Vol 17, No 8 (2017) / Mahanta

p-capture reaction cycles in rotating massive stars and their impact on elemental abundances in globular cluster stars: A case study of O, Na and Al

p-capture reaction cycles in rotating massive stars and their impact on elemental abundances in globular cluster stars: A case study of O, Na and Al

Mahanta Upakul1, , Goswami Aruna2, , Duorah Hiralal3, , Duorah Kalpana4,

Department of Physics, Bajali College, Pathsala, PIN 781325, India
Indian Institute of Astrophysics, Bengaluru, PIN 560034, India
Department of Physics, Gauhati University, Guwahati, PIN 781014, India
Department of Physics, Gauhati University, Guwahati, PIN 781014, India

† Corresponding author. E-mail: upakulmahanta@gmail.com aruna@iiap.res.in hl_duorah@yahoo.com khowang56@yahoo.co.in


Abstract: Abstract

Elemental abundance patterns of globular cluster stars can provide important clues for understanding cluster formation and early chemical evolution. The origin of the abundance patterns, however, still remains poorly understood. We have studied the impact of p-capture reaction cycles on the abundances of oxygen, sodium and aluminium considering nuclear reaction cycles of carbon-nitrogen-oxygen-fluorine, neon-sodium and magnesium-aluminium in massive stars in stellar conditions of temperature range 2×107 to 10×107 K and typical density of 102 gm cc−1. We have estimated abundances of oxygen, sodium and aluminium with respect to Fe, which are then assumed to be ejected from those stars because of rotation reaching a critical limit. These ejected abundances of elements are then compared with their counterparts that have been observed in some metal-poor evolved stars, mainly giants and red giants, of globular clusters M3, M4, M13 and NGC 6752. We observe an excellent agreement with [O/Fe] between the estimated and observed abundance values for globular clusters M3 and M4 with a correlation coefficient above 0.9 and a strong linear correlation for the remaining two clusters with a correlation coefficient above 0.7. The estimated [Na/Fe] is found to have a correlation coefficient above 0.7, thus implying a strong correlation for all four globular clusters. As far as [Al/Fe] is concerned, it also shows a strong correlation between the estimated abundance and the observed abundance for globular clusters M13 and NGC 6752, since here also the correlation coefficient is above 0.7 whereas for globular cluster M4 there is a moderate correlation found with a correlation coefficient above 0.6. Possible sources of these discrepancies are discussed.

Keywords: Galaxy: globular cluster: individual (M3, M4, M13, NGC 6752);stars: abundances;stars: massive



1 Introduction

Globular cluster (GC) stars are known to display homogeneous abundance patterns for Fe-peak elements but significant abundance variations are seen among the light elements. Na-O and Mg-Al anticorrelations are also observed among the stars in GCs (Kraft et al. 1997 ; Gratton et al. 2004 ; Carretta et al. 2009 ; Smolinski et al. 2011 and references therein). Inhomogeneities and abundance variations observed in light-elements increase with decreasing metallicity. For instance, C, N, O and Na abundances in giants of M7 with an average metallicity ([Fe/H]) of ∼ −0.7 show mild variations compared to their counterparts observed in giants of M5 with a metallicity of ∼ −1.2. Several studies, both observational and theoretical, in literature are devoted to understanding the complex abundance patterns of GCs (Cordero et al. 2015 ; Roederer et al. 2016 ; Spite et al. 2016 ; Villanova et al. 2016 ). The origins of these observed abundance anomalies, however, still remain poorly understood. GCs span a wide range in metallicities with [Fe/H] as small as −2.38 up to as large as +0.12 (table 2 in Gratton et al. ( 2004 ) and references therein). Here among the chosen GCs, GC M3 has an average metallicity of −1.39 and is more distant than M5. The scatter among light elements and the enhanced odd atomic numbered elements typically seen in metal-poor GCs are also noticed in M3, although at a lower level of star-to-star variation (Cohen & Meléndez 2005 ). GC M13 with an average metallicity of −1.50 is known to show large star-to-star differences in the abundance of Al, Mg, Na and O among its red giants (Cohen & Meléndez 2005 references therein). GC M4 is the closest to us (Dixon & Longmore 1993 ) and has an average cluster metallicity of −1.17 (Liu & Janes 1990 ; Drake et al. 1992 , 1994 ). A characteristic feature observed in the color-magnitude diagrams of GC M4 stars is the presence of broadened red-giant branches (Marino et al. 2008 ). The two giant branches appear to correlate with typical GC variations in [O/Fe] and [Na/Fe] abundances rather than variations in total [C+N+O/Fe] abundance (Martell et al. 2011 ). Detailed elemental abundances are available for a large number of stars belonging to this cluster. The largest spread in light element abundances is known to be observed in GC NGC 6752. This GC has been studied by many groups including Yong et al. ( 2005 ), and the reported average cluster metallicity due to Yong et al. ( 2005 ) is ∼ −1.61. But the common feature of these four GCs is the observed Na-O and Mg-Al anti-correlation. Proton-capture chains that convert C and O to N, Ne to Na and Mg into Al in the hydrogen-burning layers of evolved stars are believed to be responsible for these observed trends; however, the astrophysical site(s) for their occurrence is still under debate. Among the contested scenarios, one is evolutionary which considers deep mixing of the stellar envelope through the hydrogen burning shell that brings products of proton-capture chains to the surface. The other one is primordial where observed abundances originating from proton-capture synthesis that took place in some earlier generation of massive stars whose interior could easily attain a high temperature so that advanced H-burning could be sustained. However, recent spectroscopic studies have challenged the former one as it cannot possibly explain why a similar kind of abundance anomaly is present even in stars which are below and above the main sequence turn-off (Decressin et al. 2007b ). Also, the evolutionary scenario cannot explain why stars in GC ω Cen, which is the most massive GC in the Milky Way, have been found with a large spread in metallicity, thereby discarding the earlier paradigm that the GCs are examples of “Simple Stellar Populations” (D’Ercole et al. 2008 ). Under the currently favored astrophysical sites where the processing of elements takes place, as well as the mechanism by which processed material is subsequently delivered into new generations of stars are (i) massive asymptotic giant branch (AGB) stars (D’Ercole et al. 2008 ), (ii) fast-rotating massive main sequence stars (Decressin et al. 2007b ) and (iii) massive interacting binaries (de Mink et al. 2009 ). During Hot Bottom Burning (HBB) that occurs in AGB stars, the CNO elements undergo p-capture nucleosynthesis which is found to be sensitive to the metallicity of the star (Ventura et al. 2013 ). But this HBB, for any effective depletion of oxygen abundance due to p-capture reaction, requires a temperature as high as (Ventura et al. 2013 ). Again at high temperature the destruction channel for sodium is dominant compared to the production reaction by p-capture on 22Ne nuclei, thus the sodium that previously accumulated at the surface is destroyed (Denissenkov & Herwig 2003 ; Ventura et al. 2013 ). This points towards correlation between Na and O abundance which in fact has been confirmed by a recent study performed by Ventura et al. ( 2013 ) at very low metallicity unless the polluted material that is lost from the surface of the stars via slow winds suffers a certain amount of dilution (Ventura et al. 2013 ). This leads to an Na-O anticorrelation and the materials then remain trapped within the gravitational potential of the cluster. As mentioned earlier, these four GCs have shown an Na-O anticorrelation, three of which have already been reported in a recent study performed by D’Antona & Ventura ( 2007 ) who considered some new stellar models. In the third scenario, interacting massive binary stars may provide an efficient way to lose large amounts of processed material that could be incorporated into an enriched population (de Mink et al. 2009 ). This scenario is attractive because a large fraction of massive stars are indeed observed to be members of binaries that will interact during their lifetime (Sana et al. 2013 ). de Mink et al. ( 2009 ) assumed a non-conservative evolution of massive close binaries which may not be correct because the mass that is lost by the loser (the primary one) leaves the binary as a slow wind driven by rotation of the gainer (the secondary one) and takes specific orbital angular momentum of the gainer. Thus this will lead to a merger of the binary and the evolution will be different as suggested by the same author (Vanbeveren et al. 2012 ). In the “winds from fast rotating massive stars” scenario (discussed in Section 2), it is assumed that massive stars within GCs rotate near break-up velocity. Processed material is brought to the surface by rotational mixing, lost via a mechanical wind, and then accumulates in a disk around the star, where the second generation of (low-mass) stars is assumed to form (Decressin et al. 2010 ).

Reaction T 9 (ρ 2 = 1)
N 0.02 3.76 × 10−14 3.79 × 1011
0.03 1.74 × 10−11 8.21 × 108
0.05 1.31 × 10−8 1.09 × 106
0.08 2.28 × 10−6 6.26 × 103
0.1 2.03 × 10−5 7.03 × 102
N 0.02 1.89 × 10−13 7.55 × 1010
0.03 8.77 × 10−11 1.62 × 108
0.05 6.49 × 10−8 2.20 × 105
0.08 1.10 × 10−5 1.29 × 103
0.1 9.60 × 10−5 1.48 × 102
O 0.02 4.11 × 10−16 3.47 × 1013
0.03 3.85 × 10−13 3.71 × 1010
0.05 6.15 × 10−9 2.32 × 105
0.08 1.93 × 10−7 7.40 × 104
0.1 2.19 × 10−5 6.52 × 102
O 0.02 1.59 × 10−16 8.98 × 1013
0.03 1.45 × 10−13 9.85 × 1010
0.05 2.21 × 10−10 6.46 × 107
0.08 6.50 × 10−8 2.19 × 105
0.1 7.20 × 10−7 1.98 × 104
O 0.02 3.90 × 10−15 3.66 × 1012
0.03 3.70 × 10−12 3.86 × 109
0.05 5.97 × 10−9 2.39 × 106
0.08 1.89 × 10−6 7.55 × 103
0.1 2.16 × 10−5 6.61 × 102
F 0.02 5.45 × 10−47 2.62 × 1044
0.03 1.25 × 10−43 1.14 × 1041
0.05 6.66 × 10−37 2.14 × 1034
0.08 1.34 × 10−34 1.06 × 1032
0.1 4.64 × 10−33 3.07 × 1030
F 0.02 3.78 × 10−18 3.77 × 1015
0.03 6.59 × 10−15 2.16 × 1012
0.05 1.98 × 10−11 7.21 × 108
0.08 9.71 × 10−9 1.47 × 106
0.1 1.30 × 10−7 1.09 × 106
Ne 0.02 6.12 × 10−21 2.33 × 1018
0.03 2.15 × 10−17 6.64 × 1014
0.05 1.36 × 10−13 1.05 × 1011
0.08 1.24 × 10−10 1.15 × 108
0.1 2.17 × 10−9 6.58 × 106
F 0.02 2.08 × 10−18 6.86 × 1015
0.03 9.69 × 10−15 1.47 × 1012
0.05 7.62 × 10−11 1.87 × 108
0.08 1.58 × 10−8 9.04 × 105
0.1 1.39 × 10−7 1.02 × 105
Ne 0.02 5.47 × 10−17 2.61 × 1014
0.03 6.84 × 10−14 2.08 × 1011
0.05 2.57 × 10−11 5.55 × 108
0.08 3.88 × 10−9 3.68 × 106
0.1 3.93 × 10−8 3.63 × 103
F 0.02 1.2 × 10−17 1.19 × 1015
0.03 1.79 × 10−14 7.98 × 1011
0.05 1.35 × 10−10 1.05 × 108
0.08 7.11 × 10−6 2.00 × 103
0.1 3.17 × 10−4 4.50 × 101
O 0.02 3.76 × 10−17 3.79 × 1014
0.03 1.33 × 10−13 2.07 × 109
0.05 8.72 × 10−10 1.63 × 107
0.08 9.71 × 10−9 1.47 × 106
0.1 1.65 × 10−5 8.65 × 102

Table 2Estimated proton-capture reaction lifetimes for the nuclear reactions. The rate constants are in , T 9 is in the unit of 109 K and is in the unit of 102 gm cc−1. The values for 13N, 15O, 17F and 18F are 862.77, 176.39, 93.059 and 9504 s respectively. and represent the β-decay and proton-capture lifetimes respectively in second..

In this work we have considered the nuclear burning cycles carbon-nitrogen-oxygen-fluorine (CNOF), neon-sodium (NeNa) and magnesium-aluminium (MgAl) in a rotating massive star at high temperature and low density conditions and have estimated the abundances of the product elements O, Na and Al in a temperature range of 2 × 107 to 10 × 107 K since the CNO, NeNa and MgAl cycles are activated above temperatures 0.02 × 109 K, 0.035 × 109 K and 0.05 × 109 K respectively (Decressin et al. 2007b ). Although nucleosynthesis during NeNa and MgAl cycles is discussed at length in Arnould et al. ( 1999 ) and José et al. ( 1999 ), the availability of new estimates of the cross-sections for many of the reactions involved in these cycles, from recent experimental determination (Iliadis et al. 2010 ), prompted us to reinvestigate the synthesis of elements due to these cycles. Our aim is to compute the abundances of key elements O, Na and Al, which are important diagnostics for understanding the associated chemical evolution, and thereby examine the impact of proton-capture reactions on the observed abundances of GC stars. A comparison between the computed and observed data suggests contributions coming from other sources, yet to be identified, that might have possibly influenced the observed abundances.

In Section 2 we present the physical conditions for the occurrence of nuclear burning cycles and lifetime measurements for proton-capture reactions. The evolution of stable elements in the cycles at equilibrium due to hydrogen burning is presented in Section 3. Section 4 deals with CNOF and other nuclear burning cycles and the abundance calculation of selected elements. Discussion and concluding remarks are presented in Section 5.

2 The Physical Situation

Classical models of stellar evolution focus on the dominant role of various stages of nuclear burning in the stellar core. Not only is the temperature crucial in any stellar situation, but density also plays an important role. Even though the density ranges over many orders of magnitude, its response to the burning rates is only linear and hence its importance is much less significant as far as elemental synthesis is concerned unless one considers a situation where low temperature and high density are prevailing. The materials are synthesized in the inner regions near the core and are brought to the surface by means of some physical mechanisms like convection or by dredge-up. Here the stars are assumed to be spherically symmetrical with no magnetic field, no rotation and no mass loss from the surface. The process of convection is the only mixing mechanism operating in the convective regions, which are always fully mixed (Salaris et al. 2002 ). These conditions are the canonical model of a star.

But in recent years, it has become clear that stellar evolution, particularly for more massive stars, can also be profoundly influenced by the loss of mass and angular momentum from the stellar envelope and surface (Maeder & Meynet 2008 ). Models of main sequence evolution in rotating massive stars show that, at the surface, the velocity approaches a critical limit. This is induced by the internal evolution of stars, which results in transport of angular momentum from the contracting, faster rotating inner convective core to the expanding, slowed down radiative envelope (Meynet et al. 2006 ). However beyond the critical limit, any further increase in rotation rate is not dynamically allowed, hence further contraction of the interior is then balanced by a net loss of angular momentum through induced mass loss. Here this mass loss is assumed to be isotropic. Thus synthesized mass by the star will now be delivered to the interstellar medium (ISM), and hence the ISM is pre-enriched with material that has been produced by other stars. Moreover, the materials are assumed to be released into the ISM with a very low velocity and can easily be retained in the potential well of the GC. Here we assume the density of stellar matter is constant which, in general, is not a sensitive parameter unless one goes to a high value of ∼ 105 gm cc−1 or even higher where a pycnonuclear type of reaction has to be considered. We also assume the star to be in equilibrium, thereby making it maintain a balance between the produced energy due to thermonuclear fusion and loss of energy at the surface. Thus the temperature of the star can essentially be assumed to be a fixed quantity. Here we mainly focus on which type of star produces the material enriched in H-burning products and how. However, we neither comment on the physical factors that influence mass loss nor look for what physical mechanism is responsible for selecting only material bearing the signatures of H-processing that has been lost.

3 Hydrogen burning nucleosynthesis

The rate of nuclear reactions is dependent on the density of the reactants, the velocity of one reactant relative to another, and the probability of a reaction occurring. Mathematically

where i, j are two separate species and is the function preventing double counting of those two species. For computational simplification, the number density of any element with mass number can be expressed in terms of its mass fraction by . Here ρ is density and is Avogadro’s number. The thermonuclear reaction rate for a proton-capture reaction then takes the form
where is the reaction rate constant and is the mass fraction of any other heavy element. From this, the lifetime against proton-capture for elements in the enhanced state is given by the following equation

4 CNOF cycle

If heavier elements are present in some stellar condition where the temperature and density are low, then the 12C(p, γ)13N reaction can compete with the p-p reaction and thereby can initiate the CNO burning mechanism (Clayton 1983 and references therein). 13N is β unstable and decays to 13C in a time scale of 862.77 s (Audi et al. 1997 ) since its proton capture lifetime is quite large in the considered temperature and density condition. On the other hand, 13C is a stable isotope of carbon with relative abundance 1.1078 (Lodders 2003 ). This 13C forms 14N by taking a proton and then 14N to 15O by taking a proton again. 15O then decays via a β emission to 15N with an average lifetime of 176.39 s (Audi et al. 1997 ). Here 15N branching appears. Table 1 shows the branching ratios of the cycle at various temperatures.

T 9
0.02 4.615 × 10−4 1.312 × 10−1
0.03 4.232 × 10−4 5.601 × 10−3
0.05 3.652 × 10−4 4.031 × 10−3
0.08 3.134 × 10−4 5.745 × 10−3
0.1 2.857 × 10−4 7.722 × 10−3

Table 1The branching ratio ( ) at various temperatures in units of T 9. The rate constants for 15N are taken from the NACRE compilation and those for 17O are taken from Iliadis et al. ( 2010 )..

These branching ratios show that the (p, α) reaction wins over the (p, α) reaction, thus confirming the cyclic behavior forming 12C by most of the 15N nuclei. This is CN cycling. The branching ratio will guide how much 15N will go on to form 16O by taking a proton. It is from this point onwards that the oxygen-fluorine reaction network starts, for which discussions are available in many places in the literature (Bethe 1997 ; Cameron & Kahl 2013 ). Here we give a brief description relevant to this work. A proton capture by 16O leads to the formation of unstable 17F which decays into 17O. This secondary isotope of oxygen has very small relative abundance (Lodders 2003 ). This is probably due to the fact that 17O gets destroyed via both 17O (p, α)14N and 17O (p, γ)18F; the former reaction rate is higher than the latter one at the same temperature. Thus the (p, α) reaction is also important for further evolution and starts competing with (p, γ) reactions. Inclusion of the (p, α) reaction that has also been considered here introduces another branching point in the cycle, for which the branching point ratio of (p, γ) to (p, α) has been listed in Table 1. If the cycle advances forming 18F then this will be immediately followed by the 18F( , ) 18O reaction. This is because 18F is unstable against β-decay, and at low density such as ( g cm−3) = 1 and temperature , the p-capture lifetime of 18F is very large compared to its β decay mean lifetime i.e. 9504 s (Audi et al. 1997 ). 18O, the tertiary isotope of oxygen, has very low abundance compared to 16O but greater compared to 17O (Lodders 2003 ). From this point onwards, we adopt the reaction steps given in Hansen et al. ( 2004 ); Mountford ( 2013 ) i.e. 18O (p, γ) 19F. Sometimes the radiative capture on 18O cannot be neglected as compared to O18 (p, α) 15N even though the (p, α) channel is substantially stronger than (p, γ) channels. Depending upon the spin and energy of resonance, the latter can be of comparable strength (Wiescher et al. 1980 ). Still we have checked for possible alteration of abundances of oxygen by including a third branching point at 18O. But still it does not lead us to find any significant change in the mass fraction of 16O, which has been found to be consistent with the earlier report of Audouze et al. ( 1977 ) in which the author mentioned that the 18O (p, γ) 19F leak has little effect on CNO equilibrium abundance. For instance, the 16O abundance changes by only 12% at temperatures 0.03 and 0.05 (refer to Section 4.2). Moreover, the goal of this choice is because it may improve our knowledge of levels in the 19F nucleus that are relevant to nuclear astrophysics and hopefully for a possible fluorine production network. Recently, Buckner et al. ( 2012 ) studied the 18O (p, γ) 19F reaction and found that most 19F levels decay by γ γ-cascades through the first (110 keV) excited state, and all 19F levels (with known decay schemes) de-excite through the second (197 keV) excited state. It is also an interesting element in the periodic table because of the fact that though it is surrounded by some of the most abundant elements in the universe like oxygen, nitrogen and neon, it is itself very rare. Perhaps this is because it is an odd Z element with only one single stable isotope and it is very fragile with nine protons and ten neutrons (Palacois 2006 ). Then the finally produced 19F is destroyed by a (p, α) reaction forming 16O since the 19F (p, α) 16O (p, α) reaction rate is faster than the (p, γ) reaction which would have produced 20Ne. Thus the CNOF cycle is

The generalized differential equation that governs the evolution of any element in terms of number density via a proton-capture reaction or a β-decay, or both, at the enhanced condition is given by Clayton ( 1983 )
where is the decay constant of an unstable nucleus with number density . As all the three cycles involve proton-capture reactions and β-decays, the abundances will primarily depend upon the lifetime of these processes. If the β-decay lifetime for an unstable element in the cycle is shorter than the proton-capture lifetime for the same element, then the β-decay lifetime can be bypassed and thus the element can be thought of as representing the next stable element in the cycle having the same mass number. Considering that Equation (5) for each cycle takes the form of the following differential rate equations in terms of mass fraction of any element
where s are terms for the respective proton-capture reactions. and stand for mass numbers of different nuclei. Equation (6) can be expressed as a function of the hydrogen mass fraction to get a series of first order simultaneous linear differential equations for each cycle as
which are then solved for a suitable initial condition.

4.4 Calculation of Abundances

In the case of the CNOF cycle, the initial abundance of heavy elements has been chosen in such a manner that it does not affect the production of primary nitrogen from being the most abundant element in the CNO reaction, which actually gets hampered for metallicities higher than Z = 0.001. Moreover, at higher metallicities rotational mixing is not so efficient. We also do not take very or extremely low metallicities because low metallicity reveals a low content of heavy elements, thereby making the opacity lower. Hence the stars become more compact and therefore hotter, so they are going to have high luminosity. This high luminosity increases the radiation pressure and thus stellar mass loss (Meynet et al. 2009 ). Fast rotating massive stars, with an initial metallicity Z = 0.0005, with typical time averaged velocity of 500 km s−1 on the main sequence easily reach the critical velocity (Decressin et al. 2007b ) at the beginning of their evolution and remain near the critical limit during the rest of their main sequence. As a consequence, they lose a large amount of material through a mechanical wind, which probably leads to the formation of a slow outflowing Keplerian equatorial disk. Keeping all of these factors in mind, simultaneous linear first order differential equations are solved numerically for each cycle varying the hydrogen mass fraction up to X = 0.60 to get the equilibrium mass fraction abundances of the stable heavy elements with the initial condition of X = 0.70, Y = 0.298, such that X + Y + Z = 1 where Z is the sum of the mass fraction of elements 12C, 14N, 16O and 19F distributed equally; i.e. for CNOF then X = 0.70, Y = 0.2995, for NeNa and lastly X = 0.70, Y = 0.2995, for the MgAl cycle (Meynet et al. 2008 ). The equilibrium abundance by mass fraction of the stable isotopes of considered elements are taken up to the first decimal place. We have also calculated the range of uncertainty in the equilibrium mass fraction abundance of 16O, 23Na and 27Al due to uncertainty in the reaction rate constants considering both low and high rate constants for respective reactions (Iliadis et al. 2010 ) as well as the NACRE. These are presented in Table 5.

T 9 16O 23Na 27Al
Low Medium High Low Medium High Low Medium High
0.02 **** ***** *****
0.03 **** ***** *****
0.05
0.08
0.10

Table 5 The estimated abundance by mass fraction of 16O, 23Na and 27Al due to uncertainty in the reaction rate constants with low, medium (recommended) and high reaction rate constant.

These abundances by mass fraction of the stable isotopes of any element (x) with respect to Fe are then calculated using the expression

Here and The quantities and are the solar number densities taken from Lodders ( 2003 ). For oxygen we have taken the equilibrium 16O mass fraction for the abundance calculations since its relative abundance is larger compared to the other two stable isotopes of oxygen. The calculated abundances of O, Na and Al are presented in Tables 6, 7, 8 and 9, for density for the clusters M3, M4, M13 and NGC 6752 respectively. The observed abundances of a few metal poor evolved stars from Cohen & Meléndez ( 2005 ); Ivans et al. ( 1999 ); Cohen & Meléndez ( 2005 ) and Yong et al. ( 2005 ) are also presented in these tables for comparison. The calculated abundances at a particular temperature are those for which the difference between observed and calculated abundance is found to be minimum. Here most of the stars have been impacted since the influence of the wind is necessary in order to explain the GC abundance anomalies (Decressin et al. 2007b ). Moreover, the second generation of stars is formed because of the ejection of material from the rapidly rotating massive stars which are mixed with primordial material left over from the star formation process (Bastian et al. 2014 ). A recent study conducted by Bekki ( 2011 ) has shown that in massive star clusters a second generation of stars is formed from the gaseous ejecta of AGB stars if the mass of the cluster exceeds . This gives the possibility for the presence of many massive stars within the same GC. A comparison between the computed and observed abundances from Cohen & Meléndez ( 2005 ); Ivans et al. ( 1999 ); Cohen & Meléndez ( 2005 ) and Yong et al. ( 2005 ) is shown in Figures 3, 4, 5 and 6 respectively. In Figure 6 the observed [O/Fe] and [Na/Fe] values have dex (Yong et al. 2005 ).

Star a a a b b
VZ1397 −1.36 0.52 ± 0.05* −0.16 ± 0.05 +0.50 −0.20
II-46 −1.44 0.37 ± 0.05* −0.23 ± 0.07 +0.37 −0.23
VZ1000 −1.42 0.08 ± 0.10 +0.10 ± 0.06 +0.22 +0.33
III-28 −1.54 0.55 ± 0.07 −0.33 ± 0.06 +0.48 −0.07
IV-25 −1.42 0.38 ± 0.05* −0.10 ± 0.05 +0.36 −0.14
C41303-2217 −1.36 0.33 ± 0.05* −0.39 ± 0.09 +0.30 −0.25
IV-27 −1.40 0.08 ± 0.08 +0.11 ± 0.06 +0.20 +0.31
III-61 −1.34 0.60 ± 0.05 −0.45 ± 0.09 +0.48 +0.27
III-60 −1.45 0.29 ± 0.05* −0.23 ± 0.13 +0.29 −0.16
C41544-2336 −1.38 0.39 ± 0.05* +0.32 −0.19
V-30 −1.39 +0.18 +0.30
V-31 −1.45 0.35 ± 0.10 +0.35 +0.36
C41543-2334 −1.25 0.50 ± 0.07 +0.25 ± 0.05* +0.39 +0.46

Cohen & Meléndez ( 2005 )

this work

Table 6 The Estimated Abundance Ratios of [O/Fe] and [Na/Fe] for GC M3

Star a a a a b b b
L4611 −1.16 +0.06 ± 0.03 +0.33 ± 0.05 +0.60 ± 0.03 +0.05 +0.37 +0.66
L4613 −1.17 +0.05 ± 0.03 +0.26 ± 0.05 +0.63 ± 0.03 +0.06 +0.38 +0.67
L1514 −1.16 +0.41 ± 0.03 +0.01 ± 0.05 +0.44 ± 0.03 +0.40 +0.07 +0.66
L1411 −1.20 +0.20 ± 0.03 +0.43 ± 0.05 +0.79 ± 0.03 +0.19 +0.42 +0.75
L3209 −1.20 +0.27 ± 0.03 +0.23 ± 0.05 +0.71 ± 0.03 +0.23 +0.16 +0.71
L2307 −1.19 +0.17 ± 0.03 +0.39 ± 0.05 +0.73 ± 0.03 +0.17 +0.40 +0.73
L2406 −1.19 +0.19 ± 0.03 +0.31 ± 0.05 +0.55 ± 0.03 +0.17 +0.40 +0.69
L4511 −1.18 +0.23 ± 0.03 +0.44 ± 0.05 +0.83 ± 0.03 +0.21 +0.44 +0.73
L1501 −1.20 +0.10 ± 0.03 +0.42 ± 0.05 +0.81 ± 0.03 +0.09 +0.42 +0.75
L3413 −1.17 +0.40 ± 0.03 −0.04 ± 0.05 +0.61 ± 0.03 +0.41 +0.08 +0.67
L2617 −1.17 +0.01 ± 0.03 +0.50 ± 0.05 +0.84 ± 0.03 +0.03 +0.43 +0.72
L3624 −1.16 +0.29 ± 0.03 +0.10 ± 0.05 +0.69 ± 0.03 +0.32 +0.10 +0.68
L3612 −1.19 +0.10 ± 0.03 +0.47 ± 0.05 +0.77 ± 0.03 +0.08 +0.45 +0.74
L2206 −1.18 +0.31 ± 0.03 +0.25 ± 0.05 +0.69 ± 0.03 +0.34 +0.39 +0.69
L2208 −1.17 +0.36 ± 0.03 +0.55 ± 0.05 +0.90 ± 0.03 +0.37 +0.60 +0.97
L2519 −1.16 +0.37 ± 0.03 −0.19 ± 0.05 +0.50 ± 0.03 +0.36 +0.07 +0.66
L4201 −1.18 +0.41 ± 0.03 +0.31 ± 0.05 +0.55 ± 0.03 +0.42 +0.39 +0.68
L4633 −1.19 +0.33 ± 0.03 −0.03 ± 0.05 +0.59 ± 0.03 +0.35 +0.10 +0.69
L1408 −1.20 +0.24 ± 0.03 +0.18 ± 0.05 +0.47 ± 0.03 +0.23 +0.16 +0.70
L4414 −1.15 +0.16 ± 0.03 +0.21 ± 0.05 +0.70 ± 0.03 +0.14 +0.36 +0.70
L1701 −1.20 +0.49 ± 0.03 −0.02 ± 0.05 +0.62 ± 0.03 +0.44 +0.11 +0.70
L3207 −1.17 +0.45 ± 0.03 −0.23 ± 0.05 +0.37 ± 0.03 +0.41 −0.39 +0.67
L3215 −1.20 +0.27 ± 0.03 −0.03 ± 0.05 +0.47 ± 0.03 +0.23 +0.11 +0.70
L4302 −1.19 +0.20 ± 0.03 +0.31 ± 0.05 +0.48 ± 0.03 +0.17 +0.40 +0.69

Ivans et al. ( 1999 )

this work

Table 7 The Estimated Abundance Ratios of [O/Fe], [Na/Fe] and [Al/Fe] for GC M4

Star a a a a b b b
II-67 −1.30 −1.14 ± 0.10 0.32 ± 0.05 0.64 ± 0.10 +0.10 +0.26 +0.80
IV-25 −1.41 −0.33 ± 0.05* 0.41 ± 0.09 0.85 ± 0.10 +0.21 +0.37 +0.91
II-76 −1.53 0.55 ± 0.05* −0.32 ± 0.05 ≤ 0.59 +0.63 −0.08 −0.22
III-18 −1.45 0.55 ± 0.10 0.36 ± 0.06 0.74 ± 0.10 +0.55 +0.36 +0.95
K-188 −1.45 0.33 ± 0.07 −0.08 ± 0.05* ≤ 0.57 +0.35 −0.11 −0.30
III-7 −1.53 0.46 ± 0.08 0.10 ± 0.05* ≤ 0.70 +0.46 −0.03 −0.22
I-18 −1.47 0.17 ± 0.10 0.32 ± 0.05* ≤ 0.76 +0.27 +0.38 +0.97
I-49 −1.49 0.58 ± 0.05* 0.00 ± 0.05* ≤ 0.67 +0.59 −0.07 +0.99
J37 −1.48 0.15 ± 0.10 0.37 ± 0.06 ≤ 0.97 +0.28 +0.39 +0.98
C41196-2632 −1.48 −0.01 ± 0.10 0.17 ± 0.05* +0.28 +0.39
II-4 −1.55 0.26 ± 0.10 0.28 ± 0.05* +0.35 +0.46
IV-29 −1.49 0.52 ± 0.10 −0.33 ± 0.05* +0.59 −0.07
J45 −1.47 0.16 ± 0.10 0.18 ± 0.05* +0.27 +0.38
I-5 −1.55 0.49 ± 0.05* +0.49
C41155-3103 −1.50 −0.06 ± 0.10 0.35 ± 0.05* +0.30 +0.41
C41148-3103 −1.51 0.55 ± 0.05* −0.09 ± 0.05* +0.61 −0.09
C41134-3056 −1.40 0.24 ± 0.05* −0.02 ± 0.07* +0.24 −0.16
C40559-2839 −1.55 0.37 ± 0.06 0.15 ± 0.08 +0.36 −0.01
C41101-3050 −1.44 0.37 ± 0.10 0.18 ± 0.05* +0.37 +0.35
C41099-3046 −1.36 0.19 ± 0.05* 0.04 ± 0.05* +0.49 +0.27
C41135-3053 −1.47 0.35 ± 0.05* −0.05 ± 0.05* +0.37 −0.09
C41133-2750 −1.53 0.26 ± 0.05* −0.04 ± 0.10 +0.33 −0.03
C40535-2819 −1.55 0.38 ± 0.07 0.05 ± 0.06 +0.38 −0.01
C40539-2813 −1.61 0.25 ± 0.07 0.31 ± 0.05* +0.41 +0.52
C41135-2753 −1.72 0.14 ± 0.13 0.30 ± 0.10 +0.62 +0.16

Cohen & Meléndez ( 2005 )

this work

Table 8 The Estimated Abundance Ratios of [O/Fe], [Na/Fe] and [Al/Fe] for GC M13

Star a a a a b b b
NGC 6752-mg0 −1.62 0.71 0.67 1.08 0.72 0.58 1.12
NGC 6752-mg1 −1.60 0.46 0.38 0.82 0.51 0.73 1.10
NGC 6752-mg2 −1.59 0.55 0.19 0.77 0.54 0.03 1.09
NGC 6752-mg3 −1.60 0.47 0.22 0.77 0.46 0.07 1.10
NGC 6752-mg4 −1.60 0.38 0.29 0.90 0.40 0.51 1.10
NGC 6752-mg5 −1.59 0.42 0.32 0.74 0.42 0.03 1.09
NGC 6752-mg6 −1.59 0.60 0.13 0.57 0.54 0.03 1.09
NGC 6752-mg8 −1.68 0.40 0.34 0.74 0.48 0.12 1.18
NGC 6752-mg9 −1.63 0.47 0.28 0.77 0.47 0.76 1.13
NGC 6752-mg10 −1.63 0.44 0.28 0.78 0.44 0.07 1.13
NGC 6752-mg12 −1.62 0.66 −0.09 0.09 0.72 0.53 −0.13
NGC 6752-mg15 −1.60 0.40 0.31 0.72 0.40 0.51 1.10
NGC 6752-mg18 −1.60 0.46 0.19 0.59 0.46 0.04 1.10
NGC 6752-mg21 −1.60 0.01 0.57 1.18 0.40 0.56 1.15
NGC 6752-mg22 −1.61 0.19 0.63 0.99 0.41 0.57 1.11
NGC 6752-mg24 −1.63 0.65 −0.09 0.12 0.58 0.02 −0.12
NGC 6752-mg25 −1.60 0.59 0.14 0.51 0.55 0.04 −0.15
NGC 6752-0 −1.62 −0.15 0.55 1.33 0.42 0.54 1.42
NGC 6752-1 −1.58 0.57 0.08 0.32 0.53 0.02 −0.17
NGC 6752-2 −1.59 −0.09 0.60 1.18 0.39 0.55 1.14
NGC 6752-3 −1.64 0.70 −0.04 0.22 0.74 0.03 −0.11
NGC 6752-4 −1.61 −0.04 0.61 1.20 0.41 0.57 1.16
NGC 6752-6 −1.61 0.09 0.54 0.96 0.41 0.53 1.11
NGC 6752-7 −1.84 0.90 0.02 0.19 0.94 0.23 0.09
NGC 6752-8 −1.62 0.66 −0.01 0.48 0.72 0.01 1.12
NGC 6752-9 −1.63 0.65 −0.02 0.13 0.41 0.02 −0.12
NGC 6752-10 −1.60 −0.02 0.65 1.06 0.40 0.56 1.10
NGC 6752-11 −1.64 0.37 0.35 0.90 0.44 0.55 1.14
NGC 6752-12 −1.62 0.29 0.27 0.41 0.42 0.06 −0.13
NGC 6752-15 −1.61 0.65 −0.10 0.58 0.56 0.00 1.11
NGC 6752-16 −1.60 0.09 0.36 0.83 0.40 0.51 1.10
NGC 6752-19 −1.61 0.29 0.22 0.59 0.41 0.05 1.11
NGC 6752-20 −1.59 0.08 0.67 1.15 0.39 0.55 1.14
NGC 6752-21 −1.61 0.49 0.29 0.63 0.47 0.05 1.11
NGC 6752-23 −1.62 0.11 0.59 1.25 0.42 0.58 1.17
NGC 6752-24 −1.65 0.56 0.01 0.36 0.56 0.04 −0.10
NGC 6752-29 −1.64 0.51 −0.07 0.35 0.50 0.03 −0.11
NGC 6752-30 −1.62 0.61 0.15 0.56 0.57 0.06 1.12

Yong et al. ( 2005 )

this work

Table 9 The Estimated Abundance Ratios of [O/Fe], [Na/Fe] and [Al/Fe] for NGC 6752

5 Discussion and concluding remarks

Synthesis due to the proton-capture reaction has been investigated using updated reaction rates (Iliadis et al. 2010 ) for nuclear reaction cycles CNOF, NeNa and MgAl under conditions of high temperature and low density.

The first consequence we observe is the production of an important amount of primary 14N. Our calculation yields the highest production of this element with mass fraction 0.001 (Fig. 1) which is the same order of magnitude as a stellar model with mass , and initial rotational velocity 800 km s−1 (Decressin et al. 2007b ).

Fig. 1 Profiles of equilibrium abundances by mass fraction of stable isotopes with respect to temperature are shown for stable nuclides in the CNOF cycle (top panel), NeNa cycle (middle panel) and MgAl cycle (bottom panel) at density gm cc−1. Top panel: The 14N abundance is found to be highest among all cases, while the abundance of 16O is higher compared to the abundances of other isotopes of oxygen, but does not show any significant variation. This is probably because these two isotopes are circulated back by 17O (p, α) 14N and 19F (p, α) 16O respectively. Middle panel: In the NeNa cycle, Na does show variation with respect to increase in temperature. This is probably directly reflected by the increase in 20Ne abundance. Bottom panel: In the MgAl cycle, the elements presented here only show their effect at temperature greater than and they do show noticeable variation with respect to temperature. Abundance of 24Mg decreases with respect to temperature at a steeper slope. Thus at higher temperature, more 25Mg will be present which in turn will influence the production of Al, which also decreases slowly with the increase in temperature. These decrements result in an increase of the 27Al isotope.

Abundance ratios, i.e., [O/Fe], [Na/Fe] and [Al/Fe], are also calculated and compared with their counterparts observed in a sample of metal poor stars in GCs M3, M4, M13 and NGC 6752 (Figs. 2, 3, 4 and 5 respectively). This comparison demonstrates the extent to which these nuclear reaction cycles can explain the observed abundances of O, Na and Al. These ratios show similar trends for stars of all the four GCs considered here. We find that within the temperature and density conditions considered, there is a small range of temperature values at which the computed [O/Fe], [Na/Fe] and [Al/Fe] ratios agree well with the observed ratios in a sample of stars of GCs M3, M4, M13 and NGC 6752.

Fig. 2 A comparison between the calculated and observed abundance ratios of [O/Fe] and [Na/Fe] for GC M3. The data points are taken from Table 6. Here solid circles mean “this work” and open circles mean Yong et al. ( 2005 ).
Fig. 3 A comparison between the calculated and observed abundance ratios of [O/Fe], [Na/Fe] and [Al/Fe] for GC M4. The data points are taken from Table 7. Here solid circles mean “this work” and open circles mean Ivans et al. ( 1999 ).
Fig. 4 A comparison between the calculated and observed abundance ratios of [O/Fe], [Na/Fe] and [Al/Fe] for GC M13. The data points are taken from Table 8. Here solid circles mean “this work” and open circles mean Cohen & Meléndez ( 2005 ).
Fig. 5 A comparison between the calculated and observed abundance ratios of [O/Fe], [Na/Fe] and [Al/Fe] for NGC 6752. The data points are taken from Table 9. Here solid circles mean “this work” and open circles mean Yong et al. ( 2005 ).
Fig. 6 Observed Na-O relation. Computed ratios in the [Na/Fe] vs. [O/Fe] plot are compared with their counterparts observed in the evolved stars of all four GCs: M3 (top left), M4 (top right), M13 (bottom left) and NGC 6752 (bottom right). We observe signs of anticorrelation between Na and O for M3, M4, M13 and NGC 6752. Here solid circles correspond to the data points of “this work” and open circles are for Cohen & Meléndez ( 2005 ) (top left), Ivans et al. ( 1999 ) (top right), Cohen & Meléndez ( 2005 ) (bottom left) and Yong et al. ( 2005 ) (bottom right).

GC stars are known to show Na-O anti-correlation. This anti-correlation is a common feature for GCs in the range of metallicity ( ). To examine this effect, we have plotted the computed [Na/Fe] vs. [O/Fe] and compared with observations (Fig. 6). The computed abundance ratios are found to follow similar trends for all the GCs. We have also produced the Na-O anti-correlation range collectively for all the four clusters, which is well within the comparable range observed of (Decressin et al. 2007a ). Probably a more accurate result can be obtained if one considers the 23Na Ne reaction rate lowered by a factor of 2 for non-solar-scaled metallicity (Ventura & D’Antona 2006 ). Moreover, recently Buckner et al. ( 2012 ) reported an updated set of 18O (p, γ)19F reaction rate constants which may again give even more fine tuned results.

We have also found that the 24Mg mass fraction abundance is in the same order of magnitude as the values reported in table 3 of Decressin et al. ( 2007b ) which falls well within the considered temperature range.

Here also we did not find any anti-correlation between Mg and Al, which is similar to the kind of result obtained in Decressin et al. ( 2007b ). However we did not investigate the reason for such an observation. Decressin et al. ( 2007b ) reported that the 24Mg Al reaction rate increment could possibly lead to such an observation.

We also observe that the 16O mass fraction estimated in our calculation has the same order of magnitude as that of the mean mass fraction in winds reported in table 5 of Decressin et al. ( 2007b ) at the end of the central H burning. We also find that the 23Na has the same order of magnitude (Table 5), when compared with table 5 of Decressin et al. ( 2007b ), but these orders are closer to the mass fraction obtained if one uses the high reaction rate constants in the reaction networks. As far as 27Al is concerned, we find that at temperatures greater than our estimated orders are almost 10 times greater than those estimated by Decressin et al. ( 2007b ).

The major uncertainties present are the poorly known influence of molecular weight gradients on the shear instabilities and our ignorance of the effect of magnetic fields in the stellar interior on the transport of chemical and angular momentum, which are still being considered as prime candidates for affecting the abundance pattern. A star rotating axially, at low metallicity, triggers many instabilities in its stellar interior. These instabilities participate in the transport of chemical species and of angular momentum. Among these instabilities, shear mixing and horizontal turbulence are important agents for effective transport of chemical elements and angular momentum since in a low metallicity star rotational mixing plays a dominant role (Meynet 2008 ). Moreover we have considered the cyclic behavior of reaction networks, i.e., the evolution of elements is confined. However, there may also be some leakage of 19F going to 20Ne via a proton-capture, 23Na (p, γ) 24Mg (Cavallo et al. 1998 ) and finally 27Al takes a proton resulting in the formation of 28Si. This will definitely alter the abundance profile of the considered elements. The possibilities of other internal pollution mechanisms (D’Orazi et al. 2013 ) within the cluster are also likely to affect the abundance pattern. Theoretical uncertainties are also likely to remain, especially in the calculation of reaction rates (Iliadis et al. 2010 ), and the choice of the initial condition. Also, the mass loss in rotating stars is thus an asymmetric phenomenon. Very hot stars have dominant polar ejection whereas stars with temperature < 24 000 K eject most of their material through equatorial ejection, forming a disk which removes a lot of mass from the star as compared to polar ejection, even though mass loss does not play a dominant role in low metallicity stars (Meynet 2008 ). Nevertheless, the elements that are formed following proton-capture synthesis may have been brought to the surface which carry angular momentum with them and which is why the stars get their rotational motion. Once the rotational speed reaches a critical value, the stars start to lose their mass in the form of these synthesized materials into outer space, thus enriching the space by these materials from which the second generation stars are formed. Thus, the computed abundances of elements in the present set of initial conditions do not deviate much from the observed values of [O/Fe], [Na/Fe] and [Al/Fe] ratios. These results support the possibility that these nuclear cycles occur in such kind of a stellar situation at high temperature and low density conditions, and thus also supporting the primordial scenario.


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Cite this article: Mahanta Upakul, Goswami Aruna, Duorah Hiralal, Duorah Kalpana. p-capture reaction cycles in rotating massive stars and their impact on elemental abundances in globular cluster stars: A case study of O, Na and Al. Cancer Biol Med. 2017; 8:080.

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