Shape of a Rotating Star

ASTR 472: Stellar Structure & Evolution Lecture 13

Shape of a Rotating Star

Rapid rotation can cause a star to bulge significantly near its equator. If we assume thatthe star is in hydrostatic equilibrium, we can derive the shape of a rotating star.

Consider the case of solid body rotation with angular velocity Ω constant throughout thestar, and assume that density ρ is constant. We will also assume the Roche model, whichassumes that the gravitational potential behaves like a point mass.

For hydrostatic equilibrium, the sum of the forces acting on a mass element must equal thecentripetal acceleration. Note the use of the vector ~ρ, which defines the component of theposition vector perpendicular to the rotation axis.

~F pressure + ~F gravity = ~F cent

−mρ~∇P − GMm

r2r = −mΩ2~ρ

The gravitational potential Φ is

Φ = −GMr

~∇φ =GM

r2r

∇2Φ = 4πGρ

The rotating potential V can be expressed in cylindrical coordinates (with the radial coor-dinate ~ρ, not the same as density):

V = −1

2Ω2|~ρ|2

~∇V = −Ω2 ~ρ

∇2V =1

ρ

∂

∂ρ

(ρ∂V

∂ρ

)=

1

ρ

∂

∂ρ

(−Ω2ρ2

)= −2Ω2

1


Since both gravity and rotation can be described as gradients of a potential, the combinedgravitational and rotational forces can be expressed as the gradient of a potential Ψ. Fromthis, we can relate the potential to the pressure gradient and find the effective surfacegravity:

~∇Ψ = ~∇φ+ ~∇V

=GM

r2r − Ω2~ρ

= −1

ρ~∇P

1

ρ~∇P = − ~∇Ψ = ~geff

This is the equation of hydrostatic equilibrium for solid body rotation. Note that thisrelationship implies that ~∇P and ~∇Ψ are parallel.

Taking the curl of this relationship (and noting that the curl of any gradient is always 0):

~∇× ~∇P︸︷︷︸ = ~∇×(−ρ ~∇Ψ

)= −ρ ~∇× ~∇Ψ︸︷︷︸− ~∇ρ× ~∇Ψ

0 = ~∇ρ× ~∇Ψ

Therefore ~∇ρ is parallel to both ~∇Ψ and ~∇P . This implies that the surfaces of constantdensity, pressure, and potential coincide, and the star is said to be barotropic. If we findthe shape corresponding to constant Ψ, then we have found the shape of the rotating star.

The combined gravitational and rotational potential is simply

Ψ = Φ + V

= −GMr− 1

2Ω2|~ρ|2

= −GMr− 1

2Ω2r2 sin2 θ

At the stellar poles, θ = 0 so

Ψ = −GMRpole

= const. over surface

We now have a function relating the stellar radius with latitude for a rotating star. Theshape of the surface is described by

−GMRpole

= −GMR− 1

2Ω2R2 sin2 θ

1

R(θ)=

1

Rpole

− Ω2R2

2GMsin2 θ

2


Critical Velocities

The classic definition of critical velocity is defined by equating the gravitational force to thecentrifugal force at the stellar equator.

GMm

R2eq

=mV 2

crit

Req

Vcrit =

√GM

Req

=

√2GM

3Rpole

Note that Vcrit is not valid for high mass stars with a high Eddington factor (very luminoussupergiants, Wolf-Rayet stars, and luminous blue variables) since the radiation pressure alsoaffects ~geff .

In terms of the critical angular velocity,

Vcrit = ΩcritReq

Ωcrit =Vcrit

Req

=2Vcrit

3Rpole

=

(2

3

)3/2(GM

R3pole

)1/2

Ω2crit =

8

27

GM

R3pole

Note that Rpole does not change much at all, even for a rapidly rotating star (see Figure 1).It is reasonable to use the approximation Rp,crit ≈ Rpole for the non-rotating case.

In general, Veq/Vcrit < Ω/Ωcrit (see Figure 2).

We can write the shape of the rotating barotropic star in terms of the critical rotation Ω/Ωcrit.Use the substitutions

ω =Ω

Ωcrit

x =R

Rp,crit

The polynomial describing the shape can be written in terms of the critical rotation:

1

R(θ)=

1

Rpole

− Ω2R2

2GMsin2 θ

Rp,crit

R=Rp,crit

Rpole

− Ω2critω

2R2Rp,crit

2GMsin2 θ

1

x=Rp,crit

Rpole

− 8

27

GM

R3pole

ω2x2R3p,crit

2GMsin2 θ

1

x=Rp,crit

Rpole

− 4

27ω2x2 sin2 θ

Notice that as Ω/Ωcrit → 1, the equatorial radius Req → 1.5 Rpole (see Figure 3).

3


Figure 1: Variations of Rpole with Ω for standard metallicity Z = 0.02 (Ekstrom et al. 2008,A&A, 478, 467).

Figure 2: Relationship between Veq/Vcrit and Ω/Ωcrit for the Roche model (Meynet 2008,arXiv:0801.2944v1). The solid line assumes that Rp,crit = Rpole. The dot-dashed lines showthe relations for Z = 0.02 models with 1 M and 60 M.

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Figure 3: Shape of a rotating star in the Roche model with several values of ω = Ω/Ωcrit.This plot assumes Rp,crit = Rpole.

Surface Gravity of a Rotating Star

We have shown that the effective potential of the rotating star is

Ψ = −GMR

+1

2Ω2R2 sin2 θ

The effective surface gravity is

~geff = − ~∇Ψ

= −(∂Ψ

∂Rr +

1

r

∂Ψ

∂θθ +

1

r sin θ

∂Ψ

∂φφ

)=

[− GM

R2(θ)+ Ω2R(θ) sin2 θ

]r +

[Ω2R(θ) sin θ cos θ

]θ

Note that the effective gravity vector is not parallel to the radius vector. The angle between−~geff and r is

−~geff • r = |~geff ||r| cos ε

cos ε = − ~geff • r|~geff ||r|

A surface element dσ on the equipotential of a rotating star is

dσ =r2 sin θ dθ dφ

cos ε

5


The Von Zeipel Theorem

Recall that the flux of a radiative star depends on the temperature gradient within the star(see Hansen §4.2):

F (r) = − 4ac

3κρT 3 dT

dr

For a 3-dimensional rotating star, we can express this as

~F (θ,Ω) = −χ~∇T (θ,Ω) χ =4acT 3

3κρ

In the barotropic case (solid body rotation), the equipotentials and isobars also coincide witha surface of constant T and ρ:

~F (θ,Ω) = −χ dT

dP~∇P (θ,Ω)

= −ρχ dT

dP~geff(θ,Ω)

= +ρχdT

dP~∇Ψ(θ,Ω)

The effective potential of the rotating star is

Ψ = Φ + V = −GMr− 1

2Ω2r2 sin2 θ

∇2Ψ = ∇2Φ +∇2V = 4πGρ− 2Ω2

The total luminosity on a surface Σ of equipotential is

L(Ω) =

∫Σ

~F (θ,Ω) • d~σ

= +

(ρχ

dT

dP

)∫Σ

~∇Ψ(θ,Ω) • d~σ︸︷︷︸apply divergence theorem

= +

(ρχ

dT

dP

)∫V

~∇ • ~∇Ψ(θ,Ω) dV

= +

(ρχ

dT

dP

)∫V

∇2Ψ(θ,Ω) dV

= +

(ρχ

dT

dP

)∫V

(4πGρ− 2Ω2) dV

= +

(ρχ

dT

dP

)(4πGMr − 2Ω2 Mr

〈ρr〉

)

6


Thus the proportionality constant is

ρχdT

dP=

L(Ω)

4πGMr

(1− Ω2

2πG〈ρr〉

)

Using the notation

M∗ = Mr

(1− Ω2

2πG〈ρr〉

)the stellar flux of a rotating star can be written

F (θ,Ω) = − L(Ω)

4πGM∗ ~geff(θ,Ω)

The stellar flux is related to the effective temperature according to

F = σT 4eff

Thus, the Von Zeipel Theorem states that the temperature over the rotating star’s surfacevaries according to T ∼ gβeff , where β = 1/4:

Teff(θ,Ω) =

(L

4πσGM∗ geff(θ,Ω)

)1/4

Both geff and Teff vary over the surface of a rotating star. The equatorial regions are fainterand cooler than the polar regions. This effect is called gravity darkening (see Figure 4).

7


Figure 4: A predicted K-band image of the rapidly rotating star Regulus produced with theCHARA long-baseline optical interferometer. The best-fit model of spectra with interfero-metric visibilities finds a gravity darkening exponent β = 0.25± 0.11, inclination of rotationaxis i = 90 ± 15, and Veq/Vcrit = 0.86, Tpole = 15400 K, and Teq = 10314 K (McAlister etal. 2005, ApJ, 628, 439).

Figure 5: (A) Intensity image of the surface of Altair (λ = 1.65 µm) created using interfero-metric imaging with four telescopes in the CHARA array. Typical photometric errors in theimage correspond to 4% in intensity. (B) Reconstructed image convolved with a Gaussianbeam of 0.64 mas, corresponding to the diffraction limit of CHARA for these observations.For both panels, the specific intensities at 1.65 µm were converted into the correspondingblackbody temperatures; contours for 7000, 7500, and 8000 K are shown. From Monnier etal. (2007, Science, 317, 342).

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Mass Loss of Rotating Stars

Radiatively driven stellar winds are accelerated away from the stellar surface by the radiationpressure of the star’s luminosity. Because rapidly rotating stars have highly distorted shapes,the stellar winds are anisotropic due to several competing effects:

1. geff effect: Due to the higher gravity and flux at the poles, the polar mass loss rateis enhanced, while the equatorial mass loss rate is smaller. This lends to peanut-shaped outflows observed in bright stars such as η Carinae (see Maeder 2010, Physics,Formation, and Evolution of Rotating Stars).

2. κ effect: In a typical stellar photosphere, the opacities are due to more than justelectron scattering. Therefore opacity κ increases with the lower Teff near the equator.Higher κ favors more momentum absorbed by the winds, enhancing the equatorialmass loss rate.

3. Rotational effect: The higher centripetal force at the equator also enhances theequatorial mass loss rate.

Shellular Rotation

Instead of rotating as solid bodies, real stars may have differential rotation, so that Ω(r, θ)that varies with latitude and radius within the star. An important case is shellular ro-tation, in which Ω is constant on isobars (surfaces of constant pressure) but not with theradial coordinate inside the star.

In the case of shellular rotation, the centrifugal force cannot be derived from a potential,so the shape of the stellar surface is altered. The derived stellar surface is isobaric but notequipotential and the star is said to be baroclinic. See Maeder (2010, Physics, Formation,and Evolution of Rotating Stars) for more discussion about shellular rotation.

Rotational Mixing

Due to the increased equatorial radius, a rapidly rotating star begins its life on the mainsequence at a lower effective temperature and luminosity than a non-rotating star of the samemass. However, as core hydrogen burning continues, shellular rotation causes meridionalcirculation. This mixing enriches the helium and nitrogen abundances in the envelope,the opacity decreases, and the star becomes overluminous for its mass. At the same time,rotational mixing replenishes the hydrogen core and extends the main sequence lifetime (e.g.Meynet & Maeder 2000, A&A, 361, 101; Heger & Langer 2000, ApJ, 544, 1016). See Figure6.

The primary star of a short period, massive binary is particularly susceptible to these mixingeffects, so that the star remains blue and stays within its Roche lobe as it evolves (de Minket al. 2009, A&A, 497, 243). These effects on the primary’s temperature, luminosity, andradius are illustrated in Figure 7.

9


Figure 6: Schematic structure of meridional circulation in a rotating 20 M star with 5.2R and Vini = 300 km s−1 at the ZAMS. The figure is made as a function of Mr. In theupper hemisphere on the right section, matter is turning counterclockwise along the outerstream line and clockwise along the inner one. The inner sphere is the convective core withradius 1.7 R. From Meynet & Maeder (2002, A&A, 390, 561).

5.5

5.6

5.7

5.8

5.9

6

4.55 4.6 4.65 4.7 4.75

log

L (L

!)

log Teff (K)

SMC: 50M! + 25M!

1.5d

1.6d1.7d

1.8d2.0d 2.5d

3.0d 3.5d 4.0d

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4

R / R

RLO

F

age (Myr)

SMC: 50M! + 25M!

Primary star

1.5d1.6d1.7d1.8d2.0d

2.5d

3.0d3.5d4.0d

Figure 7: Left: The evolution from the onset of central H burning until the moment of Rochelobe overow (RLOF) for a 50M star in a binary with a 25M companion (not plotted) withinitial orbital periods between 1.5 and 4 days (from de Mink et al. 2009). Right: Radius ofthe same 50 M primary as a fraction of its Roche lobe radius (from de Mink et al. 2009,A&A, 497, 243).

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Shape of a Rotating Star