E Caffau

L Monaco

P Bonifacio

S Korotin

S Andrievsky

S Cristallo

M Spite

F Spite

L Sbordone

P François

G Cescutti

S Salvadori

The CEMP star SDSS J0222-0313: the first evidence of proton ingestion in very low-metallicity AGB stars? ⋆

Keywords: carbon -Stars, abundances -Stars, Population II -Galaxy, halo

The CEMP star SDSSJ0222-0313: the first evidence of proton ingestion in very low-metallicity AGB stars?

Introduction

Carbon-enhanced metal-poor (CEMP) stars are very common objects in the metal-poor regime ([Fe/H] < -2.0). According to Beers & Christlieb (2005), a metal-poor star can be defined as a CEMP when [C/Fe] > 1.0 1 , and in this work we adopt their definition. Beers & Christlieb (2005) divide CEMP stars into the following sub-classes according to the abundance ratios (implying, besides C, Ba, and Eu): (i) CEMP-r when [C/Fe] > 1.0 and [Eu/Fe] > 1.0. These stars are supposed to be enhanced in heavy elements produced in the rapid n-capture process (r-process); (ii) CEMP-s when [C/Fe] > 1.0, [Ba/Fe] > 1.0, and [Ba/Eu] > 0.5. These stars are expected to be enriched in heavy elements produced in the slow n-capture process (s-process); (iii) CEMP-r/s when [C/Fe] > 1.0 and [Ba/Eu] < 0.5. These stars are charac-⋆ Based on observations collected with Mike at the Magellan-II (Clay) telescope at the Las Campanas Observatory under programme CN2018B-5. terised by enhancements in all heavy elements; (iv) CEMP-no when [C/Fe] > 1.0 and [Ba/Fe] < 1.0. In principle these stars could show the composition of the cloud where they formed.

More recently, Hansen et al. (2019) (see their Table 6) suggested a new classification, efficient in discriminate CEMP-s and CEMP-r, based on the [Sr/Ba] ratio. Both Beers & Christlieb (2005) and Hansen et al. (2019) introduced the sub-classes to all CEMP stars, without a closer look at the C abundance. Their classifications do not subdivide CEMP stars according to the absolute C abundance, A(C) 2 . Spite et al. (2013) suggested dividing CEMP stars according to their A(C) in a high and a lowcarbon band. Later, Bonifacio et al. (2015) suggested a different nature in the C enhancement and the chemical composition of CEMP stars in the two carbon bands. They hypothesised that the stars belonging to the high-carbon band are part of multiple systems, and that their abundance is the result of a mass transfer from the asymptotic giant branch (AGB) phase of the more massive, more evolved companion. These stars also show enhance-A& A proofs: manuscript no. j0222m0313_final ment in heavy elements (see e.g. Caffau et al. 2018) that put them in the CEMP-s, CEMP-r, or CEMP-r/s sub-classes. Lucatello et al. (2005) and Starkenburg et al. (2014) derived that 100 % of the CEMP-s stars they investigated show variation in radial velocity, which supports the mass transfer scenario suggested by Bonifacio et al. (2018). The CEMP stars of the low-carbon band, mainly CEMP-no stars according to Bonifacio et al. (2015), can be part of multiple systems (see e.g. Caffau et al. 2016), but their abundances reflect the chemistry of the gas cloud in which they formed. Recently, Arentsen et al. (2019) found four binary systems in a sample of CEMP-no stars, but the fact that some of these stars are binaries is not unexpected.

To understand the early formation and evolution of old, metal-poor stars, it is of invaluable importance to understand the formation of CEMP stars and the mechanism in place that enriched their atmospheres. The majority of the most iron-poor stars known to date are CEMP stars belonging to the low-carbon band described by Spite et al. (2013). The C-rich environment could have made star formation easier (see e.g. Bromm & Loeb 2003). At higher Fe abundance there are stars on both the highcarbon and low-carbon band (see e.g. Caffau et al. 2018), and understanding their composition is extremely important.

We present here the chemical investigation of a Magellan Inamori Kyocera Echelle (Mike) spectrum of SDSS J022226.20-031338.0 (SDSS J0222-0313, for short). We confirm here the abundances derived from a low-resolution FORS spectrum and we increase the number of elements for which we derive the abundance. This is a CEMP star, rich in heavy elements that we classify as a CEMP-s star, according to the scheme of Beers & Christlieb (2005). With its high [Sr/Ba] ratio, according to Hansen et al. (2019) it should be a CEMP-no star, but taking into account the uncertainties in the Sr and Ba determination, it would fit in their CEMP-s sub-class.

Selection

The star was selected by Caffau et al. (2018) from the Sloan Digital Sky Survey (SDSS York et al. 2000;Yanny et al. 2009) for its low metallicity and its strong G-band. It was observed on MJD=57956.32784048 with FORS2 (Appenzeller et al. 1998) at the ESO VLT, in service mode during the ESO programme 088.D-0791. The spectrum was taken with GRISM 1200B, with central wavelength at 436 nm, with 0 ′′ . 29 slit width corresponding to a resolving power of 5 000 (see Caffau et al. 2018, for details). From the chemical investigation of the FORS spectrum it was clear that the star is a CEMP, with an almost solar carbon abundance, a strong overabundance in Ba ([Ba/Fe] = 1.98) and an even higher overabundance in Sr ([Sr/Fe] = 2.25). Compared to stars in the same metallicity range, enhanced and nonenhanced in carbon, this star occupies an empty region in the [Ba/Fe] versus [Sr/Ba] diagram (see Fig. 1).

Observations

We observed SDSS J0222-0313 with the Mike spectrograph [START_REF] Bernstein | Proc. SPIE[END_REF]) mounted at the Magellan-II (Clay) 6.5 m telescope of the Las Campanas observatory. We used a 0.7 ′′ slit delivering a resolving power of R=53 000 and 42 000 on the blue and red side, respectively. We further employed a 2 × 2 on chip binning and fast readout mode. Seven one-hour exposures were taken on the target during the night between October 27 and 28, 2018, under clear sky conditions and seeing variable between Spite et al. 2014). The pink star represents the position of the star J0222-0313 from the analysis of the FORS spectrum (with log g of 4.0; Caffau et al. 2018), while the black star is the result of the investigation of the Mike spectrum, taking into account the NLTE corrections for Sr and Ba. The blue squares are stars from Spite et al. (2005), the green diamonds are CEMP stars from the literature (Aoki et al. 2001;Barbuy et al. 2005;Sivarani et al. 2006;Behara et al. 2010;Spite et al. 2013;Yong et al. 2013). The black horizontal dashed line is the r-only solar value for [Sr/Ba] according to Mashonkina & Gehren (2001). Normal stars are in the upper left part of the diagram, sharing the surface with CEMP-no stars; CEMP-s stars are in the lower right part of the diagram. 0.7-1.0 ′′ . Data were reduced using the CarPy3 python pipeline (Kelson et al. 2000;Kelson 2003). Thorium-argon lamp frames taken during the night were used for the wavelength calibration. Internal quartz lamp frames were used for order localisation, while both quartz frames and observations of the bright B2V star taken with the diffuser into the optical beam ('milky-flat') were used for flat fielding. Mike spectra cover the wavelength range ≈ 333 -506 nm (blue) and ≈ 484 -941 nm (red). Individual frames have signal-to-noise ratios (S/N) in the range 20 -25 at 400 nm and in the range 30 -40 at 780 nm, for all but the last frame of the night which has a significantly lower S/N.

Analysis

Radial velocity and kinematics

The radial velocity provided by the Sloan Survey on the SDSS DR12 spectrum is V r = -124 ± 3 km/s. We were able to measure the radial velocity from the FORS2 spectrum, V r = -90 ± 32 km/s, but due to the flexures of FORS and the lack of telluric absorption in the observed wavelength range, the uncertainty is large. From the Mike spectrum we derived V r = -127.1 ± 1.2 km/s from the cross-correlation of the red Mike spectrum in the spectral range 500-680 nm with a synthetic spectrum computed using the SYNTHE code (Sbordone E. Caffau et al.: The CEMP star SDSS J0222-0313 et al. 2004;Kurucz 2005) along with an ATLAS9 model atmosphere of parameters T eff =6400 K, log g=3.00, [Fe/H] = -3 and [α/Fe]=+0.4 and broadened to the resolving power of the Mike spectrum. This radial velocity value takes into account a zero point correction which was calculated by cross-correlating the Mike spectrum with a telluric spectrum calculated with the TAPAS4 [START_REF] Bertaux | [END_REF] web service for the time and location of the observations and the target coordinates. Crosscorrelations were performed using the IRAF5 task fxcor. Even though we expect that the star is in a multiple system, the data at our disposal do not strongly support any binarity information, but we expect the companion to be a low-mass white dwarf (0.5 M ⊙ ), and further investigations would be very useful in order to confirm this binarity.

The parallax for this star in Gaia DR 2 is negative, so we were not able to derive its distance. By using the Gaia colour BP-RP, we compared it to PARSEC isochrones of close metallicity and age of 10.2 Gyr. We ruled out the main sequence solution because the star would have been close enough to allow Gaia to give a parallax and because the dwarf solution is in disagreement with the iron ionisation balance (see next section). We considered the possibility of the star being a sub-giant (log g =3.8), deriving from the isochrones a distance of 5.4 kpc, and we considered the case of surface gravity derived by matching the Fe abundance from Fe i and Fe ii lines, which corresponds to 8.6 kpc (log g =3.4, assumed in the chemical investigation), respectively. We computed the star's orbit using the web-based interface GravPot166 (Fernández-Trincado et al. 2016) and in both cases the orbit happens to be a Halo type, with retrograde motion. The star reaches a maximum distance from the Galactic plane of 19 and 39 kpc, respectively. Considering its position in the angular momentum L z -energy diagram the star could belong to the Sequoia Event recently discovered by Myeong et al. (2019).

Stellar parameters

Comparing the Gaia photometry to an isochrone, the star could belong to the main sequence (MS), sub-giant (SG), or horizontal branch (HB). Gaia is not able to help us because the parallax value is negative, although Bailer-Jones et al. (2018) provide a distance of 3.0± 0.7 kpc assuming an exponential decrease in the Galactic density. At this distance the star should be a MS star, but this is just a statistical result that is not meant to be true for a single star. The MS solution is in disagreement with the chemical investigation and with the fact that at such close distance Gaia would have been able to measure a significant parallax.

We compared the BP-RP Gaia colour to a PARSEC isochrone of metallicity -2.7 and an age of 10.2 Gyr, from Bressan et al. ( 2012) and (without taking into account any reddening) we derived an effective temperature, T eff , of 6364, 6329, and 6223 K for the three cases of MS, SG, and HB, respectively. Analysing the Mike spectrum with a temperature in this range, the Fe balance is derived in the case close to the SG solution. By fitting Hα wings in the spectral order of the observed Mike spectrum containing Hα, for the five red observations we have for this star, we derive a T eff around 6300 K when the star is a sub-giant, in perfect agreement with the value from the isochrone. Caffau et al. (2018) derived an effective temperature of T eff =6345 K from the SDSS photometry. From Gaia BP-RP colour, when applying the reddening from Pan-STARRS (Green et al. 2018) we derive a temperature of 6540 K with the conversion recommended on the PARSEC isochrones site7 ; without reddening we have T eff =6330 K. The agreement is very close, but we decided to keep the effective temperature of 6345 K from Caffau et al. (2018).

The microturbulence of 1.7 km/s was derived from the Fe i lines, as a non-slope of the Fe abundance as a function of the strength of the lines. This microturbulence is consistent with values derived from high-resolution, high S/N spectra in stars of similar parameters (see e.g. Bonifacio et al. 2009).

We adopted the parameters T eff =6345 K, log g=3.4 and ξ = 1.7 km/s, and we derived [Fe/H]=-2.82. The uncertainties we associated with the stellar parameters are 100 K for the effective temperature, 0.4 dex for the surface gravity, and 0.2 km/s for the microturbulence. The uncertainty of 100 K in T eff is that derived from the fit on the wings of Hα; we recall that this temperature and the one derived from the Gaia DR2 colour are in very good agreement. For the gravity, a 0.4 dex uncertainty allows the star to be an SG or an HB star. For the microturbulence, we would not expect a microturbulence smaller than 1.5 km/s for an evolved star, nor larger than 1.8 km/s for a warm star. With these parameters the star does not fit in the PARSEC isochrone of the metallicity of the star. A surface gravity of 3.8 would put the star in the SG branch of the PARSEC isochrone (see Fig. 2). This value of gravity would keep the Fe abundances from the Fe i and Fe ii lines still compatible within uncertainties. A difference of 0.4 dex in the surface gravity does not change the chemical inventory of the star. For the SG solution, according to the isochrone, the star has a mass of M = 0.8M ⊙ .

Abundances

The spectrum was analysed with MyGIsFOS (see Sbordone et al. 2014), an automatic code able to derive chemical abundances by comparing an observed spectrum to a grid of synthetic spectra. We computed the synthetic spectra with the code SYNTHE (see Kurucz 2005;Sbordone et al. 2004) from a grid of ATLAS 12 model (Kurucz 2005). With MyGIsFOS we derived the abundances for Mg, Ca, Sc, Ti, Cr, Mn, Fe, Co, and Ni. For the other elements we computed an ATLAS 12 model with the stellar parameters of the star and an enhancement in Mg, C, and O, as we derived from the analysis, and the nitrogen (which we could not derive) enhanced as much as the carbon. From this model we computed with SYNTHE a grid of synthetic spectra with different abundances of the elements to be derived. We compared the observed spectrum to the grid of synthetic spectra and performed a χ 2 minimisation to derive the abundances.

For oxygen we measured the equivalent widths of the three lines of the triplet at 777 nm and, by using WIDTH (Kurucz 2005), we derived the oxygen abundance. For carbon we fitted the G-band and also six atomic lines visible in the Mike spectrum, thanks to the high C in this star. The A(C) derived from the G-band is in good agreement with the abundance derived from the C i line after applying the non-local thermodynamic equilibrium (NLTE) correction. From the G-band, by fitting the range 422.9-423.2 nm, we also derived the isotopic ratio 12 C/ 13 C as 7.4 ± 1.5, where the uncertainty is the formal 1 σ provided by the χ 2 fitting. The abundances we derived are listed in Table 1.

Table 1. Chemical abundances. For the solar reference values, C, O, Fe, and Th are from Caffau et al. (2011); the others are from Lodders et al. (2009). We have a tentative detection of Th from the 401.9 nm Th ii line. The measurement is uncertain and is shown in Fig. 3. This line is blended with a 13 CH line, which is taken into account in the fit of the Th line, with 12 C/ 13 C = 7.4. The presence of this feature at the wavelength of the Th ii line could be explained with a low 12 C/ 13 C ratio of 5 ± 1.5, but this value is only marginally consistent with the result from the CH line at 423 nm (see Fig. 4). We searched for U, but due to the relatively low S/N in this range, we cannot measure but nor do we rule out the presence of the 385.9 nm U ii line in the spectrum (see Fig. 5).

Element Z Nlines

A(X) ⊙ A(X) [X/H] σ [X/Fe] σ ∆NLTE A(X)NLTE CH 6 G-
The abundances with respect to iron, [X/Fe], and the upper limits are summarised in Fig. 6. Figure 7 shows the abundances with respect to yttrium, [X/Y], of the heavy elements, highlighting a peculiar chemical pattern of this star.

NLTE computations

Atomic level populations for all studied atoms were determined using the MULTI code (Carlsson 1988) with modifications as given in Korotin et al. (1999). Proper comparison of observed and computed profiles in many cases requires a multi-element synthesis to take into account possible blending lines of other species. For this process we fold the NLTE (MULTI) calculations, specifically the departure coefficients, into the LTE synthetic spectrum code SYNTHV (Tsymbal 1996), which enables us to calculate the NLTE source function for lines of the chem-ical element under consideration. These calculations included all spectral lines from the VALD data base (Ryabchikova et al. 2015) in a region of interest. The LTE approach was applied for lines other than the lines of the chemical element under consideration. Abundances of corresponding elements were adopted in accordance with the [Fe/H] value.

The NLTE corrections for all the atomic lines of C, O, Na, Mg, Al, Ca, Sr, and Ba were determined, and in Table 1 the NLTE corrections and the abundances after applying the NLTE corrections are provided. For carbon, we used the atomic model described in Lyubimkov et al. (2015). The model consists of 47 levels of C i, 11 levels of C ii, and the ground level of C iii. In addition, we included in the model 63 levels of C i and 9 levels of C ii in order to calculate partition function in LTE. A total of 207 radiative transitions were considered in detail. For 128 weak b-b transitions, radiative rates were considered in LTE. They were unchanged in calculations of the NLTE level populations. There is a remarkable difference in NLTE corrections between the lines of different multiplets for the neutral carbon. The lines in the optical domain are practically formed in LTE, and their NLTE corrections are small (they do not usually exceed -0.1 dex). The situation with lines in the IR region is the opposite. This is visible from the line-to-line scatter, which is definitely smaller after applying the NLTE corrections. The NLTE effects lead to a significant amplification of the profiles (corrections achieve -0.6 dex).

Fig. 2. PARSEC isochrone for [M/H]=-2.75 and an age of 10.2 Gyr (small black dots) (from Bressan et al. 2012). The red symbol shows the parameters we derived (T eff from the SDSS photometry and log g from the Fe ionisation equilibrium). The blue symbol on the isochrone corresponds to a log g that is 0.4 dex higher and corresponds to a disagreement in A(Fe) of 0.13 dex between the abundance derived from Fe i lines and that derived from Fe ii lines. This value is well within the uncertainties.

Fig. 3. Observed spectrum (solid black) in the range of the 401.9 nm Th ii line, compared with the best fit (solid red) providing A(Th) = 0.12, and two synthetic spectra at A(Th) of -0.3 and 0.5 (dashed blue). The Th ii line is blended with a 13 CH line, which is taken into account.

The NLTE model of the oxygen atom was first described by Mishenina et al. (2000), and then updated by Korotin et al. (2014). The model consists of 51 O i levels of singlet, triplet, and quintet systems, and the ground level of the O ii ion. An additional 24 levels of neutral oxygen and 15 levels of ions in higher states were added for particle number conservation. Fine structure splitting was taken into account only for the ground level and the 3p5P level (the upper level of the triplet lines at 777 nm). A total of 248 bound-bound transitions were included. Accurate quantum mechanical calculations were employed for the first 19 levels to find collision rates with electrons (Barklem 2007).

The NLTE atomic model of sodium was presented by Korotin et al. (1999) and then updated by Dobrovolskas et al. (2014). The updated sodium model currently consists of twenty energy levels of Na i and the ground level of Na ii. In total, 46 radiative transitions were taken into account for the calculation of the population of all levels. Fine structure splitting was taken into account only for the 3p level to ensure reliable calculations of the D line profiles. Collisional cross-sections obtained using quantum mechanical computations (Barklem et al. 2010) were used for the nine lowest levels. For other levels, the classical formula of Drawin was utilised in the form suggested by Steenbock & Holweger (1984), with the correction factor S H = 1/3. The NLTE effects strengthen the sodium lines.

In the case of Mg, we used the model atom from Mishenina et al. (2004). It consisted of 84 levels of Mg i, 12 levels of Mg ii, and the ground state of Mg iii. In the computation of departure coefficients, radiative transitions between the first 59 levels of Mg i and the ground level of Mg ii were taken into account. All 424 b-b transitions were included in the linearisation procedure. The model then was modified by Černiauskas et al. (2017), who added collisional rates with hydrogen atoms for transitions between the eight lower levels (Barklem et al. 2012). From the ADAS database 8 effective collision strengths with electrons were taken for the transition between the 13 lower levels. This modified model was tested with the help of the solar spectrum, Procyon spectrum and spectra of three stars with metal deficiency: HD 211998 ([Fe/H]=-1.6), HD 140283 ([Fe/H]=-2.5), and HD122563([Fe/H]=-2.7). The magnesium lines we analysed have NLTE corrections that are rather small. They have different signs and do not exceed 0.10 dex.

The Al atomic model is described in detail in Andrievsky et al. (2008). This model atom consists of 76 levels of Al i and 13 levels of Al ii. The model was modified to some extent. First we added collisional rates with hydrogen atoms for transition between several lower levels (Belyaev 2013). Similarly to the magnesium model, we also used spectra of several stars to test the aluminium atomic model. The spectra of the following stars were used: the Sun, Arcturus, Pollux, HR4796, Procyon, and Canopus. Comparison of observed and synthesised profiles of lines from different multiplets show good agreement, and this confirms the reliability of our aluminium model. Unfortunately 8 Summers H.P. 2004, version 2.6 -http://www.adas.ac.uk Fig. 7. Abundances of the heavy elements with respect to Y that we derived for this star.

in our spectroscopic analysis of SDSS J0222-0313 we were limited by only one available Al i line at 396 nm strongly affected by NLTE, which translates into a large NLTE correction (see Table 1).

The calcium abundance was derived by analysis of calcium lines in the two ionisation states. Our model of Ca atom consists of 70 levels of Ca i, 38 levels of Ca ii, and the ground state of Ca iii. In addition, more than 300 levels of Ca i and Ca ii were included to keep the condition of the particle number conservation in LTE. The information about the adopted oscillator strengths, photoionisation cross-sections, collisional rates, and broadening parameters can be found in Spite et al. (2012). This model was modified later on and collisional rates between calcium and hydrogen atoms for the 20 lower levels of Ca i were added. The necessary data were taken from Belyaev et al. (2017). Similar to the atomic model of magnesium and aluminium, our Ca models were tested with the help of the spectra of well-studied stars: the Sun, Arcturus, Pollux, and Procyon.

The strontium atomic model includes 44 low levels of Sr ii with n ≤ 12 and l ≤ 4 and the ground level of Sr iii. It also accounts for the fine splitting under the terms 4d2D and 5p2P0, which is why we included 24 Sr i levels only in the equation of particle number conservation. A more detailed description of the model atom can be found in Andrievsky et al. (2011).

The barium model contains 31 levels of Ba i, 101 levels of Ba ii with n ≤ 50, and the ground level of Ba iii. The 91 b-b transitions between the first 28 levels of Ba ii (n ≤ 12 and l < 5) were also computed in detail. For two levels, 5d2D and 6p2P0, the fine structure was taken into account. The odd Ba isotopes have hyperfine splitting of their levels, and thus several hyperfine structure components for each line (Rutten 1978). This effect is most pronounced for the Ba ii lines 455.4 and 649.6 nm. The information about the adopted oscillator strengths, photoionisation cross-sections, collisional rates and broadening parameters can be found in Andrievsky et al. (2009). In the spectrum of our programme star, two resonance lines (455.4 and 493.4 nm), and one subordinate line (614.1 nm) were analysed.

Uncertainties

The uncertainties reported in Table 1 are the line-to-line scatter for both the LTE and NLTE analysis. When the abundance was derived from one single line or the G-band we looked at the uncertainty in the determination of the continuum and the S/N to estimate the error. These uncertainties are reported in the figures. In Table 2 we list the systematic uncertainties related to the uncertainties in the stellar parameters. The uncertainty on the [Sr/Ba] ratio is not very affected by the uncertainties on the stellar parameters because they react in a similar way as a consequence of changes in the stellar parameters.

Discussion

SDSS J0222-0313 is characterised by a high C abundance that places the star on the high-carbon band as defined by Spite et al. (2013) and Bonifacio et al. (2015, see their Fig. 6). In Fig. 8 we present a revisit on this plot with SDSS J0222-0313 depicted as a red star symbol. Its C abundance (A(C)=8.45) is even higher than the value we derived from the low-resolution spectrum in Caffau et al. (2018). In that paper we concluded that the stars in the high-carbon band, which showed a systematic enhancement in Ba, belong to multiple systems and that their chemical composition has been altered by a more evolved companion, while the stars in the low-carbon band, with a normal Ba abundance, have a chemical composition of the gas cloud in which they formed. In Fig. 9, we compared the abundances of C, O, Sr, and Ba with respect to Fe derived for SDSS J0222-0313 with stars analysed by Hansen et al. (2016) and Hansen et al. (2019), which are mostly CEMP stars. SDSS J0222-0313 is always in the upper part of each panel, but still consistent with the comparison sample stars.

SDSS J0222-0313 is rich in Ba (i.e. [Ba/Fe] > 1) with [Ba/Fe] = +1.86 in LTE; it is also rich in Sr with [Sr/Fe] = +2.66 in LTE, and has a significant upper limit in Eu ([Eu/Fe] < -1.0). The star is rich in heavy elements produced mainly during s-process (Sr peak and Ba peak), but not in elements produced mainly during r-process. It also belongs to the high-carbon band, so SDSS J0222-0313 could be classified as a CEMP-s star. (Bonifacio et al. 2015;Caffau et al. 2016;Bonifacio et al. 2018;Sivarani et al. 2006;Bonifacio et al. 2009;Spite et al. 2013;Behara et al. 2010;Sivarani et al. 2004;Caffau et al. 2013) and unevolved CEMP stars (light blue square) and evolved CEMP stars (double light blue squares) from the literature (Yong et al. 2013;Cohen et al. 2013;Carollo et al. 2014;Masseron et al. 2012;Jonsell et al. 2006;Thompson et al. 2008;Cohen et al. 2003;Hansen et al. 2015Hansen et al. , 2016;;Lucatello et al. 2003;Aoki et al. 2008;Aoki et al. 2002;Frebel et al. 2005;Aoki et al. 2006;Li et al. 2015;Norris et al. 2007;Christlieb et al. 2004;Keller et al. 2014;Frebel et al. 2015;et al. 2014;Ivans et al. 2005). However, it shows at least two chemical features that are rather peculiar for a CEMP-s star. The first is the very high ratio of [Sr/Ba]=0.80 in LTE, which cannot be reproduced by a standard s-process nucleosynthesis in AGB stars at low metallicities (Bisterzo et al. 2012). We note that even low-metallicity massive AGB stars, whose heavy element nucleosynthesis may be dominated by the 22 Ne(α,n) 25 Mg reaction (which tends to produce more low-mass s-elements, ls, than high-mass s-elements, hs), are not able to fit the spectrum observed in SDSS J0222-0313 (see e.g. Cristallo et al. 2015). This ratio would place this star in the CEMP-no area, according to the classification of Hansen et al. (2019, see Fig. 10), but SDSS J0222-0313 is clearly too rich in Ba and Sr to be a CEMP-no star. This misclassification, still compatible with the classification by Hansen et al. (2019) within the uncertainties, is a clear symptom that this star has an uncommon chemical pattern. The s-process nucleosynthesis in rotating massive stars is able to produce this level of [Sr/Ba] ratio (Frischknecht et al. 2016;Limongi & Chieffi 2018), but rotating massive stars eject Sr and Ba only during supernovae explosion. For this reason, a mass transfer should be excluded in this scenario (but see Choplin et al. 2017). Chemical evolution models with rotating massive stars fail to reproduce the almost solar A(C) value (Cescutti et al. 2016) and enhancement of strontium and barium (Cescutti et al. 2013). Furthermore, cosmological chemical evolution models for the formation of the Milky Way predict that CEMP stars should have [C/Fe] < 2.0 at [Fe/H] > -3 if they are enriched by primordial faint supernovae (see e.g. Fig. 8 in de Bennassuti et al. 2017), which can be produced by fast-rotating massive stars (e.g. Meynet et al. 2006).

We can thus exclude this scenario since SDSS J0222-0313 has [C/Fe] = +2.73 and [Fe/H] = -2.82. In addition, SDSS J0222-0313 has a second peculiarity. This star actually has an extremely low abundance of lanthanum compared to that of barium: the upper limit on lanthanum ([La/Fe] < 0.92) is basically 1 dex below the [Ba/Fe] ratio. Neither standard s-process nucleosynthesis in AGB stars nor s-process nucleosynthesis in rotating massive stars can reproduce this 1 dex difference between these two elements.

It is well known that, at very low metallicities, a sub-sample of CEMP stars show surface distributions enriched in both sand r-process elements (the so-called CEMP-rs stars). Even if these objects are characterised by very high [hs/ls] ratios (thus at odds with SDSS J0222-0313), it may be worth considering their pollution history. One of the most popular explanations ascribes these exotic distributions to a nucleosynthesis process, halfway between the s-process and r-process: the so-called intermediate process (i-process, characterised by neutron densities n n ∼ 10 14 ÷ 10 16 cm -3 ; Cowan & Rose 1977). The stellar site(s) where the i-process is at work has(have) not been univocally identified yet. To date, the most promising candidates are rapidly accreting white dwarfs (RAWDs) and proton ingestions in low-mass low-metallicity stars. Denissenkov et al. (2017) have proposed RAWDs in close binary systems as an astrophysical site for the i-process. They concluded that these objects are an important site for the Galactic production of elements belonging to the first-peak of the sprocess (comparable to low-mass AGB stars). RAWDs are able to reproduce the surface heavy element distribution of Sakurai objects (characterised by almost solar metallicities; see Herwig et al. 2011). However, at low metallicities these objects produce more heavy s-elements than light s-elements (see e.g. Fig. 11 of Denissenkov et al. 2018). Thus, we discard this possibility to explain the abundances observed in SDSS J0222-0313 (which has a very low [hs/ls] ratio).

In recent years, one-zone post-process calculations for the i-process have become available [START_REF] Dardelet | confE.145D[END_REF]Hampel et al. 2016). These calculations are characterised by very high neutron densities (n n > 10 15 cm -3 ) and, as a consequence, show high [Ba/La] and [hs/ls] ratios. While the first ratio closely agrees with the spectrum observed in SDSS 0222-0313, the second clashes with the low measured [hs/ls]. These theoretical explorations do not anchor the occurrence of the i-process to a specific stellar site.

A promising physical process to produce the abundance pattern as observed in SDSS 0222-0313 is the ingestion of hydrogen in a convectively unstable He-burning region (hereafter proton ingestion episode, PIE). Such a peculiar mixing episode strongly depends on the stellar mass and metallicity: the lower the two quantities, the higher the probability of having a PIE (with an increasingly higher efficiency). There is a vast amount of literature on PIEs, which may occur during off-centre Heburning flashes (before core He-burning at extremely low metallicities) or at the first fully developed thermal pulse at slightly higher Z (Hollowell et al. 1990;Fujimoto et al. 2000;Iwamoto et al. 2004;Campbell & Lattanzio 2008;Cristallo et al. 2009Cristallo et al. , 2016;;Koch et al. 2019). As a general rule, these models show [hs/ls] not compatible (i.e. too high) with SDSS J0222-0313. Nevertheless, hints that both these conditions are satisfied can be found in the model published by Cristallo et al. (2009), i.e. a 1.5 M ⊙ model at [Fe/H]=-2.44 with no alpha-element enrichment ([α/Fe]=0) 9 .

In this model, at the time of the first fully developed thermal pulse (TP), some protons are engulfed in the growing convective shell triggered by the sudden activation of the 3α process. This occurs because the entropy barrier supported by the hydrogenburning shell is small, due to the rather low CNO abundance in the envelope. As soon as protons are mixed within the shell, they are mixed downward and burn on-the-fly. Cristallo et al. (2009) calculated them by setting appropriate limits to the mixing of protons, which take into account the ratio between the mixing turnover timescale and the local proton burning timescale. Thus, protons are mixed down to a fixed mass coordinate, locally releasing a large amount of energy. On the other hand, H-burning products (in particular 13 C and 14 N) are mixed down to the bottom of the convective shell, where the temperature largely exceeds 100 MK and the 13 C(α,n) 16 O is efficiently activated. As a consequence, a rich s-process nucleosynthesis develops (see Figure 5 of Cristallo et al. 2009). When the energy released by the H-burning exceeds the energy produced at the bottom of the shell, the convective shell splits, and the two sub-shells experience a completely different nucleosynthesis. In particular, the E. Caffau et al.: The CEMP star SDSS J0222-0313 upper shell continues to ingest protons, thus further synthesising 13 C, which burns at a lower temperature and thus is only able to feed the first s-process peak (Sr-Y-Zr). Later, the envelope penetrates downwards (third dredge-up, TDU), mixing the whole upper shell within the envelope, which results strongly enriched in light s-elements only (see Figure 6 of Cristallo et al. 2009).

Actually, we also expect the envelope to be slightly Ba-rich. This derives from the fact that, during a PIE, a large amount of 135 I is produced, which later decays to 135 Cs (τ ∼9 hr), and finally to 135 Ba (τ ∼ 2 × 10 6 yr). This is a consequence of the extremely high neutron densities attained by the model (n n > 10 14 cm -3 ). Moreover, this model predicts a very low envelope 12 C/ 13 C ratio (less than 10) and large overabundances of 14 N (from H-burning) and 16 O (coming from the 13 C(α,n) 16 O reaction). A number of points, however, deserve to be discussed in more detail. First, it should be noted that the model presented by Cristallo et al. (2009) and the subsequent set published in Cristallo et al. (2016) experience further normal TPs followed by efficient TDUs. These TDUs smooth or even erase the nucleosynthesis features characterising the first TDU. Actually, the model presented in Cristallo et al. (2009) has [Fe/H]=-2.45 and [α/Fe]=0, while the models published in Cristallo et al. (2016) were computed with an initial [Fe/H]=-2.85 and an α enrichment [α/Fe]=0.5 (i.e. the value characterising halo stars, on average). The global metallicities of the two sets are equivalent (Z ∼ 5 × 10 -5 ), but they have a completely different relative element distribution. The inclusion of an α enhancement strongly influences the nucleosynthesis triggered by the PIE. As a net result, the 1.5 M ⊙ model in Cristallo et al. (2016) shows a large [hs/ls] already at the first TDU. In fact, an increase in the oxygen abundance (as in the α-enhanced case) delays the occurrence of the shell splitting. On the other hand, SDSS J0222-0313 shows an α-rich spectrum, and thus we cannot ignore it. However, one additional thought should be given to the treatment of these peculiar processes in hydrostatic 1D models. During a PIE, the nuclear energy released within the convective turnover timescale is coupled to the turbulent mixing, which occurs on different length scales. Therefore, average quantities (as calculated in 1D codes) may not be representative of the real processes occurring in stars. In the past, there have been attempts to simulate these mixing events with 3D simulations (Mocák et al. 2010;Stancliffe et al. 2011;Herwig et al. 2014;Woodward et al. 2015). However, the details (and results) of the available simulations differ widely. Moreover, current simulations are strongly dependent on the adopted resolution. As a consequence, the current available hydrodynamic 3D simulations are not ready to efficiently (and firmly) constrain the physics and nucleosynthesis of PIEs yet. Some of these 3D models, however, show that some material is mixed through the hydrogen burning region, thus relaxing the assumption that protons can only be mixed down to a well-defined layer (as done in Cristallo et al. 2009 andCristallo et al. 2016 models).

Another important point to be discussed is the effect that a PIE would have on models with an initial mass M≤ 1 M ⊙ . For these low masses, we expect that the occurrence of the PIE may trigger a dynamical expulsion of the whole envelope, due to the sudden release of energy by CNO burning at the base of the upper shell. This would avoid any further mixing in the envelope, thus preserving the PIE features in the material lost. When this material is accreted by a companion star, the nucleosynthetic pattern is fully preserved.

Far from concluding that the uncommon heavy element distribution of SDSS J0222-0313 is unequivocally ascribed to a PIE, we guess that a very low-metallicity model ([Fe/H] ∼ -2.85) with a low initial mass (M≤ 0.9 M ⊙ ) may closely reproduce all the observed features. We reserve the calculation of this model and the exploration of the adopted physical prescriptions to a future paper, but it is worthwhile to discuss this scenario. In this hypothetical binary system, the difference in mass between SDSS J0222-0313 and the more evolved companion would be small. According to the PARSEC isochrone, SDSS J0222-0313 has a mass of about 0.8 M ⊙ . If the primary companion of the system has a slightly higher mass (e.g. M∼0.9 M ⊙ ), it would arrive on the AGB with a tiny envelope (due to the mass lost during the red giant branch phase). In the case of a dynamical expulsion of the whole envelope, it would be plausible to have a consistent fraction of material accreted on the secondary companion. At that epoch, this secondary companion would still be in the main sequence phase and its envelope would be radiative. As a consequence, the accreted material would not need any dilution, apart from secular effects due to the gravitational settling for example; however, they are hard to predict a priori (see e.g. Stancliffe 2010).

The fact that no other star with a chemical pattern like that of SDSS J0222-0313 has yet been found, may be also due to a lower probability of forming stars with similar masses in binary systems than with a large difference in masses. The distribution of the mass ratio10 for solar metallicity stars peaks at about q = 0.25 (Duquennoy & Mayor 1991), while theoretically for Pop III stars it should be even more biased towards very low mass ratios (Stacy & Bromm 2013). The fact that we do not see variations in the radial velocity of SDSS J0222-0313 can be attributed to too few secured spectra, or spectra acquired at the wrong time.

Conclusions

SDSS J0222-0313 is a halo star with a retrograde orbit, compatible with the Sequoia accretion event described by Myeong et al. (2019). It is a CEMP star belonging to the high-carbon band described by Spite et al. (2013), with an almost solar C abundance and a low 12 C/ 13 C ratio of 7.4. Its atmosphere is also enriched in neutron-capture elements of the first peak and in Ba. Its chemical composition could be explained by mass transfer from a more evolved low-mass companion in its AGB phase. The conditions for the more evolved companion to discharge on its atmosphere n-capture elements of the first peak and not of the second peak can be due to a quite rare case of proton ingestion episode, as described by Cristallo et al. (2009). No variation in radial velocity is clearly evident from the acquired spectra to confirm that the star has a companion. The low mass of the white dwarf companion would change the radial velocity by a small quantity, but we can also be dealing with a pole-on system, for which we could never detect variation in radial velocity. We will try to secure further spectra in the future in order to have more radial velocity measurements to see if we are able to derive the orbit of the system and so the mass ratio. These high-resolution spectra would also allow us to determine the odd-to-even ratio of barium isotopes, thus supporting or discarding the hypothesis that the chemical distribution of this star is a fingerprint of a proton ingestion episode (which predicts a large enhancement of 135 Ba with respect to 138 Ba, the most abundant isotope in normal s-process enriched stars).

With all their limitations, classifications of CEMP stars are very useful to place stars with similar chemical properties in the same sub-class; for example, the Spite et al. (2013) divi-A& A proofs: manuscript no. j0222m0313_final sion in A(C) is very useful for distinguishing CEMP-no from other CEMP stars. With the classification by Hansen et al. (2019) we can quite confidently separate a CEMP-s from a CEMP-r star. But these classificaitons can also be used to help recognise strange objects, which can be identified by their misfits.

Fig. 1 .

 1 Fig. 1. [Ba/Fe] vs. [Sr/Ba] for a sample of metal-poor stars ( an updated version of Fig. 1 in Spite et al. 2014). The pink star represents the position of the star J0222-0313 from the analysis of the FORS spectrum (with log g of 4.0; Caffau et al. 2018), while the black star is the result of the investigation of the Mike spectrum, taking into account the NLTE corrections for Sr and Ba. The blue squares are stars from Spite et al.(2005), the green diamonds are CEMP stars from the literature(Aoki et al. 2001;Barbuy et al. 2005;Sivarani et al. 2006;Behara et al. 2010;Spite et al. 2013;Yong et al. 2013). The black horizontal dashed line is the r-only solar value for [Sr/Ba] according toMashonkina & Gehren (2001). Normal stars are in the upper left part of the diagram, sharing the surface with CEMP-no stars; CEMP-s stars are in the lower right part of the diagram.

Fig. 4 .

 4 Fig. 4. Top panel: Observed spectrum (solid black) in the range of the G-band, compared with synthetic spectra with different 12 C/ 13 C ratios (dashed red 90; dashed green 7.5, the best fit; and dashed blue 3). Bottom panel: Th ii line with the synthetic spectra of the top panel and with a solar scaled Th abundance.

Fig. 5 .

 5 Fig.5. Observed spectrum (solid black) in the range of the 385.9 nm U ii line.

Fig. 6 .

 6 Fig.6. Abundances with respect to iron that we derived for this star.

Fig. 8 .

 8 Fig.8. Plot of A(C) vs. [Fe/H] for known CEMP stars (fromSpite et al. 2013). SDSS J0222-0313 (red star) is compared to the CEMP sample stars (open red bullets) analysed in our collaboration(Bonifacio et al. 2015;Caffau et al. 2016;Bonifacio et al. 2018;Sivarani et al. 2006;Bonifacio et al. 2009;Spite et al. 2013;Behara et al. 2010;Sivarani et al. 2004;Caffau et al. 2013) and unevolved CEMP stars (light blue square) and evolved CEMP stars (double light blue squares) from the literature(Yong et al. 2013;Cohen et al. 2013;Carollo et al. 2014;Masseron et al. 2012;Jonsell et al. 2006;Thompson et al. 2008;Cohen et al. 2003;Hansen et al. 2015 Hansen et al. , 2016;;Lucatello et al. 2003;Aoki et al. 2008;Aoki et al. 2002;Frebel et al. 2005;Aoki et al. 2006;Li et al. 2015;Norris et al. 2007;Christlieb et al. 2004;Keller et al. 2014;Frebel et al. 2015; et al. 2014;Ivans et al. 2005).

Fig. 9 .

 9 Fig. 9. Abundances of the SDSS J0222-0313 in LTE (blue star) compared to the CEMP sample from Hansen et al. (2016) (black diamonds) and Hansen et al. (2019) (red squares).

Fig. 10 .

 10 Fig. 10. [Sr/Ba] ratio of SDSS J0222-0313 (blue star for the LTE ratio and open circle for the NLTE value) compared to the CEMP sample from Hansen et al. (2016, black diamonds) and Hansen et al. (2019, red squares), with [Fe/H] < -2.0. Error bars for SDSS J0222-0313 are the uncertainties in [Ba/Fe] and [Sr/Ba]. We include the division (dotted lines) suggested by Hansen et al. (2019) in their figure 7. The star SDSS J0222-0313, being a CEMP-s star, would be expected to have a lower [Sr/Ba] value due to its high [Ba/Fe].

Table 2 .

 2 Variations in the chemical abundances related to changes in the stellar parameters.

	Element			∆(A(X))
		T eff -100 T eff +100 log g-0.4 log g+0.4 ξ ∓ 0.2
	C i	-0.04	0.05	0.14	-0.12 ±0.01
	O i	-0.07	0.06	0.13	-0.13 ±0.02
	Na i	0.08	-0.07	-0.07	0.11 ±0.05
	Mg i	0.07	-0.07	-0.05	0.07 ±0.04
	Al i	0.09	-0.09	-0.01	0.00 ±0.03
	Ca i	0.03	-0.06	-0.02	0.02 ±0.03
	Ca ii	0.06	-0.05	0.05	-0.01 ±0.04
	Sc ii	0.04	-0.05	0.07	-0.14 ±0.07
	Ti ii	0.04	-0.04	0.14	-0.13 ±0.02
	Cr i	0.10	-0.09	-0.01	0.00 ±0.03
	Mn i	0.10	-0.11	-0.02	0.01 ±0.01
	Fe i	0.09	-0.08	0.00	0.01 ±0.05
	Fe ii	0.01	-0.02	0.13	-0.14 ±0.02
	Co i	0.12	-0.12	-0.01	0.01 ±0.02
	Ni i	0.10	-0.10	-0.01	0.00 ±0.03
	Sr ii	0.07	-0.07	0.03	-0.01 ±0.06
	Y ii	0.07	-0.06	0.11	-0.09 ±0.10
	Zr ii	0.05	-0.05	0.12	-0.13 ±0.10
	Ba ii	0.07	-0.08	0.14	-0.13 ±0.10
	Th ii	0.06	-0.06	0.12	-0.13 ±0.01

https://code.obs.carnegiescience.edu/ carnegie-python-distribution

http://cds-espri.ipsl.fr/tapas/

IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.

https://gravpot.utinam.cnrs.fr

http://stev.oapd.inaf.it/cgi-bin/cmd

At low metallicities an enrichment of isotopes is expected: 16 O, 20 Ne, 24 Mg, 28 S i, 32 S , 36 Ar, and 40 Ca.

defined as q = M 2 /M 1 , where M 2 ≤ M 1

Acknowledgements. EC and PB have been supported by the Programme National de Physique Stellaire of the Institut National des Sciences de l'Univers of CNRS. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

-----------------------


Mahmoudian Navid

Thomas Bidgoli

Aline Maugey

Roumy

EVALUATION FRAMEWORK FOR 360-DEGREE VISUAL CONTENT COMPRESSION WITH USER VIEW-DEPENDENT TRANSMISSION

Keywords: degree content, omnidirectional, Evaluation framework, Compression, User-dependent transmission

Immersive visual experience can be obtained by allowing the user to navigate in a 360-degree visual content. These contents are stored in high resolution and need a lot of space on the server to store them. The transmission depends on the user's request and only the spatial region which is requested by the user is transmitted to avoid wasting network bandwidth. Therefore, storage and transmission rates are both critical. Splitting the rates into storage and transmission has not been formally considered in the literature for evaluating 360-degree content compression algorithms. In this paper, we propose a framework to evaluate the coding efficiency of 360-degree content while discriminating between storage and transmission rate and taking into account user dependency. This brings the flexibility to compare different coding methods based on the storage capacity on the server and network bandwidth of users.

INTRODUCTION

With spherical visual content, also known as 360-degree or omnidirectional images and videos, users can freely observe the surrounding area and interact with the scene by means of head-mounted displays (HMD) or hand-held pointing devices as with the Google Maps Platform. These contents are compressed and stored on a server. Then, upon request of a direction by a user, only a small portion of the entire 360-degree content is displayed. Another characteristic of these contents is that each requested content is displayed with high resolution to provide a realistic immersive experience for the user. Hence these contents must be compressed efficiently in a way that the compressed stream can adapt to the user's request. This introduces naturally three criteria: storage S (the amount of data stored on the server), transmission rate R (the amount of data sent to the user) and the distortion D (quality of the view displayed to the user).

The goal of this paper is to propose a framework to evaluate the efficiency of 360-degree content compression algorithms that takes into account the three above mentioned criteria. Despite several contributions that have been made to improve user-dependent compression, there is no proper evaluation framework to compare different coding schemes. This might be explained by the fact that the proposed 360-degree content rely on classical 2D compression standards. Indeed, compressing 360-degree content generally first consists in projecting the spherical content to a new 2D representation before coding the data. Different projections are used, e.g., equirectangular [START_REF] Snyder | Flattening the Earth: Two Thousand Years of Map Projections[END_REF], cube map [START_REF] Kuzyakov | Next-generation video encoding techniques for 360 video and vr[END_REF][START_REF] Ng | Data compression and transmission aspects of panoramic videos[END_REF], dodecahedron [START_REF] Fu | The rhombic dodecahedron map: An efficient scheme for encoding panoramic video[END_REF], etc. A first This work was partially supported by the Cominlabs excellence laboratory with funding from the French National Research Agency (ANR-10-LABX-07-01) and by the Brittany Region (Grant No. ARED 9582 Inter-COR).

approach to transmit the data to the user consists in compressing and sending the projected content [START_REF] Yu | A framework to evaluate omnidirectional video coding schemes[END_REF][START_REF] Upenik | On the performance of objective metrics for omnidirectional visual content[END_REF]. A second approach avoids sending the whole content and the projection is further mapped into several entities to adapt to the user's requested direction. For instance, [START_REF] Zare | Hevc-compliant tile-based streaming of panoramic video for virtual reality applications[END_REF][START_REF] Qian | Optimizing 360 video delivery over cellular networks[END_REF] partition the equirectangular image into several tiles. Each tile is encoded independently and during transmission, the tiles related to the requested area are transmitted. In pyramid projection, the spherical content is mapped to multiple pyramids where the base of each pyramid presents a pre-defined viewport in higher resolution and the rest of the pyramid represents the non-viewport part in lower resolution. Upon user request, the pyramid whose base is closer to the user's viewing direction [START_REF] Kuzyakov | Next-generation video encoding techniques for 360 video and vr[END_REF] will be transmitted to the user.

Since the existing 360-degree content compression algorithms are based on 2D compression algorithms, the evaluation of these coding schemes still relies on classical 2D metrics. In [START_REF] Yu | A framework to evaluate omnidirectional video coding schemes[END_REF] the classical rate-distortion (RD) plots are used for the evaluation because the whole content is sent to the user (R = S). The novelty in [START_REF] Yu | A framework to evaluate omnidirectional video coding schemes[END_REF] is the computation of the average users' viewport quality, which is approximated by a weighted spherical Peak Signal-to-Noise Ratio (PSNR). When the whole content is sent, the authors in [START_REF] Upenik | On the performance of objective metrics for omnidirectional visual content[END_REF] provide a subjective quality evaluation and compare the results with a range of objective quality metrics. In [START_REF] Hosseini | Adaptive 360 vr video streaming: Divide and conquer[END_REF], a rate allocation strategy is proposed to lower the transmission rate, therefore only R is considered and not the trade-off between R and D. In [START_REF] Zare | Virtual reality content streaming: Viewport-dependent projection and tilebased techniques[END_REF], RD curves and sum of storage are used for the evaluation and the trade-off between S and D is not considered. To reduce the computational cost, they approximate the distortion using a set of discrete viewing orientations that does not depend on the users' navigation.

To the best of our knowledge, none of the existing evaluation methods consider jointly storage, transmission rate and distortion. In this paper, we propose a user-dependent evaluation framework to compare different 360-degree content compression methods in which the distortion at the user side is considered by computing the coding error in the viewport shown to users and at the same time, it differentiates the transmission rate from storage. For that, first, we propose to use Bjontegaard metric [START_REF] Bjontegaard | Calculation of average psnr differences between rd-curves[END_REF] to compute average storage and transmission rate saving for the same quality experienced by the user. Second, we propose iso points to compare methods when the system must satisfy some constraints. Finally, we extend Bjontegaard metric by combining the transmission rate and storage to take into account further constraints in the evaluation. The evaluation framework brings the ability to achieve a compromise between the capacity of the server to store the data and users' bandwidth and to have a clear understanding on the benefit of one method with respect to (w.r.t.) another one.

The rest of the paper is organized as follows. In Section 2 the 360-degree content compression with user-dependent transmission is formulated. Section 3 explains the proposed evaluation framework. We illustrate the evaluation methodology on existing coding schemes in Section 4.

COMPRESSION WITH USER-DEPENDENT TRANSMISSION

Available coding schemes are mostly able to compress 2D image/video formats. Therefore, the spherical image/video Γ is first mapped to one or several rectangular planes and then these mapped representations are compressed with existing 2D coding schemes.

To avoid suboptimal usage of the network bandwidth, the mapping also enables to extract and transmit the spatial region requested by the user's direction θ. If we denote the mapping function as m : Γ → I, where I is the corresponding 2D representation, the encoding procedure can be considered as

(f • m)(Γ) = (b1, . . . , bn)
where f is the encoding function and bi, i = 1, .., n are independently extractable bitstreams which are stored on the server. The storage size on the server is

S = |f • m(Γ)| = n i=1 |bi| (1)
A head rotation θ of the user triggers a request of a subpart γ(θ) of the spherical image Γ. Then, the server extracts parts of the stored bitstreams i ∈ I θ related to γ(θ) and sends them to the user for decoding. The transmission rate of request θ which is equal to the size of extracted bitstreams in I θ can be written as

R θ = i∈I(θ) |bi| (2)
where R θ ≤ S. After receiving the requested bitstreams, the decoder h θ decodes the requested region and maps it back on the sphere:

m -1 • h θ : (bi)i∈I θ → γ(θ).
Note that due to lossy compression γ(θ) = γ(θ). To measure a realistic quality of experience at the user side, the distortion D θ is computed in the viewport v(θ) shown to the user which is the image plane tangential to the sphere at a point defined by the radius vector with rotation θ (w.r.t. the sphere coordinate system)

D θ = ||v(θ) -v(θ)|| 2 2 .
Both distortion D θ and rate R θ vary significantly with the direction θ. Therefore, to be able to compare performance between different methods, the expected values over all users' requests are computed for these quantities. Compared to classical 2D video/image coding, the use of expected values over all users' requests for the transmission rate and distortion is new and comes from the userdependent navigation. A method which has lower expected distortion E θ (D θ ) for the same amount of storage S and same expected transmission rates E θ (R θ ) is preferred. Therefore, unlike conventional 2D image/video evaluation methodologies where 2D rate-distortion (RD) plots are used for comparison, here we need to compare them with 3D Storage-Rate-Distortion (S-R-D) curves (for simplicity from now on we omit the term expected and simply call S-R-D). By changing the Quantization Parameter (QP) of the encoder, different qualities of the 360-degree content are generated. It is worth noting that since the only free parameter that is adjustable is the QP value, the S-R-D values generate 3D curves.

PROPOSED EVALUATION FRAMEWORK

Performing a formal comparison between 3D curves is a difficult task especially when the curves have different domains as is the case for typical S-R-D curves, see Fig. 1. We propose 3 ways to compare methods based on the constraints imposed by the system.

0 1 2 3 •10 4 0 1 •10 7 30 40 E θ (R θ ) (in bits) S (in bits) PSNR Method A Method B Method C Method D Method E Fig. 1: Example of S-R-D curve.
Comparison between different methods is difficult in 3D. We propose 3 solutions to compare between methods.

Average storage and transmission rate saving

Bjontegaard-Delta (BD) rate [START_REF] Bjontegaard | Calculation of average psnr differences between rd-curves[END_REF] is a commonly used metric as it performs a fair comparison between two coding methods. More precisely, a common PSNR range is determined and the average bitrate reduction/increase is computed for this PSNR range. We extend the BD measure for storage and transmission rate by projecting S-R-D curves to R-D and S-D planes and we name them BD-R and BD-S respectively. The BD-S represents the average reduction/increase in the amount of data stored on the server for the same PSNR range omitting the transmission rate. BD-R is equivalent to classical BD-R and represents the average bit rate saving/increase for the same PSNR which is transmitted to each user and storage is not taken into account. To evaluate different methods, the pair (BD-R, BD-S) must be considered jointly.

It is worth noting that in [START_REF] Zare | Virtual reality content streaming: Viewport-dependent projection and tilebased techniques[END_REF], the BD-R is computed and the sum of storage over all stored qualities are compared. However the sum of stored qualities may not cover the same range of distortion, which leads to an unfair comparison. Introducing BD-S solves this issue because it considers the same range of PSNR for both methods.

Two problems can happen with this metric. The first one is due to the fact that sometimes it is not clear how to compare between pairs. For instance, assume (-20%, 10%) and (15%, -10%) for (BD-R, BD-S) pair between 2 methods (negative values represent percentage saving w.r.t. the reference). Here the former performs better on average in terms of transmission rate and the latter consumes less space on the server for the same PSNR. This problem will be tackled in Section 3.3. Second problem is that BD represents the average performance, but sometimes there are some constraints that the scheme must satisfy. Therefore, we propose iso points in the next subsection to evaluate methods not based on their average behavior, but rather on their performance under some constraint.

E θ (R θ ) A B C D (a) S E θ (R θ ) E θ (R θ ) + λ • S (R, S) λ • S (b) S E θ (R θ ) + λ 1 • S Method C Method D (c) S E θ (R θ ) + λ 2 • S Method C Method D (d)

Iso points

In this section, we propose a set of evaluation tools that can be used when the system must satisfy some constraints. For instance, the service might be constrained to serve the clients with a minimum distortion level d. To choose and compare the coding schemes, we introduce the iso-distorsion points:

{(Si, Ri, Di)|Di = d and i = indices of method} where linear interpolation of the S-R-D curve is used to compute the isocurve.

Similarly, iso storage (iso transmission rate) values show that at the same storage (transmission rate) what are the distortiontransmission rate (distortion-storage) values. Depending on what are the constraints of the streaming system (distorion or transmission rate or storage), one can plot iso points and compare the methods on iso values domain.

An example of iso distortion values is shown in Fig. 2a. Comparing method A and B which have iso distortion values, reveal that method A consumes less bandwidth than method B and it needs lower space on the server to store data. Therefore, method A is preferred over method B. Sometimes isocurves do not help to determine which method is better than the other. For instance, in Fig. 2a method C takes lower space on the server and method D performs better in terms of transmission rate.

Weighted BD

To solve the ambiguities occurred with isocurves and (BD-R,BD-S) pairs, there is a need to adjust the tradeoff between rate and storage. This can be done by introducing a weighted combination:

E θ (R θ ) + λ • S (3)
where λ balances the importance between storage and transmission rate and is determined by industrial constraints (a high value of λ means that storage is more important than transmission rate). With this definition (3), a point in the R-S plane is mapped onto a point on the axis depicted in blue in Fig. 2b. This mapping also yields a projection of the 3D S-R-D curve onto the 2D plane defined by the distortion D and the weighted combination [START_REF] Ng | Data compression and transmission aspects of panoramic videos[END_REF]. Examples of such curves are shown in Fig. 2c. Then, from two such curves, a BD measure can be computed. This is called weighted BD.

The choice of the regularizer parameter λ is crucial, and may yield opposite conclusions when comparing compression methods, see Figs. 2c and2d. For instance, (3) can result from

α • E θ (R θ ) + β • S ( 4
)
where the regularizers α, β can be used to impose another criterion like delay time for each method. For example, if β represents the time per byte that it takes to read the data from the hard drive in the server and α represents the inverse of the bandwidth, then (4) represents the overall delay time which consists of delay time for the server to read data from its hard drive plus the delay time for the data to be received by the users through the network. By performing BD over this new metric we can compare the delay time over the same PSNR values between two methods.

EXPERIMENTAL ILLUSTRATION

In this section, we illustrate the usefulness of the proposed framework for immersive compression scheme analysis. Four large size 8960x4480 panoramic images from the SUN360 database [START_REF] Xiao | Recognizing scene viewpoint using panoramic place representation[END_REF] are considered: 2 from indoor environments and 2 from outdoor environments. Note that the evaluation framework can also be applied in the same manner to 360-degree videos. As for the compression methods, we consider tile-based coding of equirectangular images with 3 tile sizes (640x320, 1280x640 and 4480x2240) which partition the panoramic image into 14x14, 7x7 and 2x2 tiles respectively. We denote them by E. 14x14, E. 7x7 and E. 2x2 respectively in the following. Each tile is encoded independently using HEVC intracoding [START_REF] Sullivan | Overview of the high efficiency video coding (hevc) standard[END_REF]. Upon user request, the server sends only the tiles related to the requested viewport. If any tile has been sent already in a previous request, the server avoids sending the tile again. We also implemented the cube map representation with 2 different face resolutions: 2560x2560 and 3008x3008 (the sum of six cube faces of size 2560x2560 is almost equal to the panoramic image size). We denote them by C. lower and C. higher respectively. Classically in cube representation, the server sends the whole cube. Here, in order to be similar to the tile-based approach, each face is encoded separately with HEVC intra-coding. Then, the server sends only the faces of the cube which are related to the requested viewport.

Our proposed evaluation framework does not rely on a specific user navigation model: it simply needs a set of user head position no matter how they are generated. We restrict our experiments to images, but one can perform similarly with videos. Since there is no long duration navigation dataset for this large size image database, we created one by following the recommendations in [START_REF] Yu | A framework to evaluate omnidirectional video coding schemes[END_REF]. In particular, it is observed in [START_REF] Yu | A framework to evaluate omnidirectional video coding schemes[END_REF] that users more frequently view areas around the equator (up to latitude ±30) than the poles. Moreover, when users are moving their head by changing the longitude and latitude angles, it is more likely for them to continue their previous direction rather than changing the head movement to the opposite direction. Therefore, we simulated user's navigation by defining three probabilities p1, p2, p3, for each of the longitude and latitude angles of the user's head orientation. p1 is the probability of continuing the previous head movement, p2 is the probability of staying at the current angle and p3, p1 > p3 is the probability of changing movement to the opposite direction. The latitude is initialized randomly in range ±30.

We generate navigation for 20 users where each has 600 requests (in total 12000 requests per image). For each image E θ (R θ ), E θ (D θ ) are computed by averaging over all users' requests.

Storage and transmission rate saving

Intuitively, approaches with smaller tile size should perform better in terms of transmission, but be less efficient in terms of storage. This is because small tiles brings more flexibility to send only the parts which are requested, but larger tiles result in better encoding performance. The proposed (BD-R, BD-S) criterion allows to assess quantitatively to what extent the tile size influences the tradeoff between transmission rate and storage, as shown in Table 1. For instance, the first two rows of Table 1 give the (BD-R, BD-S) pair averaged over all images, where 7x7 tiling is used as a reference (positive values show they perform worse than 7x7 tiling and negative values show they perform better). Interestingly, we observe that the tile size influences much more the transmission rate (from ∼-6% to ∼32%) than the storage (∼-4% to ∼10%).

Our proposed S-R-D characterization of the performance of an immersive coding scheme can also be plotted in terms of joint R-D and S-D curves, as in Figs. 3a and3b (for one image). In this representation, we observe that cube map approach saturates at high bit rates in both R-D and S-D curves. This can be explained by the fact that the original images are stored in equirectangular format, and there is a loss in quality in the cubemap approach, due to the extra projection from the equirectangular to the cube map representation.

Iso distortion values

The proposed iso distortion points allow to compare schemes at a given operating point and can help guiding the design of the compression scheme. For instance, one can evaluate the influence of the face resolution on the performance of the cubemap approach in Fig. 3c. At low PSNR (34 dB), lower face resolution achieves a smaller transmission rate and storage than higher face resolution, whereas, at high PSNR (39 dB), higher face resolution performs better in both storage and transmission rate. This highlights the strategy that for cube map projection, it is better to store cube faces with lower resolution at lower bitrates (low PSNRs) and at high bitrates cube faces resolution must be increased.

Weighted BD

Weighted BD was introduced in this paper to deal with cases where the comparison between two coding schemes is not straightforward. This occurs in particular, when one scheme achieves a lower transmission rate but a higher storage. By contrast, the weighted-BD (3) can help to discriminate between the schemes, by fixing λ, which is interpreted as the cost between storage and transmission rate, as shown in the last 3 rows of Table 1. For example, in our experiment the storage is about 700-1000 times larger than the transmission rate. Imposing λ = 1e -3 means that storage and transmission rate are both of the same importance. In this scenario, 7x7 tiling performs better than the other ones. Using smaller λ = 1e -4 , transmission is getting more important and 14x14 tiling should be chosen. Weighted-BD for different λs are plotted in Fig. 3d, where the lowest wighted-BD corresponds to the best method. More precisely, when λ < 0.0008, 14x14 tiling achieves the best performance, whereas the reference 7x7 tiling is better in 0.0008 ≤ λ < 0.009 and 2x2 tiling should be chosen for λ ≥ 0.009.

CONCLUSION

In this paper, we proposed an evaluation framework to compare 360degree content compression schemes for user-dependent transmission. We considered both storage and transmission rate jointly as they both affect decision making depending on the constraints we have in the streaming system. For tiling the equirectangular image, we showed that tiling the image into around 7x7 tiles, brings a good compromise between storage and transmission rate when storage and transmission rates are both of the same importance. Using iso distortion curves, we showed that for cube map we should consider a multi-resolution strategy where at lower bitrates (low PSNR values) cube faces with lower face resolution perform better in terms of both storage and transmission rate. At high bitrates (higher resolution) by increasing the cube faces resolution we can expect better performance again in both transmission rate and storage.

Fig. 2 :

 2 Fig. 2: New metric for the analysis of immersive coding schemes. (a) iso distortion points: each point corresponds to the R,S performance of a compression method for a fixed distortion. Therefore, all points have the same distortion. (b) Projection of the SR plane onto the new axis R + λS (in blue). (c) and (d) Projected S-R-D curves for different λ where λ1 < λ2.

Fig. 3 :

 3 Fig. 3: Curves for user-dependent compression and transmission of image pano aagymbthhtcgfb in SUN360 dataset. (a) R-D curve. (b) S-D curve. (c) Iso distortion points. (d) weighted BD for different λ.

Table 1 :

 1 BD measures averaged over all images. 7x7 tiling is used as a reference. The first 2 rows are (BD-R, BD-S) pair. The last 3 rows are weighted BD computed over the same PSNR range for (3).

			E. 14x14	E. 2x2	C. lower C. higher
		BD-R	-5.67 %	27.71 % 28.74 %	32.03 %
	.	BD-S λ = 0.01	9.84 % 8.10 %	-3.94 % -0.52 %	2.97 % 5.77 %	5.68 % 8.55 %
		λ = 1e -3	1.21 %	13.37 % 17.10 %	20.14 %
		λ = 1e -4 -4.52 % 25.28 % 26.77 %	30.02 %