizing the dispersion of the fitted ellipses
with respect to the chord extremities. Due to the distribution and errors of
the detected chords, a wide range of ellipses can be fitted.

The most recent and accurate work on determining Chiron’s size, available in
the literature, combined thermal fluxes with mid/far infrared fluxes to derive
Chiron’s relative emissivity at radio (mm and sub-mm) wavelengths (Lellouch et
al., 2017). We considered their spherical approach, which gives an area-
equivalent diameter of 210${}^{+11}_{-14}$ km, as we understand it to be the
most conservative one. For instance, the error bar of the size given by this
model encloses all the other scenarios (e.g., different roughness and
elliptical profile). We used the aforementioned equivalent diameter as upper
and lower limits on our fitting procedure to restrict the range of possible
elliptical limbs compatible with our observed chords.

Our elliptical limb shape fitting to the occultation chords is detailed in
Braga-Ribas et al. (2013) and Souami et al. (2020), for example. In short, we
generated a large set of ellipses, varying their semi-major axes $a^{\prime}$
from 90 to 160 km by steps of 1 km, their oblatenesses
$\epsilon=(a^{\prime}-b^{\prime})/a$ (where $a^{\prime}$ and $b^{\prime}$ are
the semi-major and semi-minor axis of the apparent ellipse) from 0 to 0.625 by
steps of 0.005, and their Position Angles (PA)888The Position Angle is the
angle between the celestial north and the minor-axis orientation, measured
positively to the east. from -90∘ to +90∘ by steps of 2∘. Moreover, in this
procedure, we imposed that the area-equivalent radius, $R_{\rm
equiv}=a\sqrt{1-\epsilon}$, of the ellipse be consistent with the Lellouch et
al. (2017)’s value within the error domain mentioned above, that is, $98\leq
R_{\rm equiv}\leq 111$ km.

For each ellipse, the free parameters of the fit were the coordinates of body
center $(f\rm_{c},g\rm_{c})$ projected in the sky plane. This fit returned the
classical $\chi^{2}$ value based on the radial residuals of the fit, that is,
the radial distance of the chord at the ingress/egress, $r_{i,obs}$, to the
tested ellipse, $r_{i,cal}$, proportional to the radial uncertainty,
$\sigma_{radial}$, of the ingress/egress time (Eq. 1). The results were sorted
from the minimum value $\chi^{2}_{\rm min}$, and for all the ellipses up to
$\chi^{2}+1$ to define the marginal 1$\sigma$ error bar of each parameter. The
results are displayed in Fig. 3 for various parameters of interest.

$\chi^{2}=\sum_{\rm i=1}^{\rm
N}\frac{({r_{i,obs}}-{r_{i,cal}})^{2}}{{\sigma_{radial}}^{2}}.$ (1)

Table 9 summarizes the results of the fit (limb’s size, shape, and
orientation) and Fig. 4 displays the best ellipse in black and the
1$\sigma$-level solutions in gray.

Figure 3:  Combinations of the limb-fitting parameters shown in each panel.
The 1$\sigma$-level solutions are plotted in black. The solutions above this
limit are shown in different colors, as defined in the color bar. Solutions
above the 3-$sigma$ are shown in yellow. Table 3: Elliptic fit to the Chiron’s
limb from the September 8, 2019 occultation.

Equivalent radius† (km) | $R_{\rm equiv}=105^{+6}_{-7}$
---|---
Semi-major axis (km) | $a=126\pm 22$
Position angle (deg) | $PA=11\pm 33$
Oblateness |  $\epsilon$= 0.27 $\pm$ 0.18
999$\leavevmode\nobreak\ {}^{\dagger}$Taken from Lellouch et al. (2017), see
text.

Figure 4: Same as Fig. 2, with the best fitting ellipse plotted in black
(e.g., the lowest ${\chi^{2}}$ value) and the 1$\sigma$-level ellipse plotted
in grey. The green spot in the center of the ellipses represents their
respective centers. The Paris chord was shifted forward by 0.9 seconds for the
best alignment with the other sites (see text for details). The green line
below the ellipses is the negative chord observed from the Forcarei site.

###  2.2 Size and shape constraints

Although challenging due to the presence of coma and ring, rotational light
curves have been observed for Chiron, giving a synodic rotation period of P =
5.917813 $\pm$ 0.000007 h (Marcialis & Buratti, 1993). Groussin et al. (2004)
derived a true light curve amplitude of $\Delta m_{0}=\leavevmode\nobreak\
$0.16 $\pm$ 0.03, which is the light curve amplitude that we would observe if
the object’s pole is perpendicular to the line of sight. This implies an axial
ratio $b/a=\beta=0.862\pm 0.023$, for a triaxial body with axes $a>b>c$.
Considering the pole direction given by Ortiz et al. (2015) to be the same as
that of the main body, the opening angle (B) with respect to the object’s
equator was B = 42∘ in 2019. So, the observed semi-major axis $a$ of the
ellipse can be considered as the actual equatorial radius of Chiron within a
factor of 1.16 (at maximum), given the known rotational variability and the
fact that the rotational phase of Chiron at the time of the occultation was
not known, implying that $b=109\pm 19\leavevmode\nobreak\ $km.

If Chiron has a Jacobi shape (i.e., fluid hydrostatic equilibrium), we can
expect $c/a=\gamma=0.53905\pm 0.00785$ for a plausible density of
$1,119\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 4\leavevmode\nobreak\
\rm{kg\leavevmode\nobreak\ m^{-3}}$ and rotation period of P = 5.917813 $\pm$
0.000007 h, which would give a semi-minor axis of $c=$ 68 $\pm$ 13 km, using
Chandrasekhar (1987) formalism. Under these assumptions, Chiron has a volume-
equivalent radius of Rvol= 98 $\pm$ 17 km.

We verified if the proposed ellipsoid agrees with the observed chords by
assuming that Chiron has the same pole as the rings proposed by Ortiz et al.
(2015) (i.e., the ring’s plane crosses the object’s equator). As the
rotational phase during the 2019 event is unknown, we must test all the
phases. Being $\omega$ the angle of the object’s major-axis ($a$) with respect
to the intersection of the object’s equatorial plane and the International
Celestial Reference System (ICRS) fundamental plane, as defined in Archinal et
al. (2018), we note that values of $\omega$ in a range of 90∘ (i.e., one forth
of a phase) provides compatible fits with the observed chords (Fig. 5). Figure
6 shows the best fit of Chiron as a Jacobi ellipsoid fitted to the 2019
occultation chords.

Figure 5: Test of all possible rotation phases, considering Chiron as a Jacobi
ellipsoid. Rotation phases within a range of 90∘ are compatible. Figure 6:
Best fit of the Jacobi ellipsoid with $a=126\pm 22$ km, $b=109\pm 19$ km, and
$c=68\pm 13$ km derived for Chiron, with a $\omega=120^{\circ}$, represented
by the big black dot, which also indicates the position of the major-axis $a$.
Here, f0 and g0 are the center of the ellipse presented in Fig.4.

##  3 Stellar occultation of November 28, 2018

Also predicted by the Lucky Star project, this event was of interest due to
the small shadow velocity of only 5.77 $\rm{km\leavevmode\nobreak\ s^{-1}}$,
relative to the geocenter, thus allowing a high spatial resolution per data
point. The occulted star was the Gaia DR3 2646598228351156352, with a
magnitude G = 17.28. Propagating the position of the star from the Gaia DR3
catalog epoch using parallax and proper motion to the time of the event, its
geocentric position at epoch is:

$\alpha=23h\leavevmode\nobreak\ 46m\leavevmode\nobreak\ 04s.33575\pm
0.2508\leavevmode\nobreak\ {\rm mas,}$ $\delta=+02^{\circ}\leavevmode\nobreak\
13^{\prime}\leavevmode\nobreak\ 05^{\prime\prime}.5270\pm
0.3445\leavevmode\nobreak\ {\rm mas.}$

Observations were arranged at the SAAO and at the Boyden Observatory, both in
the Republic of South Africa. A third site was used in Namibia, the Automatic
Telescope for Optical Monitoring (ATOM); (See Hauser et al., 2004, for details
on this facility.). Table 4 provides circumstantial details and Fig. 7
displays the post-diction map.

Table 4: Observational circumstances of the November 28, 2018 Chiron
occultation.

Site | Longitude E | Latitude S | Elevation | Telescope | Camera | Cycle∗ | $\sigma_{flux}$ | Observers
---|---|---|---|---|---|---|---|---
(∘ ’ ”) | (∘ ’ ”) | (m) | aperture (mm) | time (s)
SAAO | 20 48 37.8 | 32 22 41.9 | 1,760 | 1,016 | SHOC‡ | 1.5 | 0.08 | A. Sickafoose
Boyden | 26 24 17.0 | 29 02 19.8 | 1,372 | 1,500 | Alta-U55 | 10 | 0.08 | P. van Heerden
ATOM$\leavevmode\nobreak\ {}^{\dagger}$ | 16 30 05.6 | 23 16 19.7 | 1,800 | 700 | DU888_BV | 1.0 | 0.20 | F. Jankowsky

101010∗Read-out times are negligible for these cameras, so the exposure time
equals the cycle time. †Observation initiated after the actual local closest
approach. ‡SHOC stands for Sutherland High-speed Optical Cameras, see
Coppejans et al. (2013).

Table 5: Derived event times and chord length as observed from SAAO on November 28, 2018. Site | SAAO/SA
---|---
Ingress (UTC) | 20:50:12.51 $\pm$ 0.06 sec
Egress (UTC) | 20:50:43.80 $\pm$ 0.10 sec
Chord length | 180.5 $\pm$ 0.9 km
Figure 7: Post-diction map of the November 28, 2018 event. The black lines
delineate Chiron’s shadow path, the blue dot is the SAAO site where the
occultation was detected, the red dot is Boyden, where data were acquired but
no event was detected, and the white dot (ATOM) could only start acquisition
after the actual local closest approach.

The occultation could only be detected from SAAO. Using an exposure time of
1.5 seconds and no filter, each data point has a spatial resolution of 8.65 km
and S/R=14. Using the same procedures described in Section 2, the occultation
light curve and the ingress and egress times were obtained (see Table 5). The
stellar apparent diameter was estimated to be 0.3 km and the Fresnel scale at
$D=18.3538$ au is $L_{f}=0.945$ km, so, again, the light curve smoothing was
largely dominated by the exposure time.

The chords in the sky plane are displayed in Fig. 9. The large exposure time
(10 seconds) used at Boyden Observatory smooths any putative short event,
making it undetectable. Considering the long chord observed from SAAO, a long
chord (at least one exposure time) would be detected from Boyden as well. As
no flux drop was seen from there (Fig. 15), we can conclude the center of
Chiron was north of SAAO. This is a crucial constraint for the search of
secondary events caused by the putative rings proposed by Ortiz et al. (2015).

##  4 Constraints on the secondary events

As mentioned earlier in this work, secondary events have been observed in the
past during stellar occultations by Chiron (Bus et al., 1996; Elliot et al.,
1995; Ruprecht et al., 2015) and it has been suggested that they may have been
caused by a ring system, jets, or a dust shell (Ortiz et al., 2015; Sickafoose
et al., 2020). The only data set presented in this paper that offers useful
limits on the detection of material around Chiron, while considering the image
cadence and S/N, is the one obtained at SAAO in November 2018 (discussed
further in this section).

###  4.1 Reanalysis of 2011 data

We re-assessed the physical properties of the features observed during the
occultation by Chiron in 2011 (Ruprecht et al., 2015) when symmetrical
secondary drops were detected around Chiron. Under the assumption that they
are rings, as proposed by Ortiz et al. (2015), we followed the same procedures
as those we used to analyze (10199) Chariklo’s rings; (see Braga-Ribas et al.,
2014; Bérard et al., 2017; Morgado et al., 2021).

Using the FTN data, as presented in Fig. 4 of Ruprecht et al. (2015) and
reproduced in Fig. 2 of Sickafoose et al. (2020), we first corrected the
timing of those plots to their original values, which are presented in the
above-mentioned references with a shift forward of 7 seconds to account for
the geographic offset between FTN at Haleakala and IRTF at Mauna Kea. Using
the preferred pole position of Ortiz et al. (2015), with ecliptic coordinates
$\lambda=144^{o},\leavevmode\nobreak\ \beta=24^{o}$, we calculated the opening
angle with respect to the object’s equator (i.e., the sub-observer point),
which was B = 59.6∘ in 2011, and the relative star velocities perpendicular to
the rings, as measured in the ring plane. With these values, we fitted two
sharp-edged rings with different sizes and opacity, accounting for diffraction
(see the Extended Data in Braga-Ribas et al., 2014; Bérard et al., 2017, for
further details). We tested ring widths varying from 0.5 to 4 km and opacity
from 0 to 0.6. The resulting fits are shown in Fig. 8.

Figure 8: Reanalysis of the FTN data obtained during the November 29, 2011
stellar occultation by Chiron; (Ruprecht et al., 2015; Sickafoose et al.,
2020). Adopting the preferred pole position proposed on Ortiz et al. (2015),
we fit the light curves with two sharp-edged semi-transparent rings,
accounting for the stellar velocity perpendicular to the ring as measured in
the ring plane. Data are shown in black, the ring models are plotted in blue,
and the red curves are the synthetic light curves, accounting for diffraction,
stellar diameter, and finite integration time. The green crosses at the bottom
are the fit residuals to the synthetic light curve. The times were corrected
by -7 seconds with respect to the values published in Ruprecht et al. (2015)
and Sickafoose et al. (2020), so they are back to their original values (see
text). Table 6: Ring properties obtained from Faulkes 2011 data. We used the
same nomenclature as used by Sickafoose et al. (2020), where A1 and A2
correspond to the drops before the closest approach and A3 and A4 after the
closest approach.

Ring properties from the 2011 detection
---
| A1 | A2 | A3 | A4
$v_{\perp ring}$ (km s-1) | 7.18 | 6.99 | 8.67 | 8.73
$t_{mid}$ (sec)† |  931.01 $\pm$ 0.02 |  932.06 $\pm$ 0.02 |  985.78 $\pm$ 0.01 |  986.80 $\pm$ 0.02
W′ (km) |  1.53 $\pm$ 0.15 |  2.38 $\pm$ 0.20 |  2.07 $\pm$ 0.37 |  3.27 $\pm$ 0.39
Wr (km) |  2.22 $\pm$ 0.21 |  3.63 $\pm$ 0.29 |  1.98 $\pm$ 0.36 |  3.12 $\pm$ 0.37
$\rm{p}^{\prime}$ |  0.61 $\pm$ 0.06 |  0.47 $\pm$ 0.04 |  0.62 $\pm$ 0.06 |  0.40 $\pm$ 0.06
pN |  0.32 $\pm$ 0.04 |  0.23 $\pm$ 0.03 |  0.33 $\pm$ 0.04 |  0.20 $\pm$ 0.03
$\tau_{\rm{N}}$ |  0.41 $\pm$ 0.14 |  0.27 $\pm$ 0.08 |  0.42 $\pm$ 0.16 |  0.22 $\pm$ 0.09
Ep (km) |  0.71 $\pm$ 0.11 |  0.85 $\pm$ 0.12 |  0.65 $\pm$ 0.14 |  0.61 $\pm$ 0.12
111111† Seconds after 08:00 UT. Corrected by -7 seconds; see text.

Table 6 shows the perpendicular velocity relative to the rings, $v_{\perp
ring}$, the mid-time of each detection, ($t\rm_{mid}$), the apparent radial
width, ($\rm{W}^{\prime}$), where ”apparent” refers to the quantity measured
in the sky plane, the radial width, (Wr), in the ring plane, the apparent
opacity, ($\rm{p}^{\prime}=1-I/I_{0}$), the normal opacity to the ring plane,
(p$\rm{}_{N}=|\sin{B}|(1-\sqrt{1-\rm{p}^{\prime}})$), for a monolayer ring,
the normal optical depth to the ring plane,
($\tau\rm_{N}=|\sin{B}|(\tau^{\prime}/2),$ for a poly-layer ring, where
$\tau^{\prime}=-\ln(1-\rm{p}^{\prime})$, and the equivalent width,
($E\rm_{p}=W\rm_{r}\leavevmode\nobreak\ \rm{p_{N}}$).

They are consistent with the values of Sickafoose et al. (2020) but differ due
to i) the applied velocity to calculate the widths and ii) a factor of two
between the apparent and the actual ring optical depths due to light
diffraction by individual ring particles, as per Cuzzi (1985) and the
discussion in Bérard et al. (2017). This factor of two applies in the cases
where the Airy diffraction scale on individual particles with radius $r$
$L\rm_{A}\leavevmode\nobreak\ \sim\leavevmode\nobreak\ (\lambda/2r)D$, is
larger than the width of the ring. In visible wavelengths and assuming ring
particles with sizes of ten meters at most, this provides $L\rm_{A}\gtrsim 80$
km for $D=18.3538$ au, hence validating our assumption (i.e., the actual ring
optical depth is half of the one measured with the occultation light curve).
Consequently, our equivalent width ($E\rm_{p}$) values are almost half of
those given by Sickafoose et al. (2020).

###  4.2 Searching for rings

To search for the proposed rings in the November 28, 2018 SAAO data set, we
used the preferred pole position of Ortiz et al. (2015) ($\lambda\sim
144^{\circ},\leavevmode\nobreak\ \beta\sim 24^{\circ}$) and the ring
parameters derived from our re-analysis of the 2011 data (Table 6). From the
event geometry (Fig. 9), we can estimate the time intervals where the proposed
rings are expected (small blue segments).

Figure 9: Reconstructed geometry of 2018 event with the rings as proposed by
Ortiz et al. (2015). The large blue segment corresponds to the chord detected
from SAAO, and the small ones are the expected location of the rings. The
magenta dashed line corresponds to each exposure from the Boyden observatory
negative chord.

Using the properties of the rings found in Section 4.1, we searched a region
of 20 seconds (or $\sim$115 km) around the expected rings locations, on both
sides (before and after the main body ingress and egress). This corresponds to
about seven times the total duration of the occultations by the rings and gap.
The best fit for the ingress is obtained a few kilometers farther than the
expected location ($\sim$36 km) and has a $\chi^{2}$ value per degree of
freedom of $\chi^{2}_{\rm pdf}=0.59$ (Fig. 10, left panel). For the egress,
the best fits occur at the expected location, with a $\chi^{2}_{\rm pdf}=1.09$
(Fig. 10, right panel).

Figure 10: Search for the Chiron’s rings proposed by Ortiz et al. (2015) with
the properties given by Sickafoose et al. (2020) using the SAAO data obtained
during the November 28th 2018 occultation. The gray regions indicate the
ring’s expected positions from Ortiz et al. (2015). The left (resp. right)
panel presents the data before (resp. after) the occultation by the main body.

To assess the detection level of ring material, we analyzed the SAAO light
curve, following the same procedures as in Bérard et al. (2017). We considered
the data up to 2,000 km from Chiron’s center, excluding the main body
occultation part. The light curve was converted into flux as a function of
radial distance to the object’s center, counted in the ring plane. Accounting
for the standard deviation of the light curve, we obtained a 1$\sigma$ upper
limit of $\rm{p}^{\prime}=0.08$ for the apparent opacity
($\rm{p}^{\prime}=0.24$ at the 3$\sigma$ level), see the red (resp. green)
dashed line in Fig. 11. From the relation
$\tau^{\prime}=-\ln(1-\rm{p}^{\prime}),$ we derive an upper limit of
$\tau^{\prime}=\leavevmode\nobreak\ 0.25$ (3$\sigma$ level) for the apparent
optical depth.

The star velocity at the ring plane was 7.99 $\rm{km\leavevmode\nobreak\
s^{-1}}$, yielding a spatial resolution of 12 km per data point. If existent,
individual rings of this width would be detectable if $\rm{p}_{\rm{N}}=0.09$
or $\tau\rm_{N}>0.09$. The proposed rings would then be detected at a
2.6$\sigma$ level during ingress and at a 2.2$\sigma$ during egress (see the
yellow dashed lines in Fig. 11). This means that the expected ring signatures
were too weak to be assertively detectable in this data set with a 3$\sigma$
or higher confidence level.

Figure 11: Light curve of the flux versus the distance to Chiron’s center in
the ring plane. The dashed lines in red and green correspond to the detection
limits at the 1 and 3-$\sigma$ levels, respectively. The yellow dotted line
corresponds to the expected depth of the occultation promoted by the proposed
rings.

###  4.3 Searching for a shell of material

The SAAO data can also be used to put upper limits on the detection of a shell
of material around Chiron. Using the same procedure as the previous section,
the light curve was converted to flux versus distance to the object’s center,
but here the data were projected to the sky plane. Considering the standard
deviation of the light curve, we obtain an apparent opacity of
$\rm{p}^{\prime}=0.08$ at the 1$\sigma$ level and $\rm{p}^{\prime}=0.24$ at
the 3$\sigma$ level, which represents an apparent optical depth of
$\tau^{\prime}=0.25$ (3$\sigma$). Here, we considered the full resolution of
8.65 km allowed by the data set.

If we consider that a shell of material can be spread over a large region, we
should also search for symmetrical drops over several kilometers radially to
the object’s center. To do so, we binned the data using windows of 45 and 81
km (according to the proposed structures in Elliot et al., 1995) and obtained
apparent optical depth limits of $\tau^{\prime}=$ 0.11 and 0.08 at the
3$\sigma$ level, respectively. As we can see in Figs. 12 and 13, no
symmetrical or smooth flux drops are seen in both graph.

Figure 12: Similar as Fig. 11, but the data were smoothed to a resolution of
45 km per data point. Figure 13: Similar as Fig. 11, but the data were
smoothed to a resolution of 81 km per data point.

##  5 Conclusions

We have presented here the first multi-chord stellar occultation by Chiron.
Sizes determinations were made using different techniques before this work.For
instance, Groussin et al. (2004) combined visible, infrared, and radio
observations to derive a diameter of 142 $\pm$ 10 km; Fornasier et al. (2013)
combined Herschel and Spitzer thermal data to derive (among other solutions)
an equivalent diameter of 218 $\pm$ 20 km. Lellouch et al. (2017) reanalyzed
the previous data adding observations made with ALMA to derive an equivalent
diameter of 210 $\pm$ 10 km for a spherical model. They have also considered
an elliptical model and the influence of the putative rings, but they found
that neither help to reconcile the discrepant diameter determinations. We
decided to use their spherical model as the most conservative approach and
give reliable limits to Chiron’s apparent equatorial diameter
(R$\rm{}_{equat}=126\pm 22$ km) and oblateness ($\epsilon=0.27\pm 0.18$).

Considering the derived true rotational light curve amplitude of $\Delta
m_{0}=\leavevmode\nobreak\ $0.16 $\pm$ 0.03 (Groussin et al., 2004) and
assuming a Jacobi shape (i.e., fluid hydrostatic equilibrium), we can expect
Chiron’s semi-axis to be $a=126\pm 22$ km, $b=109\pm 19$ km, and $c=68\pm 13$
km, implying a model-dependent density of $1,119\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 4\leavevmode\nobreak\ \rm{kg\leavevmode\nobreak\
m^{-3}}$, for a rotation period of 5.917813 h, and a volume-equivalent radius
of Rvol= 98 $\pm$ 17 km. With these values, we can obtain the first estimation
of Chiron’s mass of $4.8\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 2.3\times
10^{18}$ kg. When Chiron is viewed equator-on, the average equivalent surface
radius is R${}_{Sequiv}=89\pm 16$ km, it can be used with the true absolute
magnitude of 7.28 $\pm$ 0.08 in V band obtained by Groussin et al. (2004) and
Mag${\rm{{}_{V}}}=\leavevmode\nobreak\ -26.74\leavevmode\nobreak\ $ for the
Sun, to calculate Chiron’s geometric albedo to be
$p\rm{{}_{V}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.076\pm 0.026$, which
is not in agreement with their value of 0.11 $\pm$ 0.02; however, as discussed
by Ortiz et al. (2015), they did not consider the presence of the putative
rings, which may contribute to up to 30% of the total reflected flux. Even if
Chiron does not exhibit a Jacobi equilibrium shape, which might be the case,
it is reasonable to consider that it does not diverge much from the here
proposed shape.

We reanalyzed the secondary events observed in 2011 using Faulkes Telescope
(Ruprecht et al., 2015; Sickafoose et al., 2020) and obtained the properties
of the rings using the same procedures as Braga-Ribas et al. (2014); Bérard et
al. (2017), considering the size and pole orientation proposed by Ortiz et al.
(2015).

Table 7: Chiron’s astrometric positions from 2018 and 2019 occultations. Event date | Right ascension | Astrometric
---|---|---
(dd-mm-yyyy) | (hh mm ss.s) | uncertainty
Instant (UT) | Declination | (mas)
(hh:mm:ss.s) |  (∘ ’ ”)
28-11-2018 | 23 46 04.35419 |  $\pm$ 0.26
20:50:43.14 | +02 13 05.25822 |  $\pm$ 0.35
08-09-2019 | 00 10 12.73280 |  $\pm$ 0.91
23:04:26.20 | +04 37 05.26207 |  $\pm$ 0.45

Data obtained from South Africa Astronomical Observatory allowed for a search
for signatures of the proposed ring system. No clear evidence of the presence
of the rings was found, mainly due to the long exposure time needed to record
the occultation event. If the structures observed in 2011 were present in the
2018 data, they would have been detected at 2.6 $\sigma$ during the ingress
and 2.2 $\sigma$ at the egress. These values are too low to claim any
detection. Figure 11 clearly shows that many drops are below the 1 $\sigma$
level, but none below the 3$\sigma$ level. We note that a ring such as the
C1R, the main ring of (10199) Chariklo (Morgado et al., 2021), would have been
detected above the 3 $\sigma$ level.

We also searched for broad features, such as a shell of material, and found no
evidence. The upper limits are $\tau^{\prime}=$ 0.25, 0.11, and 0.08 for
shells spread over $\sim$9 km, 45 km, and 81 km, respectively. Elliot et al.
(1995) reported a feature (F2) with width $\sim$74 km and a maximum optical
depth of $\tau^{\prime}\sim$0.11. Considering the detection limit for apparent
optical depth obtained with the 81 km-smoothed curves, the F2 feature would be
detected outside the 3-$\sigma$ level.

Figure 14: Most promising upcoming occultation event by Chiron on September
10, 2023. Solid black lines correspond to the main body shadow path, while the
blue dashed lines the projected distance of the proposed rings. Prediction
based on the latest NIMA ephemeris (NIMAv15, available on the Lucky Star web
page), using astrometric positions from all the detected occultations by
Chiron and the Gaia DR3 catalog.

Chiron is currently in a region of the sky characterized by a low density of
stars, so relevant occultation events by this object are very rare. Thus,
observations of new occultation events are needed. Adding the astrometric
positions of the events presented here (Table 7) and those presented in Rommel
et al. (2020), we searched for future events. A promising event will occur on
September 10, 2023 (Fig. 14)121212Further information can be found in the
Lucky Star event’s webpage https://lesia.obspm.fr/lucky-star/occ.php?p=123202,
which may allow for more precise size and shape determinations and contribute
further information on Chiron’s environment.

###### Acknowledgements.

This work was carried out within the Lucky Star umbrella that agglomerates the
efforts of the Paris, Granada, and Rio teams, which is funded by the European
Research Council under the European Community H2020 (ERC Grant Agreement No.
669416). This study was financed in part by the Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and the
National Institute of Science and Technology of the e-Universe project (INCT
do e-Universo, CNPq grant 465376/2014-2). The following authors acknowledge
the respective CNPq grants: F.B-R 314772/2020-0; BEM 150612/2020-6; RV-M
307368/2021-1, 401903/2016-8; J.I.B.C. 308150/2016-3; M.A. 427700/2018-3,
310683/2017-3, 473002/2013-2. The following authors acknowledge the respective
grants: C.L.P thanks the FAPERJ/DSC-10 (E26/204.141/2022); B.E.M. thanks the
CAPES/Cofecub-394/2016-05 grant; M.A. acknowledges F.A.P.E.R.J. grant
E-26/111.488/2013; ARGJr acknowledges F.A.P.E.S.P. grant 2018/11239-8; Some of
the results were based on observations taken at the 1.6 m telescope on Pico
dos Dias Observatory of the National Laboratory of Astrophysics
(L.N.A./Brazil). This research made use of SORA, a Python package for stellar
occultations reduction and analysis, developed with the support of ERC Lucky
Star and LIneA/Brazil. This work has made use of data from the European Space
Agency (E.S.A.) mission Gaia (https://www.cosmos.esa.int/gaia), processed by
the Gaia Data Processing and Analysis Consortium (D.P.A.C.,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). J.L.O., P.S-S., N.M.,
and M.K. acknowledge financial support from the grant CEX2021-001131-S funded
by MCIN/AEI/10.13039/501100011033 and they also acknowledge the financial
support by the Spanish grant nos. AYA-2017-84637-R and PID2020-112789GB-I00
and the Proyectos de Excelencia de la Junta de Andalucía grant nos.
2012-FQM1776 and PY20-01309.

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##  Appendix A Occultation Light Curves

The occultation light curves obtained from the here reported events are
presented in Fig. 15. The first four light curves from the 2019 event allowed
to constrain Chiron’s size and shape. The last light curve from the 2018 event
allowed searching for secondary events around it.

Figure 15: Black lines and dots are the data obtained from each site, the blue
lines the opaque band, and red lines the modeled light curve, convoluted to
the exposure time. Times are given in seconds after midnight of the event
date. The site names are given in each panel. The green vertical dashed lines
on Boyden’s panel indicate the theoretical ingress and egress times.