Rafel ; Simo
email: rsimo@icm.csic.es

Violeta Saló

Rafel Simó

Albert Calbet

Revisiting the dilution technique to quantify the role of microzooplankton

Keywords: DMS, DMSP, microzooplankton, dilution dimethylsulfide, dimethylsulfoniopropionate, dilution technique, microzooplankton, grazing

With the aim of re-examining the suitability of the dilution technique in studies of dimethylated sulfur cycling in the sea, we conducted two types of experiments: one with a simplified food chain of cultured organisms, and secondly with field-sampled natural communities. In the case of Isocrhysis galbana being grazed by Oxyrrhis marina, algal growth (µ) and grazing mortality (m) rates calculated using either chlorophyll a (chla) or particulate dimethylsulfoniopropionate (DMSPp) were very similar. By means of a parallel incubation of prey with no predator, the regression-derived phytoplankton µ was estimated.

Net 24-h changes in DMS concentration showed a linear trend against DMSP grazing rates, with a slope that allowed estimation of the rate of DMS production due to grazing. When the method was applied to natural communities from coastal NW Mediterranean waters, µ values of the bulk chla-containing phytoplankton assemblage were higher than that of the DMSPcontaining algae, but m were not significantly different. The grazing-mediated DMS production rate was successfully estimated, and parallel incubations with and without a specific inhibitor of bacterial DMS metabolism allowed us to estimate the gross DMS production by the whole community and the bacterial DMS consumption rate. A further experiment showed that bacterial consumption was not affected by dilution.

INTRODUCTION

The dilution technique [START_REF] Landry | Estimating the grazing impact of marine microzooplankton[END_REF][START_REF] Landry | A refined dilution technique for measuring the community grazing impact of microzooplankton, with experimental tests in the central equatorial Pacific[END_REF] has been widely used to determine the impact of microzooplankton grazing on phytoplankton biomass and production in natural waters [START_REF] Calbet | Phytoplankton growth, microzooplankton grazing, and carbon cycling in marine systems[END_REF][START_REF] Calbet | The trophic roles of microzooplankton in marine systems[END_REF]. This technique allows rates of grazing on, and growth of, natural phytoplankton assemblages to be obtained with minimal manipulation. It consists of a dilution series of whole seawater with particle-free seawater added to create a gradient of microzooplankton grazing pressure on phytoplankton.

The technique is built upon the assumptions that phytoplankton growth is exponential and independent of the dilution if nutrients occur at non-limiting concentrations, whereas phytoplankton mortality is density-dependent and proportional to microzooplankton concentration.

The dilution technique has also proved to be useful to determine some of the biogeochemical fluxes associated with microzooplankton herbivory, such as ammonium uptake and regeneration [START_REF] Neuer | Determination of ammonium uptake and regeneration rates using the seawater dilution method[END_REF]. Likewise, it has recently been explored for obtaining rates of dimethylsulfide (DMS) production derived from microzooplankton grazing [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF]Archer et al., 2001a;[START_REF] Archer | Is dimethyl sulphide production related to microzooplankton herbivory in the southern North Sea?[END_REF]. DMS is the most abundant volatile sulfur compound in surface oceanic waters, where its concentration is supersaturated, so it tends to diffuse to the atmosphere [START_REF] Andreae | Ocean-atmosphere interactions in the global biogeochemical sulfur cycle[END_REF]. This continuous emission of DMS from the ocean (estimated global flux of 15-33•10 12 gS year -1 ; [START_REF] Kettle | Flux of dimethylsulfide from the oceans: A comparison of updated data sets and flux models[END_REF][START_REF] Simó | Global ocean emission of dimethylsulfide predicted from biogeophysical data[END_REF] is the most important natural source of sulfur to the global troposphere, with potentially large implications for aerosol acidity and cloud formation and microphysics [START_REF] Charlson | Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate[END_REF][START_REF] Andreae | Ocean-atmosphere interactions in the global biogeochemical sulfur cycle[END_REF]. The concentration and emission of DMS from the surface ocean is driven by tight biogeochemical cycling involving physicochemical and biological processes. metabolite synthesized by many marine phytoplankton species [START_REF] Keller | Dimethyl sulfide production in marine phytoplankton[END_REF][START_REF] Stefels | Environmental constraints on the production and removal of the climatically active gas dimethylsulphide (DMS) and implications for ecosystem modelling[END_REF]. The enzymes responsible for the conversion of DMSP into DMS (generally designated 'DMSP lyases') occur in most DMSP-producing phytoplankters and in some bacteria. The plankton community structure and dynamics have been shown to influence the rates and yields of the DMSP-to-DMS conversion, together with a number of meteorological (solar radiation spectrum and intensity, wind speed) and chemical (inorganic nutrients, colored dissolved organic matter) parameters [START_REF] Simó | Production of atmospheric sulfur by oceanic plankton: biogeochemical, ecological and evolutionary links[END_REF][START_REF] Simó | From cells to globe: approaching the dynamics of DMS(P) in the ocean at multiple scales[END_REF][START_REF] Stefels | Environmental constraints on the production and removal of the climatically active gas dimethylsulphide (DMS) and implications for ecosystem modelling[END_REF].

A number of laboratory [START_REF] Dacey | Oceanic dimethylsulfide: Production during zooplankton grazing on phytoplankton[END_REF][START_REF] Malin | Dimethy sulphide: Productiona and atmospheric consequences[END_REF][START_REF] Christaki | Assessment of the role of copepods and ciliates in the release to solution of particulate DMSP[END_REF][START_REF] Wolfe | Grazing-activated production of dimethyl sulfide (DMS) by two clones of Emiliania huxleyi[END_REF] and field studies [START_REF] Daly | Particulate dimethylsulfoniopropionate removal and dimethylsulfide production by zooplankton int he Southern Ocean[END_REF][START_REF] Kwint | Turnover of dimethylsulfoniopropionate and dimethylsulfide in the marine environment: a mesocosm experiment[END_REF][START_REF] Archer | Is dimethyl sulphide production related to microzooplankton herbivory in the southern North Sea?[END_REF] have demonstrated that zooplankton grazing enhances DMS production, probably by facilitating the mixing of algal DMSP and algal or bacterial DMSP lyases. Protozoans and other microzooplankton are major grazers in the ocean (Burkill et al., 1993) and, as such, are expected to play a central role in DMS production.

Several authors have examined the potential application of the dilution technique to quantify the growth and grazing-mediated mortality rates of DMSP-producing phytoplankton [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF]Archer et al., 2001b;[START_REF] Archer | Is dimethyl sulphide production related to microzooplankton herbivory in the southern North Sea?[END_REF][START_REF] Fredrickson | The algal osmolyte DMSP as a microzooplankton grazing deterrent in laboratory and field studies[END_REF] and the grazing-mediated DMS production rate [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF]Archer et al., 2001a;[START_REF] Archer | Is dimethyl sulphide production related to microzooplankton herbivory in the southern North Sea?[END_REF] in plankton communities from temperate and subpolar waters. While growth and mortality rates were successfully measured, assays for DMS production did not always exhibit the significantly linear behavior required to reliably estimate production rates [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF]Archer et al., 2001a), which prompts a further assessment of the method. The limitations and pitfalls of the dilution approach that have been highlighted by several authors [START_REF] Gallegos | Microzooplankton grazing on phytoplankton in the Rhode River, Maryland: nonlinear feeding kinetics[END_REF][START_REF] Chavez | Growth rates, grazing, sinking, and iron limitation of equatorial Pacific phytoplankton[END_REF][START_REF] Evans | Precision of estimates of phytoplankton growth and microzooplankton grazing when the functional response of grazers may be nonlinear[END_REF][START_REF] Landry | A refined dilution technique for measuring the community grazing impact of microzooplankton, with experimental tests in the central equatorial Pacific[END_REF][START_REF] Dolan | Dilution effects on microzooplankton in dilution grazing experiments[END_REF] might have hampered its applicability to studies of the DMS cycle. We aimed to test the utility of the method for quantifying DMSP consumption and DMS production through microzooplankton herbivory, with a simplified food chain of cultured organisms. For this purpose we used monospecific cultures of a DMSP-producing prey (the prymnesiophyte Isochrysis galbana) and an herbivorous protist (the heterotrophic dinoflagellate Oxyrrhis marina). A similar laboratory experiment with the same species has been carried out previously by Archer et al. (Archer et al., 2001a). The special feature and novelty of our own study is that we compared growth and grazing rates obtained from chlorophyll a (chla), algal cell counts and DMSP measurements in our model system.

Furthermore, replicate bottles with prey but no predator were prepared in order to calculate the specific growth rate of phytoplankton and compare it with that estimated from the regression analysis of the dilution series.

We report also on two identical dilution experiments conducted with natural plankton communities. Additional bottles were prepared to estimate gross DMS production and bacterial DMS consumption. Our goal was to test the applicability of this technique to quantify grazing-derived rates of DMSP consumption and DMS production and, hence, to assess the role of microzooplankton herbivory in the dynamics of dimethylated sulfur in meso-to oligotrophic surface waters. Small aliquots of O. marina and exponentially growing I. galbana were mixed in 0.2 µmfiltered and autoclaved seawater enriched with f/2 medium to produce an experimental stock with a prey to predator ratio of approximately 20:1. Once the initial experimental stock (termed 100%) was obtained, 75%, 50% and 25% dilution treatments were prepared by addition of filtered, autoclaved seawater enriched with f/2 medium. The actual level of dilution was determined from the initial concentration of I. galbana in each bottle relative to the concentration in the 100% treatment (undiluted bottle). All treatments were prepared in duplicate, 620 mL Pyrex bottles. The use of f/2 medium ensured that nutrients would not be limiting over the 24 hours experimental period. In addition, two bottles containing only I. galbana at the same concentration as that of the 100% bottles were prepared, and named 100iso. The latter bottles were used to directly obtain the net growth rate of the phytoplankter and the net DMS production rate in the absence of grazing. All bottles were filled leaving no headspace, capped and rotated gently by hand prior to subsampling. Temperature and light conditions were the same as described above. The incubation time was 24 hours, with two sampling points; t 0 and t 24 . Subsamples of 350 mL from each dilution bottle were used for cell enumeration of prey, chla measurements, and DMS(P) analyses. Predators from the undiluted water were enumerated at t 0 .

Experiments with natural communities

Seawater was sampled at 41.22° N, 02.13° E, 1.5 km off the north western Mediterranean coast, from a water-column depth of ca. 40 m. Water was collected from 5 meters depth at 11:00h local time on the 14 June and 31 July 2006, with a 15 L transparent hydrographic bottle and was gently siphoned into a carboy and rapidly transported to the laboratory. Light and temperature were measured at the sampling depth with a LI-COR LI-1400 data logger and a YSI 30 portable temperature meter.

A volume of the sampled water was gravity filtered through 0.2 µm with a Pall Acropak 0.8/0.2 500 capsule previously rinsed thoroughly with HCl and distilled water. This water was used to dilute the natural whole water and prepare all the dilution bottles. Each dilution treatment (2.3 L, transparent polycarbonate bottles) was prepared in duplicate, except that the 100% was prepared in six replicates: two were used for the micrograzing dilution experiment and incubated in the light, two were spiked with dimethyldisulfide (DMDS) to a final concentration of 200 nM, and the other two were left as dark controls. The dilution bottles were amended with a nutrient mixture (15 µM NH 4 Cl and 1 µM Na 2 HPO 4 ) in order to ensure nutrients would not be limiting algal growth, as such limitation would distort the results of the Dilution bottles were incubated at the in situ temperature right under the water surface in a large (600 L) outdoor incubator with open-circuit water running from the coastal seawater intake at the Institut de Ciències del Mar, Barcelona. Natural sunlight was dimmed with one layer of neutral plastic mesh, which allowed ca. 60% of surface irradiance. The DMDSamended and corresponding control bottles were incubated in the dark. The experiments lasted 24 hours. The initial chla and DMS(P) concentrations were estimated from those measured in the whole and particle-free waters from which all the dilution bottles were prepared. The light incubated bottles were subsampled at t 24 for chla and sulfur analyses. The dark incubated bottles were subsampled at t 4 , t 6 , t 10 , t 20 and t 24 for sulfur analyses.

In order to test if bacterial DMS consumption is affected by dilution with 0.2 µm filtered (low-bacteria) water, we conducted a further experiment on 6 July. Seawater was collected from the same location and a full series of dilution bottles was prepared in duplicate as described above. One replicate of each was spiked with DMDS, and both were incubated in the dark over 24 hours. Subsamples for DMS analysis were withdrawn at t 6 , t 10 and t 24 .

Cell counts

In the laboratory experiment, a Multisizer III Coulter Counter was used to obtain cell abundances of both predator and prey. Subsamples of 20 mL were analyzed in duplicate for each experimental bottle. When predator numbers were too low to be accurately measured with the Coulter Counter (<100 cells mL -1 ), aliquots were preserved in 4% acidic Lugol's solution and counted in Sedgewick-Rafter counting chambers under the microscope. The undiluted natural community samples were examined under microscopically to count microzooplankton and phytoplankton larger than 5 µm. Bacteria and autotrophic pico-and nanoeukaryotes were counted following flow cytometry analysis procedures [START_REF] Olson | Phytoplankton analysis using flow cytometry[END_REF][START_REF] Gasol | Using flow cytometry for counting natural planktonic bacteria and understanding the structure of planktonic bacterial communities[END_REF]. Samples of 1.2 ml for bacterial abundance determination were preserved with 1% paraformaldehyde + 0.5% glutaraldehyde (final conc.), and kept frozen at -80ºC until analysis using a Becton-Dickinson FACSCalibur flow cytometer after staining with SybrGreen I (10,000x). Bacteria were detected and counted by their signature when plotting side scatter (SSC) versus green fluorescence (FL1) and FL1 vs. red fluorescence. Synechococcus, Prochlorococcus and photosynthetic picoeukaryote abundances were estimated from 2-mL samples preserved and stored as above, and analyzed in the FACSCalibur flow cytometer in unstained aliquots. The various phytoplankton groups were detected and counted by their signature when plotting side scatter (SSC) vs. red fluorescence (FL3) and versus orange fluorescence (FL2).

Chlorophyll a analyses

Aliquots of 150 to 300 mL were filtered gently through a GFF filter and kept frozen at -20ºC until analysis. Chla concentration was measured fluorometrically in 90% acetone extracts of the filters with and without acidification, following [START_REF] Strickland | A practical handbook of seawater analysis[END_REF].

Analysis of sulfur compounds

A 40 mL aliquot of seawater was withdrawn from each experimental bottle at t0 and t24 for sulfur analyses. DMS was measured immediately using a purge-and-trap/gas chromatography system modified from [START_REF] Simó | Sequential method for the field determination of nanomolar concentrations of dimethyl sulfoxide in natural waters[END_REF]. Volumes of 3 to 10 mL of GF/F filtered water were purged with He for 4 to 10 minutes, the stripped DMS was cryogenically trapped in a teflon

Calculation of growth and grazing rates

Net growth rates (r) of the measured algal variables (chla, I. galbana cell counts and DMSPp), were determined from initial (C t0 ) and final (C t24 ) values assuming an exponential growth model:

r = ln (C t24 /C t0 )/t
The r values of duplicate bottles were plotted against the level of dilution (fraction of whole water in the dilution treatment), and model I regression analysis was used to compute the specific growth rate of the algae (µ'= intercept) and the rate of mortality due to grazing (m = slope). For the field experiment, because the intercept of the equation would provide an overestimation of phytoplankton instantaneous growth rates (nutrients were added to these [START_REF] Landry | Estimating the grazing impact of marine microzooplankton[END_REF] or, in other words, the intercept µ' minus the net growth rate difference between nutrient amended and unamended 100% bottles. The errors associated with m and µ were the standard errors of the slope and the intercept, respectively, provided by the regression analysis.

The net growth rate obtained in the laboratory experiment for the 100iso incubation bottles (r iso :100% treatment with no predators) was compared to the value provided by the intercept of the regression analysis of the specific growth rates vs the dilution factor (µ'). This way the assumption that an infinite dilution would provide the growth rate of the prey in the absence of grazing was verified.

Calculation of grazing-mediated DMS production

The difference between initial and final concentrations of DMS was used to calculate net DMS production in duplicate bottles at each dilution level. In parallel, DMSP grazing rates at each dilution were calculated by scaling DMSP mortality rates m DMSP to the dilution factor and multiplying them by the mean DMSP concentration in each dilution bottle (<C DMSP >), calculated according to the equation of [START_REF] Frost | Effects of size and concentration of food particles on the feeding behaviour of the marine planktonic copepod Calanus pacificus[END_REF]:

<C DMSP > = C to [e (µ-m) (t1-t0) -1] / (t 1 -t 0 ) (µ-m)
Net DMS production rate values were paired to the corresponding DMSP grazing rates, and a model I regression analysis was conducted. The slope provided the daily DMS production per grazed DMSP (nM DMS nM DMSP -1 ), which was multiplied by the mean DMSP concentration in the control (unamended and undiluted) bottles to obtain the rate of DMS production due to grazing (Pg). The intercept (Png) corresponded to the net DMS production that resulted from processes unrelated to grazing, namely algal exudation and cell lysis,

Calculation of gross DMS production and bacterial DMS consumption

In the natural community experiments, DMDS was added to two replicate bottles of whole water as a selective inhibitor of bacterial DMS metabolism [START_REF] Wolfe | Radioisotope and chemical inhibitor measurements of dimethyl sulfide consumption rates and kinetics in estuarine waters[END_REF]. The slopes of DMS accumulation with time in each of the DMDS-amended bottles were averaged and taken as the rate of gross DMS production by the total plankton community in the dark.

The slopes of DMS concentration in the corresponding duplicate controls were averaged to provide the net DMS production rate in the dark. The errors associated with gross and net DMS production rates were the averaged standard errors of the slopes. The difference between both rates provided the rate of bacterial DMS consumption at each dilution [START_REF] Simó | Biological turnover of DMS, DMSP and DMSO in contrasting open-sea waters[END_REF], with a propagated error.

The same calculations were applied to the results of the experiment on 6 July, where DMDS was added to one replicate of each pair of bottles throughout a full dilution series. In this case, the slope of DMS accumulation with time provided the rate of gross DMS production at each dilution level. The 100iso bottle contained 9833 ± 163 I. galbana cells mL -1 , 1.59 ± 0.01 µg chla L -1 and 23.3 ± 0.3 nM DMSPp at t 0 . When plotting the net growth rate against the dilution factor, significant (p < 0.01) negative slopes were obtained for algal cell abundances, chla and DMSPp (Fig. 1).

RESULTS

Laboratory experiment

Table 1 illustrates the growth rates of I. galbana (µ') and the grazing rates (m) obtained from the regression analyses applied to either cell counts, chla concentrations, or DMSPp concentrations (Fig. 1). Grazing rates were very similar for chla and DMSPp (no significant differences between slopes, ANCOVA test p =0.699), and differed from those obtained with cell counts (p < 0.001). Growth rates based on chla, cell counts and DMSPp were statistically different (intercept of the equations significantly different, ANCOVA, p < 0.001). The algal growth rate calculated from the 100iso bottles (Table 1) fell between the rates estimated from the regression analyses of chla and cell counts, and was very similar to that of DMSPp.

The initial concentrations of DMS were similar in all dilution bottles (0.3 nM), indicating that both the I. galbana culture and the 0.2 µm filtered and autoclaved sea water contained some DMS. The initial whole water pool was prepared by highly diluting (1:125) the I. galbana culture with the filtered seawater + f/2, so, although the I. galbana culture contained more DMS than the dilution water the resulting mixture contained a DMS concentration very similar to that of the filtered seawater. The final (t 24 ) DMS concentrations were higher than the initial ones in all cases, and this increase was lower as waters were more diluted. When plotting net DMS production against DMSPp ingested, a straight line with significant positive slope (0.02 nM DMS nM DMSPp -1 ; p < 0.001) was obtained (Fig. 2). The product of this slope by the DMSPp ingestion rate in the undiluted sample yielded a rate of DMS production due to grazing (Pg) of 0.26 ± 0.03 nM DMS d -1 (Table 1). The intercept of the regression gave a Png estimate (0.02 ± 0.02 nM DMS d -1 , 95% confidence) that was indistinguishable from the value of net DMS production obtained in the 100iso bottles (Table 1).

Experiments with natural communities

The plankton composition of the Mediterranean coastal waters sampled on 14 June and 31

July can be found in [START_REF] Calbet | Impact of micro-and nanograzers on phytoplankton assessed by standard and size-fractionated dilution experiments[END_REF]. Briefly, diatoms occurred in abundances of 24 diatom cells mL -1 in June (dominant genera: Guinardia and Dactyliosolen) and 131 cells mL -1 in July (Nitzschia, Pseudo-nitzschia and Leptocylindrus). Autotrophic flagellates (5-15 µm) occurred at about 2•10 3 cells mL -1 and were dominated by haptophytes, a high-DMSP producing group. Picoeukaryotic phytoplankton amounted about 3•10 3 cells mL -1 . Prokaryotic phytoplankton contained 5•10 4 and 20•10 4 Synechococcus cells ml -1 in June and July, respectively, and no Prochlorococcus. In June, the microbial grazers were mainly heterotrophic dinoflagellates (10 µgC L -1 ), followed by heterotrophic flagellates (7 µgC L -1 ) and some ciliates (5 µgC L -1 ). In July, heterotrophic flagellates (10 µgC L -1 ) dominated among microzooplankton, followed by lower biomass numbers of dinoflagellates and ciliates 16 chla L -1 , 55 ± 2.1 nM DMSPp and 7.8 ± 0.0 nM DMS in June, and 0.40 ± 0.02 µg chla L -1 , 8.7 ± 0.6 nM DMSPp and 5.2 ± 0.2 nM DMS in July. Filtered seawater contained 9.2 ± 0.3 nM DMS in June and 6.0 ± 0.2 nM DMS in July. It could be argued that calculating the initial DMS concentrations from those measured in whole and particle-free waters could introduce uncertainty owing to potential effects of water handling (and associated mechanical stress) on DMS metabolism. To test this, we conducted a further seawater sampling and prepared a dilution series by the same procedure employed in the regular experiments. Initial DMS concentrations were measured in each. DMS levels in the whole and the 0.2-µm filtered seawater were 4.5 ± 0.1 and 3.4 ± 0.0 nM, respectively. In all dilution bottles, calculated DMS concentrations did not differ from measured concentrations: 4.2 vs. 4.2 ± 0.0 nM (75%), 3.9 vs. 3.9 ± 0.1 nM (50%), and 3.7 vs. 3.6 ± 0.1 nM (25%). [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF] had already performed a similar test and obtained the same results.

Measurements conducted in each dilution bottle after 24 hours confirmed that grazing did occur at higher rates in the concentrated than in the diluted bottles. As a result, net growth rates of both chla and DMSPp showed a significant linear decrease with the dilution factor (Figs. 3 and4). DMSPp producers, which were a subgroup of the chla-containing community, grew slower than the bulk phytoplankton assemblage in June (0.32 vs. 1.08 d -1 ) and July (0.59 vs. 0. 96 d -1 ), yet the difference was not significant in the latter case (Table 2). Mortality rates (0.65 vs. 0.82 d -1 in June; 1.45 vs. 0.99 d -1 in July) were not significantly different in either month (Table 2).

With respect to DMS production, the slope of the plots of net DMS production against DMSPp consumption allowed estimation of rates of DMS production due to grazing (Pg) of 4.0 ± 0.9 nM DMS d -1 in June and 3.6 ± 1.3 nM DMS d -1 in July. The amounts of DMSPp 2).

The bottles incubated in the dark with and without added DMDS were used to obtain the rates of gross DMS production by the total community and bacterial DMS consumption in the dark.

The DMDS amended bottles showed a faster DMS accumulation than the controls, indicating that there was an active bacterial consumption of DMS (Fig. 5). In June, gross DMS production was 6.3 ± 0.8 nM DMS d -1 . Subtraction of the slopes gave a daily bacterial DMS consumption of 0.8 ± 1.8 nM d -1 . Comparison of Pg (4.0 nM DMS d -1 ) and gross DMS production (6.3 nM DMS d -1 ) estimates indicates that a large fraction of the DMS produced inside the experimental bottles was due to grazing. In July, the same calculations gave a gross DMS production of 5.0 ± 0.5 nM DMS d -1 and a bacterial consumption of 5.0 ± 1.0 nM d -1 .

Again, most of the gross DMS production was due to grazing (3.6 ± 1.3 nM d -1 ). It must be noted, though, that gross DMS production rates were measured in the dark, while grazingmediated DMS production rates were determined in polycarbonate bottles in the light. Effects , with no differences between dilution treatments. These results strongly indicate that dilution did not affect the rates of DMS consumption by bacteria.

DISCUSSION

We revisited the use of the dilution technique to assess its utility to quantify the role of microzooplankton grazing in DMS(P) cycling in meso-to oligotrophic Mediterranean waters.

First, we wanted to corroborate experimentally the validity of the assumptions behind the design of the dilution technique: (i) the apparent growth rate of phytoplankton shows a linear variation with the dilution fraction, with an intercept that provides the growth rate of the phytoplankton prey or, more precisely, the net growth of phytoplankton in the absence of grazing and with excess nutrients; (ii) the slope of this curve provides the grazing rate on the bulk phytoplankton assemblage, if the apparent growth is measured in biomass (cell counts, chla) units, or the grazing rate on the sub-assemblage of the DMSP producers, if the apparent growth is measured in DMSP units.

With this aim, we ran an experiment with a simplified food chain with cultured DMSPproducing prey and predator. The observation that the algal growth rate measured from 100iso (no predator) bottles fell within the 95% confidence intervals of the intercept of the regression between cell counts and dilution level supported the first assumption of the dilution technique.

The same was observed with DMSPp, although chla based growth rates were slightly outside the confidence interval. It is noticeable, however, the dissimilarity between growth rates obtained based on cell counts and chla, while growth rates in 100iso bottles were almost identical irrespective of the biomass indicator. A priori one would expect both approaches (cell based and chla based) to be equivalent and provide similar results. Accumulation of undigested chla and DMSPp concentrations in grazer food vacuoles may have contributed to increase the intercepts of apparent chla and DMSPp growth, but a simple calculation assuming that every O. marina might contain up to 5 undigested I. galbana cells showed this could only account for 25% of the observed difference and therefore cannot be invoked as the sole reason for this paradoxical result. We may find an alternative explanation when observing closely the size structure of the prey community in 100iso bottles (free of predators) and in the dilution series. After 24h incubation the I. galbana cells in the predator free bottles were ca. 15 % smaller in volume (at the 99% confidence level according to a Student t test) than the ones subjected to grazing. It seems therefore, that O. marina was selectively feeding on smaller cells and shaping in this way the size-structure of the prey community. This has consequences for the estimation of algal growth rates in the experiment (higher rates on chla-based results than on cell counts). Conversion of cell counts into algal carbon by using a conversion of cell carbon to the cell volume gave a growth rate of 0.79 ± 0.04 d -1 , i.e., undistinguishable from that of chla. Obviously, the outcome of the experiment would have been the opposite if grazers had selected for larger cells.

The experiment also served to corroborate the second assumption of the dilution technique and support it's suitability to provide mortality rate measurements for the subgroup of the DMSP-producers. DMSP and chla-based estimates of grazing were very similar (Table 1, Fig. 1). In this case, where the algal assemblage was formed by a monoclonal DMSP-producing algal culture, the similarity of DMSP and chla-based estimates is indicative of the suitability of DMSP as a biomass marker, although both rate estimates were significantly different from those obtained from cell counts for the same reasons discussed above. Again, accumulation of undigested chla and DMSPp concentrations in grazer food vacuoles could hardly account for The culture-based experiment, therefore, provided confidence in the fundamentals of the method. The subsequent experiments with natural (hence complex) plankton communities confirmed the potential of the dilution technique to provide growth and grazing-mediated mortality rates for a natural phytoplankton assemblage as well as the sub-assemblage of the DMSP-producers, as already reported by Archer and collaborators (Archer et al., 2001b).

The use of the dilution technique for measuring microzooplankton-mediated DMS production was first suggested by Wolfe and colleagues [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF], and further developed and tested by Archer and colleagues (Archer et al., 2001a). The idea had been inspired by previous laboratory experiments which had shown that microzooplankton grazing on DMSPcontaining phytoplankton leads to DMS formation [START_REF] Wolfe | Release and consumption of DMSP from Emiliania huxleyi during grazing by Oxyrrhis marina[END_REF][START_REF] Christaki | Assessment of the role of copepods and ciliates in the release to solution of particulate DMSP[END_REF][START_REF] Wolfe | Grazing-activated production of dimethyl sulfide (DMS) by two clones of Emiliania huxleyi[END_REF]. If grazing by microzooplankton results in the transformation of DMSPp to DMS, DMS production should decrease with decreasing grazing pressure, i.e., with dilution of the community with filtered seawater [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF]Archer et al., 2001a). Actually, DMS production should decrease with decreasing DMSP ingestion rates with dilution. This prediction indeed matches our observations in both the lab and the natural community dilution experiments, where DMS production rates varied with the DMSP mortality rates throughout the dilution series in such a proportional way that a model I regression analysis could be applied (Figs. 234). The natural grazing-mediated DMS production rate could be calculated from the slope of the regression and the DMSP mortality in the undiluted and unamended bottle. We consider that this approach to grazing-mediated DMS production is more appropriate than that employed by Archer et al. (2001a), who normalized net DMS production rates to the relative algal concentration in each incubation in order to account for the role of phytoplankton as the primary source of DMS. We argue that this normalization procedure yields erroneous numbers. Let us formulate the DMS budget in the 100% whole seawater bottles:

daily net DMS production = DMS t24 -DMS t0 = Pg + Png (1)
where Pg is the gross DMS production due to grazing and Png is the net production not due to grazing, i.e., gross DMS production due to cell-associated and free DMSP lyases minus bacterial DMS consumption. The budget in a d dilution bottle (with d equal to either 0.75 or 0.50 or 0.25) will thus be: The plot of the normalized net DMS production against the dilution factor d would have a hyperbolic shape y = a + (b/x) that might give the false impression of a linear behaviour, where the slope would not be Pg as intended, but would change sign depending on whether there would be net DMS production or consumption in the absence of grazing.

It is to be noted that the grazing-mediated DMS production, Pg, derived from DMSP mortality, does not refer only to DMSP-to-DMS conversion during the very process of grazing, e.g., by the mix of algal DMSP and DMSP lyase upon ingestion or in grazer's vacuoles. It may include also processes dependent on grazing but indirectly related, such as the bacterial DMS production from the dissolved DMSP released during grazing. Over the 24 h period of the incubations, this process will also vary with dilution as dilution affects grazing and the associated release of dissolved DMSP (Archer et al., 2001). Being a very labile substrate, which generally turns over in a matter of hours [START_REF] Kiene | New and important roles for DMSP in marine microbial communities[END_REF][START_REF] Vila-Costa | Seasonal variability of the dynamics of dimethylated sulfur compounds in a coastal northwest Mediterranean site[END_REF], most DMSP released by grazing will be consumed (and partly converted to DMS) during the 24 hours of incubation, at all dilution levels. For our purpose of assessing the role of grazing in DMS production, this secondary production also counts as 'grazing-mediated'.

As discussed above, the intercept of the regression, Png, provides the net DMS production due to processes other than grazing. The fact that, in the laboratory experiment, the intercept of the dilution series was not significantly different from the net DMS production in the bottle with I. galbana only (a tiny net production) supports the use and meaning of the intercept. In the experiment with the natural community on 31 July, the intercept Png was negative (-0.95 ± 0.24 nM d -1 ). That is, without grazing there was a net DMS loss. Within the capped, glass experimental bottles, none or very little headspace was left, so, DMS removal due to ventilation was negligible. DMS photolysis, if any because of the absence of UVB radiation, if so, would have had little effect on estimates of algal-specific DMS production (Archer et al., 2001;[START_REF] Toole | Photolysis and the dimethylsulfide (DMS) summer paradox in the Sargasso Sea[END_REF]. Thus, processes associated with bacterial activity must be the main reason responsible for DMS consumption. The intercept Png, hence, is expected to correspond to bacterial net DMS consumption, that is, the net result of bacterial DMS consumption and production from the DMSPd already present in the initial community.

Underlying this interpretation of the intercept, though, there is the assumption that bacterial consumption is not largely affected by dilution. This has been raised as the major pitfall of the application of the dilution technique to the DMS cycle [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF], because changes in bacterial activity due to dilution could confound the estimates of grazing-mediated DMS production rates or even prevent them by breaking any linearity [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF].

Our experiment on 6 July also served to test the behavior of bacterial DMS consumption with dilution. The experiment yielded indistinguishable DMS consumption rates (Fig. 6), indicating that bacterial DMS consumption was not significantly affected by dilution. This could seem to be quite a striking result, because dilution should have decreased bacterial abundances and activities proportionally. However, there are two hypothetical explanations for such an apparently null effect: (a) By diluting with filtered seawater, we dilute flagellate numbers along with bacterial numbers. The probability of bacterivore-bacterium encounter decreases with dilution, and we should expect bacterivory and bacterial mortality to decrease accordingly. (b) Since dissolved organic matter (DOM) is released during seawater filtration, the bottles with more filtered water (more diluted) will have more DOM. Most probably DMS is not the limiting factor for bacterial growth, and most of the DMS is consumed through cometabolism with other DOM components [START_REF] Vila-Costa | Phylogenetic identification and metabolism of marine dimethylsulfide-consuming bacteria[END_REF][START_REF] Del Valle | Depth-dependent fate of biologicallyconsumed dimethylsulfide in the Sargasso Sea[END_REF]. Therefore, we can expect specific DMS consumption to increase with dilution. We argue that [START_REF] Simó | Biological turnover of DMS, DMSP and DMSO in contrasting open-sea waters[END_REF] reported that DMS accumulation after addition of 260 nM DMDS did not remain linear over a 20-h period, as though the action of DMDS as inhibitor were only effective in the shorter term (8 h). In experiments, however, DMDS amended incubations gave linear DMS production curves throughout the 24-h period, and duplicate bottles replicated very well (Fig. 5), so that community gross DMS production rates could be measured quite precisely. The further estimation of bacterial DMS consumption by subtraction of the slopes of gross and net DMS productions carries a much larger statistical error. An alternative approach would be the 35 S-DMS radiotracing method, where radio-labeled DMS is added in trace amounts and its disappearance is measured with time to obtain very precise rates of bacterial DMS consumption [START_REF] Del Valle | Depth-dependent fate of biologicallyconsumed dimethylsulfide in the Sargasso Sea[END_REF]. Then, to derive gross DMS production rates DMS consumption must be added to net DMS production, again increasing the statistical uncertainty by error propagation. The choice of one approach or the other is, therefore, dependent on whether the emphasis is placed on the gross production or the bacterial consumption. There is also the possibility of combining both. In any case, the otherwise desirable use of 35 S-DMS is limited by its availability because this is not yet a commercially available radiotracer. [START_REF] Wolfe | DMSP and DMS dynamics and microzooplankton grazing in the Labrador Sea: application of the dilution technique[END_REF]Archer et al., 2001a;Archer et al., 2001b;[START_REF] Archer | Is dimethyl sulphide production related to microzooplankton herbivory in the southern North Sea?[END_REF]. The main assumptions of the method have been tested, and the data treatment to derive grazing-mediated DMS production rates has been revised. We have also complemented the method with estimates of the gross DMS production by all processes in the whole planktonic community, to be compared with grazing-mediated DMS production. Altogether, the proposed protocol (Table 3) is intended to provide comprehensive information on the role of microzooplankton grazing in DMS(P) dynamics in natural plankton communities, at least in meso-to oligotrophic (Mediterranean-like) waters: (1) growth and grazing rates of both the bulk phytoplankton assemblage and the sub-assemblage of the DMSP-producers; (2) rates of community gross DMS production and DMS production due to grazing; and (3) bacterial DMS consumption. It is important to notice, however, that there are some processes that the dilution technique as applied here cannot discriminate and measure yet, such as bacteria and virus mediated DMS production. Further studies are required to achieve a sound quantification of these processes. Bottom: net DMS production vs DMSP grazed. In all cases, parallel datapoints correspond to replicate incubation bottles. The results of the linear regression analyses are reported in Table 2. 2.

  through the enzymatic cleavage of dimethylsulfoniopropionate (DMSP), a

  Isocrhysis galbana (obtained from the Provasoli-Guillard National Center of Culture of Marine Phytoplankton in 1988 ) was maintained in filtered, autoclaved seawater enriched with f/2 medium at 19ºC in a 100 µE m -2 s -1 14:10 hours light:dark cycle.The dinoflagellate Oxyrrhis marina (originally isolated from north-western Mediterranean coastal waters in 1995) was grown on Rhodomonas salina and incubated at the same temperature and light conditions as the prymnesiophyte. Stock cultures of both predator and prey were maintained under non-axenic conditions in 2 L Pyrex glass bottles. Before the experiment O. marina was transferred to the dark and starved for 3 days to allow grazers to remove and digest all prey.

  Two 100% bottles (i.e. undiluted seawater) were incubated in parallel without nutrient enrichment to be used as controls to correct for the natural algal growth.

  liquid nitrogen, injected by placing the trap in hot water, and separated and quantified in a Shimadzu 14A gas chromatograph equipped with a flame photometric detector. Daily calibrations were performed by injecting volumes of a gaseous DMS standard obtained from a permeation tube. Total (particulate + dissolved) DMSP (DMSPt) was measured by adding one pellet of NaOH to 10 mL of sample water gas tight vials, which were stored in the dark for 1 to 5 days. In this way all the cells were disrupted and DMSP was hydrolyzed into DMS. The samples were then analyzed as for DMS, and the result was corrected for pre-existing DMS. Dissolved DMSP (DMSPd) was determined by gravity filtering 3 mL of sample water through a GF/F filter, adding one NaOH pellet to the filtrate, keeping it in the dark for 1 to 5 days, measuring the evolved DMS, and subtracting the preexisting DMS concentration. Particulate DMSP (DMSPp) was calculated by the difference between DMSPt and DMSPd. Two replicates were run for each experimental bottle.

  growth rates (µ) were obtained from net growth in unamended 100% bottles plus mortality rate m

  initial DMSPd, and bacterial DMS metabolism. In the laboratory experiment the Png value could be checked with the net DMS production in the bottles containing 100% prey and no predator (100iso). The error associated with Png was the standard error of the intercept. The error of Pg was obtained from those of the slope and the mean DMSP concentration in replicate controls.

  water pool from which all the dilutions were prepared contained 10246 ± 458 (mean ± standard error) I. galbana cells mL -1 and 543 ± 10 O. marina cells mL -1 . The chla concentration was 1.83 ± 0.01 µg L -1 and the DMSPp concentration was 21.4 ± 0.2 nM, i.e., values similar to those commonly found in coastal and productive open-ocean waters.

( 2

 2 µgC L -1 each). Initial chla, DMSPp and DMS concentrations of each dilution bottle were calculated from the measurements made in the whole and filtered waters: whole water contained 0.49 ± 0.03 µg

  (calculated as the grazing mortality rate m times the 24h-mean DMSPp concentration in the whole seawater controls) were 28.8 nM d -1 in June and 8.3 nM d -1 in July, which resulted in yields of grazing-mediated DMS production from the ingested DMSPp of 14% and 43%, respectively. The intercepts gave rates of net DMS production due to processes other than grazing (Png) of 0.3 ± 0.3 nM DMS d -1 in June and -1.0 ± 0.2 nM DMS d -1 in July (Table

  of presence or absence of visible light on DMS production are unclear but, should they occur, the grazing-mediated DMS yields could vary from these reported.The results of the experiment conducted to test if dilution affects bacterial DMS consumption are shown in Fig.6. The time course of DMS concentrations in the incubations with and without DMDS was determined for each dilution. Resulting bacterial DMS consumption rates

  of the observed difference, the remaining being explained by apparent sizeselective grazing.

  daily net DMS production (d) = DMS t24(d) -DMS t0(d) = (Pg•d) + Png (2) with the assumption that only the gross DMS production due to grazing is reduced by the dilution. If we plot the daily net DMS production against the dilution factor d, the slope of the regression line equals the gross DMS production due to grazing (Pg) and the intercept gives the net production not due to grazing (Png). If we normalized the daily net DMS production by the algal fraction, following Archer et al. (2001), equation (2) would read: algal-specific daily net DMS production (d) = (DMS t24(d) -DMS t0(d) )/d = ((Pg•d) + Png)/d = Pg + (Png/d) (3)

  a constant proportion of the DMS production through the dilution series and,

  may help compensate for the initial dilution of the DMS consumers and eventually yield similar consumption rates over the 24-hours period. Finally, we want to point out that the initial DMS concentration was not reduced with the dilution factor because filtered seawater contained a little more DMS than whole seawater.The use of DMDS as an inhibitor of bacterial DMS metabolism to determine gross DMS production in parallel to dilution experiments is based upon the assumption that DMDS efficiently blocks bacterial DMS metabolism throughout the duration of the experiments. In a study in North Atlantic waters,

  we have revisited the dilution technique suggested by Wolfe, Archer and collaborators

  Results of the laboratory dilution experiment with O. marina grazing on I. galbana. Values for grazing mortality (m) and algal growth (µ) rates were determined with cell counts, chlorophyll a, and DMSPp after model I regression analyses of the curves of Fig. 1. The directly measured algal growth rate in the bottle with I galbana only (r iso ) is also shown. The bottom line shows the relationship between DMS production and dilution fraction (grazing pressure) as calculated from model I regression analyses of the curves of Fig. 2. Pg refers to DMS production due to grazing (slope), and Png is for DMS production due to factors other than grazing (intercept). The directly measured DMS production in the bottle with I galbana only (100iso) is also shown. Values are presented with the associated standard error. Biomass descriptor m (d -1 )

Fig. 3 .

 3 Fig. 3. Dilution experiment with a natural plankton community sampled on 14 June in coastal

Fig. 4 .

 4 Fig. 4. Dilution experiment with a natural plankton community sampled on 31 July in coastal

  DMS production vs DMSP grazed. In all cases, parallel datapoints correspond to replicate incubation bottles. The results of the linear regression analyses are reported in Table

Fig. 5 .

 5 Fig. 5. Time course of DMS concentrations in dark incubations of whole natural communities

  Fig.1

Table 2 :

 2 Summary of the growth (µ) and grazing mortality (m) rate estimates obtained with the natural community experiments. Note the comparison between the bulk phytoplankton community (chla measurements) and the sub-community of the DMSP producers (DMSPp measurements). The bottom lines show the rates of DMS production due to grazing (Pg) and not due to grazing (Png). Also included is the coefficient of model I linear regressions (r 2 ).

	µ (d -1 )	r 2	r iso (d -1 )

Table 3 :

 3 Recommended protocol for the application of the dilution technique to quantify the growth of DMSP-producing algae, the DMSPp consumption and DMS production by micro-herbivores, the community gross DMS production, and the bacterial DMS consumption in plankton communities. All treatments to be prepared at least in duplicates. : Ideally to be incubated in the same light regime as the dilution bottles. We incubated them in the dark. d: Time zero concentrations are calculated from measurements in whole and 0.2 µm filtered according to the dilution factors. e: At least one intermediate time point is convenient for the regression analysis. In our case we sub-sampled after 4, 6, 10, 20 and 24 hours. Plots of the apparent growth of Isochrysis galbana versus the dilution factor in the laboratory grazing experiment with Oxyrrhis marina. Algal biomass measured as cell counts (filled circles), chlorophyll a (open circles), and DMSPp (open squares). Parallel datapoints correspond to replicate incubation bottles. The results of the linear regression analyses are reported in Table1. Fig. 2. Relationship between DMS produced and DMSP grazed in each dilution level in the laboratory experiment with O. marina feeding on I. galbana. Parallel datapoints correspond to replicate incubation bottles. The results of the linear regression analyses are reported in

	Bottle	% of whole		Incubation	Subsampling
	label	seawater	Amendment	conditions	times (h) d	Measurements	Data analysis	Process rate calculations
	25	25	+ nutrients	light b	24	chla, DMSPp,	1. Regression of chla	1.1. Algal growth (µ chla or µ DMSP )
						DMS	or DMSPp apparent	from intercepts corrected for
							growth vs dilution	growth without nutrients
	50	50	+ nutrients	light	24	chla, DMSPp,	factor
						DMS		1.2. Mortality by grazing (m chla or
								m DMSP ) from slopes
	75	75	+ nutrients	light	24	chla, DMSPp,
						DMS	2. Regression of DMS	2.1. Net DMS production not due
							production vs	to grazing (Png) from intercept
	100	100	+ nutrients	light	24	chla, DMSPp,	DMSP grazed in
						DMS	each dilution	2.2. DMS production due to
							(scaled m DMSP x	grazing (Pg) from slope x
	100c	100	none	light	24	chla, DMSPp,	mean DMSP conc.)	mean DMSP conc. in controls
						DMS
	100d	100	none	light/dark c	8, 24 e	DMS	Regression of DMS	Community net DMS production
							concentration vs time	from slope
	100i	100	+ DMDS a	light/dark c	8, 24 e	DMS	Regression of DMS	Community gross DMS production
							concentration vs time	from slope
								Bacterial DMS consumption from
								subtraction of the above slopes f

cf: Alternatively, the 35 S-DMS radiotracing method can be used to determine DMS consumption more precisely and to derive community gross DMS production by addition to community net DMS production.

Table 1 .

 1

http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research

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Acknowledgements

We acknowledge support from the Spanish Ministry of Science and Innovation (former Education and Science) through a PhD fellowship to V.S. and through the project MICROROL (CTM2004-02575/MAR). We thank J. Movilla for helping with field sampling, E. Saiz for assistance with the setup of the dilution experiment, I. Trepat and J. Felipe for chlorophyll a analyses, and two anonymous reviewers whose comments helped to improve the manuscript.


Montserrat Corbera
email: montserrat.corbera@uvic.cat

Josep Maria Cors
email: cors@epsem.upc.edu

Jaume Llibre
email: jllibre@mat.uab.cat

On the central configurations of the planar 1 + 3 body problem

Keywords: 1 + 3 body problem, central configurations, coorbital satellites

We consider the Newtonian 4 body problem in the plane with a dominat mass M . We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision arithmetics, we provide evidence that the number of central configurations varies from five to seven.

Introduction

We study configurations with one massive central mass and several infinitesimal coorbital satellites describing the same circular orbit around the central massive mass. Such configurations are called relative equilibra, because in a rotation frame the satellites remain fixed. Recently these configurations have attracted the attention of astronomers. [START_REF] Renner | Stationary configurations for co-orbital satellites with small arbitary masses[END_REF], suggest that the presence of coorbital satellites might explain, at least partly, the confinement of Neptune's ring arcs.

When the configuration of the coorbital satellites changes its size, but keep the shape, the motion is called homographic. In that case each satellite describe a Keplerian orbit around the central mass.

A configuration that allows relative equilibria and homographic motions is called a central configuration and equals to configurations such that the total Newtonian acceleration of every mass is equal to a constant multiplied by the position vector of this mass with respect to the center mass of the configuration, see [START_REF] Lee | Central configurations of the five-body problem with equal masses[END_REF] and the references quoted there.

The central configurations in the case of one large mass and n infinitesimal arbitrary masses are called central configurations of the 1 + n body problem.

Maxwell [START_REF] Maxwell | On the stability of the motion of Saturn's rings[END_REF] studied the case of n equal masses orbiting Saturn at a common radius and uniformly distributed about a cercle of this radius. He concluded that, for large n, the ring is stable if a convenient inequality between the mass of the ring and the mass of Saturn is satisfied. More recently, Moeckel [START_REF] Moeckel | Linear stability of relative equilibria with a dominant mass[END_REF] studied the linear stability of the N -body problem when the motion is a rotation about the center of mass and under the condition that all the masses except one become vanishingly small. Notice that the 1 + n body problem is a particular case. Moeckel shows that the 1 + n-gon (when the small equal masses are uniformly distributed) is stable if and only if n ≥ 7. For n ≤ 6, he gives some exemples of the stable configurations where the small equal bodies are not uniformly distributed. Additionally Roberts [START_REF] Roberts | Linear stability of the n + 1-gon relative equilibrium, Hamiltonian[END_REF] carries the analysis a step further showing that the large mass has to grow proportionally to n 3 to ensure the linear stability. Such a criteria agrees with those given by [START_REF] Scheeres | Linear stability of a self-graviting rings[END_REF].

For large values of n, [START_REF] Hall | Central configurations in the planar 1+n body problem[END_REF] shows that if n ≥ e 27,000 , then there is a unique class of central configurations, the regular 1 + n-gon. In [START_REF] Casasayas | Central configurations of the planar 1 + n body problem[END_REF] the same result is proved under the assumption that n ≥ e 73 .

When n is small and the small masses are equal, in [START_REF] Cors | Central configurations of the planar coorbital satellite problem[END_REF] the authors obtain numerically that the 1 + n-gon is the only configuration when n ≥ 9. In the case n = 4 they proved that there are only three symmetric central configurations. Recently [START_REF] Albouy | Relative equilibria of four identical satellites[END_REF] proved that any central configuration of the 1 + 4 body problem must be symmetric.

In [START_REF] Renner | Stationary configurations for co-orbital satellites with small arbitary masses[END_REF] the authors removed the condition that the infinitesimal masses are identical and obtained results about the inverse problem, that is, given a configuration of the coorbital satellites, find the infinitesimal masses making it a central configuration. They also studied the linear stability.

In this paper we study the planar central configurations of the 1 + 3 body problem without collision between two infinitesimal satellites when two infinitesimal masses are equal. The characterization of these central configurations is summarized in Section 5.

Definitions and equations

In this section we present the equations of the central configurations of the 1 + n body problem. More details can be found in [START_REF] Casasayas | Central configurations of the planar 1 + n body problem[END_REF] and [START_REF] Cors | Central configurations of the planar coorbital satellite problem[END_REF].

Consider N particles of masses m 1 ,...,m N in R 2 subject to their mutual Newtonian gravitational attraction. In an inertial reference frame with the origin at the center of mass of the N bodies and choosing suitable units, the equations of motion of the N body problem in

R 2 are M q ′′ = -V q , ( 1
)
where M is the mass matrix M = diag(m 1 , m 1 , ..., m N , m N ), q = (q 1 , ..., q N ) is the position vector with q i ∈ R 2 , V the potential vector

V (q 1 , ..., q N ) = 1≤i<j≤N m i m j q i -q j ,
and V q = (∂V /∂q 1 , ..., ∂V /∂q N ). Note that the Newtonian gravitational constant has been taken equal to one choosing conveniently the unit of time. Taking into account the singularities of equation ( 1), the configuration space of the planar N body problem associated with the mass matrix M is

M(m 1 , ..., m N ) = {q ∈ R 2N : N i=1 q i = 0, q i = q j , for i = j}.
Given a matrix M , we say that q ∈ M represents a central configuration of the associated planar N body problem if there exists a positive constant λ 2 such that M -1 V q = λ 2 q, i.e., if the acceleration vector of every particle is directed towards the center of mass and its modulus is proportional to the distance from the particle to the center of mass. We shall denote by C the set of planar central configurations associated with a given mass matrix M . Notice that C is invariant with respect to homothetic transformations and rotations of R 2 . We shall denote by C the set of planar central configurations modulus the group SO(2) of plane rotations. Now we concentrate our interest on the central configurations of the planar 1 + n body problem with infinitesimal unequal masses. That is, we consider N = 1 + n and let q(ε) = (q 0 (ε), q 1 (ε), ...., q n (ε)) ∈ C be a central configuration of the planar 1 + n body problem with masses m 0 = 1, m i = µ i ε, i = 1, ..., n, which depends continuously on ε. We say that q = (q 0 , q 1 , ..., q N ) is a central configuration of the planar 1 + n body problem if there exists lim ε→0 q(ε) and this limit is equal to q.

The following two results can be found in the unpublished paper of [START_REF] Hall | Central configurations in the planar 1+n body problem[END_REF] and in [START_REF] Casasayas | Central configurations of the planar 1 + n body problem[END_REF].

PROPOSITION 2.1. All the central configurations of the planar 1 + n body problem lie on a circle centered at q 0 = 0. Let q = (q 0 , ..., q n ) be a central configuration of the planar 1 + n body problem. We say that q is non-collision if q i = q j for i = j. That is, we exclude the possibility that the distance between two or more masses tends to zero as ε → 0. PROPOSITION 2.2. Let q = (q 0 , ..., q n ) be a non-collision central configuration of the planar 1 + n body problem. Denoting by ϑ i the angle defined by the position of q i for i = 1, ..., n., we have n j=1,j =i

µ j sin(ϑ j -ϑ i )   1 - 1 2 √ 2 (1 -cos(ϑ j -ϑ i )) 3   = 0, (2)
for i = 1, ..., n.

We call any solution (ϑ 1 , • • • , ϑ n ) of system (2) a coorbital central configuration of the planar 1 + n body problem. Since we are interested in central configurations modulus rotations and homothetic transformations, without loss of generality we can assume that the circle has radius 1 and that ϑ 1 = 0. Then the configuration space of the coorbital central configurations of the planar 1 + n body problem is

A = {(ϑ 1 , • • • , ϑ n ) ∈ [0, 2π) n : ϑ 1 = 0, ϑ i = ϑ j for i = j}.
The set A has (n-1)! connected components, each of which corresponds to a given permutation of the n satellites on the circle. We shall consider one connected component of A, i.e. a given counterclockwise ordering of the n satellites. Without loss of generality we can assume that 0 = ϑ 1 < ϑ 2 < • • • < ϑ n < 2π. In this case it is natural to take as coordinates the angles between two consecutive satellites

θ i = ϑ i+1 -ϑ i , i = 1, ..., n -1.
Also it is convenient to work with an nth redundant coordinate angle 1+3-rev3.tex; 16/09/2010; 16:10; p.4 which measures the angular distance between particle n and particle 1.

θ n = 2π - n-1 i=1 θ i ,
In this way the configuration space will be the simplex S given by the intersection of the hyperplane n i=1 θ i = 2π and the space {(θ 1 , ..., θ n ) : θ i > 0}.

In the coordinates (θ

1 , • • • , θ n ) system (2) becomes µ 2 f (θ 1 ) + µ 3 f (θ 1 + θ 2 ) + • • • + µ n f (θ 1 + ... + θ n-1 ) = 0, µ 3 f (θ 2 ) + µ 4 f (θ 2 + θ 3 ) + • • • + µ 1 f (θ 2 + ... + θ n ) = 0, µ 4 f (θ 3 ) + µ 5 f (θ 3 + θ 4 ) + • • • + µ 2 f (θ 3 + ... + θ n + θ 1 ) = 0, • • • (3) µ 1 f (θ n ) + µ 2 f (θ n + θ 1 ) + • • • + µ n-1 f (θ n + θ 1 + • • • + θ n-2 ) = 0, θ 1 + • • • + θ n = 2π,
where the function

f (θ) = sin θ 1 - 1 2 √ 2 (1 -cos θ) 3 .
In the case of three satellites system (3) is

µ 2 f (θ 1 ) + µ 3 f (θ 1 + θ 2 ) = 0, µ 3 f (θ 2 ) + µ 1 f (θ 2 + θ 3 ) = 0, µ 1 f (θ 3 ) + µ 2 f (θ 3 + θ 1 ) = 0, θ 1 + θ 2 + θ 3 = 2π,
or equivalently

µ 2 f (θ 1 ) -µ 3 f (θ 3 ) = 0, µ 3 f (θ 2 ) -µ 1 f (θ 1 ) = 0, (4) µ 1 f (θ 3 ) -µ 2 f (θ 2 ) = 0, θ 1 + θ 2 + θ 3 = 2π, because f (2π -θ) = -f (θ).
Notice that the first three equations in (4) are linearly dependent. So one equation is irrelevant. We also chose the unit of mass so that µ 3 = 1. Under these assumptions equations (4) are equivalent to the system

µ 1 f (θ 1 ) -f (θ 2 ) = 0, µ 2 f (θ 1 ) -f (θ 3 ) = 0, θ 1 + θ 2 + θ 3 = 2π.
(5)

Next results allows us to classify the coorbital central configurations of the 1 + 3 body problem in two main classes, as we shall see latter on.

PROPOSITION 2.3. Let (θ 1 , θ 2 , θ 3 ) be a coorbital central configuration solution of system (5).

(i) If f (θ i ) = 0 for some i = 1, 2, 3, then µ j = 0 for some j = 1, 2.

(ii) The sign of f (θ i ) must be positive or negative for all i = 1, 2, 3.

Proof. Supose that f (θ 1 ) = 0, then from system (5) f (θ 2 ) = f (θ 3 ) = 0. The zeros of the function f are π/3, π and 5π/3. Clearly the sum of any choice of these three values never equals to 2π.

Consider now f (θ 2 ) = 0. Then either µ 1 = 0 or f (θ 1 ) = 0 (previous case). Finally if f (θ 3 ) = 0 implies, using the same argument, that µ 2 = 0. So statement (i) follows

On the other hand, supose that f (θ 1 ) < 0, then from system (5) clearly f (θ 2 ) < 0 and f (θ 3 ) < 0, as well. The other cases run in a similar way.

Proposition 2.3 says that any coorbital central configuration of the 1 + 3 body problem cannot posses an angle θ i equal to π/3, π or 5π/3 for i = 1, 2, 3. COROLLARY 2.4. Let (θ 1 , θ 2 , θ 3 ) be a coorbital central configuration solution of system (5). Then either θ i ∈ (π/3, π) for all i = 1, 2, 3, or two angles are less than π/3 and the third belongs to the interval (4π/3, 5π/3).

Proof. This follows from the Proposition 2.3 and the plot of the function f .

In short, the 1 + 3 body problem have two unconnected classes of coorbital central configurations. We call coorbital central configuration of class A the one having two angles less than π/3, and coorbital central configuration of class B the one having the three angles greater than π/3.

Symmetric central configurations

A coorbital central configuration (θ 1 , θ 2 , . . . , θ n ) of the planar 1 + n body problem is symmetric with respect to a straight line L containing the central body, if modulus a cyclic permutation of the angles we have, when n is even either

θ 1 = θ n , θ 2 = θ n-1 , . . . , θ n 2 = θ n+2 2 ,
(in this case the symmetry axis L contains two satellites), or (in this case the symmetry axis L contains no satellites) and when n is odd

θ 1 = θ n-1 , θ 2 = θ n-2 , . . . , θ n-2 2 = θ n+2 2 , θ n 2 ,
θ 1 = θ n , θ 2 = θ n-1 , . . . , θ n-1 2 = θ n+3 2 ,
(in this case the symmetry axis L contains one satellite).

From the definition of symmetric coorbital central configuration in the case of three satellites the configuration is symmetric if and only if two of the angles (θ 1 , θ 2 , θ 3 ) are equal. PROPOSITION 3.1. If the coorbital central configuration of the 1 + 3 body problem is symmetric, with respect to a straight line L containing the central body, then µ i = µ j for some i, j = 1, 2, 3 with i = j Proof. Suppose that θ 2 = θ 3 then from system (5) we have that µ 1 f (θ 1 ) = µ 2 f (θ 1 ) and so µ 1 = µ 2 . Similar arguments can be used to prove cases θ 1 = θ 2 or θ 1 = θ 3 .

Without loss of generality, we assume that µ 1 = µ 2 = µ. So system (5) is equivalent to system

µf (θ 1 ) -f (θ 2 ) = 0, µf (θ 1 ) -f (θ 3 ) = 0, θ 1 + θ 2 + θ 3 = 2π. ( 6
)
By setting α = θ/2 we see that f (θ) can be written as

g(α) = cos α 4 sin 2 α (8 sin 3 α -1).
Let

α 1 = θ 1 /2 and α 2 = θ 2 /2 and α 3 = θ 3 /2 = π -α 1 -α 2 .
By writing g(α 3 ) in terms of α 1 and α 2 we get

g(α 3 ) = cos(α 1 + α 2 )(1 -8(sin(α 1 + α 2 )) 3 ) 4(sin(α 1 + α 2 )) 2 ,
where α 1 ∈ (0, π) and α 2 ∈ (0, π). Since α 1 + α 2 = 0 and α 1 + α 2 = π (notice that if α 1 + α 2 = π then α 3 = 0 which is not possible), the denominator of g(α 3 ) is different from zero. Let e 1 (α 1 , α 2 ; µ) = µg(α 1 ) -g(α 2 ) and e 2 (α 1 , α 3 ; µ) = µg(α 1 ) -g(α 3 ). The system ( 6) becomes

e 1 (α 1 , α 2 ; µ) = 0, e 2 (α 1 , α 3 ; µ) = 0, (7)
where α 1 , α 2 ∈ (0, π) and

α 3 = π -α 1 -α 2 ∈ (0, π).
Under the assumption µ 1 = µ 2 = µ, the symmetric coorbital central configurations satisfy θ 2 = θ 3 , or equivalently α 2 = α 3 . Analyzing the solutions of (7) with α 2 = α 3 depending on the values of µ we obtain the following result. Proof. The proof is structured as follows. First, by using appropriate variables, we transform system (7) with α 2 = α 3 into a system of polynomial equations. Then by using resultant theory we obtain all the values of µ at which the number of solutions of that system changes, and we count the number of solutions of the system for the different values of µ. Finally we analyze the solution of the system in function of µ.

The assumption α 2 = α 3 implies that α 1 = π -2α 2 , so α 2 ∈ (0, π/2). Since cos α 1 = sin 2 α 2 -cos 2 α 2 and sin α 1 = 2 sin α 2 cos α 2 , system (7) can be written in terms of c = cos α 2 and s = sin α 2 . Then eliminating the denominators we get the following system equivalent to system (7)

-s 2 E 1 = 0, -s 2 c 2 + s 2 E 2 = 0, F = 0, (8)
where E 1 = 32s 3 c 3 -4c 3 + 64s 3 c 5 -64s 5 c 3 -c 2 + s 2 µ, E 2 = 32s 3 c 9 + 96s 5 c 7 + 96s 7 c 5 + 32s 9 c 3 -4c 3 + 64s 3 c 7 -c 4 -64s 7 c 3 + s 4 µ, F = c 2 + s 2 -1, and 0 < s 1 and 0 c < 1 because α 2 ∈ (0, π/2). Clearly c 2 + s 2 = 0 and s = 0. Therefore system (8) can be written as

E 1 = 0, E 2 = 0, F = 0. ( 9
)
We perform the following substitutions:

c 2 = 1 -s 2 , c 3 = c(1 -s 2 ), c 4 = (1 -s 2 ) 2 , c 5 = c(1 -s 2 ) 2 , c 7 = c(1 -s 2 ) 3 , and c 9 = c(1 -s 2 ) 4 .
After these substitutions E 1 = E 2 and system ( 9) becomes E = 0, F = 0 where E = -32cs 5 +32cs 3 +4cs 2 -4c+ 128cs 7 -192cs 5 + 64cs 3 + 2s 2 -1 µ. 1+3-rev3.tex; 16/09/2010; 16:10; p.8 From now on instead of writing the system E = 0, F = 0, we also shall write E = F = 0.

Using the resultant theory we can eliminate the variable c in the equations E = F = 0. The resultant of the polynomials E and F with respect to the variable c is the polynomial R = 16384s 16 -65536s 14 + 102400s 12 -77824s 10 + 28672s 8 -4096s 6 + 4s 4 -4s 2 + 1 µ 2 + -8192s 14 + 28672s 12 +1024s 11 -36864s 10 -3584s 9 + 20480s 8 + 4608s 7 -4096s 6 -2560s 5 + 512s 3 µ + 1024s 12 -3072s 10 -256s 9 + 3072s 8 +768s 7 -1008s 6 -768s 5 -48s 4 + 256s 3 + 48s 2 -16, in the variables s and µ, and R satisfies the following property: if (c, s) = (c * (µ), s * (µ)) is a common root of the polynomials E and F , then s * (µ) is a root of the polynomial R (see for instance (Lang, 1993) and [START_REF] Olver | Classical invariant theory[END_REF] for more details on the resultant theory). Therefore the solutions of equation R = 0 give at least all the values s(µ) of the common solutions (c(µ), s(µ)) of E = F = 0. Notice that equation R = 0 could also give solutions s(µ) that do not provide common solutions of E = F = 0.

We are interested in the set of values of µ at which the number of real solutions (c(µ), s(µ)) of system E = F = 0 changes. Since equation R = 0 provides all the values s(µ) of the common solutions (c(µ), s(µ)) of E = F = 0, the set of the values of µ at which the number of real solutions of R = 0 changes contains at least all the values of µ at which the number of real solutions of E = F = 0 changes.

The number of real solutions of R = 0 can change only at the values of µ that provide solutions with multiplicity greater than 1, that is, at the values of µ where R = dR/ds = 0.

Again using resultant theory we can eliminate the variable s in the system R = dR/ds = 0. The resultant of the polynomials R and dR/ds with respect to s is the polynomial

H = k 0 µ 34 (µ -4)(µ + 4)H 0 ,
where k 0 = 0 is a large constant and H 0 is the following polynomial of degree 22

H 0 = 544400220773632720896µ 22 -11491835434562035187712µ 20
+2898856085837444554752µ 19 + 45101390498475214934016µ 18 -50207178662075239133184µ 17 -3597828247551090484224µ 16 +54441252684647804362752µ 15 -90330266881601149980672µ 14 +105029467560226902200832µ 13 -87060482984162666161152µ 12 +53192685061230729315840µ 11 -25608023464518181047260µ 10 +9249906747903520991616µ 9 -1771852062223329593868µ 8 -159500845521348320736µ 7 + 119990316536124348321µ 6 +12123699015371647968µ 5 -11384652544065195120µ 4 +400296455761238280µ 3 + 345472406710968960µ 2 +2678966190976320µ -8438743501575408.

By the properties of the resultant, the solutions of equation H = 0 provide at least all the values of µ of the common solutions (s, µ) of system R = dR/ds = 0, although they also can provide values of µ that do not correspond to common solutions of the system. Therefore the real solutions of H = 0 with µ > 0 provide all the possible values of µ at which the number of real solutions of system R = 0 changes. We compute all the solutions of the polynomial equation H 0 = 0 with the help of Mathematica and we find 22 different solutions of which only 3 are real solutions with µ > 0 µ = u 1 = 0.8961616399532140 . . . µ = u 2 = 1.2506375233635711 . . . µ = u 4 = 4.0000142922001194 . . . On the other hand, equation H = 0 has the additional real solution with µ > 0 µ = u 3 = 4.

In short there are four possible values of µ at which the number of real solutions of E = F = 0 could change, µ = u 1 , u 2 , u 3 , u 4 . Since we only are interested in studying the solutions of E = F = 0 satisfying 0 < s 1 and 0 c < 1 we need to control not only the changes in the number of real solutions of the system E = F = 0 without constraints. We also need to control when a real solution (s(µ), c(µ)) starts to satisfy the conditions 0 < s 1 and 0 c < 1 for some values of µ.

Next we find values of µ where the number of real solutions of system E = F = 0 does not change but the number of real solutions satisfying 0 < s 1 and 0 c < 1 can change. For doing that, since all the solutions s(µ) of E = F = 0 are solutions of R = 0, we find the values of µ at which either s = 0 or s = 1 is a solution of R = 0. Evaluating R = 0 at s = 0 and s = 1 we get

R| s=0 = µ 2 -16 = 0, R| s=1 = µ 2 .
Therefore the unique values of µ at which the number of real solutions of system E = F = 0 satisfying 0 < s 1 and 0 c < 1 can change are µ = u 1 , u 2 , u 3 , u 4 .

Finally we compute the number of real solutions of system E = F = 0 satisfying 0 < s 1 and 0 E,F, c] = 0 with 0 < s 1 provide a solution of system E = F = 0. Using this procedure for each v i , i = 1, . . . , 5 we get one solution for µ = v 1 and three solutions for µ = {v 2 , v 3 , v 4 , v 5 }, so the number of real solutions of system E = F = 0 with 0 < s 1 and 0 c < 1 changes at µ = u 1 .

c < 1 at values µ = {v 1 , v 2 , v 3 , v 4 , v 5 } with 0 < v 1 < u 1 < v 2 < u 2 < v 3 < u 3 < v 4 < u 4 < v 5 in
Analyzing the solutions of system E = F = 0 as a function of µ we observe that the solution (s(µ), c(µ)) that we have obtained for 0 < µ < u 1 persists for all µ and it gives a family of symmetric coorbital central configurations of class A, denoted by A s . At µ = u 1 appears a new symmetric coorbital central configuration of class B, denoted by B s which bifurcates in two families of symmetric coorbital central configurations, denoted by B s 1 and B s 2 for µ > u 1 . See Figures 1 and2 where the angles 1 (µ) and θ 2 (µ) have been computed. Also see Figures 3 and4 where qualitative pictures of coorbital central configurations have been plotted for different values of µ.

Non-symmetric central configurations

In Section 3 we have analyzed the symmetric coorbital central configurations of the 1 + 3 body problem and we have found the value of µ which the number of symmetric coorbital central configurations

Here we analyze the values µ the number of coorbital central configurations change independently if such configurations are symmetric or not. In particular we prove the following result. (b) From the non-symmetric coorbital central configurations of the 1 + 3 body problem do not bifurcate any non-symmetric coorbital central configurations.

Proof. System (7) can be written in terms of c 1 = cos α 1 and s 1 = sin α 1 , c 2 = cos α 2 and s 2 = sin α 2 , then by eliminating the denominators and by doing the substitutions

c 2 1 = 1 -s 2 1 , c 3 1 = c 1 (1 -s 2 1 ), c 4 1 = (1 -s 2 1 ) 2 , c 2 2 = 1 -s 2 2 , c 3 2 = c 1 (1 -s 2 2 ), c 4 2 = (1 -s 2 2 ) 2 , system (7) is equivalent to system E 1 = E 2 = F 1 = F 2 = 0 where E 1 = -8c 2 s 2 1 s 3 + c 2 s 2 1 + 8c 1 s 3 1 s 2 2 -c 1 s 2 2 µ, E 2 = 64c 2 s 3 2 s 6 1 -32c 2 s 2 s 6 1 + 64c 1 s 4 2 s 5 1 -64c 1 s 2 2 s 5 1 + 8c 1 s 5 1 -64c 2 s 3 2 s 4 1 +24c 2 s 2 s 4 1 -32c 1 s 4 2 s 3 1 + 24c 1 s 2 2 s 3 1 + s 2 s 3 1 + 8c 2 s 3 2 s 2 1 -c 1 c 2 s 2 1 + -16c 2 s 2 s 6 1 -16c 1 s 2 2 s 5 1 + 8c 1 s 5 1 + 16c 2 s 2 s 4 1 + 8c 1 s 2 2 s 3 1 +2c 2 s 2 s 3 1 + 2c 1 s 2 2 s 2 1 -c 1 s 2 1 -2c 2 s 2 s 1 -c 1 s 2 2 µ, F 1 = c 2 1 + s 2 1 -1, F 2 = c 2 2 + s 2 2 -1.
We are interested in solutions of this system with 0 < s 1 1, 0 < s 2 1, -1 < c 1 < 1, -1 < c 2 < 1, and such that α 3 = π -α 1 -α 2 ∈ (0, π), here α i = arctan(s i /c i ) when s i /c i > 0 and α i = arctan(s i /c i ) + π when s i /c i < 0, i = 1, 2. Next we will find the values of µ at which the number of real solutions of system E 1 = E 2 = F 1 = F 2 = 0 satisfying conditions changes. First we find the values of µ at which the number of real solutions of

E 1 = E 2 = F 1 = F 2 = 0 changes.
eliminate the variable c 1 by doing the resultants

R 1 = Res [E 1 , E 2 , c 1 ], R 2 = Res [E 1 , F 1 , c 1 ], R 3 = Res [E 1 , F 2 , c 1 ].
Next we eliminate the variable c 2 by doing the resultants

S 1 = Res [R 3 , R 1 , c 2 ], S 2 = Res [R 3 , R 2 , c 2 ].
The factorization of the polynomials S 1 and S 2 contains factors with multiplicity greater than 1. We define two new polynomials S 1 and S 2 as the polynomials that contain exactly the same factors than S 1 and 1+3-rev3.tex; 16/09/2010; 16:10; p.12 S 2 but all of them with multiplicity one. Then we eliminate the variable s 2 by doing the resultant

T = Res [S 1 , S 2 , s 2 ].
The factorization of T is formed by a large constant, a power of µ and five factors in the variables s 1 and µ. We eliminate the constant, the power of µ (we are only interested in solutions with µ > 0) and the multiplicities of the factors and we obtain a new polynomial

T = T 1 . T 2 . T 3 . T 4 . T 5 ,
where

T 1 = s 1 , T 2 = 2s 1 -1, T 3 = 4s 2 1 + 2s 1 + 1, T 4 = 4096s 16 1 -8192s 14 1 -2048s 13 1 + 4096s 12 1 + 4096s 11 1 + 384s 10 1 -2048s 9 1 -768s 8 1 -32s 7 1 + 384s 6 1 + 64s 5 1 + s 4 1 -32s 3 1 -2s 2 1 +1) µ 4 + 2048s 14 1 -2048s 12 1 -512s 11 1 -768s 10 1 + 512s 9 1 +1824s 8 1 + 192s 7 1 -1056s 6 1 -448s 5 1 -12s 4 1 + 256s 3 1 + 28s 2 1 -16) µ 2 + 256s 10 1 -1280s 8 1 -32s 7 1 + 1024s 6 1 + 160s 5 1 -128s 3 1 µ + 256s 12 1 + 192s 8 1 -252s 6 1 ,
and T 5 is a polynomial depending on s 1 and µ of degree 44 in the variable s 1 . By the properties of resultants, the new equation T = 0 contains all the values s 1 (µ) of the solutions (c 1 (µ), s 1 (µ), c 2 (µ), s 2 (µ)) of the system E 1 = E 2 = F 1 = F 2 = 0 and probably values s 1 (µ) that do not provide solutions of the initial system. Therefore the values of µ at which the number of real solutions of E 1 = E 2 = F 1 = F 2 = 0 changes are also values of µ which the number of real solutions of T = 0 changes.

The number of real solutions of T = 0 can change either at the values of µ at which a real root of a factor T i has multiplicity greater than one, or at the values of µ at which a real root of a factor T i is also a root of a factor T j with i = j.

Clearly the root of the factor T 1 , s 1 = 0 is not a root of the factors T 2 and T 3 . Moreover

T 4 | s 1 =0 = µ 4 -16µ 2 , T 5 | s 1 =0 = -µ 10 -4µ 8 .
µ = u 4 = 0.2055692659990572 . . . µ = u 9 = 0.6791080539220485 . . . µ = u 12 = 0.9939202640321302 . . . µ = u 14 = 1.0072825044062513 . . .

In short, the possible values of µ > 0 at which the number of real solutions of T = 0 can change are µ = u i for i = 1, . . . , 23. We are interested in the values of µ at which the of real solutions of

E 1 = E 2 = F 1 = F 2 = 0 with 0 < s 1 1, 0 < s 2 1, -1 < c 1 < 1, -1 < c 2 < 1, and α 3 = π -α 1 -α 2 ∈ (0, π).
Using the same argument as in Section 3 we need to control not only the changes in the number of real solutions of the system E 1 = E 2 = F 1 = F 2 = 0 without constraints. We also need to control when a real solution starts to the conditions 0 < s j 1 and -1 < c j < 1, j = 1, 2 for some values of µ. Since the set of real solutions s 1 (µ) of system E 1 = E 2 = F 1 = F 2 = 0 is contained in the set of real solutions of T = 0, in order to find these last possible values of µ it is sufficient to find the values µ at which a solution s 1 (µ) of T = 0 satisfies that s 1 (µ) = 0 or s 1 (µ) = 1. Notice that s 1 = 0 is a solution of T = 0 but this solution does not depend on µ so we do not consider it. Evaluating T /s 1 at s = 0 and s = 1 we get

(T /s 1 )| s 1 =0 = -µ 4 -16µ 2 -µ 10 -4µ 8 (T /s 1 )| s 1 =1 = 92747200.
we do not obtain new possible values of µ at which the number of solutions real solutions of the system E 1 = E 2 = F 1 = F 2 = 0 with satisfying the desired conditions changes.

Finally we compute the number of real solutions of the system

E 1 = E 2 = F 1 = F 2 = 0 with 0 < s 1 1, 0 < s 2 1, -1 < c 1 < 1, -1 < c 2 < 1, and α 3 = π -α 1 -α 2 ∈ (0, π) at values µ = v i with v 1 ∈ (0, u 1 ), v i+1 ∈ ( u i , u i+1
) for i = 1, . . . , 23 and v 24 > u 23 by proceeding of the following way. System (7) can be written in terms of c 1 = cos α 1 and s 1 = sin α 1 , c 2 = cos α 2 and s 2 = sin α 2 . In order to simplify our computations we introduce two new variables z 1 = e iα 1 and z 2 = e iα 2 where i = √ -1. Using these variables, cos α j = (z 2 j + 1)/(2 z j ) and sin α j = (z 2 j -1)/(2 i z j ) for j = 1, 2. Then by doing the previous substitutions and by eliminating the denominators system ( 7) is equivalent to system e 1 = e 2 = 0 where Notice that µ = u 1 is the bifurcation value that we have obtained in Section 3. Analyzing the solutions of system e 1 = e 2 = 0 as a function of µ in a neighbourhood of µ = u 1 and µ = u 19 we observe the following. In a neighbourhood of µ = u 1 the four solutions corresponding to non-symmetric coorbital central configurations that we have found for µ < u 1 persist when µ u 1 and at µ = u 1 appears a new solution that bifurcates into two ones for µ > u 1 providing symmetric coorbital central configurations. Two of the four solutions corresponding to non-symmetric coorbital central configurations that we have found for µ < u 19 match to one of the solutions providing a symmetric coorbital central configuration at µ = u 19 and these last solution persist for µ > u 19 . In short, at µ = u 19 a symmetric coorbital central configuration bifurcates to two non-symmetric ones for µ < u 19 . This proves statements (a), and since there is no other bifurcation value (except µ = u 1 ) it follows statement (b).

e 1 = iz 6 1 z 8 2 -2iz 4 1 z 8 2 + iz 2 1 z 8 2 -2iz 6 1 z 6 2 + 4iz 4 1 z 6 2 -2iz 2 1 z 6 2 -z 6 1 z 5 2 + 2z 4 1 z 5 2 -z 2 1 z 5 2 -z 6 1 z 3 2 + 2z 4 1 z 3 2 -z 2 1 z 3 2 + 2iz 6 1 z 2 2 -4iz 4 1 z 2 2 + 2iz
Analyzing the solutions of system E 1 = E 2 = F 1 = F 2 = 0 (or equivalently of system e 1 = e 2 = 0) as a function of µ we observe that we have five families of central configurations for µ < u 1 , the symmetric family A s found in Section 3, two non-symmetric families of class A denoted by A 1 and A 2 two non-symmetric families of class B denoted by B 1 and B 2 ; seven families of coorbital central configurations for u 1 < µ < u 19 the symmetric family A s , the two symmetric families B s 1 and B s 2 found in Section 3, which match at µ = u 1 plus the four non-symmetric families A 1 , A 2 , B 1 and B 2 ; and finally five families of coorbital central configurations for µ > u 19 , the three symmetric families A s , B s 1 and B s 2 plus the two non-symmetric families A 1 , A 2 . We note that the two non-symmetric families B 1 and B 2 match at µ = u 19 to one of the symmetric families of class B, exactly B s 1 . Again see Figures 1,2, 3 and 4.

Conclusions

We have derived results concerning to the number of coorbital central configurations of the planar 1 + 3 body problem when two infinitesimal masses are equal, that is, m 1 = m 2 = µ and m 3 = 1. The problem exhibits two unconnected classes of central configurations denoted A, when two angles are less than π/3 and B, when the three angles are greater than π/3. Each class have symmetric (denoted by A s , B s 1 and B s

2 ) and non-symmetric configurations. results obtained in Sections 3 and 4 can be summarized as follows.

The coorbital central configurations of the 1 + 3 body problem with two equal infinitesimal masses are the following ones. Notice that the proof of the results stated are analytical with the exception that we have computed numerically the real roots of some polynomials of one variable. For all polynomials of one variable that we have found we have computed (with mathematica) as many roots as of its degree (of course some are complex), we have checked that every root obtained once it is substituted in its polynomial, this vanishes. We have done these computations with 100 decimals for each root. We thus provide numerical evidence that we have obtained all the real roots of the polynomials that we have studied and that we do not forget any real root.

In order to see how the classes A and B of the coorbital central configurations change in function of the parameter µ, we have computed numerically the values of the angles θ 1 (µ) and θ 2 (µ), for each family of coorbital central configurations, see Figures 1 and2.

In Figures 3 and4 qualitative pictures of the classes A and B of the coorbital central configurations are showed for different values of µ.

  PROPOSITION 3.2. Let u 1 = 0.89616 . . .. The 1 + 3 body problem has the following symmetric coorbital central configurations: (a) One symmetric coorbital central configuration A s belonging to class A for µ ∈ (0, u 1 ). (b) Two symmetric coorbital central configurations for µ = u 1 , one belonging to class A (A s ) and one belonging to class B (B s 1,2 ). (c) Three symmetric coorbital central configurations for µ > u 1 , one belonging to class A (A s ) and two belonging to class B (B s 1 , B s 2 ). The two symmetric coorbital central configurations of class B coincide at µ = u 1 .

  the following way. Let Res [P, Q, X] denote the resultant of the polynomials P and Q with respect to X. Fixed µ = v i we compute the resultants Res [E, F, and Res [E, F, c], which are polynomials of the single variable c and s Using the properties of resultants, if (c 0 , s 0 ) is a solution of system E = F = 0, then Res [E, F, s](c 0 ) = 0 and Res [E, F, c](s 0 ) = 0. We compute with the help of Mathematica all the solutions of the polynomial equations Res [E, F, s] = 0 and Res [E, F, c] = 0. Finally we check which pairs (c, s) formed by a real solution c of Res [E, F, s] = 0 with 0 c < 1 and a real solution s of Res [

PROPOSITION 4. 1 .

 1 The following statements hold. (a) At µ = 1.4238513 . . ., symmetric coorbital central configuration B s 1 given in Proposition 3.2, bifurcates to two non-symmetric 1+3-rev3.tex; 16/09/2010; 16:10; p.11 coorbital central configurations (B 1 , B 2 ), which persist for all µ ∈ (0, 1.4238513 . . .).

  (a) For µ ∈ (0, 0.89616...) there are 5 coorbital central configurations, 3 of class A (A s , A 1 and A 2 ) and 2 of class B (B 1 and B 2 ). (b) For µ = 0.89616... there are 6 coorbital central configurations, 3 of class A (A s , A 1 and A 2 ) and 3 of class B (B 1 , B 2 and B s 1,2 ). (c) For µ ∈ (0.89616..., 1.42385...) there are 7 coorbital central configurations, 3 of class A (A s , A 1 and A 2 ) and 4 of class B (B 1 , B 2 , B s 1 and B s 2 ). (d) For µ = 1.42385... there are 5 coorbital central configurations, 3 of class A (A s , A 1 and A 2 ) and 2 of class B (B s 1 and B s 2 ). (e) For µ > 1.42385... there are 5 coorbital central configurations, 3 of class A (A s , A 1 and A 2 ) and two of class B (B s 1 and B s 2 ).

Figure 2 .Figure 3 .

 23 Figure 2. The angles θ1(µ) and θ2(µ) along the class B.

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Acknowledgments

The authors have been partially supported by the grants MEC/FEDER MTM2008-03437, and the last two additionally by the grant CIRIT number 2009SGR-410.

So µ = 4 is the unique value of µ > 0 at which the root of the factor T 1 is a root of some of the other factors. Proceeding in a similar way for the root of the factor T 2 , s 1 = 1/2, we get

and

Since we are interested in solutions of ( 7) with α 1 , α 2 ∈ (0, π) and α 3 = π -α 1 -α 2 ∈ (0, π), we only consider solutions of e 1 = e 2 = 0 with

Fixed µ = v i for i = 1, . . . , 24 we solve the polynomial system e 1 = e 2 = 0 by means of resultants. With the help of Mathematica, we compute the resultants Res [e 1 , e 2 , z 1 ] and Res [e 1 , e 2 , z 2 ], which are polynomials of the single variable z 2 and z 1 respectively. Then we compute all the solutions of the polynomial equations Res [e 1 , e 2 , z 1 ] = 0 and Res [e 1 , e 2 , z 2 ] = 0. We check which pairs (z 1 , z 2 ) formed by a solution z 2 of Res [e 1 , e 2 , z 1 ] = 0 and a solution z 1 of Res [e 1 , e 2 , z 2 ] = 0 provide a solution of system e 1 = e 2 = 0. Finally we chose the solutions (z 1 , z 2 ) of system e 1 = e 2 = 0 satisfying the conditions |z 1 | = |z 2 | = 1, Im(z 1 ), Im(z 2 ) > 0 and α 3 = π -α 1 -α 2 ∈ (0, π).

After all these computations we get the following: system e 1 = e 2 = 0 has 5 solutions for µ = v 1 , . . . , v 10 , four corresponding to nonsymmetric coorbital central configuration and one corresponding to a symmetric one; it has 7 solutions for µ = v 11 , . . . , v 19 , four corresponding to non-symmetric coorbital central configurations and three corresponding to symmetric ones; it has 5 solutions for µ = v 20 , . . . , v 24 , two corresponding to non-symmetric coorbital central configurations and three corresponding to symmetric ones. So the number of solutions of system e 1 = e 2 0 satisfying the desired conditions changes at µ = u 10 = u 1 and at µ = u 19 .

We remark that we have studied all the bifurcations values using the changes of solutions in the variable s 1 , we have a similar study using the changes of solutions in the variable s 2 , obtaining exactly the same results.