## Introduction

Comet 2I/Borisov, also known as C/2019 Q4 (Borisov), was discovered on 30 August 2019 by Gennady Borisov at the MARGO observatory, Crimea (MPEC 2019-R106). Its orbital eccentricity (3.356191 ± 0.000015) shows that the object is not gravitationally bound to the Solar System, making it the first unambiguous case of a comet arriving from interstellar space. This is only the second recognised case of a small body entering the Solar System from interstellar space. The first such case, namely that of 1I/′Oumuamua, discovered on 18 October 2017 by the Pan-STARRS 1 telescope, was that of an apparently asteroidal body, lacking detectable signs of cometary activity1. In contrast, 2I/Borisov exhibited a coma, and its apparition provided a unique opportunity to glean information about a body apparently similar to Solar System comets, but otherwise unrelated to them. Most observations to-date were aimed at obtaining spectra of the comet, and at measuring its dust and gas emissions, in order to determine the material composition and compare it with those of Solar System comets. In particular, reflectance spectra in the 0.49–0.92-μm wavelength range show a reddish slope2 similar to that of other Solar System bodies, including comets, Jupiter Trojan asteroids belonging to the so-called D taxonomic class3 and several Centaurs and transneptunian objects4 (the latter two categories exhibiting a wide range of colours, including – but not limited to – bodies with extremely red spectral slope).

Measurements of cometary linear polarisation provides information about physical characteristics of the coma dust particles that is difficult to obtain by other observing techniques. Sunlight scattered by dust particles is partially polarised, i.e., the electric field associated to the radiation has a preferred plane of oscillation. The polarised fraction of the radiation varies with the scattering angle, and depends on the characteristics of the scattering medium, in particular its complex refractive index (hence its chemical composition), and morphology, that includes size distribution, shape and structure of the scattering particles. In planetary science, polarisation is measured as the flux perpendicular to the plane Sun-Object-Observer (the scattering plane) minus the flux parallel to that plane, divided by the sum of the two fluxes; this measurement is usually repeated in different viewing conditions described by the so-called phase angle (the angle between the directions to the Sun and to the observer as seen from the target). Surprisingly, surfaces of airless objects (like the Moon or asteroids) and cometary atmospheres show similar phase angle dependence of polarisation. In particular it is found that at small phase angles (≤20), the linear polarisation is directed along the scattering plane; because the way the polarisation is measured, this situation is described as negative polarisation. At larger phase angles, the linear polarisation becomes positive, that is, directed along the direction perpendicular to the scattering plane, and increases until it reaches its maximum at phase angles 90–100. The devil is in the details, and the difference between different kinds of Solar System bodies, as well as between specific comets or asteroids, can be seen from values and location of minimum and maximum polarisation, from the phase angle values where negative polarisation changes to the positive one, or from the wavelength gradient of the polarisation5,6,7,8.

Extraction of the characteristics of the scattering medium from the polarimetric results is a difficult task as it requires modelling of numerous light-scattering phenomena such as reflection, diffraction, interference, shadow hiding, etc. Specifically, the negative polarisation observed for airless bodies results mainly from so-called coherent backscattering, an effect resulted from multiple scattering between the regolith particles9,10, whereas in cometary atmospheres single scattering, and, thus, the properties of the dust particles themselves, define the observed polarisation. The composition, size and structure of the cometary particles vary not only from comet to comet, but also within each comet; the particles are different near the nucleus and in the tail, in jets and ambient coma, close to the Sun and far from it8,11. Intensive numerical computation12,13 and laboratory data14,15,16 are used for the modelling of the observations. A remarkable result of polarimetric investigations was the prediction that cometary dust is made of aggregates of submicron grains17, which was later confirmed by the in situ studies of the Rosetta mission18 and of the samples returned by the Stardust mission19.

Even without numerical modelling, some information may be readily inferred through a very simple analysis of the physics of light scattering. For instance, light scattered by a low-albedo surface or complex (aggregated) particles tends to be more polarised than the light scattered by a higher-albedo surfaces or aggregated particles (the so-called Umov effect)20, as high-albedo scatterers are more likely to produce multiple scattering, which in turn is responsible for a more efficient depolarisation. Larger positive polarisation can be also associated with smaller, and, thus, Rayleigh-like, particles, while light scattered by more complex large aggregated particles is more affected by multiple scattering and, thus, is more depolarised. In case of comets, the continuum polarisation may be strongly modified by molecular emission lines due to the gas component within the coma and tail, therefore broadband polarimetric measurements should be analysed taking also into account the spectral characteristics of the object, or even better, using filters specially designed to avoid emission lines.

Accurate polarisation-phase curves have been obtained for a number of comets and over a wide range of phase angle, from 0 up to more than 100 in some cases, depending on target distance from the Sun and from the Earth in different circumstances8. It is therefore of the highest interest to understand whether a comet coming from interstellar space shares the same kind of polarimetric behaviour exhibited by Solar System comets, since, in principle, comets that accreted in other astrophysical environments could be significantly different from Solar System bodies.

Here we report polarimetric observations obtained with the European Southern Observatory (ESO) Very Large Telescope (VLT), and we show that the polarisation of interstellar comet 2I/Borisov is quite different to what is generally observed in comets of our Solar System, with a notable exception, that of comet C/1995 O1 (Hale-Bopp). Hale-Bopp is believed to have appeared close to our Sun only once before its recent approach in 1997, therefore its material is quite pristine, but its polarimetrically homogeneous coma suggests an even more pristine nature for comet 2I/Borisov.

## Results

Data were obtained in service mode using the FORS2 instrument21 attached at the Cassegrain focus of the Unit 1 (Antu) of the ESO VLT. FORS2 was used in imaging polarimetric mode with the V_HIGH filter (centred at 557 nm with a 123-nm full-width half-maximum, or FWHM), R_SPECIAL (centred at 655 nm with a 165-nm FWHM) and I_BESS (centred at 768 nm with a 138-nm FWHM). Reflectance spectra of comet 2I/Borisov show no prominent emission lines in the range covered by the R_SPECIAL and I_BESS filters2,22 (hereafter referred to as RF and IF filters, respectively), but the V_HIGH filter (hereafter referred to as VF filter) covers exactly the second brightest emission after the CN(0-0) at 387 nm – the C2 Swan band at 512 nm. However, comet 2I/Borisov appears to be C2-depleted23, so the VF filter is also not severely contaminated by molecular emissions. Therefore we assume that the observed broadband polarisation is due to the dust properties.

The reduced Q/I and U/I Stokes parameters24 were obtained using aperture polarimetry and rotated to the reference direction perpendicular to the scattering plane (see Methods subsection Polarimetry for details). In this system of reference, Q/I represents the flux perpendicular to the Sun-Comet-Earth plane minus the flux parallel to that plane, divided by the sum of the two fluxes (that is, the total flux). For symmetry reasons, the reduced Stokes parameter U/I should be close to zero. In the following, instead of Q/I, we will adopt the often-used notation Pr.

Our data covers a fairly large interval of phase angles, including the critical transition around the inversion angle, but does not sample the negative branch: additional observations in the negative branch were scheduled at the VLT in April and May 2020, but due to COVID-19, science operations on Paranal were suspended. In addition to the aperture polarimetry described above, we also employed a slit polarimetry method25 to measure polarisation along the comet tails, and we obtained also photometric measurements of the coma.

### Aperture polarimetry

The results of our aperture polarimetry measurements are given in the observing log of Table 1, and data obtained in the RF and IF filters are shown in Fig. 1, together with the polarisation measurements of comet Hale-Bopp, of comet 1P/Halley, and of other comets, selected with the help of an online database26. Polarimetric observations of comets are obtained in filters with different effective wavelength and bandwidth, and using different apertures. In Fig. 1 we show literature data obtained with filters with different bandwidths, all centred at wavelengths between 620 and 695 nm for comparison with data taken with the RF filter (panel a), and between 715 and 810 nm for comparison with data taken with the IF filter (panel b). Some photometric filters employed by the observers may be designed to cover certain gas emission bands, or to cover only the dust continuum, or may cover both regions potentially dominated by molecular lines and regions dominated by the dust continuum, like those that we have employed for our observations. For our comparison with literature we have avoided using data obtained through filters designed to cover molecular bands, yet the presence of molecular bands may well affect measurements with broadband filters. In an attempt to minimise these effects, when comet polarisation obtained at a given epoch was reported for different apertures, we adopted the value corresponding to the smallest one, on the grounds that it should be the most representative of the inner coma dust, where usually the highest values of polarisation are observed8. Some outliers with very large error bars have also been omitted.

Our observations show that the polarisation-phase curve of 2I/Borisov demonstrates a regular behaviour, with a clearly monotonic and linear increase of polarisation for increasing phase angle. Like comets of our Solar System, also comet 2I/Borisov exhibits negative polarisation at small phase angles. The value of the inversion angle helps to distinguish some classes of Solar System objects characterised by unusual properties. For instance, inversion angle values as large as 27–30 characterise a class of spinel-rich asteroids named Barbarians after the prototype (234) Barbara27. On the lower end of the observed range are the asteroids belonging to the F-taxonomic class28,29, that have an inversion angle αinv between 15 and 17, a property shared also by the nucleus of comet 2P/Encke observed in the absence of coma30 and by active asteroid 133P/Elst-Pizarro31. Centaurs seem to have more peculiar phase-polarisation curves, with small αinv values, in particular centaur Chiron shows αinv between 6.5 and 8 (ref. 32). Active comets instead exhibit a homogeneous behaviour, with αinv generally around 22 (ref. 8). Comet 2I/Borisov is not an exception, with αinv 20. 5 (in the RF filter), a value slightly smaller than for other comets, but still consistent with the average in our Solar System. The slope at the inversion angle is another important characteristic of the polarimetric curve, and it is empirically found that for asteroids there exists a relationship between the slope of the polarimetric curve at the inversion phase angle and the albedo – the steeper the slope the lower the albedo6. From a linear interpolation of the measurements of 2I/Borisov in the RF filter we obtain a slope of 0.45 ± 0.03 % deg−1. This value is somewhat extreme for comets, that exhibit polarimetric slopes typically ranging between 0.2 and 0.4 % deg−18. If our target were an asteroid, a slope of 0.45% deg−1 around the inversion angle would correspond to an extremely low albedo (<0.05). However, the albedo-polarisation relationship cannot be strictly applied to cometary comae and tails, because of the very different physics of light scattering by rather densely packed layers of particles in the former, and by rarified clouds of particles in active comets.

Figure 1 suggests a strong similarity between the polarisation behaviour of 2I/Borisov and that of comet C/1995 O1 (Hale-Bopp), that may be better illustrated by performing a fit to the data. Polarimetric curves of the small bodies of the Solar System may be fitted with the empirical function33,34

$${P}_{{\rm{r}}}(\alpha )=b\ {(\sin \alpha )}^{c1}\ {\left(\cos \frac{\alpha }{2}\right)}^{c2}\ \sin (\alpha -{\alpha }_{0})$$
(1)

where α is the phase angle, and b, c1, c2 and α0 are free parameters, or with an exponential function with only three free parameters35, also commonly used in asteroid works29,36. Because of the small number of data points, none of these functions allow us to make a reliable extrapolation of the behaviour of the polarisation of comet 2I/Borisov in the negative branch, nor at large phase angles. However, Eq. (1) may be conveniently used to describe the polarimetric curve of comet Hale-Bopp. Figure 1 shows also the best-fits to the Hale-Bopp data obtained with filters that have effective wavelengths close to those of the RF and IF FORS2 filters (note, however, that some outliers have not been considered for the best-fit to the data).

We finally note that polarimetric measurements of comet 2I/Borisov taken in the phase angle range 12.5–28, using the Advanced Camera for Survey/Wide Field Channel (ACS/WFC) of the Hubble Space Telescope (HST), have been recently presented in another study37. These observations, obtained in a filter F606W covering the wavelength range from 480 to 710 nm, have large uncertainties (about ten times higher than that of our measurement at the critical phase angle 28). They do not allow to appreciate any similarity with comet Hale-Bopp, nor any difference in steepness when compared to the large majority of the other comets of our Solar System, but they suggest the lack of any region of locally higher positive polarisation surrounding the nucleus.

Our measurements show that in the positive branch, the polarisation increases with wavelength; in the negative branch, we obtained only one measurement in the VF and RF filters, at a phase angle of 18, and found that the absolute value of linear polarisation in the RF filter is slightly smaller than in the VF filter. Because our measurements were obtained with broadband filters, we cannot obtain a refined estimate of the polarimetric spectral gradient (PSG); the significance of a comparison with other comets is limited by the use of different filters, and also by the fact that polarimetric measurements of different comets are generally obtained at different phase angle values, but we can approximate

$${\left.\frac{{\rm{d}}| {P}_{{\rm{r}}}(\lambda )| }{{\rm{d}}\lambda }\right|}_{\lambda = {\lambda }_{0}}\simeq {{\rm{PSG}}}_{({\lambda }_{1},{\lambda }_{2})}=\frac{| {P}_{{\rm{r}}}({\lambda }_{2})| -| {P}_{{\rm{r}}}({\lambda }_{1})| }{{\lambda }_{2}-{\lambda }_{1}}\,\,\,\,{\rm{where}}\,{\lambda }_{0}=\frac{{\lambda }_{1}+{\lambda }_{2}}{2}\ ,$$
(2)

where Pr stands for the absolute value of Pr. According to this definition, PSG is positive when the fraction of linear polarisation increases with wavelength, regardless of its direction, although is not defined around zero. In Table 2 we report the polarimetric spectral gradients PSG$${\,}_{({\lambda }_{1},{\lambda }_{2})}$$ of comet 2I/Borisov for the pair λ1 = 557 nm, λ2 = 655 nm, PSG (557,655 nm), and for the pair λ1 = 655 nm, λ2 = 768 nm, PSG (655,768 nm), at various phase angles. In Fig. 2 we compare these PSGs with those of comet Hale-Bopp estimated using observations38 with filters centred at 484.5 nm (with a 65-nm wide passband), at 620 nm (with a 60-nm wide passband), at 670 nm (with a 30-nm wide passband), and at 730 nm (with a 50-nm wide passband), having calculated the PSG for the pairs λ1 = 484.5 nm, λ2 = 620 nm, and for the pair λ1 = 670 nm, λ2 = 730 nm.

### Polarimetry along the tails

Polarimetric profiles were measured along both the gas-tail direction (anti-sunward direction) and the direction in which the dust tail is oriented, half-way between the anti-sunward and the direction opposite to the heliocentric velocity vector. Figure 3 shows the unpolarised image of comet 2I/Borisov obtained on 25/12/2019, and Fig. 4 shows the corresponding intensity and polarimetry profiles in all three filters along the horizontal direction and along the anti-sunward direction. Along both directions the polarisation profiles appear generally constant out to 20,000 km from the nucleus, while beyond 40,000 km the S/N becomes too low to extract any useful information. Consistent tail behaviour is seen in the December and January data, while the presence of numerous background stars prevented us from reaching firm conclusions from the observations obtained in the negative branch in March 2020. The stability of the polarisation throughout the coma is a sign of homogeneous ejection of the material from the nucleus, that is, a sign of weak contribution of active areas (if any) to the coma formation. We note that this behaviour greatly differs from the structures observed in the polarimetric profiles and polarimetric images of Hale-Bopp with a few thousands km scale38,39 and of other comets40,41.

### Photometric colours

The optical colours derived from photometry, calculated as explained in Methods subsection Photometry, are presented in Table 2. They are in agreement with those found in previous studies2,42,43, and broadly consistent with those of both short-period (Kuiper belt) and long-period (Oort cloud) comets, which indeed do not display any significant difference among themselves4.

We note that the photometric colours of 2I/Borisov are much bluer than the ones for comet C/1995 O1 (Hale-Bopp), for which, for instance, it was reported V − R = +0.71 ± 0.07 and R − I = + 1.03 ± 0.13 as an average value on 14–15 Aug 199544. Interestingly, Hale-Bopp was observed to become bluer on 16 Aug 1995, reaching similar values to the ones observed for 2I/Borisov, suggesting that this shift in colour are explained by an outburst which ejected a large quantity of very small ice grains44. Another study17 also indicates that the blue colour was typical for the spiral structures in the Hale-Bopp coma, which decreased the average coma colour.

The photometric time-variability of the photocentre during an observing series in each filter (~15–20 min) is smaller than the standard error of each individual measurement, and no clear trend could be discerned. This could be caused by a nearly spherical shape for the nucleus or a slow rotation, but the most plausible cause is that the signal contribution from the coma prevents direct observation of the nucleus rotation45.

## Discussion

The photometric data obtained in our and other investigations have large uncertainties, and are compatible with very different classes of small bodies of our Solar System, including short and long-period comets, Centaurs, Jupiter Trojans and some main belt asteroids4, and the same is true for the reflectance spectra. In summary, photometric data do not point to 2I/Borisov as an object with distinctive characteristics. Instead, the polarimetric characteristics of 2I/Borisov are more suggestive of its interstellar origin.

Several studies46,47 have divided comets into two polarimetric classes: low- and high-polarisation comets. This different behaviour is usually ascribed to a very different dust-to-gas ratio in the coma8. Since gas contamination dilutes the polarisation produced by the dust scattering, the higher dust-to-gas ratio, the higher the comet polarisation. However, the gas contamination is a consequence of the difference in the intrinsic properties of the dust particles. In low-polarisation comets, the particles are bigger and more compact than in higher polarisation comets, hence they are concentrated near the nucleus, and the coma is dominated by gas. In high-polarisation comets, the particles are smaller and more porous, capable of populating the coma and creating a high dust-to-gas ratio environment. This picture is supported by the difference in orbital characteristics of the two groups: comets with a gas-dominated coma are old comets with small perihelia and aphelia, hence strongly affected by solar radiation and solar wind, whereas dusty comets have larger perihelia and aphelia and experience less weathering. New comets, not yet processed by solar wind, are expected to be dusty, and to show high polarisation. Other interpretations invoke a different dust composition48, or explain high-polarisation comets with a higher level of activity49 (high-polarisation comets would have for instance more jets with small particles than inactive comets, and in the jets there are smaller particles and thus, higher polarisation). However, both these interpretations fail to explain the fact that the observed polarisation tends to decrease with the distance from the nucleus.

There exists a third polarisation class50, so far represented by a single comet, C/1995 O1 (Hale-Bopp). This comet was observed up to phase angle 50, and showed an increase of polarisation with phase angle noticeably exceeding the other high-polarisation comets. At the epoch of its measurement, the polarimetric curve of comet C/1995 O1 (Hale-Bopp) was steeper and higher than that of any other comet previously observed. Hale-Bopp is a long-period comet, having a period currently >2500 ys, as determined during its perihelion passage in 1995–1997. The peculiar positive polarisation behaviour for Hale-Bopp was explained by domination of small particles in its coma, with an estimated behaviour approaching the Rayleigh scattering regime51. This was consistent with the disappearance of the negative polarisation branch in the near-IR52, but especially with thermal IR spectroscopy, which showed a strong silicate feature53 typical for small dust particles54. The Deuterium/Hydrogen (D/H) ratio in the H2O of comet Hale-Bopp inferred from spectra is >10−4, about ten times that of the value commonly assumed for the proto-solar nebula55. This was interpreted as diagnostic of a comet that originated in the outer Solar System, at temperatures ~30 ± 10 K, an astrophysical environment probably distinct but not too dissimilar from that of the interstellar medium55.

Figure 1 shows that, within the range of phase angles covered by our observations, 2I/Borisov has a polarimetric curve remarkably similar to the unique curve of comet Hale-Bopp, and different from that of any other comet: the probability the two measurements around phase angle 27–28 in the RF filter could be the outliers of an otherwise normal behaviour of Solar-System comets is <10−2 (see Methods subsection Statistical tests).

Figure 2 shows also similarities in the polarimetric spectral gradient of the two comets. In fact, the PSG characteristics of Fig. 2 are not so rare; most of the comets display, like comets Hale-Bopp and 2I/Borisov do, a positive PSG in the positive branch. In the negative branch, the PSG of comets is also generally positive at visible wavelengths56,57, although data available are still limited, and often obtained at low significance level. A negative PSG in the positive branch was found only for a small number of comets belonging predominantly to the short-period and/or Jupiter family classes, which in this respect seem to exhibit a clearly different behaviour with respect to long-period comets8.

The similarity between the polarimetric properties of the two comets must depend upon the microscopic structure and composition of the aggregates, and not on their macroscopic characteristics, as the two comets are quite different in size: the analysis of the photometric profile of the inner coma suggests that comet Hale-Bopp belongs to the class of giant comets, with the diameter of the nucleus being estimated between 20 and 35 km58, while 2I/Borisov’s nucleus size is ≤0.4 km43,59. The close similarity between the polarimetric behaviour of the comet 2I/Borisov and Hale-Bopp suggests that, whatever astrophysical environment in which comet 2I/Borisov originated in, such environment had properties which led to the formation of a body bearing significant analogies with those accreted in the outer regions of our Solar System, a remarkable result on its own. This similarity could also suggest that the dust particles of 2I/Borisov are small, like those of Hale-Bopp. We are not aware of any measurements of comet 2I/Borisov in the thermal infra-red that could be used to set a constraint on particle size; however, close to perihelion, 2I/Borisov exhibited a NIR spectrum with a negative slope, which was explained as an increase in water ice and/or decrease in dust size60. HST imaging of the comet was modelled assuming coma particle size ~100 μm61, and other models have found particles of millimetre size particles62, in contrast with our claim. However, what these models derive is actually the ratio between solar radiation pressure and gravity force β which in turns does not have a one-to-one relationship with the particle or aggregate size. In fact, the best-fits values for β could either correspond to large radii, or to dust-grains with radius ≤ 0.1 μm63, or to aggregates as small as 0.2 μm64. We note that extremely high polarisation values were measured for comet C/1999 S4 (LINEAR) after it started to break apart41; since these data were obtained when the comet was seen in a phase angle range very different than our observations of comet 2I/Borisov, it is difficult to make a direct comparison; however, it is reasonable to hypothesise that comet C/1999 S4 was releasing small particles during this event, which were responsible for the high degree of the observed polarisation. We finally note that if we assumed that the same relationship between albedo and polarisation found for asteroids also holds for comets, our observations would indicate that also 2I/Borisov has a low geometric albedo, a property shared indeed by most comets of our Solar System.

When discovered, Hale-Bopp was among the brightest comets ever seen, and displayed cometary activity at large heliocentric distances, a fact interpreted to indicate a high-volatile content. Its polarimetric images showed clear structures, revealing the presence of jets and arcs39. By contrast, at the time of our observations, comet 2I/Borisov was polarimetrically homogeneous, showing no sign of active areas contributing to the coma formation. Prior to its recent perihelion passage, comet Hale-Bopp probably was near the Sun at least once, and possibly only once, ~2250 BC65; at the time of that first approach, the original material was removed from the surface and active areas were open66, hence Hale-Bopp could manifest activity during its recent perihelion passage. Comet 2I/Borisov instead, most likely never passed close to the Sun or any other star, and may represent the first truly pristine comet that has ever been observed.

## Methods

### Polarimetry

Polarimetry was obtained using the beam-swapping technique67, setting the retarder waveplate at the eight position angles 0, 22.5, …, 157.5, although in one case (the measurement with the RF filter on 2020-01-08) some exposures had to be discarded because of the presence of background objects too close to the comet photocentre, and in one case (IF filter on 2019-12-25) we deliberately used only the four positions 0, 22.5, 45, 67.5 of the retarder waveplate to save overhead time. The reduced Stokes parameters X/I (X = Q, U) were calculated as67

$$\frac{X}{I}=\frac{1}{2N}\mathop{\sum }\limits_{j=1}^{N}\left[{\left(\frac{{f}^{\parallel }-{f}^{\perp }}{{f}^{\parallel }+{f}^{\perp }}\right)}_{{\alpha }_{j}}-{\left(\frac{{f}^{\parallel }-{f}^{\perp }}{{f}^{\parallel }+{f}^{\perp }}\right)}_{{\alpha }_{j}+4{5}^{\circ }}\right]\ .$$
(3)

where $${f}_{\alpha }^{\parallel }$$ and $${f}_{\alpha }^{\perp }$$ are the fluxes measured in the parallel and perpendicular beams, respectively, with the retarder waveplate at the position angle α; for X = Q, αj {0, 90}; for X = U, αj {22. 5, 112. 5}; N is the number of αj values of the observing series (N = 2 for most of our observations). This double difference method is shown to give practically the same results as the double ratio method67. The uncertainties are given by

$${\sigma }_{X/I}^{2}=\frac{1}{{(2N)}^{2}}\ \mathop{\sum }\limits_{j=1}^{N}\left[{\left(g({\alpha }_{j})\right)}^{2}+{\left(g({\alpha }_{j}+4{5}^{\circ })\right)}^{2}\right]$$
(4)

where

$${g}^{2}(\alpha )={\left(\frac{2{f}^{\parallel }{f}^{\perp }}{{({f}^{\parallel }+{f}^{\perp })}^{2}}\right)}^{2}\ {\left(\frac{{({\sigma }_{{f}^{\parallel }})}^{2}}{{({f}^{\parallel })}^{2}}+\frac{{({\sigma }_{{f}^{\perp }})}^{2}}{{({f}^{\perp })}^{2}}\right)}_{\alpha }$$
(5)

and $${\sigma }_{{f}^{\parallel }}$$, $${\sigma }_{{f}^{\perp }}$$ are the uncertainties of the fluxes f and f, respectively. Sky background was generally calculated in a region close to the comet but with little apparent contamination from the coma. FORS2 instrumental polarisation around the centre of the field of view is ≤0.03%, hence negligible in the context of our observations68, but background polarisation must be estimated within ~1 arcmin from the source to avoid instrument polarisation that becomes significant at the edge of the field of view69. The reduced Stokes parameters are reported adopting as a reference direction the perpendicular to the great circle passing through the object and the Sun, using the formula

$$\frac{Q}{I}= \cos \left(2(\chi +\epsilon +{{\Phi }}+\pi /2)\right)\ \frac{Q^{\prime} }{I}+\sin \left(2(\chi +\epsilon +{{\Phi }}+\pi /2)\right)\ \frac{U^{\prime} }{I}$$
(6)
$$\frac{U}{I}=-\sin \left(2(\chi +\epsilon +{{\Phi }}+\pi /2)\right)\ \frac{Q^{\prime} }{I}+\cos \left(2(\chi +\epsilon +{{\Phi }}+\pi /2)\right)\ \frac{U^{\prime} }{I}$$
(7)

where $$Q^{\prime}$$ and $$U^{\prime}$$ are the Stokes parameters measured in the instrument reference system, χ is the instrument position angle (counted counterclockwise from North to East) at the time of the observations, ϵ is an angle, filter dependent, that is introduced to correct for the chromatism of the retarder waveplate67, Φ is the angle between the direction Object-North Pole and the direction Object-Sun. This angle can be calculated applying the four parts formula to the spherical triangle defined by the object with coordinates (αT,δT), the Sun, with coordinates (α,δ) and the North celestial pole70:

$$\sin {\delta }_{{\rm{T}}}\cos ({\alpha }_{\odot }-{\alpha }_{{\rm{T}}})=\cos ({\delta }_{{\rm{T}}})\tan ({\delta }_{\odot })-\sin ({\alpha }_{\odot }-{\alpha }_{{\rm{T}}})\frac{1}{\tan ({{\Phi }})}\ .$$
(8)

Q/I estimated from Eqs. (6)–(7) represents the flux perpendicular to the plane Sun-object-Earth (the scattering plane) minus the flux parallel to that plane, divided by the sum of these fluxes. For aperture polarimetry, Stokes parameters were calculated from fluxes measured in apertures up to ~9″ size) at one or two pixel (=0.125/0.25″) increments, and the adopted polarisation value is determined by the plateau observed in the growth curve, as illustrated in Fig. 5. As a further quality check, we have also calculated the null parameters NQ and NU that are the difference between the corresponding reduced Stokes parameters calculated from consecutive pairs of measurements67 using the formulas

$${N}_{X}=\frac{1}{2N}\mathop{\sum }\limits_{j=1}^{N}{(-1)}^{(j-1)}\left[{\left(\frac{{f}^{\parallel }-{f}^{\perp }}{{f}^{\parallel }+{f}^{\perp }}\right)}_{{\alpha }_{j}}-{\left(\frac{{f}^{\parallel }-{f}^{\perp }}{{f}^{\parallel }+{f}^{\perp }}\right)}_{{\alpha }_{j}+4{5}^{\circ }}\right]\ .$$
(9)

Null parameters are expected to have a Gaussian distribution centred about 0 with the same σ given by Eq. (4), and their deviations from zero would flag the possible presence of systematics effects. Null parameters were found consistent with zero for all datasets except those obtained on 2020-02-06, where they deviate from zero by 5σ for aperture radius > 1″. The presence of systematic effects, also suggested by the deviation from zero of U/I in the IF filter, is likely due to the bad seeing conditions under which the observations were carried out (the observations were actually repeated at a later epoch), joined to a particularly crowded background. For this epoch, polarimetry was measured within a 1″ wide aperture.

In addition to aperture polarimetry described above, we also employed a slit polarimetry method25 which allows us to measure polarisation along the tail, or any other chosen direction. A slit of adjustable width was chosen around the comet nucleus and aligned with the comet tail or other features. In this way, most of the background sources can be avoided, improving the quality of the final results, although some comet signal is lost in the process. Some artefacts remain, best seen in the photometric plots of IF filter in Fig. 4. A similar method was used in the past to obtain radial profiles of the tail and jets of comet 67P/Churyumov-Gerasimenko71. The width of the slit, centred on the comet photocentre, was between 36 and 42 pixels (4.5–5.25″) across the dust tail (Fig. 4a–d) and between 18 and 20 pixels (2.25–2.5″) across the anti-sunward direction (Fig. 4e–h).

### Photometry

We measured the comet coma brightness using circular apertures of 2.5″, which were large enough to include the total flux of trailed stars for which the magnitude was known through Gaia72 observations. As a drawback for using such a large aperture size, most of the frames had to be discarded due to the crowded stellar background which contaminated the photometry of the comet. In the end, we obtained reliable photometry only from data obtained during the night of 2019-12-25 for each of the FORS2 filters used (VF, RF and IF), and during the night of 2020-02-06 for the images gathered with RF and IF filters.

In order to calculate the dust optical colours in the Johnson-Cousins UBVRI system, we calculated an empirical transformation between FORS2 filters and the former system (private communications with C. Jordi). We integrated the spectra for Johnson-Cousins R, V and I filters, as well as for VF, RF and IF filters using a virtual star with G = 15 without adding any interstellar extinction. The choice of the magnitude selected is irrelevant since we are only interested in the relative difference between values. The relation between filters resulted to be almost perfectly linear (r = 0.999) and therefore the transformation can be expressed in the form of the linear equations

$${({{V}}-{{R}})}_{{\rm{JC}}}=-0.0171+1.1101\,{(V-R)}_{{\rm{F}}}$$
(10)
$${({{R}}-{{I}})}_{{\rm{JC}}}=0.0315+0.9558\,{(R-I)}_{{\rm{F}}}$$
(11)

where (VR)JC and (RI)JC are the optical colours expressed in the Johnson-Cousins system, whereas (VR)F and (RI)F are the colours measured with the corresponding FORS2 filters.

### Statistical tests

Statistical prediction limits for the observations at significance level (1 − q) × 100% may be expressed as

$${\mathrm{P}}{ {\mathrm{L}}}_{q\times 100}(\alpha )={P}_{r}(\alpha ,\hat{p})\pm {t}_{n-4}(q/2)\sqrt{\frac{\mathop{\sum }\nolimits_{i}^{N}{({Y}_{i}-{P}_{r}({\alpha }_{i};\hat{p}))}^{2}}{n-4}}\sqrt{1+{J}^{T}(\alpha ){C}_{p}J(\alpha )}$$
(12)

where N is the sample size, Yi is the measurement at αi, p = [b, c1, c2, α0] is the parameter vector, $$\hat{p}$$ and Cp the parameter estimates and associated covariance from the fit, J = pPr(α; p) evaluated at $$p=\hat{p}$$ and tn−4(q/2) is the q/2 percentage point of Student’s distribution with n − 4 degrees of freedom. For the asymptotic case n =  we have t(0.05) = 1.645 and t(0.025) = 1.960. To quantify the disagreement between 2I/Borisov and the other comets, we fitted the ensemble cometary data to Eq. (1) using nonlinear least squares minimisation, subtracted the fit from the data and compared the predictive confidence limits (12) on the observations to the Borisov measurements. The result for the RF-filter and IF-filter measurements separately is shown in Fig. 6. Here we considered only data with available observational uncertainties and 0 < α < 60, yielding sets with N = 393 (RF filter) and N = 113 (IF filter) respectively. We then discarded the 5% of each dataset with the highest uncertainties. Our two RF-filter measurements of 2I/Borisov at α = 27. 5 and α = 28. 6 (red points, panel a) lie between PL05 and PL10, therefore the probability P that both RF-filter measurements follow the same phase-polarisation behaviour as the other comets is 2.5 × 10−3 < PR < 10−2. Our IF-filter measurement of 2I/Borisov at α = 28. 5 (panel b) lies between the same two predictive contours and therefore PI < 10−1. Though the RF and IF measurements must be correlated to some degree, in any case $${P}_{{{{R}}}_{{\rm{F}}},{{\rm{I}}}_{{{F}}}}<1{0}^{-2}$$.