Hubble Space Telescope Detection of the Nucleus of Comet C/2014 UN₂₇₁ (Bernardinelli–Bernstein)

The presence of the bright coma (Figure 1) presents an obstacle to the direct measurement of the signal from the nucleus. We used three methods of increasing power to isolate the nucleus signal.

The first method we applied was to place a circular aperture of 020 (5 pixels) in radius at the centroid of the comet in each of the five individual exposures, regard the measured signal as being all from the nucleus, and determine the sky background using a concentric annulus having inner and outer radii of 8'' and 40'', respectively. At this distance, contamination by the coma is completely negligible. We thereby obtained the apparent V-band magnitude of the region enclosed by the 020 radius aperture to be m_V = 21.11 ± 0.03, in which the reported uncertainty is the standard deviation on the repeated measurements. Because the measured signal has contributions from both the nucleus and the surrounding coma enclosed by the aperture, the true apparent magnitude of the nucleus must be fainter than the measured one. To correct for the observing geometry, we simply assumed a linear phase function with a slope of β_α = 0.04 ± 0.02 mag deg⁻¹ appropriate for comets at small phase angles α ≳ 5° (e.g., Lamy et al. 2004). Some solar system bodies show a backscattering opposition surge caused by multiple scattering in the regolith. However, cometary nuclei in general are dark, and multiple scattering effects are likely negligible (e.g., Li et al. 2013; Masoumzadeh et al. 2017). For these reasons, we do not attempt to account for the opposition surge.

We computed the absolute magnitude of the nucleus from

$$\begin{eqnarray}&&{H}_{{\rm{n}},V}={m}_{{\rm{n}},V}-5\mathrm{log}\left({r}_{{\rm{H}}}{\rm{\Delta }}\right)-{\beta }_{\alpha }\alpha ,\end{eqnarray} \tag{ 1 }$$

where the subscript "n" denotes parameters for the nucleus, r_H and Δ are, respectively, the heliocentric and cometocentric distances expressed in astronomical units (au), and α is the phase angle in degrees. Substituting, we found that the nucleus of the comet must have H_n,V > 8.09 ± 0.03, in which the uncertainty is the standard error on the mean. The geometric albedo and the radius of the nucleus are directly related to the absolute magnitude by

$$\begin{eqnarray}&&{p}_{V}{R}_{{\rm{n}}}^{2}={10}^{0.4\left({m}_{\odot ,V}-{H}_{{\rm{n}},V}\right)}{r}_{\oplus }^{2},\end{eqnarray} \tag{ 2 }$$

where p_V is the geometric albedo in the V band, R_n is the nucleus radius, and m_⊙,V = −26.76 ± 0.03 is the apparent V-band magnitude of the Sun at heliocentric distance r_⊕ = 1 au (Willmer 2018). Inserting numbers, we found ${p}_{V}{R}_{{\rm{n}}}^{2}\,\lesssim \left(2.6\pm 0.1\right)\times {10}^{2}$ km². Lellouch et al. (2022) recently reported p_V = 0.049 ± 0.011 and R_n = 69 ± 9 km using their ALMA observation in combination with the optical measurements by Bernardinelli et al. (2021). Their model assumes that (1) the geometric albedo of the nucleus is low, (2) the ALMA signal is uncontaminated by dust grains, and (3) the near-Earth asteroid thermal model is applicable. If their assumption that the coma contamination was negligible in the ALMA data is valid (but see Section 4.1), the reported size of the nucleus will be trustworthy because the thermal emission measures ${R}_{{\rm{n}}}^{2}$ and almost has no dependency upon the albedo. Therefore, assuming the aspect angles are not too different between the ALMA and HST observations, we found an upper limit to the geometric albedo of the nucleus of p_V < 0.055 ± 0.014, contingent on the nucleus size derived by Lellouch et al. (2022).

The above method provides only a coarse upper limit to ${p}_{V}{R}_{{\rm{n}}}^{2}$. As a second method, we again applied the same circular aperture of 020 radius at the centroid of the comet but instead measured the background in a contiguous annular region extending up to 028 from the centroid. The resulting flux measured by the aperture is still an upper limit to the counterpart from the nucleus. This is because this method underestimates the surface brightness of the coma in the central aperture, but it nevertheless provides a better constraint than does the first method, in which no correction was attempted whatsoever. We found the resulting apparent V-band magnitude to be m_V = 21.23 ± 0.03, corresponding to H_n,V > 8.21 ± 0.03, and p_V < 0.049 ± 0.012 if still assuming the nucleus size reported by Lellouch et al. (2022). In comparison, Lellouch et al. (2022) reported the exact geometric albedo of the comet, rather than an upper limit, to be p_V = 0.049 ± 0.011, which is indistinguishable from what we obtained from the second method. We refrain from the relevant discussion until Section 4.

Given the ultrastable point-spread function (PSF) and the supreme spatial resolution and sensitivity of the HST/WFC3 camera, we opted to employ nucleus extraction as a third technique. The latter has been successfully applied to a number of comets observed by HST (e.g., Lamy et al. 1998a, 1998b, 2009, 2011) and systematically evaluated (Hui & Li 2018). The basic idea of the technique is to remove the contamination of the coma by means of fitting its surface brightness profile and extrapolating inwards to the near-nucleus region, assuming that the coma is optically thin, such that the signal from the coma and that from the nucleus are separable. The surface brightness of the coma was fitted by an azimuthally dependent power-law model. We expressed the surface brightness of the comet as a function of the angular distance to the nucleus (ρ) and the azimuthal angle (θ) in the sky plane as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Sigma }}\left(\rho ,\theta \right) & = & \left[{k}_{{\rm{n}}}\delta \left(\rho \right)+{k}_{{\rm{c}}}\left(\theta \right){\left(\frac{\rho }{{\rho }_{0}}\right)}^{-\gamma \left(\theta \right)}\right]* { \mathcal P }\\ & = & {k}_{{\rm{n}}}{ \mathcal P }+\left[{k}_{{\rm{c}}}\left(\theta \right){\left(\frac{\rho }{{\rho }_{0}}\right)}^{-\gamma \left(\theta \right)}\right]* { \mathcal P }.\end{array}\end{eqnarray} \tag{ 3 }$$

Here, k_n and k_c are the scaling factors for the nucleus and coma, respectively, δ is the Dirac delta function, γ is the logarithmic surface brightness gradient of the coma, ${ \mathcal P }$ is the energy-normalized PSF, ρ₀ = 1 pixel is a normalization factor to guarantee that the two scaling factors share the same unit, and the symbol * is the convolution operator.

We followed the procedures detailed in Hui & Li (2018) to extract the nucleus signal from our HST data. Basically, we fitted the surface brightness profile of the coma in azimuthal segments of 1° over an annular region between 024 to 080 from the optocenter, where the contribution from the nucleus is negligible in the individual exposures. Smoothing of the best-fit parameters for the coma was carried out to alleviate fluctuations due to uncleaned artifacts caused by cosmic-ray hits (Figure 2), followed by extrapolating the surface brightness profile inwards to the near-nucleus region and convolution with the HST/WFC3 PSF model generated by TinyTim (Krist et al. 2011). Subtraction of the coma model from the observed image on a 7× resampled pixel grid revealed a well-defined stellar source around the original centroid of the comet in the residual image, which was measured to have an FWHM of 0071 ± 0004 (or 1.8 ± 0.1 pixels), in line with the FWHM of the PSF model by TinyTim, and therefore interpreted as the nucleus of the comet (Figure 3). We then fitted the PSF model to the source, whereby we obtained the scaling factor k_n as the total flux for the nucleus using aperture photometry. Comparisons between the radial brightness profiles of the observation and the models are plotted in Figure 4.

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It is known that the nucleus extraction technique produces systematic biases in the determination of the nucleus signal when the nucleus is very faint compared to the adjacent coma and that it can even fail on a few occasions (Hui & Li 2018). In order to ascertain how our result might be biased by the technique, we assessed the ratio between the nucleus flux and the total flux measured within a 060 radius circular aperture. The ratio was found to be always ≳30%, falling into a regime where the bias is totally negligible (Hui & Li 2018). Therefore, we are confident that the signal of the nucleus determined from our HST observation of comet C/2014 UN₂₇₁ is robust.

The result is that our best estimate of the apparent V-band magnitude of the nucleus is m_n,V = 21.65 ± 0.11. Substitution into Equation (1) yields H_n,V = 8.63 ± 0.11, which is clearly fainter than what Bernardinelli et al. (2021) reported based on their optical observations (H_n,V = 8.21 ± 0.05, converted from the Sloan bands; Lellouch et al. 2022), and corresponds to ${p}_{V}{R}_{{\rm{n}}}^{2}=\left(1.57\pm 0.16\right)\times {10}^{2}$ km² yielded by Equation (2). Still adopting the nucleus size reported by Lellouch et al. (2022), we determine a nucleus geometric albedo of p_V = 0.033 ± 0.009, in which the uncertainty was properly propagated from all measured and reported errors. Our result suggests a lower albedo for the nucleus surface because it is possible that the photometry by Bernardinelli et al. (2021) is contaminated by the dust environment of comet C/2014 UN₂₇₁. Nonetheless, the albedo we derived is unremarkable in comparison to those of other cometary nuclei (distributed in a narrow range of p_V ≈ 0.02–0.06; Lamy et al. 2004).

Our analysis of the HST observation of comet C/2014 UN₂₇₁ provided us with an estimate of its nucleus absolute magnitude ∼0.4 mag fainter than the result by Bernardinelli et al. (2021), presumably as a result of coma contamination in the large aperture optical photometry used by these authors. If there is no dust contamination of the 233 GHz ALMA signal, the nucleus albedo must be lower than the one derived by Lellouch et al. (2022), as shown by the hollow green and filled blue circles in Figure 5.

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However, Lellouch et al. (2022) concluded that up to ∼24% of the 233 GHz continuum flux could be from an unseen dust coma in their data. In this case, we estimate that the effective diameter of the comet would be reduced to 119 ± 15 km and the geometric albedo increased to p_V = 0.044 ± 0.012, shown as a filled purple circle in Figure 5. In order to affect the 233 GHz cross section, the dust in the comet would need to be large. Two observations suggest that the coma might indeed be rich in large grains. First, independent observations of other long-period comets (notably C/2017 K2; e.g., Jewitt et al. 2017, 2019a; Hui et al. 2018;) have convincingly revealed that submillimeter and larger dust grains are produced at great heliocentric distances. Second, based on a syndyne-synchrone computation, Farnham et al. (2021) deduced that C/2014 UN₂₇₁ has been ejecting submillimeter-sized and larger dust grains for years prior to the epoch of observation. The driver of mass loss at distances ∼20 au (and potentially at much larger distances; Bouziani & Jewitt 2022) is presumably the sublimation of carbon monoxide.

An additional factor that might affect estimates of the albedo is the rotation of the nucleus, resulting in cross sections different between the ALMA and HST observations. However, most known cometary nuclei have aspect ratios ≲2:1 (e.g., Lamy et al. 2004), and so we do not expect rotational effects in nucleus photometry larger than a factor of 2.

In short, our improved estimate of the absolute magnitude shows that the nucleus of C/2014 UN₂₇₁ is slightly smaller or slightly darker than that found by Lellouch et al. (2022) but we strongly confirm their result that the nucleus is larger than any other previously measured long-period cometary nucleus. Figure 5 is shown as a comparison of the results.

We measured the logarithmic surface brightness gradient of the coma in Section 3.2, which allows for a qualitative diagnosis of the observed activity. In steady state, the surface brightness gradient of the coma is expected to be γ = 1 but will be steepened to ∼1.5 by the solar radiation pressure (Jewitt & Meech 1987). Indeed, we found that the surface brightness gradient of the coma lies within a range between ∼1.0 and 1.7 (see Figure 2), broadly consistent with the coma produced in steady state. The result that the gradient is generally steeper around the azimuthal angles facing toward the Sun and is shallower otherwise is expected.

In addition to characterizing the nucleus of comet C/2014 UN₂₇₁, we also measured the coma using multiple concentric apertures. The background was determined in the same fashion as in the first method in Section 3.1. For correction of the observing geometry, we still assumed a linear phase function with slope β_α = 0.04 ± 0.02 mag deg⁻¹, which is also appropriate for cometary dust at small phase angles (e.g., Kolokolova et al. 2004 and citations therein). We plot the measurements in Figure 6, in which the dominant errors are systematic and due to the uncertainty in the assumed phase function. In the two smallest apertures, the measured apparent flux fluctuated marginally beyond the 1σ errors with time, possibly indicating varying cometary activity. However, we opt not to overinterpret these fluctuations.

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In the following, we estimate the mass loss of the comet using the largest circular aperture of a fixed projected radius of 20,000 km in the order-of-magnitude manner of Jewitt et al. (2021). Presuming that the total cross section of the dust particles having mean radius ${\bar{a}}_{{\rm{d}}}\sim 0.1\,\mathrm{mm}$ was ejected in steady state at speeds v_ej ∼ 10 m s⁻¹ (Farnham et al. 2021), we can then relate the mass-loss rate to the measured absolute magnitude of the dust by

$$\begin{eqnarray}&&{\overline{\dot{M}}}_{{\rm{d}}}\sim \displaystyle \frac{\pi {\rho }_{{\rm{d}}}{\bar{a}}_{{\rm{d}}}{v}_{\mathrm{ej}}{r}_{\oplus }^{2}}{{\ell }{p}_{V}}{10}^{0.4\left({m}_{\odot ,V}-{H}_{{\rm{d}},V}\right)},\end{eqnarray} \tag{ 4 }$$

in which the subscript "d" denotes parameters of the dust grains, ρ_d ∼ 1 g cm⁻³ is the nominal bulk density of the dust grains, and ℓ = 2 × 10⁴ km is the projected radius of the largest aperture we used to measure the coma. By substitution, we find the dust mass-loss rate ${\overline{\dot{M}}}_{{\rm{d}}}\sim {10}^{3}$ kg s⁻¹. In comparison, the distant comet C/2017 K2 (PANSTARRS) was estimated to exhibit a dust mass-loss rate of ∼10² kg s⁻¹ at r_H ≲ 20 au (Jewitt et al. 2017, 2021; Hui et al. 2018;), while Szabó et al. (2008) reported ∼10³ kg s⁻¹ for comet Hale–Bopp at similar heliocentric distances on the outbound leg of its orbit. None of the aforementioned values are better than order-of-magnitude estimates.