2.3 Luminosity and Radial Variation
3.1.2 Eigenfunction Expansion
Having demonstrated that a straightforward iterative series expansion is invalid for this kind of problem, we now turn to eigenfunction expansion. The most convenient basis for doing this is that of vector spherical harmonics. These are defined as
Ylm= ˆrYlm (3.28)
Ψlm=r∇Ylm (3.29)
Φlm=r×∇Ylm, (3.30)
where the gradient operators are constrained to the surface of the unit sphere, Ylm are the usual scalar spherical harmonics, and−l ≤m≤l as usual. These operators are mutually orthogonal, and their norms are 1, l(l+ 1), and l(l+ 1) respectively.
Given a fieldA, we may write A=
∞
X
l=0 l
X
m=−l
Alm,1Ylm+Alm,2Ψlm+Alm,3Φlm,
∇×A=
∞
X
l=0 l
X
m=−l
−l(l+ 1)
r Alm,3Ylm−
∂r+1 r
Alm,3Ψlm +
−Alm,1 r +
∂r+ 1 r
Alm,2
Φlm, ˆ
r×A=
∞
X
l=0 l
X
m=−l
−Alm,3Ψlm+Alm,2Φlm,
∇·A=
∞
X
l=0 l
X
m=−l
∂rAlm,1+ 2
rAlm,1− l(l+ 1) r Alm,2
!
Ylm.
(3.31)
3. HIGHER DIMENSIONAL MODELS 43 Note that the coefficients in these expansions are all functions just ofr. Expanding bothH and G in this manner yields
∇+ ˆrb hs
!
×G=
∞
X
l=0 l
X
m=−l
−l(l+ 1)
r Glm,3Ylm− ∂r+ 1 r + b
hs
!
Glm,3Ψlm + −Glm,1
r + ∂r+1 r + b
hs
!
Glm,2
!
Φlm
=− b hs
∞
X
l=0 l
X
m=−l
−Hlm,3Ψlm+Hlm,2Φlm.
(3.32)
Note that ∇×H = 0 implies that Hlm,3 = 0. The orthogonality of the vector spherical harmonics, combined with the divergence-free nature of G, then allows us to write
−l(l+ 1)
r Glm,3 = 0 (3.33)
− ∂r+ 1 r − b
hs
!
Glm,3 = 0 (3.34)
∂r+1 r − b
hs
!
Glm,2−Glm,1
r =− b
hsHlm,2 (3.35)
∂rGlm,1+2
rGlm,1− l(l+ 1)
r Glm,2 = 0. (3.36)
The first condition gives us Glm,3 = 0. The second condition is then trivially satisfied.
The third and fourth conditions must be combined to obtain a solution. Using the fourth to obtain the second coefficient, we write
∂r+1 r − b
hs
! "
r l(l+ 1)
∂rGlm,1+ 2 rGlm,1
#
−Glm,1
r =− b
hsHlm,2. (3.37) Once a solution to this is known, the value ofGlm,2 may be computed directly.
The differential equation of interest may be solved numerically without much difficulty, given Hlm, but for the purposes of our rough calculations suppose we insist thatGlm,1 changes with characteristic scale of order the stellar radius. This amounts to insisting that ∂r has eigenvalues of order 1/r. Given thaths r we may write
− b hs
r
l(l+ 1)∂rGlm,1+ 2
l(l+ 1)Glm,1
!
=− b
hsHlm,2, (3.38)
3. HIGHER DIMENSIONAL MODELS 44
or r
l(l+ 1)∂rGlm,1+ 2
l(l+ 1)Glm,1 =Hlm,2. (3.39) To simplify matters somewhat, we consider a modified version of the input heating considered earlier. In this case, the input heat takes on the form
ε= δ(r−rh)
4πr2h [Le,0Y00+Le,1(Y1,−1−Y1,1)]. (3.40) This qualitatively reproduces the expected heating behavior, with preferential heating on one side but without any net cooling, so long as Le,0 >√
6Le,1. The heating all occurs at a radiusrh, with maximum heating on the positive ˆx side. The source term
∇lnk×H is not impacted in any way by the Y0,0 term, as this term produces a radial flux field. The remaining terms give rise to a flux field which contributes to the source term. The equations governing this field are given by
∂rH1,±1,1+2
rH1,±1,1− 2
rH1,±1,2 = δ(r−rh)
4πrh2 (∓Le,1), (3.41)
H1,±1,3 = 0 (3.42)
−H1,±1,1
r +
∂r+1 r
H1,±1,2 = 0. (3.43)
The first of these relations arises from the divergence condition H, while the second two arise from the requirement that the curl ofH vanish. The general solution to this set of equations is
H1,±1,1 = A
r3 +B+ ∓Le,1(r3 + 2r3h)Θ(r−rh)
12πr3rh2 (3.44)
H1,±1,2 =− A
2r3 +B+ ∓Le,1(r3−rh3)Θ(r−rh)
12πr3r2h (3.45)
H1,±1,3 = 0, (3.46)
where Θ(x) is the Heaviside step function and the constants A and B are to be fixed by boundary condition considerations. In this case, we want the flux to drop to zero at infinity, and we want it to be finite at finite radius. As a result, both constants are zero and we have
H1,±1,1 = ∓Le,1(r3+ 2r3h)Θ(r−rh)
12πr3rh2 (3.47)
H1,±1,2 = ∓Le,1(r3−rh3)Θ(r−rh)
12πr3r2h (3.48)
H1,±1,3 = 0. (3.49)
3. HIGHER DIMENSIONAL MODELS 45 Using this to solve for G1,±1,1 in the simplified differential equation gives
G1,±1,1 = ∓Le,1(r−rh)2(r+ 2rh)Θ(r−rh)
12πr3rh2 , (3.50)
where we have already imposed the condition that this converge at the origin. To obtain the flux from one side of the star to the other from this, we note that Glm,1 doesn’t contribute to the flux through a slice of the star which cuts it in half. The only such contribution arises from the angular terms. We already know that Glm,3 = 0, so we just need to compute Glm,2. This may be done as described previously, yielding
G1,±1,2 = ∓Le,1 12πr2rh2
(r−rh)2(r+ 2rh)δ(r−rh)
2 +(r3−rh3)Θ(r−rh) r
!
. (3.51) Only the portion of the vector field directed along ˆφ contributes to the flux through the plane separating the two halves of the star, and this is given by
G±φ=±
s 3 2π
Le,1 12πr2rh2
(r−rh)2(r+ 2rh)δ(r−rh)
2 + (r3−r3h)Θ(r−rh) r
!
, (3.52) where we have set φ=π/2. Integrating this over the plane of interest then yields
L=
Z R 0
dr
Z π/2 0
d(cosθ)r(Gφ−G−φ) = 4
Z R 0
rGφ= (R−rh)2(R+ 2rh) 2√
6π3/2Rr2h Le,1, (3.53) whereR is the stellar radius and Lis the total power flowing from one side of the star to the other as a result of the circulation field. In typical situations,R−rh R, so
L≈Le,1 1 20
1− rh R
2
. (3.54)
By comparison, the flux due to the curl-free term is given by the incident flux times the ratio of the solid angle that the plane of interest sweeps as seen from the heating point, which is roughly 2π/3, to the total solid angle of 4π, so in most cases this term dominates over the circulation term. One important consequence of Eq. (3.54) is that as the heat is deposited deeper, the flux which manages to find its way to the opposing side increases as expected.
Interestingly, this result is independent of a, b. So long as they do not vary substantially on a spherical shell, this independence should hold. Additionally, note that the situation in any case is very different from that of an isotropic star, wherein half of the heating flux is present on each side. This is a result of the fact that
3. HIGHER DIMENSIONAL MODELS 46 a spherically symmetric shell of heating cannot alter the flux inside it, while an anisotropic heating shell can.
It is worth noting two effects which we have not considered here. The first is the potential for a more complex thermal conductivity structure due to convection, and the second is that of wind transport/dissipation. In the case of the former, the key effect will be the potential for significantly greater conductivity gradients misaligned with the thermal gradient. In the case of the latter, the key effect will be additional heating terms, manifesting as regions of nonzeroε, even when no heating is present at those locations. Finally, rotation plays a role in determining how these complications alter the situation. Estimating the significance of these effects is the subject of subsequent sections.