On the Structure and Properties of Differentially Rotating Main-Sequence
  Stars in the 1-2 M_sun Range
ASTROPHYSICAL JOURNAL, ACCEPTED
Preprint typeset using LATEX style emulateapj v. 10/10/03
ON THE STRUCTURE AND PROPERTIES OF DIFFERENTIALLY ROTATING, MAIN-SEQUENCE STARS IN THE
1 − 2 M⊙ RANGE
K. B. MACGREGOR, STEPHEN JACKSON, ANDREW SKUMANICH, T. S. METCALFE
High Altitude Observatory, NCAR, P. O. Box 3000, Boulder, CO 80307∗
Astrophysical Journal, Accepted
ABSTRACT
We conduct a systematic examination of the properties of models for chemically homogeneous, differentially
rotating, main-sequence stars of mass 1 − 2 M⊙. The models were constructed using a code based on a re-
formulation of the self-consistent field method of computing the equilibrium stellar structure for a specified
conservative internal rotation law. The code has recently been upgraded with the addition of new opacity, equa-
tion of state, and energy generation routines, and a mixing-length treatment of convection in the outer layers of
the stellar interior. Relative to nonrotating stars of the same mass, these models all have reduced luminosities
and effective temperatures, and flattened photospheric shapes (i.e., decreased polar radii) with equatorial radii
that can be larger or smaller, depending on the degree of differential rotation. For a fixed ratio of the axial
rotation rate to the surface equatorial rotation rate, increasingly rapid rotation generally deepens convective
envelopes, shrinks convective cores, and can lead to the presence of a convective core (envelope) in a 1 M⊙
(2 M⊙) model, a feature that is absent in a nonrotating star of the same mass. The positions of differentially
rotating models for a given mass M in the H-R diagram can be shifted in such a way as to approximate the non-
rotating ZAMS over ranges in luminosity and effective temperature that correspond to a mass interval between
M and about 0.7 M. We briefly note a few of the implications of these results, including (i) possible ambiguities
arising from similarities between the properties of rotating and nonrotating models of different masses, (ii) a
reduced radiative luminosity for a young, rapidly rotating Sun, (iii) the nuclear destruction of lithium and other
light metallic species in the layers beneath an outer convective envelope, and (iv), the excitation of solar-like
oscillations and the operation of a solar-like hydromagnetic dynamo in some 1.5 − 2 M⊙ stars.
Subject headings: stars: interiors — stars: rotation
1. INTRODUCTION
Rotation is a universal stellar physical attribute. This con-
clusion is supported by an enormous body of observational
data, accrued over a period of nearly 100 years, from which
information about rotational speeds and periods, the depen-
dence of these quantities on parameters such as mass and
age (i.e., evolutionary state), and the effects of rotation on
the shape, effective temperature, chemical homogeneity, and
other basic properties of stars have been inferred. Despite in-
creasing evidence that rotation can have a significant impact
on a variety of stellar characteristics, it is generally not in-
cluded as a component of the structural/evolutionary models
which are the primary tools for the interpretation of observa-
tions. This omission is, in part, a consequence of the increased
complexity of the problem of determining the structure and
evolution of a rapidly, differentially rotating star: a one-
dimensional model necessarily becomes two-dimensional, the
gravitational potential must be derived by solution of Pois-
son’s equation, and the uncertain physics of convective and
circulatory flows, along with other rotation-dependent mecha-
nisms that contribute to angular momentum redistribution and
chemical mixing, needs to be addressed.
Some progress toward the development of a straightforward
yet robust technique for computing the internal structure of
a rotating star has recently been made with the implementa-
tion of a new version of the self-consistent field (SCF) method
(Jackson, MacGregor, & Skumanich 2005, hereafter Paper I).
In its original form (Ostriker & Mark 1968; see also Jack-
son 1970), the SCF approach to computing a model for an
∗The National Center for Atmospheric Research (NCAR) is sponsored by the
National Science Foundation.
assumed conservative law of rotation consisted of two sep-
arate steps: (i) determination of the gravitational potential
for a given distribution of the mass density ρ, followed by
(ii) solution of the equations of stellar structure with the po-
tential from (i) to update the equilibrium distributions of ρ
and other quantities. The process was initiated with a trial
distribution for ρ, and the two steps were executed sequen-
tially and iterated upon until the densities from (i) and (ii)
agreed to within a specified tolerance. While the SCF method
was successfully used to construct detailed models of rotating
upper-main-sequence stars (Bodenheimer 1971), it failed to
converge for objects less massive than about 9 M⊙, behavior
which precluded its application to intermediate- and low-mass
stars (see, e.g., Chambers 1976; Clement 1978, 1979; Paper
As described in Paper I (see also §2 below), the refor-
mulated SCF method circumvents the difficulties responsi-
ble for nonconvergence in lower-mass, more centrally con-
densed stars with an approach that entails the specification of
trial functions for the pressure P, the temperature T , and the
shape of constant-density surfaces, together with iterative ad-
justment of both the profiles and central values of P and T .
With these modifications, the method is capable of produc-
ing self-consistent models for rotating main-sequence stars of
all masses. It has been validated through detailed compar-
isons with stellar models (both rotating and nonrotating) for
masses ≥ 2 M⊙ computed by other investigators using alter-
native techniques (Paper I), and has been applied in an exam-
ination of the photospheric shape of the Be star Achernar, an
object revealed by interferometric observations as highly flat-
tened by rapid rotation (Jackson, MacGregor, & Skumanich
2004, and references therein). More recently, the structure
http://arxiv.org/abs/0704.1275v1
2 MACGREGOR ET AL.
code based on the method has undergone considerable reno-
vation, with the replacement of routines for the equation of
state, nuclear energy generation rates, and opacities, and the
addition of a mixing length treatment of convection. It is now
equipped for use in an investigation of the effects of rotation
on the structure and properties of stars with masses ≤ 2 M⊙.
In the present paper, we use the formalism described above
to conduct a detailed survey of the structural characteristics
of differentially rotating stars having masses M in the range
1 ≤ M ≤ 2 M⊙. In the absence of rotation, this mass inter-
val encompasses considerable variation in internal properties,
with the gross structure of objects at the lower limit consist-
ing of an inner radiative core and an outer convective enve-
lope, changing to a convective core and radiative envelope for
objects at the upper limit. The influence of rotation on the
basic morphology of stellar interiors for these masses has not
received much attention, with rotation-related effects usually
treated as perturbations to the nonrotating structure, if at all
(see, e.g., Thompson et al. 2003). However, observations in-
dicate the occurrence of surface rotation speeds rapid enough
to imply non-negligible modifications to many stellar proper-
ties, particularly if the rotation is differential: the projected
rotation speeds of 2 M⊙ main-sequence stars are typically in
excess of 100 km s−1, and comparable values have been mea-
sured for near zero-age main-sequence (ZAMS) 1 M⊙ stars in
young clusters (Stauffer 1991; Wolff & Simon 1997; Tassoul
2000, and references therein).
Motivated by these considerations, we have constructed an
extensive set of self-consistent models for 1 − 2 M⊙ ZAMS
stars, to systematically study the dependence of stellar char-
acteristics on the rate and degree of differential rotation. Our
SCF models extend previous computational results for stars
in this mass range, some of which were obtained for uni-
formly rotating configurations (Faulkner, Roxburgh, & Strit-
matter 1968; Sackmann 1970; Kippenhahn & Thomas 1970;
Papaloizou & Whelan 1973; Roxburgh 2004), while others
were obtained using either approximate methods for noncon-
servative differential rotation (Endal & Sofia 1981; Pinson-
neault et al. 1989; Eggenberger, Maeder, & Meynet 2005) or
a non-SCF, finite-difference technique in the case of conser-
vative rotation (Clement 1979). In §2 we provide a synopsis
of the new SCF method, briefly describing its implementa-
tion in a code for computing the structure of chemically ho-
mogeneous, differentially rotating main sequence stars, the
improvements and extensions to the input physics that have
been made since Paper I, and the results of tests of code reli-
ability through comparisons with extant models for nonrotat-
ing ZAMS stars with masses ≤ 2 M⊙. An examination of the
properties of models spanning a wide range of internal rota-
tion characteristics is presented in §3, with particular attention
paid to the behavior of such quantitites as luminosity and ef-
fective temperature, central thermodynamic properties, the lo-
cation and extent of convective regions, and the size and shape
of the stellar photosphere. Among the consequences of these
rotation-induced changes in stellar properties is a shift in the
position of objects in the classical H-R diagram (HRD); in-
deed, for some adopted rotation laws, the resulting structural
modifications can enable a differentially rotating star to have
the same Te f f and L values as a non-rotating star of signifi-
cantly lower mass and to thereby occupy the same position in
the HRD. In the concluding section of the paper (§4), we use
the model results to consider how the changes brought on by
rapid differential rotation might affect other features of main-
sequence stars in this mass range, such as the radiative lumi-
nosity of the young Sun and associated effects on the planets,
the abundance of lithium in the surface layers of mid-F dwarf
stars, and the presence of strong, large-scale magnetic fields
in some A stars.
2. THE NEW SCF METHOD
2.1. Implementation
The new version of the SCF method (described in detail in
Paper I) is an iterative scheme that is initialized by specify-
ing (i) a pair of one-dimensional trial functions, one for the
temperature distribution and one for the pressure distribution
(each normalized by its central value and defined over a spa-
tial range that is normalized by the equatorial radius of the
star), and (ii) a two-dimensional normalized function describ-
ing the shape of the equidensity surfaces. The normalized
two-dimensional trial density distribution, which is used as
the source term in Poisson’s equation for the gravitational po-
tential, follows from the equation of state and the trial func-
tions for P and T . For a conservative law of rotation, in which
the angular velocity depends only on the perpendicular dis-
tance from the axis of rotation, it is possible to define an effec-
tive potential from which both the gravitational and centrifu-
gal forces can be derived. The surfaces of constant effective
potential (i.e., level surfaces) can then be identified and used
to solve a set of ordinary differential equations analogous to
the usual equations of stellar structure for nonrotating stars.
This solution step yields updated temperature and pressure
distributions, allowing the iterative cycle to be repeated and
the process continued until convergence is achieved. When
the input and output functions representing the normalized
temperature and pressure profiles are in agreement, the two
parameters corresponding to the central temperature (Tc) and
central pressure (Pc) are adjusted by a Newton-Raphson tech-
nique to bring them closer to the actual physical conditions
at the center of the final equilibrium model. The entire pro-
cedure (consisting of an SCF loop nested inside a Newton-
Raphson loop) is repeated until an acceptable level of agree-
ment between the input and output values of the two central
parameters is attained. When used to construct models for
differentially rotating main-sequence stars, this reformulation
of the SCF method has been shown to converge for all masses
in the range 0.6 ≤ M ≤ 30 M⊙, and for values of the dimen-
sionless rotational kinetic energy t (the ratio of the rotational
kinetic energy to the absolute value of the gravitational po-
tential energy of the configuration) as high as 0.10–0.12 for
intermediate- and high-mass models, and up to nearly 0.26 for
some fully convective, highly flattened, disk-like, 1 M⊙ mod-
els. By comparison, the largest t values among the non-SCF
models computed by Clement (1978, 1979) were ≈ 0.18 for
30 M⊙ and ≤ 0.12 for models in the range 1.5 ≤ M ≤ 5M⊙,
whereas Bodenheimer (1971) obtained a 60 M⊙ SCF model
with t ≈ 0.24.
As in Paper I (see also Jackson, MacGregor, & Skumanich
2004), the internal rotation of each of the stellar models dis-
cussed in §3 is given by an angular velocity distribution of the
Ω(ϖ) =
αϖ/Re
, (1)
where ϖ = r sin θ, and Re is the equatorial radius of the star.
For a given model, the constants Ω0 and α are prescribed
parameters that characterize, respectively, the axial rotation
rate and the ratio of the surface equatorial rotation rate to the
axial rate (a measure of the degree of differential rotation),
DIFFERENTIALLY ROTATING STARS 3
Ωe/Ω0 = 1/(1 +α2). In practice, the value of the first of these
quantities is specified through the parameter η = Ω0/Ωcr, with
Ωcr the equatorial angular velocity for which the magnitudes
of the gravitational and centrifugal forces at Re are equal.
Rigorous answers to questions about the existence and
uniqueness of solutions to complicated systems of integro-
differential equations are generally very difficult to obtain.
Although specifying the two rotational parameters α and η
for chemically homogeneous models of fixed mass and com-
position does not always lead to a converged model, when it
does, our experience with the code suggests that the model is
unique. For a given set of (α,η) values, using the SCF code to
converge models of the same mass and chemical composition
from two or more different trial models always seems to lead
to the same final converged model. On the other hand, it can
be demonstrated that specifying the parametersα and Ω0 does
not, in general, lead to a unique solution. A minor drawback
to use of the parameters (α, η) in presenting the results is that
several important global properties, including the luminosity
L, the total angular momentum J, and the dimensionless rota-
tional kinetic energy t, are not monotonic functions of η when
α is held fixed (see also §3.1). Despite this shortcoming, we
feel that the apparent uniqueness of the models corresponding
to particular (α,η) values justifies the use of these quantities
in presenting and discussing results. We emphasize that se-
quences of models obtained by holding α fixed and varying η
should not be interpreted as any sort of evolutionary sequence.
The significance of models characterized by the same value of
α is that they have the same degree of differential rotation, that
is, the same Ωe/Ω0. The “half-width” of the rotation profile,
ϖ1/2 = Re/α, does, however, change from model to model for
constant α.
2.2. Input Physics
Since the publication of Paper I, the input physics for the
SCF code described therein has been updated considerably,
with the installation of software components that at various
times were parts of the stellar-evolution code developed by
Don VandenBerg at the University of Victoria. All of the
models presented in this paper were computed for the fol-
lowing abundances by weight of H, He, and heavy elements:
X = 0.7112, Y = 0.27, and Z = 0.0188. The opacities were ob-
tained, as in VandenBerg et al. (2000), from tables of OPAL
opacities calculated by Rogers & Iglesias (1992) and from ta-
bles of low-temperature opacities calculated by Alexander &
Ferguson (1994), using interpolation subroutines written by
VandenBerg (1983). Other subroutines written by Vanden-
Berg (1992) were utilized for the following: (i) the equation
of state formulated by Eggleton, Faulkner, & Flannery (1973,
EFF); (ii) nuclear energy generation rates for hydrogen burn-
ing from Caughlan & Fowler (1988), including the effect of
electron screening as treated by Graboske et al. (1973) for the
case of equilibrium abundances of CNO isotopes; and (iii) a
standard mixing-length treatment of surface convective zones
(see, e.g., Baker & Temesvary 1966; Kippenhahn, Weigert, &
Hofmeister 1967). We have made no attempt to incorporate
any of the direct effects of rotation into the adopted convec-
tion model; instead, we have simply modified the nonrotat-
ing mixing-length description of convection by replacing the
local gravitational acceleration, g, with the effective gravita-
tional acceleration, ge f f = g −Ω(ϖ)2ϖ eϖ (i.e., g as reduced
by the local centrifugal acceleration), averaged over equipo-
tential surfaces. For all of the models, a value of 1.9 for the
ratio of the mixing length to the pressure scale height has been
adopted.
2.3. Validation
In view of the revisions and updates that have been made
to the SCF code of Paper I, it seems worthwhile to make
a careful comparison of our nonrotating models for stars on
the lower main sequence with models for the same mass ob-
tained from a standard (nonrotating) stellar evolution code
For this purpose, we have used the current version of the
evolutionary code of Christensen-Dalsgaard (1982, hereafter
referred to as the JCD code) to generate two evenly spaced
sequences of seven models each, spanning the mass range
0.8 ≤ M ≤ 2.0 M⊙ along the ZAMS. These models have
chemical composition X = 0.711, Z = 0.019, quite close to
the abundances adopted for our SCF models, and were com-
puted using the same value (1.9) of the mixing length pa-
rameter. The basic EFF equation of state was used to con-
struct one of the sequences, while the other was derived with
the so-called CEFF equation of state, a modification of the
EFF treatment that includes Coulomb corrections (see, e.g.,
Christensen-Dalsgaard & Dappen 1992). Aside from the in-
clusion or omission of Coulomb effects in the equation of
state, the package of input-physics routines used to generate
the JCD models is very similar to that installed in the SCF
code (see, e.g., Christensen-Dalsgaard, Proffitt, & Thompson
1993; Di Mauro & Christensen-Dalsgaard 2001). In addition
to the intrinsic distinctions between the mathematical tech-
niques used to obtain the two sets of models, the current ver-
sion of the SCF code differs from the JCD code in the follow-
ing ways: (i) the equation of state does not include treatment
of Coulomb corrections; (ii) the photospheric pressure in the
SCF code follows from the application of a different, sim-
plified boundary condition (see Paper I); and (iii), there is a
slight difference in the vintage of the nuclear energy genera-
tion data.
We have conducted a quantitative comparison of some of
the important properties (R, L, Te f f , Pc, Tc, ρc) of SCF,
JCD/EFF, and JCD/CEFF models for nonrotating ZAMS stars
of the same mass. A theoretical HRD indicating the positions
of the various models is displayed in Figure 1. The EFF and
CEFF main sequences (the dashed and dotted lines, respec-
tively) are very nearly coincident, with the former models
shifted shifted along the locus relative to the latter models,
toward somewhat lower luminosities and effective tempera-
tures. On the basis of the preceding discussion, we expect the
SCF models to agree better with EFF models than with CEFF
models, and inspection of the results plotted in Figure 1 in-
dicate that this is generally the case. For the luminosity, the
most sensitive of the stellar properties, the largest discrepan-
cies occur for the 0.8 M⊙ models, with the SCF L about 2%
lower than that of the EFF model of the same mass; this latter
value is, in turn, about 7% lower than L for the corresponding
CEFF model. The luminosities of the SCF and EFF mod-
els are essentially equal for 2 M⊙, while a comparison of the
EFF and CEFF models for that mass reveals that the CEFF
model is about 3% more luminous. The effective tempera-
tures and radii of the SCF models deviate from those of the
corresponding EFF models by less than 1%, except for those
models between 1.2 and 1.4 M⊙ where the the magnitudes of
the discrepancies are somewhat larger, ≈ 1% − 2%. The SCF
values for Pc and ρc differ from the EFF values of those quan-
tities by about 2%, with the relative difference in the Tc values
. 1%. There is also good qualitative agreement between the
SCF and JCD models with respect to the appearance or ab-
4 MACGREGOR ET AL.
FIG. 1.— A theoretical HR diagram showing the positions of nonrotating,
chemically homogeneous stellar models for the indicated masses, as com-
puted using different codes and equations of state. The various symbols de-
note models obtained using the SCF code of the present paper (∗), a current
version of the code of Christensen-Dalsgaard (1982, JCD) with the simpli-
fied equation of state of Eggleton, Faulkner, & Flannery (1973, EFF) (△),
and the JCD code with a version of the EFF equation of state that includes
Coulomb corrections (CEFF) (⊓⊔). The dashed and dotted lines are the com-
puted ZAMS locations derived from the JCD/EFF and JCD/CEFF models,
respectively.
sence of convective cores and envelopes, and there is good
quantitative agreement with respect to the radial extent and
the enclosed mass of the principal radiative-convective inter-
faces.
3. PROPERTIES OF DIFFERENTIALLY ROTATING STELLAR
MODELS
3.1. Convergence Properties
The properties of SCF models for rotating ZAMS stars with
masses 1 ≤ M ≤ 2 M⊙ are summarized in Figures 2-8 and
Table 1. To facilitate discussion of the computed stellar char-
acteristics, we adopt the convention, established in Paper I, of
identifying each model by its mass, M, and the two rotational
parameters, α and η, for the reasons discussed in §2. Regions
of the (α, η) parameter space in which converged 6 M⊙ SCF
models can and cannot be obtained have been illustrated in
Figure 3 of Paper I. While the (α, η) planes for the lower-
mass SCF models considered here closely resemble that for
the 6 M⊙ models, there are some important differences. Of
particular relevance to the present paper are the regions cor-
responding to Region II in Paper I, regions of relatively high
angular momentum wherein the SCF method is incapable of
producing converged models. These forbidden zones are con-
siderably expanded for the 1 and 2 M⊙ models: for the for-
mer models, the values (αt, ηt) corresponding to the lower
tip of this region are αt = 1.39, ηt ≈ 2.4, while for the latter,
αt = 2.83 and ηt ≈ 6.5. Along each constant-α sequence for
0 < α ≤ αt , quantities such as the axial angular velocity Ω0
and the total angular momentum J are non-monotonic func-
tions of η, increasing to a maximum for η ≈ ηt and decreas-
ing thereafter. As α→ αt , convergence in the vicinity of this
maximum becomes significantly slower, and fails completely
when α is large enough (i.e., > αt) to place the model within
the forbidden zone. For αt <α≤ 7 (the largest value of α we
have considered), converged models can be readily obtained
on the low-η sides of these regions. On the high-η sides,
converged models can be obtained only within the ranges
FIG. 2.— Selected properties of differentially rotating, 1 M⊙ , ZAMS stellar
models. The model characteristics are shown as functions of η for 0 ≤α≤ 5,
where the parameters α and η specify the assumed internal angular velocity
distribution given by equation (1). The quantities depicted in the various pan-
els include: (a) the surface equatorial rotation speed Ve; (b) the luminosity L
in units of L0, the luminosity of a nonrotating 1 M⊙ model; (c) the central
temperature Tc relative to the corresponding value Tc0 for the nonrotating
model; (d) the average effective temperature Te f f ; (e) the equatorial radius
Re in units of R0, the radius of the nonrotating model; (f) the maximum per-
pendicular distance Zmax from the equatorial plane to the photosphere (solid
curves), and the polar radius Rp (dotted curves), as fractions of Re; (g) the
radii of the base of the convective envelope rce and the convective core rcc,
both measured in the equatorial plane relative to Re; and, (h) the temperature
Tce at the base of the convective envelope.
1.39 ≤ α ≤ 3 for the 1 M⊙ models, and 2.83 ≤ α ≤ 3.47 for
the 2 M⊙ models. The physical, mathematical, and computa-
tional reasons for the lack of convergence of models on either
side of the forbidden zone are discussed in considerable de-
tail in Paper I. In this paper, we present results for complete
constant-α sequences of models having α < αt , but confine
our attention to just the low-η sides of forbidden zones for
models with α > αt . Models on the high-η sides of these re-
gions have highly rotationally flattened, disk-like structures
that will be the focus of a subsequent paper.
3.2. 1 M⊙ Models
In Figure 2, we show how some of the characteristics of the
1 M⊙ models depend on the dimensionless rotation parame-
ter η for 0 ≤ α ≤ 5. The relation between η and the physical
quantity Ve, the equatorial rotation speed at the stellar surface,
is given in panel (a), from which it can be seen that for each
DIFFERENTIALLY ROTATING STARS 5
TABLE 1. SELECTED MODELS
Model I II III IV V VI VII VIII IX X XI
M 0.8 1 1 1 1 1.2 1.2 1.6 2 2 2
α 0 0 1.5 3.75 5 0 4 0 0 3 4.75
η 0 0 1.55 3.58 4.15 0 3.74 0 0 5.64 5.9
t 0 0 0.044 0.079 0.052 0 0.065 0 0 0.047 0.078
J 0 0 6.44 7.27 5.49 0 8.93 0 0 23.71 23.73
Ve 0 0 219 123 81 0 112 0 0 205 127
L 0.224 0.653 0.332 0.224 0.336 1.591 0.653 6.075 15.47 7.560 6.083
Teff 4710 5540 4710 4700 5080 6110 5550 7490 9090 5890 7480
log gs 4.635 4.551 4.478 4.709 4.679 4.412 4.616 4.310 4.337 3.855 4.404
Re 0.714 0.878 0.934 0.744 0.767 1.129 0.904 1.466 1.590 2.927 1.512
Rp/Re 1 1 0.697 0.638 0.712 1 0.668 1 1 0.371 0.549
Zmax/Re 1 1 0.697 0.764 0.837 1 0.791 1 1 0.530 0.770
log Pc 17.078 17.178 17.100 16.967 17.019 17.259 17.101 17.326 17.287 17.317 17.282
log Tc 7.046 7.121 7.069 7.004 7.039 7.187 7.094 7.280 7.326 7.289 7.255
log ρc 1.892 1.918 1.893 1.823 1.841 1.935 1.869 1.911 1.827 1.893 1.892
ρc/ρ 25 40 35 17 20 73 29 114 96 446 83
rcc/Re . . . . . . . . . 0.138 0.138 0.046 0.090 0.094 0.122 0.054 0.110
mcc/M . . . . . . . . . 0.023 0.031 0.007 0.014 0.082 0.141 0.085 0.074
rce/Re 0.678 0.718 0.618 0.718 0.722 0.822 0.746 0.990 0.990 0.718 . . .
mce/M 0.921 0.968 0.905 0.935 0.950 0.997 0.972 1.000 1.000 0.995 . . .
log Tce 6.458 6.392 6.467 6.452 6.440 6.104 6.388 4.780 4.771 5.988 . . .
Fig. . . . . . . 3a 3b 3c . . . 3d . . . . . . 3e 3 f
NOTE. — Quantities listed (units in parentheses): total mass, M (M⊙); rotational parameters, α, η; dimensionless rotational kinetic energy,
t; total angular momentum, J (1050 g cm2 s−1); equatorial velocity, Ve (km s
−1); luminosity, L (L⊙); mean effective temperature, Teff (K);
mean surface gravity, gs (cm s
−2); equatorial radius, Re (R⊙); polar radius, Rp; maximum (normal) distance from the equatorial plane to the
surface of star, Zmax; central pressure, Pc (dyn cm
−2); central temperature, Tc (K); central density, ρc (g cm
−3); mean density, ρ; distance in the
equatorial plane from the center to the top of the convective core, rcc, and to the bottom of the convective envelope, rce; mass enclosed by the
upper bounding surface of the convective core, mcc., and by the lower bounding surface of the convective envelope, mce; and temperature at the
bottom of the convective envelope, Tce (K).
of the α sequences depicted, Ve increases monotonically with
η. Along the curves with α <αt in Figure 2, the plotted mod-
els span the range from η = 0 (i.e., nonrotating) to the value
η = 1 +α2 for which Ωe = Ωcr. Along the curves with α > αt ,
the last plotted model is located adjacent to the boundary of
the forbidden region in the (α,η) plane; for these differen-
tially rotating models, the centrifugal and gravitational forces
have nearly equal magnitudes in the core of the star. We note
that the degree of differential rotation increases with α, in the
sense that the configurations corresponding to larger values
of α have a greater difference between the axial and surface
equatorial rates of rotation (see §2.1).
As panel (b) of Figure 2 makes evident, the radiative lumi-
nosities of these models are diminished relative to the lumi-
nosity L0 of a nonrotating 1 M⊙ star. This is a well-known
consequence of including rotation in the determination of the
equilibrium stellar structure (see, e.g., Clement 1979; Boden-
heimer 1971). In the results shown in Figure 2, the reduc-
tion in L is larger for differentially rotating models than it
is for models that are uniformly or nearly uniformly rotat-
ing. A model with α = 0 rotating at the break-up rate (η = 1)
has L/L0 = 0.78, while an α = 2 model with η = 2.42 has
L/L0 = 0.15, a reduction of more than a factor of 6 from the
nonrotating value. Much of the reason for this behavior lies
in the effect of rotation on the thermodynamic conditions in
the deep, energy-producing regions of the stellar interior. For
these 1 M⊙ models, the contribution of the centrifugal force to
supporting material against gravity enables the star to emulate
an object of lower mass with correspondingly reduced values
of Pc, Tc, and ρc (e.g., Sackmann 1970). The results presented
in panel (c) illustrate the dependence of Tc on model rota-
tional properties; similar variations are found for both Pc and
ρc. For rigidly rotating configurations, this centrifugal sup-
port is largest in the outermost layers of the interior, which
contain only a small fraction of the stellar mass; in this case,
Pc, Tc, and ρc are little changed from the values appropri-
ate to a nonrotating star of the same mass. For the α = 0,
η = 1 model noted previously, Pc/Pc0 = 0.94, Tc/Tc0 = 0.96,
and ρc/ρc0 = 0.98, where the subscript 0 indicates the nonro-
tating value. Alternatively, in models for higher values of α,
the effects of rotation are increasingly concentrated toward the
central regions of the star, with the result that the perturbations
to the central thermodynamic quantities can be more substan-
tial; for α = 2, η = 2.42, Pc/Pc0 = 0.56, Tc/Tc0 = 0.68, and
ρc/ρc0 = 0.81. Panels (b) and (c) also indicate that the mag-
nitudes of the changes in L, Tc, and other quantities depend
on the assumed profile of internal differential rotation. The
model for α = 5, η = 4 has L/L0 = 0.54, with Pc/Pc0 = 0.72,
Tc/Tc0 = 0.84, and ρc/ρc0 = 0.85, smaller reductions relative
to the nonrotating model than those for α = 2, η = 2.42. This
behavior is an outgrowth of the structural modifications aris-
ing from the centrifugal force distributions associated with the
different rotation profiles. In the α = 5 model, the ratio of
the centrifugal to gravitational force in the equatorial plane,
2r/g (r is the radial coordinate in the equatorial plane), is
sharply peaked in the innermost portion of the stellar core,
with maximum value 0.83 at the center, decreasing to ≈ 0.1
at r/Re = 0.3. For the shallower angular velocity profile of the
α = 2 model, the force ratio decreases from a smaller central
value of 0.42 to 0.23 at the stellar surface, 10 times the value
found throughout the outer 50% of the interior of the α = 5
model.
6 MACGREGOR ET AL.
In this connection, we note that Bodenheimer (1971) found
that the luminosities of models for rotating 30 M⊙ stars, com-
puted assuming a variety of internal rotation laws, depended
primarily on the total angular momentum content of a given
stellar model and not on the details of its distribution within
the interior. Specifically, his results indicated that the lumi-
nosities of models corresponding to four different prescribed
distributions of the angular momentum per unit mass de-
creased with increasing total angular momentum J, with the
relation between L and J nearly the same for each model se-
quence. If the luminosities of our 1 M⊙ models are plotted
versus their respective J values, an analogous reduction in L
for increasing J can be discerned. However, the relation be-
tween L and J is roughly independent of α only for small J,
and exhibits a clear dependence on α that becomes increas-
ingly pronounced as J is made larger. That the luminosity can
be even approximately expressed as a function of J necessar-
ily reflects the modifications produced by rotation to the ther-
modynamic conditions in the energy-producing core of the
star, as described previously (see also Mark 1968). The na-
ture and origin of the relation between L and J for stars with
masses 1 − 2 M⊙ and higher, including its dependence on the
structural characteristics of the models, will be addressed in
detail in a subsequent paper in this series.
As in Paper I, we define an average effective temperature
through the relation Te f f =
, where σ is the Stefan-
Boltzmann constant and A the area of the stellar surface. The
results for Te f f depicted in panel (d) exhibit dependences on α
and η that are similar to those seen for the quantities plotted in
the preceding panels. Reductions by more than 1500 K from
the nonrotating value (Te f f = 5540 K) are possible as, for ex-
ample, in the case α = 1.5, η = 2.18, for which Te f f = 3880 K.
The rotation-induced variations in Te f f represent the com-
bined effects of changes in both L and A. An indication as
to the behavior of A can be gleaned from examination of the
influence of rotation on the stellar size and shape. This in-
formation is presented in panels (e) and (f), where we show,
respectively, the equatorial radius Re (in units of the radius
R0 of the nonrotating, spherical 1 M⊙ model), and the polar
radius Rp together with Zmax, the maximum perpendicular dis-
tance from the equatorial plane to the stellar surface. These
latter two quantities are both given as fractions of Re; values of
Rp/Re that are < 1 reflect a rotational flattening of the config-
uration, while values of Zmax/Re that are 6= Rp/Re are indica-
tive of a deviation from a convex spheroidal shape through the
development of a concavity in each of the two polar regions
of the star.
Note that the response of the equatorial radius to increasing
η differs greatly depending upon whetherα is . 1.5 or & 2. In
the former case, the centrifugal force attains its largest value
relative to the gravitational force at the stellar surface, conse-
quently producing a distension of the outer, equatorial layers
of the stellar interior and an overall increase in Re. Although
Rp decreases somewhat relative to the radius of the corre-
sponding nonrotating model, the net effect of the changes in
Re and Rp is usually an increase in the volume of the rotating
star. For some models with α ≈ αt , the decrease in Rp can
more than compensate for the the increase in Re, and the vol-
ume of the rotating star is reduced. In the case where α & 2,
both Re and the stellar volume shrink with increasing η along
a constant-α sequence. For these models, the centrifugal force
is largest in comparison to the gravitational force in the central
region of the stellar core, causing the central thermodynamic
conditions, luminosity, and size (i.e, radius, surface area, vol-
ume) to assume values that are characteristic of nonrotating
stars of lower mass. This raises the possibility that a differen-
tially rotating star can imitate a less massive nonrotating star
in radius and effective temperature, as well as in luminosity.
In panel (f), it can be seen that for α . 1.5, the curves for
Zmax/Re (solid lines) and Rp/Re (dotted lines) are coincident
and< 1, implying that the photospheric shape of these models
is oblate spheroidal. Such a model (α = 1.5, η = 1.55) is de-
picted in panel (a) of Figure 3. For α> 2, however, Zmax >Rp,
symptomatic of the development of a “dimple” or indentation
at either pole, as in the cases of the models shown in panels
(b) (α = 3.75, η = 3.58) and (c) (α = 5, η = 4.15) of Figure 3.
Panels (g) and (h) of Figure 2 contain results pertaining to
the location, extent, and properties of convective regions in
the models. In the absence of rotation, the internal structure
of a 1 M⊙ star consists of an inner, radiative core that encom-
passes ≈ 72% of the stellar radius and contains ≈ 97% of the
stellar mass, surrounded by an outer, convective envelope. As
is apparent in panel (g), the inclusion of rotational effects can
modify this basic morphological picture in two ways: either
by increasing the size of the convective envelope or by pro-
moting the formation of a convective core.
For lower values of α (i.e., α < 3 in panel [g]), the rota-
tionally induced changes in internal structure cause the outer
convection zone to deepen as η increases. For example, in
the model with α = 1.5, η = 1.55 (see panel [a] in Figure 3),
the radius in the equatorial plane of the base of the convective
envelope is rce/Re = 0.618, significantly deeper than the the
base radius rce/Re = 0.718 in the nonrotating model. Associ-
ated with this reduction in rce is a decrease in the mass of the
core (mc/M ≈ 0.90), and an increase in the temperature Tce
at the bottom of the envelope, as can been seen in panel (h);
for the α = 1.5, η = 1.55 model, Tce = 2.93× 10
6 K, as com-
pared with Tce = 2.46×10
6 K for the nonrotating model. Since
some chemical species (e.g., Li, Be, and B) can be destroyed
by thermonuclear reactions at temperatures& 2.5×106 K, en-
hancements of Tce of this magnitude are likely to have conse-
quences for the surface abundances of these elements. In the
case of the strongly differentially rotating models with α≥ 3,
the largest centrifugal effects are concentrated in the inner-
most portion of the core, so that the fractional thickness of
the outer envelope is little affected. However, as the magni-
tude of the centrifugal-to-gravitational force ratio in the core
increases for larger η, the decreasing pressure gradient im-
plied by the requirement of hydrostatic equilibrium forces the
otherwise stably stratified central regions of the interior to be-
come convective. Under these conditions, with the presence
of a convective core, the structure of the deep interior resem-
bles that of a higher-mass star. The models shown in panels
(b) and (c) of Figure 3 are examples of 1 M⊙ stars with con-
vective cores; in each of these models, the radial extent of this
region in the equatorial plane is about 14% of Re and contains
≈ 2 − 3% of the stellar mass. For comparison, the convective
core in a model for a nonrotating 2 M⊙ star has a radius that
is about 12% of Re and contains ≈ 14% of the stellar mass.
To illustrate the structural differences between differen-
tially and near-uniformly rotating stars, in Figure 4 we show
the profiles of several physical quantities for models with
(α,η) = (3.00,3.26), and (1.00,1.20), along with the corre-
sponding results for the nonrotating model. The luminosi-
ties and total angular momenta of the two rotating models
are L/L0 = 0.201 and J = 9.23 (α = 3), and L/L0 = 0.634 and
J = 5.30 (α = 1), respectively, where J is meaasured in units
DIFFERENTIALLY ROTATING STARS 7
FIG. 3.— Contours of level surfaces in the meridional plane for some of the nonspherical models listed in Table 1. The six rotating models shown are defined
by the total mass and the two rotational parameters (M, α, η) as follows: (a) 1 M⊙ , 1.5, 1.55; (b) 1 M⊙ , 3.75, 3.58; (c) 1 M⊙, 5, 4.15; (d) 1.2 M⊙, 4, 3.74; (e)
2 M⊙, 3, 5.64; and, (f) 2 M⊙ , 4.75, 5.9. From the surface inward, the level surfaces depicted in each panel enclose a fraction of the total mass equal to 1.000,
0.995, 0.950, and 0.500, respectively. The fractional radii in the equatorial plane of these level surfaces for the various models are: (a) 1.00, 0.88, 0.71, 0.31; (b)
1.00, 0.90, 0.75, 0.40; (c) 1.00, 0.89, 0.72, 0.37; (d) 1.00, 0.87, 0.68, 0.34; (e) 1.00, 0.71, 0.37, 0.12; and, (f) 1.00, 0.66, 0.47, 0.24. Radiative portions of the
interior are indicated in white, and convective regions are shaded gray. The fractional equatorial radii and enclosed masses for the interfaces between radiative
and convective zones in the models are listed in Table 1. The numbers at the tops of the panels denote the total mass M and equatorial radius Re of each model.
of 1050 g cm2 s−1. In panels (a) and (b), the temperature T
and mass density ρ are depicted as functions of the radial po-
sition r (measured in units of the present-day solar radius R⊙)
in the equatorial plane of the star. The central values of both
quantities exhibit rotation-induced reductions relative to the
nonrotating case, the magnitudes of these modifications being
larger for α = 3 (Tc/Tc0 = 0.68, ρc/ρc0 = 0.76) than for α = 1
(Tc/Tc0 = 0.92, ρc/ρc0 = 0.97). As noted previously, this be-
havior is a consequence of differences in the magnitudes and
distributions of the centrifugal force in the two rotating mod-
els. These distinctions can be clearly seen in the profiles of the
centrifugal-to-gravitational force ratio shown in panel (d). For
α = 3, the force ratio is largest in the deep interior, attaining
a maximum value of 0.96 at the stellar center and decreas-
ing outwards to a magnitude ≈ 0.1 in the photosphere. For
α = 1, the ratio increases monotonically throughout the inte-
rior, rising from a central value of just 0.026 to a maximum
of 0.36 at Re. The substantial contribution of the centrifu-
gal force to the support of the innermost regions of the α = 3
model is responsible for the considerable enhancement of the
density scale height there, evident in panel (b); ρ declines by
only ≈ 10% over the inner 20% of the stellar interior. The
resulting changes in the internal mass distribution (panel [c])
lead to a star with a smaller radius, Re = 0.72 R⊙ as opposed
to Re = 0.88 R⊙ for the nonrotating model. Alternatively, for
the α = 1 model, the lack of centrifugal support in the core
of the star leads to temperature, density and mass distribu-
tions therein that closely resemble those of the nonrotating
model. Closer to the surface, however, the density distribu-
tion becomes extended, a product of the increasing centrifu-
gal reduction of gravity in the outer layers of the interior; as
a result, the stellar radius is larger than that of the nonrotating
model, Re = 1.00 R⊙.
3.3. Solar Look-Alike Models
Figure 5 is a theoretical HRD for 1 M⊙ models that span a
broad range of internal rotational states, from uniform rotation
to extreme differential rotation (up to Ω0 = 50 Ωe for α = 7).
The nonrotating ZAMS is delineated by a dotted line, with the
positions of several specific models for 0.6≤ M ≤ 1.0 M⊙ in-
dicated. Clearly, as already implied by panels (b) and (c) of
Figure 2, the locations of rotating models in such a diagram
are displaced to the right of and below the position they would
occupy in the absence of rotation, toward lower values of both
the luminosity and the effective temperature. Models with
uniform or near-uniform internal rotation (i.e., α = 0, 1) lie
8 MACGREGOR ET AL.
FIG. 4.— Profiles of selected physical quantities in the interiors of 1 M⊙
models with (α,η) = (3.00,3.26) (solid curves), (1.00,1.20) (dashed curves),
and (0.00,0.00) (dotted curves). The profiles depict the dependence of each
quantity on radial position r in the equatorial plane, from the center of the
star to the surface. The various panels show (a) the temperature T , (b) the
mass density ρ, (c) the fraction m/M of the total mass contained within a
level surface that intersects the equatorial plane at r, and (d) the ratio (Ω2r/g)
of the centrifugal force to the gravitational force.
well above the nonrotating ZAMS, while those for which the
degree of differential rotation is substantial (α ≥ 3) have po-
sitions in the HRD that collectively approximate the nonrotat-
ing ZAMS over virtually the entire mass range depicted. For
example, the rotating 1 M⊙ model (α = 3.75, η = 3.58) shown
in panel (b) of Figure 3 is characterized by Te f f = 4700 K
and L/L⊙ = 0.224, which is indistinguishable from the val-
ues Te f f = 4710 K, L/L⊙ = 0.224 for a nonrotating, 0.8 M⊙
model. Such a coincidence of the positions of stars of differ-
ent mass in the classical HRD is made possible by the fact
that the largest structural changes take place in the cores of
strongly differentially rotating models. Specifically, the de-
crease in L brought about by the smaller values of Pc, Tc, and
ρc, together with the corresponding reduction in Re, enable
such a model to effectively mimic a lower mass stellar model
in which rotational effects are not included.
Figure 6 illustrates some of the internal properties of three
so-called solar look-alike models, that is, models for differ-
FIG. 5.— A theoretical HR diagram showing the positions of models for
rotating, ZAMS, 1 M⊙ stars. Luminosities are given in units of the luminos-
ity L0 = 0.653 L⊙ of a non-rotating, 1 M⊙ star. The various symbols denote
models constructed using the rotation law of equation (1) with the values of
the parameters (α,η) listed in the Figure. The ZAMS for non-rotating stars is
indicated by the dotted line, with the positions of models for masses 0.6, 0.7,
0.8, 0.9, and 1.0 along it marked by an ∗ symbol.
entially rotating stars with masses > 1 M⊙ that have many
physical attributes in common with the model for a nonrotat-
ing, 1 M⊙ star. Inspection of panels (a) and (b) reveals that
the profiles and the central values of the temperature and den-
sity for the 1.1, 1.2, and 1.3 M⊙ models depicted therein are
quite close to those of the solar-mass model with α = η = 0.
A consequence of these structural similarities is that many
of the general characteristics of the models are nearly iden-
tical. For example, the 1.2 M⊙ model which, in the absence
of rotation, would have L/L⊙ = 1.591, Re/R⊙ = 1.129, and
Te f f = 6110 K instead has L/L⊙ = 0.653, Re/R⊙ = 0.904, and
Te f f = 5546 K, compared with L/L⊙ = 0.653, Re/R⊙ = 0.878,
and Te f f = 5545 K for the nonrotating, 1 M⊙ model. A cross-
section in the meridional plane of the rotating 1.2 M⊙ inte-
rior is shown in panel (d) of Figure 3, from which it can be
seen that a solar-like convection zone having rce/Re = 0.746
is present in the layers beneath the photosphere. With a sur-
face equatorial rotation speed Ve = 112 km s
−1, such an object
could be mistakenly identified as a rapidly rotating 1 M⊙ star,
if observations were analyzed through comparison with non-
rotating stellar models. That the values of measureable or
inferrable stellar properties can be the same in rotating and
nonrotating models for different masses represents a potential
source of ambiguity in the interpretation of a variety of obser-
vations. Panels (c) and (d) reveal subtle differences in the
internal mass distributions and in the variation of the ratios of
the centrifugal to gravitational force in the equatorial planes
of these solar look-alike models. Future space-based astero-
seismological observations may be capable of exploiting such
differences in internal structure to distinguish slowly rotating
stars from more rapidly, differentially rotating higher-mass
stars that happen to have the same values of L and Te f f and
thus occupy the same position in the HRD (see, e.g., Lochard
et al. 2005).
3.4. 2 M⊙ Models
In Figure 7, we present a summary of the rotational depen-
dence of 2 M⊙ model characteristics, using the same format as
adopted for Figure 2. The model for a non-rotating star of this
DIFFERENTIALLY ROTATING STARS 9
FIG. 6.— Profiles in the equatorial plane of the temperature (a), den-
sity (b), mass fraction (c), and the ratio of the centrifugal to gravitational
acceleration (d) for a nonrotating, 1 M⊙ model (α = η = 0) and three so-
lar look-alike models. The latter models were obtained for (M,α,η) =
(1.1 M⊙,4.00,2.74), (1.2 M⊙,4.00,3.74) and (1.3 M⊙,4.00,4.39), and have
values of Te f f and L that differ from those of the non-rotating, solar model
by about 1% or less.
mass has L0 = 15.47 L⊙, R0 = 1.59 R⊙, and Te f f = 9090 K, and
a convective core with rcc/Re = 0.122, mcc/M = 0.141; such a
model has a thin, subsurface convective layer, rce/Re = 0.99,
that contains a negligible fraction of the stellar mass. The re-
sults shown in Figure 7, which span the rotational parameter
ranges 0 ≤α≤ 5, 0 ≤ η≤ 7, exhibit many of the same behav-
iors as noted previously in connection with the 1 M⊙ models
of Figure 2. In particular, along each constant-α sequence, the
luminosity (panel [b]), central temperature, (panel [c]) and ef-
fective temperature (panel [d]) all initially decrease as η is in-
creased, with the largest reductions in these quantities occur-
ring in the cases of differentially rotating models with α≥ 3.
For α = 1, 2, both L/L0 and Tc/Tc0 pass through minima and
then increase slightly with η beyond those points, whereas
Te f f continues to decrease with η for all α sequences. The
central pressure Pc (panel [e]), on the other hand, is an increas-
ing function of η for 0 ≤ α≤ 3, and displays a non-montonic
dependence on η for the model sequences corresponding to
α = 4, 5. The origin of this variation is the mass dependence
FIG. 7.— Selected properties of differentially rotating, 2 M⊙ , ZAMS stellar
models, as in Figure 2. The quantities depicted in the various panels include:
(a) Ve, (b) L, (c) Tc, (d) Te f f , (e) Pc, (f) Re, (g) Zmax and Rp, and (h) rce and
of Pc for stars on the non-rotating ZAMS. In our SCF models
without rotation, Pc has a maximum value for M near 1.6 M⊙,
and is decreasing for both larger and smaller M values. The
maximum occurs for a mass close to the value marking the in-
ternal structural transition between stars with radiative cores
and convective envelopes and stars with convective cores and
radiative envelopes. Hence, insofar as the properties of a ro-
tating star are like those of a non-rotating star of lower mass,
we expect the Pc values of these 2 M⊙ models to increase as
the effects of rotation become more pronounced. For cases in
which the central thermodynamic conditions are significantly
perturbed by rapid, differential rotation in the core of the star,
this mass-lowering effect can be large enough to shift Pc to
values characteristic of nonrotating stars with M < 1.6 M⊙
(i.e., on the other side of the central pressure peak), as in the
2 M⊙ models for α = 4, 5.
Panels (f) and (g) of Figure 7 convey information pertain-
ing to the photospheric sizes and shapes of the models. Those
that rotate nearly rigidly are equatorially distended, and, in
the direction perpendicular to the equatorial plane, have their
largest dimension along the rotation axis (Zmax = Rp); since
Rp < Re, these models (like their 1 M⊙ counterparts) have a
flattened, spheroidal shape. Models with increasing degrees
of differential rotation (i.e., with α & 3) develop polar con-
cavities (as indicated by Zmax > Rp), and ultimately (i.e., for
10 MACGREGOR ET AL.
α = 4, 5 and sufficiently large η) become more compact, with
Re < R0. Panels (e) and (f) of Figure 3 give cross-sectional
representations of the photospheric shapes of 2 M⊙ mod-
els for (α, η) = (3.00, 5.64) and 4.75, 5.90), repectively; in
the former model, Re = 2.93 R⊙ > R0, while for the latter,
Re = 1.51 R⊙ < R0.
The influence of rotation on the occurrence and extent of
convective regions within the models is explored in panel (h)
of Figure 7. The convective core, a salient feature of the
non-rotating stellar interior, generally decreases in size as the
value of η is increased. This contraction stems from a re-
duction in the magnitude of the radiative gradient ∇rad in the
core region of the star, a result of the way in which the cen-
tral thermodynamic conditions are modified by rotation (see,
e.g., MacGregor & Gilliland 1986); with ∇rad smaller, the
size of the region wherein it exceeds the adiabatic gradient
∇ad shrinks accordingly. Only for the most rapidly, differen-
tially rotating models does rcc show a modest increase com-
pared to the non-rotating model; in panel [h], the model for
(α, η) = (5.00,6.88) has rcc/Re = 0.124. In this case, as was
seen for the 1 M⊙ models having convective cores in Fig-
ure 2, the growth of rcc can be traced to a larger value of
∇rad , produced by a centrifugal-to-gravitational force ratio
that is nearly unity at the stellar center. Note also that for
some 2 M⊙ models, the convection zone underlying the stel-
lar photosphere can extend into the stellar interior by more
than the few 10−3 R0 that is the thickness of this region in the
absence of rotation. As can be seen, for example, in panel
(e) of Figure 3, the base of the convective envelope in the
equatorial plane of the model with (α, η) = (3.00,5.64) is lo-
cated at rce/Re = 0.718, a fractional depth which is the same
as in the non-rotating 1 M⊙ model. This leads to the intrigu-
ing possibility that solar-like oscillations, driven by turbulent
convection, may be excited in stars that would normally be too
massive to generate them. The results shown in panel (h) sug-
gest that such a solar-like convective envelope is most likely to
occur in 2 M⊙ models with intermediate differential rotation
(say., α ≈ αt = 2.83). For a profile of this kind, the inner and
outer portions of the interior can each rotate rapidly enough
to both significantly perturb the thermodynamic conditions in
the core and increase Re by extending the stellar envelope.
In Figure 8, we show the positions of models for differen-
tially rotating, 2 M⊙ stars in a theoretical HRD. As in Figure
5, the location of the non-rotating ZAMS is indicated by a dot-
ted line, with the positions of models for stars with masses in
the range 1.0 ≤M ≤ 2.0 M⊙ indicated along it. This HRD for
rotating 2 M⊙ models has a number of features in common
with the corresponding representation of 1 M⊙ properties.
Without exception, models are shifted from the non-rotating
location to new positions characterized by lower Te f f and L
values. For α . 3, these new positions lie to the right-hand
side of (i.e., above) the non-rotating ZAMS; for rotation that
is increasingly differential, however, the model locations ap-
proach the non-rotating ZAMS, moving just to the left-hand
side of (i.e., below) it for α = 5. For α = 4.75, the model
positions are distributed along the non-rotating ZAMS, de-
lineating its track in the HRD for masses greater than about
1.5 M⊙. This coincidence again raises the possibility that
a rapidly, differentially rotating 2 M⊙ star could effectively
masquerade as a non-rotating star of lower mass. As an ex-
ample, with L/L⊙ = 6.083, Re/R⊙ = 1.512, and Te f f = 7480 K
the model for (α, η) = (4.75,5.90) shown in panel (f) of Fig-
ure 3 closely resembles a non-rotating 1.6 M⊙ star, for which
L/L⊙ = 6.075, Re/R⊙ = 1.466, and Te f f = 7490 K.
FIG. 8.— A theoretical HR diagram showing the positions of models for
rotating, 2 M⊙ ZAMS stars, as in Figure 5. Luminosities are given in units of
the luminosity L0 = 15.468 L⊙ of a nonrotating, 2 M⊙ star. The ZAMS for
nonrotating stars is indicated by the dotted line, with the positions of models
for masses 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, and 2.0 M⊙ along it marked
by an ∗ symbol.
4. SUMMARY AND DISCUSSION
The SCF approach to computing the structure of rotating
stars is robust and efficient, capable of yielding converged
models of stars of all masses, with internal angular-velocity
distributions covering the range from uniform to extreme dif-
ferential rotation. The code described in Paper I has been up-
graded throught the introduction of opacities, energy gener-
ation rates, and an equation of state that are close to state-
of-the-art, and with the introduction of a mixing-length treat-
ment of convective energy transport. We have used this re-
vised code to conduct a systematic study of the properties of
models for rotating ZAMS stars with masses 1 ≤ M ≤ 2 M⊙,
assuming that the rotation of the stellar interior can be ad-
equately represented by the parameterized angular-velocity
profile given in equation (1).
Our results suggest that rotation affects virtually every char-
acteristic of the intermediate- and low-mass stars considered
herein, with the magnitude, and sometimes even the sense,
of the changes in fundamental attributes depending on both
the rate and degree of the differential rotation in the stellar
interior. For models that are uniformly or nearly uniformly
rotating, the centrifugal contribution to the hydrostatic sup-
port of the star is largest in the outer layers of the interior;
for models that are strongly differentially rotating, the cen-
trifugal force is largest compared to gravity in the core of the
star. Luminosities and effective temperatures are always di-
minished relative to nonrotating stars of the same mass, while
equatorial radii increase in uniformly rotating or moderately
differentially rotating models, and decrease in models where
the difference between the angular velocity at the center and
that at the equator is sufficiently large. As the degree of dif-
ferential rotation is increased, the photospheric shape of the
star changes from a convex surface closely approximating a
spheroid flattened along the rotation axis to a roughly oblate
surface with deepening polar concavities; in such cases, the
greatest perpendicular distance from the equatorial plane to
the surface occurs away from the axis of rotation, and is al-
ways less than the equatorial radius of the star. In 1 M⊙ mod-
els, the effects of rotation can either increase the thickness of
DIFFERENTIALLY ROTATING STARS 11
the outer convective envelope, or contribute to the formation
of a convective core. In rotating 2 M⊙ models, the size of the
convective core is diminished relative to that found in the ab-
sence of rotation, and an extensive, solar-like convection zone
can be present in the outer layers of what would otherwise be
a stable radiative envelope.
The extent to which the results of these computations accu-
rately depict the stellar structural modifications arising from
rotation depends on the validity of a number of simplifying
assumptions and approximations made in the course of devel-
oping the basic model, as well as on the inherent limitations
of the SCF method itself. Here, we briefly address the most
salient of the factors that might restrict the applicability of
these results, noting where improvements and extensions can
(or cannot) be made (see also Paper I).
1). The modified SCF iterative scheme utilized in the
present investigation necessarily requires the angular velocity
distribution within the star to be conservative, and thus ex-
pressible as a function of just the perpendicular distance from
the rotation axis, Ω = Ω(ϖ) (see eq. [1]). Although this speci-
fication facilitates the construction of models through the con-
siderable mathematical simplification it introduces, its effect
on some computed stellar structural characteristics, such as
the photospheric shapes and core physical properties of very
rapidly, differentially rotating models, is likely to differ from
that produced by a rotation law of (say) the form Ω = Ω(r).
There is some evidence, from both simulations and observa-
tions, for the occurrence of differential rotation of the type
given by equation (1) (see, e.g., Dobler, Stix, & Brandenburg
2006, and references therein). Yet analyses of solar acous-
tic oscillations yield a picture of the large-scale internal dy-
namics of the Sun which is not in accord with the angular
velocity being constant on cylindrical surfaces (see Thomp-
son et al. 2003). We note that use of the SCF method pre-
cludes consideration of non-conservative rotation laws (e.g.,
Ω = Ω(r)); however, detailed treatment of the hydrodynamical
and magnetohydrodynamical processes affecting the internal
rotational states of the Sun and stars is presently beyond the
scope of any of the extant structural/evolutionary models.
2). The model treats only the prescribed rotational motion
of the stellar interior, omitting any meridional circulatory flow
and its consequent effects on the internal angular momentum
distribution. For models with radiative envelopes, rotation-
ally driven circulation will cause the rotation profile to devi-
ate over time from the state given by equation (1), unless such
evolution is mitigated by additional angular momentum trans-
port mechansims (see, e.g., Maeder & Meynet 2000, and ref-
erences therein). For models with convective envelopes, the
rotation profile in the outer layers of the interior is the prod-
uct of the complex interplay between meridional circulation
and turbulent heat and angular momentum transport (Rempel
2005; Miesch, Brun, & Toomre 2006); whether Ω is solar-
like or constant on cylindrical surfaces depends on the lati-
tudinal entropy distribution in the subadiabatically stratified
layers below the convection zone.
3). The model employs a simplified treatment of con-
vection, locating unstable regions by application of the
Schwarzschild criterion and utilizing an averaged, rotationally
modified mixing-length description (see §2.2) to determine
the structure of an outer convective envelope, if present. Use
of the Solberg-Høiland condition (e.g., Ledoux 1965; Kip-
penhahn & Weigert 1990) to ascertain the onset of convective
instability would account for the direct influence of the cen-
trifugal force. Since models computed for the rotation law of
equation (1) have ∂ j/∂ϖ > 0, where j = Ωϖ2 is the specific
angular momentum (see Paper I), this change would likely de-
crease the equatorial-plane thickness of an outer convection
zone, and produce a latitudinal variation in rce. However, for
rapid rotation, there is considerable uncertainty regardless of
the convection criterion/model adopted, because of both the
rudimentary nature of extant treatments of convective energy
transport and the universal assumption of axisymmetry among
rotating stellar models.
We reiterate that the present model describes a chemi-
cally homogeneous, ZAMS star, and neglects effects associ-
ated with the structural and compositional evolution of the
stellar interior. We are presently developing a mean-field
hydrodynamics-based treatment of turbulent chemical and an-
gular momentum transport which maintains conservative ro-
tation profiles, thus making it possible to investigate the main-
sequence evolution of these SCF models. Despite the short-
comings enumerated above, we believe that the basic model
and method described herein compare favorably with alterna-
tive approaches to determining the structure and evolution of
rotating stars. The most widely used of these (e.g., Meynet
& Maeder 1997) relies on the use of approximate, Roche-
like equipotentials to represent the internal gravity of the star,
thereby strictly limiting it to describing slowly rotating stars
to ensure the accuracy of the computed models. Whatever
its drawbacks, the SCF method yields two-dimensional, ax-
isymmetric configurations that represent fully consistent so-
lutions to both the set of stellar structure equations and Pois-
son’s equation for the gravitational potential.
The results presented herein have a number of implications
for young stars with masses between 1 and 2 M⊙. Mea-
surements of projected equatorial rotation speeds in excess of
100 km s−1 for some solar-type stars in young open clusters
(see, e.g., Stauffer 1991) raise the possibility that the struc-
ture and properties of these objects could be significantly al-
tered from those of nonrotating stars of the same mass. Such
stars, provided that their interior rotation is sufficiently differ-
ential, could in actuality be somewhat more massive objects
for which surface rotation as rapid as that indicated by obser-
vations is the normally expected ZAMS state. The question of
whether or not the kind of strong differential rotation required
to produce such ambiguity is present within the interiors of
some low- and intermediate-mass stars may ultimately be re-
solved through space-based asteroseismological observations,
which should allow low-resolution inversions of the rotation
profiles in the inner ∼ 30% of the stellar radius (see Gough
& Kosovichev 1993). Asteroseismology from space may also
afford the means for identifying individual stars whose rota-
tion enables them to pose as lower-mass objects, since, when
effects associated with asphericity are not too large, the aver-
age mode frequency spacing is sensitive to the mean density,
a quantity which Table 1 reveals to be different for look-alike
models.
If rapid differential rotation is a possibility for objects in the
mass range spanned by the models of §3, then the associated
changes in structure and properties could have consequences
for a variety of important astrophysical processes that take
place within and around such stars. The reduced radiative
luminosity of a rapidly rotating young Sun would likely influ-
ence the evolution of the solar nebula, and further exacerbate
the discrepancies between the properties of nonrotating mod-
els and observational inferences indicating a higher ZAMS lu-
minosity (see, e.g., Sackmann & Boothroyd 2003). The rota-
tionally induced deepening of a sub-photospheric convection
12 MACGREGOR ET AL.
zone, together with the increase in temperature of the mate-
rial at the base, could contribute to the depletion of lithium
in the stellar surface layers by reducing the thickness of the
region through which chemical species must be transported
in order to be destroyed by nuclear processes. The formation
of a solar-like convective envelope in a young, differentially
rotating, 2 M⊙ star could excite global oscillations, and be
accompanied by the operation of a solar-like hydromagnetic
dynamo. Dynamo-generated fields that diffuse into and are
retained by the radiative interior (see, e.g., Dikpati, Gilman,
& MacGregor 2006) could enable the star to remain magnetic
long after spin-down and the elimination of nonuniform rota-
tion have led to the disappearance of both the surface convec-
tive layer and the dynamo. Each of these possibilities will be
addressed in forthcoming papers.
We wish to thank D. A. VandenBerg for providing soft-
ware from his stellar-evolution code that was used to han-
dle the input physics in our SCF code, and we wish to thank
J. Christensen-Dalsgaard for the use of his stellar-evolution
code to help us validate our models. This work was supported
in part by an Astronomy & Astrophysics Postdoctoral Fel-
lowship under award AST-0401441 (to T. S. M.) from the Na-
tional Science Foundation.
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