VanderMeer 2005: A Meta-Analysis of Melatonin for Primary Sleep Disorders

Clark A M, Hartling L, Vandermeer B, McAlister F A. Meta-analysis: secondary prevention programs for patients with coronary artery disease. Annals of Internal Medicine 2005; 143(9): 659-672. [PubMed: 16263889]

EconPapers FAQ Archive maintainers FAQ Cookies at EconPapers Format for printing The RePEc blog The RePEc plagiarism page Biodiversity, yield, and shade coffee certificationIvette Perfecto, John Vandermeer, Alex Mas and Lorena Soto PintoEcological Economics, 2005, vol. 54, issue 4, 435-446Date: 2005References: View references in EconPapers View complete reference list from CitEc Citations: View citations in EconPapers (36) Track citations by RSS feedDownloads: (external link) -8009(04)00401-XFull text for ScienceDirect subscribers onlyRelated works:This item may be available elsewhere in EconPapers: Search for items with the same title.Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/TextPersistent link: :eee:ecolec:v:54:y:2005:i:4:p:435-446Access Statistics for this articleEcological Economics is currently edited by C. J. ClevelandMore articles in Ecological Economics from ElsevierBibliographic data for series maintained by Catherine Liu (Obfuscate( 'elsevier.com', 'repec' )). var addthis_config = "data_track_clickback":true; var addthis_share = url:" :eee:ecolec:v:54:y:2005:i:4:p:435-446"Share This site is part of RePEc and all the data displayed here is part of the RePEc data set. Is your work missing from RePEc? Here is how to contribute. Questions or problems? Check the EconPapers FAQ or send mail to Obfuscate( 'oru.se', 'econpapers' ). EconPapers is hosted by the Örebro University School of Business.

VanderMeer 2005

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1 IntroductionIn the previous papers of the series on stellar evolution with rotation,the effects of rotation on the stellar evolution was studied with an emphasis on the early stages of the pre-supernova evolution (mainsequence, MS, and He-burning). Themodels reproduce many observational features at various metallicities very well, likesurface enrichment (Meynet & Maeder 2002b) and ratios between red and blue supergiants (Maeder & Meynet 2001, hereinafter Paper VII).In Meynet & Maeder (2005, Paper XI) and Meynet & Maeder (2003, Paper X), the Wolf-Rayet (WR hereinafter) star population atdifferent metallicities are studied. In view of the lower mass loss rates obtained when clumping effects inthe winds are accounted for, models without rotation do not reproducethe WR populations in galaxies.Models of rotating massive starsgive a much better fit to the populations of WR stars at differentmetallicities than do non-rotating models. They can reproduce the observed number ratio of WR to O-type stars, the observed ratio of WN to WC stars (for metallicitieslower than solar), the observed fraction of WR stars in the transitionphase WN/WC, and finally the observed ratio of type Ib and Ic to type IIsupernovae. A good fit of the observed properties of WR stars is ofparticular interest in this study, since WR stars are thought to be progenitors of GRBs.In Hirschi et al. (2004, hereinafter Paper XII), we described the recent modifications made to the Geneva codeand the evolution of the models until silicon burning.In this paper, we look at the evolution of rotation in massive starswith an emphasis on the final stages of the evolution towards neutron stars (NS), black holes (BH), and especially long soft -ray bursts (GRB).1.1 Neutron starsThere are many observations of neutron stars and pulsars available. A catalogue of observed pulsars and their properties is available at theweb page of the group ATNF (2005). The fastest young pulsars have periodslarger than 10 ms: 16 ms for J0537-6910 (Marshall et al. 1998),33 ms for the Crab pulsar (B0531+21, Staelin & Reifenstein 1968), and 50 ms for J0540-6919 (Seward et al. 1984). There are about 20 pulsars with aperiod smaller than 100 ms in the catalogue and an age less than 100 000 years.The pulsars may have slowed down slightly since their birth but theinitial period of pulsars is at least 10 ms.NS are supposed to form during the collapse of stars with an initial mass between about 10 and 25 .If we compare 10 ms with the values obtained in Tables 4-7 (Cols. 11 and 12), we see that in general our models have muchmore angular momentum than the observed pulsars. The difference canreach a factor of about 100 around 15 and a factor of about 10around 60 .In order to reconcile the model predictions with the observations,additional angular momentum has to be lost before the formation of thepulsar. This can occur during the pre-supernova stages due to theeffects of the magnetic fields not taken into account in this work or elseduring the collapse and the explosion.The existence of a primordial magnetic field in some stars (Braithwaite & Spruit 2004) may also slow down their core. Models including the effects of magnetic fields have only recently beendeveloped (Maeder & Meynet 2004; Heger et al. 2004) due to the complexity of the interplay between rotation and magnetic fields.Heger et al. (2005) show that the braking by magnetic fields, which areproduced by differential rotation during the evolution (see Spruit 2002) significantly reduce the discrepancy between the predicted and observed pulsar periods.However, these models, as well as binary models presented in Petrovic et al. (2005), contain toolittle angular momentum at the pre-supernova stage to produce GRBs.In this paper, we study the casewithout magnetic field, which is more favourable to GRB production.The case with magnetic braking will be treated in a later study.In addition, the present study offers a basis of comparison for futuremodels that include magnetic fields.It is important here to note that internal magnetic braking is most efficient during the Red SuperGiant (RSG) phase, during which the star has a fastrotating core and a slow rotating envelope.This means that magnetic braking is much less effective for stars which skip the RSG stage, as mentioned in Heger et al. (2005). These stars are the most massive WR stars, with masses larger thanabout 60 .Furthermore, more massive stars have shorterlifetimes and thus less time to be slowed down. It is also possible thatmechanisms that slow the nascent neutron star down may not be asefficient when a black hole is formed. These differences betweenstars with masses around 15 (forming a neutron star and going through the RSG stage) and 60 (not going through the RSG phase and forming blackholes) could possibly explain, with the same physics, the pulsar periodsand the GRBs. Further studies will verify this possibility. Braking between the collapse and the pulsar formation can also occurvia different mechanisms: r-Mode instability, neutrino-powered magneticstellar winds, fall-back, and the propeller mechanism. However, these mechanisms may not be efficientenough to slow the core down efficiently (see Heger et al. 2005, for adiscussion). The latest models studying gravitational waves seem to indicate thatbraking is possible during and after the collapse (Ott et al. 2005). If the cores are not slowed down during the pre-supernova stages, rotation may play an important role in supernova explosions.Rotation may in this case provide a large amount of energy for the explosion (1052 erg), but this is generally not observed (Janka et al. 2005). 1.2 Black holes and -ray burstsTheoretical models still struggle to reproduce supernova explosions (for a discussion and references, see Fryer & Heger 2005). It is therefore not possible for us to predictthe fate of the remnant of our models with certainty or precision. Nevertheless, following the estimates presented inFryer (1999) and Heger et al. (2003), we will consider in this study that the lower mass limit for blackhole formation is around 25 .The upper mass limit dependsstrongly on the mass loss and is around 100 at solar metallicity.The maximum neutron star mass is also still uncertain(see Morrison et al. 2004; Srinivasan 2002, for recent studies). Depending on the nuclear equation of state used and the rotation rate, the upper mass limit for neutron stars can vary between 1.6 and 3 although 2 to 2.5 is a more common range.BHs cannot be observed directly. Thus they are usually detected inX-ray binaries. In certain cases, observations allow the masses ofthe binary companions to be determined. If the accreting object has a derived mass larger than 2-3 ,it is considered a BH candidate. The best known candidates are LMC X-3 and CygX-1 (see Kaper & van der Meer 2005, and references therein).GRBs can be divided into two main types: a) short and hard and b) long and soft.See Piran (2005) for a recent review of GRBs.The long soft GRBs have recently been connected to supernova(SN) explosions of the type Ib,c (see for example Matheson 2003).Since then, several studies have been devoted to finding which stars canlead to GRBs. Heger et al. (2004), Heger et al. (2005), and Hirschi et al. (2004) looked at massive single star at solar metallicity as progenitors. Note that the two models differ in the treatment of meridionalcirculation, which is an advective process (Meynet & Maeder 2002a). Our models account for the advection of angular momentumduring the MS phase, while models of Heger and collaborators treatmeridional circulation as a diffusive process. Maeder & Meynet (2005) showthat the treatment of meridional circulation as an advective process iscrucial for modelling the interplay between circulation and magnetic braking.Another difference in this article is the fact that we study the effects of metallicity in detailin the context of GRBs for models of differentially rotating stars.This is important since metallicities lower than solar are expected to bemore favourable to the production of GRBs (due to weaker mass loss, see MacFadyen & Woosley 1999; Woosley & Heger 2003).Podsiadlowski et al. (2004) and Izzard et al. (2004) considered both single and binary massive stars. Note that Izzard et al. (2004) assume that the loss of angular momentum is proportional to the amountof mass lost and do not consider internal transport of angular momentum.Their approximate treatment of the evolution of angular momentum leadsthem to the conclusion that only binary stars can retain enough angularmomentum in their core to form GRBs. Even without magnetic braking,their models predict no GRBs from single massive stars which is incontradiction with our models and those of Heger et al. (2000). Petrovic et al. (2005)look at single and binary systems with and without magnetic fields atsolar metallicity. They find that in the models followed, both singleand binary models without magnetic field can produce GRBs and both single and binary models with magnetic field considered in the study cannot produce GRBs. They conclude that, if the present modelling of magneticfields is accurate, GRBs have to be produced in some exotic channels ofbinary systems.Fryer & Heger (2005) looked at massive binary star mergers, which are believedto be one of the best binary candidates for GRBs. They follow the evolution prior to and after the merger with the KEPLER code (Heger et al. 2000)and the merger process with 3D simulations, which is very interestingand which has a lot of potential for further applications. They find that, in somecases, merged helium stars can retain more angular momentum than singlemassive stars. However, they obtain these faster rotators only for models inwhich they remove angular momentum artificially without removing massfrom the merged star. If more realistic mass loss prescriptions areused after the merger, the final angular momentum contained in the central remnant issimilar in merged stars and in single massive stars. This is due to the strong mass loss that WR stars experience, especiallythe fast rotating ones. Merged stars lose a large amount of mass and somemerged systems probably form NSs instead of BHs. It is therefore not proven that binary stars retain more angularmomentum than single stars and the exact frequency of potential binarystar progenitors has not been compared quantitatively with the GRB rates.Here we discuss the possibility of single massive stars being progenitors when studyingmodels covering a large range of initial masses and metallicities. Thesemodels reproduce the WR star population very well, and many of themretain enough angular momentum to produce GRBs as shown below. We describe thecalculations in Sect. 2, the evolution of rotation in Sect. 3, and present our predictedGRB rates in Sect. 4.2 Description of the calculationsIn order to study the evolution of core rotation and its dependence on initial massand metallicity, we used the models presented in Meynet & Maeder (2003, Paper X) and Hirschi et al. (2004, Paper XII) for solar metallicityand Meynet & Maeder (2005, Paper XI) for non-solar metallicities, which span a wide range in initial masses (9 to 120 at solar metallicity) and metallicities (Z=0.004 to Z=0.040). All these models (except for one 60 model at Z=0.004) have an initial rotational velocity of 300 km s-1, which gives an average value for the MS of about 220 km s-1, the average observed rotational velocities on the MS(see for example Fukuda 1982). They were all calculated with thesame input physics: Schwarzschild criterion for convection, overshooting of 0.1 Hfor thehydrogen (H) and helium (He) burning cores, same mass loss prescriptions, andtreatment of rotationincluding meridional circulation and shear diffusion(see Papers X-XII for more details). Non-solar metallicity models were evolved until the end of core He-burning. Solar metallicity models were evolved until the end of core He-burning for the 9, 85 and 120 models, end of oxygen (O) burning for the 12 model, and end ofsilicon (Si) burning for the 15, 20, 25, 40, and 60 models.As we shall see later, the largest changes in the angular momentum of the core occurduring H and He-burnings, which is why we did not evolve non-solar metallicity modelsfurther than the end of He-burning.Several quantities are useful for studying the evolution of rotation:The angular velocity at mass coordinate m or radius r (here r is the averageradius of a shell, see Meynet & Maeder 1997, for details), [], and its ratio to the Keplerian velocity,,where .When becomes larger than about 0.9, the star gets elongated along itsequator and at break-up, the star equatorial radius is 1.5 times larger than the polarradius (value obtained using the Roche model). Therefore the star may reach break-up before approaches one, in particular for models with alarge Eddington factor, which reach break-up for much lower than (Maeder & Meynet 2000).The angular momentumof a core of mass M, [],where []is the specific angular momentum atmass coordinate m. The momentum of inertia of a core of mass M,[].The properties of the models are presented in four tables: Table 4 for the metallicity of the Small Magellanic Cloud (SMC, Z=0.004),Table 5 for the Large Magellanic Cloud (LMC, Z=0.008),Table 6 for solar metallicity (Z=0.020), andTable 7 for the Galactic centre (GC, Z=0.040).In these tables, for each model we give the initial mass and velocity, as well as the remnantmass estimated from the carbon-oxygen (CO) core mass, using the relation fromMaeder (1992). We used the value of the CO core at the end of the evolution for thecalculation (at the end of core He-burning in general and at the end ofSi-burning for the models from Paper XII).Then,Col. 1 is the evolutionary stage to which the values correspond;Col. 2 is the total mass at the given stage;Col. 3 is the angular momentum contained in the remnant;in units of ;Col. 4 is the moment of inertia of the remnant, in units of ;Col. 5 is the specific angular momentum at the remnant edge, ,in units of ;Col. 6 is the average specific angular momentum in the remnant, ,in units of ;Col. 7 is the specific angular momentum at the mass coordinate 1.56 ,j1.56, in units of ;Col. 8 is the angular momentum contained in the inner 1.56 ,in units of ;Col. 9 is the specific angular momentum at the mass coordinate 2.5 ,j2.5, in units of ;Col. 10 is the angular momentum contained in the inner 2.5 ,in units of .Assuming that a neutron star with a baryonic mass, ,and with a radius, R=12 km, wouldform from the models, we calculated the two following quantities:(Col. 11) the ratio of the NS angular velocity to its Keplerian angular velocity, (NS);(Col. 12) the initial rotation period of the neutron star, in units of milli-seconds.A proto-neutron star with a baryonic mass, ,loses a bindingenergy (BE) of 0.159 due to neutrinos emission. This value is calculated using Eq. (36) from Lattimer & Prakash (2001): BE/,in which corresponds to the gravitational mass and where ,and using the fact that.These equations give one second degree equation: 2ff7e9595c