Rotating massive stars: From first stars to gamma ray bursts

This article first reviews the basic physics of rotating stars and their evolution. The changes of the mechanical and thermal equilibrium of rotating stars are examined. An important, predicted and observed, effect is that rotating stars are hotter at the poles and cooler at the equator. The mass loss by stellar winds, which are influenced by the anisotropic temperature distribution, is discussed. These anisotropies in the interior are also driving circulation currents, which transports the chemical elements and the angular momentum in stars. Internal differential rotation, if present, creates instabilities and mixing, in particular, the shear mixing, the horizontal turbulence and their interactions. A major check of the model predictions concerns the changes of the surface abundances, which are modified by mass loss in the very massive stars and by rotational mixing in O-type and B-type stars. The observations are shown to confirm the existence of rotational mixing, with much larger effects at lower metallicities. The predictions of stellar models concerning the evolution of the surface velocities, the evolutionary tracks in the Hertzsprung-Russell diagram and lifetimes, the populations of blue, red supergiants and Wolf-Rayet stars, and the progenitors of type Ibc supernovae are discussed. In many aspects, rotating models are shown to provide a much better fit than nonrotating ones. Using the same physical ingredients as those which fit the best observations of stars at near solar metallicities, the consequences of rotating models for the status of Be stars, the progenitors of gamma ray bursts, the evolution of Pop III stars and of very metal-poor stars, the early chemical evolution of galaxies, the origin of the C-enhanced metal poor stars and of the chemical anomalies in globular clusters are explored. Rotation together with mass loss are two key physical ingredients shaping the evolution of massive stars during the whole cosmic history.

Figure 1

Probability density by $\mathrm{km}{\mathrm{s}}^{-1}$ of rotation velocities for 496 stars with types O9.5 to B8, i.e., masses between about $3M_{\odot }$ and $20M_{\odot }$. From 71.

Figure 2

Shape of the surface for various values of $\omega =\Omega /{\Omega }_{\mathrm{crit}}$ ( labeled at the bottom of each curve). The x axis is the equatorial radius, and the y axis is the polar one. Hence, this is how we would see the star equator on. $R_p$ is the polar radius. From 49.

Figure 3

The critical velocities $v_{\mathrm{crit}}$ as a function of stellar masses for different metallicities $Z$ for stars on the ZAMS. The effects of the changes of the polar radius with rotation are accounted for. From 41.

Figure 4

The mass fluxes around a rotating star of $100M_{\odot }$ with ${10}^{6.5}{L}_{\odot }$ and a ratio of the angular velocity to the break-up angular velocity $\omega =0.80$, assuming a polar ${\mathrm{T}}_{\mathrm{eff}}=30000\mathrm{K}$. From 101.

Figure 5

Schematic structure with stream lines of meridional circulation in a rotating $20M_{\odot }$ model of $5.2R_{\odot }$ with $Y=0.30$, $Z=0.02$, and $v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$ at the beginning of the MS phase. The figure is made as a function of $M_r$. In the upper hemisphere on the right section, matter is turning counterclockwise along the outer stream line and clockwise along the inner one. The inner sphere is the convective core. It has a radius of $1.7R_{\odot }$. From 126.

Figure 6

Unstable displacements with $m=1$ of the azimuthal magnetic field in the polar region. The large arcs (thin broken lines) indicate the horizontal stellar surfaces. Adapted from 163.

Figure 7

Left: Evolution of the angular velocity $\Omega$ as a function of the distance to the center in a $20M_{\odot }$ star with $v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$. $X_c$ is the hydrogen mass fraction at the center. The dotted line shows the profile when the He core contracts at the end of the H-burning phase. From 125. Right: rotation profiles at various stages of evolution (labeled by the central H content $X_{\mathrm{c}}$) of a $15M_{\odot }$ model with $X=0.705$, $Z=0.02$, an initial velocity of $300\mathrm{km}{\mathrm{s}}^{-1}$ and magnetic field from the Tayler-Spruit dynamo. From 110.

Figure 8

Evolution with mass loss (only) of the total mass and internal structure as a function of time for a star with an initial mass of $60M_{\odot }$. The cloudy regions indicate convection, the heavy diagonal hatching the regions of high nuclear production, and the light vertical hatching regions of variable composition. From 94.

Figure 9

Variation of the diffusion coefficients as a function of the radius inside a $20M_{\odot }$ model with an initial metallicity $Z=0.014$ and an initial velocity on the ZAMS equal to 40% the critical velocity. From left to right is shown the situation corresponding to central mass fraction of hydrogen equal to 0.60, 0.30 and 0.01, respectively. Courtesy of S. Ekstrom.

Figure 10

The differences in $\log (\mathrm{N}/\mathrm{H})$ as a function of the initial masses at three stages during the MS phase for models at $Z=0.02$ with the average rotation velocities (i.e., 217, 209, 197, 183, 172, $168\mathrm{km}{\mathrm{s}}^{-1}$ for, respectively, $12M_{\odot }$, $15M_{\odot }$, $20M_{\odot }$, $25M_{\odot }$, $40M_{\odot }$, and $60M_{\odot }$). The 3 stages are indicated by the value of the central H content $X_{\mathrm{c}}$. From 112.

Figure 11

The evolution of the differences in $\log (\mathrm{N}/\mathrm{H})$ during the MS phase as a function of the actual rotation velocities ($\sin i$ being equal to 1 here) for models of $20M_{\odot }$ with $Z=0.02$ and different initial velocities. From 112.

Figure 12

Evolutionary tracks in the plane surface N/C ratio, normalized to its initial value, vs the rotational period in hours for different initial mass stars, various initial velocities, and for the metallicities $Z=0.02$ and 0.002. Positions of some periods in days are indicated at the bottom of the figure. The dotted tracks never reach the critical limit during the MS phase. The short dashed tracks reach the critical limit during the MS phase. The dividing line between the shaded and nonshaded areas corresponds to the entrance into the stage when the star is at the critical limit during the MS evolution. large circles along some tracks indicate the stage when $\upsilon /{\upsilon }_{\mathrm{crit}}$ becomes superior to 0.7. If Be stars are rotating at velocities superior to 70% of the critical velocity, present models would predict that they would lie in the region comprised between the large circles and the dividing line. Beware of the different vertical scales used when comparing similar masses at different metallicities. From 41.

Figure 13

The N abundance (in a scale where $\log H=12.0$) as a function of $v\sin i$ for the MS stars (black dots) in N11 with masses between $14M_{\odot }$ and $20M_{\odot }$ according to 75. The downward arrows indicate abundance upper limits. The binaries are shown by a square. The evolved stars in a band of 0.1 dex in $\log T_{\mathrm{eff}}$ beyond the end of the MS are shown with open symbols. The grey band indicates uncertainties of $\pm 0.25$ dex. From 112.

Figure 14

Evolution of the equatorial surface velocities along the evolutionary tracks in the HR diagram starting from $v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$. For clarity, only the first part of the $40M_{\odot }$ track is shown. From 125.

Figure 15

Evolution of the surface equatorial velocity as a function of time for $60M_{\odot }$ stars with $v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$ at different initial metallicities. From 126.

Figure 16

Angular velocity as a function of the Lagrangian mass coordinate, $m_r$ inside the $25M_{\odot }$ model ($v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$) at various evolutionary stages. From 68.

Figure 17

Local specific angular momentum profiles for the $25M_{\odot }$ model ($v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$) at different evolutionary stages. From 68.

Figure 18

Evolutionary tracks for nonrotating (dotted lines) and rotating (continuous lines) models with solar metallicity. The rotating models have an initial velocity $v_{\mathrm{ini}}$ of $300\mathrm{km}{\mathrm{s}}^{-1}$. For clarity, only the first part of the tracks for the most massive stars ($\mathrm{M}\ge 40M_{\odot }$) is shown. Portions of the evolution during the WR phase for the rotating massive stars are indicated by short-dashed lines. The long-dashed track for the $60M_{\odot }$ model corresponds to a very fast-rotating star ($v_{\mathrm{ini}}\sim 400\mathrm{km}{\mathrm{s}}^{-1}$), which follows a nearly homogeneous evolution. Only the beginning of its evolution is shown. From 125.

Figure 19

Evolution of the $T_{\mathrm{eff}}$ as a function of the fraction of the lifetime spent in the He-burning phase for $20M_{\odot }$ stars with different initial velocities. From 107.

Figure 20

Comparison of the internal distribution of helium in two models of $20M_{\odot }$ at the middle of the He-burning phase. The dash-dotted line concerns the models with zero rotation and the continuous line represents the case with $v_{\mathrm{ini}}=300\mathrm{km}{\mathrm{s}}^{-1}$. From 107.

Figure 21

Variation of the number ratios of Wolf-Rayet stars to O-type stars as a function of the metallicity. The observed points are taken as in 104. The dotted line shows the predictions of the models of 132 with normal mass loss rates. The continuous and dashed lines show the predictions of the rotating and nonrotating stellar models, respectively, of 128 and 129. The pentagon shows the ratio predicted by $Z=0.040$ models computed with the metallicity dependence of the mass loss rates during the WR phase proposed by 24.

Figure 22

Variation of the number ratios of WN to WC stars as a function of metallicity. The grey area encompasses the observed ratios. Individual measures are indicated by black circles labeled with the name of the galaxy [see references in 129]. The solar (O/H) value is from 4 and the O/H values for the SMC and LMC are from 74. The dotted lines show the predictions of the rotating and nonrotating stellar models of 128 and 129. The pentagon shows the ratio predicted by $Z=0.040$ models computed with the metallicity dependence of the mass loss rates during the WR phase. The open triangle shows the WC/WN ratio obtained from the rotating models allowing an LBV phase during the WR phase. From 122.

Figure 23

Internal profile of $U(r)$ in models of $20M_{\odot }$ at various metallicities, with ${\Omega }_{\mathrm{ini}}/{\Omega }_{\mathrm{crit}}=0.5$, where $u(r,\theta )$ the vertical component of the velocity of the meridional circulation is $u(r,\theta )=U(r)P_2(\cos \theta )$ with $P_2$ the second Legendre polynomial. The radius is normalized to the outer one. All models are at the same evolutionary stage, when the central H mass fraction is about 0.40. From 40.

Figure 24

Left panels: Evolution of $Z=0$ models (with rotation: continuous lines; without rotation: dotted lines) in the Hertzsprung-Russell diagram. The grey area shows the zone of the diagram where He burns in the core of the rotating models. Right panel: Evolution of the $\Omega /{\Omega }_{\mathrm{crit}}$ ratio during the MS. All the models start the MS with ${\upsilon }_{\mathrm{eq}}=800\mathrm{km}{\mathrm{s}}^{-1}$, except the $9M_{\odot }$ which starts the MS with $500\mathrm{km}{\mathrm{s}}^{-1}$. From 40.

Figure 25

Left panel: Evolution in the Hertzsprung-Russell diagram. Right panel: Evolution of the mass of the model. The mass indicated is the mass lost at each stage, not a summation. Black line: rotating model; continuous part: beginning of MS ($X_{\mathrm{c}}=0.753$ down to 0.58; dashed part: rest of the MS; dotted part: beginning of core He-burning phase ($Y_{\mathrm{c}}=1.00$ down to 0.96); continuous part: rest of the He burning. Grey line: nonrotating model for comparison. From 40.

Figure 26

Ranges of masses of different types of supernovae at different metallicities. The shading indicates the area where formation of a black hole (BH) is expected; elsewhere, the remnant is a neutron star (NS). From 50.

Figure 27

Observed rate of type Ic SNe (triangles with error bars) and predicted rate of type Ic SNe whose progenitor is a WO star (dashed line) with respect to the total number of core-collapse SNe (CCSNe). The grey rectangle represents the extension in metallicity (133) and rate (152) of GRB events. From 50.

Figure 28

Young bi-modal shaped nebula blown from the highly aspherical, polar dominated winds around a $20M_{\odot }$ star at $Z={10}^{-5}$. The star has evolved during 21 552 yr since it has left the ZAMS. The maximal extension of the wind is $\sim 1.5\times {10}^{18}\mathrm{cm}$. This model was computed by the A-Maze code (186). From 50.

Figure 29

Stellar yields divided by the initial mass $p_{im}^{\mathrm{tot}}$ as a function of the initial mass for the nonrotating (left) and rotating (right) models at solar metallicity. The different total yields (divided by $m$) are shown as piled up on top of each other and are not overlapping. ${}^4\mathrm{He}$ yields are delimited by the top solid and long dashed lines (top shaded area), ${}^{12}\mathrm{C}$ yields by the long dashed and short-long dashed lines, ${}^{16}\mathrm{O}$ yields by the short-long dashed and dot-dashed lines and the rest of metals by the dot-dashed and bottom solid lines. The bottom solid line also represents the mass of the remnant ($M_{\mathrm{rem}}^{\mathrm{int}}/m$). The corresponding SN explosion type is also given. The wind contributions are superimposed on these total yields for the same elements between their bottom limit and the dotted line above it. Dotted areas help quantify the fraction of the total yields due to winds. Note that for ${}^4\mathrm{He}$, the total yields is smaller than the wind yields due to negative SN yields. From 70.

Figure 30

Product of the stellar yields $m{p}_{im}^{\mathrm{tot}}$ by Salpeter’s IMF (multiplied by an arbitrary constant $1000M^{-2.35}$) as a function of the initial mass for the nonrotating (left) and rotating (right) models at solar metallicity. The different shaded areas correspond from top to bottom to $m{p}_{im}^{\mathrm{tot}}\times 1000\times M^{-2.35}$ for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{O}$, and the rest of the heavy elements. The dotted areas show for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$ the wind contribution. From 70.

Figure 31

Yields comparison between the nonrotating $Z=0$ models from Chieffi and Limongi (2004, CL04, open pentagons), the rotating $Z={10}^{-8}$ models from Hirschi (2007, filled squares), and the rotating (filled triangles) and nonrotating (open triangles) $Z=0$ models from 40. Left: ${}^{12}\mathrm{C}$; center: ${}^{14}\mathrm{N}$; right: ${}^{16}\mathrm{O}$. From 40.

Figure 32

Evolution of the N/O and C/O ratios. Data points for halo stars are from 76 (2004, open squares) and of 162. The points with error bars are for DLA systems from 147. The continuous curve is the chemical evolution model obtained with the stellar yields of slow-rotating $Z={10}^{-5}$ models from 126 and 70. The dashed line includes the yields of fast-rotating $Z={10}^{-8}$ models from 67 at very low metallicity. The dotted curve is obtained using the yields of the $Z=0$ models presented in 40 up to $Z={10}^{-10}$. The chemical evolution models are from 18.

Figure 33

$s$-process distributions between ${}^{57}\mathrm{Fe}$ and ${}^{138}\mathrm{Ba}$ normalized to solar for the $25M_{\odot }$ and $[\mathrm{Fe}/\mathrm{H}]=-4$ at the end of the convective C-burning shell. The horizontal line corresponds to the ${}^{16}\mathrm{O}$ overabundance in the C shell (thick line), multiplied and divided by two (thin lines). Isotopes of the same element are connected by a line. The cases presented are the following: (i) case without additional ${}^{22}\mathrm{Ne}$ (triangles) and $X({}^{22}\mathrm{Ne}{)}_{\mathrm{ini}}=5.21\times {10}^{-5}$; (ii) case with additional ${}^{22}\mathrm{Ne}$ (open squares, full squares for the $s$-only isotopes) and $X({}^{22}\mathrm{Ne}{)}_{\mathrm{ini}}=5.0\times {10}^{-3}$; and (iii) case with additional ${}^{22}\mathrm{Ne}$ (open circles, full circles for the $s$-only isotopes) and $X({}^{22}\mathrm{Ne}{)}_{\mathrm{ini}}=1.0\times {10}^{-2}$. From 148.