Mon. Not. R. Astron. Soc. 316, 605±612 (2000) The recent star formation history of the Hipparcos solar neighbourhood X. Hernandez,1w David Valls-Gabaud2 and Gerard Gilmore3 1Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy 2Laboratoire d'Astrophysique, UMR CNRS 5572, Observatoire Midi-PyreÂneÂes, 14 Avenue E. Belin, 31400 Toulouse, France 3Institute of Astronomy, Cambridge University, Madingley Road, Cambridge CB3 0HA Accepted 2000 March 2. Received 2000 February 28; in original form 1999 September 13 ABSTRACT We use data from the Hipparcos catalogue to construct colour±magnitude diagrams for the solar neighbourhood, which are then treated using advanced Bayesian analysis techniques to derive the star formation rate history, SFR(t), of this region over the last 3 Gyr. The method we use allows the recovery of the underlying SFR(t) without the need of assuming any a priori structure or condition on SFR(t), and hence yields a highly objective result. The remarkable accuracy of the data permits the reconstruction of the local SFR(t) with an unprecedented time resolution of <50 Myr. An SFR(t) that has an oscillatory component of period <0.5 Gyr is found, superimposed on a small level of constant star formation activity. Problems arising from the non-uniform selection function of the Hipparcos satellite are discussed and treated. Detailed statistical tests are then performed on the results, which confirm the inferred SFR(t) to be compatible with the observed distribution of stars. Key words: methods: statistical ± stars: formation ± solar neighbourhood. SFR(t) from stellar kinematics (e.g. Gomez et al. 1990, Marsakov 1 INTRODUCTION et al. 1990). The problem of deducing the star formation rate history, SFR(t), of With the recent availability of the Hipparcos satellite catalogue the Milky Way has generally been attempted in terms of indirect (ESA 1997) we are now in a position to attempt recovery of the inferences, mostly through chemical evolution models. The local SFR(t) directly, without the need of any model dependent validity of these methods relies on the soundness of the assumptions. Previous direct approaches have been undertaken assignation of a `chemical age' to each of the studied stars. through the binning of observed stars into age groups according to Generally a metallicity indicator is chosen and used to measure the degree of chromospheric activity as measured through selected the metal content of a number of stars, which are then binned into emission-line features, with conflicting results depending on the age groups through the use of an age±metallicity relation derived assumed age±activity relation (e.g. Barry 1988 and Soderblom from a chemical evolution model. For example, Rocha-Pinto & et al. 1991). Using this technique, Rocha-Pinto et al. (2000a) have Maciel (1997) take a variety of age±metallicity relations (AMRs) derived a star formation history from the chromospheric activity± from the literature and use a closed box chemical evolution model age distribution of a larger sample comprising 552 stars, and have to translate the AMRs into SFR(t)s, allowing for an intrinsic found evidence for intermittency in the SFR(t) over 14 Gyr. The Gaussian spread in the AMR assumed to be constant in time. Hipparcos catalogue offers high-quality photometric data for a The advantages of these methods are that large samples of stars, large number of stars in the solar neighbourhood, which can be both in the solar neighbourhood and further away within the disc used to construct a colour±magnitude diagram (CMD) for this of the Galaxy, can be studied. An inferred SFR(t) can be region. Once a CMD is available, it is in principle possible to constructed over an ample time range and spatial extent within the recover the SFR(t) that gave rise to the observed distribution of Galaxy, which is consistent with the metallicities of the sample stars, assuming only a stellar evolutionary model in terms of a set studied and the chemical evolution model proposed. However, the of stellar tracks. In practice, the most common approach to validity of the AMR assumption can not be checked independently inverting CMDs has been to propose a certain parametrization for of the proposed chemical evolution model, and is necessarily the SFR(t), used to construct synthetic CMDs, which are dependent on the choice of the mixing physics of the interstellar statistically compared with the observed ones in order to select medium (ISM), the possible infall of primordial non-enriched gas the values of the parameters that result in a best match CMD. and the still largely unknown Galactic formation scenario in Examples of the above are found in Chiosi et al. (1989), Aparicio general. This last problem also affects attempts at inferring the et al. (1990) and Mould, Han & Stetson (1997), using Magellanic and local star clusters, and Mighell & Butcher (1992), Smecker- Hane et al. (1994), Aparicio & Gallart (1995), Tolstoy & Saha w E-mail: [email protected] (1996) and Mighell (1997), using local dwarf spheroidals (dSphs). q 2000 RAS 606 X. Hernandez, D. Valls-Gabaud and G. Gilmore We have extended these methods in Hernandez, Valls-Gabaud updated calibrations given by Bessell, Castelli & Plez (1998) does & Gilmore (1999, henceforth Paper I) by combining a rigorous not change the transformations significantly in the regime used maximum likelihood statistical approach, analogous to that here, unlike the case for giant and asymptotic giant branch stars introduced by Tolstoy & Saha (1996), with a variational calculus (see Weiss & Salaris 1999). treatment. This allows a totally non-parametric solution of the In this case we implement the method with a formal resolution problem, where no a priori assumptions are introduced. This of 15 Myr, compatible with the high resolution of the Hipparcos method was applied by Hernandez, Gilmore & Valls-Gabaud observations. It is one of the advantages of the method that this (2000, henceforth Paper II) to a set of Hubble Space Telescope resolution can be increased arbitrarily (up to the stellar model CMDs of local dSph galaxies in order to infer the SFR(t) of these resolution) with computation times scaling only linearly with the interesting systems. The result differs from what can be obtained resolution. from a chemical evolution model, in that a direct answer is Our only other inputs are the positions of, say, n observed stars available, with a time resolution which depends only on the in the HR diagram, each with a colour and luminosity ci and li, accuracy of the observations. respectively. Starting from a full likelihood model, we first Limitations on the applicability of our method to the Hipparcos construct the probability that the n observed stars resulted from a data appear in connection to the selection function of the certain SFR(t). This will be given by catalogue. The need to work only with complete volume-limited Yn t1 samples limits the age range over which we can recover the SFR(t) L SFR tGi t dt ; 2 to 0±3 Gyr, with a resolution of ,0.05 Gyr. This makes it i1 t0 impossible to compare our results with those of chemical evolu- tion models, which typically sample ages of 0±14 Gyr, with where resolutions of 0.5±1.5 Gyr. m1 r m; t 2D l ; t; m2 In Section 2 we give a summarized review of the method G t exp i i 2ps l s c 2s2 l introduced in Paper I, the sample selection and results are m0 i i i discussed in Section 3. Section 4 presents a careful statistical 2D c ; t; m2 Â exp i dm: testing of our results, and Section 5 our conclusions. 2 2s ci In the above expression r(m;t) is the density of points along the 2 THE METHOD isochrone of age t around the star of mass m, and is determined by In this section we give a summary description of our Hertz- the assumed IMF together with the duration of the differential sprung±Russell (HR) diagram inversion method, which was phase around the star of mass m. The ages t0 and t1 are minimum described extensively in our Paper I. In contrast with other and a maximum ages that need to be considered, as m0 and m1 are statistical methods, we do not need to construct synthetic CMDs a minimum and a maximum mass considered along each iso- for each of the possible star formation histories being considered. chrone, e.g. 0.6 and 20 M(. s(li) and s(ci) are the amplitudes of Rather, we use a direct approach that solves for the best SFR(t) the observational errors in the luminosity and colour of the ith star. compatible with the stellar evolutionary models assumed and the These values are supplied by the particular observational sample observations used. one is analysing. Note that for the sample we have selected The evolutionary model consists of an isochrone library and an (Section 3), there is no correlation between these errors. Finally, initial mass function (IMF). Our results are largely insensitive to D(li ; t,m) and D(ci ; t,m) are the differences in luminosity and the details of the latter, for which we use colour, respectively, between the ith observed star and a general 8 star of age and mass (m, t). We shall refer to Gi(t) as the likelihood > m21:3; 0:08 M , m < 0:5M ; matrix, since each element represents the probability that a given <> ( ( 22:2 star, i, was actually formed at time t with any mass. r m/ m ; 0:5M( , m < 1:0M(; 1 > A similar version was introduced in Paper I but was restricted to : 22:7 m ; 1:0M( , m: the case of observational errors only in one variable, which was adequate for the problem of studying the SFR(t) of local dSph The above fit was derived by Kroupa, Tout & Gilmore (1993) for a galaxies treated in Paper II.