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The Determination of Asteroid H and G Phase Function Parameters Using MNRAS 504, 761–768 (2021) doi:10.1093/mnras/stab726 The determination of asteroid H and G phase function parameters using Gaia DR2 Downloaded from https://academic.oup.com/mnras/article/504/1/761/6242886 by Consejo Superior de Investigaciones Cientificas (CSIC) user on 09 August 2021 M. Colazo ,1‹ R. Duffard2 and W. Weidmann3,4 1Instituto de Astronom´ıa Teorica´ y Experimental: Laprida 854, X5000BGR – Cordoba´ 2Instituto de Astrof´ısica de Andaluc´ıa: Apartado 3004,18080 Granada 3Observatorio Astronomico´ de Cordoba:´ Laprida 854 – Cordoba´ Capital, CPA: X5000BGR, Laprida 854, Cordoba,´ Argentina 4Consejo de Investigaciones Cient´ıficas y Tecnicas´ de la Republica,´ Buenos Aires, Argentina Accepted 2021 March 8. Received 2021 March 3; in original form 2020 December 29 ABSTRACT The Gaia mission will provide the scientific community with high-quality observations of asteroids of all categories. The second release of Gaia data (DR2) was published in 2018 and consists of 22 months of observations of 14 099 known Solar system objects, mainly asteroids. The purpose of this work is to obtain a catalogue of phase function parameters (H and G) for all the asteroids that were observed during the Gaia mission and that were published in DR2. For this purpose, we introduce an algorithm capable of building this catalogue from the magnitude and UTC epoch data present in the DR2 data base. Because Gaia will never observe asteroids with a phase angle of 0◦ (corresponding to opposition), but with phase angles higher than 10◦, we added data from ground observations (corresponding to small phase angles) and thus improved the determination of the H and G parameters of the phase function. We also built a catalogue of the parameters of the H, G1 and G2 phase function. We compared our results of the H, G functions with those of the Astorb data base and observed that the level of agreement is satisfactory. Key words: catalogues – surveys – minor planets, asteroids: general. it can do this for up to almost 27 d (Pal,´ Molnar´ & Kiss 2018). The 1 INTRODUCTION asteroids it observes are mostly those with inclinations greater than The Gaia mission will provide the Solar system scientific community 6◦ (Pal´ et al. 2020). An important difference between Gaia, K2 and with high-quality observations of about 300 000 asteroids of all TESS lies in the orbital position at which the asteroid is observed. categories. Most of these asteroids are from the main belt, but small InthecaseofTESS, the objects are observed during opposition; asteroids from the near earth asteroids (NEA) group, Jupiter trojans, K2 observes them near their stationary points (Pal´ et al. 2018); and centaurs and transneptunians have also been observed. Gaia Data Gaia observes the asteroids when their phase angle exceeds 10◦.The Release 2 (DR2)1 was published in 2018 and consists of 22 months scientific community must be prepared to analyse this large amount of observations of 14 099 known Solar system objects, mainly of data, which will lead to a more detailed knowledge of asteroid asteroids (Hestroffer et al. 2010), based on more than 1.5 million populations, which are currently poorly characterized because basic observations (Gaia Collaboration et al. 2018). In addition to the data physical properties such as mass, density, rotation properties, shape, base provided by Gaia, other space and ground-based surveys are and albedo are not yet known for most of them. generating asteroid photometry data in large volumes. As an example, We know that the brightness of asteroids varies with both their we could mention K2 (Kepler 2) (Howell et al. 2014;VanCleveetal. rotation on their axis and their movement around the Sun. In this 2016; Huber & Bryson 2016), Transiting Exoplanet Survey Satellite paper, we are interested in the second of these effects: the variation of (TESS) (Ricker et al. 2015), Vera C. Rubin Observatory (Schwamb the reduced magnitude (brightness on Johnson’s filter V normalized et al. 2019; Vera C. Rubin Observatory LSST Solar System Science at 1 au from the Sun and the observer, expressed in magnitudes) of Collaboration et al. 2020), and VISTA (Arnaboldi 2010; Popescu the objects as a function of the phase angle α. et al. 2017). K2 is the extension of the Kepler mission after one of This relationship is known as the phase function. In 1988, the its wheels failed. Because it points near to the plane of the ecliptic, it International Astronomical Union (IAU) adopted the H, G system to can observe hundreds of asteroids (Barentsen & Kepler Team 2016). represent it: H is the mean absolute magnitude in Johnson’s V band Furthermore, it can measure the brightness of a large number of with a zero phase angle, and G is the ‘slope parameter’ that describes asteroids in the main belt almost continuously (Szaboetal.´ 2016). the shape of the phase function. The absolute magnitude H is defined TESS, like K2, can observe light curves nearly uninterruptedly, and as the brightness that the investigated object would have if it was located at 1 au from both the Sun and the Earth and its phase angle was 0◦. E-mail: [email protected] 1 https://www.cosmos.esa.int/web/gaia/data-release-2 C 2021 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society 762 M. Colazo, R. Duffard and W. Weidmann As the phase angle decreases, that is, as the asteroid approaches 2.2 H, G1, G2 phase function opposition, the object will become brighter (its magnitude will The three-parameter magnitude phase function can be described as decrease). For angles below 6◦, the decrease in magnitude is non- (Muinonen et al. 2010) linear, which is termed the opposition effect (Carbognani, Cellino & ◦ = − + Downloaded from https://academic.oup.com/mnras/article/504/1/761/6242886 by Consejo Superior de Investigaciones Cientificas (CSIC) user on 09 August 2021 Caminiti 2019). By contrast, for phase angles greater than 10 ,this V (α) H 2.5log10[G11(α) G22(α) relationship is almost linear. Gaia can obtain the phase–magnitude +(1 − G1 − G2)3(α)], (5) curves precisely in this interval where the relationship between the ◦ = ◦ = ◦ = V and α parameters is linear (Gaia Collaboration et al. 2018). As where 1(0 ) 2(0 ) 3(0 ) 1. Analogous to the previous Carbognani et al. (2019) note, it is interesting to combine the data of model, the authors propose writing the reduced magnitude V as ◦ ◦ Gaia α> α< − ( 10 ) and data obtained from the ground ( 10 ) to obtain 10 0.4V (α) = a (α) + a (α) + a (α). (6) determinations of H of good quality that will serve, for example, to 1 1 2 2 3 3 obtain estimations of the diameter of thousands of asteroids. The coefficients a1, a2 and a3 can be calculated using minimum linear In addition, Belskaya & Shevchenko (2000) proposed that the squares, and then the parameters H, G1 and G2 are obtained with the slope value of the linear part of the phase function can serve as an following equations: estimate of the albedo of the objects. Another interesting investiga- H =−2.5log (a + a + a ), tion when analysing the Gaia data would be the properties of the 10 1 2 3 a phase–magnitude curves according to taxonomic type (Carbognani G = 1 , 1 a + a + a et al. 2019). 1 2 3 a In the near future, we need to be prepared to join together several G = 2 . 2 + + (7) different asteroid data sources, obtained with different filters and a1 a2 a3 from different sources/telescopes. We can reference all the data According to Muinonen et al. (2010), the H, G phase function is ◦ ◦ obtained to one reference standard such as the Sloan filters or the reasonably good, especially in the range of 10 to 60 . Furthermore, it Gaia photometric system. has the advantage of having a simple analytical form. However, there The purpose of this work is to obtain a catalogue of phase function are cases where it produces bad adjustments even for high-quality parameters (H and G) for all asteroids that were observed by the Gaia observations. In contrast, the function H, G1, G2 does not present mission and published in DR2. This paper is organized as follows. this problem. Having more parameters leads to better adjustments. In Section 2, we provide a brief description of the data used for the This function, unlike H, G, can adequately adjust the phase curves analysis of DR2. In Section 3, we introduce the algorithm used to of asteroids with very high or very low albedo. obtain the parameter catalogue. In Section 4, we discuss the results obtained, and finally, in Section 5, we present the conclusions and 3 Gaia DATA directions for future research. As mentioned in Section 1, Gaia DR2 was published in 2018 and included observations of 14 099 known objects from the Solar system 2 PHASE FUNCTION EQUATIONS (Hestroffer et al. 2010). In this work, we selected as a sample the 13 981 objects that are asteroids. Fig. 1 represents this sample in the 2.1 H, G magnitude phase function a (semimajor axis) versus I (inclination) plane, where the different families and Trojan asteroids are identified with different colours. The H, G phase function for asteroids can be described analytically We downloaded the data base from the Gaia Archive.2 It has three through the following equation (Muinonen et al. 2010): columns: number mpc, epoch UTC and g magnitude (instrumental). The magnitude is published in the data base and is obtained from the V (α) = H − 2.5log [(1 − G) (α) + G (α)], (1) 10 1 2 flux. We then also downloaded the flux and error flux to calculate the where α is the phase angle, V(α)istheV magnitude reduced to unit error in the magnitude ourselves.
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